Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf
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152 11 Phase Domains and Phase Solitons
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Fig. 11.2. The half-width x0 of the dark line |
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∆tional (11.11), as a function of the detuning ∆
11.3.2Two-Dimensional Domainsgiven by (11.10), and the calibrated 1D func-
To study analytically the motion of a domain boundary in 2D, we assume a ring-shaped form. This represents, equivalently, a circular domain centered at the origin of a polar coordinate system. Such a dark ring can be described by the ansatz
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where r0 is the radius of the ring, and the half-width x0 is given by (11.10). Owing to the cylindrical symmetry of the ansatz (11.12), the variational potential can be written as
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r dr , (11.13) |
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which has to be calibrated in the same way as in the 1D case, to avoid the background contribution.
Analytical integration of (11.13) using the ansatz (11.12) is not possible in the 2D case; however, two other complementary approaches can be used. One possibility is to evaluate approximately the integral in some limiting cases. The other possibility is to calculate the integral (11.13) numerically.
When the radius of the ring is large enough (when r0/x0 1), the order parameter A(r) is nearly an odd function with respect to the ring radius r0. Making the change of variables x = r − r0, assuming the integrated function f(x) to be even, and extending the integration limits to the whole space, we obtain
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F2D = 2π |
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f(x) dx = 2πr0F1D . (11.14) |
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11.3 Dynamics of Domain Boundaries |
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This result is not exact, because the integration limits in reality do not extend to infinity, and also because the integrated function is not even, but has a small odd part. The errors occurring because of these two assumptions give corrections to (11.14) of orders O(exp (−r0/x0)) and O(1/r0), respectively. Therefore, for an asymptotically large radius of the ring r0, and for a width of the dark line x0 = O(1), we obtain the potential
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(11.15) |
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Thus the potential in 2D is roughly equal to the potential in 1D (11.11) multiplied by the length of the weakly curved (circular) dark line. In this approximation, the sign of F1D determines the evolution of the ring. As can be seen from Fig. 11.2, around zero detuning F1D is positive, and thus the longer the domain boundary is, the larger the 2D potential is. As the solution tends to minimize the potential, the domains contract. For a detuning larger than some ∆c, the 1D potential is negative, and the domains expand. The
particular case of stationary rings occurs at ∆c = 2/7 ≈ 0.535.
The result of numerical integration of the RSH equation (11.4) shows contraction or expansion of the domains, depending on the detuning. In Fig. 11.3, an example of domain contraction for a small value of the detuning is given.
Figure 11.4, in contrast, shows domain expansion for a large detuning value. The asymptotic pattern in this case is a labyrinth structure. The numerical results indicate that rings of large radius are marginally stable at a detuning ∆c = 0.45 ± 0.05, which di ers from the analytically evaluated
Fig. 11.3. Evolution of phase domains obtained using the RSH equation for a small signal detuning, ∆ = 0.25. The integration was performed in a box of size 70 units (the integration grid contained 128×128 points), with periodic boundaries. Time increases from left to right. In the upper row the field intensity is plotted, and the lower row shows the field phase. The pictures were obtained at times t = 0 (the initial distribution), t = 20, t = 100 and t = 250. The last domains disappear at t = 370
154 11 Phase Domains and Phase Solitons
Fig. 11.4. Evolution of phase domains obtained using the RSH equation for a large signal detuning, ∆ = 0.65. Other parameters as in Fig. 11.3. The initial picture at time t = 0 was prepared by integrating the RSH equation with a detuning ∆ = 0.25 (as in the previous figure), and further calculations were then performed with the new detuning value. The other plots were obtained at times t = 20, t = 150 and t = 1000
equilibrium detuning value given above. This di erence occurs because of the inexact form of the ansatz. The domain boundaries are not exactly of hyperbolic-tangent form (with monotonically decaying tails), but show an oscillatory decay of the tails. A more accurate analysis that takes account of these spatial oscillations and leads to a better correspondence with the numerical values is performed in Sect. 11.5.
In this way, by varying the detuning ∆, one can manipulate the domain dynamics, forcing the domains to expand or to shrink. Expanding domains keep their topological properties during the evolution for moderate values of the detuning: the number of domains in a finite labyrinth pattern is equal to that in the initial pattern, as can be seen in Fig. 11.4.
