## Abstract

In a KTa_{1-x}Nb_{x}O_{3} optical beam deflector, light rays are bent by a space charge formed by electrons that are injected through an electrode and captured by crystal defects. For complete device control, it is important to evaluate the space charge. We propose an optical method for measuring charge density distributions that utilizes the electrooptic (EO) effect of the material. With this method, electron accumulation caused by a screening effect was commonly observed near the cathode. The electron accumulation region extended toward the anode as the applied voltage and permittivity increased. This method can be applied to any EO materials that exhibit space charge distributions.

© 2014 Optical Society of America

## 1. Introduction

Recently a new type of optical beam deflector was proposed [1]. It is made of a block of electrooptic (EO) material with a pair of film electrodes. A space charge is formed inside this material by injecting electrons (or holes) via the electrodes. The space charge first forms an electric field distribution and then a refractive index distribution via the EO effect. This index distribution bends a light beam. An EO beam deflector has already been reported that is a prism made of an EO material with triangular electrodes [2]. The prism device was much faster than other beam deflectors because of its EO nature. However, the deflection angle was insufficiently large because deflection occurs only at the surfaces of the prism and the EO index modulation is minute. The speed of the new type EO deflector is the same but it can accumulate the effect of the index modulation along an optical path. Thus, with this deflector, deflection angles of more than several degrees are easily attained, which are much larger than those obtained with conventional prism devices. Denser space charges and thus larger deflection angles can be realized when the permittivity of the material is large. A potassium tantalate niobate (KTa_{1-x}Nb_{x}O_{3}, KTN) single crystal is a suitable material for this deflector because it has a huge permittivity together with an extremely strong EO effect. By using a KTN deflector, we have developed a high-speed wavelength-swept light source for an optical coherence tomography (OCT) cross-sectional imaging system [3]. The speed of the KTN is attractive for the *in-vivo* observation of moving organic objects [4].

As described later it is important for this KTN deflector to adjust the spatial distributions of the charge density and thus control the light wavefront. However, there have been few reports on electron motion and electron traps in KTN single crystals other than photorefractive doped KTN crystals [5]. Therefore, we developed a method to evaluate the charge distributions and thus investigate the electronic properties of KTN. In this paper, we report this measurement technique and discuss the obtained charge distributions. For the measurement, we utilized the fact that the light wavefront is modulated by the index distribution, and is thus determined by the charge distribution. Although we use this method to evaluate our KTN optical deflector, it can be applied to any substance that exhibits the EO effect.

## 2. Theory of beam deflection and charge density evaluation

First we explain the relationship between the deflection function and the space charge. For this we employ an EO crystal block. Two opposing faces of the block are fully covered with metal films that function as electrodes. Here we assume that the low-frequency permittivity of the EO crystal is huge. Indeed it is common for the relative permittivity of a KTN single crystal to exceed 10,000 in the paraelectric phase at temperatures just above its paraelectric-ferroelectric phase transition temperature. With such a high permittivity, a low frequency electric field inside the material is almost parallel to the surface according to basic electromagnetic theory. An exception is at surfaces covered with an electrode. There, the electric field is perpendicular to the surface. As a result, we are able to assume that an electric field is perpendicular to the electrodes anywhere inside the EO block. We take the *x*-axis parallel to the electric field.

Before deflecting a light beam, we inject electrons into the EO crystal by applying a high voltage between the electrodes (charge process) [6]. The injected electrons are trapped by local electronic states in the crystal and form a stable space charge. Then Gauss’s law is written as follows.

Here*ε*is the permittivity. Both the low frequency electric field

*E*and charge density

*ρ*are spatially nonuniform. On the other hand the EO index modulation

*Δn*is given bywhere

*n*

_{0}is the original refractive index.

*g*is a 2nd order EO coefficient that depends on the light polarization [7]. It is

*g*

_{11}for

*x*-polarized lights (polarization parallel to the

*E*vector described above) but

*g*

_{12}for

*y*- or

*z*-polarized lights (polarization perpendicular to the

*E*vector). As the spatial variation of the electric field is determined by

*ρ*(

*x*) with Eq. (1), the refractive index modulation is also determined by

*ρ*(

*x*). When electrons are homogeneously trapped in the EO crystal,

*ρ*is constant and the electric field varies linearly in the crystal. Then

*Δn*(x) is given by

*d*is the thickness of the EO crystal and

*V*is an applied voltage. The origin of

_{s}*x*is at the midpoint between the electrode pair. Note that

*V*is used for deflection control and is different from the voltage used for the charge process described above. The first term, which is linearly dependent on x, indicates the beam deflection function. As it is proportional to

_{s}*V*, the deflection angle can be linearly controlled with the applied voltage. The coefficient of the deflection is proportional to

_{s}*ρ*and

*ε*. The second term in this equation also depends on

*V*. However, the refractive index is changed uniformly with

_{s}*V*by this term, which does not bend light rays. The last parabolic term has the effect of a convex lens on the wavefront. The lens power is a function of

_{s}*ρ*. In this way, the charge density controls the output wavefront or beam deflection. When deriving Eq. (3), we assumed that

*ρ*was spatially uniform. If it was not uniform, the wavefront would be additionally deformed. Controlling and evaluating the distribution of the charge density are thus important for this optical beam deflector.

