Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Mitchell, Michael Eric
Master of Applied Science - MASc
University of British Columbia
Terahertz radiation, given its unique location in the electromagnetic spectrum between radio-frequency and infrared radiation, has been highly regarded for its potential of non-invasive spectroscopy. The long wavelengths and low photon energies of terahertz radiation allow for the measurement of the rotational and vibrational modes of cellular tissue without damaging the cells themselves. However, the long wavelengths inherent to THz radiation restrict the focal spot sizes and therefore limit the size of samples that can be analyzed with THz spectroscopy systems. Spatial resolution has been a long-standing issue for THz spectroscopy systems, especially for the long wavelengths of the low frequency (0.3-to-0.7) THz band, being the focus of this thesis. The thesis highlights the validity, through theoretical and experimental testing, of using dielectric spheres to focus THz radiation to a scale that improves the spatial resolution of conventional THz spectroscopy systems. The dielectric spheres focus the THz radiation into so-called “THz microjets”, which can achieve subwavelength focusing over a broad-bandwidth. A complete optimization of the dielectric sphere, both in terms of size and material parameters, is undertaken. Optimal dielectric spheres are then manufactured out of custom THz microcomposite materials and experimentally characterized in terms of the focusing performance. Finally, the dielectric spheres are demonstrated in a spectroscopic application to show the improvement in resolution.
Terahertz Microjets for Spectroscopy on the Subwavelength Scale by Michael Eric Mitchell B.A.Sc., The University of British Columbia, 2020 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in THE COLLEGE OF GRADUATE STUDIES (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2022 © Michael E. Mitchell, 2022 ii The following individuals certify that they have read, and recommend to the College of Graduate and Studies for acceptance, a thesis/dissertation entitled: Terahertz Microjets for Spectroscopy on the Subwavelength Scale submitted by Michael Mitchell in partial fulfillment of the requirements for the degree of Master of Applied Science Examining Committee: Dr. Jonathan Holzman, School of Engineering Supervisor Dr. Christopher Collier, School of Engineering Supervisory Committee Member Dr. Thomas Johnson, School of Engineering Supervisory Committee Member Dr. Alex Hill, Irving K. Barber Faculty of Science External Examiner iii Abstract Terahertz radiation, given its unique location in the electromagnetic spectrum between radio-frequency and infrared radiation, has been highly regarded for its potential of non-invasive spectroscopy. The long wavelengths and low photon energies of terahertz radiation allow for the measurement of the rotational and vibrational modes of cellular tissue without damaging the cells themselves. However, the long wavelengths inherent to THz radiation restrict the focal spot sizes and therefore limit the size of samples that can be analyzed with THz spectroscopy systems. Spatial resolution has been a long-standing issue for THz spectroscopy systems, especially for the long wavelengths of the low frequency (0.3-to-0.7) THz band, being the focus of this thesis. The thesis highlights the validity, through theoretical and experimental testing, of using dielectric spheres to focus THz radiation to a scale that improves the spatial resolution of conventional THz spectroscopy systems. The dielectric spheres focus the THz radiation into so-called “THz microjets”, which can achieve subwavelength focusing over a broad-bandwidth. A complete optimization of the dielectric sphere, both in terms of size and material parameters, is undertaken. Optimal dielectric spheres are then manufactured out of custom THz microcomposite materials and experimentally characterized in terms of the focusing performance. Finally, the dielectric spheres are demonstrated in a spectroscopic application to show the improvement in resolution. iv Lay Summary Terahertz radiation has been highly regarded for its potential of non-invasive spectroscopy (analysis) of materials. However, the long wavelengths inherent to THz radiation restrict the focal spot sizes possible and therefore limit the size of samples that can be effectively analyzed. This thesis proposes dielectric spheres as a THz focusing element with the goal of enabling THz analysis on small size samples. The dielectric spheres focus the THz radiation into so-called “THz microjets”, which are shown to be capable of subwavelength focusing. This work outlines an optimization and manufacturing procedure allowing for optimal THz microjets to be created. Both the focusing and spectroscopic capabilities of THz microjets induced by dielectric spheres are tested and found to improve the performance of conventional terahertz spectroscopy systems. v Preface The work presented in this thesis is tied to my contributions in journal articles at varied stages. My earliest contributions were published in [1], for which I assisted in the fabrication and testing of the composite materials. This early work lays a foundation for Chapter 2 in this thesis. My subsequent contributions, having been submitted as [2], make up the bulk of the work in Chapter 2. I developed much of the theory and implemented the experiments for the work in this chapter, with assistance from J. Reich on the fabrication and M. H. Bergen on the tests and interpretations. My latest contributions were published in [3] and [4], and they relate to the development of graphene for THz technology and the study of 3D printing materials for THz applications. [1] M. H. Bergen, M. E. Mitchell, E. M. Mellors, and J. F. Holzman, “Manifestations of loss in terahertz polymer composites,” Opt. Mater. Express 11, 3469 (2021). [2] M. E. Mitchell, M. H. Bergen, J. Reich, and J. F. Holzman, “Terahertz microjets as realizations of subwavelength focusing in the terahertz spectrum,” IEEE Trans. Terahertz Sci. Technol., submitted May 7, 2022. [3] I. R. Hristovski, L. A. Herman, M. E. Mitchell, N. I. Lesack, J. Reich, and J. F. Holzman, “Manifestations of laser-induced graphene under ultraviolet irradiation of polyimide with varied optical fluence,” Nanomater. 12, 1241 (2022). [4] C. H. Brodie, I. Spotts, H. Reguigui, C. A. Leclerc, M. E. Mitchell, J. F. Holzman, and C. M. Collier, "Comprehensive study of 3D printing materials over the terahertz regime: absorption coefficient and refractive index characterizations," Opt. Mater. Express 12, 3379-3402 (2022) vi Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface ............................................................................................................................................ v Table of Contents ......................................................................................................................... vi List of Tables .............................................................................................................................. viii List of Figures ............................................................................................................................... ix List of Symbols ........................................................................................................................... xiv Acknowledgements .................................................................................................................... xvi Chapter 1: Introduction ............................................................................................................... 1 1.1 Thesis Scope ................................................................................................................... 5 Chapter 2: Optimization of Terahertz Microjets for Terahertz Spectroscopy ....................... 8 2.1 Terahertz Emission ......................................................................................................... 8 2.1.1 Photoconductive Antennas .................................................................................... 10 2.1.2 Optical Rectification ............................................................................................. 12 2.2 Geometrical Considerations .......................................................................................... 16 2.2.1 Refractive Index and Material Losses ................................................................... 18 2.2.2 Sphere Diameter .................................................................................................... 21 2.3 Material Considerations ................................................................................................ 22 2.4 Focusing Performance .................................................................................................. 25 2.5 Discussion ..................................................................................................................... 30 Chapter 3: Application of Terahertz Microjets for Terahertz Spectroscopy ....................... 33 3.1 Terahertz Spectroscopy ................................................................................................. 33 vii 3.2 Terahertz Time Domain Spectroscopy with a Focused Beam ...................................... 39 3.2.1 Sample Position Analysis ..................................................................................... 39 3.2.2 Sample Size Analysis ............................................................................................ 43 3.2.3 Aperture Analysis ................................................................................................. 46 3.3 Spatially Constricted Spectroscopy with THz microjets .............................................. 49 Chapter 4: Conclusion ................................................................................................................ 56 4.1 Summary of Conclusions .............................................................................................. 56 4.2 Limitations .................................................................................................................... 59 4.3 Future work ................................................................................................................... 59 Bibliography ................................................................................................................................ 61 Appendices ................................................................................................................................... 68 Appendix A: Derivation of Kramers-Kronig Relations ............................................................ 68 viii List of Tables Table 1.1 Performance summary of terahertz focusing elements ................................................... 5 Table 3.1 Cutoff frequencies for diminishing aperture sizes ........................................................ 51 ix List of Figures Figure 1.1 Dielectric sphere as a focusing element. Radiation is focused through a sphere with a large capture cross-section and minimal material losses, to form a tightly focused beam with a minimal focal spot size. .................................................................................................................. 6 Figure 2.1 Schematic of the experimental setup. Pump and probe beams (red) are denoted as PU and PR, respectively. The emission setup includes a microscope objective (MO) and emitter (EM). Gold parabolic mirrors (GPM1 and GPM2) are used to capture and collimate the THz radiation (yellow). The THz radiation is focused through a dielectric sphere (FS) and recollimated through a second sphere (CS) before being overlapped on the pellicle beam-splitter (PB). The detection setup includes an electro-optic crystal (EO), quarter-wave plate (QW), polarizing beam splitter (PBS), and a differential photo diode (DP). ........................................... 10 Figure 2.2 Representation of a photoconductive THz antenna. An incident ultrafast pump laser pulse, shown in red, with a pulse duration of 100 fs and repetition rate of 80 MHz strikes a SI-GaAs wafer, shown in grey. Gold plated electrodes spaced at 200 µm are externally biased at a voltage of 50 Vp-p and frequency of 30 kHz to create a bias electric field. A THz pulse, depicted in black, is transmitted from the accelerated charges. .................................................................. 12 Figure 2.3 Schematic of electro-optic THz emission. An ultrafast pump pulse, shown in red, strikes a [110] ZnTe electro-optic crystal, shown in green. Both a frequency-doubled pulse, shown in blue, and a THz pulse, shown in black, are emitted and incident onto a Si wafer, shown in purple, which only transmits the THz pulse. ............................................................................ 14 Figure 2.4 Comparison of THz emission techniques. Red and blue curves denote THz pulses that are generated via optical rectification in a [110] ZnTe electro-optic crystal and a 200-µm gap x GaAs photoconductive antenna, respectively. In (a), the normalized THz time-domain waveforms are plotted versus time. In (b), the THz amplitude is plotted versus frequency. ....... 15 Figure 2.5 Intensity profiles of dielectric spheres shown as colourmaps. The colours show the logarithm of intensity from low (in black) to moderate (in red) to high (in white) versus transverse and axial dimensions of the dielectric spheres for varied frequency, f, and refractive index, n. In (a), a frequency of f = 0.3 THz and refractive index of n = 1.50 is shown. In (b), a frequency of f = 0.7 THz and refractive index of n = 1.50 is shown. In (c), a frequency of f = 0.3 THz and refractive index of n = 1.75 is shown. In (d), a frequency of f = 0.7 THz and refractive index of n = 1.75 is shown. ........................................................................................................... 