1. Introduction

Laser peak intensity in the focal spot is a key parameter of most laser-plasma interaction experiments. The theoretical limit for the maximal value of laser peak intensity in the focal spot is achieved through ideal locking and phasing of all spatial-frequency Fourier components of laser radiation. To optimize temporal compression, elements with controllable dispersion are used [1,2]. The focal spot may be optimized by means of adaptive optical systems (AOS) [1,3–5] based on deformable mirrors (DM). These methods have proven to be reliable and robust tools in conventional OPCPA and CPA laser systems, when the spatial and temporal components of a laser pulse can be factorized and, therefore, spatial and temporal focusing can be optimized independently.

Recently, a nonlinear post-compression technique [6] called TFC (Thin Film Compression) [7] or CafCA (Compression after Compressor Approach) [8,9] has been actively developing for high-power lasers. The method relies on spectral broadening using self-phase modulation of a pulse in a cubic nonlinear medium followed by temporal compression using chirped mirrors. Impressive results have been obtained. A compression factor of 6 was attained [10]. A sub-10-fs pulse was generated [11]. The compressed pulse was used to produce laser wakefield acceleration and to generate synchrotron radiation in [12] (see also the reviews [9,13] and references therein). However, after post-compression, the pulse acquires nonlinear aberrations, resulting in spectral inhomogeneities in the spatial shape of the spectral phase and the spectrum due to spatial intensity nonuniformity of the input beam. Obviously, in this case adequate correction requires non-standard spectrally dependent methods of wavefront correction. For the same reason, the fluence measured in the focal plane is related to the intensity in a non-trivial way. Several deformable mirror-based approaches for the correction of nonlinear wavefront distortions were considered in [14]. They do not guarantee perfect correction, but lead to significant improvement of the focal spot. In particular, it was theoretically shown that when the focal intensity is maximized, the focal fluence is also close to its maximum. However, in the recent experimental works addressing the optimisation of the focal fluence of post-compressed pulses, for example [15], the focal intensity variations are bypassed.

In this paper we present results of experimental study of focusing a petawatt laser pulse at the PEARL facility after post-compression using an F/2.5 off-axis parabolic mirror and deformable mirror-based adaptive optics.

2. Nonlinear phase distortions

Nonlinear phase distortions (NPDs) occurring as a result of post-compression in a spatially inhomogeneous beam are a non-trivial combination of monochromatic and chromatic aberrations leading to a decrease in spatial coherence of the pulse and a decrease in the Strehl ratio $S$ during focusing [14]. For short pulses, $S$ means the ratio of the measured focal peak fluence to its theoretical Fourier-transform limit obtained from the measured near-field fluence distribution. The value of $S$ can be retrieved from experimental beam distribution measured by a CCD-camera with an exposure much longer than the pulse duration. $S$ determined in this way is not sensitive to the spatially uniform spectral phase of the pulse, i.e. $S$ values are the same for stretched and transform-limited laser pulses.

As for NPDs, they may be interpreted as rapid changes of the wavefront in time [13,14]. A hypothetical ideal NPD correction implies external action on the wavefront that could become flat at all time points, as a result of which the focal spot on the camera could be virtually indistinguishable from the linear case (absence of NPD) with the Strehl ratio $S$ close to unity.

Conventional AOSs, however, cannot change their DM shape on the femtosecond timescale and a decrease in $S$ is inevitable for any feasible stationary DM shape. At the same time, it was theoretically shown in [14] that the situation for peak intensity at the focal spot is not so deplorable as there is a DM shape at which the focal peak intensity recovers its aberration-free value. This shape, in the first approximation, corresponds to the correction of phase distortions at the moment of time when the pulse power reaches its maximum. The phase distortions at other moments of time are not corrected, which leads to a decrease of $S$, according to the time-averaged origin of the Strehl ratio.

The ratio of peak focal intensities in the aberrational and nonaberrational cases, by analogy with the Strehl ratio $S$, can be denoted as an intensity ratio $S_i$. The difference between $S_i$ and $S$ manifests itself only for pulses with spatiotemporal coupling such as in the case of NPD, whose wavefronts have different shapes at different moments of time. Note that $S_i$, in turn, depends on the temporal shape of the pulse and $S_i$-optimal DM shape depends on high-order spectral dispersion that is not compensated by chirped mirrors. In addition, $S_i$ value cannot be directly measured in the same way as $S$, but it can be estimated from numerical simulations of self-phase modulation similar to those performed in [14].

