2 General expression for the power contrast ratio $\mathbb{C}\left(\boldsymbol{t}\right)$

Most lasers use the Treacy compressor^[ Reference Treacy ^{29 ]}, shown in Figure 1, which has three parameters: $N$ is the groove density, L is the distance between the gratings along the normal and α is the angle of incidence on the first grating. In the general case of an out-of-plane compressor^[ Reference Osvay and Ross ^{30 ]}, there is one more parameter γ that is the angle of incidence on the first grating in the plane orthogonal to the diffraction plane. The angle of reflection $\beta$ for the central frequency ${\omega}_0=c{k}_0=2\pi c/{\lambda}_0$ is determined from the expression for the grating $\sin\beta =-{\lambda}_0N/ \cos\gamma +\sin \alpha$ . An important special case of an out-of-plane compressor is the Littrow compressor^[ Reference Khazanov ^{31 ,} Reference Vyatkin and Khazanov ^{32 ]}, in which $\alpha ={\alpha}_\mathrm{L}$ , where ${\alpha}_\mathrm{L}$ is the Littrow angle. It is convenient to start the consideration with the field ${E}_0\left(\Omega, \boldsymbol{r}\right)$ ( $\Omega =\omega -{\omega}_0$ ) incident on an imperfect optical element that may be any diffraction grating G1–G4 and input and output optics M_in and M_out, as well as gratings and stretcher mirrors. We will search for the contrast $\mathbb{C}(t)$ caused by each element separately, assuming that all other elements are perfect. After reflection, the field takes the following form:

(2) $$\begin{align}{E}_1\left(\Omega, \boldsymbol{r}\right)={E}_0\left(\Omega, \boldsymbol{r}\right)+{E}_0\left(\Omega, \boldsymbol{r}\right)\left(\Theta \left(\boldsymbol{r}\right)+\mathcal{A}\left(\boldsymbol{r}\right)+ i\varphi \left(\boldsymbol{r}\right)\right),\end{align}$$

Figure 1 Compressor scheme. G1–G4, diffraction gratings; OAP, off-axis parabola; M_in, input optics; M_out, output optics. Dotted lines represent radiation scattered by G4, where scattered pulses lag behind the main one. Dashed lines represent radiation scattered by G3, where scattered pulses lag behind (green) or overtake (red) the main pulse, depending on the sign of ${k}_x$ .

where $\mathcal{A}\left(\boldsymbol{r}\right)\ll 1$ , $\varphi \left(\boldsymbol{r}\right)\ll 1$ and $\Theta \left(\boldsymbol{r}\right)$ are real, homogeneous, ergodic random functions (fields) characterizing the fluctuations of the field amplitude and phase, as well as the presence of obscurations: $\Theta \left(\boldsymbol{r}\right)=1$ inside and $\Theta \left(\boldsymbol{r}\right)=0$ outside the obscurations. Following Ref. [Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig23], we assume that the obscurations ‘absorb’ all incident laser light, and thus they affect only the amplitude of the incident beam and not its phase. The size and coordinates of the obscurations on the surface of the optical element are random variables. The second term in Equation (2) will be further called noise for brevity, and the noise energy will be taken to be much lower than the energy of the main pulse ${W}_0=\int {\left|{E}_0\left(\Omega, \boldsymbol{r}\right)\right|}^2\mathrm{d}\boldsymbol{r}\mathrm{d}\Omega$ :

(3) $$\begin{align}{W}_X&=\int {\left|X\left(\boldsymbol{r}\right){E}_0\left(\Omega, \boldsymbol{r}\right)\right|}^2\mathrm{d}\boldsymbol{r}\mathrm{d}\Omega\nonumber\\&={\sigma}_X^2{W}_0\ll {W}_0\kern0.84em \left(X=\Theta, \mathcal{A},\varphi \right).\end{align}$$

Since in this paper we are interested in small-scale intensity fluctuations, the characteristic spatial scale of ${E}_0\left(\boldsymbol{r}\right)$ is much larger than the characteristic spatial scale of $\Theta \left(\boldsymbol{r}\right)$ , $\mathcal{A}\left(\boldsymbol{r}\right)$ and $\varphi \left(\boldsymbol{r}\right)$ , that is, the correlation length of these functions is much smaller than the beam size. In other words, the spatial spectrum of the noise is much wider than the spectrum of the main beam. From ergodicity it follows that

(4) $$\begin{align}{\sigma}_X^2=<{X}^2\left(\boldsymbol{r}\right)>,\end{align}$$

with ${\sigma}_{\Theta}^2={S}_\mathrm{ob}/{S}_0$ , where ${S}_\mathrm{ob}$ is the total area of all obscurations and ${S}_0$ is the beam area. The angle brackets denote ensemble averaging. For clarity, $\Theta \left(\boldsymbol{r}\right)$ , $\mathcal{A}\left(\boldsymbol{r}\right)$ and $\varphi \left(\boldsymbol{r}\right)$ are shown in Figure 2. Equation (2) has the most general form and includes all possible imperfections of the optical element. Note that beam clipping, which also affects the contrast^[ Reference Khazanov ^{33 ]}, is not described by Equation (2) and is outside the scope of this paper. The functions and their Fourier transforms will be designated by the same letters, but with different arguments:

(5) $$\begin{align}{E}_j\left(\Omega, \boldsymbol{k}_{\boldsymbol{\perp}} \right)&=\frac{1}{2\pi}\int {E}_j\left(\Omega, \boldsymbol{r}\right){e}^{i\boldsymbol{k}_{\boldsymbol{\perp}} \boldsymbol{r}}\mathrm{d}\boldsymbol{r},\nonumber\\{E}_j\left(\Omega, \boldsymbol{r}\right) &=\frac{1}{2\pi}\int {E}_j\left(\Omega, \boldsymbol{\kappa} \right){e}^{-i\boldsymbol{k}_{\boldsymbol{\perp}} \boldsymbol{r}}\mathrm{d}\boldsymbol{k}_{\boldsymbol{\perp}},\end{align}$$

Figure 2 Schematic representation of random functions $\Theta \left(\boldsymbol{r}\right),\ \mathcal{A}\left(\boldsymbol{r}\right)\mathrm{and}\;\varphi \left(\boldsymbol{r}\right)$ .

and analogously for the temporal Fourier transform. Hereinafter, the range of integration, if not otherwise specified, is $\pm \infty$ . From Equations (2) and (5) we obtain the following:

(6) $$\begin{align}{E}_1\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)={E}_0\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)-\frac{1}{2\pi}\int \mathrm{d}\boldsymbol{r}{e}^{i{\boldsymbol{k}}_{\boldsymbol{\perp}}\boldsymbol{r}}{E}_0\left(\Omega, \boldsymbol{r}\right)\mathcal{A}\left(\boldsymbol{r}\right).\end{align}$$

The field on the focusing parabola (point E in Figure 1) up to which the field ${E}_1\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ in Equation (6) passes through an optical system containing a pair(s) of parallel gratings and a section(s) of free space of total length ${L}_\mathrm{f}$ is considered to be the output field ${E}_\mathrm{out}\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right).$ Regardless of the order in which these elements are passed, the field ${E}_\mathrm{out}\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ has the following form:

(7) $$\begin{align}{E}_\mathrm{out}\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)={E}_1\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right){e}^{i\Psi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)},\end{align}$$

where $\Psi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ is the sum of the phase ${\Psi}_\mathrm{f}={L}_\mathrm{f}\sqrt{k_0^2-{k}_{\boldsymbol{\perp}}^2}$ introduced by the free space of length ${L}_\mathrm{f}$ and (i) by two pairs of parallel diffraction gratings from point A to point B and from point C to point D for M_in or G1; or (ii) by one pair from point C to point D for G2 or G3; or (iii) there are no other terms for G4 or M_out. It is convenient to present the function $\Psi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ as a Taylor series at the point $\Omega =0$ , ${k}_x={k}_y=0$ ( ${k}_{x,y}$ are the components of the vector ${\boldsymbol{k}}_{\boldsymbol{\perp}}$ ) by extracting the term of the first power in $\Omega$ and designating all the other terms as $\Phi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ :

(8) $$\begin{align}\Psi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)=\Omega \tau \left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)+\Phi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right).\end{align}$$

The expressions for the phase introduced by the pair of parallel diffraction gratings ${\Psi}_\mathrm{p}\left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ , as well as for ${\psi}_\mathrm{a}^b$ derivatives of ${\Psi}_\mathrm{p}$ with respect to $\omega$ , ${k}_x$ , ${k}_y$ at the point $\Omega =0$ , ${k}_x={k}_y=0$ , also needed for the Taylor series, can be found in Refs. [Reference Khazanov34,Reference Kocharovskaya, Martyanov and Khazanov35]. Using them, one can obtain the expression for $\tau \left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ in the following form:

(9) $$\begin{align}\tau \left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)&=2{\tau}_x\frac{k_x}{k_0}+2{\tau}_y\frac{k_y}{k_0}+{t}_x\frac{k_x^2}{k_0^2}+{t}_y\frac{k_y^2}{k_0^2}+2{t}_{xy}\frac{k_x{k}_y}{k_0^2},\end{align}$$

(10) $$\begin{align}{\tau}_x&=\frac{1}{2}{k}_0{\psi}_{x\omega}^{\prime \prime }=\mp \frac{AL}{2c},\kern1.92em {\tau}_y=\frac{1}{2}{k}_0{\psi}_{y\omega}^{\prime \prime }=\mp \frac{GL}{2c},\end{align}$$

