Abstract

The general case of light propagation in lossless anisotropic media occurs in crystals that are biaxial, either naturally so or by induced means (e.g., by electrooptic effect). In these crystals the optical properties, such as the refractive indices, change with propagation direction and are conveniently described by the two-sheeted wavevector surface. Most published work treats light propagation only in the principal planes of the crystal, where the wavevector surface reduces to a circle and an ellipse and the mathematics is simplified. Commonly, however, a biaxial bulk or waveguide device, especially an active device, will be oriented so that the light propagation is not in a principal plane. A complete and concise coordinate-free approach is presented for isolating each sheet, thereby providing a convenient means for calculating the directional optical properties of the two decoupled waves for arbitrary wavevector directions and birefringence levels. The versatility of this approach coupled with available graphics software is demonstrated by displaying numerous cross sections of the wavevector surfaces.

© 1991 Optical Society of America

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References

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  1. See, for example, special issue on Electro-Optic Materials and Devices, IEEE J. Quantum Electron. QE-23 (1987).
  2. W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512–1523 (1987).
    [CrossRef]
  3. E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
    [CrossRef]
  4. H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
    [CrossRef]
  5. See, for example, special issue on Optical Computing, Proc. IEEE 72 (1984).
  6. T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59–66 (1987).
    [CrossRef]
  7. M. Born, Optik (Springer-Verlag, Berlin, 1933).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  9. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).
  10. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).
  11. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1965).
  12. M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).
  13. D. R. Lovett, Tensor Properties of Crystals (Adam Hilger, Philadelphia, 1989).
  14. T. A. Maldonado, T. K. Gaylord, “Electro-Optic Effect Calculations: Simplified Procedure for Arbitrary Cases,” Appl. Opt. 27, 5051–5066 (1988).
    [CrossRef] [PubMed]
  15. F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Saunders, New York, 1961).
  16. V. Ramaswamy, “Propagation in Asymmetrical Anisotropic Film Waveguides,” Appl. Opt. 13, 1363–1371 (1974).
    [CrossRef] [PubMed]
  17. S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
    [CrossRef]
  18. F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, 1958).
  19. H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).
  20. I. V. Lindell, “Coordinate Independent Dyadic Formulation of Wave Normal and Ray Surfaces of General Anisotropic Media,” J. Math. Phys. 14, 65–67 (1973).
    [CrossRef]
  21. H. Gelman, “General Expansion of the Determinant of the Maxwell Dyadic,” J. Math. Phys. 11, 3053–3054 (1970).
    [CrossRef]
  22. K. S. Kunz, “Treatment of Optical Propagation in Crystals Using Projection Dyadics,” Am. J. Phys. 45, 267–269 (1977).
    [CrossRef]
  23. H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from a Uniaxially Anisotropic Medium,” IEEE Trans. Microwave Theory Tech. MTT-31, 331–336 (1983).
    [CrossRef]
  24. H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from an Anisotropic Medium,” Radio Sci. 16, 1213–1215 (1981).
    [CrossRef]
  25. T. W. Johnston, “Plane-Wave Dispersion in Gyrotropic Media,” Radio Sci. 4, 729–732 (1969).
    [CrossRef]
  26. For example, AutoCAD Release 9, Autodesk, Inc., Publication TD106-010 (7Jan.1988).
  27. F. I. Fedorov, Theory of Elastic Waves in Crystals (Plenum, New York, 1968).
    [CrossRef]
  28. J. W. Gibbs, E. B. Wilson, Vector Analysis (Scribner, New York, 1907).
  29. C. E. Weatherburn, Advanced Vector Analysis (Bell, London, 1928).
  30. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]
  31. A. E. Taylor, W. R. Mann, Advanced Calculus (Wiley, New York, 1972).
  32. A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
    [CrossRef]
  33. T. A. Maldonado, T. K. Gaylord, “Hybrid Guided Modes in Biaxial Dielectric Planar Waveguides,” to be published.
  34. B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
  35. G. Strang, Linear Algebra and Its Applications (Wiley, New York, 1971).
  36. C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

1988 (2)

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

T. A. Maldonado, T. K. Gaylord, “Electro-Optic Effect Calculations: Simplified Procedure for Arbitrary Cases,” Appl. Opt. 27, 5051–5066 (1988).
[CrossRef] [PubMed]

1987 (5)

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59–66 (1987).
[CrossRef]

See, for example, special issue on Electro-Optic Materials and Devices, IEEE J. Quantum Electron. QE-23 (1987).

