Abstract

An eigenvector equation within the electric octopole–magnetic quadrupole approximation is derived, from which it is possible in principle to establish the eigenvectors and the refractive indices of polarized light forms sustained by an anisotropic crystal for any chosen direction of phase propagation. It is shown that, for propagation along the crystal axes when the rays do not separate, a decomposition technique based on eigenvectors and the Jones resolution of a medium into independent differential plates substantially simplifies solution of the general problem. Precise expressions are obtained for the Jones parameters n, (nynx), (n−45n45), and (nRnL) in terms of multipole-property tensors of the crystal. The effects of crystal symmetry on these expressions are summarized.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  2. R. C. Jones, “New calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. 46, 126–131 (1956).
    [CrossRef]
  3. E. B. Graham, R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London Ser. A 430, 593–614 (1990).
    [CrossRef]
  4. A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A1988–1991 (1976).
  5. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, 1982).
  6. H. A. Lorentz, “Double refraction by regular crystals,” Proc. K. Ned. Acad. Wet. 24, 333 (1922);also in H. A. Lorentz, Collected Papers, P. Zeeman, A. D. Fokker, eds. (Nijhoff, The Hague, 1936), Vol. 3, pp. 315–320.
    [CrossRef]
  7. J. Pastrnak, K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971).
    [CrossRef]
  8. R. E. Raab, “Magnetic multipole moments,” Molec. Phys. 29, 1323–1331 (1975).
    [CrossRef]
  9. V. Devarajan, A. M. Glazer, “Theory and computation of optical rotatory power in inorganic crystals,” Acta Crystallogr. A 42, 560–569 (1986).
    [CrossRef]
  10. R. R. Birss, Symmetry and Magnetism, 2nd ed. (North-Holland, Amsterdam, 1966).
  11. R. R. Birss, “Macroscopic symmetry in space–time,” Rep. Prog. Phys.26, 307–360 (1963).
    [CrossRef]
  12. A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” in Intermolecular Forces, J. O. Hirschfelder, ed., Adv. Chem. Phys.12, 107–142 (1967).
  13. I. M. B. de Figueiredo, R. E. Raab, “A molecular theory of new differential light-scattering effects in a fluid,” Proc. R. Soc. London Ser. A 375, 425–441 (1981).
    [CrossRef]
  14. E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
    [CrossRef]
  15. L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. 2.
  16. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  17. E. U. Condon, F. Seitz, “Lorentz double refraction in the regular system,” J. Opt. Soc. Am. 22, 393–401 (1932).
    [CrossRef]
  18. E. B. Graham, R. E. Raab, “On the Jones birefringence,” Proc. R. Soc. London A 390, 73–90 (1983).
    [CrossRef]
  19. H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
    [CrossRef]
  20. H. Nakano, H. Kimura, “Quantum statistical-mechanical theory of optical activity,” J. Phys. Soc. Jpn. 27, 519–535 (1969).
    [CrossRef]
  21. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 398.
  22. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  23. P. H. Boulanger, M. Hayes, “Electromagnetic plane waves in anisotropic media: an approach using bivectors,” Phil. Trans. R. Soc. London A 330, 335–393 (1990).
    [CrossRef]

1992 (1)

E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
[CrossRef]

1990 (2)

E. B. Graham, R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London Ser. A 430, 593–614 (1990).
[CrossRef]

P. H. Boulanger, M. Hayes, “Electromagnetic plane waves in anisotropic media: an approach using bivectors,” Phil. Trans. R. Soc. London A 330, 335–393 (1990).
[CrossRef]

1989 (1)

H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
[CrossRef]

1986 (1)

V. Devarajan, A. M. Glazer, “Theory and computation of optical rotatory power in inorganic crystals,” Acta Crystallogr. A 42, 560–569 (1986).
[CrossRef]

1983 (1)

E. B. Graham, R. E. Raab, “On the Jones birefringence,” Proc. R. Soc. London A 390, 73–90 (1983).
[CrossRef]

1981 (1)

I. M. B. de Figueiredo, R. E. Raab, “A molecular theory of new differential light-scattering effects in a fluid,” Proc. R. Soc. London Ser. A 375, 425–441 (1981).
[CrossRef]

1976 (1)

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A1988–1991 (1976).

