Generalized Synchronization of Spatiotemporal Chaos in a Liquid Crystal Spatial Light Modulator

Patterns in spatiotemporal systems are evident in na-ture and have been investigated extensively for manyyears [1]. Experiments and numerical models of patternformation have revealed and explained different aspectsof these intriguing phenomena. A most interesting dy-namical aspect of pattern formation that is only begin-ning to be explored is the question of spatiotemporalsynchronization, particularly for chaotic systems. In thepast decade, a great variety of temporal synchronizationphenomena have been reported for chaotic systems, in-cluding identical, phase, lag, and generalized synchroni-zation [2]. However, it is only very recently that therehave been efforts to demonstrate identical synchroniza-tion of spatiotemporal chaotic patterns [3].Measurements of drive and response signals in ex-tended spatiotemporal systems may have important ap-plications in methods for noninvasive testing andmonitoring of structures and materials, ranging frombuildings to nanostructures such as biomolecules. Theycould also be used for computing in optical systems andencoded communications systems, including parallelcommunications. The dynamical synchronization of spa-tial patterns could have applications to spatiotemporalcommunication [4], which provides a practical motivationfor thework presented here.The dynamics and patterns ofthe drive and response systems may differ greatly incharacter when generalized synchronization occurs andmay provide an additional element of camouflage formessage information.While identical synchronization is more easily under-stood, detected, and quantified,generalized synchroni-zation presents a much greater challenge to theexperimentalist. Theoretically, generalized synchroniza-tion may be shown to exist between the drive and re-sponse systems through the predictability [5] or theexistence of a functional relationship [6]. These ap-proaches have been implemented in numerical modelsbut are often difficult to implement in experimental mea-surements, due to the presence of noise and lack of pre-cision. The auxiliary system method has been suggestedfor detecting generalized synchronization [7] when rep-licas or duplicates of the response system are available. Inthis method, the drive system is coupled to two or moreresponse systems. If, after starting from different initialconditions, the response systems display identical syn-chronization after a period of transient dynamics depen-dent on system parameters, we can conclude that theresponse signal is generally synchronized to the drive[7]. In recent experiments, generalized synchronizationhas been observed in a one-dimensional laser system withoptoelectronic feedback using the auxiliary method.Thishas been seen when systems with many degrees of free-dom are coupled together by only a few variables or alumped variable [8].In this Letter, we demonstrate generalized synchroni-zation experimentally in a spatiotemporally chaotic two-dimensional system. An optoelectronic feedback loopusing a liquid crystal spatial light modulator (SLM) is aflexible method for generating spatiotemporal patternsand has been used in this manner over the last 20 years[9–11].These patterns range from hexagonal close packeddot structures to spatiotemporal chaos. Numerical modelsof liquid crystal SLM behavior show a variety of regularand irregular patterns that correspond well to experi-ments [10]. We use the auxiliary system method to showthat generalized synchronization occurs between a re-corded and replayed sequence of drive and responsepatterns.In our experiments, the reference light beam from alinearly polarized 633 nm helium neon laser is incidenton a Hamamatsu computer controlled, electrically ad-dressed liquid crystal SLM, as shown in the experimentalsetup of Fig. 1. After reflecting off of the phase grating,the beam propagates for approximately 2.5 m before it isincident on a Pulnix TM-72EX charge coupled device(CCD) camera. The SLM uses the videographics arrayoutput to make a gray-scale pattern on an internal liquidcrystal display (LCD). Though the liquid crystal is notpixilated, the LCD consists of an array of 640 480pixels, 480 480 of which are active and of which thecentral 244 360 are used in this experiment. The imagefrom the camera is then displayed on a computer monitorV