**4. Grating compressor applied to FEL pulses**

One of the main problems faced when manipulating intense FEL radiation is the use of robust optical components to be operated with the FEL pulses with minimum risk of damaging. From this point of view, grazing incidence elements are preferable. The double-grating compres‐ sor, whose optical elements are used at grazing incidence, is very suitable for FEL pulses. We assume FEL parameters already discussed in the literature [58] that may be a test case for the application to chirped-pulse amplification (CPA) of FEL pulses.

The scheme of the CPA applied to a seeded FEL is shown in **Figure 5**. The electron beam, being generated through a chirped laser seeding pulse, originates, afterthe radiator, a chirped optical pulse with positive GDD. The chirp is corrected by the grating compressor that introduces different optical paths for different wavelengths and shortens the pulse duration close to the Fourier limit.

**Figure 5.** Schematic of the CPA applied to a seeded FEL.

Assuming a linear electron bunch energy spread at the entrance of the FEL radiator, the wavelength spread is evaluated as *Δλ*/*λ* = −2*ΔE*/*E*. The photon chirp induced by the en‐ trance energy spread is 2*α*/*E*, where *α* is the electron chirp. This has to be compensated to reduce the pulse time duration.

As a feasibility study, we want to define a configuration to compress an FEL pulse centered at 13.5 nm. **Table 1** resumes the FEL parameters used in the simulation. Using these parame‐ ters, the time to be compensated is calculated to be 310 fs for an FEL emission centered at 13.5 nm with a bandwidth *Δλ* = 0.8 nm. If the GDD introduced by the compressor is opposite to the intrinsic GDD of the chirped pulse, the pulse time duration is reduced.


**Table 1.** FEL parameters used in the simulation.

Since the rays are parallel, the spatial chirp does not influence the quality of the final spot size,

In the following, we discuss the use of the double-grating configuration to the case of

One of the main problems faced when manipulating intense FEL radiation is the use of robust optical components to be operated with the FEL pulses with minimum risk of damaging. From this point of view, grazing incidence elements are preferable. The double-grating compres‐ sor, whose optical elements are used at grazing incidence, is very suitable for FEL pulses. We assume FEL parameters already discussed in the literature [58] that may be a test case for the

The scheme of the CPA applied to a seeded FEL is shown in **Figure 5**. The electron beam, being generated through a chirped laser seeding pulse, originates, afterthe radiator, a chirped optical pulse with positive GDD. The chirp is corrected by the grating compressor that introduces different optical paths for different wavelengths and shortens the pulse duration close to the

Assuming a linear electron bunch energy spread at the entrance of the FEL radiator, the wavelength spread is evaluated as *Δλ*/*λ* = −2*ΔE*/*E*. The photon chirp induced by the en‐ trance energy spread is 2*α*/*E*, where *α* is the electron chirp. This has to be compensated to

As a feasibility study, we want to define a configuration to compress an FEL pulse centered at 13.5 nm. **Table 1** resumes the FEL parameters used in the simulation. Using these parame‐ ters, the time to be compensated is calculated to be 310 fs for an FEL emission centered at 13.5

compression of FEL pulses and of attosecond pulses generated through HHs.

since all the rays are focused on the same point.

232 234High Energy and Short Pulse Lasers

**4. Grating compressor applied to FEL pulses**

application to chirped-pulse amplification (CPA) of FEL pulses.

**Figure 5.** Schematic of the CPA applied to a seeded FEL.

reduce the pulse time duration.

Fourier limit.

The compressor parameters are summarized in **Table 2** for the two geometries. The GD of the configurations is shown in **Figure 6(a)**. The curve is almost the same for both geometries. As expected, for narrow-bandwidth pulses, the resulting GD is linear, and the GDD is constant: GDD ≈−37 fs<sup>2</sup> .


