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*ABCD* formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser

Optics Express, Vol. 15, Issue 20, pp. 12749-12754 (2007)

http://dx.doi.org/10.1364/OE.15.012749

Acrobat PDF (251 KB)

### 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

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

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

03. 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]

04. 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]

05. 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]

06. 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]

07. 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]

*ABCD*formalism [8

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

*i.e.*,

*E*(

_{s}*t*,

*z*) =

*E*

_{0}

*e*

^{i(ksZ-ωst)}e

^{-(αs+iβs)ω2s(t-z/vg,0)2}with

*E*

_{0},

*k*,

_{s}*ω*,

_{S}*α*,

_{s}*β*, and

_{s}*V*

_{g,0}=

*α*/(

_{s}*k*+ 2

_{s}*k*/3) characterizing the peak field, the pulse wavenumber, frequency, duration, chirp, and group velocity respectively. The chirp (2

_{w}*β*

_{s}*ω*

^{2}

*≡*

_{s}*dω*/

*dt*) is independent of the pulse duration

*t*,

*z*) coordinates is [7

07. 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]

*E*

_{0,FEL}(

*z*= 0) =

*E*

_{0},

*α*(

_{s,f}*z*= 0) =

*α*,

_{s}*β*(

_{s,f}*z*= 0) =

*β*, and the centrovelocity [11

_{s}11. R. L. Smith, “Velocities of light,” Am. J. Phys. **38**, 978“984 (1970). [CrossRef]

*v*(

_{c}*z*= 0) = (〈

*t*〉/

*z*)

^{-1}∣

_{z=0}=

*v*

_{g,0}In Eq. (1), ρ is the Pierce parameter [12

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]

*k*= 2

_{w}*π*/

*λ*with

_{w}*λ*being the undulator period,

_{w}*α*(

_{s,f}*z*) = [4

*σ*

^{2}

*(*

_{t,s,f}*z*)

*ω*

^{2}

*]*

_{s}^{-1},

*β*

^{2}

*(*

_{s,f}*z*) =

*α*(

_{s,f}*z*)

*σ*

^{2}

*(*

_{ω,s,f}*z*)/

*ω*

^{2}

*-*

_{s}*α*

^{2}

*with*

_{s,f}*σ*(

_{t,s,f}*z*) being the FEL pulse rms temporal duration and

*σ*(

_{ωs,f}*z*) being the FEL pulse rms frequency bandwidth. The

*ABCD*matrix can be used to transform the complex Gaussian pulse parameter,

*p*(

*z*) [8

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

*ABCD*matrix,

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

*P*> 0 and

*R*> 0, but

*Q*can be negative or positive. For

*μ*= 0, we have

*C*= 0 and

*D*= 1 [10]. In this case, the form of the

*ABCD*matrix (

*A*=

*D*= 1,

*C*= 0) and

*B*complex is characteristic of a system with group velocity dispersion (Re

*B*) and gain (Im

*B*). To illustrate the concept, imaging that a light pulse is represented by an ellipse in the

*t*-

*ω*space with

*t*as the horizontal-axis and

*ω*the vertical-axis. The group velocity dispersion is responsible for the “horizontal shearing” of the phase space ellipse. Since for

*μ*= 0, we have

*C*= 0; there is no inherent time lensing or temporal chirping effect (“vertical shearing”) in the FEL [15

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

07. 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]

*ABCD*analysis is not limited to a simple chirped Gaussian seed pulse. Arbitrary seed pulses can be constructed from a complete set of chirped Hermite-Gaussian functions characterized by a single

*p*-parameter and propagated through the system with the

*ABCD*matrix formalism [8

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

*ABCD*matrix is applied to the following coordinate transformation

*v*

^{-1}

_{g,0}=

*dk*/

*dω*∣

_{ω=ωs}= (

*k*+ 2

_{s}*k*/3)/

_{w}*ω*with or without energy chirp in the electron bunch. The vector (

_{s}*τ*,

*dτ*/

*dξ*)

*describes a time ray with*

^{T}*τ*standing for the delayed time and

*ξ*the normalized propagation distance. Its position

*τ*represents time deviation from a reference time, whereas its slope (

*dτ*/

*dξ*) represents frequency sweep. A nonzero

*B*in the system stands for a horizontal shearing along the

*t*-axis, while a nonzero

*C*stands for a vertical shearing along the

*ω*-axis. It is now clear how important a nonzero

*C*is. It is only with increment in the frequency bandwidth via a nonzero

*C*, can a net temporal pulse compression be possible. For a seeded FEL with three chirps, the

