Observation of speckle pattern formation in transparent nonlinear random media

Francisco J. Rodríguez; Can Yao; Jorge L. Domínguez-Juárez; Jorge Bravo-Abad; Jordi Martorell

doi:10.1364/OL.36.001347

When light is generated or propagates in random structures, there are several manifestations of the wave nature that survive the phase front distortion, such as speckle pattern formation [1], the appearance of a backscattering cone [2, 3], and random lasing [4]. In general, any of such phenomena occurs in a disordered distribution of one material embedded in another one with a different index of refraction. Scattering of light is also observed from transparent nonlinear materials in which the nonlinear domains are randomly distributed in space [5]. However, no clear signs of a coherent interaction have been reported from the light generation in such nonlinear crystals. In fact, it has been shown that the coherent addition of the two interacting waves is washed out by the dis order, and the growth of the nonlinear process versus the thickness of the material becomes linear instead of quadratic [5, 6, 7, 8].

In this Letter, we demonstrate that the granulation observed on the second-harmonic generation (SHG) pattern from a strontium barium niobate (SBN) crystal originates from the interference between the different second-harmonic (SH) waves independently scattered in all directions by all the nonlinear domains that compose the crystal structure. Specifically, by measuring the angular distribution of SH light emerging from a SBN crystal, we find distinct features showing speckle pattern formation at the SH frequency. The intensity distributions in the observed speckle patterns are in very good agreement with the predictions obtained from a Green’s function formalism, which explains the observed speckle in terms of the interference at the observation point among the SH light generated from the different nonlinear domains. The theoretical model used shows that a matching of the phases across the crystal among the second-order nonlinear polarization and the SH fields does not play a role in the reported phenomena. It is important to note that the observation reported in this work cannot be predicted in a straightforward manner using the conventional description of SHG in SBN based on a continuous distribution of reciprocal lattice vectors [9].

Strontium barium niobate is a transparent crystal, ferroelectric at room temperature with large second-order susceptibility when both the fundamental and SH fields are polarized parallel to the polar c axis. The ferroelectric domains form columns along the c axis with random antiparallel polarizations. Because of this optically equivalent alignment of the domains, light propagating through the crystal does not experience a refractive index change. As a result, only an extremely weak linear scattering originated by the domain walls may be observed [10]. In contrast with the linear light transmission, the SHG is strongly diffused in a plane perpendicular to the c axis when the fundamental beam propagates in a direction perpendicular to the c axis or in a cone when the fundamental beam propagates parallel to the c axis [5, 8, 9]. When using a fundamental beam with good coherence, random variations in the intensity of this diffused SHG can be observed. We show that the SHG intensity pattern from such transparent material shares the characteristic features of speckle patterns produced in the linear regime by variations in the refractive index. A linear speckle is characterized by speckle sizes inversely proportional to the illuminated area but independent of the size of the scatterers and by an intensity contrast ratio close to unity [1].

In our experiments we used a ${Sr}_{0.61} {Ba}_{0.39} {Nb}_{2} O_{6}$ crystal of cubic shape with $5 mm$ sides. SHG scattered in a plane was obtained by illuminating the crystal with a Q-switched Nd:YAG laser ( $1064 nm$ , $6 ns$ , $10 Hz$ ). The incident laser beam propagated inside the SBN in a direction normal to the c axis and with polarization parallel to the c axis. The pictures in Fig. 1 show the SH intensity patterns. Figures 1a, 1b demonstrate that the size of the speckles is inversely proportional to the beam diameter. At the same time, as long as the beam diameter was kept constant, the size of the speckles was the same in poled crystals and in annealed crystals with smaller domains, as shown in Fig. 1b and the inset. A distinct SH peak [Fig. 1c] appears at the position that coincides with the propagation direction of the fundamental beam. Such SH peak is not linked to the random distribution of domains, and as we tilt the crystal [cf. the sequence of Fig. 1c], its intensity changes periodically with the path length inside the crystal following the expected Maker fringe behavior. The position of such peak is downshifted with respect to the scattering plane, probably because the faces of the crystal are not perfectly aligned with the c axis. In Fig. 1d, the angular distribution of the speckle 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’s function formalism [8]. Specifically, if we consider a plane wave at the fundamental frequency (with amplitude $E^{(ω)}$ and wave vector $k^{(ω)}$ ) incident onto a nonlinear medium, the far-field SH electric field computed at a distance r from the sample center can be written in the following form:

