Self-Seeded Free-Electron Lasers

Abstract

Self-seeding is an active filtering method for Free-Electron Lasers (FELs) enabling the production of nearly Fourier-limited pulses in the X-ray frequency range where external seeding is not available. Schematically, it is composed of three parts: a Self-Amplified Spontaneous Emission (SASE) FEL working in the linear regime, a monochromator, and an FEL amplifier. Active filtering is achieved by letting the FEL pulse produced in the SASE FEL through the monochromator, while the electron beam is sent through a bypass, and its microbunching is destroyed due to dispersion. The filtered SASE pulse, serving as a seed, is recombined with the electron beam at the entrance of the FEL amplifier part. It is then amplified up to saturation and possibly beyond via post-saturation tapering. This allows for the production of high-brightness, nearly single-mode FEL pulses. The technique has been or will be implemented in a number of X-ray FEL (XFEL) facilities under operation or in the construction phase. In this chapter, we review the principle of self-seeding, its practical realizations, and related techniques.

Appendix: Theory of Soft X-Ray Self-Seeding (SXRSS) with Gas Monochromator

The self-seeding technique was originally devised for the soft X-ray range. Even before the successful demonstration of the HXRSS setup at the LCLS (Amann et al. 2012) and at SACLA (Inagaki et al. 2014), the community recognized the need for an extension of the method to the soft X-ray range. In fact, the use of diamond crystals to implement wake monochromators is unfortunately limited by the distance between atomic layers to X-ray energies above 3 keV. However, as discussed in section “Principles of Operation and Design”, the general principle of operation of a wake monochromator does not include exclusively the use of crystals. In fact, the overall technique, albeit difficult to implement without crystals, is only based on the possibility of obtaining narrow bandwidth “holes” in the SASE FEL spectrum. In (Geloni et al. 2011d), a possible extension of the wake monochromator method to the soft X-ray range was presented, based on a gas cell filled with resonantly absorbing gas. If the transmittance spectrum in the gas exhibits an absorbing resonance with a bandwidth narrow enough for seeding then, similarly to the hard X-ray case, the temporal waveform of the transmitted radiation pulse is characterized by a long monochromatic wake. In other words, the FEL pulse forces the gas atoms to oscillate in a way consistent with a forward-propagating, monochromatic radiation beam.

In Geloni et al. (2011d), it was proposed to take advantage of autoionizing resonances in rare gases to seed in the XUV/soft X-ray range. The phenomenon of autoionization is well known in literature (see, for instance, textbooks like Bohm (1979), Fano and Rau (1986), and references therein). Here we consider helium atomic gas as an example.⁶ Autoionizing resonances result from the decay of doubly excited Rydberg states He^∗ into the continuum, i.e., He^∗→ He⁺ + e⁻. Since the continuum can also be reached by direct photoionization, both paths add coherently, giving rise to interference. This interference is related to the typical Fano line shape for the cross section as a function of energy Fano (1961). Here we only report the cross section for these series, which can be modeled by the expression Morgan and Hederer (1984):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma(\lambda,q,\varGamma) = \sigma_b(\lambda)\left(\frac{\left(\sum_{n=2}^{\infty}(q_n/\mathscr{E}_n)+1\right)^2}{\sum_{n=2}^\infty(1/\mathscr{E}_n)^2 +1}\right), {} \end{array} \end{aligned} $$

(8)

where the energy-dependent background cross section expressed in Megabarn (1 Mb = 10⁻¹⁸ cm²) is given by σ_b(λ) = −0.05504 − 1.3624 ⋅ 10⁻⁴λ + 3.3822 ⋅ 10⁻⁵λ², with λ the radiation wavelength in Angstrom units, while the reduced energy $\mathscr {E}_n$ is defined as $\mathscr {E}_n = {2(E_{R_n}-hc/\lambda )}/{\varGamma _n}$. The asymmetry index q_n, the energy of the nth resonance $E_{R_n}$, and the resonance width Γ_n can be found, for example, in (Morgan and Hederer 1984). Using these parameters, the cross sections for the series (sp, 2n+) can be calculated from Eq. (8) and are shown in Fig. 15 as a function of the energy. The photoabsorption cross-section σ is linked to the light attenuation through a gas medium of column density n₀l, where l is the length of the cell and n₀ the gas density.

