research papers Temperature-dependent gyration tensor of LiIO 3 single crystal using the high-accuracy universal polarimeter

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1 Journal of Applied Crystallography ISSN Temperature-dependent gyration tensor of single crystal using the high-accuracy universal polarimeter Received 9 October 2001 Accepted 14 January 2002 J. Herreros-CedreÂs, C. HernaÂndez-RodrõÂguez* and R. Guerrero-Lemus Departamento de FõÂsica BaÂsica, Facultad de FõÂsica, Universidad de La Laguna, Avda. Francisco SaÂnchez s/n, La Laguna, Tenerife, Spain. Correspondence chdezr@ull.es # 2002 International Union of Crystallography Printed in Great Britain ± all rights reserved The gyration tensor of has been measured in the temperature range between 293 and 493 K at a wavelength of nm. Optical activity and birefringence for the (010) plane were determined by using a high-accuracy universal polarimeter (HAUP). Likewise, optical activity for the (001) plane was studied by using a conventional polarimeter in the same temperature range at nm wavelength. In the latter case, a modulation of the optical activity was observed. This effect can be explained by multiple re ections within the slab with a high degree of plane parallelism. 1. Introduction Lithium iodate ( ) forms a strongly birefringent crystal with many technological applications. The crystal can be grown from solution (Yang et al., 1989) and is of technological interest because of its non-linear properties (Kurtz, Perry & Bergman, 1968; Nash et al., 1969). It has been shown that lithium iodate forms a negative uniaxial crystal that is optically active (Nash et al., 1969), producing a high conversion ef ciency in second-harmonic generation (Kurtz & Perry, 1968). The crystal symmetry of is hexagonal, being of the point group 6 and space group P6 3 (see Rosenzweig & Morosin, 1966). Optical rotatory measurements were made for propagation along [001] by Vlokh et al. (1975). Stadnicka et al. (1985) determined the absolute optical chirality of lithium iodate and obtained the optical rotatory dispersion. Tebbutt (1991) studied the optical activity of as a function of wavelength. In x2 of this work, the principles of the high-accuracy universal polarimeter (HAUP) are brie y described along with the method of determining the absolute value of birefringence. In x3, we describe the experimental procedure used. Results and a discussion of the optical activity in the temperature range between 293 and 493 K are presented in x4. We have studied the optical activity as a function of temperature up to 493 K because crystals present a destructive phase transition at 529 K (Nash et al., 1969). Birefringence as a function of temperature for the (010) plane of is obtained by using the HAUP method and is described in x4.1. In x4.2 we present a method of determining the parasitic contributions of the HAUP technique when the crystal presents only an optically active phase. Once these parasitics are removed, optical activity is obtained for the (010) plane. In x4.3, the determination of the optical activity for the (001) plane of by using a conventional polarimeter is described. A modulation of this parameter is observed when the temperature is changed. 2. HAUP technique When a collimated light beam falls normally on the surface of a crystal that presents birefringence and optical activity, it splits into two elliptically polarized orthogonal components propagating with different velocities (Yariv & Yeh, 1984). These components leave the medium with a phase difference (Nye, 1985) =( 2 B + 2 G) 1/2, where B =(2/)nd is the phase delay caused by the linear birefringence n, and G = (2/)(G/n)d is the delay caused by the optical activity. and n are the free-space light wavelength and effective refractive index of the sample [n =(n 0 n 00 ) 1/2, with n 0 and n 00 being the two refractive indices that the crystal would have in the absence of optical activity]. For any given propagation direction in the crystal, the gyration coef cient G of the material is given by G = g ij l i l j, where l i and l j are the direction cosines of the normal wave and g ij are coef cients of the gyration tensor, which describes the optical activity of the crystal. In the HAUP method, the sample is placed between nearly crossed polarizers. Under conditions of small azimuths of the polarizer with respect to the fast axis of the sample, the relative intensity of transmitted light through the analyser is given by (Kobayashi & Uesu, 1983) ˆ A B C 2 D 2 ; 1 where is the analyser angle with respect to the position of the crossed polarizers when there is no sample (see Fig. 1). A, B, C and D are coef cients related to the systematic errors of the device and the optical properties of the material. The coef cients that allow the simultaneous determination of the birefringence and the optical activity are given by C ˆ 4 sin 2 =2 2 and D 0 ˆ D B=2 ˆ 2k sin 2 cos 2 =2 : Herreros-CedreÂs et al. Temperature-dependent gyration tensor J. Appl. Cryst. (2002). 35, 228±232

