7 The Restless Vortex

A single, isolated vortex in a class A or C laser, as described by the CSH equation (2.26), is stationary. However, in the case of a class B laser, the situation is di erent. Free vortices are not stationary but are in permanent motion. Also, a lattice of vortices experiences permanent self-sustained motion, leading to various oscillation modes.

For class B lasers, the population inversion decays slowly, and the following relations hold: γ /κ = O(ε) and γ /γ = O(ε), where ε is a smallness parameter. The eigenvalue associated with the population inversion D in the linear stability analysis, λ (k) = −γ , does not lie su ciently deep below the zero axis. Consequently, the population inversion is not enslaved by the fast variables, namely the field and the polarization, and cannot be adiabatically eliminated. The CSH equation is therefore an oversimplified model for a class B laser and does not describe adequately its spatio-temporal phenomena.

It is well known that single-transverse-mode class B lasers show relaxation oscillations with a frequency

ωrel =	2κγ (D0 − 1) −	γ4		,	(7.1)
		2

where the polarization is assumed to relax infinitely fast, i.e. γ → ∞. On the other hand, a single-transverse-mode laser, as described by the CSH equation, reaches its stationary state monotonically, and does not show relaxation oscillations. It is plausible to expect also that the spatio-temporal dynamics of class B lasers are di erent from those of lasers of class A and class C, which are described by the CSH equation.

7.1 The Model

An order parameter equation system describing the spatio-temporal dynamics of a class B laser close to the emission threshold can be obtained by applying the techniques presented in Chap. 2. We use here the method of adiabatic elimination used in Sect. 2.3 in the limit of a class A laser.

For a class B laser, owing to the slowness of the population inversion, one can consider the equations of the field and polarization ((2.1a) and (2.1b))

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 103–115 (2003)c Springer-Verlag Berlin Heidelberg 2003

104 7 The Restless Vortex

separately from the equation of the population inversion (2.1c), and diagonalize the system formed by (2.1a) and (2.1b), considering D(r, t) as a parameter. This leads to the following growth rate of perturbations:

λ1,2(∆ω) = −	κ	2 + i ∆ω ± 4D − ∆ω2	.	(7.2)
	2

Here, as in Chap. 2, the particular case κ = γ is considered for simplicity. Generalization to arbitrary values of κ and γ (but keeping O(κ) = O(γ )) is straightforward. The eigenvector system associated with (7.2) is, in matrix form,

1

(1/2)

i ∆ω +

√

V =

4D − ∆ω2

,

(7.3)

1

(1/2)

i ∆ω − √

4D − ∆ω2

1 T

and its adjoint transformation matrix S = (V − ) is given by

(1/2)

−i ∆ω + √

1 .

S =

√

1

4D − ∆ω2

(7.4)

2

√κ 4D − ∆ω

(1/2) i ∆ω + √

4D − ∆ω2

1

The expressions for the eigenvalues (7.2), the eigenvectors (7.3) and the transformation matrix (7.4) are analogous to the expressions (2.10), (2.15) and (2.16) for class A and class C lasers in Chap. 2. The di erence is that here

atimeand space-dependent population inversion D(r, t) appears instead of

ahomogeneous, stationary population inversion D0.

Now, we define a new set variables A = (A, B)T, related to the old ones by the transformation A = SE and viceversa, E = S−1A.

The equations for the field and polarization can now be rewritten, in terms

of the new variables, as

2 ∂A

= −i ∆ω A +

4D − ∆ω2 − 2 A ,

(7.5a)

κ ∂t

2 ∂B

= −i ∆ω B − 4D − ∆ω2 + 2 B .

(7.5b)

κ ∂t

The equation for the population inversion transforms to

∂D

κ

= −γ D − D0 +

4D − ∆ω2 |A|2 − |B|2

.

(7.6)

∂t

2

As follows from (7.5b), the stable variable B decays rapidly to zero and can be neglected. Now, expanding the term with the square root in (7.5a) and (7.6) as a Taylor series, assuming the near-threshold condition p = D − 1 = O(ε2) and the near-to-resonance condition ∆ω = O(ε), we obtain

7.2 Single Vortex

105

2 ∂A

= (D − 1) A + i

a 2 − ω A −

1

a 2

− ω

2

A ,

(7.7a)

κ ∂t

4

∂D

2

= −γ D − D0 + |A|

.

(7.7b)

∂t

If we remove the assumption of κ = γ (but assume that these two coefficients are of the same order, O(1)) we obtain the order parameter equation for the general case of a class B laser:

∂A	= (D − 1) A + i a 2 − ω A −	κ2	a 2 − ω	2	A,	(7.8a)
∂τ		(κ + γ )2
∂D	= −γ D − D0 + |A|2 ,					(7.8b)
∂τ

where we recall that the variables appearing in (7.8) are related to the original ones through the relations τ = tκγ /(κ + γ ), γ = (κ + γ )γ /κγ , and a

normalized field intensity κ |A|2 → |A|2 has been used.

