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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
Special Acknowledgements
© Copyright 2005
Sharif University of Technology
Alireza Dabirian
Amir Hossein Atabaki
Amir Hosseini
Meysamreza Chamanzar
Mohammad Ali
Mahmoodzadeh
Keyhan Kobravi
Sadjad Jahanbakht
Maryam Safari
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
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
E-polarization:
k c
r 1 r
Using Bloch theorem we obtain
LE Er 0
LE r 2 k 2
E r exp jκ r κ r
L E κ κ r 0
L E κ r 2 2 jκ κ 2 k 2
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Plane Wave Expansion
Using Discrete Fourier Expansion we have
κ r Gκ exp jG r
G
r G exp jG r
G
Here G G mn mb1 nb2 , and H H mn
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
zˆ a1
a 2 zˆ
b 2 2
b1 2
a1 a 2 zˆ
a1 a 2 zˆ
For square lattice b1 2 axˆ, b2 2 ayˆ
Finally, the eigenvalue equation for κ is
H
2
2κ H κ G HHκ k Gκ
2
H
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
2
Plane Wave Epansion
Expanding the master equation we get
N
N
2
2
κ mn κ
k
k
mnκ m p,nq pqκ
mn κ
p N q N
4
2
2
2
2
2
m x n y x y
m n
a
a
2
mn κ
where we have used
G G mn
2
mxˆ nyˆ , κ x xˆ y yˆ
mb1 nb 2
a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Plane Wave Epansion
Rewriting in matrix form we obtain
Sκ κ k κ κ
2
where is the flattened vector of square
matrix κ mn κ :
κ κ 2 N 12 N 1
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
2 N 1 1
2
Plane Wave Epansion
Similarly Sκ is the flattened matrix of a 4D
tensor:
Sκ S mnpq κ
mn κ p m ,q n
2 N 1 2 N 1 2 N 1 2 N 1
Hence
Sκ Sκ 2 N 1 2N 1
2
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
2
Plane Wave Expansion
Similarly for H-polarization we have:
LH H r 0 LH r k 2
k c
r 1 r
After applying Bloch theorem we get:
H κ G κ
G H Hκ
H
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
k Gκ
2
Plane Wave Epansion
Therefore for H-polarization:
N
N
p N q N
m p,nq pqκ k mnκ k κ mnκ
2
mnpq κ
2
2
m p x n q y x 2 y 2
mp nq
a
a
2
mnpq κ
2
where we have used
G Gmn mb1 nb2 , H H pq pb1 qb2
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Plane Wave Expansion
For Triangular-Lattice we use
a1 axˆ
a
a 2 xˆ 3 yˆ
2
2
1
b1
yˆ
xˆ
a
3
4
b2
yˆ
3a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
b2
a2
a1
b1
Plane Wave Expansion
Hence for E- and H-polarizations in
triangular lattice we respectively get
mn κ
4
4 2
2
m x 2n 3m y
m n 3m n
a
a
2
x y
2
2
3
4
m q np
m p nq
2
a
2
m p x 2n q 3 m p y x 2 y 2
a
2
mnpq κ
© 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
1: Permittivity of Holes/Rods
2: 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
Blue-Solid Line: TE mode, Red-Dashed Line: TM mode
1
N
0.8
0.6
0.4
0.2
0
0
1
© Copyright 2005
Sharif University of Technology
2
3X
4
5
6 M
7
a
1st
Workshop on Photonic Crystals
Mashad, Iran, September 2005
8
9
10
Typical Band Structures
2D Square Array of Dielectric Rods
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.8
0.7
0.6
N
0.5
0.4
0.3
0.2
0.1
L=1, r=0.25, a=11.3, b=1
0
0
1
2
3X
© Copyright 2005
Sharif University of Technology
4
5
a
6 M
7
8
9
10
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Si Rods in Air
Si=11.3
r/a=0.25
Typical Band Structures
2D Square Array of Dielectric Rods
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.8
0.7
0.6
N
0.5
0.4
PBG #1, E-polarization
0.3
0.2
0.1
L=1, r=0.25, a=11.3, b=1
0
0
1
2
3X
© Copyright 2005
Sharif University of Technology
4
5
a
6 M
7
8
9
10
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Si Rods in Air
Si=11.3
r/a=0.25
Typical Band Structures
E-polarization, first surface, L=1, r=0.25, a=11.3, b=1
Band Surface #1
Contours of first band
0.5
3
0.4
2
0.3
1
y
0.2
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
E-polarization, first two surfaces, L=1, r=0.25, a=11.3, b=1
Band Surface #2
Countours of second band
0.5
3
0.4
2
0.3
1
y
0.2
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
E-polarization, first three surfaces, L=1, r=0.25, a=11.3, b=1
Band Surface #3
Countours of third band
0.5
3
0.4
2
0.3
1
y
0.2
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
2D Square Array of Holes in Host Dielectric
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.5
N
0.4
0.3
0.2
0.1
L=1, r=0.38, a=1, b=11.3
0
0
2
X
© Copyright 2005
Sharif University of Technology
6
4
a
M
8
10
