Ann. Phys. (Leipzig) 13, No. 7–8, 391 – 402 (2004) / DOI 10.1002/andp.200410082 Transverse pattern formation and its control in photorefractive optics C. Denz 1, ∗ , Ph. Jander 1 , M. Schwab 1, ∗∗ , O. Sandfuchs 2, ∗∗∗ , M. Beli´ c 3, ∗∗∗∗ , and F. Kaiser 2 1 Institute of Applied Physics, Westf¨ alische Wilhelms-Universit¨ at M ¨ unster, 48149 M¨ unster, Germany 2 Institute ofApplied Physics, Darmstadt University of Technology, 64289 Darmstadt, Germany 3 Optique Nonlin´ eaire Th´ eorique, Universit´ e Libre de Bruxelles, 1050 Bruxelles, Belgium Received 16 February 2004, accepted 5 April 2004 Published online 17 June 2004 Key words Pattern formation, photorefractive optics. PACS 42.65.Hw, 42.65.Sf This review is dedicated to Prof. Dr. H.-G. Purwins, Institute of Applied Physics, University of M¨ unster, who is always enthousiastically supporting nonlinear sciences. A wealth of periodic transverse patterns is observed in nonlinear optical, photorefractive single-feedback systems. It is the extension of a photorefractive medium in the direction of propagation of the laser beam that allows for the formation of stable hexagons, squares, rhombuses and dodecagons above the first bifurcation of the system. In this article we review the formation of these periodic patterns and present linear and nonlinear stability analyses that predict the type of pattern to appear.Amplitude equations for the interactions of unstable sidebands are introduced. We compare numerical and experimental results on two-dimensional transverse pattern formation.We present control methods for the invasive and non-invasive manipulation of different pattern states using a Fourier filtering technique. c  2004 WILEY-VCHVerlag GmbH & Co. KGaA, Weinheim 1 Introduction Spatiotemporal complexity of nonlinear optical systems is a new and exciting field of research in optics, that has already found way into a number of review papers [1], special issues [2], and books [3,4]. Spatiotemporal patterns arise when the translational symmetry of a homogeneous state is spontaneously broken, induced by transverse coupling due to diffraction in nonlinear optical systems, and can give rise to chaotic dynamics and/or spontaneous pattern formation. The more extended an optical system is in its transverse dimensions, the greater the wealth of different transverse patterns. Among them, periodic structures are predominant for weak nonlinearities, whereas spatial solitons, spatiotemporal dynamics as well as optical turbulence may arise for higher nonlinearities. The existence and stability of these structures is strongly affected by symmetries, both geometric ones due to the optical configuration or intrinsic symmetries of the optical interactions under the influence of material properties. In photorefractive (PR) optics transverse pattern formation appears predominantly due to two-wave mixing processes in a reflection grating geometry. Under certain circumstances each beam becomes spon- taneously unstable against the excitation of satellite beams [5], whose wave vectors belong to a cone in the Fourier space, and whose initial growth is exponential. This growth is arrested either through the depletion of pump beams or through the saturable character of the medium nonlinearity. In the nonlinear stage of ∗ Corresponding author E-mail: denz@uni-muenster.de ∗∗ Present address: Charlottenstr. 39, 70825 Korntal-M¨ unchingen, Germany ∗∗∗ Present address: Sophienstr. 1, 07743 Jena, Germany ∗∗∗∗ Permanent address: Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia c  2004 WILEY-VCHVerlag GmbH & Co. KGaA, Weinheim