PWE and FDTD Methods for Analysis of Photonic Crystals

PWE and FDTD Methods for Analysis of Photonic Crystals
Integrated Photonics Laboratory School of Electrical Engineering Sharif University of Technology

Photonic Crystals Team
Faculty Bizhan Rashidian Rahim Faez Farzad Akbari Sina Khorasani Khashayar Mehrany Students & Graduates Alireza Dabirian Amir Hossein Atabaki Amir Hosseini Meysamreza Chamanzar Mohammad Ali Mahmoodzadeh Special Acknowledgements Keyhan Kobravi Sadjad Jahanbakht Maryam Safari © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

1st Workshop on Photonic Crystals
Outline Plane Wave Expansion (PWE) E- and H-Polarizations Sharif PWE Code Typical Band Structures Finite Difference Time Domain (FDTD) Description of Method Boundary Conditions Bloch Boundary Condition Perfectly Matched Layer Symmetric Boundary Condition © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Outline FDTD Sources Sharif FDTD Analysis Interface & Tool Band Structure Comparison to PWE/FEM Defective Structures Waveguide Cavity Coupled-Resonator Optical Waveguide Photonic Crystal Slab Waveguide Conclusions © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion Using Discrete Fourier Expansion we have Here , and are Inverse Lattice Vectors. © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion Inverse Lattice Vectors in 2D are given by For square lattice Finally, the eigenvalue equation for is © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Epansion Rewriting in matrix form we obtain where is the flattened vector of square matrix : © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion Similarly for H-polarization we have: After applying Bloch theorem we get: © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion Hence for E- and H-polarizations in triangular lattice we respectively get © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Sharif PWE Code Written in MATLAB Input arguments: N: Number of Plane Waves R: Number of Divisions on Each Side of BZ a: Lattice Constant (default value is 1) r: Radius of Holes/Rods e1: Permittivity of Holes/Rods e2: Permittivity of Host Medium © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Typical Band Structures
Infinitesimal perturbations in vacuum © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

2D Square Array of Dielectric Rods Si Rods in Air eSi=11.3 r/a=0.25 PBG #1, E-polarization © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

2D Square Array of Holes in Host Dielectric Air Holes in Si eSi=11.3 r/a=0.38 PBG #2, H-Polarization © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

2D Triangular Array of Holes in Host Air Holes in Si eSi=11.3 r/a=0.30 PBG #1, H-polarization © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

2D Triangular Array of Rods in Air Si Rods in Air eSi=11.3 r/a=0.35 PBG #2, E-polarization PBG #1, E-polarization © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Why FDTD ? Once run, information of the system in the whole frequency spectrum is achieved Capable of modal analysis with Fourier transforming No matrix inversion is needed, thanks to the explicit scheme This is extremely advantageous in large configurations with many components Very efficient for parallel processing © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Description of 3D FDTD Yee proposed a scheme in 1966 for time domain calculation of Maxwell’s equations FDTD was not practical until the advent of faster processors and larger memories in mid 1970s Taflove coined the acronym FDTD in 1970s © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Description of 3D FDTD Field components are discretized in each cell Maxwell’s curl equations are substituted by their difference equivalent Central difference scheme with second order accuracy Electric and magnetic field vectors interlaced in time © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Description of 3D FDTD Field components are discretized in each cell Maxwell’s curl equations are substituted by their difference equivalent Central difference scheme with second order accuracy Electric and magnetic field vectors interlaced in time Explicit Scheme No Matrix Inversion © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Description of 3D FDTD The finite difference equivalent of the z-component of Ampere’s law becomes © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Features of FDTD Maxwell’s integral equations are satisfied as the same time. Maxwell’s equations, rather than Helmholtz equation is solved Both electric and magnetic field boundary conditions are met explicitly Maxwell’s divergence equations are simultaneously satisfied, because of the location of the field components Interlacing of the electric and magnetic fields in time, makes the scheme explicit No matrix inversion is needed © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Bloch Boundary Condition
Bloch boundary Condition is used to analyze periodic structures by considering only one cell © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Bloch Boundary Condition
Bloch boundary Condition is used to analyze periodic structures by considering only one cell From Bloch’s theorem © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Symmetry Boundary Condition
If the structure is symmetric with respect to a plane, the electromagnetic field components are either even or odd with respect to the same plane. The computational efficiency is greatly enhanced Degenerate modes can be studied separately © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Perfectly Matched Layer
For transparent boundaries we need a boundary condition which should Has zero reflection to incoming waves Any frequency Any polarization Any angle of incidence Be thin Effective near sources © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

