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Photonic architectures for ultrafast all-optical switching and retro-modulation Born, Brandon 2017

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PHOTONIC ARCHITECTURES FOR ULTRAFAST ALL-OPTICAL SWITCHING AND RETRO-MODULATION   by   Brandon Born  B.A.Sc., The University of British Columbia, 2013   A THESIS SUBMITTED IN PARTIAL FULLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in   THE COLLEGE OF GRADUATE STUDIES  (Electrical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Okanagan)   December 2017  © Brandon Born, 2017     ii  The undersigned certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled:  Photonic architectures for ultrafast all-optical switching and retro-modulation    Submitted by                   Brandon Born                   in partial fulfilment of the requirements of   the degree of                   Doctor of Philosophy                  .   Dr. Jonathan Holzman, School of Engineering, The University of British Columbia Supervisor, Professor (please print name and faculty/school above the line)    Dr. Kenneth Chau, School of Engineering, The University of British Columbia  Supervisory Committee Member, Professor (please print name and faculty/school in the line above)   Dr. Loïc Markley, School of Engineering, The University of British Columbia Supervisory Committee Member, Professor (please print name and faculty/school in the line above)   Dr. Jake Bobowski, Physics, The University of British Columbia University Examiner, Professor (please print name and faculty/school in the line above)   Dr. Harry Ruda, Material Science, The University of Toronto External Examiner, Professor (please print name and university in the line above)    November 30, 2017  (Date Submitted to Grad Studies)    iii Abstract Optical communications is the backbone of the modern internet. However, the true potential of the deployed optical networks has not been reached due to a reliance on far slower electronic processing. The ultimate objective is a fully all-optical network, to circumvent electronic bottlenecks and allow for network transparency, but there are challenges to the realization of all-optical implementations in current fibre optic and emerging free-space optical (FSO) communication networks. This thesis introduces two photonic architectures to address these challenges. First, an all-optical switch (AOS) architecture is developed for implementation in fibre optic front-end systems, where all-optical switching is the fundamental building block of all-optical processing. The AOS architecture applies a nanophotonic superlens, in the form of a dielectric sphere, to form an intense non-evanescent near-subwavelength focus called a photonic nanojet. The photonic nanojet forms at the back surface of the sphere in a coating of semiconductor nanoparticles. The AOS architecture is refined using Drude theory to study the free-carrier dynamics within the semiconductor nanoparticles, as well as Mie theory and ray theory to optimize the photonic nanojet’s intensity. Experiments are conducted with both milli- and micro-scale spheres coated by Si, SiC, CdTe, InP, and CuO nanoparticles. The realized AOS architecture meets the ultimate goals of femtojoule switching energies and femtosecond switching times. Second, an all-optical retro-modulation (AORM) architecture is developed for implementation in FSO communication systems, for which there is keen interest to establish high-bandwidth aerial-ground links. All-optical retro-modulation, a novel method for passive FSO aerial-ground communication, is proposed for future laser-based links with satellites, unmanned aerial vehicles, and high-altitude platforms. The AORM architecture is implemented with high-refractive-index S-LAH79  iv hemispheres to realize effective retroreflection and an interior semiconductor thin film of CuO nanocrystals to realize ultrafast all-optical modulation. A detailed investigation is carried out on the bandstructure and ultrafast free-carrier dynamics of the CuO nanocrystals. The AORM architecture is fabricated and shown to meet the ultimate goal of multidirectional FSO communication at terabit-per-second data rates. Overall, the introduced AOS and AORM photonic architectures can lay the foundation for future all-optical networks.  v Preface The presented research was primarily conducted in the Integrated Optics Laboratory and the Applied Micro and Nanosystems Facility at the University of British Columbia, Okanagan campus under the supervision of Prof. Jonathan F. Holzman. Research was also conducted at the University of SydneyC.2, under the supervision of Prof. Benjamin J. Eggleton. A publication list is presented below. This dissertation is primarily based on the studies in B. Born et al.J.1-J.4. A description of the author contributions is given for the relevant publications. Prof. Jonathan F. Holzman provided faculty support and contributed to the writing of the manuscript for all of the articles listed. Journal Articles J.1 Born, B., Hristovski, I. R., Geoffroy-Gagnon, S. & Holzman, J. F. All-optical retro-modulation for terabit-per-second free-space optical communication. Journal of Lightwave Technology (2017). (submitted) B. Born was the principle investigator of this work, and carried out the experiments, analyses, and manuscript preparation (i.e., writing and revisions). I. R. Hristovski and S. Geoffroy-Gagnon assisted with the experiments, mentored by and under the supervision of B. Born. J.2 Born, B., Krupa, J. D. A., Geoffroy-Gagnon, S., Hristovski, I. R., Collier, C. M. & Holzman, J. F. Ultrafast Charge-Carrier Dynamics of Copper Oxide Nanocrystals. ACS Photonics 3, 2475–2481 (2016). B. Born was the principle investigator of this work, and carried out the experiments, analyses, and manuscript preparation (i.e., writing and revisions). J. D. A. Krupa, S.  vi Geoffroy-Gagnon, and I. R. Hristovski assisted with the experiments, mentored by and under the supervision of B. Born. J.3 Born, B., Geoffroy-Gagnon, S., Krupa, J. D. A., Hristovski, I. R., Collier, C. M. & Holzman, J. F. Ultrafast All-Optical Switching via Subdiffractional Photonic Nanojets and Select Semiconductor Nanoparticles. ACS Photonics 3, 1095–1101 (2016). B. Born was the principle investigator of this work, and carried out the experiments, analyses, and manuscript preparation (i.e., writing and revisions). J. D. A. Krupa, S. Geoffroy-Gagnon, and I. R. Hristovski assisted with the experiments, mentored by and under the supervision of B. Born. J.4 Born, B., Krupa, J. D. A., Geoffroy-Gagnon, S. & Holzman, J. F. Integration of photonic nanojets and semiconductor nanoparticles for enhanced all-optical switching. Nat. Commun. 6, 8097 (2015). (featured article) B. Born was the principle investigator of this work, and carried out the experiments, analyses, and manuscript preparation (i.e., writing and revisions). J. D. A. Krupa, S. and Geoffroy-Gagnon assisted with the experiments, mentored by and under the supervision of B. Born. J.5 Jin, X., Hristovski, B. A., Collier, C. M., Geoffroy-Gagnon, S., Born, B. & Holzman, J. F. Ultrafast all-optical technologies for bidirectional optical wireless communications. Opt. Lett. 40, 1583–1586 (2015). B. Born provided the first experimental evidence for the spherical modulator, one of the key concepts of this work. B. Born and S. Geoffroy-Gagnon acquired the presented results of the photonic nanojet and reflected signal intensity data, which is presented in Appendix	 vii E. X. Jin was the principle investigator, and carried out the experiments, analyses, and manuscript preparation, with the help of B. A. Hristovski and C. M. Collier. J.6 Collier, C. M., Born, B., Jin, X. & Holzman, J. F. Ultrafast charge-carrier and phonon dynamics in GaP. Appl. Phys. Lett. 103, 7–10 (2013). B. Born was the principle experimentalist for this work, which involved conceptualizing the paper's core deliverables, and designing/testing experiments to acquire the presented results. C. Collier was the principle investigator, and carried out the analyses and manuscript preparation. J.7 Collier, C. M., Born, B., Bethune-Waddell, M., Jin, X. & Holzman, J. F. Ultrafast photoexcitation and transient mobility of GaP for photoconductive terahertz emission. IEEE J. Quantum Electron. 49, 691–696 (2013). B. Born was the principle experimentalist for this work, which involved designing/testing experiments to get acquire the presented results. C. Collier was the principle investigator, and carried out additional experiments, the analyses, and manuscript preparation. J.8 Jin, X., Collier, C. M., Garbowski, J. J. A., Born, B. & Holzman, J. F. Ultrafast transient responses of optical wireless communication detectors. Appl. Opt. 52, 5042–9 (2013). B. Born was one of the experimentalist for this work, and was responsible for designing/testing the time-resolved pump-probe spectroscopy experimental setup, and acquiring the presented differential transmissivity results of the GaAs photoconductive gap. X. Jin was the principle investigator, and carried out the experiments, analyses, and manuscript preparation, with the help of J. J. A. Garbowski and C. M. Collier. J.9 Collier, C. M., Born, B. & Holzman, J. F. Ultrafast response of SiC and Si nanocomposite material systems. Electron. Lett. 48, 1618–1619 (2012).  viii B. Born was the principle experimentalist for this work, which involved conceptualizing the paper's core deliverables, and designing/testing experiments to get acquire the presented results. C. Collier was the principle investigator, and carried out the analyses and manuscript preparation. J.10 Nichols, J., Collier, C. M., Landry, E. L., Wiltshire, M., Born, B. & Holzman, J. F. On-chip digital microfluidic architectures for enhanced actuation and sensing. J. Biomed. Opt. 17, 67005 (2012). B. Born was one of the experimentalist for this work, and was responsible for designing/testing the digital microfluidic chips, acquiring the presented results of the droplet moving, and fabricating microlenses. J. Nichols was the principle investigator, and carried out the experiments, analyses, and manuscript preparation, with the help of E. L. Landry, M. Wiltshire, and C. M. Collier. J.11 Collier, C. M., Wiltshire, M., Nichols, J., Born, B., Landry, E. L. & Holzman, J. F. Nonlinear dual-phase multiplexing in digital microfluidic architectures. Micromachines 2, 369–384 (2011). B. Born was the principle experimentalist for this work, this involved designing/testing the digital microfluidic chips and acquiring the presented results of the droplet moving. C. Collier was the principle investigator, and carried out additional experiments, the analyses, and manuscript preparation. J.12 Born, B., Landry, E. L. & Holzman, J. F. Electrodispensing of Microspheroids for Lateral Refractive and Reflective Photonic Elements. IEEE Photonics J. 2, 873–883 (2010). B. Born was the principle investigator for this work, and carried out the experiments, analyses, and manuscript preparation (i.e., writing and revisions). E. L. Landry carried  ix out the experimental retroreflection tests for this work and contributed to the writing of the manuscript. Conference Proceedings C.1. Bergen, M.H., Born, B., Geoffroy-Gagnon, S., & Holzman, J. F. Terahertz microjets and graphene: Technologies towards ultrafast all-optical modulation. IEEE Summer Topicals, San Juan, Puerto Rico (Jul. 2017). C.2. Born, B., Casas Bedoya, A., Choudhary, A., Mahendra, A., Li, K., Morrison, B., Pelusi, M.D., & Eggleton, B.J. A monolithic photonic chip architecture for all-optical radio-frequency spectrum analysis. International OSA Network of Students’ Conference on Optics, Atoms and Laser Applications, Melbourne, Australia (Nov. 2016). C.3. Born, B., Krupa, J.D.A., Geoffroy-Gagnon, S., & Holzman, J. F. Ultrafast all-optical switching with photonic nanojets and semiconductor nanoparticles. SPIE Photonics West, San Francisco, USA, 9746: 974615(1-9) (Feb. 2016). C.4. Born, B., Geoffroy-Gagnon, S., & Holzman J. F. An investigation of semiconductor nanoparticles for application to all-optical switching. SPIE Photonics West, San Francisco, USA, 9746: 97460E(1-8) (Feb. 2016). C.5. Jin, X., Hristovski, B.A., Collier, C.M., Geoffroy-Gagnon, S., Born, B., & Holzman, J. F. Spherical transceivers for ultrafast optical wireless communications. SPIE Photonics West, San Francisco, USA, 9744 97440F(1-8) (Feb. 2016). C.6. Born, B., Collier, C. M., & Holzman, J. F. Practical nanophotonic architectures for ultrafast all-optical switching. SPIE Optics + Photonics, San Diego, USA, 8807: 88070G(1-7) (Sep. 2013).  x C.7. Collier, C. M., Born, B., Jin, X., & Holzman J. F. Ultrafast spectroscopy of hot electron and hole dynamics in GaP. SPIE Optics + Photonics, San Diego, USA, 8845: 88450U(1-8) (Sep. 2013). C.8. Collier, C. M., Born, B., Jin, X., Westgate, T. M., Bethune-Waddell, M., Bergen, M. H., & Holzman, J. F. Transient mobility and photoconductive terahertz emission with GaP. SPIE Optics + Photonics, San Diego, USA 8846: 884616(1-9) (Sep. 2013). C.9. Jin, X., Collier, C. M., Garbowski, J. J. A., Born, B., & Holzman, J. F. Ultrafast transient characteristics of photoconductive elements for optical wireless communications. SPIE Optics + Photonics, San Diego, USA, 8847: 884710(1-8) (Sep. 2013). C.10. Collier, C. M., Jin, X., Born, B. & Holzman, J. F. Ultrafast optical analyses and characteristics of nanocomposite media. SPIE Photonics West, San Fran., USA, 8260: 826012(1-7) (Feb. 2012). C.11. Nichols, J., Landry, E. L., Born, B., Wiltshire, M., Collier, C. M., & Holzman, J. F. Optical sensing for on-chip digital microfluidics. SPIE Photonics West, San Fran., USA, 8251: 82510L(1-10) (Feb. 2012). C.12. Collier, C. M., Nichols, J., Wiltshire, M., Born, B., Landry, E. L., Ahmadi, A., & Holzman, J.F. Optoelectronic elements for digital lab-on-a-chip systems. Canadian Society for Chemistry (CSC) Canadian Chemistry Conference and Exhibition, Montréal, Canada (Jun. 2011). (invited) C.13. Collier, C. M., Born, B., & Holzman, J. F. Voltage phase control for enhanced addressability in highly-parallel digital microfluidic architectures. ASME Proc. of International Conference on Nanochannels, Microchannels & Minichannels, Edmonton, Canada, 58055, 127-134 (Jun. 2011).  xi C.14. Nichols, J., Born, B., Landry, E. L., & Holzman, J. F. Microfabrication technologies based on electro-dispensing. ASME Proc. of International Conference on Nanochannels, Microchannels & Minichannels, Edmonton, Canada, 58056, 295-300 (Jun. 2011). C.15. Nichols, J., Born, B., Roberts, D. J., & Holzman, J. F. An ultrasensitive optical detector of waterborne pathogens. Canadian Institute for Photonic Innovations, Ottawa, Canada (May 2011). C.16. Jin, X., Collier, C. M., Born, B., Beaudoin, M., O'Leary, S. K., & Holzman, J. F. Semiconductor-polymer nanocomposite materials for ultrafast photodetectors. Canadian Institute for Photonic Innovations, Ottawa, Canada (May 2011). C.17. Landry, E. L., Born, B., Ross, G., & Holzman, J. F. Integrated photonic retroreflectors for lateral cross-connects and interconnects, IEEE Proc. of Canadian Conference on Electrical and Computer Engineering, Calgary, Canada, 1-4 (May 2010).  xii Table of Contents Abstract ................................................................................................................... iii	Preface ...................................................................................................................... v	Table of Contents .................................................................................................. xii	List of Tables ......................................................................................................... xv	List of Figures ....................................................................................................... xvi	List of Abbreviations ....................................................................................... xxviii	Acknowledgements ............................................................................................ xxix	Chapter 1	 Introduction ..................................................................................... 1	1.1.	 Background ..................................................................................................................3	1.1.1.	 All-Optical Switching For Fibre Optic Front-End Systems ................................3	1.1.2.	 All-Optical Retro-Modulation For Free-Space Optical Communication Links ..7	1.2.	 Thesis Scope and Outline ...........................................................................................11	Chapter 2	 Preliminaries on Semiconductor Free-Carrier Dynamics ........ 14	2.1.	 Theory for All-Optical Switching in Planar Semiconductors ....................................16	2.1.1.	 Reflection, Transmission, and Absorption Characteristics ................................16	2.1.2.	 Planar Free-Carrier Dynamics ...........................................................................18	2.2.	 Experimental Validation ............................................................................................27	2.2.1.	 Pump-Probe Spectroscopy Experimental Setup ................................................27	2.2.2.	 Experimental Results .........................................................................................31	2.3.	 Summary & Discussion .............................................................................................37	Chapter 3	 Preliminaries on Photonic Nanojets ............................................ 38	 xiii 3.1.	 Light Scattering Theories ...........................................................................................39	3.1.1.	 Ray Theory .........................................................................................................39	3.1.2.	 Mie Theory .........................................................................................................41	3.2.	 Localized Photoinjection Experimental Tests ...........................................................48	3.3.	 Summary & Discussion .............................................................................................53	Chapter 4	 All-Optical Switching Architecture ............................................. 54	4.1.	 Theory for All-Optical Switching in Spherical Semiconductors ...............................57	4.2.	 All-Optical Switching Experiments in Spherical Semiconductors ............................65	4.2.1.	 Milli-Scale Spheres ............................................................................................65	4.2.2.	 Micro-Scale Spheres ..........................................................................................69	4.3.	 Semiconductor Nanoparticle Optimization ................................................................72	4.4.	 Summary & Discussion .............................................................................................77	Chapter 5	 All-Optical Retro-Modulation Architecture ............................... 79	5.1.	 Semiconductor Free-Carrier Dynamics of Cupric Oxide Nanocrystals ....................81	5.1.1.	 Nanocrystal Fabrication .....................................................................................82	5.1.2.	 Optical Absorption Characterization .................................................................84	5.1.3.	 Free-Carrier Dynamics and State-Filling ...........................................................86	5.1.4.	 Experimental Validation ....................................................................................89	5.1.5.	 Summary & Discussion .....................................................................................96	5.2.	 All-Optical Retro-Modulation Architecture ..............................................................98	5.2.1.	 Link Budget Analysis ........................................................................................99	5.2.2.	 Theoretical Design ...........................................................................................104	5.2.3.	 Experimental Results .......................................................................................110	5.2.4.	 Summary & Discussion ...................................................................................117	Chapter 6	 Conclusion ................................................................................... 118	 xiv 6.1.	 Summary of Contributions .......................................................................................119	6.2.	 Future Work .............................................................................................................126	Bibliography ........................................................................................................ 128	Appendices ........................................................................................................... 145	Appendix A	 Reflection, Transmission, and Absorption Experiments ..............................145	Appendix B	 Free-Carrier Recombination in GaAs ..........................................................149	Appendix C	 Optical Encoder Setup ..................................................................................153	Appendix D	 Planar Free-Carrier Dynamics Analytical Solution .....................................159	Appendix E	 Ray Theory for a Dielectric Sphere ..............................................................162	Appendix F	 Mie Theory for a Dielectric Sphere ..............................................................165	Appendix G	 Spherical Free-Carrier Dynamics Analytical Solution .................................173	Appendix H	 Planar Free-Carrier Dynamics Code ............................................................175	Appendix I	 Retroreflected Divergence Angle Code .......................................................180	  xv List of Tables Table 1.1. Switching energies and switching times of AOS architectures. White and grey cells denote success and failure, respectively, in demonstrating femtojoule switching energies or femtosecond switching times. ....................................................................6	Table 2.1. Parameters used to generate the theoretical response for all-optical switching in  planar semiconductors. The parameters are for GaAs and they are used according    to Equation (22) and the boundary condition given in Equation (23). .......................32	  xvi List of Figures Chapters Figure 2.1. Initial conditions for the free-carrier density within the planar semiconductor. The incident control pulse photoinjects free-carriers at ! = 0 with an exponential      decay in free-carrier density along the optical axis, #. This free-carrier density distribution then evolves in time via diffusion and recombination. ...........................24	Figure 2.2. Time-resolved pump-probe spectroscopy experimental setup. (a) Schematic of the time-resolved pulsed pump-probe laser system. (b) Schematic of the experimental setup. A 780 nm control (pump) beam is used for the GaAs and Si samples, and a 390 nm control (pump) beam is used for the SiC samples. A 1550 nm signal    (probe) beam is used for all the samples. *A second harmonic generation (SHG) crystal and bandpass filter are introduced into the beam path to form the 390 nm control (pump) beam. **The focusing element takes the form of a microscope objective or dielectric spheres, according to the experiment being performed. .........30	Figure 2.3. Elementary results for all-optical switching. Experimental (negative) differential transmission of the signal beam, – Δ&(!)/&, shown normalized as a function of  delay time, for photoinjection of the bulk Si, SiC, and GaAs samples, using a microscope objective as the focusing element, as depicted in the inset. ....................31	Figure 2.4. Modelled free-carrier dynamics for a planar GaAs bulk sample. (a) The free-     carrier density, * #, ! , is shown as a function of time and space according to Equation (22) for the GaAs bulk sample. The * #, !  curves are shown with time increments of 50 ps. (b) The differential transmission response, – Δ&(!)/&,  xvii theoretical (black) and experimental (blue) curves are shown for the GaAs bulk sample. The theoretical response is obtained with Drude theory based on       Equation (21) and the time-evolving free-carrier density. The experimental   response is obtained from pump-probe spectroscopy using a 20× microscope objective. .....................................................................................................................36	Figure 3.1. Photonic nanojet intensity in the milli-scale regime. The theoretical intensity of a photonic nanojet, calculated with ray theory at the back surface of a dielectric  sphere, is shown as a function of the sphere’s refractive index, ,. An intensity colourmap is shown at the top of the figure. ..............................................................40	Figure 3.2. Photonic nanojet maximum intensities in the micro-scale regime. The theoretical intensity of the photonic nanojet, calculated with Mie theory at the back surface      of a dielectric sphere, is shown as a function of the sphere’s refractive index, ,,     and diameter, -, for photoinjection at a wavelength of 780 nm. The intensity colourmap that is the result of over a hundred thousand individual Mie theory simulations. The maximum (white) intensity is normalized for all , with constant    -. A trendline for the maximum intensity is shown as a solid grey curve. Select Mie theory simulations, with logarithmic intensities, are shown in the figure insets as (a) a sphere with - =	3 µm and , = 1.76, and (b) a sphere with - = 30 µm and , =	1.83. ...........................................................................................................................42	Figure 3.3. Characterization of photonic nanojet intensities formed by microspheres.   Normalized intensity at the microsphere’s back surface is shown versus the microsphere’s refractive index, ,, for microsphere diameters of - = 20 µm   (purple), 30 µm (green), 40 µm (black), and 50 µm (red). The peak intensities are     xviii at , = 1.80, 1.82, 1.83, and 1.86, respectively. Insets (a), (b), and (c) show the intensity at the microsphere’s back surface for - = 40 µm and , = 1.66, 1.83,       and 2.00, respectively. The results are generated from Mie theory simulations at a wavelength of 780 nm. The scale bar is 1 µm. ...........................................................45	Figure 3.4. Characterization of the photonic nanojet’s beam profile. Cross-sectional profiles     are shown for the optimal , = 1.83 microsphere at # = 0 (blue), # = 1 µm (red),   and # = 1.68 µm (yellow) from the back surface microsphere. The photonic    nanojet has a subwavelength FWHM of 540 nm, being transverse to the optical    axis at the microsphere’s back surface, and with a micro-scale propagation     distance of 1.68 µm, being the distance along the optical axis over which the      beam remains subwavelength. The scale bar is 1 µm. ................................................47	Figure 3.5. All-optical switching via photoinjection in GaAs using varying sphere refractive indices. Results for the experimental (blue) and theoretical (black) differential transmission of the signal beam, ±Δ& ! &, are shown normalized as a function     of time, for photoinjection in GaAs, using spheres with a diameter of - = 2.0 mm and refractive indices of (a) , = 1.51, (b) , =1.76, (c) , = 1.83, and (d) , = 1.98 ±0.02. The differential transmission results are shown normalized, with respect to the results in -, and the relative scaling factors are labelled in the figures. The  figures include theoretical curves derived from Equations (21) and (22),      pertaining to planar free-carrier dynamics. Mie theory simulations are shown in     the insets, with logarithmic colour maps, to illustrate the varying focal conditions    of the coincident control and signal beams. ................................................................50	 xix Figure 3.6. Photonic nanojet intensities as a function of sphere diameter. The displayed points are for the sphere’s maximum normalized intensity for refractive indices of , =    1.5 to 2.0 as a function of diameter, -. The refractive indices giving the highest intensity of a given diameter are plotted. For sphere sizes of - > 2 µm, the      relative intensity (with a fixed input power) slowly increases with decreasing    sphere size. For sphere sizes of - < 2 µm, the relative intensity dramatically increases as the sphere size decreases, approaching the wavelength (780 nm). Mie theory simulations are shown in the insets with logarithmic colourmaps. .................52	Figure 4.1. Illustration showing the general concept of the AOS architecture. Semiconductor nanoparticles of varying compositions are embedded onto the full surface of a dielectic sphere. Collimated control and signal beams propogate from left to right and are focued into an intense non-evanescent near-subwavelength photonic   nanojet. The highest intensity of the photonic nanojet (shown in white with a logarithmic scale colourmap) overlaps with the semiconductor nanoparticles on     the back surface, which enables strong and rapid interactions between the control  and signal beams. ........................................................................................................56	Figure 4.2. Modelled free-carrier dynamics for Si and SiC nanoparticles. (a) The free-carrier density, * 2, ! , is shown as a function of time and space, according to Equation (33), for Si nanoparticles with a radius of 20 nm, where * 2, !  has increments of     2 ps. (b) The free-carrier density, * 2, ! , is shown as a function of time and     space, according to Equation (33), for SiC nanoparticles with a radius of 50 nm, where * 2, !  is shown with time increments of 1 ps. ...............................................60	 xx Figure 4.3. Characterization of time constants for semiconductor nanoparticles. The time constant, 3, i.e., switching time, is displayed versus the surface recombination velocity,	45, and radius, 6, for a nanoparticle with rapid diffusion, i.e., 7 ≫ 456    and a long bulk lifetime, i.e., 39. Results are formed by solving Equations (35),   (36), (37), and (38). The red region is the desired regime, where femtosecond   optical switching is achieved. Inset (a) shows a nonuniform free-carrier density distribution, * 2, ! = 0.5 ps, in the nanoparticle for a small diffusion coefficient, i.e., 7 ≈ 456. Inset (b) shows a uniform free-carrier density distribution, * 2, !       = 0.5 ps, in the nanoparticle for a large diffusion coefficient, i.e., 7 ≫ 456. ...........64	Figure 4.4. All-optical switching with Si and SiC nanoparticles on a milli-scale sphere. Experimental (positive) differential transmission for the signal beam, Δ& ! &, is shown normalized versus time for photoinjection of (a) Si nanoparticles and (b)    SiC nanoparticles. The nanoparticles coat spheres with a diameter of -	 = 2 mm   and refractive index of ,	 = 1.98 ±0.02. The figures include insets with Mie     theory simulations and scanning electron microscope images of nanoparticles      with a 200 nm scale bar. .............................................................................................68	Figure 4.5. All-optical switching with Si and SiC nanoparticles on a micro-scale sphere. Experimental (positive) differential transmission for the signal beam, Δ& ! &, is shown normalized versus time for photoinjection of (a) Si nanoparticles and (b)    SiC nanoparticles. The nanoparticles coat microspheres with a diameter of -	 =      40 µm and refractive index of ,	 = 1.83 ± 0.02. The figures include insets with    Mie theory simulations and scanning electron microscope images of nanoparticles with a 200 nm scale bar. .............................................................................................71	 xxi Figure 4.6. All-optical switching with Si, CdTe, InP, and CuO coated microspheres having a diameter of - = 40 µm and refractive index of , = 1.83. Normalized differential transmission curves, Δ& ! &, are shown versus time for microspheres having the (a) Si, (b) CdTe, (c) InP, and (d) CuO nanoparticles coatings. The results exhibit switching energies of 1 pJ, 500 fJ, 400 fJ, and 300 fJ, with switching times of 1.8   ps, 2.3 ps, 900 fs, and 350 fs, respectively. The insets show Mie theory     simulations with logarithmic intensities and scanning electron microscope images   of nanoparticles, with a 200 nm scale bar. ..................................................................76	Figure 5.1. The CuO nanocrystal size as a function of the film thickness. The films have thicknesses of (a) 20, (b) 60, (c) 100, and (d) 200 nm, with nanocrystal sizes of      50, 130, 140, and 300 nm, respectively. The insets show scanning electron microscope images of the nanocrystals. .....................................................................83	Figure 5.2. Optical absorption characteristics for the CuO nanocrystals. The absorption coefficient, ;, is shown as a function of the wavelength, <. A Tauc plot is shown     in the inset, as ;ℎ> ? versus energy, ℎ>, in black, with a linear fit shown in red. From this Tauc plot, the direct bandgap of the CuO nanocrystals is defined to be @A = 1.55 eV. ..............................................................................................................86	Figure 5.3. Transient absorption characteristics for the CuO nanocrystals. The control and   signal beams are focused onto the layer of nanocrystals using a 40× microscope objective, as seen in the inset. The transmitted signal beam power is measured as      a function of the time delay. The results, displayed as normalized differential transmission of the signal beam, Δ& ! &, are divided into four intervals: prior to  xxii zero time delay, ! < 0 ps; at zero time delay, ! = 0 ps; within the first picosecond after zero time delay, 0 ps < ! < 	1 ps; after the first picosecond, ! >	1 ps. .............88	Figure 5.4. Illustration of the free-carrier dynamics within the bandstructure of CuO nanocrystals. The bandgap is @A = 1.55 eV. Photogenerated holes (blue) and electrons (red) are shown in the bandstructure at four time intervals: (a) prior to   zero time delay, ! < 0 ps, the VB and CB are populated by photogenerated holes   and electrons from the signal beam, which has a photon energy of @D = 0.8 eV;      (b) at zero time delay, ! = 0 ps, additional electrons and holes are photogenerated   by the control beam, which has an energy of @E = 1.6 eV; (c) within the first picosecond after zero time delay, 0 ps < ! <	1 ps, the photogenerated holes in the VB undergo momentum relaxation due to carrier-carrier scattering; and (d) after    the first picosecond, ! > 1 ps, the photogenerated holes in the VB undergo energy relaxation and trap-assisted recombination. ...............................................................90	Figure 5.5. Transient pump-probe spectroscopy results for the 50 nm CuO nanocrystals. In      (a), the experimental results for the normalized differential transmission,      Δ& ! &, of the CuO nanocrystals are given for control beam fluences increasing linearly from 0.4 J·m-2 (yellow) to 10 J·m-2 (red). Superimposed on each experimental curve is the tri-exponential decay curve-fit (black). In (b), the weighting of the first time constant’s term (top), F1, the weighting of the second  time constant’s term (middle), F2, and the weighting of the third time constant’s  term (bottom), F3, are plotted as a function of the control beam fluence, Φ. The  inset shows the rates of the first (top), 1/31, second (middle), 1/32, and third (bottom), 1/33, time-constants as a function of J. Trendlines are shown in red. ......92	 xxiii Figure 5.6. Implementation of the active UL and passive DL with the AORM architecture. (a) The overall system facilitates multidirectional communication via an active UL     and passive DL. The active UL is shown as the ground-to-aerial beam (orange),   with a solid angle of KL, being transmitted from the ith ground TRX to the aerial TRX. The passive DL is shown as a portion of the incident beam being retro-modulated over an area of MN. This retro-modulated signal beam (green), with a   solid angle KO, is transmitted from the aerial TRX back down to the ground TRX, which has a receiver area of MO. The inset depicts multidirectional communication with this implementation for an arbitrary number of ground TRXs. (b) The   proposed AORM architecture collects the aerial-to-ground beam (orange) with a glass sphere and focuses the beam through a CuO film and onto its back surface.      