Abstract

In a seeded high-gain free-electron laser (FEL), where a coherent laser pulse interacts with an ultrarelativistic electron beam, the seed laser pulse can be frequency chirped, and the electron beam can be energy chirped. Besides these two chirps, the FEL interaction introduces an intrinsic frequency chirp in the FEL even if the above-mentioned two chirps are absent. We examine the interplay of these three chirps. The problem is formulated as an initial value problem and solved via a Green function approach. Besides the chirp evolution, we also give analytical expressions for the pulse duration and bandwidth of the FEL, which remains fully longitudinally coherent in the high-gain exponential growth regime. Because the chirps are normally introduced for a final compression of the FEL pulse, some conceptual issues are discussed. We show that to get a short pulse duration, an energy chirp in the electron beam is important.

© 2007 Optical Society of America

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  1. J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron laser," Nucl. Instrum. Methods Phys. Res. A 250, 484-489 (1986).
    [CrossRef]
  2. K. J. Kim, "An analysis of self-amplified spontaneous emission," Nucl. Instrum. Methods Phys. Res. A 250, 396-403 (1986).
    [CrossRef]
  3. K. J. Kim, "Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers," Phys. Rev. Lett. 57, 1871-1874 (1986).
    [CrossRef] [PubMed]
  4. K. J. Kim, "Temporal and transverse coherence of self-amplified spontaneous emission," Lawrence Berkeley National Laboratory Report No. LBNL-40672 (Lawrence Berkeley National Laboratory, 1997).
  5. S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
    [CrossRef]
  6. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
    [CrossRef]
  7. R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
    [CrossRef]
  8. J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).
  9. A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
    [CrossRef]
  10. B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. , 30, 1951-1963 (1994).
    [CrossRef]
  11. M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
    [CrossRef]
  12. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  13. M. J. Bastiaans, "Propagation laws for the 2nd-order moments of the Wigner distribution function in 1st-order optical-systems," Optik (Jena) 82, 173-181 (1989).
  14. R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970).
    [CrossRef]
  15. D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
    [CrossRef]

2006 (1)

E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

2004 (1)

A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
[CrossRef]

2003 (1)

S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
[CrossRef]

1998 (1)

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

1994 (1)

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. , 30, 1951-1963 (1994).
[CrossRef]

1989 (1)

M. J. Bastiaans, "Propagation laws for the 2nd-order moments of the Wigner distribution function in 1st-order optical-systems," Optik (Jena) 82, 173-181 (1989).

1986 (3)

J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron laser," Nucl. Instrum. Methods Phys. Res. A 250, 484-489 (1986).
[CrossRef]

K. J. Kim, "An analysis of self-amplified spontaneous emission," Nucl. Instrum. Methods Phys. Res. A 250, 396-403 (1986).
[CrossRef]

K. J. Kim, "Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers," Phys. Rev. Lett. 57, 1871-1874 (1986).
[CrossRef] [PubMed]

1984 (1)

R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
[CrossRef]

1975 (1)

D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
[CrossRef]

1970 (1)

R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970).
[CrossRef]

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Anderson, D.

D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
[CrossRef]

Askne, J.

D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, "Propagation laws for the 2nd-order moments of the Wigner distribution function in 1st-order optical-systems," Optik (Jena) 82, 173-181 (1989).

Boller, K.-J.

A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
[CrossRef]

Bonifacio, R.

R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
[CrossRef]

Huang, Z.

S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
[CrossRef]

Khachatryan, A. G.

A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
[CrossRef]

Kim, K. J.

K. J. Kim, "An analysis of self-amplified spontaneous emission," Nucl. Instrum. Methods Phys. Res. A 250, 396-403 (1986).
[CrossRef]

K. J. Kim, "Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers," Phys. Rev. Lett. 57, 1871-1874 (1986).
[CrossRef] [PubMed]

K. J. Kim, "Temporal and transverse coherence of self-amplified spontaneous emission," Lawrence Berkeley National Laboratory Report No. LBNL-40672 (Lawrence Berkeley National Laboratory, 1997).

Kolner, B. H.

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. , 30, 1951-1963 (1994).
[CrossRef]

Krinsky, S.

S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
[CrossRef]

Kubota, H.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Lisak, M.

D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
[CrossRef]

Murphy, J. B.

J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).

Nakazawa, M.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Narducci, L. M.

R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
[CrossRef]

Pellegrini, C.

R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
[CrossRef]

Sahara, A.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Saldin, E. L.

