## Abstract

Femtosecond petawatt (fs-PW) lasers, with femtosecond pulses and sub-meter-sized beams, could be easily distorted by spatiotemporal coupling (STC). In 2016, a femtosecond terawatt pulsed beam was experimentally reconstructed in the 3-dimensional (3D) space-time domain for the first time, and showing STC induced distortions. Referring to recently developed laser techniques, traditional first-order STCs can be controlled and then removed. However, the complex STC induced by wavefront errors in a meter-sized grating compressor, where the spatial and spectral coordinates of beams and pulses are coupled, would introduce a non-negligible and complicated distortion. Herein, we theoretically simulated this complex STC in the 3D space-time/spectrum domain and presented its evolution with various factors, which opens a new perspective to analyze CPA lasers in the 3D domain.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In 2017, the record of the highest peak-power of ultra-intense lasers was constantly refreshed twice by femtosecond petawatt (fs-PW) lasers [1, 2]. Ultra-intense lasers with peak powers and focused intensities of up to ~10 PW (1 PW = 10^{15} W) and ~10^{23-24} W/cm^{2} would help to explore unprecedented physics phenomena in the relativistic regime [3,4]. The techniques of chirped pulse amplification (CPA) and optical parametric chirped pulse amplification (OPCPA) are usually used to generate such relativistic intensity lasers [5,6]. In a CPA or OPCPA laser, instead of direct amplification, the ultrashort pulse is first stretched and then amplified, before finally being recompressed. The temporal stretching and compression are obtained by chirping the pulse with positive and negative dispersions, respectively. In most cases, the output pulse would deviate from the input one due to kinds of spectral, temporal, spatial, and coupling distortions. The complex amplitude of an optical pulse is composed of two parts: phase and amplitude, and then, as shown in Fig. 1(a), the distortions can be divided into two kinds: spectral-phase and spectral-amplitude distortions. In a small-scale CPA laser with a narrow beam-aperture, the on-axis (propagation axis) and the off-axis pulses possess negligible differences, and then a ray-pulse could be utilized for simplification. However, for a large-scale CPA laser with a large beam-aperture, a beam-pulse instead of a ray-pulse needs to be considered. In this condition, the two distortions should be further classified into four: on-axis spectral phase distortion (OSPD), on-axis spectral amplitude distortion (OSAD), spatio-spectral-phase distortion (SSPD) and spatio-spectral-amplitude distortion (SSAD) [i.e., ♥, ♠, ♣ and ♦ shown in Fig. 1(a)].

In early CPA lasers with narrow beam-apertures, ray-pulses are mainly distorted by OSPD (♥) and OSAD (♠). Then, the temporal/spectral and the spatial characteristics of the ray-pulses can be directly investigated in the time-frequency and the space domains, respectively. This separate processing method is reasonable and convenient, which has accelerated the rapid development of CPA lasers in the past three decades. Based this concept, several novel active techniques for control of OSPD and OSAD were proposed, such as liquid-crystal modulators [7], mechanically deformable mirrors [8] and acousto-optic modulators [9–11]. Especially, a commercial closed-loop compensator, i.e., the combination of Wizzler and Dazzler [12], was developed and widely used in kinds of femtosecond CPA lasers.

However, with the scaling of peak powers to a PW level, beam-apertures, especially at grating compressors, need to be spatially expanded to a sub-meter size due to optical damages and nonlinearities. Then, the ray-pulse is changed into a beam-pulse, and SSPD (♣) and SSAD (♦) cannot be simply neglected anymore. In theory, femtosecond pulses are more sensitive to spectral phase rather than spectral amplitude [13]. And, in engineering, referring to current meter-sized compression gratings, the 2D-spatial uniformity of the diffraction efficiency generally usually is very good, however that of the diffraction wavefront cannot satisfy the requirement of femtosecond pulses [14]. Thereby, more attentions recently are focused on SSPD (♣) rather than SSAD (♦). In 1986 and 2001, two theoretical methods of the Kirchhoff-Fresnel integral with the first-order approximation and the Fourier angular spectrum of a pulsed beam were proposed [15,16], which present that the propagation diffraction of a finite aperture beam would induce STC (space-dependent pulse broadening and spatiotemporal curvature) even in an ideal grating compressor without any alignment and wavefront errors. After that, more and more first-order STCs, such as angular dispersion, spatial dispersion, pulse front tilt, pulse front curvature, etc., were theoretically and experimentally studied [17–22], and a general theory was finally presented by Akturk et al. [23]. Meanwhile, kinds of novel measurement methods were presented and demonstrated [24–26]. Generally, the mechanisms of above well-researched first-order STCs include inherent physical characteristics (e.g., diffraction, dispersion, etc.), optical aberrations, alignment errors, etc., and actually, they are easy to be compensated at present. One example is the cancellation of the residual angular dispersion and the pulse front tilt caused by parallelism errors of compression gratings [20]. Three methods of accurately aligning gratings by using the Littrow angle autocollimation, the far-field monitoring and the spectrographic autocollimation were proposed [27–29]. Another example is the compensation of the pulse front curvature existing in telescopes due to the difference between phase and group velocities [30, 31]. A setup consisting of a pair of negative lenses and an Offner imaging system was designed to introduce a conjugate pulse front curvature for compensation [32].

