Wave-front correction of a femtosecond laser using a deformable mirror

Transcription

1 Wave-front correction of a femtosecond laser using a deformable mirror Elizabeth Daly a, Christopher Dainty a, Gerard O Connor b and Thomas Glynn b a Applied Optics Group, Department of Experimental Physics, National University of Ireland, Galway, Ireland; b National Centre for Laser Applications, Department of Experimental Physics, National University of Ireland, Galway, Ireland ABSTRACT Typical applications of ultra-high-power femtosecond lasers include precision drilling and surface micro-machining of metals, and micro-structuring of transparent materials However, high peak-power pulsed lasers are difficult to focus close to the diffraction limit because of aberrations that induce deviations from a perfect spatial wavefront The sources of these aberrations include thermally induced and nonlinear optical distortions, as well as static distortions such as those introduced by gratings used in chirped-pulse amplification (CPA) A spatially clean beam is desirable to achieve the highest possible intensity on-target, and to minimize the energy deposited outside the central focus One way to achieve this is to correct the wave-front using an adaptive optical element such as a deformable mirror, a more cost-effective solution than increasing peak intensity by providing further pulse amplification The wave-front of the femtosecond system is measured using a Hartmann-Shack wave-front sensor, and corrected with a 37-channel deformable membrane mirror used slightly off-axis The deformable mirror has been tested with a FISBA OPTIK µphase r HR digital interferometer, which is also used to calibrate the performance of the wave-front sensor The influence of fluctuations of the laser on the measurement is minimized by averaging the centroid positions obtained from several consecutive frames The distorted wavefront is compared to a reference flat wave-front which is obtained from a collimated laser diode operating at the same wavelength as the femtosecond system The voltages on the deformable mirror actuators are then set to minimise the difference between the measured and reference wave-fronts using a simple least squares approach Wave-front sensor and correction software is implemented in Matlab Keywords: Adaptive optics, Hartmann-Shack, wave-front sensor, femtosecond laser, deformable mirror 1 INTRODUCTION The technique of chirped pulse amplification (CPA) is now routinely used to generate ultra-high peak-power pulses in femtosecond laser systems 1,2 The CPA chain however, provides many opportunities for degradation of the spatial quality of the beam due to thermal, nonlinear and other static geometrical effects 3 These wavefront aberrations all contribute to poor focusability of the laser beam: energy is spread so that the dimensions of the central focused spot grow and, consequently, the intensity is reduced Adaptive optics allows for the possibility to correct the phase front of an aberrated beam and so achieve diffraction-limited focusing Correction schemes require a sensor to measure the distortion and an active element which can apply the correction There is arguably more flexibility in the type of correction that can be achieved if it is possible to place the active element somewhere inside the laser cavity Such schemes include using a deformable mirror as a cavity mirror to provide mode control, 4 or to perform temporal pulse compression, 5 and use of acousto-optic devices to shape the phase and amplitude of short pulses 6 However, intra-cavity control is not generally an Further author information: (Send correspondence to ED) ED: elizabethdaly@nuigalwayie, Telephone: +353 (0) ext2985 CD: cdainty@nuigalwayie, Telephone: +353 (0) GO C: gerardoconnor@nuigalwayie, Telephone: +353 (0) TG: thomasglynn@nuigalwayie, Telephone: +353 (0)

