Abstract

A mathematical model for describing a holographic coupler–optical fiber system is considered. The object wave to be recorded in the hologram is assumed to be a polarized wave which is scattered by the optical fiber. In the reconstruction the optimal condition for the diffracted wavefront to be coupled to a second fiber is obtained and numerically analyzed.

© 1989 Optical Society of America

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References

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  1. P. K. Tien, “Integrated Optics and Wave Phenomena,” Rev. Mod. Phys. 49, 361–420 (1977); see also J. Gowar, Optical Communication Systems (Prentice-Hall, Englewood Cliffs, NJ, 1984).
    [CrossRef]
  2. O. D. D. Soares, “Holographic Coupler for Fiber Optics,” Opt. Eng. 20, 740–745 (1981).
    [CrossRef]
  3. E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
    [CrossRef]
  4. H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
    [CrossRef]
  5. G. Goldmann, H. H. Witte, “Holograms as Optical Branching Elements,” Opt. Quantum Electron. 9, 75–78 (1977).
    [CrossRef]
  6. T. Yoshino, T. Kubota, T. Ose, “Holographic Couplers for Monomode Fiber,” Appl. Opt. 22, 1800–1801 (1983).
    [CrossRef] [PubMed]
  7. M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
    [CrossRef]
  8. A. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1698 (1970).
  9. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).
  10. L. De Pedraza, M. L. Calvo, “Non Linear Effects in the Holographic Register: an Improved Alternative to Kozma’s Model,” Optik 72, 43–49 (1986).
  11. A. M. P. P. Leite, O. D. D. Soares, “Holographic Coupler for Fiber Optics Communications,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 10–17 (1980).
  12. A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
    [CrossRef]
  13. R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
    [CrossRef]
  14. R. J. Glauber, Lectures in Theoretical Physics, Vol. 1, W. E. Brittin, L. G. Dunham, Eds. (Interscience, New York, 1959).
  15. L. S. Watkins, “Scattering from Side-Illuminated Clad Glass Fibers for Determination of Fiber Parameter,” J. Opt. Soc. Am. 64, 767–772 (1974).
    [CrossRef]
  16. M. L. Calvo, P. Juncos, “Bidimensional Scattering of TM Waves by an Optical Fiber: Eikonal Approximation,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 35–37 (1980).
  17. O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).
  18. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  19. J. Rey Pastor et al., Análisis Matemático, Vol. 2 (Kapelusz, Buenos Aires, 1958).

1986 (1)

L. De Pedraza, M. L. Calvo, “Non Linear Effects in the Holographic Register: an Improved Alternative to Kozma’s Model,” Optik 72, 43–49 (1986).

1983 (1)

1981 (1)

O. D. D. Soares, “Holographic Coupler for Fiber Optics,” Opt. Eng. 20, 740–745 (1981).
[CrossRef]

1980 (3)

A. M. P. P. Leite, O. D. D. Soares, “Holographic Coupler for Fiber Optics Communications,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 10–17 (1980).

R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
[CrossRef]

M. L. Calvo, P. Juncos, “Bidimensional Scattering of TM Waves by an Optical Fiber: Eikonal Approximation,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 35–37 (1980).

1978 (1)

A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
[CrossRef]

1977 (3)

P. K. Tien, “Integrated Optics and Wave Phenomena,” Rev. Mod. Phys. 49, 361–420 (1977); see also J. Gowar, Optical Communication Systems (Prentice-Hall, Englewood Cliffs, NJ, 1984).
[CrossRef]

O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).

G. Goldmann, H. H. Witte, “Holograms as Optical Branching Elements,” Opt. Quantum Electron. 9, 75–78 (1977).
[CrossRef]

1975 (1)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

1974 (2)

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

L. S. Watkins, “Scattering from Side-Illuminated Clad Glass Fibers for Determination of Fiber Parameter,” J. Opt. Soc. Am. 64, 767–772 (1974).
[CrossRef]

1970 (1)

A. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1698 (1970).

1969 (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Alvarez-Estrada, R. F.

R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
[CrossRef]

Ash, E. A.

A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
[CrossRef]

O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

Calvo, M. L.

L. De Pedraza, M. L. Calvo, “Non Linear Effects in the Holographic Register: an Improved Alternative to Kozma’s Model,” Optik 72, 43–49 (1986).

