Abstract

We present a discussion of the two-color above-threshold ionization process, performed with the aim of studying strong-field continuum–continuum transitions. We extend the approach of Rzażewski et al. [ J. Opt. Soc. Am. B 7, 481 ( 1990)] to obtain analytically solvable equations for the multiple continuum model analyzed by Deng and Eberly [ J. Opt. Soc. Am. B 2, 486 ( 1985)]. Some comments on the effect of continuum–continuum transitions on the line shape of a laser-induced continuum structure are also made.

© 1996 Optical Society of America

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  1. P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
    [CrossRef]
  2. J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
    [CrossRef]
  3. Z. Bialynicka-Birula, J. Phys. B 17, 3091 (1984).
    [CrossRef]
  4. M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
    [CrossRef]
  5. Z. Deng and J. H. Eberly, Phys. Rev. Lett. 53, 1810 (1984); J. Opt. Soc. Am. B 2, 486 (1985).
    [CrossRef]
  6. H. G. Muller, H. B. van Linden van den Heuvell, and M. J. van der Wiel, J. Phys. B19, L733 (1986).
    [CrossRef]
  7. J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
    [CrossRef] [PubMed]
  8. K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
    [CrossRef]
  9. V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
    [CrossRef]
  10. B. J. Dalton and M. E. St. J. Dutton, J. Mod. Opt. 41, 1099 (1994).
    [CrossRef]
  11. S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
    [CrossRef] [PubMed]
  12. Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
    [CrossRef] [PubMed]
  13. S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
    [CrossRef] [PubMed]
  14. O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
    [CrossRef] [PubMed]
  15. N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
    [CrossRef] [PubMed]
  16. K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
    [CrossRef] [PubMed]
  17. P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
    [CrossRef]

1995 (3)

V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
[CrossRef]

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

1994 (2)

B. J. Dalton and M. E. St. J. Dutton, J. Mod. Opt. 41, 1099 (1994).
[CrossRef]

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

1993 (1)

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

1991 (4)

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
[CrossRef] [PubMed]

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
[CrossRef]

1990 (1)

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

1989 (1)

K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
[CrossRef]

1984 (3)

Z. Bialynicka-Birula, J. Phys. B 17, 3091 (1984).
[CrossRef]

M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
[CrossRef]

Z. Deng and J. H. Eberly, Phys. Rev. Lett. 53, 1810 (1984); J. Opt. Soc. Am. B 2, 486 (1985).
[CrossRef]

1979 (1)

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Agostini, P.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Antonetti, A.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Armstrong, L.

M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
[CrossRef]

Bachau, H.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

Bialynicka-Birula, Z.

Z. Bialynicka-Birula, J. Phys. B 17, 3091 (1984).
[CrossRef]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

Breger, P.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Buffa, R.

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

Cavalieri, S.

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
[CrossRef] [PubMed]

Charalambidis, D.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

Constantinescu, R. C.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Cormier, E.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

Dalton, B. J.

B. J. Dalton and M. E. St. J. Dutton, J. Mod. Opt. 41, 1099 (1994).
[CrossRef]

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Deng, Z.

Z. Deng and J. H. Eberly, Phys. Rev. Lett. 53, 1810 (1984); J. Opt. Soc. Am. B 2, 486 (1985).
[CrossRef]

Dutton, M. E. St. J.

B. J. Dalton and M. E. St. J. Dutton, J. Mod. Opt. 41, 1099 (1994).
[CrossRef]

Eberly, J. H.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
[CrossRef]

Z. Deng and J. H. Eberly, Phys. Rev. Lett. 53, 1810 (1984); J. Opt. Soc. Am. B 2, 486 (1985).
[CrossRef]

Edwards, M.

M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
[CrossRef]

Eramo, R.

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

Fabre, F.

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Faucher, O.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

Fotakis, C.

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

Grillon, G.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Harris, S. E.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

Haus, J. W.

K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
[CrossRef]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

Javanainen, J.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
[CrossRef]

Karapanagioti, N.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

Knight, P. L.

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Lambropoulos, P.

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

Lauder, M. A.

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Mainfray, G.

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Maquet, A.

V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
[CrossRef]

Matera, M.

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
[CrossRef] [PubMed]

Muller, H. G.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

H. G. Muller, H. B. van Linden van den Heuvell, and M. J. van der Wiel, J. Phys. B19, L733 (1986).
[CrossRef]

Mysyrowicz, A.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Pan, L.

M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
[CrossRef]

Pavone, F. S.

S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
[CrossRef] [PubMed]

Petite, G.