2 Here, k is the common ellipticity of the polarization ellipses, = p q, where p and q are the residual ellipticities of the polarizer and analyser, respectively, and is the error in the determination of the position of the crossed polarizers (Kobayashi et al., 1986). Thus, if it were possible to evaluate these systematic errors (p, q and ) we could determine the exact birefringence n and optical activity G = 2knn far from the optical axis directions. To obtain the absolute value of n, the condition B =(2/)nd =2(m + e) must be determined, where m is an integer and e is a fraction. The birefringence at one wavelength may be computed if the integer value at any temperature can be determined. In general, the integer value may be identi ed if an approximate value of n is known via a different measurement method. It is important to note the signi cant difference of our method from the technique used by other authors, where both polarizer and analyser are rotated around the minimum intensity positions. We have preferred to rotate the sample and analyser. In this way, the light beam falling on the sample stays unaltered, avoiding the possible variation in intensity arising from defective circular polarization at the entrance to the polarizer. For all measurements, both sample and analyser were rotated 0.5 around the minimum intensity positions, in 5 0 steps. The grid of intensity values allows experimental parameter measurement by least-squares t in terms of ve basis functions (1,, 2 +,, and 2 ), according to Moxon & Renshaw (1990). 3. Experimental (010) and (001) lithium iodate plates, of and mm thickness, respectively, were supplied by Casix, Inc. These crystals were of a very good optical quality (transparency and homogeneity, with well polished faces). The plane orientation was checked with our polarization microscope (Olympus BH2), observing in conoscopic illumination. The measurements were taken with the HAUP setup designed in our laboratory (see Fig. 2) (Go mez & HernaÂndez, 1998). The polarimetric device was illuminated with an He±Ne laser (5 mw) at a free-space wavelength of nm. The polarizers are calcite prisms of 25 mm length with an extinction ratio of They are mounted on motorized rotating stages with a resolution of grades. The light beam, passing through the polarizer, sample and analyser, is captured by a photomultiplier attached to its lock-in ampli er. The computer registers the emergent signal. The temperature of the sample in the host stage has a stability of 0.05 K. For each temperature, the relative intensity of the transmitted light as a function of the polarizer position and the de ecting angle with respect to the crossed-polarizers condition, read at the analyser, is measured in a region around the absolute minimum of the transmitted intensity, in order to avoid systematic errors in the subsequent re nement of data. 4. Results and discussion Optical activity and birefringence for the (010) plate were determined by using a high-accuracy universal polarimeter (HAUP) in the temperature range between 293 and 493 K at a wavelength of nm. In determining the ellipticity k, we have collected two sets of data using the same sample; the second graph was recorded after the sample had been rotated by 90 about the direction of the laser beam (Moxon & Renshaw, 1990; Herna ndez-rodrõâguez, Go mez- Garrido & Veintemillas, 2000). By determining the values of D 0 for the 0 and 90 orientations, the systematic errors of the HAUP method, p, q and (Kremers & Meekes, 1995), can be evaluated or eliminated. Likewise, the optical activity for the (001) lithium iodate plate was studied by using a conventional polarimeter in the same temperature range at nm wavelength, by which a modulation of this parameter was observed. Figure 1 Schematic representation of the azimuth of linearly polarized incident light propagated through polarizer P, the de ecting angle + of the crystal fast axis S, and the azimuth of analyser A, all with respect to the reference OX axis. Figure 2 Drawing of our HAUP setup. BS: beam splitter. Ch: chopper. P: polarizer. HS: heating stage. A: analyser. PM: photomultiplier. RC: rotor controller. TC: temperature controller. PC: computer. Lock-in: ampli er. J. Appl. Cryst. (2002). 35, 228±232 Herreros-CedreÂs et al. Temperature-dependent gyration tensor 229

