On modeling of distortions suppression in nonlinear optical system with delayed feedback loop

T. E. Romanenko ; A. V. Razgulin

Matematičeskoe modelirovanie, Tome 26 (2014) no. 11, pp. 123-136.

Résumé

This paper deals with the effect of suppression of phase distortions for an optical system with feedback described by parabolic functional differential equation with time delay and spatial shift. Simplified model of delayed differential equations is proposed. It allows one to perform the analytical study of suppression of harmonic distortions. The obtained results are compared with the results of numerical modelling for the initial problem, focused on how rotation and delay affect on distortion suppression.

Détail

Keywords: delay, functional differential equation, feedback optical system, stability, suppression of distortions.

Bibliographie

[1] Tyson R. K., Principles of adaptive optics, 3rd ed., CRC Press, 2010

[2] Vorontsov M. A., Koryabin A. V., Shmalgauzen V. I., Upravlyaemye opticheskie sistemy, Nauka, M., 1988

[3] Vorontsov M. A., Katulin V. A., Naumov A. F., “Wavefront control by an optical-feedback interferometer”, Optics Communications, 71:1–2 (1989), 35–38 | DOI

[4] Barnes T., Eiju T., Matsuda K., “High resolution adaptive optics using an interference phase loop”, Optics Communications, 132 (1996), 494–502 | DOI

[5] Fisher A. D., Warde C., “Technique for real-time high-resolution adaptive phase compensation”, Optics Letters, 8:7 (1983), 353–355 | DOI

[6] Vorontsov M. A., Kirakosyan M. E., Larichev A. V., “Korrektsiya fazovykh iskazhenii v nelineinom interferometre s opticheskoi obratnoi svyazyu”, Kvantovaya elektronika, 18 (1991), 117–120

[7] Vorontsov M. A., Larichev A. V., “Intelligent laser systems: adaptive compensation of phase distortions in nonlinear system with two-dimensional feedback”, Nonlinear Optics, Proc. SPIE, 1409, 1991, 260–266 | DOI

[8] Larichev A. V., Dinamicheskie protsessy v nelineinykh opticheskikh sistemakh s dvumernoi obratnoi svyazyu, Dis. ... kand. fiz.-matem. nauk po spetsialnosti 01.04.21, MGU im. M. V. Lomonosova, M., 1995, 108 pp.

[9] Vorontsov M. A., Zheleznykh N. I., Ivanov V. Yu., “Transverse interactions in the 2-D feedback non-linear optical systems”, Optical and Quantum Electronics, 2 (1990), 501–515 | DOI

[10] Vorontsov M. A., Shishakov K. V., “Phase-distortion suppression in nonlinear cavities with gain”, J. of the Optical Society of America, 9 (1992), 71–77 | DOI

[11] Dou R., Vorontsov M. A., Sivokon V. P., Giles M. K., “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers”, Optical Eng., 36:12 (1997), 3327–3335 | DOI

[12] Chesnokov S. S., Rybak A. A., “Spatiotemporal Chaotic behavior of time-delayed nonlinear optical systems”, Laser Physics, 10:5 (2000), 1061–1068

[13] Chesnokov S., Rybak A., Stadnichuk V., “Rezhimy opticheskoi turbulentnosti v nelineino-opticheskoi sisteme s zaderzhkoi v tsepi raspredelennoi obratnoi svyazi”, Optika atmosfery i okeana, 15:7 (2002), 572–578

[14] Vorontsov M. A., Iroshnikov N. G., “Nonlinear dynamics of neuromorphic optical system with spatio-temporal interactions”, Optical Memory and Neural Networks, Proc. SPIE, 1621, 1991, 292–298 | DOI

[15] Razgulin A. V., Romanenko T. E., “Vraschayuschiesya volny v parabolicheskom funktsionalno-differentsialnom uravnenii s povorotom prostranstvennogo argumenta i zapazdyvaniem”, Zhurn. vychislit. matematiki i mat. fiziki, 53:11 (2013), 42–60

[16] Romanenko T. E., “Dvumernye vraschayuschiesya volny v funktsionalno-differentsialnom uravnenii diffuzii s povorotom prostranstvennykh argumentov i zapazdyvaniem”, Differentsialnye uravneniya, 50:2 (2014), 260–263 | DOI | Zbl

[17] Razgulin A. V., “Finite-dimensional dynamics of distributed optical system with delayed feedback”, Comput. Math. Appl., 40:12 (2000), 1405–1418 | DOI | MR | Zbl

[18] Wu J., Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996 | MR

[19] Hale J., Theory of functional differential equations, Springer, New York, 1977 | MR | Zbl

[20] Pontryagin L. S., “O nulyakh nekotorykh elementarnykh transtsendentnykh funktsii”, Izvestiya akademii nauk SSSR, 1942, no. 6, 115–134 | MR

[21] Noonburg V. W., “Roots of a transcendental equation associated with a system of differential-difference equations”, SIAM J. Appl. Math., 17:1 (1969), 198–205 | DOI | MR | Zbl

[22] He Q., Kanga L., Evans D. J., “Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay”, Numerical Algorithms, 16 (1997), 129–153 | DOI | MR

[23] Pao C. V., “Numerical methods for systems of nonlinear parabolic equations with time delays”, Journal of Mathematical Analysis and Applications, 240:1 (1999), 249–279 | DOI | MR | Zbl

[24] Lekomtsev A. V., Pimenov V. G., “Convergence of the alternating direction method for the numerical solution of a heat conduction equation with delay”, Proceedings of the Steklov Institute of Mathematics, 272, 2011, S101–S118 | DOI

[25] Kamont Z., Kropielnicka K., “Implicit difference methods for evolution functional differential equations”, Numerical Analysis and Applications, 4:4 (2011), 294–308 | DOI | Zbl

[26] Razgulin A. V., “Proektsionno-raznostnaya skhema dlya parabolicheskogo funktsionalno-differentsialnogo uravneniya s dvumernym preobrazovaniem argumentov”, Zhurn. vychislit. matematiki i mat. fiziki, 45:10 (2005), 1848–1859 | MR | Zbl