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March 1, 1997 / Vol. 22, No. 5 / OPTICS LETTERS 283

Modulational instability and pattern formationin the field of noncollinear pump beams

A. V. Mamaev* and M. Saffman

Department of Optics and Fluid Dynamics, Risø National Laboratory, DK-4000 Roskilde, Denmark

Received November 6, 1996

We study pattern formation that is due to modulational instability of noncollinear counterpropagating beams.Angular misalignment of the pumps over a broad range leads to the generation of lines oriented perpendicularto the plane of the pump beams. A theoretical expression for the angular separation of the lines is inclose agreement with observations. Outside this range, for both small and large misalignments, we observesqueezed hexagonal patterns. 1997 Optical Society of America

There are several reasons for the current high level ofinterest in pattern formation in optical systems. Non-linear optics has proved to be an attractive mediumfor studying symmetry-breaking and dynamics innonequilibrium systems. Results obtained in opticsparallel corresponding observations from f luid andother systems, as in the case of formation of hexagons,1

but also reveal exotic spatiotemporal structuresnot seen elsewhere.2– 6 There are also intriguingprospects for applications of pattern formation toinformation processing.7 – 9

A generic mechanism leading to optical pattern for-mation is the transverse modulational instability ofpump beams counterpropagating in a cubic nonlin-ear medium. For collinear pumps there is cylindri-cal symmetry about the z (propagation) axis, andhexagonal patterns are typically observed.1,10– 12 Itwas shown previously that misalignment of the pumpbeams breaks the phase-matching condition in theplane that contains the pumps and leads to a collapseof the hexagonal pattern into a one-dimensional rollpattern in the plane perpendicular to the plane of thepump beams.1 If the instability is restricted to theplane of misalignment, e.g., by use of elliptically shapedpump beams, a far-field frequency shift and a corre-sponding near-f ield drift motion that are proportionalto the misalignment angle are obtained.13,14

We show here that introducing pump misalign-ment, while permitting instability in both transversedimensions, leads to the generation of previouslyunobserved patterns. For intermediate values ofthe misalignment we observe lines perpendicular tothe misalignment plane. Modulational instabilityof the lines leads to a two-dimensional transversepattern with rhombic symmetry. For small mis-alignment we observe both rolls and squeezedhexagons, and for relatively large misalignment, inwhich the angle at which the lines are generatedis comparable with the normal angle for hexagongeneration, we observe squeezed hexagons.

The generation of lines can be explained in thecontext of the simplified model of a thin nonlin-ear slice with a feedback mirror.15 Consider counter-propagating pumps given by F ­ F0szdexpsiqxdexp 3

f2isq2y2k0dzgexpsik0zd and B ­ B0szdexpsiqxdexp 3

0146-9592/97/050283-03$10.00/0

fisq2y2k0dzgexps2ik0zd, corresponding to a tilt of theinput beam by angle f ­ qyk0, where k0 ­ 2pyl isthe vacuum wave number. We consider the instabilityof F and B with respect to angularly tilted sidebandsof the form f6szdexps6ik'xd and b6szdexps6ik'xd.Specification of the boundary conditions introducesdiffraction into the linear stability analysis in the form

b6s0d ­ f6s0dexpsi2lk6d , (1)

where k6 ­ sk' 6 qd2y2k0 and l is the distance fromthe nonlinear slice to the mirror. For no misalignmentsq ­ 0d the normal roll instability appears at u0,slice ­k'yk0 ­

plys4l d in a self-focusing medium.15 In the

presence of f inite misalignment the 1 and 2 sidebandsexperience equal diffractive phase shifts modulo 2p,provided that 2lsk1 2 k2d ­ 2mp, where m is aninteger. This implies that, provided that the thresholdfor the formation of rolls in the aligned system isreached, there will be the possibility of generation oflines at angles

um ­ mp

2k0lf. (2)

Note that the angular separation of the lines isinversely proportional to the misalignment of theinput beam.

Formation of lines in the f ield of noncollinear pumpsis a general mechanism that is not restricted to thethin-slice model. The arguments leading to Eq. (2) forthe angular separation of the lines are valid for thethin-slice geometry. For a thick nonlinear mediumwith a feedback mirror (as in our experiments) or for anexternally pumped thick medium, coupled equations forthe sidebands of misaligned pumps should be solved.Based on the known similarities between pattern for-mation in thick media and in the thin-slice model, wedo not expect the presence of a thick medium to changethe situation significantly. The validity of our modelin describing an experiment with a thick nonlinearmedium is demonstrated by the good agreement be-tween Eq. (2) and experiment, demonstrated in Fig. 2below.

