Abstract

A three-dimensional analysis of bending losses in dielectric optical waveguides is presented. It constitutes a nontrivial generalization of previous two- and three-dimensional studies by other authors. Our analysis is based on homogeneous integral equations for the total radiation field and suitable asymptotic approximations for Green’s functions. A key role is played by a new three-dimensional approximation for a relevant Bessel function with large order and argument (the former being larger than the latter). A nontrivial check of the consistency of all those approximations is given. General formulas are presented for the radiated field and the energy flow and for a bending-loss coefficient in three dimensions. Numerical results are also given, in order to assess the difference between the results of other authors and ours. Such a difference is rather small for monomode behavior near cutoff, increases as the behavior of the waveguide changes from monomode to multimode, and decreases as the parameter V increases for a given core radius and propagation mode.

© 1987 Optical Society of America

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References

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  1. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).
  2. S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
    [CrossRef]
  3. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
    [CrossRef]
  4. J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
  5. E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
    [CrossRef]
  6. D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).
  7. J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
    [CrossRef]
  8. D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  9. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
    [CrossRef]
  10. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
    [CrossRef] [PubMed]
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.
  12. D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
    [CrossRef]
  13. Y. Takuma, M. Miyagi, S. Kawakami, “Bent asymmetric dielectric slab waveguides,” Appl. Opt. 20, 2291–2298 (1981); M. Miyagi, “Bending losses in hollow and dielectric tube leaky waveguides,” Appl. Opt. 20, 1221–1229 (1981).
    [CrossRef] [PubMed]
  14. A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
    [CrossRef]
  15. J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
    [CrossRef]
  16. M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).
  17. V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
    [CrossRef]
  18. W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
    [CrossRef]
  19. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.
  20. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.
  21. M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
    [CrossRef]
  22. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.
  23. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  24. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

1986 (1)

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

1982 (2)

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

1976 (3)

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
[CrossRef]

1975 (3)

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

1974 (3)

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

1973 (1)

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

1970 (1)

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

1956 (1)

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Alvarez-Estrada, R. F.

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

Calvo, M. L.

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Chang, D. C.

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

Erdelyi, A.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

Felsen, L. B.

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

Feschbach, H.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

Gambling, W. A.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Gloge, D.

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

Kawakami, S.

Kimura, T.

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Kuester, E. F.

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

Marcuse, D.

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
[CrossRef] [PubMed]

D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
[CrossRef]

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.

Matsumura, H.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Maurer, S. J.

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

Miller, M. A.

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Mitchell, D. J.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

Miyagi, M.

Morse, P. M.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Sakai, J.

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Sakai, J. I.

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

Shevchenko, V. V.

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Takuma, Y.

Talanov, V. I.

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

White, J. A.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (3)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

Electron. Lett. (2)

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. D (1)

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Radio Sci. (1)

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Radiophys. Quantum Electron. (1)

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

Z. Tekh. Fiz. (1)

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Other (6)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

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Figures (7)

Fig. 1
Fig. 1

Unbent three-dimensional waveguide with transverse cross section T. In this and the other figures O denotes the origin of coordinates, far outside the fiber.

Fig. 2
Fig. 2

Bent three-dimensional fiber. The part of the waveguide in −∞ < z < z0 (lower part) is perfectly straight. The dashed line represents one curvature radius R. The part above Z0 (upper part) is curved. Since R is large, the transverse cross sections above and below Z0 are approximately equal. Since the origin lies far from the waveguide, the range of possible values of y′ as x′ varies inside it is quite small, which justifies the approximations made in expression (8).

Fig. 3
Fig. 3

The angle formed by x ¯ ( x ¯ R ) with the y axis is θ (close to π/2). The projection of x ¯ on the xz plane is indicated by a dashed vector, and so is the angle between the xz plane and the x axis. The vector represented by (—·—·—) is the projection of x ¯ (not displayed) on the xz plane: the angle between such a projection and the x axis is φ′.

Fig. 4
Fig. 4

New local coordinates are defined with respect to some new origin O′ lying inside the transverse cross section T. The distance between the new (O′) and the former (O) origins is R. A short vector would have local coordinates (x1y′) with respect to the new origin O′.

Fig. 5
Fig. 5

Flow of radiated energy from a bent waveguide across a finite solid angle Ω on a spherical surface of very large radius | x ¯| (≫ R), the center of which is at the origin O. The solid angle is determined by π/2 − θ0 < θ < π/2 + θ0, φ1 < φ < φ2 (see the text). The double-line arrow (⇒) represents the outgoing time-averaged Poynting vector ½Re(Ē × H ¯ *).

