Abstract
The theoretical description of the composite planar waveguide based on semiconductor crystals in which the film with a linear profile of the refractive index sandwiched by two photorefractive crystals with diffusion nonlinearity is proposed. Four types of the transverse magnetic stationary waves propagating along the film waveguide are found. The guided waves propagate with oscillating and non-oscillating profile and non-oscillating symmetrically and anti-symmetrically distributed across film interfaces in the different ranges of the effective refractive index. The waves exist with the discrete spectrum of the effective refractive index. The influence of the optical and geometrical parameters of the film waveguide on the guided wave characteristics and the filed intensity distributions is analyzed. In particular, the temperature of the semiconductor crystal affects the profiles and the type of the guided waves. It is shown that it is necessary to choose the optimal film thickness sufficient for excitation of the guided wave making it possible to obtain a non-oscillating decrease in the field.
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References
S. Noda, F.T. Mahi, H. Zappe, Photonic crystals, in Reference module in materials science and materials engineering. ed. by Y. Ren (Elsevier, 2016), pp.101–112. https://doi.org/10.1016/B978-0-12-803581-8.00555-5
H.E. Ruda, N. Matsuura, A properties and applications of photonic crystals, in Optical properties of materials and their applications. ed. by J. Singh (Wiley, Hoboken, 2019), pp.251–268. https://doi.org/10.1002/9781119506003.ch9
E. Garmire, Nonlinear optics in semiconductors. Phys. Today 47, 42–48 (1994). https://doi.org/10.1063/1.881432
M.J. Adams, An introduction to optical waveguides (Wiley, Chichester, 1981)
C.-L. Chen, Foundations for guided-wave optics (Wiley, 2005), p.462. https://doi.org/10.1002/0470042222
F. Ebrahimi (ed.), Surface waves—new trends and developments (IntechOpen, London, 2018), p.154. https://doi.org/10.5772/intechopen.68840
T. A. Laine, Electromagnetic Wave Propagation in Nonlinear Kerr Media (Royal Institute of Technology (KTH), Department of Physics, Stockholm, Sweden, 2000) 47.
M. Čada, M. Qasymeh, J. Pištora, Optical wave propagation in Kerr media, in Wave propagation theories and applications. (IntechOpen, 2013), pp.175–192. https://doi.org/10.5772/51293
T.H. Zhang, X.K. Ren, B.H. Wang, C.B. Lou, Z.J. Hu, W.W. Shao, Y.H. Xu, H.Z. Kang, J. Yang, D.P. Yang, L. Feng, J.J. Xu, Surface waves with photorefractive nonlinearity. Phys. Rev. A 76, 013827 (2007). https://doi.org/10.1103/PhysRevA.76.013827
P.A. Prudkovskii, Autowaves in two-wave mixing in photorefractive media. Quantum Electron. 41, 30–33 (2011). https://doi.org/10.1070/QE2011v041n01ABEH014463
D.D. Nolte, Photorefractive materials, in Encyclopedia of materials: science and technology. (Elsevier Ltd, London, 2001), pp.6955–6961. https://doi.org/10.1016/B0-08-043152-6/01232-8
N. Kamanina (ed.), Nonlinear optics (IntechOpen, London, 2018), p.224. https://doi.org/10.5772/2073
H. Chaib, T. Otto, L. Eng, Modeling the electrical and optical properties of BaTiO3 and LiNbO3 single crystals at room temperature. Ferroelectrics 304, 93–98 (2004). https://doi.org/10.1080/00150190490457609
D. Dragoman, M. Dragoman, Advanced optoelectronic devices (Springer, 1999), p.424
J.G. Mendoza-Alvarez, F.D. Nunes, N.B. Patel, Refractive index dependence on free carriers for GaAs. J. Appl. Phys. 51(8), 4365–4367 (1980). https://doi.org/10.1063/1.328298
