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Journal of Electromagnetic Waves and Applications

Nonlinear Prism Coupler: Approximate Theory and Comparison With Rigorous Analysis

Sergey A. Darmanyan, E. Popov, M. Nevière

Published Jan 1, 2000• 2 citations
Publisher

Abstract

A prism coupler filled with nonlinear Kerr-like material is studied theoretically. A simple, approximate analysis is developed to predict its behavior in both linear and nonlinear optics. In a wide range of incident amplitude magnitude, optical bistability and multistability are found. As far as field map not undergo drastic change, the simpler approximate theory is able to quantitatively analyze both prism couplers and waveguide in nonlinear optics and to study such effects as nonlinear phase shift, existence of nonlinear asymmetric modes in symmetric environments, bi-and multistability. The predictions of the analytical theory are compared with the results obtained via numerical integration of Maxwell equations and they agree quite well.

DOI
10.1163/156939300x00905

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A prism coupler filled with nonlinear Kerr-like material is studied theoretically. A simple, approximate analysis is developed to predict its behavior in both linear and nonlinear optics. In a wide range of incident amplitude magnitude, optical bistability and multistability are found. As far as field map not undergo drastic change, the simpler approximate theory is able to quantitatively analyze both prism couplers and waveguide in nonlinear optics and to study such effects as nonlinear phase shift, existence of nonlinear asymmetric modes in symmetric environments, bi-and multistability. The predictions of the analytical theory are compared with the results obtained via numerical integration of Maxwell equations and they agree quite well.
Bistability induced in a second-harmonic signal generated in the presence of the Kerr effect in a prism coupler is investigated. Using a rigorous solution provided by Maxwell's equations, we analyze this behavior, taking into account third-order nonlinearity (Kerr nonlinearity). Comparison with a Kerr-based bistable device shows the interest of quadratic nonlinearity for integrated optics. We show that the bistable behavior depends on the sign of third-order nonlinearity. The numerical simulations are presented for both an incident plane wave and a spatially limited incident beam.
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.
A perturbation method for studying the bistable behavior of a prism or grating coupler filled with a Kerr nonlinear medium is developed. A field representation limited to its first space harmonic already gives a satisfactory agreement with the much more complicated rigorous differential method. A better accuracy can be obtained by the use of the two first space harmonics. We then obtain an almost perfect agreement with rigorous theory, always with much lower computation times. All the relevant parameters useful to characterize the bistable behavior are given by this perturbation method.
The salient question which arises when light is coupled via a grating or a prism into a waveguide exhibiting an intensity-dependent refractive index is whether switching occurs with or without bistability under appropriate detuning conditions. There are two prevailing theories. In the first1, a plane wave input is assumed leading to a uniform optically induced index change in the coupling region. In the second2, a finite width incident beam is assumed and the growth of the guided field under the coupler leads to an index change proportional to the local guided wave intensity. The first approach leads to optical bistability under certain detuning conditions, whereas the second results in switching only. Although it is known that plane wave analyses are inadequate when dealing with linear-nonlinear interfaces, bistability has been observed experimentally with nonlinear prism couplers3. Fundamentally the finite beam theoretical approaches are unable to predict bistability because no longitudinal feedback is included. Recently we showed that when non-locality is introduced into the optically induced refractive index change, longitudinal feedback occurs and bistability is predicted for the growing mode case4. However, the model used for the non-locality was artificial and resulted only in qualitative predictions. In this paper the nonlocality is modelled by the same equation used to describe heat transfer and carrier diffusion, leading to a quantitative estimate of the minimum diffusion length required for bistability.
We deal with second-harmonic generation in chi((2)) nonlinear-optical resonators such as prism or grating couplers. The theoretical study is performed within the framework of a recently developed coupled-mode analysis leading to a set of equations governing the amplitudes of pump and second-harmonic frequency fields. We predict analytically that second-harmonic generation in prism or grating couplers may lead to optical bistability. We believe this to be the first demonstration of such an effect in chi((2)) optical resonators.