Fig. 6
Fig. 6

Behavior of the ratio ξM/ξS at the upper bound (β = kn0) and lower bound (β = knc) versus the V parameter. Three values for the radius ρ of the fiber have been considered: 1.0, 2.0, and 10.0 μm, covering all the possibilities from a monomode to a multimode regime. Other parameters are R = 1.0 cm, λ = 0.9 μm, and Δ = 0.06.

Fig. 7
Fig. 7

Behavior of the ratios ξA/ξS (solid lines) and ln ξA/ξS (dashed lines) versus the V parameter at the upper bound β = kn0. Three values for the radius ρ of the fiber have been considered: 1.0 μm (almost monomode), 2.0 μm (intermediate regime), and 10.0 μm (multimode); a fixed radius of curvature R = 1.0 cm was used.

Equations (68)

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( Δ + k 2 c ) E ¯ = - k 2 ( - c ) E ¯ ,
( Δ + k 2 c ) H ¯ = - k 2 ( - c ) H ¯ ,
E ¯ pm ( x ¯ ) = exp ( i β z ) E ¯ 0 ( x ) ,             γ 2 = β 2 - k 2 c .
E ¯ 0 ( x ) = - ( 1 / 4 i ) T d x H 0 ( 1 ) ( i γ x - x ) k 2 [ ( x ) - c ] E ¯ 0 ( x ) .
H 0 ( 1 ) ( i γ x - x ) = - 4 i d l x d l y ( 2 π ) 2 exp { i [ l x ( x - x ) + l y ( y - y ) ] } l x 2 + l y 2 + γ 2 ,
( - Δ T + γ 2 ) i 4 H 0 ( 1 ) ( i γ x - x ) = δ ( 2 ) ( x - x ) ,
E ¯ ( x ¯ ) = d 3 x ¯ G ( x ¯ - x ¯ ) k 2 [ ( x ¯ ) - c ] E ¯ ( x ¯ ) ,
G ( x ¯ - x ¯ ) = ( 1 / 2 π ) 3 d 3 l ¯ exp [ i l ¯ ( x ¯ - x ¯ ) ] l ¯ 2 - ( k 2 c + i η ) = ( 1 / 4 π ) exp ( i k c 1 / 2 x ¯ - x ¯ ) x ¯ - x ¯ .
G ( x ¯ - x ¯ ) ( 1 / 4 π x ¯ ) exp ( i k c 1 / 2 x ¯ ) exp ( - i k ¯ x ¯ ) ,
k ¯ / k c 1 / 2 = x ¯ / x ¯ = ( cos φ sin θ , cos θ , sin φ sin θ ) .
z x = ( x 2 + z 2 ) 1 / 2 sin φ , z x = ( x 2 + z 2 ) 1 / 2 sin φ .
k ¯ x ¯ u cos ( φ - φ ) , x ¯ ( x 2 + z 2 ) 1 / 2 ; u = k c 1 / 2 sin θ ( x 2 + z 2 ) 1 / 2 .
exp ( - i k ¯ x ¯ ) M = - + exp [ i ( π / 2 ) M ] exp [ i M ( φ - φ ) ] J M ( u ) ,
[ x 2 + z 2 ] 1 / 2 R + x 1 .
E ¯ ( x ¯ ) exp ( i β z ) E ¯ 0 ( x 1 , y ) = E ¯ pm ;
E ¯ ( x ¯ ) ( 1 / x ¯ ) exp [ i ( k c 1 / 2 x ¯ + β R φ + π β R / 2 ) ] R e ¯ ,
e ¯ = ½ T d x 1 d y J β R ( u ) k 2 [ ( x 1 , y ) - 0 ] E ¯ 0 ( x 1 , y ) ,
J β R ( k c 1 / 2 sin θ R ) ( 1 / 2 π R ) 1 / 2 [ 1 / ( β 2 - k 2 c sin 2 θ ) 1 / 4 ] × exp { - ( β R / sin 3 θ ) × [ ( β 2 / k 2 c ) - sin 2 θ ] 3 / 2 } .
J β R exp [ β R Λ ( θ ) ] / [ 2 π R γ 1 ( θ ) ] 1 / 2 exp [ x 1 γ 1 ( θ ) ] ,
γ 1 ( θ ) = ( β 2 - k 2 c sin 2 θ ) 1 / 2
Λ ( θ ) = β - 1 γ 1 ( θ ) - tanh - 1 ] β - 1 γ 1 ( θ ) ] .
e ¯ = ½ exp [ β R Λ ( θ ) ] [ 2 π R γ 1 ( θ ) ] 1 / 2 e ¯ 1 ,
