Abstract

Analytic solutions are presented for the mode amplitudes in two-mode couplers which have a constant coupling coefficient and propagation coefficients that vary linearly. The solutions are obtained in terms of parabolic cylinder functions which reduce, asymptotically toward the ends of the coupler, to simpler forms in elementary functions. These lead to a simple analytic expression for the coupler efficiency, and also to quantitative criteria for the coupler length.

© 1976 Optical Society of America

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  1. D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).
    [Crossref] [PubMed]
  2. M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
    [Crossref]
  3. M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
    [Crossref]
  4. J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
    [Crossref]
  5. A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).
    [Crossref]
  6. W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
    [Crossref]
  7. A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).
    [Crossref] [PubMed]
  8. A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
    [Crossref]
  9. Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
    [Crossref]
  10. E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); J. Appl. Phys. 36, 875 (1965).
    [Crossref]
  11. E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
    [Crossref]
  12. S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
    [Crossref]
  13. J. L. Doane, J. Appl. Phys. 45, 2748 (1974).
    [Crossref]
  14. R. B. Smith, Electron. Lett. 10, 204 (1975).
    [Crossref]
  15. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.
  16. J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.
  17. A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
    [Crossref]
  18. J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).
    [Crossref]
  19. A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).
    [Crossref]
  20. P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).
    [Crossref]
  21. M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).
    [Crossref]
  22. G. H. Wannier, Physics 5, 251 (1965).
  23. C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).
    [Crossref]
  24. R. B. Smith (unpublished).
  25. H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
    [Crossref]
  26. N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).
    [Crossref]

1975 (6)

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
[Crossref]

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, and J. H. Harris, Appl. Opt. 14, 1860 (1975).
[Crossref] [PubMed]

A. F. Milton and W. K. Burns, Appl. Opt. 14, 1207 (1975).
[Crossref] [PubMed]

R. B. Smith, Electron. Lett. 10, 204 (1975).
[Crossref]

P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).
[Crossref]

M. Matsuhara, K. O. Hill, and A. Watanabe, J. Opt. Soc. Am. 65, 804 (1975).
[Crossref]

1974 (1)

J. L. Doane, J. Appl. Phys. 45, 2748 (1974).
[Crossref]

1973 (1)

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
[Crossref]

1971 (1)

J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).
[Crossref]

1970 (1)

A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).
[Crossref]

1969 (2)

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
[Crossref]

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
[Crossref]

1967 (1)

N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).
[Crossref]

1966 (2)

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
[Crossref]

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
[Crossref]

1965 (2)

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
[Crossref]

G. H. Wannier, Physics 5, 251 (1965).

1964 (1)

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); J. Appl. Phys. 36, 875 (1965).
[Crossref]

1962 (1)

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
[Crossref]

1955 (3)

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
[Crossref]

A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

1932 (1)

C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).
[Crossref]

Achutaramayya, G.

Burns, W. K.

Chen, Y. S.

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
[Crossref]

Cook, J. S.

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
[Crossref]

Cullen, A. L.

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
[Crossref]

Dalgoutte, D. G.

Davies, O. J.

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
[Crossref]

Doane, J. L.

J. L. Doane, J. Appl. Phys. 45, 2748 (1974).
[Crossref]

Donaghey, L. F.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
[Crossref]

Fox, A. G.

A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

Harris, J. H.

Hill, K. O.

Horowitz, P.

P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).
[Crossref]

Ishimaru, A.

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
[Crossref]

Jones, A. L.

Joseph, R. L.

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); J. Appl. Phys. 36, 875 (1965).
[Crossref]

Keller, H. B.

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
[Crossref]

Keller, J. B.

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
[Crossref]

Kirchner, E. K.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
[Crossref]

Knorr, J. B.

J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).
[Crossref]

Louisell, W. H.

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

Matsuhara, M.

Miller, J. C. P.

J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.

Milton, A. F.

Morgenthaler, F. R.

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
[Crossref]

Olson, F. A.

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
[Crossref]

Rezende, S. M.

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
[Crossref]

Schloman, E.

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); J. Appl. Phys. 36, 875 (1965).
[Crossref]

Smith, R. B.

Snyder, A. W.

A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).
[Crossref]

Teh, G. A.

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
[Crossref]

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
[Crossref]

Van Kampen, N. G.

N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).
[Crossref]

Wannier, G. H.

G. H. Wannier, Physics 5, 251 (1965).

Watanabe, A.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

Wilson, M. G. F.

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
[Crossref]

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
[Crossref]

Zener, C.

