Abstract
We solve the traveling wave solution for the explicit-time nonlinear photorefractive dynamics equation. Nonlinearity comes from the support of linear and quadratic electro-optic effects. We investigate two cases, i.e., the wave assumes to be in low amplitude and without such an assumption. In this step, we apply the direct solution method by setting the ansatz of the wave solution from the start. The first case relies on the Taylor series expansion to integrate the equations of these results and get an exact solution for the traveling wave. In the second case, we thoroughly evaluate the original dynamic equation. It reduces to a first-order differential equation that provides the initial conditions for a numerical evaluation. The exact solution shows the propagation wave traveling in the positive and negative directions in the diffraction axis direction. An angle between the traveling wave and the center of the propagation plane decreases as the value of the displacement constant in the solution increases. In addition, we also study the power of the traveling wave, and the numerical solution gives a kink-like traveling wave.
Similar content being viewed by others
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Christodoulides, D.N., Carvalho, M.I.: Bright, dark, and gray spatial soliton states in photorefractive media. JOSA B 12, 1628–1633 (1995)
Segev, M., Agranat, A. J.: Spatial solitons in centrosymmetric photorefractive media. Opt. Lett. 22, 1299–1301 (1997)
Petrović, M., Jović, D., Belić, M., Schrćder, J., Jander, Ph., Denz, C.: Two dimensional counterpropagating spatial solitons in photorefractive crystals. Phys. Rev. Lett. 95, 053901 (2005)
Shwetanshumala, S., Konar, S.: Bright optical spatial solitons in a photorefractive waveguide. Phys. Scr. 82, 045404 (2010)
Katti, A., Katti, C.P.: Gap solitons supported by an optical lattice in biased photorefractive crystals having both the linear and quadratic electro-optic effect. Zeitschrift für Naturforschung A 75, 749–756 (2020)
Motzek, K., Kaiser, F., Weilnau, C., Denz, C., McCarthy, G., Krolikowski, W., Desyatnikov, A., Kivshar, Y.S.: Multi-component vector solitons in photorefractive crystals. Opt. Commun. 209, 501–506 (2002)
Xu, Z., Kartashov, Y.V., Torner, L.: Gap solitons supported by optical lattices in photorefractive crystals with asymmetric nonlocality. Opt. Lett. 31, 2027–2029 (2006)
Zhang, T.H., Ren, X.K., Wang, B.H., Lou, C.B., Hu, Z.J., Shao, W.W., Xu, Y.H., Kang, H.Z., Yang, J., Yang, D.P., Feng, L.: Surface waves with photorefractive nonlinearity. Phys. Rev. A 76, 013827 (2007)
Katti, A.: Effect of temperature on incoherently coupled Dark–Bright soliton pair in photorefractive crystals. In: Tiwari, M., Maddila, R.K., Garg, A.K., Kumar, A., Yupapin, P. (eds) Optical and Wireless Technologies. Lecture Notes in Electrical Engineering, vol. 771 (2022). Springer, Singapore. https://doi.org/10.1007/978-981-16-2818-4_3
Chen, Z., Segev, M., Christodoulides, D.N.: Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012)
Kumari, B., Katti, A., Alvi, P.A.: Coupling of optical spatial solitons in photorefractive multiple quantum well planar waveguide. Optik 183, 1048–1060 (2019)
Konar, S., Biswas, A.: Properties of optical spatial solitons in photorefractive crystals with special emphasis to two-photon photorefractive nonlinearity. Opt. Mater. 35, 2581–2603 (2013)
Hao, L., Hou, C., Wang, X., Wang, Q., Mu, H.: Coherently coupled bright-bright screening soliton pairs in biased centrosymmetric photorefractive crystals. Optik 127, 5928–5934 (2016)
Katti, A.: Bright pyroelectric quasi-solitons in a photorefractive waveguide. Optik 156, 433–438 (2018)
Dai, C.Q., Wang, Y.Y.: Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals. Nonlinear Dyn. 102, 1733–1741 (2020)
Pierangeli, D., Di Mei, F., Conti, C., Agranat, A.J., DelRe, E.: Spatial rogue waves in photorefractive ferroelectrics. Phys. Rev. Lett. 115, 093901 (2015)
Pierangeli, D., Conti, C., DelRe, E.: Rogue waves in photorefractive media. In: Nonlinear Guided Wave Optics: A testbed for extreme waves, pp. 8-1-8-21 (2017). IOP Publishing, Bristol, UK
