LFNM 2003, 19-20 September 2003, Alushb. Cfimea. Ukralrm 97

ISING- AND BLOCH FRONTS IN 2D PARAMETRIC WAVE MIXING

Ye. Larionova'.', A. Esteban-Martin**, U. Peschel'", C.O. Weiss

Physikalisch- Technische Bundesanstalt, 381 16 Braunschweig, . Germany 'E-mail: janelari@yahoo.com

Fax: 049 531 592 4423

* Visitingfiom Laboratory of Photonics, Kharkiv State Technical University of Radio Electronics, 61 166 Kharkiv, Ukraine *" Visiting from Departament D'Optica, Universitat de ValGncia, Spain

Institut f i r Festkorpertheorte und Theoretische Optik, Friedrich-Schiller Universitdt, Jena, 0-07743, Germany.

In optical systems, in which ,light is generated by a. degenerate, phase-matched wave mixing process, a field can be described by a real-valued function of space and time. The field phase satisfying the phase-matching condition has two possible values due to the degeneracy [l]. As a consequence, domains of opposite phase (with "dark ring"'- spatial solitons [2] as a special case) can exist in systems with a large Fresnel number.

Due to the real-valued nature of the field (a consequence of the phase matching between pump and generated wave) one would expect the phase domains to be separated by "black" domain walls, across which the amplitude of the field crosses zero ("king-walls" [3]). However, we have experimentally observed that besides Ising fronts (real field) as predicted by (the approximate) SHE model in [l], there arealso Bloch fronts (complex field) in the field generated by degenerate four wave mixing in a BaTi03-resonator. We have applied the 2D Fourier transform technique (Fig.1) and a fitting procedure (Fig.2) for reconstructing the

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cture of the fronts.

X X a) b)

Figure 1 - 2D phase of four wave mixing field (a) reconstructed by a Fourier technic (X,Y are the transverse spatial coordinates) from an experimental interferogram (b).

;m\ interferogram 100 2'i[ZT/ ~

Phase. jangle 300 0 fi!

cutthrough 0 50 100 150 200 250 300 350 400

..-~ .., 0 50 100 150 200 250 300 350 400

X hpt i tude 1o

0 50 100 150 200 250 300 350 400 X

a) b) Figure 2 - a) Fit of cut of experimental interferogram of a Bloch front with analytic front solution of [3]; b)

retrieved phase and amplitude across the front.

0-7803-7709-5/03/$17.00 0 2003 IEEE

OR LFNM 2003, 19-20 September 2003, Alushta. Crimea, Ukmina

We find that the Fourier transfom technique is more generally applicable and the experiment shows that it works well. Also, the relatively good reconstruction by the fitting procedure would seem to suggest that the model given in this paper reduces in a certain limit to the 1D front model of [3]. The extended theoretical model confirms that the fronts surrounding phase domains can, contrary to the Swift-Hohenberg model, which is valid near the emission threshold, have Ising- as well as Bloch character.

Extrapolating from 1D we expect the "greyness" of the fronts to be related with front motion. However, we have found from the model that the greyness rather correlates with front curvature (Fig. 3). Even stationary fronts can be grey (if curved). Fig.4 gives an experimental confirmation.

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40

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25 a, & 0.1

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0 .8

curvature 0 5 10 15 20 25 30 35 40 45 50

a) b) Figure 3 - a) Phase structure of moving pattern (Detuning=0.2); A is a real front; B is a complex front; C is a

point like transition.between real and complex fronts; b) greyness vs. curvature of fronts constructed from phase pictures such as a); the relation of phase- and amplitude profile was assumed to obey analytical front solution from [3].

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P W

OO 2 0 4 0 6 a e curvature b)

Figure 4 - a) Recordings of a front straightening in time. Time between pictures is 2 s. The succession of 'pictures shows that the front becomes darker as it becomes less curved. b) A plot of front

"greyness" vs. a rough measured front curvature constructed from a).

Numerical solutions of the model are in a good agreement with the experiment concerning:

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presence of king and Bloch type fronts changing in phase by R across the front (of Bloch as well of king type) dependence of pattems on detuning of resonator as was observed in [4] and predicted in [2] (labyrintic pattems at a large detuning, vanishing structures at small detuning and solitons in intermediate detuning range).

So far we have not experimentally observed the moving asymmetric spatial solitons predicted by this model. The present experiments used only a rather course technique for stabilising the detuning parameter. We have now set up an improved version of the four wave mixing experiment which will allow further tests of this model.

.. REFERENCES

I . K. Staliunas, Victor J. Sanchez-Marcillo, Phys. Rev. A 57, 1454-1457, 1998. 2. K. Staliunas, Victor J. Sanchez-Marcillo, Phys. Lett. A 241,28-34, 1998. 3. P. Coullet, J. Lega, B. Houchmanzadeh, J. Lajzerowicz, "Breaking chirality in

nonequilibrium systems", Phys. Rev. Lett. 65, N 11, 1352-1355, 1990. 4. V.B. Taranenko, K. Staliunas, C.O. Weiss, Phys. Rev. Lett. 81, N 1 I , 2236-2239, 1998.