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Stochastic Maxwell Equations in Photonic CrystalModeling and Simulations

Hao-Min Zhou

School of Math, Georgia Institute of Technology

Joint work with:

Ali Adibi (ECE), Majid Badiei (ECE), Shui-Nee Chow (Math)

IPAM, UCLA, April 14-18, 2008

Partially supported by NSF

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Outline

Introduction & Motivation

Direct Method A Stochastic Model

Wiener Chaos Expansions (WCE) Numerical Method based on WCE

Simulation results Conclusion

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Stochastic PDEs :

Solutions are no longer deterministic.

Main interest: statistical properties, suchas mean, variance.

Multi-scale structures.

Fluid DynamicsEngineering

Material Sciences

Biology

Finance

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),(),(),( tyytyJtyJ jiji = Spatially incoherent source:

),( tyJ j

),( tyJ i

Such as diffuse light in optics.

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Applications in sensing

Raman spectroscopy for bio and environmental sensing

Photonic Crystal

spectrometer(in nano-scale)

Humantissue

Spatially incoherent

light

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Photonic Crystal Spectrometer

IncidentSpatially IncoherentField

Detectors

Output A

Output B

Output C

Output D

Output E

Output F

HeterogeneousPhotonic Structure

SpectrallyDiverseField

Multiplex multimodal spectrometer

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Photonic Crystal (designed) as the medium

goal: model the incoherent source and simulate output

First step in the design of Photonic Crystal spectrometers

Wave propagation is governed by Maxwell equations

a

Input Source(incoherent)

at A

Output at B(electric field

intensity)

),( tyJ ),( tyE2

Optimal design of the shapes of Photonic Crystals for largest band gap (Kao-

Osher-Yablonovitch, 05)

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Maxwell Equation

Maxwell equations:

ttt

= ),(),( rHrE

),(),(

)(),( tt

tt rJ

rErrH +

=

0= E

0= H

t

t

t

tt

=

),(),()(),(

2

2rJrE

rrE

Helmholtz wave equation:

electric field

magnetic field

Input incoherent source)J(r,t

)E(r,t)H(r,t

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Helmholtz wave equation

z-invariant structure implies

two sets of decoupled equations Transverse Magnetic (TM) (Ez,Hx,Hy)

Transverse Electric (TE) (Hz,Ex,Ey)

3D space structure reduces to 2D

Helmholtz TM wave equation:

tttyyxx tyxJtyxEyxtyxEtyxE )),,(()),,()(,()),,(()),,(( =+

PDEs are linear.

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Direct Method for PC Spectrometer

Incoherent property implies the direct (brute-force) method.

Input nonzero point source at , and

Compute output electric field at B

Total electric intensity at B:

Why point source? Non-point sources, such as plane waves,lead to coherent outputs.

Pro: correct physics (linear equations + incoherent outputs).

Con: very inefficient.

),( tyJi

piy

).(,0),( ijtyJ jp =

2 2( , , ) ( ( , , )) .pB i Bi

E x y t E x y t=

),( tyEpi

+=ji

p

j

p

i

i

p

i

i

p

i EEEE,

22 )()(

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A Stochastic Model

Spatially incoherent source

Stochastic model

More general:


)()(),,( tVyXtyxJ Az =

?

( )

=

2

0

0 exp)(sin)(T

tttttV

10ax10a Photonic Crystal asthe simulation medium

a

),()( ydWyX = )(yW Brownian Motion.

( , , ) ( , , ) ( , , ),tJ x y t f x y t dW x y t=

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Stochastic Helmholtz Wave Equations

Current density is a stochastic source.

Solution for electric field is random.

Monte Carlo simulation is slow, and hard to recover the

incoherent properties

Our strategy: WCE.


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Monte Carlo

Traditional methods, Monte Carlo (MC)

simulations,

Not many computational methods available.

Solve the equations realization by realization.

Each realization, the equations become deterministic and

solved by classical methods.The solutions are treated as samples to extract statistical

properties.

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Monte Carlo

MC can be very expensive:

Has slow convergence, governed by law of large numbers

and the convergence is not monotone.( is the number ofMC realizations)

Need to resolve the fine scales in each realization to obtainthe small scale effects on large scales, while only large scale

statistics are of interests, such as long time and large scale

behaviors.