e ¯ 1 = T d x 1 d y exp [ x 1 γ 1 ( θ ) ] k 2 [ ( x 1 , y ) - c ] E ¯ 0 ( x 1 , y ) .
F in = - + d x 1 d y u ¯ z ½ Re ( E ¯ pm × H ¯ pm * ) = + d x 1 d y ½ Re ( E 0 x H 0 y * - E 0 y H 0 x * ) .
H ¯ ( x ¯ ) ( 1 / x ¯ ) exp { i [ k c 1 / 2 x ¯ + β R φ + π / 2 β R ] } R h ¯ ,
h ¯ = ½ exp [ β R Λ ( θ ) ] [ 2 π R γ 1 ( θ ) ] 1 / 2 h ¯ 1
h ¯ 1 = T d x 1 d y exp [ x 1 γ 1 ( θ ) ] k 2 [ ( x 1 , y ) - 0 ] H ¯ 0 ( x 1 , y ) .
F rad = x ¯ 2 Ω d Ω ( x ¯ / x ¯ ) ½ Re ( E ¯ × H ¯ * ) .
F rad ~ ( R / 16 π ) T d x 1 d y T d x 1 d y k 4 [ ( x 1 , y ) - c ] × [ ( x 1 , y ) - c ] α 1 α 1 = 0 N A ¯ α α Re [ E ¯ 0 α ( x 1 , y ) × H ¯ 0 α ( x 1 , y ) ] c α c α *
A ¯ α α = φ 1 φ 2 d φ sin θ d θ [ γ 1 α ( θ ) γ 1 α ( θ ) ] - 1 / 2 × exp { R [ β α Λ α ( θ ) + β α Λ α ( θ ) ] } × exp [ γ 1 α ( θ ) x 1 + γ 1 α ( θ ) x 1 ( x ¯ / x ¯ ) .
A ¯ α α ~ 2 ( γ α γ α ) - 1 / 2 exp ( γ α x 1 - γ α x 1 ) A 1 α α A ¯ 2 ,
A 1 α α = 0 π / 2 d θ exp { R [ β α Λ α ( θ ) + β α Λ α ( θ ) ] } [ π / R ( γ α + γ α ) ] 1 / 2 exp { R [ β α Λ α ( π / 2 ) + β α Λ α ( π / 2 ) ] }
A ¯ 2 = φ 1 φ 2 d φ ( x ¯ / x ¯ ) .
F rad R 1 / 2 exp [ 2 β R Λ ( π / 2 ) ] 2 φ 0 F ,
F = [ ( 2 π γ 3 ) 1 / 2 ] T d x 1 d y T d x 1 d y k 4 [ ( x 1 , y ) - c ] × [ ( x 1 , y ) - c ] exp [ γ ( x 1 - x 1 ) ] × { u ¯ z Re [ E ¯ 0 ( x 1 , y ) × H ¯ 0 ( x 1 , y ) ] } .
τ = ( 1 / F in ) ( F rad / 2 φ 0 R ) = exp { - 2 R β [ tanh - 1 ( β - 1 γ ) - β - 1 γ ] } R 1 / 2 F F in ,
τ M = γ 2 β ( 1 + γ d ) ( - c ) k 2 - γ 2 ( - c ) k 2 exp ( 2 γ d ) exp [ - ( γ 3 / β 2 ) R ] ,
τ SL = ( π V 8 / 16 ρ R W 3 ) exp [ - / ( R / ρ ) ( Δ W 3 / V 2 ) ] × [ 0 { 1 - f ( r ) } F 0 ( r ) r d r ] / 0 F 0 2 ( r ) r d r
n 2 ( r ) = n c o 2 [ 1 - 2 Δ f ( r ) ] Δ = ( n c o 2 - n c l 2 ) / 2 n c o 2 , V = k ρ ( n c o 2 - n c l 2 ) , W = ρ ( β 2 - k 2 n c l 2 ) 1 / 2 ,
ξ M = 2 3 ( γ 3 / β 2 ) R
ξ S = / ( R / ρ ) ( Δ W 3 / V 2 ) ,
ξ A = 2 β R [ tanh - 1 ( γ / β ) - ( γ / β ) ] ,
( ξ A / ξ M ) - 1 ( W α / ρ β α ) 2 = [ 1 - ( k n c l / β α ) 2 ] .
ξ M / ξ S = ( V α / ρ β α ) 2 ( 1 / 2 Δ )
g = exp { ξ S [ ( ξ M / ξ S ) - 1 ] } ,
ξ M - ξ S = ξ S [ ( ξ M / ξ S ) - 1 ] = ln g .
J β R [ k c 1 / 2 sin ( x 2 + z 2 ) 1 / 2 ] 1 / { [ 2 π β R tanh α ] 1 / 2 } × exp [ β R ( tanh α - α ) ] } ,
cosh α = β R / k c 1 / 2 sin θ ( x 2 + z 2 ) 1 / 2 ( β / k c 1 / 2 sin θ ) [ 1 - ( x 1 / R ) ]
β R v = β R [ ( cosh α ) 2 - 1 ] 1 / 2 / cosh α = ( β R ) 2 - k 2 c sin θ ( x 2 + z 2 ) 1 / 2 R γ 1 ( θ ) .