18 Figure 2.6 Refractive index-dependent intensity gains of dielectric spheres. The colourmaps span low (in black) to moderate (in red) to high (in white) and plot intensity gain versus the frequency, f, and refractive index, n, for various values of the extinction coefficient, k. In (a), (b), and (c), extinction coefficients of k = 0.00, k = 0.01, and k = 0.02 are shown, respectively. Each figure shows the intensity gains resulting from 48,000 Mie theory simulations carried out at the discrete combinations of f and n. The combinations of f and n yielding maximal intensity gain are shown as black trendlines. For (a), (b), and (c), the colourmap spans min to max values of 400 to 4000, 100 to 250, and 5 to 65, respectively. ................................................................................. 20 Figure 2.7 Diameter-dependent intensity gains of dielectric spheres. Colourmaps span low (in black) to moderate (in red) to high (in white), and plot intensity gain versus the frequency, f, and diameter, d, for various values of the extinction coefficient, k. In (a), (b), and (c), the results are shown for extinction coefficients of k = 0.00, k = 0.0025, and k = 0.01, respectively. Each figure shows the intensity gains resulting from 5,000 Mie theory simulations carried out at xi discrete combinations of f and d. For (a), (b), and (c), the colourmap spans min to max values of 10 to 5000, 1 to 530, and 0.5 to 60, respectively. ......................................................................... 22 Figure 2.8 Refractive index of the microcomposite material, n, versus volumetric fraction of SiO2 microparticles inclusions in the UHMWPE host, V. The results are shown as experimental markers, shown in red, and a fitted theoretical curve, shown in black. The inset shows a scanning-electron-microscope image of the microcomposite material with V ≈ 30%. ................ 25 Figure 2.9 Focusing characteristics of THz microjets. In (a), the field distribution at the centre frequency, f = 0.5 THz, versus the transverse dimension of the nominal (n = 1.5) dielectric sphere is shown. In (b), the field distribution at the centre frequency, f = 0.5 THz, versus the transverse dimension of the optimal (n = 1.65) dielectric sphere is shown. In (c), the focal spot size, d, is plotted versus frequency, f, for the nominal dielectric sphere. In (d), the focal spot size, d, is plotted versus frequency, f, for the optimal dielectric sphere. In (a), (b), (c), and (d), black lines denote theoretical results for the dielectric spheres, red markers / error-bars denote experimental results for the dielectric spheres, and blue markers / error-bars denote experimental results for the (BATOP TPX-D25.4-f10) THz lens. The subwavelength regime is shaded in grey. ....................................................................................................................................................... 28 Figure 2.10 Focal span characteristics of THz microjets. The logarithm of the intensity profile at the rear of the nominal (a) and optimal (b) dielectric spheres is plotted versus the transverse and axial dimensions. In (a) and (b), the focal spot positions of the dielectric sphere (as red circles) and the THz lens (as blue circles) are plotted. The lower bound, circle and upper bound represent the focal distances of frequencies of 0.3, 0.5, and 0.7 GHz, respectively. ................................... 30 Figure 3.1 Terahertz spectroscopy of lactose. In (a), the reference pulse, shown in blue, and the sample pulse, shown in red, are plotted as a function of time. In (b), the THz amplitude of the xii reference pulse, shown in blue, and the sample pulse, shown in red, are plotted as a function of frequency. The noise floor of the system is plotted as the dashed black line. In (c), the refractive index is plotted as a function of frequency. The blue line corresponds to the experimental data with the red dashed line representing the refractive index calculated through the Kramers-Kronig relations. In (d), the extinction coefficient is plotted as a function of frequency. ........................ 36 Figure 3.2 Effect of sample position in highly-focused THz-TDS. The refractive index, averaged over 0.60 to 0.65 THz, is plotted as blue markers (with error bars) for varying sample positions relative to the focus. The error bars denote the deviation in extracted refractive index between two subsequent THz-TDS runs. The red dashed line corresponds to the expected refractive index over the averaged frequency range. The dark grey shaded region denotes the rear surface of the sphere with the light grey shading corresponding to the evanescent regime (within one wavelength from surface). ............................................................................................................ 42 Figure 3.3 Two cases of pulse and sample interactions. In (a), the focal spot size, d, is smaller than the sample diameter. In (b) the focal spot size is larger than the sample meaning a portion of the beam passes through the sample, shown in red, and a portion passes around, not interacting with the sample, shown in blue. .................................................................................................... 44 Figure 3.4 THz TDS signals for sample sizes below the focal spot size. THz pulses are shown versus time with reference and sample pulses depicted as blue and red lines, respectively. The resulting time domain pulses from a 1000 µm, 750 µm and 500 µm diameter sample placed at the focus of a THz microjet are shown in (a), (b), and (c), respectively. ..................................... 45 Figure 3.5 Aperture effect on a focused THz beam. Results for aperture diameters of 1000, 360, and 186 µm are shown in blue, red and yellow, respectively. The THz amplitude is plotted versus xiii time with each time-domain signal normalized to the time-domain signal of the 1000 µm diameter aperture. ......................................................................................................................... 48 Figure 3.6 Broadband aperture characteristics. The THz amplitude is shown in colour, with its colourmap spanning low (in black) to moderate (in red) to high (in white), versus the frequency, f, and aperture diameter. ................................................................................................................ 49 Figure 3.7 Refractive index of lactose for decreasing aperture sizes. The extracted refractive index is shown versus frequency for no aperture and aperture diameters of 1000, 715, 510, 365, 260, 186, and 132 µm. The dashed lines show frequencies where the material parameter cannot be reliably extracted with the solid circles indicating the cutoff frequencies for the aperture diameters. ...................................................................................................................................... 53 Figure 3.8 Extinction coefficient of lactose for diminishing aperture sizes. The extracted extinction coefficient is shown versus frequency for no aperture and aperture diameters of 1000, 715, 510, 365, 260, 186, and 132 µm. The dashed lines show frequencies where the material parameter cannot be reliably extracted with the solid circles indicating the cutoff frequencies for the aperture diameters. .................................................................................................................. 54 xiv List of Symbols c – Speed of light d – Sphere diameter Esam – Sample spectrum Eref – Reference spectrum f – frequency I – Intensity j – Current k – Extinction coefficient l – Sample thickness n – Refractive index ne – Electron density N – Depolarization factor q – Electron charge t – Time tx-x – Fresnel transmission coefficient V – Volumetric fraction a – Absorption coefficient d – Full-width-at-half-maximum e – Complex dielectric constant l – Wavelength xv µe – Electron mobility s – Conductivity ce – Dielectric susceptibility xvi Acknowledgements I would like to thank all the individuals who made this thesis, and the work contained within it possible. Firstly, I would like to thank my supervisor Dr. Jonathan Holzman for the endless support and mentorship he offered throughout my time at UBCO. His technical expertise and devoted attention to his graduate students is unmatched and will not be forgotten. I would also like to thank my committee members, Dr. Christopher Collier, for his knowledge and help throughout this thesis and Dr. Thomas Johnson for his insight and willingness to adjudicate my thesis. Next, I would like to thank everyone who contributed to my research and assisted me along the way. Firstly, I would like to thank Dr. Mark Bergen for his tireless efforts in training and teaching me all things THz. I would not be where I am without his initial guidance. I also have to thank Isaac Spotts for his unparalleled sense of humor and willingness to find problems and subsequently help re-build everything in the lab. The countless hours we spent re-aligning and re-building setups gave me a deeper understanding of the research we completed. I would also like to thank the other member of the Integrated Optics Lab, being Alex MacGillivray, and the members of the Collier Research Group, being Camille Leclerc and Seth Lowry, for all the help and laughs throughout the years. This made my time as a graduate student more enjoyable. Finally, I would like to thank my parents, Lori and Harry Mitchell for their continued and unwavering support throughout my academic career. 1 Chapter 1: Introduction Terahertz (THz) radiation has wavelengths between 1 mm and 10 µm and was often referred to as the THz Gap. This term came about because THz radiation lies in the electromagnetic spectrum between radio-frequency and infrared radiation, giving it unique challenges for generation and detection. At the same time, this unique position for THz radiation gives it frequencies that match the characteristic rotational and vibrational modes of cellular tissue as well as photon energies that are well below the ionization energies of cells1. This gives THz radiation the potential for highly-sensitive and non-invasive spectroscopy of biological tissue. The abilities of THz spectroscopy are now seen in the simple recognition of carbohydrates2 to cancer diagnostics3–6, wherein the THz radiation couples to a manifold of inter- and intra-molecular modes and gives heightened sensitivity to the physiological markers of cancerous tissue. This has led to notable achievements for THz spectroscopy in the detection of basal cell carcinoma7 and melanoma8, with particular interest on the lower frequencies, 0.3 to 0.7 THz, where the absorption shows its strongest correlation to the differing water content in healthy and cancerous cells. Unfortunately, the relatively long wavelengths inherent to the THz spectrum make it difficult to glean the spectroscopic details from individual cells7,9. This lack of spatial resolution is especially prevalent for the lower THz frequencies, spanning 0.3 to 0.7 THz, which have long (millimetre-scale) wavelengths and must be focused in a subwavelength regime10. Given the potential for such THz radiation, several studies have proposed many different physical structures that act as focusing elements for THz radiation. When designing such a focusing element there are three main metrics, being the capture cross-section, material loss, and focal spot size, which define the overall performance of the focusing element. The metric of capture cross-section 2 indicates the physical size of the focusing element, which should be made large to maximize the captured power. The capture cross-section is quoted throughout this work as the side-length for the focusing element (as opposed to a cross-sectional area). The metric of material loss is used to encompass the absorptive losses incurred by propagating THz radiation through a medium, and therefore needs to be minimized for an optimal focusing element. The metric of focal spot size, which is the physical size that the THz radiation can be focused to, also needs to be minimized for an optimal focusing element. It should be noted that the three focusing element metrics seen above differ from those commonly seen in the literature. First, many research groups compare their focusing element's performance to the diffraction limit. This diffraction limit is often defined in terms of the numerical aperture (NA) of the focusing element and, as such, allows for quoting of diffraction-limited focusing while not achieving subwavelength focusing. As will be discussed later, subwavelength focusing is significantly harder to achieve in the THz regime than diffraction-limited focusing. As such, the characterizations of focal spot size throughout this work will compare the full-width-at-half-maximum (FWHM) relative to the free-space wavelength, λ0. Such a metric allows for uniform comparison between the physical focusing elements and gives insight into the physical size of the focal spot. Additionally, the majority of studies apply focusing elements without consideration to maximizing their capture cross-section or minimizing their material losses. This leads to reduced signal levels in applications of THz spectroscopy. The relevant studies on focusing elements for THz radiation will be discussed below and highlighted in Table 1.1, which summarizes the focusing element metrics of capture cross-section, material loss, and focal spot size, as defined above. 