Abstract

Propagation of electromagnetic waves is considered for a medium with (x, y)-dependent locally isotropic dielectric and magnetic susceptibilities ik = (x, y)δik and μik = μ(x, y)δik, i.e., for a waveguide. In the paraxial approximation the polarization is disconnected from the propagation. We have developed a self-consistent theory of the postparaxial corrections. It allows, in particular, for the description of intrafiber geometrical rotation of polarization and its inverse phenomenon, the optical Magnus effect, which are both determined by the profile of refractive index n=μ only and constitute spin–orbit interaction of a photon. The birefringence splitting of linearly polarized modes or meridional rays on the other hand, turns out to be dependent on the gradients of impedance ρ=μ/, the quadrupole part of spin–orbit interaction. An important point of the theory is a transformation of field variables such that the z-propagation operator becomes Hermitian, in analogy with the transitions from a full relativistic Dirac equation to the Schrödinger–Pauli equation with spin–orbital corrections. A theoretical explanation is given for the phenomenon previously observed in experiment: preservation of circular polarization by an axially symmetric step-profile multimode fiber and depolarization of an input linearly polarized wave by the same fiber.

© 1996 Optical Society of America

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References

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  1. A. Snyder and J. Love, Optical Waveguide Theory (Methuen, London, 1984).
    [Crossref]
  2. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987).
  4. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).
  5. R. K. Luneberg, Mathematical Theory of Optics (Brown Press, Providence, R.I., 1944).
  6. Yu. A. Kravtsov and Yu. I. Orlov, Geometric Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
    [Crossref]
  7. F. Bortolotti, Rend. R. Acc. Naz. Linc. 6a 4, 552 (1926) [in Italian (?); cited in Ref. 3].
  8. S. M. Rytov, Dokl. Acad. Nauk SSSR 18, 2 (1938); Tr. Fiz. Inst. Akad. Nauk SSSR 2, 1 (1940). (translated into English in Ref. 9, p. 6); V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 21, 222 (1941) (translated into English in Ref. 9, p. 11).
  9. B. Markovski and V. I. Vinitsky, Topological Phases in Quantum Theory, (World Scientific, Singapore, 1989).
  10. R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
    [Crossref] [PubMed]
  11. M. V. Berry, Proc. R. Soc., London Ser. A 392, 45 (1984); Nature (London),  326, 227 (1987).
    [Crossref]
  12. M. V. Berry, Phys. Today 43(12), 34 (1990).
    [Crossref]
  13. A. A. Esayan and B. Ya. Zel’dovich, Sov. J. Quantum Electron. 18, 149 (1988).
    [Crossref]
  14. N. D. Kundikova, Sov. J. Quantum Electron. 25, 2 (1995).
  15. F. I. Fedorov, Dokl. Acad. Nauk SSSR 102, 69 (1955).
  16. C. Imbert, Phys. Rev D 5, 787 (1972); O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
    [Crossref]
  17. B. Ya. Zel’dovich and V. S. Liberman, Sov. J. Quantum Electron. 20, 427 (1990).
    [Crossref]
  18. V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
    [Crossref] [PubMed]
  19. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
    [Crossref] [PubMed]
  20. V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 4367 (1993).
    [Crossref]
  21. A. Yu. Savchenko and B. Ya. Zel’dovich, Bull. Russ. Acad. Sci. 58, Suppl. 3, 158 (1994) (in English).
  22. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [Crossref]
  23. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon, Oxford, 1984).
  24. A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2287 (1994).
    [Crossref]
  25. V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. E 49, 2389 (1994).
    [Crossref]
  26. N. B. Baranova and B. Ya. Zel’dovich, JETP Lett. 59, 681 (1994).
  27. M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).
  28. M. A. Player, J. Phys. A 20, 3667 (1987).
    [Crossref]

1995 (1)

N. D. Kundikova, Sov. J. Quantum Electron. 25, 2 (1995).

1994 (5)

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2287 (1994).
[Crossref]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. E 49, 2389 (1994).
[Crossref]

N. B. Baranova and B. Ya. Zel’dovich, JETP Lett. 59, 681 (1994).

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

A. Yu. Savchenko and B. Ya. Zel’dovich, Bull. Russ. Acad. Sci. 58, Suppl. 3, 158 (1994) (in English).

1993 (1)

V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 4367 (1993).
[Crossref]

1992 (2)

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

1990 (2)

B. Ya. Zel’dovich and V. S. Liberman, Sov. J. Quantum Electron. 20, 427 (1990).
[Crossref]

M. V. Berry, Phys. Today 43(12), 34 (1990).
[Crossref]

1988 (1)

A. A. Esayan and B. Ya. Zel’dovich, Sov. J. Quantum Electron. 18, 149 (1988).
[Crossref]

1987 (1)

