Abstract

The zero-dispersion characteristics are shown of coupled modes in twin waveguides when the medium of the substrate is lossy or active or when leakage exists. When the coupling strength of coupled modes composed of even and odd modes is critical, the phase velocities of these modes degenerate to make dispersion zero. The condition for zero dispersion is analytically obtained. The characteristics of reflection coefficients at various boundaries in the twin-waveguiding structure are studied in terms of the dual relation in the case in which the medium of the substrate is lossy or is active. The effect of the degenerate modes caused by leakage on the phase-shift characteristics of reflected waves at the prism–twin-waveguide boundaries is also studied.

© 1982 Optical Society of America

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References

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  1. V. N. Smiley, “An active interference filter as an optical maser amplifier,” Proc. IEEE 51, 120–124 (1963).
    [Crossref]
  2. H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
    [Crossref]
  3. J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
    [Crossref]
  4. S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
    [Crossref]
  5. P. K. Tien and R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [Crossref]
  6. J. E. Midwinter, “Evanescent field coupling into a thin-film waveguide,” IEEE J. Quantum Electron. QE-6, 583–590 (1970).
    [Crossref]
  7. T. Tamir and H. L. Bertoni, “Lateral displacement of optical beam at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [Crossref]
  8. M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
    [Crossref]
  9. K. O. Hill, R. J. MacDonald, and A. Watanabe, “Evanescent-wave amplification in asymmetric-slab waveguides,” J. Opt. Soc. Am. 64, 263–273 (1974).
    [Crossref]
  10. P. R. Callary and C. K. Caniglia, “Internal reflection from an amplifying layer,” J. Opt. Soc. Am. 66, 775–779 (1976).
    [Crossref]
  11. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, “Enhanced total reflection with surface plasmons,” J. Opt. Soc. Am. 69, 419–422 (1979).
    [Crossref]
  12. M. Yamada and Y. Suematsu, “Analysis of an integrated twin-guide laser with coupled-wave theory,” IEEE J. Quantum Electron. QE-13, 201–205 (1977).
    [Crossref]
  13. Y. Suematsu and K. Kishino, “Coupling coefficient of coupled dielectric waveguides,” Radio Sci. 12, 87–92 (1977).
    [Crossref]
  14. V. Shah and T. Tamir, “Brewster phenomena in lossy structure,” Opt. Commun. 23, 113–117 (1977).
    [Crossref]
  15. F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, New York, 1955).
  16. H. Kitajima and K. Hano, “Anomalies of electromagnetic waves in multilayered structures containing anisotropy,” J. Opt. Soc. Am. 68, 1693–1701 (1978).
    [Crossref]
  17. H. Kitajima and K. Fujita, “Double-tuned light-absorption characteristics in the attenuated-total-reflection structure containing a metal-clad dielectric-film waveguide with a low-refractive-index buffer layer,” Opt. Lett. 6, 392–394 (1981).
    [Crossref] [PubMed]

1981 (1)

1979 (1)

1978 (1)

1977 (3)

M. Yamada and Y. Suematsu, “Analysis of an integrated twin-guide laser with coupled-wave theory,” IEEE J. Quantum Electron. QE-13, 201–205 (1977).
[Crossref]

Y. Suematsu and K. Kishino, “Coupling coefficient of coupled dielectric waveguides,” Radio Sci. 12, 87–92 (1977).
[Crossref]

V. Shah and T. Tamir, “Brewster phenomena in lossy structure,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

1976 (1)

1974 (1)

1972 (1)

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

1971 (1)

1970 (2)

P. K. Tien and R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[Crossref]

J. E. Midwinter, “Evanescent field coupling into a thin-film waveguide,” IEEE J. Quantum Electron. QE-6, 583–590 (1970).
[Crossref]

1969 (1)

S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
[Crossref]

1964 (1)

J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
[Crossref]

1963 (2)

V. N. Smiley, “An active interference filter as an optical maser amplifier,” Proc. IEEE 51, 120–124 (1963).
[Crossref]

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

Bertoni, H. L.

Brand, F. A.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

Burlamacchi, P.

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

Callary, P. R.

Caniglia, C. K.

Chang, M. S.

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

Fujita, K.

Hano, K.

Hatkin, L.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

Hill, K. O.

Holmes, D. A.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

Hu, C.

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

Jacobs, H.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

Kishino, K.

