## Abstract

An important issue, mosaic grating compressor, is studied to recompress pulses for multiPetawatt, high energy laser systems. Alignment of the mosaic elements is crucial to control the focal spot and thus the intensity on target. No theoretical approach analyses the influence of compressor misalignment on spatial and temporal profiles in the focal plane. We describe a simple 3D numerical model giving access to the focal plane view after a compressor. This model is computationally inexpensive since it needs only 1D Fourier transforms to access to the temporal profile. We present simulations of monolithic and mosaic grating compressors.

© 2006 Optical Society of America

## 1. Introduction

Ignition scale facilities now under construction, LMJ [1] (*Laser MégaJoule*, in France) and NIF [2] (*National Ignition Facility*, in the US), constitute a remarkable opportunity in science and for an inertial fusion energy demonstration. LIL, the laser prototype of the LMJ consists of an elementary unit of the LMJ, formed of eight beams, each one able to deliver nanosecond pulses of 18 kJ at the fundamental wavelength 1.053 µm and 7 kJ at 351 nm. The LIL facility will be split into eight independent beams to obtain a symmetrical irradiation on target. Moreover, an additional line with a multi-petawatt, high energy capability will be constructed (PETAL, PETawatt Aquitaine Laser) [3]:

• to carry out integrated Fast-Ignition experiments on LIL using its target implosion capability,

• to provide LIL with a short-pulse, high-energy backlighting capability for new physics experiments under conditions that also allow the development of backlighting techniques.

The goal for this laser chain is to achieve the multi petawatt level with multikilojoule energy and few hundred femtoseconds pulse. One of the main key issues is the compression stage in terms of realization (high damage threshold, large scale grating) and alignment (phase control of segmented grating).

The technique used to amplify femtosecond pulses in Nd:glass power chain is the well known CPA technique [4] (Chirped Pulse Amplification). The CPA technique consists of stretching ultra short pulses, amplifying them in a laser media to fully extract the stored energy without inducing non-linear effects or damages, and compressing them back to the initial pulse duration. The principle of the stretcher is based on an initial frequency phase transformer, which induces a lengthening of the temporal pulse shape. The compressor provides the inverse modifications. The frequency dispersion in the stretcher is obtained, for example, with diffraction gratings or fiber Bragg gratings. The compressor uses large size diffraction gratings for nonlinear considerations.

At the end of the laser chain, in the compressor, the pulse will be both short in time and highly energetic. These conditions imply that the last grating of the compressor is subjected to high beam fluency. For example if we consider the parameters of the PETAL beam (3.6 kJ and a beam section of 370×370 mm^{2}), we reach fluency as high as 2.6 J/cm^{2} in right section. Today the best results for grating damage threshold [5] are obtained with dielectric gratings with a level of around 3 J/cm^{2} for an incident angle of 72° and a pulse duration of 10 ps. To stay under this limit, we have to use a higher incident angle on the grating. This working condition with a high angle implies that the beam projection on the grating is drastically enlarged and consequently large size gratings are required. In the case of the PETAL system, the incident angle has been fixed to 77.2° on the last grating of the compressor giving 370×1670 mm^{2} grating useful size for the first grating.

Large-scale gratings (2 m) involve technological issues and prohibited cost. Segmented mosaic grating compressors are strongly considered by different groups as an alternative solution [6, 7]. At that point, the difficulty is to analyze and control the alignment of the mosaic to insure the equivalence between a mosaic grating compressor and a monolithic grating compressor. The key issue is mainly a spatial phase problem and the question of coherent addition of elementary gratings becomes fundamental. Indeed, the final objective is to control the intensity of the beam on target, it means the encircled energy in the focal spot with the shortest pulse duration.

In previous works, different theoretical approaches have been used to understand the evolution of laser field after a compressor. The first complete theoretical approaches concern the single pass compressor [8]. A partial theoretical analysis of default induced by compressor misalignments has been made by Fiorini and al [9] without spatial approach. At our knowledge, none theoretical approach analyses the influence of misalignment of a compressor on both spatial and temporal profiles in the focal plane. Many simulations based on numerical approaches allow this analysis in the focal plane after a compressor [10–12]. Nevertheless these approach needs a computationally involved complete 3D Fourier transform. Here we have developed a simplest method of calculation with only 1D Fourier transform.