S. Ravindran, A. Datta, K. Alameh, Y.T. Lee, GaAs based long-wavelength microring resonator optical switches utilising bias assisted carrier-injection induced refractive index change. Opt. Express 20(14), 15610–15627 (2012). https://doi.org/10.1364/OE.20.015610
J.E. Zucker, T.Y. Chang, M. Wegener, N.J. Sauer, K.L. Jones, D.S. Chemla, Large refractive index changes in tunable-electron-density InGaAs/InAlAs quantum wells. IEEE Photon. Technol. Lett. 2(1), 29–31 (1990). https://doi.org/10.1109/68.47032
K. Ishida, H. Nakamura, H. Matsumura, InGaAsP/InP optical switches using carrier induced refractive index change. Appl. Phys. Lett. 50(3), 141 (1987). https://doi.org/10.1063/1.97695
S. Popov, M. Enoch, Nevière, Plasmon surface waves and complex-type surface waves: comparative analysis of single interfaces, lamellar gratings, and two-dimensional hole arrays. Appl. Opt. 46(2), 154–160 (2007). https://doi.org/10.1364/AO.46.000154
O. Takayama, L. Crasovan, S. Johansen, D. Mihalache, D. Artigas, L. Torner, Dyakonov surface waves: a review. Electromagnetics 28(3), 126–145 (2008). https://doi.org/10.1080/02726340801921403
Yu.S. Kivshar, Chapter 8—surface Plasmon polaritons in complex settings and generalized geometries, in Handbook of Surface Science, vol. 4, (Elsevier, North-Holland, 2014), pp.253–278. https://doi.org/10.1016/B978-0-444-59526-3.00008-2.E
K.F. Sergeichev, D.M. Karfidov, S.E. Andreev, Yu.E. Sizov, V.I. Zhukov, Excitation and propagation of Sommerfeld-Zenneck surface waves on a conducting strip in the centimeter-wave band. J. Commun. Technol. Electron. 63, 326–334 (2018). https://doi.org/10.1134/S1064226918040101
M. Gryga, D. Vala, P. Kolejak, L. Gembalova, D. Ciprian, P. Hlubina, One-dimensional photonic crystal for Bloch surface waves and radiation modes-based sensing. Opt. Mater. Express 9, 4009–4022 (2019). https://doi.org/10.1364/OME.9.004009
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
D.A. Shilkin, E.V. Lyubin, A.A. Fedyanin, Nonlinear excitation and self-action of Bloch surface waves governed by gradient optical forces. ACS Photonics 9(1), 211–216 (2022). https://doi.org/10.1021/acsphotonics.1c01402
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
W.-Y. Lee, S.-Y. Wang, Guided-wave characteristics of optical graded-index planar waveguides with metal cladding: a simple analysis method. J. Lightwave Technol. 13(3), 416–421 (1995). https://doi.org/10.1109/50.372436
P. Karasinski, R. Rogozinski, Influence of refractive profile shape on the distribution of modal attenuation in planar structures with absorption cover. Opt. Commun. 269(1), 76–88 (2007). https://doi.org/10.1016/j.optcom.2006.07.067
A.B. Shvartsburg, A. Maradudin, Waves in gradient metamaterials (World Scientific, Singapore, 2013), p.339. https://doi.org/10.1142/8649
S. Chatterjee, P.R. Chaudhuri, Some unique propagation characteristics of linearly graded multilayered planar optical waveguides. J. Basic Appl. Phys. 3(1), 1–9 (2014)
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, 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(11), 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.E. Savotchenko, Surface waves in linearly graded-index and intensity-dependent index layered structure. J. Opt. Soc. Am. A 39(7), 1210–1217 (2022). https://doi.org/10.1364/JOSAA.451297
S.E. Savotchenko, The surface waves propagating along the contact between the layer with the constant gradient of refractive index and photorefractive crystal. J. Opt. 24(4), 045501 (2022). https://doi.org/10.1088/2040-8986/ac51e9
