1Laboratoire pour l’Utilisation des Lasers Intenses (LULI), Ecole Polytechnique, CNRS, Commissariat a l’Energie Atomique(CEA), Université Pierre et Marie Curie (UPMC), 91128 Palaiseau, France
2Gesellschaft für Schwerionenforschung (GSI), Plankstrasse 1, 64291 Darmstadt, Germany3PHASICS, Campus de l’Ecole Polytechnique, Route de Saclay, 91128 Palaiseau CEDEX, France
For high-energy-density physics research, such asthermonuclear fusion and astrophysics laboratoryexperiments, continuous improvements of the per-formance of laser systems are required. By eitherincreasing the delivered energy as for the Le laserMégajoule (LMJ) and National Ignition Facility(NIF) projects [1,2] or reducing the pulse durationusing femtosecond front ends [3,4], petawatt rangehas nowadays been reached. In such lasers, large-scale, flash-lamp-pumped rod or disk amplifiers areused as high-energy amplification stages. However,the thermal loading in these large-size optical com-ponents dramatically degrade the quality of thelaser beam and thus decrease its focusability. More-over, the shot repetition rate is severely limited bythe dissipation time of the heat loaded on the laser
glasses; usually it is of a few hours at the kilojouleenergy level. To overcome this difficulty, wavefrontcorrection systems using a deformable mirror (DM)integrated in the laser chain have been developed[5–8].
We report in this paper results on the dynamicwavefront control of the LULI2000, which is a Nd3�:phosphate laser system having the ability to work inthe chirped pulse amplification regime to reach peta-watt level. It delivers pulses of kilojoules in the nano-second range at the wavelength of � � 1053 nm. Theamplification section of this facility consists of twolarge-scale, flash-lamp-pumped glass chains, each ofwhich contains more than 100 optical components.The design of a wavefront correction system of such acomplex laser system first requires analyzing the or-igin of each aberration.
The wavefront aberrations of the LULI2000 facil-ity are classified mainly in three categories. Theaberrations of the first category result from static
imperfections of optical elements and beam mis-alignment. The second category concerns the aber-rations due to thermal tilt. The third categorygroups together all the remaining thermally in-duced aberrations. Such a classification takes intoconsideration some coupled effects. For example,the coma aberrations can result from beam mis-alignment when the chain is “cold.” They can also beinduced by thermal tilt when the chain is in a se-quence of shots. As will be discussed, their correc-tion is different from those of the other thermalaberrations. To reduce effectively all the Zernikeaberrations, we have associated an adaptive-optics(AO) system, based on a bimorph DM and a four-wave lateral shearing interferometer, with a pre-cise alignment procedure before each full-energykilojoule shot. With such a system, the LULI2000facility is capable of producing a reproducible focalspot close to the diffraction limit (Strehl ratio �0.7)while increasing at the same time its repetition rateby a factor of 2 (one shot per hour). In Section 2, wewill present the detailed analysis of the origins ofthe different aberrations in our chain. In Section 3,we will discuss the solutions for reducing these ab-errations and present our optimized procedure. Re-sults will be given in Section 4. We will conclude inSection 5.
2. Aberration Analysis of the LULI2000 Laser System
The amplification section of the LULI2000 consists oftwo independent optical chains with flash-lamp-pumped Nd3�: phosphate amplifiers. Each is com-posed of four sequential amplification stages, withdifferent aperture diameters (50, 94, 150, and208 mm respectively; see Fig. 1). The first stage con-sists of three rod amplifiers while the other threestages use disk amplifiers. The two chains are seededby a common nanosecond oscillator and deliver �1 kJpulses. Alternatively, one of these two chains can alsobe seeded by a chirped pulse to reach the petawattpower level after compression.