1975 (1)

R. E. Raab, “Magnetic multipole moments,” Molec. Phys. 29, 1323–1331 (1975).
[CrossRef]

1971 (1)

J. Pastrnak, K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971).
[CrossRef]

1969 (1)

H. Nakano, H. Kimura, “Quantum statistical-mechanical theory of optical activity,” J. Phys. Soc. Jpn. 27, 519–535 (1969).
[CrossRef]

1956 (1)

1948 (1)

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

1932 (1)

1922 (1)

H. A. Lorentz, “Double refraction by regular crystals,” Proc. K. Ned. Acad. Wet. 24, 333 (1922);also in H. A. Lorentz, Collected Papers, P. Zeeman, A. D. Fokker, eds. (Nijhoff, The Hague, 1936), Vol. 3, pp. 315–320.
[CrossRef]

Barron, L. D.

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

Birss, R. R.

R. R. Birss, “Macroscopic symmetry in space–time,” Rep. Prog. Phys.26, 307–360 (1963).
[CrossRef]

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

Boulanger, P. H.

P. H. Boulanger, M. Hayes, “Electromagnetic plane waves in anisotropic media: an approach using bivectors,” Phil. Trans. R. Soc. London A 330, 335–393 (1990).
[CrossRef]

Buckingham, A. D.

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A1988–1991 (1976).

A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” in Intermolecular Forces, J. O. Hirschfelder, ed., Adv. Chem. Phys.12, 107–142 (1967).

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

E. U. Condon, F. Seitz, “Lorentz double refraction in the regular system,” J. Opt. Soc. Am. 22, 393–401 (1932).
[CrossRef]

de Figueiredo, I. M. B.

I. M. B. de Figueiredo, R. E. Raab, “A molecular theory of new differential light-scattering effects in a fluid,” Proc. R. Soc. London Ser. A 375, 425–441 (1981).
[CrossRef]

Devarajan, V.

V. Devarajan, A. M. Glazer, “Theory and computation of optical rotatory power in inorganic crystals,” Acta Crystallogr. A 42, 560–569 (1986).
[CrossRef]

Dunn, M. B.

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A1988–1991 (1976).

Glazer, A. M.

V. Devarajan, A. M. Glazer, “Theory and computation of optical rotatory power in inorganic crystals,” Acta Crystallogr. A 42, 560–569 (1986).
[CrossRef]

Graham, E. B.

E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
[CrossRef]

E. B. Graham, R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London Ser. A 430, 593–614 (1990).
[CrossRef]

E. B. Graham, R. E. Raab, “On the Jones birefringence,” Proc. R. Soc. London A 390, 73–90 (1983).
[CrossRef]

Hayes, M.

P. H. Boulanger, M. Hayes, “Electromagnetic plane waves in anisotropic media: an approach using bivectors,” Phil. Trans. R. Soc. London A 330, 335–393 (1990).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 398.

Jones, R. C.

Kimura, H.

H. Nakano, H. Kimura, “Quantum statistical-mechanical theory of optical activity,” J. Phys. Soc. Jpn. 27, 519–535 (1969).
[CrossRef]

Lorentz, H. A.

H. A. Lorentz, “Double refraction by regular crystals,” Proc. K. Ned. Acad. Wet. 24, 333 (1922);also in H. A. Lorentz, Collected Papers, P. Zeeman, A. D. Fokker, eds. (Nijhoff, The Hague, 1936), Vol. 3, pp. 315–320.
[CrossRef]

Nakano, H.

H. Nakano, H. Kimura, “Quantum statistical-mechanical theory of optical activity,” J. Phys. Soc. Jpn. 27, 519–535 (1969).
[CrossRef]

Pastrnak, J.

J. Pastrnak, K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971).
[CrossRef]

Pierrus, J.

E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
[CrossRef]

Raab, R. E.

E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
[CrossRef]

E. B. Graham, R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London Ser. A 430, 593–614 (1990).
[CrossRef]

E. B. Graham, R. E. Raab, “On the Jones birefringence,” Proc. R. Soc. London A 390, 73–90 (1983).
[CrossRef]

I. M. B. de Figueiredo, R. E. Raab, “A molecular theory of new differential light-scattering effects in a fluid,” Proc. R. Soc. London Ser. A 375, 425–441 (1981).
[CrossRef]

R. E. Raab, “Magnetic multipole moments,” Molec. Phys. 29, 1323–1331 (1975).
[CrossRef]

Rosenfeld, L.

