# Few-Cycle Infrared Pulse Evolving in FEL Oscillators and Its Application to High-Harmonic Generation for Attosecond Ultraviolet and X-ray Pulses

## Abstract

**:**

## 1. Introduction

## 2. Superradiance in Short-Pulse FEL Oscillators

#### 2.1. Experiments at FELIX

#### 2.2. Experiments at JAERI-FEL

#### 2.3. Experiments at KU-FEL

## 3. Scaling Law of the Superradiance in FEL Oscillators

## 4. Role of Shot Noise

## 5. Stabilization of Carrier-Envelope Phase

## 6. Proposal of FEL-HHG

## 7. Research Program for FEL-HHG

#### 7.1. Pursuing the Ultimate Extraction Efficiency

#### 7.2. Stacking FEL Pulses in an External Cavity

#### 7.3. CEP-Stable Laser for Seeding FEL Oscillators

## 8. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bonifacio, R.; Casagrande, F. The superradiant regime of a free electron laser. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**1985**, 239, 36–42. [Google Scholar] [CrossRef] - Dicke, R.H. Coherence in Spontaneous Radiation Processes. Phys. Rev.
**1954**, 93, 99–110. [Google Scholar] [CrossRef] [Green Version] - Gover, A.; Ianconescu, R.; Friedman, A.; Emma, C.; Sudar, N.; Musumeci, P.; Pellegrini, C. Superradiant and stimulated-superradiant emission of bunched electron beams. Rev. Mod. Phys.
**2019**, 91, 035003. [Google Scholar] [CrossRef] [Green Version] - Bonifacio, R.; Casagrande, F.; Cerchioni, G.; Salvo Souza, L.; Pierini, P.; Piovella, N. Physics of the high-gain FEL and superradiance. La Riv. Del Nuovo Cimento
**1990**, 13, 1–69. [Google Scholar] [CrossRef] - Bakker, R.J.; van der Geer, C.A.J.; Jaroszynski, D.A.; van der Meer, A.F.G.; Oepts, D.; van Amersfoort, P.W. Broadband tunability of a far-infrared free-electron laser. J. Appl. Phys.
**1993**, 74, 1501–1509. [Google Scholar] [CrossRef] - Knippels, G.M.H.; Mols, R.F.X.A.M.; van der Meer, A.F.G.; Oepts, D.; van Amersfoort, P.W. Intense Far-Infrared Free-Electron Laser Pulses with a Length of Six Optical Cycles. Phys. Rev. Lett.
**1995**, 75, 1755–1758. [Google Scholar] [CrossRef] [PubMed] - Piovella, N.; Chaix, P.; Shvets, G.; Jaroszynski, D.A. Analytical theory of short-pulse free-electron laser oscillators. Phys. Rev. E
**1995**, 52, 5470–5486. [Google Scholar] [CrossRef] - Chaix, P.; Piovella, N.; Grégoire, G. Superradiant, single-supermode and nonlinear regimes of short pulse free electron laser oscillators. Phys. Rev. E
**1999**, 59, 1136–1151. [Google Scholar] [CrossRef] - Jaroszynski, D.A.; Chaix, P.; Piovella, N.; Oepts, D.; Knippels, G.M.H.; van der Meer, A.F.G.; Weits, H.H. Superradiance in a Short-Pulse Free-Electron-Laser Oscillator. Phys. Rev. Lett.
**1997**, 78, 1699–1702. [Google Scholar] [CrossRef] - Nishimori, N.; Hajima, R.; Nagai, R.; Minehara, E.J. Sustained Saturation in a Free-Electron Laser Oscillator at Perfect Synchronism of an Optical Cavity. Phys. Rev. Lett.
**2001**, 86, 5707–5710. [Google Scholar] [CrossRef] - Dattoli, G.; Marino, A.; Renieri, A. A multimode small signal analysis of the single pass free electron laser. Opt. Commun.
**1980**, 35, 407–412. [Google Scholar] [CrossRef] - Hajima, R.; Nishimori, N.; Nagai, R.; Minehara, E.J. Analyses of superradiance and spiking-mode lasing observed at JAERI-FEL. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2001**, 475, 270–275. [Google Scholar] [CrossRef] - Hajima, R.; Nishimori, N.; Nagai, R.; Minehara, E.J. High-efficiency ultrashort pulse generation in a high-gain FEL oscillator near the perfect synchronism. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2002**, 483, 113–118. [Google Scholar] [CrossRef] - Hajima, R.; Nagai, R. Generation of a Self-Chirped Few-Cycle Optical Pulse in a FEL Oscillator. Phys. Rev. Lett.
**2003**, 91. [Google Scholar] [CrossRef] - Burnham, D.C.; Chiao, R.Y. Coherent Resonance Fluorescence Excited by Short Light Pulses. Phys. Rev.
**1969**, 188, 667–675. [Google Scholar] [CrossRef] - Bonifacio, R.; Piovella, N.; McNeil, B.W.J. Superradiant evolution of radiation pulses in a free-electron laser. Phys. Rev. A
**1991**, 44, R3441–R3444. [Google Scholar] [CrossRef] - Piovella, N. Transient regime and superradiance in a short-pulse free-electron-laser oscillator. Phys. Rev. E
**1995**, 51, 5147–5150. [Google Scholar] [CrossRef] - Watanabe, T.; Wang, X.J.; Murphy, J.B.; Rose, J.; Shen, Y.; Tsang, T.; Giannessi, L.; Musumeci, P.; Reiche, S. Experimental Characterization of Superradiance in a Single-Pass High-Gain Laser-Seeded Free-Electron Laser Amplifier. Phys. Rev. Lett.
