202 13 Three-Dimensional Patterns

range of the resonator. Nevertheless, variation of the resonator length (on the scale of the optical wavelength) allows one to change the detuning parameter in (13.5) and (13.13), and thus allows the manipulation of the 3D structures.

This seeming paradox can be understood in the following way. The maximum gain for a plane wave of the subharmonic field occurs when its phase has a particular value ϕ = 0, π with respect to the pump phase at the entrance of the nonlinear crystal. Tuning of the resonator length breaks the optimum phase relation for the plane wave. Therefore a modulation appears in the subharmonic field, causing a Guoy phase shift, which brings the phase to its optimum value. The Guoy phase shift is proportional to the spatial wavenumber of the modulation appearing. This modulation can appear in the transverse or longitudinal direction, or in both directions simultaneously, resulting in oblique lamellae or a tetrahedral structure.

13.4.2 Analogy Between 2D and 3D Cases

The order parameter equation derived here for a 3D DOPO is analogous to that derived for a DOPO in the 2D case [3]. The only di erence is the dimensionality of the problem. This suggests that this analogy between 2D and 3D systems is valid not only for DOPOs, but also for other nonlinear optical systems. A requirement is that the nonlinear processes should be fast compared with the time of light propagation over the typical length scales of the longitudinal modulation. In this case the order parameter equations derived for other nonlinear optical systems in 2D (e.g. externally driven nonlinear resonators containing focusing or defocusing media or saturable absorbers [5]) can be straightforwardly extended to the 3D case, and used to simulate a broad gain line in a synchronously or continuously pumped system. Instead of extended or localized structures in 2D, one should the obtain the corresponding 3D structures, propagating cyclically in the resonator.

The 3D extension of the equations results in the corresponding 3D extension of the structures. the 3D structures that have direct counterparts in 2D are phase domains, localized structures in the form of “bubbles”, and lamellar structures. However, the family of 3D structures is richer than that in 2D. An example of a 3D structure that does not have a counterpart in 2D is the resonant tetragonal pattern, which is supported by a cubic nonlinearity.

References

1.A. De Wit, G. Dewel, P. Borckmans and D. Walgraef, Three-dimensional dissipative structures in reaction–di usion systems, Physica D 61, 289 (1992). 193

2.N.L. Komarova, B.A. Malomed, J.V. Moloney and A.C. Newell, Resonant quasiperiodic patterns in a three-dimensional lasing medium, Phys. Rev. A 56, 803 (1997). 193

References 203

3.K. Staliunas, Transverse pattern formation in optical parametric oscillators, J. Mod. Opt. 42, 1261 (1995). 195, 196, 202

4.S. Longhi and A. Geraci, Swift–Hohenberg equation for optical parametric oscillators, Phys. Rev. A 54, 4581 (1996). 195

5.P. Mandel, M. Georgiou and T. Erneux, Transverse e ects in coherently driven nonlinear cavities, Phys. Rev. A 47, 4277 (1993). 202