S.E. Savotchenko, The composite planar waveguide structure consisting of the linearly graded-index layer and the nonlinear layer formed with an increasing the electric field. Optik 252, 168542 (2022). https://doi.org/10.1016/j.ijleo.2021.168542
S.E. Savotchenko, Light localization in a linearly graded-index substrate covered by intensity dependent nonlinear self-focusing cladding. J. Opt. 24, 065503 (2022). https://doi.org/10.1088/2040-8986/ac6bab
S.E. Savotchenko, Guided waves in a graded-index substrate covered by an intensity-dependent defocusing nonlinear medium. Appl. Phys. B Lasers Opt. 128(8), 153 (2022). https://doi.org/10.1007/s00340-022-07872-1
S.E. Savotchenko, Features of the bound state formation near a nonlinear defect in the presence of a homogeneous external field. Eur. Phys. J. Plus 137, 867 (2022). https://doi.org/10.1140/epjp/s13360-022-03065-z
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
Y.V. Bludov, D.A. Smirnova, Y.S. Kivshar, N.M.R. Peres, M.I. Vasilevsky, Nonlinear TE-polarized surface polaritons on grapheme. Phys. Rev. B. (2014). https://doi.org/10.1103/PhysRevB.89.035406
O. Takayama, A.A. Bogdanov, A.V. Lavrinenko, Photonic surface waves on metamaterial interfaces. J. Phys. Condens. Matter 29(46), 463001 (2017). https://doi.org/10.1088/1361-648X/aa8bdd
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. Cheng, W. Wandan, C. Pan, C. Hou, S. Chen, D. Mihalache, F. Baronio, Photonic rogue waves in a strongly dispersive coupled-cavity array involving self-attractive Kerr nonlinearity. Phys. Rev. A 105, 013717 (2022). https://doi.org/10.1103/PhysRevA.105.013717
P.F. Qi, Z.J. Hu, R. Han, T.H. Zhang, J.G. Tian, J.J. Xu, Apodized waveguide arrays induced by photorefractive nonlinear surface waves. Opt. Express 23, 31144–31149 (2015). https://doi.org/10.1364/OE.23.031144
P. Qi, T. Feng, S. Wang, R. Han, Z. Hu, T. Zhang, J. Tian, J. Xu, Photorefractive surface nonlinearly chirped waveguide arrays. Phys. Rev. A 93, 053822 (2016). https://doi.org/10.1103/PhysRevA.93.053822
L. Chun-yang, J. Ying, S. De, M. Yi-ning, Y. Ji-kai, C. Wei-jun, Guided modes in thin layer waveguide induced by photorefractive surface waves, Chinese. J. Lumin. 39, 1572–1578 (2018). https://doi.org/10.3788/fgxb20183911.1572
M.S. Hamada, A.I. Assad, H.S. Ashour, M.M. Shabat, Nonlinear magnetostatic surface waves in a ferrite-left-handed waveguide structure. J. Microw. Optoelectr. 5, 45–54 (2006)
S.E. Savotchenko, Nonlinear surface waves in a symmetric three-layer structure that is composed of optical media with different formation mechanisms of nonlinear response. Opt. Spectrosc. 128(3), 345–354 (2020). https://doi.org/10.1134/S0030400X20030170
S.E. Savotchenko, Propagation of surface waves along a dielectric layer in a photorefractive crystal with a diffusion mechanism for the nonlinearity formation. Quantum Electron. 49(9), 850–856 (2019). https://doi.org/10.1070/QEL16968
S.E. Savotchenko, Nonlinear surface TM waves in a Kerr defocusing nonlinear slab sandwiched between photorefractive crystals. Solid State Commun. 296(7), 32–36 (2019). https://doi.org/10.1016/j.ssc.2019.04.008
S.E. Savotchenko, Nonlinear surface waves at the interface between optical media with different nonlinearity induction mechanisms. J. Exp. Theor. Phys. 129(2), 159–167 (2019). https://doi.org/10.1134/S1063776119070100