Recently, with the rapid development of fs-PW CPA lasers, measurement of beam-pulses in the space-time domain and exploration of complex STCs involving high-order STCs attract researchers’ attentions. In 2014, Miranda et al. presented a method for characterizing ultrashort laser pulses in space and time, based on spatially resolved Fourier transform spectrometry [33]. The spatiotemporal characterization of an ultrashort pulse with pulse front tilt was demonstrated. In theory, the wavefront errors of meter-sized gratings in the compressor would easily introduce SSPD, and accordingly STC, due to a large beam-aperture and heavy angular dispersion. In 2015, Qiao et al. presented that the surface deformation of compression gratings would induce STC that is detrimental to the spatial focusability and the temporal compressibility of the output pulse [34]. The pulse stretching, focal spot enlarging and spatiotemporal splitting of the focused pulse in the 2D space-time domain were theoretically discussed. In the same year, Ma et al. experimentally demonstrated the spatiotemporal noise characterization in the far-field induced by the surface imperfections of optics in the stretcher and the compressor [35]. The STC in the stretcher would significantly degrade the far-field temporal contrast, especially in the off-axis positions. In 2016, Pariente et al. presented the first complete spatiotemporal experimental reconstruction of the *E*-field for a 100-TW peak-power laser, and revealed spatiotemporal distortions that can affect such beams [36]. According to their results, residual STC in TW/PW lasers is not as trivial as simple pulse front distortions (pulse front tilt and pulse front curvature). In the same year, Nagy et al. pointed out that the spatiotemporal characterization of a high-peak-power pulsed laser beam reveals previously undetected wavefront distortions, and a complete spatiotemporal characterization helps to exploit the potential of existing ultrahigh-intensity lasers [37]. In 2017, our group proposed that the SSPD induced by the wavefront errors of compression gratings would obviously reduce the actual peak power/intensity of fs-PW CPA lasers in both near- and far-fields [38], which greatly challenges the reliability of recently reported records of the highest peak powers [1,2]. Fortunately, the methods proposed by Miranda et al. and Pariente et al. provide a chance to verify this prediction experimentally.

To re-estimate fs-PW CPA lasers, in this paper we propose the first complete analysis of beam-pulses in the 3-dimensional (3D) space-time/frequency domain. Figure 1(b) shows the schematic of current popular ultrafast CPA lasers, which consists of an oscillator, a grating stretcher, pre- and main-amplifiers, spatial filters (telescopes), a grating compressor, and a closed-loop spectral phase compensator (i.e., Wizzler and Dazzler). Generally, the spatial modulation induced by imperfections at the stretcher can be directly considered as a spectral modulation due to two reasons: narrow beam-aperture and large angular dispersion. Moreover, owing to the current high quality of available optical transmission materials, the space-dependent material-dispersion-difference on the cross section of the beam-pulse is so small that can be neglected in engineering. In this case, the accumulated spectral phase distortion after the oscillator, the grating stretcher, amplifiers and spatial filters is approximate space-independent (pulse front curvature induced by spatial filters can be compensated [32]). In this case, with an ideal grating compressor, the final spectral phase distortion would be space-independent and can be conveniently removed with the help of the closed-loop spectral phase compensator. However, Fig. 1(b) shows in the grating compressor the frequency-dependent phase fronts separate spatially and locate at different positions of G2 and G3. Once G2 and G3 possess spatially distributed wavefront errors, different frequencies would possess different phase front distortions, and after G4 the final spectral phase distortion would vary across the beam-aperture and become space-dependent (i.e., SSPD). Unfortunately, due to strain of polishing, coating and mounting of substrates and errors in width and parallelism of grating grooves, the wavefront errors of current meter-sized compression gratings are far from ideal cases, and they are also much worse than those of other optics (e.g., mirrors, lenses, gain materials, etc.). Then, a non-negligible SSPD would be generated, however the problem is that it is generally ignored in current fs-PW CPA lasers. In this paper, we will mainly analyze the 3D spatiotemporal and spatio-spectral characteristics of fs-PW CPA lasers induced by this SSPD, which would be helpful to analyze CPA lasers in the 3D domain.

## 2. Theoretical model and parameters

#### 2.1 SSPD in the compressor

Before the compressor, owning to characterization of CPA lasers, advanced materials and novel compensation methods, the spatial variation (i.e., SSPD) can be neglected. However, at the compressor, it should be considered. For simplification, a square beam is used, and the beam-pulse is assumed to consist of 40 × 40 ray-pulses. A 3D data matrix is utilized to describe the complex amplitude in the 3D space-frequency domain, and the horizontal, vertical and longitudinal directions of the 3D data matrix denote the *x*-, *y*- and frequency-axes, respectively. Figure 1(b) shows a coordinate system of *x*–*y*–*z* is set up on the phase front of the center frequency (*ω*_{2}). The center ray at the center frequency [*ω*_{2} in Fig. 1(b)] is chosen as the *z*-axis, and the angular dispersion direction and its orthogonal direction are the *x*- and *y*-axes. For the location before G1 or after G4 (i.e., without spatial chirp), the coordinate of ray-pulses in the normal section of the beam-pulse are frequency-independent, which is given by

*x*-

*y*plane, and possesses spatial chirp. The coordinate of ray-pulses in the normal section becomes frequency-dependent, which is given by

*δx*(

*ω*) is the frequency-dependent

*x*-axis shift of an arbitrary frequency

*ω*with respect to that of the center frequency

*ω*

_{0}, which is given by [38]

*d*is the grating constant,

*α*is the incident angle,

*β*

_{0}and

*β*are the diffraction angles of

*ω*

_{0}and

*ω*, and

*T*/

*Δλ*is the chirp rate (i.e., the ratio between the duration

*T*and the bandwidth

*Δλ*of the corresponding chirped pulse). + and – in Eq. (3) applies to the compressor configuration with an incident angle larger and smaller than the Littrow angle, respectively.

Equations (1) and (2) show that G1/4 and G2/3 possess frequency-independent and frequency-dependent influences due to without and with spatial chirp, respectively. Mathematically, by combining “like terms”, the diffraction wavefront of G1 and G2 can be overlaid onto that of G4 and G3, respectively, and two mathematically overlaid diffraction wavefronts of *f*_{G1&4} and *f*_{G2&3} are used to describe the combinations of G1 and G4 (G1&4) and G2 and G3 (G2&3). Referring to the measured diffraction wavefronts of meter-sized gratings in our laboratory [38], a sine function distortion is assumed for general theoretical research, and to achieve a general result, a 2D distortion in the *x*-*y* plane is generated by

*H*

_{1}and

*H*

_{2}are peak-to-valley (PV) values of low and high spatial frequency wavefront errors,

*L*

_{x}and

*L*

_{y}are spatial periods of the low spatial frequency wavefront error in the

*x*- and

*y*-axes, and

*δ*(

*x*,

*y*) is a space-dependent random number.

