Numerical simulations of nanostructural photonic crystals

NUMERICAL SIMULATIONS OF NANOSTRUCTURAL PHOTONIC CRYSTALS SUN YAN NATIONAL UNIVERSITY OF SINGAPORE 2004 NUMERICL SIMULATIONS OF NANOSTRUCTURAL PHOTONIC CRYSTALS SUN YAN (B S., Peking University, China) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGPAORE 2004 Abstract I Name: Sun Yan Degree: Master of Engineering Department: Electrical & Computer Engineering Thesis Title: Numerical Simulation of Nanostructural Photonic Crystals Abstract A three-dimensional Finite-Difference Time-Domain (FDTD) technique is implemented to calculate transmission spectra of finite-size dielectric photonic-crystal slabs, including a square lattice, two triangular lattices, a square lattice with periodic backgrounds and a triangular lattice with a point defect The simulation results for the remarkable band gaps agree well with references Based on the direct-integration method, a frequency-dependent FDTD formulation is extended into a parametric and comparative study of the linearly dispersive Lorentz dielectric that shows a consequent dramatic increase in the band-gap size when the characteristic parameters of the second-order relaxation equation are varied The MIT Photonic-Bands (MPB) package is also used to calculate band structures of twodimensional periodic dielectric photonic structures, including a multi-permittivity square lattice, a multi-radius square lattice and a multi-period square lattice Compared with the three-dimensional FDTD method, the two-dimensional MPB simulation results are found to be sufficiently accurate for the design of novel three-dimensional structures Key words: Finite-Difference Time-Domain (FDTD) method, Photonic band gap (PBG), Photoniccrystal slab, Lorentz dielectric, Transmission coefficient, Band structure Numerical Simulations of Nanostructural Photonic Crystals Acknowledgements II Acknowledgements First of all, I would like to show the sincerest appreciation to my supervisors, Prof Leong Mook-Seng of the National University of Singapore (NUS), and Dr Li Er-Ping of the Institute of High Performance Computing (IHPC), for their earnest guidance and patience throughout this research as well as editorial correctness of this thesis My interest in the FDTD method for computational electromagnetics was shown in 2000 Associated Professor Li Ming-Zhi offered me an opportunity to research and complete my B.S thesis on the parallel FDTD algorithm for electromagnetic simulations when I joined the Electromagnetic Field & Microwave Technology Laboratory of the Department of Electronics at Peking University (PKU), China Two more years I had spent there as an undergraduate then a graduate were very memorable in my life I would like to thank Dr Yuan Wei-Liang, Dr Zhang Yue-Jiang, Ms Wang Sheng and Mr Liu En-Xiao for their good ideas I would also like to thank my colleagues and friends for their kind help Last, but not the least, I would like to give many thanks to my dearest parents and brother for their entire love, support and encouragement, who are constant inspiration for me to my best Sun Yan July 21, 2004 The Capricorn, Singapore Numerical Simulations of Nanostructural Photonic Crystals Table of Contents III Table of Contents Abstract ………………………………………………………………………… …… I Acknowledgements ………………………………………………………………… … II Table of Contents …………………………………………………………………… III List of Figures ……………………………………………………………………… VI List of Tables ……………………………………………………………………… XI Chapter Introduction ………………………………………………………………… 1.1 Overview of photonic crystals ……………………………………………………… 1.2 Computational methods …………………………………………………………… 1.2.1 Time-domain and frequency-domain methods …………………………………… 1.2.2 Finite-Difference Time-Domain (FDTD) method ………………………………… 1.3 Motivation ………………………………………………………… ……………… 1.4 Outline of this thesis …………………………………………………… …………… 1.5 List of contributions ………………………………………………………………… Chapter Finite-Difference Time-Domain Method …………………… ………… 2.1 Maxwell’s equations ………………………………………………………………… 2.2 General FDTD formulation ……………………………………………….……… 10 2.3 Numerical dispersion and stability ………………………………………….……… 14 2.3.1 Numerical dispersion ……………………………………………………… …… 14 2.3.2 Numerical stability ………………………………………………………… … 16 2.4 Incident wave source conditions …………………………………………… …… 17 Numerical Simulations of Nanostructural Photonic Crystals Table of Contents IV 2.4.1 Plane-wave condition ……………………………………………………… … 17 2.4.2 Pointwise E and H hard sources in one dimension …………………………….