Abstract

Within the framework of perturbation theory, the method of analytical continuation of transition matrix elements is proposed. The method is based on the Sturmian expansion of the transition matrix elements and the analytical continuation of the divergent part of the Sturmian transition matrix elements in the case of above-threshold ionization with one excess photon. Investigation of the ratio Q(N+1)/Q(N) with λ and comparison with experimental data proves the validity of perturbation theory up to I = 1013 W cm−2 for multiphoton ionization of atomic hydrogen in the ground state.

© 1990 Optical Society of America

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  1. G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).
  2. G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).
  3. P. Agostini, F. Fabre, G. Manfray, G. Petite, and N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
    [Crossref]
  4. P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
    [Crossref]
  5. H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
    [Crossref] [PubMed]
  6. D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
    [Crossref]
  7. Y. Gontier and M. Trahin, Phys. Rev. 172, 83 (1968).
    [Crossref]
  8. Y. Gontier and M. Trahin, Phys. Rev. A 4, 1896 (1971).
    [Crossref]
  9. Y. Gontier and M. Trahin, Phys. Rev. A 7, 2069 (1973).
    [Crossref]
  10. Y. Gontier and M. Trahin, J. Phys. B 13, 4383 (1980).
    [Crossref]
  11. Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
    [Crossref]
  12. E. Karule, J. Phys. B 4, L67 (1971).
    [Crossref]
  13. E. Karule, in Atomic Processes, R. K. Peterkop, ed. (Zinatne, Riga, 1975; in Russian),pp. 5–24.
  14. E. Karule, J. Phys. B 11, 441 (1978).
    [Crossref]
  15. E. Karule, in Nonlinear Processes in Two-Electron Atoms, N. B. Delone, ed. (Academy of Sciences USSR, Moscow, 1984; in Russian), pp. 209–235.
  16. E. Karule, J. Phys. B 18, 2207 (1985).
    [Crossref]
  17. E. Karule, J. Phys. B 21, 1997 (1988).
    [Crossref]
  18. R. Shakeshaft, Phys. Rev. A 34, 244 (1986).
    [Crossref] [PubMed]
  19. R. Shakeshaft, Phys. Rev. A 34, 5119 (1986).
    [Crossref] [PubMed]
  20. Gao Bo and F. Starace, Phys. Rev. Lett. 61, 403 (1988).
  21. Gao Bo and F. Starace, Phys. Rev. A 39, 4550 (1989).
    [Crossref]
  22. L. C. Hostler, J. Math. Phys. 11, 2966 (1970).
    [Crossref]
  23. M. Rotenberg, Adv. Atom. Molec. Phys. 6, 233 (1970).
    [Crossref]
  24. R. Shakeshaft, J. Phys. B 18, L611 (1985).
    [Crossref]
  25. W. Gordon, Ann. Phys. 2, 1031, (1929).
    [Crossref]
  26. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).
  27. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.
  28. Y. Gontier and M. Trahin, Europhys. Lett. 5, 595 (1988).
    [Crossref]
  29. N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
    [Crossref]
  30. R. M. Potvliege and R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
    [Crossref] [PubMed]

1989 (2)

Gao Bo and F. Starace, Phys. Rev. A 39, 4550 (1989).
[Crossref]

R. M. Potvliege and R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
[Crossref] [PubMed]

1988 (3)

Y. Gontier and M. Trahin, Europhys. Lett. 5, 595 (1988).
[Crossref]

E. Karule, J. Phys. B 21, 1997 (1988).
[Crossref]

Gao Bo and F. Starace, Phys. Rev. Lett. 61, 403 (1988).

