paper 150204235818

Abstract

We develop the dynamical as well as stochastic aspects for the polarized light transmitting the non-linear media. The central concept is the non-linear birefringence, which is realized for the two-component non-linear Schrödinger equation (NLSE). We put an ansatz: ψ = (ψ1 , ψ2 ) = (a(z), b(z))F (x), where F (x) is the single component scalar wave, which is given as the “soliton” solution for the NLSE. The parameter z is the coordinate of propagation direction, and the x R= (x, y) is that of polarized plane. By substituting this ansatz into the ∂ action function S = ψ ∗ (iλ ∂z − H)ψdxdz, where λ is the wave length for traveling wave, and H is the Hamiltonian which is written by 2 × 2 matrix, and using the variation principle, we derive the coupled equation for a(z) and b(z), which can be reduced to the equation of motion for the Stokes parameters. Taking into account of randomness and dissipation contained in the birefringent media, we derive the Langevin equation for the Stokes parameters. From this equation, we can obtain the Fokker-Planck (FP) equation for the probability distribution with respect to Stokes parameters. In order to derive the FP equation, we employ the technique of functional integral, which is based on the assumption of the Gaussian white noise for the random fluctuation arising from birefringence.

1. Landau, L. D., E. M. Lifshitz, and Allen L. King, "Electrodynamics of Continuous Media," American Journal of Physics, Vol. 29, No. 9, 647–648, September 1961.
doi:10.1119/1.1937882 Google Scholar

2. Born, , M. and E. Wolf and Principle of Optics Pergamon and Oxford and 1975., M. and E. Wolf, 1975. Google Scholar

3. Sala, Kenneth L., "Nonlinear refractive-index phenomena in isotropic media subjected to a dc electric field: Exact solutions," Physical Review A, Vol. 29, No. 4, 1944–1956, April 1984.
doi:10.1103/physreva.29.1944 Google Scholar

4. Tratnik, M. V. and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Physical Review A, Vol. 35, No. 7, 2965–2975, April 1987.
doi:10.1103/physreva.35.2965 Google Scholar

5. Kuratsuji, Hiroshi and Shouhei Kakigi, "Maxwell-Schrödinger Equation for Polarized Light and Evolution of the Stokes Parameters," Physical Review Letters, Vol. 80, No. 9, 1888–1891, March 1998.
doi:10.1103/physrevlett.80.1888 Google Scholar

6. Ade, P. A. R., Y. Akiba, A. E. Anthony, K. Arnold, M. Atlas, D. Barron, D. Boettger, J. Borrill, C. Borys, S. Chapman, Y. Chinone, M. Dobbs, T. Elleflot, J. Errard, G. Fabbian, C. Feng, D. Flanigan, A. Gilbert, W. Grainger, N. W. Halverson, M. Hasegawa, K. Hattori, M. Hazumi, W. L. Holzapfel, Y. Hori, J. Howard, P. Hyland, Y. Inoue, G. C. Jaehnig, A. Jaffe, B. Keating, Z. Kermish, R. Keskitalo, T. Kisner, M. Le Jeune, A. T. Lee, E. M. Leitch, E. Linder, M. Lungu, F. Matsuda, T. Matsumura, X. Meng, N. J. Miller, H. Morii, S. Moyerman, M. J. Myers, M. Navaroli, H. Nishino, H. Paar, J. Peloton, D. Poletti, E. Quealy, G. Rebeiz, C. L. Reichardt, P. L. Richards, C. Ross, K. Rotermund, I. Schanning, D. E. Schenck, B. D. Sherwin, A. Shimizu, C. Shimmin, M. Shimon, P. Siritanasak, G. Smecher, H. Spieler, N. Stebor, B. Steinbach, R. Stompor, A. Suzuki, S. Takakura, A. Tikhomirov, T. Tomaru, B. Wilson, A. Yadav, O. and Zahn, "Evidence for Gravitational Lensing of the Cosmic Microwave Background Polarization from Cross-Correlation with the Cosmic Infrared Background," Physical Review Letters, Vol. 112, No. 13, April 2014.
doi:10.1103/physrevlett.112.131302 Google Scholar