*ABCD*matrix is given in Eq. (3), we find that the only possibility to have a nonzero

*C*element is to have a nonzero

*μ*,

*i.e.*, there has to be an energy chirp in the electron bunch. Indeed, a seeded FEL is represented by an integral Eq. [7]

*A*(

*θ*,

*Z*) is introduced by

*E*(

*t*,

*z*) =

*A*(

*θ*,

*Z*)

*e*

^{i(θ-Z)}with dimensionless variables as

*Z*=

*k*and

_{w}z*θ*= (

*k*+

_{s}*k*)

_{w}*z*-

*ω*. This integral Eq. (8) which propagates an input seed

_{s}t*A*(

*θ*,0) through the FEL via the Green function is of the general form of an integral representation of an

*ABCD*canonical transformation, as such the phase space area and longitudinal coherence is preserved [9, 10]. Readers may refer to Ref. [7

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

*ABCD*formalism. We now study a HHG seeded FEL. We demonstrate that with properly introducing an energy chirp in the electron bunch, a few-cycle attosecond FEL pulse train can be retained by post-undulator chirped pulse compression. We will also provide the derivations which leads to this

*ABCD*matrix in Eq. (3).

*s*[5

05. 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]

*E*

_{s,0}is the peak amplitude of the electric field. Even though, the HHG electric field consists of multiple orders,

*s*with wavenumber,

*k*and angular frequency,

_{s}*ω*, and multiple pulse-lets,

_{s}*n*as a double summation, only the resonant harmonic is relevant, since FEL is a narrow bandwidth filter [5

05. 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]

*N*+ 1 ultrashort pulselets with attosecond structure which is referred as the attosecond pulse train (APT). Pulselet peaks occur at times,

*t*=

_{n}*nτ*/2 where

*τ*is the ir laser period. The duration of a single attosecond pulselet (SAP) can be less than one femtosecond. The temporal envelope of the entire HHG pulse has an rms duration

*σ*

_{t,0}. We define the much shorter rms duration,

*σ*for the single attosecond pulselet with

_{t,s}*α*

_{s}*ω*

_{s}

^{2}≡ 1/(4

*σ*

^{2}

*); and*

_{t,s}*β*characterizes the single harmonic seed chirp. The APT in Eq. (9) is simply the amplitude modulation of a single carrier frequency,

_{s}*ω*. We usest,

_{s}*σ*

_{t,0}= 10 fsec,

*σ*=

_{t,s}*τ*/10 ≈ 267 attoseconds in the Fourier transform limit, and the ir laser wavelength

*λ*= 800 nm in this paper. According to the study in Ref. [5

_{ir}**90**, 021109 (2007). [CrossRef]

*i.e.*, for

*n*= 0, so that

*t*= 0. For this single pulselet, the electric field in the (

_{n}*t*,

*z*) coordinates is given in Eq. (1). To compute the second moments, especially, the chirp, we introduce a Wigner function [7

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

*δt*=

*t*-

*z*/

*v*,

_{c}*δω*=

*ω*-

*ω*, and

_{s}*ABCD*matrix in Eq. (3) can be verified. The centrovelocity

*v*[11] is

_{c}**24**, 484 (2007). [CrossRef]

**90**, 021109 (2007). [CrossRef]

*ω*= 2

_{s}*πc*/

*λ*with

_{s}*λ*as the 27

_{s}^{th}harmonic of their laser, ρ = 6.3 × 10

^{-3}, undulator period

*λ*= 3 cm, and

_{w}*z*∈ [0,2

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

*L*≈ 4 m). Following the resonance condition, initially the seed and the electron bunch are matched,

_{sat}*i.e.*,

*μ*≈ 2

*β*. It is easy to check that with this set of parameters, we have

_{s}*μ*∣ ≪ 1 which is usually satisfied. With this set of parameters, it is difficult to manipulate the FEL pulse duration via chirps during the FEL process; however, the residual frequency chirp is apparently inherited from the initial energy chirp in the electron bunch. Hence, post-undulator compression will be possible to generate FEL pulse with attosecond duration. In the following, let us choose,

