Abstract

The physical processes involved in energy exchange between the radiation field and relativistic electrons in a free-electron laser cavity with an axial guide field have been discussed in the context of the single-particle dynamics as well as the solution of the Vlasov–Maxwell equations. Thus laser gains have been calculated at the new radiation frequency.

© 1988 Optical Society of America

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References

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  1. T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
    [CrossRef] [PubMed]
  2. H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
    [CrossRef]
  3. H. P. Freund, A. T. Drobot, Phys. Fluids 25, 736 (1982).
    [CrossRef]
  4. H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
    [CrossRef]
  5. H. P. Freund, Phys. Rev. A 27, 1977 (1983).
    [CrossRef]
  6. W. B. Colson, Phys. Lett. A 64, 190 (1977); Ph. D. dissertation (Stanford University, Stanford, Calif., 1977).
    [CrossRef]

1987

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

1983

H. P. Freund, Phys. Rev. A 27, 1977 (1983).
[CrossRef]

1982

H. P. Freund, A. T. Drobot, Phys. Fluids 25, 736 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

1981

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

1977

W. B. Colson, Phys. Lett. A 64, 190 (1977); Ph. D. dissertation (Stanford University, Stanford, Calif., 1977).
[CrossRef]

Bakshi, U.

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

Bali, L. M.

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

Colson, W. B.

W. B. Colson, Phys. Lett. A 64, 190 (1977); Ph. D. dissertation (Stanford University, Stanford, Calif., 1977).
[CrossRef]

da Jornada, E. H.

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Dillenburg, D.

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Drobot, A. T.

H. P. Freund, A. T. Drobot, Phys. Fluids 25, 736 (1982).
[CrossRef]

Freund, H. P.

H. P. Freund, Phys. Rev. A 27, 1977 (1983).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, A. T. Drobot, Phys. Fluids 25, 736 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Jha, P.

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

Liberman, B.

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Pandya, T. P.

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

Schneider, R. S.

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Sprangle, P.

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

Phys. Fluids

H. P. Freund, A. T. Drobot, Phys. Fluids 25, 736 (1982).
[CrossRef]

Phys. Lett. A

W. B. Colson, Phys. Lett. A 64, 190 (1977); Ph. D. dissertation (Stanford University, Stanford, Calif., 1977).
[CrossRef]

Phys. Rev. A

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, R. S. Schneider, B. Liberman, Phys. Rev. A 26, 2004 (1982).
[CrossRef]

H. P. Freund, Phys. Rev. A 27, 1977 (1983).
[CrossRef]

T. P. Pandya, L. M. Bali, U. Bakshi, P. Jha, Phys. Rev. A 35, 5131 (1987).
[CrossRef] [PubMed]

H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Jornada, B. Liberman, R. S. Schneider, Phys. Rev. A 24, 1965 (1981).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Relative gains Δ versus axial field strengths ωb/ω0β0 for γ0 = 3.5. Curves a, ωr ~ 2γ02β0ω0; curves b, ωr ~ 2γ02ωb. Variations of δ with axial field strength are shown by dashed curves.

Equations (81)

