Abstract

General formulas describing field distributions and eigenvalue equations are obtained for both transverse-electric and transverse magnetic modes in a multilayer slab waveguide. New results show that additional multilayers can produce useful effects, such as increasing the cutoff values and the confinement factors of guided modes.

© 1987 Optical Society of America

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References

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  1. J. McKenna, “The excitation of planar dielectric waveguides at p-n junctions I,” Bell Syst. Tech. J. 46, 1491–1566 (1967).
  2. P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,”J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  3. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  5. J. D. Love, A. W. Snyder, “Optical fiber eigenvalue equation; plane wave derivation,” Appl. Opt. 15, 2121–2125 (1976).
    [CrossRef] [PubMed]
  6. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.
  7. Y. Yamamoto, T. Kamiya, H. Yanai, “Propagation characteristics of a partially metal-clad optical guide: metal-clad optical strip line,” Appl. Opt. 14, 322–326 (1974).
    [CrossRef]
  8. P. K. Tien, R. J. Martin, G. Smolinsky, “Formation of lightguiding interconnections in an integrated optical circuit by composite tapered-film coupling,” Appl. Opt. 12, 1909–1916 (1973).
    [CrossRef] [PubMed]
  9. W. Sohler, “Light-wave coupling to optical waveguides by a tapered cladding medium,” J. Appl. Phys. 44, 2343–2345 (1973).
    [CrossRef]
  10. M. J. Sun, M. W. Muller, “Measurements on four-layer isotropic waveguides,” Appl. Opt. 16, 814–815 (1977).
    [PubMed]
  11. G. E. Smith, “Phase matching in four-layer optical waveguides,” IEEE J. Quantum Electron. QE-4, 288–289 (1968).
    [CrossRef]
  12. D. F. Nelson, J. McKenna, “Electromagnetic modes of anisotropic dielectric waveguides at p-n junctions,” J. Appl. Phys. 38, 4057–4074 (1967).
    [CrossRef]
  13. Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).
  14. S. Ruschin, E. Marom, “Coupling effects in symmetrical three-guide structures,” J. Opt. Soc. Am. A 1, 1120–1128 (1984).
    [CrossRef]
  15. A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
    [CrossRef]
  16. P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
    [CrossRef]
  17. P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,”J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  18. L. M. Walpita, “Solutions for planar waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
    [CrossRef]
  19. J. F. Revelli, “Mode analysis and prism coupling for multilayered optical waveguides,” Appl. Opt. 20, 3158–3167 (1981).
    [CrossRef] [PubMed]
  20. H. Furuta, H. Noda, A. Ihaya, “Novel optical waveguide for integrated optics,” Appl. Opt. 13, 322–326 (1974).
    [CrossRef] [PubMed]
  21. C. B. Hocker, W. K. Burus, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [CrossRef] [PubMed]
  22. S. Wang, “Proposal of periodic layered wavguide structures for distributed lasers,” J. Appl. Phys. 44, 767–780 (1973).
    [CrossRef]
  23. E. Marom, Hughes Research Laboratories, Malibu, California 90265 (personal communication, September1986).

1985 (1)

1984 (1)

1981 (1)

1977 (3)

1976 (2)

1974 (4)

Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

H. Furuta, H. Noda, A. Ihaya, “Novel optical waveguide for integrated optics,” Appl. Opt. 13, 322–326 (1974).
[CrossRef] [PubMed]

Y. Yamamoto, T. Kamiya, H. Yanai, “Propagation characteristics of a partially metal-clad optical guide: metal-clad optical strip line,” Appl. Opt. 14, 322–326 (1974).
[CrossRef]

1973 (3)

P. K. Tien, R. J. Martin, G. Smolinsky, “Formation of lightguiding interconnections in an integrated optical circuit by composite tapered-film coupling,” Appl. Opt. 12, 1909–1916 (1973).
[CrossRef] [PubMed]