Hermann-Avigliano, C., Salinas, I.A., Rivas, D.A., Real, B., Mančić, A., Mejía-Cortés, C., Maluckov, A., Vicencio, R.A.: Spatial rogue waves in photorefractive SBN crystals. Opt. Lett. 44, 2807–2810 (2019)
Jia, S., Lee, J., Fleischer, J.W., Siviloglou, G.A., Christodoulides, D.N.: Diffusion-trapped Airy beams in photorefractive media. Phys. Rev. Lett. 104, 253904 (2010)
Wiersma, N., Marsal, N., Sciamanna, M., Wolfersberger, D.: Airy beam self-focusing in a photorefractive medium. Sci. Rep. 6, 1–6 (2016)
Suarez, R.A., Vieira, T.A., Yepes, I.S., Gesualdi, M.R.: Photorefractive and computational holography in the experimental generation of Airy beams. Opt. Commun. 366, 291–300 (2016)
Zhan, K., Yang, Z., Liu, B., Xu, X., Jiao, Z., Jia, Y.: Propagations of Airy beams and nonlinear accelerating optical beams in photorefractive crystals with asymmetric nonlocality. Ann. Phys. 530, 1800033 (2018)
Bouchet, T., Marsal, N., Sciamanna, M., Wolfersberger, D.: Two dimensional Airy beam soliton. Sci. Rep. 12, 1–6 (2022)
Carvalho, M.I., Facão, M., Christodoulides, D.N.: Self-bending of dark and gray photorefractive solitons. Phys. Rev. E 76, 016602 (2007)
Hao, L., Wang, Q., Hou, C.: Spatial solitons in biased photorefractive materials with both the linear and quadratic electro-optic effects. J. Mod. Opt. 61, 1236–1245 (2014)
Su, Y., Jiang, Q., Ji, X.: Photorefractive spatial solitons supported by pyroelectric effects in strontium barium niobate crystals. Optik 126, 1621–1624 (2015)
Hao, L., Hou, C., Wang, X., Wang, Q., Mu, H.: Coherently coupled bright-bright screening soliton pairs in biased centrosymmetric photorefractive crystals. Optik 127, 5928–5934 (2016)
Hao, L., Wang, Q., Hou, C.: Temperature effects on (1+ 1)-dimensional steady-state bright spatial solitons in biased photorefractive crystals with both the linear and quadratic effects. J. Mod. Opt. 63, 1000–1008 (2016)
Segev, M., Crosignani, B., Yariv, A., Fischer, B.: Spatial solitons in photorefractive media. Phys. Rev. Lett. 68, 923 (1992)
She, W.L., Lee, K.K., Lee, W.K.: All optical quasi-steady-state photorefractive spatial solitons. Phys. Rev. Lett. 85, 2498 (2000)
DelRe, E., Palange, E.: Optical nonlinearity and existence conditions for quasi-steady-state photorefractive solitons. JOSA B 23, 2323–2327 (2006)
Keqing, L., Tiantong, T., Yanpeng, Z.: One-dimensional steady-state spatial solitons in photovoltaic photorefractive materials with an external applied field. Phys. Rev. A 61, 053822 (2000)
**-song, L.: Universal theory of steady-state one-dimensional photorefractive solitons. Chin. Phys. 10, 1037 (2001)
Zhan, K., Hou, C., Du, Y.: Self-deflection of steady-state bright spatial solitons in biased centrosymmetric photorefractive crystals. Opt. Commun. 283, 138–141 (2010)
Zhang, Y.H., Su, W., Duan, C.L., Tian, A.L.: Steady state multiple dark spatial solitons in the biased photorefractive-photovoltaic crystals. Optoelectron. Lett. 14, 367–371 (2018)
Katti, A., Yadav, R.A., Prasad, A.: Bright optical spatial solitons in photorefractive waveguides having both the linear and quadratic electro-optic effect. Wave Motion 77, 64–76 (2018)
Dai, C.Q., Wang, Y.Y.: Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals. Nonlinear Dyn. 102, 1733–1741 (2020)
Zhu, H.P., Chen, H.Y.: Parameter regulation for periodic wave pair and photovoltaic soliton pair excitations in the bias photovoltaic-photorefractive crystal. Res. Phys. 30, 104872 (2021)
DelRe, E., Crosignani, B., Di Porto, P.: Photorefractive solitons and their underlying nonlocal physics. In: Progress in optics, vol. 53, pp. 153–200. Elsevier
Tsarukyan, L., Badalyan, A., Drampyan, R.: Bending optical soliton-induced waveguide channels in a photorefractive LiNbO3 crystal. Appl. Phys. B 128, 1–12 (2022)
Peccianti, M., Conti, C., Assanto, G., De Luca, A., Umeton, C.: All-optical switching and logic gating with spatial solitons in liquid crystals. Appl. Phys. Lett. 81, 3335–3337 (2002)
