Generating carrier-envelope-phase stabilized few-cycle pulses from a free-electron laser oscillator

TL;DR

This paper introduces a method to produce CEP-stabilized few-cycle pulses using a free-electron laser oscillator, enabling advanced applications in high-harmonic generation, x-ray frequency combs, and chemical control.

Contribution

It presents a novel scheme for CEP stabilization in FEL oscillators through external seed injection synchronized with the electron bunches.

Findings

Demonstrates feasible generation of CEP-stabilized few-cycle pulses

Enables high-energy attosecond and zeptosecond science applications

Facilitates development of broadband x-ray optical frequency combs

Abstract

We propose a scheme to generate carrier-envelope-phase (CEP) stabilized few-cycle optical pulses from a free-electron laser oscillator. The CEP stabilization is realized by continuous injection of CEP-stabilized seed pulses from an external laser to the FEL oscillator whose cavity length is perfectly synchronized to the electron bunch repetition. Such CEP-stabilized few-cycle FEL pulses will be an efficient driver for exploring high-harmonic generation at energies 1-10~keV for attosecond and zeptosecond science, broad-band x-ray optical frequency combs and coherent control of chemical reaction.

Figures

Tables

Table 1. Table 1: Parameters of the FEL oscillators

	JAERI-FEL	This Letter
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 ($\mu$m)	22.3	6
Rayleigh length (m)	1.0	0.52
FEL parameter, $\rho$	0.0044	0.0052
cavity loss	6%	4%

Full text

Recent progress in ultrafast lasers has realized generation of ultra-intense optical pulses comprising only a few field oscillation cycles to open new avenues for strong-field and attosecond science Brabec-RMP ; Krausz-RMP . Stabilization of carrier-envelope phase (CEP), which is the timing of the field oscillations with respect to the pulse peak, is essential to expand applications of such few-cycle optical pulses to high-harmonic generation (HHG) Ishii-NC , broadband optical frequency combs Udem-Nature and coherent control of chemical reaction Kling-PCCP .

Free-electron laser (FEL) oscillators have been operated at a wide range of wavelengths from millimeter to VUV, but CEP stabilization has never been demonstrated because the evolution of FEL pulses is initiated by shot noise, microscopic fluctuation of longitudinal density of electrons. Seeding external laser pulses to single-pass FELs is becoming a mature technology for improving shot-to-shot wavelength stability, bandwidth and coherence of FEL pulses Zhao ; Allaria , but CEP-stabilized few-cycle pulses are difficult to generate even with CEP-stabilized seed laser pulses because of rapid phase rotation during exponential growth of FEL pulses.

In the present Letter, we propose a scheme of continuous injection of CEP-stabilized seed laser pulses to generate CEP-stabilized few-cycle pulses from FEL oscillators.

The duration of optical pulses generated in a FEL oscillator is governed by lasing dynamics through the single-pass gain, round-trip loss of the cavity, electron bunch length, slippage distance and cavity length detuning Dattoli ; Piovella-PRE-52 . The slippage distance is defined as $L_{s}=\lambda N_{u}$, the product of the FEL wavelength, $\lambda$, and the number of undulator periods, $N_{u}$. Cavity length detuning is introduced in FEL oscillators to compensate for the effect of laser lethargy, i.e., a group velocity slower than the vacuum speed of light. In a FEL oscillator in the strong-slippage regime, in which the electron bunch is shorter than the slippage distance, an electron bunch superradiantly emits a few-cycle optical pulse in the limit of small cavity length detuning Piovella-PRE-52 ; Chaix-PRE-1999 . The generation of a 6-cycle pulse at a wavelength of 10.4 $\mu$m and a 7-cycle pulse at 8.5 $\mu$m from FEL oscillators has been reported FELIX ; CLIO and discussed in the context of degenerate supermodes Chaix-PRE-1999 .

FEL lasing in a perfectly synchronized optical cavity (or zero-detuning length) was demonstrated at the Japan Atomic Energy Research Institute (JAERI); the FEL pulse was characterized to be 2.32 optical cycles at a wavelength of 23.3 $\mu$m Nishimori-PRL ; Hajima-PRL . In experimental and numerical studies, it was found that the lasing in a perfectly synchronized optical cavity only occurs in the high-gain and small-loss regime of short-pulse FEL oscillators and requires a relatively long rising time to reach saturation Nishimori-PRL ; Hajima-sase ; Ottaviani , which is supported by a superconducting linac.