By substitution of Eqs. (1) and (2) into (4), the phase difference involving SSPD induced by wavefront errors of compression gratings (G1-4) is given by

For the generally used closed-loop spectral phase compensator (e.g., Wizzler and Dazzler), as shown in Fig. 1(b), a very narrow beam (<3mm) is injected into Wizzler and Dazzler for measurement and compensation, respectively, and, consequently, they have no spatial resolution. After several cycles of the measurement-compensation operation, a “spectral phase compensation” condition would be achieved, and the phase after this process can be described by

*ϕ*

_{pd}(

*x*,

*y*,

*ω*)|

_{x}_{=}

_{y}_{= 0}is a 3D data matrix, and for each spatial position of (

*x*,

*y*) the spectral phase equals to the one on the

*z*-axis (i.e.,

*x*=

*y*= 0, the narrow beam injected into Wizzler). Equation (6) shows the spectral phase compensation only applies to the on-axis position (i.e.,

*x*=

*y*= 0), and the off-axis positions still possess residual spectral phase distortions in the 3D space-frequency domain.

#### 2.2 Fourier relationships

In the case of an ideal CPA laser without SSPD or SSAD, the input and output complex amplitudes satisfied the relationship

*ϕ*

_{CPA}(

*ω*) is the spectral phase introduced by dispersion elements which equals to zero for the dispersion-free condition, and

*M*(

*x*,

*y*) denotes the beam expanding by telescopes.

Under the plane wave approximation, after the grating compressor, the complex amplitude with SSPD *E*(*x*,*y*,*ω*) in the 3D space-frequency domain is equal to the dot product between the 3D complex amplitude *E*_{CPA}(*x*,*y*,*ω*) and the 3D SSPD phase exp[*iϕ*_{SSPD}(*x*,*y*,*ω*)], which is given by

*E*-field of the compressed pulse in the 3D space-time domain can be obtained by the inverse Fourier transform of each ray-pulse

*E*(

*x*,

*y*,

*ω*) in frequency, which is given bywhere

*iFT*denotes the inverse Fourier transform. The longitudinal direction of the 3D data matrix is changed from the frequency-axis to the time-axis.

The focusing process from the near-field (i.e., the input plane of the focusing optics) to the far-field (i.e., the focal plane) can be calculated by the 2D Fourier transform of each monochromatic field *E*(*x*,*y*,*ω*) in space, which is given by

*FT*

_{2}denotes the 2D Fourier transform. Similarly, the

*E*-field of the focused pulse in the 3D space-time domain can be calculated by the inverse Fourier transform of each ray-pulse

*E*(

*ξ*,

*η*,

*ω*) in frequency, which is given by

The compressed and focused pulses in the 3D space-time domain can be obtained based on the *E*-fields of Eqs. (9) and (11), and the 3D temporal contrast can be achieved by calculating the relative intensity.

#### 2.3 Simulation parameters

The laser pulse possesses a Gaussian spectrum with a center wavelength of 800 nm and a bandwidth (Full width at half maximum, FWHM) of 50 nm. The spectral window of the stretcher is 700-900 nm. The input beam of the compressor possesses a 400 mm square aperture and a flat-top intensity distribution. The total chirp rate is 1.5 ns/50 nm (i.e., 0.75 ns/50 nm for each pair of G1-G2 and G3-G4). The grating groove density is 1480 g/mm, and the incident angle is 46.3° (i.e., 10° larger than the Littrow angle). The PV value and the spatial period of the low spatial frequency wavefront error of compression gratings are *λ*_{m}/4 (*λ*_{m} = 633 nm) and 300 mm (i.e., spatial frequencies 3.3 m^{−1}), respectively, and the high spatial frequency wavefront error possesses a random amplitude with a maximum PV of *λ*_{m}/20 (*λ*_{m} = 633 nm) and a fixed spatial frequency of 100 m^{−1}. The focal length of the focusing optics is 2 m. For the Fourier transforms, in time, the window size and the resolution are 400 fs and 1 fs, and in space, which are 4000 mm and 10 mm.

## 3. Simulation results and analysis

#### 3.1 Phase in the 3D space-frequency domain

Generally, the spectral phase of a CPA laser is written as the Taylor expansion about the center angular frequency *ω*_{0} in the frequency domain,

*ϕ*′(

*ω*

_{0}),

*ϕ*′′(

*ω*

_{0}), and

*ϕ*′′′(

*ω*

_{0}) are the derivatives of the phase with respect to the angular frequency at the center one, and indicate, respectively, the group delay (GD), group velocity dispersion (GVD) and third-order dispersion (TOD). Before the compressor, due to the large-amount of positive chirp introduced by the stretcher,

*ϕ*′′(

*ω*

_{0}) is positive, and both

*ϕ*′(

*ω*

_{0}) and

*ϕ*′′′(

*ω*

_{0}) are negative [39]. Then, the phase of a chirped pulse in the 3D space-frequency domain is illustrated by Fig. 2(a), and the variation in the 2D-space is neglected due to the space-independence explained in the section 1. In the case of a perfect dispersion compensation (i.e., perfect pulse compression), the phase would be changed to a both space- and frequency-independent constant in the 3D domain [Fig. 2(b)]. However, once the SSPD induced by diffraction wavefront errors of compression gratings are considered, the actual cases would deviate from the ideal one.

