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

64 3 Order Parameter Equations for Other Nonlinear Resonators

5.P. Mandel, M. Georgiou and T. Erneux, Transverse e ects in coherently driven nonlinear cavities, Phys. Rev. A 47, 4277 (1993); G.J. de Valc´arcel, K. Staliunas, E. Rold´an and V.J. S´anchez-Morcillo, Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing, Phys. Rev. A 54, 1609 (1996); S. Longhi and A. Geraci, Swift–Hohenberg equation for optical parametric oscillators, Phys. Rev. A 54, 4581 (1996). 55, 63

6.V.J. S´anchez-Morcillo, E. Rold´an, G.J. de Valc´arcel and K. Staliunas, Generalized complex Swift-Hohenberg equation for optical parametric oscillators, Phys. Rev. A 56, 3237 (1996). 59, 61

7.K. Staliunas, M.F.H. Tarroja, G. Slekys, C.O. Weiss and L. Dambly, Analogy between photorefractive oscillators and class-A lasers, Phys. Rev. A 51, 4140 (1995). 60, 61

8.K. Staliunas, Laser Ginzburg-Landau equation and laser hydrodynamics, Phys. Rev. A 48, 1573 (1993); J. Lega, J.V. Moloney and A.C. Newell, Universal description of laser dynamics near threshold, Physica D 83, 478 (1995). 61

In (4.3) the super-di usion has been assumed to be small: all the terms containing the coe cient g have been neglected, except for the most sig-

nificant one (g 4Φ in (4.3b)). Normalizing again the spatial variables with

√

the change r/ 2 → r, and rewriting (4.3) in terms of the velocity in the transverse space v = v = Φ, we obtain

Equations (4.4) describe, in the hydrodynamic analogy, the evolution of a “photon fluid”. The left parts of the equations can be interpreted as the conservation of mass and momentum (the analogues of the continuity and Euler equations, respectively). The right part of (4.4a) describes the presence of sources and sinks: “mass” is created owing to the linear gain and dissipates owing to the saturation of the inversion in the laser. The right part of (4.4b) can be interpreted as the dissipation of momentum due to super-viscosity, which is proportional to the g-factor. The last term in (4.4b), called the “quantum pressure” term, has no analogue in standard fluid mechanics.

The “photon fluid” in a laser as described by the LGL equation does not possess the usual compressibility, where the internal pressure is proportional to some local function of the density (note that the quantum pressure in (4.4b) is nonlocal). However, a classical pressure can occur if we consider an additional self-focusing or self-defocusing mechanism, e.g., if a focusing– defocusing Kerr material is present in the laser resonator. In this case the LGL equation becomes

The Euler equation (4.4b) is modified because of this self-focusing/defo- cusing to

where the classical pressure is proportional to the fluid density, since p = αρ2/2. For a defocusing medium (α > 0) the compressibility relation is “normal”, while for a focusing medium (α < 0) it is “anomalous”.

Looking ahead (this topic will be treated in detail in Chap. 6), in the case of a curved (parabolic) resonator, an additional external potential appears in

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