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Air Holes in Si
Si=11.3
r/a=0.38
Typical Band Structures
2D Square Array of Holes in Host Dielectric
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.5
PBG #2, H-Polarization
N
0.4
0.3
0.2
0.1
L=1, r=0.38, a=1, b=11.3
0
0
2
X
© Copyright 2005
Sharif University of Technology
6
4
a
M
8
10
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Air Holes in Si
Si=11.3
r/a=0.38
Typical Band Structures
E-polarization, first surface, L=1, r=0.38, a=1, b=11.3
Band Surface #1
0.4
Contours of first band
3
0.3
2
0.2
y
1
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
Band Surface #2
E-polarization, first and second surfaces, L=1, r=0.38, a=1, b=11.3
0.4
Contours of second band
3
0.3
2
0.2
y
1
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
E-polarization, first three surfaces, L=1, r=0.38, a=1, b=11.3
Band Surface #3
0.4
Contours of third band
3
0.3
2
0.2
y
1
0
0.1
-1
0
-2
2
-3
-3
-2
-1
0
1
x
© Copyright 2005
Sharif University of Technology
2
3
2
0
0
-2
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
-2
Typical Band Structures
2D Triangular Array of Holes in Host
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.45
0.4
0.35
n
0.3
0.25
0.2
Air Holes in Si
Si=11.3
r/a=0.30
0.15
0.1
L=1, r=0.3, a=1, b=11.3
0.05
0
0
1
2
4
3
© Copyright 2005
Sharif University of Technology
M
6
5
a
7
8
9
K
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Typical Band Structures
2D Triangular Array of Holes in Host
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.45
0.4
0.35
n
0.3
PBG #1, H-polarization
0.25
0.2
Air Holes in Si
Si=11.3
r/a=0.30
0.15
0.1
L=1, r=0.3, a=1, b=11.3
0.05
0
0
1
2
4
3
© Copyright 2005
Sharif University of Technology
M
6
5
a
7
8
9
K
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Typical Band Structures
2D Triangular Array of Rods in Air
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.6
0.5
n
0.4
0.3
Si Rods in Air
Si=11.3
r/a=0.35
0.2
0.1
L=1, r=0.35, a=11.3, b=1
0
0
1
2
3
© Copyright 2005
Sharif University of Technology
4
M
5
a
6
7
8
9
K
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Typical Band Structures
2D Triangular Array of Rods in Air
Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization
0.6
0.5
PBG #2, E-polarization
n
0.4
0.3
PBG #1, E-polarization
Si Rods in Air
Si=11.3
r/a=0.35
0.2
0.1
L=1, r=0.35, a=11.3, b=1
0
0
1
2
3
© Copyright 2005
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4
M
5
a
6
7
8
9
K
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
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1st Workshop on Photonic Crystals
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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
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FDTD
Computational window is divided into a
cubic lattice
z
x
y
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Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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1st Workshop on Photonic Crystals
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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
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Description of 3D FDTD
The finite difference equivalent of the
z-component of Ampere’s law becomes
* i, j , k 12 t
1
2 i, j , k 12
n 1
1
E z i, j , k 2
* i, j , k 12 t
1
2 i, j , k 12
n
1
E
i
,
j
,
k
z
2
H yn 2 i 12 , j , k 12 H yn 2 i 12 , j , k 12
x
1
H
n 12
x
1
t
i, j , k 12
* i, j , k 1 t
2
1
2 i, j , k 12
1
2
i, j 12 , k 1 2 H x i , j 12 , k 12 J n1 i, j , k 1
source
2
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n
y
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
z
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
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1st Workshop on Photonic Crystals
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Stability of FDTD
The stability condition is
0 t
1
c
1
x 2
1
1
1
2
y z 2
This implies that
Numerical Phase Velocity c
© Copyright 2005
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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
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1st Workshop on Photonic Crystals
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Bloch Boundary Condition
Bloch boundary Condition is used to analyze periodic