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B m = x ^ B cos k 0 z - y ^ B sin k 0 z + z ^ b .
E r = E ( x ^ cos ξ + y ^ sin ξ ) ,
B r = E ( - x ^ sin ξ + y ^ cos ξ ) .
d d t ( γ v x ) = ( e b / m c ) v y + ( e B / m c ) v z sin k 0 z + ( c - v z ) ( e E / m c ) cos ξ ,
d d t ( γ v y ) = - ( e b / m c ) v x + ( e B / m c ) v z cos k 0 z + ( c - v z ) ( e E / m c ) sin ξ ,
d d t ( γ v z ) = - ( e B / m c ) ( v x sin k 0 z + v y cos k 0 z ) + ( e E / m c ) ( v x cos ξ + v y sin ξ ) ,
d d t ( c γ ) = ( e E / m c ) ( v x cos ξ + v y sin ξ ) .
v x ( 0 ) = c β 0 δ [ cos ( k 0 z 0 + β 0 ω 0 t ) - cos ( k 0 z 0 + ω b t ) ] ,
v y ( 0 ) = - c β 0 δ [ sin ( k 0 z 0 + β 0 ω 0 t ) - sin ( k 0 z 0 + ω b t ) ] ,
v z ( 0 ) = c β 0 + c β 0 δ 2 ( cos Ω b t - 1 ) ,
z ( 0 ) = z 0 + c β 0 t + ( c β 0 δ 2 / Ω b ) ( sin Ω b t - Ω b t ) ,
γ ( 0 ) = γ 0 ,
d d t ( γ v x ) ( 1 ) = γ 0 v ˙ x ( 1 ) + d d t [ γ ( 1 ) v x ( 0 ) ] = ω b γ 0 v y ( 1 ) + [ c - v z ( 0 ) ] γ 0 ω E cos ξ ( 0 ) + ω B γ 0 [ v z ( 1 ) sin k 0 z ( 0 ) + k 0 z ( 1 ) v z ( 0 ) cos k 0 z ( 0 ) ] ,
d d t ( γ v y ) ( 1 ) = γ 0 v ˙ y ( 1 ) + d d t [ γ ( 1 ) v y ( 0 ) ] = - ω b γ 0 v x ( 1 ) + [ c - v z ( 0 ) ] γ 0 ω E sin ξ ( 0 ) + ω B γ 0 [ v z ( 1 ) cos k 0 z ( 0 ) - k 0 z ( 1 ) v z ( 0 ) sin k 0 z ( 0 ) ] ,
d d t ( γ v z ) ( 1 ) = γ 0 v ˙ z ( 1 ) + d d t [ γ ( 1 ) v z ( 0 ) ] = - ω B γ 0 [ v x ( 1 ) sin k 0 z ( 0 ) + v y ( 1 ) cos k 0 z ( 0 ) ] + c γ ˙ ( 1 ) ,
c γ ˙ ( 1 ) = ω E γ 0 [ v x ( 0 ) cos ξ ( 0 ) + v y ( 0 ) sin ξ ( 0 ) ] ,
γ 0 v ˙ x ( 1 ) + d d t [ γ ( 1 ) v x ( 0 ) ] = ω b γ 0 v y ( 1 ) + [ c - v z ( 0 ) ] ω E γ 0 cos ξ ( 0 ) ,
γ 0 v ˙ y ( 1 ) + d d t [ γ ( 1 ) v y ( 0 ) ] = - ω b γ 0 v x ( 1 ) + [ c - v z ( 0 ) ] ω E γ 0 sin ξ ( 0 ) .
v x , y ( 1 ) = v x , y ( 1 , 0 ) + v x , y ( 1 , 2 ) ,             v z ( 1 ) = v z ( 1 , 1 ) ,             γ ( 1 ) = γ ( 1 , 1 ) ,
z ( 0 ) = z ( 0 , 0 ) + z ( 0 , 2 ) ,             v x , y ( 0 ) = v x , y ( 0 , 1 ) ,
z ( 0 , 0 ) = z 0 + c β 0 t , z ( 0 , 2 ) = ( c δ 2 β 0 / Ω b ) ( sin Ω b t - Ω b t ) ,
ξ ( 0 ) = ξ ( 0 , 0 ) + k r z ( 0 , 2 ) , ξ ( 0 , 0 ) = k r z 0 + ϕ - ω r ( 1 - β 0 ) t ,
γ 0 v ˙ z ( 1 , 1 ) + γ ˙ ( 1 , 1 ) c β 0 = c γ ˙ ( 1 , 1 ) - ω B γ 0 [ v x ( 1 , 0 ) sin k 0 z ( 0 , 0 ) + v y ( 1 , 0 ) cos k 0 z ( 0 , 0 ) ]
c γ ˙ ( 1 , 1 ) = ω E γ 0 [ v x ( 0 , 1 ) cos ξ ( 0 , 0 ) + v y ( 0 , 1 ) sin ξ ( 0 , 0 ) ] .
