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Metal waveguides for multi-axial light guiding at nanometer scales Maqsood, Muhammad Waqas 2011

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Metal Waveguides for Multi-Axial Light Guiding at Nanometer Scales by Muhammad Waqas Maqsood B.Sc., University of Engineering and Technology, 2008 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) December 2011 c Muhammad Waqas Maqsood 2011Abstract Recent advancements in nanofabrication now allow precise fabrication of devices and systems on nanometer scales. This technology is currently used in the  eld of photonics to construct optical systems possessing sub- wavelength features. A basic component of most optical systems is an optical waveguide. There has been an increased interest in nanofabricated opti- cal waveguides that incorporate metal layers due to their fabrication com- patibility with existing complimentary-metal-oxide-semiconductor (CMOS) processes and, as will be discussed in this thesis, their ability to sustain sub-wavelength-con ned electromagnetic modes. In this work, we have de- veloped analytical techniques for designing metal waveguides that achieve tailored optical functionalities. The developed techniques are applied in two design examples which address contemporary problems related to waveguid- ing at sub-wavelength and nanometer scales. Realization of complex optical circuits based on miniaturized optical waveguides requires components that can bend light around tight 90 bends. In the  rst design example, we apply our analytical technique to opti- mize a bi-axial waveguide constructed from two uni-axial metal waveguides joined together at 90 . The optimization procedure consists of mapping out wavevector values of the electromagnetic modes sustained by the two waveg- iiAbstract uides over the intended operational frequency range. The constituent mate- rials and geometry of the waveguides are selected such that each waveguide sustains only one low-loss mode. The geometry of each of the waveguides is tailored in such a way that the in-plane wavevector components for both waveguide modes are matched. The wavevector matching results in e cient coupling between the two modes, yielding 90 light bending with predicted e ciencies over 90%. In the second design example, we apply our analytical technique to opti- mize a bi-axial waveguide structure for coupling free-space light into surface plasmon polaritons (SPP), electromagnetic excitations bounded to the sur- face of a metal. One of the practical challenges in realizing devices that use SPPs is the development of e cient ways to couple in free-space plane-wave light. We study the simple SPP coupling geometry consisting of a slit in a metal  lm,  lled and covered with a dielectric. We break the con guration down into two constituent uni-axial waveguide components, modeling the slit as a metal-dielectric-metal waveguide and the adjacent metal surface as a metal-dielectric waveguide. Using similar analysis as in the  rst example, we optimize the materials and geometry of the slit so that wavevector match- ing is achieved between the light emanating from the slit and the adjacent SPP modes, resulting in predicted peak SPP coupling e ciencies over 68%. iiiPreface This work has been done under the guidance of Dr. Kenneth Chau at the School of Engineering in The University of British Columbia. My colleague Reyad Mehfuz was involved with the  nite-di erence time-domain technique (FDTD) calculations. Throughout the course of this work, Dr. Chau has played a key role in providing guidance and support. Portions of my thesis have been published in two journal articles:  M. W. Maqsood, R. Mehfuz, and K. J. Chau, \High-throughput di rac- tion assisted surface-plasmon-polariton coupling by a super-wavelength slit," Optics Express 18, 21669-21677 (2010).  M. W. Maqsood, R. Mehfuz, and K. J. Chau, \Design and optimization of a high-e ciency nanoscale  90 light-bending structure by mode selection and tailoring," Applied Physics Letters 97, 151111 (2010), A portion of the thesis has also been published in a conference proceeding:  M. W. Maqsood, K. J. Chau, \Designing a Nanometer-scale Light Bending Structure," Proceedings of SPIE (6pp), in press (2011). ivTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Designing Uni- and Multi-Axial Optical Waveguides . . . . . 4 1.3 Analytical Approach for Designing Multi-Axial Metallic Wave- guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 8 vTable of Contents 1.5.1 Constitutive Material Relations . . . . . . . . . . . . 9 1.5.2 Electromagnetic Wave Equation . . . . . . . . . . . . 10 1.6 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Limitations of the Drude Model . . . . . . . . . . . . . . . . 16 1.8 Dielectric Function for Silver . . . . . . . . . . . . . . . . . . 21 1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Metallic Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Lay- ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers . . . . . . . . . . . . . . . . . . . . . 33 2.4 Davidenko Method . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Multi-Axial Nanoscale Light Bending . . . . . . . . . . . . . 46 3.1 Material and Geometry Selection . . . . . . . . . . . . . . . . 48 3.2 Mode Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Modi ed Bi-Axial Waveguide Structure . . . . . . . . . . . . 56 3.4 Wavevector Matching . . . . . . . . . . . . . . . . . . . . . . 60 3.5 Waveguide Structure Optimization . . . . . . . . . . . . . . . 65 3.5.1 FDTD Simulation . . . . . . . . . . . . . . . . . . . . 65 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 viTable of Contents 4 Multi-Axial Surface-Plasmon-Polariton Coupling . . . . . . 73 4.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . 83 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Appendix: Fit Functions . . . . . . . . . . . . . . . . . . . . . . . 102 viiList of Tables 1.1 Constant parameters for the Drude model for Ag and Au . . 16 viiiList of Figures 1.1 The Drude model for metals . . . . . . . . . . . . . . . . . . . 13 1.2 Comparison between real permittivity values for gold obtained indirectly from the Drude model using experimentally-measured !p and  (line) with those obtained directly from permittivity measurements (squares). . . . . . . . . . . . . . . . . . . . . . 17 1.3 Comparison between imaginary permittivity values for gold obtained indirectly from the Drude model using experimentally- measured !p and  (line) with those obtained directly from permittivity measurements (squares). . . . . . . . . . . . . . . 18 1.4 Comparison between real permittivity values for silver ob- tained indirectly from the Drude model using experimentally- measured !p and  (line) with those obtained directly from permittivity measurements (squares). . . . . . . . . . . . . . . 19 1.5 Comparison between imaginary permittivity values for silver obtained indirectly from the Drude model using experimentally- measured !p and  (line) with those obtained directly from permittivity measurements (squares). . . . . . . . . . . . . . . 20 ixList of Figures 1.6 Polynomial  t function to model the real part of the dielectric function of silver. . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Polynomial  t function to model the imaginary part of the dielectric function of silver. . . . . . . . . . . . . . . . . . . . 23 2.1 Single interface formed between a metal and dielectric. . . . . 27 2.2 Geometry of a three-layered structure consisting of metallic cladding layers sandwiching a dielectric core. . . . . . . . . . 31 2.3 Uni-axial waveguide composed of an arbitrary number of metal- dielectric layers stacked along the z direction. . . . . . . . . . 35 3.1 Waveguide structure consisting of a silver slit  lled with GaP, with the complete structure covered with a semi-in nite GaP layer. GaP is chosen due to its higher refractive index of n = 3:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Dispersion curves for the asymmetric SPP modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency for various dielectric core thickness values. . . . . . 51 3.3 Dispersion curves for the symmetric SPP modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency for various dielectric core thickness values. . . . . . 53 xList of Figures 3.4 Dispersion curves for the TM1 modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM), as a function of frequency for vari- ous dielectric core thickness values. . . . . . . . . . . . . . . . 54 3.5 Dispersion curves for the SPP mode sustained by a Ag-GaP interface. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency. . . . . . . . . . . . . 57 3.6 Waveguide structure consisting of a slit in a silver  lm where the slit is  lled with GaP and the entire silver  lm is coated with a GaP layer of  nite thickness. The structure is im- mersed in air. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Modal solution for an MDD (Ag-GaP-Air) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency for the SPP mode. c) Real part of the wavevector and d)  gure of merit (FOM) as a function of frequency for the TM1 mode. . . . . . . . . . . . . . . . . . . 59 3.8 A simple example demonstrating the re ection loss due to wavevector mismatch. An electromagnetic plane wave is a) incident onto a dielectric interface and b) scattered into re-  ected and transmitted components after interacting with the interface. The relative amplitudes of the re ected and trans- mitted components are proportional to the degree of wavevec- tor mismatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 xiList of Figures 3.9 Wavevector matching applied to a bi-axial waveguide. Wavevec- tor matching is achieved when the transverse component of the wavevector in the MDM waveguide matches with the lon- gitudinal component of the wavevector in the MDD waveguide. 63 3.10 Plot of the longitudinal wavevector components of the TM1 mode sustained by the MDD waveguide ( 2) and the trans- verse wavevector components of the TM1 mode sustained by the MDM waveguide ( 1z). The blue circle shows the match- ing point at the operational frequency of w = 6 1014 Hz. . . 64 3.11 Simulation geometry of the waveguide structure designed to bend incident light at a frequency ! = 6 1014 Hz. The struc- ture has a metal thickness t = 300nm, slit width w = 150nm and dielectric cap d = 100nm. The detectors D1 and D2 mea- sure the time-averaged magnetic  eld intensity,jHyj2, of the TM1 mode in the GaP layer, and the detector D3 measures jHyj2 radiated into the air region. . . . . . . . . . . . . . . . . 68 3.12 FDTD calculated electromagnetic response of the designed structure at di erent incident electromagnetic wave frequencies 69 3.13 Coupling e ciency as a function of GaP layer thickness d. The frequency of the incident light is kept constant at ! = 6 1014 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.14 Coupling e ciency as a function of  . The slit width and layer thickness are kept constant at w = 150 nm and d = 100 nm, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 71 xiiList of Figures 4.1 Waveguide structure consisting of a slit in a metal  lm im- mersed in a dielectric. TM-polarized light is normally incident from the bottom of the structure and is con ned within the dielectric core. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Formulation of a hypothesis for di raction-assisted SPP cou- pling by a super-wavelength slit aperture. a) Figure-of-merit and b) the real transverse wavevector component versus fre- quency and wavelength for TM0 and TM1 modes sustained in slits of di erent widths. c) Di raction spectrum correspond- ing to the TM0 mode in a 200- nm-wide slit and the TM1 modes in 350- nm-wide and 500- nm-wide slits. d) Wavevector- space depiction of di raction-assisted SPP coupling from slits of width w = 200 nm, w = 350 nm, and w = 500 nm, im- mersed in a uniform dielectric of refractive index n = 1:75. . . 80 4.3 Images of the FDTD-calculated instantaneous jHyj2 distri- bution (left) and the time-averaged jHyj2 angular distribu- tion (right) for a slit of width values a) w = 200 nm, b) w = 350 nm, and c) w = 500 nm immersed in a dielectric (n = 1:75) and illuminated by a quasi-plane-wave of wave- length  0 = 500 nm. A common saturated color scale has been used to accentuate the  elds on the exit side of the slit. 84 xiiiList of Figures 4.4 a) SPP coupling e ciency as a function of optical slit width for dielectric refractive index values n = 1:0 (squares), n = 1:5 (circles), n = 1:75 (upright triangles), n = 2:0 (inverted tri- angles), n = 2:5 (diamonds). b) The measured SPP intensity (squares), radiative intensity (circles), and total intensity (di- amonds). The shaded region indicates the sub-wavelength- slit-width regime. . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Wavevector mismatch Re[ ~ spp] (Re[ ~ z]+ p) as a function of refractive index of the dielectric region for a  xed optical slit width nw = 600 nm and free-space wavelength  0 = 500 nm. . 88 xivAcknowledgements I would like to thank my supervisor, Dr. Kenneth Chau, for introducing me to the interesting topics of metal waveguides and surface waves. He has been a great person and a source of inspiration for me. His guidance and encouragement throughout my thesis were invaluable. I am also thankful to Dr. Thomas Johnson who served as an examination committee member for my thesis. The discussions I had with him were very useful in shaping my masters. I would also like to thank my colleague, Reyad Mehfuz, for all of his support. His work using FDTD technique has been very helpful and played an important role in development of this work. At the end, I would like to thank my family members. Their support throughout this time has been remarkable and has been instrumental in shaping my life and career. xvTo my family for their support and for all those who seek truth xviChapter 1 Introduction 1.1 Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures An optical waveguide is a device that can guide light. This is achieved by con ning light within a boundary such that light can only propagate along one or multiple directions de ned by the waveguide [1]. The  rst optical waveguide was demonstrated in 1840 by Daniel Colladon using a thin curved stream of falling water [2]. A beam of light shone into the water stream was shown to follow the curvature of the stream. The stream of water thus acted like a pipe for light. In 1870, John Tyndall explained this phenomenon in terms of total internal re ection (TIR) [3]. TIR occurs at an interface between two media having di erent optical refractive indices. Consider an interface formed between dielectric medium 1 with refractive index n1 and dielectric medium 2 with refractive index n2. A beam of light incident from medium 1 into medium 2 will obey Snell’s law, written as n1sin i = n2sin r; (1.1) 11.1. Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures where 0   i  90 is the angle of incidence of the beam (with respect to normal to the interface) and  r is the angle of refraction of the beam refracted into medium 2. When n1 is greater than n2, there exists an acute critical angle of incidence where the refracted beam becomes parallel to the interface. For angles of incidence greater than the critical angle, light no longer refracts, but is totally internally re ected. This forms the basis of optical waveguiding using dielectric materials. Since the 1970s, developments in glass processing technology have al- lowed the controlled manufacturing of optical  bre waveguides constructed from thin glass wires. An optical  ber consists of two concentric glass regions: a cylindrical core of higher refractive index and a surrounding cladding of lower refractive index. The optical  bre works by guiding light along the axis of the  bre and con ning light within the core by total inter- nal re ection. Fiber optics have revolutionized the  eld of communications by providing a means to transfer information with low loss, high speed, wide bandwidth, and immunity to electromagnetic interference, which has trans- lated into tangible bene ts such as fast internet speeds and instantaneous communication across continents. Along with faster information transmission via optical  bres, the last half century has witnessed dramatic miniaturization of electronic compo- nents, resulting in smaller devices with more functionalities. The modern cellular phone, for example, has more processing power than the original room-sized computers. This miniaturization has been possible due to the development of nanofabrication processes enabling controllable manufactur- ing on nanometer-size scales. In the last decades, there has been intense 21.1. Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures research interest in using nanofabrication to miniaturize optical waveguides. The motivation for these e orts has been two-fold:  rst, smaller optical waveguides means that more functionalities can be compressed into an op- tical system [4{9] and second, optical waveguides that are size-compatible with smaller electronic components could pave the way to opto-electronic systems that exploit both the advantages of electronics and optics, which could lead to signi cantly faster and more powerful computers [10{13]. A major road-block in e orts to miniaturize optical waveguides is the so-called \di raction limit". This term was originally used to describe the minimum resolution that can be achieved by an optical imaging system (on the order of the wavelength of light). The same physical constraints placed on the resolution of optical imaging systems also applies to most optical waveguiding systems. Generally, light cannot propagate in a dielectric- based waveguide system when the transverse pro le of the waveguide is less than the wavelength of light. At infrared and visible frequencies, this means that the minimum transverse pro le size of an optical waveguide is on the order of 100s of nanometers or even microns, which is several orders of magnitude greater than the typical feature sizes of electronic com- ponents [11, 14, 15]. This constraint can be overcome by using optical waveguides constructed from metals. A waveguide created using metallic components can support light waves that propagate even when the trans- verse pro le of the waveguide is signi cantly less than the wavelength of light [16{19]. In recent years, there has been tremendous research inter- est in the development of nanometer-scale optical waveguides constructed from metallic components. Metallic waveguiding systems have been realized 31.2. Designing Uni- and Multi-Axial Optical Waveguides in the form metal nanoparticle arrays [20], metal nanowires [21{24], and nano-apertures in metal  lms [14, 25{27]. Research continues to understand light interactions with these small metallic systems and to optimize their performance. 1.2 Designing Uni- and Multi-Axial Optical Waveguides With the advent of nanofabrication, optical waveguides can now be created in a wide range of geometries using dielectric, metallic, or semi- conducting materials. Given the many degrees of freedom a orded by nanofab- rication, optimization of a waveguide design requires e cient modeling tech- niques. All modeling schemes are, in some way or another, rooted in Maxwell’s equations, a set of four equations describing the relationship between elec- tromagnetic  elds and their sources. The way in which Maxwell’s equations are applied to model a given optical waveguide depends in large measure on the geometry of the waveguide. One of the simplest con gurations is the uni-axial waveguide, which guides electromagnetic waves along a single axial direction. The electromagnetic properties of a uni-axial waveguide can be modeled by analytically solving Maxwell’s equations. This is achieved by assuming a general form of the electromagnetic  elds propagating along the direction of the waveguide. Applying boundary conditions to the interfaces of the waveguide yields characteristic eigenvalue equations having solutions corresponding to the wavevector of a wave. This wavevector contains infor- mation on the wavelength and attenuation of the wave. When a uni-axial 41.2. Designing Uni- and Multi-Axial Optical Waveguides waveguide is composed of dielectric materials described by real permittivity values, the resulting eigenvalue equation is real and can be easily solved using standard root- nding algorithms like the Newton-Raphson method. When the uni-axial waveguide is composed of metallic materials described by complex permittivity values, the resulting eigenvalue equation is complex and requires more sophisticated root- nding algorithms. Multi-axial waveguides guide electromagnetic waves along more than one direction. They are challenging to model by analytical solutions to Maxwell’s equations and often require numerical techniques. One of the most popular numerical techniques is the  nite-di erence time-domain (FDTD) technique. The FDTD technique is based on a one-, two-, or three-dimensional spatial grid containing the waveguide structure and an electromagnetic wave source. Approximating the time- and space- derivatives of Maxwell’s equations using di erence equations, Maxwell’s equations are solved to determine the elec- tric and magnetic  eld values in the simulation space at each grid point as a function of time. Numerical simulation tools like the FDTD technique are powerful because they enable detailed visualization of electromagnetic  elds within complicated optical waveguides. One of the major limitations, how- ever, is the massive computational power and processing time required to complete a simulation. Because only one combination of parameters can be explored for a given simulation, hundreds or thousands of simulations are required to completely map out the frequency-dependent electromagnetic response of an optical waveguide for various material combinations and ge- ometrical con gurations. As a result, optimization using only numerical simulations is often not feasible. 51.3. Analytical Approach for Designing Multi-Axial Metallic Wave-guides 1.3 Analytical Approach for Designing Multi-Axial Metallic Waveguides The goal of this thesis is to explore a new analytical method to model the electromagnetic properties of multi-axial, metallic, nanometer-scale waveg- uides. The methodology is based on conceptually dividing a multi-axial waveguide into uni-axial waveguide sub-components. By approximating the more-complex multi-axial waveguide as a collection of simpler uni-axial waveguide parts, analytical solutions can be obtained to describe electro- magnetic wave behaviour in di erent regions of the multi-axial waveguide. To achieve electromagnetic coupling between uni-axial waveguide compo- nents, the geometries of the uni-axial waveguide components are tailored so that the in-plane electromagnetic momenta are matched. This is achieved by analytically mapping out the electromagnetic wavevectors of the uni-axial waveguide sub-components as a function of both frequency and geometry and then tailoring the geometries so that the magnitudes of the wavevectors are equal. We test the predictive power of this approach in two design exam- ples in which visible-frequency electromagnetic waves are guided by bi-axial, metallic, nanometer-scale waveguides along two directions. Optimal param- eters selected to yield maximum coupling using our analytical method are shown to match those predicted to yield maximum coupling using FDTD numerical simulations. Our analytical method thus provides a supplemental, high-level tool for the design of complicated waveguide systems and provides new physical insights into multi-axial electromagnetic waveguiding. 61.4. Thesis Outline 1.4 Thesis Outline This thesis is divided into two parts. The  rst part, consisting of Chap- ters 1 and 2, develops the theoretical background required for understanding the design technique and the second part, consisting of Chapters 3 and 4, describe the two basic examples where the technique is applied. In the remainder of Chapter 1, we will introduce classical electromag- netic theory used in the thesis. Chapter 2 describes the background and theory of metallic waveguides. We formulate the dispersion relations de- scribing electromagnetic modes in metallic waveguides and discuss the nu- merical techniques used to  nd their roots. In Chapters 3 and 4, we dis- cuss the application of a new analytical method to model bi-axial, metallic, nanoscale waveguides based on breaking down the bi-axial waveguide into two uni-axial waveguide sub-components. Predictions based on this method are compared with predictions using well-accepted FDTD simulations. The thesis concludes in Chapter 5. Appendix A lists a series of analytical func- tions that are  tted to the wavevector of electromagnetic modes in various metallic waveguide structures. The intention is to provide the interested reader with easy-to-evaluate functions and to circumvent the need to solve the complex dispersion relations, which can be tedious and time-consuming. 71.5. Maxwell’s Equations 1.5 Maxwell’s Equations In 1873, J. C. Maxwell developed the fundamental equations describing the behavior of electromagnetic waves [28]. The equations relate four vector  elds - the displacement  eld, D, the electric  eld, E, the magnetic  ux density, B, and the magnetic  eld, H - to the presence of charge density,  , and current density, J. The equations are given by r  D(r; t) =  ; (1.2) r  B(r; t) = 0; (1.3) r  E(r; t) =  @B(r; t) @t ; (1.4) and r  H(r; t) = J(r; t) + @D(r; t) @t ; (1.5) where (r,t) describes the position and time dependence of the  elds. The displacement  eld, D is related to electric  eld, E and magnetic  ux density  eld, B is related to magnetic  eld, H via D(r; t) =  oE(r; t) + P(r; t) (1.6) and B(r; t) =  oH(r; t) + M(r; t); (1.7) where  o is the free-space permittivity,  o is the free-space permeability, P is the polarization  eld and M is the magnetization  eld. 81.5. Maxwell’s Equations 1.5.1 Constitutive Material Relations The  elds P, M, and J describe the response of a medium to electro- magnetic  elds. By assuming a medium that is linear, isotropic, lossless, and instantaneously responsive, we get the relations P(r; t) =  o eE(r; t); (1.8) M(r; t) =  o mH(r; t); (1.9) and J(r; t) =  E(r; t); (1.10) where  e is the electric susceptibility,  m is the magnetic susceptibility, and  is the electric conductivity. Inserting Eq. 1.8 into Eq. 1.6, we get D(r; t) =  o rE(r; t) (1.11) where  r is the relative electric permittivity given by  r = 1 +  e: (1.12) Similarly, inserting Eq. 1.9 into Eq. 1.7, we get B(r; t) =  o rH(r; t); (1.13) 91.5. Maxwell’s Equations where  r is relative magnetic permeability given by  r = 1 +  m: (1.14) For most materials,  m ’ 0 and  r = 1. 1.5.2 Electromagnetic Wave Equation We generally deal with systems having no external charge ( = 0). Im- posing the additional constraint that J = 0, Maxwell’s equations for such systems can be written as r  D(r; t) = 0; (1.15) r  B(r; t) = 0; (1.16) r  E(r; t) =  @B(r; t) @t ; (1.17) and r  H(r; t) = @D(r; t) @t : (1.18) Inserting the constitutive relation of Eq. 1.11 in Eq. 1.15 we get: r  ( o rE(r; t)) = 0;  o r(r  E(r; t)) + E(r; t)  r( o r) = 0; which yields r  E(r; t) =  E(r; t)  r( o r)  o r : (1.19) 101.5. Maxwell’s Equations For an isotropic and homogeneous medium, r( o r) = 0, which leads to the compact relation r  E(r; t) = 0: (1.20) Using Eq. 1.13 with  r = 1 (for nonmagnetic materials), Eq. 1.17 be- comes r  E(r; t) =   o @H(r; t) @t : (1.21) Taking the curl of both sides yields r  r  E(r; t) = r     o @H(r; t) @t  =   o @ @t (r  H(r; t)): (1.22) Using Eq. 1.18, Eq. 1.22 simpli es to r  r  E(r; t) =   o @2D(r; t) @t2 =   o o r @2E(r; t) @t2 : (1.23) The left side of Eq. 1.23 simpli es to r  r  E(r; t) = r(r  E(r; t)) r2E(r; t) (1.24) Using Eq. 1.20 in Eq. 1.24 yields r  r  E(r; t) =  r2E(r; t): (1.25) 111.5. Maxwell’s Equations Combining Eq. 1.23 and Eq. 1.25, we get r2E(r; t)  o o r @2E(r; t) @t2 = 0: (1.26) Given the speed of light de ned by c = 1 p  o o ; (1.27) we arrive at the general wave equation r2E(r; t)  r c2 @2E(r; t) @t2 = 0; (1.28) which is also known as the Helmholtz wave equation. Using a similar treat- ment for the magnetic  eld yields r2H(r; t)  r c2 @2H(r; t) @t2 = 0: (1.29) The above equations describe the electric and magnetic  eld component of an electromagnetic wave as it travels in a medium. Both equations will be used for analyzing the electromagnetic wave behaviour in a medium. 121.6. Drude Model Figure 1.1: Cartoon depiction of a piece of metal. The magni ed part of the metal shows its microscopic constituents consisting of free electrons (red dots) moving around  xed positive ions (blue circles). 1.6 Drude Model In 1900, Paul Drude explained the transport of electrons in metals by applying the principles of kinetic theory [29, 30]. In this treatment, metals consist of a density of free electrons,  n, immersed in a collection of positive ions, as shown in Fig. 1.1. The electrons are unbound and free to move, surrounded by background ions without any restoring force. When an external oscillating electromagnetic  eld is applied to the metal, electrons start oscillating. These oscillations are damped by electron colli- sions with other electrons and ions. The damping rate is approximately 1= , where  is the average time between collisions [31]. To derive the equation of motion for an electron inside the metal, let’s consider an electromagnetic wave varying in the x-axis and uniform along all other directions such that 131.6. Drude Model E(r; t) = E(x; t)x^. Treating a free electron like a classical particle, we have the equation of motion [32] m d2~‘ dt2 + m  d~‘ dt =  eE(x; t); (1.30) where ~‘, m, and e are the displacement, e ective mass, and charge of an electron, respectively. Generally, the electric  eld is time-harmonic with the form ~E(!) = E0(x)e  i!t; (1.31) where ! is the frequency. Time-harmonic excitation yields time-harmonic electron motion given by ~‘(!) = ~‘oe  i!t; (1.32) where ~‘o is a complex quantity whose magnitude is equal to the peak dis- placement and whose phase describes temporal shifts between the electron displacement and the driving  eld [33, 34]. Inserting Eq. 1.32 into Eq. 1.31 yields m( !2~‘oe  i!t) + m  ( i!~‘oe  i!t) =  e ~E(!)  m!~‘oe  i!t  ! + i   =  e ~E(!) which results in the complex amplitude ~‘(!) = e ~E(!) m!(! + i= ) : (1.33) 141.6. Drude Model The displacement of individual electrons contribute towards a dipole mo- ment density. This dipole moment density yields a polarization  eld given by ~P (!) =   ne~‘(!) =   ne2 m!(! + i= ) ~E(!): (1.34) In frequency-domain notation, the polarization  eld is related to the electric  eld by ~P (!) =  o(~ m(!) 