1. Introduction

The Petawatt Aquitaine Laser (PETAL) is a large and ultrahigh intensity laser beamline operating on the principle of Chirped Pulse Amplification (CPA) [1]. It was designed and constructed by the French Commissariat à l’Energie Atomique et aux Energies Alternatives (CEA) in the Laser Megajoule (LMJ) facility [2], to deliver laser pulses in the kJ and ps range, at the wavelength of 1053 nm. It is based on a sub-aperture compression scheme whose concept was demonstrated on the Laser Integration Line facility, a prototype beamline of the LMJ laser [3]. Building and commissioning of the PETAL short pulse laser in the LMJ facility was carried out in 2014-2015 with first light and power ramp-up up to 1.15 PW (850 J with pulse duration of 700 fs) in 2015 [4]. During this high intensity campaign, laser damage sites were reported on the first leaky mirror of the vacuum transport section. Despite steadily efforts to understand and improve the laser damage resistance of vacuum mirrors to short pulses [5,6], and the upgrade of various transport mirrors based on this development, some specific damage patterns still persist on transport mirrors. These laser induced damage sites on transport mirrors severely limit the maximal energy of the facility to half the value originally demonstrated. Understanding their origin turns out to be of high importance to increase the final power of the laser pulse delivered by this complex laser chain. More generally, such impact of grating phase modulation defects on downstream optics is of major interest for high intensity petawatt to multi-petawatt laser systems spreading worldwide [7].

In this work, we show that the laser damage patterns result from a phase modulation induced by the diffraction gratings of the compression stage. We evidence the fact that the relation between the phase modulation and the over-intensity at the position of the transport mirror strongly depends on the positioning of the grating inside the compressor. This work aims at quantifying for each diffraction grating of the compressor the over-intensity induced by the phase modulation. For this purpose, we rely on the use of a full numerical calculation performed with the software Miró [8] and high resolution interferometric measurements. A validation of our approach is also obtained with experimental characterization of the damage sites on the transport mirrors. We show how this model permits to explain and retrieve the different morphologies of the damage patterns on the transport mirrors.

2. Description of the PETAL facility

In this section, we provide an overview of the PETAL chain with a focus on the final stage of the beamline. In particular, we describe the compressor stage and the transport mirrors that will be used later for identifying the influence of each diffraction grating on the streamline surintensity.

The PETAL beamline is based on the chirped pulse amplification (CPA) technique [9] combined with optical parametric amplification (OPA). In the front-end [10], the seed pulse (100 fs, 1053 nm), is stretched by a single grating in an all-reflective Offner system and enters two stages of OPA. The beam is then shaped using an apodizer to create a square beam and a phase plate combined with a spatial filtering [11] is used to shape the 4 sub-aperture beams, as required by the compression scheme. The front-end is able to deliver 60 mJ beams with a pulse duration of 4.5 ns and a spectral full width at half maximum of 8 nm.

The amplifier section has a similar architecture as the LMJ amplifier section, but it is designed to handle a single beam of 37 cm $\times$ 37 cm in a four-pass system with angular multiplexing and a reverser [4]. It uses 16 Nd:glass laser slabs and is designed to deliver more than 6 kJ. Monochromatic commissioning has demonstrated energy of 4.9 kJ in 5 ns whereas large spectrum commissioning demonstrated energy of 1.4 kJ in 2 ns with a 3.5 nm spectral bandwidth, using 14 slabs [4]. The decrease in spectral bandwidth is due to spectral gain narrowing in the amplifier. The current operating point is close to 600 J to limit damage initiation and growth in the compressor and transport sections.

