118 8 Domains and Spatial Solitons

A

a)

p

A

Domain Soliton

b)

p

space

A

Domain Soliton

c)

p

space

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

8.2 Mechanisms Allowing Soliton Formation

119

absolute bistability between two di erent extended solutions (which may or not be homogeneous). In order to show this, let us consider the complex Swift–Hohenberg equation, and analyze the properties of its simplest nontrivial solution: a traveling wave.

8.2.1 Supercritical Hopf Bifurcation

Consider the simplest form of the complex Swift–Hohenberg equation, as obtained in Chaps. 2 and 3:

	∂A			2
		= pA − A |A|2 + i a 2 − ∆ A − g	a 2 − ∆		A .	(8.1)
	∂t
Consider also a solution of (8.1) in the form of a traveling wave,
A = |A| exp(ik0x − iωt) .						(8.2)

The intensity of the traveling wave can be found from (8.1), and plotted in terms of the various parameters. Figure 8.2a shows the usual bifurcation diagram, with the intensity as a function of the criticality parameter p. Another useful representation, shown in Fig. 8.2b, is the dependence of the intensity on the squared wavenumber. It is evident that no bistability is possible here, since the trivial solution is always unstable against a traveling wave when the latter exists. Consequently, solitons cannot be stable under supercritical conditions, and subcriticality of the emerging solution is then required. In order to introduce subcriticality, an external e ect is usually added to the system, such as an intracavity saturable absorber, an intracavity focusing/defocusing material or parametric forcing. The corresponding equation modeling the dynamics is a modified Ginzburg–Landau or Swift–Hohenberg equation, with additional terms describing the subcriticality.

	A		2		A		2

0		0
	p	k02	k2

Fig. 8.2. Dependence of the intensity on the criticality parameter p and on the squared wavenumber k2, in the case of a supercritical Hopf bifurcation

120 8 Domains and Spatial Solitons

8.2.2 Subcritical Hopf Bifurcation

We consider two possible modifications of (8.1) that lead to subcritical solutions: (a) a modification of the local nonlinear terms, and (b) a modification of the nonlocal terms, containing spatial derivatives.

(a) The Quintic Complex Swift–Hohenberg Equation. In this case, a nonlinear term of fifth order is considered:

∂A∂t = pA − bA |A|2 − A |A|4 + i a 2 − ∆ A − g a 2 − ∆ 2 A . (8.3)

The traveling-wave solution behaves as depicted in Fig. 8.3.

	A		2		A		2

							0
0				p				k02	k2

Fig. 8.3. Dependence of the intensity on the criticality parameter p and on the squared wavenumber k2, in the case of a subcritical Hopf bifurcation generated by a quintic nonlinear term

The subcritical character of the bifurcation follows from the left-hand graph in Fig. 8.3 (there is a coexistence of solutions for p < 0). Di erently from the supercritical case, the traveling-wave branch is disconnected from the trivial-solution branch (see the right-hand graph in Fig. 8.3). As a consequence, the trivial solution is stable for any wavenumber, reflecting an absolute bistability (bistability is not absolute if the upper branch connects with the trivial branch, as will be discussed in the next example).

A physical system showing this behavior is a laser with a saturable absorber, whose corresponding order parameter equation is

∂A		=		pA	−	αA	− A
∂t				1 + |A|2 /Ip		1 + |A|2 /Is
			+i a 2 − ∆ A − g a 2 − ∆ 2 A .					(8.4)

The e ect of this type of nonlinearity will be analyzed in detail in the following chapter. Note that the fifth-order nonlinearity corresponds to the first terms in the Taylor expansion of the first two terms in (8.4).

In this example, the gain must be larger than the losses for stable solitons. This condition can be visualized by plotting the first and second terms in (8.4), as shown in Fig. 8.4.

8.2 Mechanisms Allowing Soliton Formation

121

loss

gain

0								A		2	Fig. 8.4. Gain and losses in a laser
											with a saturable absorber

(b) Spatial Nonlinear Resonance. Usually the resonant wavenumber, corresponding to the solution selected by the system, is constant and given by k02 = −∆/a. In some cases, however, the resonant wavenumber is intensitydependent, a phenomenon known as nonlinear resonance. The complex Swift– Hohenberg equation then takes the form

∂A

= pA

A

A

2

+ i a

2

∆

∆0

A

2 A

∂t

−

|

−

−

|

|

|

−g

a 2 − ∆ − ∆0 |A|2

2

A .

(8.5)

The properties of the traveling-wave solution of (8.5) are illustrated in Fig. 8.5: the nonlinear resonance results in a tilt of the resonance curve (a linear resonance corresponds to a symmetric parabola).

Note that, unlike the case of the quintic complex Swift–Hohenberg equation, the instability now is not absolute: there always exist a wavenumber (in fact, the band of wavenumbers between the two intersections of the resonance

A 2

0
k02	k2

A 2		A 2
0		0
k02	k2		p

Fig. 8.5. Dependence of the intensity on the criticality parameter p and on the squared wavenumber k2, in the case of a subcritical Hopf bifurcation generated by a nonlinear resonance term