2.1 Pedestal noise: amplified spontaneous emission and parametric fluorescence

In the pulse with temporal contrast degradation, a pedestal always can be seen before the main pulse, appearing as a temporally long noise background. In CPA laser system, the major source of such pedestal noise is amplified spontaneous emission (ASE)^{[Reference Braun, Kopf, Jung, Rudd, Cheng, Weingarten, Keller and Mourou14]} with the duration of nanosecond range, while in OPCPA laser system parametric fluorescence (PF)^{[Reference Devaux and Lantz15, Reference Bromage, Rothhardt, Hädrich, Dorrer, Jocher, Demmler, Limpert, Tünnermann and Zuegel16]} mostly contributes to this noise background, whose duration is equal to the temporal length of the pump. In both cases, these pedestal noises are inherent with the amplifications and the typical number of the temporal contrast between the intensities of such pedestal noise and the main pulse is ${\sim}10^{7}$ . Although the intensity of such nanosecond-scale pedestal noise might be rather low compared to the ionization threshold of the targets in some physical experiments, the accumulated energy over such a long noise duration can exceed the absorption fluence threshold of the targets, resulting ablation and melting^{[Reference Kalashnikov, Andreev and Schönnagel10]}.

2.2 Pre-pulse noise: imperfect optics and nonlinear effects

Apart from the long-duration pedestal noise, the nanosecond-scale temporal contrast can also be deteriorated by some noise peaks that precedes the main pulse, called pre-pulses. On the one hand, since the optics in the laser system can hardly be perfect, the pre-pulses as well as post-pulses can be introduced in a linear way by scattering of imperfect reflective and diffractive optics, birefringence of imperfect transmitting optics, or leaking of polarizing optics. Petawatt laser systems should be carefully designed to avoid such linear pre-pulses, but usually the post-pulses still exist. On the other hand, in a nonlinear way, the post-pulses with less attention will generate satellite pulses related to the nonlinearity of the refractive index and gain, especially in the multiple-pass amplifiers and regenerative amplifiers^{[Reference Didenko, Konyashchenko, Lutsenko and Tenyakov17]}. In petawatt pulses, the intensity of such pre-pulses from the nonlinear generation can be quite strong and easily reach the ionization threshold of the target. As a result, considerable and undesired pre-plasma will be produced and degenerate the performance of the physical experiments.

2.3 Coherent noise: spectral amplitude and phase modulation

The picosecond-scale temporal contrast is determined by the steepness of the leading edge of a pulse, which is associated with the wing-shape coherent noise around the main pulse. Many research efforts have been made to find out various influences in the laser systems responding to this coherent noise. Principally, such coherent noise is resulted from the spectral amplitude and phase modulation, but for different laser systems, the specific causes comprise spectral clipping^{[Reference Kalashnikov, Andreev and Schönnagel10]}, high order dispersion^{[Reference Osvay, Csatári, Ross, Persson and Wahlstrom18]}, pump-induced noise in OPCPA system^{[Reference Forget, Cotel, Brambrink, Audebert, Le Blanc, Jullien, Albert and Chériaux19–Reference Dorrer21]} and spatiotemporal coupling such as inhomogeneous amplification^{[Reference Papadopoulos, Zou, Le Blanc, Chériaux, Georges, Druon, Mennerat, Ramirez, Martin, Fréneaux, Beluze, Lebas, Monot, Mathieu and Audebert22]}, and spatial-to-spectral phase coupling^{[Reference Dorrer and Bromage23]}. All these influences work separately or simultaneously, flattening the steepness and degrading the picosecond-scale temporal contrast of the recompressed pulse. The resulting slow rise of the main pulse can cause the plasma hydrodynamic expansion of the sensitive target and change the supposed laser–matter interaction conditions.

3.1 Lens-based spatial filters in SG-II 5 PW laser system

SG-II 5 PW high-power laser is located at National Laboratory on High Power Laser and Physics (NLHPLP) in Shanghai Institute of Optics and Fine Mechanics (SIOM). It is designed to be a multiple-stage OPCPA system working at the central wavelength of 808 nm, which will deliver an ultra-intense pulse with 150 J energy and 30 fs pulse width after compression at focal target region. The schematic of the SG-II 5 PW laser system is shown in Figure 1. A low-energy small-diameter chirped pulse is produced from an oscillator and a stretcher. The pulse gets amplified by multiple-stage amplifiers, based on OPCPA using LBO crystals. Meanwhile the pulse is propagated and expanded through multiple-stage spatial filters. Then the chirp of the high-energy large-aperture pulse is compensated in the four-grating compressor (G1–G4). Finally, the compressed high-power pulse is focused onto the target with ultrahigh intensity, utilizing an off-axis parabolic (OAP) mirror. Therefore, high temporal contrast at the focal region is required by physical experiments of laser–matter interaction.

