Abstract

To aid in the design of electrooptic devices, a simplified procedure is introduced for determining the principal indices of refraction and the orientations of the index ellipsoid when an electric field is applied. The approach employs the general Jacobi method, and it is applicable to an arbitrary isotropic, uniaxial, or biaxial crystal class with an arbitrary direction of applied field. It includes a straightforward approach for labeling the new principal dielectric axes so as to produce the minimum global rotation of the ellipsoid from the zero-field principal dielectric axes. The necessary calculations can be easily implemented with a pocket calculator and are often found to be more accurate than those obtained with larger computers using standard library math packages. Furthermore, for analyzing device performance, simple analytic expressions for the orientations of the fast and slow polarization axes and the fast and slow indices of refraction are derived for a given arbitrary direction of optical propagation and any arbitrary direction of applied electric field. The calculational procedure is applicable to linear and quadratic electrooptic effects and the photoelastic effect in isotropic, uniaxial, and biaxial crystals. Illustrative examples for GaAs, LiNbO3, and KDP are presented.

© 1988 Optical Society of America

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References

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  1. See, for example, Special Issue on Electrooptic Materials and Devices, IEEE J. Quantum Electron. QE-23 (Dec.1987).
  2. W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512 (1987).
    [CrossRef]
  3. E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229 (1987).
    [CrossRef]
  4. H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524 (1987).
    [CrossRef]
  5. See, for example, Special Issue on Optical ComputingProc. IEEE 72 (1984).
  6. See, for example, “Special Feature on Integrated Optics: Evolution and Prospects,” Opt. News 14 (Feb.1988).
  7. T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59 (1987).
    [CrossRef]
  8. C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
    [CrossRef]
  9. C. L. Chang, C. S. Tsai, “Electro-Optic Analog-to-Digital Conversion Using Channel Waveguide Fabry-Perot Modulator Array,” Appl. Phys. Lett. 43, 22 (1983).
    [CrossRef]
  10. C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Processing,” Proc. IEEE 72, 942 (1984).
    [CrossRef]
  11. D. Eimerl, “Symmetry of the Electro-Optic Effect,” IEEE J. Quantum Electron. QE-23, 2104 (1987).
    [CrossRef]
  12. J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).
  13. W. L. Bond, “The Mathematics of the Physical Properties of Crystals,” Bell Syst. Tech. J. 23, 1 (1943).
  14. D. E. Sands, Vectors and Tensors in Crystallography (Addison-Wesley, Reading, MA, 1982).
  15. D. R. Hartree, Numerical Analysis (Clarendon, Oxford, 1952).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  17. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).
  18. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).
  19. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).
  20. A. I. Borisenko, I. E. Tarapov, Vector and Tensor Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1968), R. A. Silverman, translator and editor.
  21. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).
  22. T. C. Phemister, “Fletcher’s Indicatrix and the Electromagnetic Theory of Light,” Am. Mineral. 39, 173 (1954).
  23. W. A. Wooster, A Textbook on Crystal Physics (Cambridge University Press, Cambridge, 1938).
  24. P. Gay, An Introduction to Crystal Optics (Longsmans, Green, & Co., London, 1967).
  25. F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart, & Winston, New York, 1961).
  26. P. F. Kerr, Optical Mineralogy (McGraw-Hill, New York, 1959).
  27. A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).
  29. J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation (Springer-Verlag, New York, 1971).
    [CrossRef]
  30. G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U.P., Baltimore, MD, 1983).
  31. B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
  32. F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).
  33. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1981).
  34. R. P. Paul, Robot Manipulators (MIT Press, Cambridge, MA, 1981).
  35. S. Namba, “Electro-Optical Effect of Zincblende,” J. Opt. Soc. Am. 51, 76 (1961).
    [CrossRef]
  36. P. V. Lenzo, E. G. Spencer, K. Nassau, “Electro-Optic Coefficients in Single-Domain Ferroelectric Lithium Niobate,” J. Opt. Soc. Am. 56, 633 (1966).
    [CrossRef]
  37. B. H. Billings, “The Electro-Optic Effect in Uniaxial Crystals of the type X H2PO4. I. Theoretical,” J. Opt. Soc. Am. 39, 797 (1949).
    [CrossRef]
  38. IMSL Library Reference Manual (International Mathematical & Statistical Libraries, Inc., Houston TX, 1980), edition 9.2.
  39. R. J. Willard, “A Re-Examination of Some Mathematical Relations in Anisotropic Substances,” Trans. Am. Microsc. Soc. 80, 191 (1961).
    [CrossRef]

1988 (1)

See, for example, “Special Feature on Integrated Optics: Evolution and Prospects,” Opt. News 14 (Feb.1988).

