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Metal waveguides for multi-axial light guiding at nanometer scales 2011
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Title | Metal waveguides for multi-axial light guiding at nanometer scales |
Creator |
Maqsood, Muhammad Waqas |
Publisher | University of British Columbia |
Date Created | 2012-01-11 |
Date Issued | 2012-01-11 |
Date | 2011 |
Description | A basic component of most optical systems is an optical waveguide. There has been an increased interest in nanofabricated optical waveguides that incorporate metal layers due to their fabrication compatibility with existing complimentary-metal-oxide-semiconductor (CMOS) processes and, their ability to sustain sub-wavelength confined electromagnetic modes. In this work, we have developed 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 waveguiding at sub-wavelength and nanometer scales.In the first example, we apply our analytical technique to optimize a bi-axial waveguide that can bend light at 90 degree, constructed from two uni-axial metal waveguides. The optimization procedure consists of mapping out wavevector values of the electromagnetic modes sustained by the two waveguides over the operational frequency range. The constituent materials 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 a way that the in-plane wavevector components for both waveguide modes are matched. The wavevector matching results in efficient coupling between the two modes, yielding predicted bending efficiencies over 90%. In the second example, we apply our analytical technique to optimize a bi-axial waveguide structure for coupling free-space light into surface plasmon polaritons (SPP), electromagnetic excitations bounded to the surface of a metal. We study the simple geometry consisting of a slit in a metal film, filled and covered with a dielectric. We break the configuration down into two uni-axial waveguide components. Using our approach, we optimize the materials and geometry of the slit so that wavevector matching is achieved between the light emanating from the slit and the adjacent SPP modes, yielding coupling efficiency over 68%. |
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Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2012-01-11 |
DOI | 10.14288/1.0072527 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCO |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/40001 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0072527/source |
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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 2011 Abstract Recent advancements in nanofabrication now allow precise fabrication of devices and systems on nanometer scales. This technology is currently used in the field 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-confined 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 first 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- ii Abstract 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 efficient coupling between the two modes, yielding 90◦ light bending with predicted efficiencies 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 efficient ways to couple in free-space plane-wave light. We study the simple SPP coupling geometry consisting of a slit in a metal film, filled and covered with a dielectric. We break the configuration 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 first 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 efficiencies over 68%. iii Preface 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 finite-difference 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 diffrac- 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-efficiency 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). iv Table 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 v Table 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 Modified 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 vi Table 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 vii List of Tables 1.1 Constant parameters for the Drude model for Ag and Au . . 16 viii List 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 ωp and τ (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 ωp and τ (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 ωp and τ (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 ωp and τ (line) with those obtained directly from permittivity measurements (squares). . . . . . . . . . . . . . . 20 ix List of Figures 1.6 Polynomial fit function to model the real part of the dielectric function of silver. . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Polynomial fit 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 filled with GaP, with the complete structure covered with a semi-infinite 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) figure 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) figure of merit (FOM) as a function of frequency for various dielectric core thickness values. . . . . . 53 x List 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) figure 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) figure of merit (FOM) as a function of frequency. . . . . . . . . . . . . 57 3.6 Waveguide structure consisting of a slit in a silver film where the slit is filled with GaP and the entire silver film is coated with a GaP layer of finite 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) figure of merit (FOM) as a function of frequency for the SPP mode. c) Real part of the wavevector and d) figure of merit (FOM) as a function of frequency for the TM1 mode. . . . . . . . . . . . . . . . . . . 59 3.8 A simple example demonstrating the reflection loss due to wavevector mismatch. An electromagnetic plane wave is a) incident onto a dielectric interface and b) scattered into re- flected and transmitted components after interacting with the interface. The relative amplitudes of the reflected and trans- mitted components are proportional to the degree of wavevec- tor mismatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 xi List 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 (β2) and the trans- verse wavevector components of the TM1 mode sustained by the MDM waveguide (β1z). 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 ω = 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 field intensity,|Hy|2, of the TM1 mode in the GaP layer, and the detector D3 measures |Hy|2 radiated into the air region. . . . . . . . . . . . . . . . . 68 3.12 FDTD calculated electromagnetic response of the designed structure at different incident electromagnetic wave frequencies 69 3.13 Coupling efficiency as a function of GaP layer thickness d. The frequency of the incident light is kept constant at ω = 6× 1014 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.14 Coupling efficiency as a function of λ. The slit width and layer thickness are kept constant at w = 150 nm and d = 100 nm, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 71 xii List of Figures 4.1 Waveguide structure consisting of a slit in a metal film im- mersed in a dielectric. TM-polarized light is normally incident from the bottom of the structure and is confined within the dielectric core. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Formulation of a hypothesis for diffraction-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 different widths. c) Diffraction 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 diffraction-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 |Hy|2 distri- bution (left) and the time-averaged |Hy|2 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 λ0 = 500 nm. A common saturated color scale has been used to accentuate the fields on the exit side of the slit. 84 xiii List of Figures 4.4 a) SPP coupling efficiency 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[β̃spp]−(Re[β̃z]+κp) as a function of refractive index of the dielectric region for a fixed optical slit width nw = 600 nm and free-space wavelength λ0 = 500 nm. . 88 xiv Acknowledgements 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. xv To my family for their support and for all those who seek truth xvi Chapter 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 confining light within a boundary such that light can only propagate along one or multiple directions defined by the waveguide [1]. The first 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 reflection (TIR) [3]. TIR occurs at an interface between two media having different 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 n1sinθi = n2sinθr, (1.1) 1 1.1. Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures where 0 ≤ θi ≤ 90 is the angle of incidence of the beam (with respect to normal to the interface) and θr 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 reflected. 