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 Er   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
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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 axˆ, b2  2 ayˆ
Finally, the eigenvalue equation for  κ  is
 H
2

 2κ  H  κ G HHκ  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,nq 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 12 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,nq 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  2n  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
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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
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
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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M
5
a
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7
8
9
K
1st Workshop on Photonic Crystals
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
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
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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
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
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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 n1 i, j , k  1 
source
2 
© Copyright 2005
Sharif University of Technology
n
y
1st Workshop on Photonic Crystals
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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
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
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
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
κ r  R  exp jκ  R κ r 
Ex  0, y   exp j x Lx Ex  Lx , y 
Ex, y  0  exp j y Ly Ex, 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
Mashad, Iran, September 2005
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
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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Sharif University of Technology
1st Workshop on Photonic Crystals
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PML
Classification of Problems

Type III: Eigenvalue

Point-defects
PML

PML/SBC on all sides
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
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
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
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
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 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
Mashad, Iran, September 2005
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
Mashad, Iran, September 2005
Point Defects via FDTD

Time-domain output of probe
H-polarization
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
Mashad, Iran, September 2005
Point Defects via FDTD

FFT Spectrum near the Photonic Band Gap
© 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:
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
© Copyright 2005
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
© Copyright 2005
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
© 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

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
© 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
  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
© Copyright 2005
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
© Copyright 2005
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, …


© Copyright 2005
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

© Copyright 2005
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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
© Copyright 2005
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
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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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
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
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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Slab Photonic Crystals

TE-like
nsi  3.5
r  0.4 a
d  0.55a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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Slab Photonic Crystals

TM-like
nsi  3.5
r  0.4 a
d  0.55a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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Slab Photonic Crystal Cavity


Even mode :
3D : N  0.3005 2D + effective index :  N  0.304
QT  157
Q  161
Q  6820
© Copyright 2005
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
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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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
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Photonic Crystal Slab Waveguides

Dispersion Diagram r  0.4a nsi  3.5 d  0.55a
© Copyright 2005
Sharif University of Technology
1st Workshop on Photonic Crystals
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Photonic Crystal Slab Waveguides

Mode Profiles
A
© Copyright 2005
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 !