W. Sohler, “Light-wave coupling to optical waveguides by a tapered cladding medium,” J. Appl. Phys. 44, 2343–2345 (1973).
[CrossRef]

S. Wang, “Proposal of periodic layered wavguide structures for distributed lasers,” J. Appl. Phys. 44, 767–780 (1973).
[CrossRef]

1970 (1)

1968 (1)

G. E. Smith, “Phase matching in four-layer optical waveguides,” IEEE J. Quantum Electron. QE-4, 288–289 (1968).
[CrossRef]

1967 (2)

D. F. Nelson, J. McKenna, “Electromagnetic modes of anisotropic dielectric waveguides at p-n junctions,” J. Appl. Phys. 38, 4057–4074 (1967).
[CrossRef]

J. McKenna, “The excitation of planar dielectric waveguides at p-n junctions I,” Bell Syst. Tech. J. 46, 1491–1566 (1967).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.

Burus, W. K.

Fox, A. J.

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

Furuta, H.

Hocker, C. B.

Hong, C.-S.

Ihaya, A.

Kamiya, T.

Kawakami, S.

Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).

Love, J. D.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Marom, E.

S. Ruschin, E. Marom, “Coupling effects in symmetrical three-guide structures,” J. Opt. Soc. Am. A 1, 1120–1128 (1984).
[CrossRef]

E. Marom, Hughes Research Laboratories, Malibu, California 90265 (personal communication, September1986).

Martin, R. J.

McKenna, J.

J. McKenna, “The excitation of planar dielectric waveguides at p-n junctions I,” Bell Syst. Tech. J. 46, 1491–1566 (1967).

D. F. Nelson, J. McKenna, “Electromagnetic modes of anisotropic dielectric waveguides at p-n junctions,” J. Appl. Phys. 38, 4057–4074 (1967).
[CrossRef]

Muller, M. W.

Nelson, D. F.

D. F. Nelson, J. McKenna, “Electromagnetic modes of anisotropic dielectric waveguides at p-n junctions,” J. Appl. Phys. 38, 4057–4074 (1967).
[CrossRef]

Nishida, S.

Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).

Noda, H.

Ohtaka, Y.

Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).

Revelli, J. F.

Ruschin, S.

Smith, G. E.

G. E. Smith, “Phase matching in four-layer optical waveguides,” IEEE J. Quantum Electron. QE-4, 288–289 (1968).
[CrossRef]

Smolinsky, G.

Snyder, A. W.

Sohler, W.

W. Sohler, “Light-wave coupling to optical waveguides by a tapered cladding medium,” J. Appl. Phys. 44, 2343–2345 (1973).
[CrossRef]

Sun, M. J.

Tien, P. K.

Ulrich, R.

Walpita, L. M.

Wang, S.

S. Wang, “Proposal of periodic layered wavguide structures for distributed lasers,” J. Appl. Phys. 44, 767–780 (1973).
[CrossRef]

Yamamoto, Y.

Yanai, H.

Yariv, A.

Yeh, P.

Appl. Opt. (7)

Bell Syst. Tech. J. (1)

J. McKenna, “The excitation of planar dielectric waveguides at p-n junctions I,” Bell Syst. Tech. J. 46, 1491–1566 (1967).

IEEE J. Quantum Electron. (1)

G. E. Smith, “Phase matching in four-layer optical waveguides,” IEEE J. Quantum Electron. QE-4, 288–289 (1968).
[CrossRef]

J. Appl. Phys. (3)

D. F. Nelson, J. McKenna, “Electromagnetic modes of anisotropic dielectric waveguides at p-n junctions,” J. Appl. Phys. 38, 4057–4074 (1967).
[CrossRef]

W. Sohler, “Light-wave coupling to optical waveguides by a tapered cladding medium,” J. Appl. Phys. 44, 2343–2345 (1973).
[CrossRef]

S. Wang, “Proposal of periodic layered wavguide structures for distributed lasers,” J. Appl. Phys. 44, 767–780 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Proc. IEEE (1)

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

Trans. Inst. Electron. Commun. Eng. Jpn. (1)

Y. Ohtaka, S. Kawakami, S. Nishida, “Transmission characteristics of a multi-layer dielectric slab optical waveguide with strongly evanescent wave layers,” Trans. Inst. Electron. Commun. Eng. Jpn. 57C, 187–194 (1974).

