Abstract

In this paper we describe the methodology behind the calculation of the indicative surfaces (ISs) of the electric-field-induced optical path length change (EFIOPC) in anisotropic crystal materials accounting for the piezoelectric deformation. It is considered in detail for a particular case of 3m point group symmetry and applied to LiNbO3 single crystals doped with 7 mol. % MgO (hereafter LiNbO3:MgO). The contribution of the inverse piezoelectricity into EFIOPC appears to be considerable and, in many cases, modifying, for instance, the spherical coordinates of the extreme directions or even leading to the appearance of new directional maxima on relevant ISs. The ISs of EFIOPC are of considerable practical importance as they allow us to determine an optimal geometry for electro-optic coupling. The spatial anisotropic analysis of EFIOPC in LiNbO3:MgO crystals suggests that the lowest effective driving voltage is provided by electro-optic cells representing the rectangular slabs of X/50° crystal cut. The modulation efficiency of such electro-optic cells is about 1.5 times better than ones fabricated in the usual way (i.e., as rectangular crystal slabs with the faces parallel to the principal crystallographic directions).

© 2013 Optical Society of America

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  4. B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
    [CrossRef]
  5. A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
    [CrossRef]
  6. G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
    [CrossRef]
  7. T. Inoue and T. Suhara, “Electro-optic Bragg deflection modulator using periodically poled MgO:LiNbO3,” IEEE Photon. Technol. Lett. 23, 1252–1254 (2011).
  8. J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
    [CrossRef]
  9. S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.
  10. A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
    [CrossRef]
  11. M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.
  12. W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.
  13. H. Hirori and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1  MV/cm generated by optical rectification in LiNbO3 and applications to nonlinear optics,” Proc. SPIE 8240, 82400B (2012).
  14. Y. Chen, G. Liu, Y. Zheng, and F. Geng, “Periodically poled Ti-diffused near-stoichiometric MgO:LiNbO3 waveguide nonlinear-optic wavelength converter,” Opt. Express 17, 4834–4841 (2009).
    [CrossRef]
  15. I. P. Kaminow, Li Tingye, and A. E. Willner, Optical Fiber Telecommunications (Academic, 2008), Vol. V, p. 915.
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    [CrossRef]
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    [CrossRef]
  18. A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
    [CrossRef]
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    [CrossRef]
  22. A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
    [CrossRef]
  23. V. L. German, “Some theorems on the anisotropic environments,” Report Academy Sci. USSR 48, 95–98 (1945) [in Russian].
  24. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University, 1985), p. 333.

2012 (2)

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

H. Hirori and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1  MV/cm generated by optical rectification in LiNbO3 and applications to nonlinear optics,” Proc. SPIE 8240, 82400B (2012).

2011 (3)

T. Inoue and T. Suhara, “Electro-optic Bragg deflection modulator using periodically poled MgO:LiNbO3,” IEEE Photon. Technol. Lett. 23, 1252–1254 (2011).

S. Park and I.-K. Jeong, “Correlated thermal motion in ferroelectric LiNbO3 studied using neutron total scattering and a Rietveld analysis,” J. Korean Phys. Soc. 59, 2756–2759 (2011).
[CrossRef]

E. Saitoh, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “A design method of lithium niobate on insulator ridge waveguides without leakage loss,” Opt. Express 19, 15833–15842 (2011).
[CrossRef]

2010 (2)

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
[CrossRef]

2009 (5)

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

Y. Chen, G. Liu, Y. Zheng, and F. Geng, “Periodically poled Ti-diffused near-stoichiometric MgO:LiNbO3 waveguide nonlinear-optic wavelength converter,” Opt. Express 17, 4834–4841 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

2003 (1)

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

2002 (1)

2000 (1)

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

1997 (1)

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

1945 (1)

V. L. German, “Some theorems on the anisotropic environments,” Report Academy Sci. USSR 48, 95–98 (1945) [in Russian].

Andrushchak, A. S.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

Andrushchak, N. A.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

Baba, A.

A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
[CrossRef]

Baida, F.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Benchabane, S.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Bernal, M. P.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Cannata, J. M.

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

Chen, Wo-Hsing

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

Chen, Y.

Chernyhivsky, E. M.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

Cochran, S.

S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.

Coutaz, J.-L.

Demyanyshyn, N. M.

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

Deng, J.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Duvillaret, L.

Fu, Q.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Gaba, V. M.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Geng, F.

German, V. L.