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Rahman, N. K.

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Rzazewski, K.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
[CrossRef]

K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
[CrossRef]

Schins, J. M.

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Shao, Y. L.

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

Taieb, R.

V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
[CrossRef]

van der Wiel, M. J.

H. G. Muller, H. B. van Linden van den Heuvell, and M. J. van der Wiel, J. Phys. B19, L733 (1986).
[CrossRef]

van Linden van den Heuvell, H. B.

H. G. Muller, H. B. van Linden van den Heuvell, and M. J. van der Wiel, J. Phys. B19, L733 (1986).
[CrossRef]

Veniard, V.

V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
[CrossRef]

Wang, L.

K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
[CrossRef]

Zhang, J.

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

B. J. Dalton and M. E. St. J. Dutton, J. Mod. Opt. 41, 1099 (1994).
[CrossRef]

J. Phys. B (2)

Z. Bialynicka-Birula, J. Phys. B 17, 3091 (1984).
[CrossRef]

M. Edwards, L. Pan, and L. Armstrong, J. Phys. B 17, L515 (1984).
[CrossRef]

Phys. Rep. (2)

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991), and references therein; M. Gavrila, ed., Atoms in Strong Fields (Academic, San Diego, Calif., 1992).
[CrossRef]

P. L. Knight, M. A. Lauder, and B. J. Dalton, Phys. Rep. 190, 1 (1990), and references therein.
[CrossRef]

Phys. Rev. A (2)

S. Cavalieri, R. Eramo, R. Buffa, and M. Matera, Phys. Rev. A 51, 2974 (1995).
[CrossRef] [PubMed]

K. Rzazewski, L. Wang, and J. W. Haus, Phys. Rev. A 40, 3453 (1989); J. Opt. Soc. Am. B. 7, 481 (1990).
[CrossRef]

Phys. Rev. Lett. (9)

V. Veniard, R. Taieb, and A. Maquet, Phys. Rev. Lett. 74, 4161 (1995).
[CrossRef]

P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
[CrossRef]

Z. Deng and J. H. Eberly, Phys. Rev. Lett. 53, 1810 (1984); J. Opt. Soc. Am. B 2, 486 (1985).
[CrossRef]

Y. L. Shao, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 67, 3669 (1991).
[CrossRef] [PubMed]

S. Cavalieri, F. S. Pavone, and M. Matera, Phys. Rev. Lett. 67, 3673 (1991).
[CrossRef] [PubMed]

O. Faucher, D. Charalambidis, C. Fotakis, J. Zhang, and P. Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993).
[CrossRef] [PubMed]

N. Karapanagioti, O. Faucher, Y. L. Shao, D. Charalambidis, H. Bachau, and E. Cormier, Phys. Rev. Lett. 74, 2431 (1995).
[CrossRef] [PubMed]

K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, Phys. Rev. Lett. 73, 2180 (1994).
[CrossRef] [PubMed]

Other (1)

H. G. Muller, H. B. van Linden van den Heuvell, and M. J. van der Wiel, J. Phys. B19, L733 (1986).
[CrossRef]

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Equations (22)

