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Journal of the Optical Society of America B

Gratings in nonlinear optics and optical bistability

P. Vincent, Nicole A. Paraire, M. Nevière, A. Köster, R. Reinisch

Published Jul 1, 1985• 68 citations
Publisher

Abstract

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.

DOI
10.1364/josab.2.001106

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Dec 22, 1989•2 citations
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.
We consider grating induced optical bistability (OB) arising from the resonant excitation of a given nonlinear guided mode. In that case it can be shown, using the coupled-mode formalismi(1), that in the whole bistable domain : a) the transverse field map of the resonantly excited nonlinear guided mode corresponds to that of the associated linear one which is known. b) the Kerr-type nonlinearity only modifies the longitudinal wavevector component. Thus the rigorous theory of diffraction(2), previously used to study grating-induced plane wave OB(3), can be replaced by a much more simple one which works as follows : i) determine the complex longitudinal wavevector component of the linear leaky guided mode p, βpL, through the solution of the associated linear homogeneous grating problem(2), ii) use the rigorous theory of diffraction in linear optics(2) to determine the in-coupling efficiency, tp, of the incident plane wave, iii) use the coupled-mode formalism to derive the analytical expression(4) of the complex nonlinear longitudinal wavevector component βpNL.
Bistable optical systems are of considerable interest to produce high-speed optical signal-processing devices. Among them, those that use a nonlinear optical medium are attractive because they allow the lowest possible switching time. By the use of a suitable electromagnetic resonance, the bistability threshold can be lowered to a reasonnable value, for the incident power. Keeping in mind that a bistable optical system should be well adapted to optical integration in a planar geometry, one is naturally confronted to a corrugated waveguide filled with a nonlinear (Kerr) medium.
This paper is concerned with the interplay between electromagnetic resonances and the phase-matching condition in guided-wave nonlinear optics. Several examples are given: quasi phase-matched second-harmonic generation using a grating with a subwavelength periodicity, cascaded optical bistability, Kerr-type distributed couplers under pulse illumination with duration of the order of the cavity build-up time or shorter, surface plasmon-enhanced stimulated Raman scattering at a grating overcoated by a few monolayers of a Raman active medium.
Most of the intrinsic optical devices which have been studied in recent years consist of a Fabry-Perot resonator containing a nonlinear medium. The response time of such devices is determined by the transients associated with the high-Q resonator and the response time of the nonlinear medium. The fastest devices must be free of any Fabry-Perot cavity. The excitation of either surface plasmons at a nonlinear interface or guided waves in thin nonlinear films is a way to obtain an ultrafast bistable optical device. These two kinds of device have been studied.
The paper is concerned with second harmonic generation in optical resonators like prism or grating couplers when the depletion of the pump field is strong enough so that is can no longer be neglected. This means that we have to take into account the feedback between the signal field at 2(omega) circular frequency and the pump field at (omega) . Two different approaches are developed. The first one is a rigorous electromagnetic analysis based on the numerical integration of nonlinear Maxwell equations at both (omega) and 2(omega) frequencies, incorporated in a root-finding procedure to take the depletion into account. The second is based on a couple mode analysis which leads to a set of coupled mode equations governing the amplitudes of the pump and second harmonic frequency fields. These methods allow to predict that second- harmonic generation in prism or grating couplers can lead to optical bistability. Contrary to the Kerr-like bistability, this new type of bistability occurs not only at one side of the resonance maximum but on both sides provided the absolute value of the offset in the incident angle is above a threshold. The two approaches give numerical results in very good agreement. The coupled mode theory provides analytical predictions of the threshold value and brings physical insight on this new effect.
In order to study optical bistability in a non-linear distributed feedback structure (NLDFB), we have generalized previous work. A mono-mode rectangular waveguide is chosen as the basic configuration for the device, although other guides could be considered. The grating, providing the passive feedback mechanism, has a geometry that can be arbitrarily chosen. Linear absorption is taken into account and the non-linearity is taken to be of the Kerr type. The effect of these three perturbations is treated using the Coupled Mode Theory and the Slowly Varying Envelope Approximation, including the radiation modes. Expressions for the coupling constants due to the three fundamental perturbations are derived and we show that new terms arise due to the interplay between the latter. Our model shows that an extra term, causing a shift of the Braggfrequency occurs in the equations. The analytical expressions are integrated numerically with the proper boundary conditions. Overall reflection and transmission properties of the device are displayed as a function of the intensity, the period of the grating, its length and several other device parameters.