Abstract

In this paper we discuss the free-electron laser (FEL) dynamics, solving the Liouville equation ruling the longitudinal phase-space evolution of the electron beam. We evaluate the bunching coefficients and study the interplay between FEL dynamics and harmonic generation. The effect of the initial beam energy spread is also included.

© 1993 Optical Society of America

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References

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  1. See, e.g., W. B. Colson, C. Pellegrini, A. Renieri, eds., Laser Handbook (North-Holland, Amsterdam, 1990), Vol. VI.
  2. J. N. Elgin, Nucl. Instrum. Methods A 304, 406 (1991); J. N. Elgin, C. Penman, Nucl. Instrum. Methods A 304, 787 (1991); F. De Martini, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 195.
    [CrossRef]
  3. J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
    [CrossRef]
  4. L. H. Yu, Phys. Rev. A 44, 5178 (1991).
    [CrossRef] [PubMed]
  5. G. Dattoli, A. Renieri, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 221.
  6. Note that the basis |n〉 may be specified by|n〉=(2π)−1/2exp(inχ) and the scalar product defined as〈m|n〉=12π∫02πexp[i(m−n)χ]dχ. The choice of χ is totally arbitrary. However, if we identify χ with ψ, the function (32) is just the longitudinal phase-space distribution.
  7. The operators ʱare easily identified asEˆ±=exp(±iχ), and the number operator nˆ is simplynˆ=1i∂∂χ.
  8. A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
    [CrossRef]
  9. W. B. Colson, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 301.

1993 (1)

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

1991 (2)

L. H. Yu, Phys. Rev. A 44, 5178 (1991).
[CrossRef] [PubMed]

J. N. Elgin, Nucl. Instrum. Methods A 304, 406 (1991); J. N. Elgin, C. Penman, Nucl. Instrum. Methods A 304, 787 (1991); F. De Martini, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 195.
[CrossRef]

1990 (1)

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Ben Zvi, J.

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Colson, W. B.

W. B. Colson, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 301.

Dattoli, G.

G. Dattoli, A. Renieri, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 221.

Di Mauro, F. F.

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Doria, A.

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

Elgin, J. N.

J. N. Elgin, Nucl. Instrum. Methods A 304, 406 (1991); J. N. Elgin, C. Penman, Nucl. Instrum. Methods A 304, 787 (1991); F. De Martini, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 195.
[CrossRef]

Feinestein, J.

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

Gallerano, G. P.

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

Krinsky, S.

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Pantell, R.

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

Renieri, A.

G. Dattoli, A. Renieri, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 221.

White, M. G.

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Yu, L. H.

L. H. Yu, Phys. Rev. A 44, 5178 (1991).
[CrossRef] [PubMed]

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Doria, G. P. Gallerano, J. Feinestein, R. Pantell, “Coherent emission and gain from a bunched e-beam,” IEEE J. Quantum Electron. 29, 1428 (1993).
[CrossRef]

Nucl. Instrum. Methods A (2)

J. N. Elgin, Nucl. Instrum. Methods A 304, 406 (1991); J. N. Elgin, C. Penman, Nucl. Instrum. Methods A 304, 787 (1991); F. De Martini, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 195.
[CrossRef]

J. Ben Zvi, F. F. Di Mauro, S. Krinsky, M. G. White, L. H. Yu, Nucl. Instrum. Methods A 296, 787 (1990); R. Bonifacio, L. De Salvo Souza, P. Piccini, E. T. Scharleman, Nucl. Instrum. Methods A 304, 787 (1990); R. Barbini, F. Ciocci, G. Dattoli, L. Gianessi, G. Maino, C. Mari, A. Marino, C. Ronsivalle, A. Torre, ENEA rep. RT/INN/90.35 (Ente per le Nuove Technologie, l’Energia e l’Ambiente, Frascati, Italy, 1990).
[CrossRef]

Phys. Rev. A (1)

L. H. Yu, Phys. Rev. A 44, 5178 (1991).
[CrossRef] [PubMed]

Other (5)

G. Dattoli, A. Renieri, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 221.

Note that the basis |n〉 may be specified by|n〉=(2π)−1/2exp(inχ) and the scalar product defined as〈m|n〉=12π∫02πexp[i(m−n)χ]dχ. The choice of χ is totally arbitrary. However, if we identify χ with ψ, the function (32) is just the longitudinal phase-space distribution.