V. L. German, “Some theorems on the anisotropic environments,” Report Academy Sci. USSR 48, 95–98 (1945) [in Russian].

Gong, Sh.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Gotra, Z. Yu.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

Grabovskii, V. V.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Günter, P.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

Hirori, H.

H. Hirori and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1  MV/cm generated by optical rectification in LiNbO3 and applications to nonlinear optics,” Proc. SPIE 8240, 82400B (2012).

Hu, H.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

Inoue, T.

T. Inoue and T. Suhara, “Electro-optic Bragg deflection modulator using periodically poled MgO:LiNbO3,” IEEE Photon. Technol. Lett. 23, 1252–1254 (2011).

Jeong, I.-K.

S. Park and I.-K. Jeong, “Correlated thermal motion in ferroelectric LiNbO3 studied using neutron total scattering and a Rietveld analysis,” J. Korean Phys. Soc. 59, 2756–2759 (2011).
[CrossRef]

Kaidan, M. V.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, Li Tingye, and A. E. Willner, Optical Fiber Telecommunications (Academic, 2008), Vol. V, p. 915.

Kawaguchi, Y.

Khelif, A.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Kityk, A. V.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

Kopko, B. M.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Koshiba, M.

Kost, Y. P.

Kotopoulis, S.

S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.

Kuzminov, U. S.

U. S. Kuzminov, Electro-Optical and Nonlinear Optical Crystal of Lithium Niobate (Nauka, 1987) [in Russian], p. 264.

Laba, H. P.

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

Laude, V.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Liu, G.

Luo, W.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Maksymyuk, T. A.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

Mandracci, P.

Matkovskii, A. O.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Mytsyk, B. G.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

Narasimhamurty, T. S.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981), p. 624.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University, 1985), p. 333.

Oliinyk, V. Ya.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Park, S.

S. Park and I.-K. Jeong, “Correlated thermal motion in ferroelectric LiNbO3 studied using neutron total scattering and a Rietveld analysis,” J. Korean Phys. Soc. 59, 2756–2759 (2011).
[CrossRef]

Poberaj, G.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

Postema, M.

S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.

Pryriz, Ya. V.

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

Rakitina, L. G.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Rialland, S.

Ritter, T. A.

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

Roussey, M.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

Sahraoui, B.

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

Saitoh, E.

Saitoh, K.

Schranz, W.

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
[CrossRef]

Searfass, C. T.

A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
[CrossRef]

Shung, K. K.

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

Silverman, R. H.

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

Sohler, W.

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

Solskii, I. M.

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Stefanskii, I. V.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Sugak, D. Yu.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Suhara, T.

T. Inoue and T. Suhara, “Electro-optic Bragg deflection modulator using periodically poled MgO:LiNbO3,” IEEE Photon. Technol. Lett. 23, 1252–1254 (2011).

Tanaka, K.

H. Hirori and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1  MV/cm generated by optical rectification in LiNbO3 and applications to nonlinear optics,” Proc. SPIE 8240, 82400B (2012).

Tingye, Li

I. P. Kaminow, Li Tingye, and A. E. Willner, Optical Fiber Telecommunications (Academic, 2008), Vol. V, p. 915.

Tittmann, B. R.

A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
[CrossRef]

Vlokh, O. G.

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

Wang, Han

S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.

Willner, A. E.

I. P. Kaminow, Li Tingye, and A. E. Willner, Optical Fiber Telecommunications (Academic, 2008), Vol. V, p. 915.

Yan, G.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Yurkevych, O. V.

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

Zaritskii, I. M.

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Zheng, Y.

Zhou, D.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Baba, C. T. Searfass, and B. R. Tittmann, “High temperature ultrasonic transducer up to 1000°C using lithium niobate single crystal,” Appl. Phys. Lett. 97, 232901 (2010).
[CrossRef]

Cryst. Res. Technol. (1)

D. Yu. Sugak, A. O. Matkovskii, I. M. Solskii, B. M. Kopko, V. Ya. Oliinyk, I. V. Stefanskii, V. M. Gaba, V. V. Grabovskii, I. M. Zaritskii, and L. G. Rakitina, “Growth and optical properties of LiNbO3:MgO single crystals,” Cryst. Res. Technol. 32, 805–811 (1997).
[CrossRef]

Crystallogr. Rep. (1)

O. G. Vlokh, B. G. Mytsyk, A. S. Andrushchak, and Ya. V. Pryriz, “Spatial distribution of piezo-induced change in the optical path length in lithium niobate crystals,” Crystallogr. Rep. 45, 138–144 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Inoue and T. Suhara, “Electro-optic Bragg deflection modulator using periodically poled MgO:LiNbO3,” IEEE Photon. Technol. Lett. 23, 1252–1254 (2011).

