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 S. Mandel H. Wanare R. Grobe G. Rutherford Acknowledgements: E. Gratton, M. Wolf, V. Toronov NSF, Research Co, NCSA
Electromagnetic wave Maxwell’s eqns Light scattering in random media Photon density wave Photon diffusion Boltzmann eqn 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 methods Finite difference U(t->t+dt) 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
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
First tests : Snell’s law and Fresnel coefficients Refraction at air-glass interface 10 5 y/l n 1 l n 2 0 q 2 -5 -10 -5 0 z/l 5 10
Fresnel Coefficient
Second test Tunneling due to frustrated total internal reflection d s q n 1 n 2 n 1
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 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
20 T=8 T = 16 T = 24 T = 40 10 0 -10 y/l 10 0 -10 -20 z/l
Summary - 1 • Developed a new algorithm to produce exact spatiotemporal 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 Input light Output light D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R 2999, (1993) tumor
Boltzmann Equation for photon density wave J. B. Fishkin, S. Fantini, M. J. Vande. Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996) Studied diffusion approximation and P 1 approximation Q: How do diffusion and Boltzmann theories compare?
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)
Solution of Boltzmann equation Incident: — Transmitted: — Diffusion: —
Frequency responses reflected transmitted Exact Boltzmann: — Diffusion approximation: — Confirmed behavior obtained in P 1 approx J. B. Fishkin, S. Fantini, M. J. Vande. Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996)
Photon density wave Right going Exact Boltzmann: — Diffusion approximation: — Left going
Resonances at w = n l/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? www. phy. ilstu. edu/ILP
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