**2007**, 98, 034802. [Google Scholar] [CrossRef] [PubMed] - Yang, X.; Mirian, N.; Giannessi, L. Postsaturation dynamics and superluminal propagation of a superradiant spike in a free-electron laser amplifier. Phys. Rev. Accel. Beams
**2020**, 23, 010703. [Google Scholar] [CrossRef] [Green Version] - Nimtz, G. On superluminal tunneling. Prog. Quantum Electron.
**2003**, 27, 417–450. [Google Scholar] [CrossRef] - Chiao, R.Y. Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations. Phys. Rev. A
**1993**, 48, R34–R37. [Google Scholar] [CrossRef] [PubMed] - Jentschura, U.D.; Horváth, D.; Nagy, S.; Nándori, I.; Trócsányi, Z.; Ujvári, B. Weighing the neutrino. Int. J. Mod. Phys. E
**2014**, 23, 1450004. [Google Scholar] [CrossRef] - Aharonov, Y.; Reznik, B.; Stern, A. Quantum Limitations on Superluminal Propagation. Phys. Rev. Lett.
**1998**, 81, 2190–2193. [Google Scholar] [CrossRef] [Green Version] - Zen, H.; Suphakul, S.; Kii, T.; Masuda, K.; Ohgaki, H. Present Status and Perspectives of Long Wavelength Free Electron Lasers at Kyoto University. Phys. Procedia
**2016**, 84, 47–53. [Google Scholar] [CrossRef] - Bakker, R.J.; Knippels, G.M.H.; van der Meer, A.F.G.; Oepts, D.; Jaroszynski, D.A.; van Amersfoort, P.W. Dynamic desynchronization of a free-electron laser resonator. Phys. Rev. E
**1993**, 48, R3256–R3258. [Google Scholar] [CrossRef] - Zen, H.; Ohgaki, H.; Hajima, R. High-extraction-efficiency operation of a midinfrared free electron laser enabled by dynamic cavity desynchronization. Phys. Rev. Accel. Beams
**2020**, 23, 070701. [Google Scholar] [CrossRef] - Zen, H.; Ohgaki, H.; Hajima, R. Record high extraction efficiency of free electron laser oscillator. Appl. Phys. Express
**2020**, 13, 102007. [Google Scholar] [CrossRef] - Brau, C.A. Free-Electron Lasers; Academic Press, Inc.: San Diego, CA, USA, 1990. [Google Scholar]
- Nishimori, N.; Hajima, R.; Nagai, R.; Minehara, E.J. Systematic measurement of maximum efficiencies and detuning lengths at the JAERI free-electron laser. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2002**, 483, 134–137. [Google Scholar] [CrossRef] - Piovella, N. A hyperbolic secant solution for the superradiance in free electron lasers. Opt. Commun.
**1991**, 83, 92–96. [Google Scholar] [CrossRef] - Penman, C.; McNeil, B.W.J. Simulation of input electron noise in the free-electron laser. Opt. Commun.
**1992**, 90, 82–84. [Google Scholar] [CrossRef] [Green Version] - Brabec, T.; Krausz, F. Intense few-cycle laser fields: Frontiers of nonlinear optics. Rev. Mod. Phys.
**2000**, 72, 545–591. [Google Scholar] [CrossRef] - Kärtner, F.X.E. Few-Cycle Laser Pulse Generation and Its Applications; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Krausz, F.; Ivanov, M. Attosecond physics. Rev. Mod. Phys.
**2009**, 81, 163–234. [Google Scholar] [CrossRef] [Green Version] - Chini, M.; Zhao, K.; Chang, Z. The generation, characterization and applications of broadband isolated attosecond pulses. Nat. Photonics
**2014**, 8, 178–186. [Google Scholar] [CrossRef] - Udem, T.; Holzwarth, R.; Hansch, T.W. Optical frequency metrology. Nature
**2002**, 416, 233–237. [Google Scholar] [CrossRef] [PubMed] - Cerullo, G.; Baltuška, A.; Mücke, O.D.; Vozzi, C. Few-optical-cycle light pulses with passive carrier-envelope phase stabilization. Laser Photonics Rev.
**2011**, 5, 323–351. [Google Scholar] [CrossRef] - Hajima, R.; Kagiyama, S.; Kondo, S. Numerical analysis of high-gain free-electron laser oscillators. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**1997**, 393, 142–146. [Google Scholar] [CrossRef] - Hajima, R.; Nagai, R. Generating Carrier-Envelope-Phase Stabilized Few-Cycle Pulses from a Free-Electron Laser Oscillator. Phys. Rev. Lett.
**2017**, 119, 204802. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Corkum, P.B.; Krausz, F. Attosecond science. Nat. Phys.