The operators ʱare easily identified asEˆ±=exp(±iχ), and the number operator nˆ is simplynˆ=1i∂∂χ.

W. B. Colson, in Laser Handbook, W. B. Colson, C. Pellegrini, A. Renieri, eds. (North-Holland, Amsterdam, 1990), Vol. VI, p. 301.

See, e.g., W. B. Colson, C. Pellegrini, A. Renieri, eds., Laser Handbook (North-Holland, Amsterdam, 1990), Vol. VI.

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Figures (2)

Fig. 1
Fig. 1

(a) Re b 1 ( 1 ) versus (ν,τ), (b) Im b 1 ( 1 ) versus (ν,τ), (c) b 0 ( 2 ) versus (ν,τ). The plots are centered on ν0 = 2.6 (μ = 0.1/π).

Fig. 2
Fig. 2

Energy distribution versus (ν,τ). The plot is centered on ν0 = 2.6 (μ = 0.1/π, ΩR = 0.5).

Tables (1)

Tables Icon

Table 1 Definitions of Symbols

Equations (86)

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ψ = Ω R 2 sin ψ ,
Ω R 2 = 2 π 2 N ( e E 0 ) L u K ( JJ ) / m 0 c 2 γ 2 ,
JJ = J 0 ( ξ ) J 1 ( ξ ) , ξ = 1 2 K 2 1 + K 2 ,
H = 1 2 ν 2 Ω R 2 cos ψ ,
ρ t = ν ρ ψ + Ω R 2 ( sin ψ ) ρ ν .
ρ = 1 2 π n = + ( 1 ) n exp ( i n ψ ) b n ( ν , τ ) ,
i τ b n = n ν b n Ω R 2 2 ν ( b n 1 b n + 1 ) ,
b n ( 0 ) = f ( ν ) δ n , 0 ,
f ( ν ) = 1 ( 2 π ) 1 / 2 ( π μ ) 2 exp [ 1 2 ( ν ν 0 ) 2 ( π μ ) 2 ] ,
μ = 4 N σ .
f ( ν ) = δ ( ν ν 0 ) .
b n = m = 0 m b n ( m ) , Ω R 2 ,
i τ b n ( m ) = n ν b n ( m ) 1 2 ν [ b n 1 ( m 1 ) b n + 1 ( m 1 ) ] .
b 0 ( 0 ) ( 0 ) = f ( ν ) , b n ( 0 ) = 0 , n 0 ,
b 0 ( 0 ) ( τ ) = f ( ν ) , b n ( 0 ) = 0 , n 0 ,
b 1 ( 1 ) = 1 2 exp ( i ν τ ) [ ν f ( ν ) ] 0 τ exp ( + i ν τ ) d τ , b 1 ( 1 ) = [ b 1 ( 1 ) ] * .
i τ b 0 ( m ) = 1 2 ν [ b 1 ( m 1 ) b 1 ( m 1 ) ] .
i τ b 0 ( 1 ) = 0 , i τ b 0 ( 2 ) = 1 2 ν [ b 1 ( 1 ) b 1 ( 1 ) ] .
b 0 ( 2 ) = 1 2 ν { 1 cos ( ν τ ) ν 2 [ ν f ( ν ) ] } .
G - 1 Ω R 4 + d ν 0 2 π ( ν ν 0 ) ρ ( ν , ψ ) d ψ ,
