Observation of speckle pattern formation in transparentnonlinear random media
Francisco J. Rodríguez,1,4 Can Yao,1 Jorge L. Domínguez-Juárez,1 Jorge Bravo-Abad,2 and Jordi Martorell1,3,51ICFO—Institut de Ciències Fotòniques, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain
2Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049, Madrid, Spain3Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Terrassa, Spain
When light is generated or propagates in random struc-tures, there are several manifestations of the wave naturethat survive the phase front distortion, such as specklepattern formation [1], the appearance of a backscatteringcone [2,3], and random lasing [4]. In general, any of suchphenomena occurs in a disordered distribution of onematerial embedded in another one with a different indexof refraction. Scattering of light is also observed fromtransparent nonlinear materials in which the nonlineardomains are randomly distributed in space [5]. However,no clear signs of a coherent interaction have beenreported from the light generation in such nonlinear crys-tals. In fact, it has been shown that the coherent additionof the two interacting waves is washed out by the dis-order, and the growth of the nonlinear process versusthe thickness of the material becomes linear instead ofquadratic [5–8].In this Letter, we demonstrate that the granulation ob-
served on the second-harmonic generation (SHG) pat-tern from a strontium barium niobate (SBN) crystaloriginates from the interference between the differentsecond-harmonic (SH) waves independently scatteredin all directions by all the nonlinear domains that com-pose the crystal structure. Specifically, by measuringthe angular distribution of SH light emerging from aSBN crystal, we find distinct features showing specklepattern formation at the SH frequency. The intensity dis-tributions in the observed speckle patterns are in verygood agreement with the predictions obtained from aGreen’s function formalism, which explains the observedspeckle in terms of the interference at the observationpoint among the SH light generated from the differentnonlinear domains. The theoretical model used showsthat a matching of the phases across the crystal amongthe second-order nonlinear polarization and the SH fieldsdoes not play a role in the reported phenomena. It is im-portant to note that the observation reported in this workcannot be predicted in a straightforward manner usingthe conventional description of SHG in SBN based ona continuous distribution of reciprocal lattice vectors [9].
Strontium barium niobate is a transparent crystal, fer-roelectric at room temperature with large second-ordersusceptibility when both the fundamental and SH fieldsare polarized parallel to the polar c axis. The ferroelectricdomains form columns along the c axis with randomantiparallel polarizations. Because of this opticallyequivalent alignment of the domains, light propagatingthrough the crystal does not experience a refractive in-dex change. As a result, only an extremely weak linearscattering originated by the domain walls may be ob-served [10]. In contrast with the linear light transmission,the SHG is strongly diffused in a plane perpendicular tothe c axis when the fundamental beam propagates in adirection perpendicular to the c axis or in a cone whenthe fundamental beam propagates parallel to the c axis[5,8,9]. When using a fundamental beam with good coher-ence, random variations in the intensity of this diffusedSHG can be observed. We show that the SHG intensitypattern from such transparent material shares the char-acteristic features of speckle patterns produced in thelinear regime by variations in the refractive index. A lin-ear speckle is characterized by speckle sizes inverselyproportional to the illuminated area but independentof the size of the scatterers and by an intensity contrastratio close to unity [1].
In our experiments we used a Sr0:61Ba0:39Nb2O6 crystalof cubic shape with 5mm sides. SHG scattered in a planewas obtained by illuminating the crystal with a Q-switched Nd:YAG laser (1064 nm, 6 ns, 10Hz). The inci-dent laser beam propagated inside the SBN in a directionnormal to the c axis and with polarization parallel to the caxis. The pictures in Fig. 1 show the SH intensity pat-terns. Figures 1(a) and 1(b) demonstrate that the sizeof the speckles is inversely proportional to the beam dia-meter. At the same time, as long as the beam diameterwas kept constant, the size of the speckles was the samein poled crystals and in annealed crystals with smallerdomains, as shown in Fig. 1(b) and the inset. A distinctSH peak [Fig. 1(c)] appears at the position that coincideswith the propagation direction of the fundamental beam.
