1. Introduction

The investigation of photorefractive materials like LiNbO₃, LiTaO₃, BaTiO₃ or Sr_xBa_1−xNb₂O₃ is of special interest for their technical application in non-linear optics like frequency mixing, wave multiplexing or holographic data storage. The development of optical devices requires a comprehensive quantitative understanding of the material properties and the underlying physical processes. In non-centrosymetric photorefractive crystals local changes of the refractive index can be induced as a function of the intensity distribution and the lateral dimensions of an applied laser beam [1]. Processes like the photo-induced light scattering [2], the writing of holographic gratings [3] or photo-induced internal lenses [4] through the generation of macroscopic space charge fields are of fundamental interest and require the development of proper methods for their investigation.

Coherent diffractive imaging techniques like ptychography were developed in the last decade as novel lensless microscopy methods [5–7]. Ptychography combines the high spatial resolution of diffractive imaging with the large field of view of scanning probe techniques. The method solves the phase problem by the detection of multiple diffraction patterns at different positions of a probe beam on a sample. The complex transmission function of the sample as well as the amplitude and the phase of the probe beam are iteratively reconstructed from the diffraction data. Ptychography is applied especially in those fields where projection optics are missing, e.g., in the hard x-ray regime [8, 9]. But it is also beneficial to investigations where the knowledge of the amplitude and phase of the coherent beam are a necessary prerequisite, as it is known for photorefractive materials.

In this letter we introduce ptychography for the investigation of light-induced refractive index changes in the photorefractive material iron-doped lithium niobate (LiNbO₃:Fe). We determine the intensity and phase distribution of the writing beam and the lateral distribution of the refractive index change with a spatial resolution of 3 μm from a single ptychographic measurement. Asymmetries of the refractive index change are attributed to the intensity distribution of the writing beam.

Additionally, we propose an extension to the ptychographic algorithm and show its benefit to the spatial resolution of the reconstruction by the measurement of a test structure.

2. Ptychography

Ptychography is a lensless scanning imaging technique. Multiple diffraction patterns resulting from the interaction between a coherent probe beam and the sample are detected in the far field [5, 6]. After the detection of each diffraction pattern the position of the probe beam on the sample is changed perpendicular to the optical axis. The illuminated sample areas of neighboring diffraction patterns partially overlap, thus the different diffraction patterns partially contain redundant information. The whole set of measured diffraction patterns is processed jointly by the “ePIE” [7], an iterative algorithm to reconstruct the complex transmission function of the sample together with the complex function of the probe beam. The method of ptychography allows a detailed investigation of the transmittance and the phase shift of the investigated sample in combination with the complete determination of the intensity and phase distribution of the coherent probe beam. The highest achievable lateral resolution derived by the “ePIE” is in principle limited by the scattering angle via Abbe’s criterion. A related algorithm, called “SR-PIE”, is capable of improving the resolution beyond the maximal scattering angle of the detector [10].

2.1. Experimental setup and data acquisition

A principle ptychographic setup is sketched in Fig. 1(a). Our setup consists of a COHERENT Verdi V5 laser, operating at 532 nm wavelength, a VIS-AR-coated lens with a focal length of 40 mm, the sample on a motorised sample holder and a CCD-chip, integrated in an FLI PL50100 camera. The CCD-chip consists of 3056 × 3056 pixels and has an edge length of 36.672 mm. A logarithmically scaled example diffraction pattern is shown on the right-hand side of Fig. 1(a). The distance between the sample and the CCD-chip is 67.05 mm. Thus we are able to capture a maximum scattering vector of |q| = 6.233 μm⁻¹, corresponding to a scattering angle of 15.3°. According to Abbe’s criterion, the smallest structure that can be reconstructed at this angle is 0.973 μm, which is equal to the pixel size of the reconstruction. The sample holder is mounted on three linear stages (Newport MFA-PPD) and one rotation stage (Newport URS75).

 

Fig. 1 (a) Experimental setup and exemplary diffraction pattern (logarithmically scaled); (b) (from left to right) Reconstructed intensity, logarithmically scaled intensity and phase of the laser beam with corresponding horizontal profiles through the maximum intensity of 7.29 × 10⁻⁵ W/cm²; (c) Beam caustic of the intensity with marked positions of the investigated test structure and the LiNbO₃:Fe sample

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The whole field of view of the reconstructed sample and probe beam functions depends on the number of measured diffraction patterns and the spacing between the corresponding illuminated areas on the sample.