The PETAL compressor scheme is based on the coherent addition of 4 sub-apertures beams, independently compressed and synchronized with micro-metric precision [3]. A schematic view of the compressor is displayed in Fig. 1. To limit the volume of the vacuum vessel enclosing the compressor and the size of the gratings, the compression is applied in two stages. The first stage, under air environment, reduces pulse duration from approximately 2 ns to 350 ps in a non-split full aperture beam, unfolded double pass compression configuration. G11, G12, G13 and G14 are monolithic 830 $\times$ 420 mm$^{2}$ gratings, with a groove density of 1680 g/mm. The angle of incidence for G11 and G13 is 56$^{\circ }$ and distance between gratings of pairs G11/G12 and G13/G14 is 2.62 m. The 35 mm thick fused silica window, separating the two compression stages, adds a nonlinear phase distortion of 0.7 rad for a 3.6 kJ pulse [12]. In our numerical simulations, we considered a pulse with an energy inferior to 1 kJ, therefore we neglected the nonlinear effects induced by the window. The second stage, in vacuum, finalizes the compression down to a pulse duration of 500 fs. It consists of 4 independent pairs of gratings (G21/G22, G23/G24, G25/G26 and G27/G28) each one addressing a sub-aperture. Second stage gratings are 450 $\times$ 420 mm$^{2}$, with a groove density of 1780 g/mm. Incident angle on the first gratings of the pair (G21, G23, G25 and G27) is 77.2$^{\circ }$ and the grating distance is 2 m. The second compression stage is designed with a small compression factor to limit transverse chromatism due to the single pass configuration. Between the two compression stages, a segmented mirror compensates the spatial phase shift induced by the separation of the 4 sub-apertures in the 2^nd stage gratings. The transmission efficiencies of the first and second compression stages are 81% and 91% respectively. All compressor gratings operate in $s$-polarization. Compression diagnostics are performed after a pair of leaky mirrors, MT5 (transport mirror #5) and MT5bis, which are multi-layer dielectric coated mirrors with a 0.16% transmission in target configuration (−45$^{\circ }$, +45$^{\circ }$). Experimental results show sub-aperture beams pulse duration between 540 fs and 570 fs for a 219 J shot and 700 fs for the 1.15 PW shot (850 J) [4].

 

Fig. 1. Schematic view of the PETAL compressor. Gratings are labeled from G11 to G14 for the 1^st stage, under atmospheric pressure, and from G21 to G28 for the 2^nd stage, in the vacuum section. The beam enters the vacuum vessel through a 35 mm thick fused silica window. Transport mirrors appearing in the scheme are labeled MT4, MT5 and MT6. MS stands for segmented mirror. All the gratings as well as MS, MT4 and MT5 operate in S polarisation. MT6 operates in P polarisation, directing the beam upwards to the rest of the transport section.

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Current PETAL compressor gratings are manufactured by three different vendors, that we will call vendor 1, vendor 2 and vendor 3: G11, G12 are from vendor 1, G13, G14, G21, G22, G25, G26 are from vendor 2 and G23, G24, G27, G28 are from vendor 3. These gratings were specified and manufactured to full fill diffracted wavefront performance: a power inferior to $\lambda /3$ and planeity without power of $\lambda /3$ peak-to-valley and $\lambda /5$ RMS, over their clear aperture. Further measurements and characterisation of gratings reflected wavefronts, performed in CEA, and there effect on downstream optics will be detailed in section 3.

The final transport section, in vacuum, starts from the end of the compressor to the vacuum chamber. It is composed of 6 transport mirrors from MT5 to MT10 and a parabolic mirror (PAR), located between MT9 and MT10, which focuses the beam in the center of the LMJ target chamber. Transport mirrors MT6 to MT10, and the parabolic mirrors have highly reflective multi-layer dielectric coatings. MT5 and MT6 are represented on Fig. 1. Mirror configurations of the mirrors of the vacuum transport section are listed in Table 1. Because of the space constraints in the LMJ building, the beam travels approximately 30 m between the end of the compressor and the vacuum chamber.

Table 1. List of Optics in the Compression and Transport Sections, and Their Corresponding Optical Path Length (OPL) From the Compressor Entrance (G11).

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3. Observation of laser damage sites on transport mirrors

Visual inspections carried out after two laser shots the 1.15 PW (850 J) laser shot in 2015 revealed severe laser induced damage on the MT5 mirror and initial damage site were detected after a previous shot of approximately 0.8 PW, (570 J) [4]. The dielectric reflective coating on this component is an early design that exhibits a low laser damage threshold: measurements carried out on a short pulse laser induced damage threshold (LIDT) testing setup described in [13], exhibit an LIDT of 1.68 J/cm$^{2}$ at 770 fs in the beam normal in operating condition (45$^{\circ }$ incidence, S polarization, wavelength of 1053 nm). The low threshold is convenient for revealing the beam structure, as damage density is linked with the distribution of fluence hot-spots. Pictures of the MT5 mirror under grazing illumination, taken in an observation room, are displayed in Fig. 2. These images reveal the different damage morphologies caused by high energy shots on a large part of the mirror. This result evidences the fact that the different diffraction gratings have different effects on the distribution of the over-intensity on the MT5 mirror. The mirror has not been removed from the laser chain between the two shots that caused damage sites, therefore we do not have precise pictures of just the damage sites initiated during the first shot (0.8 PW). Nervertheless, thanks to the observations on chain, we know that damage sites were initiated on sub apertures 2, 3 and 4. After this two laser shots, this mirror was replaced by a new one, with improved coating design, with an LIDT measured at 3.73 J/cm$^{2}$ at 770 fs in the beam normal. After being exposed to equivalent shots, no damage was observed on this new mirror.