It then undergoes retroreflection and propagates back to its source. The on-board control beam (red) is focused via external optics onto the CuO film, where it has       a cross-sectional area of MP, to modulate the signal beam (green) over a cross-sectional area of MN. ..................................................................................................103	Figure 5.7. Ray tracing simulations for various spherical retroreflectors. The retro-modulated divergence solid angle, KO, of each spherical retroreflector is given as a function      of the control beam radius, rE, normalized to the sphere’s radius, 6. The following non-cladded and cladded spheres are presented: an ideal , = 2.000 sphere      (black), an , = 1.955 sphere (red), an , = 1.955 sphere with an , = 1.500    cladding (blue dash), and an , = 2.500 sphere with an , = 2.301 cladding      (green). The goal of KO < 10-10 is achieved for the portions of the curves below    the displayed horizontal dashed line. ........................................................................107	 xxiv Figure 5.8. Measurement of the impulse response of the AORM architecture. (a) Schematic of the pump-probe spectroscopy experimental setup for measuring the impulse response. The 1550 nm beam (orange) is introduced via a 50-50 beamsplitter and compressed with a telescope before incidence on the AORM architecture. The   retro-modulated signal beam is measured with an InGaAs detector. The 780 nm control beam (red) is focused with a 20× microscope objective. (b) Experimental impulse response of the AORM architecture with a CuO film comprised of nanocrystals measured for an incident control beam fluence of 5 µJ/cm2. The inset shows the AORM architecture with the 1550 nm signal and 780 nm control      beams incident on the CuO film, and a scanning-electron-microscope image of      the nanocrystals in the CuO film with a 200 nm scale bar. ......................................111	Figure 5.9. The physical implementation of the pump-probe experimental setup. The inset   shows the mounted AORM architecture, comprised of two mated S-LAH79 hemispheres, forming a full sphere, with a CuO nanocrystal thin film layer     between them. The AORM architecture is held in place by two transparent sheets. ........................................................................................................................113	 Appendices Figure A.1. Depiction of the reflection, transmission, and absorption for a slab with two interfaces. The absorption occurs over an effective length of TU, being much less  than the slab thickness. .............................................................................................146	Figure A.2. Schematic of the experimental setup for measuring the change in reflection, transmission, and absorption. The 1550 nm signal beam (orange) is introduced via     xxv a 50-50 beamsplitter and is focused with a 20× microscope objective onto the    single crystal semiconductor samples. The reflected and transmitted signal beams  are measured with InGaAs detectors on either side. The 780 nm signal beam (red)    is introduced via a dichroic beamsplitter to be collinear with the signal beam. .......147	Figure B.1. The differential transmission, Δ& ! /&, in bulk GaAs as a function of delay time between the control and signal beams for varying control beam fluences. The 780  nm control beam fluences range over 10-6~10-4 J/cm2, which corresponds to    carrier densities over 1017~1019 cm-3. This range yields with a change in the effective lifetime of roughly 25%. ............................................................................149	Figure C.1. The autocorrelation setup and chrome-on-glass knife-edge setup are depicted. A   650 nm continuous wave (CW) beam (orange) is split by a 50-50 beamsplitter      into two arms to form a Michelson interferometer. The beam power is measured     by a Si photodetector. One mirror from the linear autocorrelator and the chrome-   on-glass knife-edge are mounted on a delay stage for control by the piezo linear actuator. As the edge scans across the focal spot of the 780 nm beam (red), the autocorrelator’s fringes are recorded as a function of the mirror position. As the fringes are recorded, the transmitted power past the knife-edge is also recorded    with a second Si photodetector. ................................................................................155	Figure C.2. The LabVIEW control interface for measuring the focal spot size of focusing elements. The interface allows motion in forward and reverse directions. The step distance varies with respect to the direction of motion, the actuator’s speed, and weight of the delay stage. The approximate step (pulse) distance is updated every time the program is run with measurable interference. The steps (pulses) per     xxvi sample are the number of steps the linear actuators takes before taking a measurement. The delay time is the time allowed for the lock-in to average the signal, and the scanning range is the total measurement range, which is converted    to both femtoseconds and micrometers. ...................................................................156	Figure C.3. The LabVIEW program structure for controlling the linear actuator and lock-in amplifier. LabVIEW directs the linear actuator to move the desired number of   steps, waits the desired averaging time, and takes a measurement from the lock-in amplifier. ...................................................................................................................157	Figure C.4. The LabVIEW program structure for calibrating the linear actuator step distance. LabVIEW takes the Fourier transform of the collected data to find the frequency with respect to the linear actuator step distance and with the knowledge of the beam’s wavelength, it calculates the true step distance in nanometers and femtoseconds. ...........................................................................................................157	Figure C.5. The experimentally measured focal spot size for the 20× microscope objective   (red). The convolution of a Gaussian profile (blue) with a unit step function is     used to curve-fit the results (black). The focal spot diameter, corresponding with    the FWHM of the Gaussian profile, is found to be 6±1 µm. ....................................158	Figure E.1. Relationship between ray entrance and exit positions. The ray enters the sphere at     a height WX, with an incident angle of YXZ normal to the surface. The transmitted     ray with an angle of YXL propagates through the sphere with a slope of [. The ray then exits the sphere at the coordinates (\X, WX) with an incident angle of Y]Z and      a transmitted angle of Y]L. ........................................................................................162	 xxvii Figure E.2. Intensity at the back of the sphere as derived from ray theory as a function of the refractive index. It can be seen that ray theory predicts the intensity to be greatest    at a refractive index of , = 2.0. ................................................................................164	Figure F.1. Mie theory initial conditions of a plane wave and single sphere. The figure is a logarithmic plot of the beam intensity derived from Mie theory. A single sphere    with a radius of ; and a refractive index of ,? is surrounded by a medium with a refractive index of ,]. An incident plane wave forms internal and scattered waves inside and outside the sphere, respectively. ..............................................................169	Figure I.1. MATLAB ray tracing simulations of the retro-modulated divergence solid angle,    KO, as a function of the control beam radius, 2P, normalized to the sphere’s radius,   6, for various spherical retroreflectors. The following non-cladded and cladded spheres are presented: (a) an ideal , = 2.000 sphere, (b) an , = 1.955 sphere, (c)    an , = 1.955 sphere with an , = 1.500 cladding, and (d) an , = 2.500 sphere     with an , = 2.301 cladding. The rays are plotted in three-dimensions, propagating through the inner and outer spheres, shown in grey. The direction of propagation      is denoted with a gradient from green rays to yellow rays. The resulting retroreflected divergence solid angle functions are plotted on the right, which are also seen in Figure 5.7. .............................................................................................187	   xxviii List of Abbreviations Abbreviation: Definition: AORM All-optical retro-modulation AOS All-optical switch CB Conduction band DL Downlink FSO Free-space optical FOV Field of view FWHM Full width at half maximum TRX Transceiver UL Uplink VB Valence band     xxix Acknowledgements I am deeply grateful to Dr. Jonathan F. Holzman, my Ph.D. supervisor, for his support, guidance, encouragement, enthusiasm, and friendship. His helpful suggestions, assistance, and patience throughout my studies have made this dissertation possible. His rigorous scholarship, clarity in thinking, and professional integrity will influence me for the rest of my life.  I would like to thank Dr. Harry Ruda, from the University of Toronto, for his willingness to serve as my external examiner, as well as Dr. Kenneth Chau, Dr. Loïc Markley, and Dr. Jake Bobowski for serving as my committee members. I appreciate their time and constructive comments over the years. I also give my thanks to Prof. Benjamin Eggleton for welcoming me to his group at the University of Sydney, and for his insightful discussions and advice. I would like to thank my dear colleagues, Xian Jin, Ilija Hristovski, Simon Geoffroy-Gagnon, Jeff Krupa, Mitch Westgate, Mark Bergen, Emily Landry, Max Bethune-Waddell, Lane Henderson, Daniel Guerrero, Trevor Stirling, Jamie Garbowski, Adrian Boivin, Christopher Collier, and Jackie Nichols for sharing their expertise, assistance, insight, and camaraderie during my studies. I extend my gratitude for the technical support given to me by David Arkinstall, Tim Giesbrecht, Durwin Bossy, Marc Nadeau, David Zinz, and Emily Zhang. Also I thank the School of Engineering for providing the research facilities and funding to carry out my studies. Finally, a special thanks are owed to my parents, Gerry and Susan Born, for their understanding, patience, and encouragement over all these years of education, as well as Michelle Epp for her invaluable thoughtful support. Without their encouragement and support, none of my achievements would be possible.   1 Chapter 1  Introduction The modern internet greatly relies upon the unrivaled data rates of optical communications, with optical communications accounting for more than 73 percent of the market share for global telecommunications in 20161. However, the true potential of optical networks is not being realized due to a reliance on electronic processing with slow data rates. To lessen the effects of these electronic bottlenecks, researchers have introduced electro-optic processing via hybrid electrical-optical processors, such as the state-of-the-art solution demonstrated by Sun et al. in 20152. However, the ultimate objective is a fully all-optical network. Such a network would circumvent the electronic bottlenecks and create network transparency, i.e., low latency and independence of the communication protocol3. However, there are challenges to the realization of all-optical implementations for current fibre optic and emerging free-space optical (FSO) communication networks. This thesis develops two photonic architectures to address these challenges. The architectures are based upon similar core principles in electromagnetics and material science, as described with relevant background in the upcoming section 1.1, but their implementations differ. First, an all-optical switch (AOS) architecture is developed for implementation in fibre optic front-end systems, where electro-optic processing is typically applied. All-optical switching, being the fundamental building block of all-optical processing4,5, is introduced in section 1.1.1. Second, an all-optical retro-modulation (AORM) architecture is developed for implementation in FSO communication systems, where FSO technology is being keenly pursued for high-bandwidth  2 aerial-ground links6. All-optical retro-modulation, being a novel method for passive links in FSO communication, is introduced in section 1.1.2. An outline of the full thesis is given in section 1.2.    3 1.1. Background The relevant background information on all-optical switching and all-optical retro-modulation is given in the following sections. Section 1.1.1 introduces all-optical switching for fibre optic front-end systems, in terms of its challenges and goals. Section 1.1.2 introduces all-optical retro-modulation for FSO communication links, in terms of its own unique challenges and goals.  1.1.1. All-Optical Switching For Fibre Optic Front-End Systems Logic operations are at the core of data processing, and switching is the key mechanism for logic operations. Switching is typically realized via electronics, with transistors relaying the flow of data, but there is pressure to introduce switches that can operate directly at the data rates supported by optical fibre transmission systems4,5. The terabit-per-second data transmission rates of optical fibres far exceed the gigabit-per-second data processing rates of electronics, however, and this forms an optoelectronic bottleneck7. The optoelectronic bottleneck emerges within the fibre optic front-end systems, where optoelectronic processing and multiplexing/demultiplexing are used to relay high-speed optical data. To alleviate this optoelectronic bottleneck, all-optical switching has been proposed7,8. The deployments of all-optical switching in fibre optic front-end systems to date have taken two forms: network-on-chip architectures8,9, where electronic processing is replaced with equivalent all-optical processing, and all-optical multiplexing/ demultiplexing architectures10, where data is parallelized into multiple gigabit-per-second streams, via time-division multiplexing11 or orthogonal frequency-division multiplexing12, such that the data can be processed by contemporary electronics.  4 The network-on-chip and multiplexing/demultiplexing architectures both require the use of an purely optical architecture for switching, being the photonic analogue to the electronic transistor. To be sufficiently practical, however, the applied AOS architecture must be implemented with consideration to demands for low switching energies13 and ultrafast switching times14. For example, Miller et al. have argued that AOS architectures should target low switching energies, at or below 100 fJ/bit13, while Husko et al. have argued that AOS architectures should target ultrafast switching times at or below 10 ps14. For all-optical switching in terabit-per-second optical networks, specifically, the targeting of femtojoule switching energies is appropriate, although the timescale should be made more strict by targeting femtosecond (i.e., subpicosecond) switching times. Furthermore, any proposed AOS architectures should have micro- or nano-scale dimensions. This would enable scalability and thus allow for highly parallel implementations, without succumbing to practical issues of excessive power, heating, or size. Various all-optical switching technologies have been introduced through the years in the pursuit of femtojoule switching energies and femtosecond switching times, although it has become apparent through such work that these two demands are often mutually exclusive15,16,17. Early all-optical switching studies focused on developing semiconductor materials with ultrafast switching times—with photoinjection (excitation) and recombination (recovery) occurring ideally on a femtosecond timescale. The processes of free-carrier photoinjection and recombination were applied in semiconductors having simple planar topologies, i.e., bulk wafers, so the all-optical switching times were limited by the rates of bulk recombination and thus the free-carrier lifetimes. It was found that semiconductors with ultrashort lifetimes could establish ultrafast switching between coincident control (pump) and signal (probe) beams. For example, Ganikhanov et al.18, Lambsdorff et al.19, and Gupta et al.20, demonstrated ultrafast all-optical switching using  5 ion-implanted, low-temperature-grown, and radiation-damaged GaAs, respectively. Ultimately, it was found that ultrafast switching times could be achieved with these engineered materials, but the use of simple focusing into semiconductors demanded high beam intensities—which necessitated high switching energies for the control beam21. More recent studies on all-optical switching have focused on the development of device geometries for low switching energies—ideally on a femtojoule level. Increased beam intensities and enhanced mixing of beams were introduced by way of many (typically resonant) devices. Almeida et al.16 demonstrated micro-scale all-optical switching through the use of ring resonators. Nozaki et al.17 demonstrated nano-scale all-optical switching by using photonic crystal resonators. Volz et al.22 demonstrated atomic-scale all-optical switching via quantum dots. Such resonant devices can support low switching energies, in general, but their implementations led to prolonged switching times due to their inherent formation of resonant cavity lifetimes23. It is also important to consider issues of practicality for some of these implementations, particularly those based upon quantum dots, as handling and coupling can be difficult for structures with nano-scale dimensions. While the above studies on all-optical switching have shown that the goals of low switching energies and ultrafast switching times are often mutually exclusively, there has been some success in reaching both of these goals. The successful work was put forth by Hu et al., in the form of an organic photonic-bandgap microcavity24. The all-optical switch achieved a switching time of 1 ps and switching energy of 520 fJ. However, it is important to note here that ultra-sensitive cavities such as this cannot operate with the broad spectral inputs used in wavelength division multiplexing systems. A summary of the switching energies and switching times for the above-referenced works is given in Table 1.1. White cells denote success in demonstrating femtojoule switching energies or  6 femtosecond switching times, and grey cells denote failure in demonstrating either of these goals. The last row of the table shows the goal of this work, in demonstrating switching energies below 100 fJ and switching times below 1 ps. The proposed work’s goal of realizing femtojoule switching energies can be achieved by initiating an especially strong interaction between the incident control and signal beams. Such an interaction, being proportional to the intensities, can be strengthened by concentrating the beams within a high-intensity focal spot. The focusing is achieved by introducing a nanophotonic superlens, in the form of a dielectric microsphere, which creates an intense non-evanescent near-Table 1.1. Switching energies and switching times of AOS architectures. White and grey cells denote success and failure, respectively, in demonstrating femtojoule switching energies or femtosecond switching times.  Device Switching Energy Switching Time GaAs photonic crystal nanocavity14 120 fJ 15 ps Ion-implanted GaAs18 Not given 220-550 fs Radiation damaged GaAs19  Not given 500 fs Low-temperature-grown GaAs20 Not given 600 fs Low-temperature-grown GaAs21 10 pJ 500 fs InP photonic crystal nanocavity25 40 fJ 10 ps Silicon ring resonator16 20 pJ 500 ps InGaAsP photonic crystal nanocavity17 0.42 fJ 20 ps InAs/GaAs quantum dot–cavity system22 0.02 fJ 50 ps Organic photonic-bandgap microcavity24 520 fJ 1 ps Proposed work fJ-scale fs-scale  7 subwavelength focus called a photonic nanojet26,27. The microsphere’s diameter and refractive index are optimized to maximize the intensity of the photonic nanojet—thereby promoting strong nonlinear optical interactions at femtojoule switching energies. The proposed work’s goal of realizing femtosecond switching times can be achieved by facilitating rapid recovery of the all-optical switching material. Semiconductor nanostructures have been shown to be successful at this and are used in this work. Such materials have recovery times that are as much as six orders of magnitude shorter than those of their corresponding bulk semiconductors, due simply to the prevalence of surface states within the nanostructures28,29. In this thesis, semiconductor nanostructures are implemented in the form of semiconductor nanoparticles of varying compositions. The semiconductor nanoparticles are embedded onto the surface of the aforementioned dielectric sphere. This has the intense focus of the photonic nanojet form within the semiconductor nanoparticles. It is ultimately shown that the proposed AOS architecture with its integration of photonic nanojets and semiconductor nanoparticles can enable both femtojoule switching energies and femtosecond switching times—along with practical benefits that include omnidirectionality, a small footprint, and monolithic integration with the focusing element. 1.1.2. All-Optical Retro-Modulation For Free-Space Optical Communication Links FSO communication is a new frontier for high-speed networks. These FSO communication systems operate with optical carrier frequencies, as opposed to radio frequencies, and this has noteworthy advantages. The advantages include the potential for terabit-per-second data rates, no spectrum licensing, improved channel security, reduced power consumption, low system mass, and a 90% decrease in antenna diameter30. These attributes make FSO technology attractive for  8 emerging applications in aerial communications. The applications include military surveillance and reconnaissance, for disseminating large volumes of secure information in remote locations, as well as providing “last-mile” high-speed telecommunication service in metro or rural areas31. Such applications are being realized by way of aerial-ground links using unmanned aerial vehicles (typically less than 10 km in length)32, high-altitude platforms (typically 17-25 km in length)33,34, or satellites (typically in low Earth orbit with lengths on the order of 1,000 km)6, and are now attracting commercial interest from companies such as Google Inc., Facebook Inc., BridgeSat Inc., ViaLight Communications GmbH, and others. The major challenge for long-range FSO bidirectional links is their requirement for line-of-sight alignment with sub-micro-radian accuracy between the ground and aerial transceivers (TRXs)6. This is a particularly major issue for asymmetrical FSO links with connections between multiple ground TRXs (at distributed locations) and one aerial TRX (with limited system mass and power). Additional TRX pairs escalate the aerial system’s mass, power, costs, and complexity when used with large numbers of links. The proposed work addresses this issue, with the goal to support operation with numerous multidirectional aerial-ground links at terabit-per-second data rates. The goal to establish multidirectional communication, without escalating mass and power, can be met by implementing retro-modulation on board the aerial TRX. Retro-modulation harnesses the optical power received from the ground-to-aerial active uplink (UL). The UL beam undergoes modulation and retroreflection at the aerial TRX, to form an aerial-to-ground passive downlink (DL). Such passive DLs enable multidirectional communication, while eliminating pointing-acquisition-tracking on board the aerial vehicle, and support communication with an arbitrary number of TRXs within the retroreflector’s field of view (FOV)35. Given such operation, the  9 passive DL witnesses its greatest benefits for deployments with many distributed ground TRXs. (This is complementary to technologies that seek miniaturization for the aerial TRX, such as CubeSat36, as they see their greatest benefits for deployments with many distributed aerial TRXs.) In addition, passive DLs offer two practical benefits. One, the use of retroreflection in the aerial TRX makes the system relatively insensitive to misalignment. This ultimately negates pointing losses from vibrations and platform jitter on the aerial TRX, which is a major challenge for FSO links6. Two, the use of retroreflection reduces the aerial TRX’s power requirements. The vast majority of the power is instead applied by the ground TRXs, where the design constraints for mass and power are much less stringent. The major challenge for implementing passive DLs, and the reason they have not been commonly used to date, is the limited speed of the applied mechanical37 and liquid crystal38 modulators. Such modulators support kilobit-per-second data rates at best. In response, recent studies have investigated passive DLs with electro-optical implementations, such as multiple-quantum-well modulators combined with corner cube39,40 and multi-element (cat’s eye) retroreflectors35,41. Data rates up to 45 megabit-per-second have been demonstrated with such systems, and they have the potential for gigabit-per-second speeds42. With the evolution of the above technologies for passive DLs in mind, this work introduces all-optical retro-modulation with an all-optical modulator43 and a spherical (cat’s eye) retroreflector being used to realize multidirectional communication at terabit-per-second data rates. Such a system avoids the fundamental bandwidth bottleneck between optics and electronics, which results from the use of far slower (gigabit-per-second) electronic processing and modulation7. The all-optical modulation is implemented here through cross absorption modulation, whereby a control beam linearly modulates the absorption seen by a coincident signal beam. The proposed cross  10 absorption modulation scheme uses a control beam with a wavelength on resonance with the semiconductor’s bandgap. This generates photoexcited free-carriers, being highly mobile electron populations in the conduction band (CB) and less mobile hole populations in valence band (VB). The free-carriers induce a change to the semiconductor’s refractive index and absorption by way of free-carrier absorption and/or state-filling. This approach improves the switching response and avoids the need for phase-matched propagation over centimeter lengths4. However, such an approach leads to an additional constraint, whereby the generated free-carriers must recombine sufficiently fast to enable terabit-per-second data rates. This issue is solved by using a thin film of cupric oxide (CuO) nanocrystals, for which the generated free-carriers undergo ultrafast relaxation and recombination44. Ultimately, the CuO nanocrystalline material is implemented with a spherical retroreflector to realize an effective all-optical retro-modulation (AORM) architecture for passive DLs. This thesis shows the realization of such a concept, via theoretical and experimental studies of the proposed AORM architecture. Furthermore, the AORM architecture is analyzed via a link budget (i.e., an analysis of the received power of a link that is subject to various gain and loss mechanisms) for its application within passive DLs.     11 1.2. Thesis Scope and Outline This thesis investigates two photonic architectures for the distinct applications of fibre optic front-end systems and FSO communication links. The two architectures rely upon similar core principles of electromagnetics and material science, and these principles are introduced in Chapter 2 and Chapter 3 through preliminary studies. The application of these architectures are then shown in Chapter 4 and Chapter 5. Chapter 2, preliminaries on semiconductor free-carrier dynamics, puts forward a preliminary investigation on the underlying semiconductor free-carrier dynamics used in this thesis for all-optical switching. Section 2.1 presents the theory for all-optical switching in a planar semiconductor via cross absorption modulation. The theory shows the changes in reflection, transmission, and absorption of the signal beam (in section 2.1.1) that come about due to the free-carrier dynamics with a planar semiconductor, according to Drude theory (in section 2.1.2). Section 2.2 presents the time-resolved pump-probe spectroscopy experimental setup used throughout this thesis (in section 2.2.1) as well as preliminary results for well-known semiconductors as points of reference (in section 2.2.2). These experimental results are used to verify the theory presented in this chapter. Section 2.3 presents some key conclusions, based upon the theory and experimental results, on the enhancement of switching energies and switching times for all-optical switching. Chapter 3, preliminaries on photonic nanojets, puts forward theoretical and experimental analyses for focusing with dielectric spheres. Section 3.1 presents ray theory and Mie theory to describe light scattering for milli-scale and micro-scale regimes. Section 3.2 presents experimental results on the implementation of photonic nanojets for localized photoinjection. The focal geometry’s effects on localized photoinjection are experimentally verified with pump-probe  12 spectroscopy and theoretically verified with the theory from section 2.1 on all-optical switching in planar semiconductors. Section 3.3 presents some key conclusions on the use of photonic nanojets for all-optical switching. Chapter 4 introduces the AOS architecture for all-optical switching and processing, with the primary application being optical fibre front-end systems. To achieve practical all-optical switching, the goals of femtojoule switching energies and femtosecond switching times are pursued. The proposed architecture integrates photonic nanojets and semiconductor nanoparticles to realize these two goals, while enabling omnidirectionality, a small footprint, and monolithic integration with the focusing element. Section 4.1 presents the theory for all-optical switching in spherical semiconductor nanoparticles. Section 4.2 presents results for the proposed AOS architecture with dielectric spheres in milli-scale and micro-scale regimes. Section 4.3 shows experimental tests of semiconductor nanoparticles with varying compositions (to optimize the AOS architecture). Section 4.4 summarizes the performance of each semiconductor material and demonstrates the performance of the overall AOS architecture. Chapter 5 introduces the AORM architecture for all-optical retro-modulation, with the primary application being free-space optical communication systems. The goal for the work in this chapter is to establish numerous multidirectional FSO links at terabit-per-second data rates, without escalating the system mass and power. To realize this goal, the proposed architecture implements retro-modulation with a spherical retroreflector and a semiconductor thin film being comprised of CuO nanocrystals. Section 5.1 presents theoretical and experimental results to convey the underlying free-carrier dynamics within the CuO nanocrystals. The fabrication process for the CuO nanocrystals is shown in section 5.1.1, and the absorption characteristics are shown in section 5.1.2. The free-carrier dynamics and the bandstructure of CuO nanocrystals are interpreted in  13 section 5.1.3. The interpretation is validated with experimental time-resolved pump-probe spectroscopy measurements with varying control beam fluences in section 5.1.4. The findings are summarized in section 5.1.5. Section 5.2 then presents the proposed AORM architecture. As a proof of concept, a link budget analysis of passive DLs with the proposed AORM architecture is presented in section 5.2.1. Three link budget conditions are identified and are considered in the design of the proposed AORM architecture in section 5.2.2. Experimental results for the AORM architecture are presented in section 5.2.3. The results are summarized and compared to the link budget conditions in section 5.2.4. Chapter 6 concludes the thesis. A summary of the contributions made by this thesis is given in section 6.1 and suggestions for future work is given in section 6.2. Overall, two new photonic architectures are introduced in this thesis. The AOS architecture is developed for all-optical switching, with the primary application being optical fibre front-end systems. The AORM architecture is developed for all-optical retro-modulation, with the primary application being free-space optical communication systems. The photonic architectures are designed to meet the goals of these applications—and in doing so lay the groundwork for future implementations of all-optical networks.   14 Chapter 2  Preliminaries on Semiconductor Free-Carrier Dynamics In this chapter, the theory for all-optical switching in planar semiconductors is presented, with elementary experimental results, to lay the foundation for the proposed all-optical switch (AOS) architecture. Fundamentally, all-optical switching involves one beam inducing material changes that affect the reflected, transmitted, and absorbed optical powers of a second coincident beam. Such changes can be accomplished by optically inducing perturbations to the refractive index and/or absorption coefficient of the material. For the work in this thesis, semiconductors are used as the materials of choice because they enable strong light-material interactions. These strong interactions enable cross absorption modulation between the beams over short propagation distances within the semiconductors, which is an important characteristic for the applications in this thesis. Cross absorption modulation is similar to the more commonly known process of cross gain modulation, in that a control (pump) beam linearly modulates a coincident signal (probe) beam. However, for cross absorption modulation, the control beam modulates the losses seen by the signal beam, rather than the gain45. Cross absorption modulation is implemented with a control beam having a photon energy that is roughly resonant with the semiconductor’s bandgap. This yields photoexcited free-carriers, being highly mobile electron populations in the conduction band and less mobile hole populations in valence band. The photoexcited free-carriers perturb the semiconductor’s refractive and absorptive characteristics, via corresponding changes to the refractive index and absorption  15 coefficient, and this leads to variations in the reflected, transmitted, and absorbed power of the signal beam. Such changes from the optically-injected free-carriers are the foundation of semiconductor-based all-optical switching. For all-optical switching to be effective, the interaction between the control and signal beams should be both strong (to allow the control beam to be applied with a sufficiently small switching energy) and fast (to allow the signal beam to recover with a short switching time). To realize such effective all-optical switching, the underlying theory is developed here to describe the change in reflected, transmitted, and absorbed powers of the signal beam and the underlying free-carrier dynamics within the semiconductor following photoexcitation by the control beam. The theory is presented in section 2.1, and experimental results for its validation are presented in section 2.2. Section 2.3 puts forward the key conclusions and design recommendations from the theory for the realization of AOS architectures with reduced switching energies and reduced switching times.    16 2.1. Theory for All-Optical Switching in Planar Semiconductors This section develops a theory to describe all-optical switching between control and signal beams within a planar semiconductor. In section 2.1.1, the reflection, transmission, and absorption characteristics of the signal beam are analyzed. This is done to associate changes in the refractive and absorptive properties of the semiconductor to changes in the reflection, transmission, and absorption of the signal beam. In section 2.1.2, the free-carrier dynamics within the planar semiconductor are analyzed, via Drude theory, to define the control-beam-induced changes in the refractive and absorptive properties of the semiconductor.  2.1.1. Reflection, Transmission, and Absorption Characteristics The goal of this section is to link the control-beam-induced changes in a planar seminconductor’s refractive and absorptive properties to the reflected, transmitted, and absorbed powers of the signal beam passing through that material. In general, for a beam of light propagating across the planar interface from air into the semiconductor, portions of the beam’s optical power will be reflected, transmitted, and absorbed. The proportion of reflected power at normal incidence is defined by the reflectance,  ^ = _` − 1_` + 1 2 = _` − 1 D + cD_` + 1 D + cD, (1) where _` = _` + dc is the semiconductor’s complex refractive index46, _` is real refractive index, and c is the extinction coefficient. The non-reflected optical power propagates into the bulk of the material and undergoes absorption according to the absorptance, e. The absorptance is defined by  17 the bulk material’s absorption coefficient, f, and the length over which the absorption occurs, being defined here as the effective length gh. This gives an absorptance of   e = i-jkl, (2) for a collimated light beam propagating along the z-axis normal to the material. Note the absorption coefficient, f, is related to the extinction coefficient, c, according to f = 2mc n, where m is the signal beam’s angular frequency and n is the speed of light47. The absorptance seen within this thesis is due to induced free-carrier absorption47 and state-filling49. The overall transmittance, o, through one interface at normal incidence is then defined by  o = 1 − ^ e. (3) In general, all-optical modulation can be induced via changes to the material’s real refractive index, Δ_` p , and/or absorption coefficient, Δf p , as a function of time, p . For an induced change at the interface, there will be a change in reflectance, Δ^ p . The change in Δ^ p  due to Δ_` p  and Δf p  can be solved for using differential calculus, given by  Δ^ p = q^q_` Δ_` p + q^qf Δf p . (4) For GaAs and the other semiconductors used in this work c is known to be small at the signal beam’s wavelength of 1550 nm51. Thus, for the case where c ≪ _`, the Δf p  term in Equation (4) will be zero as q^ qf ≈ 0. This simplifies the expression to be  Δ^ p = 4 _` − 1_` + 1 3 Δ_` p . (5) With Equations (1) and (5), the normalized reflectance change, Δ^ p ^, is  Δ^ p^ = 4Δ_` p_`2 − 1 . (6)  18 With a similar set of steps, the change in transmittance, Δo p , is found to be  Δo p = qoq_` Δ_` p + qoqf Δf p = −i-jkl 4 _` − 1_` + 1 3 Δ_` p − ghi-jkl 1 − ^ Δf p . (7) With Equations (2), (3), and (7), the normalized change in transmittance is solved for and renamed as the normalized differential transmission, Δo p o, to have it be consistent with terminology in the literature. This gives a normalized differential transmission of  Δo po = − _` − 1_` _` + 1 Δ_` p − ghΔf p . (8) Equations (6) and (8) dictate the modulation depth and speed of an all-optical switch through the dynamics of Δ_` p  and Δf p . The precise nature of these induced changes, Δ_` p  and Δf p , are analyzed in the next section, 2.1.2, for a planar semiconductor interface. Appendix A shows a similar analysis to that of this section, but with contributions to reflective losses for the entire experimental system, including both interfaces of the semiconductor slab and all optical elements in the setup. It is ultimately shown that the analysis of a single planar semiconductor interface in this section is equivalent to the analysis of the entire experimental system. 