1987 (6)

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59 (1987).
[CrossRef]

See, for example, Special Issue on Electrooptic Materials and Devices, IEEE J. Quantum Electron. QE-23 (Dec.1987).

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512 (1987).
[CrossRef]

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229 (1987).
[CrossRef]

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524 (1987).
[CrossRef]

D. Eimerl, “Symmetry of the Electro-Optic Effect,” IEEE J. Quantum Electron. QE-23, 2104 (1987).
[CrossRef]

1984 (2)

C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Processing,” Proc. IEEE 72, 942 (1984).
[CrossRef]

See, for example, Special Issue on Optical ComputingProc. IEEE 72 (1984).

1983 (2)

C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
[CrossRef]

C. L. Chang, C. S. Tsai, “Electro-Optic Analog-to-Digital Conversion Using Channel Waveguide Fabry-Perot Modulator Array,” Appl. Phys. Lett. 43, 22 (1983).
[CrossRef]

1966 (1)

1961 (2)

S. Namba, “Electro-Optical Effect of Zincblende,” J. Opt. Soc. Am. 51, 76 (1961).
[CrossRef]

R. J. Willard, “A Re-Examination of Some Mathematical Relations in Anisotropic Substances,” Trans. Am. Microsc. Soc. 80, 191 (1961).
[CrossRef]

1954 (1)

T. C. Phemister, “Fletcher’s Indicatrix and the Electromagnetic Theory of Light,” Am. Mineral. 39, 173 (1954).

1949 (1)

1943 (1)

W. L. Bond, “The Mathematics of the Physical Properties of Crystals,” Bell Syst. Tech. J. 23, 1 (1943).

Billings, B. H.

Bloss, F. D.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart, & Winston, New York, 1961).

Bond, W. L.

W. L. Bond, “The Mathematics of the Physical Properties of Crystals,” Bell Syst. Tech. J. 23, 1 (1943).

Borisenko, A. I.

A. I. Borisenko, I. E. Tarapov, Vector and Tensor Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1968), R. A. Silverman, translator and editor.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Brackett, C. A.

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512 (1987).
[CrossRef]

Busch, J. R.

C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
[CrossRef]

Chang, C. L.

C. L. Chang, C. S. Tsai, “Electro-Optic Analog-to-Digital Conversion Using Channel Waveguide Fabry-Perot Modulator Array,” Appl. Phys. Lett. 43, 22 (1983).
[CrossRef]

Eimerl, D.

D. Eimerl, “Symmetry of the Electro-Optic Effect,” IEEE J. Quantum Electron. QE-23, 2104 (1987).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).

Gay, P.

P. Gay, An Introduction to Crystal Optics (Longsmans, Green, & Co., London, 1967).

Gaylord, T. K.

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59 (1987).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1981).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U.P., Baltimore, MD, 1983).

Hartree, D. R.

D. R. Hartree, Numerical Analysis (Clarendon, Oxford, 1952).

Hopf, F. A.

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

Kaminow, I. P.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).

Kenan, R. P.

C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
[CrossRef]

Kerr, P. F.

P. F. Kerr, Optical Mineralogy (McGraw-Hill, New York, 1959).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

Lenzo, P. V.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

Namba, S.

Nassau, K.

Neyer, A.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229 (1987).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

Parlett, B. N.

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).

Paul, R. P.

R. P. Paul, Robot Manipulators (MIT Press, Cambridge, MA, 1981).

Phemister, T. C.

T. C. Phemister, “Fletcher’s Indicatrix and the Electromagnetic Theory of Light,” Am. Mineral. 39, 173 (1954).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).

Reinsch, C.

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation (Springer-Verlag, New York, 1971).
[CrossRef]

Sands, D. E.

D. E. Sands, Vectors and Tensors in Crystallography (Addison-Wesley, Reading, MA, 1982).

Spencer, E. G.

Stegeman, G. I.

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

Tarapov, I. E.

A. I. Borisenko, I. E. Tarapov, Vector and Tensor Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1968), R. A. Silverman, translator and editor.

Taylor, H. F.