In our proposal, CEP-stabilized few-cycle pulses are realized by combining FEL lasing in a perfectly synchronized optical cavity and an external seed laser with CEP stabilization. In the following text, the generation of CEP-stabilized few-cycle FEL pulses is discussed based on the results of time-dependent one-dimensional FEL simulations that employ a FEL code similar to the analysis of the JAERI-FEL Hajima-PRL . An optical pulse evolved in a FEL oscillator reaches to saturation in which single-pass gain is balanced by round-trip loss. Transverse profile and phase front of the FEL pulse are primarily determined by the eigenmodes of the oscillator; thus one-dimensional simulations yield a reasonable approximation to reproduce lasing behavior in FEL oscillators Bakker-1993 ; Piovella-PRE-52 ; Nagai-2002 . Our previous study revealed that the shot noise of an electron beam plays a critical role in a perfectly synchronized cavity not only in initiating the FEL lasing but also in sustaining the lasing after saturation Hajima-NIMA-2001 . In our simulation, the shot noise is implemented according to a model proposed by Penman and McNeil Penman and coherent spontaneous emission CSE is not included. We assume a machine design similar to the JAERI-FEL but change the wavelength to 6 $\mu$m considering potential applications of the FEL to HHG as listed in Table I.

Figure 1 shows the temporal profiles of FEL pulses after 1500, 2000 and 2500 round trips in a perfectly synchronized FEL oscillator. In this plot, the longitudinal coordinate is defined such that the leading edge of the electron bunch is located at the undulator entrance, $z=0$, at the reference time, $t=0$. The pulse intensity is expressed as a dimensionless value normalized by the high-gain FEL parameter $\rho$ Bonifacio . The position and profile of the electron bunch at the entrance and exit of the undulator are also depicted to demonstrate that the FEL lasing is in the strong-slippage regime.

Figure 1 illustrates characteristics of FEL pulses evolved in a high-gain and strong-slippage FEL oscillator with a perfectly synchronized optical cavity. The optical pulse consists of an exponential lobe of leading edge and a main peak followed by ringing. The duration of the main peak, 4.4 cycles (FWHM) after 2500 round trips, is much shorter than that of the electron bunch. The pulse height and peak position are not fixed and exhibit continuous variation along with the pulse energy changes. A main peak followed by ringing is common to superradiance observed in two-level systems Burnham-PR and identical to previous results for the analysis of a high-gain FEL amplifier Bonifacio and a perfectly synchronized FEL oscillator in the transient regime Piovella-PRE-51 , both of which indicated the FEL lasing to be superradiance. The electron bunch slips backward inside the optical pulse during the motion in the undulator. In this motion, the electron bunch forms microbunch through interaction with the optical field and then emits strong radiation in the slippage region. The emission in the slippage region is accompanied by frequency down chirp to keep emission along electron energy decreasing Hajima-PRL . This down chirp contributes to a large FEL conversion efficiency, but introduces a large energy spread in the spent electron beam as well. In a conventional FEL oscillator, the optical cavity length is shortened so that the optical pulse is pushed forward every round trip to enlarge a single-pass gain by introducing strong electron bunching with a high-intensity optical field at an early section of the undulator. The cavity-length shortening achieves feedback of radiation power from the tail to the head of optical pulse, which also communicate optical phase and frequency from the tail to the head. As a result, the cavity length shortening constrains phase correlation inside the optical pulse, which prohibits the strong down chirp observed in the lasing at a perfectly synchronized cavity.

The logarithmic plot of the FEL pulse in Fig. 1 shows that the dynamic range of laser pulse intensity from the leading edge to the peak is greater than $10^{11}$. The leading part of the optical pulse contains incoherent shot noise with random amplitude and phase. The amplitude and phase of the field in the exponential envelope, the main peak and the ringing are all governed by the interaction of electrons and the radiation initiated by the shot noise in the leading part. Consequently, the carrier frequency and phase of the FEL pulses are not stabilized and vary over many round trips.

The simulation result shown in Fig. 1 suggests the possible stabilization of the optical pulse frequency and phase by fixing the amplitude and phase of the shot noise in the leading part of the FEL pulse. This stabilization is realized by overlapping the pulse head with an external seed laser pulse whose frequency and phase are stabilized. We conducted a simulation to confirm the scheme of CEP stabilization. Figure 2 shows FEL pulses obtained in a simulated FEL oscillator with injection seeding, where all the parameters are the same as in Fig. 1. The seed laser pulse is assumed to have the resonant wavelength, an intra-cavity intensity of $|A_{seed}|^{2}=1.3\times 10^{-5}$ and a temporal duration of $20\lambda$ with CEP stabilization. The seed pulse timing is chosen such that half of the seed pulse overlaps with the FEL pulse and the rest is out of the FEL pulse to indicate the seed laser intensity not affected by the FEL interaction.

In Fig.2, we can see that the seed laser efficiently stabilizes the FEL oscillator with a perfectly synchronized cavity. The FEL pulse after the saturation retains an almost identical temporal shape: the main pulse of 3.8 cycles (FWHM) followed by periodic ringing.

The effects of the CEP stabilization can be clearly seen in Fig. 3, which shows instantaneous intensity and phase of FEL pulses evolving in a perfectly synchronized cavity for the two cases without and with a seed laser. The instantaneous phase, $\phi_{L}$, is defined such that the complex field is expressed as $|A|\exp i(\omega_{r}(z/c-t)+\phi_{L})$, where $\omega_{r}$ is the FEL resonance frequency. The simulation parameters are same as Figs. 1 and 2, respectively. In the lasing without a seed laser, the phase of the leading part has fluctuation due to the shot noise, which makes pulse shape and CEP unstable over many round trips. The FEL pulse evolution with an external seed laser exhibits a quite different aspect, in which the pulse shape and CEP after the onset of saturation are stabilized. The nonlinear phase advance from the head to the tail of pulses in Fig. 3 (a) and (b) corresponds to the frequency down chirp Hajima-PRL .

