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Faraday effect in Sn2P2S6 crystals

Oleh Krupych,1 Dmytro Adamenko,1 Oksana Mys,1 Aleksandr Grabar,2

and Rostyslav Vlokh1,*1Institute of Physical Optics, 23 Dragomanov Street, 79005 Lviv, Ukraine

2Institute for Solid State Physics and Chemistry, Uzhgorod National University,54 Voloshyn Street, 88000 Uzhgorod, Ukraine

*Corresponding author: [email protected]

Received 29 May 2008; revised 3 October 2008; accepted 3 October 2008;posted 8 October 2008 (Doc. ID 96761); published 5 November 2008

We have revealed a large Faraday rotation in tin thiohypodiphosphate (Sn2P2S6) crystals, which makesthis material promising for magneto-optics. The effective Faraday tensor component and the Verdet con-stant for the direction of the optic axis have been determined by measuring the pure Faraday rotation inSn2P2S6 crystals with both the single-ray and small-angular polarimetric methods at the normal con-ditions and a wavelength of 632:8nm. The effective Verdet constant is found to be equal to 115 rad=T ×m.© 2008 Optical Society of AmericaOCIS codes: 260.1180, 260.2130, 260.5430, 230.2240, 230.3810.

1. Introduction

The tin thiohypodiphosphate crystals belong to thefamily of ferroelectric semiconductor Sn2xpb2ð1−xÞS6ySe6ð1−yÞ crystals (see [1]). These crystals are transpar-ent in a wide spectral range (0:53–8:0 μm [2]) and pos-sess ferroelectric phase transition at Tc ¼ 337K,which changes the point group of symmetry accordingto the scheme 2=mFm. These crystals attract the in-terest of researchers because of their good electro-optic [3,4], piezo-optic [5,6], acousto-optic [7] andphotorefractive [8,9] properties. Previous researchhas shown that Sn2P2S6 crystals are characterizedby very high electro-optic coefficients (r11 ¼174pm=V [4]) and an acousto-optic figure of merit(M2 ¼ ð1:7� 0:4Þ × 10−12 kg=s3 [7]).On the other hand, based on the natural assump-

tion that the values of electro-optic and electrogyra-tion coefficients of any material must be of a similarorder of magnitude (see, e.g., [10]) and taking into ac-count that Sn2P2S6 crystals are proper ferroelectrics(that means that the changes in both optical birefrin-gence and optical activity occurring at the phasetransition are directly caused by spontaneous elec-

tric polarization), it is possible to estimate roughlythe values of their electro-optic and electrogyrationcoefficients. Then, the electrogyration coefficientfor Sn2P2S6 could be estimated as ∼10−10 m=V[11],whereas, for most of the compounds studied pre-viously, it is of the order of ∼10−12 m=V. In general,all the data on the optical properties of Sn2P2S6 crys-tals obtained up to the present suggest high figuresof merit for different optical effects in these crystalsinduced by external fields. This means that the crys-tals may be applied when operating efficiently theoptical radiation. However, where the induced opti-cal activity effect and, in particular, the Faradayrotation are concerned, the experimental data havenot been presented in the literature. The reasonsfor the lack of the induced optical activity and theFaraday optical activity data for Sn2P2S6 crystalsare probably concerned with a low symmetry of thesecrystals. They are optically biaxial, thus assumingquite complicated experimental conditions for thecorresponding measurements.

The optical gyration can exist below the phasetransition temperature of Sn2P2S6 . Here, the do-mains with the opposite signs of spontaneous electricpolarization should be enantiomorphous, becausethe center of symmetry is lost during the phase tran-sition. In our recent report [11], we showed that the

0003-6935/08/326040-06$15.00/0© 2008 Optical Society of America

6040 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

natural optical rotation in Sn2P2S6 crystals mea-sured along the optic axis reaches high values atroom temperature (ρ ≈ 44� 1:5 deg =mm). The aimof the present work is to study the Faraday rotationin Sn2P2S6 crystals.