Y. Suematsu and K. Kishino, “Coupling coefficient of coupled dielectric waveguides,” Radio Sci. 12, 87–92 (1977).
[Crossref]

Kitajima, H.

MacDonald, R. J.

Marinace, J. C.

J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
[Crossref]

Michel, A. E.

J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
[Crossref]

Midwinter, J. E.

J. E. Midwinter, “Evanescent field coupling into a thin-film waveguide,” IEEE J. Quantum Electron. QE-6, 583–590 (1970).
[Crossref]

Miller, S. E.

S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
[Crossref]

Nathan, M. I.

J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
[Crossref]

Plotz, G. A.

Shah, V.

V. Shah and T. Tamir, “Brewster phenomena in lossy structure,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

Simon, H. J.

Smiley, V. N.

V. N. Smiley, “An active interference filter as an optical maser amplifier,” Proc. IEEE 51, 120–124 (1963).
[Crossref]

Suematsu, Y.

M. Yamada and Y. Suematsu, “Analysis of an integrated twin-guide laser with coupled-wave theory,” IEEE J. Quantum Electron. QE-13, 201–205 (1977).
[Crossref]

Y. Suematsu and K. Kishino, “Coupling coefficient of coupled dielectric waveguides,” Radio Sci. 12, 87–92 (1977).
[Crossref]

Tamir, T.

Terman, F. E.

F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, New York, 1955).

Tien, P. K.

Tucciarone, J. M.

Ulrich, R.

Watanabe, A.

Whinnery, J. R.

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

Yamada, M.

M. Yamada and Y. Suematsu, “Analysis of an integrated twin-guide laser with coupled-wave theory,” IEEE J. Quantum Electron. QE-13, 201–205 (1977).
[Crossref]

Appl. Phys. Lett. (1)

M. S. Chang, P. Burlamacchi, C. Hu, and J. R. Whinnery, “Light amplification in a thin film,” Appl. Phys. Lett. 20, 313–314 (1972).
[Crossref]

Bell Syst. Tech. J. (1)

S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
[Crossref]

IEEE J. Quantum Electron. (2)

J. E. Midwinter, “Evanescent field coupling into a thin-film waveguide,” IEEE J. Quantum Electron. QE-6, 583–590 (1970).
[Crossref]

M. Yamada and Y. Suematsu, “Analysis of an integrated twin-guide laser with coupled-wave theory,” IEEE J. Quantum Electron. QE-13, 201–205 (1977).
[Crossref]

J. Appl. Phys. (1)

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, “Maximum gain for forward- and backward- wave optical maser amplifiers,” J. Appl. Phys. 34, 2617–2624 (1963).
[Crossref]

J. Opt. Soc. Am. (6)

Opt. Commun. (1)

V. Shah and T. Tamir, “Brewster phenomena in lossy structure,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (2)

J. C. Marinace, A. E. Michel, and M. I. Nathan, “Triangular injection laser,” Proc. IEEE 52, 722–723 (1964).
[Crossref]

V. N. Smiley, “An active interference filter as an optical maser amplifier,” Proc. IEEE 51, 120–124 (1963).
[Crossref]

Radio Sci. (1)

Y. Suematsu and K. Kishino, “Coupling coefficient of coupled dielectric waveguides,” Radio Sci. 12, 87–92 (1977).
[Crossref]

Other (1)

F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, New York, 1955).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem and distribution of refractive indices. lj (j = 1–5) denotes each interface.

Fig. 2
Fig. 2

Dispersion characteristics of coupled mode, df0 versus sin θ3 (= β/n3), where n″ = 0. The curves labeled (1) and (2) correspond to the modes of the first and second waveguides, respectively, when dbb0 = ∞. The thicknesses normalized by λ0 of the modes at sin θ3 = 0.9578 are df10 = 0.4663 and df20 = 0.6151, respectively. When they couple with each other through the buffer of db0 = 1.6, phase matching [Eqs. (20) and (21)] is satisfied at points I and II, respectively.

Fig. 3
Fig. 3

Vector trajectories of Ř(l3) when n ^ = n′ − in″ for various thicknesses of the buffer layer, Ř(l3)(−in″) versus sin θ3 (= β/n3), where n″ = 0.0005.

Fig. 4
Fig. 4

Vector trajectories of Ř(l3) when n ^ = n′ + in″ for various thicknesses of the buffer layer, Ř(l3)(+in″) versus sin θ3 (= β/n3), where n″ = 0.0005.

