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Journal of Applied Physics

On the intrinsic optical bistability of a single Kerr nonlinear dielectric layer

Abdel-Baset M.A. Ibrahim

Published May 8, 2015• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.4919821

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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.
This paper deals with the conceptualization of Maxwell-Duffing theory to model photonic deflection sensor along with functionality, which is based on the phenomena of optical bi- and multistabilities. The sensing system is considered to be consisting of Kerr nonlinear material along with suitably positioned mirrors. The efficacy of the approach is emphasized through a series of numerical simulations, and the reliability of the system is discussed. Effects due to system memory and periodicity in the optical bistability threshold have been demonstrated. It has been found that the approach provides a powerful tool to study optical bistability in resonating structures, particularly for materials with large third-nonlinearity and for operating frequencies near the natural resonance of the material.
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 study the nonlinear optical bistability and multistability of a nonlinear ferroelectric (FE) Fabry-Perot (FP) interferometer assumed coated with partial dielectric mirrors on its interfaces. The FE material is considered to have an intensity-dependent refractive index where the third order nonlinear susceptibility χ ilmn (3) acts like Kerr coefficient. The nonlinear response of the FE medium is modeled by the Landau-Khalatnikov (LK) dynamical equation. Within a single frequency approximation, the nonlinear wave equation is written in terms of polarization P rather than the electric field E as the dependent variable. The resulting nonlinear polarization equation is numerically integrated across the FP film using the fourth order Runge-Kutta method. The behavior of the resulting polarization and transmission as a function of the field input intensity is investigated.
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude-phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the permittivity at the interface. Under total internal reflection, the spatial evolution of the permittivity is described by a first-order differential equation and decays monotonically to the linear value. The incident intensity on the evanescent branch is shown to possess a maximum when 2𝛽2 > 𝜀1 +𝜀𝑙, where 𝛽 is the conserved propagation constant, 𝜀1 the linear permittivity, and 𝜀𝑙 the low-intensity permittivity of the Kerr medium. For intensities between the threshold 𝑆𝑐 and the maximum 𝑆max, three steady states coexist: two evanescent (TIR) solutions (one stable, one unstable) and one propagating solution. The propagating branch rises with an infinite slope at 𝜀 = 𝛽2 (square-root singularity), leading to an ultra-sharp switching transition. The relation between the normalized decay constant 𝜇 and the incident intensity exhibits a folded curve, revealing an “internal” bistability in the TIR regime. Both self-focusing (𝛼 > 0) and self-defocusing (𝛼 < 0) Kerr nonlinearities are treated. The propagation constant 𝛽 appears as a free design parameter, allowing the switching intensity to be tuned through waveguide geometry. These results provide a unified picture of optical bistability at a nonlinear interface, with potential applications in all-optical switching and sensing.