Abstract

A systematic algebraic approach is presented as a preferred alternative to an iterative numerical method for deriving expressions for the principal refractive indices and dielectric axes of a nonmagnetic crystal in a uniform electric field. This approach is applicable for an arbitrary field and for any symmetry point group. The results, to the chosen order in the field, are expressed algebraically in terms of measurable crystal tensors. Illustrations are given of the linear electro-optic effect for the point groups 4̅3m, 3m, 4̅2m, and 1 and of the quadratic effect in 4̅2m. The latter serves to highlight a shortcoming in the numerical approach. Comparisons are drawn with numerical results published previously.

© 1998 Optical Society of America

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  1. I. P. Kaminow, E. H. Turner, “Electrooptic light modulators,” Appl. Opt. 5, 1612–1628 (1966).
    [CrossRef] [PubMed]
  2. S. Namba, “Electrooptical effect of zincblende,” J. Opt. Soc. Am. 51, 76–79 (1961).
    [CrossRef]
  3. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  4. M. J. Gunning, R. E. Raab, “Systematic eigenvalue approach to crystal optics: an analytic alternative to the geometric ellipsoid model,” J. Opt. Soc. Am. A 15, 2199–2207 (1998).
    [CrossRef]
  5. D. F. Nelson, “General solution of the electro-optic effect,” J. Opt. Soc. Am. 65, 1144–1151 (1975).
    [CrossRef]
  6. K.-H. Hellwege, A. M. Hellwege, eds., Elastic, Piezoelectric, Pyroelectric, Piezooptic, Electrooptic Constants, and Nonlinear Dielectric Susceptibilities of Crystals, Vol. XI, Landolt–Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1979).
  7. T. A. Maldonado, T. K. Gaylord, “Electrooptic effect calculations: simplified procedure for arbitrary cases,” Appl. Opt. 27, 5051–5066 (1988).
    [CrossRef] [PubMed]
  8. A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A, 1988–1991 (1971).
  9. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, UK, 1982).
  10. R. E. Raab, J. H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non-magnetic chiral medium,” J. Electromagn. Waves Appl. 8, 1073–1089 (1994).
    [CrossRef]
  11. M. J. Gunning, R. E. Raab, “Electric-field-induced optical activity in nonmagnetic crystals,” J. Opt. Soc. Am. B 14, 1–7 (1997).
    [CrossRef]
  12. C. Graham, R. E. Raab, “Eigenvector approach to the evaluation of the Jones N matrices of nonabsorbing crystalline media,” J. Opt. Soc. Am. A 11, 2137–2144 (1994).
    [CrossRef]
  13. A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” Adv. Chem. Phys. 12, 107–142 (1967).
    [CrossRef]
  14. E. B. Graham, R. E. Raab, “Non-reciprocal optical rotation in cubic antiferromagnetics,” Philos. Mag. B 64, 267–274 (1991).
    [CrossRef]
  15. J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, UK, 1985).
  16. A. D. Buckingham, J. A. Pople, “Theoretical studies of the Kerr effect I: deviations from a linear polarization law,” Proc. Phys. Soc. London Sect. A 68, 905–909 (1955).
    [CrossRef]
  17. A. D. Buckingham, H. C. Longuet-Higgins, “The quadrupole moments of dipolar molecules,” Mol. Phys. 14, 63–72 (1968).
    [CrossRef]
  18. R. R. Birss, Symmetry and Magnetism, 2nd ed. (North-Holland, Amsterdam, 1966).
  19. J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
    [CrossRef]
  20. M. V. Klein, Optics (Wiley, New York, 1970).
  21. G. C. Ghosh, G. C. Bhar, “Temperature dispersion in ADP, KDP and KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143–145 (1982).
    [CrossRef]
  22. W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
    [CrossRef]
  23. P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
    [CrossRef]

1998

1997

1995

W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
[CrossRef]

1994

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

C. Graham, R. E. Raab, “Eigenvector approach to the evaluation of the Jones N matrices of nonabsorbing crystalline media,” J. Opt. Soc. Am. A 11, 2137–2144 (1994).
[CrossRef]

R. E. Raab, J. H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non-magnetic chiral medium,” J. Electromagn. Waves Appl. 8, 1073–1089 (1994).
[CrossRef]

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

1991

E. B. Graham, R. E. Raab, “Non-reciprocal optical rotation in cubic antiferromagnetics,” Philos. Mag. B 64, 267–274 (1991).
[CrossRef]

1988

1982

G. C. Ghosh, G. C. Bhar, “Temperature dispersion in ADP, KDP and KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143–145 (1982).
[CrossRef]

1975

1971

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A, 1988–1991 (1971).