 

Fig. 2. Stitched images of the damage sites on the MT5 mirror after exposure to high energy shots (570 J and 850 J, with a pulse duration of 700 fs) on PETAL in 2015. Images where taken in an observation room, with white light illumination at grazing incidence. Sub-apertures are numbered from 1 to 4, corresponding respectively to beam going through grating pairs G21-G22, G23-G24, G25-G26 and G27-G28.

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Fig. 3. Examples of wavefront measurement for 1^st stage gratings manufactured by vendor 1 (a) and 2 (b) and for 2^nd stage grating fabricated by vendor 2 (c) and 3 (d). Images represent the wavefront of the diffracted beam, subtracted by piston, tilt and defocus aberations. The color-bar is the same for all images. The white line on each phase image is 1 cm long. On each image, the grating lines are vertical ($y$ direction), meaning that the dipersion direction is horizontal ($x$ direction). "rms(1-10 mm)" are average wavefront distortion root mean square, in the 1 to 10 mm period band, among the different measurements on each grating, and there standard deviation. PSD curves in (e) $x$ and (f) $y$ directions are also displayed. These PSD curves are sums of 1D PSD.

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Laser induced damage sites were also observed on other transport mirrors, especially on the second half of the transport section starting with MT7, while MT5 and MT6 exhibit no laser induced damage. This may suggest that phase defects induced by diffraction gratings on the compressor develop in intensity modulations after MT6, sparing MT5 and MT6 but exposing the subsequent mirror to laser induced damage.

4. Evolution of phase modulation along the laser chain

In the case of small modulations of wavefront, the evolution of intensity can be described with the theory of Talbot effect, which is an approximation of the diffraction theory. We carried out analytical analysis to identify the periods of wavefront modulations which can be responsible of over-intensities in the transport section and to determine our needs in wavefront metrologies on the diffraction gratings of PETAL.

When a periodic object of period $p$ is illuminated by a plane wave, images of the object form at certain distances by free-space propagation, provided that the amplitude of the modulation is small. This phenomenon is known as the Talbot effect [14] and it can be simply explained with Fresnel diffraction integrals [15]. Let us consider a monochromatic plane wave propagating in the $z$ direction, going through a phase object inducing, in $z = 0$, a phase modulation of the complex field of the form:

Assuming a small amplitude of modulation ($\alpha << 1$) and a period large compared to the wavelength ($p>>\lambda$), the amplitude of this phase modulation follows a cosine function of $z$, whereas the amplitude $A$ of the complex field follow a sine function:

From Eqs. (2)–(3), we can see that the phase pattern repeats itself after a length $L_f$, which is called the Fresnel length. Moreover, the phase modulation transforms into an amplitude modulation after $L_f/4$, and after $3L_f/4$ but with an opposite sign, and so on with a periodicity of $L_f$. With the knowledge of the position of compression gratings and mirrors in the laser beamline, we can therefore predict what periods are likely to be critical regarding the laser induced damage in the transport section of PETAL.

Talbot effect can be understood by an interference between 3 waves [16,17]. A main plane wave propagates along the z axis, and the other two are off-axis and symmetric with respect to the z axis, at angles $\pm \lambda p$. It means that, for a beam having a finite width $D$, there is a limit to the length at which we can observe the 3 waves interference pattern. The Talbot effect is present only for propagation lengths inferior to $z_c = Dp/(2\lambda )$, giving a lower limit to the periods to be considered in our case.

We can determine with Eqs. (2)–(3) the maxima of intensity on the mirror as a function of the period $p$ of the wavefront modulation induced by the grating for each pair grating - mirror, thus identifying critical periods regarding laser induced damage. If we combine all gratings and mirrors, we find continuous ranges of critical periods, with a lower limit imposed by the fact that the beam aperture is finite. Ranges of critical periods for phase modulations on 1^st and 2^nd stage gratings are displayed in Table 2. In practice, it means that periods of 90 $\mu$m to 30 mm for the 1^st compression stage gratings and 30 $\mu$m to 22 mm for the 2^nd compression stage gratings have to be considered.