Such SH peak is not linked to the random distribution ofdomains, and as we tilt the crystal [cf. the sequence ofFig. 1(c)], its intensity changes periodically with the pathlength inside the crystal following the expected Makerfringe behavior. The position of such peak is downshiftedwith respect to the scattering plane, probably becausethe faces of the crystal are not perfectly aligned withthe c axis. In Fig. 1(d), the angular distribution of thespeckle in the plane perpendicular to the c axis is shown.To explain the physical origin of the observations
discussed above, we have applied a theoretical Green’sfunction formalism [8]. Specifically, if we consider aplane wave at the fundamental frequency (with ampli-tude EðωÞ and wave vector kðωÞ) incident onto a nonlinearmedium, the far-field SH electric field computed at adistance r from the sample center can be written inthe following form:
Eð2ωÞi ¼ ð2ωÞ2eijkð2ωÞjr
c24πr
×ZVðδi;j − k̂ð2ωÞi k̂ð2ωÞj Þðχð2Þj;k;lE
ðωÞk EðωÞ
l ÞeiΔk·r0dr0; ð1Þ
where the subindices denote Cartesian coordinates andsummation over repeated subindices is assumed. Thetensor χð2Þ represents the second-order susceptibilityof the material, Δk ¼ kð2ωÞ − 2kðωÞ the wave vector mis-match, and k̂ð2ωÞ the unit vector in the direction of obser-vation of the SHG. The integral is performed over thenonlinear material volume V .If we assume that the material is composed of non-
linear domains within which χð2Þ is constant, the integral
in Eq. (1) becomes a summation over all the domains ofthe contributions made by each domain. For the case of afundamental beam propagating normally to the c axis ofSBN and with polarization parallel to it, we can restrictourselves to the two-dimensional plane perpendicular tothe c axis. If we further assume that the structure can bemodeled as a random distribution ofN square domains ofside L situated at random positions ðxi; yiÞ in the planeand with length h normal to the plane, the intensity at thedetection point is given by
Ið2ωÞ ¼ 2ω4nð2ωÞ½χð2Þz;z;z�2c5π2r2nðωÞ2ε0
½IðωÞ�2h2L4
× sinc2�ΔkxL
2
�sinc2
�ΔkyL
2
�����XNi¼1
eiðΔkxxiþΔkyyiÞ����2;
ð2Þ
where Δkx;y are the projections of the wave vector mis-match parallel and normal to the propagation direction ofthe incident fundamental light, respectively. In all the nu-merical simulations, the extraordinary refractive index ofSBN is taken as nð1064 nmÞ ¼ 2:22 and nð532 nmÞ ¼2:32, which gives a coherent length for the SHG propa-gating in the same direction as the fundamental ofLc ¼ π=Δk ¼ 2:68 μm. As in linearly generated specklepatterns, the intensity fluctuations, shown in Fig. 2,
Fig. 1. (Color online) Speckle from the 532nm wavelengthSHG from a poled crystal detected with a CCD camera locatedat the focal plane of a 150mm focal lens. Images taken usingfundamental beam diameters of (a) 2mm and (b) 4mm. Theinset on (b) shows the SHG speckle from a crystal with smallerdomains. (c) Sequence when the crystal was rotated from 0° to5° with respect to the incident beam. A 50mm focal lens and1mm fundamental beam diameter were used in this case. (d) In-tensity as a function of the angle of emission with respect to thefundamental beam. Inset: experimental scheme. (e) Intensity ofthe averaged speckle as a function of the crystal length obtainedby translating the fundamental beam across a wedge-shapedcrystal.
Fig. 2. (Color online) Simulated SHG intensity from 303 μm ×303 μm structures consisting of (a) 3 μm × 3 μm square domainswhen 2% of them are in a given polarization and (b) 1 μm × 1 μmsquare domains, 50% in each polarization. Examples of thestructures are shown in the insets on the right, using whiteor black squares depending on the polarization. Left inset on(a): intensity fluctuations when the width of the structure is re-duced to 75 μm. Left inset on (b): intensity at 5° from a similarcomposition of domains in which the length of the structureis changed. The result is averaged over 30 different randomstructures.