3 The most basic focusing elements are standard planoconvex THz lenses, with early studies by Scherger et al.11 and Wichman et al.12 investigating micro powders and plastic compounds, respectively, to create these THz lenses. By manufacturing a focusing element with a 20-mm capture cross-section out of low material loss high-density polyethylene powder, Scherger et al. achieved a focal spot size of roughly 2.6λ0 at a frequency of 0.5 THz11. Similarly by manufacturing a focusing element with a capture cross-section of 50 mm, with different blends of low material loss polypropylene, Wichman et al.12 achieved a focal spot size of roughly 2.0λ0 at 0.5 THz. These studies highlight the importance of material selection to achieve low material losses. At the same time, the studies use simple plano-convex lenses with large capture cross-sections. However, they are unable to achieve the subwavelength focusing desired for the metric of focal spot size. For this reason, as stated earlier, the authors targeted the easier diffraction-limited focusing as opposed to subwavelength focusing. To show this distinction, a state-of-the-art planoconvex THz lens with a capture cross-section of 20 mm and low material loss is experimentally characterized later in this work and shown to give focal spot sizes down to 1.1λ0 at a frequency of 0.5 GHz. In order to improve upon the focusing provided by standard planoconvex THz lenses, Neu et al. proposed the use of meta-materials as gradient refractive index (GRIN) focusing elements. By using a capture cross-section of 2.5 mm and low loss meta-materials their GRIN lenses were shown to be capable of subwavelength focal spot sizes down to 0.96λ0 at a frequency of 1.3 THz13. The GRIN characteristic was induced by changing the radius of slots cut into 200 nm thick copper unit cells that were suspended in a benzocyclobutene. Another recent study by Hernandez-Serrano et al. expanded on the work on GRIN lenses by introducing a method for 3-D printing of GRIN lenses14. They demonstrated how specifically tailored GRIN lenses can be easily printed and can 4 achieve diffraction-limited focusing below 0.7 THz. The large capture cross-section of 38.1 mm and relatively high material losses given the 3-D printing technique brought about focal spot sizes of 2.61l0, which although being diffraction limited did not achieve subwavelength focusing. These initial studies on the use of meta-materials for focusing THz radiation has prompted further research of aptly-named “THz metalenses” of other geometries 15–18. Such THz metalenses make use of specifically designed and highly periodic structures of high refractive index materials. The metalenses have shown great potential for tight focusing of THz radiation, however the designs often rely upon polarization-dependent structures16, which limit the applications. To combat this polarization dependence, a recent study by Chen et al. introduced a symmetric periodic polarization independent structure capable of achieving subwavelength focusing at specific THz frequencies18. By constructing a periodic cube shaped Si metasurface on a SiO2-Si substrate, the authors demonstrated that subwavelength focusing was possible. Namely, Chen et al. showed that a capture cross-section of 36.2 mm and low material loss could be had with focal spot sizes down to 0.61λ0 at a frequency of 2.5 THz18. It is also worth noting that the focusing performance of the above metalenses, due to their highly periodic nature, begins to significantly suffer as the incident wavelength changes slightly from the design frequency. This limits the performance of the metalenses for broadband THz spectroscopy, making them more suitable for continuous-wave THz applications. Another focusing element proposed for subwavelength focusing was the dielectric cuboid 19,20. Here, cuboid structures with dimensions at the same size as the incident wavelength were used to focus THz radiation. Pacheno-Pena et al. analyzed the effect that the refractive index of the cuboid structure has on both the focal spot size and focal position of the THz beam protruding from the 5 rear surface of the cuboid 19. By optimizing the refractive index to a value of n = 1.75 and setting the capture cross-section to 1 mm a focal spot size of 0.41λ0 was achieved at a frequency of 0.50 THz. The small focal spot sizes seen from the dielectric cuboids can be attributed to the capture cross-section being comparable to the wavelength, which in turn limits the amount of incident THz radiation that can be captured and focused. Additionally, as will be discussed in subsequent sections, achieving the refractive index of n = 1.75 required to create the optimal cuboid proposed by Pacheno-Pena et al. would yield high material losses. Table 1.1 Performance summary of terahertz focusing elements Reference Capture cross-section (mm) Material loss Frequency (THz) Focal spot size 11 20.0 Low 0.50 2.6l0 12 50.0 Low 0.50 2.0l0 13 2.5 Low 1.3 0.96l0 14 38.1 High 0.40 2.61l0 18 36.2 Low 2.5 0.61l0 19 1.0 High 0.50 0.41l0 1.1 Thesis Scope The goal of this thesis is to achieve THz spectroscopy on a subwavelength scale across the lowest band of the THz spectrum, spanning 0.3 to 0.7 THz. This targeted band is chosen as its inherent long wavelengths yield the greatest challenges for focusing and its spectral features show great promise for characterizing biological tissue7,8. The presented work builds upon a focusing element 6 that has been optimized for capture cross-section, material loss, and focal spot size in the visible/infrared spectrum. It takes the form of a dielectric sphere and brings about focusing as a photonic nanojet21–23. A simple depiction of such a dielectric sphere can be seen in Figure 1.1. In realizing photonic nanojets, the focusing element metrics are easily optimized by making the diameter of the sphere (d) far larger than the incident visible/infrared wavelengths. This gives a broad capture cross-section (at d >> l0), a subwavelength focal spot size (at n ≈ 2 from simple ray analyses21), and negligible absorption (at k ≈ 0 for common polymers and glasses). Figure 1.1 Dielectric sphere as a focusing element. Radiation is focused through a sphere with a large capture cross-section and minimal material losses, to form a tightly focused beam with a minimal focal spot size. In this work, we look to extend these implementations of dielectric spheres to the focusing of THz radiation, as THz microjets, but such focusing is far more difficult to do in the THz spectrum. The difficulty arises as the diameter needed for the capture cross-section is comparable to the THz wavelengths, the material loss is nonnegligible, and the focal spot size is subject to the combined effects of refraction and diffraction. This thesis tackles these challenges by introducing optimized 7 dielectric spheres, whose induced THz microjets are capable of subwavelength terahertz time-domain spectroscopy (THz-TDS) across the targeted 0.3-to-0.7 THz band. This is done with thought to the overall THz spectroscopy system and its demands for this spectrum. First, differing THz emission techniques and their implications across the targeted 0.3-to-0.7-THz band are discussed. Next, the optimization of the dielectric sphere, in terms of its capture cross section (diameter), refractive index, and material losses, is presented. A methodology for creating composite materials with tunable refractive indices and minimal material losses is shown in this effort enable fabrication of the optimal dielectric spheres. The focal spot sizes of these dielectric spheres, which have had both their capture cross-section and material loss optimized, are then fully characterized and compared to state-of-the-art THz lenses. Next, the theory behind THz-TDS and the implications of using a highly-focused THz microjets and/or apertures is investigated. Finally, the spectroscopic capabilities of an optimized THz-TDS system with focusing via subwavelength THz microjets is presented. 8 Chapter 2: Optimization of Terahertz Microjets for Terahertz Spectroscopy In this chapter, the main components necessary for implementing THz-TDS with THz microjets are discussed. Firstly, a comparison of THz emission techniques and the implications for the targeted 0.3-to-0.7-THz band are shown. Next, a complete optimization of the dielectric sphere used to focus THz radiation into THz microjets, including the sphere’s capture cross-section and theoretical material loss, is discussed. A method of manufacturing tunable materials, being capable of forming optimized spheres for THz microjets, is shown. The THz microjets protruding from the rear surface of both nominal and optimal spheres are rigorously characterized and compared to theoretical simulations. Finally, the focal spot size and focusing performance of the THz microjets are compared to those of state-of-the-art THz focusing optics. 2.1 Terahertz Emission In this section, a detailed comparison of differing THz generation schemes, being photoconductive antennas and electro-optic crystals, is presented and the implications for realizing spectroscopy across the targeted 0.3-to-0.7-THz band is discussed. A schematic of the THz setup used in this work can be seen in Figure 2.1. The laser beam, shown in red, is generated by a Titanium Sapphire laser (Spectra Physics Mai Tai), with 100-fs-duration laser pulses at a repetition rate of 80 MHz and a wavelength of 780 nm. The beam from this laser is split into separate pump (PU) and probe (PR) beams. The pump beam is focused through a microscope objective (MO) onto an emitter (EM), which can either be a photoconductive antenna (PC) or an electro-optic crystal (EO). Both emitter types, being the PC antenna and the electric-optic crystal, will be discussed in sections 2.1.1 and 2.1.2, respectively. The emitted THz radiation, shown in yellow, is collected and collimated by an off-axis gold parabolic mirror (GPM1). It is then incident onto the focusing sphere 9 (FS), creating the THz microjet protruding from its rear surface. The radiation from the microjet is then collected and re-collimated by collimating sphere (CS). It is subsequently overlapped with the probe beam on a pellicle beam splitter (BS). The overlapped beams are collected and focused by a second off-axis gold parabolic mirror (GPM2) into an electro-optic crystal. The electro-optic detection setup here is similar to that first proposed by Gallot et al.24 and used by others25. With this setup, the presence of the THz electric field in the electro-optic crystal rotates the polarization of the probe beam. This change in polarization is then measured, as compared to the nominal circularly-polarized state from the quarter-wave plate (QW), by passing the probe beam through a polarizing beam splitter (PBS). The PBS splits the probe beam into two orthogonally-polarized beams, which are input into a differential photodiode (DP). The DP subtracts the photocurrents from these beams, such that the THz electric field yields a difference in the two photocurrents while the common background and noise are rejected. 10 Figure 2.1 Schematic of the experimental setup. Pump and probe beams (red) are denoted as PU and PR, respectively. The emission setup includes a microscope objective (MO) and emitter (EM). Gold parabolic mirrors (GPM1 and GPM2) are used to capture and collimate the THz radiation (yellow). The THz radiation is focused through a dielectric sphere (FS) and recollimated through a second sphere (CS) before being overlapped on the pellicle beam-splitter (PB). The detection setup includes an electro-optic crystal (EO), quarter-wave plate (QW), polarizing beam splitter (PBS), and a differential photo diode (DP). 2.1.1 Photoconductive Antennas The first type of THz emission studied in this work, being THz emission via photoconductive antennas, was initially shown by Auston et al26. A simple depiction of such a PC antenna can be seen in Figure 2.2. In this work, the antenna is implemented on a semi-insulating gallium arsenide (SI-GaAs) wafer with a 200-µm-wide gap between the displayed gold electrodes. The gap is illuminated by laser pulse with above-bandgap photon energy to create a time-varying electron density, ne(t), in the SI-GaAs substrate. The ensuing time-varying conductivity then rises 11 according to s(t) = ne(t)qµe, where q is the electron charge, and µe is the electron mobility. (Holes are ignored in this simple description due to their far lower mobility.) The time-varying current density, j(t), then evolves according to j(t) s(t)Eb, where Eb is the bias electric field applied across the gold electrodes. (The potential saturation of this bias field, due to a very dense electro-hole plasma, is ignored in this simple description.) In this work, the bias voltage was set to 50 Vp-p and was modulated at a frequency of 30 kHz to facilitate lock-in detection. The subsequent THz electric field pulse from the PC emitter, ETHz(t), shown as the black time-domain signal in Figure 2.2, is proportional to the time rate of change of the current density, as expressed by . (2.1) Many studies, such as those by Collier et al. and Alfied et al., have looked to optimize the emission of PC antennas by studying the effects that residual photocurrent consumption27, semiconductor surface texture28, and emitter bias structure29 play on THz emission. The THz emission from the PC antenna emitter used in this work can be visualized in both the time and frequency domains and is compared to the upcoming electro-optic crystal emission in Figure 2.4. »( )THz( )( )d j tE tdtµ12 Figure 2.2 Representation of a photoconductive THz antenna. An incident ultrafast pump laser pulse, shown in red, with a pulse duration of 100 fs and repetition rate of 80 MHz strikes a SI-GaAs wafer, shown in grey. Gold plated electrodes spaced at 200 µm are externally biased at a voltage of 50 Vp-p and frequency of 30 kHz to create a bias electric field. A THz pulse, depicted in black, is transmitted from the accelerated charges. 