Figure 1. Schematic of the SG-II 5 PW ultrashort pulse laser system.

The SF, not only works as a beam expander, but also filters out the high spatial frequency noise in the beam, improving the near-field beam quality and avoiding laser-induced damage to the optical elements in the laser system. The traditional spatial filter design is based on the configuration of two positive lenses (lens 1 and lens 2) and a pinhole, as shown in Figure 2. The pinhole is placed at the Fourier plane of the input lens (lens 1), passing the low frequency components and blocking the high-frequency components.

Figure 2. Schematic of an SF.

According to the image relaying requirement in SG-II 5 PW system, the multiple-stage lens-based spatial filters were designed with the parameters listed in Table 1. All the lenses used in the spatial filters are singlets. The input beam is expanded from 2.5-mm-diameter round shape to 290 mm  $\times$  290 mm square shape by 5-stage spatial filters (SF1–SF5). Because the large size LBO crystals and the gratings are square shape, between SF3 and SF4, the laser beam is spatially reshaped from round into square in order to make best use of these expensive elements of large aperture, which is achieved by using a 60 mm  $\times$  60 mm square pump light to amplify a 75-mm-diameter round signal light.

3.2 Chromatic aberration and radial group delay

However, unlike narrowband monochrome laser, the ultrashort pulse has a wide spectrum, whose spectral width is ${\sim}60$  nm in SG-II 5 PW system. In such large-aperture broadband laser system, the lens-based spatial filters will introduce chromatic aberration due to the dispersion in the material of lenses. The refractive index ( $n$ ) varies at different wavelength ( $\unicode[STIX]{x03BB}$ ), which makes the focal length ( $f$ ) of the lens changes between wavelengths:

(1) $$\begin{eqnarray}f(\unicode[STIX]{x03BB})=\frac{1}{n(\unicode[STIX]{x03BB})-1}\frac{1}{\displaystyle \frac{1}{R_{1}}-\displaystyle \frac{1}{R_{2}}}=\frac{n(\unicode[STIX]{x03BB}_{0})-1}{n(\unicode[STIX]{x03BB})-1}f(\unicode[STIX]{x03BB}_{0}),\end{eqnarray}$$

where $R_{1}$ , $R_{2}$ are the radii of curvature of the front and rear surfaces of a lens, respectively, and $\unicode[STIX]{x03BB}_{0}$ is the design wavelength, which is equal to the central wavelength of the pulse. Due to this reason, while the lens-based spatial filter operates perfectly at $\unicode[STIX]{x03BB}_{0}$ outputting a plane wave with a plane-wave input, the light at other wavelengths will experience chromatic aberration after passing through the lens-based spatial filter. The aberration is mainly defocus because of the focal length variation of the lenses related to wavelengths when the distance between the two lenses is fixed.

Table 1. Design of multiple-stage spatial filters in SG-II 5 PW system (unit: mm).

Using the matrix optics method, the curvature of the wave front at various wavelengths, $K(\unicode[STIX]{x03BB})$ , can be calculated after going through the multiple-stage spatial filters in SG-II 5 PW system. The presence of curvature indicates the defocus of the wave front and the dispersion of such defocus can be represented by the slope of the wavelength-dependent curvature, $\text{d}K/\text{d}\unicode[STIX]{x03BB}$ . As plotted in Figure 3, the curvature at central wavelength 808 nm is 0 (plane wave), while the curvatures at 778 nm and 838 nm are 0.426 km $^{-1}$ (convergent wave) and $-0.398~\text{km}^{-1}$ (divergent wave), respectively. The defocus dispersion curve (violet dashed line) tells that the relationship between wave front defocus and wavelength is nonlinear associated with the dispersion of the material of lenses. The defocus dispersion at 778 nm is $-14199$  m $^{-2}$ and the defocus dispersion at 838 nm is $-13278$  m $^{-2}$ , yielding the average defocus dispersion is $-13721$  m $^{-2}$ .

