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

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

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

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

206 14 Patterns and Noise

example, the far-field distributions of hexagonal patterns show distributions of the form (k − ki)−2, where ki (i = 1, ..., 6) are the locations of singularities of the far field, arranged on the vertices of a hexagon.

•The spatial power spectra of the noise show a 1/fα distribution, where the exponent α is close to unity, and depends on the dimensionality and symmetry of the pattern.

•The stochastic drift of the patterns is sub-Brownian: it is well known that the stochastic drift of the position of a Brownian particle obeys a square root law,

x (t)2 t1/2 .

We show that the stochastic drift of nonlinear patterns is in general di erent (weaker) than the Brownian; for example the stripe pattern has a root mean wandering t1/4 in the case of one spatial dimension.

The analysis of noisy stripes is performed by solving the stochastic Swift– Hohenberg equation as the order parameter equation for a stripe pattern in a spatially isotropic system [4], or the Newell–Whitehead–Segel equation as an amplitude equation for perturbations of stripe a pattern [5]. However, we start from an analysis of a noisy homogeneous state or, in other words, of a nonzero-temperature condensate. The main results (spatio-temporal spectra) in the case of the condensate (the first part of the chapter), are then applied to calculate the noise properties of stripe patterns.

14.1 Noise in Condensates

An order–disorder transition in a condensate or, in general, in a spatially extended nonlinear system can be described in the lowest order by a complex Ginzburg–Landau equation with a stochastic term:

Here A(r, t) is a complex-valued order parameter defined in an n-dimensional space r, evolving with time t. The control parameter is p (the order–disorder transition occurs at p = 0). The Laplace operator 2A represents the nonlocality in the system, and Γ(r, t) is an additive noise, δ-correlated in space and time and of temperature T , such that

Below the transition threshold (p < 0), the CGL equation (14.1) yields a disordered state: the order parameter A(r, t) is essentially noise filtered in space and time, with an exponential (thermal) intensity distribution. Above