**Table 2.** Parameters of the grating compressor in two geometries.

The spatial chirp calculated in the full-width-at-half-maximum (FWHM) bandwidth is 0.9 mm for both geometries. The compressor is typically inserted several tens of meters after the FEL source. As a typical angular divergence of the FEL source at 13.5 nm, 30 μrad (standard deviation) is assumed. The resulting beam diameter at the compressor input, that is as‐ sumed to be 50 m farfrom the source, is 3.6 mm FWHM, therefore much largerthan the spatial chirp.

Note that the groove density required in the OPG is higher than the CDG and that the size of the instrument is longer for the OPG. On the basis of the efficiency measurements per‐ formed in the two geometries and already discussed in Ref. [54], the total efficiency in the OPG is expected to be higher than the CDG by a factor ≈2.5.

It can be shown that the FEL pulse duration is reduced by a factor of 10 at the output of the compressor, i.e., from 310 fs to about 30 fs. This gives a substantial increase in the temporal resolution of the FEL pulses when used for ultrafast experiments.

#### **4.1. Tunability in wavelength and group delay**

The compressor can be tuned in wavelength by rotating the gratings around an axis that is tangent to the surface, passes through the grating center, and is parallel to the grooves. The rotation changes the incidence angle *α* in the CDG at a constant subtended angle *K = α* + *β*, or the azimuth angle *μ* in the OPM at constant altitude angle *γ*. The tuning in wavelength changes also the GD, since it depends on the incidence (azimuth) angles. The delays introduced in the bandwidth *Δλ* = 0.8 nm when the central wavelength is tuned in the 10–18 nm interval are shown in **Figure 6(b)**.

**Figure 6.** (a) GD of the compressor having the parameters of **Table 2**; (b) change of the delay in the bandwidth when the gratings are rotated to tune the wavelength in the 10–18 nm interval.

It is clear that the simple grating rotation is not sufficient to tune simultaneously the wave‐ length and the GD. An additional degree of freedom is required that may be the changing of the subtended angle *K* (the altitude angle *γ*) in the CDG (OPG), as shown in **Figure 6**. By acting simultaneously on grating rotation and subtended (altitude) angles, users can select simulta‐ neously the wavelength and GD. This makes the design very flexible.

**Figure 7.** Delay in 0.8-nm bandwidth introduced by the compressor having the parameters of **Table 2** when the gra‐ tings are rotated to tune the wavelength in the 10–18 nm interval: (a) CDG, variable subtended angle; (b) OPG, variable altitude angle.

The optical setup of the compressor is shown in **Figure 7**. The instrument consists of two plane gratings and two plane mirrors. The two mirrors are used to deviate the FEL beam in the same direction as the input.

#### **4.2. Operation with a diverging beam**

**4.1. Tunability in wavelength and group delay**

the gratings are rotated to tune the wavelength in the 10–18 nm interval.

neously the wavelength and GD. This makes the design very flexible.

shown in **Figure 6(b)**.

234 236High Energy and Short Pulse Lasers

altitude angle.

The compressor can be tuned in wavelength by rotating the gratings around an axis that is tangent to the surface, passes through the grating center, and is parallel to the grooves. The rotation changes the incidence angle *α* in the CDG at a constant subtended angle *K = α* + *β*, or the azimuth angle *μ* in the OPM at constant altitude angle *γ*. The tuning in wavelength changes also the GD, since it depends on the incidence (azimuth) angles. The delays introduced in the bandwidth *Δλ* = 0.8 nm when the central wavelength is tuned in the 10–18 nm interval are

**Figure 6.** (a) GD of the compressor having the parameters of **Table 2**; (b) change of the delay in the bandwidth when

It is clear that the simple grating rotation is not sufficient to tune simultaneously the wave‐ length and the GD. An additional degree of freedom is required that may be the changing of the subtended angle *K* (the altitude angle *γ*) in the CDG (OPG), as shown in **Figure 6**. By acting simultaneously on grating rotation and subtended (altitude) angles, users can select simulta‐

