Third column electro-optical coefficients of monoclinic Sn 2 P 2 S 6

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 rT 113 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 OCIS codes: (160.2100) Electro-optical materials; (160.1190) Anisotropic optical materials; (120.3180) Interferometry; (120.5060) Phase modulation. 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). 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). 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). 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. 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). 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). 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). 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). 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). 10. D. Haertle, A. Guarino, J. Hajfler, G. Montemezzani, and P. Günter, “Refractive indices of Sn2P2S6 at visible and infrared wavelengths,” Opt. Express 13, 2047–2057 (2005). http://www.opticsinfobase.org/ oe/abstract.cfm?URI=oe-13-6-2047 11. D. Haertle, M. Jazbinsek, G. Montemezzani, and P. Günter, “Nonlinear optical coefficients and phase-matching conditions in Sn2P2S6,” Opt. Express 13, 3765–3776 (2005). http://www.opticsinfobase.org/oe/ abstract.cfm?URI=oe-13-10-3765 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). 13. A. Volkov, A. Shumelyuk, S. Odoulov, D. R. Evans, and G. Cook, “Anisotropic diffraction from photorefractive gratings and Pockels tensor of Sn2P2S6,” Opt. Express 16, 16923–16934 (2008). http://www. opticsexpress.org/abstract.cfm?URI=oe-16-21-16923 (C) 2012 OSA 1 July 2012 / Vol. 2, No. 7 / OPTICAL MATERIALS EXPRESS 920 #165533 $15.00 USD Received 29 Mar 2012; revised 22 Apr 2012; accepted 23 Apr 2012; published 14 Jun 2012 14. P. Günter and M. Zgonik, “Clamped-unclamped electro-optic coefficient dilemma in photorefractive phenomena,” Opt. Lett. 16, 1826–1828 (1991). 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. 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 Kristallografiya 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). 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). 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).


Introduction
Tin thiohypodiphosphate (Sn 2 P 2 S 6 ) 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][2][3][4][5][6][7][8][9]. At room temperature Sn 2 P 2 S 6 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 Sn 2 P 2 S 6 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 Sn 2 P 2 S 6 , the maximum was found for the unclamped coefficient r T 111 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 Sn 2 P 2 S 6 for field parallel to the crystallophysical 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 electrooptic coefficients of Sn 2 P 2 S 6 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 T 113 = 67 ± 7 pm/V, roughly three times smaller than the overall largest coefficient r T 111 .

Crystal samples and experimental approach
Being Sn 2 P 2 S 6 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 d 333 , 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 n 3 ) 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 r 131 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 r 133 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 Sn 2 P 2 S 6 samples grown at the University of Uzhgorod. Their dimensions are 5.7 × 4.5 × 5.0 mm 3 (SPS1), 7.8 × 10.5 × 6.0 mm 3 (SPS2) and 3.8 × 4.6 × 4.2 mm 3 (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 beamsplitter devides the beam from a He-Ne laser (λ = 633 nm) into two arms. The beam in the lower signal arm goes through the Sn 2 P 2 S 6 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 I(Δφ ) = I 1 + I 2 + 2 √ I 1 I 2 cos Δφ , where Δφ = Δφ 0 + Δφ E and I 1 and I 2 are the intensities coming from each of the two arms. Here Δφ 0 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 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 ef f 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 ef f 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 e f f and d e f f the electro-optic contribution to the phase shift is by a factor n 3 /[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 r 223 , r 333 , r f ast and r slow , which are calculated using the electro-optic phase shift alone. By putting the average phase shift Δφ 0 = π/2 (linear detection working point) the absolute value of the effective electro-optic coefficient r ef f in Eq. (1) can then be determined as 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 ΔI max = 4 √ 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 birefrin- gence (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 ∝ cos 2 (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.

Electro-optic effect and experiments
In our coordinate system (x, y, z) = (1,2,3) the third-rank electro-optic tensor of Sn 2 P 2 S 6 has the form with the inherent symmetry (r i jk = r jik ). We are interested in the coefficients of the third column in the above tensor. After applying an electric field E = (0, 0, E) along the z-direction the inverse dielectric tensor at optical frequencies, (1/n 2 ) i j ≡ (1/ε) i j , is modified as the outer-diagonal elements (1/n 2 ) 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 r 223 , r 333 , r f ast and r slow , for which the propagating wave is an eigenwave of the crystal. As will be discussed in more detail later, r f ast 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.
Since the measurement give direct information only on the sign of the product rE 3 , 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 n 1 and slow axis corresponding to n 3 ) 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 The electrically induced change of the scalar inverse refractive index squared for the eigenwaves of the configurations of Fig. 2 is obtained as Hereˆ d is the unit vector along the polarization direction of the eigenwave and Δ(1/n 2 ) i j 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 = −n 3 Δ(1/n 2 )/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 r 223 and r 333 , 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 n 3 ) and is found withˆ d = (− cos α, 0, sin α) in Eq. (5). This delivers r slow = r 113 cos 2 α + r 333 sin 2 α − 2r 133 sin α cos α ≈ 1 2 (r 113 + r 333 ) − r 133 , where the last approximated expression on the right-hand side is obtained by substituting the

Discussion and conclusions
It is seen in Table 1 that the third-column electro-optic coefficient of Sn 2 P 2 S 6 have a sizable value, the largest one (r 113 ≈ 67 pm/V) correspond to more than twice the electro-optic response of the standard electro-optic material LiNbO 3 . Nevertheless this maximum value is still about a factor of three smaller than the largest first-column electro-optic coefficient (r 111 ), 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 electrooptic 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 r 133 /r 333 and in less extent of r 223 /r 221 , the general trend for the relative magnitude of the tensor elements is maintained in the two cases. These results may suggest that in Sn 2 P 2 S 6 , unlike in BaTiO 3 or KNbO 3 [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 r 133 /r 333 and r 131 /r 111 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].
In conclusions, we have determined directly the unclamped electro-optic coefficients of Sn 2 P 2 S 6 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 r 113 ≈ 67 pm/V. Despite being smaller than all the diagonal coefficients for field in x-direction, this value is more than twice