2 Dependence of focal intensity on out-of-plane compressor parameters

Let the compressor consist of gratings with groove density N and distance between the gratings along the normal L. We will consider the OC with the angle of incidence on the first grating α in the diffraction plane and γ in the non-diffraction plane. Most labs use a compressor consisting of two identical pairs of gratings: the parameters $\alpha,\gamma, N\;\mathrm{and}\;L$ are the same for the two pairs. Such a compressor is referred to as a symmetric one (Figure 1(a)). In an asymmetric compressor^[ Reference Shen, Du, Liang, Wang, Liu and Li^27, Reference Khazanov^28, Reference Khazanov^39, Reference Huang and Kessler^42] the grating pairs differ from each other; they have at least one of the parameters $\alpha,\gamma, N\;\mathrm{or}\;L$ that differs from the others. The asymmetric compressor smooths small-scale fluence fluctuations and, hence, reduces the probability of laser-induced damage. A special case of the asymmetric compressor is a compressor consisting of just one pair of gratings – a two-grating compressor (Figure 1(b))^[ Reference Vyatkin and Khazanov^13, Reference Shen, Du, Liang, Wang, Liu and Li^27, Reference Trentelman, Ross and Danson^30]. It has some additional advantages: smoothing of large-scale fluence fluctuations, simplicity and lower cost. From the point of view of compressor (a)symmetry, we will restrict ourselves to two most interesting cases – a symmetric compressor and an asymmetric two-grating compressor (Figures 1(a) and 1(b)). They will be designated as 4OC (four-grating OC) and 2OC (two-grating OC). A plane TC is a particular case of the OC at $\gamma =0$ , so it will be designated as 4TC and 2TC. Another interesting special case is the Littrow compressor, in which $\alpha ={\alpha}_{\mathrm{L}}$ , where ${\alpha}_{\mathrm{L}}$ is the Littrow angle; the abbreviations 4LC and 2LC will be used for this compressor. The LC has a number of additional advantages^[ Reference Khazanov^21, Reference Smith, Erdogan and Erdogan^33, Reference Khazanov^41]. We assume that the beam size and the size of gratings G2 and G3 are such that all frequencies fall into the aperture of G2 and G3, that is, there are no beams that ‘miss’ the grating. This is not the case for the so-called full-aperture compressor ^[ Reference Vyatkin and Khazanov^13, Reference Trentelman, Ross and Danson^30, Reference Wang, Wang, Xu and Leng^43], which is not considered in this paper.

In the case of a perfect grating, the incident plane wave after reflection remains plane, that is, its spatial phase $\Delta \left(x,y\right)= \mathrm{const}$ , and the angle of reflection $\theta \left(\omega \right)$ is determined by the following expression for the grating:

(1) $$\begin{align}\mathrm{sin}\theta \left(\omega \right)=m\frac{2\pi c}{\omega}\frac{N}{\mathrm{cos}\gamma}+ \mathrm{sin}\alpha,\end{align}$$

where $m$ is the diffraction order. A perfect grating is understood as a grating with a perfectly flat substrate surface and perfectly parallel and equidistant grooves. As a result of an imperfect (out-of-plane) surface and imperfect (non-equidistant and non-parallel) grooves, the wavefront of the reflected wave is no longer flat and $\Delta \left(x,y\right)\ne \mathrm{const}$ . AM1,2 can compensate for distortions only at one (central) frequency ${\omega}_0$ . Since $\Delta$ depends on frequency $\omega$ (space–time coupling), this compensation cannot be complete, which leads to a decrease in the focal intensity. In Ref. [Reference Khazanov23], an expression was found for the spatial phase $\Delta \left(x^{\prime },y^{\prime },\omega \right)$ of a plane monochromatic wave after reflection from the grating:

(2) $$\begin{align}\Delta \left(x^{\prime },y^{\prime },\omega \right)&=-\frac{\omega }{c}\left({H}_{\mathrm{gr}}{h}_{\mathrm{gr}}\left(\frac{x^{\prime }}{\mathrm{cos}\theta \left(\omega \right)},\frac{y^{\prime }}{\mathrm{cos}\gamma}\right)\right.\nonumber\\&\left.\quad+\ {H}_{\mathrm{wr}}{h}_{\mathrm{wr}}\left(\frac{x^{\prime}\mathrm{cos}\Phi}{\mathrm{cos}\theta \left(\omega \right)},\frac{y^{\prime }}{\mathrm{cos}\gamma}\right)\right),\end{align}$$

