All coefficients of the third-column of the unclamped electro-optic tensor of Sn2P2S6 single crystals are determined by direct interferometric techniques. It is found that the largest coefficient for electric field parallel to the z-axis is ≈ 67 pm/V at the wavelength of 633 nm and room temperature.
© 2012 Optical Society of America
Tin thiohypodiphosphate (Sn2P2S6) is a wide bandgap semiconductor ferroelectric that has received considerable interest in the last few years, specially due to its attractive properties as a fast photorefractive material with optimum response in the near infrared [1–9]. At room temperature Sn2P2S6 belongs to the monoclinic point group m. Owing to this low level symmetry and the related large number of independent parameters, several of the characteristic physical, optical and nonlinear optical properties of this material have not been fully determined yet. Concerning optical properties, the refractive indices and their dispersion were characterized in  in the range 550–2200 nm, while the second and third order nonlinear optical properties of Sn2P2S6 were discussed in . A direct measure of the values of the coefficients of the first column of the linear electro-optic tensor was performed in . These coefficients correspond to an applied electric field nearly parallel to the material’s spontaneous polarization and are expected to be the largest in Sn2P2S6, the maximum was found for the unclamped coefficient with a value of 174 pm/V at room temperature and for the wavelength λ = 633 nm. No direct measurement of the electro-optic coefficients of Sn2P2S6 for field parallel to the crystallo-physical z-axis were performed to date. There exist however estimations of the ratios between the coefficients of the third column and those of the first column . The latter are based on the comparison of the diffraction efficiencies of photorefractive gratings in various isotropic and anisotropic diffraction geometries. However, the electro-optic response in photorefractive experiments is also strongly influenced by the fact that the space-charge electric field is periodic . Therefore, the photorefractive electro-optic response is neither fully unclamped nor fully clamped and the degree of clamping depends on the grating orientation and on various additional material parameters involving elasticity, piezoelectricity and elasto-optics [14, 15]. As a consequence, without the knowledge of all the numerous remaining parameters, thorough photorefractive investigations such as in  cannot lead alone to the determination of the the unclamped electro-optic coefficients, which require therefore a direct measurement.
In the present work we provide a direct determination of the values for the unclamped electro-optic coefficients of Sn2P2S6 for an applied field parallel to the z-axis of the crystal, corresponding to the third column of the electro-optic tensor. The measurements are performed using a Mach-Zehnder type interferometric technique for light propagation along the polar x-axis or the y-axis of the crystals as well as with the one-beam Senarmont technique. It is confirmed that the third column coefficients are generally smaller than those obtained for field in x direction (first column). Nevertheless, a sizable response is observed also for this field direction, which is nearly perpendicular to the spontaneous polarization. The corresponding largest coefficient is , roughly three times smaller than the overall largest coefficient .
2. Crystal samples and experimental approach
Being Sn2P2S6 of monoclinic symmetry, care must be payed to the coordinate system xyz in which the electro-optic and other tensors are defined. In this paper we stick to the choice discussed in detail in , which complies with the one in most earlier literature. It has to be noted that the present choice differs from the one in  and  by the convention for the sign of the piezoelectric coefficient d333, which is negative in the present case. The consequence is a different relative orientation of the positive x and z-axes, so that the major axis of the oblique optical indicatrix (associated to the largest refractive index n3) is in the second and fourth quadrant of the xz-plane (see ) instead as the first and third quadrants like in Refs. [10, 12]. For field along the x axis the diagonal elements of the electro-optic tensor (of the type rii1) are unaffected, while the nondiagonal element r131 switches sign between the two choices. On the other hand, for electric field parallel to the z-axis the diagonal elements rii3 are switching sign upon inversion of the convention for the positive z-axis, while the non-diagonal element r133 is unaffected. Therefore, for both the first and the third column coefficients, the relative sign between the diagonal and non diagonal elements depends on the choice for the axes convention.
Our experiments were performed using three nominally undoped Sn2P2S6 samples grown at the University of Uzhgorod. Their dimensions are 5.7 × 4.5 × 5.0 mm3 (SPS1), 7.8 × 10.5 × 6.0 mm3 (SPS2) and 3.8 × 4.6 × 4.2 mm3 (SPS3) along x, y and z axes, respectively. They were electrically poled by heating above the phase transition temperature TC at 338 K and slowly cooling them down to room temperature under an electric field of about 1 kV/cm applied along the x-axis, which is near to the spontaneous polarization direction. Sample SPS1 was polished on the x-surfaces to permit light propagation along the polar x-axis, sample SPS2 was polished on the surfaces perpendicular to the monoclinic y-axis, which is perpendicular to the crystallographic mirror plane. Sample SPS3 was polished on both x and y surfaces. In all our electro-optic measurements the electric field was applied along the z-axis through graphite electrodes on the z surfaces. As discussed below, the above sample combination allowed to perform each type of electro-optic measurements independently on two different samples.
