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References 79

relatively small di usion parameter g = 0.5 used in the calculations by which Fig. 4.8 was obtained. However, the vortices interact “correctly” only before the shock develops between them. After the shock develops, it begins to screen the vortices from one another, and the phase gradients caused by one vortex are no longer visible to the other vortex. The vortices then move not because of a mutual interaction mediated by phase gradients, but because of interaction with the shock. The bottom vortex then moves to the right, as rolling along the shock. The upper vortex also rolls to the right, for the same reason.

The interaction between vortices and shocks can be understood in terms of vortex motion due to amplitude gradients, as investigated in [7]. An amplitude gradient (a gradient of the fluid density) causes a gradient of the pressure, owing to the compressibility relation in the laser–hydrodynamics analogy. The pressure gradient, according to the laser hydrodynamic equations (4.4), causes a drift of the vortex. The vortex drift has, however, a component perpendicular to the force acting on the vortex. This is the Magnus drift: not only a vortex, but also every rotating object undergoes a drift perpendicular to the force acting on it.

References

1.K. Staliunas, Laser Ginzburg–Landau equation and laser hydrodynamics, Phys. Rev. A 48, 1573 (1993). 65

2.E. Madelung, Quantentheorie in hydrodynamischer form, Z. Phys. 40, 322 (1926). 65

3.P. Coullet, L. Gil and J. Lega, Defect-mediated turbulence, Phys. Rev. Lett. 62, 1619 (1989). 70

4.K. Staliunas, G. Slekys and C.O. Weiss, Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns, Phys. Rev. Lett. 79, 2658 (1997). 70

5.S. Rica and E. Tirapegui, Interaction of defects in two-dimensional systems, Phys. Rev. Lett. 64, 878 (1990). 74

6.S. Aranson, L. Kramer and A. Weber, Interaction of spirals in oscillatory media, Phys. Rev. Lett. 67, 404 (1991). 74

7.K. Staliunas, Dynamics of optical vortices in a laser beam, Opt. Commun. 90, 123 (1992). 79

5 Finite Detuning: Vortex Sheets

and Vortex Lattices

In this chapter, the spatio-temporal dynamics of the fields emitted by class A and C lasers are investigated in the case of moderate negative detuning. The transverse field dynamics in lasers are described by the CSH equation (2.26), as derived in Chap. 2. In this chapter, the same normalizations as in (4.2) are used, bringing (2.26) into the form

where the normalized detuning ∆ = −ω/p has also been defined. Note that the laser detuning ω and that of the CSH equation given by (5.1) are defined with opposite signs. As follows from the linear stability analysis in Chap. 2, laser patterns appear in the blue-detuned case ω < 0 (when the cavity resonance frequency is less than the central frequency of the atomic transition). On the other hand, it is customary to write the real and complex Swift– Hohenberg equations in the form (5.1), where patterns occur for positive values of ∆.

The simplest solution of (5.1) is a tilted (or traveling) wave (TW),

with a√wavevector k pertaining to the resonant ring (the modes obeying |k| = ∆, which, as predicted by the linear stability analysis in Chap. 2, experience the maximum amplification).

If the wavenumber of the TW is mismatched from that of the maximally amplified mode (the wavelength is shorter or longer than the resonant value), the resulting TW solution is

which has an amplitude smaller than that of the resonant TWs, and oscillates with a frequency proportional to the mismatch ∆ω = k2 − ∆, where k = |k| . The dependence of the amplitude of the TW on the mismatch ∆ω in (5.3) is similar to the usual resonance curve of a driven oscillator.

The presence of traveling waves is closely related to the existence and dynamics of vortices, as already discussed in Chap. 4. In Sect. 5.1 the motion

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 81–90 (2003)c Springer-Verlag Berlin Heidelberg 2003

82 5 Finite Detuning: Vortex Sheets and Vortex Lattices

of a single vortex superimposed on a TW is analyzed, in a similar way to the analysis in Sect. 4.3. In this case, the finite detuning introduces new features into the dynamics compared with the zero-detuning case, such as vortex stretching. In Sect. 5.2, the interaction of two TWs with di erent orientations is studied. It is shown that such an interaction may lead to the formation of TW domains. At a boundary between two domains (where two TWs merge), an array of equally charged vortices is formed, which corresponds to a vortex sheet. Finally, the interaction of four TWs at particular angles is shown to generate a stable cross-roll pattern, which can alternatively be described as a square vortex lattice.

