## Abstract

The development of phased-array grating compressor is a crucial issue for high-energy, ultra-short pulse petawatt-class lasers. We present a theoretical and experimental analysis of two-grating phasing in a broadband pulse mosaic compressor. The phase defaults induced by misaligned gratings are studied. Monochromatic grating phasing is experimentally achieved with an interferometric technique and pulse compression is demonstrated with a two-phased-array grating system.

©2007 Optical Society of America

## 1. Introduction

The application of chirped pulse amplification (CPA) technique [1] to broadband, high-energy petawatt-class lasers implies the design of efficient and large dimension pulse compressor. Multilayer dielectric (MLD) gratings used in the pulse compressor are very promising to compress high-energy pulses to the sub-picosecond regime. Because of the high diffraction efficiency, high damage threshold, good wavefront quality and large dimension, MLD gratings seem to be well-adapted [2]. However, these gratings are limited in size and cannot be used adequately for multi-kJ, short pulse laser systems. To reach the petawatt regime with a compact pulse compressor, a grating phasing can be considered. The grating phasing consists of a coherent addition of multiple gratings that will act as a monolithic large grating [3]. A theoretical analysis of the grating phasing is necessary to know the influence of phase defaults induced by grating misalignments on the spatial and temporal pulse profiles. To accomplish an accurate grating phasing in a pulse compressor, phase errors between each grating have to be measured with simple and compact diagnostics and removed by a high-precision mechanical system.

Grating phasing has been firstly demonstrated in a double-pass pulse compressor by tiling the first grating [4]. We report pulse compression experiments where only the second grating is a two-phased-grating. Within the framework of Pico2000 petawatt laser at LULI [5], we present firstly in section 2 a theoretical analysis of two-grating phasing taking into account the compressor configuration and broadband pulses. After a brief presentation of the degrees of freedom between two adjacent gratings, we express analytically the constant and linear phase defaults induced by grating misalignments. The far-field irradiance is then numerically computed for the three main phase defaults. The alignment tolerances are given for three laser parameters : peak intensity, pulse synchronization and pulse duration related respectively to the constant, linear and quadratic phase defaults. In section 3, we perform a monochromatic plane wave grating phasing with an accurate interferometric diagnostic. The phased-grating mechanical system and the fringe matching technique for grating alignment are validated. The influence of the phase defaults on the spatial laser beam profile is experimentally shown by far-field profile measurement. In section 4, the pulse compression using a two-phased grating system is demonstrated in a mJ chirped pulse amplification system. An embedded interferometric system is designed for the grating phasing in the compressor and finally the recompressed pulses are characterized in the spatial and temporal domains.

## 2. Theoretical analysis of diffraction grating phasing

#### 2.1 Degrees of freedom between two adjacent diffraction gratings

The diffraction grating phasing consists of determining the phase errors between
two adjacent gratings, which can be caused by relative translations and
rotations, and then removing these phase errors by using actuators. In the case
of a two-grating mosaic, there are five degrees of freedom following the grating
coordinates (*x*, *y*, *z*) :
longitudinal piston (*Δz*), lateral translation
(*Δx*) also called *gratings gap*,
tip (*θx*), tilt (*θy*), and
grating-plane rotation (*θz*) (Fig. 1). The translation along *y* axis,
parallel to the grating grooves, is inconsequential for the reflected wavefront.
The relative grating period difference (Δ*d*) related
to the grating manufacturing is considered as a tilt-like phase error [6–7].

However, a reduction in degrees of freedom can be realized by grouping them
together [8], thus we can compensate three phase defaults by the
three others. The lateral translation (*Δx*), the
grating period mismatch (*Δd*), and the tip
(*θx*) can be compensated by respectively
longitudinal piston (*Δz*), tilt
(*θy*) and grating-plane rotation
(*θz*).

#### 2.2 Phase errors analysis of a grating mosaic compressor

A standard pulse compressor is composed of two parallel diffraction gratings and
a roof mirror in the case of a double-pass configuration [9]. The mosaic pulse compressor developed for Pico2000
petawatt laser at LULI [5] consists of phasing the second and the third gratings
where the pulse spectrum is spread. It is composed of six 485 mm-gratings : a
first single grating G1 followed by two gratings mosaic (G_{21} -
G_{22} ; G_{31} - G_{32}) and a fourth single
grating G_{4} for a beam diameter of 200 mm. The grating groove density
is 1740 mm-1 and the grating distance is 1800 mm. The goal is to preserve the
spectral bandwidth by reducing the spectral clipping in the compressor for an
optimum temporal compression [10].