We would be able to derive the distribution of the charge density if we could obtain the
wavefront or the refractive index distribution *Δn*. There are various
methods for measuring refractive indices. The key points are that the absolute values of the
index are not required and only *Δn* and a sufficient spatial resolution
are needed. Figure 1 shows the apparatus we used, which
employs the principle of the phase shift method [8]. The
laser light beam is first split into two linearly polarized light beams at polarizing beam
splitter 1 (PBS1). The two beams are mixed again at PBS2. However, we can change the phase
retardation between the two components with the piezo mirror. The mixed beam is then expanded so
that it covers the whole area of the object to be measured. The direction of the object is set
so that one of the light components becomes the *x*-polarized light described
above and the other becomes the *y*-polarized light. The light beam modulated by
the object according to Eq. (2) then transmits the
analyzing polarizer and is observed with the camera. As with the conventional EO intensity
modulator, the polarizer is rotated 45 degrees from the *x*-axis (or from the
*y*-axis). Although the two light components have proceeded independently, they
interfere with each other at this point because they are both converted to lights with the same
polarization. The two lenses placed between the object and the camera are used to form an image
of the object on the camera plane. With this configuration, we are able to obtain
two-dimensional distributions of the phase retardation
*Δϕ*(*x, y*) between the polarized components by
changing the phase shift with the piezo mirror and simultaneously obtaining the interference
patterns with the camera. *Δϕ*(*x, y*) can be
calculated by using the following formula [8].

*I*(

*x*,

*y*,

*Φ*) is the light intensity distribution acquired with the camera and with a phase shift

*Φ*induced by the piezo mirror [8]. Hereafter we fix the

*y*coordinate at the center of the object and treat

*Δϕ*as a function of the

*x*coordinate only because we are interested in the distributions of

*ρ*between the electrodes. Thus we write it as

*Δϕ*(

*x*).

*Δϕ*(

*x*) is related to

*E*(

*x*) by Eq. (2) as follows:

*L*is the length of the object and

*λ*is the wavelength. Thus we can calculate

*ρ*(

*x*) with

The spatial resolution of this system is affected by the numerical aperture of the imaging lens. From the aperture of the lens, the resolution was estimated to be about 2 μm. However, the effective resolution is actually also limited by the object. Here, to evaluate the resolution, we consider a narrow Gaussian beam whose beam waist is located in the object. When deriving Eq. (5), we assumed that the direction changes of light rays are negligible and that the rays are almost parallel to each other. This assumption requires there to be no variation in the beam width in the object. Therefore the Rayleigh length of the beam should be long compared with the object length *L*. Then we have the minimum beam waist *w*_{0} given by the next formula.

*L*is 4 mm,

*n*

_{0}is 2.28 and

*λ*is 633 nm. We regard this beam waist as a criterion of the spatial resolution, which is sufficient for charge density evaluations.

We prepared a beam deflector made of a KTN block as the sample to be evaluated. We grew a KTN single crystal with the top seeded solution growth method [9], cut the crystal to obtain a 1.0 mm x 3.2 mm x 4.0 mm block, and formed a pair of titanium film electrodes on the 3.2 mm x 4.0 mm faces. Therefore, the electrode gap was 1 mm. In the charge process, electrons were injected through the electrode into the KTN block simply by applying high voltages between the electrodes to form a space charge. The KTN block undergoes a structural phase transition from a ferroelectric to a paraelectric phase at *T _{c}* = 41.7 °C. Hence, the composition should be about KTa

_{0.6}Nb

_{0.4}O

_{3}according to a previous report [10]. We controlled the temperature of the block at a value above the phase transition in the paraelectric phase. In this phase, the permittivity of the crystal block can be changed by changing the temperature according to the Curie-Weiss law. Strain induced birefringence up to 10

^{−4}was observed for our sample without voltage application. Because the birefringence is evaluated as phase retardation, we removed this background simply by subtracting it from all retardation data.

