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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Optical Bistability In Integrated Optics: An Extended Model For The Non-Linear Distributed Feedback Device

Annemie Empsten, Irina P. Veretennicoff

Published Dec 22, 1989• 2 citations

Abstract

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.

DOI: 10.1117/12.961394

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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.

The influence of the different parameters on the optical bistable behavior of a distributed feedback structure with an intensity-dependent refractive index is studied. Previous work is generalized toward two-dimensional waveguides, showing absorption. A general method is proposed to calculate the coupling coefficients up to a desired accuracy. This method is used to compare the feedback efficiency of sine, rectangular, and trapezoidal grating profiles. The numerical integration routine that was developed allows for the study of combinations of different nonlinear distributed feedback (NLDFB) structures. This is used to do a first investigation on the behavior of NLDFB devices with an aperiodic or nonperfect grating profile.>

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.

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.

In the last chapter, we examined the eﬀect of cubic nonlinearity on selfrefraction of an optical beam. We have also seen that the presence of a quadratic nonlinearity can also contribute to change in the proﬁle of a Gaussian beam in a crystal. In this chapter we will see how the eﬀect of nonlinearity, alongwith the proper feedback, can give rise to optical bistability and hysteresis. This is not unexpected as hysteresis and bistability are also observed in nonlinear electronic circuits with feedback, such as in the Schmitt trigger, as well as in hybrid optical devices, such as an acoustooptic device with feedback (Banerjee and Poon, 1991). In what follows, we will examine two diﬀerent types of optically bistable devices based on optical nonlinearities. The ﬁrst is a nonlinear ring resonator comprising a two-level gain medium as the nonlinear element. We will treat both absorptive and dispersive bistability. Absorptive bistability is the case when the incident optical frequency is close to or equal to the transition frequency of the atoms from one level to another. In this case, the absorption coeﬃcient becomes a nonlinear function of the incident intensity. On the other hand, if the frequencies are far apart, the medium behaves like a Kerr-type material and the system exhibits what is called dispersive bistability. In this case the material can be modeled by an eﬀective n2 or v (3), meaning that the refractive index becomes a nonlinear function of the optical intensity. The second conﬁgurationwewill examine is a linear-nonlinear interface. In this case, the interface provides the feedback due to refractive index mismatch. In fact, the reﬂection coeﬃcient (as well as the transmission coeﬃcient) is modiﬁed due to the nonlinearity, which, as we shall see later, is responsible for demonstrating optical switching and bistability.