E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

Schneidmiller, E. A.

E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

Smith, R. L.

R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970).
[CrossRef]

Tamura, K.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

van Goor, F. A.

A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
[CrossRef]

Wang, J.-M.

J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron laser," Nucl. Instrum. Methods Phys. Res. A 250, 484-489 (1986).
[CrossRef]

Wang, X. J.

J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).

Watanabe, T.

J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wu, J.

J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).

Yu, L.-H.

J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron laser," Nucl. Instrum. Methods Phys. Res. A 250, 484-489 (1986).
[CrossRef]

Yurkov, M. V.

E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

Am. J. Phys. (1)

R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970).
[CrossRef]

IEEE J. Quantum Electron. (2)

B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. , 30, 1951-1963 (1994).
[CrossRef]

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (2)

J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron laser," Nucl. Instrum. Methods Phys. Res. A 250, 484-489 (1986).
[CrossRef]

K. J. Kim, "An analysis of self-amplified spontaneous emission," Nucl. Instrum. Methods Phys. Res. A 250, 396-403 (1986).
[CrossRef]

Opt. Commun. (1)

R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free-electron laser," Opt. Commun. 50, 373-378 (1984).
[CrossRef]

Optik (Jena) (1)

M. J. Bastiaans, "Propagation laws for the 2nd-order moments of the Wigner distribution function in 1st-order optical-systems," Optik (Jena) 82, 173-181 (1989).

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, J. Askne, and M. Lisak, "Wave packets in an absorptive and strongly dispersive medium," Phys. Rev. A 12, 1546-1552 (1975).
[CrossRef]

Phys. Rev. E (1)

A. G. Khachatryan, F. A. van Goor, and K.-J. Boller, "Interaction of free charged particles with a chirped electromagnetic pulse," Phys. Rev. E 70, 067601 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

K. J. Kim, "Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers," Phys. Rev. Lett. 57, 1871-1874 (1986).
[CrossRef] [PubMed]

Phys. Rev. ST Accel. Beams (2)

S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
[CrossRef]

E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energy-chirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

Other (2)

J. B. Murphy, J. Wu, X. J. Wang, and T. Watanabe, "Longitudinal-coherence preservation and chirp evolution in a high-gain laser-seeded free-electron-laser amplifier," Brookhaven National Laboratory Report No. BNL-75807-2006-JA or Stanford Linear Accelerator Center Report No. SLAC-PUB-11852 (Brookhaven National Laboratory, Standford Linear Accelerator Center, 2006).

K. J. Kim, "Temporal and transverse coherence of self-amplified spontaneous emission," Lawrence Berkeley National Laboratory Report No. LBNL-40672 (Lawrence Berkeley National Laboratory, 1997).

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Figures (11)

Fig. 1
Fig. 1

Contour plot of the seed laser Wigner function W s ( t z c , ω ω 0 ) . The solid ellipses is for α 0 = 1 4 and β 0 = 1 2 ; the dashed ellipses for α 0 = 1 4 and β 0 = 1 2 ; dotted ellipse for α 0 = 5 4 and β 0 = 0 ; and the dotted-dashed ellipse for α 0 = 1 4 and β 0 = 0

Fig. 2
Fig. 2

Centrovelocity of the FEL with both an initial seed frequency chirp and an electron beam energy chirp for z [ 0 , 20 ] m and with ρ = 10 3 , λ 0 = 0.8 μ m , and λ w = 0.039 m . The centrovelocity v c is given in Eq. (32). The solid curve is for β 0 > 0 and μ > 0 ; the dashed curve for β 0 < 0 and μ > 0 ; the dotted curve for β 0 > 0 and μ < 0 ; and the dotted-dashed curve for β 0 < 0 and μ < 0 .

Fig. 3
Fig. 3

Contour plot of the Wigner function W ( t z v c , ω ω 0 ) of the FEL light for z [ 0 , 20 ] m . The centrovelocity v c is given in Eq. (32). The seed laser does not have a frequency chirp initially, nor does the electron beam have an energy chirp initially. In the plot, δ ω = ω ω 0 .

Fig. 4
Fig. 4

Contour plot of the Wigner function W ( t z v c , ω ω 0 ) of the FEL light for z [ 0 , 20 ] m . The centrovelocity v c is given in Eq. (32). The left (right) plot is for an initial positive (negative) seed laser frequency chirp of equal magnitude.

