2 The field at the compressor output

We will consider a compressor consisting of two pairs of parallel gratings (Figure 1), on which a laser beam with an arbitrary temporal and spatial spectrum is incident. The z-axis coincides with the beam trajectory at the compressor input with ${k}_{x,y} = 0$ and $\omega = {\omega}_0 = {ck}_0$ , where ${\omega}_0$ and ${k}_0$ are, respectively, the carrier frequency and the wave vector. The angles ${\alpha}_{1,2}$ and ${\beta}_{1,2}$ of this beam are related by the grating expression as follows:

(1) $$\begin{align}\sin {\beta}_{1,2} = m\frac{2\pi }{k_z\left({\omega}_0\right)}{N}_{1,2}+\sin {\alpha}_{1,2},\end{align}$$

where ${N}_{1,2}$ is the groove density, $m$ is the diffraction order and ${k}_z^2 = {k}_0^2-{k}_x^2-{k}_y^2$ . Hereinafter, the subscripts ‘1’, ‘2’ refer to the first and second pair of gratings, respectively. The magnitudes of $\alpha, N$ and $L$ in the second pair of gratings may be either like in the first pair or different from them. In the first case, the compressor will be called a symmetric or Treacy compressor (TC)^[ Reference Treacy ^{27 ]}. In the second case, it will be called an asymmetric compressor or AFGC^[ Reference Shen, Du, Liang, Wang, Liu and Li ^{24 ]}. The AFGC is employed in the ARC (advanced radiographic capability) picosecond laser^[ Reference Williams, Crane, Alessi, Boley, Bowers, Conder, Nicola, Nicola, Haefner, Halpin, Hamamoto, Heebner, Hermann, Herriot, Homoelle, Kalantar, Lanier, LaFortune, Lawson, Lowe-Webb, Morrissey, Nguyen, Orth, Pelz, Prantil, Rushford, Sacks, Salmon, Seppala, Shaw, Sigurdsson, Wegner, Widmayer, Yang and Zobrist ^{28 ,} Reference Crane, Tietbohl, Arnold, Bliss, Boley, Britten, Brunton, Clark, Dawson, Fochs, Hackel, Haefner, Halpin, Heebner, Henesian, Hermann, Hernandez, Kanz, McHale, McLeod, Nguyen, Phan, Rushford, Shaw, Shverdin, Sigurdsson, Speck, Stolz, Trummer, Wolfe, Wong, Siders and Barty ^{29 ]}, but asymmetry is used to make the compressor more compact, while smoothing of fluctuations is not discussed.

Figure 1 AFGC scheme. Gratings in each pair are parallel and identical. Distances between the gratings $L$ , groove densities $N$ and incidence angles $\alpha$ are different in different pairs.

The ray with arbitrary ${k}_x$ and $\omega$ in Figure 1 is shown by the red color. The angle of incidence on the first grating for this ray is $\alpha +\Delta \alpha \left(\omega, {k}_x\right)$ , where $\Delta \alpha = \mathrm{arctan}\left(\frac{k_x}{k_z\left(\omega \right)}\right)$ . The angle of the ray reflection from the grating is $\theta \left(\omega, {k}_x\right)$ . It depends on ${k}_x$ and $\omega$ , with $\theta \left(\omega = {\omega}_0,{k}_x = 0\right) = \beta$ . The angles of reflection $\theta\;\mathrm{and}\;\beta$ are counted to the right of the normal (i.e., for a mirror, $\beta = \alpha )$ . In the minus first diffraction order ( $m = -1$ ), the angles $\theta\;\mathrm{and}\;\beta$ are negative: $\beta$ < 0, $\theta$ < 0; therefore, the minus sign is indicated in the figure.

Let the real electric field at the compressor input at the point D1 be ${\varepsilon}_{\mathrm{in}}\left(t,x,y\right) = \operatorname{Re}\left\{{E}_{\mathrm{in}}\left(t,x,y\right){e}^{i{\omega}_0t-i{k}_0z}\right\}$ , where ${E}_{\mathrm{in}}\left(t,x,y\right)$ is the complex amplitude. The relationship between ${E}_{\mathrm{in}}\left(t,x,y\right)$ and the complex amplitude at the point D2 at the compressor output ${E}_{\mathrm{out}}\left(t,x,y\right)$ is more readily found in the spectral representation. Hereinafter, the spectra will be designated by the same characters but with different arguments:

(2) $$\begin{align}{E}_{\mathrm{in, out}}\left(\omega, {k}_x,{k}_{{y}}\right) &= \mathbf{F}\left\{{E}_{\mathrm{in, out}}\left(t,x,y\right)\right\}\!, \nonumber\\{E}_{\mathrm{in, out}}\left(t,x,y\right) &= {\mathbf{F}}^{-\mathbf{1}}\left\{{E}_{\mathrm{in, out}}\left(\omega, {k}_x,{k}_{{y}}\right)\right\}\!,\end{align}$$

where $\mathbf{F}\;\mathrm{and}\;{\mathbf{F}}^{-\boldsymbol{1}}$ are the forward and inverse 3D Fourier transforms. Our consideration is restricted to the case of parallel and identical gratings in each pair. This ensures that the input and output wave vectors are parallel. In other words, the input and output wave vectors are always equal, and so are the frequencies. Hence, ${E}_{\mathrm{out}}$ and ${E}_{\mathrm{in}}$ are related by the following