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i α ˙ ( t ) = ω α α ( t ) [ 0 + χ p ( ω 1 ) C 1 ( ω 1 , t ) d ω 1 ] × f p ( t ) cos ( ω p t ) , i C ˙ 1 ( ω 1 , t ) = ω 1 C 1 ( ω 1 , t ) χ p ( ω 1 ) α ( t ) f p ( t ) cos ( ω p t ) [ 0 + X 12 ( ω 1 , ω 2 ) C 2 ( ω 2 , t ) d ω 2 ] cos ( ω d t ) , i C ˙ k ( ω k , t ) = ω k C k ( ω k , t ) [ 0 + X k , k 1 ( ω k , ω k 1 ) × C k 1 ( ω k 1 , t ) d ω k 1 + 0 + X k , k + 1 ( ω k , ω k + 1 ) C k + 1 ( ω k + 1 , t ) d ω k + 1 ] × cos ( ω d t ) , i C ˙ n ( ω n , t ) = ω n C n ( ω n , t ) [ 0 + X n , n 1 ( ω n , ω n 1 ) × C n 1 ( ω n 1 , t ) d ω n 1 ] cos ( ω d t ) ,
( α ) new = ( α ) old exp ( i ω α t ) ( C k ) new = ( C k ) old exp ( i Δ t ) , ( ω k ) new = ( ω k ) old Δ , Δ = ω p ω α > 0
i α ˙ ( t ) = f p ( t ) 2 Δ + χ p ( ω 1 ) C 1 ( ω 1 , t ) d ω 1 , i C ˙ 1 ( ω 1 , t ) = ω 1 C 1 ( ω 1 , t ) χ p ( ω 1 ) f p ( t ) a ( t ) [ Δ + X 12 ( ω 1 , ω 2 ) C 2 ( ω 2 , t ) d ω 2 ] cos ( ω d t ) , i C ˙ k ( ω k , t ) = ω k C k ( ω k , t ) [ Δ + X k , k 1 ( ω k , ω k 1 ) × C k 1 ( ω k 1 , t ) d ω k 1 + Δ + X k , k + 1 ( ω k , ω k + 1 ) × C k + 1 ( ω k + 1 , t ) d ω k + 1 cos ( ω d t ) ] , i C ˙ n ( ω n , t ) = ω n C n ( ω n , t ) [ Δ + X n , n 1 ( ω n , ω n 1 ) C n 1 ( ω n 1 , t ) d ω n 1 ] × cos ( ω d t ) .
X k , m ( ω k , ω m ) = χ k , m F ( | ω k ω m | ) ( m = k ± 1 ) ,
C ˆ k ( x , t ) = + C k ( ω k , t ) exp ( i ω k x ) d ω k , χ ˆ p ( x ) = + χ p ( ω 1 ) exp ( i ω 1 x ) d ω 1 , F ˆ ( x ) = + F ( ω ) exp ( i ω x ) d ω ,
i α / t = f p ( t ) 4 π + χ ˆ p ( x ) C ˆ 1 ( x , t ) d x , i C ˆ 1 / t = i C ˆ 1 / x χ ˆ p ( x ) f p ( t ) 2 α ( t ) χ 12 F ˆ ( x ) C ˆ 2 ( x , t ) × cos ( ω d t ) , i C ˆ k / t = i C ˆ k / x [ χ k , k 1 C ˆ k 1 ( x , t ) + χ k , k + 1 C ˆ k 1 ( x , t ) F ˆ ( x ) cos ( ω d t ) , i C ˆ n / t = i C ˆ n / x χ n , n 1 F ˆ ( x ) C ˆ n 1 ( x , t ) cos ( ω d t ) .
d d η C ˆ ( η , θ ) = i [ H ¯ ( η , θ ) C ˆ ( η , θ ) + g ( η , θ ) ] ,
C ˆ ( η , θ ) = [ C ˆ 1 C ˆ 2 C ˆ k C ˆ n ] ,
H ¯ ( η , θ ) = H ¯ 0 F ˆ ( η ) cos [ ω d ( η + θ ) ] ,
g ( η , θ ) = g 0 χ ˆ p ( η ) f p ( η + θ ) 2 α ( η + θ ) ,
H ¯ 0 = [ 0 χ 12 0 0 χ 21 0 χ 23 0 0 0 0 χ n 1 , n 2 0 χ n 1 , n 0 0 χ n , n 1 0 ] ,
g 0 = [ 1 0 0 ] .
b ( η , θ ) = U ¯ C ˆ ( η , θ ) ,
U ¯ H ¯ 0 U ¯ 1 = Λ ¯ 0
d b k d η = i { λ k F ˆ ( η ) cos [ ω d ( η + θ ) ] } b k + i U k 1 χ ˆ p ( η ) f p ( η + θ ) 2 × α ( η + θ ) ,
b k ( η , θ ) = i U k 1 exp [ i Φ k ( η , θ ) ] θ η χ ˆ p ( η ) f p ( η + θ ) 2 × α ( η + θ ) exp [ i Φ k ( η , θ ) ] d η ,
Φ k ( η , θ ) = λ k θ η F ˆ ( η ) cos [ ω d ( η + θ ) d η ]
C ˆ k ( x , t ) = i π [ H ( x ) H ( x t ) ] { m 1 n U mk * U m 1 × exp [ i λ m Φ 0 ( x , t ) ] } f p ( t x ) α ( t x ) ,
Φ 0 ( x , t ) = 0 x F ˆ ( u ) cos [ ω d ( u + t x ) ] d u
α ˙ ( t ) = i f p ( t ) 2 X p ( 0 ) C ˆ 1 ( 0 , t ) = π χ p 2 ( 0 ) 4 f p 2 ( t ) α ( t ) ,
α ( t ) = exp [ π χ p 2 ( 0 ) 4 0 t f p 2 ( u ) d u ]
R = α ˙ ( t ) α ( t ) = π χ p 2 ( 0 ) 2 f p 2 ( t )

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