Fig. 5
Fig. 5

Plot of the FEL pulse duration, bandwidth, chirp, and longitudinal emittance versus z for the same conditions as in Fig. 4. Notations are explained in the text.

Fig. 6
Fig. 6

Contour plot of the Wigner function W ( t z v c , ω ω 0 ) of the FEL light for z [ 0 , 20 ] m . The centrovelocity v c is given in Eq. (32). The seed laser does not have a frequency chirp initially. The left (right) plot is for an initial positive (negative) electron beam energy chirp of equal magnitude. In the plot, δ ω = ω ω 0 .

Fig. 7
Fig. 7

Contour plot of the Wigner function W ( t z v c , ω ω 0 ) of the FEL light for z [ 0 , 20 ] m . The centrovelocity v c is given in Eq. (32). The electron beam has a positive energy chirp. The left (right) plot is for an initial positive (negative) seed laser chirp of equal magnitude.

Fig. 8
Fig. 8

Plot of the FEL pulse duration, bandwidth, chirp, and longitudinal emittance versus z for the same conditions as in Fig. 7. Notations are explained in the text.

Fig. 9
Fig. 9

Contour plot of the Wigner function W ( t z v c , ω ω 0 ) of the FEL light for z [ 0 , 20 ] m . The centrovelocity v c is given in Eq. (32). The electron beam has a negative energy chirp. The left (right) plot is for an initial positive (negative) seed laser chirp of equal magnitude.

Fig. 10
Fig. 10

Plot of the FEL pulse duration, bandwidth, chirp, and longitudinal emittance versus z for the same conditions as in Fig. 9. Notations are explained in the text.

Fig. 11
Fig. 11

Plot of the FEL pulse duration σ t , f after postundulator pulse compression versus the dimensionless frequency chirp ϵ in the seed laser (left plot) and the dimensionless energy chirp η (right plot) in the electron beam introduced in Eq. (43).

Equations (51)