**2007**, 3, 381–387. [Google Scholar] [CrossRef] - Togashi, T.; Takahashi, E.J.; Midorikawa, K.; Aoyama, M.; Yamakawa, K.; Sato, T.; Iwasaki, A.; Owada, S.; Okino, T.; Yamanouchi, K.; et al. Extreme ultraviolet free electron laser seeded with high-order harmonic of Ti:sapphire laser. Opt. Express
**2011**, 19, 317–324. [Google Scholar] [CrossRef] - Tecimer, M. High power coupled midinfrared free-electron-laser oscillator scheme as a driver for up-frequency conversion processes in the x-ray region. Phys. Rev. Spec. Top. Accel. Beams
**2012**, 15. [Google Scholar] [CrossRef] - Popmintchev, T.; Chen, M.C.; Popmintchev, D.; Arpin, P.; Brown, S.; Ališauskas, S.; Andriukaitis, G.; Balčiunas, T.; Mücke, O.D.; Pugzlys, A.; et al. Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers. Science
**2012**, 336, 1287–1291. [Google Scholar] [CrossRef] [PubMed] - Popmintchev, T.; Chen, M.C.; Bahabad, A.; Gerrity, M.; Sidorenko, P.; Cohen, O.; Christov, I.P.; Murnane, M.M.; Kapteyn, H.C. Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum. Proc. Natl. Acad. Sci. USA
**2009**, 106, 10516–10521. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nishimori, N.; Nagai, R.; Hajima, R.; Yamamoto, M.; Honda, Y.; Miyajima, T.; Uchiyama, T. Operational experience of a 500 kV photoemission gun. Phys. Rev. Accel. Beams
**2019**, 22, 053402. [Google Scholar] [CrossRef] [Green Version] - Furuya, T. Development of Superconducting RF Technology. Rev. Accel. Sci. Technol.
**2008**, 1, 211–235. [Google Scholar] [CrossRef] - Hajima, R.; Nagai, R.; Kawase, K.; Ohgaki, H.; Zen, H.; Hayakawa, Y.; Sakai, T.; Sumitomo, Y.; Shimada, M.; Miyajima, T. Application of Infrared FEL Oscillators for Producing Isolated Attosecond X-Ray Pulses via High-Harmonic Generation in Rare Gases. In Proceedings of the 39th International Free-Electron Laser Conference (FEL-19), Hamburg, Germany, 26–30 August 2019; pp. 272–275. [Google Scholar]
- Babinet, M. Memoires d’optique meteorologique. CR Acad. Sci.
**1837**, 4, 638. [Google Scholar] - Hayakawa, K.; Hayakawa, Y.; Nakao, K.; Nogami, K.; Tanaka, T.; Enomoto, A.; Fukuda, S.; Furukawa, K.; Michizono, S.; Ohsawa, S.; et al. Operation of near-infrared FEL at Nihon University. In Proceedings of the 29th International Free Electron Laser Conference, Novosibirsk, Russia, 26–31 August 2007; pp. 114–117. [Google Scholar]
- Carstens, H.; Högner, M.; Saule, T.; Holzberger, S.; Lilienfein, N.; Guggenmos, A.; Jocher, C.; Eidam, T.; Esser, D.; Tosa, V.; et al. High-harmonic generation at 250 MHz with photon energies exceeding 100 eV. Optica
**2016**, 3, 366–369. [Google Scholar] [CrossRef] - Akagi, T.; Kosuge, A.; Araki, S.; Hajima, R.; Honda, Y.; Miyajima, T.; Mori, M.; Nagai, R.; Nakamura, N.; Shimada, M.; et al. Narrow-band photon beam via laser Compton scattering in an energy recovery linac. Phys. Rev. Accel. Beams
**2016**, 19, 114701. [Google Scholar] [CrossRef] [Green Version] - Takahashi, S.; Ramian, G.; Sherwin, M.S. Cavity dumping of an injection-locked free-electron laser. Appl. Phys. Lett.
**2009**, 95, 234102. [Google Scholar] [CrossRef] [Green Version] - Smith, T.I.; Haar, P.; Schwettman, H.A. Pulse stacking in the SCA/FEL external cavity. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**1997**, 393, 245–251. [Google Scholar] [CrossRef] - Niknejadi, P.; Kowalczyk, J.M.; Hadmack, M.R.; Jacobson, B.T.; Howe, I.; Kan, S.; Smith, S.; Szarmes, E.B.; Varner, G.; Madey, J.M.J. Free-electron laser inverse-Compton interaction x-ray source. Phys. Rev. Accel. Beams
**2019**, 22, 040704. [Google Scholar] [CrossRef] [Green Version] - Reiche, S. GENESIS 1.3: A fully 3D time-dependent FEL simulation code. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**1999**, 429, 243–248. [Google Scholar] [CrossRef] - Sumitomo, Y.; Hajima, R.; Hayakawa, Y.; Sakai, T. Simulation of Short-Pulse Generation from A Dynamically Detuned IR-FEL Oscillator and Pulse Stacking at An External Cavity. J. Phys. Conf. Ser.
**2019**, 1350, 012040. [Google Scholar] [CrossRef] - Kawase, K.; Hajima, R.; Nagai, R. Development of fiber laser system for a mid infrared light source with the difference frequency generation. In Proceedings of the 17th Annual Meeting of Particle Accelerator Society of Japan, Online Meeting, 2–4 September 2020; pp. 268–271. [Google Scholar]