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 fibre waveguides constructed from thin glass wires. An optical fiber consists of two concentric glass regions: a cylindrical core of higher refractive index and a surrounding cladding of lower refractive index. The optical fibre works by guiding light along the axis of the fibre and confining light within the core by total inter- nal reflection. Fiber optics have revolutionized the field 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 benefits such as fast internet speeds and instantaneous communication across continents. Along with faster information transmission via optical fibres, 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 2 1.1. Optical Waveguides: from Water Streams to Nanometer-Scale Metallic Structures research interest in using nanofabrication to miniaturize optical waveguides. The motivation for these efforts has been two-fold: first, 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 significantly faster and more powerful computers [10–13]. A major road-block in efforts to miniaturize optical waveguides is the so-called “diffraction 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 profile of the waveguide is less than the wavelength of light. At infrared and visible frequencies, this means that the minimum transverse profile 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 profile of the waveguide is significantly 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 3 1.2. Designing Uni- and Multi-Axial Optical Waveguides in the form metal nanoparticle arrays [20], metal nanowires [21–24], and nano-apertures in metal films [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 afforded by nanofab- rication, optimization of a waveguide design requires efficient 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 fields 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 configurations 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 fields 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 4 1.2. 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-finding 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-finding 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 finite-difference 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 difference equations, Maxwell’s equations are solved to determine the elec- tric and magnetic field 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 fields 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 configurations. As a result, optimization using only numerical simulations is often not feasible. 5 1.3. 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 different 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. 6 1.4. Thesis Outline 1.4 Thesis Outline This thesis is divided into two parts. The first 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 find 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 fitted 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. 7 1.5. 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 fields - the displacement field, D, the electric field, E, the magnetic flux density, B, and the magnetic field, H - to the presence of charge density, ρ, and current density, J. The equations are given by ∇ · D(r, t) = ρ, (1.2) ∇ · B(r, t) = 0, (1.3) ∇ × E(r, t) = −∂B(r, t) ∂t , (1.4) and ∇ × H(r, t) = J(r, t) + ∂D(r, t) ∂t , (1.5) where (r,t) describes the position and time dependence of the fields. The displacement field, D is related to electric field, E and magnetic flux density field, B is related to magnetic field, H via D(r, t) = oE(r, t) +P(r, t) (1.6) and B(r, t) = µoH(r, t) +M(r, t), (1.7) where o is the free-space permittivity, µo is the free-space permeability, P is the polarization field and M is the magnetization field. 8 1.5. Maxwell’s Equations 1.5.1 Constitutive Material Relations The fields P, M, and J describe the response of a medium to electro- magnetic fields. By assuming a medium that is linear, isotropic, lossless, and instantaneously responsive, we get the relations P(r, t) = oχeE(r, t), (1.8) M(r, t) = µoχmH(r, t), (1.9) and J(r, t) = σE(r, t), (1.10) where χe is the electric susceptibility, χm is the magnetic susceptibility, and σ is the electric conductivity. Inserting Eq. 1.8 into Eq. 1.6, we get D(r, t) = orE(r, t) (1.11) where r is the relative electric permittivity given by r = 1 + χe. (1.12) Similarly, inserting Eq. 1.9 into Eq. 1.7, we get B(r, t) = µoµrH(r, t), (1.13) 9 1.5. Maxwell’s Equations where µr is relative magnetic permeability given by µr = 1 + χm. (1.14) For most materials, χm ' 0 and µr = 1. 1.5.2 Electromagnetic Wave Equation We generally deal with systems having no external charge (ρ = 0). Im- posing the additional constraint that J = 0, Maxwell’s equations for such systems can be written as ∇ · D(r, t) = 0, (1.15) ∇ · B(r, t) = 0, (1.16) ∇ × E(r, t) = −∂B(r, t) ∂t , (1.17) and ∇ × H(r, t) = ∂D(r, t) ∂t . (1.18) Inserting the constitutive relation of Eq. 1.11 in Eq. 1.15 we get: ∇ · (orE(r, t)) = 0, or(∇ ·E(r, t)) +E(r, t) · ∇(or) = 0, which yields ∇ · E(r, t) = −E(r, t) · ∇(or) or . (1.19) 10 1.5. Maxwell’s Equations For an isotropic and homogeneous medium, ∇(or) = 0, which leads to the compact relation ∇ · E(r, t) = 0. (1.20) Using Eq. 1.13 with µr = 1 (for nonmagnetic materials), Eq. 1.17 be- comes ∇ × E(r, t) = −µo∂H(r, t) ∂t . (1.21) Taking the curl of both sides yields ∇ × ∇ × E(r, t) = ∇ × ( −µo∂H(r, t) ∂t ) = −µo ∂ ∂t (∇ × H(r, t)). (1.22) Using Eq. 1.18, Eq. 1.22 simplifies to ∇ × ∇ × E(r, t) = −µo∂ 2D(r, t) ∂t2 = −µoor ∂ 2E(r, t) ∂t2 . (1.23) The left side of Eq. 1.23 simplifies to ∇ ×∇ × E(r, t) = ∇(∇ · E(r, t))−∇2E(r, t) (1.24) Using Eq. 1.20 in Eq. 1.24 yields ∇ ×∇ × E(r, t) = −∇2E(r, t). (1.25) 11 1.5. Maxwell’s Equations Combining Eq. 1.23 and Eq. 1.25, we get ∇2E(r, t)− µoor ∂ 2E(r, t) ∂t2 = 0. (1.26) Given the speed of light defined by c = 1√ µoo , (1.27) we arrive at the general wave equation ∇2E(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 field yields ∇2H(r, t)− r c2 ∂2H(r, t) ∂t2 = 0. (1.29) The above equations describe the electric and magnetic field 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. 12 1.6. Drude Model Figure 1.1: Cartoon depiction of a piece of metal. The magnified part of the metal shows its microscopic constituents consisting of free electrons (red dots) moving around fixed 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, ρn, 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 field 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/τ , where τ 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 13 1.6. Drude Model E(r, t) = E(x, t)x̂. Treating a free electron like a classical particle, we have the equation of motion [32] m d2 ˜̀ dt2 + m τ d˜̀ dt = −eE(x, t), (1.30) where ˜̀, m, and e are the displacement, effective mass, and charge of an electron, respectively. Generally, the electric field is time-harmonic with the form Ẽ(ω) = E0(x)e −iωt, (1.31) where ω is the frequency. Time-harmonic excitation yields time-harmonic electron motion given by ˜̀(ω) = ˜̀oe −iωt, (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 field [33, 34]. Inserting Eq. 1.32 into Eq. 1.31 yields m(−ω2 ˜̀oe−iωt) + m τ (−iω ˜̀oe−iωt) = −eẼ(ω) −mω ˜̀oe−iωt ( ω + i τ ) = −eẼ(ω) which results in the complex amplitude ˜̀(ω) = eẼ(ω) mω(ω + i/τ) . (1.33) 14 1.6. Drude Model The displacement of individual electrons contribute towards a dipole mo- ment density. This dipole moment density yields a polarization field given by P̃ (ω) = −ρne˜̀(ω) = − ρne 2 mω(ω + i/τ) Ẽ(ω). (1.34) In frequency-domain notation, the polarization field is related to the electric field by P̃ (ω) = o(̃m(ω)− 1)Ẽ(ω), (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(ω) = 1− ρne 2 mωo(ω + i/τ) . (1.36) Given the plasma frequency for a free electron gas ω2p = ρne 2/mo, ̃m is then given by ̃m(ω) = 1− ω2p ω(ω + i/τ) . (1.37) Re-writing Eq. 1.37 into a distinctive real part, ′(ω), and imaginary part, ′′(ω), yields ̃m(ω) = 1− ω2p (ω2 + 1/τ2) + i ω2pτ ω(1 + ω2τ2) . (1.38) 15 1.7. 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 ωp (Hz) Collision rate τ −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 16 1.7. Limitations of the Drude Model Figure 1.2: Comparison between Re[m] values for gold obtained indirectly from the Drude model using experimentally-measured ωp and τ (line) with those obtained directly from permittivity measurements (squares). 17 1.7. 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 ωp and τ (line) with those obtained directly from permittivity measurements (squares). 18 1.7. Limitations of the Drude Model Figure 1.4: Comparison between Re[m] values for silver obtained indirectly from the Drude model using experimentally-measured ωp and τ (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 19 1.7. Limitations of the Drude Model Figure 1.5: Comparison between Im[m] values for silver obtained indirectly from the Drude model using experimentally-measured ωp and τ (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, modifications to the Drude model are required to accurately describe inter- band transitions. These modifications 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- 20 1.8. 