Parameters influencing the grating formation in photorefractive BaTiO3
M. Esselbach, G. Cedilnik, A. Kiessling, and R. Kowarschik
Institute of Applied Optics
Friedrich Schiller University Jena
Max-Wien-Platz 1, D-07743 Jena, Germany
E-mail: matthias@esselbach.de
Internet: www.esselbach.de
Abstract
The interference pattern of light waves creates a refractive index modulation in photorefractive media. This process is relatively well described by the theory (Kukhtarev equations) but deviations are found. Therefore, experimental methods are used in order to characterize the processes. The influence of the absolute intensity as well as of the intensity ratio of the interfering waves on the refractive index modulation are studied in a two-wave mixing arrangement. Especially for the dependence on the absolute intensity the interesting relation Dn = f(Iabs) µ Iabsa
was found that is not predicted by the theory. Nevertheless, this experimental approximation can be used in calculation in order to minimize the error. For a four-wave mixing arrangement working as a phase conjugate mirror, the time development of the reflectivity after switching on the reading pump wave is investigated. It will be shown that always for a short time high reflectivities can be reached. For suitable intensity ratios of the interacting waves this is also possible in the steady state. This behavior will be explained theoretically. A better understanding of the grating buildup processes is reached by these studies, which allow the specific controlling of interaction parameters of the wave mixing.
Dependencies on Absolute Intensity and Modulation
Refractive index gratings can be created in photorefractive media via an intensity distribution built by interference of two waves. The strength or modulation of such a grating is influenced by the intensity of the writing beams, by their absolute intensity as well as by their intensity ratio. The theory (band transport model, Kukhtarev equations) yields the following expression for the space charge field Esc that is created inside the medium:
where m
=modulation of the intensity contribution, x = influence of positive charge carriers, kB = Boltzmann constant, T = temperature, q = unit charge,
= vector of the intensity grating, K = ê ê, k0 
, N = effective density of photorefractive charge carriers, and 
=
. The most important points of this equation are that Esc is direct proportional to the modulation m (visibility) of the intensity distribution (valid for m << 1) and that Esc does not depend on the absolute intensity Iabs. The diffraction efficiency h at the grating can be derived for the Bragg case from
where
Dn = refractive index modulation, d = thickness of the grating, l = wavelength , q = angle of the writing beams, and Esc = amplitude of the space charge field, to be proportional to m2. The relations h = f(m) and Dn = f(m) should be verified first. For this purpose the experimental arrangement shown in fig.1 was used. The grating is written by two beams with intensities I1 and I2 of an Ar-laser where the absolute intensity (Iabs = I1+I2) as well as the intensity ratio of the beams in the BaTiO3-crystal can be controlled by neutral density filters (ND). The grating is read by a probe beam from a HeNe-laser fulfilling the Bragg condition. An interaction between the beams is avoided by using two lasers.
Figure 1: Experimental arrangement
The homogeneous illumination of the crystal by the probe beam effects the erasure of the refractive index grating or at least the decrease of the modulation of the grating. In order to avoid a distortion of the results of the measurement the probe beam was switched off during the write process. Then we measured the intensity of the diffracted part of the probe beam after switching on this beam. Fig.2 shows a typical temporal course. The value of the intensity for t = 0, when the grating is undistorted, was noted.
Figure 2: Temporal course of the intensity of the diffracted probe beam
The modulation m of the intensity distribution, 
, of the writing beams is varied, whereas the total intensity Iabs is kept constant. The results are shown in fig.3. The left diagram shows the diffraction efficiency
Figure 3: Influence of the modulation of the intensity distribution on the diffraction efficiency
The relation shown by equation (2) is confirmed. The right diagram gives the same result. Here, the modulation of the refractive index
Dn as a function of the modulation m is shown. The linear dependence is obvious. Astonishingly, the dependencies are not only valid for small modulations m, as required to derive equation (1), but for all the range of m between 0 and 1. Efforts to explain this result have been done in [1, 2] by using perturbation analysis and taking higher harmonics into account.
Figure 4: Influence of the absolute intensity Iabs
In the following, the influence of the total intensity Iabs on the refractive index modulation Dn and the diffraction efficiency h is studied (fig.4). The intensity Iabs was varied over five orders of magnitude, between 100 nW/beam and 60 mW/beam (beam » 3 mm2). The relation is measured for various values of the modulation m that is used as parameter varying between different graphs.
Figure 5: Influence of the absolute intensity Iabs (logarithmic representation)
The diffraction efficiency increases with increasing intensity. Because of the large intensity range, a logarithmic representation is shown in fig.5.