(11) $$\begin{align}{t}_x&=\frac{1}{2}{k}_0^2{\psi}_{xx\omega}^{\prime \prime \prime }=\frac{EL+{L}_\mathrm{f}}{2c},\nonumber\\{t}_y&=\frac{1}{2}{k}_0^2{\psi}_{yy\omega}^{\prime \prime \prime }=\frac{FL+{L}_\mathrm{f}}{2c},\nonumber\\{t}_{xy}&=\frac{1}{2}{k}_0^2{\psi}_{xy\omega}^{\prime \prime \prime }=\pm \frac{KL}{2c},\end{align}$$

(12) $$\begin{align}A&=\frac{\cos\alpha}{{\mathit{\cos}}^3\beta}\left( \sin\alpha - \sin\beta \right),\kern0.72em\nonumber\\ F&=\frac{1+\cos \left(\alpha +\beta \right)}{\cos\beta}+\frac{{\left( \sin\beta - \sin\alpha \right)}^2}{{\mathit{\cos}}^3\beta },\end{align}$$

(13) $$\begin{align}E&=\frac{{\mathit{\cos}}^2\alpha }{{\mathit{\cos}}^3\beta }+\frac{\cos \left(\alpha +\beta \right)}{\cos\beta}\nonumber\\&\quad+\frac{\sin\alpha - \sin\beta}{{\mathit{\cos}}^3\beta}\left( \sin\alpha -3 \sin\beta \frac{{\mathit{\cos}}^2\alpha }{{\mathit{\cos}}^2\beta}\right),\end{align}$$

(14) $$\begin{align}G&=\gamma \frac{{\left( \sin\beta - \sin\alpha \right)}^2}{{\mathit{\cos}}^3\beta},\nonumber\\K&=2 A\gamma \left(1-\frac{3}{2} \sin\beta \frac{\sin\alpha - \sin\beta}{{\mathit{\cos}}^3\beta}\right).\end{align}$$

As $\gamma \ll 1$ in the general case, terms of order ${\gamma}^2$ are omitted here. From here on we will restrict ourselves to the paraxial approximation, that is, we will consider ${k}_{x,y}$ of powers not higher than 2. Thus, we take into account diffraction in the paraxial approximation and all orders of dispersion, including the dispersion of the spatial chirp, since the term $\Phi \left(\Omega, {\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ includes terms with ${\Omega}^2$ , ${\Omega}^3$ , etc. Further, we assume that the grating pairs in the compressor are identical. This is not the case for an asymmetric compressor^[ Reference Khazanov ^{34 ,} Reference Huang and Kessler ^{36 –} Reference Liang, Du, Chen, Wang, Liu, Chen, Shena, Liu and Li ^{42 ]}, but we restrict consideration to a symmetric one. The sign ∓ in Equations (10) and (11) corresponds to the first and second pairs of gratings; therefore, for two pairs of gratings, the total values of ${t}_{xx}$ , ${t}_{yy}$ are doubled, and ${\tau}_x={\tau}_y={t}_{xy}=0.$ Note that, if α is equal to the Littrow angle, then $F=E$ . This is not the case for the Treacy compressor, but usually α is chosen as close to the Littrow angle as the decoupling condition allows (the beam incident on grating G1 should not overlap with the second grating). Hence, for the compressor, the values of F and E are close, for example, in the XCELS project^[ Reference Khazanov, Shaykin, Kostyukov, Ginzburg, Mukhin, Yakovlev, Soloviev, Kuznetsov, Mironov, Korzhimanov, Bulanov, Shaikin, Kochetkov, Kuzmin, Martyanov, Lozhkarev, Starodubtsev, Litvak and Sergeev ^{43 ]}  $F=1.025E$ and we can assume that ${\tau}_y\approx {\tau}_x$ . This condition is fulfilled exactly for the Littrow compressor^[ Reference Khazanov ^{31 ,} Reference Vyatkin and Khazanov ^{32 ]}.

The quantity $\tau \left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ has a simple physical meaning – it is the time delay of the noise component with wave vector ${\boldsymbol{k}}_{\boldsymbol{\perp}}$ relative to the wave with ${\boldsymbol{k}}_{\boldsymbol{\perp}}=0$ . Since $E>0$ and $EF>{K}^2$ , the last three terms in Equation (9) are a positively defined quadratic form, that is, the only negative terms in Equation (9) are the terms with ${\tau}_x$ and ${\tau}_y$ . If ${\tau}_x={\tau}_y=0$ , then all noise components lag behind the main wave, which means that at $t<0$ the contrast is zero (a perfect case). This is true for all optical elements in Figure 1, except for gratings G2 and G3.

The power ${P}_\mathrm{out}(t)$ required for calculating $\mathbb{C}(t)$ by the definition in Equation (1) is as follows:

(15) $$\begin{align}{P}_\mathrm{out}(t)=\int {\left|{E}_\mathrm{out}\left(t,{\boldsymbol{k}}_{\boldsymbol{\perp}}\right)\right|}^2\mathrm{d}{\boldsymbol{k}}_{\boldsymbol{\perp}}.\end{align}$$

By substituting Equation (8) into Equation (7) and performing the inverse time Fourier transform we obtain ${E}_\mathrm{out}\left(t,{\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ . By substituting it into Equation (15) and the result into Equation (1) we find $\mathbb{C}(t)$ . The transformations absolutely analogous to Ref. [Reference Khazanov22] yield the following:

(16) $$\begin{align}\mathbb{C}(t)&=\frac{t_\mathrm{p}}{W_02\pi}\int \mathrm{d}{\boldsymbol{k}}_{\boldsymbol{\perp}} \mathrm{PSD2}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)\nonumber\\&\quad\cdot\int \int \mathrm{d}\Omega \mathrm{d}\Omega^{\prime }{e}^{i\varPhi \left(\Omega, {k}_x,{k}_y\right)- i\varPhi \left({\Omega}^{\prime}, {k}_x,{k}_y\right)}{e}^{i\left({\Omega}^{\prime }-\Omega \right)\left(t-\tau \right)}\nonumber\\&\quad\cdot\int {E}_0\left(\Omega, \boldsymbol{r}\right){E}_0^{\ast}\left(\Omega^{\prime },\boldsymbol{r}\right)\mathrm{d}\boldsymbol{r},\kern0.48em\end{align}$$

where ${t}_\mathrm{p}={W}_0/{P}_{0, \mathrm{out}}(0)$ is the duration of the output pulse and

(17) $$\begin{align}\mathrm{PSD}2\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)= \mathrm{PSD}{2}_{\Theta}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)+ \mathrm{PSD}{2}_{\mathcal{A}}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)+ \mathrm{PSD}{2}_{\varphi}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right),\end{align}$$

where $\mathrm{PSD}{2}_X\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ is the two-dimensional PSD of the function $X$   $\left(X=\Theta, \mathcal{A},\varphi \right)$ defined by the following:

(18) $$\begin{align}\mathrm{PSD}{2}_X\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)={\left(2\pi \right)}^2\int \mathrm{ACF}_X\left(\boldsymbol{\rho} \right){e}^{i{\boldsymbol{k}}_{\boldsymbol{\perp}}\boldsymbol{\rho}}\mathrm{d}\boldsymbol{\rho },\end{align}$$

where $\mathrm{ACF}_X\left(\boldsymbol{\rho} \right)=<X\left(\boldsymbol{r}-\boldsymbol{\rho} \right)X\left(\boldsymbol{r}\right)>$ . Here we took into account that $\Theta \left(\boldsymbol{r}\right)$ , $\mathcal{A}\left(\boldsymbol{r}\right)$ and $\varphi \left(\boldsymbol{r}\right)$ are uncorrelated with each other. Note that the noise energy is, evidently, ${W}_{\mathrm{n}}={P}_{0, \mathrm{out}}(0)\int \mathbb{C}(t) \mathrm{d}t={W}_{\Theta}+{W}_{\mathcal{A}}+{W}_{\varphi }$ . The integration of Equation (16) with respect to $t$ gives $2\pi \delta \left({\Omega}^{\prime }-\Omega \right)$ , where $\delta$ is the Dirac delta function. Next, taking into account Equation (3) we obtain an obvious relation:

(19) $$\begin{align}{\sigma}_X^2=\int \mathrm{PSD}{2}_X\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)\mathrm{d}{\boldsymbol{k}}_{\boldsymbol{\perp}}.\end{align}$$

The $\mathrm{PSD}2$ definition in Equation (18) is known in the literature, but it is not the only one. It differs from the definition used, for example, in Refs. [Reference Bromage, Dorrer and Jungquist21,Reference Khazanov22,Reference Martyanov and Khazanov44], where there is no ${\left(2\pi \right)}^2$ multiplier. We chose that in Equation (18), since in this case Equation (19) for ${\sigma}_X^2$ has a simpler form. In the special case of pure phase distortions $\left(\mathcal{A}=\Theta =0\right),$ Equation (16) coincides with Equation (14) from Ref. [Reference Khazanov22].