2. Experimental

As mentioned above, Sn2P2S6 crystals are opticallybiaxial. The axes of the Cartesian frame of reference[Fig. 1(a)] are very close to the crystallographic ones,with the only difference that the axes a and c of thelatter are slightly nonorthogonal. For room tempera-ture and a light wavelength of λ ¼ 632:8 nm, theplane of the optic axes is parallel to the b axis, beingrotated by ∼45° with respect to the a and c axes (see[12]). Under the same conditions, the angle betweenthe optic axes is equal to∼90°. The optical rotation inpure form should manifest itself if the light propa-gates along one of the optic axes [see Fig. 1(b)] [11].As one can see fromFig. 1(b), optical rotation should

have the same magnitude and opposite signs whenthe light propagates along different optic axes. Takinginto account the orientation of the plane of the opticaxes and denoting the angle between the plane of theoptic axes and the a axis as α, one can derive the rela-tions for the angles φ and Θ (where a ¼ sinΘ cosφ,b ¼ sinΘ sinφ, c ¼ cosΘ, 1↔a, 2↔b, and 3↔c):

φ ¼ arcsin�

cosVffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − sin2Vsin2α

p�;

Θ ¼ arccosðsin α sinVÞ;ð1Þ

where V means the angle between the optic axis andtheaaxis (i.e.,2V is the angle between the optic axes).For room temperature and λ ¼ 632:8nm, we haveα≃ 45 deg, V ≃ 45 deg [12], φ≃ 55 deg, and Θ≃

57 deg (the conoscopic fringes appearing in the case

when the light propagates along one of the optic axesin Sn2P2S6 crystals are presented in Fig. 2).

The Faraday effect is described by the relations

Bjk ¼ B0jk þ iejklFlmHm;Δρl ¼

π�n3

λ FlmHm;

VF ¼ π�n3

λ Flm; ð2Þ

where Δρl is the angle of the specific rotation of thelight polarization plane, Bjk is the actual optical-frequency dielectric impermeability tensor, B0

jk isthe optical-frequency dielectric impermeability ten-sor before the application of magnetic field Hm, ejklis the unit Levi–Civita tensor, �n is the mean refrac-tive index, VF is the Verdet constant, and Flm is theFaraday tensor. For the point symmetry groupm, thelatter has the following form:

H1 H2 H3

Δρ1 π�n3

λ F11 0 π�n3

λ F13

Δρ2 0 π�n3

λ F22 0

Δρ3 π�n3

λ F13 0 π�n3

λ F33

: ð3Þ

Application of a magnetic field along the optic axisleads to three nonzero components of that field H1 ¼H3 ¼ H cosΘ≃ 0:5H and H2 ¼ H cos 45° ¼ 0:707H.Let the directions of the light propagation and themagnetic field be parallel to the optic axis. Thenthe rotation of the polarization plane inducedmagne-tically is equal to

Δρ ¼ πn3m

λ F033H; ð4Þ

where nm ¼ 3:0256 is the mean refractive index[12] and F0

33 is the effective Faraday component that

Fig. 1. (Color online) (a) Cartesian and polar frames of reference and (b) shape of the gyration surface for Sn6P2S6 crystals in the crystal-lographic frame of reference. The outlets of the optic axes at the room temperature are also indicated. (k is the wave vector of light).

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6041

corresponds to the rotated reference frame, of whichthe Z0 axis coincides with the direction of the opticaxis:

F033 ¼ 1

2

�F22 þ

12F11 þ

12F33 − F13

�¼ Δρλ=πn3

mH:

ð5Þ

In our case, the Faraday rotation is not sensitive toswitching of the domains, thus permitting the studyof magnetically induced rotation on either single-domain or multidomain samples.Following from the above, we are able to determine

the increment of optical activity for Sn2P2S6 crystalswith a direct technique for measuring the polariza-tion plane rotation, applied for the light propagatingalong one of the optic axes. For studies of the Faradayrotation, we have used both a single-ray polarimetryand a small-angular imaging polarimetry.While employing a single-ray polarimetric setup, a

He–Ne laser (optical power of P ¼ 0:18W=cm2 and aradiation wavelength of 632:8nm) was used as alight source. It is known that a photorefractive effecttakes place in Sn2P2S6 at the wavelength of632:8nm[13,14]. In the single-beam scheme used,this effect could manifest itself as “optical damage”,i.e., as a formation of a photo-induced lens producingdistortion of the beam spot or of images on the CCDcamera. In order to clarify whether the photorefrac-tive effect could influence the polarimetric measure-ments, we have used a 50mW He–Ne laser and avariable attenuator. We have found that the intensityvariations of the light transmitted through aSn2P2S6 sample as large as one decade do not affectthe polarimetric results. Since in this work we usedan He–Ne laser with the cw power of 1:5mW, we

have, therefore, concluded that the influence of var-ious photo-induced effects in our case is negligi-bly small.