Table 2. Critical Period Ranges for Wavefront Modulations Induced by 1^st and 2^nd Stage Compression Gratings. Periods Are Given in the Beam Normal, Not in the Plane of the Optical Components.

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Calculations using Eqs. (2)–(3) are made considering monochromatic plane waves, which is not the case on the PETAL facility. Typical spectral width of the pulse at the entrance of compression section is 3 nm (FWHM), with a central wavelength at 1053 nm. Variations of $L_f$ within this range do not cause any noticeable change in the critical periods, but the transverse chromatism due to compression gratings [12] may induce significant smoothing of the fluence distribution of the beam. For the first stage, there is a strong lateral chromatism on gratings G12 and G13 resulting in a shift along the $x$ direction of 62 mm (in beam normal) for a wavelength variation of 3 nm. It means that defects of G12 and G13 will be smoothed in the $x$ direction. For the second stage, because of the single pass configuration, the lateral chromatism found on gratings G22-G24-G26-G28 is not compensated and is therefore present until the end of the laser chain. Accordingly, defects of G21-G23-G25-G27, as well as defects of upstream optics will be smoothed along the $x$ direction: shift of 12 mm (in beam normal) for a variation of 3 nm. In the compressor configuration presented in Fig. 1, the spectral shift induced by the second stage compressor reduces the smoothing of G12 and G13. In the numerical model presented hereafter, we took care to respect the specific orientation of the gratings.

In this study, we will address only periods greater than 1 mm, as the wavefront measuring device that we used resolve periods in the range 1-20 mm. These measured interferograms have been implemented in the numerical model described below, in order to compute the propagation of the laser pulse in the PETAL beamline.

5. Numerical model of the laser chain

Miró is a powerful numerical tool to model and predict propagation of high-energy laser beamlines [8]. It has been extensively used for the design and operational analysis of the Megajoule Laser facility nanosecond laser beams [18–21]. With its capacity to model Kerr effect, broadband amplification, spatio-temporal phase distortion and spatio-temporal coupling, it has been also used with success for modeling the short pulse PETAL Petawatt beamline [10,22,23] and more recently the APOLLON laser [24].

An extensive model of PETAL, containing the full amplification, compression, transport and focusing sections, was previously developed in order to predict the beam energy and spatio-temporal characteristics [22]. To thoroughly study the effects of compression gratings phase defects on the beam quality, we improved the compression and transport sections of the Miró PETAL model to account for high resolution (millimeter scale) spatial phase modulations. The spatial mesh step in the previous model was 2.6 mm, while it is about 0.3 mm in the new model. Calculations were carried out using a Miró mode, phase modulation, which deals with strongly stretched pulse cases by adapting the size of the simulation’s time domain. In this mode, the pulse stretching is treated with the generalized inhomogeneous wave model, applying the grating law for each propagation direction associated with a frequency of the spectrum. Unlike the original model, we consider here a linear chirp (second order in $\omega$) of the beam at the amplification section output. This approximation greatly reduces time and numerical resources needed to solve the calculations: it avoids the use of a Miró component, called dispersor, which linearly breaks down the beam into independent spectral terms.

We worked on two distinct Miró models. At first, calculations were carried out on a single sub-aperture beam model, considering a 8.3 $\times$ 8.3 cm$^{2}$ square beam propagating from the compressor input to the end of the transport. The spatial domain is therefore reduced, compared to full beam model, and the vacuum section of the compressor is simplified, requiring no separation of the beam and only one pair of gratings instead of 4. The input beam in this sub-aperture model is ideal (no phase nor amplitude defects), allowing quantifying the contribution of the compressor independently on the eventual flaws in the rest of the chain. This model allows us to quantify the influence of each individual grating on the downstream over-intensities. The second model considers a full 37 $\times$ 37 cm$^{2}$ square beam and requires more time and computing resources. It includes additionally the amplification section and the complete compressor vacuum section scheme. The fluence distribution of the input beam is taken from experimental measurements of PETAL front-end output beam, before it is injected in the amplification section. It allows us to quantify the effect of the compressor gratings phase characteristics on a beam which is representative of an experimental PETAL beam and explain the damage site patterns observed on transport mirrors. While the first model is fast and can be used for parametric studies, the second is time consuming but efficient to model full size beam apertures from low order aberrations to high spatial frequencies.

