Abstract
New peculiarities of light localization near the interface between media with an exponential profile of the dielectric permittivity and with a sharp suppression of the Kerr nonlinear response in a near-surface layer formed with an increasing light intensity are described theoretically. Two signs of nonlinear response corresponding to self-focusing and defocusing nonlinear media are considered. Exact solutions to stationary wave equation with dielectric permittivity consisting of spatial-dependent and intensity-dependent parts satisfying the boundary conditions at the medium interface and the near-surface layer boundary are found. The effect of the optical and geometric parameters on the transverse distribution of the light intensity and the features of the lightwave localization is analyzed. It is found that light localization length decreases with a decrease in the wavelength of excited radiation but the maximum of intensity remains unchanged. The wave intensity enlarges with an increase in the effective refractive index and in the threshold intensity. The near-surface domain width where a nonlinear response is suppressed enlarges with increasing the effective refractive index, the threshold intensity, and the Kerr nonlinearity coefficient. It reduces with increasing values of the parameters of the exponential graded-index layer. It is shown that the largest portion of the lightwave power is concentrated in the near-surface domain with suppressed nonlinear response, which plays the role of the main waveguide layer. It is found that the optical parameters of the exponential graded-index layer affect the redistribution of the wave energy between the contacting areas of the waveguide structure.
Similar content being viewed by others
Data Availability Statement
No Data associated in the manuscript.
References
T.M. Jordan, J.C. Partridge, N.W. Roberts, Disordered animal multilayer reflectors and the localization of light. J. R. Soc. Interface. 11, 20140948 (2014). https://doi.org/10.1098/rsif.2014.0948
D.M. Jović, M.R. Belić, C. Denz, Anderson localization of light at the interface between linear and nonlinear dielectric media with an optically induced photonic lattice. Phys. Rev. A 85, 031801 (2012). https://doi.org/10.1103/PhysRevA.85.031801
R. Asmi, N.B. Ali, M. Kanzari, Enhancement of light localization in hybrid thue-morse/periodic photonic crystals. J. Mater. 2016, 9471312 (2016). https://doi.org/10.1155/2016/9471312
A. Kahan, A. Greenbaum, M.J. Jang, J.E. Robinson, J.R. Cho, X. Chen, P. Kassraian, D.A. Wagenaar, V. Gradinaru, Light-guided sectioning for precise in situ localization and tissue interface analysis for brain-implanted optical fibers and GRIN lenses. Cell Rep. 36(13), 109744 (2021). https://doi.org/10.1016/j.celrep.2021.109744.PMID:34592157;PMCID:PMC8552649
A.K. Goyal, M. Husain, Y.Y. Massoud, Analysis of interface mode localization in disordered photonic crystal structure. J. Nanophoton. 16(4), 046007 (2022). https://doi.org/10.1117/1.JNP.16.046007
U. Langbein, F. Lederer, H.E. Ponath, Generalized dispersion relations for nonlinear plate-guided waves. Opt. Commun. 53, 417–420 (1985). https://doi.org/10.1016/0030-4018(85)90030-6
K.M. Leung, Propagation of nonlinear surface polaritons. Phys. Rev. A 31, 1189–1192 (1985). https://doi.org/10.1103/PhysRevA.31.1189
D. Mihalache, M. Bertolotti, C. Sibilia, Nonlinear wave propagation in planar structures. Prog. Opt. 27, 227–313 (1989). https://doi.org/10.1016/S0079-6638(08)70087-8
A.D. Boardman, M.M. Shabat, R.F. Wallis, TE waves at an interface between linear gyromagnetic and nonlinear dielectric media. J. Phys. D Appl. Phys. 24, 1702–1707 (1991). https://doi.org/10.1088/0022-3727/24/10/002
A.A. Sukhorukov, Y.S. Kivshar, Nonlinear localized waves in a periodic medium. Phys. Rev. Lett. 87(8), 083901 (2001). https://doi.org/10.1103/PhysRevLett.87.083901