… 18 2.4.3 The total-field/scattered-field technique ………………………… …………… 19 2.5 Absorbing boundary conditions ………………………………………………… 21 2.5.1 Outer-boundary conditions ………………………………………… …………… 21 2.5.2 Mur finite-difference scheme ……………………………………… ………… 22 2.5.3 Perfectly-matched layer absorbing boundary conditions …………… ………… 23 2.6 Numerical validations …………………………………………………… ……… 23 2.6.1 Surface electric current distributions of a perfectly conducting cube ……….…… 24 2.6.2 Bistatic RCS patterns of a spherical water droplet ………………………… …… 26 Chapter FDTD Modeling of Photonic-Crystal Slabs ……………………………… 28 3.1 Photonic-crystal slabs ………………………………………………… …………… 28 3.2 Process of the FDTD-code execution ……………………………………………… 29 3.3 Simulation results and analysis ……………………………………… ………… 30 3.3.1 Square lattice ………………………………………………………………… 31 3.3.2 Triangular lattice ……………………………………………………………… 34 3.3.3 Square lattice with periodic backgrounds …………………………………… 38 3.3.4 Triangular lattice with a point defect ………… ……………………………… 40 3.4 Conclusions ……………………………………… ……………………………… 43 Chapter A Frequency-Dependent FDTD Formulation for Dispersive Materials 44 4.1 Lorentz dielectric ………………………………………………………………… 44 4.2 A frequency-dependent FDTD formulation ……………………………………… 45 4.3 A parametric and comparative study ……………………………………………… 49 Numerical Simulations of Nanostructural Photonic Crystals Table of Contents V 4.3.1 Effect of the resonant frequency ……………………………………………… 49 4.3.2 Effect of the damping frequency ………………………………………………… 54 4.3.3 Effect of the infinite-frequency relative permittivity ……… ………………… 57 4.4 Conclusions …………………………………………………… ………………… 59 Chapter MIT Photonic-Bands ………………….…………………….…………… 60 5.1 Introduction ……………………………………………………… ……………… 60 5.2 Examples in the manual …………………………………………… …………… 61 5.2.1 Square lattice ………………………………………………………………… 62 5.2.2 Triangular lattice …………………………………………………… ………… 63 5.2.3 Comparisons with the three-dimensional FDTD simulation results ……………… 65 5.3 Simulation results and discussion … …………………………………… ……… 67 5.3.1 Triangular lattice with various dielectric constants …………………………… 67 5.3.2 Multi-permittivity square lattice ………………………………………………… 70 5.3.3 Multi-radius square lattice ….…………………………………………………… 73 5.3.4 Multi-period square lattice …………………………………………………… 76 5.4 Conclusions ………………………………………………………………………… 78 Chapter Summary ……………………………………………………………… … 79 6.1 Present work …………………………………………………………………….… 79 6.2 Future work ………………………………………………………………… …… 80 References …………………………………………………………………………… 82 Numerical Simulations of Nanostructural Photonic Crystals List of Figures VI List of Figures Figure 1.1 Schematic pictures of one-, two- and three- dimensional photonic crystals (The materials with different dielectric constants are represented by different colors.) ………………………………………………………………………….… … Figure 2.1 Positions of the E and H field vector components above a cubic cell of the Yee space lattice ……………………………………………………………….… 10 Figure 2.2 Total-field/scattered-field regions with a virtual connecting surface and lattice truncation planes (absorbing boundary conditions) …………………… … 20 Figure 2.3 Geometries of two canonical three-dimensional structures ………….……… 24 Figure 2.4 Comparison of the FDTD and EFIE-MoM results for the surface electric current distribution along the E-plane locus (a-b′-c′-d) of the perfectly conducting cube in Figure 2.3 (a) ………………………………………… 25 Figure 2.5 Comparison of the FDTD and EFIE-MoM results for the surface electric current distribution along the H-plane locus (a-b-c-d) of the perfectly conducting cube in Figure 2.3 (a) ………………………………………… 25 Figure 2.6 Comparison of the FDTD results and Mie-series solutions for the bistatic RCS patterns of the spherical water droplet in Figure 2.3 (b) …………………… 26 Figure 3.1 Two characteristic photonic-crystal slabs …………………… ……… 29 Figure 3.2 Three-dimensional FDTD model of a square lattice of dielectric rods with lattice constant a, radius 0.2a and height 2.0a …………………………… 32 Figure 3.3 Horizontal cross-section of the FDTD model of a square lattice of dielectric rods with lattice constant a, radius 0.2a and height 2.0a ………………… 32 Numerical Simulations of Nanostructural Photonic Crystals List of Figures VII Figure 3.4 Transmission spectra (The FDTD results and the reference values in [4]−[5] are indicated by a solid-line frame and a dash-line frame respectively.) ……… 33 Figure 3.5 Ez of the TM-like polarization (z = 0, t = 700∆t) … ……………………… 33 Figure 3.6 Band-gap size as a function of the slab height/thickness for two slabs in Figure 3.1 ….