11 Phase Domains and Phase Solitons

11.1 Patterns in Systems

with a Real-Valued Order Parameter

The radiation emitted by lasers and other laser-like nonlinear optical systems, such as nondegenerate optical parametric oscillators and photorefractive oscillators, has a free phase: above the generation threshold the field intensity is fixed, but the phase can take an arbitrary value. The generation threshold in laser-like systems is usually characterized by a supercritical Hopf bifurcation (Fig. 8.1a). As a consequence, the corresponding order parameter equation is the complex Ginzburg–Landau or the complex Swift–Hohenberg equation (or a generalization of one of those equations) as discussed in Chaps. 2 and 3. In Chaps. 8–10 we have seen that for some kinds of systems (e.g. in the presence of an intracavity saturable absorber or with an intracavity focusing/defocusing material), the bifurcation from the nonlasing to the lasing state can also be subcritical (Fig. 8.1b). Owing to this subcriticality, or equivalently owing to the amplitude bistability, switching waves between bistable states, amplitude domains, and spatial solitons in the form of amplitude domains of minimum size are possible.

This chapter deals with a di erent class of systems in nonlinear optics, those characterized by a real-valued order parameter. Such systems display not a subcritical or supercritical Hopf bifurcation, but a pitchfork one at the generation threshold (Fig. 8.1c). Typical examples of systems with a realvalued order parameter are degenerate optical parametric oscillator and a degenerate four-wave mixer (DFWM). The radiation in such systems prefers two values for the phase, di ering by π and associated with the two branches of the pitchfork bifurcation. Consequently, patterns associated with a realvalued order parameter such as stripes, hexagons and phase domains are favored, while laser-like patterns such as tilted waves, optical vortices and vortex lattices, of the kind studied in Chaps. 4–6, are suppressed in such systems with phase selection properties.

We now analyze patterns analogous to the amplitude domains discussed in Chap. 8, namely phase domains, and their limiting case, phase solitons [1, 2].

K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 147–167 (2003)c Springer-Verlag Berlin Heidelberg 2003

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

∂A0

(1 + iω0) A0

¯

2

2

(11.1a)

−

+ E

−

A1

+ ia0

A0

∂A1

∂t

= γ0

,

= γ1

− (1 + iω1) A1

+ A0A1 + ia1 2A1

.

(11.1b)

∂t

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

A1 = A exp(iϕ) for the subharmonic field. We obtain
A2 = −1 + ω0ω1 +			E2 − (ω0 + ω1)2	,	(11.2a)
sin (2ϕ) = −	ω0 + ω1	,			(11.2b)
	E

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

written as
	∂A		= A − A3 −	∆ + 2	2	A ,	(11.3)
	∂t

11.2 Phase Domains

149

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

detuning ∆, the RSH equation has two physically equivalent homogeneous
√
	2
solutions with equal amplitude A =	1 − ∆ , but with di erent phases ϕ =

(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.