12 Turing Patterns in Nonlinear Optics

12.1 The Turing Mechanism in Nonlinear Optics

A well-known transverse-pattern formation mechanism in broad-aperture lasers and other nonlinear resonators is o -resonance excitation. If the central frequency of the gain line of the laser ωA is larger than the resonator resonance frequency ωR, then the excess of frequency ∆ω = ωA −ωR causes a transverse (spatial) modulation of the laser fields, with a characteristic transverse wavenumber k obeying a dispersion relation ak2 = ∆ω, where a is the di raction coe cient of the resonator. The patterns that occur in such a way play the role of a “bridge” between the excitation and the dissipation, which occur at di erent frequencies, and these patterns enable maximum energy transfer through the system.

In all the previous chapters, patterns due to o -resonance excitation have been studied. These were patterns in lasers, photorefractive oscillators, degenerate and nondegenerate optical parametric oscillators, and four-wave mixers. For a degenerate OPO, the excitation frequency is equal to half of the pump radiation frequency ωA = ω0/2, and its mismatch from ωR leads to the same macroscopic pattern formation mechanism as in lasers. The o -resonance mechanism not only excites extended patterns (such as tilted waves, rolls, square vortex lattices and hexagons), but is also responsible for the stability of localized structures in the above systems.

The o -resonance pattern formation mechanism is essentially a geometrical one. It resembles the formation of rolls in Rayleigh–B´enard convection, where the width of the convection rolls is fixed mainly by the distance between the upper and lower plates. In optical resonators, the propagation angles of o -axis components are fixed by the resonance conditions. The spatial scale is thus fixed not by nonlinearity, but by linear geometric e ects.

The pattern formation mechanism discovered by Turing for reaction–di u- sion systems [1] has a di erent origin from the mechanism discussed above. Here, at the root, is an interplay between the di usions of two (or more) interacting components. The coupling between a strongly di using (lateral) inhibitor and a weakly di using (local) activator is responsible for the pattern formation.

The simplest (linearized) representation of such a reaction–di usion equations displaying a Turing instability is given by the model [2]

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 169–192 (2003)c Springer-Verlag Berlin Heidelberg 2003

170		12 Turing Patterns in Nonlinear Optics
	∂u1	2
		= a1u1 − b1u2 + d1 u1 ,	(12.1a)
	∂t
	∂u2	2
		= b2u1 − a2u2 + d2 u2 .	(12.1b)
	∂t

In this system u1 plays the role of the activator and u2 the role of the inhibitor, with di usion coe cients d1 and d2 respectively. The particular form of the cross-coupling matrix (where ai and bi have positive values) leads to maximum amplification of the wavenumbers obeying

2	=	1		a1	−	a2		(12.2)
|k|							,
		2		d1		d2

as follows from a stability analysis of (12.1).

Motivated by this analysis, one might ask the following question: is the Turing mechanism possible in nonlinear optics too? Let us take as an example

the equation for a class B laser from Chap. 7,

∂A

= (D

1) A + i a

2

ω A

g a

2

ω

2

A ,

(12.3a)

∂D

−

2−

−

−

∂t

= −γ D − D0 + |A| ,

(12.3b)

∂t

with an unsaturated population inversion D0 and a spatial-wavenumber selection factor g. Let us simplify (12.3) by assuming a very narrow gain line, i.e. g 1, which makes di raction negligible when compared with di usion, and zero detuning, ω = 0. Also, which is very significant here, let us assume that the population inversion also di uses, which results in adding a Laplacian to (12.3b). In this case, if we define the field di usion constant d1 = gd2, (12.3) converts to

∂A	= (D − 1)A − d1 4A ,	(12.4a)
∂t
∂D	= −γ D − D0 + |A|2 + d2 2D ,	(12.4b)
∂t

a system of two nonlinearly coupled di using components. The field di usion is governed not by the usual Laplace operator, but by the second power of the operator (sometimes called super-di usion, as mentioned earlier); however,

this makes no essential di erence compared with normal di usion.

¯

¯

Consider a perturbation of the stationary solution A = A + a, D = D +

¯

¯

√

d, where

A, D

=

±

D0 − 1

, 1 . Linearizing (12.4) with respect to the

perturbations leads to

∂a

= d

4a ,

(12.5a)

D0

−

1

−

d1

∂a

∂t

d + 2a

D0 − 1

+ d2 2d .

(12.5b)

∂t = −γ

The similarity to the Turing system (12.1) becomes more evident if we change the sign of the perturbation of the population inversion d. This results

12.2 Laser with Di using Gain

171

in the linear coupling matrix

√

0 − D0 − 1

L =	2γ√			,	(12.6)
		D0 − 1	−γ

and a diagonal di usion matrix

D =		−d1 4	0	=		−d1k4	0 .	(12.7)
		0	d2 2			0	−d2k2

The form of the linear coupling matrix, compared with (12.1), allows us to identify the optical field with the activator variable in a reaction–di usion system, and the population inversion with the inhibitor variable.

The main requirement for Turing pattern formation in a reaction–di usion system is that the inhibitor di uses faster than the activator. This requirement is often called the principle of “local activator and lateral inhibitor” (LALI). Consequently, it seems reasonable that for observation of similar patterns in nonlinear optics, one must require that the inhibitor (the population inversion in a laser) di uses more strongly than the optical field.