The azimuth of the incident light was chosen suchthat it always remained perpendicular to the plane ofincidence when the magnetic field was applied. Thelongitudinal magnetic field was applied using anelectromagnet (shown in Fig. 3). The sample thick-ness was d0 ¼ 1:68mm. The sample surface wasnot exactly perpendicular to the optic axis becausethe orientation of the optic axis is slightly sensitiveto environmental conditions, most of all the tempera-ture fluctuations. Therefore, the effective samplethickness was recalculated using the relation

d ¼ d0= cos�arcsin

sin βnm

�;

where β is the angle of incidence nm ¼ 3:0256.The increment of the polarization plane rotation im-posed by the Faraday effect was studied in the geo-metry when the light propagated along the optic axisand the magnetic field was applied along the samedirection.

A number of inevitable error sources exist in ourcase of magneto-optic measurements. The principalsources of errors are the angular distribution ofthe Faraday rotation and linear optical birefringencenear the optic axis, which appear even in a small-angular range of some degrees. Together with a smalldivergence of the laser beam (4 × 10�3 rad or0:23 deg), these factors could give rise to increasingmeasurement errors [15,16]. Using the small-angular imaging polarimetric technique, we are ina position to take these errors into account. Ofcourse, the accuracy of polarization measurementsin the frame of single-ray polarimetry is usuallyhigher than that typical for imaging polarimetry,which operates with a wide light beam (notice thathere we use the term “beam” in the meaning of “bun-dle of rays”). Nonetheless, the accuracy for the polar-ization azimuth achieved in this work has turned outto be not worse than 4 × 10�2 deg.

Besides, misalignments between the propagationdirection of the laser beam and the optic axis are also

Fig. 2. (Color online) Conoscopic fringes peculiar for the light pro-pagation along one of the optic axes in Sn6P2S6 crystals.

Fig. 3. Optical scheme for small-angle magnetooptic polarimetricmapping: 1, laser; 2, circular polarizer (linear polarizer combinedwith a quarter-wave plate); 3, short-focus lens; 4, coherence scram-bler; 5, long-focus lens; 6, linear polarizer (Glan prism) with mo-torized rotary stage; 7, magnetic core; 8, sample; 9, analyzer (Glanprism) with motorized rotary stage; 10, objective lens; 11, CCDcamera; 12, computer. Note that within the single-ray polarimetrictechnique, the components 3 to 5 and 10 are removed and a CCDcamera is replaced by photomultiplier.


of practical importance. Really, they could yielddifferent values of the Faraday rotation. Thus, themain problem that appears in the course of ourmeasurements is obtaining and analyzing the polari-metric data that refer to the same optical path in acrystal (i.e., the optical path measured along the op-tic axis; see Fig. 3).In our studies we used the imaging polarimetric

setup presented in Fig. 3. The difference from usualimaging polarimetry (see, e.g., [17]) is that we used aconical probing light beam instead of a parallel one.In our experiments, the divergence angle of the con-ical beam was 3:49 × 10�2 rad (approximately 2 deg).We employed a He–Ne laser (λ ¼ 632:8nm) as asource of optical radiation. The sample (8) was posi-tioned at the beam waist. An objective lens (10) pro-jected the cross-section of the light beam passedthrough an analyzer (9) onto the sensor of CCD cam-era (11). The image obtained by the camera corre-sponded to the divergence angle of the conicalbeam. That is why this technique may be called“small-angular polarimetric mapping.” In order toavoid speckle structure of the obtained images, a co-herence scrambler (4) was used. The angular diver-gence of the probing beam was limited by thedimensions of light channel in magnetic core (7).A plane-parallel crystal plate was placed between

the poles of the electromagnet. The distance betweenthe poles (65mm) was large when compare to thesample thickness, thus allowing us to reduce inhomo-geneities of the magnetic field (and so the appear-ance of the transverse component of that field)through the sample thickness to a negligibly smallvalue. The sample was positioned after aligningthe center of the conoscopic rings with the light beamcenter. The small-angular maps of the polarizationazimuth were obtained both for the zero magneticfield and the dc field of H ¼ 9:5kOe. After analyzingthese maps, we have determined the position of thecenter of the characteristic pattern corresponding tothe direction of the optic axis for each map. For thesepoints we have chosen a 2 × 2 pixel region and calcu-lated the average values of the polarization azimuth.We have considered the values obtained as the polar-ization azimuths of the light propagated exactlyalong the optic axis. The difference between theseazimuths obtained for the cases of applied magneticfield and no magnetic field gives us the Faraday ro-tation exactly for the optic axis direction.It is necessary to note that, in the case of light pro-

pagated along the optic axis, we can measure pureFaraday rotation. The point is that the linear bire-fringence vanishes in this direction and so it cannotaffect the azimuth of linearly polarized emer-ging light.