**Figure 1.**FEL efficiencies (open circles) and Ti:sapphire signals (solid circles) as a function of detuning length at the JAERI-FEL experiment. The enlargement around $\delta L=0$ is also shown. The symbols without error bars have an error less than their size. The absolute vertical scale was calibrated by an average energy loss of the electron beam over an entire macropulse (solid squares) at several detuning lengths [10].

**Figure 2.**FEL power measured as a function of detuning length with different undulator parameters at the JAERI-FEL experiment: ${a}_{w}=0.70$ (open circles), ${a}_{w}=0.49$ (open squares), and ${a}_{w}=0.31$ (crosses). The macropulse duration is 0.4 ms at a 10 Hz repetition rate. The enlargement around $\delta L=0$ for ${a}_{w}=0.31$ is also shown in the inset [10].

**Figure 3.**Numerical result for the JAERI-FEL start up simulation. Evolution of an FEL macropulse calculated with the JAERI-FEL parameters at $\delta L=0$ lasing is plotted. Whole structure of macropulse is also shown in the inset [13].

**Figure 4.**Evolution of an FEL pulse calculated with the JAERI-FEL parameters at $\delta L=0$ lasing: (Left) 100th to 200th round trips; and (Right) 4000th round trip [13].

**Figure 5.**Result of autocorrelation measurement with fringe-resolved SHG signal at the JAERI-FEL (dots). The solid line is a fitted curve by the chirped-sech${}^{2}$ pulse [14].

**Figure 6.**Evolution of the energy distribution of the spent electron beam at KU-FEL. (

**a**) Experimental result with FEL lasing for the best DCD parameter. (

**b**) Numerical result with the DCD parameter of 9.4 $\mathsf{\mu}$m. The arrow indicates the timing of the cavity detuning condition altering from $-9.4$ to 0 $\mathsf{\mu}$m [26].

**Figure 7.**Simulation results for KU-FEL. Evolution of the energy of electrons with different initial phases in the undulator at the end of the macropulse is plotted: (

**a**–

**d**) phase space plots of the two-wavelength slices of the electron beam at the center of the electron bunch and at the different longitudinal positions in undulator. The longitudinal position in the undulator for (

**a**–

**d**) are 0, 0.6, 1.2, and 1.7 m, respectively. The horizontal axis is relative position in the electron bunch normalized by the lasing wavelength $\lambda =11.6\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m [26].