2.1.2 Optical Rectification The second method of THz emission investigated in this work, being THz emission via optical rectification, is based upon a concept originally introduced by Bass et al.30 The generation of broadband THz radiation by optical rectification relies on striking an electro-optic crystal with an ultrafast laser pulse, which induces the Pockels effect. When this happens, a time-dependent polarization is induced in the material as , (2.2) 0 e( ( ) ) ( ( ) ) ( )P E t E t E te c= Ultrafast Pump Laser Pulse Emitted THz Pulse 13 where e0 is the permittivity of free space and ce is the electric susceptibility. The Pockels effect then manifests from the second-order electric susceptibility in proportion to the coefficient ce2. Assuming that the time-varying electric field from the incident laser pump pulse can be expressed as E(t) = E0sin(wt), the second-order nonlinear polarization can be expressed as . (2.3) The DC component of this nonlinear polarization follows the envelope of the incident laser pump pulse. Thus, it is similar in duration to the incident pulse and contains broadband THz radiation.31 The remaining term, while not being useful for THz generation, is used in other applications requiring frequency doubling. The frequency-doubled pulse can be seen in the setup and must be blocked from propagating through the system. In this work, an off-axis 200 µm thick wafer of high-resistivity Si was placed in the path of the THz pulse to absorb the frequency-doubled pulse (with little loss to the THz pulse). A representative electro-optic emission scheme can be seen in Figure 2.3, where an ultrafast pulse, shown in red, strikes an electro-optic crystal, shown in green, producing both a frequency-doubled pulse, shown in blue, and a THz pulse, shown in black. The frequency-doubled pulse is blocked by the Si wafer, shown in purple, which allows just the THz pulse to be transmitted. It is important to note that the Pockels effect, and subsequent THz emission via optical rectification scales with the magnitude squared of the incident electric field. This allows for optimal THz signals to be generated with high-power and short-duration laser pulses. In this work, optical rectification was carried out in a [110] ZnTe electro-optic crystal with similar laser parameters to those of PC emission, i.e., a 100-fs pump pulse duration, an 80-MHz repetition rate, and a 780-nm wavelength. However, given that the Pockels effect and subsequent THz emission scale nonlinearly with the incident pump power, this power was increased to the 2 2 22 0 e2 0 0 e2 01 cos(2 )( ) sin ( )2tP t E t E we c w e c +æ ö= = ç ÷è øcos(2 )tw14 maximum value of 2.3 W. The detection system, was identical to that for PC emission, as described in Figure 2.1, but it used lock-in signal detection to a mechanically modulated (chopped) pump beam at a frequency near 1 kHz. Figure 2.3 Schematic of electro-optic THz emission. An ultrafast pump pulse, shown in red, strikes a [110] ZnTe electro-optic crystal, shown in green. Both a frequency-doubled pulse, shown in blue, and a THz pulse, shown in black, are emitted and incident onto a Si wafer, shown in purple, which only transmits the THz pulse. Spectra from THz emission using a PC antenna and electro-optic crystal are shown in both the time and frequency domains in Figure 2.4. The red and blue curves of Figure 2.4 denote THz pulses that are generated via optical rectification in a [110] ZnTe crystal and a THz pulse generated by 200 µm gap GaAs photoconductive antenna, respectively. Ultrafast Pump Laser Pulse 15 Figure 2.4 Comparison of THz emission techniques. Red and blue curves denote THz pulses that are generated via optical rectification in a [110] ZnTe electro-optic crystal and a 200-µm gap GaAs photoconductive antenna, respectively. In (a), the normalized THz time-domain waveforms are plotted versus time. In (b), the THz amplitude is plotted versus frequency. In Figure 2.4(a), we immediately see that THz emission via the electro-optic crystal creates a shorter pulse. This shorter pulse in time corresponds to a broader frequency response, as seen in Figure 2.4(b), where the THz amplitude is plotted as a function of frequency. For our experimental setup, THz pulses generated through electro-optic crystal emission give a bandwidth out to roughly 2 THz, whereas THz pulses generated through PC antenna emission only obtain a bandwidth out to roughly 1.5 THz. However, for our targeted 0.3-to-0.7-THz band, the PC antenna emission is ideal, as it maintains a significantly higher THz amplitude across this band. This observation agrees well with the literature32, where electro-optic emission is capable of broader bandwidths with weaker signal strengths, in comparison to PC antenna emission. Given these findings, all 16 upcoming work presented in this thesis, on the characterization and application of THz microjets, was preformed using an optimized implementation of the PC antenna. 2.2 Geometrical Considerations Similar to that seen with photonic nanojets, the THz microjets considered in this work manifest via intense focusing of THz radiation through a dielectric sphere. The dielectric sphere acts as the focusing element for the THz radiation and is defined by the three aforementioned metrics, being capture cross-section, material loss, and focal spot size. The metric of capture cross-section of the dielectric sphere is defined simply by its diameter, d, whereas the metric of material loss is defined by its extinction coefficient, k. The metric of focal spot size, d(f,n), is dependent upon a combination of the diameter, d, and refractive index, n, of the dielectric sphere, and it parallels the spatial resolution that is characterized in near-field THz imaging19,33,34. For the purposes of this work, the above three metrics are combined into an overarching metric, the intensity gain, which can be optimized for the formation of THz microjets. We define the intensity gain as the ratio of intensity at the focus, I(f), to incident intensity, I0, according to . (2.4) This single, overarching metric is introduced here to recognize that dielectric spheres with broad capture cross-sections (at a large d), subwavelength focal spot sizes (at an optimal n), and low material losses (at a minimal k) will best capture and focus incident power into a well-confined THz microjet. Given that the wavelength of incident THz radiation is comparable to the dielectric sphere's diameter, rigorous electromagnetic analyses are needed in this work. Mie theory is used for these analyses to exploit the spherical symmetry35. The theoretical results for the overarching 220( ) (4 / )··e( , )I f d f c dI f np kd-»17 metric of intensity gain are compiled for THz radiation that spans the targeted 0.3-to-0.7-THz band while incident upon 20-mm-diameter dielectric spheres. This diameter was selected from the optimization process described in section 2.2.2. To illustrate the formation of THz microjets through dielectric spheres, the logarithmic intensity profiles across four representative dielectric spheres are displayed as colourmaps in Figure 2.5 for various frequencies and refractive indices. A low (n = 1.50) refractive index is shown in Figure 2.5(a) and 2.5(b) for frequencies of f = 0.3 THz and f = 0.7 THz, respectively. A high (n = 1.75) refractive index is shown in Figure 2.5(c) and 2.5(d) for frequencies of f = 0.3 THz and f = 0.7 THz, respectively. The results show THz microjets as intense focal spots with differing sizes and distances beyond the dielectric spheres. Overall, we see that an increased frequency shrinks the focal spot and drives it away from the rear surface, while an increased refractive index shrinks the focal spot and draws it closer to the rear surface. Given the effect that the refractive index of a dielectric sphere plays on the formation of THz microjets, an in-depth analysis of the material parameters, being refractive index and extinction, will be shown next. 18 Figure 2.5 Intensity profiles of dielectric spheres shown as colourmaps. The colours show the logarithm of intensity from low (in black) to moderate (in red) to high (in white) versus transverse and axial dimensions of the dielectric spheres for varied frequency, f, and refractive index, n. In (a), a frequency of f = 0.3 THz and refractive index of n = 1.50 is shown. In (b), a frequency of f = 0.7 THz and refractive index of n = 1.50 is shown. In (c), a frequency of f = 0.3 THz and refractive index of n = 1.75 is shown. In (d), a frequency of f = 0.7 THz and refractive index of n = 1.75 is shown. 2.2.1 Refractive Index and Material Losses A full characterization of the material parameters of the dielectric spheres and their resulting THz microjets are displayed in Figure 2.6 as colourmaps of intensity gain versus frequency, f, and refractive index, n, at extinction coefficients of k = 0.00, k = 0.01, and k = 0.02 in Figures 2.6(a), 19 (b) and (c), respectively. Each Figure shows the intensity gains resulting from 48,000 Mie theory simulations carried out at discrete combinations of f and n. Here, the frequencies span the targeted band, f = 0.3 to 0.7 THz, and the refractive indices span from n = 1.50, as the lower limit seen for common polymers and glasses36, to n = 2.00, as the upper limit for focusing outside the sphere according to ray analyses21. The three k values are chosen to be negligible, moderate, and high, in comparison to those of the upcoming microcomposite material. Overall, the results exhibit high intensity gains in two bands with maxima running through each band. The combinations of f and n yielding these maxima are shown as trendlines (in black). The upper trendline shows little dependence on frequency at refractive indices near n ≈ 1.86. Our analyses of the dielectric spheres along this trendline reveal that their highest intensities are confined to tight spots within a wavelength of the rear surface. We attribute this confinement to whispering gallery modes37, which deliver THz radiation along the perimeter of the dielectric spheres by total internal reflection. The modes arise at a sufficiently high refractive index contrast at the surface, which is met at n ≈ 1.86, leaving absorption as the dominant form of loss. With no absorption, the intensity gain simply increases with frequency due to the decreasing wavelength and focal spot size. However, as the extinction coefficient rises, from k = 0.00, in Figure 2.6(a), to 0.01, in Figure 2.6(b), to 0.02, in Figure 2.6(c), the absorption coefficient, 4pkf/c, grows in proportion to f and the intensity gain peaks at successively lower frequencies. This trend in absorption can be seen on both the lower and upper trendlines in Figure 2.6, although the refractive indices bringing about maximal intensity gains on the lower trendline vary with frequency. Our analyses of the dielectric spheres along this lower trendline show that their highest intensities arise in focal spots beyond the rear surface with the refractive indices yielding maximal intensity gains growing with frequency. For example, frequencies of f = 0.3, 0.5, and 0.7 THz exhibit maximal intensity gains for dielectric spheres with 20 n ≈ 1.55, 1.65, and 1.75, respectively. This trend continues towards a horizontal asymptote of n ≈ 2, as the high-frequency limit defined by ray analyses. Overall, we can conclude that THz microjets can be formed by dielectric spheres given due thought to their optimal refractive indices (being defined along the lower trendline) and minimal extinction coefficients (being as low as possible). Based on the lower trendline, operating in the non-evanescent regime, we select an optimal refractive index of n = 1.65 as it is the refractive index that maximizes intensity gain at the centre of our targeted 0.3-to-0.7-THz band. Figure 2.6 Refractive index-dependent intensity gains of dielectric spheres. The colourmaps span low (in black) to moderate (in red) to high (in white) and plot intensity gain versus the frequency, f, and refractive index, n, for various values of the extinction coefficient, k. In (a), (b), and (c), extinction coefficients of k = 0.00, k = 0.01, and k = 0.02 are shown, respectively. Each figure shows the intensity gains resulting from 48,000 Mie theory simulations carried out at the discrete combinations of f and n. The combinations of f and n yielding maximal intensity gain are shown as black trendlines. For (a), (b), and (c), the colourmap spans min to max values of 400 to 4000, 100 to 250, and 5 to 65, respectively. a b c 21 2.2.2 Sphere Diameter In addition to optimizing the refractive index, careful consideration must be given to optimizing the metric of capture cross-section. The optimal capture cross-section was found by optimizing the overarching metric of intensity gain based upon the sphere diameter. This was done by varying the sphere diameter, d, in equation 2.4 across a range of d = 8 to 28 mm. This range of sphere diameters yields the results in Figure 2.7, which shows the intensity gain as a function of sphere diameter and frequency for three representative extinction coefficient values, being k = 0.00, k = 0.0025, and k = 0.01 in Figure2.7(a), Figure 2.7(b), and Figure 2.7(c), respectively. It should be noted that the refractive index is n = 1.65 for Figure 2.7, as it is the aforementioned optimal refractive index for this work. From this optimization, it can be seen that the capture cross-section of the sphere and the frequency-dependent material loss act as opposing trends for the maximal intensity gain. In Figure 2.7(a), an extinction coefficient of k = 0.00 has the highest intensity gains arise for larger diameter spheres as the material loss is negligible. In Figure 2.7(b), with an extinction coefficient of k = 0.0025, the loss becomes non-negligible at higher frequencies, requiring a smaller diameter to achieve the maximal intensity gain. The trend of material loss dominating the intensity gain optimization is accentuated in Figure 2.7(c), for which an extinction coefficient of k = 0.01 demands that small diameter spheres be used to yield maximal intensity gain. At the centre of our 0.3-to-0.7-THz band, the extinction coefficients of k = 0.00, 0.0025, and 0.01 have the optimal sphere diameter be d = 28, 20, and 9.0 mm, respectively. As will be discussed in the upcoming section, material extinction values near k = 0.0025 are found for our microcomposite material, and so a diameter of d = 20mm is used for the dielectric spheres throughout the remainder of this thesis. 