1987 (1)

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

1986 (3)

H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[Crossref] [PubMed]

R. Shakeshaft, Phys. Rev. A 34, 244 (1986).
[Crossref] [PubMed]

R. Shakeshaft, Phys. Rev. A 34, 5119 (1986).
[Crossref] [PubMed]

1985 (2)

E. Karule, J. Phys. B 18, 2207 (1985).
[Crossref]

R. Shakeshaft, J. Phys. B 18, L611 (1985).
[Crossref]

1983 (1)

N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
[Crossref]

1981 (1)

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

1980 (2)

Y. Gontier and M. Trahin, J. Phys. B 13, 4383 (1980).
[Crossref]

Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
[Crossref]

1979 (1)

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

1978 (1)

E. Karule, J. Phys. B 11, 441 (1978).
[Crossref]

1973 (1)

Y. Gontier and M. Trahin, Phys. Rev. A 7, 2069 (1973).
[Crossref]

1971 (2)

E. Karule, J. Phys. B 4, L67 (1971).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 4, 1896 (1971).
[Crossref]

1970 (2)

L. C. Hostler, J. Math. Phys. 11, 2966 (1970).
[Crossref]

M. Rotenberg, Adv. Atom. Molec. Phys. 6, 233 (1970).
[Crossref]

1968 (1)

Y. Gontier and M. Trahin, Phys. Rev. 172, 83 (1968).
[Crossref]

1966 (1)

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).

1965 (1)

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).

1929 (1)

W. Gordon, Ann. Phys. 2, 1031, (1929).
[Crossref]

Agostini, P.

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

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

Bo, Gao

Gao Bo and F. Starace, Phys. Rev. A 39, 4550 (1989).
[Crossref]

Gao Bo and F. Starace, Phys. Rev. Lett. 61, 403 (1988).

Clement, M.

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

Delone, N. B.

N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
[Crossref]

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.

Fabre, F.

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

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

Feldman, D.

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

Gontier, Y.

Y. Gontier and M. Trahin, Europhys. Lett. 5, 595 (1988).
[Crossref]

Y. Gontier and M. Trahin, J. Phys. B 13, 4383 (1980).
[Crossref]

Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 7, 2069 (1973).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 4, 1896 (1971).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. 172, 83 (1968).
[Crossref]

Gordon, W.

W. Gordon, Ann. Phys. 2, 1031, (1929).
[Crossref]

Goreslavsky, S. P.

N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
[Crossref]

Hostler, L. C.

L. C. Hostler, J. Math. Phys. 11, 2966 (1970).
[Crossref]

Karule, E.

E. Karule, J. Phys. B 21, 1997 (1988).
[Crossref]

E. Karule, J. Phys. B 18, 2207 (1985).
[Crossref]

E. Karule, J. Phys. B 11, 441 (1978).
[Crossref]

E. Karule, J. Phys. B 4, L67 (1971).
[Crossref]

E. Karule, in Atomic Processes, R. K. Peterkop, ed. (Zinatne, Riga, 1975; in Russian),pp. 5–24.

E. Karule, in Nonlinear Processes in Two-Electron Atoms, N. B. Delone, ed. (Academy of Sciences USSR, Moscow, 1984; in Russian), pp. 209–235.

Krainov, V. P.

N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.

Manfray, G.

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

Muller, H. G.

H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[Crossref] [PubMed]

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.

Petite, G.

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

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

Poirier, M.

Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
[Crossref]

Potvliege, R. M.

R. M. Potvliege and R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
[Crossref] [PubMed]

Rahman, N. K.

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

Rotenberg, M.

M. Rotenberg, Adv. Atom. Molec. Phys. 6, 233 (1970).
[Crossref]

Shakeshaft, R.

R. M. Potvliege and R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
[Crossref] [PubMed]

R. Shakeshaft, Phys. Rev. A 34, 244 (1986).
[Crossref] [PubMed]

R. Shakeshaft, Phys. Rev. A 34, 5119 (1986).
[Crossref] [PubMed]

R. Shakeshaft, J. Phys. B 18, L611 (1985).
[Crossref]

Starace, F.

Gao Bo and F. Starace, Phys. Rev. A 39, 4550 (1989).
[Crossref]

Gao Bo and F. Starace, Phys. Rev. Lett. 61, 403 (1988).

Trahin, M.

Y. Gontier and M. Trahin, Europhys. Lett. 5, 595 (1988).
[Crossref]

Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
[Crossref]

Y. Gontier and M. Trahin, J. Phys. B 13, 4383 (1980).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 7, 2069 (1973).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 4, 1896 (1971).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. 172, 83 (1968).
[Crossref]

Tricomi, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.

van der Wiel, M. J.