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512–1523 (1987).
[CrossRef]

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
[CrossRef]

1985 (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1984 (1)

See, for example, special issue on Optical Computing, Proc. IEEE 72 (1984).

1983 (1)

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from a Uniaxially Anisotropic Medium,” IEEE Trans. Microwave Theory Tech. MTT-31, 331–336 (1983).
[CrossRef]

1981 (1)

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from an Anisotropic Medium,” Radio Sci. 16, 1213–1215 (1981).
[CrossRef]

1977 (1)

K. S. Kunz, “Treatment of Optical Propagation in Crystals Using Projection Dyadics,” Am. J. Phys. 45, 267–269 (1977).
[CrossRef]

1974 (1)

1973 (1)

I. V. Lindell, “Coordinate Independent Dyadic Formulation of Wave Normal and Ray Surfaces of General Anisotropic Media,” J. Math. Phys. 14, 65–67 (1973).
[CrossRef]

1972 (1)

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

1970 (1)

H. Gelman, “General Expansion of the Determinant of the Maxwell Dyadic,” J. Math. Phys. 11, 3053–3054 (1970).
[CrossRef]

1969 (1)

T. W. Johnston, “Plane-Wave Dispersion in Gyrotropic Media,” Radio Sci. 4, 729–732 (1969).
[CrossRef]

Bloss, F. D.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Saunders, New York, 1961).

Born, M.

M. Born, Optik (Springer-Verlag, Berlin, 1933).
[CrossRef]

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, 1512–1523 (1987).
[CrossRef]

Chen, H.

H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

Chen, H. C.

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from a Uniaxially Anisotropic Medium,” IEEE Trans. Microwave Theory Tech. MTT-31, 331–336 (1983).
[CrossRef]

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from an Anisotropic Medium,” Radio Sci. 16, 1213–1215 (1981).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, Theory of Elastic Waves in Crystals (Plenum, New York, 1968).
[CrossRef]

F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, 1958).

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Gaylord, T. K.

T. A. Maldonado, T. K. Gaylord, “Electro-Optic Effect Calculations: Simplified Procedure for Arbitrary Cases,” Appl. Opt. 27, 5051–5066 (1988).
[CrossRef] [PubMed]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59–66 (1987).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

T. A. Maldonado, T. K. Gaylord, “Hybrid Guided Modes in Biaxial Dielectric Planar Waveguides,” to be published.

Gelman, H.

H. Gelman, “General Expansion of the Determinant of the Maxwell Dyadic,” J. Math. Phys. 11, 3053–3054 (1970).
[CrossRef]

Gibbs, J. W.

J. W. Gibbs, E. B. Wilson, Vector Analysis (Scribner, New York, 1907).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1965).

Johnston, T. W.

T. W. Johnston, “Plane-Wave Dispersion in Gyrotropic Media,” Radio Sci. 4, 729–732 (1969).
[CrossRef]

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Koyamada, Y.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Kunz, K. S.

K. S. Kunz, “Treatment of Optical Propagation in Crystals Using Projection Dyadics,” Am. J. Phys. 45, 267–269 (1977).
[CrossRef]

Landau, L. D.

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

Lifshitz, E. M.

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

Lindell, I. V.

I. V. Lindell, “Coordinate Independent Dyadic Formulation of Wave Normal and Ray Surfaces of General Anisotropic Media,” J. Math. Phys. 14, 65–67 (1973).
[CrossRef]

Lovett, D. R.

D. R. Lovett, Tensor Properties of Crystals (Adam Hilger, Philadelphia, 1989).

Makimoto, T.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Maldonado, T. A.

