Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides

Th. Peschel, Ulf Peschel, F. Lederer

Published Dec 1, 1994• 19 citations
Publisher

Abstract

By using a modal approach, the distributed coupling of two inhomogeneous, transient waves with opposite angles of incidence into nonlinear waveguides is studied. We show that cross modulation and the coherent interaction via a dynamic grating, nonlinearily induced by the excited counterpropagating leaky waves, provide a longitudinal feedback. This feedback is the necessary prerequisite for bistability to occur. We show that bistability may even appear when one incident beam is weak. Consequently, modulation of the strong signal beam by a weak control pulse can be achieved. Furthermore, we demonstrate that the symmetry of the input signal can be broken beyond a certain intensity and that set-reset flip-flop operation with finite beams is feasible. We demonstrate that the mechanism responsible for this flip-flop operation differs considerably from that acting in the plane-wave case. The optical nonlinearities may be caused by either virtual or real carrier excitation, which models the behavior of semiconductors as well as nonlinear polymers or organic dyes well below or close to a resonance, respectively. For real carrier excitation the interplay of carrier relaxation, diffusion, and the propagation of the excited fields is taken into account.

DOI
10.1103/physreva.50.5153

Related papers

Grating Couplers for Kerr-Type Nonlinear Waveguides: A Simplified Theory

R. Reinisch, P. Vincent, M. Nevière +1 more

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.

Nonlinear Guided Waves and Their Applications•Jan 1, 1991•0 citations

Optical bistability in a multiple-quantum-well structure with Fabry-Pérot and distributed-feedback resonators

Gian Paolo Bava, F. Castelli, Pierluigi Debernardi +1 more

The system we analyze has the structure of a planar optical waveguide containing a multiple quantum well, with the planes of the wells parallel to the direction of propagation. We consider the two different configurations of a Fabry-P\'erot cavity and distributed-feedback device. The optical nonlinearities are described by means of the first-principles theory developed by Haug, Koch, Schmitt-Rink, and co-workers, generalized to include the finite width of the wells. The dynamics of the system is described by the equations that govern the evolution of the two counterpropagating fields, coupled with the carrier-density equation, and accompanied by the appropriate boundary conditions. The steady-state behavior is analyzed as a function of the control parameters of the system for the case of a GaAs/${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As/AlAs structure. The system develops bistability even when the input field is in resonance with the excitonic peak, but the phenomenon becomes more pronounced in the detuned configuration. The hysteresis cycles obtained in the case of distributed-feedback device turn out to be competitive with those displayed by the Fabry-P\'erot system.

Physical Review A•Apr 1, 1992•9 citations

The effect of excited leaky waves on the transient non-linear optical response of planar semiconductor resonators

F. Lederer, Thomas Peschel, Ulf Peschel

The optically non-linear response of arbitrary planar semiconductor resonators (microresonators) is studied theoretically. We use a versatile approach that takes into account the finite beam width as well as pulse length of the incident light, and incorporates the non-linearity consistently. The intensity-dependent non-linearity of semiconductors is described by appropriate phenomenological models. We show that the propagation characteristic of the leaky waves excited controls the response behaviour of the resonators. Conditions for switching and optical bistability are identified. The effect of various parameters on the response is studied numerically.

Pure and Applied Optics Journal of the European Optical Society Part A•Nov 1, 1993•8 citations

Guided-wave optical bistability in indium antimonide thin films

Jeffrey E. Ehrlich, Gaetano Assanto, G. I. Stegeman +1 more

Various grating configurations used to obtain optical bistability in a guided-wave format are discussed and experimentally demonstrated using indium antimonide planar waveguides. The theory of nonlinear diffusive distributed input grating coupling, is presented. The enhancement afforded by additional feedback mechanisms is verified, both with an externally coupled counterpropagating beam and with an integrated optics Bragg reflector.>

IEEE Journal of Quantum Electronics•Mar 1, 1991•11 citations

Nonlinear Prism Coupler with Diffusion

Guy Vitrant, R. Reinisch, Jean-Claude Paumier +2 more

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.