By substituting Equation (11) into Equation (16) we change the variables:

(20) $$\begin{align}\boldsymbol{u}&=\left(\begin{array}{cc}1& {t}_{xy}/{t}_x\\ {}0& \sqrt{T_y/{t}_x}\end{array}\right)\left|\sqrt{\left|{t}_x\right|}\right|\frac{{\boldsymbol{k}}_{\boldsymbol{\perp}}}{k_0}\nonumber\\&\quad+\mathit{\operatorname{sign}}\left({t}_x\right)\left(\begin{array}{c}{\tau}_x/\sqrt{\left|{t}_x\right|}\\ {}\left({\tau}_y-\frac{t_{xy}}{t_x}{\tau}_x\right)/\sqrt{\left|{T}_y\right|}\end{array}\right).\end{align}$$

Further, supposing that $\mathrm{PSD}2$ is an isotropic function, that is, $\mathrm{PSD}2\left({k}_x,{k}_y\right)= \mathrm{PSD}2\left({k}_{\perp}\right)$ and passing from $\left({u}_x,{u}_y\right)$ to the polar coordinates $\left(u,\theta \right)$ , we obtain

(21) $$\begin{align}\mathbb{C}(t)&=\frac{k_0^2}{2}\frac{t_\mathrm{p}}{\sqrt{\left|{t}_x{T}_y\right|}}{\int}_0^{2\pi } \mathrm{PSD}2\left({k}_0\sqrt{g\left(\theta \right)}\right) \mathrm{d}\theta, \nonumber\\&\quad \mathrm{if}\;\frac{t}{t_\mathrm{c}}>-1;\qquad\mathrm{otherwise}\;\mathbb{C}=0,\end{align}$$

where

(22) $$\begin{align}{t}_\mathrm{c}=\frac{1}{T_y{t}_x}\left({t}_y{\tau}_x^2+{t}_x{\tau}_y^2-2{t}_{xy}{\tau}_y{\tau}_x\right),\end{align}$$

(23) $$\begin{align}g\left(\theta \right)&={\kappa}_{\tau}^2{\mathit{\cos}}^2\theta \frac{\left|{t}_\mathrm{c}\right|}{\left|{t}_x\right|}+{\kappa}_{\tau}^2{\mathit{\sin}}^2\theta \frac{\left|{t}_\mathrm{c}\right|}{\left|{t}_x\right|}\left({\left(\frac{t_{xy}}{t_x}\right)}^2+\frac{\left|{t}_x\right|}{\left|{T}_y\right|}\right)\nonumber\\&-2{q}_x{\kappa}_{\tau } \cos\theta \frac{\sqrt{\left|{t}_\mathrm{c}\right|}}{\sqrt{\left|{t}_x\right|}}+{q}_x^2+{q}_y^2\nonumber\\&-{\kappa}_{\tau}^2\frac{\left|{t}_\mathrm{c}\right|}{\sqrt{\left|{t}_x\right|}}\frac{t_{xy}}{t_x}\mathit{\sin}2\theta \frac{1}{\sqrt{\left|{T}_y\right|}}\nonumber\\&-2{\kappa}_{\tau } \sin\theta \frac{\sqrt{\left|{t}_\mathrm{c}\right|}}{\sqrt{\left|{T}_y\right|}}\frac{1}{T_y{t}_x^2}\left({\tau}_y\left({t}_x^2+{t}_{xy}^2\right)-{t}_{xy}{\tau}_x\left({t}_x+{t}_y\right)\right),\end{align}$$

(24) $$\begin{align}{T}_y&=\frac{t_x{t}_y-{t}_{xy}^2}{t_x},\kern0.96em {q}_x=\frac{t_y{\tau}_x-{t}_{xy}{\tau}_y}{T_y{t}_x},\nonumber\\{q}_y&=\frac{t_x{\tau}_y-{t}_{xy}{\tau}_x}{T_y{t}_x},\kern0.84em {\kappa}_{\tau }=\sqrt{1+\frac{t}{t_\mathrm{c}}}.\end{align}$$

Here we restrict consideration to the case ${t}_x{t}_y>0$ (the diffraction has the same sign along х and у), which is always fulfilled for the compressor, but not always for the stretcher^[ Reference Khazanov ^{22 ]}, and take into account that ${t}_x{t}_y-{t}_{xy}^2>0$ (i.e., ${T}_y$ has the same sign as ${t}_x$ ). Note that only ${\kappa}_{\tau }$ is time dependent.

Next, we take into account that $\gamma \ll 1$ and, consequently, $\frac{t_{xy}}{t_x}\sim \gamma \ll 1$ and $\frac{\tau_y}{t_x}\sim \gamma \ll 1$ . By expanding $\mathbb{C}\left(t,\gamma \right)$ in a Taylor series near the point $\gamma =0$ it is easy to show that the term proportional to $\gamma$ vanishes. Consequently, the difference in the contrast $\mathbb{C}\left(t,\gamma \right)$ for an out-of-plane compressor from the contrast $\mathbb{C}\left(t,\gamma =0\right)$ for a plane compressor reduces to corrections of the order of ${\gamma}^2$ . For typical values of $\gamma =10,\dots, 15$ degrees, the correction will be about 5%, which is negligibly small for contrast. Thus, PCR $\mathbb{C}(t)$ does not depend on γ and we can use the expressions for a plane compressor. In other words, the contrast of an out-of-plane compressor practically does not differ from the contrast of a plane one. At $\gamma =0,$  Equation (21) strictly passes into Equation (19) from Ref. [Reference Khazanov22] (taking into consideration that the definition of $\mathrm{PSD}2$ in Ref. [Reference Khazanov22] differs from Equation (18) by the multiplier ${\left(2\pi \right)}^2$ ). Therefore, the results obtained in Ref. [Reference Khazanov22] for phase noise are valid in the general case if $\mathrm{PSD}{2}_{\varphi}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ is replaced by $\mathrm{PSD}2\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ in Equation (17). In particular, if we take into account that ${t}_x\approx {t}_y={t}_\mathrm{d}$ , then Equation (21) takes the following form:

(25) $$\begin{align}\mathbb{C}(t)&=\frac{k_0^2}{2}\frac{t_\mathrm{p}}{\left|{t}_\mathrm{d}\right|}{\int}_0^{2\pi } \mathrm{PSD}2\left({k}_0\left|\frac{\tau_x}{t_\mathrm{d}}\right|\sqrt{\kappa_{\tau}^2+1-2{\kappa}_{\tau } \cos\theta}\right) \mathrm{d}\theta,\nonumber\\&\quad \mathrm{if}\;\frac{t}{t_\mathrm{c}}>-1;\qquad\mathrm{otherwise}\;\mathbb{C}=0.\end{align}$$

This expression is significantly simplified in two important particular cases. Firstly, for G2 and G3 gratings, the allowance for diffraction $\left({t}_\mathrm{d}\ne 0\right)$ leads to the emergence of the cut-off time ${t}_\mathrm{c}$ and to a slight contrast asymmetry, which can be neglected, then

(26) $$\begin{align}{\mathbb{C}}_\mathrm{s}(t)&=\frac{k_0}{2}\frac{t_\mathrm{p}}{\tau_x} \mathrm{PSD}1\left(\frac{t{k}_0}{2{\tau}_x}\right),\nonumber\\& \quad \mathrm{if}\;\frac{t}{t_\mathrm{c}}>-1; \qquad \mathrm{otherwise}\;{\mathbb{C}}_\mathrm{s}=0,\end{align}$$

where $\mathrm{PSD}1\left({k}_x\right)$ is a one-dimensional PSD function related to $\mathrm{PSD}2\left({k}_{\perp}\right)$ as follows^[ Reference Khazanov, Kochetkov and Silin ^{45 ]}:

(27) $$\begin{align}\mathrm{PSD}1\left({k}_x\right)&=2\underset{k_x}{\overset{\infty }{\int }}\frac{\mathrm{PSD}2\left({k}_{\perp}\right)}{\sqrt{k_{\perp}^2-{k}_x^2}}{k}_{\perp }\mathrm{d}{k}_{\perp},\nonumber\\\mathrm{PSD}2\left({k}_{\perp}\right)&=\frac{-1}{\pi}\underset{\kappa }{\overset{\infty }{\int }}\frac{1}{\sqrt{k_x^2-{k}_{\perp}^2}}\frac{\mathrm{d}\mathrm{PSD}1\left({k}_x\right)}{\mathrm{d}{k}_x}\mathrm{d}{k}_x,\end{align}$$

with

(28) $$\begin{align}{\sigma}^2=\int \mathrm{PSD}1\left({k}_x\right)\mathrm{d}{k}_x.\end{align}$$

Secondly, there is no space chirp ${\tau}_x={\tau}_y=0$ for all optical elements in Figure 1, except for gratings G2 and G3; for them, the contrast ${\mathbb{C}}_\mathrm{d}(t)$ is determined only by the diffraction:

(29) $$\begin{align}{\mathbb{C}}_\mathrm{d}(t)&=\pi {k}_0^2\frac{t_\mathrm{p}}{\left|{t}_\mathrm{d}\right|} \mathrm{PSD}2\left({k}_0\sqrt{\frac{t}{t_\mathrm{d}}}\right),\kern0.24em \nonumber\\& \quad \mathrm{if}\;\frac{t}{t_\mathrm{d}}>0;\qquad \mathrm{otherwise}\;{\mathbb{C}}_\mathrm{d}=0.\end{align}$$

If ${t}_\mathrm{d}>0$ , then ${\mathbb{C}}_\mathrm{d}\left(t<0\right)=0$ , which explains the frequently observed^[ Reference Webb, Dorrer, Bahk, Jeon, Roides and Bromage ^{5 ,} Reference Hooker, Tang, Chekhlov, Collier, Divall, Ertel, Hawkes, Parry and Rajeev ^{15 –} Reference Lu, Zhang, Li and Leng ^{18 ,} Reference Wang, Liu, Lu, Chen, Long, Li, Chen, Chen, Bai, Li, Peng, Liu, Wu, Wang, Li, Xu, Liang, Leng and Li ^{46 ,} Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul ^{47 ]} experimental asymmetry of the contrast: at $t<0$ (pre-pulse) it is always weaker (better) than at $t>0$ (post-pulse), since all optical elements contribute to the contrast at $t>0$ , while only gratings G2 and G3 contribute to it at both $t<0$ and $t>0$ . In Ref. [Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul47] the authors experimentally proved that the contrast asymmetry cannot be explained by amplified spontaneous emission , the Raman effect, Kerr-lens mode-locking or Kerr nonlinearity in Ti:sapphire amplifiers. The hypothesis^[ Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul ^{47 ]} that phonons play a key role as Ti:sapphire lasers are vibronic contradicts recent experiments^[ Reference Webb, Dorrer, Bahk, Jeon, Roides and Bromage ^{5 ,} Reference Wang, Liu, Lu, Chen, Long, Li, Chen, Chen, Bai, Li, Peng, Liu, Wu, Wang, Li, Xu, Liang, Leng and Li ^{46 ]} with a fully OPCPA laser where the contrast asymmetry was measured. The measured post-pulse in Ref. [Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul47] did not depend on the stretching factor or on the dispersive elements (gratings or prisms), and the authors concluded that scattering on the diffraction gratings also has no impact on asymmetry of the contrast. As seen from Refs. [Reference Bonod and Neauport26,Reference Treacy29], it is true only for the scattering on the second and third gratings, but not for the scattering on all other optics, including optics in the contrast measurement beamline. Thus, this scattering is the only way to explain the contrast asymmetry. Note that it cannot be explained by the earlier developed^[ Reference Roeder, Zobus, Brabetz and Bagnoud ^{19 –} Reference Bromage, Dorrer and Jungquist ^{21 ]} diffraction-free approximation, in which $\mathbb{C}(t)$ is an even function.