Hard to estimate errors

Must involve random number generators, which need to be

carefully chosen.

),1

(

n

O

n

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Wiener Chaos Expansions

Goal: Design efficient numerical methods.

Separate the deterministic properties from

randomness.

Has better control on the errors.

Avoid random number generators, all

computations are deterministic.

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Functions depends on Brownian

motion

),,( dWxtu

.W

contains infinitely many independent

Gaussian random variables, is time and/or spatialdependent.

WCE: decompose by orthogonalpolynomials, similar to a spectral method, but

for random variables.

),,( dWxtu

W

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any orthonormal basis of , such

as harmonic functions in our computations.

)(smi ),0(2

YL

Define which are independent

Gaussian.

,)(0

=Y

sii dWsm

Let construct Wicks products),,,( 21 =

=

=1

).()(i

iiHT

is a multi-index, Hermite polynomials.)( iiH

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Cameron-Martin(1947): any

can be decomposed as

),,( dWxtu

= ),(),(),,( aTxtudWxtu

where

)).(),,((),( TxtuExtu =

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Statistics can be reconstructed from Wiener

Chaos coefficientsmean ),,(),( 0 xtuxtEu =

variance

22 ),)(( =

uxtuE

Higher order moments can be computed too.

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Properties of Wicks products:

,1))(( 0 =TE

0,0))(( = TE

= 1

0

)( TTE

=

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Wiener Chaos expansions have been

used in

Nonlinear filtering, Zakai equation (Lototsky, Mikulevicius

& Rozovskii, 97)

Stochastic media problems (Matthies & Bucher, 99)

Theoretical study of Stochastic Navier-Stokes equations

(Mikulevicius & Rozovskii, 02)

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Hermite Polynomial Expansions

A long history of using Hermite polynomials in

PDEs containing Gaussian random variables.

Random flows: Orszag & Bissonnette (67),Crow & Canavan (70),

Chorin (71,74),Maltz & Hitzl (79),

Stochastic finite element: Ghanem, et al (91,99, ).

Spectral polynomial chaos expansions:Karniadakis, Su and collaborators (a collection of papers), and

WCE for problems in fluid: Hou, Rozovskii, Luo, Zhou (04)

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WCE for stochastic Helmholtz equation

Expand the source and electric field:

Take advantage of

Equation is linear, so electric field hasexpansion

)(),(),,( ii

TtyJdWtyJ

=

=

=1

)()(i

ii ymydW

)(),,(),,,( ii TtyxEdWtyxE =

= ii ymtVdWtyJ )()(),,( Only Gaussian (linear)

= ii tyxEdWtyxE ),,(),,,(

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The stochastic equation is converted into a

collection (decoupled) of deterministic

Helmholtz equations

Standard numerical methods, such as finite

difference time domain (FDTD) in oursimulation, can be applied.

)())(()),,()(,()),,(()),,(( ymtVtyxEyxtyxEtyxE itttiyyixxi =+

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The electric fields from point sources

can be recovered by

Other moments can be computed by point

source solutions in the standard ways.

Under relative general conditions, WCE

coefficients decay quickly,

),,( tyxEpi

=j

jij

p

i tyxEymtyxE ),,()(),,(

1,

i r

E Oi

r Related to the smoothnessof the solutions.

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Simulation of a spatially incoherent source

Extremely fast convergence for 10ax10a example

Comparison of the direct method

(brute-force) simulation and the

WCE method

Convergence of the WCE method

S l f ll h

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Simulation of a spatially incoherent source (3)

Less than 1% error and more than one order of magnitude faster simulation,

Over 2 order of magnitude faster simulations for practical photonic crystals.

For 15 coefficients the gain in simulation time is 32, i.e., 32 times faster simulation

Percentage

error

Number of coefficients

20ax10a

For a 20ax10a example (doubled sized)

C l i

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Conclusion

Proposed a stochastic model for incoherent source

Design a fast numerical method based on WCE to

simulate the incoherent source for photonic crystals.

The method can be coupled with other fast Maxwell

equations solvers.

More than 2 order of magnitude faster simulations can

be achieved for practical structures.

The model and method are general and can be applied

to other types of stochastic problems involving incoherent

sources.

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