Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with nonreflecting boundary conditions

E. B. Terëshin   ; V. A. Trofimov   ; M. V. Fedotov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 161-171

Résumé

Conservative finite-difference schemes are constructed for the problems of self-action of a femtosecond laser pulse and of second-harmonic generation in a one-dimensional nonlinear photonic crystal with nonreflecting boundary conditions. The invariants of the governing equations are found taking into account these conditions. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes used in the modeling of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The numerical experiments performed show that the boundary reflects no more than 0.01% of the transmitted energy, which corresponds to the truncation error in the boundary conditions. The amplitude of the reflected pulse is less than that of the pulse transmitted through the boundary by two (and more) orders of magnitude. The simulation is based on the approach proposed by the authors for the given class of problems.

Bibliographie

[1] Hait J., United States Patent No 367443, March 3, 1992

[2] Markowicz P. P., Tiryaki H., Prasad P. N., “Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals”, Phys. Rev. Letts, 92:8 (2004), 083903 | DOI

[3] Joannopoulos J. D., Meade R. D., Winn J. N., Photonic crystals: molding the flow of light, New York, Princeton, 1995 | Zbl

[4] Scalora M., Dawling F. P., Bowden C. M., Blomer M. J., “Optical limiting and switching of ultrashort pulses in nonlinear photonic bandgap materials”, Phys. Rev. Letts, 73:10. (1994), 1368–1371 | DOI

[5] Trofimov V. A., Tereshin E. B., Fedotov M. B., “O vozmozhnosti lokalizatsii energii svetovogo impulsa v fotonnom kristalle”, Optika i spektroskopiya, 95:1 (2003), 106 | MR

[6] Alpert B., Greengard L., Hagstrom T., “Nonrefleeting boundary conditions for the time-dependent wave equation”, J. Comput. Phys., 180:2 (2002), 270–296 | DOI | MR | Zbl

[7] Fibich G., Tsynkov S., “High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering”, J. Comput. Phys., 171:2 (2001), 632–677 | DOI | MR | Zbl

[8] Hagstorm T., Hariharan S. I., “A formulation of asymptotic and exact boundary conditions using local operators”, Appl. Numer. Math., 27:4 (1998), 403–416 | DOI | MR

[9] Rowley C. W., Colonius T., “Discretely nonreflecting boundary conditions for linear hyperbolic systems”, J. Comput. Phys., 157:2 (2000), 500–538 | DOI | MR | Zbl

[10] Lysak T. M., Trofimov V. A., “TVG femtosekundnykh impulsov v usloviyakh nenulevoi amplitudy volny na udvoennoi chastote”, Optika i spektroskopiya, 93:5 (2000), 861–874

[11] Tereshin E. B., Trofimov V. A., Fedotov M. V., “Konservativnaya raznostnaya skhema dlya zadachi dvukhvolnovogo vzaimodeistviya femtosekundnykh impulsov v fotonnom kristalle”, Zh. vychisl. matem. i matem. fiz., 43:10 (2003), 1530–1535 | MR | Zbl

[12] Trofimov V. A., “Invarianty rasprostraneniya femtosekundnykh svetovykh impulsov v fotonnykh kristallakh”, Zh. vychisl. matem. i matem. fiz., 41:9 (2001), 1429–1433 | MR | Zbl

[13] Trofimov V. A., “New approach to numerical simulation of femtosecond pulse propagation in photonic crystal”, Laser Phys. and Spectroscopy, Proc. of SPIE, 4002, Bellingham, 2000, 28–36

[14] Trofimov V. A., Tereshin E. B., “Comparison of various difference schemes for the problem of femtosecond pulse propagation in nonlinear layered medium”, Proc. CMMSE-2004 (Uppsala, Sweden, June 2004), 281–287

[15] Trofimov V. A., Tereshin E. B., Fedotov M. B., “Issledovanie raznostnykh skhem dlya zadachi samovozdeistviya femtosekundnogo impulsa v fotonnom kristalle”, Vestn. MGU. Ser. 15: Vychisl. matem. i kibernetika, 2003, no. 2, 20–26 | MR | Zbl