Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

68 4 Zero Detuning: Laser Hydrodynamics and Optical Vortices

Fig. 4.1. Phase isolines around an optical vortex, shown by thin lines or curves, and flow lines (along the phase gradient), shown by thick lines with arrows. Left: optical vortex with constant radial phase. Right: optical vortex with radial phase growing monotonically away from the vortex core

component is that fluid is created at the vortex core (where the amplification (source) dominates over the saturation (sink) in (4.4a)), and flows outwards, where the sources and sinks compensate one another. Such radiating vortices are sometimes called spiral waves, since the equiphase lines are of spiral shape.

The vortex solution (4.9), however, does not posses an explicit algebraic form for the LGL equation in two spatial dimensions. In order obtain insight into the properties of the vortex (e.g. the size of the vortex core, and the vortex radiation), an analysis of a 1D analogue of a vortex is useful. The 1D analogue of a vortex is a kink wave that is equal to zero at the vortex core (x = 0), and approaches asymptotic values with constant amplitude and opposite phases at x = ±∞.

In the next section, the 1D version of (4.2) is analyzed in two limits separately: (1) in the limit of strong di raction (g 1), and (2) in the limit

A solution in the form of a kink can be found straightforwardly using an ansatz of the form A(x, t) = tanh(x/x0) exp [−iωt + iΦ(x)], with a phase gradient Φx = (α/x0) tanh(x/x0). Inserting the ansatz into (4.10), one obtains the half-width of the kink x0, the kink radiation factor α and the frequency

ω:

These values apply approximately in the 2D case for an optical vortex, where the half-width of the kink x0 plays the role of the vortex core radius r0. The

numerical values of the parameters di er slightly in the 2D case, but the scalings hold.

As follows from this analysis, the radial phase of the vortex is not uniform, since Φr = (α/r0) tanh(r/r0). According to the hydrodynamic analogy, where the phase gradient is equivalent to the velocity of the flow, the vortex creates a flow outwards or, in optical terms, the vortex radiates. The radiation is zero close to the vortex core (the radial phase variation is zero at this point), and increases away from it, saturating at a constant value. The saturation value with the initial normalizations of (4.1) is

The equal-phase lines therefore form spirals, as shown in the right-hand side of Fig. 4.1.

This provides an interpretation of the existence of “shocks” between vortices: the radiation from the neighboring vortices propagates, and collides. Owing to the collision, “shocks” appear.

The vortex core radius, expressed in terms of the initial parameters of (4.1), is proportional to the di raction coe cient in this limit of strong di raction:

In Fig. 4.2, several vortex ensembles are shown in the strong-di raction case, as obtained by numerical integration of (4.1).

It is interesting to note that the motion of the vortices is nearly chaotic. Annihilation of vortices is observed, as well as nucleation of new pairs of vortices. From a statistical point of view, the distribution of vortices remains

Fig. 4.2. The amplitude (left) and the phase (right) obtained by numerical integration of the LGL equation (4.2) with g = 0.2. The distributions in the top row are obtained at time t = 100, and those in the bottom row at time t = 300

4.2.2 Strong Di usion

Consider now the strong-di usion limit (g 1), in which (4.2) reads

Obviously, one can scale out the parameter g in this limit. However, we keep it in order to avoid misunderstandings, and keep the normalizations defined previously.

The natural guess for the solution of (4.14) in 1D is again a kink, A (r, t) = tanh(x/x0), now with a uniform phase profile. This form of kink, however, is not an exact solution of (4.14), as is easy to find by a direct test. Therefore one can use it only as an approximate solution.

Next, we use the fact that (4.14) is variational, i.e. it can be expressed in variational form as ∂tA = −δF/δA , where F(A) is a real-valued variational potential given by

This allows us to determine the half-width of the kink solution by inserting the ansatz into the variational functional, and minimizing (4.15) with respect to x0.

If we substitute the hyperbolic-tangent ansatz into (4.15), an infinite value is obtained for the potential. This is due to the contribution of the homogeneous background of amplitude |A0| = 1. Therefore we calibrate the potential (4.15) by subtracting this constant contribution:

The value of the half-width that minimizes the potential F is given by x40 = (24/5) g. The vortex core radius is therefore proportional to the di usion coe cient in this limit of strong di usion, if we use the initial normalization of (4.1):

The radial phase is constant in this limit. Vortices do not radiate when di usion is the dominating process and, consequently, they do not cause shocks in vortex ensembles.

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

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

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:

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

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

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