When only *f*_{G1&4} is considered, Fig. 2(c) shows that a frequency-independent but space-dependent phase distortion appears, which cannot introduce a spectral phase distortion but cause a space-dependent time delay. In the case of only *f*_{G2&3}, Fig. 2(d) shows that the phase is twisted along the angular dispersion direction (*x*-axis), and becomes both frequency- and space-dependent, which would induce a space-dependent spectral phase distortion. Figure 2(e) shows the 3D phase in the case of both *f*_{G1&4} and *f*_{G2&3}, which distorts dramatically in the 3D domain. By comparing two orthogonal sections (*x*-*ω* and *y*-*ω* planes for *y* = 0 and *x* = 0, respectively) [Fig. 2(f)], the phase peak/valley trajectory in the *x*-*ω* plane is tilted and in the *y*-*ω* plane is parallel to the *ω*-axis. For an *x* position in the *x*-*ω* plane, the spectral phase oscillates with a large amplitude, while for a *y* position in the *y*-*ω* plane, the oscillation amplitude is relatively small. This result illustrates that the spectral phase distortion mainly occurs along the angular dispersion direction (*x*-axis).

Next, when the spectral phase compensator (e.g., Wizzler and Dazzler) works, the compensated phase in the 3D domain is shown in Figs. 2(g) and 2(h) shows the distortion-free spectral phase only applies to the on-axis position (i.e., *x* = *y* = 0). By comparing with Figs. 2(f) and 2(h) has three major changes: (i) the OSPD is removed; (ii) the oscillation amplitudes of the off-axis (*x*≠0) spectral phases in the *x*-*ω* plane are increased [Fig. 2(h)(left)]; and (iii) those in the *y*-*ω* plane are reduced, and the mean value varies along the *y*-axis [Fig. 2(h)(right)]. Then, we can predict that: (i) the Fourier-transform-limited (FTL) pulse can be achieved only at the on-axis position; (ii) the space-dependent pulse distortion mainly occurs along the *x*-axis; and (iii) the space-dependent time delay mainly happens along the y-axis.

#### 3.2 Compressed pulse in the 3D space-time domain

In the ideal case without the 3D phase distortion [Fig. 2(b)], Figs. 3(a)-3(c) show the normalized and relative intensity of the compressed pulse in the 3D space-time domain, and FTL pulses and high temporal contrasts are obtained at every spatial position. However, when the 3D phase distortion is considered [Fig. 2(g)], Figs. 3(d)-3(f) show both the beam-pulse and the beam-temporal-contrast distort seriously. In Fig. 3(d), along the *x*-axis, the FTL pulse (i.e., the highest peak intensity) appears only at the center position (i.e., on-axis position). For other positions, pulses are spatiotemporally curved and temporally stretched, and from around ± 50 to ± 200 mm pre-pulses are observed too. The reasons to the pulse curvature, the pulse stretching and the pre-pulses are the variations of the space-dependent GD, GVD and high-order dispersion, respectively, and all of these are induced by the space-dependent spectral phase distortion shown in Fig. 2(h)(left). However, along the y-axis, no significant change is observed in Fig. 3(d). It is because that along this direction there is no angular dispersion, accordingly no spatio-spectral coupling. The wavefront error only induce a space-dependent but spectrum-independent phase difference (i.e., space-dependent time delay across the beam aperture) [Fig. 2(h)(right)]. Here let’s make a simple calculation, for a wavefront variation with a *λ*_{m}/2 (i.e., ~316 nm) PV value, the maximum time delay PV/c is only ~1.05 fs. Comparing with a ~15-30 fs pulse duration (FWHM), this time delay variation across the beam aperture is so small that cannot be conveniently observed.

Compared with the distortion of the beam-pulse, the beam-temporal-contrast degrade even more dramatically. Figures 3(e) and 3(f) show the relative intensity in the 3D domain and the details in the two orthogonal sections (*x*-*t* and *y*-*t* planes for *y* = 0 and *x* = 0, respectively). It can be found that only the on-axis pulse (i.e., *x* = *y* = 0) possesses a high temporal contrast, and at off-axis positions temporal noises appear both before and after the signals. As illustrated in Fig. 2(h), for the on-axis pulse, an ideal spectral phase could be obtained due to the spectral phase compensator (e.g., Wizzler and Dazzler), however, for off-axis pulses, the residual spectral phase distortions, especially middle and high frequency modulations, would degrade the temporal contrasts dramatically.

#### 3.3 Focused pulse in the 3D space-time domain

When the beam-pulses in Figs. 3(a) and 3(d) (i.e., without and with SSPD) are focused by an ideal focusing optics with a 2 m focal length, Figs. 4(a) and 4(d), respectively, show the focused pulses in the 3D space-time domain. By comparing these two results, the focused pulse in Fig. 4(d) possesses two major spatiotemporal distortions: the main focal spot distorts in time, and several side lobes with different spatial and temporal locations and profiles appear. The formation mechanism of the side lobes is mainly due to the low spatial frequency wavefront errors of four compression gratings (G1-4). And the spatiotemporal distortion of the main focal spot and the side lobes is mainly caused by the complex STC induced by wavefront errors of every compression grating (G1-4).

Figures 4(b)-4(f) illustrate the relative intensities of Figs. 4(a) and 4(d) in the 3D domain and the details in the two orthogonal sections (*ξ*-*t* and *η*-*t* planes for *η* = 0 and *ξ* = 0, respectively). By comparing with Figs. 4(b) and 4(c), the temporal contrast in Figs. 4(e) and 4(f), especially the on-axis temporal contrast (i.e., *ξ* = *η* = 0), degrades seriously, which is even worse than the off-axis temporal contrasts in the near-field [Figs. 3(e) and 3(f)]. In physical applications, it is a big challenge. Because the temporal contrast is a key parameter which is usually measured in the near-field (i.e., compressed pulse), however the laser is actually used in the far-field (i.e., focused pulse). Consequently, in the following sections, we will analyze the specific influences of different factors on the complex STC and the far-field temporal contrast degradation.