structures by considering only one cell
From Bloch’s theorem
κ r R exp jκ R κ r
Ex 0, y exp j x Lx Ex Lx , y
Ex, y 0 exp j y Ly Ex, y Ly
© Copyright 2005
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1st Workshop on Photonic Crystals
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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
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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
Perfectly Matched Layer
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
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Perfectly Matched Layer
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 nonmodified Maxwell’s equations.
Reflection from PML is typically -120dB, but
it can be as low as -200 dB.
© Copyright 2005
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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
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Classification of Problems
Type I: Band Structure
BBC
Perfect Lattice
CPCRA
BBC
BBC on all sides
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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Classification of Problems
Type II: Line/Plane Defect
Waveguide
CROW
BBC
Symmetry
Plane
BBC
PML
BBC on two sides
PML (and SBC) on the other sides
© Copyright 2005
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1st Workshop on Photonic Crystals
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PML
Classification of Problems
Type III: Eigenvalue
Point-defects
PML
PML/SBC on all sides
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Sharif University of Technology
1st Workshop on Photonic Crystals
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Classification of Problems
Type IV: Propagation
PML
BBC
SBC
PML
BBC
PML on all sides (or SBC if needed)
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
Gaussian in Space
Slab Waveguide Eigenmode
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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 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 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 x , y 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
X-point : x , y 0
L
© Copyright 2005
Sharif University of Technology
Probe
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
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Band-Structure via FDTD
Square lattice of dielectric rods
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Band-Structure via FDTD
Square lattice of dielectric rods
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Band-Structure via FDTD
Square lattice of air holes; FDTD vs. PWE
H-polarization
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Band-Structure via FDTD
Square lattice of air holes; FDTD vs. PWE
E-polarization
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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Band-Structure via FDTD
Square lattice of square rods; FDTD vs. FEM
E-polarization
a
L
L 0 .5 a
b 11
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Band-Structure via FDTD
Triangular lattice of air holes
Unit cell
b 7.9
r 0.3a
© 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
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Point Defects via FDTD
Time-domain output of probe
H-polarization
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Point Defects via FDTD
FFT Spectrum near the Photonic Band Gap
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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:
f1 0.2197
Monopole Mode
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September
2005
Monopole
with A1 symmetry
Point Defects via FDTD
Degenerate Dipole Modes ( f 2 0.2466)
1st Workshop on Photonic Crystals
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Double
degenerate
with E symmetry
Mashad, Iran, September
2005
Sharif University of Technology
Quality Factor of Cavities
If U(t) denotes total energy inside the
cavity then
U (t ) U (0) exp( 0t Q)
lnU (t ) lnU (0) (0 Q)t
U (t )
Q 0
P(t )
© 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
Cavity in Triangular Lattice
Eigenmode Profiles
Odd mode :
Even mode :
f = 0.297
f = 0.304
Q=83
Q=87
Small discrepancy in frequencies is due to
geometrical asymmetry of the cavity.
