1. Introduction

Iron-doped lithium niobate (LiNbO₃:Fe) crystals can be used as an excellent material to store volume holograms [1]. When illuminated, electrons from Fe²⁺-centers are excited into the conduction band. If the illumination is inhomogeneous, these electrons are redistributed and trapped by Fe³⁺-centers, due to drift, diffusion, and the bulk photovoltaic effect. The charge distribution gives rise to electric fields which modulate the index-of-refraction via the electro-optic effect. This is the so-called photorefractive effect [2].

Considering at first the case of holographic recording with two homogeneous plane waves, the dynamics can be described as follows [3]: The electric field exponentially grows and saturates at E _SC,sat = −mE _phv Here E _phv is the bulk photovoltaic field, quantifying this charge-driving force [2]. The degree of modulation m is the contrast of the interference pattern. Furthermore, the time constant is given by τ = εε ₀/(αI). Here e is the dielectric constant of the lithium niobate crystal, ε ₀ is the vacuum permittivity, α is the photoconductivity per light intensity, and I is the light intensity. The degree of modulation m has to be much smaller than 1 for this relationship to hold true. The time constant as well as the strength of the space-charge field change for higher degrees of modulation. As is commonly done, these changes are neglected in this paper. Furthermore, in most experimental cases m is indeed smaller than 1 because of internal light reflections at the crystal surfaces causing a homogeneous background illumination. The electric space charge field E _SC changes the index-of-refraction by the electro-optic-effect according to ∆n = -(1/2)n ³ rE _SC, where n is the refractive index and r is an element of the electro-optic tensor.

A serious drawback of iron-doped lithium niobate crystals is the slow response time. The build-up of the index-of-refraction changes is much slower than in other potorefractive materials. A possible way to overcome this problem is the use of focused beams and hence of larger light intensities. Response times on the order of 10 ms can be achieved by focusing [4].

However, the use of focused beams has some issues. When the power deposited due to absorption heats up the crystal by several Kelvin, the pyroelectric effect and crystal deformations limit the maximal achievable index-of-refraction changes [4]. There is another effect which limits the photorefractive effect even when the light intensity is too small for significant heating: It is known that inhomogeneous illumination leads to the build-up of charges at the borders of the illuminated section acting against the bulk photovoltaic effect [5]. Here we study the recording of gratings by focused beams considering the build-up of this compensation field and and provide an analytical solution for the dynamics of this process.

2. Experimental methods

Figure 1 shows a scheme of the setup for holography with two focused waves. Light from a frequency-doubled Nd:YAG laser with the wavelength λ = 532nm is focused with a lens of a focal length of f= 500 mm and split into two beams. The two beams interfere close to the focus inside a LiNbO₃:Fe crystal. The angle of incidence in air is γ = 24°. From the Bragg condition we find for the grating period Λ = λ/(2 sinγ) = 0.65 μm. The crystal c-axis is oriented parallel to the long side of the crystal in Fig. 1. Ordinarily polarized light is used to minimize beam coupling.

A congruently-grown lithium niobate crystal doped with 0.05weight% iron is used. These iron concentration is optimal for our cause, because decreasing the concentration would decrease the maximal refractive-index change, and increasing it would only increase the absorption without any further refractive-index-change enhancement [6]. By absorption measurements the oxidization state is found to be: c _Fe²⁺/c _Fe³⁺ = 0.09 [7].

One beam is periodically blocked with an electro-mechanical shutter. The powers P _trans and P _diff of the transmitted and diffracted beams are recorded when one beam is blocked, and the diffraction efficiency η = P _diff/(P _trans + P _diff) is obtained. This definition has the advantage that reflections at the crystal surfaces and absorption in the crystal play no role.

 

Fig. 1. Experimental setup as it is used to write holograms with two focused beams. The angle between the two recording beams is 2γ.

