pwe and fdtd methods for analysis of photonic crystals integrated photonics laboratory school of...

PWE and FDTD PWE and FDTD Methods for Methods for Analysis of Analysis of

Photonic Crystals Photonic Crystals Integrated Photonics

LaboratorySchool of Electrical

EngineeringSharif University of

Technology

© Copyright 2005Sharif University of Technology

1st Workshop on Photonic CrystalsMashad, Iran, September 2005

Photonic Crystals TeamPhotonic 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



OutlineOutline

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



OutlineOutline

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



Plane Wave ExpansionPlane Wave Expansion

E-polarization:

Using Bloch theorem we obtain

0LE rE

222

E

E

2L

0L

exp

kκj

jE

κr

r

rrκr

κ

κκ

κ

rr 1 ck

22EL k r




Using Discrete Fourier Expansion we have

Here , and are Inverse Lattice Vectors.

G

G rGr jexp

G

Gκκ rGr jexp

21 bbGG nmmn mnHH




Inverse Lattice Vectors in 2D are given by

For square lattice Finally, the eigenvalue equation

for is GκH

HκHGHκ 222 2 kκH

z

zˆ

ˆ2

21

21

aa

ab

z

zˆ

ˆ2

21

12

aa

ab

κ yaxa ˆ2,ˆ2 21 bb



Plane Wave EpansionPlane Wave Epansion

Expanding the master equation we get

where we have used

22222

22,

42yxyxmn

mnmn

N

Np

N

Nqpqqnpmmn

nma

nma

kk

κ

κκκκ κ

yxynxma

nm yxmn ˆˆ,ˆˆ2

21 κbbGG




Rewriting in matrix form we obtain

where is the flattened vector of square matrix :

κκκ κ 2kS κκ mn

112 2 N 1212 NNκκ




Similarly is the flattened matrix of a 4D tensor:

Hence

12121212,

NNNNnqmpmn

mnpqSS

κ

κκ

22 1212 NNSS κκ

κS




Similarly for H-polarization we have:

After applying Bloch theorem we get:

0LH rH

rr 1 ck

2HL k r

GκH

HκHGκGκH 2k




Therefore for H-polarization:

where we have used

222

22yxyxmnpq qnpm

anqmp

a

κ

2121 , bbHHbbGG qpnm pqmn

κκκκ κ mnmn

N

Np

N

Nqpqqnpmmnpq kk 22

,




For Triangular-Lattice we use

yxa

ˆ3

1ˆ

21

b

ya

ˆ3

42

b

xaˆ1 a

yxa

ˆ3ˆ22 a

1b

2b

1a2a




Hence for E- and H-polarizations in triangular lattice we respectively get

22

222

324

34

yx

yxmn mnma

mnnma

κ

22

2

322

2

34

yxyx

mnpq

pmqnpma

npmqnqmpa

κ



Sharif PWE CodeSharif 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



Typical Band StructuresTypical Band Structures

Infinitesimal perturbations in vacuum

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Blue-Solid Line: TE mode, Red-Dashed Line: TM mode