In the case of negative diffraction $({t}_\mathrm{d}<0)$ , the time t changes its sign to the opposite one and the contrast ${\mathbb{C}}_\mathrm{d}(t)$ is nonzero at negative times. This is possible if the scheme comprises a telescope transferring the image of the optical element beyond the focusing parabola. In addition, the contrast meter is usually located not directly in the powerful beam, but rather in its weakened replica. The meter also receives the radiation scattered throughout the optics located in the measuring path. Its contribution to the PCR is also described by Equation (29), where ${t}_\mathrm{d}={L}_\mathrm{m}/2c$ and ${L}_\mathrm{m}$ is the distance from the optical element in the path to the meter. If ${L}_\mathrm{m}>0$ , then the measured contrast will be higher than the real one at $t>0$ . However, if there is an image transfer in the path, then ${L}_\mathrm{m}$ may be less than zero, and the contrast measured will be higher than the real one at less than $0$ ; see Equation (29).

Equations (26) and (29) clearly show the $\mathrm{PSD}1\left({k}_x\right)$ mapping on ${\mathbb{C}}_\mathrm{s}(t)$ and $\mathrm{PSD}2\left({k}_{\perp}\right)$ on ${\mathbb{C}}_\mathrm{d}(t)$ , with ${k}_x=\frac{t{k}_0}{2{\tau}_x}$ in the first case and ${k}_{\perp }={k}_0\sqrt{\left|\frac{t}{t_\mathrm{d}}\right|}$ in the second case. Correspondingly, the condition of paraxial approximation ${k}_{x,\perp}\ll {k}_0$ constrains the obtained results to the conditions $\left|t\right|\ll 2\left|{\tau}_x\right|$ for ${\mathbb{C}}_\mathrm{s}(t)$ and $\sqrt{\left|t\right|}\ll \sqrt{\left|{t}_\mathrm{d}\right|}$ for ${\mathbb{C}}_\mathrm{d}(t)$ . Further investigation of $\mathbb{C}(t)$ , including quantitative comparison with ${\mathbb{C}}_\mathrm{d}(t)$ and ${\mathbb{C}}_\mathrm{s}(t)$ , is possible only for the functional forms $\mathrm{PSD}{2}_{\Theta}, \mathrm{PSD}{2}_{\mathcal{A}}\kern0.36em \mathrm{and}\ \mathrm{PSD}{2}_{\varphi}$ .

3 Power spectral density functional form PSD2 for obscurations on the grating/mirror surface

To find $\mathrm{PSD}{2}_{\Theta}\left({k}_{\perp}\right)$ for the field reflected from a surface with obscuration, we assume that the obscurations do not overlap each other and are shaped as a circle with coordinates of the centre ${\boldsymbol{R}}_{m}$ and radius ${w}_{m}$ :

(30) $$\begin{align}\Theta \left(\boldsymbol{r}\right)=\sum \limits_{m=1}^M\Pi \left(\frac{\boldsymbol{r}-{\boldsymbol{R}}_{m}}{w_{m}}\right),\end{align}$$

where $\Pi (x)=1\;\mathrm{if}\;\left|x\right|<1,\mathrm{otherwise}\;\Pi =0$ , and $M\gg 1$ is the number of obscurations. Here, ${\boldsymbol{R}}_m$ and ${w}_m$ are random quantities, with ${\boldsymbol{R}}_m$ being uniformly distributed over the beam aperture and ${w}_m$ having a probability density $f(w)$ . It is obvious that ${S}_\mathrm{ob}=\pi \sum_{m=1}^M{R}_{m}^2$ . The squared modulus of the noise spectrum ${S}_{\Theta}$ is an incoherent sum of the squared moduli of the spectra ${S}_m$ of flat-top beams having radius ${w}_m$ (Airy functions):

(31) $$\begin{align}{S}_{\Theta}\left({k}_{\perp}\right)=\sum \limits_{m=1}^M{S}_{m}={I}_0\sum \limits_{m=1}^M{w}_{m}^4\frac{J_1^2\left({k}_{\perp }{w}_{m}\right)}{{\left({k}_{\perp }{w}_{m}\right)}^2},\end{align}$$

where ${J}_1$ is the Bessel function and ${I}_0={\left|{E}_0\right|}^2$ is the incident beam intensity assumed for simplicity to be equal for all obscurations. Here we sum the spectra incoherently, since the spectral phase of each obscuration is random and the sum of a large number of terms with random phase is zero. As a consequence, ${S}_{\Theta}$ does not depend on ${\boldsymbol{R}}_{m}$ . From the definition of PSD2 in Equation (18) and of the Fourier spectrum in Equation (5), with allowance for ergodicity we obtain $\mathrm{PSD}{2}_{\Theta}\left({k}_{\perp}\right)=\frac{<{S}_{\Theta}\left({k}_{\perp}\right)>}{I_0{S}_0}$ . By averaging Equation (31) we find the following:

(32) $$\begin{align}\mathrm{PSD}{2}_{\Theta}\left({k}_{\perp}\right)=\frac{\sigma_{\Theta}^2}{\pi {k}_{\perp}^2}\frac{\underset{{w}_\mathrm{min}}{\overset{{w}_\mathrm{max}}{\int }}{w}^2{J}_1^2\left({k}_{\perp }w\right)f(w) \mathrm{d}w}{\underset{w_\mathrm{min}}{\overset{w_\mathrm{max}}{\int }}{w}^2f(w) \mathrm{d}w},\end{align}$$

which, with Equation (27) taken into account, follows

(33) $$\begin{align}\mathrm{PSD}{1}_{\Theta}\left({k}_x\right)=\frac{\sigma_{\Theta}^2}{\pi {k}_x^2}\frac{\underset{{w}_\mathrm{min}}{\overset{{w}_\mathrm{max}}{\int }} wf(w){\boldsymbol{H}}_{1}\left(2w{k}_x\right) \mathrm{d}w}{\underset{w_\mathrm{min}}{\overset{w_\mathrm{max}}{\int }}{w}^2f(w) \mathrm{d}w},\end{align}$$

where ${\boldsymbol{H}}_{1}$ is the Struve function and ${w}_{\mathit{\min},\mathit{\max}}$ are the minimum and maximum radii of the obscurations. Obviously, $2{w}_\mathrm{min}\gg {\lambda}_0$ , since the paraxial approximation does not hold otherwise. Similarly, the obscurations with radius $w$ commensurate with the beam radius ${R}_0$ cannot exist, since in this case the condition ${W}_{\Theta}\ll {W}_0$ is violated. This imposes the constraint ${w}_\mathrm{max}\ll {R}_0$ . In practice, the quantities ${w}_{\mathit{\min},\mathit{\max}}$ may have even more stringent constraints. Note that Equations (32) and (33) satisfy Equations (19) and (28) regardless of the form of the function $f(w)$ .

Figure 3 $\mathrm{PSD}{2}_{\Theta}$ is normalized to $\frac{\sigma^2{w}_\mathrm{min}^4}{\pi <{w}^2>}$ as a function of $\kappa ={k}_{\perp }{w}_\mathrm{min}$ for $\xi =3.9$ (a) and $\xi =3.1$ (b). Dotted curves represent exact Equation (36) values for Z = 100 (grey) and Z = 10 (black); dashed curves represent approximate Equation (37) values for Z = 100 (pink) and Z = 10 (red).

