Carrier-envelope phase stabilization in FEL oscillators

FEL oscillators can produce few-cycle optical pulses with a high-extraction efficiency when the oscillators are operated in the superradiant regime. Such FEL oscillators are unique light sources to explore intense light field science, especially in mid-infrared and longwave infrared where ultrashort pulses for the high-intensity applications are difficult to produce from conventional lasers. Since the laser-matter interaction in the intense field regime is described in terms of the oscillating electric field rather than the instantaneous intensity, the carrier-envelope phase (CEP) must be stabilized in many applications of few-cycle optical pulses to the intense light field science. Stabilization of CEP in FEL oscillators has been proposed with an external seed laser and coherent spontaneous emission from the electron bunches. In this paper, we study CEP stabilization in FEL oscillators assisted by coherent spontaneous emission from electron bunches with numerical simulations.


Introduction
Recent technological advances in ultrafast solid-state lasers have allowed the generation of intense optical pulses comprising only a few field oscillation cycles.Such few-cycle pulses can be applied in cutting-edge science: optical frequency comb generation and high-resolution spectroscopy, studies of ultrafast dynamics of photo-excited states of atoms and molecules, and high-harmonic generation to produce isolated attosecond VUV/X-ray pulses [1,2].In these applications, the carrier-envelope phase (CEP) defined as the difference between the optical phase of the carrier wave and the envelope position must be stabilized because the nonlinear optical process depends on the electric field rather than the pulse intensity.The CEP stabilization is realized in solid-state lasers either passively [3] or actively [4].
In a free-electron laser (FEL) oscillator, few-cycle optical pulses can be generated when the oscillator is operated in the superradiant regime.Such FEL oscillators are unique light sources to explore intense light field science, especially in mid-infrared and longwave infrared where ultrashort pulses for high-intensity applications are difficult to produce from conventional lasers [5].We have proposed the application of the few-cycle FEL pulses to the scheme of FEL-HHG, the utilization of infrared FEL pulses to drive high-harmonic generation (HHG) from gas and solid targets [6].The FEL-HHG enables one to explore ultrafast science with attosecond ultraviolet and X-ray pulses with an MHz repetition rate, which is difficult with HHG driven by solid-state lasers.A schematic view of the proposed FEL-HHG is presented in Fig. 1.
In HHG and other strong-field applications of few-cycle FEL pulses, it is necessary to stabilize the CEP.However, CEP stabilization in FEL oscillators has never been demonstrated because the evolution of FEL pulses is initiated by the uncontrollable shot noise.In our previous study, we proposed a method for full stabilization of the CEP of the few-cycle FEL pulse generated in an FEL oscillator by using a CEP-stable external seed laser [6].Stabilization of CEP in an FEL oscillator by coherent spontaneous emission (CSE) from the electron bunch was recently suggested by a simulation study [7].The CSE-assisted CEP stabilization would be an efficient approach because it works passively without any additional apparatus such as an external seed laser.In the present study, we numerically investigate the CSE-assisted CEP stabilization in FEL oscillators, which eliminates the seed laser for the CEP stabilization in the FEL-HHG.

Unaveraged FEL code
In general time-dependent simulations of FEL pulse evolution, an electron bunch is divided along the longitudinal axis into many bunch slices, each of which contains macroparticles representing electrons.The motion of the macroparticles is, then, tracked taking into account their interaction with the radiation and the undulator fields.The evolution of the radiation field is calculated by averaging the phase of macroparticles over at least one radiation wavelength.These FEL code are called 'averaged code'.
The averaged codes are, however, not appropriate for the simulation of infrared FEL oscillators operated in the superradiant regime with large extraction efficiency because the assumption that the macroparticles are bound to a specific bunch slice is not valid for the high-efficiency FEL oscillator, in which some of the electrons move across bunch slices due to large energy variation [8].We, therefore, developed an 'unaveraged code' by modifying previously developed one-dimensional code according to the paper describing the unaveraged simulation algorithm [9].The unaveraged code was benchmarked with experimental results of the FEL oscillator at Kyoto University (KU-FEL) and good agreement was confirmed [10].
The unaveraged code has another function important for infrared oscillators: the ability to calculate coherent spontaneous emission (CSE).It is known that CSE can significantly reduce the startup time and enhance the generation of high-intensity, short, superradiant radiation pulses in infrared FEL oscillators [9].Possible CEP stabilization by CSE was recently reported [7].In the following section, we present simulations of an FEL oscillator using the unaveraged code and discuss possible CEP stabilization in the superradiant FEL oscillator.