The set-up used to determine the various electro-optic tensor elements is depicted schematically on Fig. 1. For the interferometric measurement of the electro-optic coefficients a beam-splitter devides the beam from a He-Ne laser (λ = 633 nm) into two arms. The beam in the lower signal arm goes through the Sn2P2S6 crystal which is subjected to an ac-voltage of up to 10 V amplitude and frequency of few kHz. The applied voltage has no bias. By choosing the polarization of the signal beam parallel to an eigenpolarization of the crystal, the applied voltage induces a pure phase modulation on the signal beam through the electro-optic effect. The polarization of the beam in the reference arm is put parallel to the one of the signal wave by means of the upper polarizer, while its absolute phase can be set by moving the piezo-mirror, which permits so to adjust the working point for the measurement. Signal and reference wave recombine at the second beam-splitter and go through an analyzer which is put parallel to both polarizers. A small section of the resulting interference fringe is then detected by the silicon photodetector and analyzed by means of an oscilloscope and/or a lock-in analyzer.
In the interferometric set-up the detected intensity is , where Δϕ = Δϕ0 + ΔϕE and I1 and I2 are the intensities coming from each of the two arms. Here Δϕ0 is the average phase shift between the waves in the two arms of the Mach-Zehnder interferometer and is adjusted by means of the piezo-mirror, while ΔϕE is the electrically induced phase shift. The latter is given byEq. (1) gives the contribution of the piezoelectric length variation to the phase shift. This contribution is rather small, for equal values of reff and deff the electro-optic contribution to the phase shift is by a factor n3/[2(n−1)] larger than the piezoelectric one, this factor equals about 14 in our case. Therefore, by considering the known absolute value of the piezoelectric constants of the crystal , it is found that the corrections due to the second term in Eq. (1) are small and fall well within the experimental error given below for the coefficients r223, r333, rfast and rslow, which are calculated using the electro-optic phase shift alone. By putting the average phase shift Δϕ0 = π/2 (linear detection working point) the absolute value of the effective electro-optic coefficient reff in Eq. (1) can then be determined as
Some measurements were performed also by removing the upper reference arm and using the lower arm in a Senarmont-type configuration which measures the birefringence induced by the applied electric field [17, 18]. In this case both beam splitters are removed and a quarter-wave plate placed between the crystal and the analyzer allows to determine the induced birefringence (resulting from a combination of electro-optic tensor elements) by means of an intensity measurement after the analyzer. The input polarization to the measuring crystal is at 45 degrees to the two oblique main axes of the indicatrix. If the quarter-wave plate has its axis parallel to the input polarization to the electro-optic crystal, the light leaving the quarter-wave plate is linearly polarized but with its polarization rotated by half the retardation phase angle ΔΦ between the two perpendicular eigenwaves in the crystal. For an analyzer oriented at an angle β to the input polarization the transfer function of such a set-up is therefore of the kind T ∝ cos2(β − ΔΦ/2) and the effective electro-optic coefficient is determined by an expression of the kind of Eq. (2) by placing the analyzer at the half transmission point.
3. Electro-optic effect and experiments
In our coordinate system (x,y,z) = (1,2,3) the third-rank electro-optic tensor of Sn2P2S6 has the form
Figure 2 shows the crystal configurations for the measurement of the electro-optic coefficients r223, r333, rfast and rslow, for which the propagating wave is an eigenwave of the crystal. As will be discussed in more detail later, rfast is associated to the fast eigenwave for light propagation in y direction, while rslow is associated to the corresponding slow eigenwave. Each configuration was tested in two different crystal samples. Sample SPS3 was used in all configurations, while sample SPS1 was used for the configurations of Fig. 2(A) and Fig. 2(B) only, and sample SPS2 was used for the configurations of Fig. 2(C) and Fig. 2(D) only.
Since the measurement give direct information only on the sign of the product rE3, it is important to know the correct positive orientation of the axes x, y and z in order to assign the sign to the electro-optic coefficients. A combination of different techniques was used for this purpose. First, the positive direction of the y axis can be determined by observing the rotation direction of the optical indicatrix upon increasing temperature, which is counterclockwise around the positive y axis . Second, in the samples showing sufficient photorefractive effect the positive direction of the x-axis can be determined by observing the main direction of beam fanning for a x-polarized wave propagating along the z direction. Finally, the orientation of the two oblique main axes of the indicatrix (fast axis corresponding to the refractive index n1 and slow axis corresponding to n3) can be determined by observing the non specular angle of reflection on the z crystal surface of a p-polarized wave propagating in the xz-plane of the crystal. This method is described in the appendix of . A combination of the above techniques prior to the electro-optic measurements allowed to determine the crystal axes orientations depicted on Fig. 2.