2.2. Model and reconstruction algorithm

We use an iterative reconstruction algorithm, which was originally proposed by Maiden and Rodenburg in 2009 [7]. For one iteration of the extended ptychographical iterative engine (ePIE) the following procedure has to be applied for each scan point of the data acquisition sequentially:

Within the model of a thin sample the interaction of the probe beam (probe function P) and the sample (object function O) is characterised by a complex multiplication, giving the wave field Ψ behind the sample: Ψ = P · O. The propagation into the plane of the CCD-chip can be calculated by a single Fourier transformation, where in the next step the Fourier transformed wave front $\tilde{\mathrm{\psi }}$ is adapted to the measured diffraction pattern by replacing the amplitude A of $\tilde{\mathrm{\psi }}=A\cdot e^{\iota \tilde{\phi }}$ by the square root of the measured data I, giving the adapted wave field ${\tilde{\mathrm{\psi }}}^{\prime }=\sqrt{I}\cdot e^{\iota \tilde{\phi }}$. The·calculated phase $\tilde{\mathrm{\phi }}$ of the wave field remains unchanged. The wave field ${\tilde{\mathrm{\psi }}}^{\prime }$ is then propagated back into the plane of the sample by an inverse Fourier transformation, giving the updated transmitted wave field Ψ′. The splitting of the wave field ${\mathrm{\psi }}^{\prime }$ into the updated object function O′ and the updated probe function P′ is possible due to the scanning character of the data acquisition and the assumption of a constant shape of the laser beam during the scan [7].

After the procedure is done for all diffraction patterns, one iteration of the ePIE is finished. From the real and the imaginary part of the reconstructed probe function we get the intensity and the phase distribution of the laser beam in the plane of the sample. From the real and imaginary part of the object function we get the transmittance T and the phase shift Δφ of the sample.

2.3. Beam characterisation

Before the investigation of an unknown sample like a photorefractive material we characterise the laser beam with a ptychographic measurement of a test structure, whose lateral shape is known by microscopic measurements. With the ptychographically reconstructed complex wave field of the laser beam in the sample plane it is possible to numerically propagate this wave field along the optical axis. This gives us the intensity and phase distribution of the laser beam in all three spatial dimensions. The unknown sample, e.g., the photorefractive material, can now be positioned precisely at any desired position in the beam with an exact knowledge about the undisturbed intensity distribution inside the sample.

The test structure has the lateral shape of a star-like structure with integrated numerals and characters. The structures are getting smaller in the center of the star (see right-hand picture in Fig. 2(a)). The dark structures of the star are made by vapour deposition of platinum (thickness d_Pt = 4 nm) and titanium (thickness d_Ti = 1 nm), each with 0.5 nm thickness accuracy, on a glass substrate. In the bright regions there is only the glass substrate. The transmittance T = exp (−4π[κ_Tid_Ti + κ_Ptd_Pt]/λ) of the deposited layer with the applied wavelength λ = 532 nm is calculated with the extinction coefficients κ_Pt and κ_Ti for platinum and titanium from [11] to be 0:662 ± 0:037. The phase shift Δ′ of the deposited layer is calculated over the equality of the ratio of the change of the optical path length Δs to the wavelength λ and the ratio of the change of the phase Δ′ to 2π. Neglecting any thickness variation (Δd = 0) the change of the optical path length is given by the sum over the change of the refractive index Δn_i and the thickness d_i of each present layer i: Δni = ∑_i Δn_i · d_i. Thus the phase shift of the sample is given by:

 

Fig. 2 (a) Reconstructed transmittance T of the test structure, on the left-hand side without reconstruction of the background signal R, on the right-hand side with the reconstruction of R; (b) Detail of the reconstructed R; (c) Comparison of the relative error of the reconstruction versus the number of iteration, the black (red) graph is reconstructed with (without) the reconstruction of R

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With the refractive indices of platinum and titanium [11] and the corresponding thickness of both layers the calculated phase shift of the deposited layer is (0.059 ± 0.008) rad.

