## Exponential growth, superradiance, and tunability of a seeded free electron laser

Optics Express, Vol. 16, Issue 5, pp. 3255-3260 (2008)

http://dx.doi.org/10.1364/OE.16.003255

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### 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

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]

*ρ*[5

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,” Nuc. 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 De Salvo Souza, P. Pierini, and N. Piovella, “The Superradiant regime of a FEL: Analytical and numerical results,” Nucl. Inst. Meth. 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]

*i.e.*, they are two saddle points in the FEL Green function with emphasis on the FEL group velocity and slippage effect.We show the pulse shortening and give some detailed temporal and spectral information in the superradiance regime to compare with the experiment [11

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,” Nuc. 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 energy-chirped electron beam and its application for generation of attosecond x-ray pulses,” Phys. Rev. ST Accel. Beams **9**, 050702 (2006). [CrossRef]

*A*(

*θ*′, 0), convoluting with the FEL Green function [2

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]

*E*(

*t*,

*z*)=

*A*(

*θ*,

*Z*)

*e*

^{i(θ-Z)}with

*A*(

*θ*,

*Z*) being the slow varying envelope function. The evolution is [2

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]

*ρ*is the Pierce parameter,

*µ*=(

*dγ*/

*dt*)2/(

*γ*

_{0}

*ω*) is the energy chirp along the electron beam with

_{s}*γ*

_{0}being the resonance energy,

*Z*=

*k*,

_{w}z*θ*=(

*k*+

_{s}*k*)

_{w}*z*-

*ω*, where

_{s}t*k*=2

_{s}*π*/

*λ*,

_{s}*ω*=

_{s}*k*, and

_{s}c*k*=2π/

_{w}*λ*with

_{w}*λ*being the radiation wavelength,

_{s}*λ*the undulator period, and

_{w}*c*the speed of light in vacuum. The double integral in Eq. (1) can be evaluated by first performing the contour integral to obtain the Green function. Explicitly,

*g*(

*ẑ*,

*ŝ*,

*) and the corresponding phasor*α ^

*f*(

*p*,

*ẑ*,

*ŝ*,

*) defined as*α ^

7. R. Bonifacio and F. Casagrande, “The Superradiant Regime of a Free Electron Laser,” Nuc. 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 De Salvo Souza, P. Pierini, and N. Piovella, “The Superradiant regime of a FEL: Analytical and numerical results,” Nucl. Inst. Meth. A **296**, 358 (1990). [CrossRef]

*ẑ*=2

*ρZ*,

*ŝ*=

*ρθ*, and

*=-*α ^

*µ*/(2

*ρ*

^{2}). The Green function is estimated by saddle point approximation. The saddle point

*p*is found from

_{s}*df*(

*p*)/

*dp*|

_{p=ps}=0, and the Green function is approximated as

*g*(

*ẑ*,

*ŝ*,

*)≈2exp[*α ^

*f*(

*p*,

_{s}*ẑ*,

*ŝ*,

*)][2*α ^

*πf″*(

*p*,

_{s}*ẑ*,

*ŝ*,

*)]*α ^

^{-1/2}. For the un-chirped case,

*i.e.*,

*=0, the phasor is*α ^

*f*(

*p*,

*ẑ*,

*ŝ*,

*=0)=*α ^

*p*(

*ẑ*-2

*ŝ*)+2

*iŝ*/

*p*

^{2}. The saddle point is found from

*p*

^{3}-4

*iŝ*/(

*ẑ*-2

*ŝ*)=0. If

*p*is not a function of

*ŝ*, the ponderomotive phase, then

*pẑ*is not an oscillating function; hence a steady state solution. This is determined by

*ẑ*-2

*ŝ*=

*ηŝ*, with

*η*being a constant. Under this condition,

*p*

^{3}-4

*iŝ*/(

*ẑ*-2

*ŝ*)=0 becomes

*p*

^{3}

_{s}=4

*i*/

*η*. At this saddle point, the phasor is

*f*(

*p*)=(4

_{s}*i*/

*η*)

^{1/3}

*ẑ*-(4

*i*/

*η*)

^{1/3}2

*ŝ*+2

*îsη*

^{2/3}/(4

*i*)

^{2/3}. Hence, for

*f*(

*p*) not to be a function of

_{s}*ŝ*, we need -(4

*i*/

*η*)