Figure 3. Wave front defocus dispersion after the multiple-stage spatial filters.

Although the central wavelength is free of chromatic aberration, RGD exists in the lens-based spatial filter for all the wavelengths. It is resulted from the positive lens, which is always thicker in the center and thinner on the edge. The light in different radial position $(r)$ will pass different length of the material and the light on the edge will lead the light in the center. The time between these two positions is called RGD or PTD^{[Reference Bor24]}. The PTD of the multiple-stage spatial filters is equal to the sum of the PTDs of all the lenses, which can be written as below according to the previous literature^{[Reference Bor25]}.

(2) $$\begin{eqnarray}\unicode[STIX]{x0394}T_{\text{SFs}}=\sum \unicode[STIX]{x0394}T_{\text{lens}}=\sum \frac{\mathop{r}_{0}^{2}-r^{2}}{2cf(\unicode[STIX]{x03BB})\left[n(\unicode[STIX]{x03BB})-1\right]}\left(-\unicode[STIX]{x03BB}\frac{\text{d}n}{\text{d}\unicode[STIX]{x03BB}}\right),\end{eqnarray}$$

where $c$ is the speed of light and $r_{0}$ is the radius of the laser beam. The PTD curves at various wavelengths (778 nm, 808 nm, and 838 nm) are plotted in Figure 4. The PTD at 778 nm is 401.2 fs, the PTD at 808 nm is 388.2 fs, and the PTD at 838 nm is 377.9 fs, giving that the shorter wavelength has a bit larger PTD in the multiple-stage spatial filters.

Figure 4. PTD of the multiple-stage spatial filters.

Therefore, the multiple-stage spatial filters introduce chromatic aberration and RGD into the ultrashort laser system and deliver a spatiotemporally coupled complex pulse with defocus dispersion and PTD.

3.3 Temporal contrast of the compressed and focused pulse

After coming out of the multiple-stage spatial filters, the pulse will go through the compressor and focused by OAP. We simulated the temporal contrast at the focal region with the influence of chromatic aberration and RGD caused by the multiple-stage lens-based spatial filters spatiotemporally coupled with compression and focusing. Four assumptions were made for this simulation: (1) the low-energy input is a temporally clean Fourier-transform-limited Gaussian pulse with spatially uniform beam profile; (2) the stretcher and compressor match with each other perfectly without any residual chirp or high order dispersion left after compression; (3) the amplifiers are so ideal and perfect that they introduce no modulation or noise to the pulse during amplification; (4) OAP focuses the pulse onto the target without any aberration or misalignment.

In the four-grating compressor, the groove density ( $d$ ) of the diffractive gratings is 1740 lines/mm, the distance in the grating pair ( $G$ ) is 712.3 mm, and the designed angle of incidence ( $\unicode[STIX]{x1D6FE}$ ) is $56^{\circ }$ . The grating equation is given by

(3) $$\begin{eqnarray}\sin \unicode[STIX]{x1D6FE}+\sin (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FD})=\frac{m\unicode[STIX]{x03BB}}{d},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ is the angle of diffraction and $m$ is the diffraction order. A schematic of a grating pair is shown in Figure 5(a) and $Y$ direction is perpendicular to the grooves of gratings. The phase change^{[Reference Treacy26]} introduced by one grating pair is:

(4) $$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D711}=\frac{2\unicode[STIX]{x1D70B}G}{\unicode[STIX]{x03BB}\cos (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FD})}(1+\cos \unicode[STIX]{x1D6FD})-\frac{2\unicode[STIX]{x1D70B}G}{d}\tan (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FD}),\end{eqnarray}$$

where the first term is contributed by light path and the second term is the phase shift of the grating pair. The phase change of the four-grating compressor is the double of the phase change of a grating pair.

Figure 5. (a) Schematic of a grating pair; (b) residual dispersion due to compression distortion at various spatial places.

