Appl. Phys. B 72, 635–640 (2001) / Digital Object Identifier (DOI) 10.1007/s003400100534
Applied Physics B Lasers and Optics
Reconstruction of parasitic holograms to characterize photorefractive materials M.A. Ellabban∗ , R.A. Rupp, M. Fally∗∗ Institut für Experimentalphysik, Universität Wien, Strudlhofgasse 4, 1090 Wien, Austria Received: 27 September 2000/Revised version: 1 December 2000/Published online: 21 February 2001 – Springer-Verlag 2001
Abstract. Photoinduced light scattering is a serious drawback that limits the applicability of thick holographic recording media but provides valuable information on the recording medium. As long as there is no correlation between the scattering centers in the crystal, photoinduced light scattering may be explained to result from the interference pattern of the incident beam and the field scattered from a single point-like scattering center. The hologram of this ellipsoidally scattered wave field will have practically the same structure in the reciprocal space modified by a response function which reflects the anisotropic properties of the recording medium. We studied photoinduced light scattering in LiNbO3 :Fe, a model system for photorefractive materials. The transmitted intensity in the stationary state of the scattering process is investigated as a function of the reconstruction angle at different wavelengths and polarizations of the reconstructing beam. The experimental results are analyzed by a simple phenomenological model based on the Ewald construction and can be used to choose suitable conditions at which holographic scattering can be minimized as well as to extract some physical parameters of the crystal. PACS: 42.40.-i; 42.25.Fx Photorefractive crystals are interesting materials for optical storage applications, dynamic holography and optical image amplification by two-wave mixing. Holographic recording processes are usually accompanied by photoinduced light scattering, a phenomenon discovered by Ashkin et al. in 1966 [1]. It may roughly be explained as follows: Irregularities at the air–crystal boundary or inhomogeneities within the crystal cause weak initial scattering which acts as the seed for the subsequent holographic amplification process. In the first step a hologram of the scattered wave is written, i.e., the interference pattern of the incident beam and its scattered field ∗ On
leave from the Physics Department, Faculty of Science, Tanta University, Egypt ∗∗ Corresponding author. (Fax: +43-1-4277-9511, E-mail:
[email protected])
is recorded. This hologram is simultaneously read out by the same incident wave. In linear recording media the wave field reconstructed from the hologram differs from the originally scattered wave only by its intensity and a possible overall phase shift. As changes of the phase of the incident wave affect all wave fields involved equally, they do not have any consequences. The intensity and phase of the reconstructed field thus depend on the recording mechanism only. If recording is local and an absorption hologram is recorded, then the reconstructed field interferes destructively (constructively only if there is gain, i.e., negative absorption). For refractiveindex holograms the reconstructed wave is shifted by ± π2 with respect to the originally scattered wave. Hence, there is in a first approximation no change in the amplitude of the sum of both fields, which we call the interference wave in what follows. However, non-local recording mechanisms provide an additional phase shift of ± π2 . Therefore, the reconstructed and scattered wave field may lead to a weakened or amplified interference wave. If there is amplification, this intensified wave field writes a hologram with the incident wave which has again the correct readout phase for amplification. Hence, we have a positive feedback which continues to amplify the scattered field until limitations such as pump-wave depletion lead to a stationary state. Holographic scattering reduces the quality of any optical imaging system involving two-wave mixing due to the increase in background noise with time, which reduces the signal-to-noise ratio, and due to the depletion of the energy of the pump beam as well as the signal beam leading to a reduction in the net gain. In general, photoinduced light scattering limits the applicability of photorefractive crystals. Therefore, several techniques have been proposed and demonstrated to suppress optical noise. Among these are the technique of crystal rotation [2], the achromatic gratings technique [3], a pulse read-out scheme [4] and a technique that utilizes the fact that the time constant in the buildup process is large and the time constant in the erasure process of the noise gratings is small [5]. On the other hand, light-induced scattering offers possibilities for material investigations, e.g., the photoconductivity can be determined from the time dependence of the trans-