According to the grating equation, we determine the diffraction angle as a function of the wavelength :

where *α* is the incident angle on the grating,
λ the laser wavelength, and *d* the grating period.
Usually, the spectral phase introduced by the pulse compressor
*ϕ*(*ω*) can be written
as Taylor series about the central frequency *ω*
_{0} :

where *ϕ*
_{0} is the phase constant, *ϕ*
_{1} the group delay of the pulse, *ϕ*
_{2} the group velocity dispersion (GVD), and *ϕ*
_{3} is the third-order dispersion (TOD). In a standard monolithic grating
compressor, *ϕ*
_{0} and ϕ_{1} are not considered to optimize the pulse
compression. Only *ϕ*
_{2} is considered to achieve the best pulse duration and
ϕ_{3} to preserve the temporal pulse contrast. In contrast,
in a grating mosaic compressor, the constant and linear terms of the spectral
phase are really crucial for the coherent addition of the output beams and the
synchronization of the associated pulses [11]. Therefore, we have calculated the different phase
defaults induced by gratings misalignment of a two-diffraction-grating mosaic
and computed the far-field irradiance and the temporal profile for different
cases of phase errors.

Following the degrees of freedom defined in section 2.1, we calculate the phase
errors introduced by a pair of misaligned grating for a single-pass
configuration,
*Δϕ*(*ω*,
*Δx*, *Δz*,
*ϕx*, ϕ*y*,
*ϕz*), considering the reference laser beam
coordinates (*X*, *Y*, *Z*), where
*Z* is the propagation direction. We firstly study the
constant (*Δϕ*
_{0}) phase errors (Table 1 - left column) that are the major contribution on
the spatial beam profile in the focal plane and then the influence of the linear
(*Δϕ*
_{1}) (Table 1 - right column) and quadratic
(*Δϕ*
_{2}) phase errors on the temporal profile.

The constant phase defaults (*Δϕ*
_{0}) related to the far-field irradiance are computed for the three main
degrees of freedom (longitudinal piston, tip and tilt). As shown on the top of figure 2, the constant phase defaults are represented by
the phase maps from the grating mosaic (G_{21} and G_{22}).
Phase maps ϕ(*X*,*Y*), that correspond
to the phase differences introduced by two misaligned adjacent gratings, are
numerically defined by N × N matrixes composed of a N×N/2
zero matrix (G_{21}) concatenated with a N × N/2
*Δϕ*
_{0} matrix (G_{22}). A 200-mm gaussian beam lights up the grating
mosaic and the far-field intensity distribution is performed by fast Fourier
transform operation in the case of no phase default (Fig. 2(a)), differential sub-apertures piston of
π (Fig. 2(b)), a grating differential tilt and tip of
respectively *θy* = 2 μrad (Fig. 2(c)) and *θx* = 4
μrad (Fig. 2(d)). The longitudinal piston phase default has a
2π-periodic effect and causes a beam splitting in the focal plane [12]. For *Δϕ*
_{0} = 0 and *Δϕ*
_{0} = 2π, the intensity distributions are exactly the same. When
the piston phase errors appear (*Δϕ*
_{0} ≠0 modulo 2π), the peak intensity location shifts
from the center and a second peak appears. The two peaks become equal when the
piston phase error is equal to π that corresponds to
*Δz* =
λ/*(2.[cos(α)+cos(β _{0})])*.
The tolerance for a 10% peak intensity decrease corresponds to a piston of

*Δz*= 235 nm considering the Pico2000 compressor scheme (

*α*= 60° and β

_{0}= 75.5°). The tip and tilt angular phase defaults responsible of spatial chirp and focal spot depointing are evaluated. The tip and tilt phase default tolerances to have a peak intensity reduction less than 10% correspond respectively to

*θx*= 2.3 μrad and

*θy*= 0.6 μrad.