1) ~E(!); (1.35) where ~ m is the frequency-dependent relative electric permittivity of the metal. Comparing Eqs. 1.34 and 1.35, we obtain the dielectric function ~ m(!) = 1  ne2 m! o(! + i= ) : (1.36) Given the plasma frequency for a free electron gas !2p =  ne 2=m o, ~ m is then given by ~ m(!) = 1 !2p !(! + i= ) : (1.37) Re-writing Eq. 1.37 into a distinctive real part,  0(!), and imaginary part,  00(!), yields ~ m(!) = 1 !2p (!2 + 1= 2) + i !2p !(1 + !2 2) : (1.38) 151.7. Limitations of the Drude Model 1.7 Limitations of the Drude Model In this section, we examine the accuracy of the Drude model for de- scribing the dielectric functions of noble metals over visible frequencies. We select gold and silver because they both have low losses at visible frequencies (4:5 1014 8 1014 Hz) and are widely used in optical waveguides. We use experimentally-measured values of the plasma frequency and collision rates for gold and silver measured by Zeman and Sachts in [35] and summarized in Table 1.1. The experimentally-measured parameters are then inserted into the analytical function given in Eq.1.38. The resulting real and imaginary components of the dielectric function are compared to complex permittivity values experimentally measured by Johnson and Christy [36]. Table 1.1: Constant parameters for the Drude model for Ag and Au Metal Plasma frequency !p (Hz) Collision rate   1 (s 1) Silver 2:186 1015 5:139 1012 Gold 2:15 1015 17:14 1012 As shown in Fig. 1.2, values of Re[ m] for gold obtained from the Drude model closely match the experimental data over a frequency range spanning from the infrared to the ultraviolet. On the other hand, values of Im[ m] match the experimental data only at lower frequencies below 4:5 1014 Hz (Fig. 1.3). Deviations of the Drude model from experimentally-measured 161.7. Limitations of the Drude Model Figure 1.2: Comparison between Re[ m] values for gold obtained indirectly from the Drude model using experimentally-measured !p and  (line) with those obtained directly from permittivity measurements (squares). 171.7. Limitations of the Drude Model Figure 1.3: Comparison between Im[ m] values for gold obtained from indi- rectly from the Drude model using experimentally-measured !p and  (line) with those obtained directly from permittivity measurements (squares). 181.7. Limitations of the Drude Model Figure 1.4: Comparison between Re[ m] values for silver obtained indirectly from the Drude model using experimentally-measured !p and  (line) with those obtained directly from permittivity measurements (squares). values at higher frequencies are attributed to inter-band transitions which are not accounted for in the Drude model. Application of the Drude model to silver yields similar trends to those observed for gold. While there is a good agreement with experimentally- measured values of Re[ m] throughout the frequency range (Fig. 1.4), agree- ment with experimentally-measured values of Im[ m] is achieved only at lower frequencies below 4:0  1014 Hz (Fig. 1.5). Agreement between the Drude model predictions and experimental data for both gold and silver 191.7. Limitations of the Drude Model Figure 1.5: Comparison between Im[ m] values for silver obtained indirectly from the Drude model using experimentally-measured !p and  (line) with those obtained directly from permittivity measurements (squares). suggests that, at lower frequencies, the free-electron picture is an accurate description of the microscopic motion of electrons. At higher frequencies, modi cations to the Drude model are required to accurately describe inter- band transitions. These modi cations can be implemented, for instance, by restricting the free-electron approximation and modeling electrons as bound, resonant entities [37]. The imaginary part of a medium’s dielectric function describes the at- tenuation of an electromagnetic wave propagating in the medium. Silver is an ideal metal for optical waveguiding because its Im[ m] at visible fre- 201.8. Dielectric Function for Silver quencies is among the lowest of all metals [38, 39] and, as a result, exhibits the lowest visible-frequency losses. Throughout the remainder of the thesis, we will restrict our treatment of metallic waveguide systems by considering only those consisting of silver. 1.8 Dielectric Function for Silver We describe the complex dielectric function of silver by high-order poly- nomial  ts to the Johnson and Christy experimental data. The order of the polynomial is chosen such that the di erence between the  t and experimen- tal data is minimum. The polynomial  t for the real part of the dielectric function is obtained as Re[~ m(f)] =  958:72693 + (9:24279 10  12) f  (3:99243 10 26) f2 + (9:69973 10 41) f3  (1:43038 10 55) f4 + (1:30694 10 70) f5  (7:22877 10 86) f6 + (2:21643 10 101) f7  (2:88985 10 117) f8 (1.39) where f is the frequency in units of Hz. As shown in Fig. 1.6, the polyno- mial function given in Eq. 1.39 accurately models experimentally-measured values of Re[ m], with only slight deviations at higher frequencies above the visible spectrum. To obtain a polynomial  t for the imaginary part of the dielectric function of silver, we divide the visible frequency spectrum into 211.8. Dielectric Function for Silver Figure 1.6: Polynomial  t function to model the real part of the dielectric function of silver. 221.8. Dielectric Function for Silver Figure 1.7: Polynomial  t function to model the imaginary part of the dielectric function of silver. three intervals. For 1:54683 1014 Hz< f < 9:35347 1014 Hz, the  t function is Im[~ m(f)] = 48:47405 (5:22332 10  13) f + (2:29072 10 27) f2 (5:18924 10 42) f3 + (6:41697 10 57) f4  (4:12193 10 72) f5 + (1:07857 10 87) f6; (1.40) 231.8. Dielectric Function for Silver for 9:35347 1014 Hz f < 9:35347 1014 Hz, the  t function is Im[~ m(f)] = 7:36947 (7:29005 10  15) f + (5:92 10 30) f2  (2:024 10 45) f3; (1.41) and for f  9:35347 1014 Hz, the  t function is Im[~ m(f)] =  145:44771 + (2:71872 10  13) f  (1:23802 10 28) f2: (1.42) As shown in Fig. 1.7, the piece-wise polynomial  t function accurately mod- els experimentally-measured values of Im[ m]. 241.9. Summary 1.9 Summary In this chapter, we have introduced classical electromagnetic theory in the form of Maxwell’s equations and derived the general electromagnetic wave equations. We have discussed the application of the Drude model for describing the complex dielectric function of noble metals and shown that at visible frequencies, the Drude model yields inaccurate results. To de- scribe the dielectric function of silver (which will be used throughout the remainder of this thesis), we have introduced analytical high-order polyno- mial functions  tted to experimentally-measured data. In the next chapter, we will apply Maxwell’s equations to model the electromagnetic properties of uni-axial waveguides constructed from metallic constituents. 25Chapter 2 Metallic Waveguides 2.1 Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface A uni-axial waveguide is a structure that guides electromagnetic waves along one direction. One of the simplest ways to implement a uni-axial waveguide using metallic media is in the form of a single metal-dielectric interface. In this section, we will analyze the electromagnetic waves that can be sustained at a metal-dielectric interface by  rst assuming a general form of the electromagnetic  elds. Application of boundary conditions to the waveguide will then yield a characteristic eigenvalue equation from which possible wavevector values, ~ , of the electromagnetic waves are mapped as a function of !. The ~ -! relationship is known as the dispersion relation of the waveguide. To analyze a single metal-dielectric uni-axial waveguide, we consider the geometry of a semi-in nite metal occupying the region z < 0 and a semi- in nite dielectric occupying the region z > 0 as shown in Fig. 2.1. The metal is characterized by a local complex relative permittivity ~ 2, and the dielectric is characterized by a local generally-complex relative permittivity ~ 1. We 262.1. Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface Metal Dielectric z x Figure 2.1: Single interface formed between a metal and dielectric. assume a general form of the electromagnetic  elds consisting of a transverse- magnetic (TM) polarized electromagnetic wave propagating along the +x- direction, which is given by ~Hy(z) = 8 >< >: ~H1ei ~ xe~k1z z < 0 ~H2ei ~ xe ~k2z z > 0; (2.1) ~Ex(z) = 8 >< >:  i~k1 ! o~ 1 ~H1ei ~ xe~k1z z < 0 i~k2 ! o~ 2 ~H2ei ~ xe ~k2z z > 0; (2.2) and ~Ez(z) = 8 >< >:  ~ ! o~ 1 ~H1ei ~ xe~k1z z < 0  ~ ! o~ 2 ~H2ei ~ xe ~k2z z > 0: (2.3) where ~ki(i = 1; 2) correspond to the exponential decay constants along the z-axis and ~ is the wavevector. Wavevector of an electromagnetic wave 272.1. Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface traveling with velocity, v and radial frequency, ! is de ned as ~ = ! v : (2.4) The decay length of the electromagnetic  elds from the interface is given by ld = 1 j~kij : (2.5) Continuity of the tangential component of the electric  eld at z = 0 yields  ~k1 ~H1 ~ 1 = ~k2 ~H2 ~ 2 : (2.6) Given continuity of the magnetic  eld at z = 0, we arrive at ~k1 ~k2 =  ~ 1 ~ 2 : (2.7) Application of the general electromagnetic  elds to the general wave equa- tion yields ~k21 = ~ 2  k20~ 1 (2.8) and ~k22 = ~ 2  k20~ 2; (2.9) where k0 is the free space wavevector. Applying Eqs. 2.8 and 2.9 to Eq. 2.7, we obtain the dispersion relation for a single metal-dielectric interface ~ = k0 s ~ 1~ 2 ~ 1 + ~ 2 : (2.10) 282.1. Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface It is interesting to note that transverse electric (TE) polarized electro- magnetic waves cannot be sustained at a single metal-dielectric interface. This can be shown using by  rst assuming a general form of TE-polarized electromagnetic  elds given by ~Ey(z) = 8 >< >: ~E1ei ~ xe~k1z z < 0 ~E2ei ~ xe ~k2z z > 0; (2.11) ~Hx(z) = 8 >< >: i~k1 ! o ~E1ei ~ xe~k1z z < 0  i~k2 ! o ~E2ei ~ xe ~k2z z > 0; (2.12) and ~Hz(z) = 8 >< >: ~ ! o ~E1ei ~ xe~k1z z < 0 ~ ! o ~E2ei ~ xe ~k2z z > 0: (2.13) Continuity of ~Ey and ~Hx gives ~E1 = ~E2 (2.14) and ~E1(~k1 + ~k2) = 0: (2.15) For ~k1; ~k2 > 0 (required for  nite-energy solutions), ~E1 = ~E2 = 0 is the only possible solution. 292.2. Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Layers 2.2 Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Layers We next consider a more complex uni-axial metallic waveguide consists of two metal-dielectric interfaces, formed by a dielectric layer sandwiched between two metal layers (forming a metal-dielectric-metal (MDM) waveg- uide). We will analyze the electromagnetic waves that can be sustained by MDM waveguides using analytical solutions to Maxwell’s equations. Similar to the analysis used in the previous section, we assume a general form of the electromagnetic  elds and apply boundary conditions to the  elds to derive an eigenvalue equation. Consider the geometry shown in Fig. 2.2 consisting of a lower metallic region occupying z <  d, a dielectric region occupying  d < z < d, and an upper metallic region occupying z > d. The lower metallic cladding layer is characterized by a local complex relative permittivity ~ 1, the dielectric is characterized by a local real relative permittivity  3, and the upper metallic cladding layer is characterized by a local complex relative permittivity ~ 2. We assume a general exponentially decaying form of the electromagnetic  elds consisting of a transverse-magnetic (TM) polarized electromagnetic wave propagating along the +x-direction, which is given by ~Hy(z) = 8 >>>>>< >>>>>: ~H1ei ~ xe~k1z z <  d ~H31ei ~ xe~k3z + ~H32ei ~ xe ~k3z  d < z < d ~H2ei ~ xe ~k2z z > d: (2.16) 302.2. Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Layers x z d - d Metal Dielectric Metal Figure 2.2: Geometry of a three-layered structure consisting of metallic cladding layers sandwiching a dielectric core. ~Ex(z) = 8 >>>>>>>< >>>>>>>:  i ! o~ 1 ~k1 ~H1e i~ xe ~k1z z <  d  iei~ x~k3 ! o 3 ~H31e ~k3z +  iei~ x~k3 ! o 3 ~H32e  ~k3z  d < z < d i ! o~ 2 ~k2 ~H2e i~ xe ~k2z z > d: (2.17) and ~Ez(z) = 8 >>>>>>>< >>>>>>>:  ~ ! o~ 1 ~H1e i~ xe ~k1z z <  d  ~ ! o 3 ~H31e i~ xe ~k3z +  ~ ! o 3 ~H32e i~ xe ~k3z  d < z < d  ~ ! o~ 2 ~H2e i~ xe ~k2z z > d: (2.18) Continuity of ~Hy and ~Ex at the boundaries z = d and z =  d yields the 312.2. Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Layers relations ~H1e  ~k1d = ~H31e  ~k3d + ~H32e ~k3d (2.19)  ~k1 ~ 1 ~H1e  ~k1d =  ~k3  3 ~H31e  ~k3d + ~k3  3 ~H32e ~k3d; (2.20) and ~H2e  ~k2d = ~H31e ~k3d + ~H32e  ~k3d (2.21) ~k2 ~ 2 ~H2e  ~k2d =  ~k3  3 ~H31e ~k3d + ~k3  3 ~H32e  ~k3d: (2.22) Application of the generalized electromagnetic  elds into the wave equation Eq. 1.29 and free space wavevector relation, k0 = ! p  0 0 yields the relations ~k21 = ~ 2  k20~ 1; (2.23) ~k22 = ~ 2  k20~ 2; (2.24) and ~k23 = ~ 3  k20 3: (2.25) Using Eqs.2.19, 2.21, 2.23, 2.24, and 2.25, we obtain the dispersion relation e4 ~k3d = ~k3= 3  ~k1=~ 1 ~k3= 3 + ~k1=~ 1 ~k3= 3  ~k2=~ 2 ~k3= 3 + ~k2=~ 2 : (2.26) Eq. 2.26 is a general relation and can be used for any three-layered waveguide in which ~ 1 6= ~ 2 6=  3. For a symmetric structure with cladding layers 322.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers constructed using same material, ~ 1 = ~ 2 and ~k1 = ~k2, Eq.2.26 simpli es to e4 ~k3d =  ~k3= 3  ~k1=~ 1 ~k3= 3 + ~k1=~ 1 !2 : (2.27) Taking square root of both sides of Eq. 2.27 yields two families of solutions, one of which is symmetric and the other asymmetric. It can be shown that the negative solution of Eq. 2.27 gives the symmetric dispersion relation e2 ~k3d =  ~k3= 3  ~k1=~ 1 ~k3= 3 + ~k1=~ 1 ; (2.28) which can be re-expressed in the simple relation coth(~k3d) =  ~k1 3 ~k3~ 1 : (2.29) The asymmetric dispersion relation takes the form e2 ~k3d = ~k3= 3  ~k1=~ 1 ~k3= 3 + ~k1=~ 1 ; (2.30) or more simply coth(~k3d) =  ~k3~ 1 ~k1 3 : (2.31) 2.3 Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers In this section, we derive a general dispersion relation for a uni-axial waveguide composed of an arbitrary number of metal-dielectric layers using 332.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers the transfer matrix method [40{42]. The description presented here follows similar to that of Verhaugen [42]. Consider the geometry shown in Fig 2.3. The geometry consists of n alternating layers of metal and dielectric. The general form of the magnetic  elds in the bottom layer (z < (z2 = 0)) and in the uppermost layer (z > zn) is assumed as ~H1y = ~H1e i~ xe ~k1z; (2.32) and ~Hny = ~Hne i~ xe ~kn(zn z); (2.33) respectively, where ~H1 is the magnetic  eld at z = z2 and ~Hn is the magnetic  eld at z = zn. Within an arbitrary layer j, the magnetic  eld is given by ~Hjy = ~Hj(z)e i~ x: (2.34) Applying the general magnetic  elds into the electromagnetic wave equation Eq. 1.29, we obtain following relation @2 ~Hj(z) @z2 + (~k2o~ j  ~ 2) ~Hj(z) = 0: (2.35) The solution for the above equation is ~Hj(z) = ~Cj sin(~kj(zj  z)) + ~Dj cos(~kj(zj  z)) (2.36) where ~kj = q ~ 2  k2o~ j (2.37) 342.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers z x e e e e e e e zn zn-1 zn-2 zj+1 zj zj-1 z4 z3 z 2 = 0 1 2 3 j-1 j n-1 n d j Figure 2.3: Uni-axial waveguide composed of an arbitrary number of metal- dielectric layers stacked along the z direction. Di erentiating Eq. 2.36 with respect to z yields d dz ~Hj(z) =  ~Cj~kj cos(~kj(zj  z)) + ~Dj~kj sin(~kj(zj  z)): (2.38) To determine the values of ~Cj and ~Dj , we evaluate the magnetic  eld and its di erential at z = zj to yield ~Dj = ~Hj(zj) (2.39) ~Cj = 1 ~kj d dz ~Hj(zj): (2.40) 352.