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mitted intensity during recording of parasitic holograms [6]. The angle of the anisotropic scattering cone permits the calculation of the birefringence and of the Li/Nb ratio of LiNbO3 crystals [7]. Noise gratings are also used for analyzing and optimizing photochemical processes in photopolymers [8] and silver halide emulsions [9]. Recently, the subject became very interesting again, because it was found even in centrosymmetric crystals [10]. The starting point for the present work was to study the reconstruction conditions at which holographic scattering can be best minimized in the applications of imaging processes, or under which circumstances the effect can be exploited for material research. Compared with investigations using the traditional two-wave mixing techniques, the holographic scattering technique is much simpler and not sensitive to vibrations. In the present work we used a single-beam setup to study holographic scattering in a LiNbO3 crystal, a fine model crystal. The transmitted intensity as a function of the reconstruction angles around different rotation axes at different reconstruction wavelengths was studied. In addition, the dependence of transmitted intensity on the relative polarization between the recording and reconstruction beams was studied. Moreover, we used a simple phenomenological model based on the concept of the Ewald sphere, well known and widely used in X-ray crystallography, which explains the main features of holographic scattering. 1 Experiment We used an oxidized LiNbO3 :Fe crystal (742-05/Ox) in the single domain state with an iron content of 0.1 wt. % Fe2 O3 in the melt with dimensions 17.9 × 9.6 × 2.55 mm3 . The dominant charge transport mechanism in such oxidized crystals is the photovoltaic effect, because the photovoltaic field is proportional to the Fe3+ concentration. For all investigations described in this paper, holographic scattering was generated by irradiating the sample with an extraordinarily polarized ion Argon laser beam that was incident perpendicular to the sample surface. The incident intensity was 0.12 W/cm2 and the recording wavelength was either λp = 514 nm or 488 nm. Figure 1 shows the geometries used in our experimental setup for recording and reading schematically, together with a farfield scattering pattern. The transmitted intensity is detected by a silicon photodiode which is connected to a photome-
Fig. 1. General schematic of the experimental setup during recording and reconstruction, including a far field scattering pattern
Fig. 2. Time dependence of transmission during recording of parasitic holograms using an extraordinarily polarized beam at λp = 488 nm and an incident intensity of 0.12 W/cm2 . The solid line represents the fitting curve according to (6) in [6]
ter. Figure 2 shows the time dependence of the transmitted intensity IT (t) at time t, normalized to the transmitted intensity IT (0) at time 0, which represents the characteristic recording/amplification curve of the parasitic hologram for λp = 488 nm. The transmitted intensity decreases until it reaches a steady-state value. The solid line represents a fitting curve describing the recording/amplification according to Eq. (6) in [6]. For crystal rotation the sample holder is fixed on an accurately controlled rotation table (±0.001◦). We rotate the crystal during reconstruction either around its c-axis (ω-rotation) or around an axis perpendicular to the plane including the c-axis and the incident beam (φ-rotation). Experimental results for reconstruction using an extraordinary reading beam and ω -rotation have already been given in [11]. In our first investigation the polarization of the reading beam is kept unchanged with respect to the recording process, i.e. extraordinary. A very low exposure of the reading beam (≈ 0.67 mJ) is used in order to obviate any changes of the already recorded refractive-index structure. Figure 3a shows the angular dependence of the transmission for a φ-rotation at six different reading wavelengths (λr = 514, 501, 496, 488, 476 and 458 nm) for a hologram recorded at λp = 514 nm. The angular dependence of transmission for λr = λp = 514 nm is symmetric around the origin (= Bragg angle), while there are two minima at rotation angles differing from zero for all other reading wavelengths, i.e., at λr < λp . This resembles the case of the ω-rotation [11], except for the asymmetry in the depth of the minima. The rotation angle at which the minima occur increases with the difference between the construction and reconstruction wavelength. As a next step we reconstructed a parasitic hologram recorded at λp = 488 nm, as shown in Fig. 3b. At a first glance one would say that the situation is the same as compared to the ω-rotation: one minimum for all wavelengths higher than the writing wavelength. However, inspecting those curves more closely one “feels” that the transmission curve is only the envelope of a double minimum structure which cannot be resolved. This has been verified by inspecting the second derivative. For all wavelengths the position of the two minima seems to be symmetric with respect to the origin, but their depths are not the same. That asymmetry mirrors the asymmetry in the scattering pattern along the c-axis behind
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Fig. 3a,b. Angular dependence of the transmitted intensity for φ-rotation and read-out with extraordinarily polarized light at different wavelengths. a The gratings are recorded at λp = 514 nm; λr ≤ λp . b The recording wavelength is 488 nm; λr ≤ λp and λr > λp
