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We introduce a system of nonlinear coupled-mode equations (CMEs) for Bragg gratings (BGs) where the Bragg reflectivity periodically switches off and on as a function of the evolution variable. The model may be realized in a planar waveguide with the Kerr nonlinearity, where the grating is represented by an array of parallel dashed lines (grooves), aligned with the propagation direction. In the temporal domain, a similar system can be derived for matter waves trapped in a rocking optical lattice. Using systematic simulations, we construct families of gap solitons (GSs) in this system, starting with inputs provided by exact GS solutions in the averaged version of the CMEs. Four different regimes of the dynamical behavior are identified: fully stable, weakly unstable, moderately unstable, and completely unstable solitons. The analysis is reported for both quiescent and moving solitons (infact, they correspond to untilted and tilted beams in the spatial domain). Weakly and moderately unstable GSs spontaneously turn into persistent breathers (the moderate instability entails a small spontaneous change of the breather’s velocity). Stability regions for the solitons and breathers are identified in the parameter space. Collisions between stably moving solitons and breathers always appear to be elastic. © 2010 Optical Society America OCIS codes: 050.2770, 230.1480, 190.5530, 060.5530
Copyright @2021 Engineering Transactions
Faculty of Engineering and Technology
Mahanakorn University of Technology
R. Kashyap, Fiber Bragg gratings Academic Press: San Diego, 1999.
B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A.Rogers, P. Kuo, T. N. Nielsen, and B.Mikkelsen, “Integrated tunable fiber gratings for dispersion
management in high-bit rate systems,” J.Lightwave Tech. 18, 1418-1432, 2000.
H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379-381, 1979.
Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with large group retardation,” Zh. Tekh. Fiz. 51, 902 (1981) Sov. Phys. Tech. Phys. 26, 541, 1981.
W. Chen and D. L. Mills, “Gap solitons and the nonlinear-optical response of superlat-tices,” Phys. Rev. Lett. 58, 160-163, 1987.
C. M. de Sterke and J. E. Sipe, “Gap solitons”, in Progress in Optics, E. Wolf, ed.(North-Holland, Amsterdam, 1994), Vol. XXXIII, Chap. III, pp. 203-260.
Y. S. Kivshar and G. P. Agrawal. Optical Solitons Academic Press: San Diego, 2003.
D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic struc-ture,” Phys. Rev. Lett. 62, 1746-1749, 1989.
A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42, 1989.
B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796, 1994.
I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and Oscillatory Instabilities of Gap Solitons,” Phys. Rev. Lett. 80,
A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85-88, 1998.
B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630, 1996.
B. A.MALOMED: STABILITY LIMITS FOR SPATIAL GAP SOLITONS IN 11
B. J. Eggleton, C. M. De Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schr¨dinger limit: experiment and theory,” J. Opt. Soc. Am. B 16,
J. T. Mok, C. M. de Sterke, I. C. M. Litte, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nature Physics 2, 775-780, 2006.
A. A. Sukhorukov and Yu. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112-2114, 2002.
P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, “Discrete gap solitons in a diffraction-managed waveguide array,” Eur. Phys. J. D 23, 421-236, 2003.
D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902, 2003.
D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap Solitons in Waveg-uide Arrays,” Phys. Rev. Lett. 92, 093904, 2004.
F. Chen, M. Stepi´, C. E. R¨ter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M.Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Exp. 13, 4314-4324, 2005.
J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys.Rev. Lett. 90, 023902, 2003.
D. Neshev, A. A. Sukhorukov, B. Hanna, W. Kr´likowski, and Yu. S. Kivshar, “Con-trolled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905, 2004.
W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of twodimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147-150, 2003.
J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18, 1302-1304, 1993.
R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear struc-tures,” Opt. Lett. 18, 1612-1614, 1993.
W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708-6722, 1998.
F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R), 2008.
P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344-3355, 1986.
N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634-3646, 1977.
J. B. Khurgin, “Light slowing down in Moir´ fiber gratings and its implications for nonlinear optics,” Phys. Rev. A 62, 013821, 2000.
R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dualperiodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905, 2001.
D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72, 056605, 2005.
K. Levy, B. A. Malomed, “Stability and collisions of traveling solitons in Bragg-grating superstructures,” J. Opt. Soc. Am. B 25, 302 , 2008.
K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006-1012, 2006.
T. Mayteevarunyoo and B. A. Malomed, “Gap solitons in grating superstructures,” Opt.Exp. 16, 7767-7777, 2008.
P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612, 2005.
J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and selfdefocusing layers,” Phys. Lett. A 298, 140-148 , 2002.
B. A. Malomed, Soliton Management in Periodic Systems Springer: New York, 2006.
H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403, 2007.
A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Ari-mondo, “Exploring dynamic localization with a Bose-Einstein condensate,” Phys. Rev.A 79, 013611, 2009.
F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. T¨nnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling
modulation in an optical waveguide system,” Phys. Rev. Lett. 101, 143602, 2008. ENGINEERING TRANSACTIONS, VOL. 13, NO.12 1 (28) JAN-JUN 2010.
T. Mayteevarunyoo and B. A. Malomed, “Gap solitons in rocking optical lattices and waveguides with undulating gratings,” Phys. Rev. A 80, 013827, 2009.
W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and splitting of gap solitons in apodized Bragg gratings,” J. Mod. Opt. (51 2141-2158, 2004.
W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609, 2003.
D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421, 2006.
D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A: Pure Appl. Opt. 10, 085105, 2008.