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524 (1987).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).

Tomlinson, W. J.

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512 (1987).
[CrossRef]

Tsai, C. S.

C. L. Chang, C. S. Tsai, “Electro-Optic Analog-to-Digital Conversion Using Channel Waveguide Fabry-Perot Modulator Array,” Appl. Phys. Lett. 43, 22 (1983).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U.P., Baltimore, MD, 1983).

Verber, C. M.

C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Processing,” Proc. IEEE 72, 942 (1984).
[CrossRef]

C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
[CrossRef]

Verriest, E. I.

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59 (1987).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).

Voges, E.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229 (1987).
[CrossRef]

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation (Springer-Verlag, New York, 1971).
[CrossRef]

Willard, R. J.

R. J. Willard, “A Re-Examination of Some Mathematical Relations in Anisotropic Substances,” Trans. Am. Microsc. Soc. 80, 191 (1961).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Wooster, W. A.

W. A. Wooster, A Textbook on Crystal Physics (Cambridge University Press, Cambridge, 1938).

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

Am. Mineral. (1)

T. C. Phemister, “Fletcher’s Indicatrix and the Electromagnetic Theory of Light,” Am. Mineral. 39, 173 (1954).

Appl. Phys. Lett. (1)

C. L. Chang, C. S. Tsai, “Electro-Optic Analog-to-Digital Conversion Using Channel Waveguide Fabry-Perot Modulator Array,” Appl. Phys. Lett. 43, 22 (1983).
[CrossRef]

Bell Syst. Tech. J. (1)

W. L. Bond, “The Mathematics of the Physical Properties of Crystals,” Bell Syst. Tech. J. 23, 1 (1943).

Computer (1)

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

See, for example, Special Issue on Electrooptic Materials and Devices, IEEE J. Quantum Electron. QE-23 (Dec.1987).

D. Eimerl, “Symmetry of the Electro-Optic Effect,” IEEE J. Quantum Electron. QE-23, 2104 (1987).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (2)

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229 (1987).
[CrossRef]

C. M. Verber, R. P. Kenan, J. R. Busch, “Design and Performance of an Integrated Optical Digital Correlator,” IEEE/OSA J. Lightwave Technol. LT-1, 256 (1983).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. News (1)

See, for example, “Special Feature on Integrated Optics: Evolution and Prospects,” Opt. News 14 (Feb.1988).

Proc. IEEE (4)

C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Processing,” Proc. IEEE 72, 942 (1984).
[CrossRef]

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524 (1987).
[CrossRef]

See, for example, Special Issue on Optical ComputingProc. IEEE 72 (1984).

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512 (1987).
[CrossRef]

Trans. Am. Microsc. Soc. (1)

R. J. Willard, “A Re-Examination of Some Mathematical Relations in Anisotropic Substances,” Trans. Am. Microsc. Soc. 80, 191 (1961).
[CrossRef]

Other (22)

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

IMSL Library Reference Manual (International Mathematical & Statistical Libraries, Inc., Houston TX, 1980), edition 9.2.

W. A. Wooster, A Textbook on Crystal Physics (Cambridge University Press, Cambridge, 1938).

P. Gay, An Introduction to Crystal Optics (Longsmans, Green, & Co., London, 1967).

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart, & Winston, New York, 1961).

P. F. Kerr, Optical Mineralogy (McGraw-Hill, New York, 1959).

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U.P., New York, 1986).

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation (Springer-Verlag, New York, 1971).
[CrossRef]

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U.P., Baltimore, MD, 1983).

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1981).

R. P. Paul, Robot Manipulators (MIT Press, Cambridge, MA, 1981).

D. E. Sands, Vectors and Tensors in Crystallography (Addison-Wesley, Reading, MA, 1982).

D. R. Hartree, Numerical Analysis (Clarendon, Oxford, 1952).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).

A. I. Borisenko, I. E. Tarapov, Vector and Tensor Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1968), R. A. Silverman, translator and editor.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).

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Figures (9)

Fig. 1
Fig. 1

Index ellipsoids for various crystal symmetries. For isotropic crystals, the surface is a sphere. For uniaxial crystals, it is an ellipsoid of revolution. For biaxial crystals, it is a general ellipsoid.

Fig. 2
Fig. 2

Geometric relationship of the electric quantities D and E and magnetic quantities B and H to the wavevector k and to the Poynting vector S for an anisotropic dielectric.

