1. Introduction
A number of recently developed or emerging sample characterization techniques, such as coherent diffraction imaging [
1,
2], femtosecond nanocrystallography [
3], or double core-hole spectroscopy [
4], require a photon source that is capable of delivering intense (gigawatt-GW peak power), coherent femtosecond pulses at wavelengths in the extreme-ultraviolet (XUV) or x-ray spectral region. At present, free-electron lasers (FELs), which amplify the light emitted by relativistic electrons wiggling inside a periodic magnetic field of an undulator [
5], are the only light sources able to generate such pulses [
6,
7,
8,
9]. To study femtosecond dynamics with element sensitivity using a resonant pump-resonant probe approach [
10] or a nonlinear wave-mixing scheme [
11], the light source has to meet even more stringent requirements. Ideally, a FEL should produce tunable, multicolor, near transform-limited pulses, with a controlled delay, and in the best case, precisely adjustable phase profiles [
12,
13].
Recently, a significant effort has been devoted to improving the spectral properties, pulse tunability, and multipulse operation in FELs based on self-amplified spontaneous emission (SASE), where the amplification of light is triggered by spontaneous emission in an undulator [
14]. The relatively large bandwidth of the noisy SASE spectrum can be reduced using a self-seeding scheme, where light from the first half of the undulator is used to seed the second half, after having passed through a monochromator. This approach has been successful for improving the spectral brightness of SASE pulses, both in the hard and soft x-ray spectral regions [
15,
16]. Double-pulse schemes have been developed for SASE FELs that provide high-intensity two-color pulses with a variable pulse delay and flexible color separation [
17,
18,
19]. These schemes, however, might suffer from stability issues related to the intrinsically noisy SASE start-up.
The limitations of the SASE scheme can be overcome by using an external seed to trigger the FEL amplification process. One option is direct seeding with a coherent source, based on high-harmonic generation (HHG) in gas [
20,
21,
22]. However, this approach is still under development and is currently limited to wavelengths above several tens of nanometers, due to the relatively low efficiency of HHG. To reach shorter wavelengths, a frequency up-conversion technique such as high-gain harmonic generation (HGHG) is usually employed [
23]. In HGHG (see
Figure 1a, top), a seed laser imprints a periodic energy modulation at the seed wavelength
λs (typically in the UV) onto a relativistic electron beam (e-beam) in the modulator. After the e-beam passes through a dispersive section (a four-dipole magnet chicane), the energy modulation is converted into a current density modulation, with Fourier components spanning many harmonics of the seed. Such a microbunched e-beam emits coherent light at
λs/
n (
n integer) in the XUV as the electrons traverse the periodic magnetic field of the radiator. Besides having the potential to produce fully coherent, transform-limited pulses [
24], the scheme offers a natural way to control the spectrotemporal and spatial properties of the FEL pulse, by shaping the seed [
25,
26]. Furthermore, fine control over the output wavelengths in double-color operation can be achieved, either by using a single seed and a strongly chirped e-beam [
27], or by using two seed lasers operating at different wavelengths [
28].
In HGHG, the energy modulation required to produce significant bunching at the
nth harmonic is roughly
n times the initial e-beam energy spread. For high
n, this strong energy modulation severely deteriorates the FEL gain and limits the scheme to
nmax~15 (to wavelengths around 15–20 nm) [
23]. To reach higher harmonics, a double stage HGHG setup based on the fresh bunch [
29] approach can be employed (
Figure 1a, bottom). The first stage generates light at a harmonic
n1~8–13, which is then used to seed a fresh portion of the e-beam (with the original energy spread) in the second stage. The second radiator emits light at a harmonic
n2~4–8 of the first stage (i.e., at harmonic
n =
n1n2 of the seed). While this method can produce a reasonably stable FEL output at wavelengths down to 4 nm or less, it uses a large portion of the e-beam, therefore reducing the maneuvering space for double-pulse operation. As it is based on the original HGHG, the method is also expected to be prone to e-beam imperfections, especially at short wavelengths [
30].
These drawbacks of the HGHG cascade can be overcome using the echo-enabled harmonic generation (EEHG) approach [
31,
32]. EEHG uses two modulating sections to generate bunching at very high harmonics (up to
n~100). In the first modulator, the e-beam interacts with the first seed laser, as in HGHG, inducing a relatively small energy modulation. Then, after passing through a strong dispersive section, the longitudinal e-beam phase space is shredded, resulting in a finely striated pattern. This effectively decreases the initial slice energy spread and prepares the e-beam to enter the second modulating section, which acts in the same way as the standard HGHG setup. However, due to a relatively small slice energy spread within the individual striations, only a moderate energy modulation is required from the second seed laser to obtain significant bunching at high harmonics. Because of the additional degree of freedom (another chicane), EEHG can, in principle, be tuned for a nearly flat radiation phase in the output pulse, with only modest constraints on the initial longitudinal e-beam phase space profile [
30]. Recently, EEHG has produced significant microbunching at the 75th harmonic using a seed laser at 2400 nm [
33].