where

(3) $$\begin{align}{H}_{\mathrm{gr}}\left(\omega \right)= \mathrm{cos}\gamma \left( \mathrm{cos}\alpha + \mathrm{cos}\theta \left(\omega \right)\right),\ {H}_{\mathrm{wr}}\left(\omega \right)=2\frac{\omega_{\mathrm{wr}}}{\omega },\end{align}$$

where ${h}_{\mathrm{gr}}\left(x^{\prime\prime },y^{\prime\prime}\right)$ is the profile of the grating surface; ${h}_{\mathrm{wr}}\left(x^{\prime\prime },y^{\prime\prime}\right)$ characterizes groove pattern imperfection and has the sense of the difference of the total surface profiles of the optical elements on the path of two waves writing the holographic grating (Figure 2); ${\omega}_{\mathrm{wr}}=2\pi c/{\lambda}_{\mathrm{wr}}$ is the frequency of the writing waves; and $\Phi$ is the angle of incidence of the writing waves on the grating substrate, $\mathrm{sin}\Phi =N{\lambda}_{\mathrm{wr}}/2$ . Here, $\left(x^{\prime\prime },y^{\prime\prime}\right)$ are the coordinates of the grating surface and $\left(x^{\prime },y^{\prime}\right)$ are the coordinates in the plane normal to the wave vector of the beam between the first and second gratings shown in Figure 1. On reflection from the second grating, the beam changes its size, that is, to pass to the laboratory reference frame, $\left(x,y\right),$   $x^{\prime }$ in Equation (2) must be replaced by $\frac{\mathrm{cos}\theta \left(\omega \right)}{\mathrm{cos}\alpha}x$ . Thus, the phase introduced into the beam upon reflection from the first grating has the following form:

(4) $$\begin{align}{\Delta }_1\left(x,y,\omega \right)&=-\frac{\omega }{c}\left({H}_{\mathrm{gr}}{h}_{\mathrm{gr},1}\left(\frac{x}{\mathrm{cos}\alpha},\frac{y}{\mathrm{cos}\gamma}\right)\right.\nonumber\\&\quad\left.+\ {H}_{\mathrm{wr}}{h}_{\mathrm{wr},1}\left(\frac{x\cos\Phi}{\mathrm{cos}\alpha},\frac{y}{\mathrm{cos}\gamma}\right)\right).\end{align}$$

Figure 2 Scheme of writing a holographic grating. BS, beamsplitter; M1–M3, mirrors; ${\psi}_{1,2},$ phase of the beams writing the grating; ${\psi}_1-{\psi}_2=2{k}_{\mathrm{wr}}{h}_{\mathrm{wr}}$ .

To find the phase for the remaining gratings, we fix the coordinate system $\left(x^{\prime\prime },y^{\prime\prime}\right)$ on each grating (Figure 1) so that, in the case of identical gratings, the functions ${h}_{\mathrm{gr}}\left(x^{\prime\prime },y^{\prime\prime}\right)$ and ${h}_{\mathrm{wr}}\left(x^{\prime\prime },y^{\prime\prime}\right)$ should be identical for all gratings: ${h}_{\mathrm{gr}}={h}_{\mathrm{gr},n};{h}_{\mathrm{wr}}={h}_{\mathrm{wr},n};n=1,2,3,4$ is the grating number.

As can be seen from Figure 1, for even gratings the angle between the $x^{\prime\prime }$ - and $x$ -axes is obtuse; therefore, the sign of the first argument in ${h}_{\mathrm{gr}}$ and ${h}_{\mathrm{wr}}$ for them should be changed: $x\to -x$ . Accordingly, for a grating with number $n$ , the substitution $x\to {\left(-1\right)}^{n+1}x$ should be made on the right-hand side of Equation (4). As shown in Ref. [Reference Khazanov23], the expression in Equation (2) is valid if the wave vector of the incident wave makes an acute angle with the $x^{\prime\prime }$ -axis. If this angle is obtuse, then the plus sign in front of ${H}_{\mathrm{wr}}$ should be replaced with the minus sign. From Figure 1 it is clear that G3 and G4 must have the minus sign. Therefore, in front of ${H}_{\mathrm{wr}}$ the factor ${\left(-1\right)}^{\left[\frac{n-1}{2}\right]}$ should be used, where […] denotes the integer part. In addition, for the second and fourth gratings in Equation (2), obvious substitutions $\alpha \to -\theta \left(\omega \right)$ and $\theta \left(\omega \right)\to -\alpha$ should be made. Taking all this into account, the phase introduced by the reflection from the grating with number $n$ takes on the following form:

