12.3.1 Turing Instability in a DOPO

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At the threshold of the pattern-forming instability, the real part of the eigenvalue of the wavenumber with maximum growth is zero. In Fig. 12.6, the o -resonance and LALI instability regions are plotted in the parameter space (ω1, E) for a specific value of the di raction parameter. The regions are well separated in the parameter space, and therefore can be associated with di erent mechanisms. The o -resonance instability exists for all values of the pump intensity above threshold, whereas the LALI instability appears only at some critical pump value that depends on the di raction parameter a.

Fig. 12.6. Instability regions in the parameter space (ω1, E) for nonzero di raction parameter a = 5, evaluated from a linear stability analysis. There are two instability regions: for negative detuning, the traditional o -resonance instability; for positive detuning, the Turing instability

We note that pump di raction not only creates the LALI instability, but also modifies the o -resonance instability range, as can be seen from Fig. 12.6. For zero pump di raction the o -resonance instability occurs between the dashed curve and the left part of the solid curve corresponding to the neutral-stability line, as follows from the standard analysis. Pump di raction increases significantly the o -resonance instability region. However, the spatial scale of the o -resonance pattern is not modified by the presence of pump di raction.

Another important feature that reveals the di erent nature of the patterns on both sides of the resonance is the corresponding wavelength. In the case of o -resonance patterns, this wavelength depends mainly on the resonator detuning and the di raction coe cient of the signal wave. In contrast, the wavelength of the pattern in the LALI region depends essentially on the pump and on the ratio of the di raction coe cients a, and very weakly on the resonator detuning. This behavior is shown in Fig. 12.7, where the squared wavenumber of the maximally growing mode is plotted against the detuning (full line). The broken part of the curve, in the neighborhood of the resonance, corresponds to negative eigenvalues.

Some analytical expressions can be found in di erent limits. For negative detuning, the wavenumber is given by

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inset of Fig. 12.8. For comparison, the preferred patterns occurring in the o -resonance region are not hexagons, but stripes.

The threshold condition for a LALI instability can be evaluated analytically, but is in general a complicated function of the parameters. However, Fig. 12.8 indicates two main features: (i) there exists a hyperbolic relation between the pump and di raction parameters, and (ii) a minimum value of the di raction parameter am is required to reach the instability for a fixed detuning. This guided our search for an asymptotic expression, where we introduced a smallness parameter related to the deviation from the thresh-

old, given by E0 = 1 + ω21. Assuming that R = E0 (E − E0) ≈ O(ε) and D = a − am ≈ O(1/ε), and expanding the eigenvalue, we find, at leading

order in ε, that the homogeneous solution is unstable whenever

The minimum value of the di raction parameter am, which depends on the detuning, can be evaluated by analyzing the opposite limit, i.e. at large values of the pump parameter above the threshold. In this case we find that the instability can be observed only when a > am, where

where η is a positive function of the signal detuning. In the limit of small detuning, (12.25) yields an asymptotic value η = 27/8ω21.

The expression (12.27) is plotted in Fig. 12.8 (as the dashed line) for ω1 = 2 (for which η = 2). Notice the good correspondence with the exact (full line) and numerical (symbols) results.

12.3.2 Stochastic Patterns

The Turing instability in a DOPO occurs only for nonzero signal detuning, as follows from the stability analysis and also from (12.25). However, in resonance, some transverse wavenumbers are weakly damped for nonzero pump di raction. The wavenumber of the weakly damped modes can be obtained from a linear stability analysis of (12.22). In the limit of far above the threshold (E 1), this wavenumber is given by

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the spatial variables in (12.22), this length can be expressed as

which, together with (12.27), is strikingly similar to the conditions derived in [9] for the Brusselator, a paradigmatic model of chemical pattern formation.

Clearly, the scale of the pattern given by (12.30) depends on the di raction coe cients of both fields. It is interesting to compare the scale of the Turing pattern with the characteristic scales of the components, given by their spatial evolution in the absence of interaction. For this purpose, we consider first a deviation from the trivial solution, A0 = E + X, A1 = Y . In the resonant case and neglecting the nonlinear interaction, (12.22a) leads to

or, equivalently,

From the solutions of (12.32) and (12.33), we can define a characteristic spatial scale for the activator, La = a1/E, and for the inhibitor, Li = √a0, corresponding to the signal and pump fields, respectively. Now the scale of the generated pattern can be written in terms of the scales of the activator and the inhibitor, as

revealing that the characteristic spatial scale of the pattern is the geometric mean of the spatial scales of the interacting components.

It is possible to find a simple relation between La and Li in the limit of a am and E E0 (large di raction and pump parameters, and moderate detuning). In this case, the instability domain (12.27) takes the form aE > η, which can be expressed in terms of the characteristic lengths to give the threshold condition

The value of η depends on the signal detuning and can be evaluated from (12.25)√. We find that η > 1/2 always and, in particular, that η > 1 for ω1 < 3 3. Therefore, for small (and also moderate) detuning, the inhibitor range must be larger than the activator range for the occurrence of the LALI

instability. This is in accordance with the assumptions made in the derivation of (12.28).

The conditions defined by (12.34) and (12.35) are typically found in reaction–di usion systems, and are a signature of the Turing character of the instabilities described above.

12.3.3 Spatial Solitons Influenced by Pump Di raction

The spatial modulation induced by pump di raction also influences the stability of solitons [10], in accordance with the results of Chap. 11. In order to show this, we performed a numerical integration of the DOPO equations (12.22) for di erent values of a0. The amplitude along a line crossing the center of a soliton is plotted in Fig. 12.10, showing that the di raction enhances the spatial oscillations strongly.

The parameters that define the shape of a soliton are the exponent of the spatial decay and the wavenumber of the oscillating tails. These parameters can be analytically evaluated by means of a spatial stability analysis. We assume that the intensity of the field is perturbed from its stationary value in some place in the transverse space (owing to the e ects of boundaries, a spatial perturbation or a defect in the patterns), and look at how this perturbation decays (or grows) in space. For this purpose, we consider evolution in space instead of time. When the system has reached a stationary state, the solution, which we assume to have radial symmetry, can be written in the time-independent form

Fig. 12.10. Amplitude profile of a soliton across a line crossing its center, evaluated numerically for di erent pump di raction coe cients, a0 = 0.0005, 0.002 and 0.01. The amplitude of the modulation of the tails increases with increasing di raction. The other parameters are a1 = 0.001, E = 2, ω1 = −0.6, ω0 = 0

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.

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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.