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers Inserting ~Cj and ~Dj into Eq. 2.36 and Eq. 2.38 yields ~Hj(z) = 1 ~kj d dz ~Hj(zj) sin(~kj(zj  z)) + ~Hj(zj) cos(~kj(zj  z)) (2.41) d dz ~Hj(z) =  d dz ~Hj(zj) cos(~kj(zj  z)) + ~Hj(zj)~kj sin(~kj(zj  z)): (2.42) In matrix form, we can write Eq. 2.41 2 6 4 ~Hj(z) d dz ~Hj(z) 3 7 5 = 2 6 4 cos(~kj(zj  z)) 1~kj sin(~kj(zj  z)) ~kj sin(~kj(zj  z))  cos(~kj(zj  z)) 3 7 5 2 6 4 ~Hj(zj) d dz ~Hj(zj) 3 7 5 : (2.43) The corresponding electric  elds are given by ~Ejx =  i ! o~ j ei ~ x d dz ~Hj(z) (2.44) ~Ejz =  ~ ! o~ j ei ~ x ~Hj(z): (2.45) Applying boundary conditions for the tangential components of the elec- tric and magnetic  eld at the arbitrary interface z = zj yields ~Hj(zj) = ~Hj 1(zj) (2.46) and ~Ejx = ~E(j 1)x 362.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers 1 ~ j d dz ~Hj(zj) = 1 ~ j 1 d dz ~Hj(zj 1) d dz ~Hj(zj) = ~ j ~ j 1 d dz ~Hj 1(zj); (2.47) which can be re-expressed in matrix form as 2 6 4 ~Hj(zj) d dz ~Hj(zj) 3 7 5 = 2 6 4 1 0 0 ~ j~ j 1 3 7 5 2 6 4 ~Hj 1(zj) d dz ~Hj 1(zj) 3 7 5 : (2.48) At the interface z = zj+1, from Eq. 2.43 2 6 4 ~Hj(zj+1) d dz ~Hj(zj+1) 3 7 5 = 2 6 4 cos(~kj(dj)) 1~kj sin(~kj(dj)) ~kj sin(~kj(dj))  cos(~kj(dj)) 3 7 5 2 6 4 ~Hj(zj) d dz ~Hj(zj) 3 7 5 where dj = zj+1  zj is the thickness of the jth layer. Putting values from Eq. 2.48 into the above equation 2 6 4 ~Hj(zj+1) d dz ~Hj(zj+1) 3 7 5 = ~Tj 2 6 4 ~Hj 1(zj) d dz ~Hj 1(zj) 3 7 5 where ~Tj = 2 6 4 cos(~kj(dj)) 1~kj sin(~kj(dj)) ~kj sin(~kj(dj))  cos(~kj(dj)) 3 7 5 2 6 4 1 0 0 ~ j~ j 1 3 7 5 : (2.49) The magnetic  eld in layer 1 is given by Eq. 2.32. Using Eq. 2.48, we can 372.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers relate the magnetic  eld at interface z3 with the  eld in layer 1 by 2 6 4 ~H2(z3) d dz ~H2(z3) 3 7 5 = ~T2 2 6 4 ~H1 k1 ~H1 3 7 5 : (2.50) Similarly, the  eld at interface z3 can be related to the  eld at interface z2 by 2 6 4 ~H3(z4) d dz ~H3(z4) 3 7 5 = ~T3 2 6 4 ~H2(z3) d dz ~H2(z3) 3 7 5 : From above two relations, we can relate the  eld in layer 3 at interface z4 with the  eld in layer 1 by 2 6 4 ~H3(z4) d dz ~H3(z4) 3 7 5 = ~T3 ~T2 2 6 4 ~H1(z2) ~k1H1(z2) 3 7 5 : (2.51) In general, we can relate the  eld in layer j  1 at interface zj with the  eld in layer 1 by 2 6 4 ~Hj 1(zj) d dz ~Hj 1(zj) 3 7 5 = ~Tj 1 ~Tj 2::: ~T2 2 6 4 ~H1(z2) ~k1 ~H1(z2) 3 7 5 : (2.52) Further, we can relate the  eld in layer j at interface zj with the  eld in layer 1 using relation 2.48 by 2 6 4 ~Hj(zj) d dz ~Hj(zj) 3 7 5 = ~Vj ~Tj 1 ~Tj 2::: ~T2 2 6 4 ~H1(z2) ~k1 ~H1(z2) 3 7 5 (2.53) 382.4. Davidenko Method where ~Vj = 2 6 4 1 0 0 ~ j~ j 1 3 7 5 : We now de ne the quantity ~Wj = ~Vj ~Tj 1 ~Tj 2::: ~T2 = 2 6 4 ~W11 ~W12 ~W21 ~W22 3 7 5 : We can now relate the  elds at interface z2 = 0 with  elds at interface z = zn by 2 6 4 ~Hn(zn)  ~kn ~Hn(zn) 3 7 5 = ~Wj 2 6 4 ~H1(0) ~k1 ~H1(0) 3 7 5 : (2.54) The above matrix can be resolved into the simpli ed equation ~W11 + ~W12~k1 + ~W21 ~kn + ~W22 ~k1 kn = 0: (2.55) Equation 2.55 is a general dispersion relation for a multi-layered uni-axial waveguide structure. The solutions to Eq. 2.55 are complex wavevector values ~ . 2.4 Davidenko Method The analytically-derived dispersion relations obtained in the previous sections for various types of uni-axial metallic waveguides contain the neces- sary information to understand the  ow of electromagnetic waves within the 392.4. Davidenko Method waveguides. The complexity of the dispersion relation depends in large part on the complexity of the waveguide geometry. The dispersion relation for a single metal-dielectric interface is an explicit expression, but the dispersion relation for multi-layered structures is an implicit, complex, transcendental equation. In this section, we introduce a robust iterative method to solve complex transcendental equations [43]. The method is commonly ascribed to Davidenko [44{46]. The Davidenko method is widely used for solving transcendental equa- tions that cannot otherwise be solved using traditional methods such as Newton’s method and quasi-Newton methods [47]. The Davidenko method can be understood by  rst considering the equation f(x) = 0; (2.56) which has a solution determined by the Newton’s method as xn+1 = xn  f(xn) df(xn)=dx : (2.57) We can modify Eq. 2.57 to yield df(xn) dx = 0 f(xn) xn+1  xn : (2.58) Eq. 2.58 describes a straight line in which the next iteration point xn+1 lies at the intersection of this straight line with tangent to the function f(xn). This restricts the initial guess to lie in an area that is su ciently near the actual solution. This condition can be relaxed by inserting a factor, t, such 402.4. Davidenko Method that df(xn) dx =  t 0 f(xn) xn+1  xn (2.59) where 0 < t < 1. Equation 2.59 can be rearranged, using  xn = xn+1 xn, to give  xn t =  f(xn) df(xn)=dx : (2.60) In the limit t! 0,  xn transforms into a di erential. We can assume that f(x) remains continuous in the small area near xn, yielding dx dt =  f(x) df=dx =  1 df dx  1 f(x)  =  1 d(ln(f(x)))=dx : (2.61) The solution of Eq. 2.61 is given by f(x) = Ce t(x); (2.62) where C is the integration constant. Equation 2.62 suggests that f(x) = 0 is a limiting case when t ! 1. The di erential equation can also be transformed in terms of a Jacobian operator J as dx dt =  J 1f(x): (2.63) This relation is the basic form of Davidenko method. We can use the Davi- denko method to solve the dispersion relation for various uni-axial metallic 412.4. Davidenko Method waveguides. We can write the general dispersion relation as F (!; ~ ) = 0; (2.64) where the wavevector ~ is a complex quantity and can be written in its real and imaginary components as  = a+ ib: (2.65) In other words, we can write the dispersion relation 2.64 as F (!; a; b) = 0: (2.66) We can resolve Eq. 2.66 into two sets of equations given by R(!; a; b) = Re[F (!; a; b)] = 0 (2.67) I(!; a; b) = Im[F (!; a; b)] = 0: (2.68) Because the dispersion relation is a complex analytical function, we can express the Jacobian operator and its inverse in closed form. The dispersion relation then satis es the Cauchy-Riemann relations @R @a = @I @b Ra = Ib (2.69) 422.4. Davidenko Method and @R @b =  @I @a Rb =  Ia: (2.70) From the property of analytical functions, we have F = @F @ = Ra + jIa = Ra  jRb; (2.71) which yields Ra = @R @a = Re[F ] (2.72) Rb = @R @b =  Im[F ]: (2.73) The Jacobian matrix in this case will be J = 2 6 4 Ra Rb Ia Ib 3 7 5 : (2.74) Using Eqs. 2.69 and 2.70, the Jacobian matrix becomes J = 2 6 4 Ra Rb  Rb Ra 3 7 5 : (2.75) To obtain the inverse Jacobian matrix, we  rst get the determinant of 432.4. Davidenko Method Eq. 2.75 by detJ = R2a + R 2 b = jF j 2; (2.76) which yields the inverse Jacobian matrix J 1 = 1 detJ 2 6 4 Ra  Rb Rb Ra 3 7 5 = 1 jF j2 2 6 4 Re[F ] Im[F ]  Im[F ] Re[F ]: 3 7 5 : (2.77) Now we can write the Davidenko equation for the general dispersion relation as d dt 2 6 4 a b 3 7 5 =  J 1 2 6 4 R I 3 7 5 : Putting the value of the inverse Jacobian matrix yields d dt 2 6 4 a b 3 7 5 =  1 jF~ j 2 2 6 4 Re[F~ ] Im[F~ ]  Im[F~ ] Re[F~ ]: 3 7 5 2 6 4 R I 3 7 5 : (2.78) From Eq. 2.78, we can obtain separate real, ordinary di erential equations for the real and imaginary parts of the wavevector da dt =  1 jF~ j 2 (Re[F ]Re[F~ ] + Im[F ]Im[F~ ]); (2.79) and db dt = 1 jF~ j 2 (Re[F ]Im[F~ ] + Im[F ]Re[F~ ]); (2.80) which are considerably easier to solve than the original complex transcen- 442.5. Summary dental equation. 2.5 Summary In this chapter, we have explored interaction of electromagnetic waves with uni-axial metallic waveguides and derived characteristic equations that yield the dispersion relations for di erent waveguide geometries. The Davi- denko method has been introduced and will be used throughout the remain- der of the thesis to solve complex transcendental equations corresponding to the dispersion relations for various uni-axial metallic waveguides. We will next develop a method to design multi-axial waveguides by conceptually de-composing the structures into simpler uni-axial waveguide components. The dispersion relations for uni-axial waveguides will then be used to ap- proximate the electromagnetic wave behaviour in di erent sections of the multi-axial waveguide. 45Chapter 3 Multi-Axial Nanoscale Light Bending Multi-axial waveguides are structures that guide light along more than one direction. Modeling multi-axial waveguides is challenging because the electromagnetic waves sustained within these structure are no longer char- acterized by a single wavevector. Designing multi-axial waveguides gener- ally requires numerical techniques, such as the FDTD method, to model the multi-axial  ow of electromagnetic waves within the structure. Numer- ical simulations, however, are both computationally expensive and time- consuming. Because only one set of parameters can be modeled in a given simulation, full characterization and optimization of the waveguide using a numerical simulation tool only, is often not feasible. In this and coming sections, an analytical technique for designing a multi- axial waveguide designing technique is investigated. The technique is based on conceptually dividing a multi-axial waveguide into uni-axial waveguide Adapted and reprinted with permission from M. W. Maqsood, R. Mehfuz, and K. J. Chau, \Design and optimization of a high-e ciency nanoscale  90 light-bending structure by mode selection and tailoring," Applied Physics Letters 97, 151111 (2010), American In- stitute of Physics. 46Chapter 3. Multi-Axial Nanoscale Light Bending sub-components. Solving the dispersion relations of the uni-axial waveg- uide components then enables approximation of the electromagnetic wave behaviour in di erent sections of the multi-axial waveguide. By tailoring the geometry of the uni-axial waveguide sub-components so that each of them sustain a single propagative mode with matched electromagnetic wavevec- tors, e cient coupling between di erent sections of the multi-axial waveg- uide is predicted. The predictions made using this simple analytical ap- proach are con rmed using rigorous numerical simulations. In recent years, there has been a growing interest in the development of light steering structures for application in dense integrated optical systems. Achieving e cient light bending over sharp corners is generally challeng- ing due to bending losses - a general term that describes the propensity of light to scatter and escape into free-space when curvature of a waveg- uide bend is very small. Recently, several di erent approaches have been put forward which use dielectric waveguide resonators [48, 49] and dielec- tric photonic crystals [50] placed at the intersection of two waveguides, to achieve bending over 90 . An alternative method is to use waveguides com- posed of metals. One such implementation is based on joining two nanoscale metal-dielectric-metal (MDM) waveguides at 90 [51{53] to form either an L or a T-junction. When the thickness of dielectric core in these structures is reduced such that it is smaller than the wavelength of the incident light, the waveguides sustain only the lowest-order TM mode, TM0 mode (commonly referred as the surface plasmon polariton (SPP) mode), which travels along the metal surface and around the waveguide bend. There are a few limita- tions with this implementation. TM0 modes are highly dissipative and can 473.1. Material and Geometry Selection only propagate over a few microns at visible frequencies. Furthermore, there is signi cant back scattering of the TM0 mode when it encounters the bend. Back scattering can be somewhat reduced by converting the bend into a curve. Even with curved bends, waveguide bends based on SPP modes have limited e ciency. A maximum bending e ciency of 77% has been reported for a metallic waveguide bend with a bending curvature radius of 20 nm at a free-space wavelength of 632 nm [52]. 3.1 Material and Geometry Selection Here, we will explore a bi-axial, metallic, nanometer scale waveguide designed to bend light with high e ciency and low loss over right angles. We start by selecting the constituent materials of the waveguide. At visible frequencies, silver is the most suitable choice for the metal component, as indicated by the small magnitude of the imaginary part of its permittivity. For the dielectric component, we want to select a material that has low loss yet has a high refractive index to reduce the e ective wavelength of light in the structure. With this in mind, we select gallium phosphide (GaP) which is nearly transparent in the visible region and has a mostly real refractive index n ’ 3:5. We  rst consider a basic multi-axial T-junction waveguide geometry as shown in Fig. 3.1. The waveguide consists of semi-in nite silver metal  lm with a slit in the center. The slit is  lled with GaP and the complete structure is covered with GaP. The waveguide structure is illuminated from the bottom with a TM-polarized light at a frequency ! and wavelength 483.2. Mode Selection Figure 3.1: Waveguide structure consisting of a silver slit  lled with GaP, with the complete structure covered with a semi-in nite GaP layer. GaP is chosen due to its higher refractive index of n = 3:5.  . For analysis, the region before the bend is approximated as an in nite metal-dielectric-metal (MDM) waveguide and region after the bend is ap- proximated as single metal-dielectric waveguide. In this case, the MDM waveguide is composed of a GaP core surrounded by silver cladding lay- ers and the metal-dielectric waveguide is composed of an interface between silver and GaP. 3.2 Mode Selection We analyze the propagation of an electromagnetic wave in the waveguide regions by approximating each region as an independent uni-axial waveg- uide. The modes sustained by each of the uni-axial waveguides are mapped 493.2. Mode Selection out to describe the propagation characteristics of the wave inside di erent parts of the waveguide. A mode is a con ned electromagnetic wave which is described by a distinctive wavevector value at a given frequency. This wavevector value corresponds to transverse  eld amplitudes that are inde- pendent of the mode propagation. For a given waveguide geometry only a  nite number of modes are allowable and out of these only some are prop- agative. The distinction between a propagative and non-propagative mode is made through the  gure-of-merit (FOM) given by FOM = Re( ~ ) Im( ~ ) : (3.1) A mode is considered propagative if it can propagate at least one complete cycle. This condition is achieved when FOM > 2 . Using eigenvalue equa- tions derived in Chapter 2 and the Davidenko method, we obtain values of the complex wavevector, ~ , for the uni-axial waveguides. From ~ , we cal- culate FOM curves for the waveguides and identify the propagative modes having FOM > 2 . We  rst consider the lowest order asymmetric SPP mode (also known as the TM0 mode) sustained by the uni-axial MDM waveguide section. Using the asymmetric form of the dispersion relation obtained in Eq. 2.30 and experimental data of Johnson and Christy [36] for silver, complex wavevector values are numerically evaluated for the lowest order mode. Figure 3.2 depicts Re( ~ ) and FOM curves for the asymmetric SPP mode sustained by a silver-GaP-silver waveguide for slit width values varying from w = 10 nm to w = 300 nm. The Re( ~ ) curves have an in ection point near 5:5  1014 503.2. Mode Selection Figure 3.2: Dispersion curves for the asymmetric SPP modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency for various dielectric core thickness values. 