As has been mentioned in the introduction, wave-front aberrations can be classified in three catego-ries: (1) static aberrations intrinsically linked withthe quality of each optical component, its mounting,and with the alignment accuracy of the optical ele-ment with respect to the laser axis, (2) aberrationsdue to thermal tilt, and (3) all the remaining aber-rations induced by thermal effect. These aberra-tions can be easily characterized by measuring thespatial-phase deformation of the laser beam with awavefront sensor or by observing the evolution ofthe focal spot with a far-field camera. We have set afour-wave lateral shearing interferometer (SID-4)developed by PHASICS [9] at the chain output forwavefront characterization. It has a large achro-matic range �400–1100 nm�, an excellent transverseresolution (30 �m, 160 � 120 points), a high accuracy[��100 rms (rms: root-mean-square)] and a high sta-bility ���500 rms�. Such specifications adequatelyfulfill our requirements.
We use the Marechal criterion [10] to evaluate thedegradation of the Strehl ratio Rs. Thus, for a smallwavefront distortion ���
2 �� 1� relative to a rigor-ously stigmatic system, Rs can be written as
Rs � 1 � ��2 � 1 � �2
� �2
�w2, (1)
where �� and �w are, respectively, the rms phase andwavefront aberrations with respect to a perfect wave-front. In the following, we will give the characteristicsof each aberration.
A. Static Aberrations
The static aberrations (category 1) are characterizedwhen the chain is cold, i.e., before the first shot takesplace. The LULI2000 amplification chain is built upof more than 100 optical components. Before imple-mentation, the optical quality of each element wastested separately by a ZYGO interferometer. Foreach, the measured transmitted or reflected wave-front deformation (depending on if it is a transmis-sive or reflective optical component) is found to be lessthan or equal to ��4 peak-to-valley (PtV) and0.025� rms at � � 1053 nm. At the end of the chain,the total measured PtV cumulated wavefront distor-tion is approximately 0.5� and the total rms is ap-proximately 10 times smaller. The Strehl ratio Rs
amounts then to 0.9.The chain is “misaligned” when the beam axis does
not perfectly coincide with the well-defined axes ofthe optical components (especially lenses). A semiau-tomatic procedure is used for alignment of theLULI2000 chains: it is performed by centering thelaser beam on alignment targets (crosshairs) for eachamplification stage. The accuracy is typically 2% ofthe corresponding beam diameter. In parallel, beampointing of each section is controlled by a far-fieldcamera. The accuracy is different along the chain,depending on pupil conjugations. It is �3 �rad at thechain output.
Fig. 1. (Color online) Schematic of the four amplification stages ofthe LULI2000. RA, rod amplifier; DA, disk amplifier; FR, Faradayrotator; SF, spatial filter. A 100 mm diameter bimorph DM, im-plemented at the output of the second stage, is related to a wave-front sensor (SID-4) and a far-field measurement module placed atthe chain output.
The thermally induced aberrations (categories 2 and3) result from the thermal load induced by the flash-lamp light absorbed in the amplifying rods and disksas well as in their claddings. Xe-flash-lamps have abroad spectrum [11], while the absorption by the dop-ing element Nd3� has a narrow line spectrum. Thehost material and its cladding around an amplifyingdisk absorb light especially in the UV and IR [12].Heat absorption induces local dilatation and mechan-ical stress in glass, and thereafter photoelastic effectssuch as birefringence and local change of refractiveindex. Wavefront aberrations generated by thermalgradients can be distinguished in two classes: theso-called pump-shot aberrations, appearing instanta-neously during a shot due to nonuniformity of heatdeposition from pump light, and the aberrations in-duced by thermal relaxation during the followinghours.
1. Thermal Effect During and After a Single ShotTo evaluate the pump-shot aberrations, we measuredthe wavefront of a full-energy shot. Complementarily,to study the thermal relaxation aberrations we mea-sured, using the 10 Hz alignment beam as a probe,aberrations after a full-energy kilojoule shot and dur-ing the several hours of the thermalization. We haveobserved that the pump-shot aberrations are muchlower than the thermal relaxation aberrations. Thisleads us to focus on correcting the thermal relaxationaberrations. Figure 2 illustrates the typical evolutionof the peak wavefront distortion as a function of time.Three zones in this figure can be clearly distin-guished: the distortion increases rapidly during�10 min and then decreases during 50 min owing tothe active cooling effect. After 1 h, the wavefrontvaries very slowly. As a result, a recovery time of atleast 3 h is needed before a complete relaxation ofthermal aberrations. This severely limits the repeti-tion rate of the LULI2000 laser system, which will beanalyzed in the following. Of course, the cooling effi-ciency depends strongly on the cooling system used.