1968

A. D. Buckingham, H. C. Longuet-Higgins, “The quadrupole moments of dipolar molecules,” Mol. Phys. 14, 63–72 (1968).
[CrossRef]

1967

A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” Adv. Chem. Phys. 12, 107–142 (1967).
[CrossRef]

1966

1961

1955

A. D. Buckingham, J. A. Pople, “Theoretical studies of the Kerr effect I: deviations from a linear polarization law,” Proc. Phys. Soc. London Sect. A 68, 905–909 (1955).
[CrossRef]

Aillerie, M.

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, UK, 1982).

Bhar, G. C.

G. C. Ghosh, G. C. Bhar, “Temperature dispersion in ADP, KDP and KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143–145 (1982).
[CrossRef]

Birss, R. R.

R. R. Birss, Symmetry and Magnetism, 2nd ed. (North-Holland, Amsterdam, 1966).

Buckingham, A. D.

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A, 1988–1991 (1971).

A. D. Buckingham, H. C. Longuet-Higgins, “The quadrupole moments of dipolar molecules,” Mol. Phys. 14, 63–72 (1968).
[CrossRef]

A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” Adv. Chem. Phys. 12, 107–142 (1967).
[CrossRef]

A. D. Buckingham, J. A. Pople, “Theoretical studies of the Kerr effect I: deviations from a linear polarization law,” Proc. Phys. Soc. London Sect. A 68, 905–909 (1955).
[CrossRef]

Cloete, J. H.

R. E. Raab, J. H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non-magnetic chiral medium,” J. Electromagn. Waves Appl. 8, 1073–1089 (1994).
[CrossRef]

Czapla, Z.

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

Dunn, M. B.

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A, 1988–1991 (1971).

Fontana, M. D.

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

Gaylord, T. K.

Ghosh, G. C.

G. C. Ghosh, G. C. Bhar, “Temperature dispersion in ADP, KDP and KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143–145 (1982).
[CrossRef]

Górski, P.

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

Graham, C.

W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
[CrossRef]

C. Graham, R. E. Raab, “Eigenvector approach to the evaluation of the Jones N matrices of nonabsorbing crystalline media,” J. Opt. Soc. Am. A 11, 2137–2144 (1994).
[CrossRef]

Graham, E. B.

E. B. Graham, R. E. Raab, “Non-reciprocal optical rotation in cubic antiferromagnetics,” Philos. Mag. B 64, 267–274 (1991).
[CrossRef]

Gunning, M. J.

Kaminow, I. P.

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Kucharczyk, W.

W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
[CrossRef]

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

Longuet-Higgins, H. C.

A. D. Buckingham, H. C. Longuet-Higgins, “The quadrupole moments of dipolar molecules,” Mol. Phys. 14, 63–72 (1968).
[CrossRef]

Maldonado, T. A.

Mik, D.

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

Namba, S.

Nelson, D. F.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, UK, 1985).

Pople, J. A.

A. D. Buckingham, J. A. Pople, “Theoretical studies of the Kerr effect I: deviations from a linear polarization law,” Proc. Phys. Soc. London Sect. A 68, 905–909 (1955).
[CrossRef]

Raab, R. E.

M. J. Gunning, R. E. Raab, “Systematic eigenvalue approach to crystal optics: an analytic alternative to the geometric ellipsoid model,” J. Opt. Soc. Am. A 15, 2199–2207 (1998).
[CrossRef]

M. J. Gunning, R. E. Raab, “Electric-field-induced optical activity in nonmagnetic crystals,” J. Opt. Soc. Am. B 14, 1–7 (1997).
[CrossRef]

W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
[CrossRef]

C. Graham, R. E. Raab, “Eigenvector approach to the evaluation of the Jones N matrices of nonabsorbing crystalline media,” J. Opt. Soc. Am. A 11, 2137–2144 (1994).
[CrossRef]

R. E. Raab, J. H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non-magnetic chiral medium,” J. Electromagn. Waves Appl. 8, 1073–1089 (1994).
[CrossRef]

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

E. B. Graham, R. E. Raab, “Non-reciprocal optical rotation in cubic antiferromagnetics,” Philos. Mag. B 64, 267–274 (1991).
[CrossRef]

Salvestrini, J. P.

