Michael Schwab

Transverse-pattern formation in photorefractive optics

Introduction.- Light propagation in nonlinear optical media.- The photorefractive nonlinearity.- Spatial photorefractive solitons in anisotropic, saturable media.- Interaction of spatial solitons in saturable and photorefractive media.- Self-organized pattern formation in single-feedback systems.- Selection, manipulation and control of self-organized patterns.- Transverse patterns in active photorefractive oscillators.

Pattern dynamics and competition in a photorefractive feedback system

We investigate the temporal dynamics of transverse optical patterns spontaneously formed in a photorefractive single-feedback system with a virtual feedback mirror. The linear stability analysis for the system is reviewed and extended to the region of larger propagation lengths. The stationary patterns obtained experimentally are classified as a function of feedback reflectivity and feedback mirror position. Inserting masks into the feedback path permits pattern selection and control by Fourier filtering. When an asymmetry that is due to noncollinear pump beams is introduced, the otherwise stationary hexagons show several complex but periodic rotationlike motions. Furthermore, the competition of hexagonal and square patterns can be observed by the appropriate choice of feedback mirror position and coupling strength. The origin of this behavior is discussed. The temporal evolution of the patterns is illustrated by a method based on unfolding the angular distribution of the spots in the far field.

Transverse modulational instability in counterpropagating two-wave mixing with frequency-detuned pump beams

We report theoretical and experimental evidence for transverse modulational instability of two counterpropagating beams in a photorefractive medium with no external feedback. A frequency detuning is applied to one of the beams in order to drive the system to instability. We perform a linear-stability analysis that allows for detuning of the counterpropagating pump beams in addition to an additional frequency detuning of the generated sidebands relative to the main beams. The threshold condition for the general case of a complex photorefractive coupling constant is found, and instability is predicted for diffusion-dominated, drift-dominated, and mixed charge transport. We show that for the specific case of diffusion-dominated charge transport, transverse instability is always accompanied by a frequency shift of the sidebands. For frequency-degenerate pump beams the instability threshold is reached at a coupling-constant times interaction-length product of γl=5.25i. The threshold is lowered (raised) for small positive (negative) frequency shifts of one of the pump beams. The theoretical predictions were verified experimentally with a photorefractive crystal of KNbO3. A modulational instability resulting in a spatially periodic roll pattern was observed for a certain range of positive frequency detunings. Measurements of the transverse scale of the structures and the relative sideband intensities were in agreement with the theoretical analysis.

Fourier control of pattern formation in an interferometric feedback configuration

We present a control method for the minimally invasive manipulation of pattern states occurring in feedback systems with an optical nonlinearity. An interferometric feedback configuration is used to control spontaneously formed patterns by the method of Fourier filtering. This method is minimally invasive since the pattern forming feedback arm remains unchanged, while the energy removed by the filter or the control signal added to the system by the second control feedback arm tends to a small level when the desired pattern is obtained. Here, we present experimental results for a photorefractive two-arm feedback system, including switching from hexagons to rolls or squares by positive (in-phase) control and manipulation of the hexagon orientation by negative (out-of-phase) control. A comparison with results of a numerical simulation based on a thin saturable Kerr-slice model is performed.

Origin and Control of Dynamics of HexagonalPatterns in a Photorefractive Feedback System

Abstract We investigate the origin of dynamics of hexagonal pattern formation in aphotorefractive single feedback system. By introducing an asymmetry, we induced differentcomplex motion behaviours. A visualization method is presented and means to control thedynamics in the case of fixed external parameters are discussed.

STABILIZATION, MANIPULATION AND CONTROL OF TRANSVERSE OPTICAL PATTERNS IN A PHOTOREFRACTIVE FEEDBACK SYSTEM

We present an experimental realization of an almost non-invasive stabilization and manipulation method of coexisting and unstable states of pattern forming systems. In a photorefractive single feedback system, a control path is used to realize amplitude and phase-sensitive Fourier-plane filtering, utilizing only a few per cent of the systems intensity. By that means, we were able to stabilize desired but not predominantly excited patterns in parameter space regions where several patterns are present as coexisting or underlying solutions. Changing the phase of the control signal allows one to switch between different pattern states.

Light Propagation in Nonlinear Optical Media

Chapter 2 on light propagation in nonlinear optical media describes the origin of self-focusing of beams in nonlinear optical media in general. It shows how these focusing effects can be exploited to create optical spatial solitons. Starting with one-dimensional transverse soliton phenomena the general features of soliton formation which lead to a description by the nonlinear Schrodinger equation are shown. On the basis of these introductory investigations the existing conditions for two-dimensional spatial optical solitons are considered which lead to novel solitary structures with exciting interaction features.

Spatial Photorefractive Solitons

The photorefractive nonlinearity is described in Chap. 3 beginning with the origins of the space charge field arising in these materials from their interaction with light. The band transport model equations are used to formulate the basic equations for the photorefractive nonlinearity. The complexity is gradually increased, starting with one beam interacting with the nonlinear material and then taking a second beam into account to show the beamcoupling properties of photorefractive crystals. An important feature of a diffusion-dominated photorefractive material is the possibility of performing an energy exchange between interacting beams in a wave-mixing process. The two-beam coupling properties are thoroughly discussed for the special case of a counterpropagating geometry, which is the configuration of choice for the investigations of pattern formation effects described in Chaps. 8 to 10. Special care has been taken to introduce the main variables and equations that represent the starting point for subsequent considerations.

Manipulation and Control of Self-Organized Patterns by Spatio-Temporal Techniques

In Chapter 9, multiple patterns and complex pattern competition, experimental results which reveal parameter regions of simultaneously stable patterns and of pattern competition are presented. The most surprising feature of a photorefractive single-feedback system is the existence of multistability which provides a wealth of new patterns far beyond the threshold. In order to understand the complex dynamics of pattern interaction, detailed investigations of the uncontrolled system are necessary, and provide an appropriate and essential starting point for the implementation of manipulation and control schemes for active pattern selection.

Multiple Patterns and Complex Pattern Competition

Chapter 8, on self-organized pattern formation in single-feedback systems, describes how the complexity of the material-light interaction is enhanced by allowing the system to feed the beam back into the nonlinear material. A single-mirror feedback scheme with a photorefractive nonlinearity constitutes an excellent model system for studying pattern formation effects. An extensive description of the main properties of the free-running system is the topic of this chapter.