Next, it is necessary to substitute into Equations (32) and (33)  $f(w)$ of a specific form, which may be quite complex. For example, for the optics of the standard indicated in Ref. [48], $f(w)$ has the form $\log f(w)\sim {(\log w)}^2$ . We will restrict our consideration to the power form:

(34) $$\begin{align}f(w)={A}_1\frac{1}{w^{\xi }},\end{align}$$

given in Ref. [Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig23] for surface defects of optical elements after their use in a high-power laser facility. Under the almost always met condition ${w}_\mathrm{max}^{\xi -1}\gg {w}_\mathrm{min}^{\xi -1}$ , from $\underset{w_\mathrm{min}}{\overset{w_\mathrm{max}}{\int }}f(w) \mathrm{d}w=1$ we obtain ${A}_1=\left(\xi -1\right){w}_\mathrm{min}^{\xi -1}$ , that is, $f(w)$ does not depend on ${w}_\mathrm{max}$ . The data reported in Ref. [Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig23] correspond to $\xi =3.9$ and ${w}_\mathrm{min}\approx 0.025\;\mathrm{mm}$ . Before using the optical elements in the high-power laser facility, the distribution function $f(w)$ is also defined by Equation (34) with $\xi =3.9$ , but the laser damage was 7.8 times less^[ Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig ^{23 ,} Reference Stowers, Horvath, Menapace, Burnham and Letts ^{49 ]}. In what follows, we will assume $\xi$ and ${w}_\mathrm{min}$ to be arbitrary constants and use the above values only for constructing specific plots in Section 6. The moments $<{w}^n>$ are readily calculated from Equation (34):

(35) $$\begin{align}<{w}^n>=\underset{w_\mathrm{min}}{\overset{w_\mathrm{max}}{\int }}{w}^nf(w) \mathrm{d}w={w}_\mathrm{min}^n\frac{\xi -1}{\xi -n-1}\left(1-{Z}^{n-\xi +1}\right),\end{align}$$

where $Z={w}_\mathrm{max}/{w}_\mathrm{min}$ . For $\xi =n+1$ , Equation (35) is valid within the limit $\xi \to n+1$ . On substituting Equation (34) into Equations (32) and (33) and passing to dimensionless quantities $\kappa ={k}_{\perp }{w}_\mathrm{min}$ and $K={k}_x{w}_\mathrm{min}$ , we obtain the following:

(36) $$\begin{align}\frac{\pi <{w}^2>}{\sigma_{\Theta}^2{w}_\mathrm{min}^4} \mathrm{PSD}{2}_{\Theta}\left(\kappa \right)&=\frac{\xi -1}{\kappa^{5-\xi }}\underset{\kappa }{\overset{Z\kappa}{\int }}\frac{J_1^2(u)}{u^{\xi -2}} \mathrm{d}u,\nonumber\\\frac{\pi <{w}^2>}{\sigma_{\Theta}^2{w}_\mathrm{min}^3} \mathrm{PSD}{1}_{\Theta}(K)&=\frac{\xi -1}{K^{4-\xi }}\underset{K}{\overset{KZ}{\int }}\frac{{\boldsymbol{H}}_{1}(2u)}{u^{\xi -1}} \mathrm{d}u.\end{align}$$

Both integrals in Equation (36) are expressed only through the generalized hypergeometric function $_2{F}_3$ , which complicates further analytical analysis. The expression in Equation (36) can be significantly simplified in three special cases: for ${k}_{\perp }{w}_\mathrm{max}\ll 1$ and ${k}_{\perp }{w}_\mathrm{min}\gg 1$ , the functions ${J}_1$ and ${\boldsymbol{H}}_{1}$ may be replaced by their asymptotic forms, and for ${w}_\mathrm{max}^{-1}\ll {k}_{\perp, x}\ll {w}_\mathrm{min}^{-1}$ , the integration limits may be replaced by 0 and $\infty$ . In all special cases, a power-law dependence is obtained: in three sections of the spectrum ${k}_{\perp, x}{w}_\mathrm{max}\ll 1$ , ${w}_\mathrm{max}^{-1}\ll {k}_{\perp, x}\ll {w}_\mathrm{min}^{-1}$ and ${k}_{\perp, x}{w}_\mathrm{min}\gg 1,$ $\mathrm{PSD}{2}_{\Theta}$ decreases in a power-law manner with powers $0,5-\xi\;\mathrm{and}\;3$ , and $\mathrm{PSD}{1}_{\Theta}$ with powers $0,4-\xi\;\mathrm{and}\;2$ . This is demonstrated in Figure 3 where $\mathrm{PSD}{2}_{\Theta}\left({k}_{\perp}\right)$ in Equation (36) is plotted. The $\mathrm{PSD}1\left({k}_x\right)$ curves look similar. From Figure 3 it can be seen that $\mathrm{PSD}{2}_{\Theta}\left({k}_{\perp}\right)$ is well approximated by the following expression:

(37) $$\begin{align}\frac{\pi <{w}^2>}{\sigma_{\Theta}^2{w}_\mathrm{min}^4} \mathrm{PSD}{2}_{\Theta}\left(\kappa \right)&\approx \frac{1}{4}\frac{\xi -1}{\xi -5}\frac{1-{Z}^{5-\xi }}{{\left(1+{\kappa}^2/{\kappa}_\mathrm{L}^2\right)}^{\frac{5-\xi }{2}}}{e}^{-{\kappa}^2}\nonumber\\&\quad+\frac{1}{\pi}\frac{\xi -1}{\xi -2}\frac{1-{Z}^{2-\xi }}{\kappa^3}\left(1-{e}^{-{\kappa}^4}\right),\end{align}$$

where

(38) $$\begin{align}{\kappa}_\mathrm{L}=2{\left(\frac{1}{1-{Z}^{5-\xi }}\frac{\xi -5}{\xi -1}\frac{\;\varGamma \left(\xi -2\right)\varGamma \left(\frac{5-\xi }{2}\right)}{\varGamma^3\left(\frac{\xi -1}{2}\right)}\right)}^{\frac{1}{5-\xi }}.\end{align}$$

Compare the black and grey dotted curves with the red and pink dashed curves in the figure. If ${Z}^{5-\xi}\gg 1$ , which is true for $\xi =3.9$ , then ${\kappa}_\mathrm{L}$ is proportional to ${Z}^{-1}$ , that is, to ${w}_\mathrm{max}^{-1}$ (see Equation (38)). At the frequencies ${k}_{\perp}\gg {w}_\mathrm{max}^{-1}$ , the product $\left(<{w}^2> \mathrm{PSD}{2}_{\Theta}\right)$ does not depend on Z – the grey and black dotted curves in Figure 3 coincide. Since the low frequencies ${k}_{\perp}\leq {w}_\mathrm{max}^{-1}$ contribute only to contrast at small times (see Section 2), $Z$ affects the contrast only slightly.

4 Power spectral density functional form $\mathbf{PSD}_{\mathbf{r}}$ for roughness of the grating/mirror surface

The surface profile $h\left(\boldsymbol{r}\right)$ is characterized by the function $\mathrm{PSD}{2}_\mathrm{h}\left({k}_{\perp}\right)$ related to the PSD of the introduced phase fluctuations $\mathrm{PSD}{2}_\mathrm{r}\left({k}_{\perp}\right)$ by the following:

(39) $$\begin{align}\mathrm{PSD}{2}_\mathrm{r}\left({k}_{\perp}\right)={B}_0{k}_0^2 \mathrm{PSD}{2}_\mathrm{h}\left({k}_{\perp}\right),\end{align}$$

where ${B}_0={\left( \cos\alpha + \cos\beta \right)}^2$ for the grating^[ Reference Khazanov ^{50 ]} and ${B}_0=4{\mathit{\cos}}^2\alpha$ for the mirror. Much attention in the literature has been paid to measuring $\mathrm{PSD}{1}_\mathrm{h}\left({k}_x\right)$ ^[ Reference Webb, Dorrer, Bahk, Jeon, Roides and Bromage ^{5 ,} Reference Roeder, Zobus, Major and Bagnoud ^{11 ,} Reference Ranc, Blanc, Lebas, Martin, Zou, Mathieu, Radier, Ricaud, Druon and Papadopoulos ^{14 ,} Reference Roeder, Zobus, Brabetz and Bagnoud ^{19 ,} Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig ^{23 ,} Reference Khazanov, Kochetkov and Silin ^{45 ,} Reference Church ^{51 –} Reference Camus, Coic, Blanchot, Bouillet, Lavastre, Mangeant, Rouyer and Néauport ^{55 ]}. In general, it behaves differently in different spectral ranges for different optical elements. In such a case, it is necessary to substitute the experimental data into Equations (39) and (27) and the result into Equation (25). At medium and high spatial frequencies, $\mathrm{PSD}{2}_\mathrm{h}$ often has a power-law form, and at low frequencies it is equal to a constant. Therefore, $\mathrm{PSD}{2}_\mathrm{h}$ is well approximated by a Lorentz-type spectrum^[ Reference Bromage, Dorrer and Jungquist ^{21 ,} Reference Khazanov, Kochetkov and Silin ^{45 ,} Reference Harvey, Schröder, Choi and Duparré ^{56 ]} with correlation length $1/{k}_\mathrm{h}$ , surface profile dispersion ${\sigma}_\mathrm{h}^2$ and exponent $a>2$ . Then

(40) $$\begin{align}\mathrm{PSD}{2}_\mathrm{r}\left({k}_{\perp}\right)&=\frac{a-2}{2\pi }{\sigma}_\mathrm{h}^2{B}_0{k}_0^2\frac{k_\mathrm{h}^{a-2}}{{\left({k}_\mathrm{h}^2+{k}_{\perp}^2\right)}^{a/2}},\kern0.72em\nonumber\\\mathrm{PSD}{1}_\mathrm{r}\left({k}_{\boldsymbol{x}}\right)&=\frac{\varGamma \left(\frac{a-1}{2}\right)}{\sqrt{\pi}\varGamma \left(\frac{a-2}{2}\right)}{\sigma}_\mathrm{h}^2{B}_0{k}_0^2\frac{k_\mathrm{h}^{a-2}}{{\left({k}_\mathrm{h}^2+{k}_x^2\right)}^{\left(a-1\right)/2}}.\end{align}$$

Usually, $2<a<4$ . A frequently encountered value is $a\approx 2.55$ , including in Ref. [Reference Hu, Wan, Jiang, Gu, Zhang, Jin, Liu, Zhao, Cao, Wei and Shao54], where $\mathrm{PSD}{1}_\mathrm{h}\left({k}_{\perp}\right)$ was measured for an off-axis parabola used for writing holographic gratings.