22 Figure 2.7 Diameter-dependent intensity gains of dielectric spheres. Colourmaps span low (in black) to moderate (in red) to high (in white), and plot intensity gain versus the frequency, f, and diameter, d, for various values of the extinction coefficient, k. In (a), (b), and (c), the results are shown for extinction coefficients of k = 0.00, k = 0.0025, and k = 0.01, respectively. Each figure shows the intensity gains resulting from 5,000 Mie theory simulations carried out at discrete combinations of f and d. For (a), (b), and (c), the colourmap spans min to max values of 10 to 5000, 1 to 530, and 0.5 to 60, respectively. 2.3 Material Considerations Based on the optimization of intensity gain, it became clear that an optimal (n = 1.65) refractive index material with minimal material loss is needed. Unfortunately, most common materials do not meet both these criteria38 for THz frequencies. Thus, for this work, microcomposite material is explored in this section as a means to tailor the refractive indices and extinction coefficients of the dielectric spheres. The refraction and absorption of the microcomposite material are characterized by the Bruggeman model39–41. The model defines an effective dielectric constant, e, a b c 23 according to the volumetric fraction, V, of inclusions in the host material. The model can be cast as39,41 , (2.5) where e h and e i are the complex dielectric constants of the host and inclusions, respectively, and N is the depolarization factor of the inclusions. This depolarization factor is defined by the shape of the inclusions, from N = 1/3 for a sphere to lesser values for randomly-oriented oblate or prolate spheroids42. Thus, it is treated as a fitting parameter in this work. Ultimately, equation 2.5 can be solved for the complex dielectric constant of the material, and the square root, e ½ = n – jk, can be taken to define its refractive index, n, and extinction coefficient, k. The microcomposite material applied here incorporates SiO2 microparticles as inclusions in an ultra-high-molecular-weight polyethylene (UHMWPE) host. Such materials are chosen because of their contrasting refractive indices, at opposite ends of the refractive indices in Figure 2.6, and their low extinction coefficients. According to our characterizations, the UHMWPE has a refractive index of 1.52 and an extinction coefficient of 0.0025, giving eh = 2.31 – j0.006. According to the literature43, the SiO2 microparticles have a refractive index of roughly 2.05 and an extinction coefficient of 2.8×10–4, giving ei = 4.20 – j0.001. The microcomposite material was formed by mixing powdered SiO2 microparticles (US Research Nanomaterials) into powdered UHMWPE (Sigma-Aldrich, 434272) at the mass fractions needed for the specified volumetric fractions. The microcomposite material was then hot-pressed into disks for 10 minutes at a temperature of 155°C and a pressure of 20 MPa to fuse the material. Careful consideration was h ih i(1 ) 0(1 ) (1 )V VN N N Ne e e ee e e e- -- + =+ - + -24 given to the time, temperature, and pressure here to ensure that the microcomposite material was fully fused. We found that poor fusion could lead to inhomogeneity in the material, which would manifest as scattering. The refraction and absorption characteristics of the material were then measured via THz time-domain spectroscopy, following our prior work36,39. Such characteristics were acquired for volumetric fractions up to V ≈ 30%. This upper limit was set by the point in which the UHMWPE could no longer adequately fill the voids between (and fuse) the SiO2 microparticles. Higher volumetric fractions would lead to agglomerations of microparticles and scattering-induced loss at high frequencies, as discussed later. However, below this upper limit of V ≈ 30% the characteristics were found to be roughly independent of frequency, and so they are quoted in this work as values averaged over frequency. The refractive index, n, versus volumetric fraction, V, is shown in Figure 2.8 as experimental markers (in red) and a theoretical curve (in black) having been generated by the Bruggeman model and fit in accordance with our prior work39. The experimental and theoretical results show good agreement, at a root-mean-square-error of 0.012, at a depolarization factor of N = 1/8. The fact that this depolarization factor is less than that of a sphere is understandable given lack of symmetry seen for the microparticles in the inset of Figure 2.8. Unlike the refractive index, the extinction coefficient is found to be relatively independent of the volumetric fraction, with a value, k ≈ 0.0025, that is roughly equal to that of the UHMWPE host. This suggests that the microparticle inclusions at low volumetric fractions do not appreciably increase the loss via absorption or scattering. Overall, the results indicate that the microcomposite material can be tailored to form dielectric spheres with the necessary (optimal) refractive indices 25 and (minimal) extinction coefficients for THz microjets. The focusing performance of two spheres fabricated using the microcomposite materials will be discussed in the upcoming section. Figure 2.8 Refractive index of the microcomposite material, n, versus volumetric fraction of SiO2 microparticles inclusions in the UHMWPE host, V. The results are shown as experimental markers, shown in red, and a fitted theoretical curve, shown in black. The inset shows a scanning-electron-microscope image of the microcomposite material with V ≈ 30%. 2.4 Focusing Performance Two forms of dielectric spheres are implemented here and tested as realizations of THz microjets. The first (nominal) dielectric sphere is formed from pure UHMWPE. It is expected to yield THz microjets with maximal intensity gains near the lower end of the targeted 0.3-to-0.7-THz band, given the intensity gains seen in Figure 2.6 for its refractive index of n ≈ 1.52. The second (optimal) dielectric sphere is formed from microcomposite material at a volumetric fraction of V = 30%. It is expected to yield THz microjets with maximal intensity gains near the centre of the targeted 0.3-to-0.7-THz band, given the intensity gains seen in Figure 2.6 for its refractive index of n ≈ 1.65. 26 Both the nominal and optimal dielectric spheres are characterized in terms of their focusing performance. Focusing performance encompasses the metric of focal spot size, d(f, n), as well as the frequency-dependent focal spot positions (whose role will be discussed later). The focusing performance of the 20-mm-diameter dielectric spheres are contrasted to that of a conventional THz lens, as a benchmark. The selected THz lens (BATOP TPX-D25.4-f10) is specifically designed for THz applications applied at fine spatial resolutions. It has a short focal length (at 10 mm), a diameter matching our dielectric spheres (at 20 mm), and a material with low absorptive loss (being polymethylpentene, PMP/TPX). Measurements were made of focal spot sizes using knife-edge scans44 according to the work of Khosrofian et al. In doing so, THz time-domain signals were recorded while a knife-edge was translated through the focal spot of the THz beam. The THz time-domain signals were then Fourier transformed with the amplitude at each frequency fit to a cumulative distribution function. The function was then differentiated to form a field distribution of the Gaussian beam profile from which the FWHM could be extracted. The field distributions at the centre frequency, f = 0.5 THz, are shown versus the transverse dimension for the THz microjets of both the nominal and optimal dielectric spheres in Figure 2.9(a) and Figure 2.9(b), respectively. The theoretical results are displayed as red curves with the experimental results displayed as red markers with error bars denoting standard errors in slope. Overall, the theoretical and experimental results show strong agreement, at a root-mean-square-error of 0.03, with subwavelength focal spot sizes for the nominal and optimal dielectric spheres at d(f = 0.5 THz, n = 1.52) = 514 µm = 0.856l0 and d(f = 0.5 THz, n = 1.65) = 472 µm = 0.786l0, respectively. 27 By measuring the field distributions and subsequent FWHM values across our THz band, the frequency dependence the focal spot sizes could be extracted. The results from such a process are shown in Figure 2.9. The frequency-dependent focal spot size, d(f,n), is plotted versus frequency, f, for the both the nominal and optimal dielectric spheres in Figure 2.9(c) and Figure 2.9(d), respectively. The theoretical results are displayed as red curves, while the experimental results are displayed as red markers with error bars denoting standard errors in slope. For the THz lens, experimental results are displayed as blue markers with error bars. We again see strong agreement between the theoretical and experimental results, at a root-mean-square-error of 0.02 for the nominal and optimal dielectric spheres. We also see that the focal spot sizes occupy the subwavelength regime, shaded in light grey, across the targeted spectrum, although this condition is met to a greater extent by the optimal dielectric sphere. Moreover, the distinction between the dielectric spheres can be appreciated by noting their focal spot sizes at the lowest and highest frequencies. The nominal and optimal dielectric spheres at the lowest frequency give focal spot sizes of d(f = 0.3 THz, n = 1.52) = 765 µm = 0.765l0 and d(f = 0.3 THz, n = 1.65) = 689 µm = 0.689l0, respectively, and at the highest frequency they give d(f = 0.7 THz, n = 1.52) = 417 µm = 0.97l0 and d(f = 0.7 THz, n = 1.65) = 398 µm = 0.931l0, respectively. In comparing these focal spot sizes to those of the THz lens, we see that the THz lens is able to focus to subwavelength spot sizes at lower frequencies but it cannot maintain subwavelength focusing across the entire 0.3-to-0.7 THz band. 28 Figure 2.9 Focusing characteristics of THz microjets. In (a), the field distribution at the centre frequency, f = 0.5 THz, versus the transverse dimension of the nominal (n = 1.5) dielectric sphere is shown. In (b), the field distribution at the centre frequency, f = 0.5 THz, versus the transverse dimension of the optimal (n = 1.65) dielectric sphere is shown. In (c), the focal spot size, d, is plotted versus frequency, f, for the nominal dielectric sphere. In (d), the focal spot size, d, is plotted versus frequency, f, for the optimal dielectric sphere. In (a), (b), (c), and (d), black lines denote theoretical results for the dielectric spheres, red markers / error-bars denote experimental results for the dielectric spheres, and blue markers / error-bars denote experimental results for the (BATOP TPX-D25.4-f10) THz lens. The subwavelength regime is shaded in grey. When characterizing a focusing element, it is also important to note the frequency dependence of the focal spot position. Given the frequency dependence of the focal spot position observed for 29 both the nominal and optimal dielectric spheres, a modified knife-edge scan was preformed to measure the frequency dependence. The modified scan involved translating the knife-edge through the axial dimension of the focus with a THz time-domain signal being recorded at each position. The THz time-domain signals were then Fourier transformed and the amplitude of each frequency was then plotted as a function of axial distance (with the largest amplitude indicating the focal position for that frequency). Figures 2.10(a) and 2.10(b) shows the logarithm of the intensity profile at the central design frequency of 0.5 THz for the nominal and optimal dielectric spheres, respectively. The focal spot positions experimentally obtained for frequencies of 0.3, 0.5, and 0.7 THz are displayed for both the dielectric spheres, as red circles, and the THz lens, as blue circles, with the focal spot positions spanning from 11.9 to 12.9 mm, 10.7 to 11.4 mm, and 9.9 to 10.5 mm for the nominal dielectric sphere, optimal dielectric sphere, and THz lens, respectively. The implications of this frequency dependence on the focal spot position as well as why it arises in dielectric sphere focusing elements will be discussed in the following section. 30 Figure 2.10 Focal span characteristics of THz microjets. The logarithm of the intensity profile at the rear of the nominal (a) and optimal (b) dielectric spheres is plotted versus the transverse and axial dimensions. In (a) and (b), the focal spot positions of the dielectric sphere (as red circles) and the THz lens (as blue circles) are plotted. The lower bound, circle and upper bound represent the focal distances of frequencies of 0.3, 0.5, and 0.7 GHz, respectively. 