G - ν 0 [ sin ( ν 0 / 2 ) ν 0 / 2 ] 2
G Re { i 0 1 d t ( 1 t ) t exp ( i ν 0 t ) exp [ ( π μ t ) 2 2 ] }
f ( ν , τ ) = 0 2 π d ψ ρ ( ν , ψ , τ ) ,
f ( ν , τ ) f ( ν ) + Ω R 4 2 ν { 1 cos ( ν τ ) ν 2 [ ν f ( ν ) ] } .
+ f ( ν , τ ) d ν = 1
σ ν 2 = ( ν ν 0 ) 2 ( ν ν 0 ) 2 = ( π μ ) 2 + Ω R 4 { Re 0 1 d t ( 1 t ) exp ( i ν 0 t ) exp [ 1 2 ( π μ t ) 2 ] }
G ν σ ν 2 .
i τ b 1 ( m ) = ν b 1 ( m ) 1 2 ν [ b 0 ( m 1 ) b 2 ( m 1 ) ] , b 1 ( m ) = [ b 1 ( m ) ] * .
b n ( n ) ( ν , τ ) = + i 2 exp ( i n ν τ ) 0 τ [ ν b n 1 ( n 1 ) ( ν , τ ) ] exp ( i n ν τ ) d τ , b n ( n ) ( ν , τ ) = [ b n ( n ) ( ν , τ ) ] * ,
n [ b 0 ( 0 ) b 0 ( 1 ) b 0 ( 2 ) b 0 ( r ) 0 b 1 ( 1 ) b 1 ( 2 ) b 1 ( r ) 0 0 0 0 b ( r ) r ] .
b n ( m ) ( ν , τ ) = 0 , m < n .
[ b 0 ( 0 ) b 0 ( 1 ) b 0 ( 2 ) b 0 ( 3 ) b 0 ( 4 ) b 1 ( 1 ) b 1 ( 2 ) b 1 ( 3 ) b 1 ( 4 ) b 2 ( 2 ) b 2 ( 3 ) b 2 ( 4 ) b 3 ( 3 ) b 4 ( 4 ) b 4 ( 4 ) ] .
[ b 0 ( 0 ) 0 b 0 ( 2 ) 0 b 0 ( 4 ) 0 b 1 ( 1 ) 0 b 1 ( 3 ) 0 0 0 b 2 ( 2 ) 0 b 2 ( 4 ) 0 0 0 b 3 ( 3 ) 0 0 0 0 0 b 4 ( 4 ) ] .
b 0 ( 4 ) ( ν , τ ) = i 2 0 τ ν { b 1 ( 3 ) ( ν , τ ) [ b 1 ( 3 ) ( ν , τ ) ] * } d τ , b 1 ( 3 ) ( ν , τ ) = i 2 exp ( i ν τ ) 0 τ { ν [ b 0 ( 2 ) ( ν , τ ) b 2 ( 2 ) ( ν , τ ) ] } × exp ( i ν τ ) d τ .
ψ = n = + b n ( ν , τ ) | n ,
n | m = δ n , m .
i τ ψ = [ n ˆ ν Ω R 2 2 ( E ˆ + E ˆ ) ν ] ψ ,
n ˆ | n = n | n , E ˆ ± s | n = | n ± s .
i τ ψ I = H ˆ I ψ I ,
ψ I = exp ( i n ˆ ν τ ) ψ , H ˆ I = Ω R 2 2 exp ( i n ˆ ν τ ) ( E ˆ + E ˆ ) ν exp ( i n ˆ ν τ ) .
ψ I ( τ ) = { 1 i 0 τ H ˆ I ( τ ) d τ 1 2 [ 0 τ H ˆ I ( τ ) d τ 0 τ H ˆ I ( τ ) d τ ] + + i 6 [ 0 τ H ˆ I ( τ ) d τ 0 τ H ˆ I ( τ ) d τ 0 τ H ˆ I ( τ ) d τ ] + + } ψ I ( 0 ) ,
exp ( ξ A ˆ ) B ˆ exp ( ξ A ˆ ) = B ˆ + ξ [ A ˆ , B ˆ ] + ξ 2 2 [ A ˆ , [ A ˆ , B ˆ ] ] +