L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. 2.

Ross, H. J.

H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
[CrossRef]

Seitz, F.

Sherborne, B. S.

H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Stedman, G. E.

H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
[CrossRef]

Vedam, K.

J. Pastrnak, K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971).
[CrossRef]

Acta Crystallogr. A (1)

V. Devarajan, A. M. Glazer, “Theory and computation of optical rotatory power in inorganic crystals,” Acta Crystallogr. A 42, 560–569 (1986).
[CrossRef]

J. Chem. Soc. A (1)

A. D. Buckingham, M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A1988–1991 (1976).

J. Opt. Soc. Am. (3)

J. Phys. B (2)

E. B. Graham, J. Pierrus, R. E. Raab, “Multipole moments and Maxwell’s equations,” J. Phys. B 25, 4673–4684 (1992).
[CrossRef]

H. J. Ross, B. S. Sherborne, G. E. Stedman, “Selection rules for optical activity and linear birefringences bilinear in electric and magnetic fields,” J. Phys. B 22, 459–473 (1989).
[CrossRef]

J. Phys. Soc. Jpn. (1)

H. Nakano, H. Kimura, “Quantum statistical-mechanical theory of optical activity,” J. Phys. Soc. Jpn. 27, 519–535 (1969).
[CrossRef]

Molec. Phys. (1)

R. E. Raab, “Magnetic multipole moments,” Molec. Phys. 29, 1323–1331 (1975).
[CrossRef]

Phil. Trans. R. Soc. London A (1)

P. H. Boulanger, M. Hayes, “Electromagnetic plane waves in anisotropic media: an approach using bivectors,” Phil. Trans. R. Soc. London A 330, 335–393 (1990).
[CrossRef]

Phys. Rev. B (1)

J. Pastrnak, K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971).
[CrossRef]

Proc. K. Ned. Acad. Wet. (1)

H. A. Lorentz, “Double refraction by regular crystals,” Proc. K. Ned. Acad. Wet. 24, 333 (1922);also in H. A. Lorentz, Collected Papers, P. Zeeman, A. D. Fokker, eds. (Nijhoff, The Hague, 1936), Vol. 3, pp. 315–320.
[CrossRef]

Proc. R. Soc. London A (1)

E. B. Graham, R. E. Raab, “On the Jones birefringence,” Proc. R. Soc. London A 390, 73–90 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

I. M. B. de Figueiredo, R. E. Raab, “A molecular theory of new differential light-scattering effects in a fluid,” Proc. R. Soc. London Ser. A 375, 425–441 (1981).
[CrossRef]

E. B. Graham, R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London Ser. A 430, 593–614 (1990).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (7)

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

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

R. R. Birss, “Macroscopic symmetry in space–time,” Rep. Prog. Phys.26, 307–360 (1963).
[CrossRef]

A. D. Buckingham, “Permanent and induced molecular moments and long-range intermolecular forces,” in Intermolecular Forces, J. O. Hirschfelder, ed., Adv. Chem. Phys.12, 107–142 (1967).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 398.

L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. 2.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (1)

Tables Icon

Table 1 Symmetry Indications for the Existence of Birefringences for N Rays in the Crystal Groups within the Electric-Octopole–Magnetic-Quadrupole Approximation

Equations (56)

