Third column electro-optical coefficients of monoclinic Sn2P2S6

Germano Montemezzani; Michel Aillerie; Ximeng Zheng; Hafssa Remmach; Alexander A. Grabar

doi:10.1364/OME.2.000920

1. Introduction

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 [10] in the range 550–2200 nm, while the second and third order nonlinear optical properties of Sn₂P₂S₆ were discussed in [11]. A direct measure of the values of the coefficients of the first column of the linear electro-optic tensor was performed in [12]. 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 $r_{111}^{T}$ 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 [13]. 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 [14]. 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 [13] 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 $r_{113}^{T} = 67 \pm 7 pm / V$ , roughly three times smaller than the overall largest coefficient $r_{111}^{T}$ .

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 [4], which complies with the one in most earlier literature. It has to be noted that the present choice differs from the one in [12] and [10] 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 [4]) 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 r_ii₁) 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 r_ii₃ 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 T_C 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.

Fig. 1 Mach-Zehnder type experimental arrangement used for the determination of the electro-optic coefficients.

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In the interferometric set-up the detected intensity is $I (Δ ϕ) = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} cos Δ ϕ$ , where Δϕ = Δϕ₀ + Δϕ_E 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 Δϕ_E is the electrically induced phase shift. The latter is given by

Δ ϕ_{E} = \frac{2 π}{λ} L Δ n + \frac{2 π}{λ} (n - 1) Δ L = \frac{2 π L}{λ} [- \frac{n^{3}}{2} r_{eff} + (n - 1) d_{eff}] E

where E is the applied electric field to the measuring crystal, L is the length of the crystal, λ is the vacuum wavelength and n is the refractive index seen by the eigenwave propagating in the crystal. r_eff is the effective electro-optical coefficient active in the particular measuring geometry and is either equal to one of the coefficients of the electro-optic tensor or to a linear combination of them. Finally, d_eff is the effective piezoelectric coefficient causing a crystal length variation in the given measuring geometry. The second term on the right-hand-side of Eq. (1) gives the contribution of the piezoelectric length variation to the phase shift. This contribution is rather small, for equal values of r_eff and d_eff 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 [16], 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, r_fast and r_slow, 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 r_eff in Eq. (1) can then be determined as

r_{eff} = \frac{2 λ d}{π L n^{3}} \frac{Δ I_{E}}{Δ I_{\max} Δ V},

where d is the inter-electrode distance, ΔI_E is the peak-to-peak amplitude of the modulated detected signal, ΔV is the corresponding amplitude of the modulated voltage applied to the crystal, and

Δ I_{\max} = 4 \sqrt{I_{1} I_{2}}

is the peak-to-peak contrast of the intensity of the interference fringes.

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

r_{i j k} = (\begin{matrix} r_{111} & 0 & r_{113} \\ r_{221} & 0 & r_{223} \\ r_{331} & 0 & r_{333} \\ 0 & r_{322} & 0 \\ r_{131} & 0 & r_{133} \\ 0 & r_{122} & 0 \end{matrix})

with the inherent symmetry (r_ijk = r_jik). We are interested in the coefficients of the third column in the above tensor. After applying an electric field E⃗ = (0, 0, E) along the z-direction the inverse dielectric tensor at optical frequencies, (1/n²)_ij ≡ (1/ε)_ij, is modified as

{(\frac{1}{n^{2}})}_{i j} (E) = (\begin{matrix} {(1 / n^{2})}_{11} & 0 & {(1 / n^{2})}_{13} \\ 0 & {(1 / n^{2})}_{22} & 0 \\ {(1 / n^{2})}_{13} & 0 & {(1 / n^{2})}_{33} \end{matrix}) + (\begin{matrix} r_{113} E & 0 & r_{133} E \\ 0 & r_{223} E & 0 \\ r_{133} E & 0 & r_{333} E \end{matrix}),

the outer-diagonal elements (1/n²)₁₃ in the unperturbed tensor on the right-hand side account for the fact that the main axes of the index ellipsoid (optical indicatrix) are rotated (in the xz-plane) with respect to the crystallographic axes.

Figure 2 shows the crystal configurations for the measurement of the electro-optic coefficients r₂₂₃, r₃₃₃, r_fast and r_slow, for which the propagating wave is an eigenwave of the crystal. As will be discussed in more detail later, r_fast is associated to the fast eigenwave for light propagation in y direction, while r_slow 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.