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

Turner, E. H.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Adv. Chem. Phys.

A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” Adv. Chem. Phys. 12, 107–142 (1967).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

J. P. Salvestrini, M. D. Fontana, M. Aillerie, Z. Czapla, “New material with strong electro-optic effect: rubidium hydrogen selenate (RbHSeO4),” Appl. Phys. Lett. 64, 1920–1922 (1994).
[CrossRef]

IEEE J. Quantum Electron.

G. C. Ghosh, G. C. Bhar, “Temperature dispersion in ADP, KDP and KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143–145 (1982).
[CrossRef]

J. Chem. Soc. A

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A, 1988–1991 (1971).

J. Electromagn. Waves Appl.

R. E. Raab, J. H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non-magnetic chiral medium,” J. Electromagn. Waves Appl. 8, 1073–1089 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Mol. Phys.

A. D. Buckingham, H. C. Longuet-Higgins, “The quadrupole moments of dipolar molecules,” Mol. Phys. 14, 63–72 (1968).
[CrossRef]

Philos. Mag. B

E. B. Graham, R. E. Raab, “Non-reciprocal optical rotation in cubic antiferromagnetics,” Philos. Mag. B 64, 267–274 (1991).
[CrossRef]

Physica B

W. Kucharczyk, M. J. Gunning, R. E. Raab, C. Graham, “Interferometric investigation of the quadratic electro-optic effect in KDP,” Physica B 212, 5–9 (1995).
[CrossRef]

P. Górski, D. Mik, W. Kucharczyk, R. E. Raab, “On the quadratic electro-optic effect in KDP,” Physica B 193, 17–24 (1994).
[CrossRef]

Proc. Phys. Soc. London Sect. A

A. D. Buckingham, J. A. Pople, “Theoretical studies of the Kerr effect I: deviations from a linear polarization law,” Proc. Phys. Soc. London Sect. A 68, 905–909 (1955).
[CrossRef]

Other

K.-H. Hellwege, A. M. Hellwege, eds., Elastic, Piezoelectric, Pyroelectric, Piezooptic, Electrooptic Constants, and Nonlinear Dielectric Susceptibilities of Crystals, Vol. XI, Landolt–Börnstein Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1979).

M. V. Klein, Optics (Wiley, New York, 1970).

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, UK, 1985).

R. R. Birss, Symmetry and Magnetism, 2nd ed. (North-Holland, Amsterdam, 1966).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, UK, 1982).

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Equations (116)

