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

We recall that the coe cient G12 vanishes for phase-insensitive (helical) modes (6.3), and is nonzero for phase-sensitive (flower) modes (6.12).

7.4.2 Phase-Insensitive Modes

In a class A laser, the frequencies of phase-insensitive modes are not a ected by the nonlinear coupling. The vortices rotate around the optical axis with a frequency exactly equal to the di erence of the eigenfrequencies of the two corresponding modes. In the degenerate case the vortices stop rotating. We show below that for a class B laser, the vortices behave di erently.

We look for solutions of (7.17) in the form fi(t) = ni exp (iΩit). The coe cients of the population inversion become

where ∆ω = ωi − ωj is the mode degeneracy here 2, or, in other words, the di erence of the eigenfrequencies of the modes.

Fig. 7.8. The mode beat frequency ∆Ω versus the mode frequency detuning ∆ω for di erent values of γ. The mode-coupling coe cients correspond to the beating of the TEM01 and TEM01 modes (a circling vortex)

A family of curves of ∆Ω versus ∆ω for di erent values of γ is plotted in Fig. 7.8. The curves clearly show the mode-pushing phenomenon in the case of a class B laser. The pushing is weaker in the intermediate case between a class A and a class B laser (γ ≈ 1) and grows with decreasing γ. When

2 Do not confuse this ∆ω with ∆ω = ω − a 2 used previously in this chapter.

8 Domains and Spatial Solitons

This chapter is devoted to the description of the general properties of a particular class of solutions of extended nonlinear systems, namely spatial solitons, or localized structures. In contrast to extended patterns, such as rolls or hexagons, spatial solitons are solutions where the field is inhomogeneous only in a localized region of the space and is homogeneous in the rest of the space.

The stability of spatial solitons is due to a balance between linear and nonlinear e ects. In transverse nonlinear optics, these balancing e ects are usually the spreading caused by di raction and the compression (self-focusing) caused by a focusing nonlinearity.

Spatial solitons have been theoretically predicted for a variety of nonlinear optical cavities [1]. In this chapter we present some general concepts concerning spatial domains and solitons, and the next three chapters are devoted to the implementation of these ideas in concrete optical systems.

The study of optical solitons is relevant not only from a fundamental viewpoint but also because of their potential applications in information processing technology. Such practical applications will be discussed in Chaps. 9 and 11.

8.1 Subcritical Versus Supercritical Systems

The existence of spatial solitons is closely related to the character of the bifurcation from the nonlasing (trivial) to the lasing regime. It is useful to review here some basic concepts of bifurcation theory [2].

A bifurcation is supercritical if the transition from one solution to another, obtained by varying the control parameter (the pump intensity in the optical case), is continuous, and the solutions connect at the bifurcation point. Equivalently, an infinitesimal variation of the control parameter leads to an infinitesimal change in the amplitude of the solution in the case of a supercritical bifurcation. Examples of a supercritical bifurcation are shown in Figs. 8.1a,c. A supercritical bifurcation corresponds to a phase transition of type II. In contrast, a bifurcation is subcritical if, at the bifurcation point, the amplitudes of the solutions di er by a finite quantity (i.e. they are disconnected). In this case, both solutions may coexist below the bifurcation

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 117–124 (2003)c Springer-Verlag Berlin Heidelberg 2003

118 8 Domains and Spatial Solitons

A

a)

p

A

Domain Soliton

b)

A

Domain Soliton

c)

Fig. 8.1. Di erent types of bifurcation in nonlinear systems. (a) Supercritical Hopf bifurcation, (b) subcritical Hopf bifurcation, (c) supercritical pitchfork bifurcation. A is the amplitude of the solution, and is p the pump (criticality) parameter

point, the system showing bistability or hysteresis (Fig. 8.1b, center). The corresponding phase transition is of type I.

Another possible classification of bifurcations takes into account the symmetry of the phase of the emerging solution. When the phase of the solution is invariant (not fixed by the system), a Hopf bifurcation occurs, as shown in the phase diagrams at the left in Figs. 8.1a,b. If two opposite values of the phase are preferred, a pitchfork (static) bifurcation occurs instead, and a real-valued order parameter is obtained.

As discussed in the following sections, some kind of bistability is always needed for the existence of domains and of spatial solitons. The order parameter equations derived in Chaps. 2 and 3 (the Ginzburg–Landau and Swift– Hohenberg equations, either real or complex), which are representative of most nonlinear optical systems, possess a primary bifurcation of supercritical type.

8.2 Mechanisms Allowing Soliton Formation

There are two basic mechanisms that may cause subcriticality, and consequently may lead to soliton formation. Both are related to the existence of