In 1994 Bereneger constructed a boundary layer that perfectly matched to all incoming waves. It dissipates the wave within itself. It terminates to other symmetry boundary conditions, itself. It is based on a field-splitting technique, so that in 3D we get 12 equations rather than 6, therefore there is no physical insight. © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Gedney proposed another model for PML in 1996 that outperformed the Bereneger’s original model. Gendney’s PML is modeled by a lossy anisotropic media, directly explained by non-modified Maxwell’s equations. Reflection from PML is typically -120dB, but it can be as low as -200 dB. © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Classification of Problems
Photonic crystal problems with regard to the boundary conditions can be generally categorized into three groups Type I: Crystal Band-Structure Type II: Line/Plane Defect Band-Structure Type III: Eigenvalue Type IV: Propagation © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Type II: Line/Plane Defect Waveguide CROW BBC on two sides PML (and SBC) on the other sides BBC PML Symmetry Plane BBC PML © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

FDTD Sources Type I/II/III: Initial Field Type IV: Point Source Sinusoidal/Gaussian in Time Huygens’ Source (radiates only in one direction) Gaussian in Space Slab Waveguide Eigenmode © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Sharif FDTD Sharif FDTD Code Written in C++ 2D/3D Supports Initial Field, Point Source, Huygens’ Source Visual Basic Graphical Interface for 2D structures and slab waveguides (3D under development) MATLAB Graphics Post-processor © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Sharif FDTD Outputs Band-Structure Waveguide Band-Structure Probe Field Snapshots (Animations) Power-plane Integrator © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Sharif FDTD/Graphical Interface
© Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Band-Structure via FDTD
Steps to calculate the band-structure 1. Take one pair on the reciprocal lattice 2. Put an initial field in the computational grid 3. Save one field component in a low symmetry point 4. Get FFT from the saved signal 5. Detect the peaks 6. Repeat for all Bloch vectors Probe X-point : © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Point Defects via FDTD Calculating the resonance frequency: 1. Use an initial field or a Gaussian point source 2. Propagate on the FDTD grid 3. Use a probe to save field 4. Take FFT 5. Find Peaks inside PBGs © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Point Defects via FDTD Calculating the modes of the cavity: Taking Fourier transform of an Initial field propagating in the structure at each grid, at the resonant frequency. For this example: Monopole Mode © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005 Monopole with A1 symmetry

Quality Factor of Cavities
If U(t) denotes total energy inside the cavity then © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Quality Factor of Cavities
Hence for the Monopole Mode we calculate Q=315 from the slope of energy loss. © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Cavity in Triangular Lattice
This cavity has one double degenerate mode Using symmetry boundary conditions these modes are separately studied © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Eigenmode Profiles Small discrepancy in frequencies is due to geometrical asymmetry of the cavity. Even mode : f = 0.304 Q=87 Odd mode : f = 0.297 Q=83 © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Q increases exponentially with the number of the layers n Q 3 92 4 240 5 700 6 2000 7 6000 3 4 5 6 7 10 1 2 Number of layers Quality factor © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Waveguides in Square Lattice
By removing one row of rods from a bulk photonic crystal a waveguide is created © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Waveguides in Triangular Lattice
One column is removed from a bulk photonic crystal Computational cell © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Coupled Resonator Optical Waveguide
Waveguiding mechanisms: Total Internal Reflection Fibers Slab Waveguide Reflection due to Photonic Band Gap Photonic Crystal Wavegiude Evanescent Coupling Coupled Resonator Optical Waveguide © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Wave is coupled from one resonator to the adjacent through evanescent waves. Slow process Small group velocity L = 2a,3a,4a, … cavity L © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Slab Photonic Crystals
3D slab photonic crystal slabs: Confinement in the plane of slab (x-y) by PBG Confinement perpendicular to slab (z) by TIR No decoupling to TE and TM polarizations © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

TE-Like Slab Modes Even TE slab mode + Odd TM slab mode = TE-Like mode for Slab Photonic Crystal © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

TM-Like Slab Modes Even TM slab mode + Odd TE slab mode = TM-Like mode for Slab Photonic Crystal © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Symmetry boundary conditions can be applied in the middle of slab Symmetry decouples the TE-like and TM-like modes. TE-like and TM-like modes can be studied separately © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Conclusions Plane Wave Expansion method has been coded and various results were obtained. Results of MATLAB code for 2D single cell photonic crystal band structure computations are reliable and efficient enough. Performance of PWE is questionable beyond the abovementioned applications. © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Conclusions 2D and 3D FDTD codes are implemented in C++ and verified by comparing to reported results in literature in the following cases: Bandstructure of bulk photonic crystals Resonant frequencies and Q-factor of different cavities Dispersion diagram of different waveguides … © Copyright 2005 Sharif University of Technology 1st Workshop on Photonic Crystals Mashad, Iran, September 2005

Thanks for your attention !

PWE and FDTD Methods for Analysis of Photonic Crystals

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PWE and FDTD Methods for Analysis of Photonic Crystals

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