The two coupled equations (7.8) have also been derived in [1] using the multiscale expansion technique. They are the basic equations used for the study of the spatio-temporal dynamics in class B lasers throughout this chapter.

During the diagonalization procedure, the time-dependent transformation matrix S was used, which obviously does not commute with the operator of the time derivative ∂/∂t. In fact, the commutation of these operators was assumed in obtaining (7.8). Assuming the above near-threshold condition D − 1 = O(ε2), however, we obtain the result that the commutator is of higher order of smallness than the rest of the terms in (7.8), and thus can be neglected. Therefore, in the framework of the near-threshold assumption used here, the system (7.8) is valid for a class B laser.

7.2 Single Vortex

A numerical study of a single vortex in a class B laser (in the framework of (7.8), and also in the framework of the complete Maxwell–Bloch equations (2.1)) reveals a surprising result: a single, isolated vortex, placed in a homogeneous background, is not stationary as in a class A laser, but shows a self-induced, permanent motion. For some values of the parameters, the motion resembles a “stochastic meandering” of the vortex core. Sometimes the motion is circular or flower-like. In order to understand the origin of this self-induced motion we inspect the distributions of the optical field A and of the population inversion D at the vortex core.

Figure 7.1a shows the fields along a line crossing the center of a stationary vortex. At the vortex center the order parameter A, which is proportional to the optical field, is zero. As the population inversion does not decay in the

106 7 The Restless Vortex

Fig. 7.1. Electric-field modulus (solid line) and population inversion (dashed line), corresponding to (a) a stationary optical vortex, and (b) a vortex moving to the right, on the axis along which the vortex moves in (b). The changes of the field are shown by arrows

absence of an optical field, an inversion maximum builds at the location of the optical vortex.

If we perturb the envelope of the field by shifting the “optical part” of the vortex, as shown in Fig. 7.1b, restoring forces appear, represented by vertical arrows in the figure. The inversion builds up at the new location of the zero of the optical field. In this way, the inversion profile tends to follow the motion of the “optical part” of the vortex. The restoring forces on the optical fields are such that the shifted zero of the field is pushed further in the same direction: the trailing slope is amplified more since the inversion is larger there, and the leading edge is attenuated more, since the inversion is smaller there.

As a consequence, the zero of the optical field moves away from the maximum of the population inversion. Its escape velocity is proportional to κ, the buildup rate of the optical field. The population inversion follows the escaping zero of the field. Its maximum velocity is proportional to the buildup rate of the population inversion, γ . If the population inversion is faster than the optical field, i.e. γ ≥ κ, its maximum cannot escape the zero of the optical field, and thus the location of the optical vortex is stabilized. In the case of a class B laser, where γ ≤ κ, the population inversion relaxes more slowly than the field, and the race between the zero of the optical field and the maximum of the population inversion continues forever.

The velocity of the vortex motion can be estimated by assuming that the laser field at a particular location goes through half a period of a relaxation oscillation during the passage of a restless vortex through it. Since the ra-

dius of the vortex is proportional to r0 ≈ a/ (D0 − 1), and the relaxation

oscillations have a frequency ωrel = 2γ(D0 − 1) − γ2/4, the velocity of the self-induced motion estimated in this way is

	ω		2√
				2aγ			γ
|v| =	2r0 rel	=			1 −			.		(7.9)
	π		π				8 (D0 − 1)

7.2 Single Vortex

107

Fig. 7.2. Restless motion of two vortices. Two vortices with opposite topological charge have been considered, since the periodic boundary conditions require zero total charge. The positively charged vortex at the bottom right corner moves clockwise, and the other, negatively charged vortex, moves anticlockwise. The parameters are D0 = 2, γ = 0.1, a = 0.0001 and g = 1.25; the time between successive plots is t = 20

Fig. 7.3. Trajectory of a restless vortex with positive topological charge: the transient toroidal trajectory and the final circular trajectory. The parameters are as in Fig. 7.2, and time between successive points is ∆t = 3.5

Numerical calculations show that the estimation (7.9) works well con-

√

cerning the velocity of the vortex |v| ≈ 2 2aγ/π, and also concerning the threshold condition for self-induced vortex motion, γ = 8(D0 −1). In Fig. 7.2, the motion of two restless vortices, obtained numerically, is shown.

Figure 7.3 shows the trajectory of a restless vortex. The direction in which the vortex starts to move is given by an initial symmetry breaking. After a transient toroidal motion, the asymptotic vortex trajectory is circular. Apparently, the rotatory nature of the vortex motion is related to the fact that the vortex itself is a rotating object.

The vortex is squeezed in the direction of motion. Figure 7.4 shows the field amplitude (a magnified part of the first plot of Fig. 7.2), where this squeezing is visible. The field maximum behind the vortex corresponds to the maximum of the population inversion. The optical and material parts of the vortex are shifted, so that the population inversion maximum lags behind.