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ψ Z + p ψ θ 2 D 2 γ 0 2 ( A e i θ + A * e i θ ) ψ 0 p = 0 ,
( Z + θ ) A ( θ , Z ) = D 1 γ 0 e i θ d p ψ ( θ , p , Z ) ,
D 1 = e a w n 0 [ J J ] 2 2 k w ϵ 0 , D 2 = e a w [ J J ] 2 k w m c 2 ,
( Z + θ ) A ( θ , Z ) D 1 γ 0 j e i θ j + i μ Z θ j δ ( θ θ j ) + i ( 2 ρ ) 3 0 Z d Z ( Z Z ) A ( θ , Z ) e i μ ( Z Z ) θ ,
f ( θ , s ) = 0 d Z e s Z A ( θ , Z ) .
f θ + [ s i ( 2 ρ ) 3 ( s i μ θ ) 2 ] f = A ( θ , 0 ) + D 1 γ 0 j e i θ j δ ( θ θ j ) s i μ θ j .
f ( θ , s ) = θ d θ exp ( s ( θ θ ) + i ( 2 ρ ) 3 ( θ θ ) ( s i μ θ ) ( s i μ θ ) ) A ( θ , 0 ) .
A ( θ , Z ) = c d s 2 π i e s Z θ d θ exp ( s ( θ θ ) + i ( 2 ρ ) 3 ( θ θ ) ( s i μ θ ) ( s i μ θ ) ) A ( θ , 0 ) e ρ ( 3 + i ) Z 0 d ξ A ( θ ξ , 0 ) e i μ θ ( Z ξ ) e ρ ( 3 + i ) [ 9 ( ξ Z 3 ) 2 ( 4 Z ) ] e i ( μ 2 ) ( Z ξ ) ξ ,
μ 2 γ 0 ω 0 d γ d t , with γ 0 1 + a w 2 2 λ w λ 0 ,
g ( Z , θ ; μ ) e ρ ( 3 + i ) Z e ρ ( 3 + i ) [ 9 ( θ Z 3 ) 2 ( 4 Z ) ] e ( i μ 2 ) ( Z θ ) θ ,
σ t , GF ( z ) = 1 3 σ ω , GF ( z ) ,
σ ω , μ , GF ( z ) = σ ω , GF ( z ) 1 + η 9 + η 2 81 ,
η μ k w z ρ = 3 3 μ ω 0 2 σ ω , GF 2 ( z ) ,
σ ω , GF ( z ) 3 3 ρ ω 0 2 k w z ,
ϕ ( t , z ) = i [ ( 1 3 ρ 9 ρ k 0 4 k w ) k 0 z 1 2 ( k 0 + k w ) k 0 μ z 2 + ( k 0 + k w 2 ) μ ω 0 t z ( 1 3 ρ 9 ρ k 0 2 k w ) ω 0 t 1 2 ( μ + 9 ρ 2 k w z ) ω 0 2 t 2 ] .
μ k w z ρ 1 η 1 .
E s ( t , z ) = E 0 e i ( k 0 z ω 0 t ) e ( α 0 + i β 0 ) ( t z c ) 2 ,
W ( t , ω , z ) = E ( t τ 2 , z ) E * ( t + τ 2 , z ) e i ω τ d τ .
F d t d ω W ( t , ω , z ) F d t d ω W ( t , ω , z ) .
W s ( t z c , ω ω 0 ) = E 0 2 2 π α 0 exp ( [ 4 ( t z c ) 2 ( α 0 2 + β 0 2 ) 4 β 0 ( t z c ) ( ω ω 0 ) + ( ω ω 0 ) 2 ] ( 2 α 0 ) ) .
t z v c = z c = z v g ,
ω = ω 0 .
σ t , seed = ( t t ) 2 = 1 2 α 0 ,
σ ω , seed = ( ω ω ) 2 = α 0 2 + β 0 2 α 0 .
( t t ) ( ω ω ) = β 0 2 α 0 .
ϵ L ( t t ) 2 ( ω ω ) 2 ( t t ) ( ω ω ) 2 .
E ( t , z = 0 ) = E 0 e i ω 0 t ( α 0 + i β 0 ) t 2 = E 0 e i θ ( θ 2 ω 0 2 ) ( α 0 + i β 0 ) .
A ( θ , 0 ) = E 0 e ( θ 2 ω 0 2 ) ( α 0 + i β 0 ) .
A ( θ , Z ) E 0 e ρ ( 3 + i ) Z 0 d ξ e [ ( θ ξ ) 2 ω 0 2 ] ( α 0 + i β 0 ) e i μ θ ( Z ξ ) e ρ ( 3 + i ) [ 9 ( ξ Z 3 ) 2 ( 4 Z ) ] e i ( μ 2 ) ( Z ξ ) ξ E 0 , FEL e ( 3 4 ) ρ ( 3 + i ) Z + i μ θ Z ( θ 2 ω 0 2 ) ( α 0 + i β 0 ) exp ( Z { 4 i θ ω 0 2 ( α 0 + i β 0 ) + [ ( Z + 2 θ ) μ + 3 i ρ ( 3 + i ) ] } 2 { 16 Z ω 0 2 ( α 0 + i β 0 ) + 4 [ 2 i μ Z + 9 ρ ( 3 + i ) ] } ) ,
E FEL ( t , z ) = E 0 , FEL e ρ ( 3 + i ) k ω z × e i ( k 0 z ω 0 t ) e [ α ( z ) + i β ( z ) ] ( t z v c ) 2 ,