Other (4)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.

E. Marom, Hughes Research Laboratories, Malibu, California 90265 (personal communication, September1986).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the structure of an L-layer waveguide, where L = l + m + 1.

Fig. 2
Fig. 2

Geometry of a symmetrical seven-layer waveguide.

Fig. 3
Fig. 3

Index of refraction as a function of x for an L-layer symmetrical waveguide, where L = 2m + 1.

Fig. 4
Fig. 4

Cutoff values vc of the first-order mode of an L-layer waveguide as a function of layer thickness ratio D. Labeling parameter gives the value of m, where L = 2m + 1.

Fig. 5
Fig. 5

Confinement factor Γ versus v for the lowest-order mode of an L-layer symmetrical waveguide with D = 0.5. Labeling parameter gives the value of m, where L = 2m + 1.

Equations (95)

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n 0 < n i             ( i = - m , , - 1 , + 1 , , + l ) .
t ± 0 = ± d 0 ,
t ± i = ± ( d 0 + k = 1 i d ± k )             ( + : i = 1 , , l - 1 ;             - : i = 1 , , m - 1 ) ,
× E = - μ H / t ,
× H = n i 2 E / t             ( i = - m , , - 1 , 0 , + 1 , , + l ) .
E 0 ( x ) = E + 0 cos [ h 0 ( x - d 0 ) + ϕ + 0 ] = E - 0 cos [ h 0 ( X + d 0 ) - ϕ - 0 ] .
E ± 1 ( x ) = E ± 0 cos ϕ ± 0 cosh [ p ± 1 ( x - t ± 0 ) ψ ± 1 ] cosh ψ ± 1 ( + : t + 0 < x < t + 1 ;             - : t - 1 < x < t - 0 ) .
E ± i ( x ) = E ± 0 cos ϕ ± 0 [ k = 1 i - 1 cosh ( p ± k d ± k - ψ ± k ) cosh ψ ± k ] × cosh [ p ± i ( x - t ± ( i - 1 ) ) ψ ± i ] cosh ψ ± i ( + : i = 2 , 3 , , l - 1 ; t + ( i - 1 ) < x < t + i ; - : i = 2 , 3 , , m - 1 ; t - i < x < t - ( i - 1 ) ) .
E ± i ( x ) = E ± 0 cos ϕ ± 0 [ k = 1 i - 1 cosh ( p ± k d ± k - ψ ± k ) cosh ψ ± k ] × exp [ p ± i ( x - t ± ( i - 1 ) ) ] ( + : i = l ,             t + ( l - 1 ) < x < + ;             - : i = m ,             - < x < t - ( m - 1 ) ) .
E ± 0 = E 0 exp [ ± j ( h 0 d 0 - ϕ ± 0 ) ]
E ± 0 = ( - 1 ) q E - 0 ,
h 0 2 = k 2 n 0 2 - β 2 ,
p i 2 = β 2 - k 2 n i 2             ( i = - m , , - 1 , + 1 , , + l ) ,
2 h 0 d 0 = ϕ + 0 + ϕ - 0 + q π             ( 1 = 0 , 1 , 2 , ) ,