T. A. Maldonado, T. K. Gaylord, “Electro-Optic Effect Calculations: Simplified Procedure for Arbitrary Cases,” Appl. Opt. 27, 5051–5066 (1988).
[CrossRef] [PubMed]

T. A. Maldonado, T. K. Gaylord, “Hybrid Guided Modes in Biaxial Dielectric Planar Waveguides,” to be published.

Mann, W. R.

A. E. Taylor, W. R. Mann, Advanced Calculus (Wiley, New York, 1972).

Mitra, S. K.

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

Moharam, M. G.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Neyer, A.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

Parlett, B. N.

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

Ramaswamy, V.

Rao, C. R.

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

Strang, G.

G. Strang, Linear Algebra and Its Applications (Wiley, New York, 1971).

Taylor, A. E.

A. E. Taylor, W. R. Mann, Advanced Calculus (Wiley, New York, 1972).

Taylor, H. F.

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
[CrossRef]

Tomlinson, W. J.

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512–1523 (1987).
[CrossRef]

Verriest, E. I.

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59–66 (1987).
[CrossRef]

Voges, E.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

Weatherburn, C. E.

C. E. Weatherburn, Advanced Vector Analysis (Bell, London, 1928).

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1965).

Wilson, E. B.

J. W. Gibbs, E. B. Wilson, Vector Analysis (Scribner, New York, 1907).

Wolf, E.

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

Yamamoto, S.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

Yariv, A.

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. J. Phys. (1)

K. S. Kunz, “Treatment of Optical Propagation in Crystals Using Projection Dyadics,” Am. J. Phys. 45, 267–269 (1977).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Computer (1)

T. K. Gaylord, E. I. Verriest, “Matrix Triangularization Using Arrays of Integrated Optical Givens Rotation Devices,” Computer 20, 59–66 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

See, for example, special issue on Electro-Optic Materials and Devices, IEEE J. Quantum Electron. QE-23 (1987).

IEEE Trans. Microwave Theory Tech. (1)

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from a Uniaxially Anisotropic Medium,” IEEE Trans. Microwave Theory Tech. MTT-31, 331–336 (1983).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (2)

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

J. Appl. Phys. (1)

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090–5097 (1972).
[CrossRef]

J. Math. Phys. (2)

I. V. Lindell, “Coordinate Independent Dyadic Formulation of Wave Normal and Ray Surfaces of General Anisotropic Media,” J. Math. Phys. 14, 65–67 (1973).
[CrossRef]

H. Gelman, “General Expansion of the Determinant of the Maxwell Dyadic,” J. Math. Phys. 11, 3053–3054 (1970).
[CrossRef]

Proc. IEEE (3)

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
[CrossRef]

See, for example, special issue on Optical Computing, Proc. IEEE 72 (1984).

W. J. Tomlinson, C. A. Brackett, “Telecommunications Applications of Integrated Optics and Optoelectronics,” Proc. IEEE 75, 1512–1523 (1987).
[CrossRef]

Radio Sci. (2)

H. C. Chen, “A Coordinate-Free Approach to Wave Reflection from an Anisotropic Medium,” Radio Sci. 16, 1213–1215 (1981).
[CrossRef]

T. W. Johnston, “Plane-Wave Dispersion in Gyrotropic Media,” Radio Sci. 4, 729–732 (1969).
[CrossRef]

Other (19)

For example, AutoCAD Release 9, Autodesk, Inc., Publication TD106-010 (7Jan.1988).

F. I. Fedorov, Theory of Elastic Waves in Crystals (Plenum, New York, 1968).
[CrossRef]

J. W. Gibbs, E. B. Wilson, Vector Analysis (Scribner, New York, 1907).

C. E. Weatherburn, Advanced Vector Analysis (Bell, London, 1928).

F. I. Fedorov, Optics of Anisotropic Media (Izd. AN BSSR, Minsk, 1958).

H. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

T. A. Maldonado, T. K. Gaylord, “Hybrid Guided Modes in Biaxial Dielectric Planar Waveguides,” to be published.

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

G. Strang, Linear Algebra and Its Applications (Wiley, New York, 1971).

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

A. E. Taylor, W. R. Mann, Advanced Calculus (Wiley, New York, 1972).

M. Born, Optik (Springer-Verlag, Berlin, 1933).
[CrossRef]

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).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1965).