513.2. Mode Selection Hz, a frequency known as the surface plasmon resonance frequency. At the surface plasmon resonance frequency, the magnitude of the real part of the relative permittivity of the metal and dielectric are equivalent and, as a result, Re( ~ ) becomes large. From the FOM curves, the asymmetric mode is generally only propagative for frequencies below the surface plasmon resonance frequency. It is interesting to note that for the smallest slit width (w = 10 nm) and at UV frequencies above the surface plasmon resonance frequency, the asymmetric SPP mode is propagative with a negative FOM. This describes a backwards-propagating mode which has a phase velocity oriented in the opposite direction to the energy velocity. We next consider the lowest order symmetric SPP mode (also known as the TM0 mode) sustained by the MDM waveguide section. Using the sym- metric form of the dispersion relation obtained in Eq. 2.28 and experimental data of Johnson and Christy [36] for silver, complex wavevector values are numerically evaluated for the lowest order mode. The Re( ~ ) and FOM curves for the symmetric mode are presented in Figure 3.3 for slit width values varying from w = 10nm to w = 300nm. Again, the Re( ~ ) curves are characterized by an in ection point at a surface plasmon resonance fre- quency which matches that of the asymmetric SPP mode. The FOM values of the symmetric SPP mode approach zero near the resonance frequency. Below this frequency, the symmetric SPP mode is propagative with FOM values exceeding those of the asymmetric SPP mode. Based on the relative FOM values of the symmetric and asymmetric SPP modes throughout the explored frequency interval, we conclude that the symmetric SPP mode is dominant over the asymmetric SPP mode. 523.2. Mode Selection Figure 3.3: Dispersion curves for the symmetric SPP modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency for various dielectric core thickness values. 533.2. Mode Selection Figure 3.4: Dispersion curves for the TM1 modes sustained by an MDM (Ag-GaP-Ag) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM), as a function of frequency for various dielectric core thickness values. 543.2. Mode Selection We  nally consider the  rst order symmetric TM mode sustained by the MDM waveguide section, labeled as the TM1 mode. Using the symmetric form of the dispersion relation obtained in Eq. 2.28 and experimental data of Johnson and Christy [36] for silver, complex wavevector values are nu- merically evaluated for the second-lowest order mode. Figure 3.4 depicts the Re( ~ ) and FOM curves for TM1 modes sustained by the MDM waveguide for varying slit width values from w = 10nm to w = 300nm. Unlike the SPP modes, the Re( ~ ) curve of the TM1 mode is characterized by a cuto frequency - a frequency below which the wavevector value vanishes - and Re( ~ ) values that monotonically increase as a function of frequency. Near the surface plasmon resonance frequency, the FOM values of the TM1 mode are signi cantly larger than the FOM values of the SPP modes. For visible frequencies near and above the surface plasmon resonance frequency, the relative large FOM values of the TM1 mode as compared to the symmetric and asymmetric SPP modes means that the TM1 mode is the dominant mode. Next we analyze the modes sustained by the silver-GaP interface by ap- proximating the interface as an uni-axial metal-dielectric waveguide section. As discussed in Chapter 2, a uni-axial waveguide composed of a metal- dielectric interface can sustain only a single SPP mode. The Re( ~ ) and FOM curves for the SPP mode are depicted in Figure 3.5. Similar to the case of SPP mode in the MDM waveguide, the Re( ~ ) for the SPP mode at the metal dielectric interface is characterized by an in ection point at the surface plasmon resonance frequency 5:5 1014 Hz. The SPP mode is prop- agative below the surface plasmon resonance frequency and non-propagative 553.3. Modi ed Bi-Axial Waveguide Structure above it. Based on our analysis, a nanometer-scale GaP- lled slit in a silver  lm coated with a semi-in nite GaP layer can potentially sustain four types of modes. In the GaP- lled slit section of the structure, three modes are pos- sible: asymmetric and symmetric SPP modes and the  rst order TM1 mode (it can be shown that the higher order TM modes are all cuto ). Of these three modes, the TM1 mode is desirable because it has the largest FOM val- ues, which are achieved at higher frequencies above both the surface plasmon resonance frequency and the cuto frequency. In the silver-GaP section of the structure, only the SPP mode is allowable, which is propagative below the surface plasmon resonance frequency and non-propagative above it. At lower frequencies where the SPP mode is propagative, the TM1 mode is cuto and at higher frequencies where the TM1 mode in the slit section is dominant, the SPP mode at the metal interface becomes non-propagative. Thus the SPP mode of the silver-GaP interface is not compatible with the TM1 mode of the GaP- lled slit. 3.3 Modi ed Bi-Axial Waveguide Structure We next modify the structure so that the GaP layer on the metal  lm has a  nite thickness, with a dielectric region on top of the GaP layer consisting of air. The resulting structure is shown in Figure 3.6. To analyze this modi ed structure, the region on top of the metal  lm is approximated as a uni-axial metal-dielectric-dielectric (MDD) waveguide composed of a silver cladding, a GaP core, and an air cladding. This MDD waveguide is 563.3. Modi ed Bi-Axial Waveguide Structure Figure 3.5: Dispersion curves for the SPP mode sustained by a Ag-GaP interface. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of frequency. 573.3. Modi ed Bi-Axial Waveguide Structure Figure 3.6: Waveguide structure consisting of a slit in a silver  lm where the slit is  lled with GaP and the entire silver  lm is coated with a GaP layer of  nite thickness. The structure is immersed in air. advantageous to a metal-dielectric waveguide because the MDD waveguide can sustain higher-order TM modes in addition to the SPP mode. Figure 3.7 shows the Re( ~ ) and FOM curves for the SPP mode, and the  rst-order TM1 mode sustained by an MDD waveguide for core thickness values ranging from 100nm to 300nm (the higher-order TM modes are cuto for these core thickness values). Similar to the trends observed for other SPP modes, the SPP mode in the MDD waveguide is only propagative below the surface plasmon resonance frequency, 5:5 1014 Hz. As before, this SPP mode is not compatible with the TM1 mode in the MDM waveguide. The TM1 mode of the MDD waveguide, on the other hand, is compatible with the TM1 mode in the MDM waveguide because both TM1 modes are propagative at frequencies above the surface plasmon resonance frequency. Based on 583.3. Modi ed Bi-Axial Waveguide Structure Figure 3.7: Modal solution for an MDD (Ag-GaP-Air) waveguide. a) Real part of the wavevector and b)  gure of merit (FOM) as a function of fre- quency for the SPP mode. c) Real part of the wavevector and d)  gure of merit (FOM) as a function of frequency for the TM1 mode. 593.4. Wavevector Matching this analysis, we will design the light-bending structure to achieve coupling between the TM1 modes in the slit region and the TM1 mode in the GaP layer region. 3.4 Wavevector Matching Wavevector mismatch describes the di erence between the wavevector values of an electromagnetic wave in two regions of space. The concept of wavevector mismatch and its e ects on electromagnetic wave propagation can be illustrated through a simple one dimensional example. Consider an electromagnetic plane wave normally incident from medium 1, having refractive index n1, onto medium 2, having refractive index n2 6= n1. We assume that both media have positive relative permittivity values. For a homogeneous and isotropic medium, the refractive index of a region, n, is directly related to the wavevector in the region, ~ , by ~ = nko: (3.2) An electromagnetic wave incident normally at the interface between two media experiences a sudden change in its wavevector. This wavevector mis- match results in re ection of a portion of the incident electromagnetic wave and a reduction in the transmission from medium 1 to medium 2, as shown in Figure 3.8(b). The relative amplitudes of the electromagnetic wave com- 603.4. Wavevector Matching Medium1 Medium2 TransmittedlightReflectedlight Medium1 Medium2 Incidentlight a) b) Figure 3.8: A simple example demonstrating the re ection loss due to wavevector mismatch. An electromagnetic plane wave is a) incident onto a dielectric interface and b) scattered into re ected and transmitted com- ponents after interacting with the interface. The relative amplitudes of the re ected and transmitted components are proportional to the degree of wavevector mismatch. 613.4. Wavevector Matching ponents in the two media can be predicted by Fresnel equations, given as R = n1  n2 n1 + n2 (3.3) and T = 1 R = 2n2 n1 + n2 ; (3.4) where R and T are the re ection and transmission coe cients, respectively. The above relations show that as the di erence in refractive indices increases (which then yield a wavevector mismatch), a greater portion of the incident electromagnetic wave is re ected and less transmitted. We extend the concept of wavevector matching to predict the coupling e ciency of the two TM1 modes sustained by the proposed bi-axial waveg- uide structure. It is hypothesized that the degree of coupling between the TM1 mode in the slit section of the structure and the TM1 mode in the GaP layer section of the structure is maximized when the degree of mismatch be- tween the in-plane wavevector components of the two modes is minimized. This mismatch corresponds to the di erence between transverse component of the wavevector,  1z, directed along the z-axis of the TM1 mode in the slit and the longitudinal component of the wavevector, 2, directed along the x-axis of the TM1 mode in the GaP layer. The con guration is shown in Fig- ure 3.9. Thus, e cient coupling is achieved when the condition wavevector matching  1z =  2 is satis ed. 623.4. Wavevector Matching Figure 3.9: Wavevector matching applied to a bi-axial waveguide. Wavevec- tor matching is achieved when the transverse component of the wavevector in the MDM waveguide matches with the longitudinal component of the wavevector in the MDD waveguide. 633.4. Wavevector Matching Figure 3.10: Plot of the longitudinal wavevector components of the TM1 mode sustained by the MDD waveguide ( 2) and the transverse wavevector components of the TM1 mode sustained by the MDM waveguide ( 1z). The blue circle shows the matching point at the operational frequency of w = 6 1014 Hz. 643.5. Waveguide Structure Optimization 3.5 Waveguide Structure Optimization To determine an optimal con guration that yields wavevector matching, we map out values of  1z of the TM1 mode in the slit as a function of the slit width and  2 of the TM1 mode in the GaP layer as a function of the layer thickness. The wavevector components are plotted in Figure 3.10 over the visible frequency range. For a nominal operational frequency of ! = 6 1014 Hz, wavevector matching is achieved for a slit width value w = 150nm and a GaP layer thickness value d = 100nm. It is predicted that when a structure with these geometrical parameters is excited by an electromagnetic wave at a frequency of ! = 6  1014 Hz, a TM1 mode sustained in the slit couples e ciently to a TM1 mode in the GaP layer. 3.5.1 FDTD Simulation We test the predictive powers of our analytical method by performing rigorous numerical simulations of the electromagnetic response of the struc- ture using the  nite-di erence-time-domain (FDTD) technique. The FDTD technique is a widely used method to model the spatial and temporal evolu- tion of electromagnetic waves by directly solving Maxwell’s equations over a spatial grid. The temporal and spatial partial derivatives in Maxwell’s equa- tions are discretized into di erence equations using the central-di erence approximations [54]. At a particular time step, the di erence equations are solved at each grid point to determine the local amplitude of the electric and magnetic  eld components. To model the electromagnetic response of the proposed coupling struc- 653.5. Waveguide Structure Optimization ture, we use a two dimensional spatial grid in the x z plane consisting of 4000  1400 pixels with a resolution of 1 nm/pixel, resulting in a rectan- gular grid occupying a physical area of 4:0  1:4  m. A perfectly matched layer is placed at the edges of the simulation space [55{57]. This layer is impedance matched to free-space so that any electromagnetic waves inci- dent onto the layer are totally absorbed and do not re ect back into the simulation space. Within the simulation space, we de ned both an electro- magnetic wave source and our proposed structure. The source consists of a TM-polarized electromagnetic wave oscillating at a frequency of ! = 6 1014 Hz, which corresponds to a free-space wavelength  = 500nm. The TM- polarized electromagnetic wave has a magnetic  eld component Hy and two electric  eld components Ex and Ez. The wave propagates in +x-direction and is centered at z = 0 with a full-width-at-half maximum of 1200 nm. Our structure is de ned in the simulation space by assigning to the grid points occupied by the structure the relative permittivity of the local medium. For grid points corresponding to GaP, we assign a relative permittivity of 3:52. For grid points corresponding to silver, we model the relative permittivity using the Drude model with experimentally obtained Drude parameters. For grid points corresponding to air, we assign a unity relative permittivity. Based on the  eld distributions calculated by the FDTD simulations, we measure the bending e ciency of the structure. Line detectors D1, D2, and D3 are placed at di erent locations in the simulation space to measure the time-averaged values of the magnetic  eld intensity jHyj2. Detectors D1 and D2 are placed in the GaP layer to the left and right of the slit exit and the detector D3 is placed in the air region above the slit exit. The 663.5. Waveguide Structure Optimization bending e ciency is quanti ed by the ratio of jHyj2 captured by D1;D2 to the total jHyj2 emitted from the slit and captured by D1, D2, and D3. The bending e ciency therefore measures the percentage of the electromagnetic wave exiting the slit region of the structure that couples into the GaP layer. Figure 3.12 shows a snapshot of the FDTD-calculated magnetic  eld intensity distribution when the structure is illuminated at a wavelength of  = 500 nm. Light incident from the bottom of the structure channels into the slit and then emerges from the exit of the slit. There are two pathways for the light emerging from the slit: it can bend into the GaP layer or radiate into the air region above the GaP layer. At a wavelength of  = 500 nm, highly e cient light bending at the slit exit is evident from almost complete diversion of the +x-propagating TM1 mode in the slit into the  z-propagating TM1 modes in the GaP layer, with minimal light scattered into free-space plane wave modes in the air region above the GaP layer. Figures 3.12(b) and (c) show the FDTD-calculated magnetic  eld intensity distribution when the structure is illuminated at wavelengths of  = 450 nm and  = 400 nm, respectively. For these shorter wavelengths, there is a greater amount of light scattered into the air region above the bend and relatively less light coupled into the GaP layer. Comparison of the FDTD calculated  eld distributions suggests that the e ciency of light bending at the slit exit is highly sensitive to the incident electromagnetic wavelength (and hence frequency). Figure 3.13 plots the FDTD-calculated bending e ciency as a function of the GaP layer thickness d for a  xed wavelength  = 500 nm. The FDTD simulations show that the bending e ciency reaches a maximum of 92% at 673.5. Waveguide Structure Optimization Figure 3.11: Simulation geometry of the waveguide structure designed to bend incident light at a frequency ! = 6  1014 Hz. The structure has a metal thickness t = 300nm, slit width w = 150nm and dielectric cap d = 100nm. The detectors D1 and D2 measure the time-averaged magnetic  eld intensity,jHyj2, of the TM1 mode in the GaP layer, and the detector D3 measures jHyj2 radiated into the air region. 