S.E. Savotchenko, Effect of the temperature on the redistribution of an energy flux carried by surface waves along the interface between crystals with different mechanisms of formation of a nonlinear response. J. Exp. Theor. Phys. Lett. 109(11), 744–748 (2019). https://doi.org/10.1134/S0021364019110146
Y. Yuan, S. Zhou, X. Wang, Modulating properties by light ion irradiation: From novel functional materials to semiconductor power devices. J. Semicond. 43(6), 063101 (2022). https://doi.org/10.1088/1674-4926/43/6/063101
G. Eliashberg, G. Klimovitch, A. Rylyakov, On the temperature dependence of the London penetration depth in a superconductor. J. Supercond. 4, 393–396 (1991). https://doi.org/10.1007/BF00618221
V.G. Kogan, R. Prozorov, Temperature dependence of London penetration depth anisotropy in superconductors with anisotropic order parameters. Phys. Rev. B 103(5), 054502 (2021). https://doi.org/10.1103/PhysRevB.103.054502
D. Mihalache, G.I. Stegeman, C.T. Seaton, E.M. Wright, R. Zanoni, A.D. Boardman, T. Twardowski, Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface. Opt. Lett. 12, 187–189 (1987). https://doi.org/10.1364/OL.12.000187
B.A. Usievich, DKh. Nurligareev, V.A. Sychugov, L.I. Ivleva, P.A. Lykov, N.V. Bogodaev, Nonlinear surface waves on the boundary of a photorefractive crystal. Quantum Electron. 40, 437–440 (2010). https://doi.org/10.1070/QE2010v040n05ABEH014223
B.A. Usievich, DKh. Nurligareev, V.A. Sychugov, L.I. Ivleva, P.A. Lykov, N.V. Bogodaev, Surface photorefractive wave on the boundary of a photorefractive metal-coated crystal. Quantum Electron. 41, 262–266 (2011). https://doi.org/10.1070/QE2013v043n01ABEH014913
S.A. Chetkin, I.M. Akhmedzhanov, Optical surface wave in a crystal with diffusion photorefractive nonlinearity. Quantum Electron. 41, 980–985 (2011). https://doi.org/10.1070/QE2011v041n11ABEH014660
DKh. Nurligareev, B.A. Usievich, V.A. Sychugov, L.I. Ivleva, Characteristics of surface photorefractive waves in a nonlinear SBN-75 crystal coated with a metal film. Quantum Electron. 43, 14–20 (2013). https://doi.org/10.1070/QE2013v043n01ABEH014913
S.E. Savotchenko, Surface waves at the boundary of a photorefractive crystal and a medium with positive Kerr nonlinearity. Phys. Solid State 62(6), 1011–1016 (2020). https://doi.org/10.1134/S1063783420060268
S.E. Savotchenko, Effect of the dark illumination Intensity on the characteristics of surface waves propagating along the interface between photorefractive and nonlinear Kerr crystals. Russ. Phys. J. 63(1), 160–170 (2020). https://doi.org/10.1007/s11182-020-02015-5
S.E. Savotchenko, Surface waves at the boundary of a medium with a refractive index switching and a crystal with the diffusion-type photorefractive nonlinearity. Phys. Solid State 62(8), 1415–1420 (2020). https://doi.org/10.1134/S1063783420080284
G.E. Andrews, R. Askey, R. Roy, Special functions (Cambridge University Press, 1999), p.664. https://doi.org/10.1017/CBO9781107325937
W. Van Assche, Ordinary special functions, in Encyclopedia of mathematical physics. ed. by J.-P. Françoise, G.L. Naber, T.S. Tsun (Academic Press, New York, 2006), pp.637–645. https://doi.org/10.1016/B0-12-512666-2/00395-3
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Savotchenko, S.E. Temperature-controlled waveguide properties of the linearly graded-index film in the photorefractive crystal. Appl. Phys. B 129, 7 (2023). https://doi.org/10.1007/s00340-022-07950-4
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DOI: https://doi.org/10.1007/s00340-022-07950-4