The picture on the right-hand side of Fig. 2(a) shows the ptychographically reconstructed transmittance T of the test structure. The reconstructed value of the transmittance of the deposited layer is 0.685 ± 0.044. The reconstructed phase shift of the test structure has the same lateral shape like the transmittance, while the reconstructed value of the phase shift of the deposited layer is (0.053 ± 0.009) rad. Both reconstructed values are in agreement with the calculated ones within the error bars. The lateral resolution is 2 μm, because all structures with this size are visible in the reconstruction of the test structure.

The ptychographic analysis gives us, additional to the transmittance and the phase shift of the test structure, the intensity and phase distribution of the laser beam in the plane of the test structure. The intensity distribution of the beam is shown on the left-hand side of Fig. 1(b), the phase distribution on the right-hand side. The logarithmically scaled intensity distribution is plotted in the middle of Fig. 1(b). Under every image we show a horizontal profile along the y axis through the maximum of the intensity. With an optical power meter we determined the power of the laser beam to be 2.44 nW. Due to this value and the reconstructed shape of the laser beam, we derive absolute local intensity values. The spatial resolution of the intensity distribution is 2 μm, according to the spatial resolution of the reconstructed test object. The maximum of the intensity is thereby 7.29 × 10⁻⁵ W/cm². The profile of the logarithmically scaled intensity distribution in Fig. 1(b) shows a Gaussian beam shape over more than three orders of magnitude and a dynamic range of the reconstruction of at least five orders of magnitude. The full width at half maximum (FWHM) is 53 μm in horizontal (y) and 58 μm in vertical (z) direction. The reconstructed phase of the laser beam has a curved shape with multiple phase jumps, which are the consequence of restricting the phase to an interval of 2π to avoid ambiguities. The phase jumps are visible as sharp dark to bright changes in the image of the phase and can be directly seen in the horizontal profile below the image. The noise of the phase in the outer parts of the image is plausible by a comparison with the reconstructed intensity, which also drops under the noise limit of the reconstruction in these areas.

As the complex wave field of the laser beam in the plane of the test structure is completely determined, it is possible to do a numerical free space Fresnel-propagation into the positive and negative beam direction (x direction). The calculated beam caustic is shown in Fig. 1(c). The gray values encode the intensity and the plane of the test structure is marked in red. We emphasise, that with the complete knowledge of the laser beam in amplitude and phase, it is possible to characterise the quality and properties of focussing optics and laser sources like it was done for hard x-ray focussing optics [9, 12]. Thus we are able to characterise the focal point and the Rayleigh-length, i.e., the propagation length in which the area of the focus is doubled, for our setup. The two measured Rayleigh lengths are z_R1 = (2.870 ± 0.002) mm into the negative beam direction and z_R2 = (2.140 ± 0.002) mm into the positive beam direction.

2.4. Modeling instrumental contributions in the diffraction pattern and modification of the ptychographic reconstruction algorithm

In the following section we shortly present a new feature, which we introduce to the ePIE. The additional reconstruction of a so called “background signal” improves the reconstruction.

The ptychographic algorithm is used for background free signals, working with single photon counting detectors in the hard x-ray regime [9]. Since we use a normal CCD chip, we have a permanent background signal overlaying the scattered intensity. In spite of using a high dynamic range (HDR) algorithm [13] to increase the dynamic range of the detector, like it was also successfully employed in [14], the background signal of the diffraction patterns is not fully removed. To take account of this unknown background signal we include a function R to the ptychographic algorithm. R is a single field in the same size as the diffraction patterns and it is applied during the adaption of the amplitude of $\tilde{\mathrm{\psi }}$. To reconstruct the background signal per pixel we assume a constant background signal per pixel for every diffraction pattern. This constant value can be subtracted from the measured diffraction data and results in a modified adaptation step ${\tilde{\mathrm{\psi }}}^{\prime }=\sqrt{I-R^{\prime }}\cdot e^{\iota \phi }$. During the first iteration the function R is set to an estimated value. We use a self consistent update to avoid further assumptions about the background signal R. The updated background signal is $R^{\prime }=R+\gamma \cdot \left({\left|\tilde{\mathrm{\psi }}\right|}^2-I\right)$, where the parameter γ determines the update strength. To prevent that all the measured data are put into the background signal R, γ is set to a small number, typically much smaller than the inverse of the number of diffraction patterns. The update of R is done before the amplitude adaption of $\tilde{\mathrm{\psi }}$. Hence R collects all counts of the CCD-chip that cannot be explained by the interaction between the probe beam and the sample.