C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. Horowitz, Appl. Phys. Lett. 23, 306 (1975).
[Crossref]

Bell Syst. Tech. J. (3)

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
[Crossref]

A. G. Fox, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
[Crossref]

Electron. Lett. (3)

M. G. F. Wilson and G. A. Teh, Electron. Lett. 9, 453 (1973).
[Crossref]

R. B. Smith, Electron. Lett. 10, 204 (1975).
[Crossref]

A. L. Cullen and O. J. Davies, Electron. Lett. 5, 90 (1969).
[Crossref]

IEEE Trans. M.T.T. (1)

M. G. F. Wilson and G. A. Teh, IEEE Trans. M.T.T. MTT-23, 85 (1975). (This paper was published without labels for the axes and lines of the figures. Figure 4, showing the coupling efficiency, may be found correctly labeled in Ref. 2.)
[Crossref]

J. Appl. Phys. (4)

E. Schloman and R. L. Joseph, J. Appl. Phys. 35, 2382 (1964); J. Appl. Phys. 36, 875 (1965).
[Crossref]

E. K. Kirchner, L. F. Donaghey, and F. A. Olson, J. Appl. Phys. 37, 988 (1966).
[Crossref]

S. M. Rezende and F. R. Morgenthaler, J. Appl. Phys. 40, 524 (1969).
[Crossref]

J. L. Doane, J. Appl. Phys. 45, 2748 (1974).
[Crossref]

J. Opt. Soc. Am. (2)

J. Soc. Indust. Appl. Math. (1)

H. B. Keller and J. B. Keller, J. Soc. Indust. Appl. Math. 10, 246 (1962).
[Crossref]

Physica (Utr.) (1)

N. G. Van Kampen, Physica (Utr.) 35, 70 (1967).
[Crossref]

Physics (1)

G. H. Wannier, Physics 5, 251 (1965).

Proc. IEEE (Lett.) (3)

J. B. Knorr, Proc. IEEE (Lett.) 59, 1624 (1971).
[Crossref]

A. W. Snyder, Proc. IEEE (Lett.) 58, 168 (1970).
[Crossref]

Y. S. Chen and A. Ishimaru, Proc. IEEE (Lett.) 54, 1071 (1966).
[Crossref]

Proc. R. Soc. Lond. A (1)

C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932).
[Crossref]

Other (3)

R. B. Smith (unpublished).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, England, 1952), p. 323.

J. C. P. Miller, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (NBS, Washington, D. C., 1964), Chap. 19, p. 686.

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Figures (4)

FIG. 1
FIG. 1

Solutions for power in second mode for three values of small ν. Initial condition is P2 = 0 at x = −50. The horizontal lines at right indicate efficiency calculated from Eq. (16). Where only data points appear the oscillation frequency is too high to be shown.

FIG. 2
FIG. 2

Solutions for power in second mode for two values of moderate ν. Initial condition P2 = 0 at x = −50. Curves are displace vertically by 0.20 for clarity. ν = 1.0 corresponds to a minimum-length tapered coupler. For ν = 1.96 the asymptotic approach is slower.

FIG. 3
FIG. 3

Solutions for power in second mode for a large value of ν. The oscillations are due to starting-initial condition P2 = 0 at insufficiently large x = −50.

FIG. 4
FIG. 4

Taper rate γd/dz(β1β2)/β0, and coupling coefficient c, required for given values of asymptotic efficiency as z + ∞.

Tables (1)

Tables Icon

TABLE I Distances required for initial conditions and for complete mode conversion to be satisfied within 1%. For |y| ≥ y1 all the power being in a single guide is equivalent to 99% of power being in a single normal mode, and conversely. For yy2 the second oscillatory term of Eq. (12) becomes less than 1% at the output.

Equations (68)