3 4.1. Determination of birefringence for the (010) plane of Fig. 3 shows the temperature dependence of the experimental values of the C coef cient tted to equation (1) (open circles), along with the values when the sample was rotated through 90 (open squares), using our mobile platform to achieve the best perpendicularity of the crystal with respect to the incident light beam. The solid line represents the best t to equation (2) considering that the delay has a linear dependence on temperature. A displacement of the C curve is observed when the crystal is rotated by 90, possibly as a result of non-exact perpendicularity of the sample. This effect practically disappears when the sample is rotated perfectly perpendicular to the incident light beam. The refractive indices of the crystal are n o = and n e = at nm wavelength at room temperature (Yariv & Yeh, 1984). Hence, the birefringence, n = n o n e,is and m + e = with m = 120 and e = From the values of Fig. 3, we can obtain the mean birefringence in two consecutive extinction directions of the crystal for the (010) plane. The results for the birefringence, n =(/d)[m +(/2)], at a wavelength of nm taking 120 as the integer, are represented in Fig. 4, where it can be seen that, to a very good approximation, the birefringence n varies linearly with temperature, in the considered temperature range. From the adjustment carried out (line), one can extract the thermal variation coef cient of the birefringence, d(n)/dt = K Determination of optical activity for the (010) plane of Lithium iodate belongs to space group P6 3 (Stadnicka et al., 1985) and it is an optically active material with non-zero components of the gyration tensor (Yariv & Yeh, 1984) g 11 = g 22 and g 33. In order to determine the temperature dependence of these gyration components, we should rst determine the values of 2k for each temperature through equation (3). However, this equation does not permit direct calculation of 2k unless we know the values of the systematic errors and. There are different methods to determinate these parasites (Kobayashi & Uesu, 1983; Kobayashi et al., 1988). We have chosen the approach proposed by Moxon & Renshaw (1990) because we consider it to be the most appropriate technique (Herna ndez-rodrõâguez, Go mez-garrido & Veintemillas, 2000). In this method, the measurements are repeated with the crystal rotated by 90 around the beam direction. This should cause changes of sign in the delay and the ellipticity k and in when the crystal is rotated by 90. The parameter D 0 (90 ) is obtained from D 0 90 ˆ 90 2kŠ sin 2 90 cos 2 =2 ; bearing in mind the sign in k and. Equations to take this into account can be expressed as D 0 D 0 90 ˆ 90 4kŠsin 2 90 Š cos 2 =2 ' 4ksin 2 90 Š cos 2 =2 ; 5 where j 90 j 4jkj: The ellipticity k can be determined by tting equation (5) directly, using 4k and 2[ + (90 )] as adjustment parameters. Fig. 5 shows the behaviour of experimental values [ tted to equation (1)] of D 0 + D 0 (90 ) with the delay for the crystal. The solid line represents the best t to equation (5), obtaining the k, [ + (90 )] parasitics. Once 2k is obtained, the gyration tensor components can be derived from the relationship that ties this magnitude to n and G (G = 2knn). 4 6 Figure 3 Temperature dependence of parameter C for the (010) plane of. Circles and squares correspond to the two measurements for two consecutive extinction directions of the sample. The solid lines represent the t to the function of equation (2). Figure 4 Temperature dependence of the mean birefringence for the (010) plane of. 230 Herreros-CedreÂs et al. Temperature-dependent gyration tensor J. Appl. Cryst. (2002). 35, 228±232

4 The optical activity for the (010) plane of lithium iodate is represented in Fig. 6 along with the approximate value determined by Tebbutt (1991) at a temperature 298 K and at a wavelength of 600 nm. Our value for the gyrotropic coef cient g 11 = g 22 at 298 K and at nm is comparable with the value obtained by Tebbutt (1991). A linear dependence with temperature for G is observed in the considered temperature range Determination of optical activity for the (001) plane of For beam directions along the optical axis (n = 0), a conventional polarimeter has to be used to measure the angular rotation of the polarization plane of linearly polarized light emerging from the crystal. When the multiple internal re ections at the faces of the sample are taken into account, the emergent light is better described by the Jones formalism. In this case, the rotation angle of the polarization plane Figure 5 Temperature dependence of the sum of the parameters D 0 and D 0 (90 ) obtained in two consecutive extinction directions of the sample for the (010) plane of. The solid line represents the t to the function of equation (5). Table 1 Parameters tted to equation (10) for the (001) plane of. G 0 ( mm 1 ) G 1 ( mm 1 )K ' 0 (rad) ' 1 (rad) K ' 2 (rad) K 2 ( ) 10 4 through the sample is (Herna ndez-rodrõâguez & Go mez- Garrido, 2000) Here r 2, given by exp ˆ 1 2r 2 : r 2 ˆ r 2 o cos 2' ˆ ^n 1 = ^n 1 Š 2 cos 4^nd= ; is a scalar re ection parameter that takes into account the multiple re ections at the exit and entrance faces of the plate. ^n is a function of the optic axis orientation representing a similar effective refractive index near n. As can be seen, the rotation angle of the polarization plane through the sample depends on the re ection factor r 2. This scalar re ection parameter is a function of the optical path of light inside the crystal at constant wavelength and modulates the experimental parameter through the function cos(4 ^nd/). Optical activity can be obtained according to the relationship that ties this magnitude to, given by G ˆ = n =d : Thus changes in sample thickness when the temperature is increased will give rise to sinusoidal variations in the optical activity G. Fig. 7 represents the temperature dependence of the optical activity of the (001) plane of lithium iodate. The polarization plane of the incident light is rotated clockwise toward the observer and, therefore, our crystal is right-handed. The optical activity in the presence of multiple re ections of for beam directions along the optical axis veri es equation (7) Figure 6 Temperature dependence of the mean optical activity for the (010) plane of. The point marked by a square is the value according to Tebbutt (1991) at a wavelength of 600 nm. Figure 7 Temperature dependence of the mean optical activity for the (001) plane of. The solid line represents the t to the function of equation (10). Empty and lled squares denote values according to Tebbutt (1991) and Stadnicka et al. (1985), respectively. J. Appl. Cryst. (2002). 35, 228±232 Herreros-CedreÂs et al. Temperature-dependent gyration tensor 231

5 as shown by the solid line in Fig. 7. The best t according to equation (7) takes the form G ˆ G 0 G 1 T 1 2r 2 o cos ' 0 ' 1 T ' 2 T 2 Š; 10 where the tted parameters are shown in Table 1. For comparison, the value of g 33 from Tebbutt (1991) and the value of g 33 calculated from published optical rotation data (Stadnicka et al., 1985) are shown in Fig. 7. Our measurements agree closely with those obtained at the same temperature and wavelength. 5. Conclusions The HAUP technique was applied to determine the optimum birefringence from room temperature to 493 K at = nm in birefringent sections of lithium iodate. Rotation of the sample by 90 perpendicular to the light beam was used to eliminate the systematic error (Kobayashi et al., 1986) inherent to the HAUP technique and so obtain the optical activity in birefringent sections. The optical activity in the presence of multiple re ections along the non-birefringent direction was also determined by using a conventional polarimeter. The gyration tensor was determined as a function of temperature up to 493 K, obtaining values for the gyrotropic coef cients g 11 = g 22 and g 33 that are comparable with previously reported values (Tebbutt, 1991; Stadnicka et al., 1985). We would like to thank the University of La Laguna for nancial support under contract numbers (226/54/98) and (2000/57). We acknowledge the research grant (FPU) from `Ministerio de Educacio n y Cultura' awarded to JH-C. References Go mez, P. & HernaÂndez, C. (1998). J. Opt. Soc. Am. B, 15, 1147±1154. HernaÂndez-RodrõÂguez, C. & Go mez-garrido, P. (2000). J. Phys. D Appl. Phys. 33, 2985±2994. HernaÂndez-RodrõÂguez, C., Go mez-garrido, P. & Veintemillas, S. (2000). J. Appl. Cryst. 33, 938±946. Kobayashi, J., Asahi, T., Takahashi, S. & Glazer, A. M. (1988). J. Appl. Cryst. 21, 479±484. Kobayashi, J., Kumoni, H. & Saito, K. (1986). J. Appl. Cryst. 19, 337± 381. Kobayashi, J. & Uesu, Y. (1983). J. Appl. Cryst. 16, 204±211. Kremers, M. & Meekes, H. (1995). J. Phys. D Appl. Phys. 28, 1195± Kurtz, S. K. & Perry, T. T. (1968). J. Appl. Phys. 39, 3798±3813. Kurtz, S. K., Perry, T. T. & Bergman, J. G. (1968). J. Appl. Phys. Lett. 12, 186±188. Moxon, J. R. L. & Renshaw, A. R. (1990). J. Phys. Condens. Matter, 2, 6807±6836. Nash, F. R., Bergman, J. G., Boyd, G. D. & Turner, E. H. (1969). J. Appl. Phys. 40, 5201±5206. Nye, J. F. (1985). Physical Properties of Crystals, 2nd ed. Oxford: Clarendon Press. Rosenzweig, A. & Morosin, B. (1966). Acta Cryst. 20, 758±761. Stadnicka, K., Glazer, A. M. & Moxon, J. R. L. (1985). J. Appl. Cryst. 18, 237±240. Tebbutt, I. J. (1991) PhD thesis, University of Oxford. Vlokh, O. G., LazÂko, L. A. & Zheludev, I. S. (1975). Sov. Phys. Crystallogr. 20, 401±402. Yang, H.-G., Zhang, D.-F., Chen, W.-C. & Lin, Y.-Y. (1989). J. Appl. Cryst. 22, 144±149. Yariv, A. & Yeh, P. (1984). Optical Waves in Crystals. New York: Wiley. 232 Herreros-CedreÂs et al. Temperature-dependent gyration tensor J. Appl. Cryst. (2002). 35, 228±232