#### 3.4 Influence of low and high spatial frequency wavefront errors

To obtain different influences of low and high spatial frequency wavefront errors, we firstly remove the high spatial frequency wavefront errors and only keep the low spatial frequency ones, and the results are shown in Figs. 5(a)-5(f). By comparing the near-field compressed pulse [Fig. 5(a)] and the far-field focused pulse [Fig. 5(d)] with the results in Figs. 3(d) and 4(d) (both low and high spatial frequency wavefront errors are considered), no significant changes are observed. However, by comparing the near-field temporal contrast [Figs. 5(b) and 5(c)] and the far-field temporal contrast [Figs. 5(e) and 5(f)] with the results in Figs. 3(e), 3(f), 4(e) and 4(f) (both low and high spatial frequency wavefront errors are considered), both the near- and far-field temporal contrasts are slightly improved. Secondly, the low spatial frequency wavefront errors are removed, and the influences of only the high spatial frequency ones are illustrated in Figs. 5(g)-5(l). Figures 5(g) and 5(j) show that the STCs of the compressed and focused pulses disappear. But, comparing with Figs. 5(b) and 5(c), the near-field temporal contrast degrades again [Figs. 5(h) and 5(i)]. For the far-field temporal contrast, only the off-axis ones are enhanced, and the degradation of the on-axis one still exists [Figs. 5(k) and 5(l)]. We suppose the main reasons to this phenomenon include the degradation of the near-field temporal contrast and the improvement of the spatial focusability, which would spatially focus the generated temporal noises on the propagation axis.

Therefore, it is reasonable to conclude that: (i) the STCs of the compressed and focused pulses are determined by the low spatial frequency wavefront errors; (ii) the spatial focusability of the focused pulse is also determined by the low spatial frequency wavefront errors; and (iii) the near-field temporal contrast is mainly affected by the high spatial frequency wavefront errors, while the far-field temporal contrast is influenced by both the low and high spatial frequency wavefront errors.

#### 3.5 Influence of G1&4 and G2&3

To investigate the detailed influences of different compression gratings G1-4, firstly only the overlaid diffraction wavefront of *f*_{G2&3} is considered (i.e., *f*_{G1&4} is removed), and the compressed pulse, the focused pulse, and the far-field temporal contrast are illustrated in Figs. 6(a)-6(d). The STC of the compressed pulse almost remains no change [Fig. 6(a)]. However, the spatial focusability of the focused pulse is significantly improved, i.e., several side lobes completely disappear [Fig. 6(b)], but temporal stretching of the main focal spot and temporal deviations of residual side lobes still exist. For the far-field temporal contrast, especially the on-axis one, no obvious improvement is observed [Figs. 6(c) and 6(d)], and only the spatial focusability, comparing with Figs. 4(e) and 4(f), is slightly enhanced. Secondly, we only consider the overlaid wavefront of *f*_{G1&4} (i.e., *f*_{G2&3} is removed), and the corresponding results are shown in Figs. 6(e)-6(h). The STC of the compressed pulse disappears [Fig. 6(e)], and the STC of the focused pulse is improved (i.e., the number of side lobes, the temporal deviation of side lobes and the temporal length of the main focal spot are reduced) [Fig. 6(f)]. The far-field temporal contrast, comparing with Figs. 6(c) and 6(d), is enhanced in a large scale [Figs. 6(g) and 6(h)].

The major reason to the difference between Figs. 6(a)-6(d) and Figs. 6(e)-6(h) is the change of the 3D phase. Referring to Fig. 2(d), the 3D phase distorted by only *f*_{G2&3} is both frequency- and space-dependent, and then the STCs of the compressed and focused pulses [Figs. 6(a) and 6(b)] and the degradation of the far-field temporal contrast [Figs. 6(c) and 6(d)] cannot be avoided. For the spatial improvement of the focused pulse [Fig. 6(b)], it is because that the overall wavefront errors are reduced due to the cancel of *f*_{G1&4}. Similarly, the 3D phase in Fig. 2(c) only considering *f*_{G1&4} is space-dependent but frequency-independent, which means the spectral phase distortion and the STC are removed. The FTL pulse should be achieved in the 2D space [Fig. 6(e)], and only a space-dependent time delay exists. However, compared with the pulse duration (~20fs here), the time delay is too small to be observed (e. g., a 3 rad phase difference corresponds to a ~1.3 fs time delay at the 800 nm). The spatial and temporal improvements of the focused pulse [Fig. 6(f)] are mainly due to the reduction of the overall wavefront errors (the cancel of *f*_{G2&3}) and the cancel of STC. The significant enhancement of the far-field temporal contrast [Figs. 6(g) and 6(h)] benefits from the cancel of the spectral phase distortion and STC.

Thereby, it is easy to conclude that: (i) the STCs of the compressed and focused pulses are actually determined by G2&3; (ii) the spatial focusability of the focused pulse is affected by both G1&4 and G2&3; and (iii) the far-field temporal contrast is determined by G2&3.

#### 3.6 Influence of the x- and y-axes wavefront errors of G2&3

In the further investigation, keeping the negligence of the overlaid wavefront errors of G1&4, the influence of the 1D overlaid wavefront error of G2&3 along the angular dispersion direction (*x*-axis) and its orthogonal direction (*y*-axis) is individually discussed. Figures 7(a)-7(d) show the results while only the *x*-axis wavefront error is considered (i.e., the *y*-axis wavefront error is removed), by comparing with Figs. 6(a)-6(d) (both two-axes wavefront errors are considered), the STC of the compressed pulse remains no change [Fig. 7(a)]; the focused pulse is spatially improved (i.e., the number of side lobes is reduced) [Fig. 7(b)]; the far-field temporal contrast possesses no significant change [Figs. 7(c) and 7(d)]; but both the spatial focusability and the temporal compressibility in the section of *ξ* = 0 are slightly improved [Fig. 7(d)]. After that, only the *y*-axis wavefront error is considered (i.e., the *x*-axis wavefront error is removed), and Figs. 7(e)-7(h) show the results. Comparing with Figs. 6(a)-6(d) (both two-axes wavefront errors are considered), the STC of the compressed pulse disappears [Fig. 7(e)]; the focused pulse is spatially and temporally improved (i.e., the number of side lobes, the temporal deviation of side lobes and the temporal length of the main focal spot are reduced) [Fig. 7(f)]; and the far-field temporal contrast is significantly enhanced [Figs. 7(g) and 7(h)].