R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
[CrossRef]

M. L. Calvo, P. Juncos, “Bidimensional Scattering of TM Waves by an Optical Fiber: Eikonal Approximation,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 35–37 (1980).

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

De Pedraza, L.

L. De Pedraza, M. L. Calvo, “Non Linear Effects in the Holographic Register: an Improved Alternative to Kozma’s Model,” Optik 72, 43–49 (1986).

Glauber, R. J.

R. J. Glauber, Lectures in Theoretical Physics, Vol. 1, W. E. Brittin, L. G. Dunham, Eds. (Interscience, New York, 1959).

Goldmann, G.

G. Goldmann, H. H. Witte, “Holograms as Optical Branching Elements,” Opt. Quantum Electron. 9, 75–78 (1977).
[CrossRef]

Inohara, S.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

Jordan, M. P.

M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
[CrossRef]

Juncos, P.

M. L. Calvo, P. Juncos, “Bidimensional Scattering of TM Waves by an Optical Fiber: Eikonal Approximation,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 35–37 (1980).

R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
[CrossRef]

Kogelnik, A.

A. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1698 (1970).

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kompfner, R.

M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
[CrossRef]

Koyama, J.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

Kubota, T.

Leite, A. M. P. P.

A. M. P. P. Leite, O. D. D. Soares, “Holographic Coupler for Fiber Optics Communications,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 10–17 (1980).

A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
[CrossRef]

O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).

Nishihara, H.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

Ose, T.

Pennington, K. S.

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

Rey Pastor, J.

J. Rey Pastor et al., Análisis Matemático, Vol. 2 (Kapelusz, Buenos Aires, 1958).

Seaford, E.

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

Sheppard, C. J. R.

M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
[CrossRef]

Soares, O. D. D.

O. D. D. Soares, “Holographic Coupler for Fiber Optics,” Opt. Eng. 20, 740–745 (1981).
[CrossRef]

A. M. P. P. Leite, O. D. D. Soares, “Holographic Coupler for Fiber Optics Communications,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 10–17 (1980).

A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
[CrossRef]

O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

Solymar, L.

M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
[CrossRef]

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

Sosnowski, T. P.

A. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1698 (1970).

Suhara, T.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

Tien, P. K.

P. K. Tien, “Integrated Optics and Wave Phenomena,” Rev. Mod. Phys. 49, 361–420 (1977); see also J. Gowar, Optical Communication Systems (Prentice-Hall, Englewood Cliffs, NJ, 1984).
[CrossRef]

Watkins, L. S.

Witte, H. H.

G. Goldmann, H. H. Witte, “Holograms as Optical Branching Elements,” Opt. Quantum Electron. 9, 75–78 (1977).
[CrossRef]

Yoshino, T.

AGARD Conf. Proc. (1)

O. D. D. Soares, A. M. P. P. Leite, E. A. Ash, AGARD Conf. Proc. 219, 44.1–44.13 (1977).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. A. Ash, E. Seaford, O. D. D. Soares, K. S. Pennington, “Holographic Coupler for Integrated Optics,” Appl. Phys. Lett. 24, 207–208 (1974).
[CrossRef]

Bell Syst. Tech. J. (2)

A. Kogelnik, T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1698 (1970).

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Quantum Electron. (1)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a Novel Coupler for Optical Circuits,” IEEE J. Quantum Electron. QE-11, 794–796 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Microwaves Opt. Acoust. (1)

A. M. P. P. Leite, O. D. D. Soares, E. A. Ash, “Optical Fiber Bundle Holographic Coupler,” Microwaves Opt. Acoust. 2, 45–59 (1978).
[CrossRef]

Opt. Acta (1)

R. F. Alvarez-Estrada, M. L. Calvo, P. Juncos, “Scattering of TM Waves by Dielectric Fibers: Iterative and Eikonal Solutions,” Opt. Acta 27, 1367–1378 (1980).
[CrossRef]

Opt. Eng. (1)

O. D. D. Soares, “Holographic Coupler for Fiber Optics,” Opt. Eng. 20, 740–745 (1981).
[CrossRef]

Opt. Quantum Electron. (1)

G. Goldmann, H. H. Witte, “Holograms as Optical Branching Elements,” Opt. Quantum Electron. 9, 75–78 (1977).
[CrossRef]

Optik (1)

L. De Pedraza, M. L. Calvo, “Non Linear Effects in the Holographic Register: an Improved Alternative to Kozma’s Model,” Optik 72, 43–49 (1986).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

A. M. P. P. Leite, O. D. D. Soares, “Holographic Coupler for Fiber Optics Communications,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 10–17 (1980).

