Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

148 11 Phase Domains and Phase Solitons

11.2 Phase Domains

As an example of a system displaying a pitchfork bifurcation we consider again the DOPO, whose mean-field model, introduced in Chap. 3, reads

The spatially homogeneous stationary solution of (11.1) can be found by elimination of the pump field A0 from (11.1), and by using the ansatz

with an additional constraint on the phase, cos(2ϕ) > 0 [3]. The stationary intensity of the pump corresponding to (11.2) is |A0|2 = 1 + ω21. This solution is exact for the mean field-model, and coincides with (10.2) when the appropriate limits are taken.

The expressions (11.2) represent two physically equivalent solutions with the same amplitude but di erent phases, ϕ1 = − arcsin [(ω0 + ω1)/2E] and ϕ2 = ϕ1 + π. In the case of zero (or su ciently small) detuning, the numerical solution of (11.1) leads asymptotically to one of the two homogeneous distributions given by (11.2) as the final state. However, in a transient stage of the evolution, when the system starts from a random field distribution, the field shows separate domains, characterized by one of the two values of the phase inside each domain.

In Fig. 11.1, the amplitude and phase distributions of the subharmonic (signal) radiation in a DOPO are shown during a transient. The field vanishes along the lines separating the two phases, which are called domain boundaries (and also dark switching waves). The stability of domain boundaries in DOPOs was first investigated in [4]. The domains here are essentially dynamic, and can move, reconnect or disappear during the nonlinear evolution. This chapter is devoted to the nonlinear dynamics of these domains.

For the sake of simplicity, and also for the sake of generality of the results, in the following the domain dynamics are studied not by solving the meanfield DOPO model (11.1), but by solving the corresponding order parameter equation. As shown in Sect. 3.3, the dynamics of a DOPO are described, close to the threshold, by the real Swift–Hohenberg equation, which can be

Fig. 11.1. Phase domains in a DOPO for small signal detuning. The intensity (left) and phase (right) distributions are shown. The calculations started from a random distribution of the optical field (with a broadband spatial spectrum). A transient stage of the evolution is shown. The parameters used were E = 2, ω0 = 0, ω1 = −0.3, γ1 = γ0 = 1, a1 = 0.0005 and a0 = a1/2. The integration was performed using periodic boundary conditions in a region of unit size

Notice that this equation is valid for zero or moderate pump detuning. Otherwise, nonlinear resonance e ects must be taken into account.

Owing to the universal character of the Swift–Hohenberg equation as a basic pattern-forming model, the results derived from (11.3) are applicable not only to DOPOs, but also to other nonlinear optical systems such as DFWMs [5] and to physical systems of di erent natures, such as systems showing Rayleigh–B´enard convection in hydrodynamics [6].

The order parameter A(r, t) is proportional to the complex amplitude of the subharmonic field, A1(r, t). The normalization of the time τ = (E − 1) γ1t scales out the pump value E in (11.3). The coe cient of the Laplace operator is equal to unity, owing to the normalization of the spatial coordinates. The parameter ∆ is proportional to the subharmonic detuning ω1 in (11.1):

∆ = −ω1/ 2 (E − 1). Note that the detuning parameter in (11.3) is signreversed with respect to the detuning in the DOPO model (11.1). Extended patterns, such as rolls and hexagons, now occur for positive ∆ in (11.3), whereas equivalent patterns occur for negative detunings in the DOPO equations.

The solutions of (11.3) are analogous to the patterns generated by the DOPO equations discussed above. For a relatively large positive detuning, the RSH equation has a spatially modulated solution in the form of stripes with

amplitude A(r) = 4/3 cos (k · r), where the resonant wavenumber |k| =

√

∆ is dependent on the detuning. For a negative or relatively small positive

(0, π), the analogue of the domains in the DOPO (Fig. 11.1). In the following sections the evolution of domains or, equivalently, the motion of the domain boundaries is studied.

150 11 Phase Domains and Phase Solitons

11.3 Dynamics of Domain Boundaries

Before investigating the dynamics of phase domains in 2D, let us analyze the case of stationary, straight boundaries, which is actually the 1D case. Straight boundaries between su ciently large domains are stationary, since two phases (two stationary solutions) corresponding to neighboring domains are always in equilibrium here.1 Domain boundaries can move either because of curvature e ects (as investigated in Sect. 11.4), or because of their mutual interaction (Sect. 11.5).

No analytic expression exists for a straight domain boundary in the case of the RSH equation. However, such boundaries can be found analytically in the case of the real Ginzburg–Landau (RGL) equation,

Equation (11.4) has a kink-form solution in 1D, corresponding to a straight domain boundary in 2D,

which represents a solution directed along the y axis.

The RSH equation in the limit of large negative detuning actually transforms into the RGL equation (11.4). Therefore, let us assume that the domain boundary solution of the RSH equation possess a form similar to the kink

√

where 1 − ∆2 is the modulus of the homogeneous solution of the RSH equation and x0 is the (unknown) half-width of the domain boundary.

11.3.1 Variational Approach

The RSH equation is a variational equation, and thus it can be also written in the gradient form ∂A/∂t = −δF/δA , with a potential F (A) given by [7]

1This is not always true in DOPOs, since in a certain parameter range the Ising– Bloch transition can be present, leading to a drift of the walls at constant velocity. However, this transition requires a complex order parameter, and thus cannot be obtained from the Swift-Hohenberg equation.