To demonstrate the improvements of the ptychographic reconstruction, we evaluate a ptychogram of the test structure. The reconstructed transmittance T of the test structure is shown in Fig. 2(a), on the left-hand side without R and on the right-hand side with self consistent reconstructed R. The lateral resolution of the reconstruction on the right-hand side is much better, so that the numerals down to “2” are visible just like the structures with 2 μm line spacing. In the reconstruction on the left-hand side, where the algorithm is used without R, not even the numeral “8” is clearly visible.

The reconstructed background signal is displayed in Fig. 2(b), the grey value corresponds to the background signal level. Obviously R shows signals from other interactions than the sample. Dust particles on the detector entrance window are clearly visible. The larger ring-like structures are correlated to the HDR calculation process. Further investigation must be done to clarify the whole scope of the errors that can be corrected by the introduced background correction.

The use of R also reduces the calculated relative error value (equation 10 in [7]). The error graph is shown in Fig. 2(c). The black and red dots show the error value of the reconstruction with and without R, respectively. Due to the additional reconstruction of the background signal the algorithm converges slower, but the error of the converged reconstruction is reduced by one order of magnitude.

3. Experimental results

3.1. Sample

The photorefractive sample is iron doped lithium niobate (LiNbO₃:Fe), an extensively studied photorefractive material with one polar axis (z axis). The investigated crystal is an x-cut with a thickness of 0.23 mm, so that the writing of holographic gratings due to intrinsic scattering centres can be neglected. The crystal is doped with an iron concentration of 0.05 mol% and has a VIS-AR-coating on both sides. With a spectrometric measurement we evaluate a transmittance of 0.951 for light with 532 nm wavelength and a polarisation parallel to the z axis of the crystal. The position of the crystal in the beam caustic is shown as a yellow bar in Fig. 1(c). As shown in the figure, the width of the laser beam intensity changes less than 0.5 % within the crystal thickness.

3.2. Generation of the refractive index change

To generate a refractive index change via the photorefractive effect we use the same laser beam (see Fig. 1(b)) and the same setup (see Fig. 1(a)) like for the ptychographic measurements. Since the demanded energy per area for the occurrence of the effect can be reached with cw laser intensities of a few W/cm² within multiple minutes of exposure [15, 16], we adjust the laser power to 0.244 mW, which is a factor 10⁵ higher than during the ptychographic measurement. The peak laser intensity of the writing beam is I_max = 7.29 W/cm² and the polarisation of the laser light is oriented parallel to the z axis. Inside this intensity-regime the charge transport is described by the one center model [17]. We expose the crystal for 3420 s to the beam. Due to the knowledge about the laser beam intensity distribution in all three spacial dimensions (section 2.3), we are able to produce a well-defined refractive index modulation inside the LiNbO₃:Fe crystal.

3.3. Ptychographic measurement and reconstruction of the refractive index change

For the ptychographic measurement the laser power is reduced with a neutral-density filter by a factor of 10⁵ with respect to the writing power in order to avoid any changes of the refractive index during the measurement. However, the shape of the laser beam which is used for the ptychographic measurement is the same like in the writing process. This was confirmed via a measurement of the pure laser beam with the camera using different exposure times. We detect diffraction patterns at 36 × 36 positions, while we guarantee a sufficient overlap of the illuminated areas between neighbouring positions. The positions are randomly distributed over an area of about 720 μm × 720 μm to suppress reconstruction artefacts [18]. At every position we take several exposure times and combine them to a high dynamic range (HDR) diffraction pattern [13], which allows for a higher spatial resolution in the reconstruction.