In this article a simple numerical model is presented that gives access directly to the 3D focal plane view knowing the incident beam, pulse characteristics and compressor parameters. This model gives information on intensity in the focal plane (spatial, temporal and spectral profiles). The theoretical background for the model and its numerical structure are detailed. The model takes into account only the stretcher and the compressor systems and neglects the other systems of the power chain (amplifiers, spatial filtering, …). Then starting from the case of monolithic compressor we will go toward a mosaic grating compressor, showing the potential of this model to describe the evolution of the beam in the focal plane, depending on the compressor parameters.

## 2. Theoretical model and numerical approach

#### 2.1 The theoretical model

Our goal is to calculate the electric field after the compression in the focal plane in any point, for any wavelength, and time. Usually, to obtain these results, a 3D simulation is required in the compressor space. A 3D Fourier transform is used to obtain the 3D electric field in the focal plane. We propose to calculate directly the electric field in the focal plane without spatial Fourier transform. As only the spatial focal plane (far field) is wanted, the spatial representation in the compressor space (near field) is not used. Indeed the spatial domain in the focal plane corresponds to the vectorial domain in the compressor space. Thus, we use a monochromatic plane wave decomposition of the electric field in the two spatial dimensions.

The near field before the compressor can be described as:

where k_{x} (k_{y} respectively) is the wave vector in the x (y) spatial direction. After the focalization system, the focal plane corresponds to the far field:

where f is the focal length and k_{0} is the central wave vector. In this equation the normalization factor is neglected. So the far field is:

The compressor is decomposed in elementary optical components. Each optical component is considered as a boundary between two spaces. For each boundary, a transfer function is defined. Between each boundary we add a phase contribution to take into account the propagation. The evolution of the electric field between the space #1 and the space #2 is described by [Fig. 1(a)]:

The electric field in the space #2 is described as a function of the electric field in the space #1 by:

where *ω* is the frequency, *ϕ* is the phase due to the propagation from the previous boundary, *α* a normalization factor, and the function F (G respectively) is the vectorial transfer function at the boundary: ${k}_{x}^{1}$=*F*(${k}_{x}^{2}$, ${k}_{y}^{2}$, *ω*), ${k}_{y}^{1}$=*G*(${k}_{x}^{2}$, ${k}_{y}^{2}$, *ω*).

The normalization factor proceeds to the conservation of energy. The conservation of energy implied that:

$$\phantom{\rule{.6em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}=\iint \mid J\mid {\mid {\hat{E}}_{1}(F({k}_{x}^{2},{k}_{y}^{2},\omega ),G({k}_{x}^{2},{k}_{y}^{2},\omega ),\omega )\mid}^{2}d{k}_{x}^{2}d{k}_{y}^{2}$$

where J the Jacobian matrice for the variable change:

$J=\mid \begin{array}{cc}\frac{\partial F}{\partial {k}_{x}^{2}}& \frac{\partial G}{\partial {k}_{x}^{2}}\\ \frac{\partial F}{\partial {k}_{y}^{2}}& \frac{\partial G}{\partial {k}_{y}^{2}}\end{array}\mid \phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}\mid J\mid =\frac{\partial F}{\partial {k}_{x}^{2}}\frac{\partial G}{\partial {k}_{y}^{2}}-\frac{\partial G}{\partial {k}_{x}^{2}}\frac{\partial F}{\partial {k}_{y}^{2}}$

So the normalization factor is: $\alpha =\sqrt{\mid J\mid}$ and the transfer function between two spaces is:

The optical components and the propagation are only phase contributors. Only one order of diffraction is considered and the efficiency is 100% in this order for each grating. The compressor acts as a transfer function on every wave vector (k_{x}, k_{y}, ω) of the wave.

A fixed referential gives the optimal stretcher entrance beam propagation direction. Every element position and every wave vector are described in this referential.

A reference point is taken on every element of the system. Each element is considered as a boundary between two spaces where wave vector distribution of the beam might be different. Between the two reference points of the interior space we accumulate phase:
$\varphi =\overrightarrow{{P}_{1}{P}_{2}}.\overrightarrow{{k}_{1}}$
, where P_{1} is the reference point on the first boundary, P_{2} the reference point on the second boundary and *k⃗*
_{1} the wave vector between the two boundaries [Fig. 1(a)]y we use a transfer function describing the new wave vector.

The optical system used two optical components: mirror and grating. For a reflective grating the transfer function for the wave vector is described by the following equation [Fig. 1(b)]:

where
$\rho =\sqrt{{\left(\frac{\omega}{c}\right)}^{2}-{\left(\overrightarrow{n}\wedge \left(\overrightarrow{{k}_{1}}\wedge \overrightarrow{n}\right)+\overrightarrow{\kappa}\right)}^{2}}$
is deduced from the law of conservation of momentum, *k⃗*
_{1} is the incident vector on the grating, *n⃗* is the normal to the grating plane and *κ⃗* is the grating vector described as follows: *κ⃗* direction is perpendicular to the grooves of the grating and *κ⃗* modulus is equal to 2πN where N is the grating groove density.

