**5. Grating compressor for attosecond pulses**

1

= + (10)

1

*S* d

2 1

m

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

because of the higher FEL divergence.

236 238High Energy and Short Pulse Lasers

**4.3. Efficiency of the compressor**

cos cos *OP <sup>q</sup>*

 m

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

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.

The efficiency of the compressor depends on the geometry adopted for the gratings, since it is

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

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.

Attosecond pulses generated with the scheme of HHs by the use of laser pulses of few optical cycles are positively chirped as a result of the different duration of the quantum paths that contribute to the different portions of the emitted spectrum. They can be compressed by introducing a suitable device with a negative GDD.

The simplest device that exhibits negative GDD is a thin metallic filter. Anomalous disper‐ sion near absorption resonances can be exploited to compensate the positive chirp of the generated attosecond pulses. Aluminum or zirconium is normally used in the XUV range, depending on the spectralrange of operation [61]. Furthermore, the filter is also useful to block the IR laser beam since it is totally solar blind. The main drawback of the filter is the strong XUV absorption that may even exceed one order of magnitude.

Aperiodic multilayer mirrors have also been developed and successfully tested for the dispersion control in the XUV [62–65]. The multilayer coating is designed to compensate for the attosecond chirp and reduce the pulse duration. Pulse compression and focusing are demanded to the same optical element that makes the design simple and compact. For a centerphoton energy range of 100–120 eV, the mirror reflectivity is approximately 10% and the bandwidth 10–13 eV. Recently, also mirrors for the water-window region have been tested, although with reflectivity lower than 1% [66]. A metallic filter has to be inserted anyway in the optical path to block the IR laser light.

**Figure 10.** Comparison of efficiencies measured in CDG and OPG.

We discuss here the use of gratings to compress attosecond pulses by introducing a GDD that compensates for the intrinsic pulse chirp. Unfortunately, the configuration with plane gratings

discussed above is not suitable, since the G1-to-G2 distance *q* that is required to give the necessary GDD is too small to be realized in practice. Therefore, the plane-grating configura‐ tion has to be modified as shown in **Figure 10** for the OPG [67, 68], by adding an intermedi‐ ate focal point between the two gratings. The case of CDG is analogous.

The design consists of six optical elements: four identical grazing incidence parabolic mir‐ rors (P1-P4) and two identical plane gratings (G1, G2). The XUV source is located in the front focal plane of P1, and the rays are collected at the focus of the last parabolic mirror P4. The parabolic mirrors are used to collimate and refocus the XUV radiation with negligible aberrations, and the gratings are illuminated in parallel light. A spectrally dispersed image of the source is obtained in the intermediate plane. The two focusing mirrors placed between the gratings act as a telescopic arrangement. Differently from the plane-grating compressor discussed above, this makes it possible to: (i) continuously tune the GDD from negative to positive values and (ii) achieve the exceedingly small grating separations necessary to compensate the attosecond chirp.

**Figure 11.** Grating compressor for attosecond pulses.

With reference to the symbols listed in **Figure 11**, the condition for zero GDD is to have G1 imaged on G2, that is realized when *S1* + *S2 =* 2*f*. Since *f* is fixed, the GDD depends on *S1 + S2:* for *S1 +S2* < 2*f*, G1 is imaged behind G2, and the resulting GDD is positive; for *S1+S2 >* 2*f*, G1 is imaged before G2, and the resulting GDD is negative. Once the equivalent distance *q* that is required for compensation has been calculated, the effective displacement from the zerodispersion case is *ΔS* = *q/*(*ω*0 sin*γ* cos*ν*), where *ω*0 is the center angular frequency. It can be noted that the design of the compressor is simplified if *S*1 is kept fixed, and only *S*2 is tuned to change the GDD. A suitable value for *f* is in the range of 200–300 mm, giving a total length of the compressor of ≈1.5 m.

As an application to ultrashort pulses, a compressor design for the 50–100 eV region is discussed here. The characteristics of the compressor are resumed in **Table 3**. The GD introduced with the distance *S* = *S*1 + *S*2 is shown in **Figure 12**. An example of compression of a pulse with a positive GDD is presented in **Figure 13**, adapted from [69]. Note that the chirp introduced by the compressor is able to compensate the pulse chirp down to a nearly singlecycle pulse (**Figure 14**).