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d a ( z ) d z = i [ β 1 ( z ) c 12 c 21 β 2 ( z ) ] a ( z ) ,             a ( z ) = [ a 1 ( z ) a 2 ( z ) ] ,
β 1 ( z ) = β 0 ( 1 + γ z ) ,
β 2 ( z ) = β 0 ( 1 - γ z ) ,
c 12 c 21 = c ,             real .
a ( z ) A ( z ) exp ( + i β 0 z ) ,
y ( γ β 0 ) 1 / 2 z ( c / μ ) z ,
μ c / ( γ β 0 ) 1 / 2 .
d A d y = i [ + y μ μ - y ] A .
d 2 A d y 2 + ( y 2 + μ 2 i ) A = 0.
x ( 2 ) 1 / 2 y ( γ β 0 ) 1 / 2 z ,
ν 1 2 μ 2 c 2 / 2 γ β 0 .
a 1 , 2 - 1 2 ( μ 2 i ) - ν ± 1 2 i .
A 1 ( x ) = N ( ν ) E ( a 1 , - x ) N ( ν ) [ cos λ E ( a 1 , x ) + i sin λ E * ( a 1 , x ) ] ,
A 2 ( x ) = N ( ν ) E ( a 2 , - x ) N ( ν ) [ cos λ E ( a 2 , x ) - i sin λ E * ( a 2 , x ) ] ,
P 1 ( x ) A 1 2 ~ 1 - ν / x 2 1 ,             ν x 2 ,
P 2 ( x ) A 2 2 ~ + ν / x 2 0 ,             3 ν x 2 ;
P 1 ( x ) ~ cos 2 λ + ( 2 ν 1 / 2 / x ) sin λ cos λ sin [ 2 h ( ν , x ) ] + ( ν / x 2 ) ( sin 2 λ - cos 2 λ ) ,
P 2 ( x ) ~ sin 2 λ - ( 2 ν 1 / 2 / x ) sin λ cos λ sin [ 2 h ( ν , x ) ] - ( ν / x 2 ) ( sin 2 λ - cos 2 λ ) .
P 1 ( y ) ~ ( κ - y ) / 2 κ ,
P 2 ( y ) ~ ( κ + y ) / 2 κ ,
P 1 ( y ) ~ B + 2 + ( μ / κ ) B + B - sin [ 2 g ( μ , y ) ] - [ ( κ - y ) / 2 κ ] ( B + 2 - B - 2 ) ,
P 2 ( y ) ~ B - 2 - ( μ / κ ) B + B - sin [ 2 g ( μ , y ) ] - [ ( κ - y ) / 2 κ ] ( B - 2 - B + 2 ) ,
2 g ( μ , y ) 2 ϕ ( y , 0 ) + ( ψ + - ψ - ) - 1 2 π ,
ϕ ( y , 0 ) 0 y κ ( τ ) d τ 1 2 [ y κ + μ 2 ln ( κ + y ) - μ 2 ln μ ] ,
μ 2 16 κ 6 16 ( y 2 + μ 2 ) 3 .
P 1 ( y ) ( κ - y ) / 2 κ ,             P 2 ( y ) ( κ + y ) / 2 κ
[ B + ( + ) B - ( - ) ] = [ sin λ - cos λ + cos λ sin λ ] [ B + ( - ) B - ( - ) ] .
efficiency = sin 2 λ 1 - exp ( - 2 π ν ) ,
μ 2 / 4 y 1 2 = 0.01             or             y 1 = 5 μ .
y 2 μ ( 10 4 sin 2 λ cos 2 λ - 1 ) 1 / 2 .
z l β 0 = y 1 ( β 0 / γ ) 1 / 2 y 1 μ ( β 0 / c ) .
d 2 w ( x ) d x 2 + ( 1 4 x 2 - a ) w ( x ) = 0 ,             [ 19.1.2 - 3 ]
E ( a , x ) = f ( a ) U [ i a , x exp ( - 1 4 i π ) ]             [ 19.17.9 ]
E * ( a , x ) = f * ( a ) U [ - i a , x exp ( + 1 4 i π ) ] ,
f ( a ) = ( 4 exp [ π ( a + 1 2 i ) ] Γ ( 1 2 + i a ) Γ ( 1 2 - i a ) ) 1 / 4 ,             [ 19.17.9 ]
f * ( a ) = ( 4 exp [ π ( a - 1 2 i ) ] Γ ( 1 2 - i a ) Γ ( 1 2 + i a ) ) 1 / 4 .
W [ E ( a , ± x ) , E * ( a , ± x ) ] = 2 i [ 19.8.2 ]
[ 1 + exp ( 2 π a ) ] 1 / 2 E ( a , x ) exp ( π a ) E * ( a , x ) + i E * ( a , - x ) [ 19.8.3 ]
[ 1 + exp ( 2 π a ) ] 1 / 2 E * ( a , x ) exp ( π a ) E ( a , x ) - i E ( a , - x ) .
d E ( a , x ) d x + i ( 1 2 x ) E ( a , x ) - i [ ± ( a + 1 2 i ) ] 1 / 2 E ( a + i , x ) = 0 ,
d E ( a , x ) d x - i ( 1 2 x ) E ( a , x ) ± i [ ± ( a - 1 2 i ) ] 1 / 2 E ( a - i , x ) = 0.