Simulation results and discussion
A series of simulations were performed with parameters listed in Table 1 and with varying the cavity detuning length, Ld.The parameters are similar to KU-FEL operated in the photo-cathode mode [10], but we assumed a rectangular electron bunch to emphasize the effects of the CSE.In the simulations, we calculated the FEL pulse evolution up to 3000 round trips, which is only available in FEL oscillators driven by a superconducting accelerator, to investigate the variation of the carrier-envelope phase over a long period.Figure 2 shows the cavity detuning curve, the FEL extraction efficiency as a function of the cavity length detuning.The figure exhibits the feature of the detuning curve of the superradiant FEL oscillators, a steep peak at the zero-detuning length, although the position of the peak is slightly offset from the zero point, Ld = 0.006 .The offset of the peak was also reported in a simulation with a three- dimensional unaveraged code, Puffin [7].We plot the FEL pulse after the saturation, at the 2000 th round trip, for Ld = 0.006  in Fig. 3.The pulse has a typical waveform of the superradiance, the main peak followed by ringing.The duration of the main peak is 5.6 optical cycles in FWHM.The variation of the FEL pulse peak intensity, the CEP, and the peak position over 3000 round trips for different cavity-length detuning, Ld = 0.006 , -0.02 , -0.2, are shown in Fig. 4. We can see that the CEP is stabilized for Ld = 0.006 and the linear variation of the CEP, i.e., the stabilization of the carrier-envelope advance (CEA), is obtained for  = −0.02,while CEP is randomly varied for  = −0.2We examined the CEP variation at each cavity-length detuning in the simulation and found that CEP-stabilized pulses are obtained for 0 ≤ Ld ≤ 0.4 and CEA-stabilized pulses are obtained for -0.05 ≤ Ld < 0. For cavity-length detuning beyond Ld = -0.05,limit cycle and chaotic lasing appear and a stable few-cycle pulse is not realized.
In our previous study [6], we revealed by numerical simulations that the CEP in a superradiant FEL oscillator is affected by random shot noise, which dominates the leading edge of the FEL pulse even after the onset of the saturation, and the CEP can be stabilized by using CEP-stable external laser pulses as a seed laser to fix the amplitude and phase at the leading edge of the FEL pulse.
The mechanism of the CEP stabilization in the present study is considered to be similar to our previous study, the stabilization of the amplitude and phase at the leading edge of the FEL pulse.The stabilization is possible when the CSE having a fixed phase and amplitude dominates the leading edge of the FEL pulse.In our simulation, we assumed a hard-edge rectangular bunch that contains sufficient amplitude of the Fourier component equal to the FEL wavelength as shown in Fig. 5, where the pulse waveform is plotted on a logarithmic scale.
The CSE stacked in the FEL cavity at the N-th round trips can be calculated by the sum of the CSE generated at each round trip taking into account the phase shift due to the detuning and the decay due to the cavity loss:

Summary and outlook
From the unaveraged simulations, we confirmed that the CSE has a role to stabilize the CEP in fewcycle optical pulses generated from superradiant FEL oscillators.The CEP stabilization would be useful for the applications of few-cycle FEL pulses, optical frequency comb, and high-harmonic generation.
In the simulations, we assumed a rectangular bunch without any timing and amplitude jitters.For practical implementation of the CSE-assisted CEP stabilization, we need further numerical investigation with realistic electron bunch profiles including timing and amplitude jitters.It should be noted that the CSE can be enhanced by the manipulation of the electron bunch in the linac and the beam transport, insertion of a scraper at a dispersive section for example.

Figure 1 .
Figure 1.Scheme of high-harmonic generation driven by an infrared free-electron laser oscillator.

Figure 2 .
Figure 2. Calculated cavity-length detuning curve.The extraction efficiency is plotted as a function of the cavity length detuning normalized by the resonant radiation wavelength.The inset is a plot covering the entire detuning range.

Figure 3 .
Figure 3.The intensity waveform of the FEL pulse at the 2000th round trip for the detuning length of Ld = 0.006 .The profile of the electron bunch at the entrance, z = 0, and the exit, z = Lw, of the undulator is also plotted.

Figure 4 .
Figure 4.The variation of the FEL pulse peak intensity, the CEP and the peak position over 3000 round trips for different cavity-length detuning.
= ∑ (1 − /2)  exp(− ( − )) , 14th International Particle Accelerator Conference Journal of Physics: Conference Series 2687 (2024) 032013 where ECSE is the Fourier component of the CSE field corresponding to the FEL wavelength, δ is the phase shift due to the cavity length detuning and α is the cavity round trip loss.The equation shows that the Estacked has a non-zero amplitude and fixed phase regardless of the values of δ and α.The stacked CSE plays the same role as the external seed laser to generate CEP-stable few-cycle FEL pulses.Such pulses are only available at a detuning length equal to zero or small positive values.

Figure 5 .
Figure 5.The intensity waveform of the FEL pulse same as Fig. 3 plotted on a logarithmic scale.