The electrically induced change of the scalar inverse refractive index squared for the eigen-waves of the configurations of Fig. 2 is obtained asEq. (4). The Einstein summation rule over equal indices is used in the above tensor contraction. With Δn = −n3Δ(1/n2)/2 the above expression gives immediately the refractive index change Δn in Eq. (1). The configurations of Fig. 2(A) and Fig. 2(B) deliver then directly the electro-optic coefficients r223 and r333, respectively. For the configuration of Fig. 2(C) the measured effective electro-optic coefficient according to Eq. (2) corresponds to the slow eigenwave (refractive index n3) and is found with in Eq. (5). This delivers 10] by the value of 45 deg. Similarly, the effective electro-optic coefficient measured for the fast eigenwave is Eqs. (6) and (7) the coefficients r113 and r133 can then be extracted as
In the non interferometric measurement using the Senarmont configuration one uses the crystal configuration as in Fig. 2(C) and Fig. 2(D) but the input polarization is either vertical (|| z) or horizontal (|| x). One measures therefore the field-induced birefringence between the fast and the slow eigenwave, that is10]) the expression for rc becomes
We present now the results obtained for the values of the different elements of third column of the unclamped electro-optic tensor. Our experiments have shown that all the directly measured coefficients r223, r333, rfast and rslow have the same sign and are positive. The corresponding values for the wavelength of 633 nm are given in Table 1 together with measured value for the effective coefficient rc defined in Eqs. (10) and (11). The value rc (meas.) results from the measurement in the Senarmont configuration while rc (calc.) is calculated from the measured rfast and rslow using Eq. (10) or Eq. (11) with α = 43.3 deg. The two approaches are found to give a good agreement. The values of r113 and r133 in Table 1 are calculated with Eqs. (8) and (9) from the experimental data for rfast, rslow and r333. Note that the scattering of the measured data among our investigated samples and measuring positions lies within the given experimental error.
For comparison Table 1 contains also the unclamped electro-optic coefficients for electric field in x-direction from . Note that the coefficient r131 is reported in Table 1 with the opposite sign with respect to . This is a consequence of the different convention for the positive orientation of the crystallographic axes taken in this work with respect to , as discussed in Section 2 and in .
4. Discussion and conclusions
It is seen in Table 1 that the third-column electro-optic coefficient of Sn2P2S6 have a sizable value, the largest one (r113 ≈ 67 pm/V) correspond to more than twice the electro-optic response of the standard electro-optic material LiNbO3. Nevertheless this maximum value is still about a factor of three smaller than the largest first-column electro-optic coefficient (r111), confirming that a field in the x-direction, which is near to the crystal spontaneous polarization, delivers the largest response.
As mentioned in the introduction, estimations for the ratios of electro-optic coefficients were given in  on the base of photorefractive anisotropic and isotropic Bragg diffraction investigations in different configurations. Even though the two kinds of measurement are not directly comparable, we list in Table 2 the ratios of coefficients obtained from direct electro-optic measurements and those estimated in . In general these ratios are not expected to be identical in the two cases due to the fact that, as discussed in Section 1, the photorefractive estimation (associated to periodic electric fields) is related to effective coefficients, which can differ significantly from the unclamped coefficients measured here (associated to a homogeneous electric field). Nevertheless Table 2 shows that, with the exception of the ratio r133/r333 and in less extent of r223/r221, the general trend for the relative magnitude of the tensor elements is maintained in the two cases. These results may suggest that in Sn2P2S6, unlike in BaTiO3 or KNbO3 , the influence of the mechanical coupling associated to the periodic field is not such as to revolutionize too strongly the ordering of the electro-optic coefficient magnitude. Note again that in Table 2 we have reversed the signs of the ratios r133/r333 and r131/r111 with respect to . This is because the convention taken in  for the positive direction of the z axis was the same as in , but was opposite as the one taken in this work and in .
In conclusions, we have determined directly the unclamped electro-optic coefficients of Sn2P2S6 for electric field parallel to the crystallographic z axis, which is nearly perpendicular to the material’s spontaneous polarization. The measurements were performed by means of a Mach-Zehnder interferometer as well as in the one-beam Senarmont type configuration. The largest electro-optic coefficient for field in z- direction is r113 ≈ 67 pm/V. Despite being smaller than all the diagonal coefficients for field in x-direction, this value is more than twice as large than to the one of the largest coefficient (r333) of the standard electro-optic material LiNbO3. Thus, the present study confirm the interest of Sn2P2S6 for any electro-optic and/or photorefractive configuration in the xz-plane of the material.
We dedicate this work to Dr. Ivan Prits, who left us unexpectedly during these studies. We miss his friendship and competence. This work was performed in the framework of the bilateral french-ukrainian PHC DNIPRO program.
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