(3) $$\begin{align}\kern-18pt {E}_{\mathrm{out}}\left(\omega, {k}_x,{k}_{{y}}\right) = \exp \left({i}\Psi \left(\omega, {k}_x,{k}_{{y}}\right)\right)\cdot {E}_{\mathrm{in}}\left(\omega, {k}_x,{k}_{y}\right),\end{align}$$

(4) $$\begin{align}\kern7pt{E}_{\mathrm{out}}\left(t,x,y\right) = {\mathbf{F}}^{-\mathbf{1}}\left\{\exp \left({i}\Psi \left(\omega, {k}_x,{k}_{{y}}\right)\right)\cdot \mathbf{F}\left\{{E}_{\mathrm{in}}\left(t,x,y\right)\right\}\right\}\!,\end{align}$$

where $\Psi \left(\omega, {k}_x,{k}_{{y}}\right)$ is the phase incursion between the input and output planes. For simplicity, in Equation (3) it is assumed that there are no losses and the absolute value of the transmission coefficient is equal to unity. Before passing to the expression for $\Psi \left(\omega, {k}_x,{k}_{{y}}\right)$ , an important general note should be made that follows directly from Equation (3) and Parseval’s theorem. The compressor changes the near-field distribution ${E}_{\mathrm{out}}\left(\omega, {k}_x,{k}_{{y}}\right),{E}_{\mathrm{out}}\left(t,{k}_x,{k}_{{y}}\right)$ , but in no way affects the fluence in the far-field:

$$\begin{align*}\int {\left|{E}_{\mathrm{out}}\left(t,{k}_x,{k}_{{y}}\right)\right|}^2 \mathrm{d}t = \int {\left|{E}_{\mathrm{in}}\left(t,{k}_x,{k}_{y}\right)\right|}^2 \mathrm{d}t.\end{align*}$$

This result was obtained numerically for specific examples^[ Reference Shen, Du, Liang, Wang, Liu and Li ^{24 ,} Reference Du, Shen, Liang, Wang, Liu and Li ^{25 ]}. We showed analytically that this statement is true for any compressor, independent of its parameters. Analogously it can be proved that the intensity on the beam axis in the far-field ${\left|{E}_{\mathrm{out}}\left(t,{k}_x = {k}_{{y}} = 0\right)\right|}^2$ does not depend on the compressor symmetry either. The only necessary condition for this is parallel and identical gratings in each pair, otherwise Equation (3) does not hold.

The expression for $\Psi \left(\omega, {k}_x,{k}_{{y}}\right)$ (the derivation is presented in the Appendix) has the following form:

(5) $$\begin{align}\Psi \left(\omega, {k}_x,{k}_{{y}}\right) &= {L}_1\frac{\omega }{c}\left(\cos {\theta}_1+\cos \left({\alpha}_1+\frac{k_xc}{\omega}\right)\right)\nonumber\\& \quad +{L}_2\frac{\omega }{c}\left(\mathit{\cos}{\theta}_2+\mathit{\cos}\left({\alpha}_2-\frac{k_xc}{\omega}\right)\right)\nonumber\\& \quad +{L}_{\mathrm{f}}\frac{\omega }{c}-\frac{{c}}{2\omega }{k}_x^2{L}_{\mathrm{f}}-\frac{{c}}{2\omega }{k}_y^2\nonumber\\& \quad \times \left({L}_{\mathrm{f}}+\left(\mathit{\cos}\left({\theta}_1\left({k}_{x,y} = 0\right)\right)+\mathit{\cos}{\alpha}_1\right){L}_1\right.\nonumber\\& \quad \left.+\left(\mathit{\cos}\left({\theta}_2\left({k}_{x,y} = 0\right)\right)+\mathit{\cos}{\alpha}_2\right){L}_2\right),\end{align}$$

where

(6) $$\begin{align}\mathit{\sin}{\theta}_{1,2}\left(\omega, {k}_x,{k}_y\right) = m\frac{2\pi c}{\omega }{N}_{1,2}\left(1+\frac{c^2}{2{\omega}^2}{k}_y^2\right)+\sin \left({\alpha}_{1,2}\pm \frac{k_xc}{\omega}\right).\end{align}$$

Here, ${L}_{1,2}$ is the distance between the gratings along the normal and ${L}_{\mathrm{f}}$ is the distance between the input and output planes of the compressor (Figure 1). The first two terms in Equation (5) depend, among others, on the first power of ${k}_x$ . For the TC, that is, for ${L}_2 = {L}_1,\ {\alpha}_2 = {\alpha}_1, {N}_2 = {N}_1\ \mathrm{and}\ {\theta}_2\left({k}_x\right) = {\theta}_1\left(-{k}_x\right),$ this dependence disappears, while it remains for the AFGC. This is the difference between the AFGC and the TC. A compressor with a single-grating pair (single-pass single-grating pair (SSGC))^[ Reference Du, Shen, Liang, Wang, Liu and Li ^{25 ]} is a particular case of an AFGC with ${L}_2 = 0$ .

Equation (4) with allowance for Equations (5) and (6) is sufficient for numerical simulation. To continue the analytical analysis, we will expand Equation (5) in a Taylor series.