Fig. 5
Fig. 5

Dispersion characteristics of the first film waveguide for various thicknesses of the buffer layer, df10 versus sin θ3 (= β/n3) where n″ = 0.0005.

Fig. 6
Fig. 6

Curves showing thicknesses of the buffer and the gap layers for zero dispersion and zero reflection, db0 and dg0 versus n″.

Fig. 7
Fig. 7

Dispersion characteristics of the first film waveguide for various thicknesses of the buffer layer showing curves close to zero-dispersion, df10 versus sin θ3 (= β/n3), where n″ = 0.0005.

Fig. 8
Fig. 8

Curves showing the thickness relation between the gap and the buffer obtained through Eq. (45) for various values of n″, dg0 versus db0.

Fig. 9
Fig. 9

Curves showing the gap thickness normalized by λ0 obtained from Eqs. (41) for various values of n″, dg0 versus sin θ1 (= β/n1).

Fig. 10
Fig. 10

Absorption curves showing double-tuned characteristics, |B1|2 versus sin θ1, where n″ = 0.0005.

Fig. 11
Fig. 11

Phase shift of ϕl1 versus sin θ1, where B1 = exp(l1) and n″ = 0.

Tables (2)

Tables Icon

Table 1 Refractive Indices of Media at λ0 = 6328 Å

Tables Icon

Table 2 Phase Shift of ϕl3 When Δϕ2 Changes from Negative to Positivea

Equations (64)