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Diffraction on thick gratings is described by Kogelnik’s coupled wave equations [8]. The diffraction efficiency is η = sin²[λ d∆n/(λ cos θ)], where d is the thickness of the hologram, λ is the vacuum wavelength, and θ is the angle of incidence inside the crystal. For finite-sized gratings, modified formula have to be used in principle [9]. But since the differences are negligible for η < 30%, and E _SC ∝ ∆n we still take the η = sin²(const × E _SC) equation to relate η and the space charge field E _SC.

The light power in the beam without the shutter is P ₂ = 0.25mW, in the other beam it is P ₁ = 0.18mW. The beam’s 1/e ²-radius is measured to be w ₀ = 60μm if there is no crystal. So the average light intensity in the 1/e ² -disk can be estimated to be I = 38kWm^-2. This is an order of magnitude smaller than the intensities needed for relevant pyroelectric effects or crystal surface deformations [4]. In the crystal the light intensity is presumably even smaller, as absorption in the crystal and reflections at its surface have been neglected. The intensity has been chosen, such that the time constant is as small as possible, so that no long term stability is needed, without having relevant heating of the crystal.

We illuminate the crystal with one of the two beams before we start to record a hologram. In the first seconds of this pre-illumination the transmitted beam is expanding. This behavior can be explained by a photorefractive lens which quickly builds-up. After some seconds of pre-illumination, a steady state is reached, and holograms are recorded. This procedure helps to take reproducible recording curves.

3. Results

Figure 2 shows the diffraction efficiency versus time during recording. The efficiency is measured every 3 s. It increases and reaches a maximum of about η = 30% after 35 s. Afterwards it decreases and saturates at less than 5 %. The pre-illumination time is t _p = 45 s for this curve.

Now, the pre-illumination time t _p is systematically varied. Figure 3 shows recording curves for different t _p. The curves increase, reach a maximum, and drop to a smaller value. The longer the pre-illumination time, the smaller becomes the maximum. The time constant for all curves is the same within the experimental errors.

The long-term saturation values for the curves drop to 2–4%. This is not shown in Fig. 3. The reproducibility becomes worse for longer recording times. The reason is that the setup is not actively stabilized, and small phase changes can destroy previously written holograms. However, the qualitative behavior is well reproducible.

As the slope at the beginning of the recording curves is much better reproducible than the long-term behavior, we plot in Fig. 4 the slope of asin √η of the curves from Fig. 3 versus the pre-illumination time. The slope decreases with increasing pre-illumination times and saturates for longer pre-illumination times.

 

Fig. 2. The diffraction efficiency as it evolves with time. Two focused beams are used to write the hologram. A fit of the analytical formula (10) is plotted as a solid line.

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Fig. 3. Diffraction efficiency versus time while a hologram is recorded with two focused beams. One beam illuminates the crystal for different durations before starting the recording process. The pre-illumination times are: 45s, 90s, 180s, 300s, 600s.

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Fig. 4. Slope of asin(√η) for different pre-illumination times. The error at t _p = 90s is estimated from six different curves obtained under nominally identical conditions. The solid, gray line is an exponential fit.

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4. Discussion

4.1. Qualitative model

 

Fig. 5. Scheme of holographic recording with focused light. The focused light is restricted to a small area illustrated by the white spot in the center. The box shows a cross section of the light intensity. Thus charges accumulate at the borders of the illuminated area. The corresponding compensation field E _comp finally compensates the bulk photovoltaic field E _phv

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The drop of the diffraction efficiency after reaching a transient maximum can be explained by the inhomogeneous illumination caused by the two focused beams: The conductivity is close to zero outside of the illuminated area. So charges accumulate at the border of this region. The bulk photovoltaic effect redistributes electrons along the c-axis, and an electric field builds-up in the illuminated volume. This build-up saturates when the electric field compensates the bulk photovoltaic driving force. The situation is shown in Fig. 5. The bulk photovoltaic effect is the dominant charge transport process and responsible for the majority of the grating recording. Thus after some time the recorded gratings degrade.