The linear phase default (*Δϕ*
_{1}) contributes to a pulse desynchronization at the output of the mosaic
compressor. Indeed, pulses compressed respectively by G_{21} and
G_{22} will be time-delayed by the differential phase defaults. With
the (*Δϕ*
_{1}) expressions (Table 1), we establish the tolerances of piston, tip and
tilt to have a maximum desynchronisation of 10% of the Fourier limited pulse
duration (i.e. 40 fs for τ_{0} = 400 fs). A piston of
*Δz* = 1.5 μm between the neighboring
gratings G_{21} and G_{22} corresponds to a differential pulse
delay of 40 fs. Also, a tip *θx* = 15.3
μrad or a tilt *θy*= 3.8 μrad
induce a pulse desynchronisation of 10%.

The contribution of the quadratic phase defaults
(*Δϕ*
_{2}) on the recompressed pulse is evaluated in term of pulse duration
lengthening. Figure 3 presents the evolution of the pulse duration
with the piston (a), tilt (b) and tip (c) phase defaults. In the case of tilt
(resp. tip), the pulse duration is calculated for a given *X*
(resp. *Y*) and *Z* coordinates. To know the
complete evolution of pulse duration with these transverse and longitudinal
spatial coordinates, the pulse front tilt of the compressed pulses has to be
taken into account.

In the case of a single-pass on the grating mosaic, the tolerances to have a
maximum temporal lengthening of 0.1*τ_{o} is
*Δz* = 270 μm (piston),
*θx* = 0.46 mrad (tip),
*θy* = 0.22 mrad (tilt). This analysis permits to
determine that the most important effect resulting of a misaligned phase-array
grating compressor is at first a spatial effect and secondly a temporal
effect.

## 3. Monochromatic phasing experiments

Gratin9s position and grooves orientation have to be accurately controlled to provide
a phased-array grating compressor. Therefore, we have developed a mechanical system
prototype to phase two medium-scale diffraction gratings. Several motion devices
permit to have five degrees of freedom between the two 120 mm × 140 mm
diffraction gratings. Each grating reposes on two knee-joints and is fixed with
nylon screws to avoid wave front surface distortions. The lateral piston
(*Δx*) is adjusted with manual translation stage and
the longitudinal piston (*Δz*) with a closed-loop PZT
translation stage (10 nm minimum displacement). The tip and the grating-plane
rotation (*θx* and *θ;z*) are
manually controlled by micrometer devices and the tilt
(*θy*) by a precision rotation stage with PZT for high
resolution to achieve angular rotation greater than 1 μrad.

To control the grating phasing and measure the residual phase errors, some diagnostics have to be elaborated. A lot of phasing methods have been previously investigated by the astronomical community. Some of them utilize the diffraction pattern analysis [13–15], or the interferometric techniques [16, 17], or phase diversity wave front sensing [18] to achieve the phasing of multiple telescope mirror segments with an adaptive loop algorithm.

The experimental demonstration of grating phasing with our opto-mechanical system and
motion devices is achieved by using a large aperture, high fringe contrast Fizeau
interferometer. The continuous wave, monochromatic laser
(*λ* = 633 nm) with a 150 mm beam diameter lights up the
gold-coated gratings mosaic. A visible wavelength can be used to detect the
misalignment of gold-coated gratings mosaic because of high-efficiency on a large
spectral bandwidth. It is not possible in the case of multilayer dielectric gratings
due to the small spectral bandwidth (20 nm) centered at 1μm. The laser
beam is centered on the gratings gap. Figure 4 presents the fringe matching technique with five
main steps to reach the grating phasing.

