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AIP conference proceedings

Exact modelling of the optical bistability in ferroelectics via two-wave mixing: A system with full nonlinearity

Muhammad Asif Ahmad Khushaini, Abdel-Baset M.A. Ibrahim, P. K. Choudhury

Published Jan 1, 2018• 0 citations
Publisher

Abstract

In this paper, we provide a complete mathematical model of the phenomenon of optical bistability (OB) resulting from the degenerate two-wave mixing (TWM) process of laser beams interacting with a single nonlinear layer of ferroelectric material. Starting with the electromagnetic wave equation for optical wave propagating in nonlinear media, a nonlinear coupled wave (CW) system with both self-phase modulation (SPM) and cross-phase modulation (XPM) sources of nonlinearity are derived. The complete CW system with full nonlinearity is solved numerically and a comparison between both the cases of with and without SPM at various combinations of design parameters is given. Furthermore, to provide a reliable theoretical model for the OB via TWM process, the results obtained theoretically are compared with the available experimental data. We found that the nonlinear system without SPM fails to predict the bistable response at lower combinations of the input parameters. However, at relatively higher values, the solution without SPM shows a reduction in the switching contrast and period in the OB response. A comparison with the experimental results shows better agreement with the system with full nonlinearity.

DOI
10.1063/1.5034566

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In this paper, we provide a complete mathematical model of the phenomenon of optical bistability (OB) resulting from the degenerate two-wave mixing (TWM) process of laser beams interacting with a single nonlinear layer of ferroelectric material. Starting with the electromagnetic wave equation for optical wave propagating in nonlinear media, a nonlinear coupled wave (CW) system with both self-phase modulation (SPM) and cross-phase modulation (XPM) sources of nonlinearity are derived. The complete CW system with full nonlinearity is solved numerically and a comparison between both the cases of with and without SPM at various combinations of design parameters is given. Furthermore, to provide a reliable theoretical model for the OB via TWM process, the results obtained theoretically are compared with the available experimental data. We found that the nonlinear system without SPM fails to predict the bistable response at lower combinations of the input parameters. However, at relatively higher values, the solution without SPM shows a reduction in the switching contrast and period in the OB response. A comparison with the experimental results shows better agreement with the system with full nonlinearity.
Ferroelectric oxides with perovskite structure such as PbLaTiO3 (PLT), BaTiO3, PbTiO3, SrBi2Ta2O9 and LiNbO3 are very attractive class of materials which possess numerous useful properties such as high dielectric constant, large spontaneous polarization, and remarkable optical nonlinearity. Potential applications of these materials include real-time holography, correlation filtering and various novelty filter applications (Sutherland 1996 and Eaton 1991). They are also popular materials for the fabrication of nonvolatile memories (Ramesh 2001). Over the past few years, ferroelectric oxides have been widely investigated for various nonlinear optical applications (Shi 2006; Xuan 1998; Zhang 1999; Zhao 1996) especially for optical switches. Optical switches are devices invented to perform multiplexing at very fast speeds and with less delay than the customary switches works with electronic signals. A ferroelectric optical switch is expected to allow the processing of millions of signals at a speed of terahertz. To obtain optical bistability phenomenon, two ingredients are necessary, a nonlinear process and a feedback mechanism (Gibbs, 1977, 1979, and 1985). In all optical systems, the feedback can be “distributed”, “extrinsic” or “intrinsic”. In multilayer systems with alternating nonlinear materials, the feedback is “distributed”; it arises from the interaction of the propagating wave with many cross-sections of a nonlinear medium. In a Fabry-Perot (FP) resonator, the feedback is “extrinsic”; it arises as a result of reflection from the mirrors placed at its interface. In a single nonlinear layer, the feedback can be “intrinsic” or “mirrorless”; it arises in each elementary oscillator due to the strong local nonlinear response of an individual atom or molecule. In the usual or the standard analysis to study the optical bistability (Marburger 1978, Gupta 1987, Biran 1990, Danckaert 1989, Shen 1984, and Haelterman 1989) in nonlinear optics, the governing equation for optical propagation within the nonlinear medium is a nonlinear wave equation in the electric field derived from Maxwell’s equations. The usual constitutive relation between the nonlinear polarization and the electric field is then obtained by expanding the nonlinear polarization as a Taylor series in the electric field. The usefulness of this constitutive relation is that the polarization is a natural source term in the Maxwell’s
We report the observation and theoretical interpretation of two-photon double-beam optical bistability, where the nonlinearity is obtained in a two-photon transition in a sodium atomic beam. The experimental results, obtained in the dispersive regime, are compared with a simple crossed-phase-modulation model, and also with a numerical solution of the interaction between the cavity fields and three-level atoms, using the optical Bloch equations. For a wide range of parameters, both models agree with the experimental results. Oscillatory regimes may also appear, in agreement with the predictions of the complete model.
Optical bistability and switching are of great interest in the rapidly and developing field of photonics. Devices that display this behavior could potentially play a major role in the development of optical communication systems and computing. In this article we present experimental results concerning the optical bistability in ferroelectric of photorefractive BaTiO3 crystal. Two laser beams were used to interact with the photorefractive crystal which resulted in the bistability of the intensity of transmitted wave. This was achieved without the application of any optical resonator. High contrast optical bistability is found experimentally in the pump-ratio dependence of the output intensity.
A general theory of two-wave mixing (TWM) in nonlinear media is presented. Assuming a gain (or absorption) grating and a refractive index grating are generated because of the nonlinear process in a nonlinear medium, the coupled-wave equations of TWM are derived based on Maxwell's wave equation. The coupled-wave equations can be decoupled as coupled equations for the intensity and coupled equations for the phase of both beams, and these two sets of coupled equations can be solved analytically by using average total intensity in the medium instead of using the total intensity. Compared with the previous theory of TWM, the theory presented here is more general, and the applications of the theory to photorefractive materials, Kerr media, and semiconductor broad-area amplifiers are described.
In this study, we investigate the optical bistability and multistability in a pure single Kerr nonlinear layer of dielectric material using the Maxwell-Duffing approach. We employ the Duffing anharmonic oscillator as an alternative constitutive relation to express the electric field as a nonlinear function of the induced polarization. Utilizing both the electromagnetic wave equation and the Duffing constitutive relation, we obtain a nonlinear polarization equation propagates through the nonlinear layer under the slowly varying envelope approximation. With the application of boundary conditions at both interfaces, expressions for the incident, reflected, and transmitted intensities of the light wave are obtained as nonlinear functions of the polarization and other controlling parameters. For realistic input parameters, our numerical results demonstrated the existence of an intrinsic optical bistability (IOB) in a pure single dielectric layer. Furthermore, the threshold value of the observed IOB depends essentially on the value Eth, i.e., the minimum value required to invoke the third-order nonlinearity inside the material. For operating frequency near the sharp resonance region, Eth was found to be ≈105 V/cm; this is equivalent to a threshold value of the optical intensity ≈1010 W/cm2. With the developments in THz sources, it is nowadays possible to observe this IOB experimentally by tuning the operating frequency near the sharp resonance region of the material. Steps to get the threshold value of Eth are also advised.