H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[Crossref] [PubMed]

Van Linden van de Heuvell, H. B.

H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[Crossref] [PubMed]

Voronov, G. S.

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).

Welge, K. H.

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

Wemhoner, M.

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

Wolf, B.

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

Adv. Atom. Molec. Phys. (1)

M. Rotenberg, Adv. Atom. Molec. Phys. 6, 233 (1970).
[Crossref]

Ann. Phys. (1)

W. Gordon, Ann. Phys. 2, 1031, (1929).
[Crossref]

Europhys. Lett. (1)

Y. Gontier and M. Trahin, Europhys. Lett. 5, 595 (1988).
[Crossref]

J Phys. B (1)

N. B. Delone, S. P. Goreslavsky, and V. P. Krainov, J Phys. B 16, 2369 (1983).
[Crossref]

J. Math. Phys. (1)

L. C. Hostler, J. Math. Phys. 11, 2966 (1970).
[Crossref]

J. Phys. B (8)

R. Shakeshaft, J. Phys. B 18, L611 (1985).
[Crossref]

P. Agostini, M. Clement, F. Fabre, and G. Petite, J. Phys. B 14, L491 (1981).
[Crossref]

Y. Gontier and M. Trahin, J. Phys. B 13, 4383 (1980).
[Crossref]

Y. Gontier, M. Poirier, and M. Trahin, J. Phys. B 13, 1381 (1980).
[Crossref]

E. Karule, J. Phys. B 4, L67 (1971).
[Crossref]

E. Karule, J. Phys. B 11, 441 (1978).
[Crossref]

E. Karule, J. Phys. B 18, 2207 (1985).
[Crossref]

E. Karule, J. Phys. B 21, 1997 (1988).
[Crossref]

Phys. Rev. (1)

Y. Gontier and M. Trahin, Phys. Rev. 172, 83 (1968).
[Crossref]

Phys. Rev. A (7)

Y. Gontier and M. Trahin, Phys. Rev. A 4, 1896 (1971).
[Crossref]

Y. Gontier and M. Trahin, Phys. Rev. A 7, 2069 (1973).
[Crossref]

R. Shakeshaft, Phys. Rev. A 34, 244 (1986).
[Crossref] [PubMed]

R. Shakeshaft, Phys. Rev. A 34, 5119 (1986).
[Crossref] [PubMed]

H. G. Muller, H. B. Van Linden van de Heuvell, and M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[Crossref] [PubMed]

Gao Bo and F. Starace, Phys. Rev. A 39, 4550 (1989).
[Crossref]

R. M. Potvliege and R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

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

Gao Bo and F. Starace, Phys. Rev. Lett. 61, 403 (1988).

Z. Phys. D (1)

D. Feldman, B. Wolf, M. Wemhoner, and K. H. Welge, Z. Phys. D 6, 293 (1987).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).

Zh. Eksp. Teor. Fiz. Lett. (1)

G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).

Other (4)

E. Karule, in Atomic Processes, R. K. Peterkop, ed. (Zinatne, Riga, 1975; in Russian),pp. 5–24.

E. Karule, in Nonlinear Processes in Two-Electron Atoms, N. B. Delone, ed. (Academy of Sciences USSR, Moscow, 1984; in Russian), pp. 209–235.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.

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Figures (8)

Fig. 1
Fig. 1

Angular distribution of photoelectrons at λ = 355 nm (N = 4). Solid curve, S = 0 (×, experiment of Ref. 6); dashed curve, S = 1(○, experiment of Ref. 6).

Fig. 2
Fig. 2

Angular distribution of photoelectrons at λ = 532 nm (N = 6). Solid curve S = 0; dashed curve, S = 1.

Fig. 3
Fig. 3

Variation of the ratio Ql(N+1)/Ql(N) with λ at I = 1013 W cm−2 (scale linear for λ10/3).

Fig. 4
Fig. 4

Variation of the ratio Qc(N)/Ql(N) with λ and N. Dashed lines, [Qc(N)/Ql(N)]max = (2N − 1)!!/N!.

Fig. 5
Fig. 5

Variation of the ratio Qc(N+1)/Ql(N+1) with λ and N. Dashed lines, [Qc(N+1)/Ql(N+1)]max = (2N+1)!!/(2N+1)!.