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 fits to the Johnson and Christy experimental data. The order of the polynomial is chosen such that the difference between the fit and experimen- tal data is minimum. The polynomial fit 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 fit for the imaginary part of the dielectric function of silver, we divide the visible frequency spectrum into 21 1.8. Dielectric Function for Silver Figure 1.6: Polynomial fit function to model the real part of the dielectric function of silver. 22 1.8. Dielectric Function for Silver Figure 1.7: Polynomial fit 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 fit 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) 23 1.8. Dielectric Function for Silver for 9.35347× 1014 Hz≤ f < 9.35347× 1014 Hz, the fit 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 fit 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 fit function accurately mod- els experimentally-measured values of Im[m]. 24 1.9. 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 fitted 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. 25 Chapter 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 first assuming a general form of the electromagnetic fields. Application of boundary conditions to the waveguide will then yield a characteristic eigenvalue equation from which possible wavevector values, β̃, of the electromagnetic waves are mapped as a function of ω. The β̃-ω 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-infinite metal occupying the region z < 0 and a semi- infinite 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 26 2.1. 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 fields consisting of a transverse- magnetic (TM) polarized electromagnetic wave propagating along the +x- direction, which is given by H̃y(z) = H̃1e iβ̃xek̃1z z < 0 H̃2e iβ̃xe−k̃2z z > 0, (2.1) Ẽx(z) = −ik̃1 ωõ1 H̃1e iβ̃xek̃1z z < 0 ik̃2 ωõ2 H̃2e iβ̃xe−k̃2z z > 0, (2.2) and Ẽz(z) = −β̃ ωõ1 H̃1e iβ̃xek̃1z z < 0 −β̃ ωõ2 H̃2e iβ̃xe−k̃2z z > 0. (2.3) where k̃i(i = 1, 2) correspond to the exponential decay constants along the z-axis and β̃ is the wavevector. Wavevector of an electromagnetic wave 27 2.1. Uni-Axial Waveguide Composed of a Single Metal-Dielectric Interface traveling with velocity, v and radial frequency, ω is defined as β̃ = ω v . (2.4) The decay length of the electromagnetic fields from the interface is given by ld = 1 |k̃i| . (2.5) Continuity of the tangential component of the electric field at z = 0 yields − k̃1H̃1 ̃1 = k̃2H̃2 ̃2 . (2.6) Given continuity of the magnetic field at z = 0, we arrive at k̃1 k̃2 = − ̃1 ̃2 . (2.7) Application of the general electromagnetic fields to the general wave equa- tion yields k̃21 = β̃ 2 − k20 ̃1 (2.8) and k̃22 = β̃ 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 β̃ = k0 √ ̃1̃2 ̃1 + ̃2 . (2.10) 28 2.1. 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 first assuming a general form of TE-polarized electromagnetic fields given by Ẽy(z) = Ẽ1e iβ̃xek̃1z z < 0 Ẽ2e iβ̃xe−k̃2z z > 0, (2.11) H̃x(z) = ik̃1 ωo Ẽ1e iβ̃xek̃1z z < 0 −ik̃2 ωo Ẽ2e iβ̃xe−k̃2z z > 0, (2.12) and H̃z(z) = β̃ ωµo Ẽ1e iβ̃xek̃1z z < 0 β̃ ωµo Ẽ2e iβ̃xe−k̃2z z > 0. (2.13) Continuity of Ẽy and H̃x gives Ẽ1 = Ẽ2 (2.14) and Ẽ1(k̃1 + k̃2) = 0. (2.15) For k̃1, k̃2 > 0 (required for finite-energy solutions), Ẽ1 = Ẽ2 = 0 is the only possible solution. 29 2.2. 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 fields and apply boundary conditions to the fields 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 fields consisting of a transverse-magnetic (TM) polarized electromagnetic wave propagating along the +x-direction, which is given by H̃y(z) = H̃1e iβ̃xek̃1z z < −d H̃31e iβ̃xek̃3z + H̃32e iβ̃xe−k̃3z −d < z < d H̃2e iβ̃xe−k̃2z z > d. (2.16) 30 2.2. 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. Ẽx(z) = −i ωõ1 k̃1H̃1e iβ̃xek̃1z z < −d −ieiβ̃xk̃3 ωo3 H̃31e k̃3z + −ieiβ̃xk̃3 ωo3 H̃32e −k̃3z −d < z < d i ωõ2 k̃2H̃2e iβ̃xe−k̃2z z > d. (2.17) and Ẽz(z) = −β̃ ωõ1 H̃1e iβ̃xek̃1z z < −d −β̃ ωo3 H̃31e iβ̃xek̃3z + −β̃ ωo3 H̃32e iβ̃xek̃3z −d < z < d −β̃ ωõ2 H̃2e iβ̃xe−k̃2z z > d. (2.18) Continuity of H̃y and Ẽx at the boundaries z = d and z = −d yields the 31 2.2. Uni-Axial Waveguide Composed of Metal-Dielectric-Metal Layers relations H̃1e −k̃1d = H̃31e−k̃3d + H̃32ek̃3d (2.19) −k̃1 ̃1 H̃1e −k̃1d = −k̃3 3 H̃31e −k̃3d + k̃3 3 H̃32e k̃3d, (2.20) and H̃2e −k̃2d = H̃31ek̃3d + H̃32e−k̃3d (2.21) k̃2 ̃2 H̃2e −k̃2d = −k̃3 3 H̃31e k̃3d + k̃3 3 H̃32e −k̃3d. (2.22) Application of the generalized electromagnetic fields into the wave equation Eq. 1.29 and free space wavevector relation, k0 = ω √ µ00 yields the relations k̃21 = β̃ 2 − k20 ̃1, (2.23) k̃22 = β̃ 2 − k20 ̃2, (2.24) and k̃23 = β̃ 3 − k203. (2.25) Using Eqs.2.19, 2.21, 2.23, 2.24, and 2.25, we obtain the dispersion relation e4k̃3d = k̃3/3 − k̃1/̃1 k̃3/3 + k̃1/̃1 k̃3/3 − k̃2/̃2 k̃3/3 + k̃2/̃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 32 2.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers constructed using same material, ̃1 = ̃2 and k̃1 = k̃2, Eq.2.26 simplifies to e4k̃3d = ( k̃3/3 − k̃1/̃1 k̃3/3 + k̃1/̃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 e2k̃3d = − k̃3/3 − k̃1/̃1 k̃3/3 + k̃1/̃1 , (2.28) which can be re-expressed in the simple relation coth(k̃3d) = − k̃13 k̃3̃1 . (2.29) The asymmetric dispersion relation takes the form e2k̃3d = k̃3/3 − k̃1/̃1 k̃3/3 + k̃1/̃1 , (2.30) or more simply coth(k̃3d) = − k̃3̃1 k̃13 . (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 33 2.3. 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 fields in the bottom layer (z < (z2 = 0)) and in the uppermost layer (z > zn) is assumed as H̃1y = H̃1e iβ̃xek̃1z, (2.32) and H̃ny = H̃ne iβ̃xek̃n(zn−z), (2.33) respectively, where H̃1 is the magnetic field at z = z2 and H̃n is the magnetic field at z = zn. Within an arbitrary layer j, the magnetic field is given by H̃jy = H̃j(z)e iβ̃x. (2.34) Applying the general magnetic fields into the electromagnetic wave equation Eq. 1.29, we obtain following relation ∂2H̃j(z) ∂z2 + (k̃2o ̃j − β̃2)H̃j(z) = 0. (2.35) The solution for the above equation is H̃j(z) = C̃j sin(k̃j(zj − z)) + D̃j cos(k̃j(zj − z)) (2.36) where k̃j = √ β̃2 − k2o ̃j (2.37) 34 2.3. 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. Differentiating Eq. 2.36 with respect to z yields d dz H̃j(z) = −C̃j k̃j cos(k̃j(zj − z)) + D̃j k̃j sin(k̃j(zj − z)). (2.38) To determine the values of C̃j and D̃j , we evaluate the magnetic field and its differential at z = zj to yield D̃j = H̃j(zj) (2.39) C̃j = 1 k̃j d dz H̃j(zj). (2.40) 35 2.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers Inserting C̃j and D̃j into Eq. 2.36 and Eq. 2.38 yields H̃j(z) = 1 k̃j d dz H̃j(zj) sin(k̃j(zj − z)) +H̃j(zj) cos(k̃j(zj − z)) (2.41) d dz H̃j(z) = − d dz H̃j(zj) cos(k̃j(zj − z)) +H̃j(zj)k̃j sin(k̃j(zj − z)). (2.42) In matrix form, we can write Eq. 2.41 H̃j(z) d dz H̃j(z) = cos(k̃j(zj − z)) 1k̃j sin(k̃j(zj − z)) k̃j sin(k̃j(zj − z)) − cos(k̃j(zj − z)) H̃j(zj) d dz H̃j(zj) . (2.43) The corresponding electric fields are given by Ẽjx = − i ωõj eiβ̃x d dz H̃j(z) (2.44) Ẽjz = − β̃ ωõj eiβ̃xH̃j(z). (2.45) Applying boundary conditions for the tangential components of the elec- tric and magnetic field at the arbitrary interface z = zj yields H̃j(zj) = H̃j−1(zj) (2.46) and Ẽjx = Ẽ(j−1)x 36 2.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers 1 ̃j d dz H̃j(zj) = 1 ̃j−1 d dz H̃j(zj−1) d dz H̃j(zj) = ̃j ̃j−1 d dz H̃j−1(zj), (2.47) which can be re-expressed in matrix form as H̃j(zj) d dz H̃j(zj) = 1 0 0 ̃j ̃j−1 H̃j−1(zj) d dz H̃j−1(zj) . (2.48) At the interface z = zj+1, from Eq. 2.43 H̃j(zj+1) d dz H̃j(zj+1) = cos(k̃j(dj)) 1k̃j sin(k̃j(dj)) k̃j sin(k̃j(dj)) − cos(k̃j(dj)) H̃j(zj) d dz H̃j(zj) where dj = zj+1 − zj is the thickness of the jth layer. Putting values from Eq. 2.48 into the above equation H̃j(zj+1) d dz H̃j(zj+1) = T̃j H̃j−1(zj) d dz H̃j−1(zj) where T̃j = cos(k̃j(dj)) 1k̃j sin(k̃j(dj)) k̃j sin(k̃j(dj)) − cos(k̃j(dj)) 1 0 0 ̃j ̃j−1 . (2.49) The magnetic field in layer 1 is given by Eq. 2.32. Using Eq. 2.48, we can 37 2.3. Uni-Axial Waveguide Composed of an Arbitrary Number of Metal-Dielectric Layers relate the magnetic field at interface z3 with the field in layer 1 by H̃2(z3) d dz H̃2(z3) = T̃2 H̃1 k1H̃1 . (2.50) Similarly, the field at interface z3 can be related to the field at interface z2 by H̃3(z4) d dz H̃3(z4) = T̃3 H̃2(z3) d dz H̃2(z3) . From above two relations, we can relate the field in layer 3 at interface z4 with the field in layer 1 by H̃3(z4) d dz H̃3(z4) = T̃3T̃2 H̃1(z2) k̃1H1(z2) . (2.51) In general, we can relate the field in layer j − 1 at interface zj with the field in layer 1 by H̃j−1(zj) d dz H̃j−1(zj) = ˜Tj−1 ˜Tj−2...T̃2 H̃1(z2) k̃1H̃1(z2) . (2.52) Further, we can relate the field in layer j at interface zj with the field in layer 1 using relation 2.48 by H̃j(zj) d dz H̃j(zj) = Ṽj ˜Tj−1 ˜Tj−2...T̃2 H̃1(z2) k̃1H̃1(z2) (2.53) 38 2.4. Davidenko Method where Ṽj = 1 0 0 ̃j ̃j−1 . We now define the quantity W̃j = Ṽj ˜Tj−1 ˜Tj−2...T̃2 = W̃11 W̃12 W̃21 W̃22 . We can now relate the fields at interface z2 = 0 with fields at interface z = zn by H̃n(zn) −k̃nH̃n(zn) = W̃j H̃1(0) k̃1H̃1(0) . (2.54) The above matrix can be resolved into the simplified equation W̃11 + W̃12k̃1 + W̃21 k̃n + W̃22 k̃1 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 β̃. 