(5) $$\begin{align}&{\Delta }_n\left(x,y,\omega \right)=-\frac{\omega }{c}\left({H}_{\mathrm{gr}}{h}_{\mathrm{gr},n}\left({\left(-1\right)}^{n+1}\frac{x}{\mathrm{cos}\alpha},\frac{y}{\mathrm{cos}\gamma}\right)\right.\nonumber\\&\quad\left.+\ {\left(-1\right)}^{\left[\frac{n-1}{2}\right]}{H}_{\mathrm{wr}}{h}_{\mathrm{wr},n}\left({\left(-1\right)}^{n+1}\frac{x\mathrm{cos}\Phi}{\mathrm{cos}\alpha},\frac{y}{\mathrm{cos}\gamma}\right)\right).\end{align}$$

Let the input field have the following form:

(6) $$\begin{align}{E}_0\left(\omega, \boldsymbol{r}\right)={e}^{i{\varphi}_{\mathrm{in}}\left(\omega \right)+i{\varphi}_{\mathrm{D}}\left(\omega \right)}{e}^{-{\left(\frac{\omega -{\omega}_0}{\Delta \omega}\right)}^{2\mu }}{e}^{iy\frac{\omega }{c} \mathrm{sin}\gamma}\left|{E}_0\left(\boldsymbol{r}\right)\right|,\end{align}$$

where ${\varphi}_{\mathrm{in}}\left(\omega \right)$ is the spectral phase without allowance for the phase introduced by the AOPDF ${\varphi}_{\mathrm{D}}\left(\omega \right)$ . Here we assume that the wavefront of ${E}_0\left(\boldsymbol{r}\right)$ is plane, that is, ${E}_0\left(\boldsymbol{r}\right)=\left|{E}_0\left(\boldsymbol{r}\right)\right|$ . If this is not the case, AM1 can fix it. The Strehl ratio is defined by the following:

(7) $$\begin{align}St=\frac{I_{\mathrm{f}}}{I_{\mathrm{f}}\left({h}_{\mathrm{gr}}={h}_{\mathrm{wr}}=0\right)},\end{align}$$

where ${I}_{\mathrm{f}}$ is focal intensity. According to Equation (7), the Strehl ratio shows the reduction of the focal intensity compared to the case of compressor gratings with a perfectly plane surface and perfectly parallel and equidistant grooves. We assume that АМ2 provides a plane wavefront at central frequency ${\omega}_0$ , and AOPDF ensures a constant spectral phase at zero spatial frequency $\kappa =0$ , as under these conditions ${I}_{\mathrm{f}}$ is maximal^[ Reference Khazanov^21]. Following the procedure described in Ref. [Reference Khazanov21], in the ${\left(\frac{\Delta \omega }{\omega_0}\right)}^2\overline{\Psi_{4,2}^2}\ll 2$ approximation for 4OC and 2OC we find ${St}_2$ and ${St}_4$ :

(8) $$\begin{align}{St}_{4,2}=1-M\left(\mu \right){\left(\frac{\Delta \omega }{\omega_0}\right)}^2\overline{\Psi_{4,2}^2\left(x,y\right)},\end{align}$$

where $M\left(\mu \right)=\Gamma \left(\frac{3}{2\mu}\right)/\Gamma \left(\frac{1}{2\mu}\right)$ , $\Gamma$ is the gamma function and $\overline{\left(\dots \right)}$ denotes averaging over the grating surface with the weight of the laser field module:

(9) $$\begin{align}\overline{\left(\dots \right)}\equiv \frac{\int \left|{E}_0\left( x\mathrm{cos}\alpha, y\mathrm{cos}\gamma \right)\right|\left(\dots \right) \mathrm{d}x\mathrm{d}y}{\int \left|{E}_0\left( x\mathrm{cos}\alpha, y\mathrm{cos}\gamma \right)\right| \mathrm{d}x\mathrm{d}y},\end{align}$$

(10) $$\begin{align}\kern-23.5pt {\Psi}_4\left(x,y\right)&=f{g}_4\left(x,y\right)+u{w}_4\left( x\mathrm{cos}\Phi, y\right)\nonumber\\&\quad-\mathbf{FG}\left(x,y\right)-\mathbf{UW}\left( x\cos\Phi, y\right),\end{align}$$

