Abstract

We give a solution to the problem of light propagation in an inhomogeneous medium and in particular we show how to construct such an inhomogeneous medium so that it can produce quantumlike entangled functions. Using the correspondence between classical optics and quantum mechanics, we apply quantum optics methods to find the desired solution.

© 2007 Optical Society of America

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  1. M. Moshinsky, "Diffraction in time," Phys. Rev. 88, 625-631 (1952).
    [CrossRef]
  2. See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
    [CrossRef]
  3. H. M. Nussenzveigh, Introduction to Quantum Optics (Gordon & Breach, 1973).
  4. G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
    [CrossRef] [PubMed]
  5. D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
    [CrossRef]
  6. S. Danakas and P. K. Aravind, "Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom)," Phys. Rev. A 45, 1973-1977 (1992).
    [CrossRef] [PubMed]
  7. V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
    [CrossRef]
  8. S. Godoy, "Diffraction in time: Fraunhofer and Fresnel dispersion by a slit," Phys. Rev. A 65, 042111 (2002).
    [CrossRef]
  9. O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
    [CrossRef]
  10. D. Dragoman, "Phase space correspondence between classical optics and quantum mechanics," Prog. Opt. 42, 424-486 (2002).
  11. C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
    [CrossRef] [PubMed]
  12. M. A. M. Marte and S. Stenholm, "Paraxial light and atom optics: the optical Schrödinger equation and beyond," Phys. Rev. A 56, 2940-2953 (1997).
    [CrossRef]
  13. O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
    [CrossRef]
  14. J. Krug, "Optical analog of a kicked quantum oscillator," Phys. Rev. Lett. 59, 2133-2136 (2002).
    [CrossRef]
  15. M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
    [CrossRef]
  16. S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
    [CrossRef]
  17. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), p. 115.
  18. S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997), Chap. 2.
  19. S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
    [CrossRef] [PubMed]
  20. G. Arfken, Mathematical Methods for Physicists (Academic, 1985), p. 720.
  21. We could solve the equation exactly, but for the purposes of this paper, it is enough to give an approximate solution that, in the case of the conditions assumed here, has no difference with the exact solution.
  22. V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
    [CrossRef]
  23. T. Alieva and M. J. Bastiaans, "Radon-Wigner transform for optical field analysis," in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of the International Conference on Optics and Optoelectronics, O.P.Nijhawan, A.K.Gupta, A.K.Musla, and K.Singh, eds. (Narosa Publishing House, 1998), pp. 132-135.
  24. M. J. Bastiaans and K. B. Wolf, "Phase reconstruction from intensity measurements in linear systems," J. Opt. Soc. Am. A 20, 1046-1049 (2003).
    [CrossRef]

2005 (2)

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
[CrossRef]

2004 (1)

O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
[CrossRef]

2003 (1)

2002 (3)

J. Krug, "Optical analog of a kicked quantum oscillator," Phys. Rev. Lett. 59, 2133-2136 (2002).
[CrossRef]

D. Dragoman, "Phase space correspondence between classical optics and quantum mechanics," Prog. Opt. 42, 424-486 (2002).

S. Godoy, "Diffraction in time: Fraunhofer and Fresnel dispersion by a slit," Phys. Rev. A 65, 042111 (2002).
[CrossRef]

2001 (2)

M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
[CrossRef]

V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
[CrossRef]

1999 (1)

V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
[CrossRef]

1998 (1)

See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
[CrossRef]

1997 (1)

M. A. M. Marte and S. Stenholm, "Paraxial light and atom optics: the optical Schrödinger equation and beyond," Phys. Rev. A 56, 2940-2953 (1997).
[CrossRef]

1993 (2)

S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
[CrossRef] [PubMed]

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

1992 (2)

S. Danakas and P. K. Aravind, "Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom)," Phys. Rev. A 45, 1973-1977 (1992).
[CrossRef] [PubMed]

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

1983 (1)

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

1952 (1)

M. Moshinsky, "Diffraction in time," Phys. Rev. 88, 625-631 (1952).
[CrossRef]

Alieva, T.

T. Alieva and M. J. Bastiaans, "Radon-Wigner transform for optical field analysis," in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of the International Conference on Optics and Optoelectronics, O.P.Nijhawan, A.K.Gupta, A.K.Musla, and K.Singh, eds. (Narosa Publishing House, 1998), pp. 132-135.

Allen, L.

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

Aravind, P. K.

S. Danakas and P. K. Aravind, "Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom)," Phys. Rev. A 45, 1973-1977 (1992).
[CrossRef] [PubMed]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), p. 720.

Barnett, S. M.

S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997), Chap. 2.

Bastiaans, M. J.