On the LULI2000 chains, rod amplifiers are cooledby water circulation running without interruption.Compressed air at 80 millibars is fed into the disk-amplifier boxes during 20 min immediately after ashot, and then N2 gas flows for �6 min.
The main thermal aberrations, measured duringthe recovery time, are defocus and astigmatism (Fig.3). Concerning the defocus aberration, as time in-creases, the beam wavefront evolves from flat to di-vergent during 45 min, then becomes convergent. Itevolves to flat again at least 3 h after the shot. Astrong astigmatism of 0° has been measured at thechain output. Astigmatism can be produced both inrods and disks under the following circumstances. Itappears (a) when the laser beam is repolarized afterpassing through a birefringent rod; (b) if a sphericalwavefront propagates in the amplifier disks that areset up at Brewster angle; and (c) if the disks behaveas a cylindrical lens under thermal stress. To under-stand the proper origin of this aberration, we per-formed the wavefront measurements separately inrod and disk amplifiers. The experimental resultshows that astigmatism generated following (a) and(b) is negligible. However, it is induced in disks due tothe existing strong thermal gradient between thecenter and the edges of the disks. A thermal cylindri-cal lens is therefore generated, as analyzed in (c).
It is important to note that thermal effect can alsoinduce other aberrations. For example, during thethermalization time, a thermal tilt induces a shift ofthe beam axis, which gives rise to coma aberrations.This appears even if the chain is initially wellaligned. If, in addition, beam misalignment occursbefore a shot, coma aberrations after this shot will beintensified. We simulated this effect by introducing asmall axis error in the vertical direction Y, in a par-ticular plane at the chain input. This plane is one ofthe image-relay planes in the chain and is imaged bythe lenses of spatial filter from one amplificationstage to the next, until it arrives at the entranceplane of the SID-4. In Fig. 4 the time evolutions of thetilt Y [in (a)] and those of the coma Y [in (b)] areplotted for the cases where coma was induced only by
Fig. 2. (Color online) Temporal evolution of the peak wavefrontdistortion after a single full-energy kilojoule shot, measured by theSID-4. Three zones are distinguished: (1) rapid increase; (2) rapiddecay; and (3) slow variation.
Fig. 3. Temporal evolution of the Zernike aberrations after afull-energy kilojoule shot, as recorded by the SID-4, for the defocus,the astigmatisms 0° and 45°.
thermal tilt (square) and was generated by both ther-mal tilt and misalignment before shot (circle). Thesetwo evolutions are correlated. This is an importantresult because that means beam realignment beforeeach shot can indeed reduce the residual thermallyinduced tilt and coma aberrations. It is for this reasonthat they are classified separately from the other ab-errations induced by thermal effect.
2. Cumulative Thermal EffectWhen the laser chain operates in a sequence of shots,cumulative thermal effects further degrade the laserwavefront. To characterize its influence on the repe-tition rate, we carried out a sequence of five full-energy kilojoule shots with two different repetitionrates. The wavefront as well as the far-field patternwere recorded immediately before a next shot. Thebottom curve in Fig. 5 illustrates the measured wave-front distortion with a duty cycle of a shot every twohours. We see that the wavefront distortion increasesdue to heat accumulation during the sequence, andthat with this shot repetition rate, a distortion of 1.6�is accumulated after the fifth shot. By reducing theduty cycle to a shot every hour, this effect is intensi-fied. As a result, the wavefront peak distortionamounts, after the fifth shot, to more than 3� (see theupper curve in Fig. 5). The focal spot measurement
performed during these two shot sequences showsthat the wavefront distortion is directly responsiblefor the dramatic deterioration of the focal spot qualityon the target, and that this degradation worsens intime if the delay between two successive shots is tooshort. The repetition rate of the LULI2000 facility iscompromised by the laser wavefront quality (relativeto the thermal relaxation time) and laser operationeffectiveness. Typically, four shots in a day are avail-able with one shot approximately every two hours.Even so, the laser wavefront is degraded during theday under cumulative thermal effect, as shown by thebottom curve in Fig. 5.