(11) $$\begin{align}\kern-43.5pt {\Psi}_2\left(x,y\right)=fg\left(x,y\right)+ uw\left( x\mathrm{cos}\Phi, y\right),\end{align}$$

(12) $$\begin{align}\kern12pt {g}_4\left(x,y\right)&=\sum \limits_{n=1}^4{h}_{\mathrm{gr},n}\left({\left(-1\right)}^{n+1}x,y\right),\nonumber\\{w}_4\left(x,y\right)&=\sum \limits_{n=1}^4{\left(-1\right)}^{\left[\frac{n-1}{2}\right]}{h}_{\mathrm{wr},n}\left({\left(-1\right)}^{n+1}x,y\right),\end{align}$$

(13) $$\begin{align}g\left(x,y\right)&={h}_{\mathrm{gr},1}\left(x,y\right)+{h}_{\mathrm{gr},2}\left(-x,y\right),\nonumber\\ w\left(x,y\right)&={h}_{\mathrm{wr},1}\left(x,y\right)+{h}_{\mathrm{wr},2}\left(-x,y\right),\end{align}$$

(14) $$\begin{align}\mathbf{G}\left(x,y\right)&=\mathbf{\nabla}{h}_{\mathrm{gr},2}\left(-x,y\right)+\mathbf{\nabla}{h}_{\mathrm{gr},3}\left(x,y\right),\nonumber\\\mathbf{W}\left(x,y\right) &=-\mathbf{\nabla}{h}_{\mathrm{wr},2}\left(-x,y\right)-\mathbf{\nabla}{h}_{\mathrm{wr},3}\left(x,y\right),\end{align}$$

(15) $$\begin{align}f&=2\pi N\mathrm{tg}\beta, \kern0.84em u=\frac{4\pi }{\lambda_{\mathrm{wr}}},\nonumber\\ \mathbf{F}&=2\pi NL\frac{\mathrm{cos}\alpha + \mathrm{cos}\beta}{{\mathrm{cos}\gamma \mathrm{cos}}^3\beta}\left(\begin{array}{c} \mathrm{cos}\alpha \mathrm{cos}\gamma \\ {}{\lambda}_0 N\mathrm{tg}\gamma \end{array}\right),\nonumber\\ \mathbf{U}&=2\frac{\mathrm{tg}\left(\frac{\alpha -\beta }{2}\right)}{N{\lambda}_{\mathrm{wr}}}\mathbf{F},\end{align}$$

(16) $$\begin{align}\mathbf{\nabla}h\left(-x,y\right)\triangleq {\left.\mathbf{\nabla}h\left(x,y\right)\right|}_{x=-x},\end{align}$$

where $\beta =\theta \left({\omega}_0\right)$ . Hereafter, the sub-indices ‘2’ and ‘4’ correspond to the two-grating (Figure 1(b)) and the four-grating (Figure 1(a)) compressors. The sub-index ‘4, 2’ denotes either the four-grating or the two-grating compressor. Without loss of generality, hereinafter we assume that $\overline{g}=\overline{w}=\overline{g_4}=\overline{w_4}=\overline{\boldsymbol{G}}=\overline{\mathbf{W}}=0$ . In addition, the functions ${h}_{\mathrm{gr},n}$ and ${h}_{\mathrm{wr},n}$ do not contain components linear with respect to $x$ and $y$ (wedges) that are equivalent to the rotation of the grating as a whole for ${h}_{\mathrm{gr},n}$ and to the changes in the groove density N uniformly over the entire grating surface for ${h}_{\mathrm{wr},n}$ . As shown in Ref. [Reference Khazanov21], all the components in $\mathbf{\nabla}{h}_{\mathrm{gr},n}\left(x,y\right)$ linear with respect to $x$ and $y$ can be effectively compensated for by rotating one grating, for example, G4. The same is true for $\mathbf{\nabla}{h}_{\mathrm{wr},n}\left(x,y\right)$ . If this is done, then the terms linear with respect to $x$ and $y$ should be subtracted from $\mathbf{\nabla}{h}_{\mathrm{gr},n}\left(x,y\right)$ and $\mathbf{\nabla}{h}_{\mathrm{wr},n}\left(x,y\right)$ in Equation (14). These terms correspond to the aberrations of ${h}_{\mathrm{gr},n}\left(x,y\right)$ and ${h}_{\mathrm{wr},n}\left(x,y\right)$ quadratic with respect to $x$ and $y$ , that is, to defocus, vertical astigmatism and oblique astigmatism.