γ 0 v ˙ x ( 1 , 0 ) = ω b γ 0 v y ( 1 , 0 ) + ω E c γ 0 ( 1 - β 0 ) cos ξ ( 0 , 0 ) ,
γ 0 v ˙ x ( 1 , 2 ) + d d t [ γ ( 1 , 1 ) v x ( 0 , 1 ) ] = ω b γ 0 v y ( 1 , 2 ) - ω E c k r γ 0 ( 1 - β 0 ) z ( 0 , 2 ) sin ξ ( 0 , 0 ) - v z ( 0 , 2 ) ω E γ 0 cos ξ ( 0 , 0 ) ,
γ 0 v ˙ y ( 1 , 0 ) = - ω b γ 0 v x ( 1 , 0 ) + ω E c γ 0 ( 1 - β 0 ) sin ξ ( 0 , 0 ) ,
γ 0 v ˙ y ( 1 , 2 ) + d d t [ γ ( 1 , 1 ) v y ( 0 , 1 ) ] = - ω b γ 0 v x ( 1 , 2 ) + ω E c k r γ 0 ( 1 - β 0 ) z ( 0 , 2 ) cos ξ ( 0 , 0 ) - v z ( 0 , 2 ) ω E γ 0 sin ξ ( 0 , 0 ) .
v x ( 1 , 0 ) = [ c ω E ( 1 - β 0 ) / Ω r ] { sin [ k r z 0 + ϕ - ω r ( 1 - β 0 ) t ] - sin ( k r z 0 + ϕ - ω b t ) } ,
v y ( 1 , 0 ) = - [ c ω E ( 1 - β 0 ) / Ω r ] { cos [ k r z 0 + ϕ - ω r ( 1 - β 0 ) t ] - cos ( k r z 0 + ϕ - ω b t ) } ,
γ ( 1 , 1 ) v x , y ( 0 , 1 ) Δ ω = γ B ( 1 , 1 ) v x B , y B ( 0 , 1 ) ,
v x B ( 0 , 1 ) = c β 0 δ cos ( k 0 z 0 + β 0 ω 0 t ) ,
v y B ( 0 , 1 ) = - c β 0 δ sin ( k 0 z 0 + β 0 ω 0 t ) .
γ 0 v ˙ z B ( 1 , 1 ) - c ( 1 - β 0 ) γ ˙ B ( 1 , 1 ) = ω E γ 0 c δ ( 1 - β 0 ) sin ( ϕ 0 + Δ ω t ) ,
c γ ˙ B ( 1 , 1 ) = ω E γ 0 [ v x B ( 0 , 1 ) cos ξ ( 0 , 0 ) + v y B ( 0 , 1 ) sin ξ ( 0 , 0 ) ] .
γ 0 v ˙ x B ( 1 , 2 ) + d E t [ v x B ( 0 , 1 ) γ B ( 1 , 1 ) ] = ω b γ 0 v y B ( 1 , 2 ) - ω E γ 0 v z ( 0 , 2 ) cos ξ ( 0 , 0 ) Δ ω - ω E c k r γ 0 ( 1 - β 0 ) z ( 0 , 2 ) sin ξ ( 0 , 0 ) Δ ω ,
γ 0 v ˙ y B ( 1 , 2 ) + d d t [ v y B ( 0 , 1 ) γ B ( 1 , 1 ) ] = - ω b γ 0 v x B ( 1 , 2 ) - ω E γ 0 v z ( 0 , 2 ) sin ξ ( 0 , 0 ) Δ ω + ω E c k r γ 0 ( 1 - β 0 ) z ( 0 , 2 ) cos ξ ( 0 , 0 ) Δ ω ,
v z B ( 1 , 1 ) = ( ω E c δ / Δ ω γ 0 2 ) [ cos ϕ 0 - cos ( ϕ 0 + Δ ω t ) ] ,
z B ( 1 , 1 ) = ( ω E c δ / Δ ω 2 γ 0 2 ) [ sin ϕ 0 - sin ( ϕ 0 + Δ ω t ) + Δ ω t cos ϕ 0 ] ,
γ B ( 1 , 1 ) = ( ω E β 0 γ 0 δ / Δ ω ) [ cos ϕ 0 - cos ( ϕ 0 + Δ ω t ) ] ,
v x B ( 1 , 2 ) = 0 ,
v y B ( 1 , 2 ) = 0 ,
( t + 1 c t ) G B = W B ϕ 0 ,
W B = - 4 π e N c E 2 v B · E r .
W B = 4 π e N c E [ k r z B ( 1 , 1 ) ] [ v x B ( 0 , 1 ) sin ξ ( 0 , 0 ) - v y B ( 0 , 1 ) cos ξ ( 0 , 0 ) ]
W B ϕ 0 = ( ω 0 ω p 2 β 0 2 δ 2 / c γ 0 Δ ω 2 ) [ sin Δ ω t - Δ ω t cos Δ ω t ] ,
ω p = ( 4 π N e 2 / m ) 1 / 2 .
G B ( t ) = ( ω 0 ω p 2 β 0 2 δ 2 / γ 0 Δ ω 3 ) ( 2 - 2 cos Δ ω t - Δ ω t sin Δ ω t ) .
G B = ( 0.54 ω 0 ω p 2 δ 2 L 3 / 4 γ 0 β 0 c 3 ) ,
G Col = ( 0.54 ω p 2 ω B 2 L 3 / 4 γ 0 ω 0 β 0 3 c 3 ) .