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

D. R. Lovett, Tensor Properties of Crystals (Adam Hilger, Philadelphia, 1989).

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Saunders, New York, 1961).

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Figures (17)

Fig. 1
Fig. 1

(a) Light propagating in an arbitrary direction with respect to a reference coordinate system (xr,yr,zr,) through a biaxial crystal; i.e., the wavevector k ^ is not directed along a principal axis or in a principal plane. The two allowed orthogonally polarized extraor-dinarylike waves are shown. (b) A cutaway view of the index ellipsoid projection in the three planes of the device reference coordinate system. The two allowed orthogonal linear polarizations (D1,D2) and their ray vectors (S1,S2) for the waves in (a) are shown explicitly.

Fig. 2
Fig. 2

Phase velocity indices of refraction n1 and n2 for the two allowed extraordinarylike waves. The curves represent the values of the two indices as a function of the wavevector spherical coordinate angles θk and ϕk with respect to the principal dielectric axes of a biaxial crystal with principal indices nx = 1.2, ny = 1.7, and nz = 2.2. The family of curves for n1 lies between nx and ny, and the curves for n2 lie between ny and nz. The curves intersect only in the ϕk = 0° (x,z) plane at the optic axis angle θk = θO.A..

Fig. 3
Fig. 3

Geometric relationships of the electric quantities D and E and the magnetic quantities B and H to wavevector k and to ray vector S for the two allowed extraordinarylike waves propagating in an anisotropic medium. For each wave the vectors D, E, k, and S are coplanar, and BH is orthogonal to this plane. Note that H1D2, H2D1, and H1H2(D1D2).

Fig. 4
Fig. 4

(a) Light propagation in an arbitrary direction through a biaxial crystal (represented by the index ellipsoid) depicted by wavevector k. The x ^, ŷ, and z ^ axes are the principal axes, and the (xk,yk,zk) triad is the coordinate system of k with k z ^ k. The cross-sectional ellipse (cross-hatched) perpendicular to k is shown, and the two allowed linear polarizations (D1 and D2) are along the major and minor axes (xk and yk) of this ellipse. (b) The projection of the index ellipsoid containing the vector quantities for one of the extraordinarylike waves (1 or 2) propagating with wavevector k. For k, vectors E, D, and S and phase and ray velocity indices n and ns are directly extracted from the geometry of the index ellipsoid.

Fig. 5
Fig. 5

One octant of the two-sheeted wavevector surface for a biaxial crystal with one of the two optic axes shown. The surface is defined in wavevector space, and the axes are identified as kx, ky, and kz. These directions also correspond to the x, y, and z principal dielectric axes. The length (wavevector magnitude) of each sheet where the surface intersects the axes is in terms of the principal indices of refraction nx, ny, and nz and the free-space wavevector ko.

Fig. 6
Fig. 6

Sheets of the wavevector surface for a biaxial crystal together form a circle and an ellipse in the (a) (x,y), (b) (y,z), and (c) (x,z) principal planes.

Fig. 7
Fig. 7

The (x,z) principal plane projection of the two-sheeted wavevector surface in (a) technically does not consist of a circle and an ellipse but instead consists of a distinct outer sheet (b) and inner sheet (c) intersecting at four points which identify the two optic axes.

Fig. 8
Fig. 8

Orientation of the (x,y,z) principal coordinate system defined by angles ϕ, θ, and ψ relative to the reference coordinate system (xr,yr,zr).

Fig. 9
Fig. 9

Projection of the index ellipsoid onto the (x,z) principal plane. The two optic axes, denoted by Â1 and Â2, are symmetric about the z ^ principal axis by angle θO.A. The cross-sectional ellipse perpendicular to the wavevector k ^Â1 is a circle of radius ny.