Fig. 6
Fig. 6

Variation of the ratio Qc(N+1)/Qc(N) with λ at I = 1011 W cm−2.

Fig. 7
Fig. 7

Variation of the ratio Q(6+1)/Q(6) with λ at I = 3 × 1012 W cm−2. Solid curve, circularly polarized light; dashed curve, linearly polarized light.

Fig. 8
Fig. 8

Variation of the ratio Q(8+1)/Q(8) with λ at I = 1011 W cm−2. Solid curve, circularly polarized light; dashed line, linearly polarized light.

Tables (2)

Tables Icon

Table 1 Ratio of Five-Photon ATI to Four-Photon MPI at the 3p Resonance (308 nm)

Tables Icon

Table 2 Ratios of Cross Sections on the N-Photon Ionization Threshold

Equations (43)

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T rad ( K ) ( n 0 l 0 , L 1 , L 2 , , L K 1 ; E K L K | ω ) = 0 R E K L K ( r K ) r K 3 d r K j = 1 K 1 0 G L ( r j , r j + 1 ; E j ) r j 3 R n 0 l 0 ( r 1 ) d r j ,
G L ( r , r ; Ω ) = p n = L + 1 [ S n L ( 2 r / p ) S n L ( 2 r / p ) / ( n p ) ] ,
S n L ( 2 r / p ) = 2 p 1 [ ( 2 L + 1 ) ! ] 1 [ ( n L ) 2 L + 1 ] 1 / 2 × exp ( r / p ) ( 2 r / p ) L F ( n + L + 1 , 2 L + 2 ; 2 r / p ) , p = ( 2 Ω ) 1 / 2 , S n L ( 2 r / n ) = n R n L ( 2 r / n ) ,
G L ( r , r ; Ω ) = p n = L + 1 [ S n L ( 2 r / p ) S n L ( 2 r / p ) / ( n + p ) ] + 2 π i R Ω L ( r ) R Ω L ( r ) ,
S n L ( 2 r / p ) = 2 p 1 [ ( 2 L + 1 ) ! ] 1 [ ( n L ) 2 L + 1 ] 1 / 2 × exp ( r / p ) ( 2 r / p ) L F ( n + L + 1 , 2 L + 2 ; 2 r / p ) , R Ω L ( r ) = C k [ ( 2 L + 1 ) ! ] 1 ( 2 r ) L F ( i / k + L + 1 , 2 L + 2 ; 2 i k r ) , C k = 2 [ 1 exp ( 2 π / k ) ] 1 / 2 s = 1 L [ ( s k ) 2 + 1 ] 1 / 2 , k = ( 2 Ω ) 1 / 2 .
f 1 ( Z ) d Z = 1 f ( Z ) d Z , Ω < 0 ,
f ( Z ) = [ S Z L ( Ω + i , r ) S Z L ( Ω + i , r ) / ( Z 1 ) ] , 0 ,
T rad ( K ) = 0 R E K L K ( r K ) j = 1 K 1 m j = L j + 1 ( 1 m j / q j ) 1 S m j L j ( r j + 1 ) × r j + 1 3 d r j + 1 0 S m j L j ( r j ) R n 0 l 0 ( r 1 ) r 1 3 d r 1 , q = ( 2 E ) 1 / 2 .
f ( m , L , p ; n , l , q ) = 2 L + l + 2 [ ( 2 L + 1 ) ! ( 2 l + 1 ) ! ] 1 × 0 exp [ ( 1 / p + 1 / q ) r ] r L + l + 3 × F ( m + L + 1 , 2 L + 2 ; 2 r / p ) × F ( n + 1 + l , 2 l + 2 ; 2 r / q ) d r .
x F ( a , c ; x ) = ( c 1 ) F ( a , c 1 ; x ) F ( a 1 , c 1 ; x )