2.5 Discussion The THz microjets introduced in this chapter are similar in concept to the photonic nanojets realized in the visible/infrared spectrum21–23, in that they realize subwavelength focusing, but there are some key differences to note for THz microjets. For THz microjets, it is typically necessary to focus the THz radiation over a broad spectrum, and in doing so the dielectric spheres must be implemented with a refractive index, n, that is optimal (or near optimal) over the given spectrum. This point is apparent in Figure 2.9 through a comparison of the focal spot sizes of the nominal and optimal dielectric spheres. The nominal dielectric sphere exhibits its maximal intensity gains at the lower end of the spectrum, which gives a b 31 subwavelength focal spot sizes near f = 0.3 THz and weakly-subwavelength focal spot sizes near f = 0.7 THz. In contrast, the optimal dielectric sphere exhibits its maximal intensity gains near the centre of the targeted spectrum, f = 0.5 THz, which gives subwavelength focal spot sizes across the spectrum. For THz microjets, it is also necessary to consider loss. Material loss can arise via absorption, due to the nonnegligible extinction coefficients in the THz spectrum, and this can limit the capture cross-section (diameter) of the dielectric sphere. Thus, the proportional scaling up of diameter with wavelength, as is often done for photonic nanojets, must be done with thought to the rising absorption and lowering intensity gain. Loss can also arise via scattering if microcomposite material is used for the dielectric spheres. Such scattering-induced loss is strongest at high frequencies, due to its scaling with the reciprocal of the wavelength to the fourth power. It is also of greatest concern for microcomposite material near the limiting volumetric fraction, where microparticles begin agglomerating and scattering. We see this loss emerge at the higher frequencies in Figure 2.9(d), whereby the scattering-induced loss for the optimal dielectric sphere leads to spurious experimental results above the targeted band, i.e., f > 0.7 THz. When comparing the THz microjets induced by dielectric spheres to the focusing of a state-of-the-art THz lens, the advantage of a dielectric sphere with an optimized refractive index is very clear. This dielectric sphere achieves subwavelength focusing across the entire band with focal spot sizes well below that of the THz lens. Additionally, by selecting the optimal refractive index, the focal spot position span can be made comparable to that of the state-of-the-art THz lens. Here, the difference in focal spot position across the frequencies can be attributed to the manifestation of 32 chromatic aberration in the THz spectrum. This can be important for a broadband focusing, as minimal chromatic aberration will allow frequencies across the band to achieve their minimal focal spot size at the same distance away from the lens. Overall, in this chapter, the optimized dielectric sphere was shown to have a large capture-cross section, minimal material loss, and subwavelength focal spot size. This resulted in an especially high intensity gain and outperformed a state-of-the-art THz lens. To the best of our knowledge, this optimized dielectric sphere is the foremost choice for broadband focusing across the targeted 0.3-to-0.7-THz band. Based on these findings, its use in subwavelength THz spectroscopy is investigated in the next chapter. 33 Chapter 3: Application of Terahertz Microjets for Terahertz Spectroscopy In this chapter, the theory behind THz spectroscopy will be introduced and some key features will be investigated for a realization of subwavelength THz-TDS by using THz microjets. First, the method used to obtain and calculate material parameters using THz-TDS will be shown. Next, the consequences of performing THz spectroscopy with a highly-focused beam and on samples that are smaller than the focal spot will be discussed. Given these consequences, a suitable method using apertures is introduced that allows for subwavelength THz-TDS. Finally, THz-TDS is performed utilizing a THz microjet with suitable apertures. This is done to determine the minimum sample size whose spectra can be reliably measured across the targeted 0.3-to-0.7-THz band. 3.1 Terahertz Spectroscopy Terahertz time-domain spectroscopy has been used to extract both the refractive index, n(f), and extinction coefficient, k(f), of solid materials such as plastics45, glasses46, and biological samples47. Additionally, THz-TDS has been successfully applied to extract the material parameters of liquids such as water48, bovine serum49 and food products50. As shown by Jepsen and Fischer51, given a sample with a specified thickness, l, the frequency-dependent refractive index is determined by analyzing the difference between the phase of the sample spectrum, ∠Esam(f), and reference spectrum, ∠Eref(f), with the overall expression given by . (3.1) Here, the reference spectrum is captured from a THz-TDS scan with no sample present, whereas the sample spectrum is captured following transmission through the sample (at normal incidence in this work). To extract accurate material parameters of a sample, all interfaces and non-sam ref( ) ( )( ) 12c E f E fn ff lpé ùÐ -Ðë û= +34 negligible loss mechanisms introduced by the sample must be considered. Failing to do so can lead to inaccuracies in the extracted refractive index and/or extinction coefficient. These loss mechanisms include the Fresnel reflections on both the front and rear interfaces of the sample as well as absorptive loss from the material's extinction coefficient. The first Fresnel transmission coefficient, as a ratio of transmitted to incident electric fields at the (first) air-sample interface, can be written as tair-sam(f) = 2/(n+1). Similarly, the second Fresnel transmission coefficient, as a ratio of transmitted to incident electric fields at the (second) sample-air interface, can be written as tsam-air(f) = 2n/(n+1). Combining the transmission coefficients with the loss arising from the frequency-dependent extinction coefficient yields the overall expression for the which can be expressed as . (3.2) Equation 3.2 can be explicitly solved for the extinction coefficient giving the expression . (3.3) By implementing equation 3.1 and 3.3 the refractive index and extinction coefficient can be extracted from a sample with a known thickness. In this work, a test sample was made using lactose powder (Sigma-Aldrich L2643) that was pressed under an applied load of 2000 lbs for 10 minutes to create a 1-mm-thick disk with a diameter of 20 mm. This sample diameter was chosen such that the sample size is an order of magnitude larger than the focal spot size of the THz microjets, which ensured that the entire THz pulse interacted with the sample. The consequence of having sample sizes with diameters smaller than the focal spot size will be discussed in section 3.2.2. Lactose was chosen as a sample material to test the spectroscopic capabilities of THz microjets given its well-studied and unique spectral features in 4 ( ) /sam air-sam sam-air ref| ( ) | ( ) ( ) | ( ) | ef f l cE f t f t f E f p k-=samair-sam sam-air ref| ( ) |1( ) ln( ) ( ) | ( ) | 4E f cft f t f E f f lkpé ù= - ê úë û35 the 0.3-to-0.7-THz band52. Additionally, recent studies have shown how sugars, such as glucose, crystalize as a function of time53. The studies provide insight into the spatial dependence of the crystal formation in sugars where it is hoped that the high intensity gain provided by THz microjets will allow for the crystallinity of sugars such as lactose to be characterized with improved spatial resolution in future work. The improvements in spatial resolution achievable with THz microjets and the implications towards achieving such a goal will be discussed in subsequent sections. The material parameters of a lactose sample, being refractive index and extinction, are highlighted in Figure 3.1. In Figure 3.1(a), a reference pulse, shown in blue, and the sample pulse, shown in red, are plotted versus time. The sample pulse is delayed in time, owing to the increased refractive index of the sample (n > 1) as opposed to the ambient conditions of the reference scan (n @ 1). In Figure 3.1(b), the THz amplitude of both the reference spectrum, shown in blue, and the sample spectrum, shown in red, are plotted versus frequency with the noise floor of the system denoted by the dotted black line. Here, the noise floor is defined as the amplitude at which the signal is no longer detectable given the base noise level in the THz-TDS setup. As shown later in this work, the noise floor is of importance to THz-TDS systems, as the signal amplitude must be greater than the noise floor for the material parameters to be reliably extracted. Note that the sample's signal is decreased in amplitude in both the time and frequency domains because of Fresnel reflections off the interfaces as well as the material loss arising from the frequency-dependent extinction coefficient. 36 Figure 3.1 Terahertz spectroscopy of lactose. In (a), the reference pulse, shown in blue, and the sample pulse, shown in red, are plotted as a function of time. In (b), the THz amplitude of the reference pulse, shown in blue, and the sample pulse, shown in red, are plotted as a function of frequency. The noise floor of the system is plotted as the dashed black line. In (c), the refractive index is plotted as a function of frequency. The blue line corresponds to the experimental data with the red dashed line representing the refractive index calculated through the Kramers-Kronig relations. In (d), the extinction coefficient is plotted as a function of frequency. 37 The measured refractive index and extinction coefficient are plotted as a function of frequency in Figures 3.1(c) and 3.1(d), respectively. The extracted refractive index and extinction values agree well with the literature54, where the prominent spike in the extinction coefficient, centered at 0.53 THz, arises from a collective vibrational transition in the lactose crystal52,55. The large spike in the extinction coefficient leads to the inflection point seen in the refractive index centered at the same 0.53 THz. This relationship between the refractive index and extinction coefficient manifests through the Kramers-Kronig relations, which are commonly expressed in terms of the angular frequency, w = 2pf. The relation for the refractive index is56,57 , (3.4) where denotes the Cauchy principal (integral) value. A full derivation of equation 3.4 can be found in Appendix A. From equation 3.4, it can be seen that the Kramers-Kronig relations ideally require extinction coefficient values at all frequencies. This is especially challenging for THz-TDS measurements due to their limited bandwidth. Given that limited experimental data is inherent to such measurements, Bachrach and Brown58 introduced a singly subtractive Kramers-Kronig relation which uses a fixed reference point, w1, to relate the extinction coefficient and refractive index of a material. This manifests as . (3.5) Further extensions of the Kramers-Kronig relations such as the multiply subtractive method59 use multiple fixed points to further mitigate the limited bandwidth. Fortunately, the singly subtractive method seen in equation 3.5, with its single fixed point, has been successful in characterizing materials with high absorption peaks in the THz regime60. By implementing equation 3.5, we are 2 202 ' ( ')( ) 1 ''n P dw k ww wp w w¥= +-ò(·) 'P dwò2 211 2 2 2 2102( ) ' ( ')( ) ( ) '( ' )( ' )n n P dw w w k ww w wp w w w w¥-= +- -ò38 able to transform the extinction coefficient seen in Figure 3.1(d) to the theoretical refractive index denoted by the red dashed line seen in Figure 3.1(c). Here, a fixed point of w1 = 0.4 THz serves as the reference point. It is worth mentioning a common artefact that can negatively affect THz-TDS systems and achievable scale of subwavelength spectroscopy. The artefacts, arising from reflections of THz pulses, can manifest from anywhere in the experimental setup, including (and especially) the sample. The reflections created in the experimental setup can be seen in Figure 3.1(b) as periodic ringing in the reference scan, which yields prominent peaks at 0.52 and 0.60 THz. Based on the spacing of these peaks in the frequency domain, at roughly 0.08 THz, as well as the ZnTe EO crystal's refractive index, at 3.19561, and thickness, at 500 µm, we can attribute these reflections to this crystal and its cavity round-trip time of (2·3.195·500 µm)/(3×108 m/s) ≈ 0.08 THz. Such reflections occur in both the sample and reference scans and therefore do not affect the extracted material parameters to a large extent. However, as will be shown in upcoming sections, for noise-limited applications such as fine spatial resolution spectroscopy using THz microjets, the frequency nulls of these reflections can play a part in limiting the spectroscopic capabilities. The second type of reflections can arise when performing THz-TDS on thin samples with low extinction. These thin sample reflections can create erroneous results in the extracted material parameters by superimposing the initial pulse with multiple reflections slightly delayed in time by the sample. Various iterative post-processing methods such as those introduced by Lesack et al. have been shown to remove the reflections created by thin samples, allowing for greater accuracy in the material parameters62. Nonetheless, given our relatively large peak in extinction coefficient 39 at 0.53 THz, our extracted material parameters suffer minimally from reflections within the sample. Ultimately, the material parameters extracted in this section, for the refractive index and extinction coefficient, will act as a benchmark for parameters extracted in the upcoming section via tight focusing with THz microjets. 3.2 Terahertz Time Domain Spectroscopy with a Focused Beam As we target THz-TDS on a subwavelength scale, a highly-focused THz beam is needed. This allows a large fraction of the THz beam to interact with the sample. However, the use of a highly-focused THz beam can be at odds with the desire for broad spectral characteristics in THz-TDS applications. Specifically, THz-TDS with a highly-focused THz beam places constraints on both the sample position (relative to the focus) and size (relative to the focal spot size). These constraints and their implications for subwavelength THz-TDS will be discussed in section 3.2.1 and 3.2.2, respectively. 