*ω*(

*t*=

*σ*) =

_{t,s}*ω*+ 0.1

_{s}*σ*⇒

_{ω,s}*β*∼ 8.7 × 10

_{s}^{-5}. This amount of chirp for the seed of rms duration of

*σ*

_{t,0}= 10 fs translates into about 3.3 nm rms bandwidth on the 27

^{th}harmonic (

*i.e.*, 30 nm) of the ir laser. For the energy chirp, we take ∣

*μ*∣ = 2

*β*≈ 1.7 × 10

_{s}^{-4}≪ 1, which translates to ∣Δ

*∣*

_{δ}*t*=

*σ*= ∣Δ

_{t,s}_{γ}/

*γ*

_{0}∣

*t*=

*σ*≈ 1.5 × 10

_{t,s}^{-3}. Forthis parameter set, we show the FEL pulse rms duration

*σ*as a function of location

_{t,s,f}*z*into the undulator in the upper left subplot of Fig. 1. The solid (red) curve is for

*μ*= 2

*β*, the dashed (green) curve is for

_{s}*μ*= -2

*β*, and the dash-dotted (blue) curve is for

_{s}*μ*= 0. For all these three cases,

*β*≈ 8.7 × 10

_{s}^{-5}. The dotted (purple) curve is for

*μ*=

*β*= 0. However, for this set of parameters, there is very little difference for

_{s}*σ*, as we expect from the above mentioned limiting case of |

_{t,s,f}*μ*| ≪ 1,

*i.e.*,

*t*- 〈

*t*〉)(

*ω*- 〈

*ω*〉)〉 is shown as the lower right subplot in Fig. 1. With the energy chirp in the electron bunch, the final FEL pulse inherits a frequency chirp, which is crucial for an effective post-undulator chirped pulse compression. Shown as the lower left subplot in Fig. 1 is the FEL pulse rms duration after post-undulator chirped pulse compression as a function of the undulator length. Such a post-undulator compression is a final “horizontal shearing”, which results in a pulselet rms duration below the femtosecond. To illustrate this explicitly, we show as the upper row in Fig. 2 for the case of three pulselets in an HHG seed with the frequency chirp

*β*≈ 8.7 × 10

_{s}^{-5}and the energy chirp to be

*μ*= 2

*β*. Even though, the attosecond pulselets are temporally smeared out [5

_{s}**90**, 021109 (2007). [CrossRef]

*β*≈ 8.7 × 10

_{s}^{-5}, but there is no energy chirp in the electron bunch; the post-undulator chirped pulse comprssion is not adequate to restore an attosecond pulse train. Indeed,

*μ*can either be negative or positive so that the FEL frequency chirp exiting the undulator can either be negative or positive.

## References and links

01. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

02. | P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, “Femtosecond and subfem-tosecond x-ray pulses from a Self-Amplified Spontaneous-Emissionbased free-electron Laser,” Phys. Rev. Lett. |

03. | A. A. Zholents and W. M. Fawley, “Proposal for intense attosecond radiation from an x-ray free-electron Laser,” Phys. Rev. Lett. |

04. | 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 |

05. | 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. |

06. | 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. |

07. | 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 |

08. | S.P. Dijaili, A. Dienes, and J. S. Smith, “ |

09. | 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. de Fisica |

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. |

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

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

14. | J.-M. Wang and L.-H. Yu, “A transient analysis of a bunched beam free electron Laser,” Nucl. Instrum. Meth. A |

15. | B. H. Kolner, “Space-Time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

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

(140.3280) Lasers and laser optics : Laser amplifiers

(260.2030) Physical optics : Dispersion

(320.1590) Ultrafast optics : Chirping

(320.5520) Ultrafast optics : Pulse compression

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 8, 2007

Revised Manuscript: July 16, 2007

Manuscript Accepted: August 9, 2007

Published: September 20, 2007

**Citation**

Juhao Wu, Paul R. Bolton, James B. Murphy, and Kelin Wang, "ABCD formalism and attosecond few-cycle pulse via chirp manipulation of a seeded free electron laser," Opt. Express **15**, 12749-12754 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12749

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### References

- T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545 (2000). [CrossRef]
- 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]
- 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]
- 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]
- S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD Matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158 (1990). [CrossRef]
- 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).
- 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).
- R. L. Smith, "Velocities of light," Am. J. Phys. 38, 978-984 (1970). [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]
- 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]
- B. H. Kolner, "Space-Time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951 (1994). [CrossRef]

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