The purpose of this chapter is to generalize the LALI principle to arbitrary forms of nonlocalities. Indeed, both di usion and di raction are nonlocal operators responsible for the communication of fields in the transverse plane. In the original study by Turing, the usual form of di usion was considered for the two interacting components. In optics one can have more complicated situations: even the model (12.4) and (12.5) shows such complications, since besides the normal di usion of the population inversion there is a superdi usion of the optical field. One can also have a situation where the inversion is di using but the optical field is di racting (for a laser with a broad gain line). And, finally, one can have both components di racting, as in the case of optical parametric oscillators.

These cases are investigated below. In the next section, a laser with diffusing inversion is studied under subcritical and supercritical conditions, and it is shown that Turing patterns are possible in the subcritical case. It is also shown that the di usion of the population inversion stabilizes spatial solitons. In Sect. 12.3, the optical parametric oscillator is investigated. It is shown that the di raction of the pump field (playing the role of inhibitor) may lead to the excitation of Turing patterns, which are di erent from the o -resonance patterns studied in previous chapters.

12.2 Laser with Di using Gain

It is often supposed intuitively that di usion in a gain material (e.g. di usion of the population inversion in a gas laser or di usion of free charge carriers in a

172 12 Turing Patterns in Nonlinear Optics

semiconductor laser) should weaken the spatial inhomogeneity of the emitted optical field. As a consequence, gain di usion should reduce or suppress a modulational instability, and might destroy spatial solitons that would exist in its absence.

The opposite phenomenon is shown to be true in this section, namely that the di usion of a saturating gain enhances the spatial modulation of the optical field. This enhancement of modulation supports solitons and increases their stability range.

12.2.1 General Case

Consider a general model, where the mean-field equations for an optical system with saturable gain are given by [3]

∂A	= F (A, 2A) + DA ,	(12.8a)
∂t
∂D	= γ(D0 − D − D |A|2 + d 2D) ,	(12.8b)
∂t

where A(r, t) is the optical field (order parameter) and D(r, t) is the gain field (e.g. the population inversion). The operator F (A, 2A) is a given nonlinear and nonlocal function of the order parameter A(r, t), d is the di usion coe cient for the saturable gain, and γ is its relaxation rate. The complex conjugate equation of (12.8a) must also be taken into account when the optical field is complex (if di raction or focusing/defocusing nonlinearities are present in the function F (A, 2A)).

For simplicity, it is assumed below that the gain relaxation is fast (γ = O(1/ε), with ε 1), and the gain variable D can be adiabatically eliminated from (12.8b) by requiring that ∂D/∂t = 0. However, as numerical calculations show, the main conclusions are valid even for moderate gain relaxation, i.e. γ = O(1). The adiabatic elimination from (12.8b), neglecting gain di usion

(d = 0), is straightforward, and gives
D =	D0		(12.9)
		.
	2
	1 + |A|

In general (for d = 0), the adiabatic elimination requires the inversion of

the operator
N = 1 + |A|2 − d 2 ,	(12.10)

since (12.8b) can be written, in the stationary case, as ND = D0. The inversion can be performed for small di usion, assuming that d 2 = O(ε) and all the other variables are of O(1), yielding

N−1D =

D0

1 + d

2

1

+ d

2

1

D0

,

1 + |A|

1 + |A|

1 + |A|2

(12.11)

12.2 Laser with Di using Gain

173

where the Laplace operator acts on the variables to the right of it. It is easy to verify that N−1ND0 = D0 1 + O(ε2) , which confirms the validity of the inverse operator (12.11) at O(ε).

For a spatially homogeneous pump parameter D0, the last term on the right-hand side of (12.11) vanishes, and the population inversion becomes

D =	D0		+	D0			1			(12.12)
	1 + |A|	2		1 + |A|	2	d 2	1 + |A|	2	.

Inserting (12.12) into (12.8a), we finally obtain the order parameter equa-

tion

∂A

D A

1

,

= F

(A, 2A) +

0

d 2

(12.13)

∂t

1 + A

2

1 + A

2

|

|

|

|

where F (A,

2A) = F (A,

2A) + D

/(1 + A

2). The last term on the right-

0

|

|

hand side of (12.13) is due to the di usion of the saturable gain.

Equation (12.13) will be used as a basis to investigate how the gain diffusion a ects the stability of the homogeneous solutions of that equation.

Linearization of (12.13) around the homogeneous stationary solution (which now depends on the explicit form of F , and is assumed to be real-

valued		without loss	of	generality), with perturbations of the form A =
A¯ + a	1		2		−	ikr), leads to
		exp(λt+ikr) + a exp(λt
λa = La + Da,						(12.14)

where a = (a1, a2)T is the column vector of the perturbation amplitudes. L is the linear evolution matrix generated by the nondi usive part of (12.13),

L =	δF /δa1					δF /δa2			,		(12.15)
	δF /δa1					δF /δa2
and D is the perturbation matrix due to gain di usion,
D =		D0	¯2	3 d		2	1	1		.	(12.16)
			¯2
			A
		1 + A					1		1

A useful representation can be found by rewriting (12.14)–(12.16) in terms of the new basis a± = a1 ± a2 (corresponding to perturbations of the amplitude and the phase, respectively), in which one obtains, instead of (12.16),

D =	D0	¯2	3 d		2		2	0	.	(12.17)
		¯2
		A
	1 + A						0	0

From this general analysis, we can draw some conclusions: