- Email: [email protected]
Studies of errors in measurements of electrooptic coefficients in uniaxial crystals
ARTICLE IN PRESS
Optik
Optics
Optik 116 (2005) 129–132 www.elsevier.de/ijleo
Studies of errors in measurements of electrooptic coefficients in uniaxial crystals M. Izdebskia,, W. Kucharczyka, F.L. Castillo Alvaradob a
Institute of Physics, Technical University of Ło´dz´, ul. Wo´lczan´ska 219, 93-005 Ło´dz´, Poland Escuela Superior de Fisica y Matematicas, Instituto Politechnico National, Edificio 6 UPLAM, Delegado G.A. Madero, 07738 Mexico District Federal, Mexico
b
Received 17 August 2004; accepted 8 December 2004
Abstract The Jones calculus extended for optical waves that diverge and are modulated in electrooptic uniaxial crystals is applied to quadratic electrooptic coefficients. It is shown that in measurements of the coefficients the approach may be used to predict experimental errors that result from inaccuracies in the crystal cutting and alignment. The cases of static and sinusoidal modulating fields are considered. r 2005 Elsevier GmbH. All rights reserved. Keywords: Quadratic electrooptic effect; Uniaxial crystals; Jones calculus
1. Introduction Previously the Jones matrix calculus in its original form described, for example in Refs. [1–3], has been used as an approximation to analyse responses of electrooptic uniaxial crystals when the light is sent in the vicinity of the directions parallel or perpendicular to their optic axes [4–7]. Recently, the approach was extended for other directions of the light in electrooptic crystals in a way that allows to treat refracted waves and rays that diverge in a crystal and are modulated by an external low-frequency electric field [8]. In this, the effect of partial interference of overlapping refracted beams was allowed for. Since in the extended method diverging waves and rays are considered, the extended method is useful to discuss the effect of the inaccuracies in crystals cutting and alignment on the response of optical systems comprising electrooptic components. In Ref. [8] the Corresponding author.
E-mail address: [email protected] (M. Izdebski). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.12.009
extended approach has been applied to analyse the measurement conditions of linear electrooptic coefficients in LiNbO3 and KH2PO4 (KDP) crystals. The aim of this work is to apply the approach proposed in Ref. [8] to consider the effect of the inaccuracies on the results of measurements of quadratic electrooptic coefficients. This is of special interest as for some crystals a very large spread in the coefficients measured by various authors may be observed. Such discrepancies have been noted, in particular, in the KDP family of crystals. For example, for the 9g1111–g22119 coefficient the following values have been reported: 2.6 1018 [9], 2.3 1018 [10] and 3.2 1020 m2 V2 [11]. The first two values have been determined by the static polarimetric method. The quadratic electrooptic coefficients are important not only from the point view of nonlinear distortion appearing in electrooptic devices. The quadratic effect is also of interest as related to various nonlinear phenomena [12,13], including the relationship between the spontaneous birefringence and the spontaneous
ARTICLE IN PRESS 130
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antipolarization in the low-temperature phase of antiferroelectric crystals like NH4H2PO4 (ADP) [14] and measurements of the electrostriction in transmission [15].
2. Method In Ref. [8] to describe the effect of the partial interference of overlapping refracted beams the Jonestype M-matrix, proposed previously for a dichroic homogeneous nondepolarizing birefringent crystal [3], has been employed. When the fast and slow waves and/ or rays diverge in an electrooptic crystal, the Jones calculus in its original form (see, e.g. [1–3]) cannot be used. In order to describe the effect of the divergence, in Ref. [8] three aspects have been considered: (a) there are different azimuths of polarization for the fast and slow divergent waves and different distances covered by these waves, (b) two divergent beams emerging from a birefringent plate may interfere partially or not at all, and (c) in electrooptic crystals with an applied lowfrequency field the directions of the refracted waves and rays change from their field free directions according to the electric field strength. In our calculations the crystal was considered to be cut in the form of a right parallelepiped with intended orientation of its faces parallel to the x; y; and z crystallographic axes. The intended direction of the incident light was taken to agree with the +z axis. As shown in Fig. 1, the angles bc and gc describing the inaccuracies in the crystal cutting specify the rotations of the xyz system about the +x and +y axes, respectively. Thus the angles bc and gc are related to the transformation b from the intended crystal cutting orientation system xyz to the x0 y0 z0 axes related with the faces of imperfectly cut crystal. This transformation is given by [8] 2 32 3 cos gc 0 sin gc 1 0 0 6 7 6 0 cos b 1 0 sin bc 7 b ¼ 40 54 5: c sin gc 0 cos gc 0 sin bc cos bc (1) x
cry
e lan
intend
y′
z″ γa β a z′
γc
ed lig
y
t
nt ligh
incide
x′y ′p
sta
ls
am
ple
x′
ht dir
ection
in the
normal to the entrance face
βc
crysta
l
z, σ
Fig. 1. Definition of the angles bc and gc describing the inaccuracies in the crystal cutting and ba and ga related to its inaccurate alignment.
The inaccurate alignment of the crystal is described by the angles ba and ga shown also in Fig. 1. These angles describe rotations of the direction of the incident light, now along the +z00 axis, from the main +z0 axis of the parallelepiped. The rotations also perturb the zero of azimuth, now described by the axis +x00 . Thus the matrix c describing the inaccurate alignment is 2 32 3 1 0 0 cos ga 0 sin ga 6 7 6 0 cos b 1 0 sin ba 7 c ¼ 40 54 5: a sin ga 0 cos ga 0 sin ba cos ba (2) We considered electrodes deposited onto crystal faces, thus the misalignment has no effect on the direction of the modulating field, whereas the inaccuracies in the crystal cutting result in different field components from those intended. This is taken into account in our calculations. The distribution of the light amplitude in the circular cross-section of the beam we assumed to be uniform, therefore our analysis gave the upper limit of errors arising from small inaccuracies [8]. Measurements of quadratic electrooptic coefficients are very often performed employing the polarimetric method. This technique is based on measurements of the light intensity transmitted trough a system comprising of a sandwich of a polarizer, a quarter-wave plate, a crystal sample, and an analyser crossed with the polarizer (see, e.g. [14,16]). Thus, we performed our calculations considering such optical system.
3. Results The largest spread in the values of the quadratic electrooptic coefficients reported by different authors may by observed between values determined employing static and sinusoidal modulating fields. Despite the fact that the original form of the Jones calculus may be approximately applied even when the light propagates in the vicinity of the optic axis [4,5,7,8], recent results [17] indicate that sometimes differences between results obtained by the approximate method and the extended one may by noticeable. We performed our calculations based on the extended approach proposed in [8] to compare the sensitivity of measurements to the inaccuracies in the crystal cutting and alignment employing the static and sinusoidal electric fields for the intended light direction parallel to the optic axis. As an example we considered the error in the measurement of the 9g1111–g22119 coefficient in the KDP crystal for the intended configuration: r ¼ ð0; 0; 1Þ;
E ¼ ðE; 0; 0Þ;
(3)
ARTICLE IN PRESS M. Izdebski et al. / Optik 116 (2005) 129–132
where r is the unit vector describing the light direction. In our calculations we tried to simulate conditions of the previous static measurements [9,10]. When the inaccuracies are considered, contributions of other electrooptic coefficients either linear or quadratic have to be taken into account. In our calculation we employed values of the linear and quadratic electrooptic coefficients given in [18] and [11,19], respectively. The refractive indices were taken from [20]. The results obtained in our simulations of earlier measurements indicate that the inaccuracies may result in an apparent quadratic electrooptic coefficient 9g0 1111–g0 22119, different from the real 9g1111–g22119 value. The apparent coefficient would be derived from the measured changes in the light intensity when a static field is employed. The effect of the inaccuracies on the coefficient 9g0 1111–g0 22119 is illustrated in Fig. 2. Previously the approximated method based on the original Jones calculus has been used to simulate the measurements of those quadratic electrooptic coefficients that are associated with rotations of the principal axes of the optical permittivity tensor [5,7,17]. It has been found that when a static field is used and inaccuracies in crystal cutting appear, leading to unintended electric field components, a large potential for experimental errors exists, even if the exact light direction in the crystal is achieved by an adjustment of the crystal alignment. The coefficient 9g1111–g22119 is responsible for changes in the principal values of the optical frequency permittivity tensor, not rotations of its principal axes. The plots presented in Fig. 2a show that measurements of coefficients related to the changes in the principal values are much less sensitive to the deviations of the direction of electric field than those associated with rotations of the principal axes. However, when a static field is used even a small deviations of the light from the crystal optic axis resulting from crystal cutting (Fig. 2b) or misalignment may be a reason of experimental response suggesting much larger coefficients than the real ones. Previously, computer calculations showed that the use of a sinusoidal modulating field E ¼ E0 sin Ot reduces the errors related with the cutting and alignment of samples [5–7]. In order to analyse the accuracy of measurements of the 9g1111–g22119 coefficient employing a.c. electric fields we took into account the modulation index A2O ¼ I 2O =I 0 : In this I 0 is the constant component of the light transmitted by the system and I 2O is the transmitted light intensity at the second harmonic of the modulating field. Considering the influence of inaccuracies on the modulation index A2O ; we employed the following ratio: id (4) D2O ¼ A2O Aid 2O =A2O ; where Aid 2O is the ideal value of the modulation index derived on the assumption of no inaccuracies, total interference, and modulation at the linear part of the
131
Fig. 2. Effect of inaccurate cutting of the KDP crystal on the apparent quadratic electrooptic coefficient 9g0 1111g0 22119 determined by the static polarimetric method. The electric field strength is 105 V m1 and the crystal length is 1 cm. The intended configuration is r ¼ (0,0,1) and E ¼ (E,0,0). (a) Refracted light waves propagate exactly along the crystal optical axis, for small angles baEn0bc and gaEn0gc, (b) the light is sent exactly perpendicularly to the crystal face, i.e. ba ¼ ga ¼ 0.
transmission characteristic of the optical system [8]. Our results are illustrated in Fig. 3. The comparison of the plots presented in Figs. 2 and 3 shows that also for the coefficient 9g1111–g22119 the use of a sinusoidal field reduces the experimental error, especially that related with inaccuracies in the light direction. Analogous results we obtained for other members of the KDP family.
4. Conclusions Our results indicate that in measurements of quadratic electrooptic coefficients a sinusoidal electric field should be used. When inaccuracies in the crystal cutting and alignment are known, the matrices approach based on the Jones calculus described in [8] may be used to
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[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13] Fig. 3. The effect of inaccuracies in cutting the KDP sample on the relative error of measurements of the quadratic electrooptic coefficient 9g1111g22119 for a sinusoidal field of amplitude 105 V m1 in the KDP of the length 1 cm. The intended configuration is r ¼ (0,0,1) and E ¼ (E,0,0). (a) Refracted light waves propagate exactly along the crystal optical axis, for small angles baEn0bc and gaEn0gc, (b) the light is sent exactly perpendicularly to the crystal face, i.e. ba ¼ ga ¼ 0.
estimate an upper limit of the experimental error that results from the inaccuracies.
[14]
[15]
[16]
[17]
References [1] R.C. Jones, A new calculus for the treatment of optical systems. IV, J. Opt. Soc. Am. 32 (1942) 486–493. [2] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. [3] I. S´cierski, F. Ratajczyk, The Jones matrix of the real dichroic elliptic object, Optik 68 (1984) 121–125. [4] M. Izdebski, W. Kucharczyk, R.E. Raab, Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence, J. Opt. Soc. Am. A 18 (2001) 1393–1398. [5] M. Izdebski, W. Kucharczyk, R.E. Raab, Analysis of accuracy of measurement of quadratic electro-optic
[18]
[19]
[20]
coefficients in uniaxial crystals: a case study of KDP, J. Opt. Soc. Am. A 19 (2002) 1417–1421. M. Izdebski, W. Kucharczyk, R.E. Raab, Computer simulations of the measurement of quadratic electro-optic coefficients associated with rotations of the principal axes of the optical permittivity tensor, J. Opt. A: Pure Appl. Opt. 5 (2003) 147–151. M. Izdebski, W. Kucharczyk, Measurement conditions of the quadratic electrooptic coefficients along the optic axis in uniaxial crystals, Opt. Appl. 33 (2003) 213–218. M. Izdebski, W. Kucharczyk, R.E. Raab, Application of the Jones calculus for a modulated double-refracted light beam propagating in a homogeneous and non-depolarizing electro-optic uniaxial crystal, J. Opt. Soc. Am. A 21 (2004) 132–139. V.E. Perfilova, A.S. Sonin, Quadratic electrooptic effect in KDP group crystals, Izv. Akad. Nauk SSSR, Ser. Fiz (USSR) 31 (1967) 1136–1138 (English Transl.) Bull. Acad. Sci. USSR Phys. Ser. (USA) 31 (1967) 1154–1157. W. Jamroz, J. Karniewicz, J. Stachowiak, Nonlinear electrooptic effects in KDP and DKDP crystals, Kvantovaya Elektron. 6 (1979) 1365–1369 (English Transl.) Sov. J. Quantum Electron. 9 (1979) 803–805. M.J. Gunning, R.E. Raab, W. Kucharczyk, Magnitude and nature of the quadratic electro-optic effect in potassium dihydrogen phosphate and ammonium dihydrogen phosphate crystals, J. Opt. Soc. Am. B 18 (2001) 1092–1098. W. Kucharczyk, Quadratic electro-optic effect and second-order strain derivatives of electronic susceptibility, Physica B 176 (1992) 189–208. M. Izdebski, W. Kucharczyk, R.E. Raab, On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities, J. Opt. A: Pure Appl. Opt. 6 (2004) 421–424. R. Ledzion, K. Bondarczuk, W. Kucharczyk, Temperature dependence of the quadratic electrooptic effect and estimation of antipolarization of ADP, Cryst. Res. Technol. 39 (2004) 161–164. M.J. Gunning, R.E. Raab, W. Kucharczyk, Interferometric measurements of electrostrictive coefficients of KDP and ADP in transmission, Ferroelectr. Lett. Sect. 28 (2001) 93–102. R. Ledzion, K. Bondarczuk, P. Go´rski, W. Kucharczyk, Effect of deuteration on the quadratic electrooptic properties of KDP, Cryst. Res. Technol. 34 (1999) 745–749. M. Izdebski, Application of the Jones matrices calculus to electrooptic crystals, Ph.D. Dissertation, Technical University of Ło´dz´, Ło´dz´, 2004 K.-H. Hellwege, A.M. Hellwege (Eds.), Landolt-Bo¨rnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III, vols. 11 and 18; D.F. Nelson (Ed.), High Frequency Properties of Dielectric Crystals, New Series, Group III, vol. 30, Springer, Berlin, 1979, 1984, 2001. P. Go´rski, D. Mik, W. Kucharczyk, R.E. Raab, On the quadratic electro-optic effect in KDP, Physica B 193 (1994) 17–24. G.C. Ghosh, G.C. Bhar, Temperature dispersion in ADP, KDP, and KD*P for nonlinear devices, IEEE J. Quantum Electron. QE-18 (1982) 143–145.
Optik
Optics
Optik 116 (2005) 129–132 www.elsevier.de/ijleo
Studies of errors in measurements of electrooptic coefficients in uniaxial crystals M. Izdebskia,, W. Kucharczyka, F.L. Castillo Alvaradob a
Institute of Physics, Technical University of Ło´dz´, ul. Wo´lczan´ska 219, 93-005 Ło´dz´, Poland Escuela Superior de Fisica y Matematicas, Instituto Politechnico National, Edificio 6 UPLAM, Delegado G.A. Madero, 07738 Mexico District Federal, Mexico
b
Received 17 August 2004; accepted 8 December 2004
Abstract The Jones calculus extended for optical waves that diverge and are modulated in electrooptic uniaxial crystals is applied to quadratic electrooptic coefficients. It is shown that in measurements of the coefficients the approach may be used to predict experimental errors that result from inaccuracies in the crystal cutting and alignment. The cases of static and sinusoidal modulating fields are considered. r 2005 Elsevier GmbH. All rights reserved. Keywords: Quadratic electrooptic effect; Uniaxial crystals; Jones calculus
1. Introduction Previously the Jones matrix calculus in its original form described, for example in Refs. [1–3], has been used as an approximation to analyse responses of electrooptic uniaxial crystals when the light is sent in the vicinity of the directions parallel or perpendicular to their optic axes [4–7]. Recently, the approach was extended for other directions of the light in electrooptic crystals in a way that allows to treat refracted waves and rays that diverge in a crystal and are modulated by an external low-frequency electric field [8]. In this, the effect of partial interference of overlapping refracted beams was allowed for. Since in the extended method diverging waves and rays are considered, the extended method is useful to discuss the effect of the inaccuracies in crystals cutting and alignment on the response of optical systems comprising electrooptic components. In Ref. [8] the Corresponding author.
E-mail address: [email protected] (M. Izdebski). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.12.009
extended approach has been applied to analyse the measurement conditions of linear electrooptic coefficients in LiNbO3 and KH2PO4 (KDP) crystals. The aim of this work is to apply the approach proposed in Ref. [8] to consider the effect of the inaccuracies on the results of measurements of quadratic electrooptic coefficients. This is of special interest as for some crystals a very large spread in the coefficients measured by various authors may be observed. Such discrepancies have been noted, in particular, in the KDP family of crystals. For example, for the 9g1111–g22119 coefficient the following values have been reported: 2.6 1018 [9], 2.3 1018 [10] and 3.2 1020 m2 V2 [11]. The first two values have been determined by the static polarimetric method. The quadratic electrooptic coefficients are important not only from the point view of nonlinear distortion appearing in electrooptic devices. The quadratic effect is also of interest as related to various nonlinear phenomena [12,13], including the relationship between the spontaneous birefringence and the spontaneous
ARTICLE IN PRESS 130
M. Izdebski et al. / Optik 116 (2005) 129–132
antipolarization in the low-temperature phase of antiferroelectric crystals like NH4H2PO4 (ADP) [14] and measurements of the electrostriction in transmission [15].
2. Method In Ref. [8] to describe the effect of the partial interference of overlapping refracted beams the Jonestype M-matrix, proposed previously for a dichroic homogeneous nondepolarizing birefringent crystal [3], has been employed. When the fast and slow waves and/ or rays diverge in an electrooptic crystal, the Jones calculus in its original form (see, e.g. [1–3]) cannot be used. In order to describe the effect of the divergence, in Ref. [8] three aspects have been considered: (a) there are different azimuths of polarization for the fast and slow divergent waves and different distances covered by these waves, (b) two divergent beams emerging from a birefringent plate may interfere partially or not at all, and (c) in electrooptic crystals with an applied lowfrequency field the directions of the refracted waves and rays change from their field free directions according to the electric field strength. In our calculations the crystal was considered to be cut in the form of a right parallelepiped with intended orientation of its faces parallel to the x; y; and z crystallographic axes. The intended direction of the incident light was taken to agree with the +z axis. As shown in Fig. 1, the angles bc and gc describing the inaccuracies in the crystal cutting specify the rotations of the xyz system about the +x and +y axes, respectively. Thus the angles bc and gc are related to the transformation b from the intended crystal cutting orientation system xyz to the x0 y0 z0 axes related with the faces of imperfectly cut crystal. This transformation is given by [8] 2 32 3 cos gc 0 sin gc 1 0 0 6 7 6 0 cos b 1 0 sin bc 7 b ¼ 40 54 5: c sin gc 0 cos gc 0 sin bc cos bc (1) x
cry
e lan
intend
y′
z″ γa β a z′
γc
ed lig
y
t
nt ligh
incide
x′y ′p
sta
ls
am
ple
x′
ht dir
ection
in the
normal to the entrance face
βc
crysta
l
z, σ
Fig. 1. Definition of the angles bc and gc describing the inaccuracies in the crystal cutting and ba and ga related to its inaccurate alignment.
