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Journal of the Optical Society of America B

Variational analysis and angular bistability in layered nonlinear Kerr media with phase conjugation

T. A. Laine, Ari T. Friberg

Published Jun 1, 2000• 1 citation
Publisher

Abstract

The calculus of variations is applied to electromagnetic fields in a layered nonlinear structure supporting a guided wave. The system also includes a phase-conjugate mirror (PCM). By introducing a variational dimension and using a collection of plane waves as a trial function, we approximate the exact solution of the nonlinear Kerr–Maxwell equation. The formalism is new, and it involves the nonlinear interference of multiple plane waves. A simple analytical expression for the nonlinear field in the presence of the PCM is derived, and the fact that the scattered intensities may become bistable when the angle of incidence is varied is demonstrated. In particular, our theory predicts the angular bistability in the backscattering direction, where the effect of the guided waves is subtle. Our numerical results are also in good agreement with other theoretical approaches and with the experimental data.

DOI
10.1364/josab.17.000977

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The calculus of variations is applied to electromagnetic fields in a layered nonlinear structure supporting a guided wave. The system also includes a phase-conjugate mirror (PCM). By introducing a variational dimension and using a collection of plane waves as a trial function, we approximate the exact solution of the nonlinear Kerr–Maxwell equation. The formalism is new, and it involves the nonlinear interference of multiple plane waves. A simple analytical expression for the nonlinear field in the presence of the PCM is derived, and the fact that the scattered intensities may become bistable when the angle of incidence is varied is demonstrated. In particular, our theory predicts the angular bistability in the backscattering direction, where the effect of the guided waves is subtle. Our numerical results are also in good agreement with other theoretical approaches and with the experimental data.
Nonlinear optical properties in thin layers are investigated near a phase conjugate mirror. By using a rigorous electromagnetic theory of volume gratings the wave distortion correction by phase conjugation is examined in a nonlinear Kerr medium and Agarwal's theorem is numerically demonstrated. At specific incidence angles guided modes are formed inside the layered structure. When the nonlinearity is increased optical bistability is observed in waveguide resonances. An analytic solution is developed to explain the numerical data.
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.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.
Recently1, an exact theory of the irradiated nonlinear Fabry-Perot cavity and the ensuing multistability has been published. In this paper this approach is developed further and an exact treatment for the reflectivity and transmissivity of an optically isotropic structure of dielectric layers characterised by a third-order polarisation tensor at arbitrary angles of incidence is presented. The calculations use exact2 solutions of the nonlinear wave equation, rather than the commonly used ray treatment. This takes into account fully the standing wave effect and permits a precise examination of the interaction between the forward and backward travelling waves and the relationship between the transmissivity resonance condition and changes in the field profile. For an s-polarised incident wave an analytical solution is found for arbitrary angles of incidence and layer thicknesses. It is discovered that for certain ranges of layer thickness the system displays a multivalued response only above a critical angle. This bistability is achieved for irradiances much lower than those reported previously1 for normal incidence and indicates possible applications for optical processing. This plane wave treatment is supplemented by a numerical beam simulation which goes beyond the usual Fresnel approximation by utilising novel boundary conditions which permit the demostration of bistability.
In this paper we study propagation of electromagnetic waves in media with an imaginary Kerr-type nonlinearity. We show numerically that the slowly varying envelope approximation can still be used. It is proved that theoretical problems described by this type of differential equations are Hamiltonian systems. This characteristic is used to predict the variation of the electromagnetic field in the nonlinear material. Furthermore, we find a closed expression for the electromagnetic waves in a nonlinear medium with a linear absorption and an imaginary Kerr-type nonlinearity, for both a localized and a diffusive nonlinearity. Calculating the transmission probability of light through a Fabry-P\'erot structure shows that the introduction of an imaginary Kerr-type nonlinearity describes an intensity-dependent optical gain or absorption in a nonlinear medium. This system also shows bistable behavior comparable to real Kerr-type nonlinear materials. \textcopyright{} 1996 The American Physical Society.
A new use of gratings in nonlinear optics is presented, i.e., the realization of bistable components. The device studied here consist of grating couplers ruled on a Kerr nonlinear medium, which use the guided-wave resonance to increase the local field and thus the nonlinearities. The result is that they are intrinsic bistable optical systems with high-speed, low pumping thresholds, and geometry well adapted to optical integration. First, a linear electromagnetic study of the devices is presented. It follows optimizing the grating parameters in order to get the best coupling between the incident beam and the guided mode inside the corrugated waveguide. Then a graphical construction is given that demonstrates the bistable character of the system in nonlinear optics. Next a nonlinear analysis of the devices is rigorously derived from Maxwell equations. It states precisely the predictions of the graphical construction and allows comparison of the effectiveness of the guided-wave resonance with the surface plasmon resonance in order to reduce the threshold of bistability.