M. J. Bastiaans and K. B. Wolf, "Phase reconstruction from intensity measurements in linear systems," J. Opt. Soc. Am. A 20, 1046-1049 (2003).
[CrossRef]

T. Alieva and M. J. Bastiaans, "Radon-Wigner transform for optical field analysis," in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of the International Conference on Optics and Optoelectronics, O.P.Nijhawan, A.K.Gupta, A.K.Musla, and K.Singh, eds. (Narosa Publishing House, 1998), pp. 132-135.

Bélanger, P. A.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

Crasser, O.

O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
[CrossRef]

Danakas, S.

S. Danakas and P. K. Aravind, "Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom)," Phys. Rev. A 45, 1973-1977 (1992).
[CrossRef] [PubMed]

Dragoman, D.

D. Dragoman, "Phase space correspondence between classical optics and quantum mechanics," Prog. Opt. 42, 424-486 (2002).

Drummond, P. D.

See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
[CrossRef]

Dutra, S. M.

S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
[CrossRef] [PubMed]

Gagnon, L.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

Godoy, S.

S. Godoy, "Diffraction in time: Fraunhofer and Fresnel dispersion by a slit," Phys. Rev. A 65, 042111 (2002).
[CrossRef]

Gorbachev, V. N.

V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
[CrossRef]

He, H.

See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
[CrossRef]

Kheruntsyan, K. V.

See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
[CrossRef]

Knight, P. L.

S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
[CrossRef] [PubMed]

Krug, J.

J. Krug, "Optical analog of a kicked quantum oscillator," Phys. Rev. Lett. 59, 2133-2136 (2002).
[CrossRef]

Mack, H.

O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
[CrossRef]

Man'ko, M. A.

M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
[CrossRef]

Man'ko, V.

V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
[CrossRef]

Man'ko, V. I.

M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
[CrossRef]

Marte, M. A. M.

M. A. M. Marte and S. Stenholm, "Paraxial light and atom optics: the optical Schrödinger equation and beyond," Phys. Rev. A 56, 2940-2953 (1997).
[CrossRef]

Monken, C. H.

S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
[CrossRef]

Moshinsky, M.

V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
[CrossRef]

M. Moshinsky, "Diffraction in time," Phys. Rev. 88, 625-631 (1952).
[CrossRef]

Moya-Cessa, H.

S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
[CrossRef] [PubMed]

Nienhuis, G.

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

Nussenzveigh, H. M.

H. M. Nussenzveigh, Introduction to Quantum Optics (Gordon & Breach, 1973).

Pádua, S.

S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
[CrossRef]

Paré, C.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

Radmore, P.

S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997), Chap. 2.

Schleich, W. P.

O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
[CrossRef]

Sharma, A.

V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
[CrossRef]

Stenholm, S.

M. A. M. Marte and S. Stenholm, "Paraxial light and atom optics: the optical Schrödinger equation and beyond," Phys. Rev. A 56, 2940-2953 (1997).
[CrossRef]

Steuernagel, O.

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

Trubilko, A. I.

V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
[CrossRef]

Vilela Mendes, R.

M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
[CrossRef]

Walborn, S. P.

S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
[CrossRef]

Wolf, K. B.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), p. 115.

Zhiliba, A. I.

V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
[CrossRef]

Am. J. Phys. (1)

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

Fluct. Noise Lett. (1)

O. Crasser, H. Mack, and W. P. Schleich, "Could Fresnel optics be quantum mechanics in phase space?," Fluct. Noise Lett. 4, L43-L51 (2004).
[CrossRef]

J. Opt. (1)

V. N. Gorbachev, A. I. Zhiliba, and A. I. Trubilko, "Teleportation of entangled states," J. Opt. 3, S25-S29 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phys. Lett. A (1)

M. A. Man'ko, V. I. Man'ko, and R. Vilela Mendes, "Quantum computation by quantumlike systems," Phys. Lett. A 288, 132-138 (2001).
[CrossRef]

Phys. Rev. (1)

M. Moshinsky, "Diffraction in time," Phys. Rev. 88, 625-631 (1952).
[CrossRef]

Phys. Rev. A (9)

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

S. Danakas and P. K. Aravind, "Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom)," Phys. Rev. A 45, 1973-1977 (1992).
[CrossRef] [PubMed]

V. Man'ko, M. Moshinsky, and A. Sharma, "Diffraction in time in terms of Wigner distributions and tomographic probabilities," Phys. Rev. A 59, 1809-1815 (1999).
[CrossRef]

S. Godoy, "Diffraction in time: Fraunhofer and Fresnel dispersion by a slit," Phys. Rev. A 65, 042111 (2002).
[CrossRef]

S. P. Walborn, S. Pádua, and C. H. Monken, "Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion," Phys. Rev. A 71, 053812 (2005).
[CrossRef]

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: an analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

M. A. M. Marte and S. Stenholm, "Paraxial light and atom optics: the optical Schrödinger equation and beyond," Phys. Rev. A 56, 2940-2953 (1997).
[CrossRef]

S. M. Dutra, P. L. Knight, and H. Moya-Cessa, "Discriminating field mixtures from macroscopic superpositions," Phys. Rev. A 48, 3168-3173 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Krug, "Optical analog of a kicked quantum oscillator," Phys. Rev. Lett. 59, 2133-2136 (2002).
[CrossRef]

See, for instance, P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-3058 (1998).
[CrossRef]

Prog. Opt. (1)

D. Dragoman, "Phase space correspondence between classical optics and quantum mechanics," Prog. Opt. 42, 424-486 (2002).

Other (6)

H. M. Nussenzveigh, Introduction to Quantum Optics (Gordon & Breach, 1973).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), p. 115.