^{1/3}2

*ŝ*+2

*iŝη*

^{2/3}/(4

*i*)

^{2/3}=0, which sets

*η*=4,

*i.e.*, 4

*ŝ*=z

*̂*-2

*ŝ*. This gives

*p*

_{s,0}=

*i*

^{1/3}, which supports a steady state solution, the exponential growth mode. Notice that 4

*ŝ*=

*ẑ*-2

*ŝ*gives the FEL group velocity as

*ν*=

_{g}*ω*/(

_{s}*k*+2

_{s}*k*/3). When the FEL evolves into superradiance regime, the group velocity goes back to the speed of light in vacuum [10

_{w}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]

*ŝ*=

*ẑ*-2

*ŝ*should be abandoned. Indeed, for

*ν*=

_{g}*c*, we need

*ẑ*-2

*ŝ*=0. However; since

*p*~(

*ẑ*-2

*ŝ*)

^{-1/3},

*e*has an essential singularity. In summary, for

^{pẑ}*η*→4, the general condition

*ẑ*-2

*ŝ*=

*ηŝ*leads to the condition 4

*ŝ*=

*ẑ*-2

*ŝ*, which supports exponential growth mode; for

*η*→0, the general condition leads to

*ẑ*-2

*ŝ*=0 for the superradiance mode.

7. R. Bonifacio and F. Casagrande, “The Superradiant Regime of a Free Electron Laser,” Nuc. 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 De Salvo Souza, P. Pierini, and N. Piovella, “The Superradiant regime of a FEL: Analytical and numerical results,” Nucl. Inst. Meth. A **296**, 358 (1990). [CrossRef]

**68**, 369 (1988). [CrossRef]

*i.e.*,

*z*=2

_{1}*ŝ*,

*z*

_{2}=

*ẑ*-2

*ŝ*,

*z̅*=

*ẑ*

*b*

_{0}is the initial bunching. Now, according to

*p*

^{3}-4

*iŝ*/(

*ẑ*-2

*ŝ*)=0, the growth mode is

*p*=4

_{s}^{1/3}

*i*

^{1/3}

*ŝ*

^{1/3}(

*ẑ*-2

*ŝ*)

^{-1/3}. This then gives the phasor and Green function as

*f*(

*p*)=3

_{s}*i*

^{1/3}[√

*ŝ*(

*ẑ*-2

*ŝ*)

^{2/3}/2

^{1/3}⇒

*g*(

*ẑ*,

*ŝ*,

*)∝ exp{3*α ^

^{3/2}[√

*ŝ*(

*ẑ*-2

*ŝ*)

^{2/3}/2

^{4/3}}. Comparing this Green function which arises at the saddle point

*p*=4

_{s}^{1/3}

*i*

^{1/3}

*ŝ*

^{1/3}(

*ẑ*-2

*ŝ*)

^{-1/3}to the superradiance field in Eq. (28) of Ref. [8

**68**, 369 (1988). [CrossRef]

**239**, 36 (1985). [CrossRef]

**68**, 369 (1988). [CrossRef]

**296**, 358 (1990). [CrossRef]

*α*

_{0}=1/(4

*σ*

^{2}

_{t0}) with

*σ*

_{t0}being the initial seed rms pulse duration.

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]

*σ*

_{t0}≈45 fs,

*µ*=0,

*Z ∈*[0.5

*Z*

_{up},

*Z*

_{up}] with

*Z*

_{up}≡(2

*π*/0.039)10≈1611,

*ρ*=10

^{-3}, and

*λ*=0.8

_{s}*µ*m. Recall that

*Z*=

*k*and

_{w}z*k*=2

_{w}*π*/

*λ*with

_{w}*λ*=3.9 cm being the undulator period and

_{w}*L*=10 m the undulator total length. In Fig. 1(a), we plot the FEL intensity

_{w}*P*≡|

*A*(

*θ*,

*Z*)|

^{2}at

*Z*=0.5

*Z*

_{up},0.6

*Z*

_{up},0.8

*Z*

_{up},0.9

*Z*

_{up}, and

*Z*

_{up}. It is seen that the pulse duration gets shorter, if we trace the main pulselet. At

*Z*=0.5

*Z*

_{up}, the rms pulse duration

*σ*is about

_{t}*σ*≈75 fs, and at

_{t}*Z*=

*Z*

_{up},

*σ*≈35 fs. Also we observe that the FEL pulse develops pulselets. According to our convention, the main pulselet is at the head of the FEL pulse towards the right in Fig. 1. Simulation with