The gratings gap is reduced as much as possible by a manual translation stage. The
grating mosaic is firstly positioned in zero-order configuration (i.e. mirror
configuration). The flat tint allows us to have a parallel plane between the grating
mosaic and the transmission flat and therefore suppress the differential tip
(*θx*) (Fig. 4(a)). Then, the gratings are placed in -1 order in
Littrow configuration *α*
_{L}=*Arcsin(Nλ/2*) = 33.4° with
*N* = 1740 mm_{-1} and *λ* = 633
nm). The tilt (*θy*) and grating-plane rotation
(*θz*) are removed by rotating the two fringe patterns
horizontally and equalizing the fringe frequency (Fig. 4(b-d)). Finally, the differential piston induced by
lateral and longitudinal translations are adjusted to 0 (modulo 2k) by matching the
fringes (Fig. 4(e)) and thus the grating mosaic alignment is completed (Fig. 5). This technique permits to resolve a minimum
translation (piston effect) of 20 nm and a minimum rotation of 1 μrad
which are typically the resolution of the piezo-mechanical system. The alignment
stability is during one hour. Possible sources of instabilities are the temperature
variations, mechanical vibrations and PZT stability.

An analysis mask is then defined on the fringe patterns and, to reconstruct correctly the grating surface wavefront, the gap between the gratings is suppressed. With a phase-shift technique, the wavefront surface of the gratings mosaic is reconstructed by recording five interferograms and the point spread function (PSF) is calculated. The PSF is the mathematical representation of the far-field intensity of the wavefront. Figure 6 presents wavefront surface measurements and PSF calculations in the case of aligned gratings (Fig. 6(a,b)) and in the case of a p piston phase error between the gratings (Fig. 6(c,d)).

When the gratings are accurately aligned, the optical path difference (OPD) maps of each grating exhibit phase continuity and the PSF presents a single-spot far-field intensity distribution. The calculated Strehl ratio in this case is 0.91. While, when the gratings are misaligned with a differential piston of π, the OPD maps present a phase discontinuity and two beam spots appear in the PSF. The Strehl ratio decreases drastically to 0.58. Furthermore, the interferometer laser beam reflected by the grating mosaic in 0 order is focused by a 700 mm focal length lens. The far-field intensity distribution is acquired with a CCD camera (LaserCamII – Coherent) coupled with an x40 infinity-corrected microscope objective (Fig. 7). The experimental measurements of far-field intensity distribution are in good agreement with the theoretical model.

Thus, the prototype mechanical system (scale 1/3) has the potential to provide an accurate and reliable grating phasing. For the petawatt pulse compressor, the phased grating system will be installed in a vacuum chamber with an embedded phasing diagnostic. We have developed a more compact diagnostic than the Fizeau interferometer. This system is a Michelson interferometer with a continuous wave, monochromatic, monomode, Nd:YAG laser (λ=1064nm). As shown in figure 8, the cw incident laser beam of 30 mm diameter is separated by a beam splitter in two arms. The first arm is the reference and the second one lights the grating mosaic. The interference fringe pattern is recorded on a 8 bits CCD camera (Fig. 8). The grating alignment is realized by matching the fringe patterns issued from the two gratings (G1 and G2). The unengraved grating borders are clearly seen on the fringe pattern. The measurement precision of this diagnostic is a little bit less than with the Fizeau interferometer because of a smaller aperture that reduces the analyzed area.

## 4. Pulse compression with a phased-array grating system

The two-grating mosaic is installed in a double pass compressor to recompress mJ
amplified chirped pulses. The laser system is composed of a mode-locked Ti:Sapphire
oscillator (Tsunami - Spectra Physics) which can provide nJ energy, 100 fs pulses at
a central wavelength of 1057 nm (Fig. 9). Seed pulses are then frequency chirped, temporally
expanded in a single-pass Öffner stretcher with a diffraction grating
groove density of 1740mm_{-1}. The stretch factor is 89 ps/nm. The resulting
pulses having duration of 1.5 ns are amplified in a Ti:Sapphire regenerative
amplifier and recompress in the phased-array grating pulse compressor.

Initially, the monolithic compressor acting in a double-pass configuration was
composed of two gold-coated holographic diffraction gratings and a roof mirror
retroreflector. Compressed pulses were previously fully characterized in the spatial
and temporal domains. Then, the second grating has been replaced by the phased-array
(G_{21} and G_{22}) grating mosaic (280×120
mm^{2}). The incidence angle on the first grating is 72.5°, the
grating groove density is 1740 mm^{-1}, and the grating distance is 800 mm.
At the output of the mosaic compressor the spatial and temporal beam profiles are
probed and compared with the similar CPA system using the monolithic compressor. The
grating phasing is realized by using the 1-μm Michelson interferometer
embedded into the compressor. The interferometer beam path is not the same as the
CPA laser beam path that is compatible with our pulse compressor setup. We have only
one mosaic of two gratings so the interferometer diagnostic is fixed and cannot
disturb or clip the main beam path.