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D = 0 + P ,
H = μ 0 - 1 ,
P α = α α β β
α α β = 0 χ α β .
P α = α α β E β ,
α α β E = α α β + ½ β α β γ E γ .
β α β γ = β β α γ .
n i 2 = 1 + χ ii = 1 + 0 - 1 α ii ,     i = x ,   y ,   z .
0 - 1 α α β E = n x 2 - 1 + b xx γ E γ b xy γ E γ b xz γ E γ b xy γ E γ n y 2 - 1 + b yy γ E γ b yz γ E γ b xz γ E γ b yz γ E γ n z 2 - 1 + b zz γ E γ .
b ijk = 2 0 - 1 β ijk = - n i 2 n j 2 r ijk .
λ = n 2 - 1
| 0 - 1 α α β E - λ I | = n x 2 + b xx γ E γ - n 2 b xy γ E γ b xz γ E γ b xy γ E γ n y 2 + b yy γ E γ - n 2 b yz γ E γ b xz γ E γ b yz γ E γ n z 2 + b zz γ E γ - n 2 = 0 ,
α xx = α yy = α zz ,
b xyz = b yzx = b zxy = b xzy = b zyx = b yxz = b .
n x 2 = n y 2 = n z 2 = n o 2 .
n o 2 - n 2 bE z bE y bE z n o 2 - n 2 bE x bE y bE x n o 2 - n 2 = 0 .
E = 0 ,   0 ,   E .
n o 2 - n 2 bE 0 bE n o 2 - n 2 0 0 0 n o 2 - n 2 = 0 .
n 1 2 = n o 2 - bE ,   r 1 = - 1 ,   1 ,   0 / 2 ,
n 2 2 = n o 2 ,   r 2 = 0 ,   0 ,   1 ,
n 3 2 = n o 2 + bE ,   r 3 = 1 ,   1 ,   0 / 2 .
σ = n 3 2 n 1 2 - n 2 2 n 2 2 n 1 2 - n 3 2 1 / 2 ,   0 ,   ± n 1 2 n 2 2 - n 3 2 n 2 2 ( n 1 2 - n 3 2 1 / 2
= n o 2 + bE / 2 n o 2 1 / 2 ,   0 ,   ± n o 2 - bE / 2 n o 2 1 / 2 .
V = cos - 1 σ · X ˆ = cos - 1 n 3 2 n 1 2 - n 2 2 n 2 2 n 1 2 - n 3 2 1 / 2
= cos - 1 n o 2 + bE / 2 n o 2 1 / 2 .
E = E ,   E ,   0 / 2 .
n o 2 - n 2 0 bE / 2 0 n o 2 - n 2 bE / 2 bE / 2 bE / 2 n o 2 - n 2 = 0 .
n 1 2 = n o 2 - bE ,     r 2 = 1 ,   1 ,   - 2 / 2 ,
n 2 2 = n o 2 ,     r 2 = 1 ,   - 1 ,   0 / 2 ,
n 3 2 = n o 2 + bE ,     r 3 = 1 ,   1 ,   2 / 2 .
E = E ,   E ,   E / 3 .
n 1 2 = n o 2 - bE / 3 ,     r 1 = 1 ,   - 1 ,   0 / 2 ,
n 2 2 = n o 2 - bE / 3 ,     r 2 = 1 ,   1 ,   - 2 / 6 ,
n 3 2 = n o 2 + 2 bE / 3 ,   r 3 = 1 ,   1 ,   1 / 3 .
α xx = α yy ,   α zz ,
b yyy ,   b zzz ,   b xxy = b xyx = b yxx = - b yyy , b xxz = b yyz ,   b xzx = b zxx = b yzy = b zyy .
n x 2 = n y 2 = n o 2 = 1 + o - 1 α xx ,
n z 2 = n e 2 = 1 + o - 1 α zz ,
n o 2 - b 1 E y + b 3 E z - n 2 - b 1 E x b 4 E x - b 1 E x n o 2 + b 1 E y + b 3 E z - n 2 b 4 E y b 4 E x b 4 E y n e 2 + b 2 E z - n 2 = 0 ,
b 1 = b yyy ,   b 2 = b zzz ,   b 3 = b xxz ,   b 4 = b zxx .
E = 0 ,   0 ,   E .
n 1 2 = n o 2 + b 3 E ,     r 1 = 1 ,   0 ,   0 ,
n 2 2 = n o 2 + b 3 E ,     r 2 = 0 ,   1 ,   0 ,
n 3 2 = n e 2 + b 2 E ,     r 3 = 0 ,   0 ,   1 .
E = 0 ,   E ,   0 .