For the reflective element the transfer function for the wave vector is described by the following equations [Figs. 1(c)–1(d)]:

Mirror: *k⃗*′=(*k⃗*.*β⃗*
_{1})*β⃗*
_{1}+(*k⃗*.*β⃗*
_{2})*β⃗*
_{2}-(*k⃗*.*n⃗*)*n⃗*

Dihedron: *k⃗*′=-(*k⃗*.*β⃗*
_{1})*β⃗*
_{1}+(*k⃗*.*β⃗*
_{2})*β⃗*
_{2}-(*k⃗*.*n⃗*)*n⃗*

Corner cube: *k⃗*′=-*k⃗*

Where k⃗ is the input wave vector, *k⃗*′ is the output wave vector, *n⃗*, β⃗_{1} and β⃗_{2} are vectors describing the position of reflectors.

#### 2.2 The numerical approach

We start from the focal plane where we fix the meshing in 3D (k_{x}, k_{y}, ω). This meshing is adapted to optimize the representation of the beam in the focal plane. The beam propagates from the focal plane through the compressor system toward the entrance of the stretcher. After each boundary the meshing is distorted. At the entrance of the stretcher the meshing accumulates all distortions induced by misaligned optical element of the compressor. We do not take into account the system between the stretcher and the compressor.

So the algorithm propagates the meshing grid, for each wavelength, from the focal plane, through the compressor and the stretcher, to the stretcher input. Due to misalignment of the compressor the meshing is deformed. The algorithm calculates the total spectral phase. The input field is calculated on the deformed meshing. The spectral phase shift and the jacobian factor are added like described in Eq. (6). This new field is plotted on the initial meshing grid to quantify the deformation. If the system is very misaligned, the meshing can be a badly representation of the beam at the entrance of the stretcher. Nevertheless, this method allows us to have a good meshing of the beam in the focal plane.

We have implemented two beam profiles and two pulse shapes possibilities: gaussian and square. The initial field of these beams is described analytically in the focal plane by their Fourier transform. But since we only use the ideal output field distribution in 3D, we can study every shape as soon as we have their numerical repartition in the focal plane.

The compressor is a double pass compressor composed of two gratings and a reflective element as described in Fig. 2. The reflective element is a mirror, a dihedron or a corner cube. The stretcher corresponds to a double pass compressor with a negative distance between the gratings. The stretcher is perfectly aligned.

#### 2.3 Numerical approach for mosaic grating compressor

The electric field of a segmented compressor with mosaics of N gratings is calculated assuming that the N gratings are independent. It means there is no energy coupling between neighbors’ gratings. So a segmented compressor with mosaics of N gratings is simulated with N independent compressors. Since we use the same meshing of the focal plane for the N compressors we can sum the electric field in the focal plane. The initial beam is divided in N elementary beams of 1/N diameter along the perpendicular direction to the grooves. Then each beam propagates through a specific compressor and stretcher systems. Finally we sum the N output fields in the focal plane adding a phase term due to the position of every elementary beam [13]. For a segmented compressor with mosaics of two gratings, the far field in the focal plane is:

where ${E}_{\mathit{\text{FF}}}^{1}$, ${E}_{\mathit{\text{FF}}}^{2}$ are the far field of the two half monolithic compressors and D the diameter of the entire beam at the entrance of the compressor.

For a segmented compressor with mosaics of 2N gratings, the far field on the focal plane is:

where ${E}_{\mathit{\text{FF}}}^{\mathit{-}j}$ , ${E}_{\mathit{\text{FF}}}^{+j}$ are the far field of the monolithic compressor and D is the diameter of the entire beam.

For a segmented compressor with mosaics of 2N+1 grating, the far field on the focal plane is:

Our model takes care of the finite size of the beam with the monochromatic plane wave decomposition hypothesis. It takes care of the longitudinal position of the grating’s compressor allowing the observation of piston effect. The main limitation of this approach is due to the vectorial description of the beam in the compressor space. The gratings have no size limitation so no spectral clipping and licking due to the finite size of the grating. Such a vectorial description of the reflective element cannot describe the evolution of the focal spot along the propagation direction. Nevertheless, in the focal plane, angular chromatism is correctly described allowing the observation of traveling wave.