Fig. 2 Crystal configurations for the interferometric measurement of the third-column electro-optic coefficients of Sn₂P₂S₆. A) Measurement of r₂₂₃; B) r₃₃₃; C) r_slow; D) r_fast. The thick double arrow give the polarization direction of the corresponding eigenwave. For Sn₂P₂S₆ the angle α in C) and D) is α ≈ 43.3 deg for λ = 633 nm and room temperature. The positive direction of the applied electric field E₃ is always towards the bottom (+ẑ).

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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 [19]. 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 [10]. 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 as

Δ (\frac{1}{n^{2}}) = Δ {(\frac{1}{n^{2}})}_{i j} {\hat{d}}_{i} {\hat{d}}_{j} .

Here

\hat{\vec{d}}

is the unit vector along the polarization direction of the eigenwave and Δ(1/n²)_ij is the second term on the right-hand side of Eq. (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

\hat{\vec{d}} = (- cos α, 0, sin α)

in Eq. (5). This delivers

r_{slow} = r_{113} {cos}^{2} α + r_{333} {sin}^{2} α - 2 r_{133} sin α cos α \approx \frac{1}{2} (r_{113} + r_{333}) - r_{133},

where the last approximated expression on the right-hand side is obtained by substituting the polarization angle α = 43.3 ± 0.4 deg [10] by the value of 45 deg. Similarly, the effective electro-optic coefficient measured for the fast eigenwave is

r_{fast} = r_{113} {sin}^{2} α + r_{333} {cos}^{2} α + 2 r_{133} sin α cos α \approx \frac{1}{2} (r_{113} + r_{333}) + r_{133} .

From Eqs. (6) and (7) the coefficients r₁₁₃ and r₁₃₃ can then be extracted as

r_{113} = r_{fast} + r_{slow} - r_{333}

and

r_{133} = \frac{r_{fast} {cos}^{2} α - r_{slow} {sin}^{2} α - r_{333} cos 2 α}{sin 2 α} \approx \frac{1}{2} (r_{fast} - r_{slow}) .

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 is

Δ n \equiv - \frac{{\tilde{n}}^{3}}{2} r_{c} E_{3} = - (\frac{n_{1}^{3}}{2} r_{fast} - \frac{n_{3}^{3}}{2} r_{slow}) E_{3},

where r_c is the effective electro-optic coefficient determined in this configuration and ñ is an arbitrary value for the reference refractive index that can be established freely among one of the known refractive indices of the sample. If we choose ñ = n_z (see [10]) the expression for r_c becomes

r_{c} = r_{113} \frac{n_{1}^{3} {sin}^{2} α - n_{3}^{3} {cos}^{2} α}{n_{z}^{3}} + r_{333} \frac{n_{1}^{3} {cos}^{2} α - n_{3}^{3} {sin}^{2} α}{n_{z}^{3}} + r_{133} \frac{n_{1}^{3} + n_{3}^{3}}{n_{z}^{3}} sin 2 α .

Even though this expression appears complicated, it can be strongly simplified if one considers that n₁ ≈ n₃ ≈ n_z and one takes α = 45 deg. In this case one gets r_c ≅ 2r₁₃₃, therefore the effective coefficient r_c is expected to be roughly twice the non diagonal electro-optic coefficient r₁₃₃.

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, r_fast and r_slow 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 r_c defined in Eqs. (10) and (11). The value r_c (meas.) results from the measurement in the Senarmont configuration while r_c (calc.) is calculated from the measured r_fast and r_slow 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 r_fast, r_slow and r₃₃₃. Note that the scattering of the measured data among our investigated samples and measuring positions lies within the given experimental error.

Table 1. Unclamped Electro-Optical Coefficients of Sn₂P₂S₆ at λ = 633 nm

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For comparison Table 1 contains also the unclamped electro-optic coefficients for electric field in x-direction from [12]. Note that the coefficient r₁₃₁ is reported in Table 1 with the opposite sign with respect to [12]. This is a consequence of the different convention for the positive orientation of the crystallographic axes taken in this work with respect to [12], as discussed in Section 2 and in [4].