……………………………………………………………………… 34 Figure 3.7 Horizontal cross-sections of the FDTD models of two triangular lattices of dielectric rods with lattice constant a, radius 0.15a and height 2.0a ……… 35 Figure 3.8 Transmission spectra (The FDTD results and the reference values in [12] are indicated by a solid-line frame and a dash-line frame respectively.) ……… 36 Figure 3.9 Ez of the TM-like polarization (z = 0, t = 750∆t) …………………………… 36 Figure 3.10 Transmission spectra (The FDTD results and the reference values in [12] are indicated by a solid-line frame and a dash-line frame respectively.) …… 37 Figure 3.11 Ez of the TM-like polarization (z = 0, t = 650∆t) ………………………… 37 Figure 3.12 Vertical cross-sections of a photonic-crystal slab with dielectric backgrounds …………………………………………………………………………… 38 Figure 3.13 Horizontal cross-section of the FDTD model of a square lattice of high-index rods with lattice constant a, radius 0.2a and height 2.0a as well as the lowindex dielectric rods with height 2.0a extending above and below ……… 39 Figure 3.14 Transmission spectra (The FDTD results and the reference values in [4], [6] are indicated by a solid-line frame and a dash-line frame respectively.) … 40 Figure 3.15 Ez of the TM-like polarization (z = 0, t = 700∆t) ………………………… 40 Numerical Simulations of Nanostructural Photonic Crystals List of Figures VIII Figure 3.16 Horizontal cross-section of the FDTD model of a triangular lattice of dielectric rods with a central point defect, lattice constant a, radius 0.15a and height 2.0a ………………………………………………………………… 41 Figure 3.17 Transmission spectra (The FDTD results and the reference values in [12] are indicated by a solid-line frame and a dash-line frame respectively.) …… 42 Figure 3.18 Ez of the TM-like polarization (z = 0, t = 650∆t) ………………………… 42 Figure 4.1 A curve of ε∗(ω)/ε0 as a function of ω/ωr (εs = 12.0, ε∞ = 6.0 and νd/ωr = 0.1) ……………………………………………………………………………… 45 Figure 4.2 Transmission spectra (Case 1: ωr = 2π×2.0×105 GHz) ……………………… 50 Figure 4.3 Ez of the TM-like polarization (z = 0, t = 700∆t) …………………………… 50 Figure 4.4 Transmission spectra (Case 2: ωr = 2π×3.0×105 GHz) … ………… ……… 51 Figure 4.5 Ez of the TM-like polarization (z = 0, t = 700∆t) …………………………… 51 Figure 4.6 Transmission spectra (Case 3: ωr = 2π×6.0×105 GHz) ……………………… 52 Figure 4.7 Transmission spectra (Case 4: ωr = 2π×9.0×105 GHz) ….….… …………… 53 Figure 4.8 Comparison of the transmission spectra of the TM-like polarization (εs = 12.0, ε∞ = 6.0 and vd = 0.1ωr) …………………………………………………… 53 Figure 4.9 Transmission spectra (Case 1: vd = 0.05ωr) ………………………………… 55 Figure 4.10 Transmission spectra (Case 3: vd = 0.15ωr) ……………………………… 55 Figure 4.11 Comparison of the transmission spectra of the TM-like polarization (εs = 12.0, ε∞ = 6.0 and ωr = 2π×3.0×105 GHz) ……………………………………… 56 Figure 4.12 Transmission spectra (Case 1: ε∞ = 5.0) …………………………………… 57 Figure 4.13 Transmission spectra (Case 3: ε∞ = 7.0) … …………………………….… 58 Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands 72 TM modes TE modes (b) Case 2: {εr1, εr2} = {3.8, 13.0} Figure 5.13 Band structures of a two-dimensional multi-permittivity square lattice For the TM modes, the width of the low-frequency band gap increases but that of the high-frequency band gap decreases, and the two gap-bottom frequencies decrease with the increase in the dielectric constant contrast For the TE modes, only a small band gap is found with a relatively-high dielectric constant contrast The band-gap information of Figure 5.12 and Figure 5.13 is extracted in Table 5.3 2-d periodic pattern 1-d periodic pattern {εr1, εr2} TM modes TE modes Beginning End Beginning End Case 1: {3.8, 8.9} 0.429 0.458 - - - - - - Case 2: {3.8, 13.0} 0.250 0.273 0.557 0.567 0.425 0.430 - - 0.276 0.295 - - 0.383 0.460 - - 0.235 0.292 0.548 0.558 0.379 0.435 - - Case 1: {3.8, 8.9} Case 2: {3.8, 13.0} Table 5.3 Band-gap information extracted from Figure 5.12 and Figure 5.13 Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands 73 5.3.3 Multi-radius square lattice Thirdly, a multi-radius square lattice of dielectric rods is simulated with lattice constant a and dielectric constant 8.9, considering {r1, r2} = {0.20a, 0.24a} and {0.20a, 0.30a} respectively in different patterns, as shown in Figure 5.14, where the rods with r1 are represented by small dots and those with r2 are represented by large dots (a) 1-d periodic pattern (b) 2-d periodic pattern Figure 5.14 A multi-permittivity square lattice in different patterns The band structures in both cases