1、PWE and FDTD Methods for Analysis of Photonic Crystals,Integrated Photonics Laboratory School of Electrical Engineering Sharif University of Technology, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Photonic Crystals Team,Faculty Bizhan
2、 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 Te
3、chnology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,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 Sy
4、mmetric 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 Wa
5、veguide Photonic Crystal Slab Waveguide Conclusions, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Plane Wave Expansion,E-polarization:Using Bloch theorem we obtain, Copyright 2005 Sharif University of Technology,1st Workshop on Photoni
6、c Crystals Mashad, Iran, September 2005,Plane Wave Expansion,Using Discrete Fourier Expansion we haveHere , 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
7、2D are given byFor 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,Expanding the master equation we getwhere we have used, Copyright 2005 Sharif University of Tec
8、hnology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Plane Wave Epansion,Rewriting in matrix form we obtainwhere is the flattened vector of square matrix :, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Plane Wave Epan
9、sion,Similarly is the flattened matrix of a 4D tensor:Hence, 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
10、of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Plane Wave Epansion,Therefore for H-polarization:where we have used, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Plane Wave Expansion,For Triangular-Lattice
11、we use, 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 Mash
12、ad, 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 Univer
13、sity 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,Typical Band Structures,2D Square Array of Di
14、electric RodsSi Rods in Air eSi=11.3 r/a=0.25, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Square Array of Dielectric RodsSi Rods in Air eSi=11.3 r/a=0.25,PBG #1, E-polarization, Copyright 2005 Sharif Univer
15、sity of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #1, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #2, Copyright 2005 Sharif Univ
16、ersity of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #3, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Square Array of Holes in Host Dielectr
17、icAir Holes in Si eSi=11.3 r/a=0.38, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Square Array of Holes in Host DielectricAir Holes in Si eSi=11.3 r/a=0.38,PBG #2, H-Polarization, Copyright 2005 Sharif Univer
18、sity of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #1, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #2, Copyright 2005 Sharif Univ
19、ersity of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,Band Surface #3, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Triangular Array of Holes in HostAir H
20、oles in Si eSi=11.3 r/a=0.30, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Triangular Array of Holes in HostAir Holes in Si eSi=11.3 r/a=0.30,PBG #1, H-polarization, Copyright 2005 Sharif University of Techno
21、logy,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Triangular Array of Rods in AirSi Rods in Air eSi=11.3 r/a=0.35, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Typical Band Structures,2D Tri
22、angular Array of Rods in AirSi Rods in Air eSi=11.3 r/a=0.35,PBG #1, E-polarization,PBG #2, 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
23、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
24、 Photonic Crystals Mashad, Iran, September 2005,Description of 3D FDTD,Yee proposed a scheme in 1966 for time domain calculation of Maxwells 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 200
25、5 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,FDTD,Computational window is divided into a cubic lattice,x,z,y, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Description of 3D FDTD,Field
26、 components are discretized in each cell Maxwells 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 Crystal
27、s Mashad, Iran, September 2005,Description of 3D FDTD,Field components are discretized in each cell Maxwells 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 Ma
28、trix 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 thez-component of Amperes law becomes, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crys
29、tals Mashad, Iran, September 2005,Features of FDTD,Maxwells integral equations are satisfied as the same time. Maxwells equations, rather than Helmholtz equation is solved Both electric and magnetic field boundary conditions are met explicitly Maxwells divergence equations are simultaneously satisfi
30、ed, 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,Stability of FDTD,The st
31、ability condition isThis implies that,Numerical Phase Velocity, 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, Copyrig
32、ht 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 cellFrom Blochs theorem, Copyright 2005 Sharif University of Technology,1st Workshop
33、 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 s
34、tudied 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
35、 of incidence Be thin Effective near sources, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Perfectly Matched Layer,In 1994 Bereneger constructed a boundary layer that perfectly matched to all incoming waves. It dissipates the wave with
36、in 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,
37、 September 2005,Perfectly Matched Layer,Gedney proposed another model for PML in 1996 that outperformed the Berenegers original model. Gendneys PML is modeled by a lossy anisotropic media, directly explained by non-modified Maxwells equations. Reflection from PML is typically -120dB, but it can be a
38、s 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 groupsType I: Crystal Band-Structure Ty
39、pe 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,Classification of Problems,Type I: Band Structure Perfect Lattice CPCRABBC on all sides,BBC, Copyright 2005
40、 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Classification of Problems,Type II: Line/Plane Defect Waveguide CROWBBC on two sides PML (and SBC) on the other sides, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mash
41、ad, Iran, September 2005,Classification of Problems,Type III: Eigenvalue Point-defectsPML/SBC on all sides, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Classification of Problems,Type IV: PropagationPML on all sides (or SBC if needed)
42、,PML,PML,BBC,BBC,SBC, 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) Sinusoidal/Gaussian in Time
43、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 stru
44、ctures 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 I
45、ntegrator, 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,Sharif FDTD/Graphical Interface, Cop
46、yright 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,Sharif FDTD/Graphical Interface, Copyright 2005 Sh
47、arif 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,Sharif FDTD/Graphical Interface, Copyright 2005 Sharif Universit
48、y 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 fro
49、m 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,Band-Structure via FDTD,Typical spectrum obtained from the probe, Copyright 2005 Sharif University of Technology,1st Workshop on Photonic Crystals Mashad, Iran, September 2005,Band-Structure via FDTD,Square lattice of dielectric rods,