α - tanh α = n = 3 ( v ) n / n ,
( v ) n [ γ 1 ( θ ) / β ] n [ 1 - γ 1 ( θ ) ( 2 x 1 / R ) ] n / 2 [ γ 1 ( θ ) / β ] n [ 1 - n σ ( θ ) ( x 1 / R ) ] ,
σ ( θ ) = k 2 c sin θ / [ γ 1 ( θ ) ] 2 .
α - tanh α n = 3 ( 1 / n ) [ γ 1 ( θ ) / β ] n - σ ( θ ) ( x 1 / R ) n = 3 [ γ 1 ( θ ) / β ] n = tanh - 1 [ γ 1 ( θ ) / β ] - [ γ 1 ( θ ) / β ] - σ ( θ ) ( x 1 / R ) [ γ 1 ( θ ) / β ] 3 / { 1 - [ γ 1 ( θ ) / β ] 2 } .
β σ ( θ ) [ γ 1 ( θ ) / β ] 3 / { 1 - [ γ 1 ( θ ) / β ] 2 } = γ 1 ( θ )
G ( x ¯ - x ¯ ) = ( 1 / 4 π ) d l y exp [ i l y ( y - y ) ] × ( - 1 / 4 i ) M = - + exp [ i M ( φ - φ ) ] × H M ( 1 ) [ ( k 2 c - l y 2 ) 1 / 2 r > ] × J M [ ( k 2 c - l y 2 ) 1 / 2 r < ] .
E ¯ ( x ¯ ) exp ( i β R φ ) T d x 1 d y Q k 2 [ ( x 1 , y ) - c ] E ¯ 0 ( x 1 , y ) ,
Q = ( - 1 / 2 i ) R d l y 2 π exp [ i l y ( y - y ) ] × H β R ( 1 ) [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] × J β R [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] .
Q ~ ( - 1 / 4 i ) H 0 ( 1 ) { i γ [ ( x 1 - x 1 ) 2 + ( y - y ) 2 ] 1 / 2 }
J β R [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] [ 1 / ( 2 π β R tanh α ) ] × exp [ β R ( tanh α - α ) ] ,
cosh α = β R ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 [ β / ( k 2 c - l y 2 ) 1 / 2 ] [ 1 - ( x 1 / R ) ] .
R tanh α R γ 2 ;             γ 2 = ( γ 2 + l y 2 ) 1 / 2 ,             σ 1 = ( k 2 c - l y 2 ) / γ 2 2 ,
α - tanh α tanh - 1 [ γ 2 / β ] - [ γ 2 / β ] - σ 1 ( x 1 / R ) [ ( γ 2 / β ) 3 ] / [ 1 - ( γ / β ) 2 ] .
σ 1 β ( γ 2 / β ) 3 / [ 1 - ( γ 2 / β ) 2 ] = γ 2 .
J β R [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] ( exp { β R [ ( γ 2 / β ) - tanh - 1 ( γ 2 / β ) ] } ) / { [ ( 2 π R γ 2 ) 1 / 2 ] exp ( γ 2 x 1 ) } .
H β R ( 1 ) [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] - i exp [ β R ( α - tanh α ) ] / [ ( π / 2 ) β R tanh α ] 1 / 2 ,
cosh α = [ β R ] / [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ( β / [ ( k 2 c - l y 2 ) 1 / 2 ] [ 1 + ( x 1 / R ) ] .
H β R ( 1 ) [ ( k 2 c - l y 2 ) 1 / 2 ( x 2 + z 2 ) 1 / 2 ] - i exp { β R [ tanh - 1 ( γ 2 / β ) - ( γ 2 / β ) ] } / [ ( π / 2 ) R γ 2 ] 1 / 2 × exp ( - γ 2 x 1 ) .
Q = 1 / 2 π - + d l y exp [ i l y ( y - y ) ] × exp [ γ 2 ( x 1 - x 1 ) ] / 2 γ 2 = 1 / 2 π - + d l y exp [ i l y ( y - y ) ] × 1 / 2 π - + d l x exp [ i l x ( x 1 - x 1 ) ] ( l x 2 + l y 2 + γ 2 ) .

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