The reconstructed transmission function of the sample is shown in Fig. 3(a). The upper picture shows the transmittance T of the crystal and the lower picture shows the phase shift Δφ. We did not constrain the transmittance, that is why the scale bar on the right hand side of the picture shows values between 0 and 1.3. We reconstruct a maximum transmittance of 1.29 and a minimum transmittance of 0.54. The empty space transmittance of the sample is reconstructed to be T_empty = 0.95. This value corresponds to the initialized transmittance value before the first iteration of the ptychographic algorithm, which is set to this number for the whole sample, because we measured it with a spectrometer for the crystal without any induced modulation of the refractive index. The reconstructed transmittance shows a spatial structure of two arcs, one above and one underneath the area where the writing beam has been applied (center of the picture). An integration over each arc yields the same average transmittance value like T_empty. This means, that all areas with a lower transmittance than T_empty are compensated by areas with a higher transmittance. The arcs are located at the position of the strongest gradient in the reconstructed phase shift of the sample and can be attributed to a phase contrast effect due to the finite thickness of the sample. The same effect is also observed in microscopic measurements.

 

Fig. 3 (a) Reconstructed transmission function of the LiNbO₃:Fe crystal, the upper picture shows the transmittance T, the lower picture the phase shift Δφ, multiple phase jumps are visible; (b) Unwrapped representation of the phase shift in (a), the scale on the right shows the phase values as well as the refractive index change calculated over equation 1 encoded in color, the spatial structure of the refractive index change is demonstrated in a side view in the lower picture

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The reconstructed phase shift of the sample is displayed in the lower picture in Fig. 3(a) and shows multiple phase jumps. As the phase jumps occur along closed lines, it is possible to unwrap the phase and to determine phase shifts in an interval larger than 2π. The unwrapped and enlarged picture is shown in the upper image in Fig. 3(b). For a better visualisation of the spatial structure, a side view of the distribution is shown in the lower picture in Fig. 3(b).

The refractive index change Δn can be calculated with the knowledge of the phase shift Δφ, the utilized wavelength λ = 532 nm and the thickness of the crystal d = 0.23 mm with equation (1). The scale on the right-hand side of Fig. 3(b) shows next to the unwrapped phase shift Δφ the calculated refractive index change Δn and its encoding in colour. The reconstructed refractive index change shows one area with negative sign, which is along the z axis enclosed by two areas with a positive sign. The two areas with positive refractive index change have maximum values of Δn = 3.40 × 10⁻³ for the top one and Δn = 3.49 × 10⁻³ for the bottom one. The maximum negative refractive index change is Δn = −2.23 × 10⁻³. The accuracy of the refractive index change is in the order of 10⁻⁵.

From the maximum measured scattering angle we obtain a maximum reconstructable spatial resolution of 3 μm. We verified the resolution estimate with the reconstruction of the test structure in section 2.3. Additionally we applied microscopic measurements of the LiNbO₃:Fe sample and verified the existence and size of various dust particles, which are also visible in the ptychographic reconstruction.

The reconstructed laser beam intensity and phase are equivalent to the laser beam intensity and phase, which are reconstructed with the ptychogram of the test structure (see Fig. 1(b)) and are thus not shown here again. But we emphasize that the ptychogram of the test structure would not have been necessary to reconstruct the intensity and phase distribution of the writing beam, since it is also reconstructed in the ptychogram of the refractive index change. But a previous knowledge of the intensity distribution of the writing beam gives the possibility to choose the beam shape for the generation of the refractive index change.

Since we know the lateral distribution of the generated refractive index modulation as well as the lateral intensity distribution of the laser beam together with their relative position, we can compare the intensity distribution of the laser during the generation process with the resulting refractive index modulation with a high spatial resolution. This is shown in Fig. 4, where the refractive index modulation (blue line) is displayed together with the corresponding intensity of the writing beam (red line). Both profiles are drawn parallel to the z axis through the maximum of the intensity of the writing beam, the intensity profile is logarithmically scaled.

 

Fig. 4 Profile of the refractive index change (blue line) compared to the logarithmically scaled intensity of the writing beam (red line)

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The refractive index change Δn is negative in the region where the writing beam intensity is sufficiently high. The strongest changes of Δn are observed at the positions, where the intensity changes rapidly by more than two orders of magnitude (around z ≅ 0.31 mm and z ≅ 0.44 mm). At an intensity of 0.1 W/cm² Δn changes its sign from negative to positive and has maxima at intensities of about 10⁻² to 10⁻³ W/cm². While the minimum of Δn is located at the exact maximum of the intensity at z ≅ 0.40 mm, the plateau of negative Δn shows a slope on the left side between z ≅ 0.33 mm and z ≅ 0.40 mm. This is directly connected to a perturbation of the Gaussian-shape of the intensity at z ≅ 0.35 mm. The intensity distribution shows a saddle point of about 5 W/cm² at this position.