2.1.2. Planar Free-Carrier Dynamics The goal of this section is to characterize the free-carrier dynamics within a planar semiconductor and link these dynamics to the changes in refraction and absorption, being Δ_` p  and Δf p , respectively. In doing so, the key processes of free-carrier diffusion and recombination are introduced and ultimately exploited to shorten the all-optical switching times. In this work, all-optical switching is implemented through cross absorption modulation, whereby a control (pump) beam modulates the absorption of a coincident signal (probe) beam45. Essentially, the control beam amplitude is mapped onto the signal beam. A semiconductor is an  19 effective material to carry out such cross absorption modulation because it can offer strong light-material interaction. The light-matter interaction is strengthened in this work by having the photon energy of the control beam be resonant with the semiconductor’s bandgap. The resulting photoexcited free-carriers, which are predominantly considered to be electrons in the conduction band, given that holes in the valence band have a much lower mobility (i.e., a high effective mass), induce changes to the semiconductor’s refractive index, ∆_I p , and absorption coefficient, Δf p . For a signal beam with a photon energy below the semiconductor’s bandgap, the induce changes ∆_I p  and Δf p  will yield the normalized differential transmission of the signal beam, yo p o, shown in Equation (8). To model the ∆_I p  and Δf p  responses, the complex dielectric function, z p , is used in accordance with Drude theory. Drude theory is limited to describing the free-carrier response of semiconductors for only photon energies below the threshold of band transitions47, which is the case for the signal beam used in this work. Furthermore, the signal beam used in this work is far below the bandgap (i.e., the onset of resonance), such that there is negligible dispersion effects from bound electrons, as described by Lorentz theory48. Thus, the Lorentz theory is not included in this analysis. According to Drude theory47, free-carriers are subject to the signal beam’s electric field, being  { p = {0e| }h~Ä , (9) with an angular frequency of m, and a wavenumber of Å, for propagation in the Ç direction. The free-carriers are displaced by É p  according to the equation of motion  ÑÖ2É pÖp2 +ÑÜ ÖÉ pÖp = −i{ p , (10)  20 where Ñ is the effective mass, Ü is the scattering rate, and i is the electron charge. Note here that the scattering rate, Ü, is related to the mobility, á, by Ü = i Ñá . For simplicity, the effective mass is taken to be time independent, which assumes a parabolic band structure and negligible intervalley scattering (between valleys which otherwise would have differing masses). The mobility is also assumed to be time independent for this simplified model, although generally the mobility is dependent on the free-carrier density according to the empirical Caughey-Thomas relationship61. The steady-state solution for É p , given the harmonic form of the signal beam’s electric field, is then  É p = iÑ m2 + dmÜ { p . (11) This steady-state (single frequency) solution is used to approximate the ultrafast pulsed (broad spectrum) signal beam used in this work. This is a fair approximation because the pulse duration, being 150 fs, is much greater than the oscillation period of the electric field, being 5 fs. This É p  displacement for each free-carrier can be summed over the bulk to produce a macroscopic polarization in the semiconductor of  à p = −iâ p É p , (12) where â p  is the free-carrier density at the semiconductor interface. Finally, the polarization is linked to the dielectric displacement field, ä p , by  ä p = z0{ p + 	à p = z0z p { p , (13) where z0 is the permittivity of free space. The dielectric function, z p , given here can be found by substituting Equation (11) into Equation (12) and substituting the result into Equation (13). This gives  21  z p = 1 − i2â pz0Ñ 1m2 + dmÜ. (14) The real, z1 p , and imaginary, z2 p , parts of dielectric function are  z1 p = 1 − i2â pzXÑ 1m2 + Ü2, (15)  z2 p = i2â pz0Ñ Ü pm m2 + Ü2 . (16) The relationship between the refractive index and z p  is simply _` p = z p , where the magnetic permeability is taken to be equal to one, as is the case for most semiconductors. This work now deviates from the standard approach to Drude theory, taken by Maier et al.47 and others, by seeking ∆_I p  and Δf p 	for the specific case of an induced free-carrier density of Δâ p  being formed by the control beam. To find ∆_I p , it is assumed once again that the semiconductors used in this work has an extinction coefficient, c, that is small at the signal beam’s wavelength of 1550 nm. Such is the case for GaAs51. Thus, the imaginary part of the dielectric function, z2 p , is also small, as c = z2 p 2_` , resulting in _` p ≈ z] p . The derivative of z] p  with respect to â p  is taken to find ∆_I p , resulting in  ∆_I p = Ö z] pÖâ Δâ p = Öz] pÖâ Δâ p2_` = −i2Δâ p2_`z0Ñ 1m2 + Ü2. (17) Equation (17) can be simplified further by noting that the scattering time, 1 Ü, being typically hundreds of femtoseconds50, is much larger than the period of the signal beam’s electric field, being roughly 5 fs for a wavelength of 1550 nm. This simplifies the semiconductor’s refractive index change, as seen by the signal beam, to  22  ∆_` p = − i2Δâ 	p2z0_IÑm2. (18) In a similar fashion, Δf p  can be solved for by taking the derivative of the absorption coefficient, f p . It can be derived by first substituting the complex refractive index, _` = _` + dc , into Equation (9). With the definition of the wavenumber, Å = m_` n, where n is the speed of light, Equation (9) becomes  { p = {0e| ãåh ç~Ä = {0e| ãéh çè|êh ç~Ä = {0e-êh çe| ãéh ç~Ä . (19) The absorption coefficient is defined according to the exponential decay of the signal beam’s intensity, which is proportional to the square of the electric field, { p , so the absorption coefficient is given by  f p ≡ 2mcn = mn z2 p_I p , (20) where the definition of the dielectric function has been used to establish z p = _` p 2 = _`2 −c2 + d2_`c = z1 + dz2. For the case where ∆_` p ≪ _I p , the change in absorption is  Δf p = 	+ i2ÜΔâ pnz0_`Ñm2. (21) The Drude expressions for ∆_` p  and Δf p , seen in Equations (17) and (20), are manifestations of free-carrier dispersion52 and free-carrier absorption53, respectively. They can be substituted into Equation (8) to yield an expression for the normalized differential transmission, yo p o, of  Δo po = i2zX_IÑm2 _I − 12_I _I + 1 − ghÜn Δâ p . (22) Ultimately, Equation (22) describes the normalized differential transmission of the signal beam due to the free-carriers induced by the control beam. It is clear from such an expression that the  23 signal beam can be modulated with a short switching time if the free-carriers are both rapidly photoexcited, which is a valid assumption given the availability of ultrashort optical pulses for the control beam, and rapidly recombined. The recombination of free-carriers is examined in the remainder of this section. To describe the evolution in time of Δâ p 	in Equation (22), a one-dimensional differential equation is constructed to model the diffusion and recombination of the free-carriers along the Ç direction, being normal to the plane of the semiconductor. The system is illustrated in Figure 2.1 and is governed by  qâ Ç, pqp = âXg p e-h/jk	 − â Ç, pí9 + ì∇Dâ Ç, p , (23) where the three terms on the right side characterize the respective processes of photoinjection, recombination, and diffusion, in agreement with Beard et al.54. The planar semiconductor is assumed to have a thickness that is much greater than the photoinjection depth,	gh. The incident control beam photoinjects free-carriers into the planar semiconductor on the timescale of the control beam’s pulse duration. This study uses a control beam with an ultrafast pulse duration, which can be approximated as a delta-function, g p . The initial distribution for â Ç, p 	is an exponential decay into the semiconductor, with an initial free-carrier density of âX at the surface and a 1 e photoinjection depth of gh into the bulk, as shown in Figure 2.1. The photoinjection depth, gh , is defined here as a general parameter that can be less than or equal to the semiconductor’s penetration depth. Given a suitable focal geometry, with tight focusing and a short Rayleigh range penetrating into the semiconductor, gh  can be made to be less than the semiconductor’s penetration depth.  24 The bulk recombination term in Equation (23) exhibits a linear proportionality to â Ç, p  and a rate that is inversely proportional to the effective bulk recombination lifetime, í9 . This bulk recombination lifetime is due primarily to trap-assisted recombination. In general, other recombination processes may exist in the semiconductor and exhibit higher order (nonlinear) dependencies on â Ç, p , such as band-to-band recombination which would appear in Equation (22) via a dependency on	âD Ç, p . Such higher-order dependencies are investigated in Appendix B, where it is shown that trap-assisted recombination plays the dominant role in this work in depleting free-carriers. The appendix does show that non-trap-assisted recombination processes can manifest themselves through dependencies between the free-carrier lifetime and control beam fluence, i.e., â Ç, p , but these dependencies become significant only at high free-carrier densities. These fluence dependent changes create at most a 25% change to the free-carrier recombination lifetime. For this reason, and the fact that an assumption of a linear bulk recombination term allows  Figure 2.1. Initial conditions for the free-carrier density within the planar semiconductor. The incident control pulse photoinjects free-carriers at ï = 0 with an exponential decay in free-carrier density along the optical axis, ñ. This free-carrier density distribution then evolves in time via diffusion and recombination. surfaceincidentcontrol beamN(z,t) = N0Ɂ(t)eǦœȀɁzzt = 0semiconductornsn = 1 25 Equation (23) to be solved algebraically, the theory developed here only considers trap-assisted recombination. The boundary condition for â Ç, p 	in Equation (23) is applied at the semiconductor surface, and it adheres to  ì qâ Ç, pqÇ hó0 = ò5â Ç, p hó0 (24) for a diffusion coefficient of ì and surface recombination velocity of ò5. Given a suitable material system, with a sufficiently high density of surface states, this boundary condition suggests that the overall free-carrier lifetime, í, can be reduced to a value below the bulk free-carrier lifetime, í9. The solution to the one-dimensional differential equation, Equation (23), with the applied boundary condition, Equation (24), defines the free-carrier density, â Ç, p , which then defines the differential transmission, Δo p o, of the signal beam, according to Equation (22). As Equation (22) is purely a function of time and not space, an effective free-carrier density, â p , is defined here as a spatial average over the penetration depth, gh. If the bulk lifetime, í9, in Equation (23) is sufficiently long with respect to the timescales of the other processes, as is typically the case, the differential equation is simplified to  qâ Ç, pqp = âXg p e-h/jk	 + ì∇2â Ç, p , (25) which can be solved analytically to give insight into the temporal and spatial evolution of the free-carrier density. Appendix D shows the full derivation of the analytical solution. The final solution for the normalized free-carrier density is  26 â Ç, pâX = 1f − ò5/ì 1ôìp e -höõúÄ − ò5ì eùûú ùûÄèh erfc Ç + 2ò5p2 ìp+ ò5/ì + fò5/ì − f 12f ôìp e -höõúÄ − 12 el lúÄèh erfc Ç + 2fìp2 ìp− 12f ôìp e -höõúÄ − 12 el lúÄ~h erfc Ç − 2fìp2 ìp + âXe-lhelöúÄ. (26) For improved confidence in the analytical solution, Equation (23) is also solved for a finite bulk lifetime, í9, using an inverse Laplace numerical routine. Testing with this numerical routine shows that its operation with í9 = ∞ agrees with the analytical solution in Equation (23). In the subsequent section, the free-carrier density, â Ç, p , is calculated and then spatially averaged to form the theoretical response for Δâ p  in the well-known semiconductor GaAs. This result is then used to form the theoretical response for the differential transmission, ∆o p o, which is compared to the experimental response for GaAs. Such a process is used to validate the proposed theory—and reveal means to reduce the switching energy and switching times of AOS architectures.   27 2.2. Experimental Validation In this section, the procedures and results from experimental pump-probe spectroscopy are given. In section 2.1.1, the experimental setup is introduced and nominal experimental results are shown for the semiconductors GaAs, Si, and SiC. In section 2.2.2, the theory for all-optical switching in planar semiconductors from the prior section is validated against experimental results for the (well-known) semiconductor GaAs. 2.2.1. Pump-Probe Spectroscopy Experimental Setup This section presents the experimental setup that is used for time-resolved pump-probe spectroscopy in this thesis, as well as nominal experimental results generated by the setup for three semiconductors. The term pump-probe spectroscopy is used here and throughout the thesis, to be consistent with the literature, but it should be recognized that the experimental setup used for pump-probe spectroscopy is a rudimentary form of all-optical switch. Thus, the pump and control beams are equivalent, and the probe and signal beams are equivalent. The experimental setup for pump-probe spectroscopy is shown in Figure 2.2(a) and (b). The setup in Figure 2.2(a) uses the 780 nm and 1550 nm output beams from a pulsed erbium-doped fibre laser (Toptica FFS-SYS-2B) having a repetition rate of 100 MHz and pulse duration of 150 fs at the sample’s location. The synchronized 780 nm and 1550 nm output beams form the respective control and signal beams, with a moving delay stage introducing a variable time delay between the pulses in the control and signal beams. The setup in Figure 2.2(b) uses the control and signal beams to perform pump-probe spectroscopy. The beams are overlapped with a beamsplitter and focused with a 20× microscope objective onto the semiconductor sample of interest with a spot size of 6 ± 1 µm, as defined by the experimental optical encoder shown in  28 Appendix C. In this way, the ultrashort optical pulses in the pump beam impart a material-dependent impulse response onto the ultrashort optical pulses in the probe beam. The differential transmission, ∆o(p)/o, of the signal beam is then measured as a function of the relative control-signal time delay by an InGaAs detector and a lock-in amplifier (Stanford Research Systems, SR830) using a 100 ms time constant. The control beam is modulated with an optical chopper at frequency of 1 kHz. This modulation is imprinted onto the signal beam via the all-optical switching medium, and the lock-in amplifier is carries out narrowband filtering of the signal beam precisely at the frequency of the control beam. This removes the effects of ambient noise sources. An optical low pass filter is used remove the control beam and pass the signal beam before the detector. Such a setup is used with control and signal beams having respective wavelengths of 780 and 1550 nm (i.e., photon energies of 1.6 and 0.8 eV) for the GaAs and Si samples and respective wavelengths of 390 and 1550 nm (i.e., photon energies of 3.2 and 0.8 eV) for the SiC sample. A second harmonic generation crystal is used in the setup to form the control beam at a wavelength of 390 nm. The semi-insulating GaAs sample is an undoped double-sided polished wafer with a thickness of 350 µm, a high resistivity, and an orientation  of <100>. The float-zone Si sample is an undoped double-sided polished wafer with a thickness of 280 µm thick, a high resistivity, and an orientation of  <100>. The 6H-SiC sample is an n-doped single-sided polished wafer with a thickness of 330 µm, a low resistivity, and an orientation of <0001>. Figure 2.3 shows the experimental (negative) differential transmission, −∆o p /o, for the three semiconductors. The negative signal polarity witnessed here is noteworthy. It indicates that the control-beam-induced modulation to the signal beam is dominated by the second term in brackets in Equation (22), being negative and attributed to absorption in the bulk for all three  29 semiconductors. Such an observation agrees with the long (bulk-dominated) lifetimes measured in other studies and seen here. The GaAs (blue) curve exhibits a picosecond-scale lifetime of í = 350 ps. The Si (green) curve exhibits a nanosecond-scale lifetime55 of í = 20 ns. The SiC (red) curve exhibits a microsecond-scale lifetime56 of í = 20 µs. The fast transient at the onset of the SiC response is due to free-carrier scattering, as will be discussed later. Of particular note is the fact that all three semiconductors exhibit unacceptably slow recovery times for the purposes of all-optical switching. There are minimal contributions from surface recombination via traps at the semiconductor surfaces, which would serve to reduce the recovery times. This is despite the fact that the GaAs54,57, Si29, and SiC29 samples have a wide range of surface recombination velocities, being ò5 ≈ 8×103 m/s, ò5 ≈ 500 m/s, and ò5 ≈ 4000 m/s, respectively.  30  Figure 2.2. Time-resolved pump-probe spectroscopy experimental setup. (a) Schematic of the time-resolved pulsed pump-probe laser system. (b) Schematic of the experimental setup. A 780 nm control (pump) beam is used for the GaAs and Si samples, and a 390 nm control (pump) beam is used for the SiC samples. A 1550 nm signal (probe) beam is used for all the samples. *A second harmonic generation (SHG) crystal and bandpass filter are introduced into the beam path to form the 390 nm control (pump) beam. **The focusing element takes the form of a microscope objective or dielectric spheres, according to the experiment being performed.  1550 nm780 nm1550 nm probe780 nm controlBeamsplitterExperimental SetupFemtosecond LaserMoving Delay Stagedelay stagedelay780 nm control beamSHG crystal*EDQGSDVVȴOWHU*dichroicbeamsplitterfocusingelement**semiconductortargetInGaAs detectorZLWKEDQGSDVVȴOWHU1550 nm signal beamab 31  2.2.2. Experimental Results The goal of this section is to model the theoretical response of the differential transmission response, ∆o p o, based on the free-carrier density, â Ç, p , as a function of time and space, and test the response against experimental results. The well-known semiconductor GaAs is used for the test. The theory from section 2.1 is applied to GaAs using the numerical routine in Appendix H and the parameters given in Table 2.1. To this end, the theory is used to convey the processes and trends that manifest themselves at semiconductor interfaces, rather than simply extract material parameters, and this knowledge is later used to optimize the material and structure for all-optical switching.  Figure 2.3. Elementary results for all-optical switching. Experimental (negative) differential transmission of the signal beam, – °¢ ï /¢, shown normalized as a function of delay time, for photoinjection of the bulk Si, SiC, and GaAs samples, using a microscope objective as the focusing element, as depicted in the inset.   Bulk targetɒ = 350 psɒ ƉVɒ = 20 nsBulk SiBulk SiCBulk GaAs0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 50 100 150 200 250 300 Time Delay, t (ps) -)T(t)/T (a.u.) 32 The photoinjection depth, gh, is made equal to the penetration depth of GaAs, being 680 nm, for a control beam with a wavelength of 780 nm. This condition is realized experimentally by ensuring that the depth of focus of the control beam is much larger than its penetration depth58. (Shorter photoinjection depths do impact the free-carrier lifetime and this impact undergoes an extensive investigation in Chapter 3.) The values in the table for the surface recombination velocity54, bulk free-carrier lifetime54,57,59, refractive index51, and electron effective mass60 (where Ñ£ is the electron rest mass) are taken from the literature for GaAs, although it should be noted that the literature values for surface recombination velocity and bulk free-carrier lifetime can differ by an order of magnitude54,57,59. In this work, a surface recombination velocity of 8000 m/s is used, in accordance with Beard et al.54, and a bulk free-carrier lifetime of 10 ns is used because it lies Table 2.1. Parameters used to generate the theoretical response for all-optical switching in planar semiconductors. The parameters are for GaAs and they are used according to Equation (23) and the boundary condition given in Equation (24). Name Symbol Value Control (pump) beam wavelength §J 780 nm Signal (probe) beam wavelength §I 1550 nm Photoinjection depth58 gh 680 nm Surface recombination velocity54 ò5 8000 m/s Bulk free-carrier lifetime54,57,59 í9 10 ns Semiconductor refractive index51 _` 3.37 Electron effective mass60 Ñ 0.067Ñ£ kg Initial free-carrier density âX 1017~1019 cm-3 Free-carrier mobility62,63 á 400 ~ 500 cm2/V/s Diffusion coefficient ì 3 cm2/s  33 within the range of literature values54,57,59 (1 ns to 15 ns) and it allows the theoretical results to match well with the experimental results. The value for the mobility used in the table is determined from the results of Figure 3.5 in Chapter 3. The applied mobility of á ≈ 400 ~ 500 cm2/V/s replicates the polarity flip seen in the results of that chapter, which comes about when the two terms in brackets in Equation (22) sum to zero via the scattering rate Ü p = i Ñá . Note that the use of a constant mobility here is an approximation, as the mobility will in general depend upon the free-carrier density. The mobility dependence on the free-carrier density for GaAs can be defined according to the Caughey-Thomas empirical relationship61, if warranted, and has been analyzed by Mics et al.62. The value used in this work matches roughly with the mobilities measured by Joyce et al.63 and Mics et al.62 at similar free-carrier densities, based upon their quoted photoexcitation fluences. The value in the table for the initial free-carrier concentration, â0, is calculated according to its proportionality to the control pulse intensity. As such, the initial free-carrier concentration is  âX = •J¶ℏmJ®, (27) where, ¶ is the internal quantum efficiency,	ℏ = ℎ/2ô is the reduced Planck’s constant, mJ is the angular frequency of the control pulse, •J is the energy of the control beam, and ® ≈ e∅gh is the photoinjection volume set by the control beam’s cross-sectional photoinjection area, e∅ , and photoinjection depth, gh. Assuming a quantum efficiency, ¶, equal to one, Equation (27) can be applied to a pulsed control beam with an incident power of Z´, a beam spot size radius of 9¨, and a laser repetition rate of ^R≠Æ. This gives an initial free-carrier concentration of  âX = Z´§Jℎnghô 9¨2^R≠Æ. (28)  34 Using Equation (28), the experiments carried out in this work have âX  values ranging over 1017~1019 cm-3, as given in Table 2.1. The analyses of fluence dependencies for GaAs in Appendix B use âX over this range. The remaining value in the table, being the diffusion coefficient, D, is extracted by curve-fitting the theoretical and experimental differential transmission responses. In doing so, the theoretical free-carrier density, â Ç, p , is first calculated. The theoretical response for the free-carrier density, â Ç, p , is shown in Figure 2.4(a) for a bulk free-carrier lifetime of í9 = 10 ns and an initial free-carrier density of âX ≈ 5×1017 cm-3. The carrier density begins at p = 0 ps with an exponential distribution defining its depth along the z axis according to the initial conditions, which appears as the first term on the right side of Equation (23). Subsequently, at the semiconductor surface, free-carriers rapidly recombine due the high density of surface states and the appreciable surface recombination velocity in GaAs. In the semiconductor bulk, the processes of bulk recombination and diffusion are witnessed, in accordance with the latter two terms on the right side of Equation (23). Through the overall process free-carriers diffuse from the bulk toward the surface, where they undergo surface recombination. Figure 2.4(b) depicts the corresponding theoretical and experimental (negative) differential transmission responses, −∆o(p)/o . The experimental differential transmission of GaAs is obtained from time-resolved pump-probe spectroscopy, as described in section 2.2.1. The semi-insulating GaAs sample is an undoped double-sided polished wafer with a thickness of 350 µm, a high resistivity, and an orientation  of <100>. The theoretical and experimental responses show good agreement, with an R-squared value of 0.995, for a diffusion coefficient of ì ≈ 3 cm2/s. However, this diffusion coefficient is lower than expected according to the Einstein relationship and the literature54. This discrepancy is attributed to ì being used as the only fitting parameter, while basing other parameters on the literature. This constrains  35 the model’s degrees of freedom and does not allow the model to factor in the complexities of spatially varying and time varying parameters. Overall, however, this simple model accurately describes the experimental ∆o(p)/o response of GaAs and the parameters in Table 2.1 are in rough agreement with the literature values. Further theoretical investigations of the theoretical response show that that ∆o(p)/o can have its amplitude enlarged by increasing âX and its recovery time reduced by promoting surface recombination. Such observations are leveraged in Chapter 3 and Chapter 4 to bring about improved performance for all-optical switching.    36  Figure 2.4. Modelled free-carrier dynamics for a planar GaAs bulk sample. (a) The free-carrier density, Ø ñ, ï , is shown as a function of time and space according to Equation (23) for the GaAs bulk sample. The Ø ñ, ï  curves are shown with time increments of 50 ps. (b) The differential transmission response, −∆¢ ï /¢ , theoretical (black) and experimental (blue) curves are shown for the GaAs bulk sample. The theoretical response is obtained with Drude theory based on Equation (22) and the time-evolving free-carrier density. The experimental response is obtained from pump-probe spectroscopy using a 20× microscope objective.  0 50 100 150 200 250 300 Time Delay, t (ps) bGaAs Bulk target0 0.2 0.4 0.6 0.8 1.0 1.2 -)T(t)/T (a.u.)Free-carrier Density [×1017 cm-3]Depth (nm) 0 200 400 600 800 1000 012345t = 0 pst = 50 pst = 100 pst = 150 psa 37 2.3. Summary & Discussion Based on the theoretical and experimental results presented in this chapter, some conclusions can be made on enhancing the switching energies and switching times for all-optical switching. Femtojoule switching energies can be achieved by minimizing the control pulse energy, •Æ, that is needed to establish the required initial free-carrier density, âX, in the first term of Equation (23). According to Equation (27), localized photoinjection within a reduced volume, ® ≈ e∅gh, can enable such lower switching energies. In Chapter 3, localized photoinjection is applied to do this by way of a nanojet focal geometry, to yield a reduced focal spot size, i.e., a small photoinjection cross-sectional area, e∅ , and short Rayleigh range, being equal to the photoinjection depth, gh. This leads to reduced photoinjection volumes and higher initial free-carrier densities without having to raise the control pulse energy. Femtosecond switching times can be achieved by promoting surface recombination and thus minimizing the overall free-carrier lifetime, í , as a result of Equation (23) and its boundary condition in Equation (24). Localized recombination at the semiconductor surface, where free-carriers rapidly recombine, can enable ultrafast switching times. The free-carrier lifetime can be reduced according to the recombination rate relation of 1 í = 1 í9 + ò5^. Thus, an increase in the surface recombination velocity, ò5, or the surface-to-volume ratio, ^, has í decrease below the bulk free-carrier lifetime, í9. Prior studies have shown that nano-scale cylindrical28 and spherical29 semiconductor forms exhibit this trend, via their increased surface-to-volume ratios, which increases the surface state density and decreases the free-carrier lifetime. In Chapter 4, localized recombination is applied to realize such a response by way of nanoparticle material systems. The nanoparticle material systems have a high density of surface states, and they can be readily applied to the focal geometry of the proposed photonic nanojets.   38 Chapter 3  Preliminaries on Photonic Nanojets In this chapter, the concept of the photonic nanojet is investigated for use in all-optical switching. It is shown that an appropriately designed dielectric sphere can act as a superlens, focusing incident collimated light to a subwavelength focus just outside the back surface of the sphere. The protruding light can have a high intensity and is referred to as a photonic nanojet26,27. For the purposes of this work, the specific characteristics of a photonic nanojet are defined by the presence of a subwavelength full width at half maximum (FWHM) and a micro-scale propagation distance64. (Although, it should also be noted that alternative definitions do exist65.) The small focal spot of the photonic nanojet can enable exceedingly high intensities for the interacting control and signal beams—and this can be exploited to produce increased free-carrier densities and thus reduced switching energies in all-optical applications. The properties of the sphere must be chosen judiciously to attain the ideal characteristics of the photonic nanojet as there exists a complex relationship between the photonic nanojet’s intensity and the sphere’s refractive index, _, and diameter, Ö. The properties of the dielectric sphere that yield the optimal nanojet are defined in this work by theoretical analyses across two regimes: a milli-scale regime and a micro-scale regime. In section 3.1, ray theory and Mie theory are used to investigate light focusing for milli-scale and micro-scale dielectric spheres, respectively. In section 3.2, experimental tests are conducted on the use of photonic nanojets in realizing the process of localized photoinjection. In section 3.3, conclusions are presented on the application of photonic nanojets to all-optical switching.   39 3.1. Light Scattering Theories In this section, light scattering from a dielectric sphere is investigated for milli-scale and micro-scale regimes. The milli-scale regime is well-described by ray theory. Ray theory offers a means to model the propagation characteristics of Maxwell’s equations for dimensions that are much larger than the wavelength. In doing so the propagation characteristics are described by rays, being vectors that are perpendicular to the propagating wavefronts. Section 3.1.1 applies ray theory to determine the focusing characteristics of a milli-scale dielectric sphere. The micro-scale regime is well-described by wave theory, which follows Maxwell’s equations and manifests itself in various forms, depending upon the geometry of interest. For the proposed geometry of a homogenous sphere, Mie theory is particularly effective because it describes scattering of incident waves off of spheres66. Section 3.1.2 applies Mie theory to determine the focusing characteristics of a micro-scale dielectric sphere, i.e., a microsphere. Full derivations of ray theory and Mie theory are presented in Appendix E and Appendix F, respectively. 3.1.1. Ray Theory In the milli-scale regime, with sphere diameters that are much larger than the wavelength, focusing can be effectively described by ray theory. In doing so, the thick lens formula67 is applied to find the focal length, ∞ ≥ Ö 2, for a sphere with a diameter of Ö and refractive index of _. The thick lens formula yields a focal length of  ∞ = _Ö4 _ − 1 . (29) At the same time, it can be shown that the diameter of the focal spot for a focused Gaussian beam scales is proportion to 4§0 ô ∞ Ö , which means that it is desirable to minimize the ∞-number,  40 defined as ∞ Ö, to form a small and intense focal spot. The smallest ∞-number for a sphere is 1 2, because this yields focusing on the sphere’s back surface, and so according to Equation (28) the refractive index of the sphere must be _ = 2.0. The focal conditions brought about by such a sphere, with a focal length of ∞ = Ö 2, are attractive from a practical standpoint. The required semiconductor can be located on the sphere’s back surface, where it will be subject to intense focusing. Moreover, the spherical architecture provides omnidirectionality, ease of alignment, a small device footprint, and reduced need for external focusing elements. Using ray theory, for the spherical geometry, the relative intensity at the back surface of the sphere can be found as a function of the sphere’s refractive index, 1.5 < _ < 2.5. This relative intensity is shown in Figure 3.1. An intensity colourmap for this milli-scale regime is generated from the ray theory curve and is shown at the top of the figure. The ray theory calculations for this  Figure 3.1. Photonic nanojet intensity in the milli-scale regime. The theoretical intensity of a photonic nanojet, calculated with ray theory at the back surface of a dielectric sphere, is shown as a function of the sphere’s refractive index, ≤. An intensity colourmap is shown at the top of the figure. Refractive index (n)1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.510-210-1 10010-3Intensity [arb. units]Intensity Colourmap 41 intensity map, which apply Snell’s law to the given geometry, are shown in Appendix E. The resulting curve shows that high-intensity focusing comes about on the back surface of the sphere precisely at a refractive index of _ = 2.0. The peak intensity, corresponding with _ = 2.0 and shown in white, agrees with the optimal intensity given by the thick lens formula in Equation (29). The closest available material to exhibit the desired refractive index of _ = 2.0 is S-LAH79 glass, with a refractive index of _ = 1.98 ± 0.02 at a wavelength of 780 nm. This material is used for preliminary studies in section 3.2. 3.1.2. Mie Theory In the micro-scale regime, with sphere diameters that are comparable to the wavelength, i.e., microspheres, it is necessary to implement a rigorous three-dimensional electromagnetic analysis. This can be done efficiently for a spherical geometry using Mie theory26,27. Mie theory is implemented here using the formulation in Appendix F. The formulation that is applied agrees with the algorithm implemented by Lecler et al.68. The Mie theory simulations are carried out to identify the maximum intensities at the exit interface of the microsphere, intersecting with the optical axis, as a function of the microsphere’s refractive index, _, and diameter, Ö. The distribution of intensities resulting from over a hundred thousand individual Mie theory simulations is shown as an intensity colourmap in Figure 3.2. Each pixel is a separate simulation in the micro-scale regime, with diameters and refractive indices in the range of Ö < 30 µm and 1.5 < _ < 2. Overall, it is worth commenting on three characteristics within this figure: the general trend, resonance, and spherical aberration. The general trend of Figure 3.2 manifests itself as the displayed trendline of maximum intensities, shown as a solid grey curve. The trendline rises from a low refractive index at small diameters toward a high refractive index at larger diameters. Specifically, the maximum intensities  42  Figure 3.2. Photonic nanojet maximum intensities in the micro-scale regime. The theoretical intensity of the photonic nanojet, calculated with Mie theory at the back surface of a dielectric sphere, is shown as a function of the sphere’s refractive index, ≤, and diameter, ≥, for photoinjection at a wavelength of 780 nm. The intensity colourmap that is the result of over a hundred thousand individual Mie theory simulations. The maximum (white) intensity is normalized for all ≤ with constant ≥. A trendline for the maximum intensity is shown as a solid grey curve. Select Mie theory simulations, with logarithmic intensities, are shown in the figure insets as (a) a sphere with ≥ =	3 µm and ≤ = 1.76, and (b) a sphere with ≥ = 30 µm and ≤ =	1.83.  