M. L. Calvo, P. Juncos, “Bidimensional Scattering of TM Waves by an Optical Fiber: Eikonal Approximation,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 35–37 (1980).

Rev. Mod. Phys. (1)

P. K. Tien, “Integrated Optics and Wave Phenomena,” Rev. Mod. Phys. 49, 361–420 (1977); see also J. Gowar, Optical Communication Systems (Prentice-Hall, Englewood Cliffs, NJ, 1984).
[CrossRef]

Other (4)

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

M. P. Jordan, R. Kompfner, C. J. R. Sheppard, L. Solymar, “Coupling of Optical Fibers by Means of Volume Holograms,” in Proceedings, Sixth European Microoptic Conference, Rome, Italy (1976), pp. 438–442.
[CrossRef]

J. Rey Pastor et al., Análisis Matemático, Vol. 2 (Kapelusz, Buenos Aires, 1958).

R. J. Glauber, Lectures in Theoretical Physics, Vol. 1, W. E. Brittin, L. G. Dunham, Eds. (Interscience, New York, 1959).

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Figures (6)

Fig. 1
Fig. 1

Lateral illumination of the optical fiber and physical parameters.

Fig. 2
Fig. 2

Validity of the eikonal approximation for a TM wave scattered by an optical fiber: (a) interval defined as R1 > |X2| ≥ 0, R2 > |ρ| > R1 with X1 > 0. (b) Interval defined as |ρ| < R1, R1 > |X2| ≥ 0.

Fig. 3
Fig. 3

Holographic coupler: schematic representation of the experimental setup for the recording process.

Fig. 4
Fig. 4

Coupling condition inside the core of the optical fiber: (a) X2 = 2 μm; (b) X1 = 1 μm; (c) X2 = 0 μm. The upper curve represents the S function, the lower the P function.

Fig. 5
Fig. 5

Coupling condition inside the cladding and outside the core: (a) X2 = 50 μm; (b) X2 = 40 μm; (c) X2 = 30 μm.

Fig. 6
Fig. 6

Integrals of functions S (top) and P (bottom) describing the coupling condition across the X2 variable at the entrance face of fiber 2 (as in the scheme displayed in Fig. 3). The condition has been studied for γ = 10°.

Equations (54)