3 Expanding $\Psi(\omega, k_x, k_y)$ in a Taylor series

By expanding Equation (5) in a Taylor series with respect to ${k}_x,{k}_y$ and $\Omega = \omega -{\omega}_0$ , we obtain the following:

(7) $$\begin{align}\Psi \left(\omega, {k}_x,{k}_y\right) &= {\psi}_{x\omega}^{\prime \prime}\Omega {k}_x+\frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{\Omega}^2{k}_x+\frac{1}{2}{\psi}_{x x}^{\prime \prime }{k}_x^2+\frac{1}{2}{\psi}_{yy}^{\prime \prime }{k}_y^2\nonumber\\& \quad +\frac{1}{2}{\psi}_{x x\omega}^{\prime \prime \prime}\Omega {k}_x^2+\frac{1}{2}{\psi}_{yy\omega}^{\prime \prime \prime}\Omega {k}_y^2,\end{align}$$

where

(8) $$\begin{align}{\psi}_{x\omega}^{\prime \prime} = -\frac{1}{\omega_0}\left({A}_1{L}_1-{A}_2{L}_2\right),\ A = \frac{\cos\alpha}{{\mathit{\cos}}^3\beta}\left( \sin\alpha - \sin\beta \right),\end{align}$$

(9) $$\begin{align}&\frac{1}{2}\cdot {\psi}_{x\omega \omega}^{\prime \prime \prime} = \frac{1}{\omega_0^2}\left({B}_1{L}_1-{B}_2{L}_2\right),\nonumber\\& \quad B = A\left(1-\frac{3}{2} \sin\beta \frac{\sin\alpha - \sin\beta}{{\mathit{\cos}}^2\beta}\right),\end{align}$$

(10) $$\begin{align}&\frac{1}{2}\cdot {\psi}_{xx}^{\prime \prime} = -\frac{c}{\omega_0}\frac{1}{2}\left({C}_1{L}_1+{C}_2{L}_2+{L}_{\mathrm{f}}\right),\nonumber\\& \quad C = \frac{{\mathit{\cos}}^2\alpha }{{\mathit{\cos}}^3\beta }- \tan\beta \sin\alpha + \cos\alpha,\end{align}$$

(11) $$\begin{align}\frac{1}{2}\cdot {\psi}_{yy}^{\prime \prime} = -\frac{c}{\omega_0}\frac{1}{2}\left({D}_1{L}_1+{D}_2{L}_2+{L}_{\mathrm{f}}\right),\kern0.36em D = \frac{1+\cos \left(\alpha +\beta \right)}{\cos\beta}\end{align}$$

(12) $$\begin{align}&\frac{1}{2}\cdot {\psi}_{xx\omega}^{\prime \prime \prime} = \frac{c}{\omega_0^2}\frac{1}{2}\left({E}_1{L}_1+{E}_2{L}_2+{L}_{\mathrm{f}}\right),\nonumber\\& \quad E = C+\frac{\sin\alpha - \sin\beta}{{\mathit{\cos}}^3\beta}\left( \sin\alpha -3 \sin\beta \frac{{\mathit{\cos}}^2\alpha }{{\mathit{\cos}}^2\beta}\right),\end{align}$$

(13) $$\begin{align}&\frac{1}{2}\cdot {\psi}_{yy\omega}^{\prime \prime \prime} = \frac{c}{\omega_0^2}\frac{1}{2}\left({F}_1{L}_1+{F}_2{L}_2+{L}_{\mathrm{f}}\right),\nonumber\\& \quad F = D +\frac{{\left( \sin\beta - \sin\alpha \right)}^2}{{\mathit{\cos}}^3\beta },\end{align}$$

(14) $$\begin{align}{A}_{1,2} = A\left({\alpha}_{1,2},{\beta}_{1,2}\right);\ B,C,D,E,F-\mathrm{analogously}.\end{align}$$

The lower-case letter $\psi$ designates the values of the corresponding derivatives for ${k}_x = {k}_y = 0\ \mathrm{and}\ \omega = {\omega}_0$ ; for example:

(15) $$\begin{align}\left.{\psi}_{x\omega}^{\prime \prime} = \frac{\partial^2\Psi \left(\omega, {k}_x,{k}_{\mathrm{y}}\right)}{\partial {k}_x\partial \omega}\right|{\displaystyle \begin{array}{c}\\ {}\\ {}{k}_{x,y} = 0;\omega = {\omega}_0\end{array}}.\end{align}$$

Some terms are omitted in Equation (7). These include terms with the first derivative with respect to ${k}_y$ that are equal to zero, since $\Psi$ depends only on ${k}_y^2$ . Next, $\Psi \left(\omega = {\omega}_0,{k}_x = {k}_{{y}} = 0\right)$ is a constant that may also be put equal to zero. We omit ${\psi}_{\omega}^{\prime}\Omega$ , that is, the time delay, which does not depend on ${k}_{x,y}$ , and ${\psi}_x^{\prime }{k}_x$ , that is, the $\Omega$ -independent shift along the x-axis. These terms are equivalent to the shift of the origin of the t- and x-axes, which is of no importance. Also, we omit all the terms containing derivatives only with respect to $\omega$ : $\frac{1}{2}{\psi}_{\omega \omega}^{\prime \prime }{\Omega}^2$ , $\frac{1}{6}{\psi}_{\omega \omega \omega}^{\prime \prime \prime }{\Omega}^3$ and so on. These terms correspond to different degrees of time dispersion for a field with ${k}_x = 0$ . In this work aimed at studying the space–time effects, we will omit the above-mentioned terms and will assume that the dispersion introduced by the compressor corresponds to the pulse dispersion at its input. Therefore, for the zero spatial frequency ${k}_x = 0$ , the output pulse is Fourier-transform-limited. Note that for compensating higher order dispersion, the AFGC provides even more opportunities than the TC, since the AFGC has six degrees of freedom: L _1,2, N _1,2 and ${\alpha}_{1,2}$ , whereas the TC has only three: L, N and $\alpha$ . Finally, we neglect all terms with derivatives with respect to ${k}_{x,y}$ higher than the second one, as well as all derivatives higher than the third one in view of their smallness.