Fig. 3
Fig. 3

(a) The index ellipsoid cross section (crosshatched) that is normal to wavevector k. The principal axes of the crosshatched ellipse represent the directions of the allowed polarizations D1 and D2. D1, D2, and k are orthogonal to each other. (b) The plane through the index ellipsoid that contains D1 (or D2) and k. This plane also contains E1 (or E2) and S1 (or S2). The length AC represents the ray velocity index.

Fig. 4
Fig. 4

(a) Cross-section ellipse transverse to the wavevector k along the y axis with no field applied to the crystal. (b) With an applied electric field the index ellipsoid is reoriented and the eigenpolarizations in the plane transverse to k are rotated, indicated by x‴ and y‴.

Fig. 5
Fig. 5

Coordinate system (x″,y″,z″) associated with the wavevector direction and its polar coordinate relationship (ϕk,θk) to the unperturbed principal dielectric axis coordinate system (x,y,z).

Fig. 6
Fig. 6

Coordinate system (x‴,y‴) associated with the fast and slow axes (directions of D1 and D2) and its relationship to the wavevector coordinate system (x″ and y″). The wavevector k and the axes z″ and z‴ are normal to the plane of the figure.

Fig. 7
Fig. 7

Transformation of the (x,y,z) coordinate system to the (x′,y′,z′) coordinate system by one clockwise global rotation of angle Φ about the global rotation axis R viewing down the axis toward the origin. The case shown is for a 4 ¯ 3 crystal (such as GaAs) with an electric field applied along [1 1 0].

Fig. 8
Fig. 8

Jacobi rotations (a) by an angle ϕ in the (x,y) plane about the +z axis, (b) by an angle θ in the (x,z) plane about the −y axis, and (c) by an angle ψ in the (y,z) plane about the +x axis. Angles are positive as measured counterclockwise.

Fig. 9
Fig. 9

(a) Two circular cross sections through the index ellipsoid together with the (y′,z′) principal plane. Normals to the circular cross sections are optic axes (dashed lines), and these make angles ± V with respect to the z′ axis. For the case shown, nz> nx> ny. (b) The (y′,z′) plane of the index ellipsoid showing the optic axes and the circular cross sections (bold lines).

Tables (4)

Tables Icon

Table I Calculated Values for Normalized Eigenvector [ - 1 / 2 , + 1 / 2 , 0 ] T for GaAs for an Applied Field of Ex = Ey = Ez = 106 V/m Calculated by Various Methods

Tables Icon

Table II Calculated Values for Normalized Eigenvector [ + 1 / 3 , 1 / 3 , + 1 / 3 ] T for GaAs for an Applied Field of Ex = Ey = Ez = 105 V/m Calculated by the General Jacobi Method Presented In this Paper with Various Levels of Precision

Tables Icon

Table III Calculated Values for the Normalized Eigenvector [ - 1 / 6 , - 1 / 6 1 / 1.5 ] T for GaAs for an Applied Field of Ex = Ey = Ez Calculated Using a 10-Digit Precision Calculator for Various Electric Field Amplitudes

Tables Icon

Table IV Calculated Values for the Normalized Eigenvector [ - 1 / 6 - 1 / 6 1 / 1.5 ] T for GaAs for an Applied Field of Ex = Ey = Ez Calculated Using a 12-Digit Precision Calculator for Various Electric Field Amplitudes

Equations (63)