For the CEP stabilization, the seed laser intensity must be sufficiently large compared to the shot noise intensity. A set of simulations were performed to determine the amount of CEP fluctuations and pulse energy fluctuation as a function of the seed laser intensity. We plot the simulation result, i.e. variation of CEP and pulse energy after saturation against the intra-cavity seed laser intensity in Fig. 4. The average intensity of the shot noise at the leading edge of the pulse, $-20<(z-ct)/\lambda<0$, for the simulation parameters is found to be $|A_{noise}|^{2}=1.83\times 10^{-8}$, as indicated by the broken line. The CEP is uniformly random when the seed laser intensity is less than the shot noise. However, the CEP is stabilized for a seed intensity exceeding the shot noise level and the rms error of CEP, $\Delta\phi$, monotonically decreases with the scale of $\Delta\phi\propto(|A_{seed}|^{2})^{-0.56}$, which is almost consistent with a phase error equal to the vector sum of the seed laser and the random shot noise: $\Delta\phi_{sum}\propto(|A_{seed}|^{2})^{-0.5}$. The injection seeding also stabilizes the FEL pulse energy. The FEL conversion efficiency with injection seeding was found to be 10%, which corresponds to an intra-cavity pulse energy of 13 mJ and an extracted pulse energy of 0.5 mJ neglecting diffraction and absorption losses in the optical cavity.

The fluctuations in the CEP caused by FEL gain variations were also numerically evaluated for $|A_{seed}|^{2}=1.3\times 10^{-5}$. In the simulations, we introduced random jitter into the electron peak current to vary the FEL gain within a macro pulse and evaluated the fluctuations in the CEP. Figure 5 plots the calculated CEP errors as a function of the amount of jitter in the peak current; the results indicate that the rms error of CEP becomes 0.15 rad for a peak current jitter of 20%. In a practical design of FELs, accelerator parameters can be optimized to minimize the jitter, energy spread and emittance at a FEL undulator Wang-JMEA .

We comment on the implementation of a seed laser for the proposed scheme. A seed laser must generate a train of CEP-stabilized optical pulses at a repetition rate matching that of a train of electron bunches. The role of the seed laser is inducing an electron energy modulation with a stabilized wavelength, amplitude and phase. The interaction of the electron bunch and seed pulse can be conducted with various configurations. The simplest case involves overlapping a seed laser and an electron bunch in an early section of the FEL undulator. We can also use two seed lasers with different wavelengths such that the sum-frequency or difference-frequency of the two lasers is equal to the FEL wavelength. Alternatively, a dedicated seeding undulator installed upstream of the FEL undulator also works for CEP stabilization. In the dedicated undulator, we can control polarization of the seed laser and the FEL independently and use a sub-harmonic of the FEL as a seed laser wavelength.

In the above simulations, we assumed a single-undulator configuration, in which the seeding and the FEL lasing share the undulator and the optical cavity. The FEL cavity length should be controlled such that the round-trip time of the FEL pulse matches the electron bunch interval. Moreover, the cavity length must satisfy the condition for intra-cavity coherent stacking of successive seed pulses. Such control has been established for laser Compton scattering experiments employing a laser enhancement cavity Akagi . The intensity of the seed pulse in Fig. 2, $|A_{seed}|^{2}=1.3\times 10^{-5}$, corresponds to an intra-cavity pulse energy of 0.34 nJ for a 400-fs pulse.

In conclusion, we have proposed a scheme to generate CEP-stabilized few-cycle laser pulses in a FEL oscillator with a CEP-stabilized external seed laser. The proposed scheme is applicable to a wide range of wavelengths from infrared to terahertz as far as a high-gain strong-slippage FEL oscillator and a CEP-stabilized seed laser are available. Such FEL pulses can be delivered to either extra-cavity or intra-cavity experiments FELICE . Few-cycle mid-infrared FEL pulses with mJ-class energy are attractive because they can enable HHG, thereby producing vacuum ultraviolet and X-ray pulses with attosecond duration Tecimer . Because the HHG cutoff energy under the phase-matched condition depends on the drive laser wavelength as $h\nu_{cutoff}\propto\lambda^{1.7}$, many efforts are focused on extending the drive laser wavelength from the near-infrared to the mid-infrared Popmintchev-Science ; Cavaletto-Nat-Phot . The few-cycle mid-infrared FEL pulses at 10-MHz repetition proposed here could be a unique method for exploring HHG beams at energies of 1-10 keV to push ultrafast laser science to the zeptosecond regime and for realizing X-ray optical frequency combs.

The authors thank Xiao-Ming Tong and Jiro Itatani for discussion regarding HHG and optical frequency combs. This research was partially supported by the Research Foundation for Opto-Science and Technology.