張宏鈞臺灣大學：光電工程學研究所陳銘鋒Chen, Min-FengMin-FengChen2007-11-252018-07-052007-11-252018-07-052004http://ntur.lib.ntu.edu.tw//handle/246246/50675In this thesis, the finite-difference time-domain (FDTD) method with the uniaxial perfectly matched layer boundaries is employed to model several types of channel drop filters in two dimensions. The signal switching mechanisms of different kinds of filters are expressed. Microcavity resonator based filters including micro-ring and micro-disk for both transverse electric (TE) and transverse magnetic (TM) modes are analyzed. Various ring-waveguide widths are considered and the two-stage ring filter is also studied. Photonic crystal based filters involving the monopole cavity and the doubly degenerate hexapole cavity are then investigated. We first analyze the photonic band gap materials using the plane wave expansion method. The electromagnetic wave behavior in photonic crystal is also discussed, including negative refraction, waveguide bending and splitters. The essential properties for the channel drop filters, such as resonant frequencies, linewidth, and quality factor, are determined by the FDTD simulation. In order to obtain reliable results of these quantities, the curved dielectric interfaces are treated by the index average scheme. With the index average scheme, the resonant frequencies given by the the FDTD simulation are stabilized. Since the properties of photonic devices obtained by the FDTD method with the index average scheme are different from those with the staircase approximation, several published photonic crystal based resonators are redesigned to achieve high performance for the channel drop filters.Contents 1 Introduction 1 1.1 Motivations . 1 1.2 Chapter Outline . 3 2 The Finite-Difference Time-Domain Method 5 2.1 Overview . 5 2.2 Uniaxial Perfectly Matched Layer Absorbing Boundary Condition . 9 2.3 The Index Average Scheme . 17 3 Modeling of Channel Drop Filters Based on Microcavity Resonators.29 3.1 Overview. 29 3.2 Micro-ring Filters . 32 3.2.1 Signal SwitchingMechanism . 32 3.2.2 Resonant Frequency, FWHM, FSR, Q-factor, and Extinction Ratio . 33 3.2.3 Various Types ofMicro-ring Filters . 35 3.3 Micro-disk Filters . 36 4 Modeling of Channel Drop Filters Based on Photonic Crystals .63 4.1 Overview . 63 4.2 The Electromagnetic Wave Behavior in Photonic Band Gap Materials . 66 4.3 Photonic Crystal Based Channel Drop Filters . 70 4.3.1 Signal SwitchingMechanism . 71 4.3.2 Filters withMonopole Cavities . 73 4.3.3 Filters with Doubly Degenerate Hexapole Cavity . 75 5 Conclusion 95 List of Figures 2.1 The arrangement of electric and magnetic field vector components within a cubic cell, known as the Yee cell . 21 2.2 The arrangement of electric and magnetic field vector components for the TE mode (Ez, Hx, Hy) and the TM mode (Hz, Ex, Ey). 22 2.3 Structure plot of a 3-D FDTD grid employing the PEC-backed.22 UPML absorbing boundary condition with σ assignment . 23 2.4 Structure plot of a 2-D FDTD grid employing the PEC-backed UPML absorbing boundary condition with σ assignment . 24 2.5 Visualization of the Hz-field with a point source at the center of a 2-D square free space with the UPML boundary for the TMmode. . 25 2.6 Visualization of the Ez-field with a point source at the center of a cubic free space with UPML boundary. 26 2.7 Partially filled FDTD cells with ε1 and ε2. The electric field is (a) parallel to the dielectric boundary, (b) normal to the dielectric boundary, and (c) tilted to the dielectric boundary. 27 2.8 Illustration of slight anisotropy at a curved interface for the TM mode. (a) Mesh for the TE mode. (b) Mesh for the TM mode . 28 3.1 Schematic of a microcavity ring/disk filter coupled to two straight waveguides . 45 3.2 Transmittance spectra of the micro-ring filter for (a) the TE mode and (b) the TMmode . 46 3.3 Snapshots of the computed sinusoidal E-field in themicro-ring filter for the TE mode for (a) the off-resonance signal at 200 THz and (b) the on-resonance signal at 186.05 THz. 47 3.4 Snapshots of the computed sinusoidal H-field in the microring filter for the TM mode for (a) the off-resonance signal at 200 THz and (b) the on-resonance signal at 186.50 THz. 48 3.5 Transmittance spectrum of the micro-ring filter with 0.6-µm ring-waveguide width for the TEmode. 49 3.6 Snapshot of the computed sinusoidal E-field in the micro-ring filter with 0.6-µm ring-waveguide width for the TE mode at the on-resonance frequency of 199.18 THz (second-order resonance). 50 3.7 Transmittance spectrum of the micro-ring filter with 0.9-µm ring-waveguide width for the TEmode. 51 3.8 Schematic of a two-stage micro-ring filter where the centers of the two rings are apart from each other by 3.604 µm in both x- and y-directions . 52 3.9 Transmittance spectrum of a two-stage micro-ring filter of Fig. 3.8 for the TE mode. 52 3.10 Illustration of linear combinations of two energy states to form composite modes. 53 3.11 Snapshot of the computed E-field in the two-stage micro-ring filter for the TE mode at the on-resonance frequency of 185.57 THz. 54 3.12 Transmittance spectrum of the micro-disk filter for the TE mode. 55 3.13 Transmittance spectrum of the micro-disk filter for the TM mode. 56 3.14 (a) Geometry of the ring waveguide in Cartesian coordinates along with the correspondent index profile. (b) The transformed waveguide in the u-v plane along with the correspondent index profile. 