We can conclude that: (i) the STCs of the compressed and focused pulses are mainly introduced by the *x*-axis wavefront error of G2&3; (ii) the spatial focusability is influenced by both the *x*- and *y*-axes wavefront errors of G2&3; and (iii) the far-field temporal contrast is determined by the *x*-axis wavefront error of G2&3. This result agrees well with the conclusion by Bromage *et al*. in 2012 that phase noise, added across the near-field of a spectrally dispersed beam, produces space–time coupling in the far-field or focal plane [40].

## 4. Discussion and a quantitative result

In the section 3, to reduce the computation load and the complexity, the plane wave approximation is used, that is, the beam-pulse is considered as a combination of a series of monochromatic fields *E*(*ω*), and each monochromatic field *E*(*x*,*y*,*ω*) is a plane wave. Using Treacy's method of describing the dispersion of a grating compressor [41], optical paths for plane waves are computed to determine the spectral phase *ϕ*(*ω*). In the case of an ideal grating compressor, *ϕ*(*ω*) has nothing to do with the spatial position (*x,y*) in the normal plane of the propagation direction *z*. The plane wave approximation neglects the propagation diffraction and accordingly the STC presented by Martinez and Wang [15, 16]. If it is considered, the Fourier angular spectrum method can be utilized [16]. Then, each monochromatic field *E*(*x*,*y*,*ω*) is further decomposed by a series of the plane-wave angular spectrum *A*(*f*_{x},*f*_{y},*ω*). The monochromatic filed *E*(*x*,*y*,*ω*) and its angular spectrum *A*(*f*_{x},*f*_{y},*ω*) satisfy the 2D Fourier relationship. The propagation diffraction can be added into each angular spectrum *A*_{z}(*f*_{x},*f*_{y},*ω*), and the monochromatic filed *E*_{z}(*x*,*y*,*ω*) after the propagation can be reconstructed by the 2D Fourier transform of *A*_{z}(*f*_{x},*f*_{y},*ω*). Finally, the *E*-field of the compressed beam-pulse *E*_{z}(*x*,*y*,*t*) in space and time can be obtained by the space-dependent inverse Fourier transform in frequency. Based on the parameters given in the section 2.3, let’s compare the pulse compression and the beam propagation in an ideal grating compressor using these two methods. Figures 8(a)-8(d) and Figs. 8(e)-8(h) show the results using the methods of plane wave approximation and Fourier angular spectrum, respectively. For the compressed [Figs. 8(a) and 8(e)] and focused [Figs. 8(b) and 8(f)] pulses in the space-time domain obtained by two methods, we cannot find any differences. The relative intensities, which denote the temporal contrasts, are shown in Figs. 8(c), 8(d), 8(g) and 8(h). By comparing Figs. 8(c) and 8(g), the propagation diffraction induces weak intensity modulations at the beam edges in the near-field. However, compared with the noises before and after the main pulse in time, it is so weak that can be neglected. In the far-field [Figs. 8(d) and 8(h)], no obvious difference can be observed. Like Ref [16], we calculate GD, GVD and TOD versus the angular spectrum *f*_{x}, which determine the direction-dependent pulse front curvature, pulse stretching and temporal contrast, respectively, (i.e., STC distortion). Figures 8(i)-8(k) show that, for an angular spectrum width of 40 m^{−1}, the largest variation of GD, GVD, and TOD is only 1.4 × 10^{−3} fs, 7.2 × 10^{−3} fs^{2}, and 3.0 × 10^{−2} fs^{3}, respectively, and, comparing with a beam-pulse with a 10-30 fs duration, this direction-dependent variation of GD, GVD and TOD is negligible. Figures 8(i)-8(k) also illustrate the angular spectrum distribution of the beam-pulse after the non-ideal grating compressor with wavefront errors of gratings (400 mm aperture and 50 nm bandwidth, see other parameters in the section 2.3). The width of the angular spectrum *f*_{x} of the distorted beam-pulse is only around 6 m^{−1}, which is why the propagation diffraction induced STC distortion cannot be observed. Actually, for a 20fs, 1-10 PW CPA laser, according to the reported 0.4-0.8 J/cm^{2} femtosecond laser damage threshold for broadband gratings [42], if the energy fluence is set to 0.2 J/cm^{2} for considering both the beam fill factor and the system’s safety coefficient, the aperture of a square and flat-top beam is 100-320 mm, and the width of angular spectrum of the beam-pulse is only 6.25-20 m^{−1}. The propagation diffraction induced STC distortion in the grating compressor is still negligible. Besides that, for the amplitude modulation caused by the spatial-phase modulation and the propagation diffraction, when the overlaid diffraction wavefronts are distributed to each grating evenly, Fig. 8(l) shows the intensity variation of the center frequency, which is very tiny (<0.1%) and negligible. Then, according to above discussions, it can be concluded that for an fs-PW CPA laser the plane wave approximation method is reasonable, and the analysis in the section 3 is reliable.