**Figure 8.**Simulation results of FEL efficiency for the lasing at $\delta L=0$ varying the number of undulator periods, ${N}_{u}$, the bunch length and slippage distance, ${L}_{b}/{L}_{s}$, and the normalized cavity loss $\alpha $. The normalized extraction efficiency, $4\pi {N}_{u}\eta $, is plotted as a function of the normalized cavity loss $\alpha $. Experimental results at the JAERI-FEL are also plotted.

**Figure 9.**Cavity-length detuning curve calculated for the JAERI-FEL experiment with two different noise-factor: shot noise determined by the standard formula (solid line) and ${10}^{-12}$ of that (dashed line) [12].

**Figure 10.**Temporal shapes of FEL pulses in a perfectly synchronized optical cavity simulated by one-dimensional code. Profile of the electron bunch at the entrance, $z=0$, and the exit, $z={L}_{w}$, of the undulator is also plotted. The inset is the same FEL pulses plotted with a linear scale [39].

**Figure 11.**Temporal shapes of FEL pulses in a perfectly synchronized optical cavity with an external seed laser after 1500, 2000, and 2500 round trips simulated by one-dimensional code [39].

**Figure 12.**Contour plots of instantaneous phase of simulated FEL pulses in units of $\pi $ rad for: (

**a**) without injection seeding; and (

**b**) with injection seeding. Contour plots of instantaneous intensity of FEL pulses normalized to the maximum intensity for: (

**c**) without injection seeding; and (

**d**) with injection seeding [39].

**Figure 13.**Schematic view of high-harmonics generation driven by an infrared FEL oscillator (FEL-HHG).

JAERI-FEL | This Study | |
---|---|---|

Electron beam | ||

energy (MeV) | 16.5 | 50 |

bunch charge (pC) | 510 | 100 |

norm. emittance ($x/y$) | 40/22 | 12/12 |

(mm-mrad) | ||

bunch length (*) (ps) | 5 | 0.4 |

peak current (A) | 200 | 250 |

bunch repetition (MHz) | 10 | 10 |

undulator | ||

undulator parameter (rms) | 0.7 | 1.25 |

pitch (cm) | 3.3 | 4.5 |

number of periods | 52 | 40 |

FEL | ||

wavelength ($\mathsf{\mu}$m) | 22.3 | 6 |

Rayleigh length (m) | 1.0 | 0.52 |

FEL parameter, $\rho $ | 0.0044 | 0.0052 |

cavity loss | 6% | 4% |

(A) | (B) | |
---|---|---|

Electron beam | ||

energy (MeV) | 85 | 50 |

bunch charge (pC) | 60 | 100 |

norm. emittance ($x/y$) | 12/12 | 12/12 |

(mm-mrad) | ||

bunch length (ps) | 0.27 | 0.4 |

peak current (A) | 220 | 250 |

bunch repetition (MHz) | 10 | 10 |

undulator | ||

undulator parameter (rms) | 1.34 | 1.25 |

pitch (cm) | 4.0 | 4.5 |

number of periods | 80 | 40 |

FEL | ||

wavelength ($\mathsf{\mu}$m) | 2 | 6 |

Rayleigh length (m) | 0.92 | 0.52 |

FEL parameter, $\rho $ | 0.0030 | 0.0052 |

cavity loss | 6% | 4% |

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**MDPI and ACS Style**

Hajima, R.
Few-Cycle Infrared Pulse Evolving in FEL Oscillators and Its Application to High-Harmonic Generation for Attosecond Ultraviolet and X-ray Pulses. *Atoms* **2021**, *9*, 15.
https://doi.org/10.3390/atoms9010015

**AMA Style**

Hajima R.
Few-Cycle Infrared Pulse Evolving in FEL Oscillators and Its Application to High-Harmonic Generation for Attosecond Ultraviolet and X-ray Pulses. *Atoms*. 2021; 9(1):15.
https://doi.org/10.3390/atoms9010015

**Chicago/Turabian Style**

Hajima, Ryoichi.
2021. "Few-Cycle Infrared Pulse Evolving in FEL Oscillators and Its Application to High-Harmonic Generation for Attosecond Ultraviolet and X-ray Pulses" *Atoms* 9, no. 1: 15.
https://doi.org/10.3390/atoms9010015