57 3.15 Snapshots of the computed sinusoidal E-field in the microdisk filter for the TE mode at (a) the off-resonance frequency of 200 THz and (b) the on-resonance frequency of 195.73 THz for the first-order radial-mode. 58 3.16 Snapshots of the computed sinusoidal E-field in the microdisk filter for TE mode at (a) the on-resonance frequency of 191.48 THz for the second-order radial-mode and (b) the onresonance signal of 188.02 THz for the third-order radial-mode. 59 3.17 Snapshots of the computed sinusoidal H-field in the microdisk filter for the TM mode at (a) the off-resonance frequency of 200 THz and (b) the on-resonance frequency of 195.30 THz for the first-order radial-mode.60 3.18 Snapshots of the computed sinusoidal H-field in the microdisk filter for TM mode at (a) the on-resonance frequency of 190.94 THz for the second-order radial-mode and (b) the onresonance frequency of 187.34 THz for the third-order radialmode. . 61 3.19 Snapshot of the computed sinusoidal H-field in the micro-disk filter for the TM mode at the on-resonance frequency of 194.70 THz for the fourth-order radial-mode. 62 4.1 (a) Schematic of a photonic crystal with square lattice and (b) the correspoding Brillouin zone.76 4.2 The photonic band diagrams for a square lattice of circular dielectric rods (ε = 11.56) with 0.2a radius embedded in air (ε=1) for (a) TE mode and (b) TMmode. 77 4.3 (a) Schematic of photonic crystal with hexagonal lattice and (b) the corresponding Brillouin zone.78 4.4 The photonic band diagram for a hexagonal lattice of circular air rods (ε = 1) with 0.48a radius embedded in dielectric substrate (ε=13) for (a) TEmode and (b) TMmode.79 4.5 (a) Schematic of the considered structure which is a dielectric waveguide surrounded by air placed before the photonic crystal embodied in a dielectric substrate. (b) Schematic of the structure without the air holes.80 4.6 Snapshots of the computed E-field (a) for the structure of Fig. 4.5(a) and (b) for Fig. 4.5(b) .81 4.7 Snapshot of the computed CW E-field in the PC-based Lshaped waveguide at 0.353(c/a).82 4.8 Snapshot of the computed CW E-field in the PC-based Tshaped waveguide with rt = 0.07a at 0.41(c/a).83 4.9 Schematic of two continua coupled through a resonator system that supports localized states.84 4.10 Tunneling process for a PC-based resonator system that supports two resonant modes containing different symmetry with respect to the mirror plane perpendicular the waveguide. (a) The even resonant mode with respect to the mirror plane. (b) The odd resonant mode with respect to the mirror plane (c) The linear combinations of the two modes.85 4.11 (a) The resonant mode which is odd with respect to the mirror plane parallel the waveguide. (b) The resonant mode which is odd with respect to the mirror plane perpendicular the waveguide. (c) The linear combinations of the two modes.85 4.12 Schematic of the PC-based CDF structure with two waveguides and two cavities. The black circles correspond to rods with ε = 11.56, while the gray circles correspond to rods with ε = 9.5. The two smaller rods have ε = 6.6, and a radius of 0.06a.86 4.13 Transmittance spectrum in the forward direction of the PCbased CDF with two single-mode cavities of Fig. 4.12 .86 4.14 Snapshots of the computed E-field in the PC-based CDF with two single-mode cavities of Fig. 4.12 at (a) the off-resonance frequency of 0.375(c/a) and (b) the on-resonance frequency of 0.3701(c/a) . 87 4.15 Transmittance spectra of the PC-based CDF with two singlemode cavities of Fig. 4.12 in (a) the transmission direction and (b) the backward direction. 88 4.16 Schematic of the PC-based channel drop filter structure with two waveguides and four cavities. The black circles correspond to rods with ε = 11.56, while the gray circles correspond to rods with ε = 7.5. Each of the smaller rods has ε = 6.5 and a radius of 0.06a. 89 4.17 (a) Transmittance spectrum of the PC-based CDF with four single-mode cavities of Fig. 4.16 in the forward direction. (b) Snapshot of the computed E-field at the on-resonance frequency of 0.3702(c/a).90 4.18 Transmittance spectra of the PC-based CDF with four singlemode cavities of Fig. 4.16 in (a) the transmission direction and (b) the backward direction .91 4.19 Schematic of the PC-based CDF structure with two waveguides and one doubly degenerate hexapole cavity. The black circles correspond to rods with ε = 11.56, while the gray circles correspond to rods with ε = 11.9. The defect rod has a radius of 0.7a .92 4.20 Transmittance spectra of the PC-based CDF with one doubly degenerate hexapole cavity of Fig. 4.19 in (a) the forward direction and (b) the backward direction .93 4.21 Snapshots of the computed E-field at the on-resonance frequency of 0.342(c/a) where the field pattern in the cavity is anti-symmetrical with respect to (a) the mirror plane parallel the waveguides and (b) the mirror plane perpendicular the waveguides. (c) The linear combination of the two cavity modes. 94 List of Tables 3.1 Resonance data of the micro-ring filter for (a) the TE mode and (b) the TMmode.39 3.2 Resonance data of the micro-ring filter with 0.6-µm ring-waveguide width for the TE mode. 40 3.3 Resonance data of the micro-ring filter with 0.9-µm ring-waveguide width for the TE mode.41 3.4 Resonance data of the two-stage micro-ring filter for the TE mode.42 3.5 Resonance data of the micro-disk filter for the TE mode.43 3.6 Resonance data of the micro-disk filter for the TM mode.444267842 bytesapplication/pdfen-US有限差分時域法濾波器Finite-Difference Time-DomainChannel Drop Filtersthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/50675/1/ntu-93-R91941028-1.pdf