On the matrix Fourier filtering problem for a class of models of nonlinear optical systems with a feedback

A. V. Razgulin   ; S. V. Sazonova

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 9, pp. 1403-1420

Résumé

A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert-Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.

Bibliographie

[1] Goodman J. W., Introduction to Fourier optics, McGraw Hill, New York, 1968

[2] Degtiarev E. V., Vorontsov M. A., “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression”, J. Opt. Soc. Amer. Ser. B, 12:7 (1995), 1238–1248 | DOI

[3] Justh E. W., Vorontsov M. A., Garhart G., Beresnev L. A., Krishnapasad P. S., “Adaptive optics with advanced phase contrast techniques. Part II: High resolution wavefront control”, J. Opt. Soc. Amer. A, 18:6 (2001), 1300–1311 | DOI

[4] Larichev A. V., Nikolaev I. P., Shmalgauzen V. I., “Opticheskie dissipativnye struktury s upravlyaemym prostranstvennym periodom v nelineinoi sisteme s fure-filtrom v konture obratnoi svyazi”, Kvant. elektron., 23:10 (1996), 894–899

[5] Larichev A. V., Nikolaev I. P., Violino P., “LCLV-based system for high resolution wavefront correction: phase knife as a feedback intensity producer”, Opt. Commun., 138 (1997), 127–135 | DOI

[6] Nikolaev I. P., Larichev A. V., Shmal'gauzen V. I., “Controlled optical structures in a nonlinear system involving the suppression of low spatial frequencies in the feedback loop”, Quantum Electronics, 30:7 (2000), 617–622 | DOI

[7] Larichev A. V., Nikolaev I. P., Costamagna S., Violino P., “Advanced phase knife technique”, Opt. Commun., 121 (1995), 95–102 | DOI

[8] Heise B., Reinhardt M., Schausberger S., Hauser S., Bernstein S., Stifter D., “Fourier plane filtering revisited — analogies in optics and mathematics”, Sampling Theory In Signal Image Proc., 13:3 (2014), 231–248 | MR

[9] Martin R., Oppo G.-L., Harkness G. K., Scroggie A. J., Firth W. J., “Controlling pattern formation and spatio-temporal disorder in nonlinear optics”, Optics Express, 1:1 (1997), 39–44 | DOI

[10] Jensen S. J., Schwab M., Denz C., “Manipulation, stabilization, and control of pattern formation using Fourier space filtering”, Phys. Rev. letters, 81:8 (1998), 1614–1617 | DOI

[11] Schwab M., Saffman M., Denz C., Tschudi T., “Fourier control of pattern formation in an interferometric feedback configuration”, Opt. Commun., 170 (1999), 129–136 | DOI

[12] Harkness G. K., Oppo G.-L., Benkler E., Kreuzer M., Neubecker R., Tschudi T., “Fourier space control in an LCLV feedback system”, J. Opt. B.: Quantum Semiclass. Opt., 1 (1999), 177–182 | DOI

[13] Oppo G.-L., Martin R., Scroggie A. J., Harkness G. K., Lord A., Firth W. J., “Control of spatio-temporal complexity in nonlinear optics”, Chaos, Solitons Fractals, 10:4–5 (1999), 865–874

[14] Pesch M., Westhoff E. G., Ackermann T., Lange W., “Direct measurement of multiple instability regions via a Fourier filtering method in an optical pattern forming system”, Phys. Rev. Ser. E, 68 (2003), 016209 | DOI

[15] Gutlich B., Neubecker R., Kreuzer M., Tschudi T., “Control and manipulation of solitary structures in a nonlinear optical single feedback experiment”, Chaos, 13:1 (2003), 239–246 | DOI

[16] Poyneer L. A., Macintosh B. A., Veran J.-P., “Fourier transform wavefront control with adaptive prediction of the atmosphere”, J. Opt. Soc. Am. Ser. A, 24 (2007), 2645–2660 | DOI

[17] Nagashima M., Agrawal B., “Application of complex-valued FXLMS adaptive filter to Fourier basis control of adaptive optics”, American Control Conference (San Francisco, CA, USA, 2011), 2933–2944

[18] Potapov M. M., Chechkina K. A., “Ob odnoi modeli amplitudno-fazovoi filtratsii v nelineinoi opticheskoi sisteme s obratnoi svyazyu”, Vestn. mosk. un-ta. Ser. 15. Vychisl. matem. i kibernetika, 1997, no. 4, 31–36

[19] Razgulin A. V., Chushkin V. A., “O zadache optimalnoi Fure-filtratsii dlya odnogo klassa modelei nelineinykh opticheskikh sistem s obratnoi svyazyu”, Zh. vychisl. matem. i matem. fiz., 44:9 (2004), 1608–1618

[20] Razgulin A. V., “Projection-difference method for controlled Fourier filtering”, Comput. Math. Modeling, 23:1 (2012), 56–71 | DOI | MR