This compensation mechanism differs from the mechanisms in thermal or electrical fixing experiments [10, 11]. In fixing experiments the space charge gratings are locally compensated by ions or ferroelectric domain walls. Whereas in the case for recording with two focused beams the bulk photovoltaic current is compensated by a uniform macroscopic field with charge accumulations at the border of the illuminated region.

4.2. Build-up of the compensation field

To be able to quantitatively discuss the results, we make an one-dimensional calculation of the grating build-up with inhomogeneous illumination. First, we calculate how the compensation field evolves when the crystal – to be found in the region −L _crys/2 <z<L _crys/2 – is illuminated homogeneously with light of the intensity I = I _illu in the region −L _illu/2 <z<L _illu/2 and kept in darkness in all other regions. Thus, the crystal is assumed to be illuminated at the center, z = 0, and left in darkness at the borders of the crystal. The equations for the evolution of the electric field in iron-doped LiNbO₃ are [2]:

Here j is the current density, β is the bulk photovoltaic coefficient, α is the photoconductivity per light intensity, E is the electric field, ρ is the charge density, and ε is the dielectric constant of LiNbO₃. Nonlinear effects are neglected, because the intensities in the experiment are still sufficiently low. From the two last equations it follows:

In regions with no illumination we have j = 0, and thus: j|_z=−Lcrys/2 = 0. After integrating (2) over z it follows:

We now assume that the electric field E = E _SC ⁽⁰⁾ is spatially constant in −L _illu < z < L _illu, and E = E _dark = E|_z=−Lcrys/2 in the remaining crystal. If the crystal is in a short-circuited situation it is: ∫^L_crys/2 _−Lcrys/2 E = 0. Thus, it is: 0 = ∫^L_crys/2 _−Lcrys/2 E = L _illu E _SC ⁽⁰⁾+(L _crys−L _illu)E _dark. Inserting this into (3) we get in −L _illu<z<L _illu:

In the beginning, no hologram has been recorded, and thus E _SC ⁽⁰⁾ = 0 at t = 0. With

the solution to Eq. (4) is:

The saturation value of the electric space-charge field E _SC ⁽⁰⁾|_t→∞ is the so-called photovoltaic field E _phv = β/α. If there is some background illumination in the dark region, then E _SC ⁽⁰⁾ will saturate at a smaller field, and the bulk photovoltaic field E _phv will be partially compensated only. We split this field in a compensated part E _phv,c and an uncompensated part E _phv,u: E _phv = E _phv,c+E _phv,u. For L _illu ≪ L _crys it is E _phv,c = [1 −(I _dark/I _illu)] (β/α), and E _phv,u = [I _dark/I _illu](β/α), where I _dark is the light intensity in the darker region.

At first glance it might be surprising that τ goes to infinity for L _illu → L _crys. This comes from the fact that the dark parts become thinner and thinner, which increases their electrical capacity, making more and more charges necessary to build E _SC ⁽⁰⁾ up and to maintain the short-circuited condition of the outside crystal faces.

4.3. Grating recording

Second, let us assume that by the interference of two focused beams a sinusoidal modulation of the light intensity can be found in the region -L _illu/2 < z < L _illu/2: I = I _illu [1+m cos (Kz)]. The degree of modulation should fulfill m ≪ 1. If the focal area is much smaller than the crystal, τ becomes:

Inserting E = E _SC(⁰⁾ + cos(Kz)E _SC ⁽¹⁾ into Eq. (2) we find that the equation, which has to be solved, reads:

Here values with the index (1) are amplitudes of sinusoidally modulated components. The zero-order compensation field is E _SC ⁽⁰⁾ = -E _phv,c [ 1 - exp(-t/τ)] is given by Eq. (6).