Fig. 10
Fig. 10

Projection of the ray ellipsoid onto the (x,z) principal plane. The two ray axes, denoted by R ^ 1 and R ^ 2, are symmetric about the z ^ principal axis by angle θR.A. The cross-sectional ellipse perpendicular to the wavevectork k ^ R ^ 1 is a circle of radius 1/ny.

Fig. 11
Fig. 11

Example wavevector surface cross sections for a biaxial medium with principal indices of refraction nx = 1.2, ny = 1.7, and nz = 2.2. The projection plane angles θN and ϕN are the spherical coordinate angles of the vector z ^ N normal to each projection plane. The bottom row depicts the (x,y) principal plane, the upper left corner depicts the (y,z) principal plane, and the upper right corner depicts the (x,z) principal plane.

Fig. 12
Fig. 12

One sheet of a general cross section of the biaxial wavevector surface. Wavevector k is positioned at angle β with respect to x ^ N. The corresponding ray vector S is normal to the surface and is oriented at an angle α with respect to the wavevector. Angle α is decomposed into angles α and α, identifying the positions of the projection of S(S) onto the plane relative to k and of the normal component of S(S) relative to S, respectively.

Fig. 13
Fig. 13

Plots illustrating the angular separation between the ray vector and wavevector as a function of β for each sheet of the following projection planes: (a) (x,y) plane, where α1 = α2 = 0 and α = α for both sheets; (b) (x,z) plane, with the same properties as in (a); and (c) θN = 70°, ϕN = 57.68° plane, where both components of α are nonzero for both sheets.

Fig. 14
Fig. 14

General cross section with normal z ^ having projection plane angles θN = 70° and ϕN = 57.68°. Three pairs of directions are identified where the ray vector and wavevector are parallel (i.e., α = 0). These directions, denoted by k ^ y , z , k ^ x , y, and k ^ x , z, correspond to the intersection of the cross section with the (y,z), (x,y), and (x,z) principal planes. A fourth pair of directions exists where α = 0 but α ≠ 0, and it is denoted by k ^ d.

Fig. 15
Fig. 15

Array of wavevector cross sections in Fig. 11 reduces to this array for a uniaxial crystal with nx = ny = 1.7 and nz = 2.2. Note that the columns are identical, indicating independence in ϕN.

Fig. 16
Fig. 16

Array of wavevector surface cross sections further reduces to this array for an isotropic media with nx = ny = nz = 1.7. All the cross sections are identical, indicating no directional dependence.

Fig. 17
Fig. 17

(a) The two families of curves in Fig. 2 for the phase velocity indices of refraction of the two allowed waves propagating in a biaxial crystal reduce to these two curves for a uniaxial medium. One curve shows the θk directional dependence of one wave (extraordinary wave), and the second curve shows no dependence on direction (ordinary wave). (b) The curves in (a) further reduce to one straight line for isotropic media. Both allowed waves have the same phase velocity for any direction of propagation.

Equations (108)

