Multiple-pattern stability in a photorefractive feedback system

We report on the observation of a multiple-pattern stability region in a photorefractive single-feedback system. Whereas hexagonal patterns are predominant for feedback with positive diffraction length we show that a variety of stable non-hexagonal patterns are generated for certain negative diffraction lengths. For the same values of the control parameters square, rectangular, or squeezed hexagonal patterns are found alternating in time. Besides these pure states, we found a number of different mixed-pattern states. We review the linear stability analysis for this system and show that the special shape of the threshold curves in the investigated parameter region gives a first explanation for the occurrence of a multiple-pattern region. PACS: 42.65.Sf; 42.65.Hw; 47.54.+r The spontaneous formation of periodic spatial patterns is well known for a variety of nonlinear optical materials, for example atomic vapours [1], liquid crystals (Kerr slices) [2, 3], organic films [4], or photorefractives [5], where squares and squeezed hexagons were first observed in experiment [6]. Photorefractive materials are well suited for pattern observation since their intrinsically slow dynamics offers the opportunity to perform real-time measurements and observations. Moreover, low cw powers in the range of milliwatts are required and in the case of a diffusion-dominated crystal such as KNbO3, no external voltage has to be supplied providing an all-optical pattern formation system. In all these systems, a single-feedback configuration creating two counterpropagating beams in the nonlinear optical medium gives rise to transverse modulational instabilities above a certain threshold. These instabilities generally lead to the formation of hexagonal patterns, which were first reported for a photorefractive system by Honda [5]. Following this pioneering work, various other publications offered improved insight into the stages of pattern formation in these photorefractive materials [7, 8]. A first approach to a nonlinear stability analysis [9] and studies of pattern dynamics due to angular misalignment and competition behaviour were published recently [10–12]. Our focus of interest is to investigate more complex patterns that may arise in the same configuration for a certain range of the diffraction length without changing the basic interaction geometry. Although some of these patterns were observed previously [6], the appropriate region of instability has not yet been investigated. Besides pure pattern states, such as squares, rectangles, or squeezed hexagons, we could observe a large number of different mixed states where two or more patterns coexisted. This variety of patterns leads to the question of manipulation, stabilization and control of these different pattern states which is currently of high interest [13– 16]. The aim is to get defined access to different patterns, an essential condition to make use of spontaneous pattern formation in the growing field of optical information processing [17]. To make use of the different control schemes, it is absolutely necessary to gain knowledge about the different stability (and instability) regions of the present system. Thus, our aim in this paper is to get improved insight into the stages of pattern formation in this system and the parameter regions for different pattern types. A review of the linear stability analysis is given together with new results from this analysis. We combine these results with the occurrence of multiple patterns, thus giving an explanation for our experimental observations. 1 Linear stability analysis The basic interaction geometry is depicted in Fig. 1. A plane wave of complex amplitude F is incident on a thick photorefractive medium with length l. The backward beam B is produced by reflection at a mirror at a certain position L behind the medium. Our analysis is not restricted to positive diffraction lengths since the 4 f −4 f configuration enables us experimentally to produce negative diffraction lengths which are essential for observing multiple-pattern stability. The principle function of the diffraction length L is to introduce a phase lag of the generated sidebands relative to the central beam. A diffusion-dominated medium such as KNbO3 offers beam-coupling properties which are essential for pattern formation in this configuration. In this