[ E ˆ + , E ˆ ] = 0 , [ E ˆ ± , n ˆ ] = E ˆ ± .
b 5 i Ω R 10 τ 5 5 ! 2 5 exp ( i 5 ν τ ) 5 ν 5 f ( ν ) .
ρ 5 Ω R 10 τ 5 5 ! 2 5 sin [ 5 ( ψ ν τ ) ] 5 ν 5 f ( ν ) .
+ ρ 5 ( ν , ψ ) d ν 5 5 5 ! 2 4 ( Ω R τ ) 10 cos [ 5 ( ψ ν 0 τ ) ] .
+ ρ n ( ν , ψ ) d ν ( 1 ) n n n n ! 2 n 1 ( Ω R τ ) 2 n cos [ n ( ψ ν 0 τ ) ] ,
n Ω R τ < ( n ! 2 n 1 ) 1 / 2 n ,
Ω R τ < ( π n / 2 ) 1 / 4 n ( 2 e ) 1 / 2 .
+ ρ 1 ( ν , ψ , μ ) Ω R 2 cos ( ψ ν 0 ) exp [ 1 2 ( π μ ) 2 ] .
[ b 0 ( 0 ) b 0 ( 1 ) b 0 ( 2 ) b 0 ( 3 ) b 0 ( 4 ) b 1 ( 0 ) b 1 ( 1 ) b 1 ( 2 ) b 1 ( 3 ) b 1 ( 4 ) 0 b 2 ( 1 ) b 2 ( 2 ) b 2 ( 3 ) b 2 ( 4 ) 0 0 b 3 ( 2 ) b 3 ( 3 ) b 3 ( 4 ) 0 0 0 b 4 ( 3 ) b 4 ( 4 ) ] .
b 1 ( 0 ) ( 0 ) = f 1 ( ν ) ,
b 0 ( 1 ) ( ν , τ ) = ν [ 1 cos ( ν τ ) ν f 1 ( ν ) ] .
g 0 ( 1 ) Ω R 2 + ( ν ν 0 ) b 0 ( 1 ) ( ν , 1 ) d ν = Ω R 2 1 cos ν 0 ν 0 .
F ( ν ) f ( ν ) Ω R 2 ν [ 1 cos ν ν f 1 ( ν ) ] + Ω R 4 2 ν { ( 1 cos ν ν 2 ) [ ν f ( ν ) ] } + .
i τ b n = n ν b n 1 2 ( a ν b n 1 a * ν b n + 1 ) , a ˙ = i j + d ν b 1 ( ν , τ ) ,
i τ b n = n ν b n 1 2 [ a ( z Δ τ , τ ) ν b n 1 ( z , τ ) a * ( z Δ τ , τ ) ν b n 1 ( z , τ ) ] , a ˙ ( z Δ τ , τ ) = i j z + d ν b 1 ( z , ν , τ ) ,
b 1 ( 1 ) = i 2 exp ( i ν τ ) [ ν f ( ν ) ] 0 τ a ( τ ) exp ( i ν τ ) d τ ,
a ˙ = + j 2 + d ν [ ν f ( ν ) ] 0 τ a ( τ ) exp [ i ν ( τ τ ) ] d τ .
a ˙ = i j 2 0 τ ( τ τ ) exp [ i ν 0 ( τ τ ) ] a ( τ ) d τ .
a ˙ ( z , τ ) = i j ( z + Δ τ ) 2 + d ν [ ν f ( ν ) ] × 0 τ ξ exp ( i ν ξ ) a ( z + Δ ξ , τ ξ ) d ξ .
a ˙ ( τ ) = i j + b 1 ( ν , 0 ) exp ( i ν τ ) d ν + j 2 + d ν [ ν f ( ν ) ] 0 τ a ( τ ) exp [ i ν ( τ τ ) ] d τ .
a ( τ ) + 2 i ν 0 a ¨ ( τ ) ν 0 2 a ˙ i j 2 a ( τ ) = + i j + ( ν ν 0 ) 2 b 1 ( ν , 0 ) exp ( i ν τ ) d ν .
a a 0 + j a 1 + j 2 a 2 ,
