The split operator numerical solution of Maxwell’s equations

Q. Su

Intense Laser Physics Theory UnitIllinois State University

LPHY 2000 Bordeaux France July 2000

Acknowledgements: E. Gratton, M. Wolf, V. ToronovNSF, Research Co, NCSA

S. Mandel R. Grobe H. Wanare G. Rutherford

Electromagnetic wave

Maxwell’s eqns

Lightscattering in

random media

Photon density wave

Boltzmann eqn

Photon diffusion

Diffusion eqn

Outline• Split operator solution of Maxwell’s eqns

• Applications• simple optics

• Fresnel coefficients• transmission for FTIR

• random medium scattering

• Photon density wave• solution of Boltzmann eqn

• diffusion and P1 approximations

• Outlook

Numerical algorithms for Maxwell’s eqns

Frequency domain methods

Time domain methods U(t->t+t)Finite difference

A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995)

Split operatorJ. Braun, Q. Su, R. Grobe, Phys. Rev. A 59, 604 (1999)

U. W. Rathe, P. Sanders, P.L. Knight, Parallel Computing 25, 525 (1999)

Exact numerical simulation of Maxwell’s Equations

Initial pulse satisfies :

Time evolution given by :

r E 0

B 0

E

t

c2

r

B

B

t

E

H v

0

0

Split-Operator Technique

H m

,r 1

r 1

0

0 0

E r , t t

cB r , t t

U

E r , t

cB r , t

Effect of vacuum

Effect of medium

ct

E

cB

01

r

0

E

cB

H v H m

E

cB

U eH v

H m

,r t

U 12

m U1v U1

2

m O t3

F

E r , t t

cB r , t t

˜ U 1

2

m ˜ U 1v ˜ U 1

2

m F

E r , t

cB r , t

˜ U 1

2

m e1

2tF H m

,r F -1

˜ U 1

v etF H v

F - 1

and

Numerical implementation of evolution in Fourier space

where

Reference: “Numerical solution of the time-dependent Maxwell’s equations for random dielectric media” - W. Harshawardhan, Q.Su and R.Grobe, submitted to Physical Review E

n1

n2

-10 100 5-5 0

10

-10

0

-5

5

z/

y/

First tests : Snell’s law and Fresnel coefficientsRefraction at air-glass interface

0.2

0.3

0.4

0.5

0.6

0.7

0 20 40 60 80

fig2(n1=1,n2=2).d

1

Et / E

i

Fresnel Coefficient

d

n1

s

n2

n1

Second testTunneling due to frustrated total internal reflection

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2d/

Et/E

i

Amplitude Transmission Coefficient vs Barrier Thickness

Light interaction with random dielectric spheroids

• Microscopic realization• Time resolved treatment• Obtain field distribution at every point in space

• 400 ellipsoidal dielectric scatterers• Random radii range [0.3 , 0.7 ]• Random refractive indices [1.1,1.5]• Input - Gaussian pulse

One specific realization

20

0

10

-10

0

10

-10

y/

-20 z/

T = 8 T = 16

T = 24 T = 40

Summary - 1

• Developed a new algorithm to produce exact spatio-temporal solutions of the Maxwell’s equations

• Technique can be applied to obtain real-time evolution of the fields in any complicated inhomogeneous medium

» All near field effects arising due to phase are included

• Tool to test the validity of the Boltzmann equation and the traditional diffusion approximation

Photon density wave

Infrared carrier

penetration but incoherent due to diffusion

Modulated wave 100 MHz ~ GHz

maintain coherencetumor

Input light

Output light

D.A. Boas, M.A. O’Leary, B. Chance, A.G. Yodh, Phys. Rev. E 47, R2999, (1993)

1

c

t

I r,, t s d' p ,' I r,' , t s a I r,, t

Boltzmann Equation for photon density wave

J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)

Q: How do diffusion and Boltzmann theories compare?

Studied diffusion approximation and P1 approximation

Bi-directional scattering phase function

Mie cross-section: L. Reynolds, C. Johnson, A. Ishimaru, Appl. Opt. 15, 2059 (1976)Henyey Greenstein: L.G. Henyey, J.L. Greenstein, Astrophys. J. 93, 70 (1941)Eddington: J.H. Joseph, W.J. Wiscombe, J.A. Weinman, J. Atomos. Sci. 33, 2452 (1976)

Other phase functions

p ,' 1

21 g cos 1 1

21 g cos 1

1

c

tx

R x, t r a R x, t r L x, t

1

c

tx

L x, t r R x,t r a L x, t

t

(R L) 0

r 1

2 s cos 1

Diffusion approximation

Incident: —Transmitted: —

Diffusion: —

Solution of Boltzmann equation

0

0.5

1

1.5

2

-30 -20 -10 0 10 20 30

Inci

dent

inte

nsit

y

Position (cm)

0.00

0.05

0.10

0.15

0.20

-30 -20 -10 0 10 20 30

Tra

nsm

itte

d in

tens

ity

Position (cm)

J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)

Confirmed behavior obtained in P1 approx

Exact Boltzmann: —Diffusion approximation: —

Frequency responses

-2.4

-2

-1.6

-1.2

-0.8

1 10 100

Log

Tra

nsm

issi

on

(GHz)

reflected transmitted-2.5

-2

-1.5

-1

-0.5

0

1 10 100

Log

Ref

lect

ion

(GHz)

Photon density wave

Right going Left going

0

0.5

1

1.5

2

0 0.5 1 1.5 2

R (

x)

x (cm)

Exact Boltzmann: —Diffusion approximation: —

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2

L (

x)

x (cm)

-0.1

-0.098

-0.096

-0.094

-0.092

-0.09

1 10 100 1000 10 4

Log

Tra

nsm

issi

on

(GHz)

-0.1

-0.0995

-0.099

-0.0985

-0.098

0 0.5 1 1.5 2 2.5 3

Log

Tra

nsm

issi

on (mm)

Resonancesat w = n /2 (n = integer)

Exact Boltzmann: —Diffusion approximation: —

Summary

Numerical Maxwell, Boltzmann equations obtainedNear field solution for random medium scatteringDirect comparison: Boltzmann and diffusion theories

Outlook

Maxwell to Boltzmann / Diffusion?Inverse problem?

www.phy.ilstu.edu/ILP