5.1 Gratings reflected wavefront

High resolution interferometric measurements of the diffracted wavefront in the first reflected order of pulse compression gratings were performed at CEA. Measurements were carried out on spare gratings of each type used in the PETAL compressor: a 1^st stage grating from vendor 1, a sample 1^st stage grating from vendor 2 (50 mm diameter), and two 2^nd gratings, from vendor 2 and 3. A Fizeau interferometer with an aperture of 80 mm, which is typically used to address the [0.1 - 1 mm⁻¹] frequency range [25], was used to qualify various areas of the clear aperture of the grating. The wavelength of the device is 1064 nm and tested gratings are placed in Littrow configuration, which corresponds to an incidence angle of 63.3$^{\circ }$ for the 1^st stage gratings and 71.2$^{\circ }$ for the 2^nd stage gratings. We will assume that the phase characteristics of the grating’s reflected wavefront measured in these conditions are representative of the reflected wavefront of the grating in operation on the laser chain. In the numerical models, phase maps obtained from the interferometric measurements are projected in the plane of the grating.

The clear aperture of diffraction gratings being much larger than the aperture of the interferometer, we need multiple measurements targeting different areas to take into consideration the phase characteristics variations across the surface. The different areas of measurement cover a horizontal band (orthogonal to the grating lines) from edge to edge, except for the 1^st stage from vendor 2 sample which is entirely covered by the interferometer aperture. Fig. 3 shows an example of measured phase map for each of the 4 spare gratings.

We cannot directly use the measured phase map in the numerical model since their sizes are smaller than those needed in the model: the interferometer sensor diameter is 80 mm (in the beam normal), the sizes of the gratings (in the beam normal) in the full beam model are 400 $\times$ 400 mm$^{2}$ (1^st stage grating) or 100 $\times$ 400 mm$^{2}$ (2^nd stage grating) and in the sub-aperture model, the size of gratings is 100 $\times$ 100 mm$^{2}$. Instead, we use synthetic phase data generated from the power spectral density (PSD) map which is the squared modulus of the Fourier transform of the measured phase map. The method is the following: the PSD map is interpolated to the appropriate resolution (the inverse of the PSD resolution is the dimension of the signal in the space domain), then a random phase map with the same resolution is generated, while making sure that the new Fourier transform has Hermitian symmetry ($F(-k) = \bar {F}(k$)). Finally an inverse Fourier transform returns a spatially extended phase map, with the same PSD as the interferometric measurement.

5.2 Sub-aperture beam model

The sub-aperture scheme as seen in the Miró software is presented in Fig. 4, where each block represents an optical component and links between blocks represent propagation sections. Results are saved on MT5, MT7 and parabolic mirror, as these elements are located at the beginning, the middle and the end of the transport section (see Table 1). Since the 2^nd stage of the compressor consists of only one pair of gratings in this sub-aperture model, we implemented 2 schemes in Miró: scheme A using a combination of vendor 2 phase maps on 2^nd stage gratings to be equivalent to G21/G22 and scheme B combining vendor 3 phase maps to mimic the G23/G24 pair.

 

Fig. 4. View of the sub-aperture scheme A in Miró. In this model example, the second stage compression is made of only the first pair of gratings G21/G22. Scheme B is essentially the same, but with the gratings pair G23/24. Results of fluence maps are saved on yellow blocks: MT5, MT7 and PAR, which are respectively located at the beginning, in the middle and at the end of the transport section. Temporal domain at the entrance of the compressor is [$\pm$8 ns] (256 points) and changes to [$\pm$16.5 ps] (256 points) at the end of the compressor. Spatial domain of the simulation is [$\pm$0.14 m, $\pm$0.7 m] (x,y), with [1120x560] points.

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The individual sub-aperture compressor is illuminated by an ideal square shaped flat-top profile, with no intensity nor phase modulations. The width of the beam is equivalent to the width of one of the four vertical sub-apertures of the PETAL beam. At the compressor entrance, the pulse duration is set to 1.82 ns and the frequency chirp is set to 1.52 ps⁻². The central wavelength is 1053 nm and the full width at half maximum (FWHM) of the spectrum is 3.25 nm (experimental value of 3.7 nm for the 1.15 PW shot [4]). After compression, the pulse duration is approximately 590 fs, which is close to the experimental values reported in [4] (autocorrelation measurement of 540-570 fs for sub-apertures of a 219 J shot). The energy injected in the compressor is 85.5 J and 61 J reach the exit. The mean fluence at the compressor exit is 0.92 J/cm$^{2}$ (Fig. 5), which is equivalent to the mean fluence of a 1 kJ nominal PETAL pulse.