683.5. Waveguide Structure Optimization Figure 3.12: FDTD calculations to determine the bending e ciency for the designed waveguide. For calculations, TM-polarized wave centered at z = 0 with a full-width-at-half-maximum of 1200nm is normally incident on the bottom surface of the slit. Snapshots of the instantaneous magnetic  eld intensity for normally-incident, free-space, TM-polarized illumination of the structure at (a) ! = 6  1014 Hz,  = 500 nm (b) ! = 6:67  1014 Hz,  = 450 nm, and (c) ! = 7:5  1014 Hz,  = 400nm for  xed parameters w = 150nm and d = 100nm. The inset in (a) highlights the TM mode emanating from the slit to split symmetrically into TM1 modes con ned inside the GaP layer. 693.5. Waveguide Structure Optimization Figure 3.13: Coupling e ciency as a function of GaP layer thickness d. The frequency of the incident light is kept constant at ! = 6 1014 Hz. an optimal layer thickness d = 100 nm. This optimal value matches the layer thickness value predicted to yield wavevector matching. The bending e ciency drops o for layer thickness values both smaller and larger than the optimal value. Figure 3.14 depicts the FDTD-calculated bending e ciency as a func- tion of the wavelength of incident light for  xed slit width, w = 150 nm and GaP layer thickness, d = 100 nm. The FDTD calculations show that the bending e ciency is greater than 90% for incident wavelengths  = 450 nm and  = 500 nm. Increasing or decreasing the wavelength outside of these 703.5. Waveguide Structure Optimization Figure 3.14: Coupling e ciency as a function of  . The slit width and layer thickness are kept constant at w = 150 nm and d = 100 nm, respectively. values, yields a dramatic drop in the bending e ciency of the structure. The FDTD calculations therefore support our hypothesis that in-plane wavevec- tor matching, which is achieved for a single combination of slit width, layer thickness and wavelength, yields highly e cient mode coupling. For param- eters departing from these optimal parameters, the coupling e ciency drops dramatically. 713.6. Summary 3.6 Summary In this chapter we have presented a technique for designing a bi-axial nanoscale metal waveguide that can bend light around a tight  90 bend. The design technique consists of  rst breaking the bi-axial waveguide into two region, each of which is then approximated as an independent uni- axial waveguide. The modes sustained by each of the uni-axial waveguides are mapped out to determine the propagative modes. The geometrical pa- rameters for each of the uni-axial waveguides are designed such that the in-plane wavevector components of the propagative modes in the waveg- uides are matched. The wavevector matching condition is predicted to yield maximum coupling between the electromagnetic modes, even when they propagate along perpendicular directions, and thus achieve highly e cient light bending. The performance of the designed structure is modeled using a  nite-di erence-time-domain (FDTD) technique which solves Maxwell’s equations directly over a discretized spatial grid. The FDTD-calculated bending e ciency as a function of the GaP layer thickness shows a pro- nounced peak for parameters corresponding to the wavevector matching condition, supporting the hypothesis that wavevector matching yields highly e cient coupling. Our design methodology is useful because it can poten- tially circumvent the need for tedious numerical simulations and may  nd application in the design of highly-integrated, miniaturized optical circuits. 72Chapter 4 Multi-Axial Surface-Plasmon-Polariton Coupling Surface plasmon polariton (SPP) modes are electromagnetic excitations at an interface between a metal and dielectric, hold promise for the minia- turization of optical devices [58, 59]. Due to the lack of readily-available sources directly emitting SPP modes, designing methods to couple plane- wave modes to SPP modes with high e ciency and throughput remains an important objective. Plane-wave modes directly incident onto a metal- dielectric interface cannot e ciently couple into SPP modes due to a mis- match between the SPP wavevector and the component of the plane-wave wavevector along the interface. Scatterers have been used to bridge the wavevector mismatch between plane-wave and SPP modes. When a scat- terer is illuminated, enhancement of the incident plane-wave wavevector along the metal-dielectric interface by the Fourier spatial frequency compo- nents of the scatterer geometry in the plane of the interface enables wavevec- tor matching between the incident light and the SPP mode. 73Chapter 4. Multi-Axial Surface-Plasmon-Polariton Coupling A widely-used scatterer-based SPP coupling technique is to illuminate a slit in a metal  lm. Slit based SPP couplers have an inherent advantage that they can be easily incorporated in integrated optical devices. When a slit is illuminated with a TM-polarized plane wave, a small portion of the incident wave excites a guided mode in the slit. The guided mode propagates through the slit and subsequently di racts at the slit exit. The total light intensity leaving the slit exit de nes the total throughput, the SPP intensity leaving the slit exit de nes the SPP throughput, and the ratio of the SPP throughput to the total throughput de nes the SPP coupling e ciency. The throughput and e ciency of a slit are highly dependent on the width of the slit relative to the wavelength of the incident plane wave. A slit of width less than the wavelength has inherently low total throughput and low SPP throughput, but is capable of high SPP coupling e ciency. It has already been demonstrated that the SPP coupling e ciency of a sub-wavelength slit is increased up to ’ 80% by coating the slit with a nanoscale dielectric layer[60]; the dielectric layer, however, does not signi cantly a ect the SPP throughput. The objective of this chapter is to design a SPP coupling scheme capable of both high throughput and high e ciency. Increasing the total throughput of a slit can be achieved by simply in- creasing the slit width. Increasing the SPP throughput, on the other hand, is more challenging because the SPP throughput is dependent on the cou- pling between the guided mode in the slit and the SPP mode, which has not been fully explored yet. Recently, several theoretical [61{63] and semi- analytical [60, 64{66] models have been developed to describe coupling from a guided mode in a single slit to a SPP mode. The SPP mode is de ned 74Chapter 4. Multi-Axial Surface-Plasmon-Polariton Coupling by a single, unique solution to Maxwell’s equations when the magnetic  eld boundary conditions are imposed at the metal surface. The guided mode in the slit, on the other hand, generally consists of a superposition of in nite TM-polarized waveguide eigenmodes, and each of these modes correspond to a unique solution to the Maxwell’s equations when the magnetic  eld boundary condition are imposed at the slit edges. The width of the slit dictates the eigenmode composition of the guided mode in the slit. All pre- vious models [60{66] describing SPP coupling by a slit have assumed a slit width less than the wavelength. When the slit width is less than the wave- length, all eigenmodes are attenuated except the zeroth-order TM mode, TM0 (also known as SPP mode) and the guided mode is accurately and simply approximated as the TM0 mode. The TM0 mode approximation becomes increasingly inaccurate [62, 64] as the slit width is increased to values comparable to and/or larger than the wavelength and higher-order eigenmodes become predominant. To date, accurate models of SPP coupling from super-wavelength slits sustaining higher-order eigenmodes have yet to be realized. In this work, we propose and characterize a new SPP coupling technique for multi-axial metal waveguide structures. The technique is applied on a bi-axial metal waveguide constructed using slit of super-wavelength width immersed in a uniform dielectric. The width of the super-wavelength slit is selected to sustain a  rst-order TM1 eigenmode, TM1 mode, just above cut- o , which then couples to the SPP mode at the slit exit. This is in contrast to previously explored SPP coupling con gurations using sub-wavelength slits that sustain only the lowest-order TM0 eigenmode [60{66]. We show 754.1. Hypothesis that the TM1 mode just above cuto is advantageous for SPP coupling be- cause it possesses a transverse wavevector component (lying in the plane of the metal surface) that is larger than that achievable with a TM0 mode in a slit of sub-wavelength width. It is proposed that if the transverse wavevector component of the TM1 mode, added with the peak Fourier spatial frequency component (due to di raction at the slit exit), equals to the wavevector of the SPP mode on the metal surface, high SPP coupling e ciency is achievable. The hypothesis is tested by numerical simulation of visible light propagation through a slit as a function of the slit width and refractive index. An opti- mized geometry is discovered that satis es the predicted wavevector match- ing condition, yielding a peak SPP coupling e ciency of ’ 68% and an SPP throughput that is over an order of magnitude greater that achieved with a sub-wavelength slit. Compared to a sub-wavelength slit, the optimized super-wavelength slit geometry is easier to fabricate, has comparable SPP coupling e ciency and an over order-of-magnitude greater SPP throughput. 4.1 Hypothesis Consider a semi-in nite layer of metal (silver) with relative complex per- mittivity ~ m extends in nitely in the y- and z-directions and having thickness t. A slit of width w oriented parallel to the x-axis and centred at z = 0 is cut into the metal  lm. The metal  lm is immersed in a homogeneous dielectric medium with relative permittivity  d and refractive index n = p  d. The complete structure is presented in Fig. 4.1. The slit is illuminated from the region below it with a TM-polarized electromagnetic plane wave of wave- 764.1. Hypothesis length  =  0=n and wavevector  i =  ix^, where  i = 2 = . The +x-axis de nes the longitudinal direction, and the z-axis de nes the transverse axis. The electromagnetic wave couples into a guided mode in the slit having complex wavevector ~ = ~ xx^ + ~ z z^, where ~ x and ~ z are the longitudinal and transverse components of the complex wavevector, respectively. The attenuation of the guided mode in the slit can be characterized by a  g- ure of merit (FOM) as described in previous chapter using eq. 3.1. When the guided mode exits the slit, electromagnetic energy is coupled into plane- wave modes and  z-propagating SPP modes. The SPP modes have complex wavevector  ~ sppz^, where Re[ ~ spp] and Im[ ~ spp] describe the spatial peri- odicity and attenuation, respectively, of the SPP  eld along the transverse direction. We treat the bi-axial waveguide SPP coupler by  rst dividing the struc- ture into two uni-axial waveguides and mapping out the modes in each of the waveguides. The slit region is approximated as a metal-dielectric-metal (MDM) waveguide and the top of the metal  lm is approximated as a metal- dielectric (MD) waveguide. The waveguide parameters are tuned such that a propagating TM1 mode in the MDM waveguide couples to a SPP mode in the MD waveguide. We map out the wavevector values of the TM0 and TM1 modes sustained in the slit (approximated as a MDM waveguide) for varying slit width by solving the complex eigenvalue equation, Eq. 2.26 us- ing the Davidenko method. ~ m is modeled by  tting to experimental data of the real and imaginary parts of the permittivity of silver as obtained in Section 1.8, and  d is assumed to be real and dispersion-less. Figure 4.2(a) shows FOM curves for TM0 and TM1 modes in slits of varying width for the 774.1. Hypothesis Silver Silver w t Dielectric Dielectric z x y Figure 4.1: Waveguide structure consisting of a slit in a metal  lm immersed in a dielectric. TM-polarized light is normally incident from the bottom of the structure and is con ned within the dielectric core. 784.1. Hypothesis representative case where the slit is immersed in a dielectric with a refractive index n = 1:75. The FOM values for the TM0 modes are largely insensitive to variations in the slit width and gradually decrease as a function of in- creasing frequency. FOM curves for the TM1 modes are characterized by a lower-frequency region of low  gure of merit and a higher-frequency region of high  gure of merit, separated by a kneel located at a cuto frequency. The cuto slit width wc for the TM1 mode at a given frequency ! is the threshold slit width value below which the TM1 mode is attenuating. At a  xed visible frequency ! = 6:0  1014 Hz ( = 285 nm), wc  300 nm. The dominant mode in the slit can be identi ed at a particular frequency and slit width by the mode with the largest FOM. The TM0 mode is dominant for w < wc, and the TM1 mode is dominant for w > wc. The real part of the transverse wavevector component, Re[ ~ z], of the guided mode in the slit describes the component of electromagnetic momen- tum in the transverse plane parallel to the plane of the metal surface. Values of ~ z are obtained from the relation ~ z = q  i2  ~ 2x; (4.1) where  i = n 0 is the magnitude of the wavevector in the dielectric core of the slit. Figure 4.2(b) shows Re[ ~ z] values over the visible-frequency range for the TM0 mode in a slit of width w = 200 nm and for the TM1 mode in slits of widths w = 350 nm and w = 500 nm. At the frequency ! = 6:0  1014 Hz, Re[ ~ z] for the TM0 mode in the w = 200 nm slit is nearly two orders of magnitude smaller than Re[ ~ z] for the TM1 mode in 794.1. Hypothesis bspp bspp bb b b b b b b b b b b b b Figure 4.2: Formulation of a hypothesis for di raction-assisted SPP cou- pling by a super-wavelength slit aperture. a) Figure-of-merit and b) the real transverse wavevector component versus frequency and wavelength for TM0 and TM1 modes sustained in slits of di erent widths. c) Di raction spec- trum corresponding to the TM0 mode in a 200- nm-wide slit and the TM1 modes in 350- nm-wide and 500- nm-wide slits. d) Wavevector-space depic- tion of di raction-assisted SPP coupling from slits of width w = 200 nm, w = 350 nm, and w = 500 nm, immersed in a uniform dielectric of refractive index n = 1:75. 