I.V. Shadrivov, A.A. Sukhorukov, Yu.S. Kivshar, A.A. Zharov, A.D. Boardman, P. Egan, Nonlinear surface waves in left-handed materials. Phys. Rev. E 69, 016617–016621 (2004). https://doi.org/10.1103/PhysRevE.69.016617
H.S. Ashour, A.I. Assad, S-polarized surface waves in ferrite bounded by nonlinear nonmagnetic negative permittivity metamaterial. J. Al Azhar Univ. Gaza Natl. Sci. 13, 93–108 (2011)
I.V. Shadrivov, A.A. Sukhorukov, Yu.S. Kivshar, Guided modes in negative-refractive-index waveguides. Phys. Rev. E 67, 057602 (2003). https://doi.org/10.1103/PhysRevE.67.057602
A.A. Sukhorukov, Yu.S. Kivshar, Nonlinear guided waves and spatial solitons in a periodic layered medium. J. Opt. Soc. Am. B 19, 772–781 (2002). https://doi.org/10.1364/JOSAB.19.000772
B.A. Malomed, D. Mihalache, Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)
D. Mihalache, Localized structures in optical and matter-wave media: a selection of recent studies. Rom. Rep. Phys. 73, 403 (2021)
D.N. Christodoulides, M.I. Carvalho, Bright, dark, and grey spatial soliton states in photorefractive media. J. Opt. Soc. Am. B 12, 1628–1633 (1995). https://doi.org/10.1364/JOSAB.12.001628
B. Alfassi, C. Rotschild, O. Manela, M. Segev, D.N. Christodoulides, Nonlocal surface-wave solitons. Phys. Rev. Lett. 98, 213901 (2007). https://doi.org/10.1103/PhysRevLett.98.213901
Y.V. Kartashov, B.A. Malomed, L. Torner, Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247 (2011). https://doi.org/10.1103/RevModPhys.83.247
V. N. Kadantsev, A. N. Goltsov, M. A. Kondakov. Electrosoliton dynamics in a thermalized molecular chain. Russ. Technol. J. 8(1), 43–57 (2020). https://doi.org/10.32362/2500-316X-2020-8-1-43-57
M.J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, 1981)
C.-L. Chen, Foundations for guided-wave optics (Wiley, New York, 2005)
A.B. Shvartsburg, A. Maradudin, Waves in gradient metamaterials (World Scientific, Singapore, 2013)
M. Bednarik, M. Cervenka, Electromagnetic waves in graded-index planar waveguides. J. Opt. Soc. Am. B 37, 3631–3643 (2020). https://doi.org/10.1364/JOSAB.408679
T. Touam, F. Yergeau, Analytical solution for a linearly graded-index-profile planar waveguide. Appl. Opt. 32, 309–312 (1993). https://doi.org/10.1364/AO.32.000309
A.B. Shvartsburg, Dispersion of electromagnetic waves in stratified and nonstationary media (exactly solvable models). Phys. Uspekhi 43, 1201–1228 (2000). https://doi.org/10.1070/PU2000v043n12ABEH000827
M. Dalarsson, Y. Ivanenko, S. Nordebo, Wave propagation in waveguides with graded plasmonic obstacles. J. Opt. Soc. Am. B 38, 104–113 (2021). https://doi.org/10.1364/JOSAB.410092
B. Rana, B.B. Svendsen, M. Dalarsson, TE-wave propagation over an impedance-matched RHM to LHM transition in a hollow waveguide. Progress Electromagn. Res. M 110, 1–10 (2022). https://doi.org/10.2528/PIERM22022505
B.B. Svendsen, M. Söderström, H. Carlens, M. Dalarsson, Analytical and numerical models for TE-wave absorption in a graded-index GNP-treated cell substrate inserted in a waveguide. Appl. Sci. 12, 7097 (2022). https://doi.org/10.3390/app12147097
H. Wang, Q. Zhou, W. Liu, Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation. J. Adv. Res. 38, 179–190 (2022). https://doi.org/10.1016/j.jare.2021.09.007
H. Wang, X. Li, Q. Zhou, W. Liu, Dynamics and spectral analysis of optical rogue waves for a coupled nonlinear Schrödinger equation applicable to pulse propagation in isotropic media. Chaos, Solitons Fractals 166, 112924 (2023). https://doi.org/10.1016/j.chaos.2022.112924
Z. Cao, Y. Jiang, Q. Shen, X. Dou, Y. Chen, Exact analytical method for planar optical waveguides with arbitrary index profile. J. Opt. Soc. Am. A 16(9), 2209–2212 (1999). https://doi.org/10.1364/JOSAA.16.002209