X M a

N




2D Square Array of Dielectric Rods

Si Rods in AirSi=11.3

r/a=0.250 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization

L=1, r=0.25, a=11.3, b=1

X M a

N




2D Square Array of Dielectric Rods


r/a=0.250 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7


L=1, r=0.25, a=11.3, b=1

X M a

N

PBG #1, E-polarization




Band Surface #1

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

0.5

E-polarization, first surface, L=1, r=0.25, a=11.3,

b=1

Contours of first band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




Band Surface #2

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

0.5

E-polarization, first two surfaces, L=1, r=0.25, a=11.3,

b=1

Countours of second band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




Band Surface #3

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

0.5

E-polarization, first three surfaces, L=1, r=0.25, a=11.3,

b=1

Countours of third band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




2D Square Array of Holes in Host Dielectric

Air Holes in Si Si=11.3

r/a=0.380 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

Blue-Solid Line: E-polarization, Red-Dashed Line: H-polarization

L=1, r=0.38, a=1,

b=11.3

X M a

N




2D Square Array of Holes in Host Dielectric


r/a=0.380 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5


L=1, r=0.38, a=1,

b=11.3

X M a

N

PBG #2, H-Polarization




Band Surface #1

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

E-polarization, first surface, L=1, r=0.38, a=1,

b=11.3

Contours of first band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




Band Surface #2

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

E-polarization, first and second surfaces, L=1, r=0.38, a=1,

b=11.3

Contours of second band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




Band Surface #3

-2

0

2

-2

0

2

0

0.1

0.2

0.3

0.4

E-polarization, first three surfaces, L=1, r=0.38, a=1,

b=11.3

Contours of third band

x

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3




2D Triangular Array of Holes in Host


r/a=0.300 1 2 3 4 5 6 7 8 9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


L=1, r=0.3, a=1,

b=11.3

M K a

n




2D Triangular Array of Holes in Host


r/a=0.300 1 2 3 4 5 6 7 8 9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


L=1, r=0.3, a=1,

b=11.3

M K a

n PBG #1, H-polarization




2D Triangular Array of Rods in Air


r/a=0.350 1 2 3 4 5 6 7 8 9

0

0.1

0.2

0.3

0.4

0.5


L=1, r=0.35, a=11.3,

b=1

M K a

n



Why FDTD ?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



Description of 3D FDTDDescription 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



FDTDFDTD

Computational window is divided into a cubic lattice

x

z

y




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




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




The finite difference equivalent of the

z-component of Ampere’s law becomes

2112

1212

1

21

21

21

21

21

21

21*

21

21

21

21*

21

21*

211

,,,,21,,

,,,,

,,2

,,1

,,,,

,,2

,,1

,,2

,,1

,,

21

21

21

kjiJy

kjiHkjiH

x

kjiHkjiH

kji

tkji

kjit

kjiE

kji

tkji

kji

tkji

kjiE

nsource

n

xnx

ny

ny

nz

nz

z



Features of FDTDFeatures 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



Stability of FDTDStability of FDTD

The stability condition is

This implies that

222

111

110

zyx

ct

Numerical Phase Velocity

c



Bloch Boundary Bloch Boundary ConditionCondition

Bloch boundary Condition is used to analyze periodic structures by considering only one cell



Bloch Boundary Bloch Boundary ConditionCondition

Bloch boundary Condition is used to analyze periodic structures by considering only one cellFrom Bloch’s theorem

rRRr κκ κ jexp

yLxELjyxE xxx ,exp,0

yyy LyxELjyxE ,exp0,



Symmetry Boundary Symmetry Boundary ConditionCondition

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

enhancedDegenerate modes can be studied separately



Perfectly Matched LayerPerfectly 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




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.




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.



Classification of Classification of ProblemsProblems

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




Type I: Band Structure Perfect Lattice CPCRA

BBC on all sides

BBC

BBC




Type II: Line/Plane Defect Waveguide CROW

BBC on two sides PML (and SBC) on the other sides

BBC

PMLSymmetry Plane

BBCPML




Type III: Eigenvalue Point-defects

PML/SBC on all sides

PML




Type IV: Propagation

PML on all sides (or SBC if needed)

PML

PML

BBC

BBC

SBC



FDTD SourcesFDTD 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



Sharif FDTDSharif 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



Sharif FDTDSharif FDTD

Outputs Band-Structure Waveguide Band-Structure Probe Field Snapshots (Animations) Power-plane Integrator



Sharif FDTD/Graphical Sharif FDTD/Graphical InterfaceInterface



Band-Structure via FDTDBand-Structure via FDTD Steps to calculate the band-structure1. Take one pair on the reciprocal lattice 2. Put an initial field in the computational grid3. Save one field component in a low symmetry

point4. Get FFT from the saved signal 5. Detect the peaks6. Repeat for all Bloch vectors

yx ,

Probe0, yx L

X-point :



Band-Structure via FDTDBand-Structure via FDTD

Typical spectrum obtained from the probe




Square lattice of dielectric rods




Square lattice of air holes; FDTD vs. PWE

H-polarization




Square lattice of air holes; FDTD vs. PWE

E-polarization




Square lattice of square rods; FDTD vs. FEM

L

a

aL 5.011b

E-polarization




Triangular lattice of air holes

ar 3.0

9.7b

Unit cell



Point Defects via FDTDPoint Defects via FDTD

Calculating the resonance frequency:

1. Use an initial field or a Gaussian point source

2. Propagate on the FDTD grid3. Use a probe to save field4. Take FFT5. Find Peaks inside PBGs




Time-domain output of probe

H-polarization




FFT Spectrum near the Photonic Band Gap




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

2197.01 f

Monopole with A1 symmetry




Degenerate Dipole Modes ( )

2466.02 f

Double degenerate with E symmetry



Quality Factor of CavitiesQuality Factor of Cavities

If U(t) denotes total energy inside the cavity then

)(exp)0()( 0 QtUtU

tQUtU )()0(ln)(ln 0

)(

)(0 tP

tUQ



Quality Factor of CavitiesQuality Factor of Cavities

Hence for the Monopole Mode we calculate Q=315 from the slope of energy loss.