0 2015105 25 302.01.91.81.71.61.5Mie TheoryDiameter, d (µm)Refractive index, nabn = 1.83 spherea n = 1.83 sphereb76 43 form a broad band of intensities that peak at _ ≈ 1.75 for small Ö and then rise toward _ ≈ 2.0 for sufficiently large Ö, where the Mie theory and ray theory results merge. The resonance seen in Figure 3.2 manifests itself as two patterns of interferometric fringes. The first fringe pattern comes about from longitudinal modes that resonate along the optical axis and form the steep high-spatial-frequency fringes observed in the intensity map. The second fringe pattern comes about from coupled longitudinal and whispering-gallery modes69 that exhibit interferometric beating along the perimeter of the sphere and form the sloped low-spatial-frequency fringes observed in the intensity map. Although these fringes result from complex patterns, which are in general represented by vector spherical harmonics66, one can interpret the periodicity of the high-spatial-frequency fringes. The periodicity of these fringes comes about when the phase change from a round-trip of the resonant cavity yields an integer multiple of 2ô. Thus, for a wavenumber of Å = 2ô_ §0 and a sphere diameter of Ö, the resonance condition is 2ÅÖ = 2ô¥, where ¥ is an integer. This condition can be reformulated as ¥§0 2 = _Ö. The partial derivatives of ¥ can be taken with respect to _ and Ö to relate the differentials of each variable, according to Δ¥§0 2 = _ΔÖ + ÖΔ_. It is understood that Δ¥ = 1 for the integer change between adjacent fringes, and so the resonance condition becomes  _ΔÖ + ÖΔ_ = §0 2. (30) Such a result enables a rough description of what is seen for the high-spatial-frequency fringes in Figure 3.2. As an example, for a refractive index near _ = 1.8 and diameter near Ö = 10 µm, the horizontal fringe spacing is ΔÖ = 0.18 µm for a constant _, and the vertical fringe spacing is Δ_ = 0.035 for a constant Ö . These spacings agree with Equation (30) within a 20% error for the wavelength of 780 nm, which is due mainly to the idealization of the spherical cavity as a planar cavity. It is worth noting that the resonance seen here can be avoided, if there is a desire for reduced  44 sensitivity to structural and thermal fluctuations, by applying laser pulses with sufficiently short durations (being much shorter than the cavity lifetime) or by applying spheres with sufficiently large diameters (being at the upper end of the micro-scale regime or anywhere within the milli-scale regime). The spherical aberration in Figure 3.2 manifests itself as imperfect focusing at the back surface of the sphere. This leads to an increased demand on the switching energy, as the incident beam power must be increased to compensate for the imperfect focusing and reduced intensity. It should be noted, however, that spherical aberration is compensated to some degree using a full sphere. A full sphere can be considered to be two back-to-back plano-convex lenses separated by a plane-parallel plate. As such, the full sphere performs surprising well because spherical aberration will be under-corrected by the lenses and over-corrected by the plane-parallel plate70. The spherical aberration that does remain decreases as the refractive index increases. For example, using ray theory, we have found that a sufficiently high refractive index sphere, with _ ≈ 2.0, has a 42% reduction in (transverse) spherical aberration, compared to that of a sphere with a refractive index of _ ≈ 1.5. The degree of spherical aberration is quantified here by finding the difference in focal points for paraxial rays (close to the optical axis) versus marginal rays (far from the optical axis) for spheres have _ = 1.5 and _ = 2.0. The procedure used for applying ray theory in this analysis is shown in Appendix F. The theoretical results of Figure 3.2 suggest that the nanojet geometry can provide localized photoinjection in a scalable manner. High-intensity operation is achieved for a desired sphere diameter, Ö, given the appropriate selection of its refractive index, _, along the Mie theory curve. As the highest refractive indices reached by conventional polymers71 only approach _ ≈ 1.7, glass spheres are considered here for the creation of photonic nanojets. In particular, the commercially  45 available glass N-LASF9 has a refractive index of _ = 1.83, which is close to the desired region. To find the corresponding diameter for N-LASF9 glass, four microsphere diameters of Ö = 20 µm, 30 µm, 40 µm, and 50 µm, are analyzed. Figure 3.3 shows curves for the intensities at the back surfaces of these microspheres as a function of the refractive index, _. Each data point on the curves is an individual Mie theory solution. It is readily apparent from Figure 3.3 that microspheres with differing diameters have optimal (i.e., peak) intensities at differing refractive indices. For microsphere diameters of Ö = 20 µm, 30 µm, 40 µm, and 50 µm, peak intensities are achieved at the back surface of the microsphere for refractive indices of _ = 1.80, 1.82, 1.83, and 1.86, respectively. Clearly, greater refraction is  Figure 3.3. Characterization of photonic nanojet intensities formed by microspheres. Normalized intensity at the microsphere’s back surface is shown versus the microsphere’s refractive index, ≤ , for microsphere diameters of ≥ = 20 µm (purple), 30 µm (green), 40 µm (black), and 50 µm (red). The peak intensities are at ≤ = 1.80, 1.82, 1.83, and 1.86, respectively. Insets (a), (b), and (c) show the intensity at the microsphere’s back surface for ≥ = 40 µm and ≤ = 1.66, 1.83, and 2.00, respectively. The results are generated from Mie theory simulations at a wavelength of 780 nm. The scale bar is 1 µm. 1.7 2.0 2.11.6 1.8 1.9n = 2.00cn = 1.83bn = 1.66aabcd = 40 μmd = 50 μmd = 30 μmd = 20 μm0 0.2 0.4 0.6 0.8 1.0 Refractive Index (n)Normalized Intensity 46 needed for microspheres of increasing diameters. This is a result of the interplay between paraxial and non-paraxial focusing. The paraxial region of the beam forms a focal plane farther from the centre of the microsphere, in comparison to that formed by the non-paraxial region of the beam. Thus, for the desired focusing at the microspheres back surface, a large-diameter sphere with a large paraxial region requires a high refractive index, while a small-diameter sphere with a smaller paraxial region requires a low refractive index. In the limiting case of increasing sphere diameters, the refractive index that is required for focusing at the microsphere’s back surface is the limit quote earlier via geometrical optical analyzes at _ ≈  2.0. Ultimately, it is seen that a microsphere diameter of Ö	 = 40 µm, corresponding to the solid black curve in Figure 3.3, has its photonic nanojet intensity peak at the refractive index of N-LASF9 glass, _ = 1.83. The intensity profiles at the microsphere’s back surface are shown as insets in Figure 3.3 for an incident wavelength of 780 nm. Inset (a) shows the intensity profile for a refractive index of _ = 1.66. This relatively low refractive index leads to focusing beyond the microsphere and a low intensity at the microsphere’s back surface. Inset (b) shows the intensity profile for a refractive index of _ = 1.83. Inset (c) shows the intensity profile for a refractive index of _ = 2.0. This relatively high refractive index leads to focusing within the microsphere and a low intensity at the microsphere’s back surface. It can be concluded that a microsphere with a diameter of Ö = 40 µm and refractive index of _ = 1.83 is optimal for the formation of the desired high-intensity photonic nanojet. Figure 3.4 shows cross-sectional profiles for the optimal microsphere (with	Ö = 40 µm and _ = 1.83) at Ç = 0, Ç = 1 µm, and Ç = 1.68 µm from the microsphere’s back surface. The selected refractive index leads to pronounced focusing, with a subwavelength FWHM of 540 nm, being transverse to the optical axis at the microsphere’s back surface, and with a micro-scale  47 propagation distance of 1.68 µm, being the distance along the optical axis over which the beam remains subwavelength. Such characteristics meet the defined criteria for a photonic nanojet64.    Figure 3.4. Characterization of the photonic nanojet’s beam profile. Cross-sectional profiles are shown for the optimal ≤ = 1.83 microsphere at ñ = 0 (blue), ñ = 1 µm (red), and ñ = 1.68 µm (yellow) from the back surface microsphere. The photonic nanojet has a subwavelength FWHM of 540 nm, being transverse to the optical axis at the microsphere’s back surface, and with a micro-scale propagation distance of 1.68 µm, being the distance along the optical axis over which the beam remains subwavelength. The scale bar is 1 µm. -1.5 1.5 2.0-2.0 -0.5 0.50-1.0 1.0n = 1.83 1 μmz = 1.0 μmz = 0 μmz = 1.68 μm780 nmFWHMy (μm)yz0 0.2 0.4 0.6 0.8 1.0 Normalized Intensity 48 3.2. Localized Photoinjection Experimental Tests To test the capabilities of the photonic nanojet for localized photoinjection, in terms of switching energy and switching time, the pump-probe spectroscopy experimental setup introduced in Figure 2.2 is configured for operation with coincident and collimated control (pump) and signal (probe) beams focused through the 2.00-mm-diameter sphere into a GaAs sample. A control beam fluence of 280 µJ/cm2 is used. The measured differential transmission, Δo p o, is shown in Figure 3.5, as a function of time, for the spheres having refractive indices of (a) _ = 1.51, (b) _ = 1.76, (c) _ = 1.83, and (d)	_ = 1.98. The experimental curves are shown with corresponding theoretical curves, having been generated by the theory in section 2.1 on planar free-carrier dynamics. The corresponding Mie theory simulations are shown in the figure insets. It is clear from these simulations that increasing refractive indices lead to decreasing distances between the focal plane and the back surface of the sphere. The experimental (blue) and theoretical (black) results of Figure 3.5 show that increasing sphere refractive indices lead to increasing free-carrier densities at the GaAs surface, rather than within the bulk. This means that the spheres with higher refractive indices can better realize rapid surface recombination of the free-carriers. The transition from bulk- to surface-dominated recombination as the refractive index is increased can be seen by noting the changing polarity of the differential transmission curves. The negative curve of the _ = 1.51 sphere exhibits decreased signal beam transmission, and this is due to the dominance of a control-beam-induced increase to the absorption coefficient within the bulk. This is a similar observation and interpretation to those of the nominal experimental results in section 2.2, which used a microscope objective to focus into GaAs.  In contrast, the positive curves of the _ = 1.76, 1.83, and 1.98 spheres exhibit increased signal beam transmission, which is due  49 to the dominance of a control-beam-induced decrease to the refractive index near the surface. This change in polarity of Δo p o is also seen for high magnification objectives when moving the sample into and out of the focal plane. This is attributed to the low ∞-number and resulting shortened focal length and Rayleigh range, being half the depth of focus. The shortening Rayleigh range leads to the paraxial rays preferential depositing free-carriers at the semiconductor surface, resulting in a short photoinjection depth, δU, which reduces the contribution from bulk absorption according to Equation (8). The transition from bulk- to surface-dominated recombination as the refractive index is increased can also be seen by noting the reducing free-carrier lifetimes. Free-carrier lifetimes of í = 210 ps, 120 ps, 60 ps, and 10 ps are measured for the sphere refractive indices of _ = 1.51, 1.76, 1.83, and 1.98, respectively. The decreasing free-carrier lifetimes that are seen as the refractive index is increased come about from the shortening focal length and Rayleigh range, being half the depth of focus. The shortening Rayleigh range leads to the preferential deposition of free-carriers at the semiconductor surface—where there exists a high density of surface states and an appreciable surface recombination velocity, S5 = 8000 m/s. The theoretical results are generated by the theory on planar free-carrier dynamics, developed in section 2.1 with the parameters in Table 2.1. All of the parameters are kept constant while only changing the photoinjection depth, δU, of the control beam, to curve-fit the experimental results. For sphere refractive indices of _ =  1.51, 1.76, 1.83, and 1.98, the photoinjection depths into the semiconductor decrease according to gh = 490, 370¸ 230, and 80 nm, respectively. The theoretical results also show that the decreasing photoinjection depths yield increasing signal levels for the differential transmission, and this too is seen in the experimental results of Figure 3.5. Overall, the experiment and theoretical results are in good agreement. The discrepancies seen in Figure 3.5(a)  50 and (c) are likely due to the strict demands being made for curve-fitting here. The theoretical curves are generated by adjusting only the photoinjection depth, as the sole curve-fitting parameter, and taking the remaining values for the parameters from the nominal test results in section 2.1. Improved curve-fits can be generated by allowing the mobility and diffusion coefficient to vary  Figure 3.5. All-optical switching via photoinjection in GaAs using varying sphere refractive indices. Results for the experimental (blue) and theoretical (black) differential transmission of the signal beam, ±°¢ ï ¢, are shown normalized as a function of time, for photoinjection in GaAs, using spheres with a diameter of ≥ = 2.0 mm and refractive indices of (a) ≤ = 1.51, (b) ≤ =1.76, (c) ≤ = 1.83, and (d) ≤ = 1.98 ±0.02. The differential transmission results are shown normalized, with respect to the results in ≥, and the relative scaling factors are labelled in the figures. The figures include theoretical curves derived from Equations (22) and (23), pertaining to planar free-carrier dynamics. Mie theory simulations are shown in the insets, with logarithmic colour maps, to illustrate the varying focal conditions of the coincident control and signal beams. 210 psn = 1.51 sphere× -0.86 aGaAs targetd = 2.0 mm0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 50 100 150 200 250 300 Time Delay, t (ps))T(t)/T (a.u.)120 psn = 1.76 sphere× 0.22 bGaAs targetd = 2.0 mm0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 50 100 150 200 250 300 Time Delay, t (ps))T(t)/T (a.u.)n = 1.83 sphere60 ps× 0.33 cGaAs targetd = 2.0 mm0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 50 100 150 200 250 300 Time Delay, t (ps))T(t)/T (a.u.)0 Time Delay, t (ps))T(t)/T (a.u.)0 50 100 150 200 250 300 n = 1.98 sphere10 ps× 1.0 dGaAs targetd = 2.0 mm0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6  51 for each of the curves, and such variations are plausible given the differing cross-sectional areas and thus free-carrier densities in the tests. However, we wish to test the model under the strictest possible conditions, and this can best be achieved by applying only one adjustable parameter (the photoinjection depth). Given these experimental findings, it is apparent that a nanojet focal geometry, with a sphere diameter of Ö =  2.00 mm and refractive index of _ =  1.98 ±  0.02, can enable all-optical switching with femtojoule switching energies and picosecond switching times. The implementation tested here, with a GaAs sample behind the sphere, yields a roughly 10 fJ switching energy and 10 ps switching time. This switching energy is defined for a unity signal-to-noise ratio. With such a definition, the switching energy (quoted here and in the remainder of the thesis) is highly sensitive to the experiment setup and the alignment. The experiment is aligned multiple times, with two micro-positioning stages, and the quoted switching energies are stated for the best results (noting that multiple tests with the same alignment yield the same results). Further reductions in switching energies can be achieved if the sphere diameter is reduced. According to scaling calculations made from Mie theory, a sphere with a diameter of approximately one micron can achieve a switching energy below one femtojoule. This is because a smaller sphere with the same incident beam power, i.e., equivalent switching energy, will produce larger relative intensities, mostly due to resonance within the structure. This relationship between switching energy and sphere size is illustrated in Figure 3.6. However, it should be noted that such performance improvements would have practical constraints related to both coupling (beam alignment, directionality, and capture cross-section) and stability (physical and temperature sensitivity). Through a series of tests, we have found that the nanojet focal geometry, operating in the milli-scale regime offers both effective coupling (with inherent beam self-alignment,  52 omnidirectionality, and wide capture cross-sections for incident collinear control and signal beams) and good stability (with low sensitivity to structural and thermal fluctuations). Further reductions in the switching time can also be achieved by promotion of surface recombination, and such enhancements in speed are explored in the following section.    Figure 3.6. Photonic nanojet intensities as a function of sphere diameter. The displayed points are for the sphere’s maximum normalized intensity for refractive indices of ≤ = 1.5 to 2.0 as a function of diameter, ≥. The refractive indices giving the highest intensity of a given diameter are plotted. For sphere sizes of ≥ > 2 µm, the relative intensity (with a fixed input power) slowly increases with decreasing sphere size. For sphere sizes of ≥ <  2 µm, the relative intensity dramatically increases as the sphere size decreases, approaching the wavelength (780 nm). Mie theory simulations are shown in the insets with logarithmic colourmaps. Sphere Diameter, d ƉP0 5 10 15 20 25 30Normalized Intensity0.20.40.60.81.01.20 53 3.3. Summary & Discussion This chapter investigated the near-subwavelength focal characteristics of photonic nanojets for application to all-optical switching. A nanophotonic superlens was realized in the form of a dielectric microsphere, and it was optimized for the formation of high intensities, to enable low all-optical switching energies. A milli-scale regime was studied, within which the optimal characteristics were defined by ray theory. This led to the selection of a S-LAH79 glass sphere with a diameter of Ö = 2.0 mm and refractive index of _ = 2.00 as the desired structure. A micro-scale regime was also studied, within which the optimal characteristics were defined by Mie theory. This led to the selection of an N-LASF9 glass microsphere with a diameter of Ö = 40 µm and refractive index of _ = 1.83 as the desired structure. The selected dielectric microsphere was found to establish a photonic nanojet at its back surface with an especially high intensity. This yielded enhanced interaction between the control and signal beams via localized photoinjection within an adjacent GaAs sample. The theoretical and experimental results for the control-beam-induced changes to the differential transmission of the signal beam through the sample showed that the selected N-LASF9 glass microsphere could dramatically reduce the free-carrier lifetime, from that of the nominal lifetime of í = 350 ps, measured in section 2.2.2 to í = 10 ps. This was due to the reduced Rayleigh range formed from this microsphere and its corresponding ability to preferentially deposit free-carriers at the semiconductor surface, where they can undergo rapid surface recombination. In the next chapter, the abilities of the dielectric sphere to generate high intensities and enable rapid surface recombination are exploited in the application of all-optical switching.    54 Chapter 4  All-Optical Switching Architecture The goal of this chapter is to develop a practical all-optical switch (AOS) architecture for application to optical fibre front-end systems. All-optical switching in such systems must operate with femtosecond switching times to enable terabit-per-second data rates, and it must operate with femtojoule switching energies to enable future all-optical networks13,14,22,24. However, these demands are often mutually exclusive15,16,17. To develop an AOS architecture that meets both of these demands, while still being practical, it is necessary to establish especially strong and rapid interactions between the coincident control and signal beams. The onset of the control pulse must initiate a strong interaction between the control and signal beams (to facilitate operation with femtojoule switching energies). As the strength of the interaction is proportional to the intensities, this condition can best be met by concentrating the control and signal beams within a high-intensity focal spot. This results in localized photoinjection, as characterized in section 2.3, with the control-beam-induced free-carriers being deposited within a small volume for pronounced modulation of the probe beam. In Chapter 3, it was shown that these high-intensity conditions can be realized by a dielectric sphere, which acts as a superlens and forms an intense non-evanescent near-subwavelength photonic nanojet at its back surface. The concept of the photonic nanojet is used in this chapter for the AOS architecture.  The termination of the control pulse must then be followed by rapid recovery of the all-optical switching medium (to facilitate operation with femtosecond switching times). This condition can best be met by applying localized recombination, as characterized by section 2.3, with the control- 55 beam-induced free-carriers undergoing rapid recombination and the signal beam witnessing faster recovery. In Chapter 2, it was shown that surface recombination can establish this rapid recombination, given the correct selection of the semiconductor material and structure. In this chapter, the all-optical switching recovery times are enhanced, by as much as six orders of magnitude, by transitioning from the previously-studied bulk semiconductors to semiconductor nanoparticles, having far higher surface-to-volume ratios. Semiconductor nanoparticles of varying compositions are embedded onto the full surface of the aforementioned microspheres, as illustrated in Figure 4.1. This has the intense focus of the photonic nanojet overlap with the semiconductor nanoparticles. If suitably designed, such an architecture can facilitate all-optical switching with incident illumination across the full 4ô steradians solid angle of the sphere. It is ultimately shown that the proposed AOS architecture with integrated photonic nanojets and semiconductor nanoparticles can enable both femtojoule switching energies and femtosecond switching times—along with the practical benefits of omnidirectionality, a small footprint, and monolithic integration with the focusing element. In section 4.1, the theory is presented for all-optical switching in spherical semiconductor nanoparticles. In section 4.2, the proposed AOS architecture is tested with dielectric spheres in milli-scale and micro-scale regimes. In section 4.3, the developed AOS architecture is optimized using the results of further experimental tests with semiconductor nanoparticles having varying compositions. Lastly, section 4.4 summarizes the performance of each semiconductor material and the performance of the overall architecture.  56    Figure 4.1. Illustration showing the general concept of the AOS architecture. Semiconductor nanoparticles of varying compositions are embedded onto the full surface of a dielectic sphere. Collimated control and signal beams propogate from left to right and are focued into an intense non-evanescent near-subwavelength photonic nanojet. The highest intensity of the photonic nanojet (shown in white with a logarithmic scale colourmap) overlaps with the semiconductor nanoparticles (illustrated as a scanning electron microscopy inset) on the back surface, which enables strong and rapid interactions between the control and signal beams. photonic nanojetsemiconductornanoparticles 57 4.1. Theory for All-Optical Switching in Spherical Semiconductors Localized recombination is considered in this section, for enhanced all-optical switching times, by introducing a nanoparticle material system. The introduction of (roughly spherical) semiconductor nanoparticles here increases the specific surface area, being ^ = 6/Ö with a small sphere diameter, Ö , and this hastens the free-carrier recombination rate, according to the generalized approximation72 of 1 í 	= 1 í9 + ò5^ . Thus, the free-carrier lifetime, í , can be reduced to a value that is below the bulk semiconductor lifetime, í9. The spherical semiconductor nanoparticles are modeled with a similar method to that of the planar case in section 2.1, albeit with a spherical boundary condition. In this case, the differential transmission, ∆o p /o, through the nanoparticles is related to the free-carrier density, â ¨, p , according to   yo po = − _I − 1_I _I + 1 ∆_I p . (31) The differential transmission equation above is similar to Equation (8), which applies to the planar semiconductor case and includes the effects of free-carrier dispersion via	Δ_I p  and free-carrier absorption via Δf p . However, for this case the radial photoinjected depth, gµ, in the nanoparticle is especially small, being approximately equal to the nanoparticle’s radius. Thus, the effects of free-carrier absorption and Δf p  are assumed to be negligible here. To continue assembling the theory, the free-carrier density distribution is defined. The free-carrier density, â ¨, ∂, p , evolves through time, p, along the radial dimension of the semiconductor nanoparticles, ¨, according to the differential equation  58  qâ ¨, ∂, pqp = âXg p + ì∇2â ¨, ∂, p − â ¨, ∂, pí9 , (32) where the free-carrier generation is quantified by the delta function, g p , diffusion is defined by the diffusion coefficient, ì, and bulk recombination is defined by the bulk recombination lifetime, í9. The diffusing free-carriers ultimately reach the radius, ∑, and undergo surface recombination in accordance with the boundary condition of  −ì qâ ¨, ∂, pq¨ µó∏ = ò5â ¨, ∂, p µó∏. (33) The solutions to Equation (32) are found by a Laplace- (or frequency-) domain approach. The Laplace-domain solution, â ¨, π , is formed by way of the derivation shown in Appendix G, given independence from the polar angle, ∂. The free-carrier density distribution, â ¨, p , is then found by using an inverse Laplace-domain transform, according to  â , p = ℒ-1 â ¨, π= ℒ-1 â0π + 1/í9 1 − ∑¨	 ò5 sinh Ũò5 − ì/∑ sinh Å∑ + ìÅ cosh Å∑ , (34) where Å = π/ì + 1/í9. Given the complexity of Equation (34), a numerical routine is used to compute the inverse Laplace transform for varying values of the parameters ∑,	ì, and	ò5. The free-carrier density distribution, â ¨, p , is then averaged over the volume of the sphere to form the free-carrier density, â p , and differential transmission of the signal beam, ∆o p /o.  Theoretical analyses are carried out for spherical semiconductor nanoparticles using the Laplace-domain solution for the transient free-carrier density. This is done for Si nanoparticles, with ∑ = 20 nm, and SiC nanoparticles, with ∑ = 50 nm. Si and SiC are chosen because they exhibit a large range in surface recombination velocities, being ò5 ≈ 500 m/s and ò5 ≈ 4000 m/s  59 for Si and SiC, respectively, and they underwent investigations as nominal planar semiconductors in section 2.2.2. (Section 2.2.2 also showed results for GaAs, but the corresponding analyses of GaAs nanoparticles are not conducted in this chapter, due to their high toxicity and limited knowledge in the literature of safety procedures for laser excitation and handling73,74.) The solution to the free-carrier density, â ¨, p , evolving through time along the nanoparticle’s radial cross-section, is seen in Figure 4.2 for the Si and SiC nanoparticles. It is apparent that the free-carriers in the SiC nanoparticles recover more quickly than the Si nanoparticles due to their more rapid surface recombination. These free-carrier dynamics, while somewhat complex, can be greatly simplified by making educated assumptions to reach a more physical understanding. For a more physical understanding of the solutions to Equation (32), a time-domain solution can be sought, in contrast to the prior Laplace-domain solution. The differential form of Equation (32) yields a general solution that varies with the radial dimension, ¨, polar angle, ∂, and time, p. The general solution is75  â ¨, ∂, p = ºΩ,|æΩ §Ω,|	¨/∑ Ω´ cos ∂ e-Ä/ø¿,¡¬|ó1¬Ωó0 , (35) where æΩ ∙  and Ω´ ∙  are the spherical Bessel functions of the first kind and the Legendre polynomials, respectively, both of order Ñ. The eigenvalues,	§Ω,|, decay coefficients, ºΩ,|, and time constants, íΩ,| , are also seen here. The discrete eigenvalues, §Ω,| , are generated by transforming the aforementioned boundary condition into the characteristic equation, æΩ §Ω,| æΩƒ §Ω,| = −ì§Ω,| ò5∑ , and solving this characteristic equation.  The decay coefficients, γΩ,|, are found by applying the initial condition of a uniform free-carrier density, âX. The time constants, íΩ,|, are defined by   60  Figure 4.2. Modelled free-carrier dynamics for Si and SiC nanoparticles. (a) The free-carrier density, Ø ∆, ï , is shown as a function of time and space, according to Equation (34), for Si nanoparticles with a radius of 20 nm, where Ø ∆, ï  has increments of 2 ps. (b) The free-carrier density, Ø ∆, ï , is shown as a function of time and space, according to Equation (34), for SiC nanoparticles with a radius of 50 nm, where Ø ∆, ï  is shown with time increments of 1 ps.  Free-carrier Density [arb. units]Radial Cross Section (nm) -20 -10 0 10 20 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 at = 0 pst = 2 pst = 4 pst = 6 pst = 8 psFree-carrier Density [arb. units]Radial Cross Section (nm) -50 -25 0 25 50 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 t = 0 pst = 1 pst = 2 pst = 3 pst = 4 psb 61  1íΩ,| = 1í9 + ì§Ω,|2∑2 . (36) For the present study, the nanoparticle is assumed to be uniformly illuminated, with ∂ independence, and the nanoparticle radius is assumed to be much less than the diffusion length. With these (relatively safe) assumptions, the general solution in Equation (35) has a finite term only for Ñ	 = 0, and the d = 1 term of the remaining expansion dominates over the subsequent terms. This allows Equation (35) to be rewritten as  â ¨, p ≈ γ sin § ¨ ∑§¨ ∑ e-Ä/ø, (37) where the eigenvalue is § = §0,1, the decay coefficient is   γ = γ0,1 = 2â0 sin § − § cos §§ − sin § cos § , (38) and the time constant is í = í0,1. With these definitions, the characteristic equation can be cast in the simplified form of  tan §§ 	= 11 − ò5∑ ì (39) The transcendental form of this simplified characteristic equation can be accommodated by computing its § with a numerical routine and applying this § in Equations (37) and (38) to form the free-carrier density distribution, â ¨, p . The time-domain solutions are then formed by substituting the numerical routine’s results for Equation (39) into Equations (36), (37), and (38). The time-domain results with a variety of values for ò5, ∑, and ì are compared to the Laplace-domain solutions from Equation (34), and it is found that the solutions match within 1% error. It is also worth noting that the transcendental form of the simplified characteristic equation shown above can be accommodated with further approximations. For a nanoparticle radius that is much  62 less than the diffusion length, i.e., a large diffusion coefficient, ì ≫ ò5∑, the eigenvalue will become small, i.e., § → 0. This allows the left and right sides of the simplified characteristic equation to be expanded, yielding ì§2 ≈ 3ò5∑. With this result, Equation (36) can be rewritten as the generalized expression  1í ≈ 1í9 + ò5^. (40) This generalized expression, which is often applied in the literature for planar surfaces72, can be applied to the present study having a spherical geometry by introducing the surface-to-volume ratio of a sphere, being ^ = 3/∑. In the limiting case of small §, the decay coefficient becomes γ ≈ âX, the cardinal sine term in Equation (37) becomes sin §	¨/∑ / §¨/∑ ≈ 1, and the free-carrier density distribution simply manifests itself as the spatially-averaged free-carrier density of  â p ≈ â0e-Ä/ø. (41) The overall responses for time constants of semiconductor nanoparticles are computed in accordance with Equations (36), (37), (38), and (39), and the results are illustrated in Figure 4.4. The figure shows the time constant, í, as a function of the surface recombination velocity, ò5, and radius, ∑, for a nanoparticle having a large diffusion coefficient, i.e., ì ≫ ò5∑, and long bulk lifetime, i.e., í9 → ∞. It is immediately apparent from this figure that í decreases linearly with both ∑  and 1/ò5 , which is to be expected. Such scaling is of importance in applying these nanoparticles to all-optical switching, as it is desirable to have these time constants be as short as possible. In particular, the red region of Figure 4.4 is demarcated to show the regime for which femtosecond switching times are achieved. From the bounds of this red region, it is concluded that the semiconductor nanoparticle composition should be selected to have a sufficiently high surface recombination velocity, being greater than ò5 = 2000 m/s, and the semiconductor nanoparticle size  63 should be selected to have a sufficiently small radius, being less than ∑ = 30 nm. At the same time, the diffusion coefficient of the semiconductor nanoparticle should be sufficiently high, i.e., ì ≫ ò5∑, to maintain rapid diffusive transport of free-carriers from the interior to the surface. If the diffusive transport is impeded by a low diffusion coefficient, i.e., ì ≈ ò5∑, the recombination of free-carriers witnesses a bottleneck. This low-diffusion scenario is illustrated in inset (a) of Figure 4.4, which shows the free-carrier density distribution within the spherical semiconductor nanoparticle as a colourmap with ì ≈ ò5∑ and a time of 0.5 ps. The colourmap is nonuniform because there is a concentration (i.e., bottleneck) of free-carriers within the interior of the semiconductor nanoparticle. In contrast, if the diffusion coefficient is high, i.e., ì ≫ ò5∑, the diffusive transport is sufficiently rapid to have the recombination of free-carriers be readily facilitated by surface states—leading to a relatively uniform free-carrier density distribution within the semiconductor nanoparticle. This high-diffusion scenario is illustrated in inset (b) of Figure 4.4. The free-carrier density distribution of this latter semiconductor nanoparticle, for ì ≫ ò5∑ and a time of 0.5 ps, is seen to exhibit a uniform colourmap. In general, it is advantageous to have the diffusion coefficient be as large as possible for rapid diffusive transport. The above spherical free-carrier dynamics will ultimately yield a differential transmission response, ∆o p /o, for semiconductor nanoparticles, as seen next in section 4.2. As a first step in this process, the semiconductor nanoparticles Si and SiC are investigated experimentally to confirm that a high surface recombination velocity, ò5, or a large surface-area-to-volume ratio, i.e., a small radius ¨ = ∑, can yield fast recombination.  64    Figure 4.3. Characterization of time constants for semiconductor nanoparticles. The time constant, », i.e., switching time, is displayed versus the surface recombination velocity,	… , and radius, À, for a nanoparticle with rapid diffusion, i.e., ä ≫ … À and a long bulk lifetime, i.e., »Ã. Results are formed by solving Equations (36), (37), (38), and (39). The red region is the desired regime, where femtosecond optical switching is achieved. Inset (a) shows a nonuniform free-carrier density distribution, Ø ∆, ï = Õ. Œ	œ– , in the nanoparticle for a small diffusion coefficient, i.e., ä ≈ … À. Inset (b) shows a uniform free-carrier density distribution, Ø ∆, ï = Õ. Œ	œ– , in the nanoparticle for a large diffusion coefficient, i.e., ä ≫ … À. ΅ (ps)Sv (m/s) a (nm)01000 10051015704000202530407000351010000D » SvabD ≈ Sva a N(r,t)N00 65 4.2. All-Optical Switching Experiments in Spherical Semiconductors The spheres that are implemented in the proposed AOS architecture must be selected with careful consideration to their refractive index, _, and diameter, Ö. This is because a high-intensity focus is required at the back surface of the sphere, and the manifestation of this high intensity depends upon both _ and Ö, as was shown in Figure 3.2. Spheres in the milli-scale regime, with diameters above 0.1 mm, offering compatibility with optical fibres, will simply require that the sphere have a refractive index of _ ≈ 2. Whereas spheres in the micro-scale regime, with diameters ranging from 1 to 30 µm, offering compatibility with integrated network-on-chip devices, will require the sphere to have a refractive index along the grey curve of Figure 3.2, between _ ≈ 1.75 and _ ≈ 1.83. Given the considerations above for nanoparticles and dielectric spheres, the AOS architecture is tested with sphere sizes for both the milli-scale and micro-scale regimes, in sections 4.2.1 and 4.2.2, respectively. Two semiconductor materials are tested, in the form of Si and SiC nanoparticles. The nanoparticle sizes are an order of magnitude smaller than the control and signal wavelengths to yield negligible scattering of the beams entering and exiting the spheres (as is confirmed experimentally). The semiconductor nanoparticles are embedded on the surface of the spheres with a dry-coating process78. This process involves mechanical rubbing the nanoparticles into the surface of the glass spheres, whereby Van der Waals forces provide sufficient adhesion. 4.2.1. Milli-Scale Spheres To test the capabilities of the nanoparticle material system and nanojet focal geometry, in terms of switching energy and switching time, pump-probe spectroscopy is carried out for milli-scale  66 spheres embedded with Si and SiC nanoparticles. The spheres have a diameter of Ö =	2.0 mm, being well within the milli-scale regime, and are comprised of S-LAH79 glass, with a refractive index of _	 = 1.98 ± 0.02. The pump-probe spectroscopy results are shown in Figure 4.4. The figure insets show a Mie theory simulation and scanning electron microscope image of the nanoparticles. (Note, the scanning electron microscope images in this thesis are for isolated the nanoparticle powders fixed to a substrate and sputter coated for imaging.) Figure 4.4(a) shows the (positive) differential transmission, ∆o p /o, for all-optical switching between control and signal beams with the Si nanoparticles. While Si has a low surface recombination velocity, a noteworthy decrease in recovery time is seen for these nanoparticles, being approximately two thousand times faster than that of bulk Si. This rapid recovery is due to the promotion of surface recombination from the increased surface-to-volume ratio of the nanoparticles. Overall, the nanojet focal geometry and Si nanoparticle material system yield a switching time of 10 ps and switching energy of 200 fJ. Figure 4.4(b) shows the (positive) differential transmission, ∆o p /o, for all-optical switching between control and signal beams with the SiC nanoparticles. Given that SiC has a high surface recombination velocity, a dramatic decrease in recovery time is seen for these nanoparticles, being approximately sixty million times faster than that of bulk SiC. Overall, the nanojet focal geometry and SiC nanoparticle material system yield a switching time of 350 fs and switching energy of 100 fJ. It is important to note here that the bulk SiC results of Figure 2.3 and the nanoparticle results of Figure 4.4(b) both exhibit recovery with two distinct time constants. Long recovery time constants are observed at approximately 22 µs for the bulk and 5 ps for the nanoparticles. The long recovery time constants are due solely to recombination on the free-carrier density, â Ç, p , and it  67 is these time constants that witness pronounced effects from the increased surface-to-volume ratio of the nanoparticles. Short recovery time constants are observed at approximately 3 ps for the bulk and 350 fs for the nanoparticles. The short recovery time constants are attributed to ultrafast scattering in the SiC bandstructure76, as photoinjected free-carriers populate excited states in the M-sidevalley and then undergo intervalley and intravalley scattering. These dynamics are seen as a transient effective mass, Ñ, and scattering rate, Ü, through Equation (22). For the bulk SiC response of Figure 2.3, free-carrier recovery is governed by bulk absorption over a long photoinjection depth, gh, and this has the (negative) second term in the brackets of Equation (22) dominate. In this case, photoinjected free-carriers undergo energy relaxation in the M-sidevalley of the conduction band, via intravalley cooling and intervalley scattering on a picosecond scale77. The transition from an initial high scattering rate, with a dense and energetic free-carrier population, to a low scattering rate, with a broad and thermalized free-carrier population, decreases the scattering rate in Equation (22). The decreasing scattering rate, Ü, forms the 3 ps increase in the (negative) differential transmission curve seen for bulk SiC in Figure 2.3. For the nanoparticle SiC response of Figure 4.4(b), free-carrier recovery is governed by surface effects, due to the high surface-to-volume ratio of the nanoparticles, and this has the (positive) first term in the brackets of Equation (22) dominate. Here, intervalley scattering of free-carriers in the M-sidevalley76, from the upper subband to the lower subband, contributes to an increasing effective mass, Ñ, over a femtosecond timescale. The increasing effective mass, Ñ, is seen as the 350 fs decrease in the (positive) differential transmission curve for SiC nanoparticles in Figure 4.4(b).  