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

η = 2 π n λ ;
ω = π λ ( n R n L ) ;
g 0 = π λ ( n y n x ) ;
g 45 = π λ ( n 45 n 45 ) .
θ 1 = η [ i 0 0 i ] , θ 2 = ω [ 0 1 1 0 ] , θ 3 = g 0 [ i 0 0 i ] , θ 4 = g 45 [ 0 i i 0 ] ,
electric dipole { electric quadrupole magnetic dipole { electric octopole magnetic quadrupole .
electric dipole p α = Σ q r α , electric quadrupole q α β = Σ q r α r β , electric octopole q α β γ = Σ q r α r β r γ , magnetic dipole m α = Σ ( q / 2 m ) l α , magnetic quadrupole m α β = Σ ( q / 3 m ) ( r β l α + l α r β ) .
P α = α α β E β + ½ a α β γ γ E β + b α β γ δ δ γ E β + + ω 1 G α β B ˙ β + ( 2 ω ) 1 H α β γ γ B ˙ β + , Q α β = a α β γ E γ + ½ d α β γ δ δ E γ + + ω 1 L α β γ B ˙ γ + , Q α β γ = b α β γ δ E δ + , M α = χ α β B β + + ω 1 G α β E ˙ β + ( 2 ω ) 1 α β γ γ B ˙ β + , M α β = ω 1 α β γ E ˙ γ + .
α α β = α β α , a α β γ = a γ α β = a γ β α , b α β γ δ = b δ α β γ = b δ β α γ = b δ β γ α = b δ γ β α , d α β γ δ = d γ δ α β = d δ γ α β = d γ δ β α , G α β = G β α , χ α β = χ β α , α β γ = H γ α β α β γ = L β γ α = L γ β α .
electric dipole : α α β electric quadrupole magnetic dipole : a α β γ , G α β electric octopole magnetic quadrupole : b α β γ δ , d α β γ δ , χ α β , H α β γ , L α β γ ,
D α = ɛ 0 E α + P α ½ β Q α β + γ β Q α β γ , H α = µ 0 1 B α M α + ½ β M α β .
E = E 0 exp [ i ω ( t n r · σ / c ) ] ,
× E = B ˙ ,
[ n 2 σ α σ β ( n 2 1 ) ] δ α β + ɛ 0 1 α α β + i n ɛ 0 1 c 1 U α β + n 2 ɛ 0 1 c 2 V α β ] E 0 β = 0 .
U α β = σ γ [ ɛ β γ δ G α δ ɛ α γ δ G β δ + ½ ω ( a α β γ a β α γ ) ] ,
V α β = σ γ σ δ [ ω 2 ( b α β γ δ + b β α γ δ ) + ¼ ω 2 d α γ β δ ½ ω ( ɛ α γ ɛ H β ɛ δ + ɛ β γ ɛ H α ɛ δ ) + ½ ω ( ɛ α γ ɛ L β δ ɛ + ɛ β γ ɛ L α δ ɛ ) + ɛ α γ ɛ ɛ β δ ϕ χ ɛ ϕ ] ,
| a x x a x y a x z a y x a y y a y z a z x a z y a z z | = 0 ,
a i j = [ n 2 σ i σ j ( n 2 1 ) δ i j ] + ( ɛ 0 1 α i j ) + ( i n ɛ 0 1 c 1 U i j ) + ( n 2 ɛ 0 1 c 2 V i j ) .
U i i = 0 , U j i = U i j , V j i = V i j .
[ n 2 ( σ x 2 1 ) α x x ɛ 0 n 2 V x x ɛ 0 c 2 n 2 σ x σ y α x y ɛ 0 i n U x y ɛ 0 c n 2 V x y ɛ 0 c 2 n 2 σ x σ z α x z ɛ 0 i n U x z ɛ 0 c n 2 V x z ɛ 0 c 2 n 2 σ x σ y α x y ɛ 0 + i n U x y ɛ 0 c n 2 V x y ɛ 0 c 2 n 2 ( σ y 2 1 ) α y y ɛ 0 n 2 V y y ɛ 0 c 2 n 2 σ y σ z α y z ɛ 0 i n U y z ɛ 0 c n 2 V y z ɛ 0 c 2 n 2 σ x σ z α x z ɛ 0 + i n U x z ɛ 0 c n 2 V x z ɛ 0 c 2 n 2 σ y σ z α y z ɛ 0 + i n U y z ɛ 0 c n 2 V y z ɛ 0 c 2 n 2 ( σ z 2 1 ) α z z ɛ 0 n 2 V z z ɛ 0 c 2 ] [ E 0 x E 0 y E 0 z ] = [ E 0 x E 0 y E 0 z ] ,