3.2.1 Sample Position Analysis For typical THz-TDS systems, the THz pulse is incident upon the sample while it is collimated, allowing the position of the sample along the optical axis to be of little consequence to the extracted material parameters. However, performing THz-TDS with the highly-focused beam of a THz microjet places constraints on where the sample can be positioned (relative to the focus). This was experimentally tested by performing THz-TDS with a lactose sample at varying positions along the optical axis. The material parameters were extracted at each of these positions. The 40 experimental results can be seen in Figure 3.2 where the extracted refractive index, averaged over 0.60 to 0.65 THz, is plotted as blue circles with error bars. The error bars represent the deviation in refractive index extracted between two consecutive THz-TDS scans of the same sample. Such errors can be attributed to the noise inherent to the experimental setup as well as the variation in sample thickness given the manufacturing procedure. The red dashed line represents the expected refractive index of the lactose sample averaged across the same 0.60 to 0.65 THz bandwidth, as verified by the literature54 and our previous work63. The dark grey line of Figure 3.2 indicates the rear surface of the dielectric sphere with the light grey shaded region indicating the evanescent regime, at a wavelength away from the rear surface of the sphere. By analyzing the trends seen outside of this evanescent regime, it can be seen there is a region of sample positions, denoted by the vertical dashed lines, where the refractive index can be reliably extracted. Here, 'reliably extracted' refers to the region in which a spline-fitted curve through the experimental data is no more than the experimental error (the height of the displayed error bars) away from the expected refractive index. This region of sample positions allows for accurate material parameter extraction as the microjet that is formed is able to remain well focused and collimated for several wavelengths while passing through the sample. (This agrees with the theory of photonic nanojets21,22, in the visible/infrared spectrum, where the nanojets created by dielectric spheres can extend for multiple wavelengths.) Such focusing has the extracted material parameters agree well with the expected values. In contrast, by moving the sample further from the focal spot, out of the “jet”, the THz pulse begins to strike the sample when it is either quickly converging (before the focal spot) or quickly diverging (after the focal spot). In these regimes, the pulse strikes the sample with non-normal incidence, yielding an increased propagation distance though the sample. An increase in propagation distance subsequently increases the delay between the sample and reference pulse, 41 which lowers the measured refractive index of the sample. It is also interesting to note the leftmost data point in Figure 3.2, which was obtained with the sample touching the rear surface of the dielectric sphere. This point deviates from the general trend as it has the sample located within the evanescent region of the dielectric sphere, where reflections and standing waves perturb the beam. From these results, we can conclude that the THz microjet resulting from a dielectric sphere can be used to reliably extract the material parameters of a sample, given its careful positioning within the THz microjet. The forthcoming work in this thesis makes use of this careful positioning. 42 Figure 3.2 Effect of sample position in highly-focused THz-TDS. The refractive index, averaged over 0.60 to 0.65 THz, is plotted as blue markers (with error bars) for varying sample positions relative to the focus. The error bars denote the deviation in extracted refractive index between two subsequent THz-TDS runs. The red dashed line corresponds to the expected refractive index over the averaged frequency range. The dark grey shaded region denotes the rear surface of the sphere with the light grey shading corresponding to the evanescent regime (within one wavelength from surface). 43 3.2.2 Sample Size Analysis As mentioned, typical THZ-TDS involves measuring the THz transmission characteristics for both a reference spectrum and sample spectrum. The vast majority of these THz-TDS studies have large cross-sectional sizes for their samples, through which THz beams can easily pass. However, this work differs from these studies in that spectroscopic spectra are sought for samples with far smaller sizes. Terahertz microjets are used for this as they have focal spot sizes on the scale of hundreds of microns. Further, THz microjets are investigated for use with sample diameters that are smaller than the focal spot size of the THz beam. The two potential cases of THz-beam/sample interactions are shown in Figure 3.3. First, in Figure 3.3(a), the sample diameter is larger than the focal spot size, d, which ensures that all of the incident THz beam interacts with the sample. This is the best-case scenario, but it cannot always be achieved. In Figure 3.3(b), the sample diameter is smaller than the focal spot size, allowing only a fraction of the incident THz beam to interact with the sample. The remainder of the THz beam passes around the sample and does not have spectroscopic characteristics of the sample imparted onto it. The implications of operating in the second case, where the sample diameter is smaller than the focal spot size, will be discussed below. 44 Figure 3.3 Two cases of pulse and sample interactions. In (a), the focal spot size, d, is smaller than the sample diameter. In (b) the focal spot size is larger than the sample meaning a portion of the beam passes through the sample, shown in red, and a portion passes around, not interacting with the sample, shown in blue. The implications of having sample diameters that are smaller than the focal spot sizes were tested in this work via experimental THz-TDS studies of borosilicate glass discs of varying diameter. The THz beams were focused as THz microjets, and the discs were positioned in the focus of the THz microjets. Borosilicate glass was chosen due to its familiarity in spectroscopic studies and its ease of manufacturing. Discs with diameters of 1000, 750, and 500 µm were milled out of a 250-µm-thick borosilicate glass plate (Fisher Scientific 12-540-C). Each disc was used as a sample, centred in the focus of a THz microjet, with the time-domain waveform recorded for the sample spectrum. The time-domain waveform was also recorded without the sample for the reference spectrum. Figure 3.4 shows time-domain waveforms for the reference spectra, in blue, and time-domain waveforms for the sample spectra, in red. 45 Figure 3.4 THz TDS signals for sample sizes below the focal spot size. THz pulses are shown versus time with reference and sample pulses depicted as blue and red lines, respectively. The resulting time domain pulses from a 1000 µm, 750 µm and 500 µm diameter sample placed at the focus of a THz microjet are shown in (a), (b), and (c), respectively. In Figure 3.4(a), for a 1000-µm sample diameter, we can see strong effects from the sample, whereby the sample time-domain waveform is a shifted and scaled version of the reference time-domain waveform. The shifting in time and scaling in amplitude come about from the sample's refractive index and extinction coefficient, respectively. However, for a sample diameter of 750 µm, as seen in Figure 3.4(b), only a fraction of the THz beam passes through the sample. This is due to the sample diameter being notably smaller than the focal spot size. We see this largely as two overlapped THz pulses with little time shifting for one of the pulses. Figure 3.4(c) displays this overlapping to an even greater extent, where it can be inferred that a large fraction of the beam passes around the 500-µm-diameter disc (and is unperturbed). This overlapping of pulses creates artefacts and erroneous results in the spectra extracted for material parameters. As a point of reference, the refractive indices extracted for the sample diameters of 1000, 750, and 500 µm had average percent errors of 1.1%, 8.6%, and 30.1%, respectively, across the targeted 0.3-to-0.7-THz 46 band. These errors are based off of our results from a large borosilicate glass plate. Such a finding highlights the need for great overlap between the THz beam and the sample for effective subwavelength spectroscopy and reliable parameter extraction. In the following sections of this thesis, an alternative method for achieving this overlap, in the form of apertures, will be analysed. 3.2.3 Aperture Analysis Within this section, apertures will be investigated as a means to have the THz beam pass only through the sample. In doing this, it is necessary to quantify the effects that apertures can have on THz microjets and their spectrum. These effects are studied experimentally using apertures with diameters of 1000, 715, 510, 365, 260, 186, 132, and 98 µm. The apertures were laser-milled as circular holes with a tolerance of approximately ±4 µm in 15-µm-thick aluminum foil, as this thickness of aluminum completely blocks the THz radiation64. The overall effects of the THz microjets having passed through these apertures can be visualized in Figures 3.5 and 3.6, which show the aperture characteristics in both the time and frequency domains, respectively. Figure 3.5 shows the THz amplitude from aperture diameters of 1000, 360, and 186 µm, which are displayed as blue, red, and yellow curves, respectively. The THz amplitude is plotted versus time with each time-domain signal normalized to the results from the 1000 µm diameter aperture. Similar to that seen in the literature64 the decreasing aperture diameter decreases both the signal amplitude and acts as a high-pass filter. The high-pass filter effect can be seen by shrinking of the pulse duration in time as the aperture size is decreased. This is a result of the inherently large spot sizes of the lower frequencies impeding the ability for the low THz frequencies to be transmitted through the aperture. Conversely, higher frequencies and therefore smaller focal spot sizes allow for easier transmission of the higher frequency THz waves through the aperture. This frequency-dependent 47 effect can be seen explicitly in Figure 3.6, which maps out the normalized THz amplitudes in colour versus the frequency, f, and aperture diameter. The colourmap spans low signal transmission (in black) to moderate signal transmission (in red) to high signal transmission (in white) with the THz amplitude at each frequency being normalized to that of the 1000 µm diameter aperture. The high-pass filter effect can be seen by noting the large black band, which indicates weak signal transmission through the aperture. At low frequencies, the black band is large, indicating weak signal transmission through apertures up to roughly 400 µm in diameter. The black band then tapers off at lower diameters for increasing frequency. This illustrates how high frequencies are better able to achieve smaller spot sizes and higher signal transmission through smaller apertures. Figure 3.6 also highlights how the introduction of apertures decreases the transmitted and detectable signal at every frequency. This decrease is especially relevant for the long wavelengths and large focal spot sizes associated with our targeted 0.3-to-0.7-THz band, as it imposes a limit on the spatial resolution of THz-TDS with THz microjets. Additionally, the introduction of a sample for THz-TDS, and its associated reflections and material absorption also act to decrease the signal amplitude. These additional losses further limit the spatial resolution achievable. Quantifying the spatial resolution of THz-TDS of a sample using an optimized THz microjet and appropriate apertures will be examined in the next section. 48 Figure 3.5 Aperture effect on a focused THz beam. Results for aperture diameters of 1000, 360, and 186 µm are shown in blue, red and yellow, respectively. The THz amplitude is plotted versus time with each time-domain signal normalized to the time-domain signal of the 1000 µm diameter aperture. 49 Figure 3.6 Broadband aperture characteristics. The THz amplitude is shown in colour, with its colourmap spanning low (in black) to moderate (in red) to high (in white), versus the frequency, f, and aperture diameter. 3.3 Spatially Constricted Spectroscopy with THz microjets In this section, the spatial resolution of THz-TDS achievable using THz microjets will be tested. Building upon the previous analysis, apertures will be used to ensure that the acquired signals will have passed through (and interacted with) the sample. Apertures ranging in diameter from 1000 µm to 132 µm were placed in the THz microjet, i.e., within the focus protruding from the rear surface of the dielectric sphere. Based upon the transmission characteristics studied in section 3.2.3 the spatial position of the apertures was optimized for maximum signal transmission. For each of the diameters tested, a reference scan was taken followed by sample scans with a 600-µm-thick 50 lactose disc. The lactose sample was manufactured under identical conditions to those in section 3.1. The extracted material parameters, being refractive index, n, and extinction coefficient, k, are plotted as a function of frequency in Figures 3.7 and 3.8, respectively. Here, frequencies just outside of our targeted 0.3-to-0.7-THz band are included to further highlight the impacts of apertures on material parameter extraction in THz-TDS measurements. The material parameters for no aperture and aperture diameters of 1000, 715, 510, 365, 260, 186, and 132 µm are depicted by the solid black, dark-blue, orange, yellow, purple, green, light-blue, and red lines, respectively. The solid circles depict the cutoff frequency for each aperture, which is defined as the lowest frequency at which the material parameters can be extracted to within 5.0% deviation from the case with no aperture. The cutoff frequency identifier of 5.0% was selected as it was the maximal error in the extracted refractive index seen between two consecutive scans with no aperture present. Frequencies below each cutoff frequency are depicted by dashed lines. The cutoff frequencies are listed in Table 3.1 for the varying aperture diameters along with the cutoff resolution, which is calculated as the wavelength of the cutoff frequency divided by the aperture diameter. As will be discussed later, the cutoff resolution denotes the best spatial resolution possible for a given frequency. 