68  Figure 4.4. All-optical switching with Si and SiC nanoparticles on a milli-scale sphere. Experimental (positive) differential transmission for the signal beam, °¢ ï ¢, is shown normalized versus time for photoinjection of (a) Si nanoparticles and (b) SiC nanoparticles. The nanoparticles coat spheres with a diameter of ≥	 = 2 mm and refractive index of ≤	 = 1.98 ±0.02. The figures include insets with Mie theory simulations and scanning electron microscope images of nanoparticles with a 200 nm scale bar.  n = 1.98 sphere10 psSinanoparticlesa0 5 10 15 20 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time Delay, t (ps) )T(t)/T (a.u.)n = 1.98 sphere350 fs0 1 2 3 SiCnanoparticlesb0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time Delay, t (ps) )T(t)/T (a.u.) 69 Overall, the AOS architecture based on the nanojet focal geometry and SiC nanoparticle material system meets the demands for femtojoule switching energies and femtosecond switching times—with a 100 fJ switching energy and 350 fs switching time. Moreover, it is theorized that further reductions in the switching energy can be realized by using micro-scale spheres, and this assertion is tested in the upcoming section. 4.2.2. Micro-Scale Spheres Experimental results from pump-probe spectroscopy of milli-scale spheres embedded with Si and SiC nanoparticles are shown in Figure 4.5. Figure 4.5(a) and (b) show results for Si and SiC nanoparticle-coated microspheres, respectively. The microspheres have a diameter of Ö =	30-40 µm, being at the upper limit of the micro-scale regime. The microspheres are comprised of N-LASF9 glass, with a refractive index of _	 = 1.83 ± 0.02. The figure insets show a Mie theory simulation and a scanning electron microscope image of the nanoparticles. Figure 4.5(a) shows the (positive) differential transmission, ∆o t /o, for all-optical switching between control and signal beams, with Si nanoparticles. The applied Si nanoparticles again produce a noteworthy decrease in the recovery time—being even faster than that of the Ö = 2.0 mm sphere experiment in Figure 4.4(a). The enhancement in speed for this micro-scale AOS architecture is attributed to its reduced focal spot area, compared to that of the milli-scale AOS architecture. The reduced focal spot area allows the micro-scale AOS architecture to avoid aggregates of larger, i.e., 100+ nm, particles that are seen to be interspersed among the 20 nm nanoparticles in scanning electron microscope images. The milli-scale AOS architecture, with its larger focal spot area is unable to avoid the aggregates of larger particles, which have a smaller surface-area-to-volume ratio and longer free-carrier lifetime, and this leads to its longer recovery  70 time. Ultimately, the micro-scale AOS architecture with the Ö =  40 µm microsphere and Si nanoparticles yields a switching time of 2 ps and switching energy estimated to be 1 pJ. Figure 4.5(b) shows the (positive) differential transmission, ∆o p /o, for all-optical switching between control and signal beams, with SiC nanoparticles. The applied SiC nanoparticles produce a dramatic decrease in recovery time, being even faster than the results of the Ö = 2.0 mm sphere experiment shown in Figure 4.4(b). The enhancement in speed for this micro-scale AOS architecture is brought about from its reduced focal spot area, as stated above. Ultimately, the micro-scale AOS architecture with the Ö = 40 µm microsphere and SiC nanoparticles meets the demands for all-optical switching, as it yields a switching time of 270 fs and switching energy that is estimated to be 20 fJ.    71  Figure 4.5. All-optical switching with Si and SiC nanoparticles on a micro-scale sphere. Experimental (positive) differential transmission for the signal beam, °¢ ï ¢, is shown normalized versus time for photoinjection of (a) Si nanoparticles and (b) SiC nanoparticles. The nanoparticles coat microspheres with a diameter of d = 40 µm and refractive index of n = 1.83 ± 0.02. The figures include insets with Mie theory simulations and scanning electron microscope images of nanoparticles with a 200 nm scale bar.  n = 1.83 sphere2 psSinanoparticlesa0 5 10 15 20 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time Delay, t (ps) )T(t)/T (a.u.)n = 1.83 sphere270 fs0 1 2 3 SiCnanoparticlesb0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time Delay, t (ps) )T(t)/T (a.u.) 72 4.3. Semiconductor Nanoparticle Optimization The proposed AOS architecture, integrating the photonic nanojet with semiconductor nanoparticles, is further tested with Si (as a baseline of comparison), CdTe, InP, and CuO nanoparticles on microspheres. The microspheres are made of N-LASF9 glass having a diameter of Ö = 40 µm and refractive index of _ = 1.83. The semiconductor nanoparticles are embedded on the surface of the microspheres with the aforementioned dry-coating process78. The type of semiconductor used for the semiconductor nanoparticles is constrained by the control and signal beams’ wavelengths. The signal beam’s wavelength is chosen to be at the telecommunications wavelength of 1550 nm (0.8 eV) for practicality, low propagation losses, and increased sensitivity to free-carrier dispersion, which scales with the wavelength-squared79. The control beam is generated at a wavelength of 780 nm (1.6 eV), by second harmonic conversion of the signal beam. Based on these two wavelengths, the semiconductor bandgap must fall in the range of 0.8 eV to 1.6 eV. Si (having a 1.11 eV indirect bandgap), CdTe (having a 1.56 eV direct bandgap), InP (having a 1.27 eV direct bandgap), and CuO (having a 1.21 eV indirect bandgap80) are selected, as their bandgaps are within the desired range. Note that the bandgap of SiC does not fall within this range, and is no longer pursued for this reason. The investigations are carried out with the 780-1550 nm pump-probe spectroscopy experimental setup shown in Figure 2.2 to characterize the impulse response of the AOS architecture. The differential transmission, ∆o(p)/o, of the signal beam is measured as a function of the time-delay the control (pump) and signal (probe) pulses, using a lock-in amplifier with a 100 ms time constant. Representative all-optical switching results for each semiconductor are shown in Figure 4.6, as measurements of the differential signal beam’s transmission, ∆o(p)/o. The results for the Si,  73 CdTe, InP, and CuO nanoparticles are given in Figure 4.6(a), (b), (c), and (d), respectively. The minimum switching energies, defined for a signal-to-noise ratio of one, are estimated based on the relative strength of each waveform while the signal and control beam powers are held constant. The switching time (or time constant) is measured via curve-fitting from each ∆o(p)/o waveform. The Si nanoparticles, shown in the inset of Figure 4.6(a), have an average radius of 40 nm. The nanoparticles yield a switching energy of 1 pJ and switching time of 1.8 ps. The large (picojoule) switching energy is attributed to the indirect bandgap of Si and the resulting poor quantum efficiency. The long (picosecond) switching time is attributed to the large radius of the Si nanoparticles. According to the theory from section 4.1 on spherical free-carrier dynamics, the Si nanoparticles have a surface recombination velocity of 7400 m/s. Such Si nanoparticles would support the desired femtosecond operation, in the red region of Figure 4.3, given a smaller radius, but such changes would not address the unacceptably high switching energy. The CdTe nanoparticles, shown in the inset of Figure 4.6(b), have an average radius of 30 nm. The nanoparticles yield a switching energy of 500 fJ and switching time of 2.3 ps. The femtojoule switching energy seen here is lower than that of Si, which is attributed to the direct bandgap of CdTe and the correspondingly high quantum efficiency. The picosecond switching time for CdTe is slower than that of Si, however, which implies that the CdTe surface recombination velocity is lower than that of Si. According to the theory from section 4.1 on spherical free-carrier dynamics, the CdTe nanoparticles have a surface recombination velocity of 4300 m/s. With this surface recombination velocity, the radius could be reduced to enable the desired femtosecond operation in the red region of Figure 4.3. However, the radius would need to be reduced below 13 nm, and this would encroach upon CdTe’s Bohr radius of 7.3 nm81. The resulting quantum confinement would likely increase the bandgap beyond the 1.6 eV photon energy of the control beam.  74 The InP nanoparticles, shown in the inset of Figure 4.6(c), have an average radius of 20 nm. The nanoparticles yield a switching energy of 400 fJ and switching time of 900 fs. The femtojoule switching energy seen here for InP is attributed to its direct bandgap. At the same time, the femtosecond switching time seen here is the result of its high surface recombination velocity. According to the theory from section 4.1 on spherical free-carrier dynamics, the InP nanoparticles have a surface recombination velocity of 7400 m/s. This value, along with the radius, does establish operation in the red region of Figure 4.3. Thus, InP nanoparticles can meet the concurrent demands for femtojoule switching energies and femtosecond switching times. The CuO nanoparticles, shown in the inset of Figure 4.6(d), have an average radius of 20 nm. The nanoparticles yield a switching energy of 300 fJ and switching time of 350 fs. The switching energy seen here is especially low. The switching time is the fastest seen, but it is important to note that the recovery exhibits multiple distinct time constants. An in depth analysis of these time constants and free-carrier dynamics in CuO is given in section 5.1, but at this point it will simply be said that the signal beam responds with a rapid change in transmission over the 350 fs time constant and this is due to the introduction of state-filling. This 350 fs time constant seen for the CuO nanoparticles is in good agreement with the 400 fs time constant recently reported for relaxation (and its effects on state-filling) within the valence band of CuO nanowires82. Ultimately, the CuO nanoparticles can meet the concurrent demands for femtojoule switching energies and femtosecond switching times. It is worth noting that the semiconductor nanoparticles employed in this investigation have sufficiently high standard deviations in size to allow for preferential photoexcitation and probing of nanoparticles with varying sizes on the back surface of the sphere. The results presented above for Si, CdTe, and InP were chosen to have the smallest particles and the shortest time constants. It  75 is interesting to note, however, that the 350 fs time constant of the CuO nanoparticles was largely independent of the particle size, which supports its interpretation on the predominance of free-carrier relaxation seen via state-filling. Moreover, the surface recombination velocities quoted here are within the range of those quoted in the literature28,83,84, although they are on the higher side. We attribute any heightening of surface recombination velocities to the high curvature of the nanoparticles, as such curvature would lead to more pronounced lattice defects, and hastened surface recombination, in comparison to the planar surfaces that are typically studied.  76    Figure 4.6. All-optical switching with Si, CdTe, InP, and CuO coated microspheres having a diameter of ≥ = 40 µm and refractive index of ≤ = 1.83. Normalized differential transmission curves, °¢ ï /¢, are shown versus time for microspheres having the (a) Si, (b) CdTe, (c) InP, and (d) CuO nanoparticles coatings. The results exhibit switching energies of 1 pJ, 500 fJ, 400 fJ, and 300 fJ, with switching times of 1.8 ps, 2.3 ps, 900 fs, and 350 fs, respectively. The insets show Mie theory simulations with logarithmic intensities and scanning electron microscope images of nanoparticles, with a 200 nm scale bar.  n = 1.83 Si nanoparticlesa1.8 ps0 1 2 3 -1 0 0.2 0.4 0.6 0.8 1.0 Time Delay, t (ps) )T(t)/T (a.u.)n = 1.832.3 psCdTe nanoparticlesb0 1 2 3 -1 0 0.2 0.4 0.6 0.8 1.0 Time Delay, t (ps) )T(t)/T (a.u.)n = 1.83900 fsInP nanoparticlesc0 1 2 3 -1 0 0.2 0.4 0.6 0.8 1.0 Time Delay, t (ps) )T(t)/T (a.u.)n = 1.83350 fsCuO nanoparticlesd0 1 2 3 -1 0 0.2 0.4 0.6 0.8 1.0 Time Delay, t (ps) )T(t)/T (a.u.) 77 4.4. Summary & Discussion The geometrical and material characteristics inherent to semiconductor-based all-optical switching were studied in this chapter. An AOS architecture was developed, according to a guiding principle of localization, to facilitate femtojoule switching energies and femtosecond switching times. A focal geometry based upon focusing through a dielectric sphere was used to promote localized photoinjection—by directing high-intensity photonic nanojets into a peripheral semiconductor. A material system based upon nanoparticles was introduced to promote localized recombination—by embedding nanoparticles onto the surface of the sphere. The AOS architecture was implemented with milli-scale and micro-scale spheres and various types of semiconductor nanoparticles. For the milli-scale spheres, with Si and SiC nanoparticles, it was found that all-optical switching could be carried out with switching energies of 200 fJ and 100 fJ, and switching times of 10 ps and 350 fs, respectively. For the micro-scale spheres, with Si and SiC nanoparticles, it was found that all-optical switching could be carried out with switching energies of 1 pJ and 20 fJ and switching times of 2 ps and 270 fs, respectively. From these experimental observations, and the supporting theoretical model of diffusion and recombination in a spherical semiconductor, it was concluded that an AOS architecture with of an N-LASF9 glass microsphere and embedded SiC nanoparticles can meet the demands of all-optical switching, in terms of its femtojoule switching energies and femtosecond switching times, albeit with a larger bandgap than desired. The AOS architecture was then tested further with the micro-scale spheres and other types of semiconductor nanoparticles. All-optical switching results were collected for Si, CdTe, InP, and CuO nanoparticles, and they were found to exhibit switching energies of 1 pJ, 500 fJ, 400 fJ, and 300 fJ, with switching times of 1.8 ps, 2.3 ps, 900 fs, and 350 fs, respectively. The optimal  78 semiconductor witnessed in this study was CuO, which readily achieved the goals of femtojoule switching energies and femtosecond switching times. These results supersede the top results described in section 1.1.1, achieved by the organic photonic-bandgap microcavity24. It is also worth noting that the proposed AOS architecture can meet the performance demands of emerging optical fibre front-end operations, such as optical multiplexing4,12. At the same time, from a practical standpoint, the monolithic AOS architecture, with a compact device footprint and an integrated focusing element, allows for ease of coupling and omnidirectional operation. Such an architecture can be readily applied in cascaded on-chip implementations, via daisy-chained spheres85, which have proven to be challenging for all-optical devices86. Future work could improve upon such all-optical switching by tuning the nanoparticles to the conditions for Mie resonance, as this would increase the modulation depth87,88. Overall, the proposed AOS architecture can become a fundamental element in future all-optical implementations.   79 Chapter 5  All-Optical Retro-Modulation Architecture The goal of this chapter is to develop an all-optical retro-modulation (AORM) architecture for application to free-space optical (FSO) communication systems. For such systems, there is a desire to establish numerous multidirectional FSO links at terabit-per-second data rates, without escalating the system mass and power for the transceivers (TRXs). To realize this goal, the proposed architecture implements retro-modulation with a dielectric sphere as the retroreflector, using the knowledge gained on such structures from Chapter 3, and a thin film of copper oxide (CuO) nanocrystals as the modulator, given its strong performance shown in Chapter 4. Section 5.1 presents a detailed investigation on CuO nanocrystals, which are fabricated as a thin film with nanocrystalline grains, in contrast to the isolated nanoparticles used in section 4.3. The CuO nanocrystals fabrication process is characterized in section 5.1.1. The absorption characteristics, shown in section 5.1.2, are measured via pump-probe spectroscopy using an above-bandgap control beam and a below-bandgap signal beam to gain insight on the free-carrier dynamics of the CuO nanocrystals. An interpretation of the free-carrier dynamics within the bandstructure of CuO nanocrystals is given in section 5.1.3. The free-carrier dynamics are investigated further, for various control beam fluences, in section 5.1.4, and the findings are summarized in section 5.1.5. Section 5.2 demonstrates the proposed AORM architecture, with consideration to its deployment in long-range links between ground and aerial TRXs. The proposed system targets the  80 challenges of these long-range links by applying direct laser transmission for the ground-to-aerial active uplink (UL) and applying AORM for the aerial-to-ground passive downlink (DL). A link budget analysis of passive DLs is presented in section 5.2.1. In section 5.1.2, three link budget conditions are identified, for consideration in the design of the proposed AORM architecture. Experimental results are then presented and interpreted with respect to these link budget conditions in 5.2.3. The results are summarized in section 5.2.4. The AORM architecture is ultimately shown to support multidirectional FSO communication at terabit-per-second data rates with the multiple ground TRXs and an aerial TRX having low demands on system mass and power.    81 5.1. Semiconductor Free-Carrier Dynamics of Cupric Oxide Nanocrystals Cupric oxide (CuO), or tenorite, is a semiconductor of growing interest. The interest in CuO has emerged within applied physics largely from its potential for photovoltaics. Its absorption characteristics span the visible spectrum89,90, and its formation of copper vacancies and oxygen interstitials supports its use as a p-type lattice according to the literature82. It has recently been shown that p-type copper oxide films can be integrated with n-type metal oxide films to form cost-effective photovoltaic p-n heterojunctions91,92,93. At the same time, CuO has recently been shown to exhibit ultrafast relaxation and recombination dynamics for its free-carriers, which can support its use in all-optical switching94. However, an adequate understanding of its optical absorption characteristics and subsequent ultrafast relaxation and recombination dynamics has proven to be challenging. A major challenge to the understanding of CuO is the nanocrystalline morphology that the material typically forms. The differing nanocrystal sizes and phases of the nanocrystalline morphology have led to disparities for the optical absorption characteristics of the CuO nanocrystals—with reports of an indirect bandgap95,96 of approximately 1.2 eV and/or a direct bandgap95,97,98,99,100,101,102,103,104 ranging from 1.3 to 3.0 eV. The nanocrystalline morphology also leads to complexities in the evolution of the ultrafast free-carrier dynamics within the CuO nanocrystals, due to its formation of trap states within the bandgap (resulting from the nanocrystals’ large surface-to-volume ratio). Ultimately, an accurate understanding of the ultrafast free-carriers dynamics in CuO nanocrystals—while challenging—is critical to its future applications.  82 The work of Othonos et al. was a major step in gaining an understanding of the ultrafast free-carrier dynamics of CuO nanostructures82. The authors applied transient pump-probe spectroscopy to investigate CuO nanowires with diameters of 200 nm and lengths of 10 µm. It was proposed that the absorption and subsequent free-carrier dynamics are defined largely by the photoexcitation of electrons into the conduction band (CB) and the resulting relaxation of a transient population of holes within the valence band (VB).  In this work, CuO nanocrystals are explored. Fabrication of the nanocrystalline structure is discussed in section 5.1.1. The results of optical absorption spectroscopy for the CuO nanocrystals are given in section 5.2.2 to define the bandstructure. The results of transient pump-probe spectroscopy for the CuO nanocrystals are given in sections 5.1.3 and 5.1.4 to define the ultrafast free-carrier dynamics.  5.1.1. Nanocrystal Fabrication A fabrication process is developed here to create tailored CuO nanostructures, with control over the material properties and refinement of these properties to ultrafast all-optical modulation. The CuO nanocrystals are fabricated as a thin film with nanocrystalline grains. The fabrication technique developed here improves the nanostructure’s uniformity and allows for control over the nanocrystalline grain size, which was not possible with the purchased commercial nanoparticles and dry-coating adhesion process used in section 4.3. The CuO nanocrystals are characterized using scanning electron microscopy, and the fabrication process is optimized to reliably produce sufficiently small nanocrystals. The CuO nanocrystals in this study are fabricated using a similar technique to that used by Johan et al.99. Glass slides are cleaned using isopropyl and deionized water, dried with clean dry air, and DC sputtered (Nexdep Angstrom Engineering) with 99.999% pure copper, at a base  83 pressure less than 1.0 mTorr, to form thin films. The copper thin films are annealed in an oven shortly after sputtering. High temperature annealing is then applied at 600°C for two hours in ambient air to create the CuO phase—as opposed to the Cu2O (cuprous oxide) phase that is formed at temperatures99,105 lower than 300°C. A physical colour change is witnessed during this process, with the CuO thin film becoming brown in colour, in contrast to green for films left in air at room temperature (becoming Cu2O). To prevent Cu2O phase formation, the time between sputtering and annealing is minimized. The annealing process described here is the result of exhaustive fabrication and characterization tests, and it was chosen because it produced the best structural uniformity and spectral absorption characteristics for the CuO films, with thicknesses ranging from 20 nm to 200 nm. Films with sufficient quality and reliability could not be formed with thicknesses below 20 nm.  Figure 5.1. The CuO nanocrystal size as a function of the film thickness. The films have thicknesses of (a) 20, (b) 60, (c) 100, and (d) 200 nm, with nanocrystal sizes of 50, 130, 140, and 300 nm, respectively. The insets show scanning electron microscope images of the nanocrystals. 0 80 20012040 160Film Thickness (nm)0100300600400200500Nanocrystal Size (nm)a b c d500 nmab cd 84 A key finding was that the CuO nanocrystal size is a function of the original sputtered copper film thickness. Figure 5.1 shows the trend for CuO film thicknesses of 20, 60, 100, and 200 nm, which yield CuO nanocrystal sizes (i.e., diameters) of roughly 50, 130, 140, and 300 nm, respectively. The corresponding scanning electron microscope images are shown in the figure insets. For the purposes of this study, and its application to all-optical modulation, the CuO films with a nanocrystal size of 50 nm, corresponding with film thickness of 20 nm, are selected. They are sufficiently small to yield a strong contribution from trap states—and thus the potential for rapid recombination. At the same time, they are (slightly) above the size for weak exciton confinement, which can be estimated at three times the Bohr exciton radius106, ∑X. Here, the Bohr exciton radius is estimated to be ∑0 = zRz0ℎ2/ôá“i2 ≈ 6 ± 4 nm for CuO, where ℎ is Planck's constant, i is the electronic charge,	zX is the permittivity of free space, zR ≈ 12.3 is the dielectric constant98,107, á“ ≈ 	0.1Ñ£ is the reduced mass, and Ñ£ is the electron rest mass. It is important to note that the reduced mass of CuO varies significantly in the literature98,100,108, and its Bohr exciton radius has previously been reported to range from 6.6 to 28.7 nm109. 5.1.2. Optical Absorption Characterization The optical absorption characteristics of the CuO nanocrystals are shown in Figure 5.2. The absorption coefficient, f, is displayed as a function of the wavelength, §, which is calculated from the percent power absorbed, as given by 1 − e-l” , where Ö  is the film thickness. The power absorbed is measured via reflection-transmission measurements110 with a Yokogawa AQ6370 optical spectrum analyzer. For this process, the reflected and transmitted spectra are recorded for the sample, being the CuO film on its glass substrate, with a broad spectral source. These two recorded spectra are normalized with respect to the incident spectrum. These spectra are used to calculate the corresponding wavelength-dependent absorption coefficient, while accounting for all  85 optical elements and multiple reflections within the glass substrate. The results were repeated, and found to be reliably reproduced, for various positions across the CuO films and for various CuO films having the same nanocrystalline morphology. The results show finite absorption across the displayed spectrum, which is attributed to a high density of trap states between the VB and CB of the nanocrystalline CuO material. A Tauc plot, shown in the inset, is used to extract the bandgap, •F, via the direct bandgap relation, fℎ‘ D =’D(ℎ‘ − •F), where ’ is an arbitrary constant, ‘ is the photon frequency, and ℎ‘ is the photon energy110. An extrapolation of the linear region from 1.67 eV to 1.80 eV of the Tauc plot down to the horizontal axis identifies the bandgap to be •÷ = 1.55 ± 0.05 eV. The error here is quantified through many repeated measurements with different samples and spectral sources. Both direct and indirect Tauc relations were examined. The direct (squared) relation gave the best fit and a measured bandgap within the range seen in the literature for CuO99,100,101,102,103,104. The indirect (square root) relation had a poorly resolved step response as a function of photon energy, and so it was not possible to quote an indirect bandgap. It is worth noting that the optical absorption characteristics show an Urbach tail extending out to lower energies, which is known to manifest itself in disordered semiconductor materials as a decreasing bandgap due to the encroachment of defect states into the bandgap111. Thus, in general, the bandgap measured for CuO nanocrystals will depend upon the nanocrystal size. Decreasing the nanocrystal size increases the density of trap states, which encroach into the bandgap from the band edges. This is seen as a red-shift on •F112. However, the bandgap will eventually begin to increase when the nanocrystal size approaches the Bohr exciton radius due to quantum confinement. This is seen as a blue-shift112 on •F. Rehman et al. tested CuO nanocrystals with radii of 11–29 nm and observed such effects113. The nanocrystals for our study, having a radius of  86 25 nm, fall within this range and exhibit some degree of a blue-shift from the bulk crystal’s theoretically calculated bandgap of 1.25 eV114. 5.1.3. Free-Carrier Dynamics and State-Filling The free-carrier dynamics that evolve within the CuO nanocrystals are revealed by way of transient pump-probe spectroscopy43,50. The experimental setup, described in section 2.2.1, focuses the 780 nm control (pump) beam and 1550 nm signal (probe) beam using a 40× microscope objective onto the CuO nanocrystals. Translation across the sample, perpendicular to the optical axis, shows negligible changes to the results, indicating the nanocrystals are sufficiently uniform. Translation parallel to the optical axis (out of the focal plane) is critical, and is controlled via nano-actuators (Newport, 8302) to ensure that the beam spot sizes and applied fluences are reproducible.  Figure 5.2. Optical absorption characteristics for the CuO nanocrystals. The absorption coefficient, ◊, is shown as a function of the wavelength, ÿ. A Tauc plot is shown in the inset, as ◊Ÿ⁄ € versus energy, Ÿ⁄, in black, with a linear fit shown in red. From this Tauc plot, the direct bandgap of the CuO nanocrystals is defined to be {‹ = 1.55 eV. 800 1550650 950 12501100950 1400Wavelength, λ (nm)024681210$EVRUSWLRQ&RHɝFLHQWž (104 FP-1)1.3 1.91.2 1.5 1.71.61.4 1.8(QHUJ\KƊ (eV)01234(žKƊ)2  (1010 FP-1 eV)2 87 Given that the CuO nanocrystals have a measured bandgap of •F = 1.55 eV, the experiments of transient pump-probe spectroscopy use an above-bandgap photon energy for the control beam of •J = 1.6 eV, at a wavelength of 780 nm, and a below-bandgap photon energy for the signal beam of •I = 0.8 eV, at a wavelength of 1550 nm. The selected photon energies, being •J > •F and •I < •F, are critical to the study of this material. The high control beam photon energy allows it to photoexcite electrons from the VB to the CB. The low signal beam photon energy allows it to be preferentially sensitive to free-carrier dynamics evolving within the VB. This is because the signal beam will be predominantly subject to absorption from the top of the VB to gap states. Gap states are known to play a major role in semiconductor nanocrystal films115. Representative experimental results are shown in Figure 5.3 as normalized differential transmission, ∆o p /o, of the signal beam as a function of the time delay, p, between the control and signal pulses. The time delay axis has been divided into four segments. The interval prior to zero time delay is defined by p < 0 ps. The point of time at zero time delay is defined by p = 0 ps. The interval within the first picosecond after zero time delay is defined by 0 ps < p <	1 ps. The interval following the first picosecond is defined by p >1 ps. The evolution of the free-carrier dynamics in the bandstructure of the CuO nanocrystals is illustrated in Figure 5.4. Prior to zero time delay, p < 0 ps, the bandstructure is populated by electrons (red) and holes (blue) in the manner depicted by Figure 5.4(a). Given the low photon energy of the signal beam, •I ≈ •F/2 , the signal beam in this time interval predominantly photoexcites electrons from near the top of the VB to gap states in the bandgap. Such a proposition is similar to the proposed dynamics within CuS nanocrystals116. The process is labelled in the figure by •I . (Note that Figure 5.4 only depicts gap states at one energy level for simplicity, however there likely exists many states within the gap that also contribute to absorption of the  88 signal beam.) Precisely at zero time delay, p = 0 ps, the CuO bandstructure is populated by free-carriers in the manner depicted by Figure 5.4(b). The system at this point in time is subject to the aforementioned signal beam absorption as well as the newly-introduced control beam absorption. Given the large energy of the control beam, •J > •F, the control beam photoexcites a high density of electrons into the CB, leaving a high density of holes within the VB, as labelled by •J. These photogenerated holes in the VB, which initially have a narrow distribution in the energy-momentum bandstructure, yield a large decrease in the signal beam’s absorption due to state-filling. In other words, a large increase is seen in the signal beam’s transmission due to the control- Figure 5.3. Transient absorption characteristics for the CuO nanocrystals. The control and signal beams are focused onto the layer of nanocrystals using a 40× microscope objective, as seen in the inset. The transmitted signal beam power is measured as a function of the time delay. The results, displayed as normalized differential transmission of the signal beam, ∆¢ ï /¢, are divided into four intervals: prior to zero time delay, ï < 0 ps; at zero time delay, ï = 0 ps; within the first picosecond after zero time delay, 0 ps < ï < 	› ps; after the first picosecond, ï >	1 ps. -1 8-2 2 530 74 61Time Delay, t (ps)00.20.40.60.81.21.0ŢT(t)  / T (a.u.)0 ps < t < 1 ps t > 1 pst < 0 ps40xCuO Nanocrystals200 nm 89 beam-induced increase of holes within the VB and the corresponding reduction in the number of electrons available in the VB to undergo signal-beam-induced absorption. Within the first picosecond after zero time delay, 0 ps < p <1 ps, free-carriers in the VB undergo momentum relaxation on a femtosecond timescale, from their initially narrow distribution in the energy-momentum bandstructure, as depicted by Figure 5.4(c).  The distribution of free-carriers broadens due to carrier-carrier scattering, which rapidly decreases the number of holes that occupy the VB states undergoing signal beam absorption. This is seen as a rapid decrease in the signal beam’s transmission. In the following picoseconds, for p >	1 ps, the holes in the VB undergo energy relaxation on a picosecond timescale and relax upwards towards the VB edge due to carrier-phonon interactions, as depicted by Figure 5.4(d). This further decreases the signal beam’s transmission. Once the holes reach the VB edge, trap-assisted electron-hole recombination takes place to return the system to equilibrium. 5.1.4. Experimental Validation The proposed free-carrier dynamics are explored next via experimental means to substantiate the interpretations of momentum relaxation (via carrier-carrier scattering), energy relaxation (via carrier-phonon interaction), and recombination (via gap states). Given that the rates at which free-carriers undergo relaxation can vary with the photogenerated free-carrier density117,118,119, â, the transient pump-probe spectroscopy measurements are repeated with varying control beam fluences to witness variations in rates (if any). The experimental results are shown in Figure 5.5. Figure 5.5(a) shows the differential transmission of the signal beam, Δo p /o, for 13 different control beam fluences, increasing linearly from 0.4 J·m-2 to 10 J·m-2. Approximately 10% of the energy for each of these control beam fluences is absorbed by the 20 nm thick nanocrystal film, according to measured power levels and calculations based on the results of Figure 5.2. All of the  90  Figure 5.4. Illustration of the free-carrier dynamics within the bandstructure of CuO nanocrystals. The bandgap is {‹ = 1.55 eV. Photogenerated holes (blue) and electrons (red) are shown in the bandstructure at four time intervals: (a) prior to zero time delay, ï < 0 ps, the VB and CB are populated by photogenerated holes and electrons from the signal beam, which has a photon energy of {– = 0.8 eV; (b) at zero time delay, ï = 0 ps, additional electrons and holes are photogenerated by the control beam, which has an energy of {fi = 1.6 eV; (c) within the first picosecond after zero time delay, 0 ps < ï <	1 ps, the photogenerated holes in the VB undergo momentum relaxation due to carrier-carrier scattering; and (d) after the first picosecond, ï > 1 ps, the photogenerated holes in the VB undergo energy relaxation and trap-assisted recombination.   Prior to Zero Time Delay (t < 0 ps)aEgEsCBVBAt Zero Time Delay (t = 0 ps)EcbEgEsCBVBWithin the First Picosecond after Zero Time Delay (0 ps < t < 1 ps)cEgEsCBVBPicoseconds after Zero Time Delay (t  > 1 ps)dEgEsCBVB 91 Δo p /o curves are normalized with respect to response with the highest fluence. Overall, the observed Δo p /o recovery follows a tri-exponential decay that is dependent upon the control beam fluence and independent of the signal beam fluence. Curve-fitting is applied to the results with a tri-exponential function, Δo/o(p)	= fl1e-Ä/ø1 	+ 	fl2e-Ä/ø2 + fl3e-Ä/ø3 , where	í1, í2, and í3 are time constants, and fl1 , fl2 , and fl3  are weightings (i.e., amplitudes). The curve-fitting parameters are extracted by convolving the tri-exponential function with the autocorrelation of  the control and signal pulses and employing iterative least-squares minimization. The curve-fitting shows strong agreement to the experimental results, with an average R-squared value of 0.999. The fitted curves are shown superimposed (in black) on the experimental results. The three rates for the tri-exponential decay are characterized in Figure 5.5(b). This figure shows the rates for the time constants, 1/í1, 1/í2, and 1/í3, and the respective weightings, fl1, fl2, and fl3 , as a function of the control beam fluence, ‡ . Trendlines are shown in red. The dependencies of the rates and weightings on ‡ give insight into their underlying mechanisms. The first time constant, í1, is characterized for increasing control beam fluences by a weighting, fl1, that shows linear growth (albeit with some superlinearity) and a rate, 1/í1, that is nearly constant (albeit with some linearity). We will first consider the more standard linear growth of fl1 and constant response of 1/í1. Such responses are indicative of momentum relaxation, which in this case involves carrier-carrier scattering in the VB. The first time constant í1 varies from 330 fs to 630 fs, over the range of control beam fluences. Such values are comparable to those seen for carrier-carrier scattering and momentum relaxation in other materials50. Moreover, these observations and interpretations agree with those proposed by Othonos et al. for their investigate CuO nanowires, for which a time constant of 400 fs was seen at lower fluence levels with a 770 nm signal beam and was attributed to carrier-carrier scattering and momentum relaxation82.  