Taking into account that $\left|\mathbf{\nabla}{h}_n\left(x,y\right)\right|\approx \frac{h_n\left(x,y\right)}{d}$ , where $d<\frac{L_{\mathrm{g}}}{2}$ is a typical transverse scale of ${h}_i\left(x,y\right)$ variation, and that ${L}_{\mathrm{g}}<L$ , we obtain the following:

(17) $$\begin{align}\frac{f{g}_4\left(x,y\right)}{\mathbf{FG}\left(x,y\right)}&\preccurlyeq \frac{\mathrm{sin}\beta {\mathrm{cos}}^2\beta }{2\left( \mathrm{cos}\alpha + \mathrm{cos}\beta \right) \mathrm{cos}\alpha}\frac{L_{\mathrm{g}}}{L}\ll 1,\nonumber\\\frac{u{w}_4}{\mathbf{UW}}&\preccurlyeq \frac{\mathrm{ctg}\left(\frac{\alpha -\beta }{2}\right){\mathrm{cos}}^3\beta }{2\left( \mathrm{cos}\alpha + \mathrm{cos}\beta \right) \mathrm{cos}\alpha}\frac{L_{\mathrm{g}}}{L}\ll 1,\end{align}$$

and the expression in Equation (10) reduces to the following:

(18) $$\begin{align}{\Psi}_4\left(x,y\right)=-\frac{\omega_0}{c}\left(\mathbf{FG}\left(x,y\right)+\mathbf{UW}\left( x\mathrm{cos}\Phi, y\right)\right).\end{align}$$

Let us now consider this case and use Equation (18) for 4OC. It should be noted, however, that the above approximation is violated if the quadratic components in ${h}_{\mathrm{gr},n}\left(x,y\right)$ and ${h}_{\mathrm{wr},n}\left(x,y\right)$ are significantly larger than all the others taken together. Then, if the quadratic distortions are compensated for by rotating the fourth grating^[ Reference Khazanov^21], the approximation $\left|\mathbf{\nabla}{h}_n\left(x,y\right)\right|\approx \frac{h_n\left(x,y\right)}{d}$ is invalid and Equation (10) must be used instead of Equation (18).

As seen from Equation (18), the focal intensity for 4OC is determined by the total value of the ${h}_{\mathrm{gr}}$ and ${h}_{\mathrm{wr}}$ gradients ( $\mathrm{functions}\kern0.24em \mathbf{G}\left(x,y\right)$ and $\mathbf{W}\left(x,y\right)$ ), where only gratings G2 and G3 are significant, whereas the contribution of G1 and G4 is negligible. Contrariwise, for 2OC, according to Equation (11), the gradients are of no importance, and the focal intensity is determined only by the total values of ${h}_{\mathrm{gr}}$ and ${h}_{\mathrm{wr}}$ (functions $g\left(x,y\right)$ and $w\left(x,y\right)$ ).

Analogous to Equation (17), we obtain $fg\left(x,y\right)\ll \mathbf{FG}\left(x,y\right)$ and $uw\left(x,y\right)\ll \mathbf{U}\boldsymbol{W}\left(x,y\right)$ . With allowance for Equations (11) and (18), this follows that a decrease in the Strehl ratio in 2OC is much smaller than in 4OC: $\left(1-{St}_2\right)\ll \left(1-{St}_2\right)$ , which is a significant advantage of the two-grating compressor along with its other merits^[ Reference Khazanov^21, Reference Smith, Erdogan and Erdogan^33, Reference Khazanov^41].

3 Discussion of results

As can be seen from Equation (8), one function ${\Psi}_{4,2}\left(x,y\right)$ is responsible for all distortions of all compressor gratings. It has the meaning of the effective phase (effective wavefront), which characterizes all imperfections of all compressor gratings. The decrease in the Strehl ratio $\left(1-{St}_{4,2}\right)$ is proportional to the squared root mean square (rms) of this phase and to the squared $\Delta \omega$ . Therefore, a decrease/increase in $\Delta \omega$ proportionally reduces/increases the requirements for the rms of both the surface of the gratings and the surfaces of the optics used for their writing. To determine the Strehl ratio it is sufficient to know only $\overline{\Psi_{4,2}^2\left(x,y\right)}$ , that is, the dispersion (rms squared) of the function ${\Psi}_{4,2}\left(x,y\right)$ , with averaging being performed according to Equation (9). The laser beam profile ${E}_0\left(x,y\right)$ affects ${St}_{4,2}$ only through this averaging. It is obvious that the flat-top profile is better than the Gaussian one, especially in the presence in ${\Psi}_{4,2}\left(x,y\right)$ of Zernike polynomials with large radial indices. The pulse spectrum profile has a similar effect: for a Gaussian spectrum $M\left(\mu =1\right)=0.5$ and for $\mu \ge 6,M\approx 0.32$ , that is, for a super-Gaussian spectrum, ${St}_{4,2}$ is larger than for a Gaussian one.