804.1. Hypothesis the w = 350 nm and w = 500 nm slits. Values of Re[ ~ z] for the TM1 mode generally increase for decreasing slit width. Given the parameters in Fig. 4.2(b) and for a  xed ! = 6:0  1014 Hz, Re[ ~ z] for the TM1 mode increases from 8:5  106 m 1 to 1:3  107 m 1 as the slit width decreases from 500 nm to 350 nm. Di raction at the slit exit generates transverse spatial frequency com- ponents,  . The di raction spectrum is a distribution of transverse spatial frequencies generated by di raction at the slit exit. We calculate the di rac- tion spectrum by Fourier transformation of the transverse  eld pro les of the guided mode [67]. Figure 4.2(c) shows the normalized di raction spectrum for slit widths w = 200 nm, w = 350 nm, and w = 500 nm at a  xed fre- quency ! = 6:0 1014 Hz. The peak transverse spatial frequency component,  p, is the spatial frequency at which the di raction spectrum peaks. For the parameters in Fig. 4.2(c),  p shifts from 1:6 107 m 1 to  p = 8:3 106 m 1 as the slit width increases from w = 200 nm to w = 500 nm. It is noteworthy that  p < Re[ ~ spp] for all slit width values. A simple picture of di raction-assisted SPP coupling based on the data in Figs. 4.2(a)-(c) for w = 200 nm, w = 350 nm and w = 500 nm at a  xed ! = 6:0  1014 Hz is presented in Fig. 4.2(d). SPP coupling at the slit exit is mediated by di raction of the guided mode, yielding a net real transverse wavevector component Re[ ~ z] + p. Coupling from the di racted mode at the slit exit to the SPP mode adjacent to the slit exit is opti- mized when the wavevector-matched condition Re[ ~ z] +  p = Re[ ~ spp] is satis ed. Because Re[ ~ spp] is generally larger than both Re[ ~ z] and  p, large and commensurate contributions from both Re[ ~ z] and  p are required to 814.2. Methodology ful ll wavevector matching. In a sub-wavelength slit, the TM0 mode has Re[ ~ z] <<  p and SPP coupling at the slit exit requires a su ciently small slit width to generate large di racted spatial frequency components to match with Re[ ~ spp]. On the other hand, a super-wavelength slit sustains a TM1 mode with Re[ ~ z] ’  p. The large contributions of Re[ ~ z] to the net real transverse wavevector component reduces the required contributions from  p needed for wavevector matching. As a result, wavevector matching with the SPP mode adjacent to the slit exit can be achieved with a relatively large slit aperture. 4.2 Methodology SPP coupling e ciency of a slit immersed in a dielectric is modeled using  nite-di erence-time-domain (FDTD) simulations of Maxwell’s equations. The simulation grid has dimensions of 4000 1400 pixels with a resolution of 1 nm/pixel and is surrounded by a perfectly-matched layer to eliminate re ections from the edges of the simulation space. The incident beam is centered in the simulation space at z = 0 and propagates in the +x-direction, with a full-width-at-half-maximum of 1200 nm and a waist located at x = 0. The incident electromagnetic wave has a free-space wavelength  0 = 500 nm and is TM-polarized such that the magnetic  eld, Hy, is aligned along the y-direction. Control variables of this study include the type of metal (chosen as sil- ver), the thickness of the metal layer (set at t = 300 nm), the polarization of the incident electromagnetic wave (TM), the angle of incidence of the 824.3. Results and Discussion incident electromagnetic wave (normal), and the wavelength of the incident electromagnetic wave ( 0 = 500 nm). The independent variables include the width of the slit, w, which is varied from 100 nm to 800 nm, and the refractive index of the surrounding dielectric n, which varies from 1:0 to 2:5. The dependent variables are the time-averaged intensity of the SPP modes coupled to the metal surface at the slit exit, Ispp, the time-averaged inten- sity of the radiated modes leaving the slit region, Ir, and the SPP coupling e ciency,  . The dependent variables are quanti ed by placing line detec- tors in the simulation space to capture di erent components of the intensity pattern radiated from the exit of the slit, similar to the method employed in chapter 3. The Ispp detectors straddle the metal/dielectric interface, ex- tending 50 nm into the metal and  0=4 nm into the dielectric region above the metal, and are situated adjacent to the slit exit a length  0 away from the edges of the slit. The Ir detector captures the intensity radiated away from the slit that is not coupled to the surface of the metal. The coupling e ciency is then calculated by the equation  = 1 1 + Ir=Ispp : (4.2) 4.3 Results and Discussion The numerical simulations provide evidence of high-throughput and high- e ciency SPP coupling from a slit of super-wavelength width. Figure 4.3 displays representative snap-shots of the instantaneous jHyj2 intensity and time-averaged jHyj2 angular distribution calculated from FDTD simula- 834.3. Results and Discussion Figure 4.3: Images of the FDTD-calculated instantaneous jHyj2 distribu- tion (left) and the time-averaged jHyj2 angular distribution (right) for a slit of width values a) w = 200 nm, b) w = 350 nm, and c) w = 500 nm im- mersed in a dielectric (n = 1:75) and illuminated by a quasi-plane-wave of wavelength  0 = 500 nm. A common saturated color scale has been used to accentuate the  elds on the exit side of the slit. 844.3. Results and Discussion tions for plane-wave, TM-polarized, normal-incidence illumination of a slit immersed in a dielectric (n = 1:75) for slit width values w = 200 nm, w = 350 nm, and w = 500 nm. Radiative components of the  eld in the dielectric region above the slit propagate away from the metal-dielectric interface, and plasmonic components propagate along the metal-dielectric interface. For w = 200 nm [Fig. 4.3(a)], the incident plane wave couples into a propagative TM0 mode in the slit, which is characterized by intensity max- ima at the dielectric-metal sidewalls. Di raction of the TM0 mode at the exit of the slit yields a relatively strong radiative component with an angular intensity distribution composed of a primary lobe centred about the longi- tudinal axis and a relatively weak plasmonic component. A lobe describes the concentration of electromagnetic energy in a region. For w = 350 nm [Fig. 4.3(b)] and w = 500 nm[Fig. 4.3(c)], the incident plane wave couples primarily into the TM1 mode in the slit, which is characterized by an inten- sity maximum in the dielectric core of the slit. The high-throughput SPP coupling is evident by the large SPP intensities observed for w = 350 nm. Di raction of the TM1 mode at the w = 350 nm slit exit yields a relatively weak radiative component with an angular intensity distribution skewed at highly oblique angles and a relatively strong plasmonic component. Further increasing the slit width to w = 500 nm increases the total throughput of the slit, but reduces the e ciency of SPP coupling. Di raction of the TM1 mode at the w = 500 nm slit exit yields a strong radiative component with an angular intensity distribution composed of two distinct side lobes and a relatively weak plasmonic component. Trends in the SPP coupling e ciencies calculated from the FDTD simu- 854.3. Results and Discussion Figure 4.4: a) SPP coupling e ciency as a function of optical slit width for dielectric refractive index values n = 1:0 (squares), n = 1:5 (circles), n = 1:75 (upright triangles), n = 2:0 (inverted triangles), n = 2:5 (diamonds). b) The measured SPP intensity (squares), radiative intensity (circles), and total intensity (diamonds). The shaded region indicates the sub-wavelength- slit-width regime. 864.3. Results and Discussion lations are compared to qualitative predictions from the model of di raction- assisted SPP coupling described in Fig. 4.4. Figure 4.4(a) plots the FDTD- calculated SPP coupling e ciencies as a function of the optical slit width nw for dielectric refractive index values ranging from n = 1:0 to n = 2:5. For sub-wavelength slit width values nw <  0, highest SPP coupling e ciency is observed for the smallest optical slit width. This trend is consistent with di raction-dominated SPP coupling predicted to occur for sub-wavelength slit widths, in which small slit width is required to yield large di racted spa- tial frequencies to achieve wavevector matching. For super-wavelength slit width values nw >  0, the SPP coupling e ciencies exhibit periodic mod- ulations as a function of optical slit width, qualitatively agreeing with the general trends observed in experimental data measured for a slit in air [64] and theoretical predictions based on an approximate model for SPP cou- pling from a slit [62]. The data in Fig. 4.4 reveals that the magnitude of the  uctuations in the SPP coupling e ciencies are highly sensitive to the dielectric refractive index. For refractive index values n = 1:0, 1:5, 1:75, and 2:0, the SPP coupling e ciency rises as nw increases above  0 and reaches local maxima of  = 14%, 44%, 68%, and 48% at a super-wavelength opti- cal slit width nw ’ 600 nm, respectively. The rapid increase in  as the slit width increases from sub-wavelength slit width values to super-wavelength slit width values is attributed to the disappearance of the TM0 mode in the slit and the emergence of the TM1 mode in the slit, which boosts the net real transverse wavevector component at the slit exit to enable wavevector matching. It is interesting to note that the SPP coupling e ciency peak at nw = 600 nm observed for lower refractive index values is absent for n = 2:5. 874.3. Results and Discussion Figure 4.5: Wavevector mismatch Re[ ~ spp] (Re[ ~ z] +  p) as a function of refractive index of the dielectric region for a  xed optical slit width nw = 600 nm and free-space wavelength  0 = 500 nm. 884.3. Results and Discussion Figure 4.4(b) displays the time-averaged radiative intensity Ir, SPP in- tensity Ispp, and total intensity It = Ispp + Ir, as a function of the optical slit width for n = 1:75. Although the smallest optical slit width gener- ally yields high SPP coupling e ciency, the total throughput and the SPP throughput is low. As the optical slit width increases to w ’  0 from sub- wavelength values, an increase in Ir and a decrease in Ispp yield low SPP coupling e ciency. In the super-wavelength range of optical slit width val- ues, 520 nm < nw < 700 nm, concurrently high SPP throughput and high SPP coupling e ciency ( > 50%) are observed. For the optical slit width value nw ’ 600 nm, Ispp is about an order of magnitude larger than Ispp for the smallest slit width value nw = 175 nm. As the optical slit width is further increased nw > 700 nm, Ir is signi cantly greater than Ispp, resulting again in low SPP coupling e ciencies. Variations in the peak SPP coupling e ciency at a  xed optical slit width nw = 600 nm for varying n can be qualitatively explained by the mismatch between the net real transverse wavevector component Re[ ~ z] +  p and the real SPP wavevector Re[ ~ spp]. Figure 4.5 plots the transverse wavevector mismatch Re[ ~ spp]  (Re[ ~ z] +  p) as a function of the dielectric refractive index at a constant optical slit width value nw = 600 nm. The wavevector mismatch increases monotonically from  0:4  107 m 1 to 2:4  107 m 1 as the refractive index increases from n = 1:0 to n = 2:5, crossing zero at n = 1:75. Coincidence between the n value that yields peak SPP coupling e ciency at nw = 600 nm and that which yields zero wavevector mismatch supports the hypothesis that optimal SPP coupling e ciency occurs when Re[ ~ spp] = (Re[ ~ z] +  p), and that this condition can be achieved using a 894.4. Summary super-wavelength slit aperture immersed in a dielectric. The relatively large wavevector mismatch for n = 2:5 is also consistent with the noted absence of a SPP coupling e ciency peak at nw = 600 nm. 4.4 Summary In this chapter, we have designed a bi-axial waveguide SPP coupler that converts a TM1 mode in a slit to a SPP mode on an adjacent metal sur- face with high-throughput and high-e ciency. The crux of the design is a super-wavelength slit aperture immersed in a uniform dielectric sustaining a TM1 mode just above cuto . High SPP coupling e ciency is achieved when the transverse wavevector component of the TM1 mode, added with the peak di racted spatial frequency component, equals to the wavevector of the SPP mode on the metal surface. Based on numerical simulations using FDTD of light propagation through a slit of varying slit width and varying surrounding dielectric refractive index, an optimal slit width and refractive index combination is found that provides high e ciency coupling. The parameters also match the parameters predicted to yield wavevector matching. 90Chapter 5 Conclusion Optical waveguides are structures that guide light. The advent of nanofab- rication techniques now allows controlled manufacturing of optical waveg- uides with nanometer scale features using dielectric, metallic, or semi-conducting materials. Given the degrees of freedom a orded by nanofabrication, opti- mization of a waveguide design requires e cient modeling techniques. All modeling schemes are based on Maxwell’s equations. The way in which Maxwell’s equations are applied to model a given optical waveguide depends on the complexity of the waveguide geometry. One of the simplest con-  gurations is the uni-axial waveguide, which guides electromagnetic waves along a single axial direction. The electromagnetic properties of a uni-axial waveguide are easily determined and can be modeled by analytically solv- ing Maxwell’s equations. Multi-axial waveguides, on the other hand, guide electromagnetic waves along more than one direction. They are challenging to model by analytical solutions to Maxwell’s equations and often require numerical techniques. One of the most popular numerical techniques is the  nite-di erence time-domain (FDTD) technique. Numerical simulation tools like the FDTD technique enable detailed visualization of electromag- netic  elds within complicated optical waveguides systems. A limitation 915.1. Limitations of the FDTD technique is the massive computation power and processing time required to complete a simulation. Because only one combination of parameters can be explored for a given simulation, hundreds or thousands of simulations are required to completely map out the frequency-dependent electromagnetic response of an optical waveguide for various material com- binations and geometrical con gurations. As a result, optimization using only numerical simulations is often not feasible. In this thesis, we have explored a new analytical technique to analyze and optimize the performance of multi-axial waveguides. The technique consists of breaking the multi-axial waveguide down into simpler uni-axial waveguide sub-components. The uni-axial waveguides are modeled by an- alytically solving Maxwell’s equations and mapping out the wavevector of the modes sustained by the waveguides. We hypothesized that e cient cou- pling between the uni-axial waveguide sub-components is achieved when the in-plane wavevector of the modes sustained by the waveguides are matched. In Chapters 3 and 4, we presented two design examples in which bi-axial waveguides are optimized, using the proposed methodology, to achieve high coupling e ciency. In both examples, the optimized parameters selected using our methodology is shown, through FDTD simulations, to yield max- imum coupling e ciency. 