72 4 Zero Detuning: Laser Hydrodynamics and Optical Vortices

Fig. 4.4. The amplitude and phase, as obtained by numerical integration of (4.1) with g = 10. The distributions calculated at t = 8 and t = 150 are shown in the top and bottom rows respectively

Figure 4.4 illustrates the dynamics of a vortex ensemble in the strongly di usive case, as obtained by numerical integration of (4.1).

The number of vortices decreases with time in this di usive limit. This is in contrast with Fig. 4.2, which shows no variation in the number of vortices, in a statistical sense, in the di ractive limit.

Annihilation of vortices, and a plane wave as the final state can be expected, since the di usive limit is a variational one. Variational systems develop in such a way that they reach the minimum of the variational potential along the shortest path (along the gradient of the potential).

In experiments, the di usive case is obtained when the resonator length is precisely tuned to the self-imaging length, as shown in Chap. 6. An example of a pattern recorded under such conditions is shown in Fig. 4.5. An obvious di erence from the distribution shown in Fig. 4.3 is the absence of shocks.

4.2.3 Intermediate Cases

It is di cult to perform analytical evaluations of the vortex parameters for arbitrary di raction–di usion ratios. However, the vortex behavior in this intermediate case can be extrapolated from the two limits. Qualitatively, one can expect that the closer the parameters are to the di ractive limit (small g), the more the vortices radiate, and the more prominent the shocks in vortex ensembles are. Also, the dynamics are more chaotic. On the other hand, the closer the parameters are to the di usive limit (large g), the more the vortices tend to annihilate and disappear.

Some quantitative evaluations can be performed with a simplified version of the LGL equation, namely the ordinary CGL equation, which contains normal di usion. In two dimensions, this equation reads

∂A	= A + iaIm 2A + aRe 2A − |A|2 A ,	(4.19)
∂t

where aIm and aRe are the di raction and di usion coe cients, respectively.

4.2 Optical Vortices

73

Fig. 4.5. Vortices for zero detuning, for a perfectly confocal resonator, when the di usion dominates over the di raction. The pump field intensity was twice the threshold value. Shocks are absent because di usion dominates

In one transverse dimension, (4.19) possesses an exact kink solution in the form A(x, t) = β tanh(x/x0) exp [−iωt + iΦ(x)], where the gradient of the phase is given by Φx = (α/x0) tanh(x/x0). Substitution in (4.19) allows us to evaluate the parameters of the kink:

x02 = −

5

aRe

+

9

aRe2 + 18aIm2 ,

(4.20a)

2

2

3 aRe

3aRe

2

α = −

+

+ 2 .

(4.20b)

2

aIm

2aIm

In two transverse dimensions, no algebraic vortex solution exists. However, the parameters obtained for the kink solution can again be supposed to be valid also for the vortex solution, with the width of the kink x0 corresponding now to the radius of the vortex core r0.

Consider now the two limiting cases of (4.20).

In the di ractive case, where aIm aRe, the vortex radius is given by

2	√			5
r0	= 3	2aIm −		2	aRe ,	(4.21)

and is mainly determined by di raction. The vortex radiation parameter is

√			3 aRe
α =	2 −					,	(4.22)
			2		aIm

which asymptotically approaches its maximum value as the purely di ractive case is approached.