 

Fig. 5. Examples of results of two different configurations calculated with Miró: scheme A (a, b), with the G21-G22 pair of vacuum compressor (phases maps from vendor 2 2^nd stage grating) and scheme B (c, d), with G23-G24 pair of vacuum compressor (phase maps from vendor 3 2^nd stage grating). Results examples are (a,c) a fluence map on the MT7 mirror and (b,d) fluence histograms on MT5, MT7 and parabolic mirrors. Fluence histograms, with heigths in number of pixels, represent the fluence distribution inside a 6 $\times$ 6 cm$^{2}$ square centered on the beam, represented by the white square in the image (a). Standard deviations of normalised fluence distributions ($\sigma _{f}$) are displayed on the histograms

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Spectral dispersion induced by the gratings leads to widening of the beam in the horizontal direction: with a spectral width of 3 nm, the angular dispersion is 0.85$^{\circ }$ for the 1^st stage and 0.70$^{\circ }$ for the 2^nd stage. The spatial domain of the simulations is wider in the horizontal direction ([$\pm$0.14 m, $\pm$0.7 m], 1120 by 560 points) to take into account for this effect and keep the impact of edge effects negligible. Also, in our sub-aperture model, grating dimensions are set so that the spectral transmission of the compressor is equivalent to that of the PETAL compressor gratings. Sizes of the gratings are adjusted in the numerical model with apodizers (apo. in Fig. 4), setting the transmission to 0 outside the spatial limits of the gratings.

To each type of grating, we can attribute multiple generated phase maps obtained from interferometric measurements. We ran 30 configurations randomly selected, avoiding the use of the same phase map twice in a calculation. Figure 5 shows a typical result for a configuration of each scheme A (a, b) and B (c, d). First, we can see the pattern difference between the fluence of a beam coming out of vendor 2 or 3 2^nd stage compressor: MT7 fluence on scheme A (Fig. 5(a)) has specific profile with millimeter periodic vertical lines, while MT7 fluence on scheme B (Fig. 5(c)) has no specific period feature, which is consistent with the observation of transport mirror damage site (see section 3.). We can also see horizontal lines, more visible on the scheme A results, which can be explained by the smoothing effect of the gratings, occurring in the horizontal direction. Moreover, the fluence inhomogeneity of the beam is greater for scheme B, as the maximum fluence is higher and normalised fluence standard deviation values ($\sigma _{f}$) on the different outputs are higher: average $\sigma _{f}$ in the transport section is 0.069 for scheme A and 0.137 for scheme B. Finally, in both cases, there is an increase of $\sigma _{f}$ for MT7 and parabolic mirror versus MT5: about 12% for scheme A and up to 41% in the case of scheme B. In terms of fluence, the maximum is 1.17 J/cm$^{2}$ (average on the different configurations) for scheme A and 1.50 J/cm$^{2}$ for scheme B, while the mean fluence of the beam is 0.92 J/cm$^{2}$. The summary of all the configuration outputs are reported in Fig. 6 (left) in the form of a graphic displaying the $\sigma _{f}$ values and average $\sigma _{f}$ values. This graphic demonstrates the consistency of the phase measurements, as basically the same conclusions can be drawn from any configuration tested.

 

Fig. 6. (a) $\sigma _{f}$ values for the sub-aperture Miró model with random sets of phase maps on gratings. Each configuration results in three values of $\sigma _{f}$ on MT5, MT7 and parabolic mirror outputs. Blue circles represent results of the model A (vendor 2 2^nd stage grating phase maps) and red triangles represent results of the model B (vendor 3 2^nd stage grating phase maps). Biggest markers are average values. (b) Contribution of each grating to the $\sigma _{f}$ values on the MT5 (blue circle) and MT7 (red triangle) outputs. 6 (or 8 for gratings G11- G12, from vendor 1) phase maps per grating were tested.