92  Figure 5.5. Transient pump-probe spectroscopy results for the 50 nm CuO nanocrystals. In (a), the experimental results for the normalized differential transmission, ∆¢ ï /¢, of the CuO nanocrystals are given for control beam fluences increasing linearly from 0.4 J·m-2 (yellow) to 10 J·m-2 (red). Superimposed on each experimental curve is the tri-exponential decay curve-fit (black). In (b), the weighting of the first time constant’s term (top), ·1, the weighting of the second time constant’s term (middle), ·2, and the weighting of the third time constant’s term (bottom), ·3, are plotted as a function of the control beam fluence, ‚. The inset shows the rates of the first (top), ›/»1, second (middle), ›/»2, and third (bottom), ›/»3, time-constants as a function of ‚. Trendlines are shown in red. 1 100 3 642 85 7 9Control Beam Fluence, Ɯ (J·m-2)00.20.40.60.8Weighting (%)bK1K2K3100 642 80123Control Beam Fluence, Ɯ (J·m-2)Rate (ps-1)/τ11/τ21/τ314.0-1.0 2.01.00 3.0Time Delay, t (ps)00.20.40.60.81.0ŢT(t)/T (a.u.)aIncreasing control beam fluence 93 (However, our measurements with a 1550 nm signal beam, having a photon energy of only 0.8 eV, suggests that the state-filling experienced by the signal beam is associated primarily with transitions from the VB to gap states.) We next consider the more unusual superlinear growth of fl1	and linear growth of 1/í1. The onset of these unusual characteristics for CuO were hinted at in the work of Othonos et al., stating that a slight decrease in the recovery time was seen with increasing fluence82. Moreover, these characteristics of an initial rapid decay with a superlinear trend in the amplitude (and increasing rate) with respect to the control beam fluence have been seen in other semiconductor nanocrystals, including ZnCdS120, ZnO121, TiO2122, CdS123, CdSe124, and CuxS125. The superlinear dependence seen in these nanocrystals is attributed to exciton-exciton annihilation, which emerges following the saturation of trap states at sufficiently high control beam fluence72,132. Trap state saturation leads to the preferential populating of free-carriers at the band edges, where they experience rapid recombination via exciton-exciton annihilation112,123,124,125,126. Evidence for the higher-order kinetics of exciton-exciton annihilation in CuO is provided by the linear trend in 1/í1. In general, highly photoexcited semiconductor nanocrystals exhibit second-order kinetics via two-body recombination and third-order kinetics via three-body recombination127. The rate of two-body recombination124 is proportional to â, and the rate of three-body recombination128 is proportional to â2 . For two-body recombination, the underlying mechanisms can be exciton-exciton (non-radiative) annihilation and/or band-to-band (radiative) recombination. (Note that other names for these processes do exist in the literature132.) For three-body recombination, the underlying mechanism is Auger recombination, which is brought about by electron-electron-hole or electron-hole-hole interactions in which the electron and hole recombine to give kinetic energy to the remaining electron or hole. (See Appendix B for more details on band-to-band and Auger recombination.) Our observed linearity between 1/í1 and â,  94 and thus control beam fluence, suggests that there is a strong contribution from two-body recombination. In the literature, such trends in other semiconductor nanostructures were first attributed to band-to-band radiative recombination133. However, further studies with time-resolved fluorescence showed that exciton-exciton annihilation dominates in semiconductor nanostructures following trap state saturation132. Such a claim is especially feasible for CuO, given that it is known to establish highly-correlated free-carriers, i.e., excitons, with large binding energies and remarkably long lifetimes129, and exciton-exciton annihilation has been reported in the literature for Cu2O130. Ultimately, the underlying mechanism for í1 is attributed to a combination of momentum relaxation via carrier-carrier scattering within the VB and exciton-exciton annihilation. The second time constant, í2, is constant at 2 ps over the range of control beam fluences, and its underlying mechanism is relatively straightforward. This time constant is associated with energy relaxation involving carrier-phonon scattering in the VB and scattering to trap states. The value for í2 is comparable to those seen for energy relaxation in other materials50,131 and is in agreement with that seen for energy relaxation by Othonos et al.82. Moreover, the rate of energy relaxation, corresponding to the 1/í2 curve in the inset of Figure 5.5(b), is constant with respect to ‡, which indicates that the carrier-phonon scattering is independent of ‡ and thus â as well119. The associated weighting of the second time constant’s term, denoted by fl2, as a function of ‡, is seen in Figure 5.5(b). The linear trend seen here is indicative of the standard interpretation of state-filling49, which is proportional to â.  The third time constant, í3, is constant at 50 ps over the range of control beam fluences, and it is associated with trapping and recombination of free-carriers. The long duration of this process matches typical time constants seen for trap-assisted free-carrier recombination in semiconductor nanocrystals.132,134,135 Though the weighting fl3 is small, this term is essential for the curve fit to  95 match over longer timescales. It is also important to note here that the weighting of the third time constant’s term fl3 is positive, which indicates that the observed response results from state-filling of signal beam transitions from the VB up to gap states (to a large extent) and free-carrier dispersion (to a lesser extent). This is in contrast to the typical free-carrier absorption seen in bulk semiconductors, which shows a negative change in transmission as the free-carriers that are generated by the control beam lead to increased absorption of the signal beam. We do not see a transition from positive to negative polarities of the signal beam transmission for any of our conditions, like those seen by Othonos et al. for a few of their signal beam energies82. Thus, for our signal beam energy, secondary photoexcitation of electrons from gap states into the CB or from the CB up to higher states in the CB is small, if at all present. Furthermore, as multiple gap states with differing energies likely exist (e.g., shallow, deep, and mid-gap states116), í3 represents the effective rate for transitions from these states to the valence band. Ultimately, the weighting fl3 is linearly proportional to ‡, as seen in the inset of Figure 5.5(b), and this is further evidence that the signal beam is subject to absorption between states at the edge of the VB up into gap states. Given the importance of trap states for the third time constant, the theory from section 2.1 on free-carrier dynamics in spherical semiconductors is applied here to characterize surface recombination in the CuO nanocrystals. It is assumed that the nanocrystals are uniformly illuminated to form an initial photogenerated free-carrier density of â0. The free-carriers then undergo diffusion and recombination in accordance with Equation (32) and the boundary condition given in Equation (33). According to section 4.1, this theory can be greatly simplified by assuming that the system has no dependence on the polar angle, ∂, and by assuming that the nanocrystal’s radius is much less than the diffusion length, i.e., the diffusion coefficient is sufficiently large to ensure that ì ≫ ò5∑. With these simplifications, the solution for the free-carrier density becomes  96 â(p) ≈ â0i-Ä/ø3, where the recombination rate is 1/í3 ≈ 1/í9 + ò5^ and ^ = 3/∑ is the surface-to-volume ratio of the spherical nanocrystal. Given the small radius of the nanocrystals, ∑ = 25 nm, and the measured recombination lifetime of í3 = 50 ps, the surface recombination velocity is estimated to be ò5 ≈ 2×104 m/s. To the best knowledge of the authors, a value for the surface recombination velocity of CuO has not been reported in literature, but the value reported here is reasonable, given its similarity to those of CdS72 and InP28. It is worth noting that a comparison of these transient pump-probe spectroscopy results to analogous results for the other (larger) nanocrystals in this work shows that the time constant í3 increases as the radius increases. Such a trend agrees with the interpretation of trap-assisted recombination being the underlying mechanism for this time constant, given that the larger nanocrystals would have decreased densities of trap states and thus increased time constants. Moreover, the other two time constants, í1 and í2, exhibit negligible changes as the radius increases, and this constancy agrees with the interpretations of momentum and energy relaxation for their underlying mechanisms. 5.1.5. Summary & Discussion Within this section, a detailed investigation was carried out on the bandstructure and ultrafast free-carrier dynamics of CuO nanocrystals. The bandgap was measured to be 1.55 ± 0.05 eV. This aligns with recent experimental studies that shown a direct bandgap at 1.4–1.8 eV99,100,101,102,103,104, and emerging theoretical studies104,108, going beyond Standard Density Functional theory, to show a direct bandgap at 1.25 eV for bulk CuO114. Both theoretically and experimentally, there appears to be a higher direct transition104,113,114,136 near 3.0 eV, which may be the cause for the dissimilar interpretations of the absorption characteristics in the literature. Transient pump-probe spectroscopy was carried out with an above-bandgap control beam photon energy of 1.6 eV and a below-bandgap signal beam photon energy of 0.8 eV. The differential transmission of the signal  97 beam through the CuO nanocrystals was measured for various control beam fluences, and three time constants were seen in the recovery. The first time constant, ranging from 330 fs to 630 fs, had a rate that scaled linearly with the control beam fluence and a weighting that scaled superlinearly with the control beam fluence. The underlying mechanisms for this first time constant were attributed to momentum relaxation via carrier-carrier scattering within the VB and (what the authors believed to be) the first observation of exciton-exciton annihilation in CuO. The second time constant was constant at 2 ps, with a weighting that scaled linearly with the control beam fluence. Its underlying mechanism was attributed to energy relaxation via carrier-phonon scattering within the VB. The third time constant was constant at 50 ps, with a weighting that scales linearly with the control beam fluence. Its underlying mechanism was attributed to trapping and recombination, due to the high density of trap states within the bandstructure. The bandstructure and free-carrier dynamics put forward in this work lay the foundation for future applications of the CuO material system—and they will be the foundation for the all-optical modulator developed in section 5.2.    98 5.2. All-Optical Retro-Modulation Architecture The goal of this section is to develop a practical all-optical architecture for FSO communication. The work is motivated by a long-standing challenge in the telecommunication industry to provide high-speed telecommunication service to remote or dense urban locations. Such a challenge is often referred to as the “last-mile” problem31. To address the challenge, this section puts forward an all-optical architecture that can offer high-bandwidth connections between multiple ground TRXs (with distributed locations) and one aerial TRX (with limited system mass and power). It is shown that the goal of multidirectional communication can be met by implementing retro-modulation on board the aerial TRX. At the same time, such an implementation avoids escalating the mass and power at the aerial TRX, because retro-modulation harnesses the abundant optical power made available by the ground-to-aerial active UL in forming the aerial-to-ground passive DL. The all-optical architecture that is developed in this section for use in passive DLs integrates an all-optical modulator and a spherical retroreflector. All-optical modulation is used here to avoid the fundamental bottleneck that accompanies far slower (gigabit-per-second) electronic processing and modulation at the aerial TRX7. Based upon the findings of section 5.1, the all-optical modulation is implemented via cross absorption modulation in CuO nanocrystals. Free-carriers induced by the control beam change the CuO nanocrystal’s refractive index, Δ_, and absorption, Δf, by way of state-filling (predominantly) and free-carrier absorption (to a lesser extent). Such an approach benefits from the strong nonlinearity associated with generation and recombination of free-carriers, and thereby avoids the need for phase-matched propagation over centimeter lengths4. At the same time, it benefits from the fact that free-carriers in the CuO nanocrystals undergo ultrafast relaxation and recombination, as shown in section 5.1. Ultimately, CuO  99 nanocrystals are implemented with an appropriate spherical retroreflector to realize an effective AORM architecture for passive DLs. A link budget analysis of passive DLs is considered in the design of the AORM architecture, as shown in section 5.2.1. Experimental studies are conducted and the AORM architecture is evaluated in section 5.2.3 with respect to the link budget analysis. The results are summarized in section 5.2.4. 5.2.1. Link Budget Analysis A link budget analysis (i.e., quantifying the received power of a link that is subject to various gain and loss mechanisms) is employed here to test the effectiveness of passive DLs, implementing the proposed AORM architecture, with respect to contemporary active DLs. Active DLs are typically implemented with a collimated line-of-sight laser beam directed from the aerial TRX to the ground TRX. The detected signal power, d´et, for this active DL is described by the Friis transmission equation137  d´et = ¶ Ä´ eµ„Ä^2, (42) where Ä´ is the transmitted aerial-to-ground beam power, eµ is the receiver’s area, „Ä is the solid angle of the diverging beam, ^ is the link length, and	¶ is the link loss, which is predominantly due to turbulence for links greater than 1 km138. Reflection losses are ignored in Equation (42) given that suitable anti-reflection coatings can be implemented at the aerial and ground TRXs. The solid angle of the beam is defined by the laser source, and can be described by the traditional antenna equation, being the square of the wavelength divided by the area of the source’s aperture137. A typical 10-cm-diameter optical beam, operating within the telecommunication C band, i.e., 1530-1565 nm, will have a divergence-limited solid angle on the order of „Ä ≈ 10-10 steradians (sr), which has been achieved in FSO systems33,139. With this narrow FOV, the beam  100 diameter at the receiver will be on the centimeter- to meter-scale. Clearly, given this small beam size, links to additional receivers being separated by meter- to kilometer-scale distances cannot be established. The only way for a conventional aerial transceiver to establish multiple connections is to have an independent laser and beam steering elements for each link. However, this can be impractical for large numbers of links due to the aforementioned mass and power constraints. To meet the need for operation with multiple ground TRXs and minimal escalation of the system mass and power at the aerial TRX, the active DL could be configured for broadcasting over a wide solid angle—as is typically done for radio-frequency links. However, in realizing an active DL with a wide hemispherical field of view (FOV), i.e., subtending half of the sphere by having „Ä = 2ô sr, the detected signal power decreases by more than 1010, compared to a collimated active DL. Given the strict power constraints of aerial vehicles and their TRXs, the increases in laser powers that would be necessary to compensate for these large losses are not viable. To resolve the conflicting demands seen above for the collimated and broadcasted active DLs, the passive DL depicted in Figure 5.6(a) can be used. Its multidirectional operation is illustrated in the inset. For each link, a collimated beam is directed from the ith ground TRX to the aerial TRX (orange), where a fraction of the beam area is modulated and retroreflected back to the ground TRX (green). The detected signal power for such a retroreflected passive DL is  d´et = ¶ Ä´‰ e`„Ä^2 eµ„µ^2, (43) where Ä´ is the transmitted ground-to-aerial beam power, ‰ is the all-optical modulation depth, e` is the retro-modulated area of the signal beam, and „µ is the retroreflected divergence solid angle. Reflection losses are ignored in Equation (43), as before, because it is assumed that suitable reflective and anti-reflective coatings can be implemented. Moreover, the retro-modulated area, e`, and the receiver area, eµ, are assumed to be smaller than the incident beams’ areas. Based on  101 Equation (43), the passive DL is superior to the broadcasted active DL for multidirectional communication as long as the power penalties associated with a small modulated signal beam area, e` < „Ä^2, a large retroreflected divergence solid angle, „µ > eµ ^2, and a small modulation depth, ‰ < 1, do not exceed the penalty from broadcasting the active DL with a large „Ä. These three link budget conditions for the passive DL must be considered in the design of an effective AORM architecture. The proposed AORM architecture, depicted in Figure 5.6(b), incorporates a spherical retroreflector, realized as two glass hemispheres in the form of a full sphere. The glass is chosen to have a sufficiently high refractive index to have incident light rays focus onto the back surface140, leading to retroreflection over a FOV of 2ô sr. The incoming continuous wave ground-to-aerial beam (orange) is collected by the sphere, and the outgoing signal beam (green) is retro-modulated over a cross-sectional area of e`. Between the two hemispheres there exists a thin film of CuO semiconductor nanocrystals. The incident beam is focused through this CuO film on its way to and from the sphere’s back surface. At the same time, the control beam enters from the sphere’s back surface and is focused onto the CuO film over an area of eç. The CuO film brings about the desired strong and ultrafast interactions between the signal and control beams via cross absorption modulation with a modulation depth of ‰. Ultimately, the retroreflected and modulated signal beam propagates back down to the ground TRX to realize the passive DL shown in Figure 5.6(a). Note that the FOV for the AORM architecture will not cover the full 2p sr of the hemisphere, being the solid angle for effective retroflection, as modulation must also be considered. Specifically, less signal beam power will pass through the modulated area eç for large incident angles with respect to surface normal (i.e., z axis). If the FOV is defined by the solid angle over  102 which the modulated signal power is equal to or greater than one half the modulated signal power at normal incidence, the FOV for effective retro-modulation is p sr. The envisioned passive DL, with the proposed AORM architecture, can be implemented with a variety of formats for modulation and detection. However, amplitude-shift keying modulation with phase-locked self-homodyne coherent detection is particularly appropriate format. Such modulation and detection has a proven track record in passive optical networks141,142,143. Moreover, coherent homodyne detection offers unrivalled sensitivity, due to its inherent spatial and wavelength selectivity, and this has supported its use in long-range terrestrial links (e.g., spanning 142 km144) and intersatellite links (e.g., spanning 5,000 km145). It is also worth noting that the proposed passive downlink architecture is ideally suited to phase-locked self-homodyne detection because the laser on the ground can be used for both the signal beam and the phase-locked local-oscillator beam146. With such phase-locked self-homodyne detection, the phase-locked local oscillator beam can be used to introduce deconstructive interference on the carrier. Thus, any residual unmodulated carrier can be suppressed, if the modulation depth is small, ‰ ≪ 1, thereby eliminating the DC background power and its associated shot noise146,147. The detection is also typically implemented with balanced detectors to lessen relative intensity noise and amplifier noise147,148. For these reasons, phase-locked self-homodyne detection is ideally suited to passive DLs.  103  Figure 5.6. Implementation of the active UL and passive DL with the AORM architecture. (a) The overall system facilitates multidirectional communication via an active UL and passive DL. The active UL is shown as the aerial-to-ground beam (orange), with a solid angle of Âï, being transmitted from the ith ground TRX to the aerial TRX. The passive DL is shown as a portion of the incident beam being retro-modulated over an area of ÊÁ. This retro-modulated signal beam (green), with a solid angle Â∆, is transmitted from the aerial TRX back down to the ground TRX, which has a receiver area of Ê∆. The inset depicts multidirectional communication with this implementation for an arbitrary number of ground TRXs. (b) The proposed AORM architecture collects the ground-to-aerial beam (orange) with a glass sphere and focuses the beam through a CuO film and onto its back surface. It then undergoes retroreflection and propagates back to its source. The on-board control beam (red) is focused via external optics onto the CuO film, where it has a cross-sectional area of ÊË , to modulate the signal beam (green) over a cross-sectional area of ÊÁ.  Arȡtground TRXiaerial TRXRAsȡr ...TRX1 TRX2 TRX3aAsȡrAcȡtOAxzyab 104 5.2.2. Theoretical Design The three link budget conditions stated in the prior section are considered here for the proposed AORM architecture. The first condition defined by the link budget for AORM requires maximizing the modulated signal beam area, e`, and this can be accomplished by careful placement of the CuO film within the spherical retroreflector. In doing so, it becomes apparent that increasing e` will require that the control beam’s modulation area, eç , be increased, as shown in Figure 5.6(a). This has repercussions for the control beam’s power,	 ç´, and intensity, ç´ eç. In general, it is desirable to reduce eç for the same control beam power,	 ç´, given that the modulation depth, ‰, is linearly proportional to the intensity, ç´ eç . Thus, there are conflicting demands between eç  and e` . These conflicting demands can be minimized, however, by placing the CuO film at the sphere’s centre, in the Ç = 0 plane, as seen in Figure 5.6(b). At this location, the control beam has an area of eç = ô∑2 4 and it modulates the signal beam over an area of e` = ô∑2, where ∑ is the radius of the sphere. For comparison, the CuO film could be made to cover the entire back surface of the sphere, but for this case the control beam would need to have a larger area, being eç = 2ô∑2, to have a uniform intensity and modulate the same signal beam over an area of e` = ô∑2. Ultimately, it can be said that positioning the CuO film in the Ç = 0 plane offers a factor of eight improvement to ‰ , for the same ç´ , in comparison to positioning of the CuO film at the back surface. Furthermore, with respect to the FOV, being defined as the solid angle over which the signal beam is modulated, the CuO film in the Ç = 0 plane has a FOV that is independent of the control beam area, eç. In contrast, having the CuO film on the back surface yields an FOV that decreases as eç decreases. (In contrast, for the case with a CuO thin film on the back curved surface, the FOV will decrease for a smaller eç. This same challenge would apply to the aforementioned multi-element  105 (cat’s eye) retroreflector, where the signal beam is focused onto the modulator at a location off of the optical axis35,41.) This is a subtle but important benefit for the AORM architecture with the CuO film in the z = 0 plane because its control beam area, eç, be varied with no penalty to FOV. It will be shown shortly in this work that there is a benefit to decreasing eç because it enhances the retroreflected divergence angle. The second condition defined by the link budget analysis for AORM requires minimizing the retroreflected divergence solid angle, „µ. Specifically, there is a goal to achieve the same level of collimation for the retroreflected aerial-to-ground beam as the transmitted ground-to-aerial beam, which is assumed to be on the order of „µ = 10-10 sr. To reach this level of collimation for the retroreflected beam, the effects of spherical aberration introduced by the spherical retroreflector must be eliminated. Spherical aberration manifests itself as increasing retroreflected divergence angles for rays that are increasingly far from the optical axis. There are two solutions to mitigate this aberration: decrease the control beam area, eç , or design the spherical retroreflector to compensate for spherical aberration. The first solution to mitigate spherical aberration is to decrease the control beam area, eç . Decreasing eç decreases the area over which the incoming ground-to-aerial beam is modulated, as seen in Figure 5.6(b). In doing so, the modulation is preferentially applied to the paraxial rays, being close to the optical axis, which has the retro-modulated aerial-to-ground beam be subject to less spherical aberration and thus exhibit a lower divergence solid angle, „µ. Figure 5.7 shows this trend by depicting the divergence solid angle, „µ, as a function of the control beam radius, ç¨, normalized to the sphere’s radius, ∑, for various spherical retroreflectors. The results of Figure 5.7 are calculated via three-dimensional ray tracing simulations. The code for these ray tracing simulations is given in Appendix I, and three-dimensional ray-tracing diagrams are given in Figure  106 I.1. Beginning with an ideal spherical retroreflector, implemented as a glass sphere with a refractive index of _ = 2.000 (black), it is seen that a small ç¨ achieves a low „µ. Specifically, a low beam divergence of „µ < 10-10 sr can be achieved with ç¨ being approximately 1.4% of ∑. However, it would be difficult or impossible to find a material with a refractive index of exactly _ = 2.000 for use in the telecommunication C band. The closest material is S-LAH79 glass with a refractive index of _ = 1.955 at a wavelength of 1550 nm. The corresponding „µ  for the S-LAH79 retroreflecting sphere is shown in Figure 5.7 (red). It can be concluded from this result that such a sphere is impractical for long-range links because ç¨ would need to be approximately 0.006% of ∑. However, it is important to note that the impracticality of reducing eç	comes from practical constraints, such as diffraction, laser-induced damage, and thermal fluctuations, rather than performance. (The details of such constraints are discussed in the upcoming section.) Reducing eç	will not affect the retroreflector’s FOV, simply because the CuO film is at the sphere’s centre. Nor will reducing eç affect the overall detected signal power, d´et. This claim is based on Equation (43), where one might expect d´et to decrease for a small eç, because of its correspondingly small signal beam area, e`. However, the modulation depth, ‰, in the equation is proportional to the control beam intensity, ç´ eç, so d´et decreases in proportion to 1 eç. Thus, there is no performance penalty from reducing eç. The second solution to mitigate spherical aberration is to use a spherical retroreflector with a gradient refractive index149 or a concentric cladding150. Such retroreflectors have been tested in space151. The concentric cladding retroreflector is of particular interest here due to its lesser manufacturing complexity and improved performance—but only given careful attention to material selection150. A sphere with a single concentric cladding, with both materials having refractive indices below _ < 2, can be used to attain the same divergence as an ideal _ = 2 sphere.  107 Based on two chosen refractive indices, being for the outer cladding, _1, and inner sphere, _2, the corresponding ratio of radii can be determined from lens theory150 to be 2¨ 1¨ =1 _2 − 1 _1 1 2 − 1 _1 , where 1¨ = ∑. It is preferable for the interior material to have a refractive index close to _ = 2.000, such as S-LAH79 glass with _2 = 1.955, and the cladding material to be a typical polymer or glass with a refractive index near _1 = 1.5. (Curable polymer would be particularly advantageous, as it can be shaped to form a spherical shell/layer with near-molecular smoothness, while immersed in a filler solution, due to its liquid-based surface tension152.) The resulting „µ  for these two materials, with _ < 2 and 2¨ 1¨ = 0.9309 ± 2×10-4,  Figure 5.7. Ray tracing simulations for various spherical retroreflectors. The retro-modulated divergence solid angle, Â∆, of each spherical retroreflector is given as a function of the control beam radius, ∆Ë, normalized to the sphere’s radius, À. The following non-cladded and cladded spheres are presented: an ideal ≤ = 2.000 sphere (black), an ≤ = 1.955 sphere (red), an ≤ = 1.955 sphere with an ≤ = 1.500 cladding (blue dash), and an ≤ = 2.500 sphere with an ≤ = 2.301 cladding (green). The goal of Â∆ < 10-10 is achieved for the portions of the curves below the displayed horizontal dashed line. 10010-5 10-3 10-2 10-110-4Normalized Control Beam Radius, rc / a10-1210-1110-1010-910-710-510-610-8Divergence Solid Angle, ȡr (sr)n = 1.955spheren = 2.0spheren1 = 1.955n2 = 1.50claddedspheren1 = 2.301n2 = 2.50claddedspherercŶrcladdingspheren2n1 108 shown in Figure 5.7 (blue dash), matches that of the ideal _ = 2.000 sphere. The tolerance given here is defined such that half the signal power achieves the desired „µ < 10-10 for the same control beam radius. Such a structure, combined with a small eç, is a simple way to reduce spherical aberration with minimal manufacturing complexity. This structure can be further improved to nearly eliminate spherical aberration by using materials with refractive indices above _ > 2. For this case, the refractive index of the interior material depends upon the refractive index of the exterior cladding material and vice versa. For a hypothetical pair of materials, with _1 = 2.500 and _2 = 2.301 ± 1×10-3 and 2¨ 1¨ = 0.34594 ± 4×10-5, the resulting „µ is shown in Figure 5.7 (green). For this structure, a low beam divergence of „µ < 10-10 sr can be achieved with ç¨ being approximately 5% of ∑. However, to our best knowledge, materials with these sufficiently high and precise refractive indices do not exist. A structure with a third cladding layer can be implemented to alleviate this constraint, as demonstrated by J. P. Oakley150. The main benefit for such a sphere with two claddings is that it can allow for the use of a smaller sphere, with a four times smaller radius, to achieve the same retroreflection capabilities as the single cladding structure with _ < 2. However, this reduction in size and mass comes at the expense of greater manufacturing complexity, and so the choice between single cladding and dual cladding structures will depend upon the application. Specifically, smaller spheres could increase the diffraction-limited divergence solid angle above ~10-10 sr, if the sphere radius is less than ∑ = 5 cm. Thus, a spherical retroreflector with ∑ = 5 cm is recommended here, with the control beam applied to its central region with ç¨ = 0.7 mm (for the _1 = 1.955, _2 = 1.50 cladding) or ç¨ = 2.5 mm (for the _1 = 2.301, _2 = 2.50 cladding). This yields low spherical aberration and low divergence. Diffraction from the induced modulation over the control beam area can also considered. Assuming that amplitude shift keying is used, the control beam will be on for only one of the two  109 binary symbols. Thus, when the control beam is off, the signal beam will undergo the standard diffraction-limited divergence that is defined by the spherical retroreflector. However, when the control beam is on, the signal beam will undergo increased diffraction. This response is advantageous because the control beam being on both decreases the signal beam power and increases its divergence (because it is missing the central area of its beam profile). Thus, the modulation depth of the received signal beam will be improved beyond that expected solely from the decreased signal power. Such a finding is interesting and could warrant future investigations that tailor the profile of the modulated area to optimize the far-field pattern of the signal beam for enhanced modulation. The third condition defined by the link budget for AORM requires increasing the all-optical modulation depth, ‰. In this work, ‰ is defined as the relative change in signal beam power, ‰ =Δ´` ´` . In general, ‰ is linearly proportional to the material’s change in refractive index, y_, and change in absorption, Δf , induced by the control beam. Fibre and waveguide structures typically employ a Δ_  material response via nonresonant nonlinearity, which requires phase-matched propagation over centimeter lengths due to a low third-order nonlinear coefficient4. As well, in previous work from our laboratory, for short-range optical wireless communication, a nonlinear glass was used to retro-modulate the signal beam153. The refractive index of its glass-air interface was modulated via a nonresonant nonlinearity, yielding a small Δ_ and negligible change to the absorption coefficient Δf, and this gave a modulation depth of ‰ = 4Δ_ _2 − 1 . These findings suggested that a stronger nonlinearity is required—and this work does so by introducing the CuO film. It is shown in the next section that the control-beam-induced modulation of the signal beam can be enhanced via resonant photoexcitation in CuO nanocrystals to yield an improvement of 103 in ‰.  110 5.2.3. Experimental Results The proposed AORM architecture is experimentally tested to find the modulation depth and the maximum data rate. Cross absorption modulation is implemented via the proposed CuO film because it has a strong change in its absorption coefficient, Δf, via state-filling. This leads to a large modulation depth, according to ‰ = gΔf , where g  is the 1 e  penetration depth. Our measurements have shown that the penetration depth of CuO is g = 200 nm and that the control-beam-induced change in refractive index, Δ_, is small. For films thinner than the 1 e penetration depth, g is assumed to be equal to the film thickness. Note that the modulation due to the control-beam-induced change in refractive index, Δ_ , was measured and found to be negligible, at approximately 2% of the modulated power due to Δf. Ultimately, CuO is selected for use in the AORM architecture because the control beam is on-resonance with its bandgap and the resulting photogenerated free-carriers produce a strong Δf material response via state-filling, as shown in section 5.1.1. Experimental tests are carried out for a 20 nm thick CuO film with nanocrystal diameters of 50 ± 20 nm. To create the CuO phase, a 20 nm film of sputtered copper is annealed at 600°C for one hour in ambient air. The thickness was chosen as it provides the best structural uniformity, spectral absorption characteristics, and signal-to-noise ratio. The CuO thin film is deposited on the flat face of an S-LAH79 glass hemisphere, which is then coupled to another S-LAH79 hemisphere to form a full sphere. The S-LAH79 glass is chosen here as a proof of concept architecture for retro-modulation. The aforementioned more complex retroreflectors, having multiple cladding layers, could also be used with this fabrication process if there is a need for especially long links. The AORM architecture is characterized by measuring the ultrafast impulse response to determine ‰  and the maximum achievable data rate. Figure 5.8(a) depicts the pump-probe  111  Figure 5.8. Measurement of the impulse response of the AORM architecture. (a) Schematic of the pump-probe spectroscopy experimental setup for measuring the impulse response. The 1550 nm beam (orange) is introduced via a 50-50 beamsplitter and compressed with a telescope before incidence on the AORM architecture. The retro-modulated signal beam is measured with an InGaAs detector. The 780 nm control beam (red) is focused with a 20× microscope objective. (b) Experimental impulse response of the AORM architecture with a CuO film comprised of nanocrystals measured for an incident control beam fluence of 5 µJ/cm2. The inset shows the AORM architecture with the 1550 nm signal and 780 nm control beams incident on the CuO film, and a scanning-electron-microscope image of the nanocrystals in the CuO film with a 200 nm scale bar. delaystagedetectorlenstelescope50-50beamsplitterall-opticalretro-modulationarchitecture20x microscopeobjectivefree-spacelink1550 nmsignal beam780 nmcontrol beamab6-2 0 31 52 4-1Time Delay (ps)00.40.81.22.42.01.6Modulation Depth, M = |ŢPs / Ps| (× 10-4 )770 fs200 nmCuO nanocrystals 112 spectroscopy experimental setup used to measure these ultrafast characteristics. The signal (probe) beam with a wavelength of 1550 nm is chosen because it is eye safe, compatible with fibre optic telecommunication systems in the C band, and lies within the atmospheric transmission window from 700 nm to 1600 nm6. The control (pump) beam is generated at a wavelength of 780 nm by second-harmonic conversion of the signal beam. The laser source for both beams is an erbium-doped fibre laser with a measured pulse duration of 150 fs and repetition rate of 100 MHz. The 1550 nm beam (orange) is introduced via a 50-50 beamsplitter and compressed with a telescope before incidence on the AORM architecture, as depicted in Figure 5.8(a). (Note that for a real-time communication system, the signal beam would be continuous wave source.) The AORM architecture is tilted at five degrees to ensure that reflections not due to retroreflection are eliminated. The antiparallel 780 nm control beam (red) is focused with a 20× microscope objective onto the CuO film. The retro-modulated signal beam is measured using an InGaAs detector and a lock-in amplifier (Stanford Research Systems, SR830) with a 300 ms time constant. Pictures of the physical implementation of the pump-probe spectroscopy experimental setup and the AORM architecture are given in Figure 5.9. The layout of optical components is analogous to those depicted in Figure 5.8(a). The inset shows the mounted AORM architecture, comprised of two mated S-LAH79 hemispheres, forming a full sphere, with the CuO film between them. The AORM architecture is held in place with two transparent sheets. Micro-milled holes in the transparent sheets, with a diameter slightly smaller than the sphere diameter, pinch the hemispheres in place under a microscope. The measured modulation depth is presented in Figure 5.8(b) as a function of the time delay between the control and signal pulses. A modulation depth of ‰ = 2×10-4 is measured for an incident control beam fluence of 5 µJ/cm2, with a beam radius of approximately ç¨ = 15 µm.  113 Approximately 10% of the incident control beam power is absorbed by the 20 nm thick film, resulting in an absorbed fluence of 0.5 µJ/cm2. This fluence was chosen due to thermal effects that were observed for absorbed control beam fluences above 1 µJ/cm2, due to a slight heat-induced expansion of the mated glass hemispheres. Such expansion manifests itself as a slow transient drift in the measured signal power (over many seconds) following the introduction of the control beam. This impacts the pump-probe experimental measurements, as the signal is acquired with a time-resolved measurement technique, as a function of the pump-probe delay, over many minutes. However, for steady-state (real-time) operation, this transient drift would not be a concern, as it could be eliminated by a simple high-pass electronic filter. Furthermore, the CuO thin film was tested on an isolated glass substrate and no thermal effects or laser induced damage were observed up to the maximum incident control beam fluence of our ultrafast laser source with a 20×  Figure 5.9. The physical implementation of the pump-probe experimental setup. The inset shows the mounted AORM architecture, comprised of two mated S-LAH79 hemispheres, forming a full sphere, with a CuO nanocrystal thin film layer between them. The AORM architecture is held in place by two transparent sheets. 