the crystal. By inspecting Fig. 3b again, it becomes surprising clear that the relative depth of the two minima, when rotating either positively or negatively, depends on whether λr < λp or λr > λp . The minima for +φ-rotation are deeper than the others for λr > λp , but shallower for λr < λp . In our second investigation the parasitic hologram is investigated using ordinarily polarized light. Figure 4a and b show the transmission as a function of the reconstruction angle φ using an ordinarily polarized beam at different wavelengths for writing wavelengths of λp = 488 nm and λp = 514 nm, respectively. At first glance the results resemble the previously discussed case of reconstruction using extraordinarily polarized light. However, there are two main differences: First, the transmission values are much higher in the case of reading with ordinarily polarized light. This becomes apparent from a comparison of the transmission curves for reconstruction at λr = λp = 514 nm for extraordinarily and ordinarily polarized beams (Fig. 5). Second, the lowest minimum value of transmission does not occur at λr = λp but at λr < λp . It appears at λr = 476 nm for λp = 488 nm and at λr = 496 nm for λp = 514 nm. Note that the position of the minima deviates from the recording angle towards the positive direction of rotation. To complete our studies we measured the transmittance performing an ω-rotation using an ordinarily polarized beam. Figure 6a and b show the angular dependence of transmission at different wavelengths for holograms recorded at
Fig. 4a,b. Transmission as a function of the rotation angle φ at different wavelengths using an ordinarily polarized beam for reconstruction and a λp = 488 nm and b λp = 514 nm
Fig. 5. Comparison of the transmitted intensity as a function of the rotation angle φ for extraordinarily and ordinarily polarized beams. λr = λp = 488 nm
λp = 488 nm and λp = 514 nm, respectively. The transmission curves seem to be similar to the case of reconstruction performing a φ-rotation using ordinarily polarized light with respect to the fact that the lowest minimum value of transmission appears at λr < λp . However, the position of the minima deviates from Bragg angle towards the negative direction of rotation. This suggests an asymmetry in the distribution of k vectors of the structure factor with respect to the +k y and −k y direction.
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Fig. 6a,b. Angular dependence of the transmission for ω-rotation at different wavelengths using an ordinarily polarized beam for reconstruction. a λp = 488 nm. b λp = 514 nm
2 The model and its predictions As mentioned in the introduction, the stationary state intensity pattern may be considered to be the consequence of the amplified, originally scattered wave field. Let us assume first that only one scattering centre of sub-wavelength size is present from which a spherical wave would emerge in isotropic crystals and an ellipsoidal phase front in anisotropic crystals. In the reciprocal space such a wave front is represented by the spherical or an ellipsoidal surface |k| = nω/c, i.e., by the Ewald sphere or ellipsoid, respectively. Now any unknown source of scattering in photorefractive samples may be considered as a collection of such point sources which results in the far field in a rather wild interference pattern of rapidly varying intensity as a function of the spherical coordinates Θ and Φ as long as there is not any correlation between the scattering centres. The latter we may regard as absence of coherence of the refractive-index pattern in the real space. As a consequence, a smoothed average of that wild interference pattern reduces to be the ellipsoidal wave front of a single scattering centre again. Since we only monitor the transmission loss in our experiments, this corresponds to an integration which naturally includes a smoothing. The hologram of this ellipsoidally scattered wave field will practically have the same structure in the reciprocal space, modified by a response function which reflects the anisotropic properties of the recording medium. Let us assume that m(Q) is the modulation resulting from the interference of the pump beam with
the scattered wave field. In the limit of small modulations the relationship ∆n(Q) = R(Q)m(Q) holds, where ∆n(Q) is the refractive index change and R(Q) is a linear response function. According to Bragg’s law ∆n(Q) differs from zero only for Q = K = qs − qp. Here qp and qs are the wave vectors of the pump and scattered beams, respectively. The surface with nonzero ∆n(Q) forms two ellipsoids, the primary and conjugate images. The major and minor axes of the ellipsoids are 2πn o /λp and 2πn e /λp , respectively. From the scattering pattern of our crystal we know that the intensity is mainly distributed in the k x –k z plane. Therefore, the loci of the largest ∆n(Q) are the intersections of this plane with the two ellipsoids, which will be called the writing ellipses. In reconstruction, the totally scattered intensity is given by the integral value of the intersection region containing intensity (writing ellipses) and simultaneously satisfying the Bragg condition (reading ellipsoid or sphere). In the case of reconstruction