M. A. Player, J. Phys. A 20, 3667 (1987).
[Crossref]

1986 (1)

R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, Proc. R. Soc., London Ser. A 392, 45 (1984); Nature (London),  326, 227 (1987).
[Crossref]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[Crossref]

1972 (1)

C. Imbert, Phys. Rev D 5, 787 (1972); O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[Crossref]

1955 (1)

F. I. Fedorov, Dokl. Acad. Nauk SSSR 102, 69 (1955).

1938 (1)

S. M. Rytov, Dokl. Acad. Nauk SSSR 18, 2 (1938); Tr. Fiz. Inst. Akad. Nauk SSSR 2, 1 (1940). (translated into English in Ref. 9, p. 6); V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 21, 222 (1941) (translated into English in Ref. 9, p. 11).

1926 (1)

F. Bortolotti, Rend. R. Acc. Naz. Linc. 6a 4, 552 (1926) [in Italian (?); cited in Ref. 3].

Baranova, N. B.

N. B. Baranova and B. Ya. Zel’dovich, JETP Lett. 59, 681 (1994).

Berestetskii, V. B.

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon, Oxford, 1984).

Berry, M. V.

M. V. Berry, Phys. Today 43(12), 34 (1990).
[Crossref]

M. V. Berry, Proc. R. Soc., London Ser. A 392, 45 (1984); Nature (London),  326, 227 (1987).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987).

Bortolotti, F.

F. Bortolotti, Rend. R. Acc. Naz. Linc. 6a 4, 552 (1926) [in Italian (?); cited in Ref. 3].

Chiao, R. Y.

R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[Crossref] [PubMed]

Darsht, M. Ya.

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

Dooghin, A. V.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

Esayan, A. A.

A. A. Esayan and B. Ya. Zel’dovich, Sov. J. Quantum Electron. 18, 149 (1988).
[Crossref]

Fedorov, F. I.

F. I. Fedorov, Dokl. Acad. Nauk SSSR 102, 69 (1955).

Imbert, C.

C. Imbert, Phys. Rev D 5, 787 (1972); O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[Crossref]

Kravtsov, Yu. A.

Yu. A. Kravtsov and Yu. I. Orlov, Geometric Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
[Crossref]

Kundikova, N. D.

N. D. Kundikova, Sov. J. Quantum Electron. 25, 2 (1995).

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[Crossref]

Liberman, V. S.

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. E 49, 2389 (1994).
[Crossref]

V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 4367 (1993).
[Crossref]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

B. Ya. Zel’dovich and V. S. Liberman, Sov. J. Quantum Electron. 20, 427 (1990).
[Crossref]

Lifshitz, E. M.

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon, Oxford, 1984).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[Crossref]

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Methuen, London, 1984).
[Crossref]

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown Press, Providence, R.I., 1944).

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

Markovski, B.

B. Markovski and V. I. Vinitsky, Topological Phases in Quantum Theory, (World Scientific, Singapore, 1989).

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[Crossref]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometric Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
[Crossref]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon, Oxford, 1984).

Player, M. A.

M. A. Player, J. Phys. A 20, 3667 (1987).
[Crossref]

Rytov, S. M.

S. M. Rytov, Dokl. Acad. Nauk SSSR 18, 2 (1938); Tr. Fiz. Inst. Akad. Nauk SSSR 2, 1 (1940). (translated into English in Ref. 9, p. 6); V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 21, 222 (1941) (translated into English in Ref. 9, p. 11).

Savchenko, A. Yu.

A. Yu. Savchenko and B. Ya. Zel’dovich, Bull. Russ. Acad. Sci. 58, Suppl. 3, 158 (1994) (in English).

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2287 (1994).
[Crossref]

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Methuen, London, 1984).
[Crossref]

Vinitsky, V. I.

B. Markovski and V. I. Vinitsky, Topological Phases in Quantum Theory, (World Scientific, Singapore, 1989).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987).

Wu, Y. S.

R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[Crossref] [PubMed]

Zel’dovich, B. Ya.

A. Yu. Savchenko and B. Ya. Zel’dovich, Bull. Russ. Acad. Sci. 58, Suppl. 3, 158 (1994) (in English).

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2287 (1994).
[Crossref]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. E 49, 2389 (1994).
[Crossref]

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

N. B. Baranova and B. Ya. Zel’dovich, JETP Lett. 59, 681 (1994).

V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 4367 (1993).
[Crossref]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

B. Ya. Zel’dovich and V. S. Liberman, Sov. J. Quantum Electron. 20, 427 (1990).
[Crossref]

A. A. Esayan and B. Ya. Zel’dovich, Sov. J. Quantum Electron. 18, 149 (1988).
[Crossref]

Zhirgalova, I. V.