5.1 Limitations One of the major limitations of this technique is that the uni-axial waveg- uide sub-components are assumed to be in nitely-long, which is typically not 925.2. Future Work the case. For waveguides that extend over short distances comparable to or less than the wavelength, the uni-axial waveguide solutions may not accu- rately describe the electromagnetic properties of the actual system. Another major limitation of our methodology is that it cannot quantify the perfor- mance of multi-axial waveguides. The methodology only selects waveguide parameters that can potentially yield maximum coupling, but still requires numerical methods like the FDTD technique to rigorously model the waveg- uide performance. We thus envision that this technique can be potentially used as a supplement to numerical methods to provide  rst order estimates of optimal parameters. 5.2 Future Work Future work aims to explore the application of this technique to design multi-axial waveguides composed of more than two uni-axial waveguides connected with each other non-perpendicular angles. Furthermore, we have thus far restricted our treatment to simple cases where the uni-axial waveg- uide components only sustain a single mode. Further work is needed to verify that this methodology still holds for waveguide systems that sustain more than one mode. 93Bibliography [1] K. Okamoto, Fundamentals of Optical Waveguides. Elsevier Inc., sec- ond ed., 2006. [2] D. Colladon, \On the re ections of a ray of light inside a parabolic liquid stream," Comptes Rendus, vol. 15, pp. 800 { 802, 1842. [3] R. J. Bates, Optical Switching and Networking Handbook. New York: McGraw Hill, 2001. [4] M. 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These iterative techniques are usually time consuming and their convergence to a stable solution requires a good initial guess value. Here, we present explicit analytical piece-wise  t functions for the solutions to dispersion re- lations of various metallic waveguide structures. These explicit expressions can be used to approximate the complex wavevector or serve as an initial guess for iterative solution implementations. SPP Mode in a Silver-Dielectric-Silver Waveguide We obtain piece-wise  t functions to the dispersion relations of the sym- metric SPP mode over frequencies where it is propagative. The  t functions for the symmetric SPP mode can be given for di erent dielectric cores as: 102SPP Mode in a Silver-Dielectric-Silver Waveguide Dielectric Core n = 1:0 ! = 8:5 1014  8:75 1014 (Hz) and t = 150nm 300nm  = 1:72173 107  8:2303 10 8!  1:32669 1014t +1:20818 10 22!2 + 9:33943 1019t2 + 0:09494!t ! = 8:5 1014  8:75 1014 (Hz) and t = 50nm 150nm  =  2:09062 107  3:46651 10 8! + 1:09635 1014t +1:39199 10 22!2 + 8:8537 1020t2  0:45034!t ! = 3:5 1014  8:5 1014 (Hz) and t = 150nm 300nm  = 6:12927 106 + 3:20113 10 8! + 6:05407 10 23!2 +2:00357 10 37!3 + 3:16232 10 52!4 + 1:93617 10 67!5  9:32617 1013t+ 5:71877 1020t2  1:95536 1027t3 +3:57259 1033t4  2:7277 1039t5 ! = 4:75 1014  8:2 1014 (Hz) and t = 50nm 150nm  = 89293:84003 1:72137 10 7! + 6:77262 10 22!2  1:08811 10 36!3 + 7:76656 10 52!4  1:62355 10 67!5 +1:21543 1015t 2:30847 1022t2 + 2:04098 1029t3  8:71403 1035t4 + 1:45453 1042t5 103SPP Mode in a Silver-Dielectric-Silver Waveguide Dielectric core n = 1.5 ! = 3:5 1014  7:5 1014 (Hz) and t = 120nm 300nm  =  8:18148 107 + 1:09905 10 6!  4:73456 10 21!2 +1:04218 10 35!3  1:13622 10 50!4 + 4:93144 10 66!5  2:19077E14t+ 1:63401 1021t2  6:56117E27t3 +1:36474 1034t4  1:15377 1040t5 ! = 3:5 1014  6 1014 (Hz) and t = 10nm 80nm  =  5:75014 107 + 3:15718 1011e e  1:96374 1015 ! 1:11855 1015  +6:34943 107e e   7:35261 10 8 t 1:43303 10 8   3:15122 1011e 0 @ e  1:96374 1015 ! 1:11855 1015   e   7:35261 10 8 t 1:43303 10 8  1 A ! = 6 1014  7:5 1014 (Hz) t = 50nm 80nm  = 8:3505 106 + 6:23672 108e e  1:14896 1015 ! 4:84616 1014  +6:66111 106e e  5:69646 10 8 t 7:42459 10 8   6:82193 108e 0 @ e  1:14896 1015 ! 4:84616 1014   e  5:69646 10 8 t 7:42459 10 8  1 A ! = 6 1014  7:5 1014 (Hz) 104SPP Mode in a Silver-Dielectric-Silver Waveguide t = 5nm 30nm  = 3:6674 107 + 1:90183 109e e  1:14951 1015 ! 3:07971 1014  +1:36918 108e e  4:09602 10 9 t  7:24051 10 9  +2:03376 1010e 0 @ e  1:14951 1015 ! 3:07971 1014   e  4:09602 10 9 t  7:24051 10 9  1 A ! = 6 1014  7:5 1014 (Hz) t = 30nm 50nm  = 1:07351 106 + 1:27678 109e e  1:14951 1015 ! 4:57803 1014  +2:49982 107e e  2:83143 10 8 t 3:64897 10 8   1:74087 109e 0 @ e  1:14951 1015 ! 4:57803 1014   e  2:83143 10 8 t 3:64897 10 8  1 A ! = 3:5 1014  6 1014 (Hz) t = 80nm 120nm  =  814849:18221 + 4:41699E  10 8! + 1:16474 1013t +1:06179 10 23!2 + 1:32263 1014t2  0:10634!t ! = 6 1014  7:5 1014 (Hz) t = 80nm 120nm  = 2:28031 107  9:72998 10 8! + 1:75929 1014t 105SPP Mode in a Silver-Dielectric-Silver Waveguide +1:78671 10 22!2  7:58382 1014t2  0:37664!t Dielectric core n = 2.0 ! = 3 1014  6 1014 (Hz) t = 120nm 300nm  =  1:97964 107 + 1:6671 109e e  2:85129 1015 ! 1:97609 1015  +7:20147 106e e   8:48997 10 9 t 6:252 10 8   6:73854 108e 0 @ e  2:85129 1015 ! 1:97609 1015   e   8:48997 10 9 t 6:252 10 8  1 A ! = 6:5 1014  7:1 1014 (Hz) t = 120nm 300nm  = 3:51039 107 + 7:55202 109e e  1:00296 1015 ! 1:76148 1014   1:86391 106e e  4:00369 10 7 t 2:17345 10 9  6:00459 107e 0 @ e  1:00296 1015 ! 1:76148 1014   e  4:00369 10 7 t 2:17345 10 9  1 A ! = 6:0 1014  6:5 1014 (Hz) t = 120nm 300nm  = 2:82033 107 + 1:52588 108e e  7:86251 1014 ! 1:46422 1014  6:00435 109e e   6:37561 10 7 t  3:68951 10 7  106SPP Mode in a Silver-Dielectric-Silver Waveguide  3:31606 1010e 0 @ e  7:86251 1014 ! 1:46422 1014   e   6:37561 10 7 t  3:68951 10 7  1 A ! = 3:0 1014  6:0 1014 (Hz) t = 30nm 120nm  =  3:53696 107 + 3:13797 109e e  4:46772 1015 ! 2:98473 1015  1:13447 107e e  4:94075 10 8 t 2:12237 10 8   9:25342 108e 0 @ e  4:46772 1015 ! 2:98473 1015   e  4:94075 10 8 t 2:12237 10 8  1 A ! = 6:0 1014  6:5 1014 (Hz) t = 30nm 120nm  = 3:83428 107 + 1:87655 108e e  7:82176 1014 ! 1:65181 1014   9:72658 106e e  5:01303 10 8 t 1:94477 10 8   6:92474 107e 0 @ e  7:82176 1014 ! 1:65181 1014   e  5:01303 10 8 t 1:94477 10 8  1 A ! = 6:5 1014  7:1 1014 (Hz) t = 30nm 120nm  = 5:28234 107 + 1:21942 109e e  8:78901 1014 ! 1:36591 1014   1:55391 107e e  5:07127 10 8 t 1:661 10 8  107SPP Mode in a Silver-Dielectric-Silver Waveguide  2:37427 108e 0 @ e  8:78901 1014 ! 1:36591 1014   e  5:07127 10 8 t 1:661 10 8  1 A ! = 3:0 1014  5:5 1014 (Hz) t = 5nm 30nm  = 2:83643 106 + 1:89939 109e e  2:22083 1015 ! 1:28216 1015   7:36171 107e e  3:56358 10 9 t  7:72401 10 9  1:32238 1010e 0 @ e  2:22083 1015 ! 1:28216 1015   e  3:56358 10 9 t  7:72401 10 9  1 A ! = 5:5 1014  7:1 1014 (Hz) t = 5nm 30nm  = 4:82025 107 + 1:71088 1010e e  1:32937 1015 ! 3:69429 1014  2:17774 108e e  4:15853 10 9 t  7:20353 10 9  1:70486 1011e 0 @ e  1:32937 1015 ! 3:69429 1014   e  4:15853 10 9 t  7:20353 10 9  1 A Dielectric core n= 2.5 ! = 2:5 1014  5:5 1014 (Hz) t = 100nm 300nm  =  7:2073 106 + 2:59213 109e e  2:8552 1015 ! 1:74298 1015  108SPP Mode in a Silver-Dielectric-Silver Waveguide  1:79057 106e e  1:3725 10 8 t 6:04451 10 8   5:96339 108e 0 @ e  2:8552 1015 ! 1:74298 1015   e  1:3725 10 8 t 6:04451 10 8  1 A ! = 5:5 1014  6:0 1014 (Hz) t = 100nm 300nm  = 3:4993 107 + 1:27119 108e e  6:79906 1014 ! 1:08599 1014  1:70174 109e e   3:33857 10 7 t  2:34925 10 7   6:89123 109e 0 @ e  6:79906 1014 ! 1:08599 1014   e   3:33857 10 7 t  2:34925 10 7  1 A ! = 6 1014  6:5 1014 (Hz) t = 100nm 300nm  = 5:61033 107 + 1:78816 109e e  8:13426 1014 ! 1:15717 1014   8:70104 106e e  6:83944 10 8 t 2:04979 10 8  5:80717 108e 0 @ e  8:13426 1014 ! 1:15717 1014   e  6:83944 10 8 t 2:04979 10 8  1 A ! = 2:5 1014  5:5 1014 (Hz) t = 50nm 100nm  =  6:45697 107 + 1:37518 109e e  3:10394 1015 ! 2:75231 1015  109SPP Mode in a Silver-Dielectric-Silver Waveguide 1:43777 107e e  8:76514 10 8 t 4:37752 10 8   3:05341 108e 0 @ e  3:10394 1015 ! 2:75231 1015   e  8:76514 10 8 t 4:37752 10 8  1 A ! = 5:5 1014  6:0 1014 (Hz) t = 50nm 100nm  = 3:39926 107 + 2:51503 108e e  7:99154 1014 ! 2:85374 1014   3:40492 107e e  8:81644 10 8 t 1:69038 10 7   4:69014 107e 0 @ e  7:99154 1014 ! 2:85374 1014   e  8:81644 10 8 t 1:69038 10 7  1 A ! = 6:0 1014  6:5 1014 (Hz) t = 50nm 100nm  = 3:73922 107 + 5:68088 108e e  7:9886 1014 ! 1:65266 1014   4:98983 107e e  1:50844 10 7 t 1:18004 10 7  4:13079 108e 0 @ e  7:9886 1014 ! 1:65266 1014   e  1:50844 10 7 t 1:18004 10 7  1 A ! = 1:5 1014  5 1014 (Hz) t = 5nm 50nm  =  6:19853 108 + 4:75358 1010e e  1:25148 1015 ! 7:88071 1014  110SPP Mode in a Silver-Dielectric-Silver Waveguide 6:25779 108e e   1:0823 10 8 t 4:90041 10 9   4:69103 1010e 0 @ e  1:25148 1015 ! 7:88071 1014   e   1:0823 10 8 t 4:90041 10 9  1 A ! = 5 1014  6:5 1014 (Hz) t = 5nm 50nm  = 4:62771 107 + 5:12298 1010e e  1:57816 1015 ! 4:94713 1014  2:02722 1011e e   7:59034 10 8 t  3:93686 10 8  4:89833 1014e 0 @ e  1:57816 1015 ! 4:94713 1014   e   7:59034 10 8 t  3:93686 10 8  1 A Dielectric core n = 3.0 ! = 2:5 1014  5 1014 (Hz) t = 100nm 300nm  = 2:88107 107 + 2:13426 109e e  1:95942 1015 ! 1:04187 1015   2:47784 107e e   1:87313 10 8 t 5:74129 10 8  2:14085 108e 0 @ e  1:95942 1015 ! 1:04187 1015   e   1:87313 10 8 t 5:74129 10 8  1 A ! = 5 1014  6 1014 (Hz) 111SPP Mode in a Silver-Dielectric-Silver Waveguide t = 100nm 300nm  = 4:67772 107 + 3:21912 109e e  8:07685 1014 ! 1:50667 1014  402975:77697e e  4:50521 10 7 t 3:73193 10 8  1:06579 108e 0 @ e  8:07685 1014 ! 1:50667 1014   e  4:50521 10 7 t 3:73193 10 8  1 A ! = 2:5 1014  5 1014 (Hz) t = 50nm 100nm  =  5:89053 107 + 1:51844 109e e  2:60327 1015 ! 2:18067 1015  1:10074 107e e  8:7235 10 8 t 4:68116 10 8   3:0804 108e 0 @ e  2:60327 1015 ! 2:18067 1015   e  8:7235 10 8 t 4:68116 10 8  1 A ! = 5:5 1014  5:8 1014 (Hz) t = 50nm 100nm  = 3:43401 107 + 5:50512 109e e  8:34557 1014 ! 2:81694 1014   4:59404 107e e  1:32879 10 7 t 1:28984 10 7  3:23191 108e 0 @ e  8:34557 1014 ! 2:81694 1014   e  1:32879 10 7 t 1:28984 10 7  1 A ! = 5:8 1014  6:0 1014 (Hz) 112SPP Mode in a Silver-Dielectric-Silver Waveguide t = 50nm 100nm  = 5:54605 107 + 5:88966 108e e  6:81188 1014 ! 9:463 1013   5:96491 107e e  1:29969 10 7 t 2:57514 10 7  5:94126 108e 0 @ e  6:81188 1014 ! 9:463 1013   e  1:29969 10 7 t 2:57514 10 7  1 A ! = 2:5 1014  4:0 1014 (Hz) t = 5nm 50nm  =  4:87592 109 + 2:31914 1011e e  4:98913 1014 ! 3:06452 1014  4:88724 109e e   2:6278 10 8 t 5:00396 10 9   2:31762 1011e 0 @ e  4:98913 1014 ! 3:06452 1014   e   2:6278 10 8 t 5:00396 10 9  1 A ! = 4:0 1014  6:0 1014 (Hz) t = 20nm 50nm  =  1:12192 106 + 8:92744 109e e  1:63305 1015 ! 7:77546 1014  1:71474 107e e  2:54566 10 8 t 3:20063 10 8   9:24941 109e 0 @ e  1:63305 1015 ! 7:77546 1014   e  2:54566 10 8 t 3:20063 10 8  1 A ! = 4:0 1014  6:0 1014 (Hz) 113SPP Mode in a Silver-Dielectric-Silver Waveguide t = 5nm 20nm  = 5:5263 107 + 5:33629 1010e e  1:52435 1015 ! 5:03452 1014  2:58337 108e e  4:26971 10 9 t  7:10258 10 9  4:69963 1011e 0 @ e  1:52435 1015 ! 5:03452 1014   e  4:26971 10 9 t  7:10258 10 9  1 A Dielectric core n = 3.5 ! = 2:7 1014  5:0 1014 (Hz) t = 100nm 300nm  = 7:70898 106 + 8:58092 109e e  2:04069 1015 ! 9:57004 1014   4:78262 106e e  3:83384 10 7 t 1:54557 10 8  5:36255 107e 0 @ e  2:04069 1015 ! 9:57004 1014   e  3:83384 10 7 t 1:54557 10 8  1 A ! = 5:0 1014  5:5 1014 (Hz) t = 100nm 300nm  = 6:63587 107 + 1:37356 109e e  6:35074 1014 ! 7:67264 1013   7:41799 107e e  3:61797 10 7 t 1:37614 10 9  1:03749 109e 0 @ e  6:35074 1014 ! 7:67264 1013   e  3:61797 10 7 t 1:37614 10 9  1 A 114SPP Mode in a Silver-Dielectric-Silver Waveguide ! = 2:5 1014  4:8 1014 (Hz) t = 40nm 100nm  =  5:45379 106 + 1:20082 1010e e  2:82914 1015 ! 1:46163 1013  1:0073 106e e  4:9367 10 8 t 1:86263 10 8   2:86019 109e 0 @ e  2:82914 1015 ! 1:46163 1015   e  4:9367 10 8 t 1:86263 10 8  1 A ! = 4:8 1014  5:5 1014 (Hz) t = 40nm 100nm  = 7:1167 107 + 4:42601 109e e  7:71803 1014 ! 1:53727 1014   1:71814 107e e  5:16071 10 8 t 1:40188 10 8  5:46782 108e 0 @ e  7:71803 1014 ! 1:53727 1014   e  5:16071 10 8 t 1:40188 10 8  1 A ! = 2:5 1014  5 1014 (Hz) t = 20nm 40nm  =  1:16087 108 + 4:21399 109e e  1:86309 1015 ! 1:3616 1015  1:55129 108e e  2:81664 10 8 t 4:13305 10 8   5:06593 109e 0 @ e  1:86309 1015 ! 1:3616 1015   e  2:81664 10 8 t 4:13305 10 8  1 A 115SPP Mode in a Silver-Dielectric-Silver Waveguide ! = 5 1014  5:5 1014 (Hz) t = 20nm 40nm  = 1:06796 108 + 1:6759 109e e  6:72104 1014 ! 1:53218 1014   1:49326 108e e  2:93069 10 8 t 5:06187 10 8   1:87427 109e 0 @ e  6:72104 1014 ! 1:53218 1014   e  2:93069 10 8 t 5:06187 10 8  1 A ! = 2:5 1014  4:5 1014 (Hz) t = 20nm 40nm  = 9:25716 106 + 1:19191 109e e  1:06392 1015 ! 6:14568 1014   9:9057 107e e  4:14344 10 9 t  7:18321 10 9  9:86027 109e 0 @ e  1:06392 1015 ! 6:14568 1014   e  4:14344 10 9 t  7:18321 10 9  1 A ! = 4:5 1014  5:5 1014 (Hz) t = 20nm 40nm  = 8:62397 107 + 6:48528 1010e e  1:05886 1015 ! 2:73221 1014  5:25007 108e e  4:28125 10 9 t  7:07869 10 9  5:17244 1011e 0 @ e  1:05886 1015 ! 2:73221 1014   e  4:28125 10 9 t  7:07869 10 9  1 A 116TM1 Mode in a Silver-Dielectric-Silver Waveguide TM1 Mode in a Silver-Dielectric-Silver Waveguide We obtain piece-wise  t functions to the dispersion relations of the TM1 mode over frequencies where it is propagative. The  t functions for the TM1 mode can be given for di erent dielectric cores as: Dielectric core n = 1.0 For, ! = 4:5 1014  9 1014 (Hz) t = 660 800 nm  =  6:93061 107 + 9:10218 10 8! + 1:06405 1014t  1:48057 10 23!2  4:17409 1019t2  0:05735!t: ! = 6 1014  9 1014 (Hz) t = 600 660 nm  =  9:92251 107 + 1:33067 10 7! + 1:60785 1014t  2:94254 10 23!2  6:51506 1019t2  0:0959!t: ! = 7:5 1014  9 1014 (Hz) t = 400 660 nm 117TM1 Mode in a Silver-Dielectric-Silver Waveguide  =  9:92251 107 + 1:33067 10 7! + 1:60785 1014t  2:94254 10 23!2  6:51506 1019t2  0:0959!t: Dielectric core n = 1.5 For, ! = 4 1014  9 1014 (Hz) t = 550 750 nm  =  3:97487 107 + 8:22517 10 8! + 5:71795 1013t  1:84235 10 23!2  2:3324 1019t2  0:03228!t: ! = 6 1014  9 1014 (Hz) t = 350 550 nm  =  9:96598 107 + 1:63938 10 7! + 1:89559 1014t  4:68169 10 23!2  9:59265 1019t2  0:12007!t: Dielectric core n = 2.0 For, ! = 4 1014  9 1014 (Hz) t = 450 750 nm 118TM1 Mode in a Silver-Dielectric-Silver Waveguide  =  2:86733 107 + 7:95334 10 8! + 4:39631 1013t  1:4343 10 23!2  1:93033 1019t2  0:02472!t: ! = 4 1014  9 1014 (Hz) t = 350 450 nm  =  1:14088 108 + 1:82421 10 7! + 3:03427 1014t  4:46102 10 23!2  2:1498 1020t2  0:17968!t: Dielectric core n = 2.5 For, ! = 4 1014  9 1014 (Hz) t = 450 800 nm  =  1:55081 107 + 7:18463 10 8! + 2:4086 1013t  7:46259 10 24!2  1:08697 1019t2  0:0126!t: ! = 4 1014  9 1014 (Hz) t = 350 450 nm 119TM1 Mode in a Silver-Dielectric-Silver Waveguide  =  4:44726 107 + 1:09333 10 7! + 1:05957 1014t  2:16556 10 23!2  7:31737 1019t2  0:05808!t: Dielectric core n = 3.0 For, ! = 4 1014  9 1014 (Hz) t = 400 800 nm  =  1:22715 107 + 7:75821 10 8! + 2:04492 1013t  5:59765 10 24!2  9:94216 1018t2  0:00996!t: ! = 4 1014  9 1014 (Hz) t = 300 400 nm  =  3:53474 107 + 1:0746 10 7! + 9:04739 1013t  1:71373 10 23!2  6:74423 1019t2  0:04795!t: Dielectric core n = 3.5 For, ! = 4 1014  9 1014 (Hz) t = 450 800 nm 120TM1 Mode in a Silver-Dielectric-Silver Waveguide  =  8:39088 106 + 8:26824 10 8! + 1:43673 1013t  3:46659 10 24!2  7:20406 1018t2  0:00639!t: ! = 4 1014  9 1014 (Hz) t = 300 450 nm  =  2:24798 107 + 9:9966 10 8! + 5:86262 1013t  1:00665 10 23!2  4:49514 1019t2  0:02861!t: 121

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