[ n 2 α x x ɛ 0 n 2 V x x ɛ 0 c 2 α x y ɛ 0 i n U x y ɛ 0 c n 2 V x y ɛ 0 c 2 α x y ɛ 0 + i n U x y ɛ 0 c n 2 V x y ɛ 0 c 2 n 2 α y y ɛ 0 n 2 V y y ɛ 0 c 2 ] × [ E 0 x E 0 y ] = [ E 0 x E 0 y ] .
[ n 2 1 ɛ 0 [ ( α x x + α y y 2 ) ( α y y α x x 2 ) ] n 2 ɛ 0 c 2 [ ( V x x + V y y 2 ) ( V y y V x x 2 ) ] α x y ɛ 0 i n U x y ɛ 0 c n 2 V x y ɛ 0 c 2 α x y ɛ 0 + i n U x y ɛ 0 c 2 n 2 V x y ɛ 0 c 2 n 2 1 ɛ 0 [ ( α x x + α y y 2 ) + ( α y y α x x 2 ) ] n 2 ɛ 0 c 2 [ ( V x x + V y y 2 ) + ( V y y V x x 2 ) ] ] .
[ n 2 1 ɛ 0 ( α x x + α y y 2 ) n 2 ɛ 0 c 2 ( V x x + V y y 2 ) 0 0 n 2 1 ɛ 0 ( α x x + α y y 2 ) n 2 ɛ 0 c 2 ( V x x + V y y 2 ) ] .
n 2 1 ɛ 0 ( α x x + α y y 2 ) n 2 ɛ 0 c 2 ( V x x + V y y 2 ) = 1 .
n = [ 2 ɛ 0 c 2 2 ɛ 0 c 2 ( V x x + V y y ) + c 2 ( α x x + α y y ) 2 ɛ 0 c 2 ( V x x + V y y ) ] 1 / 2 .
n = [ 1 + ( α x x + α y y 2 ɛ 0 ) ] 1 / 2 [ 1 + ( V x x + V y y 4 ɛ 0 c 2 ) ] ,
[ n 2 + 1 ɛ 0 ( α y y α x x 2 ) + n 2 ɛ 0 c 2 ( V y y V x x 2 ) 0 0 n 2 1 ɛ 0 ( α y y α x x 2 ) n 2 ɛ 0 c 2 ( V y y V x x 2 ) ] ,
( 1 , 0 ) with n 2 + 1 ɛ 0 ( α y y α x x 2 ) + n 2 ɛ 0 c 2 ( V y y V x x 2 ) , ( 0 , 1 ) with n 2 1 ɛ 0 ( α y y α x x 2 ) n 2 ɛ 0 c 2 ( V y y V x x 2 ) .
( n y n x ) = ( α y y α x x 2 ɛ 0 ) + ( V y y V x x 2 ɛ 0 c 2 ) ( V y y V x x ) ( α y y α x x ) 2 64 ɛ 0 3 c 2 ,
[ n 2 α x y ɛ 0 n 2 ɛ 0 c 2 V x y α x y ɛ 0 n 2 ɛ 0 c 2 V x y n 2 ] .
( 1 , 1 ) with n 2 α x y ɛ 0 n 2 ɛ 0 c 2 V x y , ( 1 , 1 ) with n 2 + α x y ɛ 0 + n 2 ɛ 0 c 2 V x y .
( n 45 n 45 ) = α x y ɛ 0 V x y ɛ 0 c 2 .
[ n 2 i n U x y ɛ 0 c i n U x y ɛ 0 c n 2 ] ,
( 1 , i ) with n 2 + n U x y ɛ 0 c , ( 1 , i ) with n 2 n U x y ɛ 0 c .
( n R n L ) = U x y ɛ 0 c .
( n y n x ) = 1 2 ɛ 0 ( α y y α x x ) + 1 64 c 2 ɛ 0 3 × { [ 32 ɛ 0 2 ( α y y α x x ) 2 ] [ ( χ x x χ y y ) + ω ( L y z x + L x z y ) ω ( H x y z + H y x z ) + ω 2 3 ( b x x z z b y y z z ) + ω 2 4 ( d y z y z b x z x z ) ] } ,
( n 45 n 45 ) = 1 ɛ 0 c 2 [ α x y c 2 χ x y + ω c ( L x z x L y z y ) ω 2 ( H x x z H y y z ) ω 2 6 ( b y x z z + b x y z z ) + ω 2 4 d x z y z ] ,