The situation is di erent for larger values of the detuning, in particular
when ∆ > 2/3. In this case, as follows from the linear stability analysis, the homogeneous solution is modulationally unstable. Domain growth and labyrinth formation are now accompanied by the appearance of new domains (nucleation), as shown in Fig. 11.5. Another peculiarity of the large-detuning case is that the dark lines in the labyrinths can break and reconnect, which also leads to topological changes of the domains.
Fig. 11.5. Evolution of phase domains obtained using the RSH equation a for large signal detuning, ∆ = 0.85. Other parameters as in Fig. 11.3. The plots were obtained at times t = 0, 20, 150 and 1000
11.4 Phase Solitons |
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An expression for the velocity of the moving fronts in a potential system can also be derived. In the case of a cylindrically symmetric ring, the radial velocity v = dr0(t)/dt is [7]
∂F/∂r0
v = − , (11.16) 2π (∂A/∂r)2r dr
and is proportional to the force acting on the ring (the gradient of the variational potential) and inversely proportional to the “mass”, or inertia, of the ring. In the limit of large ring radius, the velocity can be evaluated analytically. In this case, using (11.14) for the potential, and evaluating the integral in the denominator in (11.16) in the same way as above, we obtain
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that is, the velocity is inversely proportional to the ring radius.
The behavior of a circular domain boundary can be generalized to domains of arbitrary form. Assuming that the curvature of the dark line is su ciently small, the equation for the local motion of the curve is ∂R/∂t = −vc, where the local motion of the dark line is directed along its normal and is proportional to the local curvature c = ∂2R/∂l2, with the proportionality coe cient given by (11.17).
11.4 Phase Solitons
The above analysis predicts either the contraction or the expansion of domains. This conclusion is valid, however, only for a su ciently large radius of a domain or, equivalently, for a su ciently small curvature of a dark line. In other words, when diametrically opposite segments of the domain boundary do not interact. For small domains, when the diameter is of the same order of magnitude as the width of the domain boundary, the situation may be di erent. Indeed, numerical integration of the RSH equation sometimes shows that the dark rings stop contracting at some small radius. Figure 11.6 shows such a scenario, where an ensemble of stable dark rings of fixed radius evolves. We call these stationary small domains phase solitons, by analogy with the amplitude solitons studied in Chap. 9. A comparison of the profiles of the two types of solitons was shown in Figs. 8.1 and 8.7.
The variational approach of the previous section cannot predict analytically the existence of solitons, since the assumption r0/x0 1 is no longer valid. Numerical integration of (11.3), however, shows that phase solitons exist for detuning values in the range 0.287 ± 0.001 < ∆ < 0.460 ± 0.001.
The interaction of opposite segments of the ring results in a repulsive force that balances the attraction between the fronts due to the tendency to contraction, allowing soliton formation.
156 11 Phase Domains and Phase Solitons
Fig. 11.6. Evolution of phase domains obtained using the RSH equation for an intermediate signal detuning, ∆ = 0.35. Other parameters as in Fig. 11.3. The plots were obtained at times t = 0, 20, 150 and 1000
To demonstrate the repulsive e ect of the interaction, the potential (11.13) has been integrated numerically using the ansatz (11.12). Three characteristic plots are given in Fig. 11.7, showing the dependence of the potential on the radius of the ring for di erent values of the detuning ∆, together with the analytical approximation (11.14) (dashed lines).
As predicted from the 1D analytical calculations, for small detuning, the potential increases with the radius, leading to a contraction of the ring. Correspondingle, for large detuning, the potential decreases with increasing radius, leading to an expansion However, for some intermediate values of the detuning, the potential exhibits a minimum at some radius of the ring (the middle curve in Fig. 11.7; see also the inset). This potential minimum indicates the existence of phase solitons, with a radius corresponding to the potential minimum. The final distribution in the series shown in Fig. 11.6 is an ensemble of such solitons. These solitons are similar to those found in systems showing optical bistability, whose order parameter equation is also of Swift–Hohenberg type [9].
Although a variational analysis using the ansatz (11.12) yields a potential minimum at some radius of the dark ring, thereby predicting its stability, the evaluated stability range 0.39±0.01 < ∆ < 0.52±0.01 does not coincide with the numerically calculated stability range 0.287 ±0.001 < ∆ < 0.460 ±0.001.
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Fig. 11.7. The potential obtained by evaluating (11.13) numerically with the ansatz (11.12), for a small detuning ∆ = 0.3, for a large detuning ∆ = 0.6 and for an intermediate value of detuning ∆ = 0.45. The 1D potentials calculated analytically (11.11) are shown by dashed lines. The case of ∆ = 0.45 is magnified in the inset