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

Bull. Russ. Acad. Sci. (1)

A. Yu. Savchenko and B. Ya. Zel’dovich, Bull. Russ. Acad. Sci. 58, Suppl. 3, 158 (1994) (in English).

Dokl. Acad. Nauk SSSR (2)

F. I. Fedorov, Dokl. Acad. Nauk SSSR 102, 69 (1955).

S. M. Rytov, Dokl. Acad. Nauk SSSR 18, 2 (1938); Tr. Fiz. Inst. Akad. Nauk SSSR 2, 1 (1940). (translated into English in Ref. 9, p. 6); V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 21, 222 (1941) (translated into English in Ref. 9, p. 11).

J. Phys. A (1)

M. A. Player, J. Phys. A 20, 3667 (1987).
[Crossref]

JETP Lett. (2)

N. B. Baranova and B. Ya. Zel’dovich, JETP Lett. 59, 681 (1994).

M. Ya. Darsht, I. V. Zhirgalova, B. Ya. Zel’dovich, and N. D. Kundikova, JETP Lett. 59, 763 (1994).

Phys. Rev D (1)

C. Imbert, Phys. Rev D 5, 787 (1972); O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972).
[Crossref]

Phys. Rev. A (3)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[Crossref]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992).
[Crossref] [PubMed]

Phys. Rev. E (2)

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2287 (1994).
[Crossref]

V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. E 49, 2389 (1994).
[Crossref]

Phys. Rev. Lett. (1)

R. Y. Chiao and Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[Crossref] [PubMed]

Phys. Today (1)

M. V. Berry, Phys. Today 43(12), 34 (1990).
[Crossref]

Proc. R. Soc., London Ser. A (1)

M. V. Berry, Proc. R. Soc., London Ser. A 392, 45 (1984); Nature (London),  326, 227 (1987).
[Crossref]

Pure Appl. Opt. (1)

V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 4367 (1993).
[Crossref]

Rend. R. Acc. Naz. Linc. 6a (1)

F. Bortolotti, Rend. R. Acc. Naz. Linc. 6a 4, 552 (1926) [in Italian (?); cited in Ref. 3].

Sov. J. Quantum Electron. (3)

A. A. Esayan and B. Ya. Zel’dovich, Sov. J. Quantum Electron. 18, 149 (1988).
[Crossref]

N. D. Kundikova, Sov. J. Quantum Electron. 25, 2 (1995).

B. Ya. Zel’dovich and V. S. Liberman, Sov. J. Quantum Electron. 20, 427 (1990).
[Crossref]

Other (8)

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Pergamon, Oxford, 1984).

B. Markovski and V. I. Vinitsky, Topological Phases in Quantum Theory, (World Scientific, Singapore, 1989).

A. Snyder and J. Love, Optical Waveguide Theory (Methuen, London, 1984).
[Crossref]

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

R. K. Luneberg, Mathematical Theory of Optics (Brown Press, Providence, R.I., 1944).

Yu. A. Kravtsov and Yu. I. Orlov, Geometric Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Skew ray in a multimode fiber.

Fig. 2
Fig. 2

Bagel (top) and and hedgehog (bottom) modes for radial index n = 2.

Equations (46)