1550 nmsignal beam780 nmcontrol beamdetector detectorsampleAORM architecturedelay stage 114 microscope objective, being 26 µJ/cm2, with a beam radius of approximately ç¨ = 7.5 µm. At this fluence, a maximum modulation depth of ‰ = 1×10-2 can be realized, and it is likely that larger modulation depths can be achieved with higher control beam fluences. If thermal effects or laser induced damage become an issue at higher repetition (i.e., data) rates than the 90 MHz experimentally tested here, ç¨ can be increased to decrease the control beam fluence. This would not decrease the received signal power because the decrease in modulation depth due to the decreased control beam fluence would be compensated for by a proportional increase in the modulated area of the signal beam. The experimental impulse response, presented in Figure 5.8(b), characterizes the maximum speed of the AORM architecture. The retro-modulated signal beam is shown to have an effective recovery time constant of 770 fs. This experimental result indicates that the AORM architecture could support high aggregate data rates, with multiple channels, where the data rate per channel is limited by the detected power and relevant noise sources. The multiple channels could be established via all-optical time-division multiplexing and demultiplexing at each ground transceiver—in an analogous manner to that applied in fibre optic links154,155. The maximum achievable aggregate data rate, for time-division multiplexing, is ultimately limited by the material’s response time, being 770 fs here, although link budget constraints would also limit the data rate. Equations (42) and (43) can be populated with data from the experimental results and reformulated to compare AORM passive DLs to broadcasted active DLs in general. For Equation (42), the transmitted aerial-to-ground power, Ä´, can be made equal to the AORM control beam power, Ä´ = ç´ , as they are both limited by the available aerial TRX power. Also, for a fair comparison, the broadcasted divergence solid angle can be made equal to the FOV provided by the AORM system, „Ä = „FOV = 2ô sr. For Equation (43),	 Ä´ is the transmitted ground-to-aerial  115 power, and this power quantifies the asymmetry of the AORM passive DL. The signal beam area is approximately e` ≈ 4eç , as a result of the sphere’s 2× magnification. The divergence solid angle of the ground-to-aerial beam, „Ä, is set to a limit of 10-10 sr. The retro-modulated aerial-to-ground beam, „µ, can also be set to 10-10 sr, assuming the control beam radius, ç¨, is sufficiently small, as defined by the region below the horizontal dashed line in Figure 5.7. The modulation depth can be reformulated as ‰ = gyf = gÈ•ç eç, where È is the modulation constant for cross absorption modulation and •ç is the energy of a bit/pulse in the control beam. The control beam power is related to •ç by ç´ = •ç’, where ’ is the bit rate. Based on the measured modulation depth, CuO film thickness of 20 nm, and control beam fluence, the modulation constant is È =200 cm/µJ. These parameters are applied to (42) and (43) to give d´et = ¶ ç´eµ „FOV^2  for broadcasting and  d´et = 4¶gÈ Ä´ ç´eµ ’„Ä„µ^4  for AORM. By taking the ratio of these two equations, it becomes apparent that the transmitted ground-to-aerial beam power, Ä´, must meet the condition  Ä´ > ’„Ä„µ^24„FOVgÈ = 10-20’^24πgÈ , (44) for the AORM passive DL to be more efficient than the broadcasted active DL (while establishing numerous multidirectional links). Ultimately, this equation describes the required asymmetry for the link to be practical. To quantify the performance of the AORM architecture in this work, the experimentally measured result of È is substituted into (44). Assuming the implemented architecture has a CuO thin film greater or equal to the penetration depth of g = 200 nm, the condition from (44) becomes Ä´ > 2 fJ km2 ⋅ ’^2. Thus, for a ground-to-aerial link operating at an aggregate bit rate of 1 terabit-per-second with distances of 1 km, 10 km, and 100 km, Ä´ must be greater than 2 mW, 200  116 mW, and 20 W, respectively. Ultimately, this work’s AORM architecture is feasible at a variety of link lengths, provided that the link is sufficiently asymmetrical and the signal power can be detected. The overall detected signal and noise powers are calculated for a 20 km link, for application to high-altitude platform33,34. The link is assumed to use the aforementioned phase-locked self-homodyne coherent detection method to remove the shot noise associated with the large carrier background power. The resulting detected signal power at the ground transceiver would be d´et =4gÈ ç´ Ä´eµ ’„Ä„µ^õ . The ground-to-aerial transmitted power is set to Ä´ =  10 W with a transmitted beam diameter of 10 cm, corresponding to „µ ≈ 3×10-10 sr for diffraction-limited performance6,33,139. The average control beam power is set to ç´ =  10 W. A 10-cm-diameter spherical retroreflector is used to give diffraction-limited performance, with the same divergence solid angle as the transmitted uplink beam. With such low divergence over a distance of 20 km, it is assumed that the entire retro-modulated signal power can be collected such that eµ („µ^D) ≈ 1 (with a maximum detector diameter of 1 meter). At a single channel bit rate of 1 gigabit-per-second, the detected power is calculated to be d´et = 4gÈ ç´ Ä´ ’„Ä^D ≈ 1.3 µW. If shot-noise-limited performance is assumed, the signal-to-noise ratio (SNR) is ℛ d´et D 2iℛ d´ety∞ = 36 dB, where ℛ = 1 is the responsivity, i is the elementary charge of an electron, and y∞ is the bandwidth, which is approximated here by the bit rate ’. This is a strong SNR, but it would be reduced if the single channel bit rate is increased or other noise sources are introduced, such as relative intensity noise (RIN) and shot noise due to residual power from the carrier. For this link budget, the maximum achievable single-channel bit rate would be 20 gigabits-per-second with an SNR of 10 dB.  117 5.2.4. Summary & Discussion The presented work put forward new all-optical technologies for long-range FSO links between ground and aerial TRXs. The proposed system applied direct laser transmission for the ground-to-aerial active uplink and a new AORM architecture for the aerial-to-ground passive DL. It was shown that such a system can function with multiple ground TRXs, over wide service coverage, and one aerial TRX, with low demands for its mass and power. The AORM architecture that was applied for the passive uplink used high-refractive-index hemispheres to realize effective retroreflection and an interior CuO film to realize ultrafast all-optical modulation between coincident control and signal beams. The AORM architecture was fabricated and tested, and it was shown to enable multidirectional FSO communication at terabit-per-second data rates. Such findings can lay the foundation for emerging long-range FSO communication systems and aid in realizing a fully all-optical network.   118 Chapter 6  Conclusion Concluding remarks are given within this chapter. The remarks include a summary of the contributions in section 6.1 and suggestions for future work on all-optical switching and all-optical retro-modulation in section 6.2.    119 6.1. Summary of Contributions This thesis introduced two photonic architectures for circumventing the optoelectronic bottlenecks in current fibre optic and free-space optical (FSO) communication systems, in an effort to achieve fully all-optical networks. An all-optical switch (AOS) architecture was developed for implementation in fibre optic front-end systems, and an all-optical retro-modulation (AORM) architecture was developed for implementation in FSO systems. These two architectures rely upon similar principles, which were introduced in Chapter 2 and Chapter 3 through preliminary studies. In Chapter 4 and Chapter 5, these architectures were developed and characterized through experimental tests. The major conclusions of these chapters are summarized here.  In Chapter 2, preliminary theoretical and experimental analyses were put forward to investigate the underlying semiconductor free-carrier dynamics of all-optical switching. The changes in reflection, transmission, and absorption of the signal beam were derived for a changing refractive index and absorption coefficient. Subsequently, the changes in refractive index and absorption coefficient, due to control-beam-induced free-carriers, were derived according to Drude theory for a planar semiconductor. Preliminary time-resolved pump-probe spectroscopy experimental results were obtained for Si, SiC, and GaAs as points of reference. The major conclusions of Chapter 2 are as follows: i. The experimental results for pump-probe spectroscopy of GaAs agreed with the theory presented in this chapter, with an R-squared value of 0.995 for a curve-fit diffusion coefficient of ì ≈ 3 cm2/s. All other parameters were obtained from the literature, as summarized in Table 2.1, where it was concluded that the presented theory accurately describes the free-carrier dynamics and the reflection, transmission, and absorption at the semiconductor interface.  120 ii. From the theory presented in this chapter, conclusions were made on how to enhance the all-optical switching energy. According to the first term of Equation (23), switching energies could be reduced by minimizing the control pulse energy needed to establish the required initial free-carrier density. The control pulse energy can be lowered by localized photoinjection, with a reduced volume in Equation (27), leading to higher initial free-carrier densities for the same control pulse energy. iii. From the theory presented in this chapter, conclusions were be made on how to enhance the all-optical switching times. According to Equation (23) and its boundary condition in Equation (24), switching times could be reduced by promoting surface recombination. Localized recombination at the semiconductor surface, where free-carriers rapidly recombine, can be enhanced by either increasing the surface recombination velocity or the surface-to-volume ratio, ^. Localized recombination would reduce the free-carrier lifetime and thus the switching time. In Chapter 3, theoretical and experimental analyses were put forward on focusing with dielectric spheres. A description for light scattering from milli-scale dielectric spheres was developed using ray theory (for compatibility with optical fibres). A description for light scattering from micro-scale dielectric spheres was developed using Mie theory (for compatibility with integrated network-on-chip devices). Also, experimental results were presented for localized photoinjection with pump-probe spectroscopy using dielectric spheres focusing into an adjacent GaAs sample. The major conclusions of Chapter 3 are as follows: i. Using ray theory, for the milli-scale regime, led to the selection of a S-LAH79 glass sphere with a diameter of Ö = 2.0 mm and refractive index of _ ≈ 2 as the desired structure for the highest intensity at the back surface of the sphere.  121 ii. Using Mie theory, for the micro-scale regime, led to the selection of an N-LASF9 glass microsphere with a diameter of Ö = 40 µm and refractive index of _ ≈ 1.83 as the desired structure for establishing a photonic nanojet at the back surface of the sphere, with an especially high intensity. iii. The theoretical and experimental results for	 the	differential transmission	 of	 the	 signal	beam	showed that the selected N-LASF9 glass microsphere could dramatically reduce the free-carrier lifetime from that of the nominal lifetime of í = 350 ps	to í = 10 ps. This was due to the reduced Rayleigh range formed from this microsphere and its corresponding ability to preferentially deposit free-carriers at the semiconductor surface, where they can undergo rapid surface recombination. For sphere refractive indices of _ = 1.51, 1.76, 1.83, and 1.98, the photoinjection depths into the semiconductor decreased according to gh = 490, 370¸ 230, and 80 nm, respectively. iv. The experiment with a sphere diameter of Ö = 2.00 mm, a refractive index of _ = 1.98 ± 0.02, and a GaAs sample behind the sphere, was found to enable all-optical switching with a roughly 10 fJ switching energy (defined for a unity signal-to-noise ratio) and 10 ps switching time. In Chapter 4, the AOS architecture was introduced to meet the goals of femtojoule switching energies and femtosecond switching times for application to fibre optic front-end systems. Following the guiding principle of localization, the proposed architecture integrated photonic nanojets and semiconductor nanoparticles to realize these two goals. The differential transmission of the signal beam, due to control-beam-induced free-carriers, was derived according to Drude theory for spherical semiconductor nanoparticles. The AOS architecture was tested for both the milli-scale and micro-scale regimes using Si and SiC nanoparticles. Further experimental tests  122 were conducted with CdTe, InP, and CuO nanoparticles to find the optimal semiconductor nanoparticles. The major conclusions of Chapter 4 are as follows: i. From theory for all-optical switching in spherical semiconductors, it was concluded that the semiconductor nanoparticle composition should be selected to have a sufficiently high surface recombination velocity, being greater than ò5 = 2000 m/s, and the semiconductor nanoparticle size should be selected to have a sufficiently small radius, being less than ∑ = 30 nm.  ii. For the milli-scale spheres, with Si and SiC nanoparticles, it was found that all-optical switching could be carried out with switching energies of 200 fJ and 100 fJ, and switching times of 10 ps and 350 fs, respectively. iii. For the micro-scale spheres, with Si and SiC nanoparticles, it was found that all-optical switching could be carried out with switching energies of 1 pJ and 20 fJ and switching times of 2 ps and 270 fs, respectively. Thus, it can be concluded that an AOS architecture embedded SiC nanoparticles can meet the demands of all-optical switching, in terms of its femtojoule switching energies and femtosecond switching times, albeit with a larger bandgap than desired. iv. The AOS architecture was tested further with the micro-scale spheres and Si, CdTe, InP, and CuO nanoparticles. They were found to exhibit switching energies of 1 pJ, 500 fJ, 400 fJ, and 300 fJ, with switching times of 2 ps, 2.3 ps, 900 fs, and 350 fs, respectively. The optimal semiconductor witnessed in this study was CuO, which readily achieved the goals of femtojoule switching energies and femtosecond switching times. These results supersede the top results described in section 1.1.1, achieved by the organic photonic-bandgap microcavity24.  123 In Chapter 5 the AORM architecture was introduced to meet the demand of establishing numerous multidirectional FSO links operating at terabit-per-second data rates, without escalating mass and power, for FSO communication systems. To realize this goal, the proposed architecture implements retro-modulation with a spherical retroreflector and a semiconductor thin film being comprised of CuO nanocrystals. The CuO nanoparticles were studied extensively to interpret the underlying free-carrier dynamics. The proposed AORM architecture was examined for use in a passive DL with a link budget analysis. Theoretical and experimental results were ultimately presented for the AORM architecture. The major conclusions of Chapter 5 are as follows: i. The optimal CuO film thickness, in terms of performance and structural uniformity, was found to be 20 nm, resulting in nanocrystal sizes of 50 nm. ii. A bandgap was measured to be 1.55 ± 0.05 eV for the CuO nanocrystals, aligning with recent experimental studies in the literature, and providing evidence that the bandgap is indeed a direct transition. iii. The differential transmission of the signal beam through the CuO nanocrystals was measured for various control beam fluences, and three time constants were seen in the recovery. The underlying mechanisms for the first time constant, ranging from 330 fs to 630 fs, were attributed to momentum relaxation via carrier-carrier scattering within the VB and (what the authors believed to be) the first observation of exciton-exciton annihilation in CuO. The underlying mechanism for the second time constant, being 2 ps, was attributed to energy relaxation via carrier-phonon scattering within the VB. The underlying mechanism for the third time constant, being 50 ps, was attributed to trapping and recombination, due to the high density of trap states within the bandstructure. It is  124 concluded that these three mechanisms fully describe the free-carrier dynamics of CuO nanocrystals. iv. Three link budget conditions were identified and considered in the design of the proposed AORM architecture. These conditions were met by the three power penalties associated with a small modulated signal beam area, e` < „Ä^2, a large retroreflected divergence solid angle, „µ > eµ ^2, and a small modulation depth, ‰ < 1, not exceeding the penalty from broadcasting the active DL with a large „Ä. v. According to the first link budget condition, the positioning of the CuO film in the Ç = 0 plane was found to offer noteworthy improvements in modulation. vi. According to the second link budget condition, a cladded spherical retroreflector was found to sufficiently correct for spherical aberration for application to long-range FSO links. vii. According to the third link budget condition, a modulation depth of ‰ = 2×10-4  was measured for an absorbed control beam fluence of 0.5 µJ/cm2, which showed that the proposed AORM architecture is feasible at a variety of link lengths, provided that the link is sufficiently asymmetrical according to Equation (44). The maximum modulation depth was measured to be 1×10-2  corresponding to the highest (experimentally achievable) absorbed control beam fluence, being 26 µJ/cm2. viii. The AORM architecture was fabricated and tested, and it was shown to have an impulse response of 770 fs, which can enable future multidirectional FSO links with high aggregate data rate approaching a terabit-per-second. Overall, two new photonic architectures were introduced in this thesis. The AOS architecture was developed for all-optical switching, with the primary application being optical fibre front-end systems. The AORM architecture was developed for all-optical retro-modulation, with the primary  125 application being free-space optical communication systems. The photonic architectures were designed to meet the goals of the two aforementioned applications. It is hoped that these findings can play a key role in realizing future all-optical networks.    126 6.2. Future Work The photonic architectures presented in this thesis were implemented as prototypes, with a technology readiness of level 4 (component and/or breadboard validation in laboratory environment), according to the National Aeronautics and Space Administration. As future work, these photonic architectures could be progressed for implementation in contemporary systems. Towards this goal, some recommendations for system integration and further improvements are given here.  For the AOS architecture to be implemented in fibre front-end systems, it would be necessary to design a cascaded configuration for multiple AOS operations. Such implementations have proven to be challenging for most other all-optical devices86. However, the AOS architecture has been shown to cascade well with daisy-chained spheres85, which could prove to be an effective implementation. Also, the AOS architecture could be further improved by tuning the nanoparticles to the conditions for Mie resonance, as this would increase the modulation depth87,88. Afterwards, as future work, the AOS architecture could be integrated either as an optical multiplexing scheme, or into network-on-chip architectures. For the AORM architecture to be implemented in FSO communication systems, it would be necessary to design the aerial receiver system, such that the aerial TXR can generate the control beam needed for all-optical modulation. This could be done either passively, in an all-optical format, or actively in an electronic format. The all-optical format would require collecting the ground-to-aerial signal with a free-space-to-fibre coupler, then regenerating the control beam with second harmonic generation and an erbium-doped fibre amplifier. This implementation would also require some all-optical logic to multiplex the signals from various distributed ground locations. The electronic format would require ultrafast detection and demultiplexing of the terabit-per- 127 second ground-to-aerial signal. The control beam would then be regenerated by way of multiplexing with contemporary electro-optic modulators. Also, the AORM architecture could be further improved upon by incorporating resonance to enhance the modulation depth. Such implementations could be in the form of a resonant antenna156 or Salisbury screen design157. 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Furthermore, all of the losses and reflections in the experimental setup are accounted for and verified here by using the technique to measure the refractive indices of the semiconductors GaAs and Si and comparing them the literature values. For this analysis, it is assumed the absorption occurs over an effective length of gh, being much less than the slab thickness as illustrated in Figure A.1. It is apparent from Figure A.1 that the summation of reflections, ^IÌ“ and the total transmission, oIÌ“, are  ^IÌ“ = ^ 1 + e2 1 − ^ 2 1 + e2^2 +⋯ = ^ + e2^ 1 − ^ 21 − e2^2 , (45) and oIÌ“ = e 1 − ^ 2 1 + e2^2 +⋯ = e 1 − ^ 21 − e2^2 . (46) These equations are simplified using the series identity Ô2} = 1 1 − Ô2  for the variable Ô and integer values Å.   146 For a given experimental configuration, where ^IÌ“ and oIÌ“ are measured, Equations (45) and (46) can be used to solve for the refractive index of the semiconductor, _`. To do this, Equation (45) is manipulated into an explicit expression for the absorption, e, in terms of ^IÌ“, to give  e = ^ 1 − ^ 2^IÌ“ − ^ + ^2 -1 	2. (47) Then Equation (47) is substituted into Equation (46) with measured results for ^IÌ“ and oIÌ“ to give a fourth-order polynomial expression for _`. The solution for _` can then be found with the desired numerical or algebraic method. The experimental setup for measuring the reflection, transmission, and absorption is depicted in Figure A.2. The refractive index of a semiconductor sample is determined for a 1550 nm signal beam (orange). The beam is introduced via a 50-50 beamsplitter and is focused with a 20× microscope objective onto the semiconductor sample. The experimental losses between the sample and detectors are defined as Ò = dÒ ℛ Ò´  and Ú = dÚ ℛ Ú´ , where the actual reflected and transmitted powers after the sample are Ò´ and Ú´, and the measured reflected and transmitted powers at the detectors are dÒ ℛ and dÚ ℛ, based on the detector’s current, d, and the detector’s responsivity, ℛ. To find the losses for the reflected signal, first, the incident power, Û´ = dÛ ℛ, is  Figure A.1. Depiction of the reflection, transmission, and absorption for a slab with two interfaces. The absorption occurs over an effective length of Ùñ, being much less than the slab thickness.  A(1 – R)2PIA2(1 – R)2R2PIPIRPIA2(1 – R)2RPI{Absorbing region 147 measured at the sample’s location. Second, a gold slide, with a known reflectivity is inserted at the sample’s location and the reflected power is measured to determine Ò = dÒ ℛ^ıÌ Û´ . Similarly, the losses for the transmitted signal are determined by measuring the transmitted power at the detector with no sample inserted. Following this method, and using Equation (47), the values of _` are measured to be 3.385 and 3.388 for the GaAs and Si samples, respectively. These values are within 0.4% of the literature values for these semiconductors. The accuracy of using one interface, described in section 2.1.1, and using two interfaces, described here, to model differential transmission and reflection experiments can now be  Figure A.2. Schematic of the experimental setup for measuring the change in reflection, transmission, and absorption. The 1550 nm signal beam (orange) is introduced via a 50-50 beamsplitter and is focused with a 20× microscope objective onto the single crystal semiconductor samples. The reflected and transmitted signal beams are measured with InGaAs detectors on either side. The 780 nm signal beam (red) is introduced via a dichroic beamsplitter to be collinear with the signal beam. semiconductorsample20× microscope objectives lenslens50-50beamsplitterdetectordetector780 nm control beam1550 nmsignal beamdichroicbeamsplitter 148 compared. For a slab material with two interfaces, the change to the sum of reflections, y^IÌ“ p , will be altered from the one interface solution given by Equation (5). Based on the assumption that absorption occurs over an effective length of gh , being much less than the slab thickness, the control beam pulse duration will be shorter than the time for reflections to return to the first surface. Thus, the change can be found by taking the derivative of (45) for only the first surface reflection, ^, and transmission on the first pass, o = e 1 − ^ . The result is  y^`ˆΩ p = 1 − e2^ 1 − ^e2 − ^2 ^Δ_` p − e2^ 1 − ^ 2e2 − ^2 ghΔf p . (48) This equation for y^`ˆΩ p  now includes a Δf p  term, as the transmitted beam is partially reflected. This is in contrast to Equation (5) for the single interface case. Likewise, the change to the total transmission, ΔoIÌ“ p , based on Equation (46), is  ΔoIÌ“ p = e 1 − ^1 − e2^2 ^Δ_` p − e 1 − ^ 21 − e2^2 ghΔf p . (49) From Equations (46) and (49), the normalized total change in transmission, ΔoIÌ“ p oIÌ“ p , is found to be equal to Equation (8), which is defined for the single reflection case. This shows that the normalized transmission change is the same for a single reflection and multiple reflections, and so the analyses can be carried out simply with Equation (8). As such, this work uses transmission based experiments and analyzes the results while considering one interface. As an aside, the equations and the experimental setup shown here can be used to get experimental curves for Δ_` p  and Δf p  describing a semiconductor wafer. This can be accomplished by solving Equations (48) and (49) with an experimentally measured y^`ˆΩ p  and ΔoIÌ“ p . Future work could make use of this measurement technique.   149 Appendix B Free-Carrier Recombination in GaAs The following experiments use an optical attenuator to control the 780 nm control beam fluence over 10-6~10-4 J/cm2, which corresponds to carrier densities in GaAs ranging from 1017~1019 cm-3, according to Equation (28). The recombination lifetime in bulk GaAs as a function of the control beam fluence are measured using time-resolved pump-probe spectroscopy experiments. The experiments are conducted using a 20× microscope objective, having a long depth of focus, to ensure the experimental photoinjection depth can be accurately described by the reported penetration depth58. The differential transmission, yo p o , for GaAs is measured using the experimental setup shown in Figure A.2. The results are shown in Figure B.1, where it is evident that the lifetime does depend, to a certain extent, on control beam fluence.   Figure B.1. The differential transmission, °¢ ï /¢, in bulk GaAs as a function of delay time between the control and signal beams for varying control beam fluences. The 780 nm control beam fluences range over 10-6~10-4 J/cm2, which corresponds to carrier densities over 1017~1019  cm-3. This range yields with a change in the effective lifetime of roughly 25%. 350-50 50 200100 300150 2500Time Delay, t (ps)00.20.40.61.01.20.8-ŢT(t) / T (a.u.)Ɖ-FP2Ɖ-FP2Ɖ-FP2Ɖ-FP2Ɖ-FP2Ɖ-FP2Ɖ-FP2 150 Band-to-band radiative recombination is proposed as the mechanism underlying this fluence dependency. In general, the recombination kinetics in semiconductors can be described by  qâ pqp = −Å1â p − Å2â2 p − Å3â3 p , (50) where the three terms on the right side characterize the respective processes of trap-assisted non-radiative recombination, band-to-band radiative recombination, and Auger recombination159,160. Trap-assisted recombination is a two-particle non-radiative recombination process, which occurs in most semiconductors, and is dominant in semiconductor nanostructures due to their high densities of trap states. Band-to-band recombination is a two-particle radiative process, whereby electron and hole pairs directly recombine to radiate a photon, and it is often seen in direct semiconductors such as GaAs158. Auger recombination is a non-radiative three-particle interaction, whereby the energy from an annihilated electron-hole pair excites another free-carrier, and it is a likely process only at high carrier densities158. The terms Å1, Å2, and Å3 are constants, and the process of diffusion is omitted here for simplicity. To focus on the process of band-to-band recombination, the Auger term in Equation (50) is neglected (and is discussed later). With the two non-radiative and radiative recombination terms only, Equation (50) takes the form of the Bernoulli equation, which can easily be solved. With the initial condition of â p = 0 = â0, the solution is  â p = âXÅ]e-}˜ÄÅ] + ÅDâX(1 − e-}˜Ä). (51) With this solution, we can now consider the cases of low and high injected free-carrier densities. For low free-carrier densities, where ÅDâX ≪ Å], Equation (51) becomes  â p ≈ âXe-}˜Ä. (52)  151 For this case, the recombination lifetime is independent of the free-carrier density and the rate 1 Å] is equal to the bulk lifetime, í9, by definition. Note, this case is used in the analyses in sections 2.1.2 and 3.2. For high free-carrier densities, where ÅDâX ≫ Å], Equation (51) becomes  â p ≈ Å]ÅD e}˜Ä − 1 . (53) For this case, the recombination lifetime is not defined by an exponential function. The function that is formed can be best understood by looking at it over short and long timescales. For short timescales, where p ≪ í9, Equation (51) can be simplified to  â p ≈ 1ÅDp. (54) Thus, the lifetime for short timescales is í ≈ e ÅDâX, where the lifetime is defined as the 1 e point according to âXe-] = â p . This short timescale lifetime depends upon the free-carrier density, given that an increasing free-carrier density decreases the lifetime. Furthermore, for long timescales, where p ≫ í9, Equation (51) can be simplified to  â p ≈ Å]ÅD e-}˜Ä. (55) This lifetime for long timescales approaches the bulk lifetime, with 1 Å] = í9 , and it is independent of the control beam fluence. Overall, the trends described above for the short and long timescales match the trends seen in Figure B.1. Moreover, the trends seen here are also seen in the work by Strauss et al.159, which show band-to-band recombination in GaAs at free-carrier densities up to 1019 cm-3. Their work also shows contributions from Auger recombination, which is a lower probability three-body interaction, at free-carrier densities above 1019 cm-3. For exceedingly high free-carrier densities, Auger recombination can be analyzed in a similar fashion as above, and in doing so it exhibits a  152 lifetime over short timescales in proportion to 1 â02 . However, for the range of free-carrier densities used here, 1017~1019  cm-3, it is concluded that band-to-band recombination will dominate over Auger recombination as the mechanism for recombination exhibiting dependence on the control beam fluence. Furthermore, the results in Figure B.1 show that the decreasing lifetime saturates at higher free-carrier densities, as seen by Bourdon et al. and attributed to filling of CB and VB states161. Overall, the lifetime enhancements seen in section 3.2 are found to result from changes to the focal conditions, which leads to changes in gh and varying contributions from surface recombination, rather than changes in control beam fluence and free-carrier density. Given that goal of this work is to design an AOS architecture for low switching energies, i.e., low control beam fluences, it is advantageous to have the recovery be defined by surface recombination (which is independent of the control beam fluence) rather than radiative recombination (which requires a large control beam fluence to make it sufficiently fast). It is also worth noting that the dependency of the results in Figure B.1 with respect to the control beam fluence is small. The effective lifetime, í, changes by 25% for a two-order-of-magnitude change in control beam fluence, where í¯	˘˙ J“ö − í¯XX	˘˙ J“ö í¯	˘˙ J“ö = 25%. In the fluence independent theory from sections 2.1.2 and 3.2, this deviation manifests itself as error in the curve-fit parameters. This error is sufficiently small for the purposes of these sections to introduce the reader to the processes of free-carrier diffusion and recombination and show how surface recombination can be used a tool to decrease the all-optical switching time.     153 Appendix C Optical Encoder Setup To measure the focal spot size of a focused beam, a scanning knife-edge measurement scheme is used. The scheme is described here. Sputtered chrome is deposited on a glass microscope slide and patterned to have a sharp vertical edge. This chrome-on-glass knife-edge is scanned across the beam’s spatial profile to find the full width at half maximum (FWHM). To measure micron-scale distances with high accuracy, a precise optical encoder is built and used for this. The optical encoder is composed of a linear autocorrelation setup with a precise piezo linear actuator (Thorlabs 8302) with a positioning resolution of approximately 30 nm. The autocorrelation setup and scanning knife-edge setup, depicted in Figure C.1, are used resolve the interference fringes and thereby precisely measure distances. The setup uses a 650 nm continuous wave (CW) beam (orange), which is split by a 50-50 beamsplitter into two arms to form a Michelson interferometer. The beam power is measured with a Si photodetector (Thorlabs DET36A). One mirror from the linear autocorrelator and the chrome-on-glass knife-edge are mounted on a delay stage for control by the piezo linear actuator. As the knife-edge scans across the focal spot of the focused beam (red), the autocorrelator’s fringes are recorded as a function of the mirror position. Based on the 650 nm wavelength, the mirror’s change in position can be determined to within a precision of approximately 30 nm. As the fringes are recorded, the transmitted power past the knife-edge is also recorded with a second Si photodetector. Due to the constraint of having only one lock-in amplifier, both signals are recorded together via the amplifier’s A – B input, where the results are superimposed together. This works because the autocorrelation signal has a constant average power. The data is recorded with a LabVIEW program, according to the control interface shown in Figure C.2 and the block diagram given in Figure C.3 and Figure C.4. The program in Figure C.3 directs the linear actuator to move  154 the desired number of steps, waits the desired averaging time, and takes a measurement. Once complete, the program in Figure C.4 takes the Fourier transform of the collected data to find the frequency with respect to the linear actuator step distance, and with knowledge of the beam’s wavelength, it calculates the true step distance in nanometers and femtoseconds. Figure C.5 shows the focal spot size measurement (red) for the 20× microscope objective. The results are obtained for a variety of positions in the vicinity of the focal spot, to find the minimum spot size. Both the 780 nm and 1550 nm beams are tested. The convolution of a Gaussian profile (blue) with a unit step function is used to curve-fit the results (black). The focal spot diameter, corresponding to the FWHM of the Gaussian profile, is found to be approximately 6±1 µm for the 780 nm beam, and 14±3 µm of the 1550 nm beam. Further measurements show that the 780 nm beam and 1550 nm beam focus at different locations, due to the microscope objective’s wavelength dependence. The focal positions of the beams are separated by a distance of 20±10 µm. The pump-probe spectroscopy setup’s optimal signal-to-noise ratio is achieve where these two beams are approximately the same size, which will be a focal spot diameter of approximately ~15 µm. These results are used to calculate the fluences for the collinear experiments in this work.    155   Figure C.1. The autocorrelation setup and chrome-on-glass knife-edge setup are depicted. A 650 nm continuous wave (CW) beam (orange) is split by a 50-50 beamsplitter into two arms to form a Michelson interferometer. The beam power is measured by a Si photodetector. One mirror from the linear autocorrelator and the chrome-on-glass knife-edge are mounted on a delay stage for control by the piezo linear actuator. As the edge scans across the focal spot of the beam (red) the autocorrelator’s fringes are recorded as a function of the mirror position. The focal spot is measured for both the 780 nm and 1550 nm beams. As the fringes are recorded, the transmitted power past the knife-edge is also recorded with a second Si photodetector. sputtered chrome edgedelay stage 20× microscope objectivelenslens50-50beamsplittermirrormirrordetectordetectorfocused beam650 nmCW beam 156   Figure C.2. The LabVIEW control interface for measuring the focal spot size of focusing elements. The interface allows motion in forward and reverse directions. The step distance varies with respect to the direction of motion, the actuator’s speed, and weight of the delay stage. The approximate step (pulse) distance is updated every time the program is run with measurable interference. The steps (pulses) per sample are the number of steps the linear actuators takes before taking a measurement. The delay time is the time allowed for the lock-in to average the signal, and the scanning range is the total measurement range, which is converted to both femtoseconds and micrometers.  