are comparable with the average intensity. In the sameway, the size of the speckles depends inversely on thewidth of the simulated structure but not on the size ofthe domains. Note that a peak in the direction of propa-gation of the fundamental beam (θ ¼ 0) is also obtainedin the simulations. This peak is the effect of the bulk ofthe crystal on top of the disorder in the structure. Whenthe length of the simulated structure is changed, the in-tensity of the peak follows the oscillatory Maker fringebehavior as observed experimentally in Fig. 1(c).We checked both experimentally and with numerical
simulations using the above theoretical model that theSHG intensity growth is linear instead of quadratic withthe number of domains. The average intensity measuredas a function of the sample length is shown in Fig. 1(e),and the inset of Fig. 2(b) demonstrates that this result isalso correctly predicted with our theoretical model.Thus, although the averaged intensity behaves like an in-coherent addition of SHG from multiple sources, thespeckle pattern demonstrates that the interference ofSHG from the domains is in fact coherent.The envelope of the speckle, or, in other words, the
angular distribution of the average intensity over manyspeckle realizations, depends on the shape and size ofthe domains. In Eq. (2) the sinc functions determinethe angular distribution of the envelope. The generaliza-tion to three dimensions is straightforward just by addinganother similar sinc function in the z direction. Takinginto account the different tensor components that canplay a role for different propagation directions of the fun-damental beam, all the observed SH emission patternsfrom SBN are correctly predicted by the emission froma single domain [8]. This is possible because all the do-mains have similar shapes and orientations. The varia-tions in the size of the domains just make the zeros ofthe sinc function disappear. Possible random variationsin the shape of the domains can also be accounted for bymodeling themwith a cylindrical symmetry instead of thesquare section. In that case the sinc functions in the x andy directions are substituted by a first-order Bessel func-tion [8]. For both square and cylindrical domains, the SHintensity for the fundamental beam propagating normalto the c axis spreads evenly at all angles on the planewhen the domains are much smaller than Lc. On the otherhand, when the domains are on the order of or larger thanLc, the SH becomes more concentrated at small anglesfrom the propagation direction of the fundamental beam.Other observed features, such as two lobes appearing atboth sides of the propagation direction of the fundamen-tal beam or concentric circles in the case of the funda-mental beam propagating parallel to the c axis, [8] canalso be reproduced with the calculated emission by a cy-lindrical domain of different sizes.The theoretical description given above indicates
that the statistical properties of the speckles are not de-pendent on the microstructure of the sample, a featureshared with the typical speckle patterns produced bychanges in the refractive index. However, important in-
formation about the microstructure can be obtained if weconsider the angular dependence of the averaged inten-sity. The envelope of the speckle obtained with the fun-damental beam propagating normal to the c axis providesa way to obtain an estimation of the mean size of the do-mains. The angular intensity distribution in the plane per-pendicular to the c axis will give the domain diameter,while the distribution in the direction parallel to the caxis will give the domain length. Using this method weestimated the mean domain diameter and length in acrystal annealed at 250 °C as 0.15 and 100 μm, respec-tively, while after the crystal was poled with a400V=mm electric field, the dimensions increased to 6and 580 μm, respectively (see Fig. 3).
In summary, we have demonstrated, both experimen-tally and theoretically, SH speckle pattern formation by atransparent SBN crystal. Because of the lack of light scat-tering in the linear regime, we believe our findings repre-sent a novel instance of coherent optical phenomenaoccurring in random nonlinear optical materials.
We acknowledge financial support from Ministerio deCiencia e Innovacion under grants MAT2008-00910/NANand CSD2007-00046, as well as grants JCI-2009-04860 (F.J. R.) and RyC-2009-05489 (J. B.-A.).
References
1. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).2. M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55,
2692 (1985).3. P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).4. N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E.
Sauvain, Nature 368, 436 (1994).5. S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S.
Feigelson, Appl. Phys. Lett. 73, 768 (1998).6. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph.
Lemasson, and E. Rosencher, Nature 432, 374 (2004).7. X. Vidal and J. Martorell, Phys. Rev. Lett. 97, 013902 (2006).8. J. Bravo-Abad, X. Vidal, J. L. Domínguez-Juárez, and J.
Martorell, Opt. Express 18, 14202 (2010).9. A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90,
243901 (2003).10. T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A.
Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
Fig. 3. (Color online) SHG intensity as a function of the angleof emission with respect to the fundamental beam in the direc-tion (a) perpendicular and (b) parallel to the c axis. The thinlines correspond to the experimental data and the thick linesto the fittings with the theoretical model for cylindrical domainsof 6 μm diameter and 580 μm length.