The function ${\Psi}_{4,2}\left(x,y\right)$ is determined by the total distortions of gratings G2 and G3 for 4OC (Equation (14)) and G1 and G2 for 2OC (Equation (13)). The rotation of one grating by 180 degrees around its normal changes the sign of the arguments of the functions ${h}_{\mathrm{gr}}\left(x,y\right)$ and ${h}_{\mathrm{wr}}\left(x,y\right)$ . In addition, with such a rotation, the angle between the wave vector of the incident wave and the $x^{\prime\prime }$ -axis changes from acute to obtuse or vice versa, that is, ${h}_{\mathrm{wr},n}\left(x,y\right)$ changes its sign^[ Reference Khazanov^23]. Thus, the rotation corresponds to the replacements:

(19) $$\begin{align}&{h}_{\mathrm{gr},n}\left(x,y\right)\to {h}_{\mathrm{gr},n}\left(-x,-y\right)\kern0.36em \mathrm{and}\nonumber\\&{h}_{\mathrm{wr},n}\left(x,y\right)\to -{h}_{\mathrm{wr},n}\left(-x,-y\right).\end{align}$$

In 2OC the gratings may be arranged in four non-equivalent ways: each grating may be placed in two ways. There are significantly more options in 4OC. The values of $\overline{\Psi_{4,2}^2\left(x,y\right)}$ and, hence, of ${St}_{4,2}$ will be different for different variants. Knowing ${h}_{\mathrm{gr},n}\left(x,y\right)$ and ${h}_{\mathrm{wr},n}\left(x,y\right)$ for each grating, for any compressor it is easy to choose the best option – the one that gives the smallest value of $\overline{\Psi_{4,2}^2\left(x,y\right)}$ .

The grating surface profiles ${h}_{\mathrm{gr},n}\left(x,y\right)$ are, as a rule, independent for different gratings. In contrast, it is reasonable to conjecture that ${h}_{\mathrm{wr},n}\left(x,y\right)={h}_{\mathrm{wr}}\left(x,y\right)$ , if the gratings are written by the same optics (see Figure 2). Then, from Equations (13) and (14) we obtain the following:

(20) $$\begin{align}w\left(x,y\right)&={h}_{\mathrm{wr}}\left(x,y\right)+{h}_{\mathrm{wr}}\left(-x,y\right),\nonumber\\ \mathbf{W}\left(x,y\right)&=-\mathbf{\nabla}{h}_{\mathrm{wr}}\left(-x,y\right)-\mathbf{\nabla}{h}_{\mathrm{wr}}\left(x,y\right),\end{align}$$

that is, the Zernike polynomials that are odd in $x$ will make a zero contribution to $w\left(x,y\right)$ , and even ones a zero contribution to $\mathbf{W}\left(x,y\right)$ (see Equation (16)). If the second grating is rotated by 180 degrees around its normal, then we have the following:

(21) $$\begin{align}w\left(x,y\right)&={h}_{\mathrm{wr}}\left(x,y\right)-{h}_{\mathrm{wr}}\left(x,-y\right),\nonumber\\ \mathbf{W}\left(x,y\right)&=-\mathbf{\nabla}{h}_{\mathrm{wr}}\left(-x,y\right)+\mathbf{\nabla}{h}_{\mathrm{wr}}\left(-x,-y\right),\end{align}$$

and the Zernike polynomials that are even in $y$ will make a zero contribution to $w\left(x,y\right)$ , and the odd ones to $\mathbf{W}\left(x,y\right)$ . From this it is clear that identical gratings do not allow for compensating for the imperfection of the grooves of each other, since in any case $w\left(x,y\right)\ne 0$ and $\mathbf{W}\left(x,y\right)\ne 0$ . However, $w\left(x,y\right)=0$ if any of the conditions

