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164 11 Phase Domains and Phase Solitons

Fig. 11.20. A photo of an airplane (left) and a reduced-resolution version (on a grid of (128×128 points). The reduced-resolution version was processed (see Fig. 11.21)

we give just an example that shows the potential of this technique to locate and track stationary and moving objects. In Fig. 11.20, an original photo of an airplane (left) and its reduced-resolution version on a grid of 128×128 points (right) are shown. The DOPO equations (11.1) were solved using the distribution in Fig. 11.20 as the initial condition for the subharmonics. In practice, this could be achieved by a short injection of a field with a distribution of subharmonic frequency corresponding to the image to be processed.

The temporal evolution of the spatial distribution of subharmonics is shown in Fig. 11.21, as obtained from the numerical integration of (11.1). The contour of the object (the airplane) is automatically reproduced by dark curves (the domain boundaries). In the nonlinear evolution (which in a experiment using a DOPO could take just few picoseconds), the dark curves contract into a phase soliton positioned at the center of the object (targeting).

The detailed scenario is as follows:

1.t = 1. There is essentially an image of the injected field distribution. The amplitude of the subharmonics at the beginning of the processing is 0.1% of the saturation value of the subharmonic amplitude (the corresponding value in terms of intensities).

2.t = 3. Linear stage of amplification: a smoothing of the field due to spatial filtering is observed. The form of the spatial filter is a ring or a central spot in the far field, depending on the detuning.

3.t = 10. Nonlinear saturation is reached: the amplitude distribution in the domains becomes more and more regular (more “flat”). The nonlinear evolution of domains begins.

4.t = 30. The domain begins contracting.

5.t = 90. The domain boundary becomes smoother, and contraction continues.

6.t = 130. At a particular size of the domain, the contraction slows down. This particular radius is roughly 3r0, where r0 is the radius of the final soliton. The evolution, however, does not stop at this size of the domain, but the domain continues contracting.

Fig. 11.21. Domain evolution as obtained by numerical integration of DOPO equations (11.1). Spatial intensity distributions are shown. The initial distribution of subharmonics is taken (injected) from Fig. 11.21. Parameters are: E = 2, ω0 = 0, ω1 = −0.2, γ1 = γ0 = 1 . Di raction coe cients are: a1 = 0.001 and a0 = 0.0005. Integration was performed with periodic boundary conditions in unit size region

7.t = 190. The contraction again slows down, this time at a radius 2r0.

8.Finally, at t = 250, the domain contracts to the final state, a stable soliton, at the geometrical center of the object.

An interesting point is that the evolution of the domain does not exactly follows the velocities (11.16) and (11.17), but slows down at some stages of evolution. This slowing down can be explained by the modulation of the slope of the variational potential shown in Fig. 11.11. At some radii of the domain, roughly equal to integer multiples of the radius of the fundamental soliton (r = nr0, where n = 1, 2, 3 ...), the slope of the potential is minimum (it is maximum at radii r = (n + 1/2) r0). The force causing contraction (the derivative of the potential) is minimum at radii that are integer multiples of r0, and here the contraction velocity is also minimum. The modulation of the slope of the potential is due to the spatial modulation of the tails of the domain boundaries, as discussed above in Sect. 11.5.

For su ciently strong modulation of the tails, the potential can have multiple local minima, at radii r = nr0. In this case a domain can stop contracting at those radii. Figure 11.22 shows some snapshots of the evolution of domains with larger spatial modulation of the tails of the domain boundaries (e.g. because of larger pump di raction, as will be discussed in the next chapter).