5 Power spectral density functional form for obscuration and roughness of the optics used for writing gratings

When writing a holographic grating, even with a perfectly plane substrate surface, the grating is not ideal due to the non-equidistance and non-parallelism of the grooves, as well as due to groove profile fluctuations.

The non-equidistance and non-parallelism of the grooves are determined by spatial fluctuations in the phase difference of two writing beams and give rise to fluctuations in the phase of the beam reflected from the grating $\Delta \varphi \left(\boldsymbol{r}\right)\ne 0$ ^[ Reference Bienert, Röcker, Dietrich, Graf and Ahmed ^{25 –} Reference Kochetkov, Shaykin, Yakovlev, Khazanov, Cheplakov, Wang, Jin, Liu and Shao ^{28 ]}, without affecting the reflection coefficient: $\mathcal{A}\left(\boldsymbol{r}\right)=~0$ . For a holographic grating this phase is equal to the phase of the writing field ${\phi}_\mathrm{wr}$ scaled along the x-axis^[ Reference Khazanov ^{27 ]}: $\Delta \varphi \left(\boldsymbol{r}\right)={\phi}_\mathrm{wr}\left( x\cos\Phi, y\right)$ . Here, $\Phi$ is the angle of incidence on the substrate (Figure 4) and $\sin\Phi =N{\lambda}_\mathrm{wr}/2$ . The phase $\Delta \varphi \left(\boldsymbol{r}\right)$ is added to the phase $\varphi \left(\boldsymbol{r}\right)$ associated with the non-flat surface of the grating that was discussed in Section 4. Since these phases are uncorrelated, their PSDs are summed. Considering that $\cos\Phi$ is usually close to unity, we will assume that the PSD functions for $\Delta \varphi$ and ${\phi}_\mathrm{wr}$ are equal and designate them as $\mathrm{PSD}{2}_\mathrm{phase}$ .

Figure 4 Scheme of writing holographic diffraction grating by two laser beams. BS, beamsplitter; M1–M5, mirrors; SF, spatial filters.

The fluctuations in the groove shape are caused by spatial fluctuations in the intensity of the radiation used to write the grating and give rise to the fluctuations in the modulus of the grating reflectivity $\mathcal{A}\left(\boldsymbol{r}\right)\ne 0$ ^[ Reference Koch, Lehr and Glaser ^{24 ]}, without affecting its phase: $\Delta \varphi \left(\boldsymbol{r}\right)=0$ . Rigorous calculation of the PSD of the grating reflectivity fluctuations $\mathrm{PSD}{2}_{\mathcal{A}}$ is a complex problem that requires knowledge of the exact relationship between these fluctuations and the intensity fluctuations of the writing field ${\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|}^2$ . However, taking into account the smallness of both fluctuations relative to their mean values, we will assume that they are proportional to each other. Then $\mathrm{PSD}{2}_{\mathcal{A}}= q\mathrm{PSD}{2}_\mathrm{dose}$ , where $q>0$ is a dimensionless coefficient and $\mathrm{PSD}{2}_\mathrm{dose}$ is $\mathrm{PSD}2$ for the radiation intensity on the substrate normalized to the average value. Now we need to find $\mathrm{PSD}{2}_\mathrm{phase}$ and $\mathrm{PSD}{2}_\mathrm{dose}$ .

Since all dynamic noises are averaged during grating exposure, only the ‘frozen’ noises caused by the imperfection of the transport optics are significant. Obviously, the fewer optical elements are used, the better. A scheme for writing a holographic grating that is optimal from this point of view is presented in Figure 4 ^[ Reference Hu, Wan, Jiang, Gu, Zhang, Jin, Liu, Zhao, Cao, Wei and Shao ^{54 ]}. Since the spatial filter ensures good filtering of high-frequency noise, only the transport optics located after the filters are of fundamental importance. The scheme in Figure 4 contains a minimum number of such elements – two off-axis parabolic mirrors (OAPs). In the classical work^[ Reference Boyd, Britten, Decker, Shore, Stuart, Perry and Li ^{57 ]} and other^[ Reference Koch, Lehr and Glaser ^{24 ]} papers, collimators are additionally used, which increases fluctuations. When reflecting the beam ${E}_\mathrm{wr}\left(\boldsymbol{r}\right)$ these two OAPs introduce into it both amplitude noise caused by scratches and defects (obscuration) and phase noise caused by the imperfection/roughness of the surface. Let us denote by $\mathrm{PSD}{2}_{\mathrm{m},\Theta}$ and $\mathrm{PSD}{2}_{\mathrm{m},\varphi }$ the PSD of the amplitude and phase of the field reflected from the OAP. We assume that the statistical properties of the OAP surface are the same as the properties of the grating surface, that is, they have equal $\mathrm{PSD}{2}_\mathrm{h}\left({k}_{\perp}\right)$ . Then $\mathrm{PSD}{2}_{m,\varphi }=\frac{B_\mathrm{wr}{k}_\mathrm{wr}^2}{B_0{k}_0^2} \mathrm{PSD}{2}_\mathrm{r}$ , Equation (40), where ${k}_\mathrm{wr}$ is the wave vector of the writing radiation and ${B}_\mathrm{wr}=4{\mathit{\cos}}^2\theta$ , where $\theta$ is the angle of incidence on the OAP (Figure 4). Analogously, we assume that the statistical properties of obscuration on these mirrors are the same as on the gratings, that is, $\mathrm{PSD}{2}_{m,\Theta}=\frac{\sigma_\mathrm{wr}^2}{\sigma_{\Theta}^2} \mathrm{PSD}{2}_\mathrm{ob}$ , Equation (37). For unused gratings it is reasonable to assume that ${\sigma}_{\Theta}^2={\sigma}_\mathrm{wr}^2$ , but with long-term use of gratings in high-power radiation the number of defects increases and ${\sigma}_{\Theta}^2\gg {\sigma}_\mathrm{wr}^2$ ; see Section 3.

The field on the substrate differs from the field on the mirror, since when the light is propagating from the mirror to the substrate over distance ${L}_\mathrm{wr}$ , the fluctuations of the amplitude and phase of the field transform into each other. An expression for $\mathrm{PSD}{2}_\mathrm{dose}$ when a beam propagates in free space was obtained in Ref. [Reference Martyanov and Khazanov44], from which, in the particular case of monochromatic radiation of interests to us, we find the following:

(41) $$\begin{align}&\mathrm{PSD}{2}_\mathrm{dose}=4T\left( \mathrm{PSD}{2}_{\mathrm{m},\Theta}+ \mathrm{PSD}{2}_{\mathrm{m},\varphi}\right)\nonumber\\&\quad+4\mathit{\operatorname{Re}}\left({T}_2{e}^{i{L}_\mathrm{wr}{k}_{\perp}^2/{k}_0}\right)\left( \mathrm{PSD}{2}_{\mathrm{m},\Theta}- \mathrm{PSD}{2}_{\mathrm{m},\varphi}\right),\end{align}$$

where

(42) $$\begin{align}T\left(\boldsymbol{\rho} \right)&=\frac{\int {\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|}^2{\left|{E}_\mathrm{wr}\left(\boldsymbol{r}-\boldsymbol{\rho} \right)\right|}^2\mathrm{d}\boldsymbol{r}}{\int {\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|}^4\mathrm{d}\boldsymbol{r}},\kern0.72em\nonumber\\{T}_2\left(\boldsymbol{\rho} \right)&=\frac{\int {E}_\mathrm{wr}^{\ast}\left(\boldsymbol{r}-\boldsymbol{\rho} \right){E}_\mathrm{wr}^{\ast}\left(\boldsymbol{r}+\boldsymbol{\rho} \right){E}_\mathrm{wr}^2\left(\boldsymbol{r}\right)\mathrm{d}\boldsymbol{r}}{\int {\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|}^4\mathrm{d}\boldsymbol{r}},\kern0.72em\nonumber\\\boldsymbol{\rho} &=\frac{L_\mathrm{wr}{\boldsymbol{k}}_{\boldsymbol{\perp}}}{k_0}.\end{align}$$

Here, we took into account that two OAPs with uncorrelated noises contribute to $\mathrm{PSD}{2}_\mathrm{dose}$ , that is, the $\mathrm{PSD}{2}_\mathrm{dose}$ in Equation (41) is twice as large as that for one beam reported in Ref. [Reference Martyanov and Khazanov44]. It can be readily shown that the $\mathrm{PSD}$ of field amplitude fluctuations $\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|$ is four times smaller than the $\mathrm{PSD}{2}_\mathrm{dose}$ , and the sum of the $\mathrm{PSD}$ of the field amplitude fluctuations $\left|{E}_\mathrm{wr}\left(\boldsymbol{r}\right)\right|$ and the $\mathrm{PSD}$ of the field phase ${\phi}_\mathrm{wr}\left(\boldsymbol{r}\right)$ is $4T\left( \mathrm{PSD}{2}_{\mathrm{m},\Theta}+ \mathrm{PSD}{2}_{\mathrm{m},r}\right)$ . Thus, $\mathrm{PSD}{2}_\mathrm{phase}=\frac{1}{4} \mathrm{PSD}{2}_\mathrm{dose}$ .

