138 9 Subcritical Solitons I: Saturable Absorber

snapshots here were taken at equally spaced times, and demonstrate the constant transverse velocity of the soliton under the linear gradient.

On the other hand, changing the length of the resonator away from the precise self-imaging length creates a phase trough with a minimum at the resonator center. The soliton then moves towards the center of the phase trough, and becomes trapped at the cavity axis.

References

1.V.B. Taranenko, K. Staliunas and C.O. Weiss, Spatial soliton laser: localized structures in a laser with a saturable absorber in self-imaging resonator, Phys. Rev. A 56, 1582 (1997). 125

2.H.R. Brand and R.J. Deissler, Stable localized solutions in nonlinear optics with large dissipation, Physica A 204, 87 (1994); N.N. Rosanov, Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, Progress in Optics, vol 35, ed. by E. Wolf (North-Holland, Amsterdam, 1996). 125

3.G. Slekys, K. Staliunas and C.O. Weiss, Spatial solitons in optical photorefractive oscillators with saturable absorber, Opt. Commun. 149, 113 (1998). 125

4.K. Staliunas, M.F.H. Tarroja, G. Slekys, C.O. Weiss and L. Dambly, Analogy between photorefractive oscillators and class-A lasers, Phys. Rev. A 51, 4140 (1995). 126

5.W.J. Firth and A. Scroggie, Optical bullet holes: robust controllable localized states of a nonlinear cavity, Phys. Rev. Lett. 76, 1623 (1996). 132

6.N. Hampp, C. Brauchle and D. Oesterhelt, Biophys. J. 58, 83 (1990). 133

10 Subcritical Solitons II:

Nonlinear Resonance

In this chapter we study the formation of bright solitons in an optical system where amplitude bistability occurs because of a nonlinear resonance mechanism. One system showing this property is a degenerate optical parametric oscillator with a detuned pump [1, 2]. The model equations for this system in the mean-field limit were presented in Chap. 3. The existence of a nonlinear resonance in a DOPO was also shown in Chap. 3, by the derivation of an order parameter equation in the limit of large pump detuning (3.24). This equation is the real version of (8.5), and therefore the general ideas presented in the previous chapter are applicable in the case of a DOPO. In particular, bright solitons can also be expected in DOPOs.

Throughout this chapter we analyse the degenerate case of an OPO only. However, since the order parameter equation for a nondegenerate OPO shows a nonlinear resonance too, the main conclusions of this chapter (about solitons) are easily extendable to the nondegenerate case [3].

10.1 Analysis of the Homogeneous State.

Nonlinear Resonance

As stated earlier, the mean-field model of a DOPO can be reduced to an order parameter equation in the form of a modified Swift–Hohenberg equation (3.24), which we rewrite here for convenience:

∂A		= pA − A3 −	1	ω1 − 2−ω0A2	2	A .	(10.1)
∂t			2

This equation describes the evolution of the order parameter A, which proportional to the signal field, close to the oscillation threshold; p = E −1 is the amount by which the pump intensity is above the threshold, and ω0 and ω1 are the pump and signal detunings, respectively.1 For nonzero ω0, (10.1) possesses a nonlinear resonance, since the frequency-selection operator (the

1Remember that the pump and signal fields appearing in the order parameter equation (10.1) are scaled with respect to their original values as defined in the mean-field model of the DOPO (3.1); the changes are given by (3.3).

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 139–146 (2003)c Springer-Verlag Berlin Heidelberg 2003

140 10 Subcritical Solitons II: Nonlinear Resonance

last term) is intensity-dependent. Corresponding plots of the spatial nonlinear resonance e ect were given in Figs. 8.5 and 8.6.

In order to find the necessary conditions for the existence of solitons in a DOPO, we proceed as in the laser case considered in the previous chapter. First, we analyze the properties of the homogeneous nontrivial solution, which for (10.1) is given by

A±2 = ω2				1 + ω0ω1		1 2ω0ω1 + 2pω02	.	(10.2)
	1
			−		± −
0

Depending on the values of the detunings, the solution (10.2) can appear via a supercritical bifurcation (when ω0ω1 < 1) or via a subcritical one (when ω0ω1 > 1). In the latter case, the system shows bistability between (10.2) and the trivial solution A = 0 for pump values in the range

	ω0ω1 − 1/2	≤	p	≤	ω12	,	(10.3)
	ω02				2

as follows from (10.2).

The lower branch of (10.2) (the solution with the minus sign) is unstable, as usual. The stability of the upper branch against space-dependent perturbations can be analyzed by substituting A = A+ +δA exp(λt+ik ·r) in (10.2), and linearizing in the perturbations δA. The growth rate of a perturbation is governed by the real part of the eigenvalue λ, given by

	1		2			2		A2			2			2 2
								+				+ 3ω1
λ(k) = p −		k		− ω1			−		6 − 2ω0	2k				+ 5ω0A+	, (10.4)
	2							2

where k2 = kx2 + ky2. Note that the following analysis is valid for either one or two transverse dimensions, owing to the rotational symmetry of the problem.

From (10.4) it follows that a perturbation can grow (λ can be positive), and develop into a pattern only when ω1 > 0, which, together with the bistability condition ω0ω1 > 1, requires that the pump detuning must be positive.

The growth rate (10.4) is maximal at a wavenumber kmax, which, as found by setting ∂λ/∂k = 0 in (10.4), is

k2

= ω

1

+ 2ω

A2

,

(10.5)

max

0

+

or, using (10.2),

ω .

kmax2

= ω1

+ 2

1 + 1

2ω0ω1

+ 2pω02

(10.6)

1

−

−

0

This corresponds to the characteristic spatial-modulation wavenumber of the pattern. It is clear from (10.5) that, for ω0 = 0, the modulation wavenumber depends on the intensity of the solution, indicating the nonlinear resonance mentioned above.