Read and DisplayC:\Users\branborn\Google Drive\Work\IOL Data\Picomotor Controller\Autocorrelation.viLast modified on 23/03/2017 at 2:09 PMPrinted on 08/09/2017 at 5:44 PMPage 1 Autocorrelation Experimental Notes: pump = 650 nm, 1 mW, ~0 deg angle of incidence  probe = 650 nm, 1 mW, ~0 deg angle of incidence electronic gain = SR570 Current Preamplifier electronic acquisition = SR830 lockin amplifier, autophased signal detector = Thorlabs Si detector Beam focusing = 5x Micro-objective chopping = 1.0 kHz on both  101Samples Taken:4-4-3-2-10123Motor Steps23242224 2240 2250 2260 2270 2280 2290 2300 2310Channel  1:0.2350.190.1950.20.2050.210.2150.220.2250.23Motor Steps23242224 2240 2250 2260 2270 2280 2290 2300 2310Channel 2:3.5-3.5-3-2.5-2-1.5-1-0.500.511.522.53Delay Time (fs)260 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25Channel 1 Results:0.23250.19250.1950.19750.20.20250.2050.20750.210.21250.2150.21750.220.22250.2250.22750.23Delay Time (fs)260 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25Channel 2 Results:ScanningC:\Users\branborn\Google Drive\Work\IOL Data\Autocorrelations\data.txtSaving  as:Successful  SaveONOFFSave Data COM3Resource Name0Time Remaining (sec):1Sample Resolution  (Pulses/Sample)100Scanning Range  (Total Samples)200Sample Delay Time (ms)  (Lock-In Time Constant)DirectionForwardReverse [nm][ms][pulses][samples]0.27 [fs]27.133.879[fs][µm]Forward Reverse38.79Approximate Pulse Distance30[nm] 157    Figure C.3. The LabVIEW program structure for controlling the linear actuator and lock-in amplifier. LabVIEW directs the linear actuator to move the desired number of steps, waits the desired averaging time, and takes a measurement from the lock-in amplifier.   Figure C.4. The LabVIEW program structure for calibrating the linear actuator step distance. LabVIEW takes the Fourier transform of the collected data to find the frequency with respect to the linear actuator step distance and with the knowledge of the beam’s wavelength, it calculates the true step distance in nanometers and femtoseconds.  Read and DisplayC:\Users\branborn\Google Drive\Work\IOL Data\Picomotor Controller\Autocorrelation.viLast modified on 23/03/2017 at 2:09 PMPrinted on 08/09/2017 at 5:46 PMPage 1Program:     PicomotorController.vi Author:       Brandon Born and Jamie Garbowski   Date:           17 May 2016 Operation:                      1. Ensure "Approximate Pulse Distance" is set to 30 nm/pulse, unless otherwise calibrated                      2. Enter the "Sample Resolution" in number of pulses. (This is the desired pulses between samples. Motor will move said amount of pulses, wait desired sample delay time, record value, and repeat)                      3. Enter the desired "Sample Delay Time" in ms. (This is the duration that the motor waits at each step. By waiting at each step for longer periods of time, and adjusting the time-constant of the data acquisition system,  the noise is averaged out.)                      4. Enter the desired "Scanning Range" in number of samples. (This is the total samples that will be measured.)                      5. Select the motor direction.                       6. Press arrow button to begin program. Note: To quickly move the motor, set the desired "Sample Resolution" to your desired travel distance, and set "Scanning Range" = 1.  3200Channel  1: 2 [0..4]Samples Taken:ScanningSaving  as:Successful  SaveSignalsSaving the experiment dataSave Data Time Remaining (sec):USER............ABS A1= ?-1 g False Set distance to travel and initiate   0 [0..1]Take Data Samples and Display Live 3 [0..5]Sample Resolution  (Pulses/Sample)Scanning Range  (Total Samples)Sample Delay Time (ms)  (Lock-In Time Constant)Direction-1 False 1E-9Resource Name440Pulse Velocity (Hz) Between SamplesValueApprox Pulse Dist Reverse False 650Laser WavelengthValueApproximate Pulse Distance True 1 True Read and DisplayC:\Users\branborn\Google Drive\Work\IOL Data\Picomotor Controller\Autocorrelation.viLast modified on 23/03/2017 at 2:09 PMPrinted on 08/09/2017 at 5:48 PMPage 1Program:     PicomotorController.vi Author:       Brandon Born and Jamie Garbowski   Date:           17 May 2016 Operation:                      1. Ensure "Approximate Pulse Distance" is set to 30 nm/pulse, unless otherwise calibrated                      2. Enter the "Sample Resolution" in number of pulses. (This is the desired pulses between samples. Motor will move said amount of pulses, wait desired sample delay time, record value, and repeat)                      3. Enter the desired "Sample Delay Time" in ms. (This is the duration that the motor waits at each step. By waiting at each step for longer periods of time, and adjusting the time-constant of the data acquisition system,  the noise is averaged out.)                      4. Enter the desired "Scanning Range" in number of samples. (This is the total samples that will be measured.)                      5. Select the motor direction.                       6. Press arrow button to begin program. Note: To quickly move the motor, set the desired "Sample Resolution" to your desired travel distance, and set "Scanning Range" = 1.  Tone Meas 1Tone Meas 2 True 30 Channel 1 Results:Channel 2 Results:0.06 fs tol True  True Operand 1.06 fs tolApproximate Pulse Distance True 1LogLogCh 1: Ch 2: Val(Sgnl)Total Pulse Distance 5 [0..5]Sample Resolution  (Pulses/Sample)Scanning Range  (Total Samples)Sample Delay Time (ms)  (Lock-In Time Constant)Direction-1 False 1E-9Resource Name440Pulse Velocity (Hz) Between SamplesValueApprox Pulse Dist Reverse False 650Laser WavelengthValueApproximate Pulse Distance True 1 True  158     Figure C.5. The experimentally measured focal spot size for the 20× microscope objective (red) with 780 nm. The convolution of a Gaussian profile (blue) with a unit step function is used to curve-fit the results (black). The focal spot diameter, corresponding with the FWHM of the Gaussian profile, is found to be 6±1 µm.  Figure C.6. The experimentally measured focal spot size for the 20× microscope objective (red) with 1550 nm. The convolution of a Gaussian profile (blue) with a unit step function is used to curve-fit the results (black). The focal spot diameter, corresponding with the FWHM of the Gaussian profile, is found to be 14±3 µm. 20-20 -10 5-5 150 10-15Distance (ƉP)-0.200.20.40.81.21.00.6ƉPDistance (ƉP)20-20 -10 5-5 150 10-15Distance (ƉP)-0.200.20.40.81.21.00.6ƉPDistance (ƉP) 159 Appendix D Planar Free-Carrier Dynamics Analytical Solution  The analytical solution to the one-dimensional planar diffusion equation is found here. The diffusion equation describing the diffusion and recombination of the free-carrier density, â Ç, p , in a semi-infinite one-dimensional volume along the Ç-axis, is subject to the following: Differential Equation: ì qDâ Ç, pqÇD = qâ Ç, pqp , (56)  Boundary Condition: ì qâ Ç, pqÇ hóX = ò5â Ç, p hóX, (57) Initial Condition: â Ç, p = 0 = NXe-h jk = NXe-lkh. (58) Here, ì is the diffusion coefficient, ò5 is the surface recombination velocity, âX is the initial free-carrier density at p = 0	and Ç = 0, and fh = 1 gh is the reciprocal of the penetration depth. The bulk recombination time, í9, is assumed to be infinitely long in Equation (23). Figure 2.1 illustrates the boundary conditions and initial conditions used. Taking the Laplace transform of Equations (56) and (57) with respect to p gives  ì qDâ Ç, πqÇD = πâ Ç, π − â Ç, p = 0= πâ Ç, π − âXe-lkh, (59)  ì qâ Ç, πqÇ hóX = ò5â Ç, π hóX. (60) Solving the differential Equation (59) with respect to Ç, the homogeneous and particular solutions together are  160  â π = ee- ú`h + ’eè ú`h + âXe-lkhπ − fDì. (61) The constant ’ is taken to be zero in order to have an non-infinite solution at Ç → ∞. The constant e is found using the boundary condition in Equation (60) and is substituted into Equation (61) to become  â π = âXπ − fDì e-lkh − ò5 + fìò5 + πì e- ú`h . (62) Equation (62) agrees with the results from Ahrenkiel et al.162 Taking the inverse Laplace of this requires that the equation be separated into proper identities. The first term is trivial; the second term requires partial fraction expansion. Defining Ô = πì , ∑ = fì , simplifies the second term,	âD Ô , to be  âD Ô = − âXì(ÔD − ∑D) ò5 + ∑ò5 + Ô e-ú¸h, (63) which expands to become  âD Ô = −âXìe-ú¸h 1ò5 − ∑ Ô + ò5 + ò5 + ∑2∑ ∑ − ò5 Ô + ∑+ 12∑ Ô − ∑ . (64) The Laplace transform identity is   ℒ~] e-} `π + ˝ = ℒ~] ìe-}˛¸π + ˝ƒ = 1ôp e-}ö/õÄ − ˝eˇ ˇÄè} erfc Å + 2˝p2 p . (65) For this case, Å = Ç/ ì and ˝ = ò5/ ì, f ì,−f ì, according to the three respective terms of Equation (64). This results in the final solution of   161 â Ç, pâX = 1f − ò5/ì 1ôìp e -höõúÄ − ò5ì eùûú ùûÄèh erfc Ç + 2ò5p2 ìp+ ò5/ì + fò5/ì − f 12f ôìp e -höõúÄ − 12 el lúÄèh erfc Ç + 2fìp2 ìp− 12f ôìp e -höõúÄ − 12 el lúÄ~h erfc Ç − 2fìp2 ìp + âXe-lhelöúÄ. (66) This analytical solution describing the diffusion and recombination of the free-carrier density, â Ç, p , can be coupled to Drude theory with Equation (8) to find the differential transmission response, ∆o p /o.    162 Appendix E Ray Theory for a Dielectric Sphere The intensity of the focused beam by a dielectric sphere with a given refractive index can estimated by measuring the density of rays at the back surface of the sphere. This gives a relative comparison between spheres of varying refractive indices. The results can be normalized to the case with tightest focusing. The following is a derivation of each ray’s position at the back surface of the sphere, at Ô],!] , for a plane wave incident on the entrance of the sphere, at ÔX,!X , as seen in Figure E.1. At the entrance, we have ÔX = − ¨D − !D, where ! = !X. Thus, the entrance incident angle is found to be ∂X| = tan-] −!X/ÔX = tan-] !/ ¨D − !D	 = sin-] !/¨ . (67) With Snell’s Law, 1 ∙ sin ∂X| = _ ∙ sin ∂"Ä, the entrance transmitted angle is found to be  Figure E.1. Relationship between ray entrance and exit positions. The ray enters the sphere at a height #Õ, with an incident angle of $Õ% normal to the surface. The transmitted ray with an angle of $Õï propagates through the sphere with a slope of &. The ray then exits the sphere at the coordinates ÉÕ,#Õ  with an incident angle of $›% and a transmitted angle of $›ï. xyž = radiusnƅ0iƅ1tƅ0tƅ1i(x0 , y0)(x1 , y1)y = y0m = slope 163 ∂XÄ = sin-] !_¨ . (68) In the sphere, the slope, Ñ, in terms of both the entrance and exit interfaces is defined as Ñ = − ! − !]ÔX − Ô] = ! − ¨D − Ô]DÔ] + ¨D − !D. (69) The slope, Ñ, in terms of just the entrance components, is solved to be Ñ = tan ∂X| − ∂XÄ = tan ∂X| − tan ∂XÄ1 + tan ∂X| tan ∂XÄ = ! _D¨D − !D − ¨D − !D¨D − !D _D¨D − !D + !D . (70) At the position Ô],!] , Equation (69) can be rearranged to find Ô] according to ÑD + 1 Ô]D − 2Ñ ! −Ñ ¨D − !D Ô] + ! −Ñ ¨D − !D D − ¨D = 0. (71) Using the quadratic formula, Ô] is Ô] = Ñ ! −Ñ ¨D − !D + ÑD + 1 ¨D − ! −Ñ ¨D − !D DÑD + 1 . (72) with !] = ¨D − Ô]D. The exiting incident angle is ∂]| = tan-] Ñ + tan-] !]/Ô] . (73) Thus, using Snell’s Law again the exiting transmitted angle is ∂]Ä = sin-] _ sin tan-] Ñ + tan-] !]/Ô] . (74) This expression can be transformed using trigonometric identities to become  164 ∂]Ä = sin-] _ ÑÔ] + !]¨ ÑD + 1 . (75) Finally, the angle that the exiting transmitted ray makes with respect to the Ô-axis is found to be ∂ = ∂]Ä − tan-] !]/Ô] . Computing the maximum intensity as the maximum density of the rays (being the inverse of the distance between rays) as a function of the refractive index gives the function seen in Figure E.2.    Figure E.2. Intensity at the back of the sphere as derived from ray theory as a function of the refractive index. It can be seen that ray theory predicts the intensity to be greatest at a refractive index of ≤ = 2.0. Refractive index (n)1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.510-210-1 10010-3Intensity [arb. units]n 165 Appendix F Mie Theory for a Dielectric Sphere Appendix F is an integration of pertinent theoretical derivations from the publications Bohren and Huffman66, Mishchenko et al.163, and Fu et al.164. It is presented with notation that is consistent to the thesis. The photonic nanojet is a high intensity beam that forms by focusing through an appropriately designed dielectric sphere. The photonic nanojet is characterized by its tight transverse constriction and jet-like protrusion just beyond the sphere. This phenomenon is best modeled by Mie theory. The theory derives the solution to the Helmholtz equation in spherical coordinates with the boundary conditions of a single dielectric sphere. A simplified derivation for a single sphere is given here with the generalized transition matrix (T-matrix) method. The T-matrix method derives the scattering theory in tensor form, which is conducive to computational analysis. Three good references are Bohren and Huffman66, Mishchenko et al.163, and Fu et al.164, which give a more detailed derivation of Mie theory.  From Maxwell’s equations the wave equation can be derived as a starting point to describe the electromagnetic waves inside and outside the sphere. Maxwell’s Equations are as follows:   '×{ = −q(qp , '×) = *+ qäqp , ∇̇ ∙ ä = ,, ∇̇ ∙ ( = 0. (76) These can be rewritten by using a time harmonic travelling wave of the form { = {Õe-| Ä~-∙∆  and ) = )Õe-| Ä~-∙∆ . If also using , = 0 and * = 0, then Maxwell’s equation’s become  166  '×{ = dmá), '×) = −dm.{, ∇̇ ∙ ä = 0, ∇̇ ∙ ( = 0. (77) Taking the curl of both sides of the first two equations from (77) and using the relationships ( =á) and ä = z{ yields  '× '×{ = −'D{ = dmá '×) = −mDzá{, '× '×) = −'D) = −dmz '×{ = −mDzá). (78) By substituting the wave number Å = m_ n 	= m zá, the Helmholtz wave equations are derived as  'D{ + ÅD{ = 0, 'D)+ ÅD) = 0. (79) Now the wave equation will be examined in spherical coordinates. The difficult task of finding the solutions in spherical coordinates to the above vector field { and )equations can be reduced to finding the solutions to the scalar wave / ¨, ∂,0  equation, being  'D/ + ÅD/ = 0. (80) The vector field equations then can be reconstructed from the scalar solution, as seen later. To find the general scalar spherical wave solutions, we can separate the variables for / according to  / ¨, ∂,0 = ^ ¨ Θ θ Φ 0 . (81) The wave equations can then be re-expressed using the Laplacian operator in spherical coordinates, being  'µD = 1¨D qq¨ ¨D qq¨ + 1¨D sin ∂ qq∂ sin ∂ qq∂ + 1¨D sinD ∂ qDq0D. (82)  167 The substitution yields three equations for the wave equation:  qq¨ ¨D q^ ¨q¨ + (ÅD¨D − §)^ ¨ = 0, § = 2 2 + 1  2 = 1, 2, 3,… (83) 1sin ∂ qq∂ sin ∂ qΘ ∂q∂ + § − ÑDsinD ∂ Θ ∂ = 0, Ñ = −2, − 2 − 1 ,	… , (2 − 1), 2 (84) qDΦ 0q0D + ÑDΦ 0 = 0. 	 (85) The solutions to these differential equations are as follows:  ^ ¨ = Ç4 , = æ4 , ,!4 , , ℎ4] , , ℎ4D , ,  (86) Θ ∂ = 4´Ω cos ∂ , 4´Ω Ô = −1 Ω 1 − ÔD Ω/D qΩ 4´ ÔqÔΩ , (87) Φ 0 = e|Ω5 = ∑Ω cosÑ0 + ˝Ω sinÑ0. (88) Here, 4´ Ô  are the Legendre polynomials and , = Ũ. The variable Ç4 ,  is one of the four spherical Bessel/Hankel functions, depending on whether the wave is inside or outside the sphere, which are defined as   æ4 , = ô2Ũ 64è]D , , !4 , = ô2Ũ 74è]/D , , ℎ4] , = æ4 , + d!4 , , ℎ4D , = æ4 , − d!4 , . (89) Here 6¸  and 7¸  are the Bessel functions of the first and second kind, respectively. Thus, the general solution for / is solved for as  168 	 / ¨, ∂,0 = Ç4 , 4´Ω cos ∂ e|Ω54Ωó~4¬4ó] .	 (90) To recover the vector field components { and ), they must be represented by a series expansion of vector spherical harmonics, according to  { = ∑4Ω8 94Ω : + ∑4Ω; Ø4Ω :4Ωó~4 ,¬4ó]  ) = − Åmá ∑4Ω; 94Ω : + ∑4Ω8 Ø4Ω :4Ωó~4¬4ó] . (91) Then 94Ω :  and Ø4Ω :  can be related to / with  94Ω : = ∇× ∆/ , Ø4Ω : = ∇×94Ω Å∆Å , (92)  where : = Å∆ and ∆ is the radial vector of the field at any given location. By applying the curl operator in Equation (92), the general form for any spherical wave can be known. To find the particular solution and fully solve the problem, the initial/boundary conditions, as illustrated in Figure F.1, are used to determine the coefficients ∑4Ω8  and ∑4Ω; . The simplest case is for an incident plane wave upon a single non-conducting homogeneous sphere. For a plane wave, it can be proven that ∑4Ω8  and ∑4Ω;  only produce non-zero terms when Ñ = 1, which simplifies Equations (90) and (91) to 	 / ¨, ∂,0 = Ç4 , 4´] cos ∂ e|5¬4ó] ,	 (93)  { = ∑4894 : + ∑4;Ø4 :¬4ó] = 9 : Ø : À8À; , (94)  169 ) = − Åmá ∑4;94 : + ∑48Ø4 :¬4ó] = − Åmá 9 : −Ø : À8À; . The two equations above can also be stated in matrix notation for computational implementation, where 9 : , Ø : , À8, and À; are matrices of all the 2	components, as seen below: 9 : Ø : = ‰]µ ‰]µ ⋯ ‰4¿<=µ â]µ âDµ ⋯ â4¿<=µ‰]> ‰D> ⋯ ‰4¿<=> â]> âD> ⋯ â4¿<=>‰]5 ‰D5 ⋯ ‰4¿<=5 â]5 âD5 ⋯ â4¿<=5 , À8À; =∑]8∑D8⋮∑4¿<=8∑];∑D;⋮∑4¿<=;. (95) As an infinite series is impossible to compute, the matrix is truncated to a maximum 2Ω∏¸ ≥ Åf, where f is the sphere radius. Each 2 coefficient for a plane wave can be derived to be ∑48∑4; = •X d4 22 + 12 2 + 1 1−d . (96)   Figure F.1. Mie theory initial conditions of a plane wave and single sphere. The figure is a logarithmic plot of the beam intensity derived from Mie theory. A single sphere with a radius of ◊ and a refractive index of ≤2 is surrounded by a medium with a refractive index of ≤1. An incident plane wave forms internal and scattered waves inside and outside the sphere, respectively.  incidentplane wave single sphereinternal wavesscatteredwavesž UDGLXVn2n1 170 Now, using the T-matrix method the incident wave can be transformed to become the internal and scattered waves. The electric and magnetic field components of the incident, internal, and scattered waves can be written as  {Z@JZA≠@P = 9BAA : Ø≠5≠@ : À8À; , )Z@JZA≠@P = − Åmá 9≠5≠@ : −ØBAA : À8À; , (97)  {Z@P≠R@CD = 9BAA :D Ø≠5≠@ :D Ë 00 ≥ À8À; , )Z@P≠R@CD = − Åmá 9≠5≠@ :D −ØBAA :D ≥ 00 Ë À8À; , (98)  {IJCPP≠R≠A = − 9BAA : Ø≠5≠@ : E 00 À À8À; , )IJCPP≠R≠A = Å•Xmá 9≠5≠@ : −ØBAA : À 00 E À8À; , (99) where :D = ÅD∆ = m_D∆/n. The T-matrix coefficients À, E, Ë, and ≥ are comprised of all the 2	components as see here:  À 00 E =∑] 0 0 0 0 00 ∑D 0 0 0 00 0 ⋱ 0 0 00 0 0 ˝] 0 00 0 0 0 ˝D 00 0 0 0 0 ⋱. (100) The even and odd coefficients, found through Equation (92) and the identity from (88), are as follows:  94≠5≠@ : = 0Ç4 , 4´] cos ∂sin ∂ cos0−Ç4 , Ö 4´] cos ∂Ö∂ sin0 , (101)  171 94BAA : = 0−Ç4 , 4´] cos ∂sin ∂ sin0−Ç4 , Ö 4´] cos ∂Ö∂ cos0 ,  Ø4BAA : =2 2 + 1 Ç4 ,, 4´] cos ∂ sin01Ũ Ö ,Ç4 ,Ö, Ö 4´] cos ∂Ö∂ sin01Ũ Ö ,Ç4 ,Ö, 4´] cos ∂ cos0, Ø4≠5≠@ : =2 2 + 1 Ç4 ,, 4´] cos ∂ cos01Ũ Ö ,Ç4 ,Ö, Ö 4´] cos ∂Ö∂ cos0− 1Ũ Ö ,Ç4 ,Ö, 4´] cos ∂ sin0, (102) for the incident wave and internal wave cases Ç4 , = æ4 ,  and for the scattered case Ç4 , =ℎ4] , . Using the boundary conditions of the single sphere’s radius, f, and refractive index, _D, with the variables of º = Åf and ºD = ÅDf = m_Df/n, the T-matrix transformation coefficients for a single sphere are given as  ∑4 = _D ,æ4 º ƒæ4 ºD − _]æ4 º ,æ4 ºD ′_D ,ℎ4] º ƒæ4 ºD − _]ℎ4] º ,æ4 ºD ′		, (103)  ˝4 = _Dæ4 º ,æ4 ºD ′− _] ,æ4 º ƒæ4 ºD_Dℎ4] º ,æ4 ºD ′− _] ,ℎ4] º ƒæ4 ºD 		, (104)  n4 = _Dℎ4] º ,æ4 ºD ′− _D ,ℎ4] º ƒæ4 ºD_Dℎ4] º ,æ4 ºD ƒ − _] ,ℎ4] º ƒæ4 ºD 		, (105)  172  Ö4 = _D ,ℎ4] º ƒæ4 ºD − _Dℎ4] º ,æ4 ºD ′_D ,ℎ4] º ƒæ4 ºD − _]ℎ4] º ,æ4 ºD ′		. (106) These field vector equations can now be used to simulate photonic nanojets.    173 Appendix G Spherical Free-Carrier Dynamics Analytical Solution The diffusion equation describing the diffusion and recombination of the free-carrier density, â ¨, p , in a spherical volume (i.e., a semiconductor nanoparticle) along the radial cross-section is subject to the following expressions: Differential Equation: ì¨D qq¨ ¨D qâ ¨, pq¨ = qâ ¨, pqp , (107) Boundary Conditions: ∓ì qâ ¨, pq¨ µó±∏ = ò5â ¨, p µó±∏, (108) Initial Condition: â ¨, p = 0 = âX. (109) Here, ì is the diffusion coefficient, ò5 is the surface recombination velocity, and ∑ is the sphere’s radius. The semiconductor nanoparticle is assumed to be fully illuminated with a uniform free-carrier density, âX, at p = 0. The bulk recombination time, í9, is assumed to be infinitely long in Equation (32). Taking the Laplace transform of Equations (107) and (108) with respect to p gives  qDâ ¨, πq¨D + 2¨â ¨, π − πìâ ¨, π = −âXì , (110)  ∓ì qâ ¨, πq¨ µó±∏ = ò5â ¨, π µó±∏. (111) Solving the differential Equation (110) with respect to ¨, the homogeneous and particular solutions together are  â ¨, π = e¨ e-µ `/ú + ’¨ eµ `/ú + âXπ , (112)  174 The constant e and ’ are found using the boundary conditions in Equation (111) to be  e = −’ = âXπ ∑ìò5 πì + 1∑ − 1 e-∏ `/ú + ìò5 πì − 1∑ + 1 e∏ `/ú. (113) Substituted the constants e  and ’  back into Equation (112) and simplifying the result using hyperbolic trigonometric identities gives  â ¨, π = âXπ 1 −	 ìò5 ∑¨ sinh ¨ π/ìò5ì − 1∑ sinh ∑ π/ì + π/ì cosh ∑ π/ì . (114) A numerical inverse Laplace routine is used to solve Equation (114) to find â ¨, p .    175 Appendix H Planar Free-Carrier Dynamics Code The MATLAB program used to solve for the results seen in Figure 2.4 is given here. % ****************************************************** % Surface diffusion and recombination free-carrier dynamics % ****************************************************** close all; clear all; clc; h1 = subplot(1,2,1); SimulationType = 'Numerical';   % 'Algebraic' or 'Numerical' depth = [84 234 374 490 680]; for jd = 1:5 % -------------- Physical parameters ------------------- nm = 1E-9;                      % nanometers ps = 1E-12;                     % picoseconds q = 1.602E-19;                  % Electronic charge in C. k = 1.3806488e-23;              % Boltzmann constant in m^2*kg/s^2/K eo = 8.8541878E-12;             % Permittivity of free space in F/m.  c = 299792458;                  % speed of light in m/s. h = 6.62606957E-34;             % Planck's constant in m^2*kg/s. Sv = 8E3;                       % Surface recombination velocity in m/s. D  = 2.8E-4;                    % Diffusion coefficient in m^2/s. d = depth(jd)*nm;               % Initial photo-injected carrier depth in m. a  = 1/d;                       % Absorption coefficient in m^-1. tbulk = 10000*ps;               % Bulk carrier lifetime in s. % ------------------------------------------------------   % --------------- Simulation Resolution ---------------- dt = 1*ps;                      % Time resolution in s. dz = 5*nm;                      % Distance resolution in m.     tmax = 400*ps;                  % Max time in s. zmax = 4E3*nm;                  % Max depth in m. t = dt:dt:tmax;                 % Time in s. z = 0:dz:zmax;                  % Depth dimension in m. plotdt = 50*ps;                 % Plot time spacings in s. % ------------------------------------------------------   % ----- Carriers generated from the control beam -------  176 Pi = 1.5E-3;                    % Incident control power in W. RR = 90E6;                      % Laser repetition rate in Hz. r  = 2.5E-6;                    % Beam spot size radius in m. Ep = h*c/(780*nm);              % Energy per photon in J. ni = 2.1E12;                    % Intrinsic carrier concetration in m^-3. n0 = Pi/(Ep*RR*pi*r^2*d);       % Carrier density at t = 0, z = 0 assuming quantum eff. = 1 in m^-3. % ------------------------------------------------------   % -------- Algebraic Solution (without tbulk) ---------- if SimulationType == 'Algebraic'     [Z,T] = meshgrid(z,t);     % Dimensionless carrier density, normalized with respect to the carrier density no at z = 0, t = 0.     nc = (Sv+D*a)/(Sv-D*a)*( 1/2/a./sqrt(pi*D*T).*exp(-Z.^2/4/D./T) - 0.5.*exp( a^2*D*T +  a*Z).*erfc((Z+2*D*a*T)./2./sqrt(D*T)) ) ...        -                   ( 1/2/a./sqrt(pi*D*T).*exp(-Z.^2/4/D./T) + 0.5.*exp( a^2*D*T -  a*Z).*erfc((Z-2*D*a*T)./2./sqrt(D*T)) ) ...        -        1/(Sv-D*a)*(     D./sqrt(pi*D*T).*exp(-Z.^2/4/D./T) -  Sv.*exp(Sv^2/D*T+Sv/D*Z).*erfc((Z+ 2*Sv*T)./2./sqrt(D*T)) ) ...        +                   ( exp(-a*Z+D*a^2*T) );     t = [0 t]; nc = [ni+n0*exp(-a*z); ni+n0*nc]; % Add t = 0 point end % ------------------------------------------------------   % ---------- Numerical Solution (with tbulk) ----------- if SimulationType == 'Numerical'     % Dimensionless carrier density, normalized with respect to the carrier density no at z = 0, t = 0.     nc = zeros(length(t),length(z));        % Carrier density vs. x and t in m^-3.       % Laplace transform (Nxs) and numerical inverse Laplace transform (nxt) computational parameters.     NC='-n0*(Sv+D/d)/(s+tbulk^-1-D/d^2)/(Sv+sqrt(D*(s+tbulk^-1)))*exp(-sqrt((s+tbulk^-1)/D)*z(iz)) + n0*exp(-z(iz)/d)/(s+tbulk^-1-D/d^2)';     aa=6; ns=40; nd=20; cnt0 = 0;       177     % Kernel for numerical inverse Laplace transform executed within an x for-loop.     for iz = 1:length(z)       FF=strrep(strrep(strrep(NC,'*','.*'),'/','./'),'^','.^');       for n=1:ns+1+nd         alfa(n)=aa+(n-1)*pi*1i;         beta(n)=-exp(aa)*(-1)^n;       end;       n=1:nd;       bdif=fliplr(cumsum(gamma(nd+1)./gamma(nd+2-n)./gamma(n)))./2^nd;       beta(ns+2:ns+1+nd)=beta(ns+2:ns+1+nd).*bdif;       beta(1)=beta(1)/2;       for kt=1:length(t)         tt=t(kt);         s=alfa/tt;         bt=beta/tt;         btF=bt.*eval(FF);         ft(kt)=sum(real(btF));       end;       nc(:,iz) = ft;              cnt = round(iz/length(z)*1000)/10;       if (cnt>=cnt0+1) clc; disp([num2str(cnt,3) ' %']); cnt0 = cnt; end;     end;     t = [0 t]; nc = [ni+n0*exp(-a*z); ni+nc]; % Add t = 0 point end % ------------------------------------------------------   % Plot nc(t,z) results delete(h1); h1 = subplot(1,2,1); hold on;  for ind = 1:round(plotdt/dt):round(tmax/dt)    plot(z*1E6,nc(ind,:)*1e-6,'Color',[(tmax/dt-ind)/(tmax/dt),0,ind/(tmax/dt)]); end xlabel('z (um)'); ylabel('Free-Carrier Density (cm^-3)'); title('Free-Carrier Density Distribution'); legend(strcat('dt = ', num2str(plotdt/ps),' ps')); xlim([0 2]);  178   % ****************************************************** % GaAs Drude model of photo-injected plasma changes to a material's reflectivity, transmissivity and absorption. % ******************************************************   % -------- Carriers seen by the signal ----------- P0 = 20E-3;                     % Incident signal power in W. w = 2*pi*c/(1550*nm);           % Frequency of incident signal in rad/s. n = 3.37;                       % Background (constant) refractive index. % ------------------------------------------------------   % -------- Surface and bulk carrier densities ----------  for it = 1:numel(t)     N(it) = sum(nc(it,1:round(d/dz)))*dz./d;                  % Effective surface photo-injected carrier density in m^-3*s^-1. end   % ----- Absorption coefficient ab & change in refractive index dn ----- m = 0.067*9.11E-31;             % Effective mass in kg. mu = 0.045;                     % Effective gamma valley electron mobility in m^2/V/s. G = q/(m*mu);                   % Transient scattering/momentum relaxation rate (gamma) in s^-1.   % ---------- Change in percent transmission ------------ dT_T = N*q^2/(eo*n*w^2*m)*((n-1)/(2*n*(n+1))-d*G/c); % ------------------------------------------------------   % Redefine time in ps. tmin = -50*ps; t0 = [tmin:dt:-dt]; t =  [t0 t]; dT_T = [zeros(1,numel(t0)) dT_T]; % ------------------------------------------------------   % Normalize dT_T dT_T = abs(dT_T)/max(abs(dT_T));    179 % Plot dT_T results h2 = subplot(1,2,2); hold on;  t = t(isfinite(dT_T)==1); dT_T = dT_T(isfinite(dT_T)==1); plot(t/ps,dT_T), xlabel('Time (ps)'), ylabel('Normalized Differential Transmission'), title('GaAs Pump-Probe (775-1550nm)'); hold on; clc; drawnow; end    180 Appendix I Retroreflected Divergence Angle Code The MATLAB program used to solve for the results seen in Figure 5.7 is given here. The plotted output of the program is shown in Figure I.1, at the end of this section. % ****************************************************** % Calculates the retroreflected divergence angle as a function of control beam radius % ****************************************************** close all; clear all; clc; r1 = 1;                 % outer sphere radius n0 = 1;                 % outside n c = [0,0,0];            % origin n = [0,0,1];            % aperature normal vector rs = .6;                % signal beam radius d = 1.5*r1;             % signal beam starting dist rho = logspace(-6, 0, 1000)/2; phi = 0:0.1:pi; theta_i = 0; n1 =  [2 1.955 1.5   2.5];   n2 =  [2 1.955 1.955 2.301]; rsize = numel(rho); psize = numel(phi); for i = 1:numel(n1)     r2 = r1*(1/n2(i)-1/n1(i))/(1/2-1/n1(i));    % inner sphere radius     if (r2 > 0) && (r2 < 1)         [P,Q,Px] = Retro_Cladded_Sphere(r1,r2,n0,n1(i),n2(i),c,n,rs,r2,d,rho,phi,theta_i,1);     else         [P,Q,Px] = Retro_Sphere(r1,n0,n1(i),c,n,rs,r1,d,rho,phi,theta_i,1);     end     [ctrl_radius(:,i), max_div(:,i)] = Max_Divergence_Angle(Q,Px,rho,phi);    subplot(1,2,2); loglog(ctrl_radius,max_div); xlim([10^-5 10^0]); ylim([10^-12 10^-4]);    xlabel('Control beam radius (m)'); ylabel('Divergence Solid Angle (sr)');     hold on; drawnow; clc; end  181  function [P,Q,Px] = Retro_Cladded_Sphere(r1,r2,n0,n1,n2,c,n,rs,rc,d,rho,phi,theta_i,plot_on) % r1: sphere 1 radius % r2: sphere 2 radius % n0: outside n % n1: sphere 1 n % n2: sphere 2 n % c: origin % n: aperature normal vector % rs: signal beam radius % rc: aperature radius % d: signal beam starting dist % create beam [P,Q] = Beam(rs,d,rho,phi,theta_i); % intersect & refract with sphere outer  [P,N] = Sphere(P,Q,c,r1); [Q] = Refract(Q,N,n0,n1); % intersect & refract with sphere inner [P,N] = Sphere(P,Q,c,r2); [Q] = Refract(Q,N,n1,n2); % intersect & aperture at z = 0 plane [P,~] = Plane(P,Q,c,n); [Q] = Aperture(P,Q,rc); Px = P(1,:,end); % intersect & refract with sphere inner [P,N] = Sphere(P,Q,c,r2); [Q] = Refract(Q,N,n2,n1); % intersect & reflect at sphere outer [P,N] = Sphere(P,Q,c,r1); [Q] = Reflect(Q,N); % intersect & refract with sphere inner [P,N] = Sphere(P,Q,c,r2); [Q] = Refract(Q,N,n1,n2); % intersect & refract with sphere inner [P,N] = Sphere(P,Q,c,r2); [Q] = Refract(Q,N,n2,n1); % intersect & refract with sphere outer  182 [P,N] = Sphere(P,Q,c,r1); [Q] = Refract(Q,N,n1,n0);  if (plot_on == 1)     clf;     subplot(1,2,1);     Plot_Rays(P,Q);     Plot_Axis(2*r1);     Plot_Sphere(r1,c);     Plot_Sphere(r2,c);     drawnow end end function [P,Q,Px] = Retro_Sphere(r,n0,n1,c,n,rs,rc,d,drho,dphi,theta_i,plot_on) % r1: sphere 1 radius % r2: sphere 2 radius % n0: outside n % n1: sphere 1 n % n2: sphere 2 n % c: origin % n: aperature normal vector % rs: signal beam radius % rc: aperature radius % d: signal beam starting dist   [P,Q] = Beam(rs,d,drho,dphi,theta_i);   [P,N] = Sphere(P,Q,c,r); [Q] = Refract(Q,N,n0,n1);   [P,~] = Plane(P,Q,c,n); [Q] = Aperture(P,Q,rc); Px = P(1,:,end);   [P,N] = Sphere(P,Q,c,r); [Q] = Reflect(Q,N);    183 [P,N] = Sphere(P,Q,c,r); [Q] = Refract(Q,N,n1,n0);   if (plot_on == 1) clf; subplot(1,2,1); Plot_Rays(P,Q); Plot_Axis(2*r); Plot_Sphere(r,c); drawnow end function [ctrl_radius, max_div] = Max_Divergence_Angle(Q,Px,rho,phi)     solid_angle = Solid_Angle(Q);     solid_angle = reshape(solid_angle,[numel(phi),numel(rho)]);     Px = reshape(Px,[numel(phi),numel(rho)]);     ctrl_radius = Px(1,:);     for k = 1:numel(rho)         max_div(k) = max(solid_angle(1,1:k)); %find max solid angle from 0 to ctrl_radius     end end function [solid_angle] = Solid_Angle(Q) solid_angle = 2*pi*(1-dot(Q(:,:,end),-Q(:,:,1))); end function [P,Q] = Beam(r,d,rho,phi,theta_i) % r: radius of circular beam with at an angle % d: distance from 1st element % theta_i: incident angle % rho, phi: ray spacings % P: ray point locations % Q: ray directions %% Make circular beam in body reference frame [R, PHI, Z] = meshgrid(rho,phi,0); X = R.*cos(PHI);             % Position Y = R.*sin(PHI); U = 0*X; V = 0*Y; W = 0*Z-1; % Orientation %% Transform beam vector field into global frame Rx = [1 0 0; 0 cos(theta_i) -sin(theta_i); 0 sin(theta_i) cos(theta_i)];  184 Rx = [Rx zeros(3,1); zeros(1,3) 1]; Dz = [0; 0; d]; %move plane away from origin Dz = [eye(3) Dz; zeros(1,3) 1]; Pf = [X(:).'; Y(:).'; Z(:).';]; P = Rx*Dz*[Pf; 0*Z(:).'+1]; P = P(1:3,:); Q = Rx*[U(:).'; V(:).'; W(:).'; 0*Z(:).'+1]; Q = Q(1:3,:); end function [P,N] = Sphere(P,Q,c,r) % P: incoming ray points % Q: incoming ray directions % c: lens centre location % r: lens radius of curvature C = kron(c',ones(1,size(P,2))); P0 = P(:,:,end); Q0 = Q(:,:,end); a = dot(Q0,Q0); b = 2*dot(Q0,P0-C); c = dot(P0-C,P0-C)-r^2; if (max(c) > 1e-10) % intersect with sphere outside     t = (-b-sqrt(b.^2-4*a.*c))./(2*a);     P1 = P0 + Q0.*[t; t; t];     N = P1-C; else % intersect with sphere inside     t = (-b+sqrt(b.^2-4*a.*c))./(2*a);     P1 = P0 + Q0.*[t; t; t];     N = C-P1; end n = sqrt(dot(N,N)); N = N./[n; n; n]; P = cat(3,P,P1); end function [Q] = Refract(Q,N,na,nb) % Q: ray directions % N: normal vector % na: refractive index rays are coming from  185 % nb: refractive index rays are going to Q0 = Q(:,:,end); a = na/nb; b = dot(-N,Q0); c = (a*b-sqrt(1-a^2*(1-b.^2))); Q1 = a*Q0 + [c; c; c].*N; Q = cat(3,Q,Q1); end function [Q] = Reflect(Q,N) % Q: ray directions % N: normal vector Q0 = Q(:,:,end); b = dot(-N,Q0); Q1 = Q0 + 2*[b; b; b].*N; Q = cat(3,Q,Q1); end function [Q] = Aperture(P,Q,r) f = sqrt(P(1,:,end).^2+P(2,:,end).^2)<=r; F = [f; f; f]; Q1 = Q(:,:,end).*F; Q = cat(3,Q,Q1); end function [P,N] = Plane(P,Q,c,n) N = kron(n',ones(1,size(P,2))); C = kron(c',ones(1,size(P,2))); P0 = P(:,:,end); Q0 = Q(:,:,end); t = dot(C-P0,N)./dot(Q0,N); P1 = P0 + Q0.*[t; t; t]; P = cat(3,P,P1); end function Plot_Rays(P,Q) % P: ray point locations % Q: ray directions %% Set default plot window set(0,'units','pixels'); screen_size_p = get(0,'ScreenSize'); set(0,'defaultfigureposition',[0 0 screen_size_p(3) screen_size_p(4)]'); %% Get vector lengths for j = 1:size(P,3)-1  186     L(:,:,j) =sqrt(dot(P(:,:,j+1)-P(:,:,j),P(:,:,j+1)-P(:,:,j))); end L = cat(3,L,1+0*L(:,:,1)); %% Plot rays  hold on; for j = 1:size(P,3)     %scatter3(P(1,:,j),P(2,:,j),P(3,:,j));     quiver3(P(1,:,j),P(2,:,j),P(3,:,j),L(1,:,j).*Q(1,:,j),L(1,:,j).*Q(2,:,j),L(1,:,j).*Q(3,:,j),0,...     'color',[j/size(P,3) .5 .2],'LineWidth',1.5,'ShowArrowHead','off'); end end function Plot_Axis(lim) hold on; axis([-1 1 -1 1 -1 1]*lim); axis square; view([90 0]) quiver3(-lim,0,0,2*lim,0,0,'k','LineWidth',2) quiver3(0,-lim,0,0,2*lim,0,'k','LineWidth',2) quiver3(0,0,-lim,0,0,2*lim,'k','LineWidth',2) text(lim,0,0,'X','FontWeight','Bold') text(0,lim,0,'Y','FontWeight','Bold') text(0,0,lim,'Z','FontWeight','Bold') end function Plot_Sphere(r,c) [x,y,z] = sphere(100); x = r*x+c(1); y = r*y+c(2); z = r*z+c(3); hold on;  surf(x,y,z); shading flat; colormap([.8 .8 .8]); alpha(.5); end  187  Figure I.1. MATLAB ray tracing simulations of the retro-modulated divergence solid angle, Â∆, as a function of the control beam radius, ∆Ë, normalized to the sphere’s radius, À, for various spherical retroreflectors. The following non-cladded and cladded spheres are presented: (a) an ideal ≤ = 2.000 sphere, (b) an ≤ = 1.955 sphere, (c) an ≤ = 1.955 sphere with an ≤ = 1.500 cladding, and (d) an ≤ = 2.500 sphere with an ≤ = 2.301 cladding. The rays are plotted in three-dimensions, propagating through the inner and outer spheres, shown in grey. The direction of propagation is denoted with a gradient from green rays to yellow rays. The resulting retroreflected divergence solid angle functions are plotted on the right, which are also seen in Figure 5.7. adcb

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