The feasibility of improving $S_i$ up to unity by means of a DM looks promising, but experimental implementation is challenging due to a priori unknown DM shape and impossibility of direct $S_i$ measurements.

The post-compression induced NPDs imply a more complex interpretation of wavefront sensor data. A standard reconstruction procedure for wavefront shape is based on measuring the centroid position of lens-grid beamlets focal spots of a Shack-Hartmann wavefront sensor [16]. For NPD, spot positions for different wavelengths (or different moments of time) are shifted relative to each other. The wavefront sensor cannot be used for time-resolved (or wavelength-resolved) measurements; as a result, the time-integrated signal is captured, which corresponds to a certain "effective" wavefront shape. In general, the "effective" wavefront correction corresponds neither to $S$ nor to $S_i$ optimization.

However, numerical simulations [14] showed that satisfactory results of focal spot optimization can still be achieved when AOS feedback is realized via effective wavefront shape or via the one measured in a narrow band close to the laser central wavelength. Both approaches were used in our experiments and were based on measurements from two independent wavefront sensors WFS1 and WFS2, one of which was equipped with a band-pass filter (see Fig. 1). Note that, according to [14], the band at the edge of the spectrum will lead to worse results.

 

Fig. 1. Layout of the experiment. KDP - nonlinear element, CM1-CM2 - chirped mirrors, W1-W2 - attenuation wedges, DM - deformable bimorph mirror, OAP - off-axis parabolic mirror, P - periscope, ML - microscopic lens, L1-L4 - achromatic lenses, BPS - band-pass filter 910$\pm$5 nm, WFS1-WFS2 - wavefront sensors, CCD1-CCD2 - far-field cameras.

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3. Experimental setup

The experiments were carried out on the PEARL laser facility [17] the layout scheme of which is shown in Fig. 1. A linearly polarized (in the plane of the figure) laser pulse (central wavelength 910 nm, duration about 60 fs, energy up to 15 J in a circular aperture $\approx$16 cm in diameter) coming from an optical compressor was directed to a 4-mm thick nonlinear KDP crystal, where the spectral bandwidth was broadened by several times due to self-phase modulation. Then the pulse was compressed down to 10 fs by CM1 and CM2 chirped mirrors, see [10] for details. An appreciable advantage of post-compression is absence of significant pulse energy losses, which are reduced to losses due to reflections from optical surfaces. Thus, the laser energy in the focal plane is virtually indistinguishable from the energy at the input of the non-linear crystal. The energy for this particular experimental series ranged from 7 J to 12 J. The maximal expected intensity on CM2 reached 7 $TW/cm^{2}$. The B-integral in CM2 estimated not to exceed 0.1. The absence of nonlinearity in CM2 is confirmed by the identity of the experimental spectra before and after the chirped mirrors.

Further, the beam was directed to uncoated glass wedge W1, deformable mirror DM, another uncoated wedge W2, and F/2.5 (F = 40 cm, off-axis distance - 175 mm) off-axis parabolic mirror OAP. The converging beam after OAP was reflected twice from uncoated wedges forming a vertical (not in the plane of the picture) periscope P, after which it was focused into a spot with a diameter of about 5 $\mu$m. A 96-channel DM (AKA DM2-200-96) was part of a 200-mm AOS (AKA Optics SAS [18]) and had a high reflectivity close to unity in the 910 $\pm$ 50 nm band, for the OAP – in the 910 $\pm$40 nm band.