3. Optimization of the Wavefront-Control System
In Section 2, we identified the origins of the differentdominant aberrations, each of which contributes tothe degradation of the wavefront and to the limita-tion of the repetition rate. This study enabled us toadopt an optimized procedure for our wavefront-control system. It is characterized by precise beamrealignment between two successive shots and an AOclosed-loop operation. Its main features are as fol-lows. (1) We perform a beam alignment before thefirst shot, so that the aberrations of the first categoryare reduced to a minimum. The AO system is alsoactivated at this stage. This way, the residual wave-front error is reduced from 0.5� to 0.2� PtV. (2) Tocorrect the aberrations of the second category, whichappear between two successive shots, beam pointingand centering are checked and readjusted systemat-ically with the semiautomatic alignment system be-fore a next shot. This way, thermally induced tilt andcoma are minimized. (3) The correction of the aber-rations of the third category, which are generated bythermal effect, is made by the AO system.
We use a bimorph DM with 32 actuators distrib-uted on three rings [13]. It has a large aperture (di-ameter of 100 mm) and a high damage threshold. Theapplied voltages on the actuators range from �300 to�300 V, corresponding to a dynamics for wavefrontcorrection of more than 6�. To calibrate the mirror,we measure the wavefront deformation induced by
Fig. 4. Temporal evolutions of the (a) tilt Y and (b) coma Y, aftera full-energy kilojoule shot, when the chain is initially well-aligned(square) and misaligned (circle).
Fig. 5. Wavefront distortion under cumulative thermal effect,measured before each full-energy kilojoule shot, for the repetitionrates of (a) one shot every two hours and (b) one shot per hour.
each actuator. Thirty-two phase-map influence func-tions are obtained by applying a voltage of 150 V toeach of them one after another. These 32 phase mapsare then projected on a polynomial basis composed of32 Zernike polynomials. We generate the so-calledlinear-response matrix B. Its columns are the resultof these projections. To evaluate the capacity of thisDM, we performed a singular-value decomposition(SVD) of the matrix B [14]. After this decomposition,B can be written as
B � UVT, (2)
where U and V are in our case 32 � 32 orthogonalmatrices whose columns represent the mirror modes.U contains the phase maps and V the correspondingcommand sets. � is a diagonal matrix whose maxi-mum 32 nonzero elements give the singular values.The larger the singular values, the more responsivethe DM to create a mode. The first deformation modesof the DM are the defocus, the astigmatisms 45° and0°, and the comas 90° and 0°. It follows that all thedominant aberrations that appear in our chain can bedecomposed readily on this basis and then dealt withseparately.
The DM was placed between the second and thethird amplification stages and was associated withthe SID-4 wavefront sensor located at the chain out-put, as shown in Fig. 1. The measurement and cor-rection planes have to be optically conjugated to eachother in order to have a linear response betweenthem. The correction loop was applied using a low-energy 10 Hz pulsed beam �� � 1053 nm�. The wave-front error is defined as the difference between themeasured aberrated wavefront and a reference,which is a flat wavefront in order that its far-fieldintensity distribution approaches to an Airy pat-tern. Assuming that the mirror response is linearand by using the SVD method, the matrix B couldbe inverted. It links the measured phase and volt-ages applied to the bimorphic actuators. We cantherefore determine the voltages to compensate thegiven wavefront deformation. In our experience, theclosed-loop converged in a few iterations in severalseconds.