Let us now address the remaining six terms in Equation (7) and the respective physical effects. The first two terms are proportional to the first power of ${k}_x$ : ${\psi}_{x\omega}^{\prime \prime}\Omega {k}_x$ is the time lag by ${\psi}_{x\omega}^{\prime \prime }{k}_x$ or the shift along the x-axis by distance ${\psi}_{x\omega}^{\prime \prime}\Omega$ ; $\frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{\Omega}^2{k}_x$ is the pulse stretching ( $\mathrm{GVD} = \frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{k}_x$ ) or the shift along the x-axis by distance $\frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{\Omega}^2$ . As seen from Equations (8) and (9), these terms in the TC are equal to zero. The second two terms in Equation (7), $\frac{1}{2}{\psi}_{xx}^{\prime \prime }{k}_x^2\kern0.24em \mathrm{and}\kern0.36em \frac{1}{2}{\psi}_{yy}^{\prime \prime }{k}_y^2$ , correspond to diffraction that is different along the x- and y-axes, as well as to spatial self-filtering, that is, to the shift along the x- and y-axes by distance $\frac{1}{2}{\psi}_{yy}^{\prime \prime }{k}_{x,y}$ . Finally, the last two terms in Equation (7) correspond to temporal self-filtering, that is, the time lag by $\frac{1}{2}{\psi}_{xx\omega, yy\omega}^{\prime \prime \prime }{k}_{x,y}^2$ . Note that the last four terms and the corresponding effects occur not only in a compressor but also in free space. Unlike the case of the compressor, in free space they are isotropic. The expressions in parentheses for $\psi$ in Equations (10)–(13) may be interpreted as the effective compressor length in terms of the corresponding effect. For example, ${C}_1{L}_1+{C}_2{L}_2+{L}_{\mathrm{f}}$ is the effective length in terms of diffraction along the x-axis, and ${F}_1{L}_1+{F}_2{L}_2+{L}_{\mathrm{f}}$ is the effective length in terms of temporal self-filtering along the y-axis. Values of the $A\ \mathrm{to}\ F$ constants for compressors borrowed from some works are given by way of example in Table 1. Let us go into the details of these effects in different compressors.

Table 1 Parameters of the compressors: TC ^[ Reference Khazanov, Shaykin, Kostyukov, Ginzburg, Mukhin, Yakovlev, Soloviev, Kuznetsov, Mironov, Korzhimanov, Bulanov, Shaikin, Kochetkov, Kuzmin, Martyanov, Lozhkarev, Starodubtsev, Litvak and Sergeev ^{13 ]}, AFGC^[ Reference Shen, Du, Liang, Wang, Liu and Li ^{24 ]} and SSGC^[ Reference Du, Shen, Liang, Wang, Liu and Li ^{25 ]}.

4 The Treacy compressor as a filter of spatial frequencies

The derivatives in Equations (8) and (9) vanish in the TC, and only two effects remain in Equation (7) (spatial self-filtering and temporal self-filtering that are proportional to ${k}_{x,y}^2$ ). Thus, from the point of view of spatial effects, the TC is similar to free space, but it introduces astigmatism, because the x- and y-axes are no longer equivalent. It is seen from Equations (10) and (11) that, in terms of spatial self-filtering and diffraction, the TC supplements the free space length ${L}_{\mathrm{f}}$ with an additional length that is different for the x- and y-axes: $2{C}_1{L}_1$ for the x-axis and $2{D}_1{L}_1$ for the y-axis. With this astigmatism taken into account, the compressor is equivalent to free space with different lengths on the x- and y-axes:

(16) $$\begin{align}{L}_{sx} = 2{C}_1{L}_1+{L}_{\mathrm{f}},\kern0.48em {L}_{sy} = 2{D}_1{L}_1+{L}_{\mathrm{f}}.\end{align}$$

Thus, the TC is an anisotropic transmissive filter. The transmission coefficient of such a filter can be obtained by generalizing expressions from Refs. [Reference Martyanov and Khazanov21,Reference Kochetkov, Martyanov, Ginzburg and Khazanov22] to the anisotropic case.

