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

In Fig. 12.11 a comparison between analytical results (dashed curve) and numerical results (continuous curve) for the spatial oscillations of the decaying tail of a domain boundary is given. The peak of the localized structure is omitted. Note that the correspondence is very good, even close to the domain boundary (the line of zero intensity). In this particular case, four minima of the intensity are visible. The opposite segment of a dark ring can be locked by each of the minima. Obviously, the soliton of minimum size, locked by the first maximum, which is the strongest, is the most stable one. However, dark rings with larger radii can also be stable.

Fig. 12.11. Spatial oscillations of the field outside a soliton, as evaluated numerically (continuous line) and analytically from the spatial stability analysis (dashed line), for E = 2.5, ω1 = 0.5, ω0 = 0, a1 = 0.00025 and a0 = 0.00125 (a = 5)

The stability range of solitons is limited on one hand by the contraction and annihilation of domains, and on the other hand by either the presence of modulational instabilities (the modulations grow, and fill the whole space) or expansion of domains. Since modulational instabilities are favored by di raction, it may seem that di raction has a negative e ect on the stability of solitons. However, for pump values at which instabilities are absent, the increase in the modulation of the tails could prevent full contraction, thus contributing to an enhancement of the stability range. Numerical calculations performed for a large pump di raction parameter show that the stability is always enhanced, at least up to some value of the pump parameter.

The presence of strong modulations in the tails also allows the formation of more complex structures, in the form of bound states of single solitons, or “molecules” of light. Some examples of molecules of varying complexity are shown in Fig. 12.12. Examples with two and three maxima are shown in Figs. 12.12a,b, and a chain composed of five maxima is shown in Fig. 12.12c.

190 12 Turing Patterns in Nonlinear Optics

Fig. 12.12. Several bound states (molecules) of solitons, obtained for a = 5, E = 2.5, ω1 = 0.5, ω0 = 0 : (a) Double; (b) triple; (c) a chain. The field amplitude along a cross section y = 0 of the chain is shown the graph by the solid line. The dashed line represents a section across the space outside the dark line

The internal structure of the chain shown in Fig. 12.12c is more clearly visible in a section along the middle (y = 32). Five maxima at equidistant points are seen. The field along a line outside the domain boundary is given by the dashed line, evaluated at y = 20.

In all cases, the large value of the pump di raction parameter is responsible for the stability of such complex structures, by amplifying the spatial oscillations and thus preventing their collapse. To show this, we have followed the evolution of the soliton “molecules” shown in Figs. 12.12b,c by decreasing the di raction parameter to a = 1 while keeping the other parameters unchanged. The resulting scenario is shown in Fig. 12.13, where the pictures have been taken at equally spaced times. The final state corresponds to a single soliton.

References 191

Fig. 12.13. Temporal evolution showing the decay to a single soliton of the molecules shown in Figs. 12.12b,c when the di raction parameter is decreased to a = 1

References

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194 13 Three-Dimensional Patterns

Synchronously pumped OPO:

cw-pumped OPO:

Fig. 13.1. Schematic illustration of a synchronously pumped (left ) and continuously pumped (right ) degenerate optical parametric oscillator

is also shown in Fig. 13.1. Therefore the analysis applies for synchronously and continuously pumped DOPOs.

The interaction of the three-dimensional slowly varying envelopes of the 3D pump and subharmonic pulses, A0(r , τ, z) and A1(r , τ, z), respectively, is described by the following set of equations:

Here vj = ∂kj /∂ωj are the group velocities for the pump (j = 0) and subharmonic (j = 1) waves, a ,j = ∂kj /∂ωj are the longitudinal dispersion coe cients, a ,j = 1/2kj are the transverse di raction coe cients, and χ is the nonlinear coupling coe cient. Evolution occurs along z, the longitudinal coordinate. The fields are defined in the 2D transverse space r = (x, y), in which the Laplace operator 2 = ∂2/∂x2 + ∂2/∂y2 acts, and in the longitudinal space τ, representing a retarded time in a frame propagating with the group velocity of the pump pulses.

The changes of the fields during one resonator round trip are assumed to be small. This allows us, first, to obtain a mapping describing the discrete changes of the subharmonic pulse in successive resonator round trips. Second, it allows us to replace the discrete mapping by a continuous evolution, and thus to obtain an order parameter equation in the form of a partial di erential equation.

13.1.1 Order Parameter Equation

Di ractive and dispersive changes of the pump are neglected during the propagation over the crystal length ∆l, which is assumed to be small com-