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Numerical methods for solving photonic crystal slabs
Photonic crystal slabs:
periodic structures of finite thickness
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Outline
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Photonic crystal (PhC) slabs
Layered media (special case of PhC slabs)
Transfer and scattering matrices
Numerical stability of T and S-matrices
Fourier Modal Method (FMM)
Factorization rules
FMM examples
Nonrectangular unit cell
Scattering matrix for software combination
C method
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Layered stack (special case of PhC slabs)
Goal: reflection, transmission, absorption, field distribution, dispersion diagrams
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Solution procedure:
4x4
1) Solve Maxwell’s equation in each layer separately
2) Compute interface and layer transfer matrices
3) Assemble the matrix (S-matrix) which links the input amplitudes with output
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Maxwell’s equations in an individual layer:
Maxwell’s equations in CGS units
constitutive relations
  xx

ˆ    yx

 zx
  xx
 xy  xz 
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 yy  yz  ; ˆ    yx

 zy  zz 
 zx
 xy
 yy
 zy
 xz 

 yz 
 zz 
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
ε, μ are generally tensors
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Maxwell’s equations in an individual layer:
Maxwell’s equations for curl operator
6 unknowns
1) All fields have time dependence
2) Look for a solution as a plane wave E,H~
3) Eliminate Ez, Hz-components
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
, k y=0
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Maxwell’s equations in an individual layer:
eigenvalue problem for kz
4x4 matrix
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Maxwell’s equations in an individual layer:
4 eigenvalues
4 eigenvectors
For isotropic medium (μ, ε) – scalars: K+1=K+2=-K-3=-K-4
Ex ( x, z)  E  exp(iK1 z  ikx x)  E  exp(iK4 z  ikx x), K1  K4 one can solve 2x2 matrices
Solution of the Maxwell’s equations in a layer
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Transfer matrix: links the amplitudes at z1 and z2
z1
z2
tij are 2x2 block matrices
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Interface and layer matrices:
layer transfer matrix through the layer dp
interface transfer matrix
total transfer matrix of the structure
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Transfer matrix is numerically unstable:
Reason: loss of precision when adding/subtracting big and small numbers
The error grows with the increase of the layer thicknesses
S-matrix or R-matrix algorithms are numerically stable
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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S-matrix recursive algorithm:
initial condition
recursive formula
If a layer of thickness L is added
no mixing of small and large exponents
Tikhodeev et al. 2002
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Transfer and Scattering matrices:
T-matrix fails with absorptive materials
4 periods of slabs:
 1  6.7;
d1  0.1m
 2  1.24  0.06i;
d 2  0.25m
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Example: dual band omnidirectional mirror
N=14 layers
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Fourier modal method (FMM):
some layers have 1D or 2D periodicity ε(x,y), μ=1
1) Solve Maxwell’s equation in each layer separately (by Floquet-Fourier decomposition)
2) Compute interface and layer transfer matrices
3) Assemble the matrix (S-matrix) which links the input amplitudes with output
IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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Maxwell’s equation in an individual layer
T. Weiss, Master's thesis, 4th Physics Institute University of Stuttgart (2008).
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Periodic layer:
u (r) periodic function
Ng=(2gx,max+1)(2gy,max+1) – truncation order
decomposition of permittivity function
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Periodic layer:
2Ng eigenvalues
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Compute interface and layer transfer matrices
Assemble the matrix (S-matrix) which links the input amplitudes with output
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Plane wave convention and space discretization
Single frequency analysis removes time from equations
Rectangular
discretization
is proved to be better
Toeplitz matrix
For the sake of simplicity we assume a permeability to be
constant
Fourier transform of functions
multiplication
Function “g” will be represented by vector and function “f” by Toeplitz matrix
Derivatives are represented
by diagonal matrices
Factorization rules
Diploma thesis: Thomas Weiss (University of Stuttgart) (2008)
L. Li, J. Opt. Soc. Am. A 13, 1870 (1996)
It often happens that functions “g” and “h”
have complementary jumps at certain points
Factorization rules
From a continuity of functions E_z, D_x and D_y one can derive:
Factorization rules
Factorization in x and y directions
FMM simulations
The accuracy depends on a number of modes and layers
FMM simulations
Simulation of nanowires array
Non-rectangular grid
L. Li, J. Opt. Soc. Am. A, Vol. 14, No. 10 (1997)
For some cases non-rectangular unit cell is the one with a smallest area
Fourier decomposition in
corresponding coordinates
Non-rectangular grid
Co- and contravariant coordinates
Incident light
Non-rectangular grid
Modes in photonic crystal
Field equations
Non-rectangular grid
Matrix representation
Non-rectangular grid
Material-matrix assembly
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Order the modes
Modes normalization on a field magnitude
Discretization is made for x and y components but modes are TE and TM
For non-rectangular grid covariant components are used
Scattering matrix and software combinations
CST-studio suite FEM solver
The same structure with FEM and FMM
Scattering matrix and software combinations
Connection between software
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Order of modes should be the same
Modes normalization on the same field magnitude
Phase of modes depend on the normalization
FEM-FMM combination
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FEM is more suitable for complex geometries
FEM is volume dependable
FMM is good for rectangular geometries
FMM does not depend on layers thickness
Scattering matrix and software combinations
Scattering matrix and software combinations
Ray tracing technique
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Matrix with embedded photonic elements
Coupling wave with geometrical optics
Could be used for structures too big for wave analysis
Embedded photonic structures could have various shapes
Scattering matrix and software combinations
Numerical optimization
Evolutionary strategy
Mirror optimization
The optimizer found a configuration with 3
Bragg reflectors
Methods for solving diffraction gratings
Departement/Institut/Gruppe
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C method: curved coordinates
Multilayer approximation in FMM
Introduce new coordinates
No multilayer approximation, matching of boundary conditions is easy!
Li et al. 1999 “Rigorous and efficient grating-analysis method made easy for optical engineers”
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C method: curved coordinates
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C method: curved coordinates
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