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

[ D x D y D z ] = [ x 0 0 0 y 0 0 0 z ] [ E x E y E z ] ,
ω e = 1 2 E · D = 1 2 i j E i i j E j = 1 2 0 E T [ ɛ ] E = ½ 0 ( E x 2 ɛ x + E y 2 ɛ y + E z 2 ɛ z ) ,
( x 2 / n x 2 ) + ( y 2 / n y 2 ) + ( z 2 / n z 2 ) = 1.
Δ ( 1 / n 2 ) i = j r i j E j             i = 1 , , 6 j = x , y , z = 1 , 2 , 3 ,
[ Δ ( 1 / n 2 ) 1 Δ ( 1 / n 2 ) 2 Δ ( 1 / n 2 ) 3 Δ ( 1 / n 2 ) 4 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 6 ] = [ r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 r 41 r 42 r 43 r 51 r 52 r 53 r 61 r 62 r 63 ] [ E x E y E z ] .
[ 1 / n 2 ] = [ 1 / n x 2 + Δ ( 1 / n 2 ) 1 Δ ( 1 / n 2 ) 6 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 6 1 / n y 2 + Δ ( 1 / n 2 ) 2 Δ ( 1 / n 2 ) 4 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 4 1 / n z 2 + Δ ( 1 / n 2 ) 3 ] .
( 1 / n 2 ) 1 x 2 + ( 1 / n 2 ) 2 y 2 + ( 1 / n 2 ) 3 z 2 + 2 ( 1 / n 2 ) 4 y z + 2 ( 1 / n 2 ) 5 x z + 2 ( 1 / n 2 ) 6 x y = 1 ,
( x / n x ) 2 + ( y / n y ) 2 + ( z / n z ) 2 = 1.
[ 1 / n 2 ] m = [ a ] m [ 1 / n 2 ] m - 1 [ a ] m T .
tan ( 2 Ω ) = 2 ( 1 / n 2 ) i j / { ( 1 / n 2 ) i i - ( 1 / n 2 ) j j } ,             i , j = 1 , 2 , 3 ,
[ cos Ω sin Ω - sin Ω cos Ω ] .
[ a ] T = [ a ] 1 T [ a ] 2 T [ a ] n T .
[ a ] 1 T [ a ] 2 T [ a ] n T [ 1 0 0 ] = [ a 11 a 12 a 13 ] ,
[ [ a ] - [ I ] ] R = 0 ,
[ a ] = [ cos Φ sin Φ 0 - sin Φ cos Φ 0 0 0 1 ] .
i = 1 3 a i i = 1 + 2 cos Φ ,
A x 2 + B y 2 + C z 2 + 2 F y z + 2 G x z + 2 H x y = 1 ,
x = x cos θ k cos φ k - y sin φ k + z sin θ k cos φ k , y = x cos θ k sin φ k + y cos φ k + z sin θ k sin φ k , z = - x sin θ k + z cos θ k .
A x 2 + B y 2 + 2 H x y = 1 ,
A = cos 2 θ k [ A cos 2 φ k + B sin 2 φ k + H sin 2 φ k ] + C sin 2 θ k - sin 2 θ k [ F sin φ k + G cos φ k ] , B = A sin 2 φ k + B cos 2 φ k - H sin 2 φ k , 2 H = sin 2 φ k cos θ k [ B - A ] + 2 sin θ k [ G sin φ k - F cos φ k ] + 2 H cos θ k cos 2 φ k .
β 1 = ½ tan - 1 [ 2 H / ( A - B ) ] .
x = x cos β 1 - y sin β 1 , y = x sin β 1 - y cos β 1 .
n x = [ A cos 2 β 1 + B sin 2 β 1 + 2 H cos β 1 sin β 1 ] - 1 / 2 .
n y = [ A sin 2 β 1 + B cos 2 β 1 - 2 H cos β 1 sin β 1 ] - 1 / 2 .
1 / n 0 2 ( x 2 + y 2 + z 2 ) + 2 r 41 ( E x y z + E y x z + E z x y ) = 1.
[ 1 / n 2 ] = [ 0.087041972 8.256108 × 10 - 7 8.256108 × 10 - 7 8.256108 × 10 - 7 0.087041972 8.256108 × 10 - 7 8.256108 × 10 - 7 8.256108 × 10 - 7 0.087041972 ] .
( 1 / n 2 ) 11 = 0.087042797             ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = 0 , ( 1 / n 2 ) 22 = 0.087041147             ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = 1.167590 × 10 - 6 ( 1 / n 2 ) 33 = 0.087041972             ( 1 / n 2 ) 23 = ( 1 / n 2 ) 23 = 0.