f ( m , L , p ; n , l , q ) = 2 L + 1 p 2 [ ( 2 l + 1 ) ! ] 1 × s = 0 2 ( 1 ) s C 2 s J 2 l + 2 10 ( m + l + s , n + 1 + l ) .
J 2 l + 2 10 ( m + l + s , n + 1 + l ) = 0 exp [ ( 1 / p + 1 / q ) r ] r 2 l + 2 F ( m + l + s , 2 l + 2 ; 2 r / p ) × F ( n + 1 + l , 2 l + 2 ; 2 r / q ) d r = 2 p 1 ( 2 l + 1 ) ! ( l ) L m s + 1 [ p q / ( q p ) ] 2 l + 4 Z m n + s 2 × ( m + 1 s n p / q ) × F 2 1 ( m + l + s , n + 1 + l ; 2 l + 2 ; 1 z 2 ) ,
f ( m , L , p ; n , l , q ) = ( 1 ) L m 2 2 l + 2 p L 1 [ p q / ( q p ) ] 2 l + 4 [ ( 2 l + 1 ) ! ] 1 z m n × s = 0 2 C 2 s ( 1 + m s n p / q ) z s 2 × F 2 1 ( l m + s , 1 n + l ; 2 l + 2 ; 1 z 2 ) .
c F ( a , c ; x ) = a F ( a + 1 , c + 1 ; x ) ( a c ) F ( a , c + 1 ; x ) .
f ( m , L , p ; n , l , q ) = 2 L + l + 2 [ ( 2 l + 1 ) ! ] 2 × s = 0 2 ( 1 ) s C 2 2 ( L m + 1 ) s × ( s m L 2 ) 2 s J 2 l + 2 10 ( m + l + s , n + 1 + l ) .
f ( m , L , p ; n , l , q ) = f ( n , l , q ; m , l , p ) = ( 1 ) L m 2 2 l + 2 p L l × [ p q / ( q p ) ] 2 l + 4 [ ( 2 l + 1 ) ! ] 1 Z m n s = 0 2 [ ( L m + 1 ) s × ( s m L 2 ) 2 s δ L , l 1 + δ L , 1 + l ] ( 1 + m s n p / q ) × C 2 s z s 2 2 F 1 ( m + l + s , n + 1 + l ; 2 l + 2 ; 1 z 2 ) .
f ( m , L , p ; n , l , q ) = 2 L + l + 2 ( 2 L + 2 ) l L + 2 [ ( 2 l + 1 ) ! ] 1 × [ p q / ( p + q ) ] L + l + 4 F 2 ( L + l + 4 , m + L + 1 , n + 1 + l , 2 L + 2 , 2 l + 2 ; 1 + z 1 , 1 z 1 ) .
I λ s ( a , c ; k 1 ; b , d ; k 2 ) = 0 exp ( s r ) r λ 1 F ( a , c ; k 1 r ) F ( b , d ; k 2 r ) d r ,
F ( a , c ; k 1 r ) = [ Γ ( c ) / Γ ( a ) Γ ( c a ) ] × 0 1 exp ( k 1 r t ) t a 1 ( 1 t ) c a 1 d t .
I λ s ( a , c , k 1 ; b , d , k 2 ) = [ Γ ( c ) / Γ ( a ) Γ ( c a ) ] × 0 1 t a 1 ( 1 t ) c a 1 I ( t ) d t .
I ( t ) = 0 exp [ ( s k 1 t ) r ] r λ 1 F ( b , d ; k 2 r ) d r = Γ ( λ ) ( s k 1 t ) λ [ s / ( s k 2 ) ] b × { ( 1 k 1 t / s ) / [ 1 k 1 t / ( s k 2 ) ] } b × m = 0 λ d { ( b ) m ( d λ ) m ( k 2 ) m / ( d ) m × ( k 2 s ) m [ 1 k 1 t / ( s k 2 ) ] m m ! } .
I = 0 1 t a 1 ( 1 t ) c a 1 [ 1 k 1 t / ( s k 2 ) ] b m ( 1 k 1 t / s ) b λ d t .
I = Γ ( a ) Γ ( c a ) / Γ ( c ) F 1 [ a , b + m , λ b , c ; k 1 / ( s k 2 ) , k 1 / s ] .