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ɛ ¯ = ( ɛ x 0 0 0 ɛ y 0 0 0 ɛ z ) ,
ɛ ¯ = ( ɛ x r x r ɛ x r y r ɛ x r z r ɛ x r y r ɛ y r y r ɛ y r z r ɛ x r z r ɛ y r z r ɛ z r z r ) ,
ɛ ¯ = ɛ x x ^ x ^ + ɛ y y ^ y ^ + ɛ z z ^ z ^ ,
ɛ ¯ = ɛ y I ¯ + ( ɛ x - ɛ y ) x ^ x ^ + ( ɛ z - ɛ y ) z ^ z ^ ,
I ¯ = x ^ x ^ + y ^ y ^ + z ^ z ^ .
ɛ i i = ɛ y I ¯ + ( ɛ x - ɛ y ) ( x i ) 2 + ( ɛ z - ɛ y ) ( z i ) 2 ,             i = x r , y r , z r ,
ɛ i j = ( ɛ x - ɛ y ) ( x i ) ( x j ) + ( ɛ z - ɛ y ) ( z i ) ( z j ) ,             i , j = x r , y r , z r .
T r ( ɛ ¯ ) = ɛ x + ɛ y + ɛ z ,
ɛ ¯ = ɛ x ɛ y ɛ z ,
adj ɛ ¯ = ɛ x ɛ z I ¯ - ɛ z ( ɛ x - ɛ y ) x ^ x ^ - ɛ x ( ɛ z - ɛ y ) z ^ z ^ ,
T r ( adj ɛ ¯ ) = ɛ y ( ɛ x + ɛ z ) + ɛ x ɛ z ,
adj ɛ ¯ - T r ( adj ɛ ¯ ) I ¯ = - ɛ y ( ɛ x + ɛ z ) I ¯ - ɛ z ( ɛ x - ɛ y ) x ^ x ^ - ɛ x ( ɛ z - ɛ y ) z ^ z ^ ,
ɛ ¯ = ɛ O I ¯ + ( ɛ E - ɛ O ) z ^ z ^ ,
T r ( ɛ ¯ ) = 2 ɛ O + ɛ E ,
ɛ ¯ = ɛ O 2 ɛ E ,
adj ɛ ¯ = ɛ O [ ɛ E I ¯ - ( ɛ E - ɛ O ) z ^ z ^ ] ,
T r ( adj ɛ ¯ ) = ɛ O ( ɛ O + ɛ E ) + ɛ O ɛ E ,
adj ɛ ¯ - T r ( adj ɛ ¯ ) I ¯ = - ɛ O ( ɛ O + ɛ E ) I ¯ - ɛ O ( ɛ E - ɛ O ) z ^ z ^ = - ɛ O ( ɛ E I ¯ + ɛ ¯ ) ,
x 2 n x 2 + z 2 n z 2 = 1
( n y 2 n x 2 ) cos 2 θ O . A . + ( n y 2 n z 2 ) sin 2 θ O . A . = 1.
tan 2 θ O . A . = ( 1 n x 2 - 1 n y 2 ) ( 1 n y 2 - 1 n z 2 ) Φ 1 2 Φ 3 2 ,
Φ 1 = ( 1 n x 2 - 1 n y 2 ) ,             Φ 3 = ( 1 n y 2 - 1 n z 2 ) .
A 1 = Φ 1 x ^ + Φ 3 z ^
A 2 = - Φ 1 x ^ + Φ 3 z ^ .
A ^ 1 = Φ 1 x ^ + Φ 3 z ^ , A ^ 2 = Φ 1 x ^ + Φ 3 z ^ ,
Φ 1 = Φ 1 M ,             Φ 3 = Φ 3 M ,
M = ( 1 n x 2 - 1 n z 2 ) .
x ^ = 1 2 Φ 1 [ A ^ 1 - A ^ 2 ] ,             z ^ = 1 2 Φ 3 [ A ^ 1 + A ^ 2 ] .
n 2 ¯ = n y 2 I ¯ + ( P + Q ) ( A ^ 1 A ^ 1 + A ^ 2 A ^ 2 ) + ( P - Q ) ( A ^ 1 A ^ 2 + A ^ 2 A ^ 1 ) ,
P = ( n z 2 - n y 2 ) 4 Φ 3 2 ,             Q = ( n x 2 - n y 2 ) 4 Φ 1 2 .
( n x 2 n y 2 ) cos 2 θ R . A . + ( n z 2 n y 2 ) sin 2 θ R . A . = 1.
tan 2 θ R . A . = n x 2 n z 2 tan 2 θ O . A . = ( n y 2 - n x 2 ) ( n z 2 - n y 2 ) Ψ 1 2 Ψ 3 2 ,
R ^ 1 = Ψ 1 x ^ + Ψ 3 z ^ , R ^ 2 = - Ψ 1 x ^ + Ψ 3 z ^ ,