Peccianti, M., Conti, C., Assanto, G., De Luca, A., Umeton, C.: Routing of anisotropic spatial solitons and modulational instability in liquid crystals. Nature 432, 733–737 (2004)
Zhan, K., Hou, C., Pu, S.: Temporal behavior of spatial solitons in centrosymmetric photorefractive crystals. Opt. Laser Technol. 43, 1274–1278 (2011)
Grandpierre, A.G., Christodoulides, D.N., Coskun, T.H., Segev, M., Kivshar, Y.S.: Gray spatial solitons in biased photorefractive media. JOSA B 18, 55–63 (2001)
Katti, A.: Coupling of separate solitons in a series circuit of two photon photorefractive crystals exhibiting simultaneous quadratic and linear nonlinearities. Optik 206, 164212 (2020)
Fressengeas, N., Maufoy, J., Kugel, G.: Temporal behavior of bidimensional photorefractive bright spatial solitons. Phys. Rev. E 54, 6866 (1996)
Lu, K., Zhao, W., Chen, Y., Yang, Y., Zhang, L., Yang, Y., Li, J., Zhang, Y., Xu, J.: Temporal development of spatial solitons in biased photorefractive-photovoltaic materials. J. Mod. Opt. 55, 1571–1585 (2008)
Lu, K., Zhao, W., Zhang, L., Li, K., Zhang, Y., Liu, X., Zhang, Y., Xu, J.: Temporal behavior of dark spatial solitons in closed-circuit photovoltaic media. Opt. Commun. 281, 2913–2917 (2008)
Zhan, K., Hou, C., Pu, S.: Temporal behavior of spatial solitons in centrosymmetric photorefractive crystals. Opt. Laser Technol. 43, 1274–1278 (2011)
Ziółkowski, A.: Temporal analysis of solitons in photorefractive semiconductors. J. Opt. 14, 035202 (2012)
Katti, A.: Temporal behaviour of bright solitons in photorefractive crystals having both the linear and quadratic electro-optic effect. Chaos Solitons Fractals 126, 23–31 (2019)
Kukhtarev, N.V., Markov, V.B., Odulov, S.G., Soskin, M.S., Vinetskii, V.L.: Holographic storage in electrooptic crystals. I. Steady state. ferroelectrics 22, 949–960 (1978)
He, J.H.: The variational iteration method for eighth-order initial-boundary value problems. Phys. Scr. 76, 680–682 (2007)
Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2248–2253 (2012)
Malwe, B.H., Betchewe, G., Doka, S.Y., Kofane, T.C.: Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation map** method. Nonlinear Dyn. 84, 171–177 (2016)
Gunter, P., Huignard, J.P. (eds.): Photorefractive Materials and Their Applications 1. Springer Science + Business Media Inc, New York (2006)
Kapitula, T., Sandstede, B.: Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Phys. D 124, 58–103 (1998)
Kivshar, Y.S., Agrawal, G.: Optical Solitons: From Fibers to Photonic Crystals. Academic press, Cambridge (2003)
Liu, X., Jiao, Y., Wang, Y., Zhou, Q., Wang, W.: Kink soliton behavior study for systems with power-law nonlinearity. Results Phys. 33, 105162 (2022)
Acknowledgements
We thank Hanifah Azzaura Musyayy-adah for the helpful discussion.
Funding
This work supported by the Lembaga Penelitian dan Pengabdian Masyarakat (LPPM), Universitas Andalas under a fundamental research grant (No.Ref.T/66/UN.16.17. PT.01.03/IS-RD/2021).
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis performed by ZA, AR, MS, and WH. The first draft of the manuscript written by AR and all authors commented on previous versions of the manuscript. The final manuscript has been read and approved by all authors.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
To prove the solution of \(q(s,\xi )\) in Eq. (25) as the traveling wave exact solution of the explicit-time nonlinear photoreactive dynamics equation in the low amplitude case, substitute Eq. (25) into Eq. (6). Thus, each of the terms can be written in the form
where \(T_{1}=iq_{\xi }\), \(T_{2}=1/2~q_{ss}\), \(T_{3}=-(\beta _{1}+\beta _{2})q\), and \(T_{4}=-\left( \beta _{1}+2\beta _{2} \right) \left( \exp \left[ -\tau \right] -1\right) \left| q \right| ^{2}q\). If we add up all the terms, then \(T_{1}+T_{2}+T_{3}+T_{4}=0\) (valid as an exact solution).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Abdullah, Z., Ripai, A., Syafwan, M. et al. Traveling wave solutions for explicit-time nonlinear photorefractive dynamics equation. Nonlinear Dyn 111, 16515–16526 (2023). https://doi.org/10.1007/s11071-023-08610-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08610-8