I λ s ( a , c , k 1 ; b , d , k 2 ) = Γ ( λ ) s λ [ s / ( s k 2 ) ] b × m = 0 λ d [ ( b ) m ( d λ ) m k 2 m / ( d ) m ( k 2 s ) m m ! ] × F 1 [ a , λ b , b + m , c ; k 1 / s , k 1 / ( s k 2 ) ] .
I λ s ( a , c , k 1 ; b , d , k 2 ) = Γ ( λ ) s λ ( 1 k 2 / s ) b × ( 1 k 1 / s ) a m = 0 λ d [ ( b ) m ( d λ ) m k 2 m / ( d ) m × ( k 2 s ) m m ! ] F 1 { a , c λ m , b + m , c ; k 1 / ( k 1 s ) , [ k 1 k 2 / ( s k 1 ) ( s k 2 ) ] } .
f ( m , L , p ; n , l , q ) = 2 L + l + 2 ( 2 L + 2 ) l L + 2 × [ ( 2 l + 1 ) ! ] 1 [ p q / ( p + q ) ] L + l + 4 ( 1 ) m + L + 1 Z m n + L + 2 × s = 0 L l + 2 [ ( n + 1 + l ) s ( l L 2 ) s ( 1 z ) s / ( 2 l + 2 ) s S ! ] × F 1 ( m + L + 1 , L l 2 s , m 1 l + s , 2 L + 2 ; 1 + z , 1 z 2 ) .
T ( N ) ( n 0 l 0 , E N L N | ω ) = C N L N 1 = L N ± 1 B ( L N , L N 1 ) × m N 1 = 1 f ( q N , L N , q N ; m N 1 + L N 1 , L N 1 , q N 1 ) X N 1 ( m N 1 , L N 1 ) ,
X j ( m j , L j ) = q j 2 L j 1 ( m j ) 2 L j + 1 / ( m j + L j q j ) × L j 1 = L j ± 1 B ( L j , L j 1 ) m j 1 = 1 f ( m j + L j , L j , q j ; m j 1 + L j 1 , L j 1 , q j 1 ) X j 1 ( m j 1 , L j 1 ) .
X 0 ( m 0 , L 0 ) = X 0 ( n 0 l 0 , l 0 ) = n o 2 1 0 [ ( n 0 l 0 ) 2 l 0 + 1 ] 1 / 2 .
T ( N + 1 ) = C N + 1 L N = L N + 1 ± 1 B ( L N + 1 , L N ) × L N 1 = L N ± 1 B ( L N , L N 1 ) Y ( m N 1 , L N + 1 , L N , L N 1 ) × X N 1 ( m N 1 , L N 1 ) , .
Y ( m N 1 , L N + 1 , L N , L N 1 ) = q N 2 L N 1 m N = 0 ( m N + 1 ) 2 L N + 1 / ( m N + L N + 1 q N ) × f ( q N + 1 , L N + 1 , q N + 1 ; m N + L N + 1 , L N , q N ) × f ( m N + L N + 1 , L N , q N ; m N 1 + L N 1 , L N 1 , q N 1 ) .
f ( q , l , q ; m + L + 1 , L , p ) = 2 l + L + 2 [ ( 2 l + 1 ) ! ( m + 1 ) 2 L + 1 ] 1 × 0 exp [ ( 1 / p + 1 / q ) r ] × r L + l + 3 L m 2 L + 1 ( 2 r / p ) F ( q + 1 + l , 2 l + 2 ; 2 r / q ) d r .
f ( m + L + 1 , L , p ; m N 1 + L N 1 , L N 1 , q N 1 ) = 2 L + L N 1 + 2 ( 2 L N 1 + 2 ) L L N + 2 / ( 2 L + 1 ) ! × [ p q N 1 / ( p q N 1 ) ] L + L N 1 + 4 × F 2 ( L + L N 1 + 4 , m N 1 + 1 , m , 2 L N 1 + 2 , 2 L + 2 ; 1 z 1 1 , 1 + z 1 1 ) .
( 2 L + 2 + m ) n / ( m + L + 1 q ) = ( q + L + 1 ) [ k = 0 n 1 ( 2 L + 2 + m ) k / ( q + L + 1 ) k + 1 + ( m + L + 1 q ) 1 ] .