Nonlinear Guided Waves and Their Applications•Jan 1, 1989•0 citations

Optical bistability with the waveguide mode: the case of a finite-width incident beam

Victor J. Montemayor, Robert T. Deck

We develop an approximate analytical theory for the reflection of a finite-width square-wave light beam at a nonlinear interface and make use of the results to assess the effect of the finiteness of the incident beam on the bistability of the signal reflected from a triple boundary interface under conditions for excitation of a waveguide mode. The nonlinearity of the interface is assumed to result from a substrate medium that has a Kerr-type quadratic dependence on the local electric-field amplitude. Our results indicate that the finiteness of the width of the incident beam substantially reduces the magnitude of the reflectance jump at the bistable switching point in comparison with that predicted by an exact plane-wave theory. Nevertheless, we conclude that a bistable reflectivity should still exist for the geometry that we consider at the switching intensity predicted by the plane-wave theory.

Journal of the Optical Society of America B•Sep 1, 1986•20 citations
By using a modal approach, the distributed coupling of two inhomogeneous, transient waves with opposite angles of incidence into nonlinear waveguides is studied. We show that cross modulation and the coherent interaction via a dynamic grating, nonlinearily induced by the excited counterpropagating leaky waves, provide a longitudinal feedback. This feedback is the necessary prerequisite for bistability to occur. We show that bistability may even appear when one incident beam is weak. Consequently, modulation of the strong signal beam by a weak control pulse can be achieved. Furthermore, we demonstrate that the symmetry of the input signal can be broken beyond a certain intensity and that set-reset flip-flop operation with finite beams is feasible. We demonstrate that the mechanism responsible for this flip-flop operation differs considerably from that acting in the plane-wave case. The optical nonlinearities may be caused by either virtual or real carrier excitation, which models the behavior of semiconductors as well as nonlinear polymers or organic dyes well below or close to a resonance, respectively. For real carrier excitation the interplay of carrier relaxation, diffusion, and the propagation of the excited fields is taken into account.
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.
The system we analyze has the structure of a planar optical waveguide containing a multiple quantum well, with the planes of the wells parallel to the direction of propagation. We consider the two different configurations of a Fabry-P\'erot cavity and distributed-feedback device. The optical nonlinearities are described by means of the first-principles theory developed by Haug, Koch, Schmitt-Rink, and co-workers, generalized to include the finite width of the wells. The dynamics of the system is described by the equations that govern the evolution of the two counterpropagating fields, coupled with the carrier-density equation, and accompanied by the appropriate boundary conditions. The steady-state behavior is analyzed as a function of the control parameters of the system for the case of a GaAs/${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Gax{\mathrm{Ga}}_{\mathit{x}}Gax​As/AlAs structure. The system develops bistability even when the input field is in resonance with the excitonic peak, but the phenomenon becomes more pronounced in the detuned configuration. The hysteresis cycles obtained in the case of distributed-feedback device turn out to be competitive with those displayed by the Fabry-P\'erot system.
The optically non-linear response of arbitrary planar semiconductor resonators (microresonators) is studied theoretically. We use a versatile approach that takes into account the finite beam width as well as pulse length of the incident light, and incorporates the non-linearity consistently. The intensity-dependent non-linearity of semiconductors is described by appropriate phenomenological models. We show that the propagation characteristic of the leaky waves excited controls the response behaviour of the resonators. Conditions for switching and optical bistability are identified. The effect of various parameters on the response is studied numerically.
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 develop an approximate analytical theory for the reflection of a finite-width square-wave light beam at a nonlinear interface and make use of the results to assess the effect of the finiteness of the incident beam on the bistability of the signal reflected from a triple boundary interface under conditions for excitation of a waveguide mode. The nonlinearity of the interface is assumed to result from a substrate medium that has a Kerr-type quadratic dependence on the local electric-field amplitude. Our results indicate that the finiteness of the width of the incident beam substantially reduces the magnitude of the reflectance jump at the bistable switching point in comparison with that predicted by an exact plane-wave theory. Nevertheless, we conclude that a bistable reflectivity should still exist for the geometry that we consider at the switching intensity predicted by the plane-wave theory.