_{t}*Genesis*[15

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

*P*

_{in}=1 MW, shows the evolution of the FEL pulse in Fig. 1(b) for

*Z*

*∈*[0,

*Z*

_{up}]. In both results, we find the pulse shortening and pulselet development. This agrees with the experimental observation that the FEL pulse temporal duration increases in the exponential growth regime and decreases in the superradiance regime [11

**98**, 034802 (2007). [CrossRef] [PubMed]

*Z*=

*Z*

_{up}is shown in Fig. 2(a) with the amplitude as the red curve, the real part the blue, the imaginary part the green, and the phase the yellow. To get the sign of the chirp in the superradiance mode, recall that,

*E*(

*t*,

*z*)=

*A*(

*θ*,

*Z*)

*e*

^{i(θ-Z)}=|

*A*(

*θ*,

*Z*)|

*e*

^{iϕ(θ, Z)+i(θ-Z)}≡|

*A*(

*θ*,

*Z*)|

*e*

^{-iΦ(t,z)}, where

*t*stands for the pulse centroid. With this, we have ∂Φ/(∂

_{c}*t*)=

*ω*+

_{s}*ω*

_{s}*∂ϕ*/(

*∂θ*), and

*∂*

^{2}

*Φ*/(

*∂t*

^{2})=-

*ω*

^{2}

_{s}

*∂*

^{2}

*ϕ*/(

*∂θ*

^{2}). From Fig. 2(b), we find that

*∂ϕ*(

*θ*,

*Z*)/(

*∂θ*)<0 in the main pulselet, hence, the entire pulse gets redshifted. This agrees with Fig. 1(b) of Ref. [11

**98**, 034802 (2007). [CrossRef] [PubMed]

*∂*

^{2}

*ϕ*(

*θ*,

*Z*)/(

*∂θ*

^{2})<0, hence a positive chirp; and from the center to the tail a negative chirp. The second pulselet has a small positive chirp. To further compare with the experiment [11

**98**, 034802 (2007). [CrossRef] [PubMed]

**24**, 484 (2007). [CrossRef]

*W*(

*t*,

*ω*,

*z*)≡∫

^{∞}

_{-∞}

*E*(

*t*-τ/2,

*z*)

*E**(

*t*+τ/2,

*z*)

*e*

^{-iωτ}

*d*τ at

*Z*=

*Z*

_{up}is shown in Fig. 3. In Fig. 3(a), we set

*µ*=0,

*i.e.*, no energy chirp. It is seen that the second pulselet,

*i.e.*, from the pulse center to the pulse tail, bears a positive chirp, which agrees with that in Fig. 2(b). For the main pulselet, it is complicated, similar to Fig. 3 of Ref. [11

**98**, 034802 (2007). [CrossRef] [PubMed]

*µ*=±2.0×10

^{-6}, respectively. Phase space rotation due to this energy chirp is seen.

*δ*-function pulses along the electron bunch, and these spikes travel towards the head of the electron bunch. During this process, the coherence length increases and the temporal duration of the spikes increases. For a seeded FEL, as long as the seed is much short compared to the electron bunch, the seed will travel through the electron bunch in a similar way as the above described case. Even though, the seed can be so strong that the electron bunch gets energy modulated quickly and FEL process starts from these energy modulated electrons quickly. To explore this more clearly, let us take the limit of a

*δ*-function seed,

*i.e.*,

*A*(

*θ*, 0)~

*δ*(

*θ*). In this limit, the evolution in Eq. (4) is reduced to

**24**, 484 (2007). [CrossRef]

*µ*=0. The

*δ*-function seed passes through the electron bunch with speed of light in vacuum, hence we have

*Z*-

*θ*→0. This supports the superradiance growth due to the singular point of (

*Z*-

*θ*)

^{2/3}in the denominator; but the exponential growth mode is not excited since

*σ*, the rms frequency bandwidth is

_{t}*σ*=1/(2

_{ω}*σ*). For a seeded FEL, the initial seed then supports a relative bandwidth of