E_{i}^{(2 ω)} = \frac{(2 ω)^{2} e^{i | k^{(2 ω)} | r}}{c^{2} 4 π r} \times \int_{V} (δ_{i, j} - {\hat{k}}_{i}^{(2 ω)} {\hat{k}}_{j}^{(2 ω)}) (χ_{j, k, l}^{(2)} E_{k}^{(ω)} E_{l}^{(ω)}) e^{i Δ k \cdot r^{'}} d r^{'},

where the subindices denote Cartesian coordinates and summation over repeated subindices is assumed. The tensor

χ^{(2)}

represents the second-order susceptibility of the material,

Δ k = k^{(2 ω)} - 2 k^{(ω)}

the wave vector mismatch, and

{\hat{k}}^{(2 ω)}

the unit vector in the direction of observation of the SHG. The integral is performed over the nonlinear material volume V.

If we assume that the material is composed of nonlinear domains within which $χ^{(2)}$ is constant, the integral in Eq. (1) becomes a summation over all the domains of the contributions made by each domain. For the case of a fundamental beam propagating normally to the c axis of SBN and with polarization parallel to it, we can restrict ourselves to the two-dimensional plane perpendicular to the c axis. If we further assume that the structure can be modeled as a random distribution of N square domains of side L situated at random positions $(x_{i}, y_{i})$ in the plane and with length h normal to the plane, the intensity at the detection point is given by

I^{(2 ω)} = \frac{2 ω^{4} n^{(2 ω)} [χ_{z, z, z}^{(2)}]^{2}}{c^{5} π^{2} r^{2} n^{(ω) 2} ε_{0}} [I^{(ω)}]^{2} h^{2} L^{4} \times {sinc}^{2} [\frac{Δ k_{x} L}{2}] {sinc}^{2} [\frac{Δ k_{y} L}{2}] | \sum_{i = 1}^{N} e^{i (Δ k_{x} x_{i} + Δ k_{y} y_{i})} |^{2},

where

Δ k_{x, y}

are the projections of the wave vector mismatch parallel and normal to the propagation direction of the incident fundamental light, respectively. In all the numerical simulations, the extraordinary refractive index of SBN is taken as

n (1064 nm) = 2.22

and

n (532 nm) = 2.32

, which gives a coherent length for the SHG propagating in the same direction as the fundamental of

L_{c} = π / Δ k = 2.68 μm

. As in linearly generated speckle patterns, the intensity fluctuations, shown in Fig. 2, are comparable with the average intensity. In the same way, the size of the speckles depends inversely on the width of the simulated structure but not on the size of the domains. Note that a peak in the direction of propagation of the fundamental beam (

θ = 0

) is also obtained in the simulations. This peak is the effect of the bulk of the crystal on top of the disorder in the structure. When the length of the simulated structure is changed, the intensity of the peak follows the oscillatory Maker fringe behavior as observed experimentally in Fig. 1c.

We checked both experimentally and with numerical simulations using the above theoretical model that the SHG intensity growth is linear instead of quadratic with the number of domains. The average intensity measured as a function of the sample length is shown in Fig. 1e, and the inset of Fig. 2b demonstrates that this result is also correctly predicted with our theoretical model. Thus, although the averaged intensity behaves like an incoherent addition of SHG from multiple sources, the speckle pattern demonstrates that the interference of SHG from the domains is in fact coherent.