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n j = ( j / 0 ) 1 / 2             ( j = 1 - 5 ) , n 6 = n ^ = n i n             ( - , lossy ; + , active ) , ( μ j / μ 0 ) 1 / 2 = 1             ( j = 1 - 6 ) , k 0 = ω ( μ 0 0 ) 1 / 2 = 2 π / λ 0             ( λ 0 , wavelength in vacuum ) .
H y j = exp [ i ( ω t - β x ) ] [ A j exp ( - i γ j z ) + B j exp ( i γ j z ) ] ,             j = 1 - 6 ,
A 1 = 1 ( incidence ) ,             B 6 = 0 ,             β / k 0 = n j sin θ j ,
γ j / k 0 = n j cos θ j ,             Re ( γ j ) 0 ,             Im ( γ j ) 0 ,             n ^ = n + i n ,             Re ( γ 6 ) < 0.
B 1 = Ř ( l 1 ) = R 12 exp ( i ϕ 12 ) + Ř ( l 2 ) exp ( - i 2 γ 2 d 1 ) 1 + R 12 exp ( i ϕ 12 ) Ř ( l 2 ) exp ( - i 2 γ 2 d 1 ) ,
R j k exp ( i ϕ j k ) = γ j / j - γ k / k γ j / j + γ k k , exp ( i ϕ k j ) = - exp ( i ϕ j k ) .
Ř ( l j ) = B j A j exp ( i 2 γ j l j ) = R j k exp ( i ϕ j k ) + Ř ( l k ) exp ( - i 2 γ k d j ) 1 + R j k exp ( i ϕ j k ) Ř ( l k ) exp ( - i 2 γ k d j ) ,
R ^ ( l j ) = A k B k exp ( - i 2 γ k l j ) = R k j exp ( i ϕ k j ) + R ^ ( l j - 1 ) exp ( - i 2 γ j d j - 1 ) 1 + R k j exp ( i ϕ k j ) R ^ ( l j - 1 ) exp ( - i 2 γ j d j - 1 ) .
d 1 d g ( gap ) ,             d 2 d f 1 ( first film ) ,             d 3 d b ( buffer ) ,             d 4 d f 2 ( second film ) ,
i γ 2 d g = a g ,             i γ 4 d b = a b ,             γ 3 d f 1 = ϕ f 1 ,             γ 5 d f 2 = ϕ f 2 .
exp ( i ϕ 12 ) + Ř ( l 2 ) exp ( - 2 a g ) = 0 ,
1 + exp ( i ϕ 12 ) Ř ( l 2 ) exp ( - 2 a g ) = 0 ,
Ř ( l 2 ) = - exp ( i ϕ 32 ) + Ř ( l 3 ) exp ( - i 2 ϕ f 1 ) 1 - Ř ( l 3 ) exp [ i ( ϕ 32 - 2 ϕ f 1 ) ] ,
Ř ( l 3 ) = exp ( i ϕ 34 ) + Ř ( l 4 ) exp ( - 2 a b ) 1 + Ř ( l 4 ) exp ( i ϕ 34 ) exp ( - 2 a b ) ,
Ř ( l 4 ) = - exp [ i ϕ 54 ] + R 56 exp [ i ( ϕ 56 - 2 ϕ f 2 ) ] 1 - R 56 exp [ i ( ϕ 56 + ϕ 54 - 2 ϕ f 2 ) ] .
{ 1 - exp [ i ( ϕ 32 + ϕ 34 - 2 ϕ f 1 ) ] } { 1 - exp [ i ( ϕ 54 + ϕ 56 - 2 ϕ f 2 ) ] } = exp ( - 2 a b ) { exp [ i ( ϕ 32 - 2 ϕ f 1 ) ] - exp ( i ϕ 34 ) } × { exp [ i ( ϕ 56 - 2 ϕ f 2 ) ] - exp ( i ϕ 54 ) } .
Δ ϕ 1 = ϕ 32 + ϕ 34 - 2 ϕ f 1 ,             Δ ϕ 1 1 ,
Δ ϕ 2 = ϕ 54 + ϕ 56 - 2 ϕ f 2 ,             Δ ϕ 2 1 ,
Δ ϕ 1 Δ ϕ 2 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) .
2 ϕ f 1 = ϕ 32 + ϕ 34 ± Δ ϕ 1 , 0 + 2 m π             ( m = 0 , 1 , 2 , ) ,
2 ϕ f 2 = ϕ 54 + ϕ 56 ± Δ ϕ 2 , 0 + 2 m π             ( m = 0 , 1 , 2 , ) ,
Δ ϕ 1 , 0 = 2 sin ϕ 34 exp ( - a b ) ,
Δ ϕ 2 , 0 = 2 sin ϕ 54 exp ( - a b ) .
2 ϕ f 1 = ϕ 32 + ϕ l 3 + 2 m π             ( m = - 1 , 0 , 1 , 2 , ) ,
d f 1 / λ 0 = ϕ 32 + ϕ l 3 + 2 m π 4 π n 3 cos θ 3 ,
Ř ( l 3 ) = Ř ( l 3 ) exp [ i ϕ l 3 ] .
Ř ( l 4 ) = 2 sin ϕ 54 / Δ ϕ 2 , Ř ( l 3 ) = exp ( i ϕ l 3 )
[ exp ( i ϕ 34 ) + a / Δ ϕ 2 ] / [ 1 + exp ( i ϕ 34 ) a / Δ ϕ 2 ] ,
a = 2 sin ϕ 54 exp ( - 2 α b ) .