_{t}*σ*/

_{ω}*ω*=1/(2

_{s}*σ*), which in turn supports a relative energy detuning of

_{t}ω_{s}*Δγ*/

*γ*

_{0}=

*σ*/(2

_{ω}*ω*)=1/(4

_{s}*σ*). Hence, with an ultrashort seed, a seeded FEL acts like a single spike SASE FEL, and exponential growth mode is excited. Indeed, a short seed having a broad bandwidth increases the tunability of the seeded FEL. For the experiment [11

_{t}ω_{s}**98**, 034802 (2007). [CrossRef] [PubMed]

*λ*=0.8

_{s}*µ*m,

*ρ*=2.5×10

^{-3}, so in order for Δ

*γ*/

*γ*

_{0}>

*ρ*, we need

*σ*<42.5 fs. We define the tunability as

_{t}*χ*≡1/(4

*ρσ*), which should be compared to the detuning

_{t}ω_{s}*D*≡(

*γ*

^{2}-

*γ*

^{2}

_{0})/(2

*ργ*

^{2}

*0*). To further explore this tunability concept, we invoke

*Genesis*simulation with results shown in Fig. 4. For the upper row,

*σ*=42.5 fs, which gives

_{t}*χ*=1. The resonance is shown in Fig. 4(b) with

*γ*

_{0}=199.6. For Fig. 4(a) and (c),

*D*=∓2, respectively, and the peak power starts to decrease. Decreasing the seed pulse duration to

*σ*=8.5 fs, the results are in the lower row. There is essentially no difference for

_{t}*D*∈[-2,2]. In summary, the FEL peak power as a function of the electron bunch initial energy is shown in Fig. 5(a) for both the case of

*σ*=42.5 fs and 8.5 fs. Indeed, this amplification bandwidth, or the tunable range is determined mostly by the initial seed bandwidth. For a Gaussian seed, the power is

_{t}*p*

_{s,0}=

*i*

^{1/3}; while for the superradiance regime,

## References and links

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 |

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. |

4. | J. Wu, P. R. Bolton, J. B. Murphy, and K. Wang, “ |

5. | R. Bonifacio, C. Pellegrini, and L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. |

6. | J. B. Murphy, C. Pellegrini, and R. Bonifacio, “Collective instability of a free electron laser including space charge and harmonics,” Opt. Commun. |

7. | R. Bonifacio and F. Casagrande, “The Superradiant Regime of a Free Electron Laser,” Nuc. Instrum. Methods Phys. Res. A |

8. | R. Bonifacio, C. Maroli, and N. Piovella, “Slippage and Superradiance in the high-gain FEL: Linear theory,” Opt. Commun. |

9. | R. Bonifacio, L. De De Salvo Souza, P. Pierini, and N. Piovella, “The Superradiant regime of a FEL: Analytical and numerical results,” Nucl. Inst. Meth. A |

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. |

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. |

12. | J.-M. Wang and L.-H. Yu, “A transient analysis of a bunched beam free electron laser,” Nuc. Instrum. Methods Phys. Res. A |

13. | S. Krinsky and Z. Huang, “Frequency chirped self-amplified spontaneous-emission free-electron lasers,” Phys. Rev. ST Accel. Beams |

14. | 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 |

15. | S. Reiche, “GENESIS 1.3: a fully 3D time-dependent FEL simulation code,” Nucl. Instrum. Methods Phys. Res., Sect. A |

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 |

**OCIS Codes**

(140.2600) Lasers and laser optics : Free-electron lasers (FELs)

(140.3280) Lasers and laser optics : Laser amplifiers

(140.6630) Lasers and laser optics : Superradiance, superfluorescence

(320.1590) Ultrafast optics : Chirping

(320.5520) Ultrafast optics : Pulse compression

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 10, 2007

Revised Manuscript: January 11, 2008

Manuscript Accepted: January 13, 2008

Published: February 25, 2008

**Citation**

Juhao Wu, James B. Murphy, Xijie Wang, and Kelin Wang, "Exponential growth, superradiance, and tunability of a seeded free electron laser," Opt. Express **16**, 3255-3260 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3255

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### References

- 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).
- 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]
- 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]
- 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. Maroli, and N. Piovella, "Slippage and Superradiance in the high-gain FEL: Linear theory," Opt. Commun. 68, 369 (1988). [CrossRef]
- 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]
- 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]
- 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.-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]
- S. Krinsky and Z. Huang, "Frequency chirped self-amplified spontaneous-emission free-electron lasers," Phys. Rev. ST Accel. Beams 6, 050702 (2003). [CrossRef]
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