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 flow of electromagnetic waves within the 39 2.4. 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 first 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 sufficiently near the actual solution. This condition can be relaxed by inserting a factor, t, such 40 2.4. 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 differential. 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 → ∞. The differential 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 41 2.4. Davidenko Method waveguides. We can write the general dispersion relation as F (ω; β̃) = 0, (2.64) where the wavevector β̃ is a complex quantity and can be written in its real and imaginary components as β = a+ ib. (2.65) In other words, we can write the dispersion relation 2.64 as F (ω; a, b) = 0. (2.66) We can resolve Eq. 2.66 into two sets of equations given by R(ω; a, b) = Re[F (ω; a, b)] = 0 (2.67) I(ω; a, b) = Im[F (ω; 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 satisfies the Cauchy-Riemann relations ∂R ∂a = ∂I ∂b Ra = Ib (2.69) 42 2.4. Davidenko Method and ∂R ∂b = −∂I ∂a Rb = −Ia. (2.70) From the property of analytical functions, we have Fβ = ∂F ∂β = Ra + jIa = Ra − jRb, (2.71) which yields Ra = ∂R ∂a = Re[Fβ] (2.72) Rb = ∂R ∂b = −Im[Fβ]. (2.73) The Jacobian matrix in this case will be J = Ra Rb Ia Ib . (2.74) Using Eqs. 2.69 and 2.70, the Jacobian matrix becomes J = Ra Rb −Rb Ra . (2.75) To obtain the inverse Jacobian matrix, we first get the determinant of 43 2.4. Davidenko Method Eq. 2.75 by detJ = R2a + R 2 b = |Fβ|2, (2.76) which yields the inverse Jacobian matrix J−1 = 1 detJ Ra −Rb Rb Ra = 1 |Fβ|2 Re[Fβ] Im[Fβ] −Im[Fβ] Re[Fβ]. . (2.77) Now we can write the Davidenko equation for the general dispersion relation as d dt a b = −J−1 R I . Putting the value of the inverse Jacobian matrix yields d dt a b = − 1|Fβ̃|2 Re[Fβ̃] Im[Fβ̃] −Im[Fβ̃] Re[Fβ̃]. R I . (2.78) From Eq. 2.78, we can obtain separate real, ordinary differential equations for the real and imaginary parts of the wavevector da dt = − 1|Fβ̃|2 (Re[F ]Re[Fβ̃] + Im[F ]Im[Fβ̃]), (2.79) and db dt = 1 |Fβ̃|2 (Re[F ]Im[Fβ̃] + Im[F ]Re[Fβ̃]), (2.80) which are considerably easier to solve than the original complex transcen- 44 2.5. 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 different 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 different sections of the multi-axial waveguide. 45 Chapter 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 flow 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-efficiency nanoscale ±90◦ light-bending structure by mode selection and tailoring,” Applied Physics Letters 97, 151111 (2010), American In- stitute of Physics. 46 Chapter 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 different 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, efficient coupling between different sections of the multi-axial waveg- uide is predicted. The predictions made using this simple analytical ap- proach are confirmed 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 efficient 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 different 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 47 3.1. Material and Geometry Selection only propagate over a few microns at visible frequencies. Furthermore, there is significant 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 efficiency. A maximum bending efficiency 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 efficiency 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 effective 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 first consider a basic multi-axial T-junction waveguide geometry as shown in Fig. 3.1. The waveguide consists of semi-infinite silver metal film with a slit in the center. The slit is filled 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 ω and wavelength 48 3.2. Mode Selection Figure 3.1: Waveguide structure consisting of a silver slit filled with GaP, with the complete structure covered with a semi-infinite GaP layer. GaP is chosen due to its higher refractive index of n = 3.5. λ. For analysis, the region before the bend is approximated as an infinite 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 49 3.2. Mode Selection out to describe the propagation characteristics of the wave inside different parts of the waveguide. A mode is a confined electromagnetic wave which is described by a distinctive wavevector value at a given frequency. This wavevector value corresponds to transverse field amplitudes that are inde- pendent of the mode propagation. For a given waveguide geometry only a finite 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 figure-of-merit (FOM) given by FOM = Re(β̃) Im(β̃) . (3.1) A mode is considered propagative if it can propagate at least one complete cycle. This condition is achieved when FOM > 2pi. Using eigenvalue equa- tions derived in Chapter 2 and the Davidenko method, we obtain values of the complex wavevector, β̃, for the uni-axial waveguides. From β̃, we cal- culate FOM curves for the waveguides and identify the propagative modes having FOM > 2pi. We first 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(β̃) 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(β̃) curves have an inflection point near 5.5 × 1014 50 3.2. 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) figure of merit (FOM) as a function of frequency for various dielectric core thickness values. 51 3.2. 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(β̃) 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(β̃) 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(β̃) curves are characterized by an inflection 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. 52 3.2. 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) figure of merit (FOM) as a function of frequency for various dielectric core thickness values. 53 3.2. 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) figure of merit (FOM), as a function of frequency for various dielectric core thickness values. 54 3.2. Mode Selection We finally consider the first 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(β̃) 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(β̃) curve of the TM1 mode is characterized by a cutoff frequency - a frequency below which the wavevector value vanishes - and Re(β̃) values that monotonically increase as a function of frequency. Near the surface plasmon resonance frequency, the FOM values of the TM1 mode are significantly 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(β̃) 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(β̃) for the SPP mode at the metal dielectric interface is characterized by an inflection 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 55 3.3. Modified Bi-Axial Waveguide Structure above it. Based on our analysis, a nanometer-scale GaP-filled slit in a silver film coated with a semi-infinite GaP layer can potentially sustain four types of modes. In the GaP-filled slit section of the structure, three modes are pos- sible: asymmetric and symmetric SPP modes and the first order TM1 mode (it can be shown that the higher order TM modes are all cutoff). 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 cutoff 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 cutoff 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-filled slit. 3.3 Modified Bi-Axial Waveguide Structure We next modify the structure so that the GaP layer on the metal film has a finite 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 modified structure, the region on top of the metal film 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 56 3.3. Modified 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) figure of merit (FOM) as a function of frequency. 57 3.3. Modified Bi-Axial Waveguide Structure Figure 3.6: Waveguide structure consisting of a slit in a silver film where the slit is filled with GaP and the entire silver film is coated with a GaP layer of finite 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(β̃) and FOM curves for the SPP mode, and the first-order TM1 mode sustained by an MDD waveguide for core thickness values ranging from 100nm to 300nm (the higher-order TM modes are cutoff 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 58 3.3. Modified Bi-Axial Waveguide Structure Figure 3.7: Modal solution for an MDD (Ag-GaP-Air) waveguide. a) Real part of the wavevector and b) figure of merit (FOM) as a function of fre- quency for the SPP mode. c) Real part of the wavevector and d) figure of merit (FOM) as a function of frequency for the TM1 mode. 59 3.4. 