The one center model, which is the commonly accepted microscopic model for the photorefractive effect in LiNbO₃:Fe [17,19], directly connects an asymmetry of the intensity distribution of the writing beam I(r) to an asymmetry of the induced refractive index change Δn r. The model describes the refractive index change as the result of a charge redistribution inside the crystal, leading to space charge fields which result in a refractive index change over the electro optic effect. The charge redistribution is the result of the photoexcitation of electrons from Fe²⁺ ions in the lattice into the valance band by the writing beam. The excitation of the electrons is followed by a movement inside the valance band and a subsequent recombination with Fe³⁺ ions in the lattice. The movement of electrons is dominated by three charge driving currents: the drift current, the diffusion current and the photovoltaik current. The photovoltaik current is directly proportional to the local intensity I(r) and to the local Fe²⁺ concentration $N_{{\text{Fe}}^{2+}}(\mathbf{r},t)$ [17], while t is the time of writing. The drift current is proportional to the electron density in the conduction band N(r, t) and the diffusion current is proportional to the gradient of the electron density in the conduction band grad N(r, t) [19]. However, the electron density in the conduction band N(r, t) is, under negligence of thermal excitations and the excitation time of electrons, expressed to be proportional to the local intensity I(r and the fraction of Fe²⁺ ions to Fe³⁺ ions $N_{{\text{Fe}}^{2+}}(\mathbf{r},t)/N_{{\text{Fe}}^{3+}}(\mathbf{r},t)$ [20]. Thus the lateral shape of the intensity distribution of the writing beam I(r) has great influence on all three charge driving currents and consequential the lateral shape of the refractive index change Δn (r).

This is evident from the comparison of the two profiles in Fig. 4. At the position of the saddle point of the writing beam at z ≅ 0.35 mm the refractive index change is not as strong as at the position of the intensity maximum. This is the result of less electron excitation at positions with less writing beam intensity. To reach the saturation of the refractive index change for the position of the saddle point the writing time has to be longer than at the position at the intensity maximum. The described slope of the plateau of the negative refractive index change is thus a result of the finite writing time. We suggest that the difference of the two positive maxima of Δn can also be ascribed to this asymmetry of the intensity distribution.

4. Conclusion

We have presented the application of ptychography to the investigation of light induced refractive index changes in the photorefractive material LiNbO₃:Fe. We reconstructed the transmittance and the refractive index modulation of an induced refractive index change together with the intensity and phase distribution of the writing beam with a spatial resolution of 3 μm. Asymmetries in the distribution of the refractive index change where explained by a comparison with the intensity distribution of the writing beam. In contrast to other measurement techniques of induced refractive index changes like digital holography [21, 22] and interferometry [15, 16, 23] ptychography additionally determines the intensity and phase distribution of the writing beam. This is a clear advantage concerning the investigation of the nonlocal response of the photorefractive material to the spatial distribution of the writing beam. Small deviations from a Gaussian beam profile can cause non-negligible structuring of the refractive index profile. The reason for this lies in the dependency of all three charge driving forces from the local intensity I (r) of the writing beam. Therefore the spatially resolved intensity distribution of the writing beam is of great interest for the separation between effects of the charge driving forces and simple geometric effects due to the shape of the intensity distribution to the resulting spatial distribution of the refractive index change. The knowledge about the intensity distribution I (r) of the writing beam is an excellent prerequisite for the analysis of the refractive index change with assistance of numerical simulations [24,25] where both, the intensity distribution of the writing beam I (r) and the resulting refractive index change Δn(r) for every pixel r of the ptychographic reconstruction, can be applied together with charge migration models like the one or two center model to fit microscopic parameters of the charge migration itself.

That is why we plan to combine the benefit of ptychography, which is the reconstruction of the spatial distribution of the writing beam together with the induced spatial distribution of the refractive index change from one ptychographic scan, with numerical simulations of the charge migration. This will enable us to fit microscopic charge migration parameters and to thoroughly investigate the nonlocality of the photorefractive response.