**Figure 7.** Delay in 0.8-nm bandwidth introduced by the compressor having the parameters of **Table 2** when the gra‐ tings are rotated to tune the wavelength in the 10–18 nm interval: (a) CDG, variable subtended angle; (b) OPG, variable The formulas calculated above assume to work with a collimated beam. In this case, the number of illuminated groove is the same for the two gratings when they are parallel, giving a corrected pulse-front tilt at the output. Indeed, in the real case of an FEL-divergent beam, this would require the use of an additional mirror at the input of the compressor to collimate the beam, namely a grazing-incidence parabola that makes the design complex. Indeed, the compres‐ sor, as presented in the previous paragraph, can be used in a divergent beam if the second grating is operated slightly out from the parallel condition, to have the same number of illuminated grooves.

**Figure 8.** Optical setup of the compressor: (a) CDG; (b) OPG.

The geometry with a divergent beam is shown in **Figure 8** for CDG. The number of illuminat‐ ed grooves is the same for both gratings if G2 is operated at a lower subtended angle, *k*2 < *k*1:

$$\cos\alpha\_2 = \cos\beta\_1 + \frac{q\_{\text{CD}}\delta\_1}{S\_1}\frac{\cos^2\alpha\_1}{\cos\beta\_1} \tag{9}$$

where *δ*1 is the divergence of the incoming beam, *S*1 is the beam cross section at G1, *α*1 and *β*<sup>1</sup> the incidence and diffraction angles on G1, and *α*2 the incidence angle on G2.

In the case of OPM, the compensation of the pulse-front tilt is expressed by

$$\cos \mu\_z = \cos \mu\_\natural + \frac{q\_{\partial \nu} \delta\_\natural}{S\_\natural} \tag{10}$$

where *μ*1 and *μ*2 are the azimuth angles on G1 and G2.

The asymmetry between G1 and G2 depends on the actual divergence of the FEL beam and on the distance between the two gratings. Let us assume the parameters of **Table 2**, with 30 μrad divergence (standard deviation). The compressor stage is supposed to be installed 50 m far from the source. In the case of CDG, the residual pulse-front tilt at 13.5 nm is 5 fs FWHM if the gratings are operated parallel and the correction to be applied to *α*2 is 0.1°. If the compressor is used at longer wavelengths, the asymmetry to be applied is more remarkable because of the higher FEL divergence.

The spatial chirp, that does not influence the quality of the final spot size in case of a parallel beam, has to be evaluated in case of a divergent beam, since different wavelengths are focused in different points in the direction of the spectral dispersion. This gives a slight asymmetry in the spot profile that is broadened in the direction of spectral dispersion. Let us define *M* as the total demagnification of the FEL beamline, *M* ≈ 50–100. The spatial chirp SC gives a limit to the minimum focal spot that can be achieved in the direction of the spectral dispersion, as SC/ M. Assuming the same parameters of **Table 2** and *M* = 75, the minimum spot size that can be achieved in the direction of the spectral dispersion is 12 μm FWHM. The broadening due to the spatial chirp is generally negligible for spot sizes in the 20–50 μm range. However, the use of the compressor may degrade the quality of the final focus if the beamline is tailored to give micro-focusing. In such cases, the insertion of the compressor has to be carefully evaluated.

#### **4.3. Efficiency of the compressor**

The efficiency of the compressor depends on the geometry adopted for the gratings, since it is well known that the OPG gives efficiency higher than the CDG [59].

**Figure 9.** Grating compressor operated in divergent beam. The pulse-front tilt is corrected for l2 = l1.

To compare the two geometries, the first-order diffraction efficiency has been measured in the 25–35 nm (35–50 eV) region in the two different geometries. Both gratings are plane, goldcoated, and have 600 gr/mm groove density. The grating used in the CDG is blazed at 2°, and the grating used in the OPG is blazed at 7°. The results are shown in **Figure 9**. The efficiency in the OPG is a factor ≈2 higher than the CDG that gives a factor 4 in the total efficiency of the compressor. The latter is expected to be ≈5% in the CDG and ≈20% in the OPG [60].