N.A. Kudryashov, Optical solitons of mathematical model with arbitrary refractive index. Optik 224, 165391 (2020). https://doi.org/10.1016/j.ijleo.2020.165391
G. Akram, M. Sadaf, I. Zainab, The dynamical study of Biswas-Arshed equation via modified auxiliary equation method. Optik 255, 168614 (2022). https://doi.org/10.1016/j.ijleo.2022.168614
N.A. Kudryashov, Stationary solitons of the model with nonlinear chromatic dispersion and arbitrary refractive index. Optik 259, 168888 (2022). https://doi.org/10.1016/j.ijleo.2022.168888
D.J. Robbins, TE modes in a slab waveguide bounded by nonlinear media. Opt. Commun. 47, 309–312 (1983). https://doi.org/10.1016/0030-4018(83)90034-2
C. Seaton, J. Valera, R. Shoemaker, G. Stegeman, J. Chilwell, S. Smith, Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media. IEEE J. Quantum Electron. 21, 774–783 (1985). https://doi.org/10.1109/JQE.1985.1072762
K. Ogusu, TM waves guided by nonlinear planar waveguides. IEEE Trans. Microw. Theory Tech. 37, 941–946 (1989). https://doi.org/10.1109/22.25394
F. Dios, L. Torner, F. Canal, Self-consistent solution for general nonlinear slab waveguides. Opt. Commun. 72, 54–59 (1989). https://doi.org/10.1016/0030-4018(89)90255-1
R.K. Varshney, M.A. Nehme, R. Srivastava, R.V. Ramaswamy, Guided waves in graded-index planar waveguides with nonlinear cover medium. Appl. Opt. 25, 3899–3902 (1986). https://doi.org/10.1364/AO.25.003899
S.J. Al-Bader, H.A. Jamid, Graded-index optical waveguides with nonlinear cladding. J. Opt. Soc. Am. A 5, 374–379 (1988). https://doi.org/10.1364/JOSAA.5.000374
S.A. Taya, H.M. Kullab, I.M. Qadoura, Dispersion properties of slab waveguides with double negative material guiding layer and nonlinear substrate. J. Opt. Soc. Am. B 30, 2008–2013 (2013). https://doi.org/10.1364/JOSAB.30.002008
A.J. Hussein, Z.M. Nassar, S.A. Taya, Dispersion properties of slab waveguides with a linear graded-index film and a nonlinear substrate. Microsyst. Technol. 27(7), 2589–2594 (2021). https://doi.org/10.1007/s00542-020-05016-z
S.A. Taya, A.J. Hussein, I. Colak, An exact solution of a slab waveguide dispersion relation with a linear graded-index guiding layer (TM case). Microsyst. Technol. 28, 1213–1219 (2022). https://doi.org/10.1007/s00542-022-05281-0
A.H.M. Almawgani, S.A. Taya, A.J. Hussein, I. Colak, Dispersion properties of a slab waveguide with a graded-index core layer and a nonlinear cladding using the WKB approximation method. J. Opt. Soc. Am. B 39, 1606–1613 (2022). https://doi.org/10.1364/JOSAB.458569
J.M. Kubica, Analysis of planar waveguides with a thin overlayer and nonlinear cladding. Opt. Quant. Electron. 55, 137 (2023). https://doi.org/10.1007/s11082-022-04390-4
S.A. Taya, A.J. Hussein, O.M. Ramahi, I. Colak, Y.B. Chaouche, Dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film (transverse magnetic case). J. Opt. Soc. Am. B 38, 3237–3243 (2021). https://doi.org/10.1364/JOSAB.439034
A.J. Hussein, S.A. Taya, D. Vigneswaran, R. Udiayakumar, A. Upadhyay, T. Anwar, I.S. Amiri, Universal dispersion curves of a planar waveguide with an exponential graded-index guiding layer and a nonlinear cladding. Results Phys. 20, 103734 (2021). https://doi.org/10.1016/j.rinp.2020.103734
S.-Y. Huang, S. Wang, Ray optics of a planar waveguide with an exponential index profile. J. Appl. Phys. 55(4), 647–651 (1984). https://doi.org/10.1063/1.333117
A.M. Shutyi, D.I. Sementsov, A.V. Kazakevich, D.G. Sannikov, Waveguide regimes of a graded-index planar waveguide with cladding. Tech. Phys. 44(11), 1329–1333 (1999). https://doi.org/10.1134/1.1259518
D.G. Sannikov, D.I. Sementsov, A.M. Shutyi, A.V. Kazakevich, Beam model of waveguide regimes in a multilayer graded-index waveguide. Tech. Phys. Lett. 25, 977–979 (1999). https://doi.org/10.1134/1.1262699