ϕ ± 0 = tan - 1 ( p ± 1 h 0 tanh ψ ± 1 ) ,
ψ ± i = p ± i d ± i + tanh - 1 ( p ± ( i + 1 ) p ± i tanh ψ ± ( i + 1 ) )             ( + : i = 1 , 2 , , l - 2 ;             - : i = 1 , 2 , , m - 2 ) ,
ψ ± i = p ± i d ± i + tanh - 1 ( p ± ( i + 1 ) p ± i )             ( + : i = l - 1 ;             - : i = m - 1 ) .
H x = - β ω μ E y ,
H z = i ω μ E y x .
H 0 ( x ) = H + 0 cos [ h 0 ( x - d 0 ) + ϕ + 0 ] = H - 0 cos [ h 0 ( x + d 0 ) - ϕ - 0 ] .
H ± 1 ( x ) = H ± 0 cos ϕ ± 0 cosh [ p ± 1 ( x - t ± 0 ) ψ ± 1 ] cosh ψ ± 1 ( + : t + 0 < x < t + 1 ;             - : t - 1 < x < t - 0 ) .
H ± i ( x ) = H ± 0 cos ϕ ± 0 [ k = 1 i - 1 cosh ( p ± k d ± k - ψ ± k ) cosh ψ ± k ] × cosh [ p ± i ( x - t ± ( i - 1 ) ) ψ ± i ] cosh ψ ± i ( + : i = 2 , 3 , , l - 1 :             t + ( i - 1 ) < x < t + i ; - : i = 2 , 3 , , m - 1 ;             t - i < x < t - ( i - 1 ) ) .
H ± i ( x ) = H ± 0 cos ϕ ± 0 [ k = 1 i - 1 cosh ( p ± k d ± k - ψ ± k ) cosh ψ ± k ] × exp [ p ± i ( x - t ± ( i - 1 ) ) ] ( + : i = l ;             t + ( l - 1 ) < x < +             - : i = m ;             - < x < t - ( m - 1 ) ) .
H ± 0 = β β H 0 exp [ ± j ( h 0 d 0 - ϕ ± 0 ) ]
H + 0 = ( - 1 ) q H - 0 .
2 h 0 d 0 = ϕ + 0 + ϕ - 0 + q π             ( q = 0 , 1 , 2 , ) .
ϕ ± 0 = tan - 1 ( n 0 2 n ± 1 2 p ± 1 h 0 tanh ψ ± 1 ) ,
ψ ± i = p ± i d ± i + tanh - 1 ( n ± i 2 n ± ( i + 1 ) 2 p ± ( i + 1 ) p ± i tan ψ ± ( i + 1 ) )             ( + : i = 1 , 2 , , l - 2 ;             - : i = 1 , 2 , , m - 2 ) ,
ψ ± i = p ± i d ± i + tanh - 1 ( n ± i 2 n ± ( i + 1 ) 2 p ± ( i + 1 ) p ± i )             ( + : i = l - 1 ;             - : i = m - 1 ) ,
E x i = β n i 2 ω H i ,
E z i = - j n i 2 ω H i x ( i = - m , , - 1 , 0 , + 1 , , + l ) .
E - 0 cos ( h 0 d 0 - ϕ - 0 ) exp [ - p 1 ( x - t + 0 ) ]             ( t + 0 < x < ) ,
E - 0 cos [ h 0 ( x - t - 0 ) - ϕ - 0 ]             ( t - 0 < x < t + 0 ) ,
E - 0 cos ϕ - 0 exp [ p - 1 ( x - t - 0 ) ]             ( - < x < t - 0 ) .
ϕ ± 0 = tan - 1 ( p ± 1 / h 0 ) .
H - 0 cos ( h 0 d 0 - ϕ - 0 ) exp [ - p 1 ( x - t + 0 ) ]             ( t + 0 < x < ) ,