Y = ( 2 ω ) 2 ( 2 L N 1 + 2 ) L N L N 1 + 2 [ ( 2 L N + 1 ) ! ] 1 ( T 1 + T 2 ) .
T 1 = 2 2 ω 1 q q N 1 L N 1 L N + 1 ( 2 L N + 1 + 2 ) L N L N + 1 + 2 / ( q N + L N + 1 ) z 3 q N + 1 L N + 1 ( 1 z 1 ) L N 1 L N + 2 × ( 1 + z 3 ) L N + L N + 1 2 s = 0 m N 1 1 [ ( 1 m N 1 ) × ( L N + L N 1 + 4 ) s ( 1 z 1 ) s / ( 2 L N 1 + 2 ) s s ! ] × n = 1 s + L N 1 L N + 2 [ ( L N L N 1 s 2 ) n ( q N + L N + 1 ) n × ( 1 + z 1 1 ) n / ( 2 L N + 2 ) n n ! ] k = 0 n 1 ( 2 L N + 2 ) k / ( q N + L N + 2 ) k ( 1 + 1 / z 1 ) k j = 0 k [ ( k ) j ( L N + L N + 1 + 4 ) j / j ! ( 2 L N + 2 ) j ( 1 + q N 1 / q N + 1 ) j ] ( 1 q N 1 / q N ) j × F 2 1 ( q N + 1 + L N + 1 + 1 , L N + 1 L N 2 j ; 2 L N + 1 + 2 ; 1 z 3 ) ,
z 1 = ( q N 1 + q N ) / ( q N 1 q N ) , z 3 = ( q N + 1 + q N 1 ) / ( q N + 1 q N 1 ) .
T 2 = 2 L N + L N + 1 + 4 q N 1 L N 1 L N q N 1 ( L N + 1 q N ) 1 × z 2 q N + L N + 3 ( q N + 1 + L N + 1 + 1 ) L N L N + 1 + 2 × ( 1 / q N + 1 + 1 / q N ) L N L N + 1 4 ( 1 + z 1 1 ) 2 L N × ( 1 z 1 ) L N 1 L N ( 1 z 2 1 ) L N L N + 1 2 × h ( M N 1 , L N 1 , L N , q N ) s = 0 L N L N + 1 + 2 [ ( L N + 1 L N 2 ) s × ( q N + L N + 1 + 1 ) s / z 2 s ( q N + 1 L N 2 ) s s ! ] × k = 0 L N + 1 L N + 2 [ ( L N L N + 1 2 ) k / z 2 k k ! ] × F 1 ( L N + 1 q N , q N + 1 + s + k + L N 1 , q N + 1 s k + L N + 3 , L N + 2 q N ; 1 / z 1 z 2 , z 2 / z 1 ) ,
z 2 = ( q N + 1 + q N ) / ( q N + 1 q N ) , h ( m , L , 1 , p ) = n = 0 m 1 [ ( 1 m ) n ( L + l + 4 ) n ( 1 z 1 ) n / ( 2 L + 2 ) n n ! ] 2 F 1 ( l L n 2 , p + 1 + l ; 2 l + 2 ; 1 + z 1 1 ) .
F 1 ( a , b , b , c ; 1 / z 1 z 2 , z 2 / z 1 ) = m = 0 n = 0 [ ( a ) m + n ( b ) m ( b ) n ( z 2 / z 1 ) n / ( c ) m + n m ! n ! ( z 1 z 2 ) m ] .
F 1 ( a , b , b , c ; 1 / z 1 z 2 , z 2 / z 1 ) = n = 0 [ ( a ) n ( b ) n / ( c ) n n ! ( z 1 z 2 ) n ] × F 2 1 ( a + n , b , c + n ; z 2 / z 1 ) .
d Ω W ( N + S ) = 4 π 2 α a 0 ω ( I / I 0 ) N + S 1 × | l ( i ) l exp ( i η l ) y 10 ( cos θ ) T 1 ( ω ) | 2 ,
η l = arg Γ ( 1 + l i / k ) , Y 10 ( cos θ ) = [ ( 2 l + 1 ) / 4 π ] 1 / 2 P 1 ( cos θ ) ,

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