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. M. Cannata, T. A. Ritter, Wo-Hsing Chen, R. H. Silverman, and K. K. Shung, “Design of efficient, broadband single-element (20–80 MHz) ultrasonic transducers for medical imaging applications,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1548–1557 (2003).
[CrossRef]

J. Appl. Phys. (2)

A. S. Andrushchak, E. M. Chernyhivsky, Z. Yu. Gotra, M. V. Kaidan, A. V. Kityk, N. A. Andrushchak, T. A. Maksymyuk, B. G. Mytsyk, and W. Schranz, “Spatial anisotropy of the acousto-optical efficiency in lithium niobate crystals,” J. Appl. Phys. 108, 103118 (2010).
[CrossRef]

A. S. Andrushchak, B. G. Mytsyk, H. P. Laba, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and B. Sahraoui, “Elastic and photoelastic constants of pure and MgO doped lithium niobate crystals,” J. Appl. Phys. 106, 073510(6) (2009).
[CrossRef]

J. Korean Phys. Soc. (1)

S. Park and I.-K. Jeong, “Correlated thermal motion in ferroelectric LiNbO3 studied using neutron total scattering and a Rietveld analysis,” J. Korean Phys. Soc. 59, 2756–2759 (2011).
[CrossRef]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

G. Poberaj, H. Hu, W. Sohler, and P. Günter, “Lithium niobate on insulator (LNOI) for micro-photonic devices,” Laser Photon. Rev. 6, 488–503 (2012).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: I. Experimental determination of electro-optic tensor by means of interferometric technique,” Opt. Lasers Eng. 47, 31–38 (2009).
[CrossRef]

Opt. Lasers. Eng. (1)

A. S. Andrushchak, B. G. Mytsyk, N. M. Demyanyshyn, M. V. Kaidan, O. V. Yurkevych, I. M. Solskii, A. V. Kityk, and W. Schranz, “Spatial anisotropy of linear electro-optic effect for crystal materials: II. Indicative surfaces as efficient tool for electro-optic coupling optimization,” Opt. Lasers. Eng. 47, 24–30 (2009).
[CrossRef]

Proc. SPIE (1)

H. Hirori and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1  MV/cm generated by optical rectification in LiNbO3 and applications to nonlinear optics,” Proc. SPIE 8240, 82400B (2012).

Report Academy Sci. USSR (1)

V. L. German, “Some theorems on the anisotropic environments,” Report Academy Sci. USSR 48, 95–98 (1945) [in Russian].

Other (7)

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University, 1985), p. 333.

U. S. Kuzminov, Electro-Optical and Nonlinear Optical Crystal of Lithium Niobate (Nauka, 1987) [in Russian], p. 264.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981), p. 624.

I. P. Kaminow, Li Tingye, and A. E. Willner, Optical Fiber Telecommunications (Academic, 2008), Vol. V, p. 915.

M. P. Bernal, M. Roussey, F. Baida, S. Benchabane, A. Khelif, and V. Laude, “Photonic and phononic band gap properties of lithium niobate,” Ferroelectric Crystals for Photonic Applications (Springer, 2009), Vol. 91, pp. 307–336.

W. Luo, J. Deng, Q. Fu, G. Yan, D. Zhou, and Sh. Gong, “An integrated passive impedance-loaded SAW sensor,” 14th International Meeting on Chemical Sensors, Nuremberg, Germany, 20–23 May (2012), pp. 1403–1406.

S. Kotopoulis, Han Wang, S. Cochran, and M. Postema, “Lithium niobate ultrasound transducers for high-resolution focused ultrasound surgery,” 2010 IEEE Ultrasonics Symposium, San Diego, California, 11–14 October (2010), pp. 72–75.

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Figures (4)

Fig. 1.
Fig. 1.

Spatial position of the rotating coordinate system X1, X2, and X3 with respect to the crystallophysical coordinate system X1, X2, and X3 for uniaxial crystals.

Fig. 2.
Fig. 2.

ISs of pure EOE [panels (a) rii(θ,φ), (c) ri(i)(θ,φ), (e) rj()(θ,φ), (g) ri(k)(θ,φ)] and relevant inverse PEE [panels (b) dk(), (d) dk(i), (f) dk(), (h) d(k)] of LiNbO3:MgO crystals (wavelength, λ=633nm, temperature, T=295K). All label values are in units of 1012m/V. Blue and red colors correspond to negative and positive regions, respectively.