The inaccurate alignment of the crystal is described by the angles ba and ga shown also in Fig. 1. These angles describe rotations of the direction of the incident light, now along the +z00 axis, from the main +z0 axis of the parallelepiped. The rotations also perturb the zero of azimuth, now described by the axis +x00 . Thus the matrix c describing the inaccurate alignment is 2 32 3 1 0 0 cos ga 0 sin ga 6 7 6 0 cos b 1 0 sin ba 7 c ¼ 40 54 5: a sin ga 0 cos ga 0 sin ba cos ba (2) We considered electrodes deposited onto crystal faces, thus the misalignment has no effect on the direction of the modulating field, whereas the inaccuracies in the crystal cutting result in different field components from those intended. This is taken into account in our calculations. The distribution of the light amplitude in the circular cross-section of the beam we assumed to be uniform, therefore our analysis gave the upper limit of errors arising from small inaccuracies [8]. Measurements of quadratic electrooptic coefficients are very often performed employing the polarimetric method. This technique is based on measurements of the light intensity transmitted trough a system comprising of a sandwich of a polarizer, a quarter-wave plate, a crystal sample, and an analyser crossed with the polarizer (see, e.g. [14,16]). Thus, we performed our calculations considering such optical system.
3. Results The largest spread in the values of the quadratic electrooptic coefficients reported by different authors may by observed between values determined employing static and sinusoidal modulating fields. Despite the fact that the original form of the Jones calculus may be approximately applied even when the light propagates in the vicinity of the optic axis [4,5,7,8], recent results [17] indicate that sometimes differences between results obtained by the approximate method and the extended one may by noticeable. We performed our calculations based on the extended approach proposed in [8] to compare the sensitivity of measurements to the inaccuracies in the crystal cutting and alignment employing the static and sinusoidal electric fields for the intended light direction parallel to the optic axis. As an example we considered the error in the measurement of the 9g1111–g22119 coefficient in the KDP crystal for the intended configuration: r ¼ ð0; 0; 1Þ;
E ¼ ðE; 0; 0Þ;
(3)
ARTICLE IN PRESS M. Izdebski et al. / Optik 116 (2005) 129–132
where r is the unit vector describing the light direction. In our calculations we tried to simulate conditions of the previous static measurements [9,10]. When the inaccuracies are considered, contributions of other electrooptic coefficients either linear or quadratic have to be taken into account. In our calculation we employed values of the linear and quadratic electrooptic coefficients given in [18] and [11,19], respectively. The refractive indices were taken from [20]. The results obtained in our simulations of earlier measurements indicate that the inaccuracies may result in an apparent quadratic electrooptic coefficient 9g0 1111–g0 22119, different from the real 9g1111–g22119 value. The apparent coefficient would be derived from the measured changes in the light intensity when a static field is employed. The effect of the inaccuracies on the coefficient 9g0 1111–g0 22119 is illustrated in Fig. 2. Previously the approximated method based on the original Jones calculus has been used to simulate the measurements of those quadratic electrooptic coefficients that are associated with rotations of the principal axes of the optical permittivity tensor [5,7,17]. It has been found that when a static field is used and inaccuracies in crystal cutting appear, leading to unintended electric field components, a large potential for experimental errors exists, even if the exact light direction in the crystal is achieved by an adjustment of the crystal alignment. The coefficient 9g1111–g22119 is responsible