(22) $$\begin{align}&{h}_{\mathrm{wr},2}\left(x,y\right)=-{h}_{\mathrm{wr},1}\left(-x,y\right)\kern0.36em \mathrm{or}\nonumber\\ &{h}_{\mathrm{wr},2}\left(x,y\right)={h}_{\mathrm{wr},1}\left(x,-y\right)\end{align}$$

is met, and $\mathbf{W}\left(x,y\right)=0$ if any of the conditions

(23) $$\begin{align}\mathbf{\nabla}{h}_{\mathrm{wr},2}\left(x,y\right)&=-\mathbf{\nabla}{h}_{\mathrm{wr},1}\left(-x,y\right)\kern0.36em \mathrm{or}\nonumber\\\mathbf{\nabla }{h}_{\mathrm{wr},2}\left(-x,-y\right)&=\mathbf{\nabla}{h}_{\mathrm{wr},1}\left(-x,y\right)\end{align}$$

is fulfilled. Thus, to completely nullify the influence of an imperfect groove pattern for 2OC or 4OC, a pair of grooves satisfying Equation (22) or (23), respectively, should be written. By interchanging the beamsplitter (BS) and M3, as well as M1 and M2 (Figure 2), it is possible to change the sign of ${h}_{\mathrm{wr}},$ but this is insufficient to fulfill Equations (22) and (23).

Table 1 Compressor parameters.

Figure 3 Here, ${St}_{4, \mathrm{gr}}\left(\Sigma \right)$ (solid curves) and ${St}_{4, \mathrm{wr}}\left(\Sigma \right)$ (dashed curves) are plotted by Equation (25) for 4TC (red) and 4LC (blue) for the compressor parameters (see Table 1) proposed for XCELS (a) and SEL-100PW (b).

The Strehl ratio $St$ depends on (i) the total grating profile described by the functions $\mathbf{G}\left(x,y\right)$ and $g\left(x,y\right)$ and (ii) the total groove imperfection which, in turn, is determined by the total profile of the optics used for writing the grooves that is described by the functions $\mathbf{W}\left(x,y\right)$ and $w\left(x,y\right)$ . Compare the influence of these two reasons assuming that the profiles are uncorrelated, that is, $\overline{\mathbf{GW}}=\overline{gw}=0$ . Then, from Equations (8), (11) and (18) we find that the Strehl ratio $St$ is a product of the Strehl ratio caused by the grating surface imperfection ${St}_{\mathrm{gr}}$ and the Strehl ratio caused by the groove pattern imperfection ${St}_{\mathrm{wr}}$ :

(24) $$\begin{align}{St}_4={St}_{4, \mathrm{gr}}{St}_{4, \mathrm{wr}},\quad {St}_2={St}_{2, \mathrm{gr}}{St}_{2, \mathrm{wr}},\end{align}$$

(25) $$\begin{align}{St}_{4, \mathrm{gr}}&=1-M{\left(\frac{\Delta \omega }{\omega_0}\right)}^2\left({F}_x^2\overline{G_x^2}+{F}_y^2\overline{G_y^2}\right),\nonumber\\{St}_{4, \mathrm{wr}}&=1-M{\left(\frac{\Delta \omega }{\omega_0}\right)}^2\left({U}_x^2\overline{W_x^2}+{U}_y^2\overline{W_y^2}\right),\end{align}$$

(26) $$\begin{align}{St}_{2, \mathrm{gr}}&=1-M{\left(\frac{\Delta \omega }{\omega_0}\right)}^2{f}^2\overline{g^2},\nonumber\\{St}_{2, \mathrm{wr}} &=1-M{\left(\frac{\Delta \omega }{\omega_0}\right)}^2{u}^2\overline{w^2}.\end{align}$$

Note that for the $St$ close to unity, the contribution of these two factors to the reduction of the Strehl ratio is additive: $1- St\approx 1-{St}_{\mathrm{gr}}-{St}_{\mathrm{wr}}.$ Supposing that the quality of grating substrate polishing is the same as the quality of polishing the optics used for writing the grating, we have $\overline{g^2}=\overline{w^2}={\sigma}^2$ and $\overline{G_x^2}=\overline{W_x^2}=\overline{G_y^2}=\overline{W_y^2}={\Sigma}^2$ . We also assume that $x$ and $y$ are equivalent. Then it is readily found that

$$\begin{align*}\frac{1-{St}_{2, \mathrm{gr}}}{1-{St}_{2, \mathrm{wr}}}=\frac{1}{2} \mathrm{tg}\beta N{\lambda}_{\mathrm{wr}},\kern0.72em \frac{1-{St}_{4, \mathrm{gr}}}{1-{St}_{4, \mathrm{wr}}}=\frac{1}{2} \mathrm{ctg}\left(\frac{\alpha -\beta }{2}\right)N{\lambda}_{\mathrm{wr}}.\end{align*}$$