H - 0 cos [ h 0 ( x - t - 0 ) - ϕ - 0 ]             ( t - 0 < x < t + 0 ) ,
H - 0 cos ϕ - 0 exp [ p - 1 ( x - t - 0 ) ]             ( - < x < t - 0 ) .
ϕ ± 0 = tan - 1 ( n 0 2 n ± 1 2 p ± 1 h 0 ) .
E - 0 cos ( 2 h 0 d 0 - ϕ - 0 ) cosh ( p + 1 d + 1 - ψ + 1 ) cosh ψ + 1 exp [ - p + 2 ( x - t + 1 ) ]             ( t + 1 < x < ) ,
E - 0 cos ( 2 h 0 d 0 - ϕ - 0 ) cosh [ p + 1 ( x - t + 0 ) - ψ + 1 ] cosh ψ + 1             ( t + 0 < x < t + 1 ) ,
E - 0 cos [ h 0 ( x - t 0 ) - ϕ - 0 ]             ( t - 0 < x < t + 0 ) ,
E - 0 cos ϕ - 0 exp [ p - 1 ( x - t - 0 ) ]             ( - < x < t - 0 ) .
ϕ - 0 = tan - 1 ( p - 1 / h 0 ) ,
ϕ + 0 = tan - 1 [ ( p + 1 / h 0 ) tanh ψ + 1 ] ,
ψ + 1 = p + 1 d + 1 + tanh - 1 ( p + 2 / p + 1 ) .
2 h 0 d 0 = 2 ϕ 0 + q π             ( q = 0 , 1 , ) ,
tan ( h 0 d 0 ) = tan ϕ 0
- cot ( h 0 d 0 ) = tan ϕ 0
tan ϕ 0 = ( p 1 / h 0 ) tanh ψ 1 ,
tanh ψ i = tanh ( p i d i ) + ( p i + 1 / p i ) tanh ψ i + 1 1 + ( p i + 1 / p i ) tanh ( p i d i ) tanh ψ i + 1 ( i = 1 , 2 , , l - 2 ) ,
tanh ψ l - 1 = tanh ( p l - 1 d l - 1 ) + ( p l / p l - 1 ) 1 + ( p l / p l - 1 ) tanh ( p l - 1 d l - 1 ) .
h 0 p 1 A { 1 + p 3 p 1 tanh ( p 1 d 1 ) + [ p 3 p 2 + p 2 p 1 tanh ( p 1 d 1 ) ] tanh ( p 2 d 2 ) } = p 3 p 1 + tanh ( p 1 d 1 ) + [ p 2 p 1 + p 3 p 2 tanh ( p 1 d 1 ) ] tanh ( p 2 d 2 ) ,
A = { tan ( h 0 d 0 ) ( for even modes ) - cot ( h 0 d 0 ) ( for odd modes ) .
n 0 , n 2 > β > n 1 , n 3
p ¯ i = β 2 - k 2 n i 2 1 / 2             ( i = 1 , 2 , 3 ) ,
p 1 = p ¯ 1 , p 2 = j p ¯ 2 , p 3 = p ¯ 3 .
h 0 p 1 A { 1 + p 3 p 1 tanh ( p 1 d 1 ) + [ p 3 p ¯ 2 - p ¯ 2 p 1 tanh ( p 1 d 1 ) ] tanh ( p ¯ 2 d 2 ) } = p 3 p 1 + tanh ( p 1 d 1 ) - [ p ¯ 2 p 1 - p 3 p ¯ 2 tanh ( p 1 d 1 ) ] tanh ( p ¯ 2 d 2 ) .
h 0 p 1 A [ 1 + p 3 p 1 tanh ( p 1 d 1 ) ] = p 3 p 1 + tanh ( p 1 d 1 ) .
A = p 3 / h 0 ,
v 2 = k 2 d 0 2 ( n 0 2 - n - m 2 ) ,
u i 2 = k 2 d 0 2 ( n i 2 - N 2 ) .
w i 2 = k 2 d 0 2 ( N 2 - n i 2 )             ( i = - m , , - 1 , 0 , + 1 , , + l ) ,
N = β / k .
u i 2 = u 0 2 - v 2 c i 2 ,
w i 2 = - u i 2 = v 2 c i 2 - u 0 2 ,
c i 2 = n 0 2 - n i 2 n 0 2 - n - m 2 .