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Cavity in Triangular Lattice
Q increases exponentially with the number of the
layers
n
Q
3
92
4
240
5
700
6
2000
7
6000
104
Quality factor
103
102
101
3
4
5
Number of layers
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
6
7
Waveguides in Square Lattice
By removing one row of rods from a bulk
photonic crystal a waveguide is created
nrod 3.4
r 0.18a
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Square Lattice
Dispersion of waveguide; single even mode
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Square Lattice
Dispersion of waveguide; single even mode
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Square Lattice
Two rows of rods are removed from a
bulk photonic crystal
nrod 3.4
r 0.18a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Square Lattice
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Square Lattice
Even 2
Odd
Even 1
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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
2.65
r 0.3a
Computational cell
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Triangular Lattice
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Waveguides in Triangular Lattice
Even
Odd
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Coupled Resonator Optical
Waveguide
Waveguiding mechanisms:
Total Internal Reflection
Reflection due to Photonic Band Gap
Fibers
Slab Waveguide
Photonic Crystal Wavegiude
Evanescent Coupling
Coupled Resonator Optical Waveguide
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Coupled Resonator Optical
Waveguide
Wave is coupled from one resonator to the
adjacent through evanescent waves.
cavity
Slow process
Small group velocity
L = 2a,3a,4a, …
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
L
Coupled Resonator Optical
Waveguide
Odd Mode
L=2
PML
Bloch
BC
Bloch
BC
Symmetry
BC
Computational cell
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Coupled Resonator Optical
Waveguide
Even Mode
L=2
PML
Bloch
BC
Bloch
BC
Symmetry
BC
Computational cell
© 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
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
TE Slab Modes
For a simple slab waveguide mode profiles
are as below
Even mode
Odd mode
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
TM Slab Modes
For a simple slab waveguide mode profiles are as
below
Even mode
Odd mode
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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
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Slab Photonic Crystals
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
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Slab Photonic Crystals
TE-like
nsi 3.5
r 0.4 a
d 0.55a
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Slab Photonic Crystals
TM-like
nsi 3.5
r 0.4 a
d 0.55a
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Slab Photonic Crystal Cavity
O. Painter et al., J. Opt. Soc. Am B. 16, 275 (1999)
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Slab Photonic Crystal Cavity
Even mode :
3D : N 0.3005 2D + effective index : N 0.304
QT 157
Q 161
Q 6820
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
1
1
1
QT Q Q||
Slab Photonic Crystal Cavities
Odd mode :
3D : N 0.2995 2D + effective index : N 0.297
QT 157
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Photonic Crystal Slab Waveguides
M. Loncar et al., J. Lightwav Tech. 18, 1402 (2000)
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Photonic Crystal Slab Waveguides
Dispersion Diagram r 0.4a nsi 3.5 d 0.55a
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Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Photonic Crystal Slab Waveguides
Mode Profiles
A
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Sharif University of Technology
1st Workshop on Photonic Crystals
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B
Photonic Crystal Slab Waveguides
Parameters : r 0.3a
nInGaAsP 3.4
d 0 .5 a
Triangular Lattice
Slab Photonic
Crystal Waveguide
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Photonic Crystal Slab Waveguides
Parameters : r 0.3a nInGaAsP 3.4 d 0.5a neff 2.65
Even Mode
Excellent agreement
between 3D and
2D Effective Index
methods
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Photonic Crystal Slab Waveguides
Parameters : r 0.3a nInGaAsP 3.4 d 0.5a neff 2.65
Odd Mode
Excellent agreement
between 3D and
2D Effective Index
methods
© 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 !