## 3. Presentation of simulations

We present two simulations of monolithic compressor and two simulations of segmented mosaic compressor. The compressor parameters are detailed before each simulation presentation. The Fig. 3 present the parameters involved in the alignment of the compressor. In case of a monolithic compressor, we find the distance between gratings (compared to the stretcher), the groove rotation, the tip of the grating, the tilt (incident angle) and the groove density. In case of segmented compressor, two new parameters need to be considered: the piston and the in-plane shift of the grooves. These two parameters correspond to the relative position of one grating to its neighbor.

The algorithm is implemented on a Matlab platform. The electric field contains 31*31 spatial points and 128 spectral points. The run time of the algorithm on a PC laptop is 10 seconds.

## 4. Numerical simulations for monolithic compressor

Simulations using a monolithic compressor are made to validate our model. Two cases are studied: a grating distance mismatch between the stretcher and the compressor systems (example #1) and non parallel gratings in the compressor (example #2). These two examples are well-known cases. The first one induces an increase of the compressed pulse duration [9] and the second one gives rise to an inhomogeneous wave [14]. For these examples, the simulations are made with the characteristics detailed in the two references [9, 14].

#### 4.1 Example #1: grating distance mismatch between the stretcher and the compressor systems[9]

We first consider the consequences of a grating distance mismatch between the stretcher and the compressor systems. Their perfect parameters are: an incident angle of 67.77° (littrow angle), a groove density of 1740 g/mm, and a grating distance of 52.5 cm. The output beam from the compressor has a diameter of 1 cm, a central wavelength of 1.064 µm, an initial pulse duration of 100 fs. This beam is focused with a focal length of 1 m. The stretching factor is also 13000 for a double pass system.

A distance of 40 µm along the propagation direction for the central wavelength is added on the second grating of the compressor, which is equivalent to 80 µm since the system is a double-pass configuration. We compare the spatial and temporal profiles between the perfect alignment and the case with a misaligned compressor. Figure 4(a) (spatial fluency profiles) shows no change in the spatial characteristics of the beam in the focal plane. But looking at the intensity versus time representation [Fig. 4(b)], we see clear differences between optimal alignment and misaligned system. First we observe a delay of 267 fs between the two maxima of intensity due to optical path. Second we observe an enlargement of the pulse for the misaligned case from 100 fs to 150 fs due to residual phase term. These two simulations are conforming to the theoretical predictions which are a delay of 266.7 fs and a stretched pulse duration of 141 fs (the resolution of the temporal space is 11 fs). [9]

#### 4.2 Example #2: non parallel gratings in the compressor system [14]

Now we consider the consequences of non parallel gratings in the compressor system. Their perfect parameters are: an incident angle of 72.5°, a groove density of 1740 g/mm, and a grating distance of 1 m. The output top hat profile beam from the compressor has a diameter of 87 µm, a central wavelength of 1.057 µm, initial pulse duration of 330 fs. This beam is focused with a focal length of 0.3 m. The distance between the second grating and the mirror (to implement the double pass) is 0.7 m.

A tilt of 0.01° is added on the second grating. Figure 5 shows the consequences on the focal spot of this non-parallel grating compressor. The classical inhomogeneous wave is obtained in the X-axis direction (diffraction direction) both in time [Fig. 5(b)] and spectrum [Fig. 5(a)]. The integrated focal spot becomes elliptical with an extension by a factor 5 in the X-axis. Moreover, the pulse duration is increased and we observe a transverse delay in the energy deposition in the focal plane. This traveling wave is very useful to realize soft X-ray lasers [14].

## 5. Numerical simulations for a segmented mosaic compressor

We consider now the case of a segmented compressor with mosaics of four gratings (Fig. 6). As explain before, to study a segmented compressor with mosaics of 4 gratings the initial beam is divided in 4 elementary beams of 1/4 diameter along the perpendicular direction to the groove. Then each beam propagates through a specific compressor and stretcher systems. Finally the 4 output fields in the focal plane are summed after adding a spatial phase term due to the position of each elementary beam.

For this segmented compressor with a mosaic of four gratings, we will present two simulations. The first one (example #3) simulates a difference (nanometric range) in the grating distance (piston). This piston error has been studied [7] and shows the strong decrease of the alignment tolerances in case of segmented compressor reaching the interferometric domain. The second simulation (example #4) shows a complete case with grating distance mismatches and angle misalignments. These simulations are made with the beam characteristic of the PETAL laser [3, 13].