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 [13] 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 [13]. 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 [20], 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 [13]. This is because the convention taken in [13] for the positive direction of the z axis was the same as in [12], but was opposite as the one taken in this work and in [4].

Table 2. Ratio of Electro-Optic Coefficients from Direct Electro-Optic Measurements and Estimated from Photorefractive Diffraction Experiments

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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.

Acknowledgments

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.

References and links

1. S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, A. A. Grabar, and I. M. Stoyka, “Photorefraction in tin hypothiodiphosphate in the near infrared,” J. Opt. Soc. Am. B 13, 2352–2360 (1996). [CrossRef]

2. A. A. Grabar, I. V. Kedyk, M. I. Gurzan, I. M. Stoika, A. A. Molnar, and Y. M. Vysochanskii, “Enhanced photorefractive properties of modified Sn₂P₂S₆,” Opt. Commun. 188, 187–194 (2001). [CrossRef]

3. M. Jazbinsek, D. Haertle, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Wavelength dependence of visible and near-infrared photorefraction and phase conjugation in Sn₂P₂S₆,” J. Opt. Soc. Am. B 22, 2459–2467 (2005). [CrossRef]

4. A. A. Grabar, M. Jazbinsek, A. Shumelyuk, Yu. M. Vysochanskii, G. Montemezzani, and P. Günter, “Photorefractive effect in Sn₂P₂S₆,” in Photorefractive Materials and Their Applications, J. P. Huignard and P. Günter, eds. (Springer, New York, 2007),Vol. 2, pp. 327–362. [CrossRef]

5. T. Bach, M. Jazbinsek, G. Montemezzani, P. Günter, A. A. Grabar, and Yu. M. Vysochanskii, “Tailoring of infrared photorefractive properties of Sn₂P₂S₆ crystals by Te and Sb doping,” J. Opt. Soc. Am. B 24, 1535–1541 (2007). [CrossRef]

6. R. Mosimann, P. Marty, T. Bach, F. Juvalta, M. Jazbinsek, P. Günter, and A. A. Grabar, “High-speed photorefraction at telecommunication wavelength 1.55 μm in Sn₂P₂S₆:Te,” Opt. Lett. 32, 3230–3232 (2007). [CrossRef] [PubMed]

7. I. V. Kedyk, P. Mathey, G. Gadret, O. Bidault, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Enhanced photorefractive properties of Bi-doped Sn₂P₂S₆,” J. Opt. Soc. Am. B 25, 180–186 (2008). [CrossRef]

8. A. Shumelyuk, A. Volkov, S. Odoulov, G. Cook, and D. R. Evans, “Coupling of counterpropagating light waves in low-symmetry photorefractive crystals,” Appl. Phys. B 100, 101–108 (2010). [CrossRef]

9. S. Farahi, G. Montemezzani, A. A. Grabar, J.-P. Huignard, and F. Ramaz, “Photorefractive acousto-optic imaging in thick scattering media at 790 nm with a Sn₂P₂S₆:Te crystal,” Opt. Lett. 35, 1798–1800 (2010). [CrossRef] [PubMed]

10. D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn₂P₂S₆ at visible and infrared wavelengths,” Opt. Express13, 2047–2057 (2005).http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-2047 [CrossRef] [PubMed]

11. D. Haertle, M. Jazbinsek, G. Montemezzani, and P. Günter, “Nonlinear optical coefficients and phase-matching conditions in Sn₂P₂S₆,” Opt. Express13, 3765–3776 (2005).http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3765 [CrossRef] [PubMed]

12. D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Yu. M. Vysochanskii, “Electro-optical properties of Sn₂P₂S₆,” Opt. Commun. 215, 333–343 (2003). [CrossRef]

13. A. Volkov, A. Shumelyuk, S. Odoulov, D. R. Evans, and G. Cook, “Anisotropic diffraction from photore-fractive gratings and Pockels tensor of Sn₂P₂S₆,” Opt. Express16, 16923–16934 (2008).http://www.opticsexpress.org/abstract.cfm?URI=oe-16-21-16923 [CrossRef] [PubMed]

14. P. Günter and M. Zgonik, “Clamped-unclamped electro-optic coefficient dilemma in photorefractive phenomena,” Opt. Lett. 16, 1826–1828 (1991). [CrossRef] [PubMed]

