11.7 Domain Boundaries and Image Processing

163

Fig. 11.17. Spatial localized structures. Left: the transient stage is shown, where one domain (the upper one) is contracting, and three other domains have already contracted to minimum radius. Right: two stationary solitons are visible

1

1

1

1

a )

2

b ) 2

c ) 2

d ) 2

Fig. 11.18. A contracting domain boundary (2), and a stable soliton (1). Time increases from (a) to (d)

Length of domain boundary

	25
	20	2
	15
	10	1
		Fig. 11.19. Length of the domain bound-
		aries of (1) and (2) from Fig. 11.18 as a
		function of time. The lines are to guide

0	10	20	30	time	the eye

of the domain boundaries are plotted as a function of time in Fig. 11.19. This evidences that the solitons are not marginally stable small domains, but really are stable formations (with a finite stability range).

11.7 Domain Boundaries and Image Processing

Phase domains and spatial solitons in the form of dark rings could be a useful tool for parallel analog information processing [13], which could be applied to all-optical artificial vision, optical neural networking or optical sensing. Here

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.

11.7 Domain Boundaries and Image Processing

165

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.

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