51 Table 3.1 Cutoff frequencies for diminishing aperture sizes Aperture diameter (µm) Cutoff frequency (THz) Cutoff resolution 1000 ~ ~ 715 ~ ~ 510 0.32 0.547l0 365 0.35 0.429l0 260 0.43 0.375l0 186 0.60 0.374l0 132 0.79 0.365l0 The high-pass effects of apertures in spatially constricting THz-TDS measurements can be seen in Table 3.1 by the increasing cutoff frequencies for decreasing aperture sizes. Again, this high-pass effect arises from higher frequencies having smaller focal spot sizes and a better ability to pass through the aperture, as compared to the low frequencies. For the larger diameter apertures of 1000 µm and 715 µm, we are able to extract accurate material parameters across the entire 0.3-to-0.7-THz band, signifying that there is no cutoff frequency for these apertures. However, for the aperture diameters of 510 µm and 365 µm, there arises cutoff frequencies of 0.32 THz and 0.35 THz, respectively, which encroach into the 0.3-to-0.7-THz band. However, the strong peak characterizing lactose, centered near 0.53 THz, can still be reliably extracted for both of these aperture diameters. Further decreasing the aperture diameter down to 260 µm pushes the cutoff frequency up to 0.43 THz, which then begins to encroach upon the peak. As the aperture size is 52 further decreased, to diameters of 186 µm and 132 µm, the cutoff frequencies are pushed up even higher, to 0.60 THz and 0.79 THz, respectively. This further emphasizes the importance of how seeking better spatial resolution using apertures decreases the resolvable bandwidth achievable. At the same time, it is worth noting the impact that the apertures have on the frequencies above the cutoff frequency in relation to THz-TDS. Radiation at the frequencies above the cutoff frequency can be focused through the aperture, for extraction of material parameters, but the aperture still decreases the signal strength. This decreasing signal strength can be seen in Figures 3.7 and 3.8 by way of the increasing fluctuations in material parameters at frequencies above the cutoff frequency. These increasing fluctuations above the cutoff frequency are amplified for decreasing aperture diameters, given the restriction of signals passing through, until the aperture diameter of 132 µm. This is the smallest aperture diameter from which a spectroscopic signal could be extracted in this work. The 132 µm aperture diameter has a cutoff frequency outside of our targeted 0.3-to-0.7-THz band, but it is included here to help emphasize the trends. 53 Figure 3.7 Refractive index of lactose for decreasing aperture sizes. The extracted refractive index is shown versus frequency for no aperture and aperture diameters of 1000, 715, 510, 365, 260, 186, and 132 µm. The dashed lines show frequencies where the material parameter cannot be reliably extracted with the solid circles indicating the cutoff frequencies for the aperture diameters. 54 Figure 3.8 Extinction coefficient of lactose for diminishing aperture sizes. The extracted extinction coefficient is shown versus frequency for no aperture and aperture diameters of 1000, 715, 510, 365, 260, 186, and 132 µm. The dashed lines show frequencies where the material parameter cannot be reliably extracted with the solid circles indicating the cutoff frequencies for the aperture diameters. We can also compare the cutoff spatial resolution, which defines the best spatial resolution achievable for a given frequency (before the material parameters are no longer reliably resolved). 55 For the aperture diameters of 510, 365, 260, 186, and 132 µm, the cutoff spatial resolutions are 0.547l0, 0.429l0, 0.375l0, 0.374l0, and 0.365l0, which occur at frequencies of 0.32, 0.35, 0.43, 0.60, and 0.79 THz, respectively. Under ideal conditions, with equal THz emission amplitude across all frequencies, we would expect to achieve similar cutoff spatial resolutions for all of the frequencies. The higher frequencies exhibit this as consistent cutoff resolutions of (0.370 ± 0.05)l0. However, the lower frequencies deviate from this trend. The deviation can be attributed to the lower THz amplitude emitted at low frequencies by the GaAs photoconductive antenna used in this work. This decreases the signal amplitude at the lower frequencies, which then worsens both the cutoff frequency and the cutoff spatial resolution. Ultimately, this chapter shows that THz-TDS can be implemented with THz microjets given sufficient attention to the (related) spectral and spatial characteristics. By correctly analyzing and implementing apertures within a THz microjet, it was shown that spectroscopic signals could be acquired with subwavelength spatial resolution across the entire 0.3-to-0.7-THz band. This was evidenced by the extraction of material parameters at cutoff spatial resolutions down to approximately 0.37l0, which to the best of our knowledge is the finest spatial resolution for far-field THz-TDS achieved to date. Concluding comments for these THz microjets are given in next chapter. 56 Chapter 4: Conclusion The invention of THz-TDS has prompted significant research on the use of THz radiation for non-invasive characterizations of samples—and particularly biological samples. Nonetheless, the long wavelengths inherent to THz radiation, have led to large spot sizes and have hindered the technology. Hindrances are especially prevalent at the lower THz frequencies, in the 0.3-to-0.7-THz band targeted in this work. This thesis has responded to this challenge and introduced a focusing element that is capable of broadband THz-TDS on a subwavelength scale. Dielectric spheres were introduced as a method to focus the THz radiation into an aptly-named THz microjet. These THz microjets were then optimized over the targeted 0.3-to-0.7-THz band and implemented with apertures for the realization of THz-TDS on a subwavelength scale. 4.1 Summary of Conclusions The core of this thesis was divided into two main chapters, with sections introducing factors related to the creation of THz microjets and their implementation in subwavelength spectroscopy across the targeted 0.3-to-0.7-THz band. Chapter 2 introduced the methodology, optimization, and characterization of THz microjets. Chapter 3 investigated the spectroscopic capabilities of these THz microjets. The conclusions from these chapters are given here. Chapter 2 and its subsections showed how the THz emission scheme along with the dielectric sphere size and material parameters affect the formation of THz microjets. A method to create optimized THz microjets was implemented with the resulting microjets being characterized and compared to a state-of-the-art THz lens. This led to the following major conclusions: 57 • Both photoconductive antennas and electro-optic crystals can be used to generate the THz radiation needed for creating THz microjets. However, across the targeted 0.3-to-0.7-THz band a photoconductive antenna with a gap of 200 µm fabricated on SI-GaAs is the preferred emitter. This is because it achieves higher THz signal strength than a 500-µm-thick [110] ZnTe EO crystal. • Three metrics, being capture cross-section, material loss, and focal spot size, define the performance of the investigated focusing elements. These three metrics combine to define an overarching metric of intensity gain for dielectric spheres, whereby selecting a refractive index of n = 1.65 and diameter of d = 20 mm gave a maximal intensity gain (at an extinction coefficient of k = 0.0025). • Microcomposite materials can be used to create THz materials with relatively low material loss. By varying the inclusions of SiO2 microparticles in a host material of UHMWPE, it is possible to create microcomposite materials with a refractive index that can be varied from n = 1.52 to 1.65 while achieving minimal material loss (at k = 0.0025). • An optimized dielectric sphere (n = 1.65) is able to form THz microjets that have subwavelength focal spot sizes across the entire 0.3-to-0.7-THz band and are smaller than those of a nominal (n = 1.52) dielectric sphere. The decrease in focal spot size can be attributed to the increased refractive index, which pulls the focus closer to the sphere and allows evanescent waves to constrict the focal spot. Such dielectric spheres can form THz microjets with focal spot sizes that are smaller than those of state-of-the-art THz lenses. These results show promise for implementations of dielectric spheres in THz-TDS setups. 58 Chapter 3 and its subsections showed the procedure and results of using THz microjets for subwavelength spectroscopy. The implications of using these highly-focused THz beams for spectroscopy were discussed, with consideration given to the use of apertures. This led to the following major conclusions: • Preforming THz-TDS with a highly-focused THz beam requires careful consideration to the position of the sample relative to the focus. If the sample is placed outside of the focus, deviations in the results can arise. Given the “jet” like forms of the THz microjets, which exist for several wavelengths, a sample can be placed up to roughly 1 mm from the focus while having the material parameters reliably extracted. • Despite the tight focusing of THz microjets, cases can still arise where the sample diameter is smaller than that of the focal spot size. This can cause erroneous spectroscopic results as a fraction of the THz beam will pass unperturbed around the sample. Apertures can be implemented in this regard to ensure the measured THz beam has passed through the sample. However, in doing this, it is necessary to recognize that the apertures act as high-pass filters and decrease the signal strength. • By implementing apertures at the focus of a THz microjet, the refractive index and extinction coefficient of a sample can be reliably obtained with a subwavelength spatial resolution. While the acquisition can be source limited at low frequencies, i.e., less than 0.4 THz, it is possible to witness a consistent cutoff spatial resolution near 0.37l0 at higher frequencies. 59 4.2 Limitations This section gives a brief overview of the limitations of THz microjets. The first major limitation is the focal spot size achieved and the subsequent spatial resolution that is possible with the THz microjets tested in this work. Although a cutoff spatial resolution down to 0.37l0 was achieved in this work, this only corresponds to spatial resolutions of roughly 370 µm to 160 µm across the targeted 0.3-to-0.7-THz band. Thus, biological applications of THz-TDS on the cellular scale are not yet possible. The second limitation of this work is the high scattering loss exhibited by the microcomposite material at frequencies near and above 0.7 THz. This loss, which is attributed to the agglomeration of SiO2 microparticles at high volumetric fractions, limits the potential of fully optimized (n = 1.65) THz microjets over broader bandwidths. 4.3 Future work There are two main extensions of this work that could greatly benefit and improve upon the THz microjet technology presented here. These extensions, being improved microcomposite materials and near-field THz imaging, are discussed here. In order to create THz microjets that are optimized across broader bandwidths, it is necessary to reoptimize the dielectric spheres. However, this requires forethought. The shift to higher frequencies could be done by using the microcomposite material applied in this work with a decreased diameter for the dielectric sphere, but this would shrink the capture cross-section. Alternatively, the shift to higher frequencies could be done by using the applied diameter of the dielectric sphere with its refractive index increased. However, doing so would require a new 60 material for the microparticles, with a higher refractive index. This is because it would become necessary to avoid the agglomerating and scattering seen in this work at high volumetric fractions. The next extension of this work would be to implement it in the improvement THz near-field imaging techniques. Most near-field imaging systems use conventional THz lenses to focus incident radiation onto the sample, which leads to relatively weak evanescent waves protruding from the sample. 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For the electric susceptibility, , its frequency response can be written, assuming a linear time-invariant and causal system, through the Fourier transform as , (A.1) where u(t) is the Heaviside step function. This expression can be written through the convolution theorem as , (A.2) where is the dirac-delta function. Performing the convolution yields , (A.3) which can then be solved for the susceptibility as (A.4) and . (A.5) These integral bounds in equation A.5 can be modified by first splitting the integral, to give (A.6) ( )e tc( ) 1 ( ) ( )2j te e t u t e dtwc w cp¥--¥= ò( )1 1( ) ( )2e e jc w c w pd wp wæ öæ ö= * +ç ÷ç ÷è øè ø( )d w( ')1 1( ) ' ( ') ( ')( '2 ( ') 2ee ed djc wc w w pc w w w wp w w p¥ ¥-¥ -¥= + ¶ --ò ò( ')1( ) ' ( )2 ( ')ee edjc wc w w pc wp w w¥-¥= +-ò( ')1( ) '( ')ee djc wc w wp w w¥-¥=-ò00( ') ( ')1 1( ) ' '( ') ( ')e ee d dj jc w c wc w w wp w w p w w¥-¥= +- -ò ò69 and then applying the relation to equation A.6 yields . (A.7) The electric susceptibility can then be re-written in terms of its real, , and imaginary,, parts to give . (A.8) The real and imaginary parts of equation A.8 can then be related giving the well-known Kramers-Kronig relations: (A.9) and . (A.10) The refractive index and extinction coefficient of a material are related to the electric susceptibility through the relation , which for small susceptibilities can be expanded through the binomial approximation as . Equating the real and imaginary components then gives and . By substituting the above expressions into equations A.9 and A.10 the refractive index and extinction coefficient can be related by the expressions *( ') ( ')e ec w c w- =*2 20( ')( ') ( ')( ')1( ) '( ' )e ee djc w w w c w w wc w wp w w¥ + + -=-ò( )Reec w( )Imec w( ) ( )( )( )( )( )ReReIm2 2 2 2Im0 0'' '2 2' '' 'eej d j de ewc ww c wc w c w w wp pw w w w¥ ¥+ = -- -ò ò( )( )( )ReIm2 20' '2 ''e dew c wc w wp w w¥=-ò( )( )( )Re2 2Im0'2 ''e dewc wc w wp w w¥= --ò( )( ) ( ) 1n j ew k w c w+ = +Re Im( ) ( ) 12 2e en j jc cw k w+ = + +Re( ) 12encw = + Im( )2eck w =70 (A.11) and . (A.12) ( ) ( )( )2 20' '21 ''n dw k ww wp w w¥= +-ò( ) ( )( )2 20' 12 ''nk dwww wp w w¥ -= --ò
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Holzman, Jonathan
Terahertz microjets for spectroscopy on the subwavelength scale
Text