Abstract

An ABCD formalism is identified to characterize a seeded Free Electron Laser (FEL) with three chirps: an initial frequency chirp in the seed Laser, an energy chirp in the electron bunch, and an intrinsic frequency chirp due to the FEL process. A scheme of generating attosecond few-cycle pulses is proposed by invoking an FEL seeded by high-order harmonic generation (HHG) from an infrared laser. The HHG seed has generic attosecond structure. It is possible to manipulate these three chirps to maintain the attosecond structure via post-undulator chirped pulse compression.

© 2007 Optical Society of America

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  1. T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000).
    [CrossRef]
  2. P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
    [CrossRef] [PubMed]
  3. A. A. Zholents and W. M. Fawley, "Proposal for intense attosecond radiation from an x-ray free-electron Laser," Phys. Rev. Lett. 92, 224801 (2004).
    [CrossRef] [PubMed]
  4. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, "Self-amplified spontaneous emission FEL with energychirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
    [CrossRef]
  5. J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
    [CrossRef]
  6. B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
    [CrossRef]
  7. J. Wu, J. B. Murphy, P. J. Emma, X. J. Wang, T. Watanabe, and X. Zhong, "Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser," J. Opt. Soc. Am. B 24, 484 (2007).
    [CrossRef]
  8. S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
    [CrossRef]
  9. R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).
  10. 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 BNL-75807-2006-JA, and SLAC-PUB-11852 (2006).
  11. R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970).
    [CrossRef]
  12. R. Bonifacio, C. Pellegrini, and L. M. Narducci, "Collective instabilities and high-gain regime in a free electron Laser," Opt. Commun. 50, 373 (1984).
    [CrossRef]
  13. J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
    [CrossRef]
  14. J.-M. Wang and L.-H. Yu, "A transient analysis of a bunched beam free electron Laser," Nucl. Instrum. Methods A 250, 484 (1986).
    [CrossRef]
  15. B. H. Kolner, "Space-Time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951 (1994).
    [CrossRef]

2007 (3)

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

J. Wu, J. B. Murphy, P. J. Emma, X. J. Wang, T. Watanabe, and X. Zhong, "Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser," J. Opt. Soc. Am. B 24, 484 (2007).
[CrossRef]

2006 (1)

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

2004 (2)

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

A. A. Zholents and W. M. Fawley, "Proposal for intense attosecond radiation from an x-ray free-electron Laser," Phys. Rev. Lett. 92, 224801 (2004).
[CrossRef] [PubMed]

2002 (1)

R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).

2000 (1)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

1994 (1)

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

1990 (1)

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
[CrossRef]

1986 (1)

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

1985 (1)

J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
[CrossRef]

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 (1984).
[CrossRef]

1970 (1)

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

Bane, K.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Bolton, P. R.

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

Bonifacio, R.

J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
[CrossRef]

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

Brabec, T.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

Clarke, J. A.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Cornacchia, M.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
[CrossRef]

Dijaili, S.P.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
[CrossRef]

Dunning, D. J.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Emma, P.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Emma, P. J.

Fawley, W. M.

A. A. Zholents and W. M. Fawley, "Proposal for intense attosecond radiation from an x-ray free-electron Laser," Phys. Rev. Lett. 92, 224801 (2004).
[CrossRef] [PubMed]

Hirst, G. J.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Huang, Z.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Kolner, B. H.

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

Krausz, F.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

McNeil, B. W. J.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Murphy, J. B.

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

J. Wu, J. B. Murphy, P. J. Emma, X. J. Wang, T. Watanabe, and X. Zhong, "Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser," J. Opt. Soc. Am. B 24, 484 (2007).
[CrossRef]

J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
[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 (1984).
[CrossRef]

Ortega-Martinez, R.

R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).

Owen, H. L.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Pellegrini, C.

J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
[CrossRef]

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

Rivera, A. L.

R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).

Roman-Moreno, C. J.

R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).

Saldin, E. L.

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

Schlarb, H.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Schneidmiller, E. A.

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

Sheehy, B.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
[CrossRef]

Smith, R. L.

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

Stupakov, G.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Thompson, N.R.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Walz, D.

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

Wang, J.-M.

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

Wang, X. J.

Watanabe, T.

Williams, P. H.

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Wu, J.

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

J. Wu, J. B. Murphy, P. J. Emma, X. J. Wang, T. Watanabe, and X. Zhong, "Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser," J. Opt. Soc. Am. B 24, 484 (2007).
[CrossRef]

Yu, L.-H.

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

Yurkov, M. V.

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

Zholents, A. A.

A. A. Zholents and W. M. Fawley, "Proposal for intense attosecond radiation from an x-ray free-electron Laser," Phys. Rev. Lett. 92, 224801 (2004).
[CrossRef] [PubMed]

Zhong, X.

