Abstract

Exponential growth and superradiance regimes in a high-gain free electron laser (FEL) are studied in this paper for both a seeded FEL and a Self-Amplified Spontaneous Emission (SASE) FEL. The results are compared to the earlier superrdaince theory and the recent experimental observation. The influence of an initial energy chirp along the electron bunch on the superradiance mode is explored for the first time. With a short seed to increase the initial seed bandwidth, a tunable seeded FEL is possible.

© 2008 Optical Society of America

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  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).
  2. J. Wu, J. B. Murphy, P. J. Emma, X. 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]
  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]
  4. J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
    [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. R. Bonifacio and F. Casagrande, "The Superradiant Regime of a Free Electron Laser," Nucl. Instrum. Methods Phys. Res. A 239, 36 (1985).
    [CrossRef]
  8. R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
    [CrossRef]
  9. R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
    [CrossRef]
  10. L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
    [CrossRef]
  11. T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
    [CrossRef] [PubMed]
  12. 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 (1986).
    [CrossRef]
  13. S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003).
    [CrossRef]
  14. 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]
  15. S. Reiche, "GENESIS 1.3: a fully 3D time-dependent FEL simulation code," Nucl. Instrum. Methods Phys. Res. A 429, 243 (1999).
    [CrossRef]
  16. X. J. Wang, Y. Shen, T. Watanabe, J. B. Murphy, J. Rose, and T. Tsang, "The first lasing of 193 nm SASE, 4th harmonic HGHG and ESASE at the NSLS SDL," in Proceedings of the 28th internatinal Free Electron Laser conference (Berlin, Germany, 2006), pp. 18.

2007 (4)

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, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

J. Wu, J. B. Murphy, P. J. Emma, X. 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]

2005 (1)

L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
[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]

1999 (1)

S. Reiche, "GENESIS 1.3: a fully 3D time-dependent FEL simulation code," Nucl. Instrum. Methods Phys. Res. A 429, 243 (1999).
[CrossRef]

1990 (1)

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

1988 (1)

R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
[CrossRef]

1986 (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 (1986).
[CrossRef]

1985 (2)

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 and F. Casagrande, "The Superradiant Regime of a Free Electron Laser," Nucl. Instrum. Methods Phys. Res. A 239, 36 (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]

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]

J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

Bonifacio, R.

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
[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]

R. Bonifacio and F. Casagrande, "The Superradiant Regime of a Free Electron Laser," Nucl. Instrum. Methods Phys. Res. A 239, 36 (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]

Casagrande, F.

R. Bonifacio and F. Casagrande, "The Superradiant Regime of a Free Electron Laser," Nucl. Instrum. Methods Phys. Res. A 239, 36 (1985).
[CrossRef]

De Salvo Souza, L.

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

Emma, P. J.

Giannessi, L.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
[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]

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]

Maroli, C.

R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
[CrossRef]

Murphy, J. B.

J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

J. Wu, J. B. Murphy, P. J. Emma, X. 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]

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]

Musumeci, P.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
[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]

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]

Pierini, P.

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

Piovella, N.

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
[CrossRef]

Reiche, S.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

S. Reiche, "GENESIS 1.3: a fully 3D time-dependent FEL simulation code," Nucl. Instrum. Methods Phys. Res. A 429, 243 (1999).
[CrossRef]

Rose, J.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

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]

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]

Shen, Y.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

Spampinati, S.

L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
[CrossRef]

Tsang, T.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[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 Phys. Res. A 250, 484 (1986).
[CrossRef]

Wang, K.

J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

Wang, X.

Wang, X. J.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

Watanabe, T.

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[CrossRef] [PubMed]

J. Wu, J. B. Murphy, P. J. Emma, X. 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]

Wu, J.

J. Wu, J. B. Murphy, P. J. Emma, X. 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]

J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

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 (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]

Zhong, X.

J. Wu, J. B. Murphy, P. J. Emma, X. 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]

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]

J. Appl. Phys. (1)

L. Giannessi, P. Musumeci, and S. Spampinati, "Nonlinear pulse evolution in seeded free-electron laser amplifiers and in free-electron laser cascades," J. Appl. Phys. 98, 043110 (2005).
[CrossRef]

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

Nuc. 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 (1986).
[CrossRef]

R. Bonifacio and F. Casagrande, "The Superradiant Regime of a Free Electron Laser," Nucl. Instrum. Methods Phys. Res. A 239, 36 (1985).
[CrossRef]

Nucl. Inst. Meth. A (1)

