We consider a nonlinear, passive optical system contained in an appropriate cavity, and driven by a coherent, plane-wave, stationary beam. Under suitable conditions, diff'raction gives rise to an instability which leads to the emergence of a stationary spatial dissipative structure in the transverse profile of the transmitted beam.A large variety of unstable phenomena have been reported in optics which lead to the appearance of organized behavior in time or both in time and in space. For example, it is well known that some optical systems, when subjected to stationary control parameters, may exhibit a pulsed, an oscillatory, or a chaotic output; it has been found also in optical bistability that spatial patterns of transverse and longitudinal type may occur in the switching process from the lower to the upper branch of the hysteresis curve. To our knowledge, however, the possibility of soft-mode symmetry-breaking instabilities leading to the spontaneous formation of stationary spatial patterns (dissipative structures) in an initially uniform system has never been pointed out in the field of optics. Such instabilities have drawn considerable interest in chemistry and in developmental biologỹ here they are commonly known as Turing instabilities.In these fields they arise generally from the coupling between nonlinear chemical reactions and diAusion. We show here on a simple optical model that analogous phenomena may arise from the coupling between light dispersion and diAraction in an appropriate optical cavity.We call z the longitudinal coordinate and x, y the transverse coordinates. We consider a cavity formed by four mirrors, two orthogonal to the axis z with a distance L and transmission coeScient T «1, and two orthogonal to the axis x with a distance b and 100% reflectivity.The cavity is filled with a medium with a nonlinear refractive index. A coherent, stationary, plane-wave field EI is longitudinally injected into the cavity. We assume that both the input and the internal cavity field are linearly polarized in the y direction; hence, because of the transversality condition, the internal field is independent of y. We assume that it has the structure E (x )cos(K, z )exp( -i toot ) +c.c. , where coo is the frequency of the input field Et and K, =trn, /L, with n, being a positive integer. The field transmitted by the system is proportional to the normalized envelope function E(x), which obeys the evolution equation BE . 2 . BE = -E+Et+iriE(~E~-0)+la .(1)The variable E* obeys the complex-conjugate equation.EI is taken real and positive for definiteness. The in-The parameter a is defined as a =1/2trT9, where 7 '=b /XL is the Fresnel number and k is the wavelength. The quantity g is defined as +1 or -1 in the case of self-focusing or self-defocusing nonlinearity, respectively, and gO is the detuning parameter. This model can be derived from the Maxwell-Bloch equations for a two-level system by introduction of the mean-field limit T«1, which reduces the dynamics to the single longitudinal mode n" the purely dispersi...

The thermodynamic theory of symmetry breaking instabilities in dissipative systems is presented. Several kinetic schemes which lead to an unstable behavior are indicated. The role of diffusion is studied in a more detailed way. Moreover we devote some attention to the problem of occurrence of time order in dissipative systems. It is concluded that there exists now a firm theoretical basis for the understanding of chemical dissipative structures. It may therefore be stated that a theoretical basis also exists for the understanding of structural and functional order in chemical open systems.

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%e study numerically a Swift-Hohenberg equation describing, in the weak dispersion limit, nascent optical bistability with transverse efFects. We predict that stable localized structures, and organized clusters of them, may form in the transverse plane. These structures consist of either kinks or dips. The number and spatial distribution of these localized structures are determined by the initia1 conditions while their peak (bottom) intensity remains essentia1ly constant for fixed values of the system's parameters.PACS numbers: 42.65.Pc, 42.60.Mi At the onset of optical bistability, there is a critical point where the output versus input characteristics have an infinite slope. The vicinity of this critical point is characterized by critical slowing down [1]. This implies that the dynamics of the system is dominated by a characteristic decay time which is of geometrical origin. It is inversely proportional to the deviation from the critical point and diverges at criticality. Thus in the vicinity of the critical point, all atomic and cavity decay times are associated with fast decays. Let (X, Y, C) be the deviations of the cavity field, of the injected field, and of the cooperativity parameter with respect to the values of these quantities at the critical point:Swift-Hohenberg equation [3], though some of them have been reported for other (nonvariational) models studied in chemistry and hydrodynamics [4,5], and for the cornplex Ginzburg-Landau equation [6 -9]; see [10], for a review on this topic.The situation which interests us requires that 6 ) 0 (or equivalently that a, )~, ) a,). Notably, in that case, the transverse Laplacian term in (2) is destabilizing and allows for the formation of stationary, spatially periodic patterns characterized by an intrinsic wavelength solely determined by dynamical parameters and not by the system's physical dimensions or geometrical constraints (Turing instability, [ll]). Using the t) expansion, based on the distance from the Turing bifurcation point as the smallness parameter [12], we have analytically determined the variety and the stability properties of the patterns which are solutions of Eq. (2) in the weakly nonlinear regime where the Turing bifurcation is supercritical [13]. This analysis restricted the values of the cooperativity parameter to the range X, = v3(1+id), Y, = 3V3(1+ 6'), C, = 4(1+ 6').(1)In these expressions, 6-: (u, -u, )/p~= -8 = -(~, -a, )/r, where~(~"v, ) is the atomic (external, cavity) frequency while p~a nd K are the atomic polarization and cavity decay rates. It has been demonstrated recently [2] that in the double limit of weak dispersion ([ 6 [(( 1) and nascent bistability ([ C [(( 1), the spatiotemporal evolution of the electric field X obeys an equation of the Swift-Hohenberg type: in which, furthermore, the homogeneous steady states necessarily are monostable, whatever the value of the input field y is, We proved that under those conditions, the only stable patterns forming in bidimensional transverse systems are those which either have th...