Ř ( l j ) ( + i n ) = { 1 Ř * ( l j ) ( - i n ) = 1 Ř ( l j ) ( - i n ) exp [ + i ϕ l j ] ,             j = 1 , 3 , 5 ( where the wave in the j th layer is propagating ) , Ř * ( l j ) ( - i n ) = Ř * ( l j ) ( - i n ) exp [ - i ϕ l j ] ,             j = 2 , 4 ( where the wave in the j th layer is evanescent ) .
1 + exp [ i ϕ 12 ] Ř * ( l 2 ) ( - i n ) exp [ - 2 a g ] = 0.
ρ 56 = 1 - R 56 ,             ρ 56 1.
Ř ( l 4 ) - 2 sin ϕ 54 ( ρ 56 2 + Δ ϕ 2 2 ) 1 / 2 exp [ i ( ϕ l 4 + π 2 ) ] ,
ϕ l 4 = tan - 1 ( Δ ϕ 2 / ρ 56 ) .
2 sin ϕ 54 exp [ - 2 a b ] / ( ρ 56 2 + Δ ϕ 2 2 ) 1 / 2 = 1 - ρ l 4 ,             ρ l 4 1 ,
Δ ϕ 34 ϕ l 4 + π 2 - ϕ 34 ,             Δ ϕ 34 1 ,
Ř ( l 3 ) ( ρ l 4 2 + Δ ϕ 34 2 ) 1 / 2 2 sin ϕ 34 × exp { ± i [ π 2 - tan - 1 ( Δ ϕ 34 / ρ l 4 ) ] } ,
4 exp ( - 2 a b ) / ρ 56 1 ,
Ř ( l 3 ) ( 1 - ρ l 3 ) exp [ i ( ϕ 34 - Δ ϕ l 3 ) ] ,
Δ ϕ l 3 = 4 sin ϕ l 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) / ( ρ 56 2 + Δ ϕ 2 2 ) 1 / 2 ,
ρ l 3 = 4 cos ϕ l 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) / ( ρ 56 2 + Δ ϕ 2 2 ) 1 / 2 .
Ř ( l 2 ) 2 sin ϕ 32 exp [ i ( ϕ l 2 - π 2 ) ] [ ρ l 3 2 + ( Δ ϕ 1 - Δ ϕ l 3 ) 2 ] 1 / 2 ,
ϕ l 2 = tan - 1 ( Δ ϕ 1 - Δ ϕ l 3 ρ l 3 ) .
2 sin ϕ 32 [ ρ l 3 2 + ( Δ ϕ 1 - Δ ϕ l 3 ) 2 ] 1 / 2 exp ( - 2 a g ) = 1 ,
ϕ l 2 = ϕ 12 - π 2 .
1 = - ρ l 3 Δ ϕ 1 cot ϕ 12 = - 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) ρ 56 Δ ϕ 1 cos ϕ 12 sin ϕ 12 .
R ^ ( l 2 ) ( 1 - ρ 32 ) exp [ i ( ϕ 32 + Δ ϕ l 2 ) ] ,
Δ ϕ l 2 = 2 cos ϕ 12 sin ϕ 32 exp ( - 2 a g ) ,
ρ 32 = 2 sin ϕ 12 sin ϕ 32 exp ( - 2 a g ) , R ^ ( l 1 ) = exp ( i ϕ 21 ) = - exp ( i ϕ 12 ) ,
1 = 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) ρ 56 ρ 32 .
d g / λ 0 = 1 4 π n 2 cos θ 2 log ( 2 sin ϕ 12 sin ϕ 32 ρ 56 ) ,
d b / λ 0 = 1 4 π n 4 cos θ 4 log ( 4 sin ϕ 34 sin ϕ 54 ρ 32 2 ) .
κ 2 > 1 Q 1 Q 2             ( overcoupling ) , κ 2 = 1 Q 1 Q 2             ( critical coupling ) , κ 2 < 1 Q 1 Q 2             ( undercoupling ) .
4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) > ρ 56 ρ 32 ( overcoupling ) , 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) = ρ 56 ρ 32 ( critical coupling ) , 4 sin ϕ 34 sin ϕ 54 exp ( - 2 a b ) < ρ 56 ρ 32 ( undercouplig ) .
2 sin ϕ 32 exp ( - 2 a g ) / ( ρ l 3 2 + Δ ϕ l 1 2 ) 1 / 2 = 1 - ρ l 1 ,             ρ l 1 1
Δ ϕ l 1 = ϕ l 2 + π 2 - ϕ 12 ,
B 1 ( ρ l 1 2 + Δ ϕ l 1 2 ) 1 / 2 2 sin ϕ 12 exp { ± i [ π 2 + tan - 1 ( Δ ϕ l 1 / ρ l 1 ) ] } .
R ^ ( l 3 ) 2 sin ϕ 34 exp [ i ( ϕ l 4 - π 2 ) ] / [ ρ 32 2 + ( Δ ϕ 1 + Δ ϕ l 2 ) 2 ] 1 / 2 ,
ϕ l 4 = tan - 1 ( Δ ϕ 1 + Δ ϕ l 2 ρ 32 ) ,
R ^ ( l 4 ) ( 1 - ρ l 4 ) exp [ i ( ϕ 54 - Δ ϕ l 4 ) ] ,             ρ l 4 1 ,
ρ l 4 = 4 cos ϕ l 4 sin ϕ 54 sin ϕ 34 exp ( - 2 a b ) / [ ρ 32 2 + ( Δ ϕ 1 + Δ ϕ l 2 ) 2 ] 1 / 2 ,
Δ ϕ l 4 = 4 sin ϕ l 4 sin ϕ 54 sin ϕ 34 exp ( - 2 a b ) / [ ρ 32 2 + ( Δ ϕ 1 + Δ ϕ l 2 ) 2 ] 1 / 2 .
2 ϕ f 2 = ϕ 56 + ϕ 54 - Δ ϕ l 4 + 2 m π             ( m = 0 , 1 , 2 , ) .
ρ l 4 > ρ 32 overcoupling ( anomalous dispersion ) , ρ l 4 = ρ 32 critical coupling ( zero dispersion ) , ρ l 4 < ρ 32 undercoupling ( normal dispersion ) .