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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. S. Voronov, N. B. Delone, Zh. Eksp. Teor. Fiz. Lett. 1, 42 (1965).
  2. G. S. Voronov, N. B. Delone, Zh. Eksp. Teor. Fiz. 50, 78 (1966).
  3. P. Agostini, F. Fabre, G. Manfray, G. Petite, N. K. Rahman, Phys. Rev. Lett. 42, 1127 (1979).
    [CrossRef]
  4. P. Agostini, M. Clement, F. Fabre, G. Petite, J. Phys. B 14, L491 (1981).
    [CrossRef]
  5. H. G. Muller, H. B. Van Linden van de Heuvell, M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
    [CrossRef] [PubMed]
  6. D. Feldman, B. Wolf, M. Wemhoner, K. H. Welge, Z. Phys. D 6, 293 (1987).
    [CrossRef]
  7. Y. Gontier, M. Trahin, Phys. Rev. 172, 83 (1968).
    [CrossRef]
  8. Y. Gontier, M. Trahin, Phys. Rev. A 4, 1896 (1971).
    [CrossRef]
  9. Y. Gontier, M. Trahin, Phys. Rev. A 7, 2069 (1973).
    [CrossRef]
  10. Y. Gontier, M. Trahin, J. Phys. B 13, 4383 (1980).
    [CrossRef]
  11. Y. Gontier, M. Poirier, 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, F. Starace, Phys. Rev. Lett. 61, 403 (1988).
  21. Gao Bo, 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, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).
  27. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York. 1953), Vol. 1.
  28. Y. Gontier, M. Trahin, Europhys. Lett. 5, 595 (1988).
    [CrossRef]
  29. N. B. Delone, S. P. Goreslavsky, V. P. Krainov, J Phys. B 16, 2369 (1983).
    [CrossRef]
  30. R. M. Potvliege, R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
    [CrossRef] [PubMed]

1989 (2)

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

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

1988 (3)

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

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

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

1987 (1)

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

1986 (3)

H. G. Muller, H. B. Van Linden van de Heuvell, 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, V. P. Krainov, J Phys. B 16, 2369 (1983).
[CrossRef]

1981 (1)

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

1980 (2)

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

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

1979 (1)

P. Agostini, F. Fabre, G. Manfray, G. Petite, 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, M. Trahin, Phys. Rev. A 7, 2069 (1973).
[CrossRef]

1971 (2)

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

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

1966 (1)

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

1965 (1)

G. S. Voronov, 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, G. Petite, J. Phys. B 14, L491 (1981).
[CrossRef]

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

Bo, Gao

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

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

Clement, M.

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

Delone, N. B.

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

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

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

Erdelyi, A.

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

Fabre, F.

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

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

Feldman, D.

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

Gontier, Y.

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

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

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

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

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

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

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

Krainov, V. P.

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

Landau, L. D.

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

Lifshitz, E. M.

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

Magnus, W.

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

Manfray, G.

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

Muller, H. G.

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

Oberhettinger, F.

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

Petite, G.

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

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

Poirier, M.

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

Potvliege, R. M.

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

Rahman, N. K.

P. Agostini, F. Fabre, G. Manfray, G. Petite, 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, R. Shakeshaft, Phys. Rev. A 39, 1545 (1989).
[CrossRef] [PubMed]

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

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

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

Starace, F.

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

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

Trahin, M.

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

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

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

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

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

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

Tricomi, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, 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, 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, M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[CrossRef] [PubMed]

Voronov, G. S.

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

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

Welge, K. H.

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

Wemhoner, M.

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

Wolf, B.

D. Feldman, B. Wolf, M. Wemhoner, 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, M. Trahin, Europhys. Lett. 5, 595 (1988).
[CrossRef]

J Phys. B (1)

N. B. Delone, S. P. Goreslavsky, 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, G. Petite, J. Phys. B 14, L491 (1981).
[CrossRef]

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

Y. Gontier, M. Poirier, 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, M. Trahin, Phys. Rev. 172, 83 (1968).
[CrossRef]

Phys. Rev. A (7)

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

Y. Gontier, 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, M. J. van der Wiel, Phys. Rev. A 34, 236 (1986).
[CrossRef] [PubMed]

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

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

Phys. Rev. Lett. (2)

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

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

Z. Phys. D (1)

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

Zh. Eksp. Teor. Fiz. (1)

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

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

G. S. Voronov, 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, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1959).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

Equations on this page are rendered with MathJax. Learn more.

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 ) ,

Metrics