S.E. Savotchenko, Surface waves propagating along the interface separating an exponential graded-index medium and the medium with a step change in the dielectric constant. Optik 271(12), 170092 (2022). https://doi.org/10.1016/j.ijleo.2022.170092
S.E. Savotchenko, Waveguide properties of interface separating a photorefractive crystal with diffusion nonlinearity and an exponential graded-index medium. Phys. Lett. A 455(12), 128516 (2022). https://doi.org/10.1016/j.physleta.2022.128516
S.E. Savotchenko, Nonlinear guided wave propagation along layers with the exponential index profile, constant index and the Kerr nonlinearity. Optik 276(4), 170689 (2023). https://doi.org/10.1016/j.ijleo.2023.170689
S.E. Savotchenko, The reduction of nonlinear response in near-surface layers by adjusting the electric field amplitude. J. Opt. 23(4), 045503 (2021). https://doi.org/10.1088/2040-8986/abeb2c
S.E. Savotchenko, Modes of suppression of a negative nonlinear response in near-surface layers controlled by an electric field strength in a crystal with a screening coating. Roman. J. Phys. 67(1–2), 202 (2022)
S.E. Savotchenko, Suppression of a self-focusing nonlinearity in near-surface layers by an electric field in a crystal with a fully shielding coating. Opt. Quant. Electron. 54(5), 305 (2022). https://doi.org/10.1007/s11082-022-03696-7
S.E. Savotchenko, Nonlinear waves in a waveguide with a linear spatial profile of the refractive index and a near-surface layer with disappearing nonlinearity. Optik 272(2), 170373 (2023). https://doi.org/10.1016/j.ijleo.2022.170373
O.V. Korovai, P.I. Khadzhi, Nonlinear surface waves in a symmetrical three-layer structure caused by the generation of excitons and biexcitons in semiconductors. Phys. Solid State 45, 386–390 (2003). https://doi.org/10.1134/1.1553548
S. Gatz, J. Herrmann, Soliton propagation in materials with saturable nonlinearity. J. Opt. Soc. Am. B 8, 2296–2302 (1991)
J. Herrmann, Propagation of ultrashort light pulses in fibers with saturable nonlinearity in the normal-dispersion region. J. Opt. Soc. Am. B 8, 1507–1511 (1991). https://doi.org/10.1364/JOSAB.8.001507
S. Bian, J. Frejlich, K.H. Ringhofer, Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals. Phys. Rev. Lett. 78, 4035–4038 (1997). https://doi.org/10.1103/PhysRevLett.78.4035
J.M. Christian, G.S. McDonald, P. Chamorro-Posada, Bistable Helmholtz bright solitons in saturable materials. J. Opt. Soc. Am. B 26(12), 2323–2330 (2009). https://doi.org/10.1364/JOSAB.26.002323
J. Wu, P. Jia, S. Wang, X. Wang, J. Yuan, L. Wang, Y. Hu, Z. Chen, J. Xu, Measuring saturable nonlinearity in atomic vapor via direct spatial mapping. Opt. Express 30, 43012–43020 (2022). https://doi.org/10.1364/OE.472652
P. Li, D. Mihalache, B.A. Malomed, Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential. Phil. Trans. R. Soc. A 376, 20170378 (2018). https://doi.org/10.1098/rsta.2017.0378
P.I. Khadzhi, L.V. Fedorov, Nonlinear surface waves for the simplest model of nonlinear medium. Phys. Tech. Lett. 61, 110–113 (1991)
N.N. Beletsky, E.A. Hasan, Closed dispersion curves for electromagnetic TE waves in a nonlinear film. Phys. of the Sol. St. 36, 647–652 (1994)
K.D. Lyakhomskaya, P.I. Hadji, self-reflection effect in the simplest non-linear medium. Tech. Phys. 70, 86–90 (2000)
A. Schuzgen, N. Peyghambarian, S. Hughes, Doppler shifted self reflection from a semiconductor. Phys. Stat. Sol. (b) 206, 125–130 (1999). https://doi.org/10.1002/(SICI)1521-3951(199803)206:1%3C125::AID-PSSB125%3E3.0.CO;2-8
E.C. Jarque, V.A. Malyshev, Nonlinear reflection from a dense saturable absorber: from stability to chaos. Opt. Commun. 142, 66–70 (1997). https://doi.org/10.1016/S0030-4018(97)00275-7
N. N. Rosanov, Nonlinear radiation reflection. In: Spatial hysteresis and optical patterns. Springer Series in Synergetics. Springer, Berlin (2002) 177–205. https://doi.org/10.1007/978-3-662-04792-7_5.