[21] Vorontsov M. A., Zheleznykh N. I., Ivanov V. Yu., “Transverse interactions in 2-D feedback non-linear optical systems”, Opt. and Quant. Electron., 22 (1990), 501–515 | DOI

[22] Ivanov V. Yu., Larichev A. V., Vorontsov M. A., “One-dimensional rotatory waves in the optical systems with nonlinear large-scale field interactions”, Proc. SPIE, 1402, 1991, 145–153 | DOI

[23] Akhmanov S. A., Vorontsov M. A., Ivanov V. Yu., Larichev A. V., Zheleznykh N. I., “Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures”, J. Opt. Soc. Amer. Ser. B, 9:1 (1992), 78–90 | DOI

[24] Razgulin A. V., “Ob avtokolebaniyakh v nelineinoi parabolicheskoi zadache s preobrazovannym argumentom”, Zh. vychisl. matem. i matem. fiz., 33:1 (1993), 69–80

[25] Razgulin A. V., “Rotatsionnye volny v opticheskoi sisteme s dvumernoi obratnoi svyazyu”, Matem. modelirovanie, 5:4 (1993), 105–119

[26] Skubachevskii A. L., “O bifurkatsii Khopfa dlya kvazilineinogo parabolicheskogo funktsionalno-differentsialnogo uravneniya”, Differents. ur-niya, 34:10 (1998), 1394–1401

[27] Skubachevskii A. L., “Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics”, Nonlinear analysis: TMA, 32:2 (1998), 261–278 | DOI | MR

[28] Chushkin V. A., Razgulin A. V., “Statsionarnye struktury v funktsionalno-differentsialnom uravnenii diffuzii s otrazheniem prostranstvennogo argumenta”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibernetika, 2003, no. 2, 13–20

[29] Belan E. P., Lykova O. B., “Vraschayuschiesya struktury v parabolicheskom funktsionalno-differentsialnom uravnenii”, Differents. ur-niya, 40:10 (2004), 1348–1357

[30] Potapov M. M., “Uravnenie nelineinoi optiki s preobrazovaniyami prostranstvennoi nezavisimoi peremennoi v roli upravlyayuschikh vozdeistvii”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 1997, no. 3, 13–16

[31] Razgulin A. V., “Ob odnom klasse funktsionalno-differentsialnykh parabolicheskikh uravnenii nelineinoi optiki”, Differents. ur-niya, 36:3 (2000), 400–407

[32] Razgulin A. V., “O parabolicheskikh funktsionalno-differentsialnykh uravneniyakh s upravlyaemym preobrazovaniem prostranstvennykh argumentov”, Dokl. AN, 403:4 (2005), 448–451

[33] Razgulin A. V., “Zadacha upravleniya dvumernym preobrazovaniem prostranstvennykh argumentov v parabolicheskom funktsionalno-differentsialnom uravnenii”, Differents. ur-niya, 42:8 (2006), 1078–1091

[34] Razgulin A. V., Roganovich I. B., “Convergence of the projection difference scheme for the nonlinear parabolic equation with transformed spatial argument”, Comput. Math. Modeling, 12:3 (2001), 262–270 | DOI | MR

[35] Razgulin A. V., “Approksimatsiya zadachi upravleniya preobrazovaniem argumentov v nelineinom parabolicheskom uravnenii”, Zh. vychisl. matem. i matem. fiz., 41:12 (2001), 1844–1856

[36] Razgulin A. V., “Proektsionno-raznostnaya skhema dlya parabolicheskogo funktsionalno-differentsialnogo uravneniya s dvumernym preobrazovaniem argumentov”, Zh. vychisl. matem. i matem. fiz., 45:10 (2005), 1848–1859

[37] Grebennikov V. A., Razgulin A. V., “Vesovaya otsenka skorosti skhodimosti proektsionno-raznostnoi skhemy dlya kvazilineinogo parabolicheskogo uravneniya”, Zh. vychisl. matem. i matem. fiz., 51:7 (2011), 1294–1307

[38] Brezis H., Functional analysis. Sobolev spaces and functional differential equations, Springer, London, 2010 | MR

[39] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972

[40] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971, 71 pp.

[41] Danford N., Shvarts Dzh. T., Lineinye operatory. Spektralnaya teoriya, Mir, M., 1966

[42] Lavrentev M. M., Savelev L. Ya., Teoriya operatorov i nekorrektnye zadachi, Izd-vo In-ta matematiki, Novosibirsk, 2010

[43] Vasilev F. P., Metody optimizatsii, Faktorial Press, M., 2002

[44] Razgulin A. V., “O metode proektsii gradienta dlya kvazidifferentsiruemykh funktsionalov s gelderovym gradientom”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 2006, no. 1, 12–15

[45] Alber Ya. I., “O minimizatsii funktsionalov klassa $C^{1,\mu}$ na ogranichennykh mnozhestvakh”, Ekonomika i matem. metody, 16 (1980), 185–190