3. Experimental Results

In Fig. 4 we present the dependence of increment ofthe optical activity on the magnetic field, obtainedwith the single-ray polarimetric method. It is seenthat the Faraday rotation depends linearly upon

the magnetic field for our case of the light propaga-tion and magnetic field directed along the optic axis.

In spite of all the precautions taken, the error of de-termination of the Faraday rotation by single-ray po-larimetry remains quite large (∼70%). It is necessaryto note that the light behind the sample has not beenexactly linearly polarized. This is caused by the pre-sence and some distribution of the linear birefrin-gence and the Faraday rotation within the angularaperture of the laser beam. The angular maps ofthe Faraday rotation, obtained by small-angular po-larimetry, are displayed in Fig. 5. It is seen that, forthe direction exactly coinciding with the optic axis,the application of themagnetic field (H ¼ 9:5kOe) in-duces the polarization plane rotation by the angleΔφ ¼ 12:0 deg (or Δρ ¼ 7:1 deg =mm). This valueagrees well with that obtained by the single-ray po-larimetry (see Fig. 4). On the other hand, therotation of the azimuth of the polarization ellipse in-duced by the magnetic field, averaged over the angleof divergence of the laser beam, is smaller than theexperimental error. Of course, such estimation ofpossible errors for the single-ray case is quite roughsince the distribution of polarization states withinthe Gaussian beam profile behind the sample is notaccounted for. Anyway, we demonstrate that applica-tion of the single-ray polarimetry alone is not suffi-cient for achieving high enough accuracy in thecomplicatedmeasurements dealing with optical rota-tionalong thedirection of optic axis inbiaxial crystals.

Using Eq. (6), one can calculate the Faradaycoefficient. The effective value of this coefficient isequal to

F033 ¼ 0:84 × 10�10 Oe�1: ð6Þ

The Verdet constant for Sn2P2S6 crystals is thusequal to VF ¼ 1:15 × 10�2 rad=Oe ×m (or 115 rad=T ×m). Let us compare this value to those knownfor some magneto-optic materials. It should be notedthat, for example, ZnTeðV ¼ 187 rad=T ×mÞ and

Fig. 4. Dependence of optical activity on the magnetic field forSn6P2S6 crystals (open circles correspond to the data obtainedby single-ray polarimetry and diamonds to the data obtained bysmall-angular polarimetry).


Cu2OðV ¼ 147 rad=T ×mÞ [18] belong tocompounds having very high Verdet constants. Forone of the best magneto-optic materials (Tb3Ga5O12crystals) the latter constant is equal to 134 rad=T ×mfor a wavelength of 632:8nm and 36:4 rad=T ×m for1053nm [19,20]. This is comparable with that ob-tained for Sn2P2S6 crystals. Only the magneto-opticparameters characteristic for the diluted magneticsemiconductors Cd1�xMnxTe are higher [21] (for ex-ample, for Cd0:62Mn0:38Te crystals, we have V ¼3:38 rad=G ×m [22]). Hence, Sn2P2S6 crystals maybe referred to as one of the most promising magne-tically nonordered crystalline materials for magneto-optic applications.

4. Conclusions

Following from the results presented above, we statethat Sn2P2S6 crystals manifest very large Faraday

rotation. Their Verdet constant is found to be equalto 115 rad=T ×m for a light wavelength of 632:8nm.This is comparable to the corresponding valuesobtained for the known magneto-optic materials.However, the low symmetry and optical biaxialityof these crystals hinder their applications and re-quire employing some additional techniques for,e.g., finding the exact orientation of the optic axisand ensuring temperature stabilization of the opticaxis orientation.

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Fig. 5. (Color online) Angular maps of azimuth of polarizationstate for Sn6P2S6 crystals in the case of light propagation in thevicinity of the optic axis: (a) H ¼ 0kOe and (b) H ¼ 9:5kOe.The angular areas occupied by the laser ray are denoted by circles.


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