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With the aim of precisely quantifying the impact of each grating regarding the degradation of the fluence homogeneity in the transport section, we ran a set of configurations considering flawless gratings except for one of them. Figure 6-b displays the resulting $\sigma _{f}$ values on elements MT5, MT7 and parabolic mirror for each configuration. As a reference, $\sigma _{f}$ for an entirely flawless compressor is 0.01, as there is some intensity modulation due to diffraction and the chromatism due to the compressor shifts the different components of the spectrum. Among the air section gratings, we see that the first pair, G11-G12, from vendor 1, has on average the highest impact, especially G11, which is not smoothed throughout the air section. Among the vacuum section gratings, the second grating G24 (vendor 3) has clearly the highest impact on the deterioration of the fluence homogeneity. For both schemes A and B, we can see the smoothing effect of the vacuum compressor, as the first grating of each pair has less impact than the second.

5.3 Full beam model

The full beam Miró model has been developed in order to study the impact of gratings phase defects in the case of a realistic beam, representative of an experimental PETAL shot. Therefore, instead of considering an ideal flawless beam, as in the sub-aperture model, the full beam model input is derived from an experimental measurement from he front-end diagnostic table. Data used come from the 1.15 PW (850 J) shot, since we can compare results from the model to damage sites observed on MT5 [4]. The spatial distribution of fluence measured at the front-end output is represented at Fig. 7(a). The input beam is approximately 4 by 4 cm$^{2}$, with an energy of 20.3 mJ and a pulse duration of approximately 4.5 ns. Measured data from the amplification section diagnostic are not directly used due to non uniform transmission of the diagnostic system that still have to be corrected. Moreover, the Miró model of the amplifier section gives us access to the calculated spatial phase of the beam.

 

Fig. 7. (a) Fluence map and profiles of the beam measured at the front-end output. (b) Calculated fluence map and profiles of the beam at the end of the amplifier section Miró model.

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The front-end beam, with its measured fluence and temporal distributions and theoretical spectral law, is injected in the amplification section Miró model described in [22]. The temporal domain of this model is [$\pm$13 ns] (256 points) and the spatial domain is [$\pm$33 cm] in both x and y direction (256 $\times$ 256 points). This model takes account for the different configurations of spatial gain distribution of the 16 amplifier slabs, as well as there static and thermal phase maps. In the present case, the number of slabs pumped is 14. The computed fluence distribution at the amplifier section output is presented at Fig. 7(b). We observe a deformation of the spatial profile due to the amplifier gain distribution. A new phase plate has been designed to compensate for this shape but was not implemented in the front-end of PETAL when the laser shot considered here was done. The use of an adequate fluence profile at the amplifier input provides a top hat spatial profile which allows to have more energetic shots while maintaining the same maximum fluence at the amplifier output. The pulse duration is 2 ns and the spectrum width is 3.4 nm, for total beam energy of 1240 J.

The beam calculated from the amplifier section model is then injected in the compression section model. The full beam compressor scheme is similar to the sub-aperture scheme presented in Fig. 4, but with 4 pairs of 2^nd stage gratings instead of 1, separated in four segments calculated independently and coherently recombined at the end of the compressor. The spatial domain of the full beam model is [$\pm$28 cm] in both directions (1792 $\times$ 1792 points) and the time domain is [$\pm$8 ns] (256 points) at the entrance of the compressor and [$\pm$16.5 ps] (256 points) at the end of the compressor. Generated phase maps described in section 5.1 are randomly attributed to the corresponding gratings in the model. The energy at the end of the compressor is 890 J, with a pulse duration of 630 fs (FWHM), which is close to the experimental shot measurements (850 J, 700 fs), in view of the fact that there is a perfect recombination of the four sub-apertures in the full beam model [4]. The resulting fluence maps on mirrors MT5 and MT7 are presented in Figs. 8(a,c). We can see that maximum fluence is higher on MT7 than on MT5 (+ 11%), which is consistent with the observations of laser induced damage sites observation of the PETAL transport section (see section 3.).

 

Fig. 8. Results of two calculations carried out with the full beam Miró model: one with a random set of phase maps on the compressor gratings (a,c) and the other one with flawless gratings, with no phase noise (b,d). Results showed here are fluence maps on MT5 (a,b) and MT7 (c,d). The white line in the top left corner of each image is 5 cm long and the color scale is the same for every images. Maximum values of fluence are noted F^max.