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( Δ p + K 2 ) E = { 0 , ρ outside Ω , K 2 ( 1 ) E , ρ inside Ω ,
E ( ρ ) = E 0 exp ( i K ρ ) ( 1 4 i ) Ω d 2 ρ H 0 ( 1 ) × ( K , ρ ρ ) K 2 [ ( ρ ) 1 ] E ( ρ )
E eik ( ρ ) = E 0 exp ( i K ρ ) ( 1 4 i ) × Ω d 2 ρ H 0 , eik ( 1 ) ( K , ρ ρ ) K 2 [ ( ρ ) 1 ] E eik ( ρ ) ,
H 0 , eik ( 1 ) ( K , ρ ρ ) = ( 2 / K ) exp [ i k ( X 1 X 1 ) ] × θ ( X 1 X 1 ) δ ( X 2 X 2 ) ,
E ( ρ ) E 0 { exp ( i K ρ ) + [ exp ( i K | ρ | ) / | ρ | 1 / 2 ] · T ( K , K ) } .
T eik ( K , K ) = ( K / 2 π ) 1 / 2 exp ( i π / 4 ) + d X 2 exp [ i K sen ( θ ) X 2 ] × { exp [ i φ eik ( X 1 = + , X 2 ) ] 1 } ,
φ eik ( X 1 , X 2 ) = ( K / 2 ) X 1 d X 1 [ ( X 1 , X 2 ) 1 ]
φ eik = 0 ; | X 2 | > R 2 ,
φ eik = K ( 2 1 ) ( R 2 2 X 2 2 ) 1 / 2 ; R 2 > | X 2 | > R 1 ,
φ eik = K { ( 2 1 ) [ ( R 2 2 X 2 2 ) 1 / 2 ( R 1 2 X 1 2 ) 1 / 2 ] + ( R 1 2 X 2 2 ) 1 / 2 [ ( 2 1 ) ( Δ 1 / 3 R 1 2 ) × ( R 1 2 + 2 X 2 2 ) ] } ; R 1 > | X 2 | > 0 ,
Δ = ( 1 2 ) / 2 1 .
T eik ( K , K ) = ( 2 K / π ) 1 / 2 exp ( i π / 4 ) ( 0 + R 1 d X 2 cos [ K sen ( θ ) X 2 ] × { exp [ i φ eik ( X 2 ) ] 1 } + + R 1 + R 2 d X 2 cos [ K sen ( θ ) X 2 ] × { exp [ i φ eik ( X 2 ) ] 1 } ) ,
ν = ( 1 4 ) ( 2 1 ) for R 2 > | X 2 | > R 1 , | ρ | < R 2 or for R 1 > | X 2 | 0 , R 2 > | ρ | > R 1 , X 1 < 0 ;
ν = N 2 = ( 1 4 ) ( 2 1 ) + [ 1 Δ / K R 1 ( 2 1 ) ] × [ λ ( R 2 , X 2 ) + λ ( R 1 , X 2 ) ] + [ 1 2 Δ 2 ( X 2 / R 1 ) 2 / ( 2 1 ) ] [ λ ( R 2 , X 2 ) + λ ( R 1 , X 2 ) ] 2
ν = N 1 = ( 1 4 ) [ 1 1 Δ ( | ρ | / R 1 ) 2 ] + { 1 Δ ( | X 1 | / R 1 ) / K R 1 [ 1 1 Δ ( | ρ | / R 1 ) 2 ] } + { 1 Δ [ λ ( R 2 , X 2 ) + X 1 / R 1 ] / K R 1 [ 1 1 Δ ( | ρ | / R 1 ) 2 ] } + { 1 2 Δ 2 ( X 2 / R 1 ) 2 [ λ ( R 2 , X 2 ) + X 1 / R 1 ] 2 / [ 1 1 Δ ( | ρ | / R 1 ) 2 ] }
U F 1 = O . O . ( 1 ) , U R 1 = O . R . ( 1 ) ,
U F 2 = O . O . ( 2 ) , U R 2 = O . R . ( 2 ) ,
U c = U F 1 T 1
T 1 | U F 1 + U R 1 | 2
U c T 2 = U F 2 * ,
U R 2 = U R 1 * ,
T 2 | U F 2 + U R 2 | 2
U c U F 1 U R 1 U F 1 * ,
( U R 2 * ) 2 | U F 1 | 2 U F 2 = ( 1 R 2 | U F 1 | 2 ) U F 2 * ,
U F j exp [ ( 1 ) j + 1 i φ eik ( X 2 j ) ] 1 ; j = 1 , 2 ,
| U F j | 2 2 { 1 cos [ φ eik ( x 2 j ) ] } ,
cos [ φ eik ( X 2 ) ] + cos [ φ eik ( X 2 ) + 2 K R · r ] ( 1 2 ) { cos 2 [ φ eik ( X 2 ) + K R · r ] + cos ( 2 K R · r ) } + 1 .
P = cos [ φ eik ( X 2 ) ] + cos [ φ eik ( X 2 ) + 2 K R · r ] ,
S = ( 1 2 ) { cos 2 [ φ eik ( X 2 ) + K R · r ] + cos [ 2 K R · r ] } + 1 .
= | R 1 R 2 d X 2 U F 2 | 2 ,