The theory (Kukhtarev) does not explain the measured relation between
Dn and Iabs. Therefore, it thoroughly seems to be appropriate to fit the measured points (fig.5) by linear functions in order to get a first rough approximation.Using ln (
Dn) ~ ln (Iabs) we get:This function describes the physical relation but cannot be explained by the theory. Other obviously possible functions (e.g. exponential) gave no satisfactory results. Nevertheless, the error in any further calculations based on intensity dependencies is much smaller if this approximation is used than if the dependence is ignored at all. Moreover, fig.6 makes clear that the deviation between approximation and real values is not too important and that the description by the derived function is rather good.
Figure 6: Fit of the derived function
Dn ~ Iabsa to the measured values shown in fig.4 for m = 0.94
Methods to Reach High Reflectivities with Four-Wave Mixing
In the following we studied the four-wave mixing concerning the reachable reflectivities. For applications it is often desirable to realize phase conjugation with high reflectivity. Using FWM, the problem is that the reading pump wave is erasing the grating inside the crystal at the same time as it reads it and therefore the reflectivity cannot be unlimited increased with a higher intensity of the read wave. Here, an arrangement (fig.7) is studied where first the grating is written by signal (S) and pump wave (P) and only then the read wave (L) is switched on by a shutter. The intensities and intensity ratios can be varied. Unexpanded laser beams are used.
Figure 7: Experimental arrangement. Intensities: S, P, L
So, it should be possible to reach reflectivities higher than in the steady state at least for a short time. Fig.8 shows the course in time of the reflectivity after switching on the read wave. The reflectivity starts at a certain value R(t = 0) and then it decreases till it reaches the steady state value R(t
® ¥). At t = 0, the read wave is diffracted at an existing grating. Then the grating is erased by the read wave, that means that the depth of the refractive index distribution is decreased. The steady state is reached because writing and erasing of the grating proceed simultaneously. In fig.8 the ratio P/S and therefore the modulation of the intensity distribution is constant for all curves. The sum intensity S+P is the parameter varied between the curves. In fig.9 only the signal intensity S is the varied parameter, whereas P and L are constant.
Figure 8: Reflectivity when the read wave L is switched on. P and S are variable with a constant ratio.
Figure 9: Reflectivity when the read wave L is switched on. S is variable.
Both figures, fig.8 and fig.9, show qualitatively the same courses in time. The graphs correspond well with our assumption. The course in time can be well described by the decreasing exponential function
R
µ 1-e-atthat is shown (for curves A and B from fig.9) in fig.10.
Figure 10: The course in time described by a decreasing exponential function according equation (4).
The switched reflectivity as well as the steady state reflectivity in dependence on the ratio L/(S+P), read wave intensity over the intensities of the writing beams, are shown in fig.11.
The course of the graph for R(0) can be explained as follows: The read wave L is diffracted at an existing and uninfluenced grating. Therefore, R increases with increasing L. The deviation for very small S+P can be attributed to thermal relaxation.
Figure 11: Switched and steady state reflectivity.
The graph R(
¥) can be explained as follows. The interference of S and P builds the intensity distribution
The influence of the additional read wave L reduces the value of the visibility from m to k, where
The diffraction efficiency
h at a thick grating is given by
This yields
The course of the function R = f(L) is shown in fig.12.
Figure 12: Function R = f(L).
An qualitative correspondence to the first part of the curve for R(
¥) in fig.11 is obvious. The increase in the second part of this curve can be explained by TWM between the initially created pc-wave (Win) and the read wave L. The TWM gain for Win is as higher as smaller Win is with respect to L. The pc-wave is stronger amplified and therefore R(¥) is higher for smaller S+P or higher L.Results of reflectivity measurements where only the intensity of the signal wave S is varied but L and P are kept constant are shown in fig.13.
It is important to note that in the switched regime a real amplified reflection (R > 1, R = output/input) is possible (for small S up to 150), whereas the steady state reflectivity is much lower (maximum: R
» 3).In both cases the reflectivity increases with decreasing signal intensity S. The limit for the measurements is given by the sensitivity of the sensor (L/S
®106). In the steady state real amplification is possible but only for very small signal intensities S.
Figure 13: Switched and steady state reflectivity. The signal intensity S is varied.
Conclusions
We have studied the dependence of the refractive index modulation with two-wave mixing in a photorefractive medium on the modulation and on the intensity. For the modulation we could confirm the relation known from the literature and predicted by the theory. We observed a strong dependence on the absolute intensity and can describe this mathematically.
The switched four-wave mixing regime enables to reach high reflectivity, but just for a short time. It could be used for applications where only the intensity just in a moment is important.
Acknowledgment
This research has been partially supported by the Deutsche Forschungsgemeinschaft (DFG) within the project "Innovationskolleg Optische Informationstechnik".
References