**2. Grating geometries for ultrashort pulses**

Grazing incidence reflective gratings may be used either in the classical diffraction geometry (CDG) or in the off-plane geometry (OPG) [50].

The CDG is shown in **Figure 1(a)**. The grating equation is sin*α* + sin*β* = *mλσ*CD, where *α* and *β* are, respectively, the incident and diffracted angles and *σ*CD the groove density.

The OPG is shown in **Figure 1(b)**. The grating equation is sin*γ* (sin*μ* + sin*ν*) = *mλσ*OP, where *γ* is the altitude angle, *μ* and *ν* are the azimuth angles as defined in the figure, and *σ*OP is the groove density. The OPG, although seldom used, gives higher throughput than the classical mount, since it has been theoretically demonstrated and experimentally measured that the peak diffraction efficiency is close to the reflectivity of the coating at the altitude angle [51, 52]. Therefore, the OPG is suitable for the design of XUV grating instruments with high efficien‐ cy [53, 54].

**Figure 1.** (a) Classical diffraction geometry; (b) off-plane geometry.

When realizing a grating compressor for ultrafast pulses, the main problem faced with is the pulse-front tilt given by the diffraction, as shown in **Figure 2** in the case of the CDG. Furthermore, different wavelengths are diffracted in different directions. The pulse-front tilt and the spectral angular dispersion have to be corrected by a second grating in a compensat‐ ed configuration to fulfill the two following conditions: (1) the differences in the path lengths of rays with the same wavelength within the beam aperture that are caused by the diffrac‐ tion from the first grating have to be compensated by the second grating; that is, the pulsefront tilt is corrected; (2) the angular spectral dispersion caused by the first grating has to be canceled by the second grating, that is, all the rays at different wavelengths exit the second grating with parallel directions. Both these conditions are satisfied by a scheme with two equal gratings mounted with opposite diffraction orders; that is, the incidence (incoming azimuth) angle on the second grating is equal to the diffraction angle (outcoming azimuth) from the first grating. The phase chirp introduced by the system is calculated as the difference in the optical paths of rays at different wavelengths. This principle is well known for the realization of stretchers and compressors in the visible and near infrared [55, 56].

**Figure 2.** Pulse-front tilt of an ultrashort pulse diffracted by a grating in the CDG. *Δτ*in and *Δτ*out are the pulse duration at input and output, respectively. At the first diffracted order, the pulse-front tilt is *Δ*OP = *Nλ*, where *N* is the number of illuminated grooves.

#### **3. Grazing incidence grating compressor**

**2. Grating geometries for ultrashort pulses**

(CDG) or in the off-plane geometry (OPG) [50].

**Figure 1.** (a) Classical diffraction geometry; (b) off-plane geometry.

cy [53, 54].

228 230High Energy and Short Pulse Lasers

Grazing incidence reflective gratings may be used either in the classical diffraction geometry

The CDG is shown in **Figure 1(a)**. The grating equation is sin*α* + sin*β* = *mλσ*CD, where *α* and *β*

The OPG is shown in **Figure 1(b)**. The grating equation is sin*γ* (sin*μ* + sin*ν*) = *mλσ*OP, where *γ* is the altitude angle, *μ* and *ν* are the azimuth angles as defined in the figure, and *σ*OP is the groove density. The OPG, although seldom used, gives higher throughput than the classical mount, since it has been theoretically demonstrated and experimentally measured that the peak diffraction efficiency is close to the reflectivity of the coating at the altitude angle [51, 52]. Therefore, the OPG is suitable for the design of XUV grating instruments with high efficien‐

When realizing a grating compressor for ultrafast pulses, the main problem faced with is the pulse-front tilt given by the diffraction, as shown in **Figure 2** in the case of the CDG. Furthermore, different wavelengths are diffracted in different directions. The pulse-front tilt and the spectral angular dispersion have to be corrected by a second grating in a compensat‐ ed configuration to fulfill the two following conditions: (1) the differences in the path lengths of rays with the same wavelength within the beam aperture that are caused by the diffrac‐ tion from the first grating have to be compensated by the second grating; that is, the pulsefront tilt is corrected; (2) the angular spectral dispersion caused by the first grating has to be canceled by the second grating, that is, all the rays at different wavelengths exit the second grating with parallel directions. Both these conditions are satisfied by a scheme with two equal gratings mounted with opposite diffraction orders; that is, the incidence (incoming azimuth) angle on the second grating is equal to the diffraction angle (outcoming azimuth) from the first grating. The phase chirp introduced by the system is calculated as the difference in the optical

are, respectively, the incident and diffracted angles and *σ*CD the groove density.

When applying the double-grating configuration to the phase manipulation of XUV pulses, all the optics have to be operated at grazing incidence. The simplest arrangement consists of two identical plane gratings mounted in the compensated configuration, as shown in **Figures 3** and **4**. Due to the symmetry of the configuration, the angular dispersion at the output is canceled, and the output rays are parallel to the input for all the wavelengths [57].

**Figure 3.** Double-grating compressor in the CDG. The diffraction angle from G2 is constant with the wavelength and equal to the incidence angle on G1, *β*2 = *α*1.

**Figure 4.** Double-grating compressor in the OPG. The outcoming azimuth from G2 is constant with the wavelength and equal to the incoming azimuth on G1, *ν*2 = *μ*1.

