Dispersion properties of slab waveguides with a linear graded-index film and a nonlinear substrate
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TECHNICAL PAPER
Dispersion properties of slab waveguides with a linear graded-index film and a nonlinear substrate Aya J. Hussein1 • Zaher M. Nassar1 • Sofyan A. Taya1 Received: 31 July 2020 / Accepted: 22 August 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Three-layer slab waveguide with a graded-index film and a nonlinear substrate is investigated. The nonlinear substrate is considered of Kerr-type and the film index is assumed to change linearly as we move from the film-clad to the filmsubstrate interfaces. The solutions of the fields in the guiding film contains Airy functions which can be written as Bessel functions of order 1/3. The dispersion relation is derived in terms of three well-known normalized parameters V, a and b and the dispersion properties are plotted and studied. Many interesting features are obtained such as the nonexistence of cut-off thickness due to the nonlinear substrate and the value of the normalized guide index does not exceed unity which means they correspond to guided modes.
1 Introduction There was substantial interest in the analysis of waveguides and the properties of waves guided by dielectric films since a long time, which led up to an enormous number of devices. The refractive indices associated with all of the media forming a waveguide are often assumed field-intensity independent. Lately, structures of slab waveguides comprising one or more materials, whose index of refraction depends on the field intensity, have received a huge interest (Akhmediev 1982; Maradudin 1983; Lederer et al. 1983; Kaplan 1977). The motivation has been the generation of analytical solutions to the wave equation characterized by nonlinear index of refraction. A theory was formulated for the refraction and reflection of electromagnetic waves (EMs) from the interface of a medium with nonlinear dielectric function (Kaplan 1977; Kaplan 1976). It was proved that when the wave incident angle is altered, hysteresis hops should be noticed from the regime of transmission to the total internal reflection regime and back. For small nonlinearity, the closeness of the linear refractive index of the nonlinear medium and that of the linear medium is the essential condition for the presence of the hysteresis consequence. Such papers, which handled & Sofyan A. Taya [email protected] 1
Physics Department, Islamic University of Gaza, Gaza, Palestine
refraction and reflection of EMs at an interface between linear and nonlinear media, motivated supplemental experimental and theoretical work on waves both refracted and guided by such an interface. The nonlinear waves confined to a thin film and guided by this film can be launched at the interface between a nonlinear and linear media. This was noticed in such a structure (Vach et al. 1984). Physicists have obtained many analytical solutions for the dispersion relation and fields for transverse electric (TE) guided waves (Boardman and Egan 1984; Stegeman et al. 1984; Milhalache and Totia 1984).
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