are plotted in Figure 5.15 respectively when the lattice is arranged in the first pattern TM modes TE modes (a) Case 1: {r1, r2} = {0.20a, 0.24a} Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands 74 TM modes TE modes (b) Case 2: {r1, r2} = {0.20a, 0.30a} Figure 5.15 Band structures of a one-dimensional multi-radius square lattice For the TM modes, the width and gap-bottom frequency of the band gap both decreases with the increase in the radius contrast For the TE modes, only a small band gap is found with a relatively-high radius contrast The band structures in both cases are plotted in Figure 5.16 respectively when the lattice is arranged in the second pattern TM modes TE modes (a) Case 1: {r1, r2} = {0.20a, 0.24a} Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands 75 TM modes TE modes (b) Case 2: {r1, r2} = {0.20a, 0.30a} Figure 5.16 Band structures of a two-dimensional multi-radius square lattice For the TM modes, one large band gap with a relatively-low radius contrast is split into two with a relatively-high value, and the gap-bottom frequency of the high-frequency band gap decreases with the increase in the radius contrast For the TE modes, only a small band gap is found with a relatively-high radius contrast The band-gap information of Figure 5.15 and Figure 5.16 is extracted in Table 5.4 TE modes Beginning End Beginning End 1-d periodic pattern TM modes Case 1: {0.20a, 0.24a} 0.314 0.413 - - - - - - Case 2: {0.20a, 0.30a} 0.310 0.358 0.501 0.506 - - - - 2-d periodic pattern {r1, r2} Case 1: {0.20a, 0.24a} 0.310 0.416 - - - - - - Case 2: {0.20a, 0.30a} 0.222 0.227 0.500 0.507 0.299 0.368 - - Table 5.4 Band-gap information extracted from Figure 5.15 and Figure 5.16 Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands 76 5.3.4 Multi-period square lattice Lastly, a multi-period square lattice of dielectric rods is simulated with radius 0.1a (a is a constant) and dielectric constant 8.9, as shown in Figure 5.17, considering {a1, a2} = {0.30a, 1.70a}, (0.50a, 1.50a), {0.70a, 1.30a} and {0.90a, 1.10a} respectively in one dimension whereas a1 = a2 = a in another dimension Figure 5.17 A one-dimensional multi-period square lattice The band structures in the four cases are plotted in Figure 5.18 respectively, and the bandgap information of which is extracted in Table 5.5 TM modes TE modes (a) Case 1: {a1, a2} = {0.30a, 1.70a} Numerical Simulations of Nanostructural Photonic Crystals Chapter MIT Photonic-Bands TM modes TE modes (b) Case 2: {a1, a2} = {0.50a, 1.50a} TM modes TE modes (c) Case 3: {a1, a2} = {0.70a, 1.30a} TM modes TE modes (d) Case 4: {a1, a2} = {0.90a, 1.10a} Figure 5.18 Band structures of a one-dimensional multi-period square lattice Numerical Simulations of Nanostructural Photonic Crystals 77 Chapter MIT Photonic-Bands 78 For the TM modes, one band gap is split into two when the different periods are cascaded It suggests that multi-gap photonic structures of interest can be built in such a manner For the TE modes, no gap is found {a1, a2} Case 1: {0.30a, 1.70a} Case 2: {0.50a, 1.50a} Case 3: {0.70a, 1.30a} Case 4: {0.90a, 1.10a} TM modes TE modes Beginning End Beginning End 0.323 0.353 - - 0.504 0.593 - - 0.339 0.349 - - 0.502 0.606 - - 0.418 0.453 - - 0.516 0.608 - - 0.406 0.486 - - 0.543 0.584 - - Table 5.5 Band-gap information extracted from Figure 5.18 5.4 Conclusions By calculating the band structures and field distributions of some novel two-dimensional periodic dielectric photonic structures, the basic function of the MPB package is described in this chapter Compared with the three-dimensional FDTD method, the simulation results for the twodimensional MPB application are found to be sufficiently accurate for the design of threedimensional structures that can provide more useful information prior to fabrication and experimental measurements Numerical Simulations of Nanostructural Photonic Crystals Chapter Summary 79 Chapter Summary This chapter will conclude the present work in this thesis and propose the possible work in the future 6.1 Present work In this thesis, the three-dimensional FDTD code is executed to model some typical finitesize dielectric photonic-crystal slabs, and calculate the transmission spectra and field distributions for the cases of a square lattice, two triangular lattices, a square lattice with periodic backgrounds and a triangular lattice with a point defect, respectively The simulation results for the remarkable band gaps agree well with those obtained by the preconditioned