**Table 3.** Parameters of the compressor for the 50–100 eV region.

discussed above is not suitable, since the G1-to-G2 distance *q* that is required to give the necessary GDD is too small to be realized in practice. Therefore, the plane-grating configura‐ tion has to be modified as shown in **Figure 10** for the OPG [67, 68], by adding an intermedi‐

The design consists of six optical elements: four identical grazing incidence parabolic mir‐ rors (P1-P4) and two identical plane gratings (G1, G2). The XUV source is located in the front focal plane of P1, and the rays are collected at the focus of the last parabolic mirror P4. The parabolic mirrors are used to collimate and refocus the XUV radiation with negligible aberrations, and the gratings are illuminated in parallel light. A spectrally dispersed image of the source is obtained in the intermediate plane. The two focusing mirrors placed between the gratings act as a telescopic arrangement. Differently from the plane-grating compressor discussed above, this makes it possible to: (i) continuously tune the GDD from negative to positive values and (ii) achieve the exceedingly small grating separations necessary to

With reference to the symbols listed in **Figure 11**, the condition for zero GDD is to have G1 imaged on G2, that is realized when *S1* + *S2 =* 2*f*. Since *f* is fixed, the GDD depends on *S1 + S2:* for *S1 +S2* < 2*f*, G1 is imaged behind G2, and the resulting GDD is positive; for *S1+S2 >* 2*f*, G1 is imaged before G2, and the resulting GDD is negative. Once the equivalent distance *q* that is required for compensation has been calculated, the effective displacement from the zerodispersion case is *ΔS* = *q/*(*ω*0 sin*γ* cos*ν*), where *ω*0 is the center angular frequency. It can be noted that the design of the compressor is simplified if *S*1 is kept fixed, and only *S*2 is tuned to change the GDD. A suitable value for *f* is in the range of 200–300 mm, giving a total length of the

As an application to ultrashort pulses, a compressor design for the 50–100 eV region is discussed here. The characteristics of the compressor are resumed in **Table 3**. The GD

ate focal point between the two gratings. The case of CDG is analogous.

compensate the attosecond chirp.

238 240High Energy and Short Pulse Lasers

**Figure 11.** Grating compressor for attosecond pulses.

compressor of ≈1.5 m.

**Figure 12.** Operation of the attosecond compressor: (a) GDD = 0; (b) GDD < 0.

**Figure 13.** GD of the compressor with parameters listed in **Table 3**.

**Figure 14.** Simulation of the compression of a XUV pulse with the parameters of **Table 3**.

#### **5.1. Example of application to attosecond pulses**

A schematic view of an experiment using compressed attosecond pulses is shown in **Figure 15**. The XUV attosecond pulses are generated on a gas jet in a vacuum chamber and are intrinsically chirped. The XUV radiation is generated with the intrinsic attosecond chirp. Different wavelengths travel in different paths inside the compressorthat compensates forthe chirp and reduces the time duration of the pulse.

**Figure 15.** Schematic of the attosecond compressor.

Attosecond pulses are generated in different XUV spectral windows, depending on the interacting gas. Using argon and ≈2⋅1014 W/cm<sup>2</sup> ultrafast laser intensity, radiation is generat‐ ed in the 25–55 eV region, while with neon and ≈6⋅1014 W/cm<sup>2</sup> intensity, radiation is generat‐ ed in the 50–120 eV region. Attosecond pulses are generated from the short trajectory

components, since this is the part of the generated radiation that survives propagation in the generating medium. Aluminum and zirconium filters can be used, respectively, in the low- (i.e., argon) and high-energy range (i.e., neon) to introduce negative chirp to compensate for the intrinsic chirp of the attosecond pulses and compress them close to the Fourier limit.

Here, we discuss the parameters to be used for a grating compressor in the two different XUV regions. For the lower intensity case, the parameters are *σ* = 100 grooves/mm, *γ* = 1.5°, *μ* = 3.7°, *ΔS* = 23 mm. For the higher intensity case, the parameters are *σ* = 200 grooves/mm, *γ* = 1°, *μ* = 4.1°, *ΔS* = 29 mm.

Simulations of the generated and compressed pulses have been performed starting from 25-fs driving pulses at 790 nm [70]. The generated pulses have a duration of 270 as in argon and 200 in neon, while the Fourier-limited duration would be, respectively, 115 and 35 as. At the output of the grating compressor, the pulse duration results, respectively, 160 and 60 as, much closer to the Fourier limit. It is also shown that the optimal Al filter would reduce the pulse dura‐ tion to 130 as, therefore shorter than the grating compressor, while the optimal Zr filter would reduce the pulse duration to 90 as, therefore longer than the grating compressor.

In general, the grating compressor is more versatile than metal filters and can be continuous‐ ly tuned from negative to positive GDD with constant throughput.