Consequently, from the section 3, we should emphasize that the wavefront errors of gratings in the compressor would induce SSPD, which cannot be removed by the currently used closed-loop spectral phase compensator (e.g., the combination of Wizzler and Dazzler). This SSPD could introduce a complex STC (including both low and high order STCs), and distort the compressed and focused pulses in the 3D space-time domain. According to the classification analysis, as given in Table 1, we can conclude that: (i) the STCs of the compressed and focused pulses are mainly introduced by the *x*-axis low spatial frequency wavefront error of G2&3; (ii) the spatial focusability of the focused pulse is affected by the low spatial frequency wavefront error (both *x*- and *y*-axes) of each compression grating (G1-4); and (iii) the far-field temporal contrast is mainly influenced by the *x*-axis wavefront error (both low and high spatial frequencies) of G2&3. In this case, the *x*-axis (i.e., angular dispersion direction) wavefront error of G2&3 is a key parameter to the degradation of CPA beam-pulses. However, the problem is that, comparing with the strain of polishing, coating and mounting of substrates, the errors in width and parallelism of grating grooves are more serious, which induce the major distortions of diffraction wavefronts. Figure 9(a) shows the wavefronts of compression gratings measured in our Laboratory, and the typical PV and spatial period for different gratings is ~0.075-0.32 wave (1064 nm) and ~35-170 mm, respectively. The significant periodical variations are along the *x*-axis (angular dispersion direction). In this condition, the degradation of CPA beam-pulses in the 3D domain cannot be avoided in engineering.

To obtain a quantitative result, we neglect the wavefront errors of G1&4 and the *y*-axis wavefront error of G2&3, and investigate the detailed influences of the PV and the spatial frequency of the *x*-axis wavefront error of G2&3 (colored in Table 1) on two key parameters: the focused peak intensity (FPI) and the on-axis far-field temporal contrast (OFTC). OFTC is detected 800 fs before the main pulse (~20 fs FWHM pulse duration), and the window size and the resolution of the Fourier transform here are changed into 1600 fs and 4 fs. The initial beam aperture is 400 mm. Figure 9(b)(i) shows FPI decreases with increasing PV, and the sensitivity to the spatial frequency is very low. Figure 9(b)(ii) shows OFTC degrades with increasing PV, and the degradation is more sensitive to high spatial frequencies rather than low ones. When PV is fixed as 0.2 ( × 633 nm), Fig. 9(b)(iii) shows FPI gradually decreases with increasing beam diameter for a low spatial frequency of 3.3 m^{−1}, but for high spatial frequencies of 10 and 20 m^{−1}, the sensitivities of FPI to both beam diameter and spatial frequency are very low. Figure 9(b)(iv) shows OFTC degrades with increasing spatial frequency, and it also degrades with increasing beam diameter for a low spatial frequency of 3.3 m^{−1}. However, for high spatial frequencies of 10 and 20 m^{−1}, the sensitivity of OFTC to beam diameter is very low. Then, for an actual fs-PW CAP laser, as shown in Table 2, if the decreased FPI is set as >50%, PV should be controlled less than 0.3 ( × 633 nm); and if the required OFTC is <-50 dB, for various spatial frequencies of 3.3, 10 and 20 m^{−1}, PV should below 0.3, 0.25 and 0.1 ( × 633 nm), respectively. To improve OFTC, the PV of a higher spatial frequency should be further reduced, and this requirement just agrees well with the characteristic of wavefront errors: the PV of high spatial frequencies generally is less than that of low ones. Besides that, the quantitative requirement corresponds to the combination of G2 and G3 (i.e., G2&3), and it should be further divided into individual G2 or G3 for actual superposition conditions.

## 5. Conclusion

In conclusion, we theoretically simulate the complete 3D spatiotemporal and spatio-spectral characteristics of ultra-intense CPA lasers induced by wavefront errors of compression gratings. 3D distortions of phases, compressed pulses and focused pulses are simulated and compared in different cases. Detailed influences of diffraction wavefront errors with different spatial frequencies (low and high), at different grating positions (G1-4), and along different spatial axes (*x* and *y*) are investigated and discussed. A general relationship between major distortions and key factors is concluded, and a quantitative requirement of wavefront errors to avoid significant degradations of FPI and OFTC is presented as an example. The result proves useful for improving and developing fs-PW CPA lasers, especially for recent 10 PW and future 100 PW laser projects. And even more importantly, we believe this work provides a new perspective to analyze CPA lasers, especially fs-PW CPA lasers, in the 3D space-time domain.

## Funding

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI) (JP25247096).

## References and links

**1. **X. Zeng, K. Zhou, Y. Zuo, Q. Zhu, J. Su, X. Wang, X. Wang, X. Huang, X. Jiang, D. Jiang, Y. Guo, N. Xie, S. Zhou, Z. Wu, J. Mu, H. Peng, and F. Jing, “Multi-petawatt laser facility fully based on optical parametric chirped-pulse amplification,” Opt. Lett. **42**(10), 2014–2017 (2017). [CrossRef] [PubMed]

**2. **J. H. Sung, H. W. Lee, J. Y. Yoo, J. W. Yoon, C. W. Lee, J. M. Yang, Y. J. Son, Y. H. Jang, S. K. Lee, and C. H. Nam, “4.2 PW, 20 fs Ti:sapphire laser at 0.1 Hz,” Opt. Lett. **42**(11), 2058–2061 (2017). [CrossRef] [PubMed]

**3. **G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. **78**(2), 309–371 (2006). [CrossRef]

**4. **Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. **427**(2–3), 41–155 (2006). [CrossRef]

**5. **D. Strickland and G. A. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. **56**(3), 219–221 (1985). [CrossRef]

**6. **A. Dubietis, G. Jonušauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. **88**(4), 437–440 (1992). [CrossRef]

**7. **A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. **28**(4), 908–920 (1992). [CrossRef]

**8. **E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. **24**(7), 493–495 (1999). [CrossRef] [PubMed]

**9. **M. A. Dugan, J. X. Tull, and W. S. Warren, “High-resolution acousto-optic shaping of unamplified and amplified femtosecond laser pulses,” J. Opt. Soc. Am. B **14**(9), 2348–2358 (1997). [CrossRef]