2 u 0 = ϕ + 0 + ϕ - 0 + q π             ( q = 0 , 1 , 2 , ) ,
ϕ ± 0 = tan - 1 ( w ± 1 u 0 tanh ψ ± 1 ) ,
ψ ± i = d ± i d 0 w ± i + tanh - 1 ( w ± ( i + 1 ) w ± i tanh ψ ± ( i + 1 ) )             ( + : i = 1 , 2 , , l - 2 ;             - : = 1 , 2 , , m - 2 ) ,
ψ ± i = d ± i d 0 w ± i + tanh - 1 ( w ± ( i + 1 ) w ± i )             ( + : i = l - 1 ;             - : i = m - 1 ) .
T 0 = E + 0 ,
T ± 1 = E ± 0 cos ϕ ± 0 ,
T ± i = E ± 0 cos ϕ ± 0 k = 1 i - 1 cosh ( d ± k d 0 w ± k ) × [ 1 - tanh ( d ± k d 0 w ± k ) tan ψ ± k ]             ( + : i = 2 , 3 , , l ;             - : i = 2 , 3 , , m ) ,
E 0 ( x ) = T 0 cos [ u 0 d 0 ( x - d 0 ) + ϕ + 0 ]             ( t - 0 < x < t + 0 ) ,
E ± i ( x ) = T ± i cosh [ ( w ± i / d 0 ) ( x - t ± ( i - 1 ) ) ψ + i ] cosh ψ ± i ( + : i = 1 , 2 , , l - 1 ;             t + ( i - 1 ) < x < t + i ; - : i = 1 , 2 , , m - 1 ;             t - i < x < t - ( i - 1 ) ) ,
E ± i ( x ) = T ± i exp [ ( w ± i / d 0 ) ( x - t ± ( i - 1 ) ) ] ( + : i = l ;             t + ( l - 1 ) < x < + ; - : i = m ;             - < x < t - ( m - 1 ) ) ,
P 0 = R 0 [ 2 + ( sin 2 ϕ + 0 + sin 2 ϕ - 0 ) / 2 u 0 ] ,
P i = R i { ( d i / d 0 ) + [ sinh 2 ( d i d 0 w i - ψ i ) + sinh 2 ψ i ] / 2 w i } [ i = - ( m - 1 ) , , - 1 , + 1 , , + ( l - 1 ) ] ,
P ± i = R ± i / w ± i             ( + : i = l ;             - : i = m ) ,
R 0 = d 0 ( β 2 ω μ ) ( E 0 2 2 ) ,
R i = d 0 ( β 2 ω μ ) ( T i 2 2 cosh 2 ψ i )             [ i = - ( m - 1 ) , , - 1 , + 1 , , + ( l - 1 ) ] ,
R ± i = d 0 ( β 2 ω μ ) ( T ± i 2 2 )             ( + : i = l ;             - : i = m ) ,
u 0 = ϕ 0 + ( q / 2 ) π             ( q = 0 , 1 , 2 , ) ,
ϕ 0 = tan - 1 ( w 1 u 0 tanh ψ 1 ) ,
ψ i = d i d 0 w i + tanh - 1 ( w i + 1 w i tanh ψ i + 1 )             ( i = 1 , 2 , , m - 2 ) ,
ψ m - 1 = d m - 1 d 0 w m - 1 + tanh - 1 ( w m w m - 1 ) .
w m = 0 ,
u 0 = v c ,
u i 2 = v c 2 ( 1 - c i 2 ) .
n i = { n 1 ( when i is odd ) n 2 ( when i is even ) ,
n 2 > n 1 .
D = d 1 d 0 = = d m - 1 d 0 = d d 0 .
c i 2 = { 2 ( when i is odd ) 1 ( when i is even ) .
Γ = P 0 P i .

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