using extraordinarily polarized light, we have to consider the intersection of the ellipsoid with the k x –k z plane, which will be called the reading ellipse. At ω = φ = 0, the major and minor axes of the reading ellipse are 2πn o /λr and 2πn e /λr , respectively. When reconstructing using ordinarily polarized light, the intersection with the plane will be a circle with radius 2πn o /λr at ω = φ = 0. Figure 7 schematically shows the writing ellipses in addition to the reading ellipse or circles at φ = ω = 0 for different wavelengths in reciprocal space. The gray part represents the writing ellipses with a strongly exaggerated width and ellipticity for better representation. The gray scale gives the intensity distribution in the k x –k z plane, i.e., the structure factor of the parasitic holograms, which is assumed to be constant along the writing ellipse. This is a rather coarse approximation, which is, however, basically justified by the observed scattering pattern. The solid line represents the intersection of the reading ellipse with that plane, i.e., for reading with extraordinary polarization at a wavelength of λr = λp . The bold solid, dashed and dotted lines
Fig. 7. Geometrical representation of the writing ellipses and either reading ellipses or circles at φ = ω = 0 at different reading wavelengths in the reciprocal space. The gray part represents the writing ellipses. The reading ellipse (solid line) corresponds to the case of e–e polarization and λr = λp . The reading circles correspond to the case of e–o polarization at λr = λp (bold solid line), λr < λp (dashed line) and λr > λp (dotted line). The shortdashed line represents the reading ellipse for e–e polarization, λr = λp and φ = 0
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represent the intersection of the reading spheres of ordinary polarization at λr = λp , λr < λp and λr > λp , respectively. To evaluate the total scattered intensity as a function of the rotation angles φ and ω, the wavelength and the polarization state of the light, we integrate the structure factor along the reading circle (ellipse). A comparison between the simulated and the measured transmission curves allows information about the structure factor of the parasitic holograms to be extracted. The details of the phenomenological model and the calculation of the transmittance are discussed in [11]. For our experimental conditions the model predicts the following behaviour for the transmitted intensity: 2.1 φ-rotation using extraordinarily polarized light Performing a φ-rotation, the lengths of the major and minor axes of the reading ellipse remain constant, rotating around the origin of the reciprocal space with a radius 2πn e /λr . – λr = λp : The largest integral value of the intersection will be at the Bragg angle (Fig. 7, solid line). The larger the deviation from the Bragg angle, the smaller the integral value of intersection, i.e., the lower the diffracted intensity (Fig. 8, bold solid line, and Fig. 3). – λr > λp : The principal axes of the reading ellipse are smaller than that of the writing ellipse. Hence the highest diffraction will not occur at the Bragg angle. This leads to the double minima structure (Fig. 8, dotted line, and Fig. 3). – λr < λp : The principal axes of the reading ellipse are larger than the axes of the writing ellipse. Again the optimal intersection will not be at the Bragg angle, resulting in two minima (Fig. 8, dashed line, and Fig. 3). – The difference in the values of the two minima as shown in Fig. 3 can be understood by inspecting the asymmetric scattering pattern with respect to the +c-axis and the −c-axis; different structure factors are evident, in contrast to our simplistic assumption that the structure factor is equally distributed along c-axis. Now we are also able to understand why the minima values are much deeper in one of the rotation directions for λr > λp and much shallower for λr < λp . Let us rotate the crystal in a positive
Fig. 8. Calculated transmission as a function of the reconstruction angle φ using extraordinarily polarized light at λr = λp (bold solid line), λr < λp (dashed line) and λr > λp (dotted line) for a hologram written at λp = 488 nm. The sample thickness is 2.55 mm
sense, +φ. For λr > λp , the coincidence will be between the two upper halves of the reading and writing ellipses (see Fig. 7) where the k-vector components have a high structure factor. In the case of λr < λp , the coincidence will stem from the lower halves of the two ellipses, where the k-vector components have a much smaller structure factor. The gain direction is marked out by the electrooptic effect in the crystal and by the sign of the carriers contributing to the charge transport processes, i.e., electrons in our case. More intensity is therefore scattered in the positive direction on the c-axis. This means that the deeper minimum in the case of λr > λp represents the direction of the +c-axis in our crystal. 2.2 φ-rotation and ω-rotation in reconstruction with ordinarily polarized light In this case we are looking for the integral value of the intersection region between the writing ellipse and a circle. – λr = λp and φ = ω = 0: According to the fact that our crystal is birefringent, 2πn e /λp = 2πn o /λp (Fig. 7, bold solid line). Therefore, a complete coincidence between the writing ellipse and the reading circle is not possible in any case. By rotating the crystal, the scattered intensity will even be lower (Fig. 9, bold solid line, and Fig. 4 for φ-rotation; Fig. 10, bold solid line, and Fig. 6 for ω-rotation). – λr < λp : There will be a certain wavelength, λrm , at which highest diffraction will occur at the Bragg angle (Figs. 9 and 10, solid line). This wavelength depends on the birefringence of the sample. Comparing the curvatures of the writing ellipse and the reading circle, λrm can be estimated by the following equation: λrm (α) =