The image of the focal plane was relayed with 20x magnification to the CCD-cameras CCD1 and CCD2 using a microscopic lens ML (20X Edmund Optics M Plan Apo Long Working Distance Infinity Corrected) and the telescope comprised lenses L1 and L2. To split the radiation between the channels 50:50 Thorlabs UV fused silica broadband plate beam splitters (coating: 700 - 1100 nm) were used. The radiation aperture in the plane of the microscopic lens was only a few mm, so the attenuation was chosen to keep the B-integral in the diagnostic line much less than unity. The wedges W1, W2 and the periscope P attenuated the pulse by about a factor of $10^{5}$. Additional attenuation needed to avoid damage of the CCDs with full energy shots was provided by means of additional neutral optical densities placed between L2 and cameras, in the areas where the beam was collimated. The image from the DM plane was relayed to the Shack-Hartmann-type wavefront sensors WFS1 and WFS2 (model WFS-1-3.2-136G manufactured by AKA Optics [18]), which were capable of measuring not only wavefront shape but the near-field distribution as well. Both sensors are based on the CMOS camera with resolution 2048x2048 pixels, sensor size 11.2x11.2 mm; equipped with a lenslet array 80x80 (in full frame), pitch 136 $\mu$m and focal length 3.2 mm; wavefront measurement accuracy less than 15 nm. In front of WFS 2 and CCD 2, a 910 $\pm$ 5-nm band pass filter BPS was installed. With this setup it was possible to simultaneously carry out measurements both with a full spectrum (using WFS 1 and CCD 1) and in a narrow spectral band (using WFS 2 and CCD 2). The experimental S values were extracted from the focal spot measurements performed with cameras CCD1 and CCD2. The extraction algorithm contains an original Fourier filtering based procedure of accuracy enhancement [19]. It was shown in [14] that data from both WFS 1 and WFS 2 can be used as a feedback for AOS, leading to similar results (see Sec.4. for details).

4. Experimental results

First of all, the AOS performance was tested without non-linear pulse compression. For this, the KDP crystal was removed from the beam. The AOS operation algorithm [20] was aimed at optimizing beam fluence at the focal spot. The results of optimization of the focal spot Strehl ratio $S$ are shown in Fig. 2(a), presenting the far field, residual (uncompensated) wavefront distortions, and the near field distribution of the beam. The Strehl ratio was found to be $S=0.73$, which is larger than the value obtained earlier [19,20] at the PEARL facility. Hereinafter all the presented values of $S$ were obtained under similar conditions, i.e. at a high energy of 10 J with a low repetition rate of 1 shot per 20 minutes. To obtain the mentioned values, it was necessary to maintain the laser energy at a given level with 10% accuracy. These conditions were met within two independent experimental days. On these days the RMS of the wavefront shot-to-shot variations was about 60 nm.

 

Fig. 2. Images of the far field (first column), RMS of the wavefront (middle column) and near field (right column) of the beam in different cases: (a) without post-compression, (b, c, d, e) with post-compression; (b,c) active compensation of linear fraction of distortions, (d,e) active compensation of "effective" wavefront; (b,d) in entire spectrum, (c,e) in a narrow spectral band of 910$\pm$5 nm. S and RMS values are: a) $S$=0.73, RMS=60 nm; b) $S$=0.16, RMS=214 nm; c) $S$=0.17, RMS=175 nm; d) $S$=0.43, RMS=91 nm; e) $S$=0.46, RMS=108 nm.

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The beam at the output of the PEARL laser compressor has no angular dispersion features or other types of spatiotemporal coupling. Thus it can be assumed that in the absence of KDP (without post-compression), the Strehl ratio $S$ was equal to the corresponding intensity ratio $S_i$. However, for a post-compressed laser pulse, $S_i\neq S$ and optimization of both parameters resulted in different DM shapes. Moreover, since $S_i$ could not be measured directly, the only value characterizing the quality of focusing was $S$ measured in different spectral ranges.

During post-compression (i.e. when a KDP crystal was placed into a beam), NPDs led to focal spot degradation, see Fig. 2(b): the Strehl ratio $S$ decreased down to 0.16, and the RMS of the wavefront distortions increased from 60 nm to 214 nm. The DM had been preshaped to correct the initial aberrations of the optical path, so this case may be addressed as a compensation of the linear fraction of the aberrations.

Next, we turned on the DM control algorithm, which after a few shots resulted in $S=0.43$ and RMS=91 nm, see Fig. 2(d). The result required 1-2 correction acts of the AOS. For comparison, Figs. 2(c,e) show the corresponding results of measurements in a narrow spectral band of 910 $\pm$5 nm obtained with WFS 2 and CCD 2. It is important to note that the same algorithm for data processing and DM control was used in both approaches.