W ( t , ω , z ) E FEL ( t τ 2 , z ) E FEL * ( t + τ 2 , z ) e i ω τ d τ = E 0 , FEL 2 e 2 3 ρ k w z 2 π α ( z ) exp ( 1 2 [ 1 r ( z ) 2 ] { [ t z v c ( z ) ] 2 σ t 2 ( z ) 2 r ( z ) [ t z v c ( z ) ] ( ω ω 0 ) σ t ( z ) σ ω ( z ) + ( ω ω 0 ) 2 σ ω 2 ( z ) } ) ,
σ t ( z ) = ( [ 4 β 0 μ ω 0 2 + μ 2 ω 0 4 + 3 σ ω , GF 4 + 4 α 0 σ ω , seed 2 + ( 6 α 0 + 2 3 β 0 3 μ ω 0 2 ) σ ω , GF 2 ] { 3 σ ω , GF 2 [ 4 β 0 μ ω 0 2 + μ 2 ω 0 4 + 4 α 0 ( σ ω , GF 2 + σ ω , seed 2 ) ] } ) 1 2 ,
σ ω ( z ) = σ ω , GF 3 2 3 [ ( 9 [ 4 α 0 ( σ ω , GF 2 + σ ω , seed 2 ) + μ ω 0 2 ( 4 β 0 + μ ω 0 2 ) ] 2 σ ω , GF 4 + { 12 β 0 + 2 μ ω 0 2 [ 4 α 0 σ ω , seed 2 + μ ω 0 2 ( 4 β 0 + μ ω 0 2 ) ] σ ω , GF 4 + 3 [ μ 2 ω 0 4 + 4 α 0 ( σ ω , seed 2 + 3 μ ω 0 2 ) ] σ ω , GF 2 } 2 ) ( [ 4 α 0 ( σ ω , GF 2 + σ ω , seed 2 ) + μ ω 0 2 ( 4 β 0 + μ ω 0 2 ) ] [ 3 σ ω , GF 4 + 4 α 0 σ ω , seed 2 4 β 0 μ ω 0 2 + μ 2 ω 0 4 + σ ω , GF 2 ( 6 α 0 + 2 3 β 0 3 μ ω 0 2 ) ] ) ] 1 2 .
( t t ) ( ω ω ) β ( z ) 2 α ( z ) = 1 6 σ ω , GF 2 [ 4 σ ω , GF 2 ( 3 β 0 σ ω , GF 2 + 3 α 0 σ ω , seed 2 ) + 4 α 0 μ ( 3 σ ω , GF 2 + 2 σ ω , seed 2 ) ω 0 2 μ 2 ( 8 β 0 + 3 σ ω , GF 2 ) ω 0 4 + 2 μ 3 ω 0 6 ] [ 4 α 0 ( σ ω , GF 2 + σ ω , seed 2 ) + μ ω 0 2 ( 4 β 0 + μ ω 0 2 ) ] .
r ( z ) ( t t ) ( ω ω ) σ t ( z ) σ ω ( z ) .
v c 1 ( z ) t z = ( k 0 + 2 3 k ω ) ω 0 + μ ω 0 k w ( 2 α 0 2 3 β 0 + 3 μ ω 0 2 ) 2 3 [ 4 α 0 ( σ ω , GF 2 + σ ω , seed 2 ) 4 β 0 μ ω 0 2 + μ 2 ω 0 4 ] .
v g ω 0 k 0 + ( 2 3 ) k w .
v c 1 ( z ) v g 1 + μ ω 0 k w ( 2 α 0 2 3 β 0 + 3 μ ω 0 2 ) 2 3 [ 4 ( α 0 2 + β 0 2 ) 4 β 0 μ ω 0 2 + μ 2 ω 0 4 ] .
μ a p 2 ρ ( ω 0 σ t ) ,
σ ω ( z ) μ = 0 = 1 1 σ ω , seed 2 + 1 σ ω , GF 2 ( z ) .
γ ( t ) 2 γ 0 2 = ω ( t ) ω 0 d γ d t γ 0 2 ω 0 d ω d t = γ 0 β 0 2 ω 0 .
μ m β 0 ω 0 2 .
1 γ 0 d γ d t 2 λ 0 ( z λ w ) c ρ .
β 0 ρ ω 0 2 k w z ,
α 0 = σ ω , seed 2 + σ ω , seed 4 4 β 0 2 2 = 1 + 1 64 β 0 2 σ t , i 4 8 σ t , i 2 ,
α 0 = 1 1 64 β 0 2 σ t , i 4 8 σ t , i 2 .
ϵ 3 3 β 0 σ ω , GF 2 , η 3 3 μ ω 0 2 σ ω , GF 2 .
σ t , f = 1 σ ω , GF { 729 [ 3 3 B ( 1 + 4 C 2 ) + 8 C 4 η ( 4 ϵ + η ) ] { 3 B ( 1 + 6 C 2 ) + 8 3 C 4 [ 81 + ϵ ( 18 4 η ) + ( η 9 ) η ] } ( 729 [ 3 3 B ( 1 + 4 C 2 ) + 8 C 4 η ( 4 ϵ + η ) ] 2 + { 9 B [ 9 + 2 ( 1 + 6 C 2 ) η ] + 8 3 C 4 [ 324 ϵ ( 9 + 8 ϵ ) η 2 + 2 η 3 ] } 2 ) } 1 2 ,
B = 3 3 + 27 64 ϵ 2 C 4 , C = σ t , i σ ω , GF .
σ t , f μ = 0 = 1 2 σ ω μ = 0 = σ t , i 1 + 1 4 C 2 ,
σ t , f σ t , i [ 1 + 1 4 C 2 1 + 4 C 2 36 C 2 η 4 27 C 2 ϵ η 1 + 2 C 2 54 η 2 + 2 ( 3 + 4 C 2 ) 243 C 2 ϵ η 2 + 16 729 C 6 ϵ 2 η 2 ] 1 2 .

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