## 3. Results and discussions

Figure 2(a) shows the observed distributions of the
phase retardation induced by the KTN block and (b) shows the charge density distributions
calculated from (a) with Eq. (6) during the charge
process. Here, to evaluate *ρ*(*x*) formed by the charge
process, we acquired each curve in the figure with a continuously applied voltage. As the light
source, we used the 633 nm line of a He-Ne laser. The sample temperature was 63.7 °C,
which was 22 °C higher than the phase transition temperature, and the relative
permittivity was 5,600. The horizontal axis shows the position between the electrode pair, where
the left end is the cathode and the right end is the anode. We fixed the *y*
coordinate at the center of the KTN block as described above. The retardation of the vertical
axis of (a) is measured as the optical path length difference between the *x*-
and *y*-polarizations, that is, *Δn _{x}L* -

*Δn*. As the second order EO coefficient

_{y}L*g*, we used the values reported by Geusic et al.;

*g*

_{11}: 0.136 C

^{2}/m

^{4}and

*g*

_{12}: −0.038 C

^{2}/m

^{4}[7]. Thus

*Δn*is negative but

_{x}*Δn*is positive. This is why the retardations are negatively plotted in Fig. 2(a).

_{y}As expected the absolute values of the retardation increased with the applied voltage. In addition the retardation was almost zero near the cathode and became even lower with distance from the cathode. The gradients of these curves approximate the deflection angles. However, for a low voltage, the gradient of the curve becomes gentler at around a certain location and tends to level out, which means that the deflection was weak in that crystal region. Such profiles are explained by Fig. 2(b). It can be seen that only a limited area near the cathode is negatively charged with trapped electrons and the charge density is close to zero for the remainder of the crystal. With an optical beam deflector, a large absolute value for *ρ* is required for a large deflection angle as indicated by Eq. (3). Thus it is preferable that the negatively charged region near the cathode spread deeply towards the anode and the entire crystal region be charged so that beam deflection occurs effectively everywhere. Figure 2(a) shows that we are able to attain this by charging a KTN deflector with a higher voltage.

In fact, we can extend the negatively charged region also by increasing the permittivity.
Figure 3 shows the charge distribution for different
permittivities. We increased the permittivity by lowering the temperature toward the phase
transition temperature. The operation temperature was 54.6 °C for
*ε _{r}*: 9,000 and 45.1 °C for

*ε*: 19,600. The voltage was 150 V. This shows that increasing the permittivity has a similar effect to increasing the voltage and is beneficial for beam deflection. In contrast, reducing the permittivity shrinks the charged region near the cathode. Ordinary dielectrics have relative permittivities of less than 100. Figure 3 indicates that the charged region would be negligible for ordinary dielectrics and space charge effects as described in this paper are rarely observed.

_{r}To determine why the width of the electron-trapping region is limited, we have to note that the trapped electrons are very stable in our KTN crystals. Once we inject electrons into a crystal, the charge distribution lasts more than a few days at room temperature. Wemple et al. reported an electron mobility of about 3 cm^{2}/(Vs) for KTN at room temperature [11], which suggests that an electron passes through 1-mm-thick KTN in less than 10 μs. We have not determined the difference between Wemple’s and our crystals. However, in our KTN single crystals, the electron motion is qualitatively changed by the fact that the electrons are tightly bound to local states. It is also noteworthy that in Fig. 2(a) the retardation is negligible near the cathode. This means that the electric field is also minute there because of the screening effect provided by the trapped electrons. This prevents the electrons from drifting toward the anode and stops additional electron injection. Equation (1) indicates that an electric field is reduced by an integrated charge density. For a low voltage, the electric field can be screened by a small total charge. Thus electrons trapped in a limited region near the cathode sufficiently suppress further electron motion. A higher voltage requires more trapped electrons to slow the electron motion down. The permittivity dependence shown in Fig. 3 is also explained with Eq. (1), which shows that the effect of the screening charge is reduced by a large permittivity.

The origin of the electron trapping is yet to be identified. For KTN, copper is an efficient dopant for photorefractive holograms in terms of forming a stable space charge [5]. However we did not intentionally dope the KTN single crystal with any impurities. In addition to impurities, oxygen vacancies might play an important role. Our next work will be devoted to an investigation of this electron trap.

## 4. Conclusion

We proposed a method for evaluating charge density distributions in an EO material such as KTN. The evaluation procedure consisted of a retardation distribution measurement with a phase shift method and the calculation of charge densities by using an electrooptic formula. For a KTN block, the measured charge density distributions showed that electrons are trapped by localized states immediately after being injected from the cathode especially with a low applied voltage. The width of the electron accumulation region increased with the applied voltage and permittivity. This behavior can be explained by a screening effect, which reduces the driving force of electron drift.

## References and links

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