a ˙ 0 = 0 , a ˙ 1 = i b ˜ ( τ ) + i 2 0 τ a 0 ( τ ξ ) ξ exp ( i ν 0 ξ ) d ξ , a ˙ 2 = i 2 0 τ a 1 ( τ ξ ) ξ exp ( i ν 0 ξ ) d ξ ,
a ˙ = A , a ¨ = B ,
d d τ ( A B a ) = [ 2 i ν 0 ν 0 2 + 1 2 i j 1 0 0 0 1 0 ] ( A B a ) i j ( + ( ν 2 ν 0 2 ) b 1 ( ν , 0 ) exp ( i ν τ ) d ν 0 0 ) ,
( A ( τ ) B ( τ ) a ( τ ) ) = U ˆ ( τ ) [ ( A ( 0 ) B ( 0 ) a ( 0 ) ) i j 0 τ U ˆ 1 ( τ ) f ( τ ) d τ ] ,
f ( τ ) = ( + ( ν 2 ν 0 2 ) b 1 ( ν , 0 ) exp ( i ν τ ) d ν 0 0 )
U ˆ ( τ ) = [ a ¨ 1 a ¨ 2 a ¨ 3 a ˙ 1 a ˙ 2 a ˙ 3 a 1 a 2 a 3 ] ,
a α = n = 1 3 a n ( α ) exp [ i ( λ n + ν 0 ) τ ] , α = 1 , 2 , 3 ,
λ 2 ( λ + ν 0 ) = 1 2 j ,
n = 1 3 a n ( α ) = δ α , 3 , n = 1 3 a n ( α ) λ n = i δ α , 2 ν 0 δ α , 3 , n = 1 3 a n ( α ) λ n 2 = ν 0 2 δ α , 3 + 2 i ν 0 δ α , 2 δ α , 1 .
b n ( ν , τ ) = m b n , m ( τ ) u m ( ν ) ,
a = 1 2 ( ν + ν ) , a + = 1 2 ( ν ν ) ,
i τ b n , m = m 2 [ ( m + 1 ) 1 / 2 b n , m + 1 + m b n , m 1 ] Ω R 2 2 2 [ ( m + 1 ) 1 / 2 b n 1 , m + 1 m b n 1 , m 1 ( m + 1 ) 1 / 2 b n + 1 , m + 1 + m b n + 1 , m 1 ] .
b n , m ( 0 ) = δ 0 , m δ n , 0 .
J 0 ( 2 ) = + ( ν ν 0 ) b 0 ( 2 ) ( ν , τ ) d ν
[ sin c ( ν 2 ) ] 2 = 2 0 1 ( 1 t ) cos ( ν t ) d t .
J 0 ( 4 ) = + ( ν ν 0 ) b 0 ( 4 ) ( ν , τ ) d ν ,
J 0 ( 4 ) = Im [ 0 τ d τ + d ν b 1 ( 3 ) ( τ ) ] .
J 0 ( 4 ) = 1 30 0 τ d τ τ ( τ τ ) 4 ( τ 6 τ ) × exp { [ 2 i ν 0 τ + 2 ( π μ τ ) 2 ] } 1 1920 0 τ d τ τ ( τ τ ) 2 × exp { [ i ν 0 τ + ( π μ τ ) 2 2 ] } × [ 40 τ τ 2 16 ( τ τ ) 3 ] i 1920 0 τ d τ ( τ τ ) 5 τ × exp [ + i ν 0 τ ( π μ τ ) 2 2 ] + i 1440 0 τ d τ ( τ τ ) 2 τ exp { [ i ν 0 τ + ( π μ τ ) 2 2 ] } × [ 240 τ 3 + 40 τ 2 ( τ τ ) + 10 τ ( τ τ ) 2 ( τ τ ) ] .
|n=(2π)1/2exp(inχ)
m|n=12π02πexp[i(mn)χ]dχ.
Eˆ±=exp(±iχ),
nˆ=1iχ.

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