Experimentally, a fitted higher-order-mode filter-ing after a calibration procedure is necessary for anoptimized closed-loop performance. We first mea-sured the residual rms phase distortion with respectto the mode number of the DM used for correction. Asshown in Fig. 6(a), as the mode number increases, theresidual phase distortion decreases continuouslyfrom more than 0.2 rad until reaches its minimumvalue �0.03 rad� with 24 modes (eight modes filtered).We then configured the DM with different values ofthe mode number [Fig. 6(b)]. We see from this figurethat the 24-mode correction configuration corre-sponds to voltages of approximately �50 V for thesemodes, while voltages applied to the outer ring ac-tuators should be much higher when higher-ordermodes are activated. In fact, the noise is more impor-
tant in higher-order modes because of their smalleigenvalues. This result implies that an adequatehigher-order-mode filtering allows the AO system towork not only with a higher dynamics, but also witha very stable convergence that is insensitive to noiseperturbation.
4. Wavefront-Correction Results
As has been discussed previously, a minimum of 1 hdelay must be respected before the temporal evolu-tion of the phase stabilizes (zone 3 in Fig. 2). Dynamicwavefront correction is thus performed only after thistime delay. We first measured both phase distortionsand focal spots before performing a semiautomaticalignment to correct the residual thermal tilt. Such analignment allows a very accurate beam centering andpointing on the DM so that its response matrix isalways the same from shot to shot. Then we startedthe correction system convergence loop. With theloop, the degraded focal spot, initially formed of anenlarged dissymmetrical pattern surrounded by sev-eral side lobes, is transformed into a single spot thatis almost diffraction limited. The Strehl ratio deducedfrom the measured residual phase is greater than 0.9.Under this condition, we performed a full-energy ki-lojoule shot. Figure 7 shows the measured focal spotsusing a low-energy 10 Hz pulsed probe beam before
Fig. 6. (Color online) (a) Residual rms phase error via the modenumber used for correction. (b) Voltage dynamics of the 32 actua-tors with different mode numbers.
and after the use of the correction loop during a se-quence of one shot per hour. As can be seen from Fig.8(a), the focal-spot quality during the fifth shot hasbeen considerably improved, compared with the caseof no wavefront correction. The encircled energy ofthe focal spot is similar to that of the correspondingAiry spot [Fig. 8(b)]. The Strehl ratio deduced fromthe measured focal spot is thus increased from 0.2[without loop, Fig. 8(a), left] to more than 0.7 [withloop, Fig. 8(a), right]. We think that the differencebetween 0.9 (probe beam) and 0.7 (full-energy kilo-joule beam) comes essentially from pump-shot in-duced aberrations. By use of the AO system in whichthe operation has been optimized, the LULI2000 la-ser system is therefore capable of firing full-energy
kilojoule shots every hour while maintaining excel-lent shot-to-shot focal spot on target.
5. Conclusion and Prospects
In this paper, we have demonstrated efficient wave-front correction in the amplification section of theLULI2000 laser system. The aberrations and theirorigins, either static or induced by the thermal stress,have been identified. An optimized procedure fortheir minimization while firing the laser in sequencehas been implemented. We have shown that by use ofan AO setup associated with an elaborate alignmentsystem, all the low-order Zernike aberrations can beeffectively corrected. As a consequence, reproduciblefocal spots close to the diffraction limit for full-energykilojoule shots fired at a repetition rate of one shotper hour have already been achieved. The focal in-tensity can therefore reach 2.5 � 1018 W�cm2 in thekilojoule per nanosecond range. Similarly, with sucha performance, intensities as high as 1021 W�cm2 canbe foreseen in the near future using the laser chain inthe CPA mode, i.e., in subpicosecond petawatt re-gime.
Future investigations include, in addition to thepresent work, studying more efficient pumping andcooling systems and using hybrid systems, e.g., thosecomposed of a DM and a liquid-crystal spatial lightmodulator, as wavefront correctors. Regarding theCPA operation regime, efforts should be devoted toreduce wavefront distortions induced by the pulsecompressor in the petawatt chain.
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