( 1 / n 2 ) 11 = λ 1 = 0.087043622             ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = 0 , ( 1 / n 2 ) 22 = λ 2 = 0.087041147             ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = 0 , ( 1 / n 2 ) 33 = λ 3 = 0.087041147             ( 1 / n 2 ) 23 = ( 1 / n 2 ) 32 = 0 ,
n x = 1 / λ 1 = 3.3894679 , n y = 1 / λ 2 = 3.3895161 , n z = 1 / λ 3 = 3.3895161.
x = [ 0.5773352 0.5773352 0.5773803 ] T , y = [ - 0.7071068 0.7071068 0 ] T , z = [ - 0.4082696 - 0.4082696 0.8164753 ] T ,
[ a ] = [ 0.5773503 0.5773503 0.5773503 - 0.7071068 0.7071068 0 - 0.4082483 - 0.4082483 0.8164966 ] .
R = [ 0.2445131 - 0.5903068 0.7692537 ] T .
n x = 1 / λ 1 = 3.3894839 ,             x = [ 1 / 2 1 / 2 0 ] T , n y = 1 / λ 2 = 3.3895161 ,             y = [ - 1 / 2 1 / 2 0 ] T , n z = 1 / λ 3 = 3.3895000 ,             z = [ 0 0 1 ] T .
n x = 3.3894722 ,             x = [ 1 / 2 1 / 2 1 / 2 ] T , n y = 3.3895005 ,             y = [ - 1 / 2 1 / 2 0 ] T , n z = 3.3895279 ,             z = [ - 1 / 2 1 / 2 1 / 2 ] T .
( 1 / n o 2 - r 22 E y + r 13 E z ) x 2 + ( 1 / n o 2 + r 22 E y + r 13 E z ) y 2 + ( 1 / n E + r 33 E z ) z 2 + 2 r 42 E y y z + 2 r 42 E x x z - 2 r 22 E x x y = 1 ,
n x = 2.288520374 ,             x = [ 1 0 0 ] T , n y = 2.288479930 ,             y = [ 0 0.999998348 - 0.001817593 ] T , n z = 2.201399727 ,             z = [ 0 0.001817593 0.999998348 ] T .
[ 1 / n 2 ] = [ 0.190940549 - 3.4 × 10 - 6 2.8 × 10 - 5 - 3.4 × 10 - 6 0.190940549 0 2.8 × 10 - 5 0 0.206348861 ] .
( 1 / n 2 ) 11 = 0.190940498             ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = - 3.399994 × 10 - 6 , ( 1 / n 2 ) 22 = 0.190940549             ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = 0 , ( 1 / n 2 ) 33 = 0.206348912             ( 1 / n 2 ) 23 = ( 1 / n 2 ) 32 = 6.178453 × 10 - 9 .
( 1 / n 2 ) 11 = 0.190937123             ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = 0 , ( 1 / n 2 ) 22 = 0.190943924             ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = 4.352451 × 10 - 9 , ( 1 / n 2 ) 33 = 0.206348912             ( 1 / n 2 ) 23 = ( 1 / n 2 ) 32 = 4.385140 × 10 - 9 .
n x = 2.288520530             x = [ 0.7097461 0.7044564 - 0.0012895 ] T , n y = 2.288479774             y = [ - 07044552 0.7097547 0.0012804 ] T , n z = 2.201339727             z = [ 0.0018172 0 0.9999983 ] T .
x 2 / n o 2 + y 2 / n o 2 + z 2 / n E 2 + 2 r 41 E x y z + 2 r 41 E y x z + 2 r 63 E z x y = 1.
[ 1 / n 2 ] = [ 0.439799689 0 6.081118 × 10 - 6 0 0.439799689 6.081118 × 10 - 6 6.081118 × 10 - 6 6.081118 × 10 - 6 0.464411505 ] .
( 1 / n 2 ) 11 = 0.439799687 ( 1 / n 2 ) 22 = 0.439799689 ( 1 / n 2 ) 33 = 0.464411507 ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = - 1.50253003 × 10 - 9 , ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = 0 , ( 1 / n 2 ) 23 = ( 1 / n 2 ) 32 = 6.0811178 × 10 - 6 .