At the FERMI FEL in Trieste, Italy, two HGHG schemes are currently in operation. FEL-1 is based on a single HGHG stage that can deliver stable pulses at wavelengths as short as ~20 nm [
8], and can operate in the two-color mode [
10,
28]. FEL-2 consists of two HGHG stages and can operate at wavelengths down to 4 nm [
34]; however, the double-stage design makes it difficult to fully exploit two-color operation. Furthermore, due to the fact that the final harmonic number is the product of harmonic numbers for the individual HGHG stages (i.e., not all harmonics of the seed are allowed), the FEL wavelength tuning is limited for fixed e-beam energy, unless the seed can support operation across a broad wavelength range. The shortcomings of the HGHG cascade have sparked an initiative that is currently underway to modify the FEL-2 line so that it is able to operate in the EEHG configuration. The setup was preliminarily studied in [
35] and the experiment is planned for 2018.
In the manuscript, we present the EEHG experimental layout suited to the FERMI FEL-2 beamline, followed by numerical simulations of the EEHG performance that can be expected in the wavelength range from 5 to 3 nm. We study the effects of the microbunching (μB) instability [
36,
37,
38], which can have a significant influence on the FEL spectrum, inducing strong sidebands near the central wavelength in the worst-case scenario. The results are compared to simulations for the current performance of the two-stage HGHG cascade.
2. Layouts and Methods
The current layout of the two FERMI FEL beamlines is shown in
Figure 1a. For the FEL-2 line (
Figure 1a, bottom) to be compatible with the requirements of the EEHG scheme, a few modifications are necessary. While the first modulator (MOD1) and the injection system for the first seed laser (seed1) can be used as is, the momentum compaction R56 (dispersive strength) of the first chicane (disp1) is not strong enough to produce a highly striated e-beam phase space, which is required for EEHG. For this purpose, we will make use of the delay line chicane, which currently serves to delay the electrons with respect to the light generated in the first stage, in order to seed a fresh part of the electron bunch in the second modulator (MOD2). Currently, the R56 of the delay line is limited to ~1 mm (1.4 GeV e-beam energy). Moving the dipole magnets further apart and upgrading the power supply will allow the dispersive strength to reach values above 2 mm, which will be sufficient to perform the first EEHG experiment at wavelengths close to 5 nm. The EEHG experiment will be carried out in a configuration where the two seed lasers have equal wavelengths. Therefore, a new, longer period (11 cm) second modulator (MOD2) will have to be installed, because the current one cannot be made resonant to an external seed at 260 nm. We will also have to design an injection system for the second laser (seed2). For this purpose, we will make use of the delay line vacuum chamber, which already has the required input ports for seed injection. No upgrades will be necessary for the second dispersive section (disp2) and the radiator line (RAD2). The gap in the first radiator (RAD1) will be fully opened (i.e., magnetic field strength set to zero) during the first EEHG experiment.
While the pilot EEHG experiment at 5.2 nm will be able to employ the existing FEL-2 beamline with minor component modifications, operation at shorter wavelengths will require a significant redesign of the layout, according to
Figure 1b. We will remove the three radiator sections in the first stage (RAD1) and combine the delay line and the first dispersive section (disp1) into a single chicane (disp1*). This will make the beamline more compact and will allow the installation of additional undulators. The new radiator (RAD2*) will therefore have up to 12 sections (with a possibly smaller period for improved performance at short wavelengths), to ensure GW power levels at wavelengths around 3 nm and below. In addition, the second modulator (MOD2*) may be split into two parts, resonant to two different wavelengths. In this way, by introducing a two-color second seed (seed 2*) laser system and working in a split-radiator configuration (i.e., first few radiator sections resonant to one wavelength, while the rest are resonant to a different wavelength), the advanced EEHG layout at FERMI will enable two-color x-ray pump–x-ray probe experiments.
The e-beam parameters in the “nominal” configuration, which were used for simulations of the EEHG performances at 5.2 nm, are included in
Table 1 (middle column). Such an e-beam can be routinely produced in the FERMI linear accelerator (linac). Operations at shorter wavelengths (~3 nm) will benefit from a higher peak current and e-beam energy. A peak current of 1 kA is achievable and can be obtained with the present FERMI layout by increasing the charge compression factor; the associated increased energy spread makes this option unsuitable for HGHG, but it can be used for EEHG, which is less sensitive to energy spread [
30,
32]. Extending the operating energy of the linac above 1.5 GeV requires an upgrade with new accelerating sections; a plan for increasing the e-beam energy up to 1.8 GeV is currently under consideration. Furthermore, to obtain significant bunching for such high harmonics (
n~86), a major upgrade of the first chicane (disp1* in
Figure 1b) will be necessary to reach dispersions in the range of 6 to 10 mm. The e-beam parameters of this “advanced configuration”, which were used for simulating the EEHG performances at 3 nm, are included in the right column of
Table 1.