In free space, the time lag $\tau$ of the pulse propagating at an angle to the z-axis is proportional to ${k}_x^2+{k}_y^2$ . In a symmetric compressor, the coefficients of proportionality for ${k}_x^2$ and ${k}_y^2$ are not the same (see Figure 2). From Equations (12) and (13) we obtain the following:

(17) $$\begin{align}\tau \left({k}_x,{k}_y\right) = \frac{1}{2}{\psi}_{xx\omega}^{\prime \prime \prime }{k}_x^2+\frac{1}{2}{\psi}_{yy\omega}^{\prime \prime \prime }{k}_y^2 = \frac{L_{tx}}{2c}\frac{k_x^2}{k_0^2}+\frac{L_{ty}}{2c}\frac{k_y^2}{k_0^2},\end{align}$$

Figure 2 Schematic representation of the field after propagation in free space (a), the TC (b) and the AFGC (c), (d) in three domains. Solid magenta lines depict the intensity front.

where

(18) $$\begin{align}{L}_{tx} = 2{E}_1{L}_1+{L}_{\mathrm{f}},\kern0.36em {L}_{ty} = 2{F}_1{L}_1+{L}_{\mathrm{f}}.\end{align}$$

Thus, from the point of view of temporal self-filtering, the TC adds to the free space length ${L}_{\mathrm{f}}$ a supplementary length that is different for the x- and y-axes: 2EL for the x-axis and 2FL for the y-axis. The ${L}_{tx}-{L}_{ty}$ difference may be quite significant. Contrary to the case of free space propagation, these curvatures do not coincide after the TC. As shown in Figure 2, all other fronts are always convex because $C,D,E,F>0$ .

5 The asymmetric four-grating compressor as a filter of spatial frequencies

The effects considered in Section 4 for the TC are exactly the same as those in the AFGC. The only difference is that the effective lengths (Equations (16) and (18)) have the following form:

(19) $$\begin{align}{L}_{sx} = {C}_1{L}_1+{C}_2{L}_2+{L}_{\mathrm{f}},\quad {L}_{sy} = {C}_1{L}_1+{C}_2{L}_2+{L}_{\mathrm{f}},\end{align}$$

(20) $$\begin{align}{L}_{tx} = {E}_1{L}_1+{E}_2{L}_2+{L}_{\mathrm{f}},\quad {L}_{ty} = {F}_1{L}_1+{F}_2{L}_2+{L}_{\mathrm{f}},\end{align}$$

which directly follows from Equations (10)–(13). The most important distinctive feature of the AFGC is that Equations (8) and (9) are not zeroed, and there appear in Equation (7) two terms and, hence, two effects proportional to the first power of ${k}_x$ : ${\psi}_{x\omega}^{\prime \prime}\Omega {k}_x$ , that is, the time lag proportional to ${k}_x$ or shift along the x-axis proportional to $\Omega$ (spatial chirp), and also $\frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{\Omega}^2{k}_x$ , that is, GVD (pulse stretching) proportional to ${k}_x$ or shift along the x-axis quadratic with respect to $\Omega$ . As seen from Equations (8) and (9), the measure of compressor asymmetry is two parameters having the dimension of the following length:

$$\begin{align*}\Delta L = {A}_1{L}_1-{A}_2{L}_2,\quad \Delta {L}^{\prime} = {B}_1{L}_1-{B}_2{L}_2.\end{align*}$$

5.1 Time lag proportional to  ${{k}}_{{x}}$  or shift along the x-axis proportional to  ${\varOmega}$ :  ${{\psi}}_{{x}{\omega}}^{\prime \prime}{\varOmega} {{k}}_{{x}}$

The physical meaning of this term becomes transparent if we write ${\psi}_{x\omega}^{\prime \prime}\Omega {k}_x = X\left(\Omega \right)\kern0.24em {k}_x = {\tau}_A\left({k}_x\right)\cdot \Omega$ , where $X\left(\Omega \right) = {\psi}_{x\omega}^{\prime \prime}\Omega$ is the beam shift along the x-axis (spatial chirp) and ${\tau}_A\left({k}_x\right) = {\psi}_{x\omega}^{\prime \prime }{k}_x$ is the time lag. These two representations are equivalent and are illustrated in Figure 2 for clarity. From the point of view of spatial noise, this effect reduces to temporal filtering, and it is convenient to consider it in terms of the pulse time lag with ${k}_x\ne 0$ :

(21) $$\begin{align}{\tau}_A\left({k}_x\right) = -\frac{k_x}{k_0}\cdot \frac{\Delta L}{c}.\end{align}$$

Unlike temporal self-filtering, the time lag here is proportional to the first rather that the second (Equation (17)) power of ${k}_x$ , which makes the effect much stronger. However, for $\Delta L\preccurlyeq {L}_x\frac{k_x}{k_0}$ , the effects are comparable. The second difference is that the sign of the lag depends on the sign of ${k}_x$ , that is, the spatial noise for which ${k}_x\Delta L>0$ will overtake the main pulse rather than lag behind it (Figure 2). Lagging and overtaking have an identical impact on fluence smoothing. At the same time, the pulses propagating ahead of the main pulse reduce the time contrast. Note that this parasitic effect is especially strong in an SSGC. For instance, for a compressor with the parameters listed in Table 1, even large-scale (usually highly energetic) noise with ${k}_x = {10}^{-4}{k}_0$ overtakes the main pulse by 0.77 ps. According to the contrary assertion made in Ref. [Reference Du, Shen, Liang, Wang, Liu and Li25], the integrated contrast does not degrade, which is evidently due to the neglect of the space–time coupling effects considered above.

It is important to note that the mentioned contrast decrease concerns only the power contrast, that is, integrally throughout the beam. From the practical point of view, contrast at the focal point is important. This contrast will not deteriorate, as the noise components overtaking the main pulse in the focal plane will be far away from the beam axis. Consequently, these components will damage the target at the beam periphery, whereas on the beam axis the target may remain intact up to the arrival of the main pulse. A quantitative study of this effect is outside the scope of this paper.