5.1 Vortices “Riding” on Tilted Waves

The evolution of an arbitrary perturbation of a TW can be studied using the

change of variable A(r, t) = exp(ikr) B(r, t). In the case of a resonant TW

√

(|k| = ∆), this change brings (5.1) into the form

which is the evolution equation for the modulation of a resonant TW. The terms on the left-hand side describe the advection of vortices (and of every structure in general), with the velocity of the TW given by v = 2k. The terms on the right-hand side describe the evolution of the envelope in a basis propagating with the underlying TW. The dynamics are similar to those of the nondetuned laser studied in the previous chapter, but not completely identical. Inspecting the right-hand side of (5.4), one can identify local gain and saturation terms, and also di raction terms identical to those for a laser with zero detuning. The di usion in (5.4) is, however, di erent. Expanding the third term on the right-hand side,

we notice that the vortices di use in di erent ways in the directions parallel and perpendicular to the direction of the advecting TW. The parallel di usion is stronger (the first order of the Laplacian) than the perpendicular di usion (the second order of the Laplacian). Therefore vortices can be expected to stretch along the TW direction.

Figure 5.1 shows two oppositely charged vortices advected by a TW, as obtained by solving numerically the CSH equation (5.1). As expected from (5.4), the vortices are stretched along the direction of advection. For periodic boundary conditions, as in Fig. 5.1, the vortices move periodically through the interaction region. The situation is di erent for zero boundary conditions. In this case vortices can appear at one boundary, be advected by the tilted wave and be destroyed at the other boundary. Such a case is illustrated by

84 5 Finite Detuning: Vortex Sheets and Vortex Lattices

perfectly tuned in the lateral direction (see Fig. 6.6). The defect in the center nucleates the vortices, which are advected by a radially spreading tilted wave.

5.2 Domains of Tilted Waves

Another natural choice for a basic solution of the CSH equation (5.1) is in the form of two counterpropagating TWs,

which corresponds to a standing wave (SW) in the case of equal amplitudes

|a+| = |a−|.

One can easily check the stability of the solution (5.6) by inserting it into (5.1), gathering the terms with the exponents exp(ik · r) and exp(−ik · r), and neglecting the terms containing the third spatial harmonics (those

with the exponents exp(3ik · r) and exp(−3ik · r)). In the case of resonant

√

counterpropagating TWs, where |k| = ∆, the equations for the amplitudes read

A linear stability analysis of (5.7) yields the result that the SW solution |a+| = |a−| is unstable against one of the two TW solutions, |a+| = 1, |a−| = 0 or |a−| = 1, |a+| = 0. The system (5.7) possesses a variational potential that has two minima, which correspond to two TWs, and a saddle point, which corresponds to a SW. Therefore, (5.7) leads to a competition between counterpropagating TWs, with the survival of the stronger one.

The competition of counterpropagating TWs can also result in their separation in space. In this case spatial domains appear, each domain characterized by a particular direction of a TW.

TW domains commonly appear in the numerical integration of the CSH equation with periodic boundary conditions, but they exist only as transients: the strongest domain finally wins, and a pure TW remains as the final pattern. With zero (Dirichlet) lateral boundaries, however, coexisting TW domains can be stationary.

Figure 5.4 shows the formation and evolution of TW domains. From the resonant ring, some spots emerge (right column, showing the spectra), corresponding to TW domains (left column). In what follows, the strongest domains survive. However, the domains are not uniform (second and third rows), but contain defects in the form of optical vortices. The optical vortices are advected by the TWs and finally disappear at the domain boundaries.

A domain boundary, as can be seen from the left column in Fig. 5.4, is an array of equally charged vortices, or a vortex sheet. This is so because of the particular phase variation at the boundary between two TWs (center