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

Electromagnetic wave Maxwell’s eqns Light scattering 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 P 1 approximations Outlook

Numerical algorithms for Maxwell’s eqns Frequency domain methods Time domain methodsU(t->t+  t) 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)

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

Split-Operator Technique Effect of vacuum Effect of medium

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

 n1n1 n2n z/ y/ First tests : Snell’s law and Fresnel coefficients Refraction at air-glass interface

Fresnel Coefficient

d n 1 s  n 2 n 1 Second test Tunneling due to frustrated total internal reflection

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

y/ -20 z/ T = 8T = 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 coherence tumor Input light Output light D.A. Boas, M.A. O’Leary, B. Chance, A.G. Yodh, Phys. Rev. E 47, R2999, (1993)

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 P 1 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 Diffusion approximation

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

J.B. Fishkin, S. Fantini, M.J. VandeVen, and E. Gratton, Phys. Rev. E 53, 2307 (1996) Confirmed behavior obtained in P 1 approx Exact Boltzmann: — Diffusion approximation: — Frequency responses reflected transmitted

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

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

Summary 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?