For typical compressor parameters, both ratios are <<1, that is, the imperfection of the optics used to write the grating exerts a greater influence. For example, for the 4TC and 4LC parameters proposed in Ref. [Reference Khazanov41] for the XCELS and SEL-100PW projects (Table 1), ${St}_{4, \mathrm{gr}}\left(\Sigma \right)$ and ${St}_{4, \mathrm{wr}}\left(\Sigma \right)$ are plotted in Figure 3, from which it is clear that ${St}_{4, \mathrm{gr}}\left(\Sigma \right)\approx {St}_{4, \mathrm{wr}}\left(\left(2.5{-}3\right)\Sigma \right)$ , that is, the requirements for the optics used to write the grating are approximately 2.5–3 times higher. The curves are plotted for ${\lambda}_{\mathrm{wr}}=413\;\mathrm{nm}$ ; for ${\lambda}_{\mathrm{wr}}=266\;\mathrm{nm}$ this coefficient will be even 1.55 times larger.

From Equation (15) it is clear that $u$ does not depend on N and the dependence of $\mathbf{U}$ on N is very weak. Consequently, ${St}_{\mathrm{wr},4}$ weakly depends on N, and ${St}_{\mathrm{wr},2}$ does not depend on it at all. Contrariwise, $f$ and $\mathbf{F}$ and, hence, ${St}_{\mathrm{gr}}$ , depend on N. In Figure 4, ${St}_{\mathrm{gr}}(N)$ and ${St}_{\mathrm{wr}}(N)$ are plotted for four compressor configurations: 4TC, 4LC, 2TC and 2LC. For the TC, $\gamma =0$ and the value of $\left|\alpha -{\alpha}_{\mathrm{L}}\right|$ for each N was chosen to be minimal, ensuring decoupling (grating G2 does not overlap the beam incident on grating G1). Analogously, for LC the angle $\gamma$ was chosen to be minimal and $\alpha ={\alpha}_{\mathrm{L}}$ . For all points in Figure 4 the distance between the gratings $L$ corresponds to the group velocity dispersion of 4.4 ps². For identical $\Sigma$ and $\sigma$ , the values of ${St}_{\mathrm{gr}}(N)$ differ little from unity. Therefore, for clarity, the curves for ${St}_{\mathrm{gr}}(N)$ are plotted a factor of 10 larger for $\Sigma$ (Figure 4(a)) and a factor of 5 larger for $\kern0.24em \sigma$ (Figure 4(b)) than for ${St}_{\mathrm{wr}}(N)$ . This once again emphasizes that the contribution of ${h}_{\mathrm{wr}}$ is much larger than that of ${h}_{\mathrm{gr}}$ . As expected, ${St}_{\mathrm{wr}}$ (green symbols) is virtually independent of N, except for a small drop at large N in Figure 4(a). For the same quality of the optics, ${St}_{\mathrm{gr}}(N)\approx 1$ and $St(N)\approx {St}_{\mathrm{wr}}(N)$ . However, if Equation (22) is met or quadratic components (defocus, vertical astigmatism and oblique astigmatism) dominate in ${h}_{\mathrm{wr}}$ , then ${St}_{\mathrm{wr}}(N)\approx 1$ and $St(N)\approx {St}_{\mathrm{gr}}(N)$ . In this case, as can be seen from Figure 4, for 4TC and 4LC it is more advantageous to have large N, and for 2TC and 2LC small N. The comparison of TC and LC (triangles versus squares) shows that $St$ is larger for TC, but the difference is insignificant, and for the two-grating compressor ${St}_{\mathrm{wr}}$ is the same for 2TC and 2LC (circles in Figure 4(b)).

Figure 4 Here, ${St}_{\mathrm{gr}}(N)$ (red) and ${St}_{\mathrm{wr}}(N)$ (green) for 4TC and 4LC are plotted by Equation (25) (a); 4TC and 2TC are plotted by Equation (25) and 2LC is plotted by Equation (26) (b). The incidence angles $\alpha$ and $\gamma$ as well as the distance between the gratings $L$ are described in the text. For clarity, the curves for ${St}_{\mathrm{gr}}(N)$ are plotted for larger values of distortion than for ${St}_{\mathrm{wr}}(N)$ : a factor of 10 for (а) and a factor of 5 for (b).