the laser hydrodynamic equation. The curved mirrors result in an additional term in the right-hand side of the LGL equation (4.2), given by −icr2A, where c is the curvature of the mirrors. The form of this new term arises from the fact that the curved mirrors correspond to a spatial dependence of the resonator detuning parameter, which is parabolic in a first approximation. Taking the curvature into account leads to the corresponding Euler equation,

where V (r) is a potential holding the laser fluid inside the resonator (in the lateral direction).

Summarizing, the transverse dynamics of the laser radiation are analogous to the dynamics of a compressible, quantized fluid. The parabolic mirrors of the laser resonator result in a parabolic potential, which localizes the laser fluid. The compressibility law of the laser fluid is nonlocal, and a local compressibility/anticompressibility term appears when additional selfdefocusing/focusing e ects are included. Finally, the super-viscosity of the laser fluid pκ2/ [2(κ + γ )]2 is inversely proportional to the width of the gain line.

4.2 Optical Vortices

The radiation of lasers, being similar to fluids and superfluids, can be expected to show vortices. Optical vortices can indeed be obtained by integrating the LGL equation (4.2) numerically. In the first approximation, an optical vortex has a helical wavefront, and therefore can be described asymptotically (at the core) by

where m = ±1 is the topological charge of the vortex, and (r, ϕ) are polar coordinates centered at the vortex core. In general, the distribution of the field in the presence of an optical vortex is

where the amplitude R(r) saturates far away from the vortex core, and Φ(r) is the radial phase responsible for the radiation from the vortex.

When Φ(r) is a constant (corresponding to a uniform radial phase), the lines of equal phase are directed radially from the vortex core (Fig. 4.1, left), and the flow of the photon fluid around the vortex is purely azimuthal. The vortex does not radiate in this case. In general, however, the flow has a radial component, as shown in Fig. 4.1 on the right. The origin of the radial flow

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.