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 difference between the wavevector values of an electromagnetic wave in two regions of space. The concept of wavevector mismatch and its effects 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, β̃, by β̃ = 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 reflection 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- 60 3.4. Wavevector Matching Medium1 Medium 2 Transmitted lightReflected light Medium 1 Medium 2 Incident light a) b) Figure 3.8: A simple example demonstrating the reflection loss due to wavevector mismatch. An electromagnetic plane wave is a) incident onto a dielectric interface and b) scattered into reflected and transmitted com- ponents after interacting with the interface. The relative amplitudes of the reflected and transmitted components are proportional to the degree of wavevector mismatch. 61 3.4. 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 reflection and transmission coefficients, respectively. The above relations show that as the difference in refractive indices increases (which then yield a wavevector mismatch), a greater portion of the incident electromagnetic wave is reflected and less transmitted. We extend the concept of wavevector matching to predict the coupling efficiency 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 difference between transverse component of the wavevector, β1z, directed along the z-axis of the TM1 mode in the slit and the longitudinal component of the wavevector,β2, directed along the x-axis of the TM1 mode in the GaP layer. The configuration is shown in Fig- ure 3.9. Thus, efficient coupling is achieved when the condition wavevector matching β1z = β2 is satisfied. 62 3.4. 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. 63 3.4. Wavevector Matching Figure 3.10: Plot of the longitudinal wavevector components of the TM1 mode sustained by the MDD waveguide (β2) and the transverse wavevector components of the TM1 mode sustained by the MDM waveguide (β1z). The blue circle shows the matching point at the operational frequency of w = 6× 1014 Hz. 64 3.5. Waveguide Structure Optimization 3.5 Waveguide Structure Optimization To determine an optimal configuration that yields wavevector matching, we map out values of β1z of the TM1 mode in the slit as a function of the slit width and β2 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 ω = 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 ω = 6 × 1014 Hz, a TM1 mode sustained in the slit couples efficiently 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 finite-difference-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 difference equations using the central-difference approximations [54]. At a particular time step, the difference equations are solved at each grid point to determine the local amplitude of the electric and magnetic field components. To model the electromagnetic response of the proposed coupling struc- 65 3.5. 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 µm. 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 reflect back into the simulation space. Within the simulation space, we defined both an electro- magnetic wave source and our proposed structure. The source consists of a TM-polarized electromagnetic wave oscillating at a frequency of ω = 6×1014 Hz, which corresponds to a free-space wavelength λ = 500nm. The TM- polarized electromagnetic wave has a magnetic field component Hy and two electric field 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 defined 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 field distributions calculated by the FDTD simulations, we measure the bending efficiency of the structure. Line detectors D1, D2, and D3 are placed at different locations in the simulation space to measure the time-averaged values of the magnetic field intensity |Hy|2. 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 66 3.5. Waveguide Structure Optimization bending efficiency is quantified by the ratio of |Hy|2 captured by D1,D2 to the total |Hy|2 emitted from the slit and captured by D1, D2, and D3. The bending efficiency 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 field intensity distribution when the structure is illuminated at a wavelength of λ = 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 λ = 500 nm, highly efficient 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 field intensity distribution when the structure is illuminated at wavelengths of λ = 450 nm and λ = 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 field distributions suggests that the efficiency 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 efficiency as a function of the GaP layer thickness d for a fixed wavelength λ = 500 nm. The FDTD simulations show that the bending efficiency reaches a maximum of 92% at 67 3.5. Waveguide Structure Optimization Figure 3.11: Simulation geometry of the waveguide structure designed to bend incident light at a frequency ω = 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 field intensity,|Hy|2, of the TM1 mode in the GaP layer, and the detector D3 measures |Hy|2 radiated into the air region. 68 3.5. Waveguide Structure Optimization Figure 3.12: FDTD calculations to determine the bending efficiency 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 field intensity for normally-incident, free-space, TM-polarized illumination of the structure at (a) ω = 6 × 1014 Hz, λ = 500 nm (b) ω = 6.67 × 1014 Hz, λ = 450 nm, and (c) ω = 7.5 × 1014 Hz, λ = 400nm for fixed parameters w = 150nm and d = 100nm. The inset in (a) highlights the TM mode emanating from the slit to split symmetrically into TM1 modes confined inside the GaP layer. 69 3.5. Waveguide Structure Optimization Figure 3.13: Coupling efficiency as a function of GaP layer thickness d. The frequency of the incident light is kept constant at ω = 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 efficiency drops off for layer thickness values both smaller and larger than the optimal value. Figure 3.14 depicts the FDTD-calculated bending efficiency as a func- tion of the wavelength of incident light for fixed slit width, w = 150 nm and GaP layer thickness, d = 100 nm. The FDTD calculations show that the bending efficiency is greater than 90% for incident wavelengths λ = 450 nm and λ = 500 nm. Increasing or decreasing the wavelength outside of these 70 3.5. Waveguide Structure Optimization Figure 3.14: Coupling efficiency as a function of λ. 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 efficiency 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 efficient mode coupling. For param- eters departing from these optimal parameters, the coupling efficiency drops dramatically. 71 3.6. 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 first 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 efficient light bending. The performance of the designed structure is modeled using a finite-difference-time-domain (FDTD) technique which solves Maxwell’s equations directly over a discretized spatial grid. The FDTD-calculated bending efficiency 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 efficient coupling. Our design methodology is useful because it can poten- tially circumvent the need for tedious numerical simulations and may find application in the design of highly-integrated, miniaturized optical circuits. 72 Chapter 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 efficiency and throughput remains an important objective. Plane-wave modes directly incident onto a metal- dielectric interface cannot efficiently 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. 73 Chapter 4. Multi-Axial Surface-Plasmon-Polariton Coupling A widely-used scatterer-based SPP coupling technique is to illuminate a slit in a metal film. 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 diffracts at the slit exit. The total light intensity leaving the slit exit defines the total throughput, the SPP intensity leaving the slit exit defines the SPP throughput, and the ratio of the SPP throughput to the total throughput defines the SPP coupling efficiency. The throughput and efficiency 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 efficiency. It has already been demonstrated that the SPP coupling efficiency 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 significantly affect the SPP throughput. The objective of this chapter is to design a SPP coupling scheme capable of both high throughput and high efficiency. 