x ^ = ( 1 2 Ψ 1 ) [ R ^ 1 - R ^ 2 ] ,             z ^ = ( 1 2 Ψ 3 ) [ R ^ 1 + R ^ 2 ] .
n 2 ¯ = n y 2 I ¯ + 1 2 ( n z 2 - n x 2 ) [ R ^ 1 R ^ 2 + R ^ 2 R ^ 1 ] .
k × k × E + ω 2 μ o o ɛ ¯ · E = 0 ,
k × I ¯ = I ¯ × k = k o 2 n 2 ( 0 - k z k y k z 0 - k x - k y k x 0 ) ,
[ k o 2 ɛ ¯ + kk - k 2 I ¯ ] · E = 0 ,
A n 4 + B n 2 + C = 0 ,
A = k ^ T · ɛ ¯ · k ^ ,
B = k ^ T · [ adj ɛ ¯ - T r ( adj ɛ ¯ ) I ¯ ] · k ^ ,
C = ɛ ¯ ,
n = k / k o .
n 1 , 2 2 = - B ( B 2 - 4 A C ) 1 / 2 2 A
A = n x 2 k x 2 + n y 2 k y 2 + n z 2 k z 2 ,
B = - n x 2 n y 2 ( 1 - k z 2 ) - n x 2 n z 2 ( 1 - k y 2 ) - n y 2 n z 2 ( 1 - k x 2 ) ,
C = n x 2 n y 2 n z 2 ,
( k T · ɛ ¯ · k ) k 2 - k o 4 ɛ ¯ = 0 ,
( k ^ T · ɛ ¯ · k ^ ) n 4 - ɛ ¯ = 0.
k ^ T · [ T r ( adj ɛ ¯ ) I ¯ - adj ɛ ¯ ] · k ^ n 2 = 2 ɛ ¯ ,
adj [ W ¯ ( k ) = k o 4 adj ɛ ¯ - k o 2 ( k T · ɛ ¯ · k ) I ¯ + [ k 2 - k o 2 T r ( ɛ ¯ ) ] kk + k o 2 ( kk · ɛ ¯ + ɛ ¯ · kk ) ,
( 1 / k o 4 ) adj [ W ¯ ( k ^ ) = k ^ k ^ n 4 + [ k ^ k ^ · ɛ ¯ + ɛ ¯ · k ^ k ^ ] n 2 - [ ( k ^ T · ɛ ¯ · k ^ ) I ¯ + T r ( ɛ ¯ ) k ^ k ^ ] n 2 + adj ɛ ¯ ,
[ k o 2 ɛ ¯ - k 2 I ¯ ] · E = - kk · E = - ( k · E ) k ,
E = [ k o 2 ɛ ¯ - k 2 I ¯ ] · E .
e 1 , 2 = adj [ J ¯ ( n 1 , 2 ) ] · k ^ .
J ¯ · E = [ ɛ ¯ - n 1 , 2 2 I ¯ ] · E = 0.
d 1 , 2 = o ɛ ¯ · e ^ 1 , 2 .
h 1 , 2 = ( 1 / ω μ o ) k ^ × e 1 , 2 = ( w / k 2 ) k ^ × d 1 , 2 .
k o S 1 , 2 = [ n 1 , 2 2 ( A I ¯ + ɛ ¯ ) + ( adj ɛ ¯ - T r ( adj ɛ ¯ ) I ¯ ) ] A n 1 , 2 4 - ɛ ¯ · n 1 , 2 k ^ ,
( 2 A n 1 2 + B + B 2 - 4 A C ) · ( 2 A n 2 2 + B - B 2 - 4 A C ) = 0 ,
( k ^ T · ɛ ¯ · k ^ ) n 2 2 = n O 2 n E 2 ,
k ^ · k ^ n 1 2 = n O 2 ,
k ^ N = G ¯ T · k ^ ,
k o S 2 = 1 n O n E A · ɛ ¯ k ^ ,
k o S 1 = 1 n O k ^ ,
n s 1 , 2 = n 1 , 2 / cos α 1 , 2 .
ab = ( x a x b x a y b x a z b y a x b y a y b y a z b z a x b z a y b z a z b ) ,
ba = ( x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z a ) .
ɛ ¯ = al + bm + cn .
ɛ ¯ = ɛ x x ^ x ^ + ɛ y y ^ y ^ + ɛ z z ^ z ^ ,
x x r = cos ϕ cos θ cos ψ - sin ϕ sin ψ ,
x y r = sin ϕ cos ϕ cos ψ + cos ϕ sin ψ ,
x z r = sin θ cos ψ ,
y x r = - cos ϕ cos θ sin ψ - sin ϕ cos ψ ,
y y r = - sin ϕ cos θ sin ψ + cos ϕ cos ψ ,