v x b ( 0 , 1 ) = - c β 0 δ cos ( k 0 z 0 + ω b t ) ,
v y B ( 0 , 1 ) = c β 0 δ sin ( k 0 z 0 + ω b t ) ,
γ b ( 1 , 1 ) = ( ω E δ γ 0 β 0 / Ω r ) [ cos ( ϕ 0 + Ω r t ) - cos ϕ 0 ]
z b ( 1 , 1 ) = [ ( 1 - β 0 ) ω E δ c β 0 / Ω r 2 ] [ sin ( ϕ 0 + Ω r t ) - sin ϕ 0 - Ω r t cos ϕ 0 ] ,
v x b ( 1 , 2 ) = ω E ω b c β 0 2 δ 2 2 Ω r 2 ( 2 Ω r t { cos ϕ 0 sin ( k 0 z 0 + ω b t ) - sin [ ϕ 0 + ( Ω r - ω b ) t - k 0 z 0 ] } + 3 cos ( ϕ 0 - k 0 z 0 - ω b t ) - 3 cos [ ϕ 0 + ( Ω r - ω b ) t - k 0 z 0 ] ) ,
v y b ( 1 , 2 ) = ω E ω b c β 0 2 δ 2 2 Ω r 2 ( 2 Ω r t { cos ϕ 0 cos ( k 0 z 0 + ω b t ) + cos [ ϕ 0 + ( Ω r - ω b ) t - k 0 z 0 ] } + 3 sin ( ϕ 0 - k 0 z 0 - ω b t ) - 3 sin [ ϕ 0 + ( Ω r - ω b ) t - k 0 z 0 ] ) .
W b = - 4 π N e c E 2 v b · E r = 4 π N e c E ( W b + W b + W b ) ,
W b = [ k r z b ( 1 , 1 ) ] [ v x b ( 0 , 1 ) sin ξ ( 0 , 0 ) - v y b ( 0 , 1 ) cos ξ ( 0 , 0 ) ] ,
W b = [ k r z ( 0 , 2 ) ] [ v x ( 1 , 0 ) sin ξ ( 0 , 0 ) - v y ( 1 , 0 ) cos ξ ( 0 , 0 ) ] ,
W b = - [ v x b ( 1 , 2 ) cos ξ ( 0 , 0 ) + v y b ( 1 , 2 ) sin ξ ( 0 , 0 ) ] ,
G b ( t ) = G b ( t ) + G b ( t ) + G b ( t ) ,
G b ( t ) = ( ω b ω p 2 β 0 δ 2 / 2 γ 0 Ω r 3 ) ( 2 - 2 cos Ω r t - Ω r t sin Ω r t ) ,
3 G b ( t ) = - [ G b ( t ) + G b ( t ) ] .
G b = ( 0.54 ω b ω p 2 δ 2 L 3 / 4 γ 0 β 0 2 c 3 )
d d t f ( z , p , t ) = f t + v z f z + p ˙ f p = 0 ,
f = f ( 0 ) + f ( 1 ) + ,
p = p ( 0 ) + p ( 1 ) +
( t + v z z ) f ( 0 ) + p ˙ ( 0 ) f ( 0 ) p = 0 ,
( t + v z z ) f ( 1 ) + p ˙ ( 0 ) f ( 1 ) p = - p ˙ ( 1 ) f ( 0 ) p
f ( 0 ) = N [ δ ( p z - p 0 ) ] - N p z ( 0 , 2 ) p z [ δ ( p z - p 0 ) ]
f ( 1 ) = - N p z ( 1 , 1 ) p z [ δ ( p z - p 0 ) ] ,
J = e - d p z [ f ( 0 ) + f ( 1 ) ] v T ,
J B , b = J B , b ( 0 , 1 ) + J B , b ( 1 , 0 ) + J B , b ( 1 , 2 ) ,
J B , b ( 0 , 1 ) = N e - d p z v B , b ( 0 , 1 ) [ δ ( p z - p 0 ) ] ,
J B ( 1 , 0 ) = J b ( 1 , 0 ) = N e - d p z v ( 1 , 0 ) [ δ ( p z - p 0 ) ] ,
J B ( 1 , 2 ) = - N e - d p z v B ( 0 , 1 ) p z B ( 1 , 1 ) p z [ δ ( p z - p 0 ) ] ,
J b ( 1 , 2 ) = J b ( 1 , 2 ) + J b ( 1 , 2 ) ,
J b ( 1 , 2 ) = - N e - d p z v b ( 0 , 1 ) p z b ( 1 , 1 ) p z [ δ ( p z - p 0 ) ] ,
J b ( 1 , 2 ) = N e - d p z v b ( 1 , 2 ) [ δ ( p z - p 0 ) ] ,
( z 2 - 1 c 2 2 t 2 ) A r ( z , t ) = - 4 π c J ( z , t ) ,
( z + 1 c t ) G ( z , t ) = - 4 π c E ( J x cos ξ + J y sin ξ )

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