Spatially, the laser beam at the output of the mosaic compressor is focused with a 700 mm focal length lens and analyzed in far-field with a CCD camera (LaserCamII - Coherent) coupled with an x6.3 infinity-corrected microscope objective (Fig. 10). The gaussian beam shape and the diameter are correctly retrieved by grating alignment.

Temporally, the laser pulse is compressed to the sub-picosecond regime by the phased
grating mosaic compressor. The pulse duration is evaluated by a single-shot
second-order autocorrelator. A temporal lengthening by a factor of 1.4 at the output
of the mosaic compressor is observed by comparison with the autocorrelation obtained
with the standard compressor (Fig. 11(a)). The initial pulse duration of 300 fs deconvolved
from the autocorrelation measurement without gratings mosaic
(*Δλ* = 5.5 nm FWHM) is enlarged to 420 fs.
The calculated temporal profile by pulse spectrum Fourier transform without spectral
phase leads to a pulse duration of 310 fs (*Δλ*
= 5 nm FWHM) (Fig. 11(b)). The pulse broadening is not induced by gratings
mosaic misalignments otherwise the far-field spatial profile will be affected. The
pulse broadening can have several origins. Firstly, the spectral phase mismatch
between the stretcher and the compressor can affect the recompressed pulse duration.
A possible solution to check this effect is to measure the spectral phase (FROG,
SPIDER) at the output of the compressor. Secondly, a residual piston phase default
can cause a pulse lengthening. Indeed, the relative piston between the adjacent
gratings is measured modulo 2π and corrected with the monochromatic
interferometer but the absolute piston cannot be measured with this system. To do
that, we could use a white-light interferometer [16]. Finally, a spectral clipping induced by the gratings gap
appears in the mosaic compressor. The pulse spectrum is spread on the two-phased
grating mosaic and clipped for the central wavelength by the gratings gap. This
effect is related to the small beam size (∼5mm) in the compressor. The
spectral clipping modifies the temporal pulse shape with sidebands creation and
affects the pulse duration by a small lengthening (< 10 fs) (Fig. 11(b)). To reduce this effect, some solutions are
actually under study : the magnification of the beam size in the compressor and the
use of diffraction gratings etched until the edges.

## 5. Conclusion and perspectives

In conclusion, we report on a complete theoretical and experimental analysis of the grating phasing for high energy petawatt-class lasers. A theoretical model taking into account compressor design and broadband pulses has been developed to predict the spatial and temporal effects. Phase defaults induced by the mosaic grating misalignments are responsible of far-field interference, spatial chirp and focusing errors in the spatial domain and pulse desynchronization and pulse duration lengthening in the temporal domain. The grating phasing has been firstly demonstrated with a cw, monochromatic laser coupled to a large aperture Fizeau interferometer. The fringe matching technique permitted a simple and reliable grating alignment diagnostic. A chirped pulse amplification system with a phased-array grating compressor has been performed to compress mJ pulses and study the temporal effects. As a perspective, some grating phasing improvements are necessary to provide a clean and sharp temporal profile. The current experiments were performed with large unengraved edges gold-coated stock gratings. The use of large dimension, high-efficiency multilayer dielectric gratings engraved until edges can overcome the gratings gap effect and allow the compression of energetic pulses [19].

## Acknowledgments

This work was performed under the auspices of the European contract LASERLAB Europe RII3-CT-2003-506350, Centre National de la Recherche Scientifique, Ecole Polytechnique, Commissariat à l’Energie Atomique, Université Paris VI and the contract Plan Etat Region E-1258 with the collaboration of Horiba Jobin Yvon Group. We would like to thank N. Blanchot, G. Marre, S. Montant and C. Rouyer from CEA-CESTA for fruitful discussions on this subject and the Optics and Imaging Sciences Group of LLE and especially T. J. Kessler. Thanks to S. Dorrard, H. Timsit, C. Sauteret and C. Le Bris for scientific and technical supports at LULI.

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