y z r = sin θ sin ψ ,
z x r = cos ϕ sin θ ,
z y r = sin ϕ sin θ ,
z z r = cos θ ,
ɛ x r x r = ɛ y + ( ɛ x - ɛ y ) x x r 2 - ( ɛ z - ɛ y ) z x r 2 ,
ɛ y r y r = ɛ y + ( ɛ x - ɛ y ) x y r 2 - ( ɛ z - ɛ y ) z y r 2 ,
ɛ z r z r = ɛ y + ( ɛ x - ɛ y ) x z r 2 - ( ɛ z - ɛ y ) z z r 2 ,
ɛ x r y r = ( ɛ x - ɛ y ) x x r x y r + ( ɛ z - ɛ y ) z x r z y r ,
ɛ x r z r = ( ɛ x - ɛ y ) x x r x z r + ( ɛ z - ɛ y ) z x r z z r ,
ɛ y r z r = ( ɛ x - ɛ y ) x y r x z r + ( ɛ z - ɛ y ) z y r z z r .
W ¯ ( k ^ ) = ɛ ¯ + n 2 ( k ^ k ^ - I ¯ ) .
W x r x r = ɛ x r x r - n 2 ( k y r 2 + k z r 2 ) ,
W y r y r = ɛ y r y r - n 2 ( k x r 2 + k z r 2 ) ,
W z r z r = ɛ z r z r - n 2 ( k x r 2 + k y r 2 ) ,
W x r y r = ɛ x r y r + n 2 k x r k y r ,
W x r z r = ɛ x r z r + n 2 k x r k z r ,
W y r z r = ɛ y r z r + n 2 k y r k z r ,
k ^ k ^ = ( k x r 2 k x r k y r k x r k z r k x r k y r k y r 2 k y r k z r k x r k z r k y r k z r k z r 2 ) .
F x r x r = ɛ x r x r k x r 2 + ɛ x r y r k x r k y r + ɛ x r z r k x r k z r ,
F x r y r = ɛ x r y r k x r 2 + ɛ y r y r k x r k y r + ɛ y r z r k x r k z r ,
F x r z r = ɛ x r z r k x r 2 + ɛ y r z r k x r k y r + ɛ z r z r k x r k z r ,
F y r x r = ɛ x r x r k x r k y r + ɛ x r y r k y r 2 + ɛ x r z r k y r k z r ,
F y r y r = ɛ x r y r k x r k ^ y r + ɛ y r y r k y r 2 + ɛ y r z r k y r k z r ,
F y r z r = ɛ x r y r k x r k y r + ɛ y r z r k y r 2 + ɛ z r z r k y r k z r ,
F z r x r = ɛ x r x r k x r k z r + ɛ x r y r k y r k z r + ɛ x r z r k z r 2 ,
F z r y r = ɛ x r y r k x r k z r + ɛ y r y r k y r k z r + ɛ y r z r k z r 2 ,
F z r z r = ɛ x r z r k x r k z r + ɛ y r z r k y r k z r + ɛ z r z r k z r 2 .
J x r x r = ( ɛ y r y r - n 2 ) ( ɛ z r z r - n 2 ) - ɛ y r z r 2 ,
J y r y r = ( ɛ x r x r - n 2 ) ( ɛ z r z r - n 2 ) - ɛ x r z r 2 ,
J z r z r = ( ɛ x r x r - n 2 ) ( ɛ y r y r - n 2 ) - ɛ x r y r 2 ,
J x r y r = ɛ x r z r ɛ y r z r - ɛ x r y r ( ɛ z r z r - n 2 ) ,
J x r z r = ɛ x r y r ɛ y r z r - ɛ x r z r ( ɛ y r y r - n 2 ) ,
J y r z r = ɛ x r y r ɛ x r z r - ɛ y r z r ( ɛ x r x r - n 2 ) .

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