In EEHG, the e-beam exiting the second chicane is density-modulated at a wavenumber
k = 2π/
λ, which is related to the input wavenumbers of the two seed lasers,
k1 = 2π/
λ1 and
k2 = 2π/
λ2, according to:
where
p and
q are integers. Typically, only one set (
p,
q) significantly contributes to the density modulation (bunching) at a given
k [
30]. In our case,
k1 =
k2 and therefore
k = (
p +
q)
k1, where
p +
q =
n is the harmonic number. Maximum bunching occurs for
p = −1 [
31,
32]; however, this configuration requires either a strong first dispersive section (R56~8 mm, available only after a major upgrade) or a strong energy modulation from the second seed laser, to obtain significant bunching. As in the case of HGHG for high
n, a strong energy modulation will deteriorate the FEL gain. Therefore, during the first EEHG experiment at FERMI (using the nominal parameters from
Table 1), we will work in the
p = −2 configuration, for which values of the first R56 around 2 mm are sufficient to produce bunching of the order of 5% for
n = 50 (
λ = 5.2 nm), using only a moderate energy modulation. For simulations at 3 nm, we assumed a stronger dispersion and could therefore work at
p = −1, allowing us to reach higher bunching compared to the
p = −2 case.
In all cases, EEHG was first optimized for maximum bunching using the equations in Ref. [
32]. The first and second modulators were resonant to 260 nm and had a length (period) of 3 m (10 cm) and 1.54 m (11 cm), respectively. Both seeds had a waist size of 500 μm. The first seed laser power was set for an energy modulation equal to 450 keV, while for the first chicane, we chose R56 = 2.1 mm (nominal configuration,
λ = 5.2 nm) or R56 = 8 mm (advanced configuration,
λ = 3 nm). The free parameters were the second chicane strength and the second seed power, which were set to optimize the bunching at the radiator entrance. The final optimization of the parameter space was carried out using time-independent FEL simulations (simulating only a single slice of the e-beam with a length
λ1) with GENESIS [
39] and GINGER [
40] FEL codes. The optimized EEHG parameters for both the nominal and advanced configurations are listed in
Table 1.
The performance of the EEHG scheme at 5.2 nm was evaluated by carrying out time-dependent FEL simulations. The e-beam current, energy, and energy spread profiles at the end of the FERMI linac (just before the first modulator) were obtained using the particle tracking code ELEGANT [
41], which takes into account collective effects such as longitudinal space charge (LSC) and coherent synchrotron radiation (CSR), which grow in bunch compressors and are at the origin of the μB instability. As the input we used the e-beam at the exit of the electron gun, generated with the General Particle Tracer (GPT) [
42] package.
Additional FEL simulations were performed with electron beams whose energy profile was modified with a superposed periodic modulation at 5 µm and two different modulating amplitudes (50 keV or 100 keV). Such energy-modulated e-beams were used to study the effects of possible strong μB instabilities, that might result, e.g., from residual modulations in the photoinjector laser longitudinal profile, and to test the robustness of EEHG within the optimized parameter space of
Table 1. Comparing the sensitivity of EEHG and HGHG to a controlled, periodic modulation is also important, because it can be experimentally studied by exploiting the possibility to induce periodic long wavelength modulations onto the e-beam with the laser heater available at FERMI [
43,
44].
For all EEHG simulations, an external script was used to reshuffle the macroparticles leaving the first dispersive section [
45]. This allowed taking into account the fact that the large dispersion of the strong chicane can convert long wavelength energy modulations into current modulations. An external script was employed because the versions of the FEL codes that were used for simulations use a “quasi-static” approximation and do not correctly take into account the motion of macroparticles over lengths longer than the seed wavelength.
Both seed laser peak powers were the same as in the time-independent case (for optimizing the bunching at the radiator entrance), and the pulses had a Gaussian shape with a full width at a half maximum (FWHM) duration of 100 fs.
For EEHG simulations at 5.2 nm, the layout of the FEL-2 beamline was used, where the magnetic field in the first radiator section (RAD1) was set to zero, i.e., these radiators only served as a drift section. The second modulator period and the dispersive strength of the delay line were modified according to
Table 2 below, while the R56 of the first chicane (disp1) was set to zero.
For simulations of the HGHG cascade currently running at FERMI, the first stage radiation at n1 = 10 was used to seed the second stage, operating at n2 = 5. The first and second dispersive strengths were set to 14 µm and 8.7 μm, respectively, while the seed (260 nm) was a Gaussian pulse with a FWHM of 100 fs and peak power of 130 MW; sufficient for producing 20 MW peak power pulses at 26 nm from the first stage, that were then used to seed the second stage, generating light at 5.2 nm.
The expected performance of the EEHG scheme at shorter wavelengths (3 nm) was evaluated by carrying out time-independent simulations using e-beam parameters for the advanced configuration, as reported in
Table 1.