n e n 2o
3/2 λp , n 2e − (n 2e − n 2o ) cos (α)
(1)
where α is the polar angle. As the main contribution to the scattered intensity is provided by small k-vectors, we estimate the optimal wavelength λrm by equating the curvature of the writing ellipse with the curvature of the
Fig. 9. Calculated transmission as a function of the reconstruction angle φ using ordinarily polarized light at λr = λp (bold solid line), λr = λrm (solid line), λr < λp (dashed line) and λr > λp (dotted line) for a hologram written at λp = 488 nm. The sample thickness is 2.55 mm
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Fig. 10. Calculated transmission as a function of the reconstruction angle ω using ordinarily polarized light at λr = λp (bold solid line), λr = λrm (solid line), λr < λp (dashed line) and λr > λp (dotted line) for hologram written at λp = 514 nm. The sample thickness is 2.55 mm
reading circle at α = 0. Equation (1) reduces to λrm =
ne λp . no
(2)
Reading with other wavelengths λr < λrm will result in two minima, similar to the case of extraordinary reading (Fig. 9, dashed line, and Fig. 4 for φ-rotation; Fig. 10, dashed line, and Fig. 6 for ω-rotation). In the case of positive birefringent crystals, maximum scattered intensity can be observed at λr > λp . Therefore, from reconstruction using ordinarily polarized light we can learn whether our crystal is either positively or negatively birefringent. In addition, the birefringence of the crystal can be estimated. – λr > λp : There will be two minima with low values of diffraction (Fig. 9, dotted line, and Fig. 4) for φ-rotation, and one minimum for ω-rotation (Fig. 10, dotted line, and Fig. 6).
(k y ) which should be considered. Work on this problem is in progress. In the case of reconstruction using an ordinarily polarized beam, the model predicts that the optimum wavelength for minimum transmission λrm ∼ = 495 nm for holograms recorded at λp = 514 nm, and λrm ∼ = 470 nm for the case of recording at λp = 488 nm. As we can employ only discrete wavelengths from our laser source, we cannot reach this particular minimum, which would be useful in determining the birefringence of our crystal. The deviation of the position of the lowest minimum value of transmission from the Bragg angle in the case of φ-rotation comes from the asymmetry of the scattered intensity with respect to the positive and negative c-axis, whereas in the case of ω-rotation, it indicates an asymmetric structure in the +k y and −k y directions. In summary, the holographic scattering technique can be used to extract the following information: – Knowing the sign of charge carriers and the sign of the effective electro-optic coefficient, we can determine the direction of the +c-axis of our crystal. – From reconstruction with ordinarily polarized light, we can discriminate negatively and positively birefringent crystals, and the birefringence can be estimated. – From the recording curve, rough estimates for various physical parameters, such as the photoconductivity, can be determined. – Comparing the simulated transmission curves to the measured data, information about the structure factor of the parasitic holograms can be extracted. To minimize the effect of parasitic holograms we recommend reconstruction with ordinary polarization and wavelengths higher than the recording wavelength. Acknowledgements. The authors thanks Prof. Dr. E. Krätzig for providing the sample. Discussions with Dr. Theo Woike are gratefully acknowledged. We thank Prof. Dr. E. Tillmanns for his continuous support. M.A.E. is grateful for a grant from the Austrian Academic Exchange Service (ÖAD).
References 3 Discussion Despite of its extreme simplification, our phenomenological model explains the main features of our results concerning parasitic hologram reconstruction for different rotation directions and polarization states. It reveals the qualitative behavior, i.e., the characteristic shape of the transmission curves, and hence predicts correctly if one or two minima appear, depending on the ratio λr /λp , the polarization state and the rotation axis. Quantitative predictions, however, do not hold. In particular the calculated position of the minima moves much faster than measured in the experiment when changing the ratio λr /λp . In addition, the correct depth of the minima cannot be calculated. This might be due to the fact that the model approximates the real scattering pattern with a twodimensional (k x –k z ) scattering distribution only. In reality there is a small component of diffraction in the third direction
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