conjugate-gradient minimization of the Raleigh quotient in a plane-wave basis in [4]−[6] and the scattering-matrix method in [12], though the differences that the finite size in the in-plane directions and the frequency shift from the two-dimensional results introduced by the finite slab height in the third dimension result in must be taken into account Then based on the direct-integration method, a frequency-dependent FDTD formulation is improved to model a finite-size square-lattice photonic-crystal slab of dielectric rods that are composed of the linearly dispersive Lorentz dielectric, and analyze the effects on the transmission spectra when the characteristic parameters of the second-order relaxation equation are varied The simulation results that show a consequent dramatic increase in the band-gap size suggest that the resonant frequency, damping frequency and infinite- Numerical Simulations of Nanostructural Photonic Crystals Chapter Summary 80 frequency relative permittivity play important roles, as well as the zero-frequency relative permittivity, in the achievement of the desired transmission properties The MPB package is also used to model some novel two-dimensional periodic dielectric photonic structures, and calculate the band structures and field distributions for the cases of a square lattice, a triangular lattice, a triangular lattice with various dielectric constants, a multi-permittivity square lattice, a multi-radius square lattice and a multi-period square lattice, respectively Compared with the three-dimensional FDTD method, the simulation results for the two-dimensional MPB application are found to be sufficiently accurate for the design of three-dimensional structures that can provide more useful information prior to fabrication and experimental measurements 6.2 Future work Many objects of electromagnetic interest, for example, photonic crystals, antenna arrays and frequency selective surface (FSS) structures [39], possess a periodicity in one or multiple dimensions The fact that the overall structure consists of the infinite replications of a basic element may lead to an unmanageable electrical size One way to alleviate such a computational burden is to model only the individual element with fine details when periodic boundary conditions (PBCs) [40] are used for the truncation of the space lattice instead of conventional ABCs To simulate the large-scale components with a high degree of accuracy and efficiency, the FDTD method in conjunction with other techniques will be constructed as well as the parallel implementation [41] Since dielectric and lattice parameters are significant to Numerical Simulations of Nanostructural Photonic Crystals Chapter Summary 81 determine the properties of photonic crystals, numerical simulations of a wide variety of materials and structures will be still under investigation The fabrication and experimental measurements of optical circuits and devices must be taken as the complementary parts of numerical simulations in the further research Numerical Simulations of Nanostructural Photonic Crystals References 82 References [1] E Yabolonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Physical Review Letters, vol 58, no 20, pp 2059-2062, 1987 [2] S John, Strong localization of photons in certain disordered dielectric superlattices, Physical Review Letters, vol 58, no 23, pp 2486-2489, 1987 [3] S G Johnson, A Mekis, et al, Molding the flow of light, IEEE Computational Science and Engineering, vol 3, no 6, pp 38-47, 2001 [4] S G Johnson and J D Joannopoulos, Photonic crystals: The road from theory to practice, pp 19, 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electromagnetic pulses, Optics Letters, vol 16, no 18, pp 1412-1414, 1991 [38] S G Johnson, The MIT Photonic-Bands manual, pp 15-19, 25-28, Massachusetts Institute of Technology [39] P Harms, R Mittra, et al, Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures, IEEE Transactions on Antennas and Propagation, vol 42, no 9, pp 1317-1324, 1994 [40] W J Tsay and D M Pozar, Application of the FDTD technique to periodic problems in scattering and radiation, IEEE Microwave and Guided Wave Letters, vol 3, no 8, pp 250-252, 1993 [41] Y Sun, The primary research of the FDTD parallel algorithm, B S Thesis, Peking University, 2001 Numerical Simulations of Nanostructural Photonic Crystals [...]