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 defined 74 Chapter 4. Multi-Axial Surface-Plasmon-Polariton Coupling by a single, unique solution to Maxwell’s equations when the magnetic field boundary conditions are imposed at the metal surface. The guided mode in the slit, on the other hand, generally consists of a superposition of infinite TM-polarized waveguide eigenmodes, and each of these modes correspond to a unique solution to the Maxwell’s equations when the magnetic field 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 first-order TM1 eigenmode, TM1 mode, just above cut- off, which then couples to the SPP mode at the slit exit. This is in contrast to previously explored SPP coupling configurations using sub-wavelength slits that sustain only the lowest-order TM0 eigenmode [60–66]. We show 75 4.1. Hypothesis that the TM1 mode just above cutoff 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 diffraction at the slit exit), equals to the wavevector of the SPP mode on the metal surface, high SPP coupling efficiency 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 satisfies the predicted wavevector match- ing condition, yielding a peak SPP coupling efficiency 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 efficiency and an over order-of-magnitude greater SPP throughput. 4.1 Hypothesis Consider a semi-infinite layer of metal (silver) with relative complex per- mittivity ̃m extends infinitely 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 film. The metal film is immersed in a homogeneous dielectric medium with relative permittivity d and refractive index n = √ 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- 76 4.1. Hypothesis length λ = λ0/n and wavevector βi = βix̂, where βi = 2pi/λ. The +x-axis defines the longitudinal direction, and the z-axis defines the transverse axis. The electromagnetic wave couples into a guided mode in the slit having complex wavevector β̃ = β̃xx̂ + β̃z ẑ, where β̃x and β̃z 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 fig- 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 ±β̃sppẑ, where Re[β̃spp] and Im[β̃spp] describe the spatial peri- odicity and attenuation, respectively, of the SPP field along the transverse direction. We treat the bi-axial waveguide SPP coupler by first 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 film 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 fitting 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 77 4.1. Hypothesis Silver Silver w t Dielectric Dielectric z x y Figure 4.1: Waveguide structure consisting of a slit in a metal film immersed in a dielectric. TM-polarized light is normally incident from the bottom of the structure and is confined within the dielectric core. 78 4.1. 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 figure of merit and a higher-frequency region of high figure of merit, separated by a kneel located at a cutoff frequency. The cutoff slit width wc for the TM1 mode at a given frequency ω is the threshold slit width value below which the TM1 mode is attenuating. At a fixed visible frequency ω = 6.0 × 1014 Hz (λ = 285 nm), wc ∼ 300 nm. The dominant mode in the slit can be identified 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[β̃z], 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 β̃z are obtained from the relation β̃z = √ βi 2 − β̃2x, (4.1) where βi = nβ0 is the magnitude of the wavevector in the dielectric core of the slit. Figure 4.2(b) shows Re[β̃z] 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 ω = 6.0 × 1014 Hz, Re[β̃z] for the TM0 mode in the w = 200 nm slit is nearly two orders of magnitude smaller than Re[β̃z] for the TM1 mode in 79 4.1. Hypothesis bspp bspp bb b b b b b b b b b b b b Figure 4.2: Formulation of a hypothesis for diffraction-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 different widths. c) Diffraction 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 diffraction-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. 80 4.1. Hypothesis the w = 350 nm and w = 500 nm slits. Values of Re[β̃z] for the TM1 mode generally increase for decreasing slit width. Given the parameters in Fig. 4.2(b) and for a fixed ω = 6.0 × 1014 Hz, Re[β̃z] 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. Diffraction at the slit exit generates transverse spatial frequency com- ponents, κ. The diffraction spectrum is a distribution of transverse spatial frequencies generated by diffraction at the slit exit. We calculate the diffrac- tion spectrum by Fourier transformation of the transverse field profiles of the guided mode [67]. Figure 4.2(c) shows the normalized diffraction spectrum for slit widths w = 200 nm, w = 350 nm, and w = 500 nm at a fixed fre- quency ω = 6.0×1014 Hz. The peak transverse spatial frequency component, κp, is the spatial frequency at which the diffraction spectrum peaks. For the parameters in Fig. 4.2(c), κp shifts from 1.6×107 m−1 to κp = 8.3×106 m−1 as the slit width increases from w = 200 nm to w = 500 nm. It is noteworthy that κp < Re[β̃spp] for all slit width values. A simple picture of diffraction-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 fixed ω = 6.0 × 1014 Hz is presented in Fig. 4.2(d). SPP coupling at the slit exit is mediated by diffraction of the guided mode, yielding a net real transverse wavevector component Re[β̃z] +κp. Coupling from the diffracted mode at the slit exit to the SPP mode adjacent to the slit exit is opti- mized when the wavevector-matched condition Re[β̃z] + κp = Re[β̃spp] is satisfied. Because Re[β̃spp] is generally larger than both Re[β̃z] and κp, large and commensurate contributions from both Re[β̃z] and κp are required to 81 4.2. Methodology fulfill wavevector matching. In a sub-wavelength slit, the TM0 mode has Re[β̃z] << κp and SPP coupling at the slit exit requires a sufficiently small slit width to generate large diffracted spatial frequency components to match with Re[β̃spp]. On the other hand, a super-wavelength slit sustains a TM1 mode with Re[β̃z] ' κp. The large contributions of Re[β̃z] to the net real transverse wavevector component reduces the required contributions from κp 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 efficiency of a slit immersed in a dielectric is modeled using finite-difference-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 reflections 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 λ0 = 500 nm and is TM-polarized such that the magnetic field, 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 82 4.3. Results and Discussion incident electromagnetic wave (normal), and the wavelength of the incident electromagnetic wave (λ0 = 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 efficiency, η. The dependent variables are quantified by placing line detec- tors in the simulation space to capture different 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 λ0/4 nm into the dielectric region above the metal, and are situated adjacent to the slit exit a length λ0 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 efficiency is then calculated by the equation η = 1 1 + Ir/Ispp . (4.2) 4.3 Results and Discussion The numerical simulations provide evidence of high-throughput and high- efficiency SPP coupling from a slit of super-wavelength width. Figure 4.3 displays representative snap-shots of the instantaneous |Hy|2 intensity and time-averaged |Hy|2 angular distribution calculated from FDTD simula- 83 4.3. Results and Discussion Figure 4.3: Images of the FDTD-calculated instantaneous |Hy|2 distribu- tion (left) and the time-averaged |Hy|2 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 λ0 = 500 nm. A common saturated color scale has been used to accentuate the fields on the exit side of the slit. 84 4.3. 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 field 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. Diffraction 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. Diffraction 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 efficiency of SPP coupling. Diffraction 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 efficiencies calculated from the FDTD simu- 85 4.3. Results and Discussion Figure 4.4: a) SPP coupling efficiency 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. 86 4.3. Results and Discussion lations are compared to qualitative predictions from the model of diffraction- assisted SPP coupling described in Fig. 4.4. Figure 4.4(a) plots the FDTD- calculated SPP coupling efficiencies 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 < λ0, highest SPP coupling efficiency is observed for the smallest optical slit width. This trend is consistent with diffraction-dominated SPP coupling predicted to occur for sub-wavelength slit widths, in which small slit width is required to yield large diffracted spa- tial frequencies to achieve wavevector matching. For super-wavelength slit width values nw > λ0, the SPP coupling efficiencies 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 fluctuations in the SPP coupling efficiencies 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 efficiency rises as nw increases above λ0 and reaches local maxima of η = 14%, 44%, 68%, and 48% at a super-wavelength opti- cal slit width nw ' 600 nm, respectively. The rapid increase in η 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 efficiency peak at nw = 600 nm observed for lower refractive index values is absent for n = 2.5. 