If the input light into the compressor is plane wave, the angles of incidence at all the positions are the same. But the light from the multiple-stage spatial filters suffers from chromatic aberration and the defocus dispersion makes the actual angle of incidence $\unicode[STIX]{x1D6FE}$ change with the different position and wavelength. Therefore the angle of diffraction $\unicode[STIX]{x1D6FD}$ and the phase change $\unicode[STIX]{x0394}\unicode[STIX]{x1D711}$ are depended on position and wavelength, resulting compression distortion and adding residual high order spatial dispersion into the output pulse. The residual dispersion curves at various spatial places are shown in Figure 5(b). The variate $y$ is the position in $Y$ direction, which is perpendicular to the grooves of gratings. The spectral phase change is directly relatedly to the angle change in $Y$  direction, and the angle change in $X$  direction does not play any role in the phase change of the compressor. The mount of residual dispersion is larger where the position is closer to the edge and only the central position has no dispersion.

With the assumptions mentioned above, the spatiotemporal pulse after compression can be expressed by^{[Reference Trebino27, Reference Born and Wolf28]}

(5) $$\begin{eqnarray}\displaystyle & & \displaystyle E(x,y,t)=F^{-1}\{\tilde{E}(x,y,\unicode[STIX]{x03BB})\}=F^{-1}\left\{F\{E_{0}\}\exp \phantom{\frac{i\unicode[STIX]{x1D714}}{2c}}\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.\left[\frac{i\unicode[STIX]{x1D714}}{2c}K(\unicode[STIX]{x03BB})(x^{2}+y^{2})-i(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{0})\unicode[STIX]{x0394}T_{\text{SFs}}-i2\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D711}\right]\right\},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $E_{0}$ is the input pulse, the angular frequency $\unicode[STIX]{x1D714}=2\unicode[STIX]{x1D70B}/\unicode[STIX]{x03BB}$ and $\unicode[STIX]{x1D714}_{0}$ is the central angular frequency.  $F$ is the Fourier transform between time $t$ and angular frequency $\unicode[STIX]{x1D714}$ and $F^{-1}$ is the corresponding inverse Fourier transform. The input pulse is spatially uniformed and not spatiotemporally coupled. Its spectrum and spectral phase are shown in Figure 6(a). The pulses at different spatial positions after compression are shown in Figure 6(b). The pulse in the central position remained high quality while the pulse on the edge position got stretchered and went ahead of that in the central position by 800 fs. The temporal contrast of the pulse on the edge position was degraded from $10^{20}$ to $10^{7}$ . The spatial distribution of temporal contrast is shown in Figure 6(c). Thus, the large-aperture output pulse after compression is so complex that pulse distortion and temporal contrast degradation depend on spatial position.

Figure 6. (a) Spectral information of input pulse; (b) pulses at different spatial positions after compression; (c) spatial distribution of temporal contrast.

The temporal contrast at the focal region is what we are interested in, since it determines the success of the physical experiments. Focusing of the complex spatiotemporal pulse described by Equation (5) was calculated with geometric-optical method under the OAP assumption we mentioned before. The results are shown in Figure 7. The black dashed line is the idea situation without any spatiotemporal distortions, no chromatic aberration or RGD, serving as a reference for comparison. The blue line is the actual situation with both chromatic aberration and RGD. There was an obvious gentle slope in the lead edge since 1 ps ahead of the peak and the temporal contrast dropped sharply from $10^{25}$ to $10^{7}$ . The temporal profile at the focal region has an asymmetric long intensity tail. It is because the PTD of the spatial filters and residual dispersion of the compressor resulted from the chromatic aberration make the light on the edge get ahead of the light in the center and become the long tail after focusing. If we look into the two spatiotemporal distortions separately, the red line is only chromatic aberration (defocus) situation and the yellow line is only RGD (PTD) situation. Compared with the two lines, we can tell that RGD is responsible for the gentle slope in the lead edge while the chromatic aberration causes worse temporal contrast degradation.

Figure 7. (a) Pulse shapes in the focal region; (b) temporal contrast in the focal region.

Hence, we have proved that systematic spatiotemporal coupling in the high-power ultrashort laser could bring the final compressed and focused pulse a high noise level and a low steepness of the lead edge, causing temporal contrast degradation at the focal region. In order to obtain high temporal contrast pulse at the focal region in SG-II 5 PW laser system, chromatic aberration and RGD introduced by multiple-stage lens-based spatial filters should be compensated by additional compensation device or avoided by replacing lens-based spatial filters with mirror-based spatial filters in the further. If only the function of beam expanding is required and spatial filtering is not necessary, using achromatic beam expanders is a good option to minimize the chromatic aberration and PTD as well^{[Reference Jeong, Ko and Lee29]}.