5.2 Chirp (GVD) proportional to  ${{k}}_{{x}}$ :  ${{\psi}}_{{x}{\omega } {\omega}}^{\prime \prime \prime }{{k}}_{{x}}{{\varOmega}}^{{2}}$

The physical meaning of this term is quite obvious – pulses with a wave vector ${k}_x$ acquire, in addition to the principal wave with ${k}_x = 0$ , a chirp (GVD) proportional to ${k}_x$ :

$$\begin{align*}\frac{1}{2}{\psi}_{x\omega \omega}^{\prime \prime \prime }{k}_x{\Omega}^2 = \mathrm{GVD}\left({k}_x\right)\cdot {\Omega}^2 = {k}_x\Delta {L}^{\prime}\cdot \frac{\Omega^2}{\omega_0^2}.\end{align*}$$

If the main pulse (with ${k}_x = 0$ ) at the compressor output is Fourier-transform-limited, then the noise with ${k}_x\ne 0$ will be extended in time independent of the sign of ${k}_x$ (Figure 2). This reduces filtering caused by the noise time lag or overtaking, as stretching of the lagging or overtaking noise pulse increases its time overlap with the main pulse.

A proper choice of ${L}_{1,2},\ {N}_{1,2}\;\mathrm{and}\;{\alpha}_{1,2}$ allows controlling the values of $\Delta L$ and $\Delta {L}^{\prime }$ , thus creating complex space–time distributions of the field in the focal plane. In particular, the combinations $\Delta {L}^{\prime} = 0$ , $\Delta L\ne 0$ or $\Delta {L}^{\prime}\ne 0$ , $\Delta L = 0$ demonstrated in Figure 2 can be implemented. Such distributions may be useful for charged particle acceleration or other applications. For instance, in the case depicted in Figure 2(d), the pulse duration in the focal plane at the points above and below the $z$ -axis ( $x\ne 0$ ) will be longer than the duration of the Fourier-transform-limited pulse on the $z$ -axis ( $x = 0$ ).

6 Reducing fluence fluctuations at the asymmetric four-grating compressor output

As mentioned above, all the effects proportional to the second power of ${k}_{x,y}$ in a TC as well as in an AFGC may be reduced to analogous effects in free space. The suppression of fluence fluctuations at free space propagation was analyzed analytically in Refs. [Reference Martyanov and Khazanov21,Reference Kochetkov, Martyanov, Ginzburg and Khazanov22]. These earlier results may be generalized taking into account Equations (10)–(13). Here, we will not dwell on this; instead, to clearly demonstrate the effect of compressor asymmetry, we will focus on the effects proportional to the first power of ${k}_x$ , specific for the AFGC. To do this, in Equation (7) we will neglect the last four terms proportional to ${k}_{x,y}^2$ and leave only the first two terms. Let us compare fluence fluctuations at the AFGC output with a reference, namely a TC that is a particular case of an AFGC with $\Delta L = \Delta {L}^{\prime} = 0$ . In this case, the field ${E}_{\mathrm{ref}}\left(t,x,y\right)$ at the TC output may be written as a sum of the main ${E}_0\left(x,y\right)U(t)$ and noise ${E}_{\mathrm{ref},\mathrm{n}}\left(t,x,y\right)$ fields:

(22) $$\begin{align}{E}_{\mathrm{ref}}\left(t,x,y\right) = {E}_0\left(x,y\right)U(t)+{E}_0\left(x,y\right)U(t)f\left(x,y\right),\end{align}$$

where $f\left(x,y\right)$ is a complex function and $\mid\!\! f\left(x,y\right)\!\!\!\mid\ \ll 1;\int f\left(x,y\right) \mathrm{d}S = 0$ . From Equations (3) and (7)–(9) we obtain the following:

(23) $$\begin{align}{E}_{\mathrm{out}}\left(\omega, {k}_x,{k}_y\right) = {E}_{\mathrm{ref}}\left(\omega, {k}_x,{k}_y\right)\exp \left(i\left(-\frac{\Delta L}{\omega_0}\Omega +\frac{\Delta {L}^{\prime }}{\omega_0^2}{\Omega}^2\right){k}_x\right). \end{align}$$

The fluence ${w}_{\mathrm{out}}\left(x,y\right)$ may also be represented as a sum of the main ${w}_{\mathrm{out},0}\left(x,y\right)$ and noise ${w}_{\mathrm{out},\mathrm{n}}\left(x,y\right)$ fluences:

(24) $$\begin{align}{w}_{\mathrm{out}}\left(x,y\right) = \int {\left|{E}_{\mathrm{out}}\left(t,x,y\right)\right|}^{{2}} \mathrm{d}t = {w}_{\mathrm{out},0}\left(x,y\right)+{w}_{\mathrm{out},\mathrm{n}}\left(x,y\right).\end{align}$$

By substituting Equations (22) and (2) into Equation (23) and the result into Equation (24) we find ${w}_{\mathrm{out},\mathrm{n}}\left(x,y\right)$ , from which we obtain an expression for the spectrum of fluence fluctuations ${w}_{\mathrm{out}, \mathrm{n}}\left({k}_x,{k}_y\right)$ :