87 4.3. Results and Discussion Figure 4.5: Wavevector mismatch Re[β̃spp]− (Re[β̃z] + κp) as a function of refractive index of the dielectric region for a fixed optical slit width nw = 600 nm and free-space wavelength λ0 = 500 nm. 88 4.3. 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 efficiency, the total throughput and the SPP throughput is low. As the optical slit width increases to w ' λ0 from sub- wavelength values, an increase in Ir and a decrease in Ispp yield low SPP coupling efficiency. In the super-wavelength range of optical slit width val- ues, 520 nm < nw < 700 nm, concurrently high SPP throughput and high SPP coupling efficiency (η > 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 significantly greater than Ispp, resulting again in low SPP coupling efficiencies. Variations in the peak SPP coupling efficiency at a fixed optical slit width nw = 600 nm for varying n can be qualitatively explained by the mismatch between the net real transverse wavevector component Re[β̃z] + κp and the real SPP wavevector Re[β̃spp]. Figure 4.5 plots the transverse wavevector mismatch Re[β̃spp] − (Re[β̃z] + κp) 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 efficiency at nw = 600 nm and that which yields zero wavevector mismatch supports the hypothesis that optimal SPP coupling efficiency occurs when Re[β̃spp] = (Re[β̃z] + κp), and that this condition can be achieved using a 89 4.4. 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 efficiency 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-efficiency. The crux of the design is a super-wavelength slit aperture immersed in a uniform dielectric sustaining a TM1 mode just above cutoff. High SPP coupling efficiency is achieved when the transverse wavevector component of the TM1 mode, added with the peak diffracted 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 efficiency coupling. The parameters also match the parameters predicted to yield wavevector matching. 90 Chapter 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 afforded by nanofabrication, opti- mization of a waveguide design requires efficient 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- figurations 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 finite-difference time-domain (FDTD) technique. Numerical simulation tools like the FDTD technique enable detailed visualization of electromag- netic fields within complicated optical waveguides systems. A limitation 91 5.1. 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 configurations. 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 efficient 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 efficiency. In both examples, the optimized parameters selected using our methodology is shown, through FDTD simulations, to yield max- imum coupling efficiency. 5.1 Limitations One of the major limitations of this technique is that the uni-axial waveg- uide sub-components are assumed to be infinitely-long, which is typically not 92 5.2. 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. 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Kihm et al., “Control of surface plasmon generation efficiency by slit-width tuning,” Applied Physics Letters, vol. 92, p. 051115, 2008. [65] O. T. A. Janssen et al., “On the phase of plasmons excited by slits in a metal film,” Optics Express, vol. 14, pp. 11823 – 11832, 2006. [66] Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Physical Review Letters, vol. 86, pp. 5601 – 5603, 2001. [67] M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Applied Optics, vol. 34, pp. 3055 – 3063, 1995. 101 Appendix: Fit Functions In Chapters 3 and 4, we discussed the design of multi-axial metallic waveguides using the solutions to dispersion relations for uni-axial metallic waveguides. Dispersion relations for uni-axial metallic waveguides are gen- erally complex transcendental equations which are difficult to solve using simple algebraic techniques and require sophisticated iterative techniques. 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 fit 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 fit functions to the dispersion relations of the sym- metric SPP mode over frequencies where it is propagative. The fit functions for the symmetric SPP mode can be given for different dielectric cores as: 102 SPP Mode in a Silver-Dielectric-Silver Waveguide Dielectric Core n = 1.0 ω = 8.5× 1014 − 8.75× 1014 (Hz) and t = 150nm−300nm β = 1.72173× 107 − 8.2303× 10−8ω − 1.32669× 1014t +1.20818× 10−22ω2 + 9.33943× 1019t2 + 0.09494ωt ω = 8.5× 1014 − 8.75× 1014 (Hz) and t = 50nm−150nm β = −2.09062× 107 − 3.46651× 10−8ω + 1.09635× 1014t +1.39199× 10−22ω2 + 8.8537× 1020t2 − 0.45034ωt ω = 3.5× 1014 − 8.5× 1014 (Hz) and t = 150nm−300nm β = 6.12927× 106 + 3.20113× 10−8ω +−6.05407× 10−23ω2 +2.00357× 10−37ω3 +−3.16232× 10−52ω4 + 1.93617× 10−67ω5 −9.32617× 1013t+ 5.71877× 1020t2 − 1.95536× 1027t3 +3.57259× 1033t4 − 2.7277× 1039t5 ω = 4.75× 1014 − 8.2× 1014 (Hz) and t = 50nm−150nm β = 89293.84003− 1.72137× 10−7ω + 6.77262× 10−22ω2 −1.08811× 10−36ω3 + 7.76656× 10−52ω4 − 1.62355× 10−67ω5 +1.21543× 1015t− 2.30847× 1022t2 + 2.04098× 1029t3 −8.71403× 1035t4 + 1.45453× 1042t5 103 SPP Mode in a Silver-Dielectric-Silver Waveguide Dielectric core n = 1.5 ω = 3.5× 1014 − 7.5× 1014 (Hz) and t = 120nm−300nm β = −8.18148× 107 + 1.09905× 10−6ω − 4.73456× 10−21ω2 +1.04218× 10−35ω3 − 1.13622× 10−50ω4 + 4.93144× 10−66ω5 −2.19077E14t+ 1.63401× 1021t2 − 6.56117E27t3 +1.36474× 1034t4 − 1.15377× 1040t5 ω = 3.5× 1014 − 6× 1014 (Hz) and t = 10nm−80nm β = −5.75014× 107 + 3.15718× 1011e−e ( 1.96374×1015−ω 1.11855×1015 ) +6.34943× 107e−e ( −7.35261×10−8−t 1.43303×10−8 ) −3.15122× 1011e −e ( 1.96374×1015−ω 1.11855×1015 ) − e ( −7.35261×10−8−t 1.43303×10−8 ) ω = 6× 1014 − 7.5× 1014 (Hz) t = 50nm−80nm β = 8.3505× 106 + 6.23672× 108e−e ( 1.14896×1015−ω 4.84616×1014 ) +6.66111× 106e−e ( 5.69646×10−8−t 7.42459×10−8 ) −6.82193× 108e −e ( 1.14896×1015−ω 4.84616×1014 ) − e ( 5.69646×10−8−t 7.42459×10−8 ) ω = 6× 1014 − 7.5× 1014 (Hz) 104 SPP Mode in a Silver-Dielectric-Silver Waveguide t = 5nm−30nm β = 3.6674× 107 + 1.90183× 109e−e ( 1.14951×1015−ω 3.07971×1014 ) +1.36918× 108e−e ( 4.09602×10−9−t −7.24051×10−9 ) +2.03376× 1010e −e ( 1.14951×1015−ω 3.07971×1014 ) − e ( 4.09602×10−9−t −7.24051×10−9 ) ω = 6× 1014 − 7.5× 1014 (Hz) t = 30nm−50nm β = 1.07351× 106 + 1.27678× 109e−e ( 1.14951×1015−ω 4.57803×1014 ) +2.49982× 107e−e ( 2.83143×10−8−t 3.64897×10−8 ) −1.74087× 109e −e ( 1.14951×1015−ω 4.57803×1014 ) − e ( 2.83143×10−8−t 3.64897×10−8 ) ω = 3.5× 1014 − 6× 1014 (Hz) t = 80nm−120nm β = −814849.18221 + 4.41699E × 10−8ω + 1.16474× 1013t +1.06179× 10−23ω2 + 1.32263× 1014t2 − 0.10634ωt ω = 6× 1014 − 7.5× 1014 (Hz) t = 80nm−120nm β = 2.28031× 107 − 9.72998× 10−8ω + 1.75929× 1014t 105 SPP Mode in a Silver-Dielectric-Silver Waveguide +1.78671× 10−22ω2 − 7.58382× 1014t2 − 0.37664ωt Dielectric core n = 2.0 ω = 3× 1014 − 6× 1014 (Hz) t = 120nm−300nm β = −1.97964× 107 + 1.6671× 109e−e ( 2.85129×1015−ω 1.97609×1015 ) +7.20147× 106e−e ( −8.48997×10−9−t 6.252×10−8 ) −6.73854× 108e −e ( 2.85129×1015−ω 1.97609×1015 ) − e ( −8.48997×10−9−t 6.252×10−8 ) ω = 6.5× 1014 − 7.1× 1014 (Hz) t = 120nm−300nm β = 3.51039× 107 + 7.55202× 109e−e ( 1.00296×1015−ω 1.76148×1014 ) −1.86391× 106e−e ( 4.00369×10−7−t 2.17345×10−9 ) 6.00459× 107e −e ( 1.00296×1015−ω 1.76148×1014 ) − e ( 4.00369×10−7−t 2.17345×10−9 ) ω = 6.0× 1014 − 6.5× 1014 (Hz) t = 120nm−300nm β = 2.82033× 107 + 1.52588× 108e−e ( 7.86251×1014−ω 1.46422×1014 ) 6.00435× 109e−e ( −6.37561×10−7−t −3.68951×10−7 ) 106 SPP Mode in a Silver-Dielectric-Silver Waveguide −3.31606× 1010e −e ( 7.86251×1014−ω 1.46422×1014 ) − e ( −6.37561×10−7−t −3.68951×10−7 ) ω = 3.0× 1014 − 6.0× 1014 (Hz) t = 30nm−120nm β = −3.53696× 107 + 3.13797× 109e−e ( 4.46772×1015−ω 2.98473×1015 ) 1.13447× 107e−e ( 4.94075×10−8−t 2.12237×10−8 ) −9.25342× 108e −e ( 4.46772×1015−ω 2.98473×1015 ) − e ( 4.94075×10−8−t 2.12237×10−8 ) ω = 6.0× 1014 − 6.5× 1014 (Hz) t = 30nm−120nm β = 3.83428× 107 + 1.87655× 108e−e ( 7.82176×1014−ω 1.65181×1014 ) −9.72658× 106e−e ( 5.01303×10−8−t 1.94477×10−8 ) −6.92474× 107e −e ( 7.82176×1014−ω 1.65181×1014 ) − e ( 5.01303×10−8−t 1.94477×10−8 ) ω = 6.5× 1014 − 7.1× 1014 (Hz) t = 30nm−120nm β = 5.28234× 107 + 1.21942× 109e−e ( 8.78901×1014−ω 1.36591×1014 ) −1.55391× 107e−e ( 5.07127×10−8−t 1.661×10−8 ) 107 SPP Mode in a Silver-Dielectric-Silver Waveguide −2.37427× 108e −e ( 8.78901×1014−ω 1.36591×1014 ) − e ( 5.07127×10−8−t 1.661×10−8 ) ω = 3.0× 1014 − 5.5× 1014 (Hz) t = 5nm−30nm β = 2.83643× 106 + 1.89939× 109e−e ( 2.22083×1015−ω 1.28216×1015 ) −7.36171× 107e−e ( 3.56358×10−9−t −7.72401×10−9 ) 1.32238× 1010e −e ( 2.22083×1015−ω 1.28216×1015 ) − e ( 3.56358×10−9−t −7.72401×10−9 ) ω = 5.5× 1014 − 7.1× 1014 (Hz) t = 5nm−30nm β = 4.82025× 107 + 1.71088× 1010e−e ( 1.32937×1015−ω 3.69429×1014 ) 2.17774× 108e−e ( 4.15853×10−9−t −7.20353×10−9 ) 1.70486× 1011e −e ( 1.32937×1015−ω 3.69429×1014 ) − e ( 4.15853×10−9−t −7.20353×10−9 ) Dielectric core n= 