**10. **F. Verluise, V. Laude, Z. Cheng, C. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. **25**(8), 575–577 (2000). [CrossRef] [PubMed]

**11. **F. Tavella, Y. Nomura, L. Veisz, V. Pervak, A. Marcinkevičius, and F. Krausz, “Dispersion management for a sub-10-fs, 10 TW optical parametric chirped-pulse amplifier,” Opt. Lett. **32**(15), 2227–2229 (2007). [CrossRef] [PubMed]

**12. **http://www.fastlite.com/en/

**13. **Z. Li, S. Li, C. Wang, Y. Xu, F. Wu, Y. Li, and Y. Leng, “Stable and near Fourier-transform-limit 30fs pulse compression with a tiled grating compressor scheme,” Opt. Express **23**(26), 33386–33395 (2015). [CrossRef] [PubMed]

**14. **D. Alessi, “High-average-power diffraction pulse-compression gratings enabling next-generation ultrafast laser systems,” LDRD Annual Report, LLNL-TR-707794, Nov. 2, 2016.

**15. **O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B **3**(7), 929–934 (1986). [CrossRef]

**16. **Z. Wang, Z. Xu, and Z. Zhang, “A new theory for the treatment of a pulsed beam propagating through a grating pair,” IEEE J. Quantum Electron. **37**(1), 1–11 (2001). [CrossRef]

**17. **Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular dispersion,” Opt. Eng. **32**(10), 2501–2504 (1993). [CrossRef]

**18. **C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. **30**(7), 1662–1670 (1994). [CrossRef]

**19. **G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B **70**(1), 1–9 (2000). [CrossRef]

**20. **K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. **10**(1), 213–220 (2004). [CrossRef]

**21. **X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. **242**(4-6), 599–604 (2004). [CrossRef]

**22. **S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. **12**(9), 093001 (2010). [CrossRef]

**23. **S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams,” Opt. Express **13**(21), 8642–8661 (2005). [CrossRef] [PubMed]

**24. **K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High-precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B **74**(S1), S259–S263 (2002). [CrossRef]

**25. **S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express **11**(5), 491–501 (2003). [CrossRef] [PubMed]

**26. **A. Börzsönyi, L. Mangin-Thro, G. Cheriaux, and K. Osvay, “Two-dimensional single-shot measurement of angular dispersion for compressor alignment,” Opt. Lett. **38**(4), 410–412 (2013). [CrossRef] [PubMed]

**27. **M. J. Guardalben, “Littrow angle method to remove alignment errors in grating pulse compressors,” Appl. Opt. **47**(27), 4959–4964 (2008). [CrossRef] [PubMed]

**28. **F. Liu, X. Liu, Z. Wang, J. Ma, X. Liu, L. Zhang, J. Wang, S. Wang, X. Lin, Y. Li, L. Chen, Z. Wei, and J. Zhang, “Compression grating alignment by far-field monitoring,” Appl. Phys. B **101**(3), 587–591 (2010). [CrossRef]

**29. **S. Li, Z. Li, C. Wang, Y. Xu, Y. Li, Y. Leng, and R. Li, “Broadband spectrographic method for precision alignment of compression gratings,” Opt. Eng. **55**(8), 086105 (2016). [CrossRef]

**30. **Z. Bor, Z. Gogolak, and G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. **14**(16), 862–864 (1989). [CrossRef] [PubMed]

**31. **B. Sun, P. S. Salter, and M. J. Booth, “Pulse front adaptive optics: a new method for control of ultrashort laser pulses,” Opt. Express **23**(15), 19348–19357 (2015). [CrossRef] [PubMed]

**32. **S. W. Bahk, J. Bromage, and J. D. Zuegel, “Offner radial group delay compensator for ultra-broadband laser beam transport,” Opt. Lett. **39**(4), 1081–1084 (2014). [CrossRef] [PubMed]

**33. **M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort laser pulses using spatially resolved Fourier transform spectrometry,” Opt. Lett. **39**(17), 5142–5145 (2014). [CrossRef] [PubMed]

**34. **J. Qiao, J. Papa, and X. Liu, “Spatio-temporal modeling and optimization of a deformable-grating compressor for short high-energy laser pulses,” Opt. Express **23**(20), 25923–25934 (2015). [CrossRef] [PubMed]

**35. **J. Ma, P. Yuan, J. Wang, Y. Wang, G. Xie, H. Zhu, and L. Qian, “Spatiotemporal noise characterization for chirped-pulse amplification systems,” Nat. Commun. **6**(1), 6192 (2015). [CrossRef] [PubMed]

**36. **G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space–time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics **10**(8), 547–553 (2016). [CrossRef]

**37. **T. Nagy and G. Steinmeyer, “Ultrafast optics: A closer look at ultra-intense lasers,” Nat. Photonics **10**(8), 502–504 (2016). [CrossRef]

**38. **Z. Li, K. Tsubakimoto, H. Yoshida, Y. Nakata, and N. Miyanaga, “Degradation of femtosecond petawatt laser beams: spatio-temporal/spectral coupling induced by wavefront errors of compression gratings,” Appl. Phys. Express **10**(10), 102702 (2017). [CrossRef]

**39. **S. Backus, C. G. Durfee III, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. **69**(3), 1207–1223 (1998). [CrossRef]

**40. **J. Bromage, C. Dorrer, and R. K. Jungquist, “Temporal contrast degradation at the focus of ultrafast pulses from high-frequency spectral phase modulation,” J. Opt. Soc. Am. B **29**(5), 1125–1135 (2012). [CrossRef]

**41. **E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. **5**(9), 454–458 (1969). [CrossRef]

**42. **P. Poole, S. Trendafilov, G. Shvets, D. Smith, and E. Chowdhury, “Femtosecond laser damage threshold of pulse compression gratings for petawatt scale laser systems,” Opt. Express **21**(22), 26341–26351 (2013). [CrossRef] [PubMed]