n o 2 - b 1 E - n 2 0 0 0 n o 2 + b 1 E - n 2 b 4 E 0 b 4 E n e 2 - n 2 = 0 .
n o 2 + b 1 E - n 2 b 4 E b 4 E n e 2 - n 2 = 0 ,
n 1 2 = n o 2 - b 1 E ,   r 1 = 1 ,   0 ,   0 ,
n 2 2 = n o 2 + b 1 E ,   r 2 = 0 ,   1 ,   b 4 E n o 2 - n e 2 ,
n 3 2 = n e 2 ,           r 3 = 0 ,   - b 4 E n o 2 - n e 2 ,   1 .
n 1 > n 2 > n 3 .
V = cos - 1 - 2 n e 2 b 1 E n o 2 + b 1 E n o 2 - b 1 E - n e 2 1 / 2 .
E = E ,   0 ,   0 .
n o 2 - n 2 - b 1 E b 4 E - b 1 E n o 2 - n 2 0 b 4 E 0 n e 2 - n 2 = 0 .
n 1 2 - n o 2 + a 1 E + a 2 E 2 = 0 ,
n 2 2 - n o 2 + a 3 E + a 4 E 2 = 0 ,
n 3 2 - n e 2 + a 5 E + a 6 E 2 = 0 .
a 1 = - b 1 ,     a 3 = b 1 ,     a 5 = 0 .
n 1 2 = n o 2 - b 1 E , r 1 = k 1 1 - b 4 2 E 2 b 1 n o 2 - n e 2 ,   1 ,   b 4 E n o 2 - n e 2 2 ,
n 2 2 = n o 2 + b 1 E , r 2 = k 2 - 1 - b 4 2 E 2 b 1 n o 2 - n e 2 ,   1 ,   - b 4 E n o 2 - n e 2 2 ,
n 3 2 = n e 2 , r 3 = - b 4 E n o 2 - n e 2 ,   0 ,   1 ,
k 1 = 1 - b 4 2 E 2 b 1 n o 2 - n e 2 - 1 / 2 ,
k 2 = 1 + b 4 2 E 2 b 1 n o 2 - n e 2 - 1 / 2
α xx = α yy ,     α zz ,
b xyz = b yxz ,     b xzy = b yzx = b zxy = b zyx .
n o 2 - n 2 b 1 E z b 2 E y b 1 E z n o 2 - n 2 b 2 E x b 2 E y b 2 E x n e 2 - n 2 = 0 .
b 1 = b xyz ,     b 2 = b xzy .
E = 0 ,   0 ,   E .
n 1 2 = n o 2 - b 1 E ,     r 1 = 1 ,   - 1 ,   0 / 2 ,
n 2 2 = n o 2 + b 1 E ,     r 2 = 1 ,   1 ,   0 / 2 ,
n 3 2 = n e 2 ,     r 3 = 0 ,   0 ,   1 .
E = 0 ,   E ,   0 .
n o 2 - n 2 0 b 2 E 0 n o 2 - n 2 0 b 2 E 0 n e 2 - n 2 = 0 .
n 1 2 = n o 2 ,     r 1 = 1 ,   0 ,   b 2 E n o 2 - n e 2 ,
n 2 2 = n o 2 ,     r 2 = 0 ,   1 ,   0 ,
n 3 2 = n e 2 ,     r 3 = - b 2 E n o 2 - n e 2 ,   0 ,   1 .
E = E ,   0 ,   0 .
n 1 2 = n o 2 ,     r 1 = 1 ,   0 ,   0 ,
n 2 2 = n o 2 ,     r 2 = 0 ,   1 ,   b 2 E n o 2 - n e 2 ,
n 3 2 = n e 2 ,     r 3 = 0 ,   - b 2 E n o 2 - n e 2 ,   1 .
E = E ,   E ,   0 / 2 .
n o 2 - n 2 0 b 2 E / 2 0 n o 2 - n 2 b 2 E / 2 b 2 E / 2 b 2 E / 2 n e 2 - n 2 = 0 .
n 1 2 = n o 2 ,     r 1 = 1 ,   - 1 ,   0 / 2 ,
n 2 2 = n o 2 ,     r 2 = 1 ,   1 ,   2   b 2 E n o 2 - n e 2 2 ,
n 3 2 = n e 2 ,   r 3 = - b 2 E n o 2 - n e 2 ,   - b 2 E n o 2 - n e 2 ,   2 2 .
E = E ,   E ,   E / 3 .
n o 2 - n 2 b 1 E / 3 b 2 E / 3 b 1 E / 3 n o 2 - n 2 b 2 E / 3 b 2 E / 3 b 2 E / 3 n e 2 - n 2 = 0 .
a 1 = - b 1 / 3 ,     a 3 = b 1 / 3 ,     a 5 = 0 .
n 1 2 = n o 2 - b 1 E / 3 , r 1 = 1 ,   - 1 ,   0 / 2 ,
n 2 2 = n o 2 + b 1 E / 3 , r 2 = 1 ,   1 ,   2 b 2 E 3 n o 2 - n e 2 2 ,