2.5 ω = 2.5× 1014 − 5.5× 1014 (Hz) t = 100nm−300nm β = −7.2073× 106 + 2.59213× 109e−e ( 2.8552×1015−ω 1.74298×1015 ) 108 SPP Mode in a Silver-Dielectric-Silver Waveguide −1.79057× 106e−e ( 1.3725×10−8−t 6.04451×10−8 ) −5.96339× 108e −e ( 2.8552×1015−ω 1.74298×1015 ) − e ( 1.3725×10−8−t 6.04451×10−8 ) ω = 5.5× 1014 − 6.0× 1014 (Hz) t = 100nm−300nm β = 3.4993× 107 + 1.27119× 108e−e ( 6.79906×1014−ω 1.08599×1014 ) 1.70174× 109e−e ( −3.33857×10−7−t −2.34925×10−7 ) −6.89123× 109e −e ( 6.79906×1014−ω 1.08599×1014 ) − e ( −3.33857×10−7−t −2.34925×10−7 ) ω = 6× 1014 − 6.5× 1014 (Hz) t = 100nm−300nm β = 5.61033× 107 + 1.78816× 109e−e ( 8.13426×1014−ω 1.15717×1014 ) −8.70104× 106e−e ( 6.83944×10−8−t 2.04979×10−8 ) 5.80717× 108e −e ( 8.13426×1014−ω 1.15717×1014 ) − e ( 6.83944×10−8−t 2.04979×10−8 ) ω = 2.5× 1014 − 5.5× 1014 (Hz) t = 50nm−100nm β = −6.45697× 107 + 1.37518× 109e−e ( 3.10394×1015−ω 2.75231×1015 ) 109 SPP Mode in a Silver-Dielectric-Silver Waveguide 1.43777× 107e−e ( 8.76514×10−8−t 4.37752×10−8 ) −3.05341× 108e −e ( 3.10394×1015−ω 2.75231×1015 ) − e ( 8.76514×10−8−t 4.37752×10−8 ) ω = 5.5× 1014 − 6.0× 1014 (Hz) t = 50nm−100nm β = 3.39926× 107 + 2.51503× 108e−e ( 7.99154×1014−ω 2.85374×1014 ) −3.40492× 107e−e ( 8.81644×10−8−t 1.69038×10−7 ) −4.69014× 107e −e ( 7.99154×1014−ω 2.85374×1014 ) − e ( 8.81644×10−8−t 1.69038×10−7 ) ω = 6.0× 1014 − 6.5× 1014 (Hz) t = 50nm−100nm β = 3.73922× 107 + 5.68088× 108e−e ( 7.9886×1014−ω 1.65266×1014 ) −4.98983× 107e−e ( 1.50844×10−7−t 1.18004×10−7 ) 4.13079× 108e −e ( 7.9886×1014−ω 1.65266×1014 ) − e ( 1.50844×10−7−t 1.18004×10−7 ) ω = 1.5× 1014 − 5× 1014 (Hz) t = 5nm−50nm β = −6.19853× 108 + 4.75358× 1010e−e ( 1.25148×1015−ω 7.88071×1014 ) 110 SPP Mode in a Silver-Dielectric-Silver Waveguide 6.25779× 108e−e ( −1.0823×10−8−t 4.90041×10−9 ) −4.69103× 1010e −e ( 1.25148×1015−ω 7.88071×1014 ) − e ( −1.0823×10−8−t 4.90041×10−9 ) ω = 5× 1014 − 6.5× 1014 (Hz) t = 5nm−50nm β = 4.62771× 107 + 5.12298× 1010e−e ( 1.57816×1015−ω 4.94713×1014 ) 2.02722× 1011e−e ( −7.59034×10−8−t −3.93686×10−8 ) 4.89833× 1014e −e ( 1.57816×1015−ω 4.94713×1014 ) − e ( −7.59034×10−8−t −3.93686×10−8 ) Dielectric core n = 3.0 ω = 2.5× 1014 − 5× 1014 (Hz) t = 100nm−300nm β = 2.88107× 107 + 2.13426× 109e−e ( 1.95942×1015−ω 1.04187×1015 ) −2.47784× 107e−e ( −1.87313×10−8−t 5.74129×10−8 ) 2.14085× 108e −e ( 1.95942×1015−ω 1.04187×1015 ) − e ( −1.87313×10−8−t 5.74129×10−8 ) ω = 5× 1014 − 6× 1014 (Hz) 111 SPP Mode in a Silver-Dielectric-Silver Waveguide t = 100nm−300nm β = 4.67772× 107 + 3.21912× 109e−e ( 8.07685×1014−ω 1.50667×1014 ) 402975.77697e−e ( 4.50521×10−7−t 3.73193×10−8 ) 1.06579× 108e −e ( 8.07685×1014−ω 1.50667×1014 ) − e ( 4.50521×10−7−t 3.73193×10−8 ) ω = 2.5× 1014 − 5× 1014 (Hz) t = 50nm−100nm β = −5.89053× 107 + 1.51844× 109e−e ( 2.60327×1015−ω 2.18067×1015 ) 1.10074× 107e−e ( 8.7235×10−8−t 4.68116×10−8 ) −3.0804× 108e −e ( 2.60327×1015−ω 2.18067×1015 ) − e ( 8.7235×10−8−t 4.68116×10−8 ) ω = 5.5× 1014 − 5.8× 1014 (Hz) t = 50nm−100nm β = 3.43401× 107 + 5.50512× 109e−e ( 8.34557×1014−ω 2.81694×1014 ) −4.59404× 107e−e ( 1.32879×10−7−t 1.28984×10−7 ) 3.23191× 108e −e ( 8.34557×1014−ω 2.81694×1014 ) − e ( 1.32879×10−7−t 1.28984×10−7 ) ω = 5.8× 1014 − 6.0× 1014 (Hz) 112 SPP Mode in a Silver-Dielectric-Silver Waveguide t = 50nm−100nm β = 5.54605× 107 + 5.88966× 108e−e ( 6.81188×1014−ω 9.463×1013 ) −5.96491× 107e−e ( 1.29969×10−7−t 2.57514×10−7 ) 5.94126× 108e −e ( 6.81188×1014−ω 9.463×1013 ) − e ( 1.29969×10−7−t 2.57514×10−7 ) ω = 2.5× 1014 − 4.0× 1014 (Hz) t = 5nm−50nm β = −4.87592× 109 + 2.31914× 1011e−e ( 4.98913×1014−ω 3.06452×1014 ) 4.88724× 109e−e ( −2.6278×10−8−t 5.00396×10−9 ) −2.31762× 1011e −e ( 4.98913×1014−ω 3.06452×1014 ) − e ( −2.6278×10−8−t 5.00396×10−9 ) ω = 4.0× 1014 − 6.0× 1014 (Hz) t = 20nm−50nm β = −1.12192× 106 + 8.92744× 109e−e ( 1.63305×1015−ω 7.77546×1014 ) 1.71474× 107e−e ( 2.54566×10−8−t 3.20063×10−8 ) −9.24941× 109e −e ( 1.63305×1015−ω 7.77546×1014 ) − e ( 2.54566×10−8−t 3.20063×10−8 ) ω = 4.0× 1014 − 6.0× 1014 (Hz) 113 SPP Mode in a Silver-Dielectric-Silver Waveguide t = 5nm−20nm β = 5.5263× 107 + 5.33629× 1010e−e ( 1.52435×1015−ω 5.03452×1014 ) 2.58337× 108e−e ( 4.26971×10−9−t −7.10258×10−9 ) 4.69963× 1011e −e ( 1.52435×1015−ω 5.03452×1014 ) − e ( 4.26971×10−9−t −7.10258×10−9 ) Dielectric core n = 3.5 ω = 2.7× 1014 − 5.0× 1014 (Hz) t = 100nm−300nm β = 7.70898× 106 + 8.58092× 109e−e ( 2.04069×1015−ω 9.57004×1014 ) −4.78262× 106e−e ( 3.83384×10−7−t 1.54557×10−8 ) 5.36255× 107e −e ( 2.04069×1015−ω 9.57004×1014 ) − e ( 3.83384×10−7−t 1.54557×10−8 ) ω = 5.0× 1014 − 5.5× 1014 (Hz) t = 100nm−300nm β = 6.63587× 107 + 1.37356× 109e−e ( 6.35074×1014−ω 7.67264×1013 ) −7.41799× 107e−e ( 3.61797×10−7−t 1.37614×10−9 ) 1.03749× 109e −e ( 6.35074×1014−ω 7.67264×1013 ) − e ( 3.61797×10−7−t 1.37614×10−9 ) 114 SPP Mode in a Silver-Dielectric-Silver Waveguide ω = 2.5× 1014 − 4.8× 1014 (Hz) t = 40nm−100nm β = −5.45379× 106 + 1.20082× 1010e−e ( 2.82914×1015−ω 1.46163×1013 ) 1.0073× 106e−e ( 4.9367×10−8−t 1.86263×10−8 ) −2.86019× 109e −e ( 2.82914×1015−ω 1.46163×1015 ) − e ( 4.9367×10−8−t 1.86263×10−8 ) ω = 4.8× 1014 − 5.5× 1014 (Hz) t = 40nm−100nm β = 7.1167× 107 + 4.42601× 109e−e ( 7.71803×1014−ω 1.53727×1014 ) −1.71814× 107e−e ( 5.16071×10−8−t 1.40188×10−8 ) 5.46782× 108e −e ( 7.71803×1014−ω 1.53727×1014 ) − e ( 5.16071×10−8−t 1.40188×10−8 ) ω = 2.5× 1014 − 5× 1014 (Hz) t = 20nm−40nm β = −1.16087× 108 + 4.21399× 109e−e ( 1.86309×1015−ω 1.3616×1015 ) 1.55129× 108e−e ( 2.81664×10−8−t 4.13305×10−8 ) −5.06593× 109e −e ( 1.86309×1015−ω 1.3616×1015 ) − e ( 2.81664×10−8−t 4.13305×10−8 ) 115 SPP Mode in a Silver-Dielectric-Silver Waveguide ω = 5× 1014 − 5.5× 1014 (Hz) t = 20nm−40nm β = 1.06796× 108 + 1.6759× 109e−e ( 6.72104×1014−ω 1.53218×1014 ) −1.49326× 108e−e ( 2.93069×10−8−t 5.06187×10−8 ) −1.87427× 109e −e ( 6.72104×1014−ω 1.53218×1014 ) − e ( 2.93069×10−8−t 5.06187×10−8 ) ω = 2.5× 1014 − 4.5× 1014 (Hz) t = 20nm−40nm β = 9.25716× 106 + 1.19191× 109e−e ( 1.06392×1015−ω 6.14568×1014 ) −9.9057× 107e−e ( 4.14344×10−9−t −7.18321×10−9 ) 9.86027× 109e −e ( 1.06392×1015−ω 6.14568×1014 ) − e ( 4.14344×10−9−t −7.18321×10−9 ) ω = 4.5× 1014 − 5.5× 1014 (Hz) t = 20nm−40nm β = 8.62397× 107 + 6.48528× 1010e−e ( 1.05886×1015−ω 2.73221×1014 ) 5.25007× 108e−e ( 4.28125×10−9−t −7.07869×10−9 ) 5.17244× 1011e −e ( 1.05886×1015−ω 2.73221×1014 ) − e ( 4.28125×10−9−t −7.07869×10−9 ) 116 TM1 Mode in a Silver-Dielectric-Silver Waveguide TM1 Mode in a Silver-Dielectric-Silver Waveguide We obtain piece-wise fit functions to the dispersion relations of the TM1 mode over frequencies where it is propagative. The fit functions for the TM1 mode can be given for different dielectric cores as: Dielectric core n = 1.0 For, ω = 4.5× 1014 − 9× 1014 (Hz) t = 660− 800 nm β = −6.93061× 107 + 9.10218× 10−8ω + 1.06405× 1014t −1.48057× 10−23ω2 − 4.17409× 1019t2 − 0.05735ωt. ω = 6× 1014 − 9× 1014 (Hz) t = 600− 660 nm β = −9.92251× 107 + 1.33067× 10−7ω + 1.60785× 1014t −2.94254× 10−23ω2 − 6.51506× 1019t2 − 0.0959ωt. ω = 7.5× 1014 − 9× 1014 (Hz) t = 400− 660 nm 117 TM1 Mode in a Silver-Dielectric-Silver Waveguide β = −9.92251× 107 + 1.33067× 10−7ω + 1.60785× 1014t −2.94254× 10−23ω2 − 6.51506× 1019t2 − 0.0959ωt. Dielectric core n = 1.5 For, ω = 4× 1014 − 9× 1014 (Hz) t = 550− 750 nm β = −3.97487× 107 + 8.22517× 10−8ω + 5.71795× 1013t −1.84235× 10−23ω2 − 2.3324× 1019t2 − 0.03228ωt. ω = 6× 1014 − 9× 1014 (Hz) t = 350− 550 nm β = −9.96598× 107 + 1.63938× 10−7ω + 1.89559× 1014t −4.68169× 10−23ω2 − 9.59265× 1019t2 − 0.12007ωt. Dielectric core n = 2.0 For, ω = 4× 1014 − 9× 1014 (Hz) t = 450− 750 nm 118 TM1 Mode in a Silver-Dielectric-Silver Waveguide β = −2.86733× 107 + 7.95334× 10−8ω + 4.39631× 1013t −1.4343× 10−23ω2 − 1.93033× 1019t2 − 0.02472ωt. ω = 4× 1014 − 9× 1014 (Hz) t = 350− 450 nm β = −1.14088× 108 + 1.82421× 10−7ω + 3.03427× 1014t −4.46102× 10−23ω2 − 2.1498× 1020t2 − 0.17968ωt. Dielectric core n = 2.5 For, ω = 4× 1014 − 9× 1014 (Hz) t = 450− 800 nm β = −1.55081× 107 + 7.18463× 10−8ω + 2.4086× 1013t −7.46259× 10−24ω2 − 1.08697× 1019t2 − 0.0126ωt. ω = 4× 1014 − 9× 1014 (Hz) t = 350− 450 nm 119 TM1 Mode in a Silver-Dielectric-Silver Waveguide β = −4.44726× 107 + 1.09333× 10−7ω + 1.05957× 1014t −2.16556× 10−23ω2 − 7.31737× 1019t2 − 0.05808ωt. Dielectric core n = 3.0 For, ω = 4× 1014 − 9× 1014 (Hz) t = 400− 800 nm β = −1.22715× 107 + 7.75821× 10−8ω + 2.04492× 1013t −5.59765× 10−24ω2 − 9.94216× 1018t2 − 0.00996ωt. ω = 4× 1014 − 9× 1014 (Hz) t = 300− 400 nm β = −3.53474× 107 + 1.0746× 10−7ω + 9.04739× 1013t −1.71373× 10−23ω2 − 6.74423× 1019t2 − 0.04795ωt. Dielectric core n = 3.5 For, ω = 4× 1014 − 9× 1014 (Hz) t = 450− 800 nm 120 TM1 Mode in a Silver-Dielectric-Silver Waveguide β = −8.39088× 106 + 8.26824× 10−8ω + 1.43673× 1013t −3.46659× 10−24ω2 − 7.20406× 1018t2 − 0.00639ωt. ω = 4× 1014 − 9× 1014 (Hz) t = 300− 450 nm β = −2.24798× 107 + 9.9966× 10−8ω + 5.86262× 1013t −1.00665× 10−23ω2 − 4.49514× 1019t2 − 0.02861ωt. 121
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