Fig. 3.
Fig. 3.

ISs of EIOPLC [panels (a), (c), (e), (g)] and relevant stereographic projections [panels (b), (d), (f), (h)] for LiNbO3:MgO crystals (wavelength, λ=633nm, temperature, T=295K): (a) and (b) longitudinal effect [δΔiik(θ,φ)]; (c) and (d) transverse effect with respect to light polarization [δΔik(i)(θ,φ)]; (e) and (f) transverse effect with respect to applied electric field [δΔjk()(θ,φ)]; (g) and (h) transverse effect with respect to wave vector [δΔi(k)(θ,φ)]. All label values are in units 1012m/V. Blue and red colors correspond to negative and positive regions, respectively.

Fig. 4.
Fig. 4.

Examples of two different geometries of the EO cells made of LiNbO3:MgO crystals. The faces colored in gray would correspond to deposited electrodes. or i and k correspond to the directions of applied electric field or light polarization and propagation, respectively. (a) is the cell used in most standard applications. (b) is the cell in the best optimized geometry for EFIOPC. The modulation efficiency of cell (b) is about 1.5 times better compared to the modulation efficiency of the standard cell (a). The axes X1, X2, and X3 correspond to the crystallophysical coordinate system.

Tables (2)

Tables Icon

Table 1. Electro-Optic (ri) and Piezoelectric (dk) Tensor Constants of LiNbO3:MgO Crystals (in Units of 1012m/V) and Refractive Indices (Ordinary no and Extraordinary ne) [18,22]

Tables Icon

Table 2. Spherical Coordinates of the Directional Maxima on the ISs of EFIOPC δΔik(θ,φ), Pure EOE, ri(θ,φ), and Inverse PEE, dk(θ,φ), in LiNbO3:MgO Crystalsa

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

δΔ=2(tkδni+(ni1)δtk)=rini3Etk+2(ni1)dkEtk,
δΔik=δΔEtk=rini3+2(ni1)dk,
δΔik(θ,φ)=ri(θ,φ)ni3(θ,φ)+2[ni(θ,φ)1]dk(θ,φ).
rii(θ,φ)=r22sin3θsin3φ+(r13+2r51)sin2θcosθ+r33cos3θ,
ri(i)(θ,φ)=r22sin2θcos3φ,
rj()(θ,φ)=r22sinθsin3φ+r13cosθ,
ri(k)(θ,φ)=r22sinθcos2θsin3φ+(r332r51)sin2θcosθ+r13cos3θ,
dk()(θ,φ)=d22sinθcos2θsin3φ+(d33d15)sin2θcosθ+d31cos3θ,
dk(i)(θ,φ)=d22cos2θcos3φ,
d(k)(θ,φ)=d22sin3θsin3φ+(d31+d15)sin2θcosθ+d33cos3θ,
ni(θ,φ)=(no2sin2θ+ne2cos2θ)1/2.
δΔiik(θ,φ)=rii(φ,θ)ni3(φ,θ)+2(ni(φ,θ)1)dk()(φ,θ)=(r22sin3θsin3φ(r13+2r51)sin2θcosθr33cos3θ)(no2sin2θ+ne2cos2θ)3/2+2((no2sin2θ+ne2cos2θ)1/21)(d22sinθcos2θsin3φ+(d33d15)sin2θcosθ+d31cos3θ);
δΔik(i)(θ,φ)=ri(i)(φ,θ)ni3(φ,θ)+2(ni(φ,θ)1)dk(i)(φ,θ)=r22sin2θcos3φ(no2sin2θ+ne2cos2θ)3/22((no2sin2θ+ne2cos2θ)1/21)d22cos2θcos3φ;
δΔjk()(θ,φ)=rj()(φ,θ)nj3(φ,θ)+2(nj(φ,θ)1)dk()(φ,θ)=(r22sinθ  sin3φ+r13cosθ)no3+2(no1)(d22sinθcos2θsin3φ+(d33d15)sin2θcosθ+d31cos3θ);
δΔi(k)(θ,φ)=ri(k)(φ,θ)ni3(φ,θ+90°)+2(ni(φ,θ+90°)1)d(k)(φ,θ)=(r22sinθcos2θsin3φ(r332r51)sin2θcosθr13cos3θ)(no2cos2θ+ne2sin2θ)3/2+2((no2cos2θ+ne2sin2θ)1/21)(d22sin3θsin3φ+(d31+d15)sin2θcosθ+d33cos3θ).

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