Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf
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12.3 Optical Parametric Oscillator with Di racting Pump |
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Fig. 12.7. The wavenumber of the pattern, given by a linear stability analysis for a = E = 10. The exact value is given by the solid line. The dashed lines correspond to analytical expressions given in the text
which clearly corresponds to the o -resonance patterns selected by the cavity. For positive detuning,
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(12.24) |
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The asymptotic expressions (12.23) and (12.24) are represented by dashed curves in Fig. 12.7, to be compared with the exact result (full line).
Turing patterns were found by numerical integration of (12.22). In Fig. 12.8 we show the threshold for the emergence of spatial patterns, for a fixed value of signal detuning. Results obtained from the linear stability analysis described above (full line) are shown, together with numerical results for some values of the di raction parameter (represented by symbols). In all cases, the final LALI patterns have hexagonal symmetry, such as the one shown in the
Fig. 12.8. Critical pump value for Turing instability as a function of the di raction parameter, for fixed signal detuning ω1 = 2. The symbols represent the result of numerical integration of the DOPO equations. The solid curve represents the result of a semianalytical calculation based on a linear stability analysis, and the dashed curve corresponds to the boundary of the instability domain given by (12.27). The inset shows a hexagonal pattern obtained numerically for a = 5, E = 3
184 12 Turing Patterns in Nonlinear Optics
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
27D2R2 − ω12 (2DR − 1)3 < 0 . |
(12.25) |
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
am (E0 − 1) = 1 . |
(12.26) |
From (12.26) it follows that am grows monotonically when the detuning is |
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decreased, and that a > 1 (and consequently a0 > a1) when ω1 < |
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Finally, the instability domain (12.25) in the original variables is |
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(a − am) (E − E0) < η , |
(12.27) |
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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
k2 = |
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(12.28) |
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12.3 Optical Parametric Oscillator with Di racting Pump |
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which is valid also for small values of the detuning ω1, where pattern formation is expected.
Equation (12.28) corresponds to a ring of weakly damped wavevectors, in the spatial Fourier domain. To check the existence of the ring numerically one must introduce a permanent noise. We can expect that the homogeneous solution will then be weakly modulated by a filtered noise, with a characteristic wavenumber given by (12.28).
In order to incorporate the noise, we have modified (12.22) by adding a term √γiΓi to the evolution equation for each field component Ai. These terms, introduced phenomenologically, represent stochastic Langevin forces defined by
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δ(r1 − r2)δ(t1 − t2) |
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(12.29) |
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where Ti are the corresponding temperatures.
A typical result of numerical integration of the Langevin equations is shown in Fig. 12.9, where a snapshot of the amplitude and the corresponding
2| k |
averaged spatial power spectrum A( ) are shown. As expected, no spatial wavenumber selection was visible in the case of zero pump di raction. For nonzero pump di raction, the DOPO filters the o -axis noise components, and a ring emerges in the far field (Fig. 12.9, right). If the pump di raction parameter is increased, the induced wavenumber ring decreases in radius and becomes more dominant, in accordance with (12.28).
The above calculations were performed for zero detuning for both waves. Therefore all possible pattern formation mechanisms due to o -resonance excitation are excluded.
The expression (12.28) for the wavenumber, although evaluated at resonance, is a good approximation to the wavenumber of the patterns for moderate values of the signal detuning, and corresponds to a characteristic length of the emerging pattern Lp = kc−1. Returning to the initial normalizations of
Fig. 12.9. Stochastic spatial distribution (left) and averaged spatial Fourier power spectrum (right) obtained by numerical integration of the DOPO Langevin equations, for ω1 = 0, E = 2, a0 = 0.005, a1 = 0.0005 (a = 10). The averaging time was t = 300. The zero spectral component has been removed
186 12 Turing Patterns in Nonlinear Optics
the spatial variables in (12.22), this length can be expressed as
Lp2 = |
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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
−Y + EY + ia1 2Y = 0 , |
(12.31) |
or, equivalently,
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Y = 0 . |
(12.32) |
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Similarly, from (12.22b) we find |
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−X + ia0 2X = 0 . |
(12.33) |
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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
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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
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ηLa . |
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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
12.3 Optical Parametric Oscillator with Di racting Pump |
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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
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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