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

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the derivation procedure for the 2D case from [3] to the 3D parametrically driven Ginzburg–Landau equation, we obtain

which is a real Swift–Hohenberg equation in 3D.

The spatio-temporal structure of the pump pulses is included in P (r, t), and therefore (13.7) is valid for both synchronously and continuously pumped OPOs. The boundary conditions in the lateral coordinates depend on the details of the experiment: for example, the aperture of the resonator implies boundaries where the fields are zero, and systems with an infinitely broad aperture (and pump profile) require no lateral boundaries at all. In the longitudinal direction, periodic boundaries must be used, corresponding to a periodic repetition of the pattern.

Further, in the analytical treatment of the patterns, a pump that is homogeneous in 3D is assumed. This assumption is legitimate when the typical size of the spatial structures is much smaller than the spatial size of the pump pulse. This occurs for a su ciently broad pump beam (|∂P/∂X|, |∂P/∂Y | |P |) and also for a su ciently long pump pulse (|∂P/∂η| |P |). Under these conditions, one can consider the pump parameter to be constant in the central region of the pulse. This allows us to scale out the pump parameter and write (13.7) in the form

which has only one free parameter, the detuning ∆.

The validity of the linear part of (13.7) can be tested by comparison of the spectrum of the Lyapunov growth exponents calculated from (13.7), with the round-trip increments of the fields calculated from (13.1) and (13.3). Such a comparison shows that (13.7) describes well the linear pattern-forming properties (transverse wavenumber selection) of a DOPO not only near the threshold, where (P − 1) 1, where (13.7) is strictly mathematically valid, but also moderately above the threshold, where (P − 1) ≈ O(1).

13.2 Patterns Obtained

from the 3D Swift–Hohenberg Equation

In the limit of small detuning, a homogeneous distribution with amplitude

√

|A| = ± 1 − ∆2 and one of two phase values, ϕ = (0, π), is a stable solution of the 3D SH equation (13.8). However, in a transient stage of the evolution, if one starts from a random field distribution, the subharmonic field can consist of separated domains, each with one of two phase values. The phase domains in 2D patterns are separated by domain boundaries (or dark switching waves),

Fig. 13.2. Phase domains as obtained by numerical integration of (13.8), depicted by surfaces of zero field. At the bottom a 2D section is shown, showing the field intensity (left ), and the field phase (right ). The detuning is ∆ = 0.4. Periodic boundaries were used on a box of size ∆x = ∆y = ∆η = 20

as analyzed in Chap. 11. Analogously, similar 3D domains exist, separated by 2D domain walls.

A numerical integration of the 3D SH equation (13.8) was performed to test the idea of domains in 3D. A split-step technique was used on a spatial grid of 32 × 32 × 32 points. The result is showm in Fig. 13.2 for a particular time in the transient evolution. Two domains of uniform phase, embedded in a background of the opposite phase, are apparent.

The dynamics of the 3D domains depend on the detuning parameter in a similar way to those of 2D domains (see Chap. 11). A negative or small positive detuning leads to the contraction and eventual disappearance of domains. A large positive detuning leads to the growth of domains and formation of a 3D “labyrinth” structure, as discussed below. However, in a particular detuning range the contracting domains can stabilize at a particular size. In this case we obtain spherically symmetric, stable “bubbles”, which are the localized structures (spatial solitons) of the 3D SH equation. Such an ensemble of stable bubbles is shown in Fig. 13.3, as obtained numerically.

The stability limits of the spatial solitons were analyzed, by solving the 3D SH equation (13.8) numerically. The bubbles are stable in the interval

198 13 Three-Dimensional Patterns

Fig. 13.3. 3D spatial phase solitons (bubbles). The same conditions as in Fig. 13.2 were used, except for the detuning value ∆ = 0.45

0.430 < ∆ < 0.460. This stability range is much narrower than that for the corresponding dark rings in 2D found in Chap. 11, which is 0.287 < ∆ < 0.460.

Large detuning values lead to periodic patterns with a dominant nonzero

√

spatial wavenumber |k| = ∆. In two dimensions, a parallel stripe pattern

occurs and has a spatial distribution A(r) ≈ 4/3 cos(kr). Hexagonal patterns are not supported by the SH equation in 2D, since the nonlinearity is purely cubic here. (It is known that a square nonlinearity is necessary for supporting stable hexagons, unless an additional neutral mode is included.) A direct continuation to the 3D case gives the analogue of a stripe pattern, a standing-wave pattern also called “lamellae”. However, besides lamellae another stable periodic structure is possible in 3D, a structure made up of four resonant standing waves, with wavevectors as illustrated in Fig. 13.4a,

√

The four k-resonant standing waves, for which |kj | = ∆, do not lie in the same plane, and thus such a tetrahedral structure can exist only in 3D space. The phases of the four nonplanar standing waves, with complex

Fig. 13.4. (a) k-resonant wavevectors forming a tetrahedral structure. (b) Isolines at 85% of maximum field intensity, and isolines at (c) 93% and (d) −93% of maximum amplitude, obtained by numerical integration of the 3D SH equation for ∆ = 1.2 and a box size ∆x = ∆y = ∆η = 10

amplitudes Aj = |Aj |eiϕj , obey

A stability analysis shows that both lamellae and tetrahedral structure are stable. For the stability analysis, a variational potential for (13.8) was calculated, namely

Note that Laplace operators do not appear in the variational potential, if we are dealing with the k-resonant structures.

Calculation of the variational potential yields the potential minima associated with these structures in the parameter space of Aj . The minimum values of the potential are F1 = −1/6 = −0.1666... for lamellae, and F4 = −2/15 = −0.1333... for the tetrahedral structure. Lamellae are thus more stable than the tetrahedral structure. For comparison, the 3D continuations of the resonant square pattern, and of the hexagonal pattern have potentials F2 = −1/10 = −0.1 and F3 = −1/9 = −0.1111..., respectively.

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However, these unstable patterns correspond not to local potential minima in the parameter space of Aj but to saddle points.

Numerical integration of (13.8) confirms the stability of the tetrahedral structure. The numerical results are given in Fig. 13.4b, in the form of isolines at 85% of the maximum field intensity. This intensity structure actually consists of two nested structures, shown in Fig. 13.4c and Fig. 13.4d, where the isolines at 93% of the maximum and minimum amplitude are plotted.

13.3 The Nondegenerate OPO

In the case of a nondegenerate OPO, the interaction between the slowly varying envelope of the 3D pump, signal and idler pulses, A0(r , τ, z), A1(r , τ, z) and A2(r , τ, z), respectively, must be considered. This is described by the following set of equations:

Here the coe cients are analogous to those in (13.1), but now correspond to the pump (j = 0), signal (j = 1) and idler (j = 2) waves. The assumption that the changes in the fields during one resonator round trip are small may be made as in (13.1), which allows us to obtain a mapping describing the discrete changes of the subharmonic pulse in successive resonator round trips, and to derive equations of continuous evolution (the order parameter equation).

which is a system of two coupled Ginzburg–Landau equations for the variables