5. Discussion

The results shown in Fig. 2(d) were obtained using the first approach – the "effective" wavefront shape. The differences in the quality of focusing when using the second approach – a narrow-band wavefront (Fig. 2(e)) – were insignificant and led to the value of the Strehl ratio close to $S = 0.43$. Thus, we experimentally confirmed that both approaches demonstrate similar results. The advantage of the second approach is that there is no need to take into account the spectral sensitivity of the CCD sensor in WFS and use broadband mirrors along the diagnostic line. The disadvantage is that the narrow-band filter significantly attenuates the signal and, given large overall attenuation in the line, this complicates AOS calibration. It is important to note that if the diagnostic path cuts off the broadened spectrum of the post-compressed pulse, then the WFS data might be significantly distorted.

It is worth noting that the spatial phase distortions in our experiment were extremely strong: the nonlinear phase (B-integral) was highly inhomogeneous across the beam due to beam nonuniformity in the near field, and its estimated maximum value reached $B=10$. Nevertheless, significant compensation of nonlinear distortions was demonstrated: the experimental Strehl ratio $S$ increased from 0.16 to 0.43. At the same time, we failed to achieve $S=0.73$, like in the linear case without post-compression (Fig. 2(a)). In addition to nonlinear distortions, AOS operation was significantly complicated by the low pulse repetition rate (1 shot per 20 minutes) and, respectively, low pulse-to-pulse near field distribution stability of the PEARL laser. Due to the difficulty of maintaining the desired energy, we also had to deviate from the values of B-integral of about 15 that proved to be optimal in terms of post-compression [10]. Moreover, as shown in [14], it is impossible to reach $S=1$ with a deformable mirror-based AOS after post-compression even in theory, as the wavefront is dynamically changing over the pulse. Even an ideally shaped DM can only correct the wavefront for a certain moment of time, meanwhile the Strehl ratio $S=1$ requires the wavefront to be corrected at all time instants along the pulse. Numerical simulation of the beam focusing shown in Fig. 2(d) set the Strehl ratio limit of $S = 0.75$ for the S-optimized DM shape. At the same time, the theoretical maximum for $S_i$ is still close to unity, since the wavefront can be optimized at the time of peak laser power.

To correctly compare focusing efficiency in cases with and without post-compression, we introduced in the numerical simulations a residual wavefront distortions and near field distribution measured in experiment (Fig. 2(a)). The numerical simulation included the calculation of self-phase modulation in the KDP crystal and the calculation for the focal field distribution, with the contribution of the AOS, as was done in [14], taking into account the actual distribution in the near field and spectral ranges of the feedback. The results of this simulation are summarized in Table 1. In this case, the maximum calculated value of the Strehl ratio was $S=0.52$. The value $S=0.43$ obtained in experiment was only 16% lower. Using the same proportion, we can estimate $S_{i,exp}$ as $0.52$, which is 16% lower than the theoretical value $S_i=0.62$ .

Table 1. Comparison of experimental results with theoretical values

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6. Conclusion

After nonlinear post-compression, the wavefront shape is dynamically changing in time along the pulse and its time-resolved measurement is currently unfeasible experimentally. We experimentally confirmed the prediction made in [14] that the "effective" wavefront shape retrieved either from a conventional wavefront sensor for the entire pulse spectrum or from the same sensor combined with a narrow band-pass filter around the laser central wavelength can be used as a feedback for a deformable mirror-based adaptive optical system.

The performed experiments on F/2.5 focusing of a pulse after nonlinear post-compression showed that adaptive optics can increase the Strehl ratio from $S=0.16$ to $S=0.43$. This value reaches 60% of the $S=0.73$ value obtained without post-compression and corresponds to 84% of the theoretical limit. Direct measurement of the intensity ratio $S_i$ after post-compression is currently unfeasible, but our simulations demonstrated that the value $S_i=0.52$ can be estimated in focus. This value corresponds to 70% of the $S_i=S=0.73$ value obtained in the linear case without post-compression.

The experimental results were obtained for very strong nonlinear distortions of the wavefront (B-integral = 10), for an inhomogeneous laser near-field distribution, and for one shot per 20 minutes repetition rate. All these circumstances significantly complicated the operation of the adaptive optical system. Under milder conditions, the Strehl ratio and the focal spot peak intensity would be even higher.