J. Wu, J. B. Murphy, P. J. Emma, X. J. Wang, T. Watanabe, and X. Zhong, "Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser," J. Opt. Soc. Am. B 24, 484 (2007).
[CrossRef]

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

Am. J. Phys. (1)

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

Appl. Phys. Lett. (1)

J. Wu, P. R. Bolton, J. B. Murphy, and X. Zhong, "Free electron laser seeded by ir laser driven high-order harmonic generation," Appl. Phys. Lett. 90, 021109 (2007).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

New J. Phys. (1)

B. W. J. McNeil, J. A. Clarke, D. J. Dunning, G. J. Hirst, H. L. Owen, N. R. Thompson, B. Sheehy, and P. H. Williams, "An XUV-FEL amplifier seeded using high harmonic generation," New J. Phys. 9, 82 (2007).
[CrossRef]

Nucl. Instrum. Methods A (1)

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

Opt. Commun. (2)

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

J. B. Murphy, C. Pellegrini, and R. Bonifacio, "Collective instability of a free electron laser including space charge and harmonics," Opt. Commun. 53, 197 (1985).
[CrossRef]

Phys. Rev. Lett. (2)

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, "Femtosecond and subfemtosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser," Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

A. A. Zholents and W. M. Fawley, "Proposal for intense attosecond radiation from an x-ray free-electron Laser," Phys. Rev. Lett. 92, 224801 (2004).
[CrossRef] [PubMed]

Phys. Rev. ST Accel. Beams (1)

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

Rev. Mex. Fis. (1)

R. Ortega-Martinez, C. J. Roman-Moreno, and A. L. Rivera, "The Wigner function in paraxial optics I. Matrix methods in Fourier optics," Rev. Mex. Fis. 48, 565 (2002).

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000).
[CrossRef]

Other (1)

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 BNL-75807-2006-JA, and SLAC-PUB-11852 (2006).

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Figures (2)

Fig. 1.
Fig. 1.

The FEL pulse rms duration (upper left), the rms bandwidth (upper right), the rms duration after post-undulator compression (lower left), and the time-frequency correlation (lower right) as a function of the location into the undulator. The solid (red) curve is for μ = 2βs , the dashed (green) for μ = -2βs , and the dash-dotted (blue) for μ = 0. For all these three cases, βs ≈ 8.7 × 10-5. The dotted (purple) curve is for μ = βs = 0.

Fig. 2.
Fig. 2.

The evolution of the Wigner function ellipse as a function of the location into the undulator is shown in the left subplots. For clarity, only three pulselets are shown. In the right subplots, each ellipse stands for experiencing a postundulator compression. In the upper row, the energy chirp in the electron bunch is μ = 2βs . In the lower row, μ = 0. In all the plots, βs ≈ 8.7 × 10-5.

Equations (13)

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

E FEL ( t , z ) = E 0 , FEL e ρ ( 3 + i ) k w z e i ( k s z ω s t ) e [ α s , f ( z ) + i β s , f ( z ) ] ω s 2 ( t z v c ) 2 ,
1 p ( z ) 2 β s , f ( z ) ω s + i 2 α s , f ( z ) ω s , as p ( z ) = Ap ( 0 ) + B Cp ( 0 ) + D ,
M ABCD = ( A B C D ) = ( 1 2 ik w z 9 ( i + 3 ) ρ ω s C D ) ,
C = ( iV W ) ω s 2 U ( iV μ = 0 W μ = 0 ) ω s 2 U μ = 0 , and D = 1 + BC ,
{ U 3 + P 2 [ Q + ( 6 + 4 R 2 ) α s + 3 ( 2 β s μ ) ] V 3 [ Q + 4 ( 1 + R 2 ) α s ] W 4 ( 3 R 2 α s + 3 β s ) + P 2 μ [ 2 Q + 4 ( 3 + 2 R 2 ) α s 3 μ ]
{ P ω s σ ω , GF Q μ ( μ 4 β s ) ω s 2 σ ω , 2 , R σ ω , s σ ω , GF with μ 2 γ 0 ω s d γ dt , and σ ω , GF ( z ) 3 3 ρω s 2 k w z ,
( τ d τ d ζ ) 2 = ( A B C D ) 1 2 ( τ d τ d ζ ) 1 , where { τ = t z dk d ω ω = ω s = t z v g , 0 ζ = ω s z d 2 k d ω 2 ω = ω s ,
A ( θ , Z ) 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 ,
E s ( t , z ) = E s , 0 e i ( k s z ω s t ) e i β s ω s 2 t 2 n = N N e t n 2 4 σ t , 0 2 e α s ω s 2 [ ( t t n ) z c ] 2 .
W ( t , ω , z ) = E FEL ( t τ 2 , z ) E FEL * ( t + τ 2 , z ) e i ω t d τ = 2 2 π σ t , s , f e δ t 2 σ t , s , f 2 2 r δ t δ ω σ t , s , f σ ω , s , f + δ ω 2 σ ω , s , f 2 2 ( 1 r 2 ) ,
r 2 β s , f σ t , s , f ω s 2 σ ω , s , f = β s , f 2 α s , f 1 σ t , s , f σ ω , s , f = ( t t ) ( ω ω ) σ t , s , f σ ω , s , f .
{ σ t , s , f 2 ( z ) = U ω s 2 V σ ω , s , f 1 ( z ) = ω s 2 ( V 2 + W 2 ) 4 UV ( t t ) ( ω ω ) β s , f ( z ) 2 α s , f ( z ) = W 2 V .
γ ( t ) 2 γ 0 2 = ω ( t ) ω s d γ dt γ 0 2 ω s d ω dt = γ 0 β s ω s .

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