S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997), Chap. 2.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), p. 720.

We could solve the equation exactly, but for the purposes of this paper, it is enough to give an approximate solution that, in the case of the conditions assumed here, has no difference with the exact solution.

T. Alieva and M. J. Bastiaans, "Radon-Wigner transform for optical field analysis," in Optics and Optoelectronics, Theory, Devices and Applications, Proceedings of the International Conference on Optics and Optoelectronics, O.P.Nijhawan, A.K.Gupta, A.K.Musla, and K.Singh, eds. (Narosa Publishing House, 1998), pp. 132-135.

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Equations (28)

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2 i k 0 E z = 2 E + k 2 ( x , y ) E ,
n n 0 + n 2 E p 0 2 ( n 2 E p 0 2 w p 2 ) r 2 .
[ x s y s ] = [ cos θ sin θ sin θ cos θ ] [ x p y p ] ,
k 2 ( x , y ) = k 0 2 ( α x x 2 + α y y 2 ) + λ x y ,
i E z = 2 2 k 0 E + [ k 0 2 1 2 ( k 0 α ̃ x 2 x 2 + k 0 α ̃ y 2 y 2 ) + λ x y ] E ,
i ψ z = 2 2 k 0 ψ 1 2 ( ( k 0 α ̃ x 2 x 2 + k 0 α ̃ y 2 y 2 ) + λ x y ) ψ .
a x = k 0 α ̃ x 2 + 1 2 k 0 α ̃ x d d x , a x = k 0 α ̃ x 2 1 2 k 0 α ̃ x d d x ,
a y = k 0 α ̃ y 2 + 1 2 k 0 α ̃ y d d y , a y = k 0 α ̃ y 2 1 2 k 0 α ̃ y d d y .
u m ( x ) = ( k 0 α ̃ x π ) 1 4 1 2 m m ! H m ( k 0 α ̃ x x ) e k 0 α ̃ x x 2 2 ,
a x u m ( x ) = m u m 1 ( x ) ,
a x u m ( x ) = m + 1 u m + 1 ( x ) .
i ψ z = [ α ̃ x ( n x + 1 2 ) + α ̃ y ( n y + 1 2 ) + λ x y ] ψ ,
i ϕ z = [ α ̃ x n x + α ̃ y n y + λ ( a x a y + a x a y ) ] ϕ H ϕ .
A = a x cos η + a y sin η , B = a x sin η + a y cos η ,
A = a x cos η + a y sin η , B = a x sin η + a y cos η .
A A = cos 2 η a x a x + sin 2 η a y a y + sin η cos η ( a y a x + a x a y ) ,
B B = cos 2 η a x a x + sin 2 η a y a y sin η cos η ( a y a x + a x a y ) ,
H = μ A A + ν B B ,
μ = 1 2 [ α ̃ x ( 1 + cos 2 η ) + α ̃ y ( 1 cos 2 η ) ]
ν = 1 2 [ α ̃ x ( 1 cos 2 η ) + α ̃ y ( 1 + cos 2 η ) ] ,
η = 1 2 tan 1 2 λ α ̃ x α ̃ y .
ϕ ( z ; x , y ) = exp [ i z ( μ A A + ν B B ) ] ϕ ( 0 ) .
u 0 ( x ) u 0 ( y ) = U 0 A ( x , y ) U 0 B ( x , y ) ,
u 1 ( x ) u 0 ( y ) = cos η U 1 A ( x , y ) U 0 B ( x , y ) sin η U 0 A ( x , y ) U 1 B ( x , y ) ,
u 0 ( x ) u 1 ( y ) = sin η U 1 A ( x , y ) U 0 B ( x , y ) + cos η U 0 A ( x , y ) U 1 B ( x , y ) ,
ϕ ( 0 ) = u 0 ( x ) u 1 ( y ) ,
ϕ ( z ; x , y ) = u 0 ( x ) u 1 ( y ) sin η cos η ( e i z μ e i z ν ) + u 1 ( x ) u 0 ( y ) ( cos 2 η e i z μ sin 2 η e i z ν ) .
ϕ = α 0 1 + β 1 0 ,

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