( 1 / n 2 ) 11 = 0.439799687 ( 1 / n 2 ) 22 = 0.439799687 ( 1 / n 2 ) 33 = 0.464411508 ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = - 1.5025299 × 10 - 9 , ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = - 3.71246916 × 10 - 11 , ( 1 / n 2 ) 23 = 0.
( 1 / n 2 ) 11 = 0.439799689 ( 1 / n 2 ) 22 = 0.439799686 ( 1 / n 2 ) 33 = 0.464411508 ( 1 / n 2 ) 12 = ( 1 / n 2 ) 21 = 0 , ( 1 / n 2 ) 13 = ( 1 / n 2 ) 31 = - 2.62511212 × 10 - 13 , ( 1 / n 2 ) 23 = ( 1 / n 2 ) 32 = - 2.62511211 × 10 - 13 .
n x = 1.507899998 n y = 1.507900003 n z = 1.467399995 x = [ 0.7071068 - 0.7071068 0 ] T y = [ 0.7071067 0.7071068 - 0.0003494 ] T , z = [ 0.0002471 0.0002471 0.9999999 ] T .
x 2 / n o 2 + y 2 / n o 2 + z 2 / n E 2 + 2 r 63 E z x y = 1.
1 / n o 2 ( x 2 + y 2 ) + 2 r 63 E z x y = 1 ,
n x = [ 1 / n o 2 + r 63 E z ] - 1 / 2 = n o - ½ n 0 3 r 63 E z ,
n y = [ 1 / n o 2 + r 63 E z ] - 1 / 2 = n o + ½ n 0 3 r 63 E z .
ϕ = ½ tan - 1 [ 2 A 12 / ( A 11 - A 22 ) ] .
[ a ] ϕ = [ cos ϕ sin ϕ 0 - sin ϕ cos ϕ 0 0 0 1 ] .
A ϕ 11 = A 11 cos 2 ϕ + A 22 sin 2 ϕ + 2 A 12 cos ϕ sin ϕ , A ϕ 22 = A 11 sin 2 ϕ + A 22 cos 2 ϕ - 2 A 12 cos ϕ sin ϕ , A ϕ 33 = A 33 , A ϕ 12 = ( A 22 - A 11 ) cos ϕ sin ϕ + A 12 ( cos 2 ϕ - sin 2 ϕ ) = A 21 = 0 , A ϕ 13 = A 13 cos ϕ + A 23 sin ϕ = A 31 , A 23 = - A 13 sin ϕ + A 23 cos ϕ = A 32 .
θ = ½ tan - 1 [ 2 A 13 / ( A 11 - A 33 ) ] .
[ a ] θ = [ cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ ] .
A θ 11 = A 11 cos 2 θ + A 33 sin 2 θ + 2 A 13 cos θ sin θ , A θ 22 = A 22 , A θ 33 = A 11 sin 2 θ + A 33 cos 2 θ - 2 A 13 cos θ sin θ , A θ 12 = A 12 cos θ + A 23 sin θ = A 21 , A θ 13 = ( A 33 - A 11 ) cos θ sin θ + A 13 ( cos 2 θ - sin 2 θ ) = A 31 = 0 , A θ 23 = - A 12 sin θ + A 23 cos θ = A 32 .
ψ = ½ tan - 1 [ 2 A 23 / ( A 22 - A 33 ) ] .
[ a ] ψ = [ 1 0 0 0 cos ψ sin ψ 0 - sin ψ cos ψ ] ,
A ψ 11 = A 11 , A ψ 22 = A 22 cos 2 ψ + A 33 sin 2 ψ + 2 A 23 cos ψ sin ψ , A ψ 33 = A 22 sin 2 ψ + A 33 cos 2 ψ - 2 A 23 cos ψ sin ψ , A ψ 12 = A 12 cos ψ + A 13 sin ψ = A 21 , A ψ 13 = - A 12 sin ψ + A 13 cos ψ = A 31 , A ψ 23 = ( A 33 - A 22 ) cos ψ sin ψ + A 23 ( cos 2 ψ - sin 2 ψ ) = A 32 = 0.
V = ± cos - 1 { ( n α / n β ) [ ( n γ 2 - n β 2 ) ] / ( n γ 2 - n α 2 ) ] 1 / 2 } .
[ a ] = [ cos ω sin ω 0 - sin ω cos ω 0 0 0 1 ] [ cos η 0 - sin η 0 1 0 sin η 0 cos η ] [ cos ζ sin ζ 0 - sin ζ cos ζ 0 0 0 1 ] .
[ cos ω - sin ω 0 sin ω cos ω 0 0 0 1 ] [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] = [ cos η 0 - sin η 0 1 0 sin η 0 cos η ] [ - cos ζ sin ζ 0 - sin ζ cos ζ 0 0 0 1 ]
a 11 cos ω - a 21 sin ω = cos η cos ζ , a 12 cos η - a 22 sin ω = cos η sin ζ , a 13 cos ω - a 23 sin ω = - sin η , a 11 sin ω + a 21 cos ω = - sin ζ , a 12 sin ω + a 22 cos ω = cos ζ , a 13 sin ω + a 23 cos ω = 0 , a 31 = sin η cos ζ , a 32 = sin η sin ζ , a 33 = cos η .

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