7. Ade, P. A. R., Y. Akiba, A. E. Anthony, K. Arnold, M. Atlas, D. Barron, D. Boettger, J. Borrill, S. Chapman, Y. Chinone, M. Dobbs, T. Elleflot, J. Errard, G. Fabbian, C. Feng, D. Flanigan, A. Gilbert, W. Grainger, N. W. Halverson, M. Hasegawa, K. Hattori, M. Hazumi, W. L. Holzapfel, Y. Hori, J. Howard, P. Hyland, Y. Inoue, G. C. Jaehnig, A. Jaffe, B. Keating, Z. Kermish, R. Keskitalo, T. Kisner, M. Le Jeune, A. T. Lee, E. Linder, E. M. Leitch, M. Lungu, F. Matsuda, T. Matsumura, X. Meng, N. J. Miller, H. Morii, S. Moyerman, M. J. Myers, M. Navaroli, H. Nishino, H. Paar, J. Peloton, E. Quealy, G. Rebeiz, C. L. Reichardt, P. L. Richards, C. Ross, I. Schanning, D. E. Schenck, B. Sherwin, A. Shimizu, C. Shimmin, M. Shimon, P. Siritanasak, G. Smecher, H. Spieler, N. Stebor, B. Steinbach, R. Stompor, A. Suzuki, S. Takakura, T. Tomaru, B. Wilson, A. Yadav, O. and Zahn, "Measurement of the Cosmic Microwave Background Polarization Lensing Power Spectrum with the POLARBEAR Experiment," Physical Review Letters, Vol. 113, No. 2, July 2014.
doi:10.1103/physrevlett.113.021301 Google Scholar

8. Zaldarriaga, Matias and Uroš Seljak, "Gravitational lensing effect on cosmic microwave background polarization," Physical Review D, Vol. 58, No. 2, June 1998.
doi:10.1103/physrevd.58.023003 Google Scholar

9. Seljak, Uros̆ and Matias Zaldarriaga, "Signature of Gravity Waves in the Polarization of the Microwave Background," Physical Review Letters, Vol. 78, No. 11, 2054–2057, March 1997.
doi:10.1103/physrevlett.78.2054 Google Scholar

10. Kamionkowski, Marc, Arthur Kosowsky, and Albert Stebbins, "A Probe of Primordial Gravity Waves and Vorticity," Physical Review Letters, Vol. 78, No. 11, 2058–2061, March 1997.
doi:10.1103/physrevlett.78.2058 Google Scholar

11. Chandrasekhar, S., "Stochastic Problems in Physics and Astronomy," Reviews of Modern Physics, Vol. 15, No. 1, 1–89, January 1943.
doi:10.1103/revmodphys.15.1 Google Scholar

12. Kubo, Ryogo, Morikazu Toda, and Natsuki Hashitsume, Statistical Physics II, Springer Berlin Heidelberg, 2nd Edition, 1991.
doi:10.1007/978-3-642-58244-8 Google Scholar

13. Chiao, R. and E. Garmire, "Self-trapping of optical beams," IEEE Journal of Quantum Electronics, Vol. 2, No. 4, 126–127, April 1966.
doi:10.1109/jqe.1966.1073894 Google Scholar

14. Swartzlander, G. A. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Physical Review Letters, Vol. 69, No. 17, 2503–2506, October 1992.
doi:10.1103/physrevlett.69.2503 Google Scholar

15. Kuratsuji, Hiroshi, "A Field Equation of the Stokes Parameters –Spin Texture of Light Polarization–," Journal of the Physical Society of Japan, Vol. 77, No. 4, 044403, April 2008.
doi:10.1143/jpsj.77.044403 Google Scholar

16. Maker, P. D., R. W. Terhune, and C. M. Savage, "Intensity-Dependent Changes in the Refractive Index of Liquids," Physical Review Letters, Vol. 12, No. 18, 507–509, May 1964.
doi:10.1103/physrevlett.12.507 Google Scholar

17. Kuratsuji, H., R. Botet, and R. Seto, "Electromagnetic Gyration: -- Hamiltonian Dynamics of the Stokes Parameters --," Progress of Theoretical Physics, Vol. 117, No. 2, 195–217, February 2007.
doi:10.1143/ptp.117.195 Google Scholar

18. Lischitz, E. M. and L. P. Pitaevski and Statistical Physics; Part 2 and Course of Theoretical Physics and Vol. 9, "E. M. and L. P. Pitaevski,", 1986. Google Scholar

19. Tsuchida, , S. and H. Kuratsuji and under Submission for Publication., S. and H. Kuratsuji. Google Scholar

Hiroshi Kuratsuji

Satoshi Tsuchida