We have observed this instability by using thegeometry shown in Fig. 1. A crystal of KNbO3:Fe

1997 Optical Society of America

284 OPTICS LETTERS / Vol. 22, No. 5 / March 1, 1997

Fig. 1. Experimental setup. Similar near- and far-fieldpictures were observed from either end of the crystal.

measuring 5.2 mm along the c axis was illuminated bya 30-mW beam at 532 nm focused to a spot of Gaussiandiameter 0.8 mm. We polarized the light parallel tothe a axis to take advantage of the r13 electro-opticcoeff icient. The counterpropagating pump beam wasgenerated by the 16% ref lection from the uncoated backsurface of the crystal. The instability in this geometryis due to the formation of ref lection gratings with kg ­2k0n lying along the c axis, where n ­ 2.3 is the indexof refraction of the crystal. The crystal was orientedsuch that the ref lected beam B was amplified by pumpbeam F . The estimated intensity gain coefficient forthe crystal that we used was Gl ø 12.

Figure 2 shows the far-field patterns and themeasured dependence of the angle u1 on the input mis-alignment f. Despite the simplicity of the theoreticalmodel there is good agreement with the experimen-tally observed angular separation of the lines. Thethreshold for modulational instability of the linesat 6u1 is higher than for the central line because asmaller fraction of the total light intensity participatesin the generation of the extra lines compared with thecentral line.

The vertical lines have a clearly visible ring struc-ture. The rings are due to the presence of Fabry–Perot resonances in the crystal, which was preparedwith the c faces parallel to ,0.2 mrad. Small changesin pump-beam intensity, and hence in the amount ofabsorption-driven heating in the crystal, change thecrystal’s length and the angular position of the Fabry–Perot resonances of the cavity.6 When a Fabry–Perotresonance coincides with u0 ø

pnlyl, which is the

angle for roll instability in a thick medium with non-linear energy coupling,16,17 the effective threshold isreduced18 and the extra lines also become modulation-ally unstable, as Fig. 3 shows. The roll instability hasangular scale u0, whereas the lines are angularly sepa-rated by u1, so the resulting two-dimensional pattern isnot rotationally symmetric. The relative phase of theinstability along the three lines is not fixed, so two-dimensional tilings with rhombic symmetry, as seenin Fig. 3(a), but also rectangular symmetry can beobserved. The near-field pattern shown in Fig. 3(b)moved along x, as was observed previously.13

At small values of f the angle between the lines isso large that the overlap with the pump beams overthe length of the crystal is reduced, so the excitationof the lines becomes weaker. Under these conditionsthe possibility of excitation of the lines also dependson the cavity tuning, as shown in Fig. 4, obtained forfixed f ­ 0.83 mrad but variable crystal temperature.Changes both in the degree of excitation of the linesand in the symmetry of the central pattern are seen.Note that the hexagons are squeezed by ,5% along thex axis.

At large values of f where u1 becomes compara-ble with u0 the lines disappear, and instead we ob-tain squeezed hexagonal structures, as shown in Fig. 5.These hexagons were squeezed by , 10%, and thesqueezing took place along either the x or the y axes,depending on fine adjustment of f. The reappear-ance of hexagons, despite large pump-beam misalign-ment, can be interpreted as a cooperative instabilityarising from the similarity of u0 and u1. Previousexperiments with hexagon formation in KNbO3 haveshown that the hexagonal pattern can rotate about theaxis. It has been shown by Honda19 that the rotationis due to, and can be controlled by, gradients in the

Fig. 2. Measured dependence of the instability angle u1on the misalignment angle f. The solid curve is Eq. (2)[assuming that the condition 2lsk1 2 k2d ­ 2mp holdsat the input end of the crystal, Eq. (2) becomes um ­mlnys4lfd, where um and f are external angles], and thedashed line shows the angle of the normal roll instability.The insets show the far-field patterns observed for f ­ 0.63and f ­ 2.6 mrad.

Fig. 3. (a) Far-field and (b) near-f ield rhombic patternsobserved for f ­ 1.9 mrad.

March 1, 1997 / Vol. 22, No. 5 / OPTICS LETTERS 285

Fig. 4. Variation of the far-f ield pattern for f ixed f ­0.83 mrad and different resonator tunings.

Fig. 5. Hexagons squeezed along (a) y and (b) x observedfor large values of f.

pump-beam intensity. The squeezed hexagons havea preferred angular orientation that inhibits rotation.

We observed instead a periodic rocking motion consist-ing of initial formation of a pattern with the alignmentseen in Fig. 5, rotation by ,15 deg, followed by a rapidshift back to the initial orientation, after which the se-quence was repeated.

In summary, we have observed some new opticalpatterns obtained with misaligned pump beams. Wenote f inally that very similar results are obtained whenthe misaligned second pump beam is generated by atilted high-ref lectivity mirror placed directly behindthe crystal.

This research was supported by a grant from theDanish Natural Science Research Council.

*Permanent address, Institute for Problems inMechanics, Russian Academy of Sciences, VernadskyProspect 101, Moscow, 117526 Russia.

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