(25) $$\begin{align}{\left|{w}_{\mathrm{out},\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^2 = T\left({k}_x\right)\cdot {\left|{w}_{\mathrm{ref},\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^2,\end{align}$$

where

(26) $$\begin{align}T\left({k}_x\right) = {\left|\frac{\int {\left|U\left(\Omega \right)\right|}^2\cdot \mathit{\exp}\left(i\left(-\frac{\Delta L}{\omega_0}\Omega +\frac{\Delta {L}^{\prime }}{\omega_0^2}{\Omega}^2\right){k}_x\right)\mathrm{d}\Omega}{\int {\left|U\left(\Omega \right)\right|}^{\mathbf{2}}\mathrm{d}\Omega}\right|}^2\!,\end{align}$$

where $T\left({k}_x\right)$ has the meaning of the spectrum filter transmittance. Two terms in the exponent correspond to two effects – lag (overtaking) and GVD. If we neglect the latter, that is, if $\frac{\Omega}{\omega_0}\ll \frac{\Delta L}{\Delta {L}^{\prime }}$ , then from Equation (26) we obtain a result fully analogous to temporal self-filtering^[ Reference Martyanov and Khazanov ^{21 ,} Reference Kochetkov, Martyanov, Ginzburg and Khazanov ^{22 ]}: the filter transmittance is equal to the square of the modulus of the autocorrelation function of the field ${A}_t$ with the argument equal to the time lag ${\tau}_A$ (Equation (21)):

(27) $$\begin{align}{\left|{w}_{\mathrm{out},\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^{{2}} = {\left|{A}_t\left({\tau}_A\right)\right|}^2\cdot {\left|{w}_{\mathrm{ref},\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^{{2}},\end{align}$$

where

(28) $$\begin{align}{A}_t\left(\tau \right) = \frac{\int U(t){U}^{\ast}\left(t-\tau \right) \mathrm{d}t}{\int {\left|U(t)\right|}^2 \mathrm{d}t}.\end{align}$$

The difference from the temporal self-filtering is that the time lag ${\tau}_A$ (Equation (21), as distinct from $\tau$ (Equation (17), is proportional to the first (rather than the second) power of ${k}_x$ ; besides, ${\tau}_A$ does not depend on ${k}_y$ . As a reference noise spectrum ${w}_{\mathrm{ref},\mathrm{n}}\left({k}_{{x}},{k}_y\right)$ , we take the frequently used model^[ Reference Harvey, Schröder, Choi and Duparré ^{30 ,} Reference Asadchikov, Kozhevnikov, Krivonosov, Mercier, Metzger, Morawe and Ziegler ^{31 ]}:

(29) $$\begin{align}{\left|{w}_{\mathrm{ref},\mathrm{n}}\left({k}_{{x}},{k}_y\right)\right|}^{\mathbf{2}} = \frac{\mathrm{const}}{{\left({h}^2+{k}_x^2+{k}_y^2\right)}^{\gamma }},\end{align}$$

where $\gamma$ is a constant that takes on a value from 1 to 2 (usually 1.28), and the spatial scale $\Lambda = 2\pi /h$ is smaller but commensurable with the beam size. By integrating Equation (27) with respect to ${k}_y$ , with Equation (29) taken into account, we obtain for a Gaussian pulse $U(t) = \exp \left(-\Big({t}^2/{\tau}_{\mathrm{p}}^2\right)$ a 1D spectrum:

(30) $$\begin{align}S\left({k}_x\right) &= {\int}_{-\mathbf{\infty}}^{\mathbf{\infty}}{\left|{w}_{\mathrm{out},\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^{{2}}{k}_y\notag\\ &= \mathrm{const}\cdot \frac{B\left(\frac{{1}}{{2}};\gamma -\frac{{1}}{{2}}\right)}{{\left({h}^2+{k}_x^2\right)}^{\gamma -\frac{1}{2}}}\cdot \mathit{\exp}\left(-{\left(\frac{\Delta L}{c{\tau}_{\mathrm{p}}}\cdot \frac{k_x}{k_0}\right)}^2\right),\end{align}$$

where $B\left(a;b\right)$ is a B-function. The $S\left({k}_x\right)$ spectrum for ${\tau}_{\mathrm{p}} = 30\;\mathrm{fs}$ is plotted in Figure 3 at different values of $\Delta L$ . Figure 3(a) clearly demonstrates filtering of high spatial frequencies in the AFGC (cf. the colored and the black curves). Knowing the fluence fluctuation spectrum ${\left|{w}_{\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^2$ , one can find the root mean square (rms) of the fluctuations:

(31) $$\begin{align}\mathrm{rms} = {\sigma}_{\mathrm{out}, \mathrm{ref}} = \sqrt{\int_{-\infty}^{\infty }{\left|{w}_{\mathrm{n}; \mathrm{ref}, \mathrm{out}}\left({k}_x,{k}_y\right)\right|}^2\mathrm{d}{k}_x\mathrm{d}{k}_y},\end{align}$$

Figure 3 One-dimensional fluence fluctuation spectrum $S\left({k}_x\right)$ at the AFGC output for a pulse having duration ${\tau}_{\mathrm{p}} = 30\;\mathrm{fs}$ ( $\gamma = 1.28$ ). The difference between the colored and black curves shows the efficiency of fluence smoothing at a given ${k}_x$ . The efficiency of small-scale self-focusing suppression for a pulse with duration ${\tau}_{\mathrm{p}} = 21.2\;\mathrm{fs}$ will be equal to the efficiency of fluence smoothing $S\left({k}_x\right)$ (see the text).