U R 1 = | R | exp ( i K R · r ) ,
K R = K ( sin γ , 0 , cos γ ) ; r = ( X , 0 , Z ) .
P , S = f ( K R · r ) .
INT P = R 2 + R 2 d X 2 P ( X 2 ) ,
INT S = R 2 + R 2 d X 2 S ( X 2 ) .
K R · r = K X sin γ,
K R · r = K Z cos γ,
U R 1 = | R | exp ( i K R · r ) .
2 R 2 exp [ 2 i K R · r ] { 1 cos [ φ eik 1 ( X 21 ) ] } × { exp [ i φ eik 2 ( X 22 ) ] 1 } = { 1 2 R 2 [ 1 cos [ φ eik 1 ( X 21 ) ] ] } { exp [ i φ eik 2 ( X 22 ) ] 1 } ,
F ( X 21 , X 22 ) = 0 , G ( X 21 , X 22 ) = 0 ,
F ( X 21 , X 22 ) = 2 R 2 { 1 cos [ φ eik 1 ( X 21 ) ] } × { cos [ φ eik 2 ( X 22 ) + 2 K R · r ] cos ( 2 K R · r ) + cos [ φ eik 2 ( X 22 ) ] 1 } cos [ φ eik 2 ( X 22 ) ] + 1 ,
G ( X 21 , X 22 ) = 2 R 2 { 1 cos [ φ eik 1 ( X 21 ) ] } × { sin [ φ eik 2 ( X 22 ) + 2 K R · r ] sin ( 2 K R · r ) + sin [ φ eik 2 ( X 22 ) ] } sin [ φ eik 2 ( X 22 ) ] .
J = | F / X 21 F / X 22 G / X 21 G / X 22 | 0 .
R 2 , d φ eik 1 ( X 21 ) / d X 21 , d φ eik 2 ( X 22 ) / d X 22 , | U F 1 | 2 0
cos [ φ eik 1 ( X 2 ) ] 1 | U F 1 | 2 4
2 R 2 { 1 cos [ φ eik 1 ( X 21 ) ] } { cos [ φ eik 2 ( X 22 ) + 2 K R · r ] cos ( 2 K R · r ) + cos [ φ eik 2 ( X 22 ) + 2 K R · r ] cos [ φ eik 2 ( X 22 ) ] cos ( 2 K R · r ) cos [ φ eik 2 ( X 22 ) ] + sin [ φ eik 2 ( X 22 ) + 2 K R · r ] sin ( 2 K R · r ) sin [ φ eik 2 ( X 22 ) ] sin ( 2 K R · r ) cos [ φ eik 2 ( X 22 ) ] { 1 + cos [ φ eik 2 ( X 22 ) + 2 K R · r ] } sin [ φ eik 2 ( X 22 ) ] sin [ φ eik 2 ( X 22 ) + 2 K R · r ] + cos [ φ eik 2 ( X 22 ) + 2 K R · r ] 1 0 .
cos [ φ eik 2 ( X 22 ) ] = 2 R 2 { 1 cos [ φ eik 1 ( X 21 ) ] } × { cos [ φ eik 2 ( X 22 ) + 2 K R · r ] cos ( 2 K R · r ) + cos [ φ eik 2 ( X 22 ) ] 1 } + 1 ,
sin [ φ eik 2 ( X 22 ) ] = 2 R 2 { 1 cos [ φ eik 1 ( X 21 ) ] } × { sin [ φ eik 2 ( X 22 ) + 2 K R · r ] sin ( 2 K R · r ) + sin [ φ eik 2 ( X 22 ) ] } .
R 2 , | U F 1 | 2 0 ,
cos [ φ eik 2 ( X 2 ) ] + cos [ φ eik 2 ( X 2 ) + 2 K R · r ] ( 1 / 2 ) { cos 2 [ φ eik 2 ( X 2 ) + K R · r ] + cos ( 2 K R · r ) } + 1 ,
INT P = R 2 + R 2 P ( X 2 ) d X 2 = 2 [ 0 + R 1 P ( X 2 ) d X 2 + + R 1 + R 2 P ( X 2 ) d X 2 ] ,
INT S = R 2 + R 2 S ( X 2 ) d X 2 = 2 [ 0 + R 1 S ( X 2 ) d X 2 + + R 1 + R 2 S ( X 2 ) d X 2 ] .
0 + R 1 P ( X 2 d X 2 = 0 + R 1 cos [ φ eik 2 ( X 2 ) ] d X 2 + 0 + R 1 cos [ φ eik 2 ( X 2 ) + 2 K R · r ] d X 2 ,
0 + R 1 S ( X 2 ) d X 2 = 0 + R 1 ( 1 / 2 ) cos 2 [ φ eik 2 ( X 2 ) + K R · r ] d X 2 + 0 + R 1 { ( 1 / 2 ) cos [ 2 K R · r ] + 1 } d X 2 ,

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