15. G. Montemezzani and M. Zgonik, “Space-charge driven holograms in anisotropic media,” in Photorefractive Materials and Their Applications, J. P. Huignard and P. Günter, eds. (Springer, New York, 2006),Vol. 1, pp. 83–118. [CrossRef]

16. Yu. M. Vysochanskii, M. I. Gurzan, M. M. Maior, E. D. Rogach, F. I. Savenko, and V. Yu. Slivka, “Piezoelectric properties of single crystals of Sn₂P₂S₆,” Sov. Phys. Crystallogr. 35, 459–461 (1990), (Translated from Kristal-lografiya 35, 784 (1990)).

17. R. O’B. Carpenter, “The electro-optic effect in uniaxial crystals of the Dihydrogen Phosphate type. III. Measurement of coefficients,” J. Opt. Soc. Am 40, 225–229 (1950). [CrossRef]

18. M. Aillerie, N. Theofanous, and M. D. Fontana, “Measurement of the electro-optic coefficients: description and comparison of the experimental techniques,” Appl. Phys. B 70, 317–334 (2000). [CrossRef]

19. A. A. Grabar, Yu. M. Vysochanskii, S. I. Perechinskii, L. A. Salo, M. I. Gurzan, and V. Yu. Slivka, “Thermooptic investigations of ferroelectric Sn₂P₂S₆,” Sov. Phys. Solid State 26, 2087–2089 (1984), (Translated from Fiz. Tverd. Tela 26, 3469–3472 (1984)).

20. G. Montemezzani, “Optimization of photorefractive two-wave mixing by accounting for material anisotropies: KNbO₃ and BaTiO₃,” Phys. Rev. A 62, 053803 (2000). [CrossRef]

Coefficient ratio	Direct electro-optic measurement (This work and [12])	Photorefractive measurement ([13])
r₁₁₃/r₃₃₃	2.7 ± 0.7	1.7 ± 0.6
r₂₂₃/r₃₃₃	0.87 ± 0.28	0.69 ± 0.03
r₁₃₃/r₃₃₃	0.58 ± 0.20	1.8 ± 0.4
r₂₂₃/r₂₂₁	0.24 ± 0.06	0.14 ± 0.02
r₃₃₁/r₁₁₁	0.80 ± 0.15	0.87 ± 0.02
r₂₂₁/r₁₁₁	0.53 ± 0.08	0.53 ± 0.02
r₁₃₁/r₁₁₁	0.14 ± 0.09	0.12 ± 0.07

Coefficient ratio	Direct electro-optic measurement (This work and [12])	Photorefractive measurement ([13])
r₁₁₃/r₃₃₃	2.7 ± 0.7	1.7 ± 0.6
r₂₂₃/r₃₃₃	0.87 ± 0.28	0.69 ± 0.03
r₁₃₃/r₃₃₃	0.58 ± 0.20	1.8 ± 0.4
r₂₂₃/r₂₂₁	0.24 ± 0.06	0.14 ± 0.02
r₃₃₁/r₁₁₁	0.80 ± 0.15	0.87 ± 0.02
r₂₂₁/r₁₁₁	0.53 ± 0.08	0.53 ± 0.02
r₁₃₁/r₁₁₁	0.14 ± 0.09	0.12 ± 0.07

Third column electro-optical coefficients of monoclinic Sn₂P₂S₆

Abstract

1. Introduction

2. Crystal samples and experimental approach

3. Electro-optic effect and experiments

4. Discussion and conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Tables (2)

Equations (11)

Optical Materials Express

Coefficient	Value [pm/V]	Reference
r₂₂₃	+21.7 ± 3.8	This work
r₃₃₃	+24.7 ± 3.7	This work
r_fast	+58.9 ± 5.1	This work
r_slow	+32.6 ± 2.3	This work
r_c (meas.)	+26.5 ± 1.5	This work
r₁₁₃	+67 ± 7	This work
r₁₃₃	+14.4 ± 3.0	This work
r_c (calc.)	+23.1 ± 5.6	This work
r₁₁₁	+174 ± 10	[12]
r₂₂₁	+92 ± 8	[12]
r₃₃₁	+140 ± 18	[12]
r₁₃₁	+25 ± 15	[12]