The authors of Ref. [Reference Hu, Wan, Jiang, Gu, Zhang, Jin, Liu, Zhao, Cao, Wei and Shao54] derived an expression similar to Equation (41) with $T=1$ , that is, neglecting the finite beam aperture ${E}_\mathrm{wr}\left(\boldsymbol{r}\right)$ (see below). The term with ${T}_2$ corresponds to the Talbot effect. Considering that ${L}_\mathrm{wr}$ is of the order of 10 m, at large ${k}_{\perp }$ this term rapidly oscillates around zero. In addition, since the contribution to $\mathrm{PSD}{2}_{\mathrm{phase, dose}}$ is made by two surfaces with close but not identical distances ${L}_\mathrm{wr}$ , the total phase fluctuations at large ${k}_{\perp }$ are averaged. Therefore, this term can be approximately replaced by its average value, which is obviously equal to zero, that is, we can set ${T}_2=0$ . The function $T\left(\boldsymbol{\rho} \right)$ corresponds to the spatial self-filtering of noise during propagation in free space^[ Reference Martyanov and Khazanov ^{44 ,} Reference Mironov, Lozhkarev, Luchinin, Shaykin and Khazanov ^{58 ,} Reference Khazanov, Mironov and Mourou ^{59 ]}, which is associated with the fact that part of the high-frequency noise escapes from the beam aperture. The transmittance $T\left(\boldsymbol{\rho} \right),$ that is, the normalized autocorrelation function of the writing beam intensity, is close to unity at $\rho \ll {R}_0$ , that is, for $\frac{L_\mathrm{wr}}{R_0}\ll \frac{k_0}{k_{\perp }}$ . Since ${L}_\mathrm{wr}\approx 10{R}_0$ , then $T\left(\boldsymbol{\rho} \right)\approx 1$ for ${k}_{\perp}\ll 0.1{k}_0$ . This condition is more rigorous than the paraxial approximation condition ${k}_{\perp}\ll {k}_0$ , so self-filtering cannot be completely neglected. At the same time, $T\left(\rho ={R}_0\right)\approx 1/2$ , that is, for ${k}_{\perp }=0.1{k}_0$ (the paraxial approximation boundary), the error will be about two. Therefore, assuming $T\left(\boldsymbol{\rho} \right)\approx 1$ we obtain a slightly overestimated noise value at high ( ${k}_{\perp}\approx 0.1{k}_0$ ) frequencies and, consequently, a slightly overestimated contrast value at large times. From Equation (41), with $T=1$ and ${T}_2=0$ taken into account, we have the following:

(43) $$\begin{align}\mathrm{PSD}{2}_\mathrm{phase}&=\frac{\sigma_\mathrm{wr}^2}{\sigma_{\Theta}^2} \mathrm{PSD}{2}_{\Theta}+\frac{B_\mathrm{wr}{k}_\mathrm{wr}^2}{B_0{k}_0^2} \mathrm{PSD}{2}_\mathrm{r},\kern0.84em\nonumber\\\mathrm{PSD}{2}_\mathrm{dose}&=4 \mathrm{PSD}{2}_\mathrm{phase},\end{align}$$

where $\mathrm{PSD}{2}_{\Theta}$ is defined by Equations (37) and (36), and $\mathrm{PSD}{2}_\mathrm{r}$ by Equation (40). Taking into account the linear relationship of $\mathrm{PSD}1$ and $\mathrm{PSD}2$ in Equation (27), an expression for $\mathrm{PSD}1$ is analogous to Equation (43). The results presented in Sections 3–5 are summarized in Table 1, from which we obtain the final expression for the total $\mathrm{PSD}$ for one grating:

(44) $$\begin{align}\mathrm{PSD}2&=\left(1+\left(1+4q\right)\frac{\sigma_\mathrm{wr}^2}{\sigma_{\Theta}^2}\right) \mathrm{PSD}{2}_{\Theta}\nonumber\\&\quad +\left(1+\left(1+4q\right)\frac{B_\mathrm{wr}{k}_\mathrm{wr}^2}{B_0{k}_0^2}\right) \mathrm{PSD}{2}_\mathrm{r}.\end{align}$$

By substituting Equation (44) into Equation (26) we obtain an expression for the contrast ${\mathbb{C}}_\mathrm{s}(t)$ for gratings G2 and G3, and the substitution of Equation (44) into Equation (29) gives an expression for the contrast ${\mathbb{C}}_\mathrm{d}(t)$ for gratings G1 and G4, as well as for the mirrors on the input and output optics. Note that for the mirrors the expressions in parentheses in Equation (44) should be replaced by 1, since in this work we neglect fluctuations in the mirror reflectivity.

Table 1 $\mathrm{PSD}$ for one grating (for a mirror, the columns ‘writing’ would contain zeros).

6 Example of contrast $\mathbb{C}\left(\boldsymbol{t}\right)$  calculations

Above we considered the contrast introduced by one of the optical elements shown in Figure 1. In what follows we will assume that the total contrast is equal to the sum of the contrasts introduced by each element. This is obvious for obscurations and surface imperfections (both of the grating and of the mirror), since the obscurations are randomly located on different elements and the $\Theta \left(\boldsymbol{r}\right)$ functions of different elements are uncorrelated. The same applies to the roughness of beamline optics. For the contrast associated with writing the gratings, the issue is more complicated. It is reasonable to assume that all gratings are written under the same conditions; then their functions $\mathcal{A}\left(\boldsymbol{r}\right)$ and $\Delta \varphi \left(\boldsymbol{r}\right)$ are correlated. A strong correlation for $\Delta \varphi \left(\boldsymbol{r}\right)$ was measured in Ref. [Reference Kochetkov, Shaykin, Yakovlev, Khazanov, Cheplakov, Wang, Jin, Liu and Shao28]. However, since G2 and G3 (as well as G1 and G4) are mirror-like in the compressor, their identity is lost by virtue of the uncorrelated $\mathcal{A}\left(x,y\right)$ and $\mathcal{A}\left(-x,y\right)$ . For definiteness, we will assume that both the input and output optics consist of four mirrors. We will use the geometric parameters of the compressor from the XCELS project^[ Reference Khazanov, Shaykin, Kostyukov, Ginzburg, Mukhin, Yakovlev, Soloviev, Kuznetsov, Mironov, Korzhimanov, Bulanov, Shaikin, Kochetkov, Kuzmin, Martyanov, Lozhkarev, Starodubtsev, Litvak and Sergeev ^{43 ]}; see Table 2.

Table 2 Compressor parameters.

We have neither theoretical nor experimental data on the value of q, but we can make the following estimate. For the dispersion of the power reflectivity of the grating ${\sigma}_\mathrm{R}^2=2{\sigma}_\mathrm{a}^2$ , we obtain ${\sigma}_\mathrm{R}^2=8q{\sigma}_\mathrm{h}^2{B}_\mathrm{wr}{k}_\mathrm{wr}^2$ . We will take the minimum value of ${\sigma}_\mathrm{h}$ available in the literature, which at ${k}_\mathrm{h}=0.1/\mathrm{mm}$ is ${\sigma}_\mathrm{h}=0.38\;\mathrm{nm}$ ^[ Reference Hu, Wan, Jiang, Gu, Zhang, Jin, Liu, Zhao, Cao, Wei and Shao ^{54 ]}. Then at ${\lambda}_\mathrm{wr}=413\;\mathrm{nm}$ , ${B}_\mathrm{wr}=3.7$ and $q=1$ we obtain ${\sigma}_\mathrm{R}=0.03$ , which is an unreasonably large value for such high-quality optics. For the National Ignition Facility (NIF) standard optics, ${\sigma}_\mathrm{h}=3.8\;\mathrm{nm}$ and ${\sigma}_\mathrm{R}=0.03$ is obtained at $q=0.01$ . Although these estimates are very rough, we can conclude from them that $q\ll 0.1$ and set $q=0$ in Equation (44). In other words, the impact of the writing optics imperfections on the fluctuations of the grating reflectivity may be neglected and only the diffracted wave phase fluctuations will be taken into account.

The solid curves in Figure 5 show the PCR introduced by gratings G2 and G3, and the dotted curves show the PCR introduced by all the other optical elements taken together: gratings G1 and G4, as well as the input M_in and output M_out optics. As stated above, such a partitioning is made because only gratings G2 and G3 have a spatial chirp $\left({\tau}_x\ne 0\right)$ and it is necessary to use either the exact Equation (25) or the approximate Equation (26), from which it follows that the cut-off time ${t}_\mathrm{c}$ in Equation (22) is 249 ps for G3 and 338 ps for G2, that is, there is both a pre-pulse and a post-pulse. There is no spatial chirp $\left({\tau}_x=0\right)$ on all the other elements, so one can use Equation (29), from which it follows that the cut-off time is zero, that is, there is no pre-pulse (if there is no negative diffraction, see Section 2).

Figure 5 Contrast ${\mathbb{C}}_\mathrm{ob}(t)$ (red) and ${\mathbb{C}}_\mathrm{r}(t)$ (blue) for gratings G2 and G3 (solid curves), for other optical elements (dashed curves) and total contrast for the entire compressor (green dotted curve). Here, (a) and (b) differ only by the scale of the horizontal axis.

Figure 6 Schematic representation of the target illumination dynamics in the focal plane. (a) For gratings G2 and G3. (b) For all other optical elements depicted in Figure 1.