( n R n L ) = 1 ɛ 0 c [ ω 2 ( a x y z a y x z ) + ( G x x + G y y ) ] .
M = exp ( i η z ) × [ cos Γ z + i g 0 sin Γ z Γ ( ω + i g 45 ) sin Γ z Γ ( ω + i g 45 ) sin Γ z Γ cos Γ z i g 0 sin Γ z Γ ] .
Γ 2 = g 0 2 + g 45 2 + ω 2 .
[ n 2 1 ɛ 0 [ ( α x x + α z z 2 ) ( α z z α x x 2 ) ] n 2 ɛ 0 c 2 [ ( V x x + V z z 2 ) ( V z z V x x 2 ) ] α x z ɛ 0 i n U x z ɛ 0 c n 2 V x z ɛ 0 c 2 α x z ɛ 0 + i n U x z ɛ 0 c n 2 V x z ɛ 0 c 2 n 2 1 ɛ 0 [ ( α x x + α z z 2 ) ( α z z α x x 2 ) ] n 2 ɛ 0 c 2 [ ( V x x + V z z 2 ) ( V z z V x x 2 ) ] ] .
n = [ 1 + ( α x x + α z z 2 ɛ 0 ) ] 1 / 2 [ 1 + ( V x x + V z z 4 ɛ 0 c 2 ) ] ,
( n z n x ) = ( α z z α x x 2 ɛ 0 ) + ( V z z V x x 2 ɛ 0 c 2 ) ( V z z V x x ) ( α z z α x x ) 2 64 ɛ 0 3 c 2 ,
( n 45 n 45 ) = α x z ɛ 0 V x z ɛ 0 c 2 ,
( n R n L ) = U x z ɛ 0 c ,
( n z n x ) = 1 2 ɛ 0 ( α z z α x x ) + 1 64 c 2 ɛ 0 3 × { [ 32 ɛ 0 2 ( α z z α x x ) 2 ] [ ( χ x x χ z z ) ω ( L z y x + L x y z ) + ω ( H z x y + H x z y ) + ω 2 3 ( b x x y y b z z y y ) + ω 2 4 ( d z y z y d x y x y ) ] } ,
( n 45 n 45 ) = 1 ɛ 0 c 2 [ α x z c 2 χ x z ω 2 ( L x y x + L z y z ) + ω 2 ( H x x y H z z y ) ω 2 6 ( b x z y y + b z x y y ) + ω 2 4 d x y z y ] ,
( n R n L ) = 1 ɛ 0 c [ ω 2 ( a x z y a z x y ) ( G x x + G z z ) ] .
[ n 2 1 ɛ 0 [ ( α y y + α z z 2 ) ( α z z α y y 2 ) ] n 2 ɛ 0 c 2 [ ( V y y + V z z 2 ) ( V z z V y y 2 ) ] α y z ɛ 0 i n U y z ɛ 0 c n 2 V y z ɛ 0 c 2 α y z ɛ 0 + i n U y z ɛ 0 c n 2 V y z ɛ 0 c 2 n 2 1 ɛ 0 [ ( α y y + α z z 2 ) + ( α z z α y y 2 ) ] n 2 ɛ 0 c 2 [ ( V y y + V z z 2 ) + ( V z z V y y 2 ) ] ] ,
n = [ 1 + ( α y y + α z z 2 ɛ 0 ) ] 1 / 2 [ 1 + ( V y y + V z z 4 ɛ 0 c 2 ) ] ,
( n z n y ) = ( α z z α y y 2 ɛ 0 ) + ( V z z V y y 2 ɛ 0 c 2 ) ( V z z V y y ) ( α z z α y y ) 2 64 ɛ 0 3 c 2 ,
( n 45 n 45 ) = α y z ɛ 0 V y z ɛ 0 c 2 ,
( n R n L ) = U y z ɛ 0 c .
( n z n y ) = 1 2 ɛ 0 ( α z z α y y ) + 1 64 c 2 ɛ 0 3 × { [ 32 ɛ 0 2 ( α z z α y y ) 2 ] [ ( χ y y χ z z ) + ω ( L z x y + L y x z ) ω ( H z y x + H y z x ) + ω 2 3 ( b y y x x b z z x x ) + ω 2 4 ( d z x z x d y x y x ) ] } ,
( n 45 n 45 ) = 1 ɛ 0 c 2 [ α y z c 2 χ y z + ω 2 ( L y x y L z x z ) ω 2 ( H y y x H z z x ) ω 2 6 ( b y z x x + b z y x x ) + ω 2 4 d y x z x ] ,
( n R n L ) = 1 ɛ 0 c [ ω 2 ( a y z x a z y x ) + ( G y y + G z z ) ] .

Metrics