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

ˆ ik = δ ik ( x , y ) , μ ˆ ik = δ ik μ ( x , y ) .
n ( x , y ) = [ ( x , y ) μ ( x , y ) ] 1 / 2
ρ ( x , y ) = [ μ ( x , y ) ( x , y ) ] 1 / 2
W ik = E ˜ i ( R + r / 2 , z ) E ˜ k * ( R r / 2 , z ) exp ( i qr ) d x d y .
× E = i ω c μ H ( r , z ) , × H = i ω c E ( r , z ) .
2 E i z 2 + x k x k E i + x i ( E k ln x k ) + ω 2 c 2 μ E i + ln μ x k E k x i ln μ x k E i x k = 0 .
a ω c μ 1 .
ln μ x i Δ μ μ 1 a , ln x i Δ 1 a .
E ( r , z ) = E slow ( r , z ) exp ( i k 0 z ) ,
i E slow z = 1 2 k 0 x k x k E slow 1 2 k 0 [ ω 2 c 2 ( r ) μ ( r ) k 0 2 ] E slow ,
i E z = H ˆ 0 E , H ˆ 0 = 1 2 k 0 x k x k ω c [ n ( r ) n 0 ]
2 i k 0 E j z = x k x k E j + ω 2 c 2 [ n 2 ( r ) n 0 2 ] E j 1 4 k 0 2 ( 2 x m 2 2 x n 2 E j + ω 2 c 2 2 x m 2 × { E j [ n 2 ( r ) n 0 2 ] } ) + x j ( E k ln x k ) + ln μ x k E k x j ln μ x k E x k .
S z d x d y = c 8 π ρ 0 E ˜ * ( r , z ) · E ˜ ( r , z ) d x d y ,
E ˜ j ( r , z ) = [ 1 ρ ( r ) ρ 0 2 ρ 0 + 1 4 k 0 2 Δ ] E j 1 2 k 0 2 x j x k E k .
i E ˜ j z = 1 2 k 0 { Δ + 2 k 0 [ k ( r ) k 0 ] } E ˜ j + 1 8 k 0 3 Δ Δ E ˜ j + n ( r ) n 0 2 k 0 n 0 Δ E ˜ j + 1 2 k 0 ln n x k x k E ˜ j + 1 2 k 0 ( ln n x j x k ln n x k x j ) E ˜ k + 1 2 k 0 ( 2 ln ρ x k x j δ jk 2 2 ln ρ x m x m ) E ˜ k .
k z = k 0 + { [ ω 2 c 2 n 2 ( r ) + Δ ] 1 / 2 k 0 }
( δ ˆ δ μ ˆ μ ) ik = 1 k 2 ( 2 ln ρ x i x k ln ρ x i ln n x k ln ρ x k ln n x i ;
2 W ik ( q , R , z ) = W 0 [ 1 0 0 1 ] + W 1 [ 0 1 1 0 ] + W 2 [ 0 i i 0 ] + W 3 [ 1 0 0 1 ] .
W 0 z = C ˆ W 0 + M ˆ W 2 ,
W 1 z = C ˆ W 1 B 12 W 2 + R 13 W 3 ,
W 2 z = C ˆ W 2 + B 23 W 3 + B 12 W 1 + M ˆ W 0 ,
W 3 z = C ˆ W 3 B 23 W 2 R 13 W 1 ,
C ˆ W i = θ k ( W i ln n R k ) R k { θ k [ 1 δ n ( R ) n 0 ] W i } ,
R 13 W i = W i ( θ y ln n X θ x ln n Y ) ,
M ˆ W i = 1 k 0 Y ( W i ln n X ) 1 k 0 X ( W i ln n Y ) ,
B 23 W i = W i 1 k 0 2 ln ρ X Y , B 21 W i = W i 1 2 k 0 ( 2 ln ρ X 2 2 ln ρ Y 2 ) .
d X i d z = θ i [ 1 δ n ( R ) n ( R ) ] + σ e zki e z ln n X k ,
d θ i d z = ln n X i ,
d ϕ d z = R 13 2 = 1 2 ( θ x d θ y d z θ y d θ x d z ) ,
φ E y φ H y 2 ρ co ρ cl ρ n n co n cl ψ ( ψ TIR 2 ψ 2 ) 1 / 2 .
φ E y φ H y 2 ψ ( ψ TIR 2 ψ 2 ) 1 / 2 ,
N ( hits ) = π 2 α π a 0 4 b .
φ E φ H π a 0 b ψ ( ψ TIR 2 ψ 2 ) 1 / 2
b max a 0 δ n / n a 0 θ 2 .
Ψ mn ( r , φ ) exp ( i β mn z ) = R mn ( r ) exp ( im φ ) exp ( i β mn z ) ,
V = ω a 0 c ( n co 2 n cl 2 ) ω a 0 c 2 n δ n ,
ω δ n c V | β m β m + 1 | | β m β n + 1 | a 0 1 ( δ n / n ) 1 / 2 .
Φ + + = ( e x + i e y ) exp ( i | m | φ ) , Φ + = ( e x i e y ) exp ( i | m | φ ) , Φ + = ( e x + i e y ) exp ( i | m | φ ) , Φ = ( e x i e y ) exp ( i | m | φ ) ,
β + + β + θ 2 θ a 0 ( δ n / n ) 3 / 2 a 0 1 ,
Φ hedgehog , n ( r , φ ) = R 1 , n ( r ) ( Φ + + Φ + ) / 2 = R 1 , n ( r ) ( e x cos φ + e y sin φ ) ,
Φ bagel , n ( r , φ ) = R 1 , n ( r ) ( Φ + Φ + ) / 2 i = R 1 , n ( r ) ( e x sin φ + e y cos φ ) ,
ћ ( e x k 1 x + e y k 1 y ) = ћ ( e x k 2 x + e y k 2 y ) .
ћ ω n 1 x 1 sin α 1 / c + ћ σ cos α 1 = ћ ω n 2 x 2 sin α 2 / c + ћ σ cos α 2 .
x 2 x 1 = σ c ω n 1 cos α 1 cos α 2 sin α 1 .
x 2 x 1 σ c ω n ( α 2 α 1 )
r 2 r 1 = σ c ω n [ s 1 × ( s 2 s 1 ) ] ,

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