R. Bonifacio, L. De Salvo Souza, P. Pierini, and N. Piovella, "The Superradiant regime of a FEL: Analytical and numerical results," Nucl. Instrum. Methods Phys. Res. A 296, 358 (1990).
[CrossRef]

Opt. Commun. (3)

R. Bonifacio, C. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988).
[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]

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]

Opt. Express (1)

J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express 15, 12749 (2007).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

T. Watanabe, X. J. Wang, J. B. Murphy, J. Rose, Y. Shen, T. Tsang, L. Giannessi, P. Musumeci, and S. Reiche, "Experimental characterization of superradiance in a single-pass high-gain laser-seeded free-electron laser amplifier," Phys. Rev. Lett. 98, 034802 (2007).
[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 energychirped electron beam and its application for generation of attosecond x-ray pulses," Phys. Rev. ST Accel. Beams 9, 050702 (2006).
[CrossRef]

Sect. A (1)

S. Reiche, "GENESIS 1.3: a fully 3D time-dependent FEL simulation code," Nucl. Instrum. Methods Phys. Res. A 429, 243 (1999).
[CrossRef]

Other (2)

X. J. Wang, Y. Shen, T. Watanabe, J. B. Murphy, J. Rose, and T. Tsang, "The first lasing of 193 nm SASE, 4th harmonic HGHG and ESASE at the NSLS SDL," in Proceedings of the 28th internatinal Free Electron Laser conference (Berlin, Germany, 2006), pp. 18.

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 (5)

Fig. 1.
Fig. 1.

(a) FEL intensity P≡|A(θ, Z)|2 at Z=0.5Z up (red), 0.6Z up (yellow), 0.8Z up (green), 0.9Z up (blue), and Z up (purple). (b) Genesis simulation of the FEL power at Z=0 (red), 0.2Z up (yellow), 0.4Z up (green), 0.6Z up (blue), 0.8Z up (dark blue), and Z up (purple).

Fig. 2.
Fig. 2.

(a) FEL field envelope |A(θ, Z)| (red), the real part of A(θ, Z) (blue), the imaginary part of A(θ, Z) (green), and the phase ϕ(θ, Z) of A(θ, Z) (yellow) at Z=Z up. (b) Plot of the FEL field envelope |A(θ, Z)| (red), the phase ϕ(θ, Z) of A(θ, Z) (blue), the ∂ϕ(θ, Z)/(∂θ) (green), and the 2 ϕ(θ, Z)/(∂θ 2) (yellow) at Z=Z up.

Fig. 3.
Fig. 3.

Wigner function at Z=Z up for µ=0(a), 2.0×10-6(b), and -2.0×10-6(c).

Fig. 4.
Fig. 4.

Genesis simulation of the FEL power at Z=0 (red), 0.2Z up (yellow), 0.4Z up (green), 0.6Z up (blue), 0.8Z up (dark blue), and Z up (purple) for γ=198.6(a), 199.6(b), and 200.6(c). The upper row is for σt ~42.5 fs, and the lower row σt ~8.5 fs.

Fig. 5.
Fig. 5.

(a) FEL power as a function of the electron energy and the seed duration. The dot (red)∙is for σt =42.5 fs with the solid curve (red) as the spectrum form factor, and the triangle (green)△for σt =8.5 fs with the dashed curve (green) as the spectrum form factor. (b) FEL power as a function of the seed duration.

Equations (4)

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A ( θ , Z ) = c ds 2 π i e s Z θ d θ e s ( θ θ ) + i ( 2 ρ ) 3 ( θ θ ) ( s i μ θ ) ( s i μ θ ) A ( θ , 0 ) ,
A ( s ̂ , z ̂ ) = 0 d ξ ̂ e i α ̂ s ̂ ( z ̂ 2 ξ ̂ ) A ( s ̂ ξ ̂ , 0 ) g ( z ̂ , ξ ̂ , α ̂ ) ,
g ( z ̂ , s ̂ , α ̂ ) 2 c d p 2 π i exp { p ( z ̂ 2 s ̂ ) + 2 i s ̂ [ p ( p i α ̂ s ̂ ) ] } 2 c d p 2 π i exp [ f ( p , z ̂ , s ̂ , α ̂ ) ] .
A ( θ , Z ) i 1 6 2 1 6 ρ 1 2 3 π 0 d ξ ξ 1 6 A ( θ ξ , 0 ) ( Z ξ ) 2 3 e i μ ( θ ξ 2 ) ( Z ξ ) + 3 i 1 3 2 1 3 ρ ( Z ξ ) 2 3 ξ 1 3 .

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