#### 5.1. Example #3:grating distance mismatches (piston errors)

We simulate the consequences of grating distance mismatches between the 4 gratings of the second stage compressor of PETAL [13]. Their perfect parameters are: an incident angle of 77.2°, a groove density of 1780 g/mm, and a distance between the gratings of 2 m. The output beam from the compressor has a diameter of 40 cm, a central wavelength of 1.053 µm, and an initial pulse duration of 500 fs. This beam is focused with a focal length of 3.5 m.

A piston is added on different grating of the mosaic #1 (Fig. 7). Piston acts on all orders of the phase term. However, its effect is particularly important on the zero order of the spatial phase. At the output of the segmented compressor, the beam can be considered as 4 elementary beams. So an optical path difference of half the central wavelength will produce a zero order phase term equal to π and induce destructive interferences in the focal plane. In this simulation it corresponds to a piston of $\frac{\lambda \mathrm{cos}\alpha}{2}=230\mathrm{nm}$ .

In Fig. 7(a), we see the perfect case with the sum of the four elementary beams in the focal plane. In Fig. 7(b), a piston equivalent to 230 nm on one external and one internal grating chosen next to each other induces a hole in the center of the focal spot. If a piston equivalent to 230 nm is put on one of the external grating [Fig. 7(c)], two spots pattern is observed. The maximum intensity has been divided by a factor two. If we now put a piston equivalent to 230 nm on one of the internal grating [Fig. 7(d)], three spots pattern is produced in the focal plane. The maximum intensity is now divided by a factor three.

These simulations show the very high alignment precisions required for a segmented compressor, mainly governed by the spatial zero order phase term. The tolerances are divided by a factor of 1000 compared to the case of monolithic compressor (example #1).

#### 5.2 Example #4: complete case with grating distance mismatches and angle misalignments

For this last example, we simulate a system with different defects on the first gratings mosaic:

- on one of the external grating a piston equivalent to 230 nm is added,
- on the internal pair of gratings we put:
- a piston proportional to the wavelength so the zero order phase term is equal to zero modulus 2π but the first order phase term will produce a delay of few hundred fs.
- angle (tilt, tip and rotation) in the three directions of 1 µrad.

On Fig. 8(a), we see the perfect case with the sum of the four elementary beams in the focal plane. On Fig. 8(b) is presented the integrated focal spot in the case of a misaligned system. This focal spot presents many spots on the X-axis due to the piston (destructive interference) similar to the simulation plotted on Fig. 7(c). On the lower part of the focal spot, we observe default of pointing both on the X and Y-axis related to the angle defaults. Default of pointing on the Y-axis is due to the tip and groove rotation, and on the X-axis to the tilt.

More, with this numerical approach, the focal spot can be analyzed versus time. The movie on Fig. 9 shows an example of intensity evolution of the focal spot. The delay between the intensity maxima (related time between the central and the lower part) is 100 fs in good agreement with the 93 fs theoretical value. This delay is equal to the first order term of the phase induced by the piston on the internal pair of gratings.

## 6. Conclusion

A simple 3D model has been developed to describe the focal plane for monolithic or segmented compressor systems. This vectorial approach, using the plane wave decomposition, associated with a spectral phase calculation, allows us to observe the evolution of the focal spot both in time and space. The model is based on the deformation of the numerical meshing. The meshing is optimized to obtain a good representation of the beam in the focal plane. The stretcher and compressor systems are decomposed in elementary optical component. Each optical element is considered as a boundary. Misalignments of optical element distort the meshing and a phase term is added for the propagation between each boundary. This numerical scheme needs only one temporal Fourier transform and limited numerical requirements.

We confirm the high precision tolerances for a segmented compressor compared to a monolithic compressor, mainly due to spatial considerations. For example, the piston and tilt alignments have to be improved by two decades in case of segmented compressor. They reach the interferometric domain with nanometric and micro-radian precisions. The use of segmented compressors required the development of new alignment techniques associated with new diagnostics and high precision mechanics [15–16].

The next step is to explore the potential of this model, in terms of spatial aberrations and spectral distortions. First, we plan to make complete simulations with the experimental wave front distortions of the LIL beam. Then, the spectral clipping on the gratings of the compressor will be implemented using analytical expression [17]. This approach is valid only for small misalignment of compressor system. So this modification enables us to observe the time evolution of the focal spot, in terms of temporal contrast ratio.

## Acknowledgments

This work is supported by the Conseil Régional d’Aquitaine and the LASERLAB-Europe consortium and is performed under the auspices of the Institut Lasers et Plasmas.

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