Considering E _SC ⁽¹⁾(0) = 0, the solution then is

The second term in Eq. (9) describes the uncompensated part of the photovoltaic field which behaves like a plane-wave-recorded grating in a completely illuminated crystal. The first term grows due to t/τ, as a grating is written, and later decreases with e ^−t/τ, as the compensation field builds up. We thus use the function

for a fit to the experimental data.

4.4. Influence of pre-illumination

If the crystal is pre-illuminated between t = -t _p and t = 0, a field is created exponentially with a time constant τ _p. Thus the homogeneous field becomes

And for t > 0 we find:

For the slope we get:

Thus, experimental data for the slope at the beginning of the recording is described by the function:

4.5. Comparison with experimental data

Now, we compare the predictions of this model to the experimental data. The solid line in Fig. 2 is a fit of Eq. (10). Fitting this function to two different curves with nominally the same conditions we find: A ₁ = 1.3 ± 0.1, τ = (36±7) s, and b = 0.16±0.06. The curves show some irregularities. These are probably due to instabilities in the setup, as it is not actively stabilized. However, some differences between the data and the calculated model curves are expected since a rectangular intensity profile and not a Gaussian curve was used to model the situation. For a Gaussian curve many different light intensities occur, making the time constants spatially dependent and causing smearing out of the evolution of the electric field.

The slopes at the beginning of the recording curves are not that much affected by instabilities. The small error bar in Fig. 4, which is deduced from several independent measurements, proves this. A fit of Eq. (14) results in: A ₂ = (0.09±0.003 )s^-1, τ _p = (86±8)s, and b = (0.14±0.03) s^-1. Comparing this with the fit to a single curve we see that the time constant τ _p is approximately twice as high as τ. This, however, is expected as only one beam and thus half the intensity is used for pre-illumination compared to recording. The long-term saturation value b is similar to that of the single-curve fit.

The time constant is τ = εε ₀/(α I _illu) (Eq. 7), where ε = 28 [12] is the dielectric constant of lithium niobate crystals, I _illu = 38kWm^-2 is the average light intensity, and α is the photoconductivity per light intensity. From [6] and using c _Fe²⁺ /c _Fe³⁺ = 0.09 we find α = 9.9×10^-16 mV^-2 for 514nm wavelength. Thus the expected time constant is: τ = 7s. The light intensity is certainly overestimated as reflections at the crystal surface, absorption and distortions enlarging the focus have been neglected. Furthermore, we use 532-nm light, where α drops significantly. Thus the time constant should be larger in reality which is in accordance to the experimental value τ ≈ 40 s.

In the analysis above we assumed that background illumination leads to an uncompensated portion of the bulk photovoltaic fields, which result in the long-term saturation value. One may think that diffusion currents can explain the long-term value as well. The electric field E _SC for holography with plane-wave gratings saturates at E _SC = E _phv+iE _diff. The complex number indicates a 90° phase shift of the diffusion grating. The bulk photovoltaic field is in our case supposed to be E _phv = β/α = 9.9MVm^-1. The diffusion field is E _diff = (k _B T/e)K = 0.37MVm^-1, where K = 2π/Λ is the grating vector length, and T = 300K is the room temperature. The fraction E _diff/E _phv ≈ 0.04 is too small to explain the long-term saturation value b. Thus we conclude that the long-term behavior has to be attributed indeed to some background illumination.

5. Conclusions

When holograms with focused beams are recorded in iron-doped lithium niobate crystals, the diffraction efficiency decreases after some time. This effect can be attributed to a compensation field which builds up as the crystal is not completely illuminated. If one of the two beams is used to pre-illuminate the crystal before recording, the maximal diffraction efficiency is further diminished. This is because the compensation field is already partially established when the recording begins. Both effects can be analytically modeled. Agreement between the theoretical equations and experimental data is found. Practical relevance lies in the fact that the effect described herein limits the maximal diffraction efficiency in the case of inhomogeneous illumination of the crystals.