E ( a , x ) = ( 2 / x ) g ( a ) exp ( 1 4 i x 2 - i a ln x ) s ( a , x ) ,
g ( a ) [ Γ ( 1 2 - i a ) / Γ ( 1 2 + i a ) ] 1 / 4
a = a 1 - ν + 1 2 i             and             a = a 2 - ν - 1 2 i ,
[ E ( a 1 , x ) ] * E * ( a 2 , x )             and             [ E ( a 2 , x ) ] * E * ( a 1 , x ) .
E 1 ( x ) E ( a 1 , x )             and             E 2 ( x ) E ( a 2 , x ) ,
d E 1 ( x ) d x - i ( 1 2 x ) E 1 ( x ) - i ν 1 / 2 E 2 ( x ) = 0 , d E 2 ( x ) d x + i ( 1 2 x ) E 2 ( x ) - i ν 1 / 2 E 1 ( x ) = 0.
E 1 ( - x ) + exp ( - π ν ) E 1 ( x ) + i [ 1 - exp ( - 2 π ν ) ] 1 / 2 E 1 * ( x ) , E 2 ( - x ) - exp ( - π ν ) E 2 ( x ) + i [ 1 - exp ( - 2 π ν ) ] 1 / 2 E 2 * ( x ) .
E 1 ( x ) = 2 ν - 1 / 4 exp [ + i h ( ν , x ) ] s 1 ( x ) , E 1 * ( x ) = ( 2 / x ) ν + 1 / 4 exp [ - i h ( ν , x ) ] s 1 * ( x ) , E 2 ( x ) = ( 2 / x ) ν + 1 / 4 exp [ + i h ( ν , x ) ] s 2 ( x ) , E 2 * ( x ) = 2 ν - 1 / 4 exp [ - i h ( ν , x ) ] s 2 * ( x ) ;
h ( ν , x ) 1 4 x 2 + ν ln x + 1 2 ϕ + 1 8 π
E 1 ( x ) 2 E 2 * ( x ) 2 E 1 ( x ) E 2 * ( x ) = 2 ν - 1 / 2 s 1 ( x ) s 2 * ( x ) ~ 2 ν - 1 / 2 [ 1 - ν / x 2 ]             for             ν x 2 .
E 2 ( x ) 2 E 1 * ( x ) 2 E 1 * ( x ) E 2 ( x ) = ( 2 ν + 1 / 2 / x 2 ) s 1 * ( x ) s 2 ( x ) ~ 2 ν + 1 / 2 / x 2             for             3 ν x 2 .
E 1 ( x ) E 2 ( x ) [ E 1 * ( x ) E 2 * ( x ) ] * = ( 2 / x ) exp [ + i 2 h ( ν , x ) ] s 1 ( x ) s 2 ( x ) ~ ( 2 / x ) exp [ + i 2 h ( ν , x ) ]             for             ( 1 + ν 2 ) x 2 .
E 1 E 2 * + E 1 * E 2 E 1 2 + E 2 2 = 2 / ν .
d d x [ A 1 ( x ) A 2 ( x ) ] = i [ + 1 2 x ν 1 / 2 ν 1 / 2 - 1 2 x ] [ A 1 ( x ) A 2 ( x ) ] .
[ E 1 ( x ) E 2 ( x ) ] , [ - E 1 * ( x ) + E 2 * ( x ) ] , [ E 1 * ( - x ) E 2 * ( - x ) ] , [ + E 1 ( - x ) - E 2 ( - x ) ] .
[ A 1 ( x ) A 2 ( x ) ] = α [ E 1 ( x ) E 2 ( x ) ] + β [ - E 1 * ( x ) + E 2 * ( x ) ] ,
[ A 1 ( x ) A 2 ( x ) ] = ν 1 / 4 2 [ + E 1 ( - x ) - E 2 ( - x ) ]
ν 1 / 4 2 { cos λ [ E 1 ( x ) E 2 ( x ) ] + i sin λ [ + E 1 * ( x ) - E 2 * ( x ) ] } ;
d A d t A = i C ( t ) A ,             A ( t 0 ) = A 0 ;
C ( t ) = [ + Δ c c * - Δ ] .
A ( t ) T ( t ) B ( t ) ,             A 0 T ( t 0 ) B 0 .
T = [ cos δ - sin δ + sin δ cos δ ] ,             T - 1 = [ cos δ + sin δ - sin δ cos δ ] .
d B d t = [ + i κ ( t ) + δ ( t ) - δ ( t ) - i κ ( t ) ] B ,
B ( t ) = [ exp ( + i ϕ ) 0 0 exp ( - i ϕ ) ] B 0 ,
P ( t ) = [ B + 2 cos 2 δ + B - 2 sin 2 δ - B + B - sin 2 δ cos 2 ψ ( t ) B + 2 sin 2 δ + B - 2 cos 2 δ + B + B - sin 2 δ cos 2 ψ ( t ) ] ,
c 2 / 4 Δ 2 α 1 ,
ϕ ( y , 0 ) = 0 y κ ( τ ) d τ = 1 2 ( y κ + μ 2 ln [ ( κ + y ) / μ ] ) .