Fig. 15

Fano profiles for the (sp, 2n+) autoionizing series of helium. The cross sections are calculated following Morgan and Hederer (1984)

If a monochromatic electromagnetic pulse of intensity I₀ and frequency ω impinges on a cell of length l, filled with a gas with density n₀, the transmitted intensity obeys the Beer-Lambert law

$$\displaystyle \begin{aligned} \begin{array}{rcl} I(\omega) = I_0 \exp[-n_0 l \sigma(\omega)]. {} \end{array} \end{aligned} $$

(9)

As a result, by comparison with Eq. (9), the modulus of the transmissivity can be defined as $|T| = \exp [-n_0 l \sigma (\omega )/2]$. We may then write

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{ln}[T(\omega)] = \mathrm{ln}[|T(\omega)|] + i \varPhi(\omega) = - n_0 l \sigma(\omega)/2 + i \varPhi(\omega).{} \end{array} \end{aligned} $$

(10)

In the following, we will choose a column density equal to n₀l = 10¹⁸ cm⁻², and we will restrict ourselves to the third (n = 4) resonant line of the (sp, 2n+)¹P₀ Rydberg series for helium. The cross sections can be calculated with the help of Eq. (8). This fixes |T| as a function of frequency. Then, following the general treatment in section “Self-Seeding XFELs with a Wake Monochromator”, one can use Kramers-Kronig relation to recover the phase according to Eq. (2). The final result in terms of modulus and phase of the transmissivity T is shown in Fig. 16.

Fig. 16

Modulus and phase of the transmissivity of helium around the n = 4 line of the (sp, 2n+)¹P₀ Rydberg series. The phase is recovered from the modulus with the help of the Kramers-Kronig relation according to Eq. (2)

The filter described here⁷ can substitute the single crystal considered in the previous section for HXRSS. Therefore, the self-seeding setup just consists of the gas cell, to be filled with noble gas, and a short magnetic chicane. Figure 17 shows an example of an FEL pulse filtered with the help of the gas monochromator. It refers to a case study described in detail in Geloni et al. (2011d). Note that the shape of the wake is now completely different compared to the case of a diamond wake monochromator, but this is not important. In fact, therefore the radiation power within the wake is much larger than the equivalent shot-noise power in the electron bunch and can be used as seed pulse.

Fig. 17

Example of filtered SASE spectrum (top) and power (bottom) after a gas monochromator. The bottom plot illustrates the wake. Even if it is different from that created by a crystal monochromator, it can be used as a seed exactly in the same way

The scheme discussed here, while remaining of theoretical interest, has several practical limitations. In fact, although the availability of a series of resonances and of different gases allows for seeding at different frequencies, the scheme lacks continuous tunability. Moreover, not all resonances can be used. In fact, they should be located further away from each other than the FEL bandwidth and be narrow enough to guarantee the applicability of the temporal windowing process. In the case in Fig. 17, a rule of thumb requires a width narrower than 10 meV. In this case, for example, only the second and the third helium lines studied here are suitable, as the first one is too wide, and starting from the fourth, there is not enough separation.⁸

Finally, it should be stressed that the autoionizing resonance that we considered here as an example is in the 20-nm range, which is outside the region of interest of the self-seeding technique. A possible way to generate shorter wavelengths, down to the few-nanometer range, is to use a two-stage output undulator, with the second stage resonant to one of the harmonics of the first one (Geloni et al. 2011d). In this way, the second part of the output undulator acts effectively as a radiator, where one exploits the nonlinear bunching present in the electron beam.