The envelope of the speckle, or, in other words, the angular distribution of the average intensity over many speckle realizations, depends on the shape and size of the domains. In Eq. (2) the sinc functions determine the angular distribution of the envelope. The generalization to three dimensions is straightforward just by adding another similar sinc function in the z direction. Taking into account the different tensor components that can play a role for different propagation directions of the fundamental beam, all the observed SH emission patterns from SBN are correctly predicted by the emission from a single domain [8]. This is possible because all the domains have similar shapes and orientations. The variations in the size of the domains just make the zeros of the sinc function disappear. Possible random variations in the shape of the domains can also be accounted for by modeling them with a cylindrical symmetry instead of the square section. In that case the sinc functions in the x and y directions are substituted by a first-order Bessel function [8]. For both square and cylindrical domains, the SH intensity for the fundamental beam propagating normal to the c axis spreads evenly at all angles on the plane when the domains are much smaller than $L_{c}$ . On the other hand, when the domains are on the order of or larger than $L_{c}$ , the SH becomes more concentrated at small angles from the propagation direction of the fundamental beam. Other observed features, such as two lobes appearing at both sides of the propagation direction of the fundamental beam or concentric circles in the case of the fundamental beam propagating parallel to the c axis, [8] can also be reproduced with the calculated emission by a cylindrical domain of different sizes.

The theoretical description given above indicates that the statistical properties of the speckles are not dependent on the microstructure of the sample, a feature shared with the typical speckle patterns produced by changes in the refractive index. However, important information about the microstructure can be obtained if we consider the angular dependence of the averaged intensity. The envelope of the speckle obtained with the fundamental beam propagating normal to the c axis provides a way to obtain an estimation of the mean size of the domains. The angular intensity distribution in the plane perpendicular to the c axis will give the domain diameter, while the distribution in the direction parallel to the c axis will give the domain length. Using this method we estimated the mean domain diameter and length in a crystal annealed at $250 ° C$ as 0.15 and $100 μm$ , respectively, while after the crystal was poled with a $400 V / mm$ electric field, the dimensions increased to 6 and $580 μm$ , respectively (see Fig. 3).

In summary, we have demonstrated, both experimentally and theoretically, SH speckle pattern formation by a transparent SBN crystal. Because of the lack of light scattering in the linear regime, we believe our findings represent a novel instance of coherent optical phenomena occurring in random nonlinear optical materials.

We acknowledge financial support from Ministerio de Ciencia e Innovacion under grants MAT2008-00910/NAN and CSD2007-00046, as well as grants JCI-2009-04860 (F. J. R.) and RyC-2009-05489 (J. B.-A.).

Fig. 1 Speckle from the $532 nm$ wavelength SHG from a poled crystal detected with a CCD camera located at the focal plane of a $150 mm$ focal lens. Images taken using fundamental beam diameters of (a) $2 mm$ and (b) $4 mm$ . The inset on (b) shows the SHG speckle from a crystal with smaller domains. (c) Sequence when the crystal was rotated from $0 °$ to $5 °$ with respect to the incident beam. A $50 mm$ focal lens and $1 mm$ fundamental beam diameter were used in this case. (d) Intensity as a function of the angle of emission with respect to the fundamental beam. Inset: experimental scheme. (e) Intensity of the averaged speckle as a function of the crystal length obtained by translating the fundamental beam across a wedge-shaped crystal.

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Fig. 2 Simulated SHG intensity from $303 μm \times 303 μm$ structures consisting of (a) $3 μm \times 3 μm$ square domains when 2% of them are in a given polarization and (b) $1 μm \times 1 μm$ square domains, 50% in each polarization. Examples of the structures are shown in the insets on the right, using white or black squares depending on the polarization. Left inset on (a): intensity fluctuations when the width of the structure is reduced to $75 μm$ . Left inset on (b): intensity at $5 °$ from a similar composition of domains in which the length of the structure is changed. The result is averaged over 30 different random structures.

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Fig. 3 SHG intensity as a function of the angle of emission with respect to the fundamental beam in the direction (a) perpendicular and (b) parallel to the c axis. The thin lines correspond to the experimental data and the thick lines to the fittings with the theoretical model for cylindrical domains of $6 μm$ diameter and $580 μm$ length.

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Observation of speckle pattern formation in transparent nonlinear random media

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