P. Roussignol, D. Ricard, J. Lukasik, C. Flytzanis, New results on optical phase conjugation in semiconductor-doped glasses. J. Opt. Soc. Am. B 4, 5–13 (1987). https://doi.org/10.1364/JOSAB.4.000005
J.-L. Coutaz, M. Kull, Saturation of the nonlinear index of refraction in semiconductor-doped glass. J. Opt. Soc. Am. B 8, 95–98 (1991). https://doi.org/10.1364/JOSAB.8.000095
T. Catunda, L.A. Cury, Transverse self-phase modulation in ruby and GdAlO3:Cr+3 crystals. J. Opt. Soc. Am. B 7, 1445–1455 (1990). https://doi.org/10.1364/JOSAB.7.001445
S. Wang, C. Zhang, R.B. Gross, R.R. Birde, The intensity-dependent refractive index of chemically enhanced bacteriorhodopsin. Opt. Commun. 112, 296–301 (1994). https://doi.org/10.1016/0030-4018(94)90634-3
Q. WangSong, X. Wang, R.R. Birge, J.D. Downie, D. Timucin, C. Gary, Propagation of a Gaussian beam in a bacteriorhodopsin film. J. Opt. Soc. Am. B 15, 1602–1609 (1998). https://doi.org/10.1364/JOSAB.15.001602
Y.V. Kartashov, F. Ye, V.A. Vysloukh, L. Torner, Surface waves in defocusing thermal media. Opt. Lett. 32, 2260–2262 (2007). https://doi.org/10.1364/OL.32.002260
Y.V. Kartashov, V.A. Vysloukh, L. Torner, Ring surface waves in thermal nonlinear media. Opt. Express 15, 16216–16221 (2007). https://doi.org/10.1364/OE.15.016216
D.P. Yang, Z.P. Chen, F. Zhao, H.Y. Yu, T.H. Zhang, J.G. Tian, J.J. Xu, Observation of photorefractive surface waves in self-defocusing LiNbO3: Fe crystal. Opt. Lett. 38(16), 153093 (2013). https://doi.org/10.1364/OL.38.003093
V.K. Chaubey, K.K. Dey, P. Khastgir, Field intensity and power confinement of four-layer slab waveguides with various refractive index profiles in the guiding region. J. Optic. Commun. 15, 95–100 (1994). https://doi.org/10.1515/JOC.1994.15.3.95
Y.-Z. Huang, Z. Pan, R.-H. Wu, Analysis of the optical confinement factor in semiconductor lasers. J. Appl. Phys. 79, 3827–3830 (1996). https://doi.org/10.1063/1.361809
M. Aellen, D.J. Norris, Understanding optical gain: which confinement factor is correct? ACS Photon. 9(11), 3498–3505 (2022). https://doi.org/10.1021/acsphotonics.2c01222
Acknowledgements
The study was carried using equipment of the Center of High Technologies of the Belgorod V. G. Shukhov State Technological University.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interest
The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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
Savotchenko, S.E. Light localization near an interface between media with an exponential permittivity profile and a sharply vanishing Kerr nonlinearity with an increasing light intensity. Eur. Phys. J. Plus 139, 155 (2024). https://doi.org/10.1140/epjp/s13360-024-04967-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-024-04967-w