Since different wavelengths are diffracted by G1 at different angles, the rays do not make the same optical paths. In the case of the CDG, the optical path is analytically expressed (for less than a constant term) as

$$OP\_{cp}(\lambda) \equiv q\_{cp} \frac{\cos \beta\_c}{\cos \beta(\lambda)} [1 - \sin \alpha \sin \beta(\lambda)] \tag{1}$$

where *α* is the incidence angle on G1, *β*(*λ*) and *β*c are, respectively, the diffraction angles from G1 at the generic wavelength *λ* and at the central wavelength of the interval of operation *λ*c, and *q*CD is the G1-G2 distance. The bandwidth of the pulse *Δλ* is limited between *λ*min and *λ*max, *Δλ* = *λ*max − *λ*min, and *λ*c = (*λ*min + *λ*max)/2. In case of a narrow-band pulse with *λ*/*Δλ* < 20%, Eq. (1) is linearized in *λ* as

$$OP\_{co}(\lambda) = \mathbf{q}\_{co}\lambda\_c \left(\frac{m\sigma\_{cp}}{\cos\beta\_c}\right)^2 \lambda \tag{2}$$

Similarly, in the case of the OPM, the optical path is expressed as

$$OP\_{op}(\lambda) \equiv q \frac{\dot{\sin}^2 \mathcal{Y} \cos \mu}{\cos \nu} (1 + \sin \mu \sin \nu) \tag{3}$$

where *μ* is the incoming azimuth on G1 that has been chosen to have the central wavelength *λ*c diffracted at *ν*c = *μ*, i.e., 2 sin*γ* sin*μ* = *λ*c *σ*OP, and *q*OP is the G1-G2 distance. In case of a narrowband pulse, Eq. (3) is linearized as

$$OP\_{op}\left(\lambda\right) = \mathbf{q}\_{op}\lambda\_c \left(\frac{m\sigma\_{op}}{\cos\mu}\right)^2 \lambda\tag{4}$$

Note thatin both cases the optical path increases with the wavelength, and this forces the group delay dispersion introduced by the double-grating configuration to be negative.

As usual, the group delay (GD) and the group delay dispersion (GDD) are expressed as a function of *ω* = 2*πc*/*λ*: GD(*ω*) = *∂φ*(*ω*)/*∂ω*.= OP(*ω*)/*c* and GDD(*ω*) = ∂GD(*ω*)/∂*ω*. The central pulse frequency *ω*c is defined as *ω*c = 2*πc*/*λ*c.

For narrow-band pulses, the GD is also linear in frequency, and the GDD is constant and negative

$$\text{GDD}\_{\text{cp}} = -\frac{q\_{\text{cp}} \mathcal{L}}{\text{op}\_{\text{c}}^{\text{3}}} \left(\frac{2\pi \sigma\_{\text{cp}}}{\cos \beta\_{\text{c}}}\right)^{2} \tag{5}$$

and

**Figure 4.** Double-grating compressor in the OPG. The outcoming azimuth from G2 is constant with the wavelength

Since different wavelengths are diffracted by G1 at different angles, the rays do not make the same optical paths. In the case of the CDG, the optical path is analytically expressed (for less

where *α* is the incidence angle on G1, *β*(*λ*) and *β*c are, respectively, the diffraction angles from G1 at the generic wavelength *λ* and at the central wavelength of the interval of operation *λ*c, and *q*CD is the G1-G2 distance. The bandwidth of the pulse *Δλ* is limited between *λ*min and *λ*max, *Δλ* = *λ*max − *λ*min, and *λ*c = (*λ*min + *λ*max)/2. In case of a narrow-band pulse with *λ*/*Δλ* < 20%, Eq.

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and equal to the incoming azimuth on G1, *ν*2 = *μ*1.

than a constant term) as

230 232High Energy and Short Pulse Lasers

(1) is linearized in *λ* as

band pulse, Eq. (3) is linearized as

$$GDD\_{op} = -\frac{q\_{op}\mathcal{L}}{\rho \sigma\_c^3} \left(\frac{2\pi\sigma\_{op}}{\cos\mu}\right)^2\tag{6}$$

Once the required GDD to manipulate the pulse has been defined, the above equations define the parameters of the grating compressor in both geometries.

Once the required GDD has been fixed, the two geometries give equivalent answerfor *q*OP *σ*OP<sup>2</sup> / cos2 *μ* = *q*CD *σ*CD2 /cos2 *β*c. In case of equal arms, i.e., *q*OP = *q*CD, since *μ* is typically below 20° and *β*<sup>c</sup> above 80°, the groove density that would be required in the OPM is much higherthan the CDM and may be not available from grating providers. Therefore, a compressor in the OPM is typically longer than the corresponding CDM, i.e., *q*OP > *q*CD.

The compressor introduces a spatial chirp of the pulse, i.e., rays with different wavelengths have the same output direction, but they are not exactly superimposed. In the conventional design of compressors for IR pulses, the spatial chirp is canceled by making the beam passing two additional times though the same gratings, so the output spatial dispersion is zero. This cannot be realized in grazing incidence, since it would require the insertion of two addition‐ al gratings that would make the configuration complex and inefficient. The spatial chirp SC(*λ*) is expressed, in case of a narrow-band pulse, as

$$\text{SC}\_{\text{CD}}\left(\lambda\right) = q\_{\text{CD}} \sigma\_{\text{CD}} \frac{\cos \alpha}{\cos^2 \beta\_c} \Delta \lambda \tag{7}$$

$$\text{SC}\_{\text{op}}\left(\mathcal{\lambda}\right) = q\_{\text{op}} \sigma\_{\text{op}} \frac{1}{\cos^2 \mu} \Delta \mathcal{\lambda} \tag{8}$$

Since the rays are parallel, the spatial chirp does not influence the quality of the final spot size, since all the rays are focused on the same point.

In the following, we discuss the use of the double-grating configuration to the case of compression of FEL pulses and of attosecond pulses generated through HHs.