The red and blue curves in Figure 5 correspond to the contrast ${\mathbb{C}}_\mathrm{ob}(t)$ caused by obscurations and the contrast ${\mathbb{C}}_\mathrm{r}(t)$ caused by surface roughness. To obtain ${\mathbb{C}}_\mathrm{ob}(t),$ only the first term from Equation (44) (i.e., the term with $\mathrm{PSD}{2}_{\Theta}$ from Equation (37)) should be substituted into Equations (25) and (29). The analysis shows that ${\mathbb{C}}_\mathrm{ob}(t)$ is almost independent of Z; it depends only on two dimensionless quantities – $\xi$ and ${k}_0{w}_\mathrm{min}$ . Further, we will assume that ${w}_\mathrm{min}=0.25\;\mathrm{mm}$ , $\xi =3.9$ ^[ Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig ^{23 ]} and that the compressor has already been used in a large number of shots, as a result of which grating G4 and the output optics ‘accumulated’ obscurations in the amount measured in Ref. [Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig23], which corresponds to ${\sigma}_{\Theta}^2=6.1\times {10}^{-4}$ . The input optics, gratings G1–G3 and writing optics are not exposed to powerful femtosecond radiation, so we will assume that for them ${\sigma}_{\Theta}^2=6.1\times {10}^{-6}$ , that is, it is 100 times smaller. We remind the reader that ${\sigma}_{\Theta}^2$ is equal to the fraction of the surface occupied by obscurations, as well as to the fraction of the noise energy ${W}_{\Theta}/{W}_0$ ; see Section 2.

To obtain ${\mathbb{C}}_\mathrm{r}(t)$ , it is necessary to substitute into Equations (25) and (29) only the second term from Equation (44) (the term with $\mathrm{PSD}{2}_\mathrm{r}$ from Equation (40)). The analysis of these expressions shows that ${\mathbb{C}}_\mathrm{r}(t)$ depends only on two dimensionless quantities – $a$ and ${\sigma}_\mathrm{r}^2={B}_0{k}_0^2{\sigma}_\mathrm{h}^2{\left({k}_\mathrm{h}/{k}_0\right)}^{a-2}$ , with the dependence on ${\sigma}_\mathrm{r}^2$ being linear. In other words, for a fixed a, the shape of ${\mathbb{C}}_\mathrm{r}(t)$ does not change. Further, we will assume that $a=2.55$ , and ${\sigma}_\mathrm{h}^2$ of all optical elements, including mirrors used for writing the gratings (see Figure 4), is 200 times less than the data in Ref. [Reference Camus, Coic, Blanchot, Bouillet, Lavastre, Mangeant, Rouyer and Néauport55], 10 times less than the NIF standard^[ Reference Spaeth, Manes, Widmayer, Williams, Whitman, Henesian, Stowers and Honig ^{23 ]}, but 10 times more than the data reported in Ref. [Reference Hu, Wan, Jiang, Gu, Zhang, Jin, Liu, Zhao, Cao, Wei and Shao54]. At ${k}_\mathrm{h}=0.1/\mathrm{mm}$ this corresponds to ${\sigma}_\mathrm{h}\approx 1.2\;\mathrm{nm}$ . It is easy to show that the fraction of the noise energy ${\sigma}_{\varphi}^2={W}_{\varphi}/{W}_0$ caused by scattering on a rough surface is equal to ${\sigma}_\mathrm{h}^2{B}_0{k}_0^2$ and at ${B}_0=2.8$ we obtain ${\sigma}_{\varphi}^2\approx 1.9\times {10}^{-4}$ . This is approximately 30 times more than ${\sigma}_{\Theta}^2=6.1\times {10}^{-6}$ presented above. However, at long times ${\mathbb{C}}_\mathrm{r}\ll {\mathbb{C}}_\mathrm{ob}$ , since ${\mathbb{C}}_\mathrm{ob}$ decreases very slowly. As a result, at long times the total PCR is determined by ${\mathbb{C}}_\mathrm{ob},$ and at short times by ${\mathbb{C}}_\mathrm{r}$ . This qualitative conclusion is valid for a wide range of ${\sigma}_{\varphi}^2$ and ${\sigma}_{\Theta}^2$ . Any quantitative comparison of ${\mathbb{C}}_\mathrm{ob}$ and ${\mathbb{C}}_\mathrm{r}$ strongly depends on the quality of the optical elements, because ${\mathbb{C}}_{\mathrm{ob},\mathrm{r}}$ are proportional to ${\sigma}_{\Theta, \varphi}^2$ and the ${\sigma}_{\varphi}^2/{\sigma}_{\Theta}^2$ ratio can vary over a wide range in practice. For example, the boundary is the instant of time at which ${\mathbb{C}}_\mathrm{r}={\mathbb{C}}_\mathrm{ob}$ is ${t}_{\ast }=-60\;\mathrm{ps}$ in Figure 5(a). If the optics roughness ${\sigma}_{\varphi}^2$ increases by a factor of 10, ${\mathbb{C}}_\mathrm{r}$ also increases by a factor of 10 and ${t}_{\ast }=-12\;\mathrm{ps}$ .

The small deviation of the contrast from parity observed in Figure 5 for curves G2 and G3 constructed according to Equation (25) may be neglected in practice and a simpler Equation (26) can be used instead of Equation (25), that is, $\mathbb{C}(t)$ is proportional to $\mathrm{PSD}1\left({k}_x\right)$ and the mapping is linear: $t={\tau}_x{k}_x/{k}_0$ . Since in the focal plane the distance from the beam axis (along the x-axis) is $x=F{k}_x/{k}_0$ , then for the pre-pulse, $x$ is related to the time of its appearance on the target as $x= Ft/{\tau}_x$ . Thus, the radiation reaches the target at the time $t=-{t}_{\mathrm{c}}$ , illuminating a vertical strip at a distance $x=F{t}_{\mathrm{c}}/{\tau}_x$ ; see Figure 6(a). Then this strip splits into two and its replicas move in opposite directions at speed $F/{\tau}_x$ . At $t=0$ , one of them reaches the axis and continues to move in the same direction at the same speed. The intensity and, hence, the fluence change proportionally to $\mathbb{C}(t)$ . For any given fluence value (for example, at which the target is destroyed), this simple reasoning allows us to calculate for each point of the target the time at which this value will be reached.

For all the other optical elements, except for G2 and G3, PCR $\mathbb{C}(t)$ is proportional to $\mathrm{PSD}2\left({k}_{\perp}\right)$ in Equation (29) and the mapping is quadratic: $t={t}_\mathrm{d}{k}_{\perp}^2/{k}_0^2$ . The radiation reaches the target at $t=0$ , exactly at the beam axis. At all subsequent instants of time, the radiation illuminates a ring of radius $r=F{k}_{\perp }/{k}_0=F\sqrt{t/{t}_\mathrm{d}}$ , which increases in proportion to $\sqrt{t}$ (Figure 6(b)).

The laws of $\mathbb{C}(t)$ decrease are different; see Figure 5(a). The ${\mathbb{C}}_\mathrm{ob}(t)$ curve for G2 and G3 clearly features the boundary between the two ranges, in which the law of ${\mathbb{C}}_\mathrm{ob}(t)$ decrease changes from ${\left|t\right|}^{-0.1}$ to ${\left|t\right|}^{-2}$ . Unfortunately, quantitative comparison with the experiments is impossible due to lack of PSD functions for the gratings and optics used in the experiments. There is a good qualitative agreement between ${\mathbb{C}}_\mathrm{ob}(t)$ and the results of measurements^[ Reference Roeder, Zobus, Major and Bagnoud ^{11 ]}, where a plateau (an area where ${\mathbb{C}}_\mathrm{ob}(t)\sim {\left|t\right|}^{-0.1}$ ) caused by scattering at stretcher mirrors, which is equivalent to scattering at gratings G2 and G3^[ Reference Khazanov ^{22 ]}, was also observed. The theory (in particular the dotted curve in Figure 5) explains the frequently observed experimental asymmetry of the contrast: the post-pulse is larger than the pre-pulse^[ Reference Webb, Dorrer, Bahk, Jeon, Roides and Bromage ^{5 ,} Reference Hooker, Tang, Chekhlov, Collier, Divall, Ertel, Hawkes, Parry and Rajeev ^{15 –} Reference Lu, Zhang, Li and Leng ^{18 ,} Reference Wang, Liu, Lu, Chen, Long, Li, Chen, Chen, Bai, Li, Peng, Liu, Wu, Wang, Li, Xu, Liang, Leng and Li ^{46 ,} Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul ^{47 ]}. Also, Figure 5(a) shows the typical ‘triangular’ $\mathbb{C}(t)$ shapes observed in many experiments^[ Reference Webb, Dorrer, Bahk, Jeon, Roides and Bromage ^{5 ,} Reference Schanz, Roth and Bagnoud ^{8 ,} Reference Roeder, Zobus, Major and Bagnoud ^{11 ,} Reference Ranc, Blanc, Lebas, Martin, Zou, Mathieu, Radier, Ricaud, Druon and Papadopoulos ^{14 –} Reference Roeder, Zobus, Brabetz and Bagnoud ^{19 ,} Reference Khodakovskiy, Kalashnikov, Gontier, Falcoz and Paul ^{47 ]}.

Despite the larger number of elements (two gratings and eight mirrors), the ${\mathbb{C}}_\mathrm{r}(t)$ value for G2 and G3 is close to the ${\mathbb{C}}_\mathrm{r}(t)$ value for all the other elements. The point is that the contribution of ${\mathbb{C}}_\mathrm{r}(t)$ for gratings is much larger than for mirrors, since in mirrors there is no dominant (due to ${k}_\mathrm{wr}^2\gg {k}_0^2$ ) contribution from writing optics.

For the PCR caused by the roughness of gratings and mirrors of the input and output optics, generalizations to advanced compressors and stretchers were proposed in Ref. [Reference Khazanov22]. Without repeating them here, we just note that they are valid for all the results obtained above, since they are based on the Equation (19) in Ref. [Reference Khazanov22], which coincides with Equation (17) if $\mathrm{PSD}{2}_{\varphi}\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ is replaced by $\mathrm{PSD}2\left({\boldsymbol{k}}_{\boldsymbol{\perp}}\right)$ .