for changes in the principal values of the optical frequency permittivity tensor, not rotations of its principal axes. The plots presented in Fig. 2a show that measurements of coefficients related to the changes in the principal values are much less sensitive to the deviations of the direction of electric field than those associated with rotations of the principal axes. However, when a static field is used even a small deviations of the light from the crystal optic axis resulting from crystal cutting (Fig. 2b) or misalignment may be a reason of experimental response suggesting much larger coefficients than the real ones. Previously, computer calculations showed that the use of a sinusoidal modulating field E ¼ E0 sin Ot reduces the errors related with the cutting and alignment of samples [5–7]. In order to analyse the accuracy of measurements of the 9g1111–g22119 coefficient employing a.c. electric fields we took into account the modulation index A2O ¼ I 2O =I 0 : In this I 0 is the constant component of the light transmitted by the system and I 2O is the transmitted light intensity at the second harmonic of the modulating field. Considering the influence of inaccuracies on the modulation index A2O ; we employed the following ratio: id (4) D2O ¼ A2O Aid 2O =A2O ; where Aid 2O is the ideal value of the modulation index derived on the assumption of no inaccuracies, total interference, and modulation at the linear part of the
131
Fig. 2. Effect of inaccurate cutting of the KDP crystal on the apparent quadratic electrooptic coefficient 9g0 1111g0 22119 determined by the static polarimetric method. The electric field strength is 105 V m1 and the crystal length is 1 cm. The intended configuration is r ¼ (0,0,1) and E ¼ (E,0,0). (a) Refracted light waves propagate exactly along the crystal optical axis, for small angles baEn0bc and gaEn0gc, (b) the light is sent exactly perpendicularly to the crystal face, i.e. ba ¼ ga ¼ 0.
transmission characteristic of the optical system [8]. Our results are illustrated in Fig. 3. The comparison of the plots presented in Figs. 2 and 3 shows that also for the coefficient 9g1111–g22119 the use of a sinusoidal field reduces the experimental error, especially that related with inaccuracies in the light direction. Analogous results we obtained for other members of the KDP family.
4. Conclusions Our results indicate that in measurements of quadratic electrooptic coefficients a sinusoidal electric field should be used. When inaccuracies in the crystal cutting and alignment are known, the matrices approach based on the Jones calculus described in [8] may be used to
ARTICLE IN PRESS 132
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[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13] Fig. 3. The effect of inaccuracies in cutting the KDP sample on the relative error of measurements of the quadratic electrooptic coefficient 9g1111g22119 for a sinusoidal field of amplitude 105 V m1 in the KDP of the length 1 cm. The intended configuration is r ¼ (0,0,1) and E ¼ (E,0,0). (a) Refracted light waves propagate exactly along the crystal optical axis, for small angles baEn0bc and gaEn0gc, (b) the light is sent exactly perpendicularly to the crystal face, i.e. ba ¼ ga ¼ 0.
estimate an upper limit of the experimental error that results from the inaccuracies.
[14]
[15]
[16]
[17]
References [1] R.C. Jones, A new calculus for the treatment of optical systems. IV, J. Opt. Soc. Am. 32 (1942) 486–493. [2] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. [3] I. S´cierski, F. Ratajczyk, The Jones matrix of the real dichroic elliptic object, Optik 68 (1984) 121–125. [4] M. Izdebski, W. Kucharczyk, R.E. Raab, Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence, J. Opt. Soc. Am. A 18 (2001) 1393–1398. [5] M. Izdebski, W. Kucharczyk, R.E. Raab, Analysis of accuracy of measurement of quadratic electro-optic
[18]
[19]
[20]
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