Cavity in Triangular Cavity in Triangular LatticeLattice

This cavity has one double degenerate mode

Using symmetry boundary conditions these modes are separately studied



Eigenmode Profiles

Small discrepancy in frequencies is due to geometrical asymmetry of the cavity.


Odd mode :

f = 0.297

Q=83

Even mode :

f = 0.304

Q=87




Q increases exponentially with the number of the layersn Q

3 924 2405 7006 20007 6000 3 4 5 6 7

101

102

103

104

Number of layers

Quality factor



Waveguides in Square Waveguides in Square LatticeLattice

By removing one row of rods from a bulk photonic crystal a waveguide is created

ar 18.0

4.3rodn




Dispersion of waveguide; single even mode




Two rows of rods are removed from a bulk photonic crystal

ar 18.0

4.3rodn




Even 2

Even 1

Odd



Waveguides in Waveguides in Triangular LatticeTriangular Lattice

One column is removed from a bulk photonic crystal

ar 3.0

65.2

Computational cell




Even

Odd



Coupled Resonator Coupled Resonator Optical WaveguideOptical Waveguide

Waveguiding mechanisms: Total Internal Reflection

Fibers Slab Waveguide

Reflection due to Photonic Band Gap Photonic Crystal Wavegiude

Evanescent Coupling Coupled Resonator Optical Waveguide




Wave is coupled from one resonator to the adjacent through evanescent waves.

Slow process Small group velocity

L = 2a,3a,4a, …L

cavity




BlochBC

PML

BlochBC

Symmetry BC

Computational cell

L=2

Odd Mode




Even Mode

BlochBC

PML

BlochBC

Symmetry BC

Computational cell

L=2



Slab Photonic CrystalsSlab 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



TE Slab ModesTE Slab Modes

For a simple slab waveguide mode profiles are as below

Even mode

Odd mode



TM Slab ModesTM Slab Modes

Even mode

Odd mode

For a simple slab waveguide mode profiles are as below



TE-Like Slab ModesTE-Like Slab Modes

Even TE slab mode+

Odd TM slab mode=

TE-Like mode forSlab Photonic

Crystal



TM-Like Slab ModesTM-Like Slab Modes

Even TM slab mode+

Odd TE slab mode=

TM-Like mode forSlab Photonic

Crystal




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




TE-like

ar 4.0ad 55.0

5.3sin




TM-like

ar 4.0ad 55.0

5.3sin



Slab Photonic Crystal Slab Photonic Crystal CavityCavity

O. Painter et al., J. Opt. Soc. Am B. 16, 275 (1999)



Slab Photonic Crystal Slab Photonic Crystal CavityCavity

Even mode : 3D : 2D + effective index : 3005.0N 304.0N

157TQ

6820Q

161Q||

111

QQQT



Slab Photonic Crystal Slab Photonic Crystal CavitiesCavities

Odd mode : 3D : 2D + effective index : 2995.0N 297.0N

157TQ



Photonic Crystal Slab Photonic Crystal Slab WaveguidesWaveguides

M. Loncar et al., J. Lightwav Tech. 18, 1402 (2000)




Dispersion Diagram ar 4.0 ad 55.05.3sin



Mode Profiles

BA





Parameters : ar 3.0 ad 5.04.3InGaAsPn

Triangular Lattice Slab

Photonic Crystal

Waveguide



Photonic Crystal Slab Photonic Crystal Slab Waveguides Waveguides

Parameters : Even Mode

Excellent agreementbetween 3D and 2D Effective Indexmethods

ar 3.0 ad 5.04.3InGaAsPn 65.2effn



Photonic Crystal Slab Photonic Crystal Slab Waveguides Waveguides

Parameters : Odd Mode

Excellent agreementbetween 3D and 2D Effective Indexmethods

ar 3.0 ad 5.04.3InGaAsPn 65.2effn



ConclusionsConclusions

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.



ConclusionsConclusions

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 …

Thanks for your attention !

pwe and fdtd methods for analysis of photonic crystals integrated photonics laboratory school of...

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