13.4 Conclusions

201

Unlike the case of (13.5), two 3D Laplace operators must be defined if the two waves have di erent di raction and/or di usion coe cients.

A further simplification of (13.14) is possible for equal group velocities of the signal and idler waves. This leads to

∂A		= (P − 1) A + i( −2	+ ∆−)A −	1	( +2 + ∆+)2A − |A|2 A , (13.16)
∂t				2

which is the complex Swift–Hohenberg equation in 3D. The resulting detunings depend on the detunings of the signal and idler field components: ∆± = ∆1 ±∆2. The same is true for the resulting components of the Laplace operators.

A numerical integration of (13.16) has been performed. The extended patterns obtained consisted of 3D tilted waves, completely analogous to those in 2D studied in Chap. 5, and also the 3D analogue of the square vortex lattice. The latter consists of a grid of parallel vortex lines with alternating directions. The localized structures obtained here correspond to vortex rings. These vortex rings can stabilize at some equilibrium radius dependent on the value of the detuning parameter. Sometimes these vortex rings can form complicated structures, two of which are shown in Fig. 13.5.

Fig. 13.5. Stable vortex rings, as obtained by numerical integration of the 3D complex Ginzburg–Landau equation (13.16)

13.4 Conclusions

This investigation of synchronously pumped OPOs leads to the following conclusions.

13.4.1 Tunability of a System with a Broad Gain Band

A DOPO in reality has a very broad gain line width (the line of phase synchronism), typically many orders of magnitude larger than the free spectral