1. LPHY 2000 Bordeaux France July 2000
The split operator numerical solution of Maxwell’s equations Q. Su Intense Laser Physics Theory Unit Illinois State University
LPHY Bordeaux France July 2000
S. Mandel R. Grobe H. Wanare G. Rutherford
Acknowledgements: E. Gratton, M. Wolf, V. Toronov
NSF, Research Co, NCSA

2. Light scattering in random media
Electromagnetic wave
Maxwell’s eqns
Light
scattering in
random media
Photon density wave
Boltzmann eqn
Photon diffusion
Diffusion eqn

3. 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

4. Numerical algorithms for Maxwell’s eqns
Frequency domain methods
Time domain methods U(t->t+dt)
Finite difference
A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995)
Split operator
J. Braun, Q. Su, R. Grobe, Phys. Rev. A 59, 604 (1999)
U. W. Rathe, P. Sanders, P.L. Knight, Parallel Computing 25, 525 (1999)

5. Exact numerical simulation of Maxwell’s Equations
Initial pulse satisfies :
Time evolution given by :

6. Split-Operator Technique
Effect of vacuum
Effect of medium

7. Numerical implementation of evolution in Fourier space
where
and
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

8. Refraction at air-glass interface
First tests : Snell’s law and Fresnel coefficients
Refraction at air-glass interface 10
-10 -5 5 n1 l n2
y/l q2
-10 10 5 -5
z/l

9. Fresnel Coefficient

10. Tunneling due to frustrated total internal reflection
Second test
Tunneling due to frustrated total internal reflection d s q n 1 n 2 n 1

11. Amplitude Transmission Coefficient vs Barrier Thickness

12. Light interaction with random dielectric spheroids
Microscopic realization
Time resolved treatment
Obtain field distribution at every point in space
One specific realization
400 ellipsoidal dielectric scatterers
Random radii range [0.3 l, 0.7 l]
Random refractive indices [1.1,1.5]
Input - Gaussian pulse

13. 20
T = 8
T = 16 10
-10
y/l
T = 24
T = 40 10
-10
-20
z/l

14. 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

15. Photon density wave Input light Output light Infrared carrier
penetration but incoherent due to diffusion
Modulated wave 100 MHz ~ GHz
maintain coherence
tumor
Input light
Output light
D.A. Boas, M.A. O’Leary, B. Chance, A.G. Yodh, Phys. Rev. E 47, R2999, (1993)

16. for photon density wave
Boltzmann Equation
for photon density wave
J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Studied diffusion approximation and P1 approximation
Q: How do diffusion and Boltzmann theories compare?

17. Bi-directional scattering phase function
Diffusion approximation
Other phase functions
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)

18. Solution of Boltzmann equation
Incident: —
Transmitted: —
Diffusion: —

19. Confirmed behavior obtained in P1 approx
Frequency responses
reflected
transmitted
Exact Boltzmann: —
Diffusion approximation: —
Confirmed behavior obtained in P1 approx
J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996)

20. Photon density wave Right going Left going Exact Boltzmann: —
Diffusion approximation: —

21. Resonances at w = n l/2 (n = integer) Exact Boltzmann: —
Diffusion approximation: —

22. Summary Outlook Numerical Maxwell, Boltzmann equations obtained
Near field solution for random medium scattering
Direct comparison: Boltzmann and diffusion theories
Outlook
Maxwell to Boltzmann / Diffusion?
Inverse problem?