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In order to determine the contribution of the gratings phase defects compared to the other beam defects originating from the front-end and the amplifier section, a calculation has been done without any phase defect on the compression gratings. The resulting fluence maps on MT5 and MT7 are displayed on Figs. 8(b,d). Maximum fluence on MT5 and MT7 are respectively 40% and 45% higher when gratings phase defects are on.

The contribution of gratings to the quantity of laser induced damage sites depends on the LIDT: with a threshold of 1.8 J/cm$^{2}$ in the beam normal, the surface of MT5 exposed to laser induced damage is about 5 cm$^{2}$ when gratings phase defects are on and 0.25 cm$^{2}$ when they are off. With a threshold of 1.59 J/cm$^{2}$ in the beam normal, which corresponds to the measured LIDT of the mirror showed at Fig. 2 corrected for the pulse duration (using a $\tau ^{0.33}$ temporal scaling law for SiO² as proposed in [26]), surfaces are respectively 33.3 cm$^{2}$ and 18.11 cm$^{2}$. Based on the observation of MT5 after exposure to 1.15 PW, the total surface of the damage sites has been estimated to 120 cm$^{2}$ in the beam normal. In the case of the full beam Miró model, the surface on MT5 exposed to fluences superior to 1.42 J/cm$^{2}$ is approximately 120 cm$^{2}$. This 12% LIDT gap is within the standard deviations of measurements carried out on laser damage testing setups from different laboratories in spite of identical samples and assumed identical experimental conditions [27]. Further analysis evidenced that this LIDT deviations across laboratories would come from discrepancies in beam spatial and temporal profile determination, underestimation of non linear effects and control of environmental conditions [27].

In the case of the present work, the difference in LIDT may also be explained by the approximations made. First, we considered a Gaussian temporal pulse profile, which is not exactly the case on PETAL [4,22], while significant differences in LIDT for different temporal shapes has been reported by Olle et al. [28]. Sub-millimeter periods were ignored for phase defects of the gratings and other optics in the compressor (mirrors, window) are considered flawless. Nevertheless, damaged area predicted from Miró model and measured damage threshold on sample is fairly close to the damaged surface measured on the MT5 mirror.

Figure 9 shows the comparison between (a) the threshold image of the fluence distribution on MT5 in the high resolution numerical Miró model and (b) the picture of the surface of MT5 from PETAL facility. The numerical model succeeds to reproduce the fluence distribution of the 1.15 PW shot. In both images we can see the high frequency structures originating from the gratings phase defects and the low frequency distribution of energy in the beam, originating from the front-end and the amplifier section.

 

Fig. 9. (a) Threshold image of fluence computed on MT5 with the full beam Miró model and (b) picture of the damage sites on MT5 after exposure to high energy shots (570 and 850 J). The threshold is placed to 1.48 J/cm$^{2}$ so that the total surface with fluence above this value approximately matches the experimental surface of damages sites on MT5. Both images are displayed in the beam normal

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6. Conclusion

With the objective to better understand the initiation and growth of damage sites on transport mirrors in the PETAL facility, Miró software has been used to describe a 37 by 37 cm$^{2}$ beam to a sub-millimeter scale. With a focus on the phase defects of compression gratings, relying on interferometric measurements of diffracted wavefront addressing the 1 to 20 mm periods range, we manage to evidence that the millimeter scale patterns of damage sites on transport mirrors are caused by the compression gratings. The sub-aperture model allowed to quantify the effect of each grating of the PETAL compressor, independently from the rest of the laser chain. Gratings from different vendors induce fluence hot-spots in the transport section, raising the maximum fluence to about 27% for scheme A and 63% for scheme B, compared to a nominal square flat-top beam. The model also highlight the importance the gratings position, on one hand because of the propagation length and on the other hand because of the smoothing of the beam due to spectral angular dispersion. The full beam Miró model takes account of the fluence inhomogeneity of the front-end beam and computes also the amplitude and phase defects from the amplifier section. The results successfully reproduce the fluence pattern observed during a high energy shot on the PETAL facility, down to the millimeter scale. The LIDT estimated from the comparison of numerical results and pictures of damages sites (1.42 J/cm$^{2}$) is 12% smaller than the LIDT measured on a laser induce damage testing setup, which is in concordance with discrepancies reported during a damage testing round-robin with different groups [27]. Investigation of sub-millimeter phase modulations and a more rigorous description of the experimentally observed damage sites could provide further insights for the understanding of laser induced damage sites.