which is a convenient quantitative characteristic of filtering efficiency: the ${\sigma}_{\mathrm{out}}/{\sigma}_{\mathrm{ref}}$ ratio shows how many times the fluence fluctuation rms will decrease by the AFGC compared to the TC. The substitution of Equation (30) into Equation (31) and integration with respect to $\mathrm{d}{k}_x$ yield the following:

(32) $$\begin{align}{\left(\frac{\sigma_{\mathrm{out}}}{\sigma_{\mathrm{ref}}}\right)}^2 = \frac{2}{B\left(\frac{1}{2};\gamma -1\right)}{\int}_0^{\infty}\frac{\mathit{\exp}\left(-{\left(\frac{\lambda }{\Lambda}\frac{\Delta L}{c{\tau}_{\mathrm{p}}}\cdot x\right)}^2\right)}{{\left(1+{x}^2\right)}^{\gamma -1/2}} \mathrm{d}x. \end{align}$$

For a given $\gamma$ , the $\frac{\sigma_{\mathrm{out}}}{\sigma_{\mathrm{ref}}}$ ratio depends only on one parameter, $\frac{\lambda }{\Lambda}\frac{\Delta L}{c{\tau}_{\mathrm{p}}}$ . The corresponding dependences are plotted in Figure 4. One can see from the figure that the value of $\gamma$ is not very important and at large $\frac{\lambda }{\Lambda}\frac{\Delta L}{c{\tau}_{\mathrm{p}}}$ the value of $\frac{\sigma_{\mathrm{out}}}{\sigma_{\mathrm{ref}}}$ is proportional to $\sqrt{\frac{\lambda }{\Lambda}\frac{\Delta L}{c{\tau}_{\mathrm{p}}}}$ . Note that for beams of a smaller diameter (smaller $\Lambda$ ), the high-frequency components have larger values and, hence, filtering is more efficient. It is clear from Equation (32) and Figures 3 and 4 that the filtering efficiency is determined by the ratio of time $\Delta L/c$ to the pulse duration at the compressor output ${\tau}_{\mathrm{p}}$ , which once again emphasizes the relevance of the used term ‘temporal filtering’.

Figure 4 Reducing the rms of fluence fluctuation $\frac{\sigma_{\mathrm{out}}}{\sigma_{\mathrm{ref}}}$ .

To conclude, we note that to find the probability for the fluence to exceed the threshold value, it is sufficient to know the fluctuation spectrum ${\left|{w}_{\mathrm{n}}\left({k}_x,{k}_y\right)\right|}^2$ ^[ Reference Kochetkov, Kocharovskaya and Khazanov ^{17 ]}.

7 Suppressing small-scale self-focusing by means of the asymmetric four-grating compressor

Exact computation of the efficiency of small-scale self-focusing demands solution of a nonlinear nonstationary Schrödinger equation in which the field at the compressor output (Equation (4)) is used as a boundary condition at z = 0. This requires numerical simulation, which is outside the scope of the present work. However, the results obtained allow us to assess the efficiency of temporal filtering. For suppressing small-scale self-focusing it is necessary to reduce noise intensity  ${I}_{\mathrm{n}}$ (rather than noise fluence) at the time of the maximum main pulse, that is, at the time of maximum nonlinearity. Consequently, the reduction of ${I}_{\mathrm{out},\mathrm{n}}\left(t = 0,{k}_x,{k}_y\right)$ will be a good estimate. After the corresponding computations for ${I}_{\mathrm{out},\mathrm{n}}\left(t = 0,{k}_x,{k}_y\right),$ similarly to Section 6 we will obtain an expression analogous to Equation (27):

(33) $$\begin{align}{I}_{\mathrm{out},\mathrm{n}}\left(t = 0,{k}_x,{k}_y\right) = {\left|U\left(t = -\frac{k_x}{k_0}\frac{\Delta L}{c}\right)\right|}^{\mathbf{2}}{I}_{\mathrm{ref},\mathrm{n}}\left(t = 0,{k}_x,{k}_y\right).\end{align}$$

Thus, from the point of view of suppressing small-scale self-focusing, an AFGC is a filter with the transmittance equal to the normalized pulse intensity at the time $-\frac{k_x}{k_0}\frac{\Delta L}{c}$ . For a Gaussian pulse, ${\left|U(t)\right|}^2 = {\left|{A}_t\left(t/\sqrt{2}\right)\right|}^2$ . Therefore, with Equations (27) and (33) taken into account, the filtering efficiency for the intensity will be the same as for the fluence shown in Figure 3, but for a $\sqrt{2}$ times shorter pulse duration, that is, 21.2 fs instead of 30 fs. As mentioned above, of primary importance is the reduction of the value of the noise intensity spectrum ${I}_{\mathrm{n}}\left(t = 0,{k}_x,{k}_y\right)$ at spatial frequencies ${k}_{\mathrm{max}}$ , for which the instability increment is the largest. For high-power femtosecond lasers, ${k}_{\mathrm{max}}/{k}_0$ is of the order of 0.03^[ Reference Mironov, Lozhkarev, Ginzburg, Yakovlev, Luchinin, Shaykin, Khazanov, Babin, Novikov, Fadeev, Sergeev and Mourou ^{3 ,} Reference Khazanov, Mironov and Mourou ^{4 ]}, that is, as seen in Figure 3, a small compressor asymmetry $\Delta L$ is quite sufficient for effective filtering.