This simple example shows the unique possibilities of phase domains in analog image processing. Domain boundaries may simulate the margins of an object. Discrete solitons can count the objects of interest in the field of vision. A (stroboscopic) array of solitons can be left behind to track a moving object and the trajectory of the moving object can be recorded by discrete positions

166 11 Phase Domains and Phase Solitons

Fig. 11.22. Some snapshots from domain evolution as obtained by numerical integration of the DOPO equations. The parameters and initial conditions are as in Fig. 11.21 except for the pump di raction coe cient, a0 = 0.0015 (three times as large as in the previous case). The larger di raction of the pump enhances spatial modulation (see Chap. 12), and consequently enables not only a stable fundamental soliton (of minimum radius), but also stable solitons with twice (or even three or more times) the minimum radius

of solitons. In general, solitons can discretize the properties of objects. And, finally, moving (inertial) solitons may be employed to forecast the trajectory of an object or the evolution of an image in general by all-optical means.

References

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2.K. Staliunas and V.J. S´anchez-Morcillo, Spatial localized structures in degenerate optical parametric oscillators, Phys. Rev. A 57, 1454 (1998). 147, 160

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6.J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319 (1977). 149

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12 Turing Patterns in Nonlinear Optics

12.1 The Turing Mechanism in Nonlinear Optics

A well-known transverse-pattern formation mechanism in broad-aperture lasers and other nonlinear resonators is o -resonance excitation. If the central frequency of the gain line of the laser ωA is larger than the resonator resonance frequency ωR, then the excess of frequency ∆ω = ωA −ωR causes a transverse (spatial) modulation of the laser fields, with a characteristic transverse wavenumber k obeying a dispersion relation ak2 = ∆ω, where a is the di raction coe cient of the resonator. The patterns that occur in such a way play the role of a “bridge” between the excitation and the dissipation, which occur at di erent frequencies, and these patterns enable maximum energy transfer through the system.

In all the previous chapters, patterns due to o -resonance excitation have been studied. These were patterns in lasers, photorefractive oscillators, degenerate and nondegenerate optical parametric oscillators, and four-wave mixers. For a degenerate OPO, the excitation frequency is equal to half of the pump radiation frequency ωA = ω0/2, and its mismatch from ωR leads to the same macroscopic pattern formation mechanism as in lasers. The o -resonance mechanism not only excites extended patterns (such as tilted waves, rolls, square vortex lattices and hexagons), but is also responsible for the stability of localized structures in the above systems.

The o -resonance pattern formation mechanism is essentially a geometrical one. It resembles the formation of rolls in Rayleigh–B´enard convection, where the width of the convection rolls is fixed mainly by the distance between the upper and lower plates. In optical resonators, the propagation angles of o -axis components are fixed by the resonance conditions. The spatial scale is thus fixed not by nonlinearity, but by linear geometric e ects.

The pattern formation mechanism discovered by Turing for reaction–di u- sion systems [1] has a di erent origin from the mechanism discussed above. Here, at the root, is an interplay between the di usions of two (or more) interacting components. The coupling between a strongly di using (lateral) inhibitor and a weakly di using (local) activator is responsible for the pattern formation.

The simplest (linearized) representation of such a reaction–di usion equations displaying a Turing instability is given by the model [2]

K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 169–192 (2003)c Springer-Verlag Berlin Heidelberg 2003

In this system u1 plays the role of the activator and u2 the role of the inhibitor, with di usion coe cients d1 and d2 respectively. The particular form of the cross-coupling matrix (where ai and bi have positive values) leads to maximum amplification of the wavenumbers obeying

as follows from a stability analysis of (12.1).

Motivated by this analysis, one might ask the following question: is the Turing mechanism possible in nonlinear optics too? Let us take as an example

with an unsaturated population inversion D0 and a spatial-wavenumber selection factor g. Let us simplify (12.3) by assuming a very narrow gain line, i.e. g 1, which makes di raction negligible when compared with di usion, and zero detuning, ω = 0. Also, which is very significant here, let us assume that the population inversion also di uses, which results in adding a Laplacian to (12.3b). In this case, if we define the field di usion constant d1 = gd2, (12.3) converts to

a system of two nonlinearly coupled di using components. The field di usion is governed not by the usual Laplace operator, but by the second power of the operator (sometimes called super-di usion, as mentioned earlier); however,

The similarity to the Turing system (12.1) becomes more evident if we change the sign of the perturbation of the population inversion d. This results