n 3 2 = n e 2 , r 3 = - b 2 E 3 n o 2 - n e 2 ,   - b 2 E 3 n o 2 - n e 2 ,   1 ,
V = cos - 1 n o 2 - b 1 E / 3 - 2 b 1 E / 3 n o 2 + b 1 E / 3 n o 2 - b 1 E / 3 - n e 2 1 / 2 .
E = 0 ,   E ,   0 ,
α xx α yy α zz ,
n x 2 + b 1 E - n 2 b 4 E b 5 E b 4 E n y 2 + b 2 E - n 2 b 6 E b 5 E b 6 E n z 2 + b 3 E - n 2 = 0 .
b 1 = b xxy ,     b 2 = b yyy ,     b 3 = b zzy ,   b 4 = b xyy ,     b 5 = b xzy ,     b 6 = b yzy .
a 1 = b 1 ,     a 2 = b 2 ,     a 3 = b 3 .
n 1 2 = n x 2 + b 1 E ,     r 1 = 1 ,   b 4 E n x 2 - n y 2 ,   - b 5 E n z 2 - n x 2 ,
n 2 2 = n y 2 + b 2 E ,     r 2 = - b 4 E n x 2 - n y 2 ,   1 ,   b 6 E n y 2 - n z 2 ,
n 3 2 = n z 2 + b 3 E ,     r 3 = b 5 E n z 2 - n x 2 ,   - b 6 E n y 2 - n z 2 ,   1 .
α α β E = α α β + ½ β α β γ E γ + γ α β γ δ E γ E δ .
γ α β γ δ = γ β α γ δ = γ α β δ γ .
n x 2 + b xx γ E γ + c xx γ δ E γ E δ - n 2 b xy γ E γ + c xy γ δ E γ E δ b xz γ E γ + c xz γ δ E γ E δ b xy γ E γ + c xy γ δ E γ E δ n y 2 + b yy γ E γ + c yy γ δ E γ E δ - n 2 b yz γ E γ + c yz γ δ E γ E δ b xz γ E γ + c xz γ δ E γ E δ b yz γ E γ + c yz γ δ E γ E δ n z 2 + b zz γ E γ + c zz γ δ E γ E δ - n 2 = 0 ,
c ijkl = 6 0 - 1 γ ijkl = - n i 2 n j 2 g ijkl ,
c xxxx = c yyyy ,   c xxyy = c yyxx ,   c xyxy ,   c zzxx = c zzyy .
n o 2 + c 1 E 2 - n 2 c 2 E 2 b 2 E / 2 c 2 E 2 n o 2 + c 1 E 2 - n 2 b 2 E / 2 b 2 E / 2 b 2 E / 2 n e 2 + c 3 E 2 - n 2 = 0 ,
c 1 = ½ c xxxx + c xxyy ,   c 2 = c xyxy ,   c 3 = c zzxx .
n 1 2 = n o 2 + b 2 2 n o 2 - n e 2 + c 1 + c 2 E 2 , r 1 = k 1 ,   1 ,   2   b 2 E n o 2 - n e 2 2 ,
n 2 2 = n o 2 + c 1 - c 2 E 2 ,     r 2 = - 1 ,   1 ,   0 / 2 ,
n 3 2 = n e 2 + c 3 - b 2 2 n o 2 - n e 2 E 2 , r 3 = k - b 2 E n o 2 - n e 2 ,   - b 2 E n o 2 - n e 2 ,   2 2 ,
k = 1 + b 2 2 E 2 n o 2 - n e 2 2 - 1 / 2
n o = 1.5075 ,   n e = 1.4670 ,   r xzy = 8.7 × 10 - 12   m   V - 1 , g xxxx = - 4.0 × 10 - 20   m 2   V - 2 ,   g xxyy = 0.2 × 10 - 20   m 2   V - 2 , g zzxx = - 0.6 × 10 - 20   m 2   V - 2 ,   g xyxy = 1.4 × 10 - 20   m 2   V - 2 .
b 2 2 / n o 2 - n e 2 = 1.5 ,   c 1 = 9.8 ,   c 2 = - 7.2 ,   c 3 = 2.8 .
n 1 = 3.3895278 ,     3.3895161 , n 2 = 3.3895000 ,     3.3895000 , n 3 = 3.3894722 ,     3.3894839 .
n o = 2.2885 ,     n e = 2.2014 , r xxz = 8.6 × 10 - 12 mV - 1 ,     r zzz = 30.8 × 10 - 12 mV - 1 .
n 1 = n 2 = 2.2884484 ,     n 3 = 2.2012357 .

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