... analogous numerical wave propagation stored in the computer memory Numerical Simulations of Nanostructural Photonic Crystals Chapter 1 Introduction 5 1.3 Motivation The heart of the subject of photonic crystals is the propagation of electromagnetic waves in a periodic dielectric medium [7] Numerical simulation plays a particularly important role in examining electromagnetic wave propagation in a photonic. .. the FDTD modeling with the PML medium such as [25] have been presented 2.6 Numerical validations On the basis of a general description above, a set of three-dimensional FDTD code for the numerical simulations of photonic crystals, in this thesis, is written and compiled by the Numerical Simulations of Nanostructural Photonic Crystals ... Y J Zhang and M S Leong, Design of multi-band gap photonic crystals by using numerical techniques, WSEAS Transactions on Electronics, vol 1, no 1, pp 40-44, 2004 4 Y Sun, E P Li and M S Leong, Three-dimensional FDTD modeling of finite-size photonic- crystal slabs, International Journal of RF and Microwave Computer-Aided Numerical Simulations of Nanostructural Photonic Crystals Chapter 1 Introduction... Figure 5.18 ……………… ………… 78 Numerical Simulations of Nanostructural Photonic Crystals Chapter 1 Introduction 1 Chapter 1 Introduction This chapter begins with an overview of photonic crystals including the backgrounds, concepts and applications The computational methods are then presented, followed by a highlight on the motivation and outline of this thesis 1.1 Overview of photonic crystals Recent years... properties [3] A photonic crystal can be constructed, consisting of a macroscopic periodic dielectric or metallic array, in which photons can be described in terms of a band structure as in the case of electrons The schematic pictures of one-, two- and three- dimensional photonic crystals are shown in Figure 1.1 Of particular interest is a complete photonic band gap (PBG) in the band structure of a photonic. .. permittivity of a photonic crystal is periodic on a scale comparable to the wavelength of the forbidden photons Hence, the manufacture of photonic crystals that operates in the visible region of electromagnetic spectra requires the innovative fabrication techniques In view of high quality and high expenses, accurate and efficient numerical models are necessary to obtain the desirable optical properties of photonic. .. in terms of a leapfrog arrangement Figure 2.1 Positions of the E and H field vector components above a cubic cell of the Yee space lattice Numerical Simulations of Nanostructural Photonic Crystals Chapter 2 Finite-Difference Time-domain Method 11 For a particular time point, all of the E computations in the space are completed and stored in memory using the previously-stored H values Then all of the... challenge in the field of photonic crystals is to fabricate the composite structures that possess the spectral band gaps at frequencies up to the optical region There have been intensive efforts to build and measure photonic band-gap structures Fabrication can be either easy or difficult depending on the desired wavelength of band gaps and the Numerical Simulations of Nanostructural Photonic Crystals Chapter... as electromagnetic band gap, a range of frequencies for which light is forbidden to propagate inside unless any defect is in the perfect crystal [3] Numerical Simulations of Nanostructural Photonic Crystals Chapter 1 Introduction (a) 1-d periodicity 2 (b) 2-d periodicity (c) 3-d periodicity Figure 1.1 Schematic pictures of one-, two- and three- dimensional photonic crystals (The materials with different... direct-integration method is improved to model a finite-size square-lattice photonic- crystal slab of dielectric rods that are composed of the linearly dispersive Lorentz dielectric, and analyze the Numerical Simulations of Nanostructural Photonic Crystals Chapter 1 Introduction 6 effects on the transmission spectra when the characteristic parameters of the second-order relaxation equation are varied In Chapter 5, ... …………………………………………………………………………… 82 Numerical Simulations of Nanostructural Photonic Crystals List of Figures VI List of Figures Figure 1.1 Schematic pictures of one-, two- and three- dimensional photonic crystals (The... computer memory Numerical Simulations of Nanostructural Photonic Crystals Chapter Introduction 1.3 Motivation The heart of the subject of photonic crystals is the propagation of electromagnetic... cross-section of the FDTD model of a square lattice of dielectric rods with lattice constant a, radius 0.2a and height 2.0a ………………… 32 Numerical Simulations of Nanostructural Photonic Crystals List of Figures