2 option with commercial laser systems Extra-cavity wave-front correction has been demonstrated by several groups using various active elements and wave-front sensors Chanteloup et al 7 used a optically addressed light valve to correct the wave-front of CW and Q-switched lasers based on the output of an achromatic three-wave lateral shearing interferometer They proposed to integrate the light valve into the CPA chain of a femtosecond laser Druon et al 8 used the same type of wave-front sensor but employed deformable mirrors as the correcting devices at the output of a femtosecond laser In the work described here we demonstrate extra-cavity wave-front correction of a commercial femtosecond laser using an inexpensive micromachined deformable mirror (MMDM) The wave-front was measured using a Hartmann-Shack (HS) wave-front sensor (WFS) which utilized light leaking from the system The speed of the simple correction loop provided a good improvement in the wave-front in approximately 10s 2 THEORY Matrix multiplication can be used to express the linear relationship between the wave-front sensor (WFS) signals and the control bytes applied to the mirror actuators: IFM c = s, (1) where c is the column vector of n squared control bytes applied to the mirror (c = [v 2 1,v 2 2,,v 2 n] T ) and s is the column vector of wave-front x- and y- slopes measured by the HS WFS (s = [s 1x,,s mx,s 1y,,s my ] T ) The Influence Function Matrix (IFM), (or response matrix) of the system is obtained by poking each mirror actuator in turn and recording the resulting wave-front slopes Its size is 2m rows by n columns For example, column number 1 of the IFM contains the m x- slopes and m y- slopes that result on poking actuator number 1 For wave-front correction, the inverse of Eq (1) is required: on measuring a set of slopes s, we want to find the control bytes c required to produce those slopes CM s = c, (2) where CM is called the control matrix However, the inverse of Eq 1 is not generally available because the IFM is normally singular and non-square Some other technique, such as Singular Value Decomposition (SVD) 9 must be used to invert Eq 1 and find the required c With SVD, the IFM can be written as a product of three matrices IFM = USV T (3) Here, U (2m n) and V (n n) are matrices with orthogonal columns and S (n n) is a diagonal matrix containing the singular values λ i U represents the n mirror modes that correspond to to the n singular values, while V represents the mirror control bytes that are related to these mirror modes through the singular values On application of a given υ i the sensor produces signals λ i u i, which means that this sensor signal u i can be corrected for by applying control byte λ 1 υ i We therefore require the pseudoinverse of the IFM, and this is formed from IFM 1 = CM = VΓU T, (4) where Γ is found by inverting each singular value in S, Γ = λ 1 i Matrix CM is often called the least-squares (LSQ) reconstruction matrix or controller, in the sense that it minimizes the measurement error ε = s IFMc 2 One attraction of the SVD method is that it allows filtering of the singular values so that modes with small sensitivities (small λ i ) and therefore large system gains (large λ 1 i ) can be filtered out by setting these singular values equal to zero in the pseudoinverse As the mirror is not perfectly linear, we cannot apply all of the correction in one single step 10 Instead, the correction is applied to the existing mirror control bytes via some gain factor Supposing sensor signal s results from the application of control vector c old = [vold 2 ] to the mirror The correction is found from the LSQ controller to be c cor = CMs, and the control bytes for correction are given by v cor = c cor The new mirror control bytes are then calculated from v new = v old g d v cor, (5)

3 where g is the loop gain and d is used to store information about the sign of c cor ; d = (+1,0, 1) for c cor (> 0,= 0,< 0) We found experimentally that a good value of g was about 004 Larger values caused significant oscillation in the wave-front rms and clipping of the control bytes at the limits (0-2), while smaller values meant that many iterations were necessary to achieve wave-front correction It is not necessary to perform wave-front reconstruction in order to do wave-front correction as one can monitor the error ε, instead of looking at the wave-front, on each loop iteration This speeds up the correction loop We chose however to reconstruct the wave-front each time around the loop as we wished to see the Zernike components and identify if mirror stroke was being wasted correcting, for example, tip and tilt, when these could be minimized using a simple tip-tilt mirror mount instead The wave-front can be represented by an expansion of polynomials, the most commonly used ones being the Zernike polynomials, 11 which are orthogonal over the unit circle W(x,y) = a j Z j (x,y), (6) j=0 where a j are the Zernike coefficients and Z j (x,y) are the Zernike polynomials expressed in Cartesian coordinates It is possible to relate the HS WFS slopes to the Zernike polynomials through s ix = s iy = a j Z jx (x,y), (7) j=0 a j Z jy (x,y), (8) j=0 where s ix are the average x- wave-front slopes over lenslet i, and Z jx (x,y) is the average of the jth Zernike derivative, with respect to x, over the same lenslet (similarly for y) These relationships can be expressed in matrix form as s = Za (9) for a finite number, j, of Zernike terms The slopes column vector s is identical to that of Eq 1, a = [a 1,,s j ] T is the column vector of Zernike coefficients, and Z is the 2m rows by j cols matrix of averaged Zernike derivatives Z = The Zernike coefficients can be found from Z 1x(x 1,y 1 ) Z 2x (x 1,y 1 ) Z jx (x 1,y 1 ) Z 1x(x 2,y 2 ) Z 2x (x 2,y 2 ) Z jx (x 2,y 2 ) Z 1x(x m,y m ) Z 2x (x m,y m ) Z jx (x m,y m ) Z 1y(x 1,y 1 ) Z 2y (x 1,y 1 ) Z jy (x 1,y 1 ) (10) Z 1y(x 2,y 2 ) Z 2y (x 2,y 2 ) Z jy (x 2,y 2 ) Z 1y(x m,y m ) Z 2y (x m,y m ) Z jy (x m,y m ) a = Z + s (11) where Z + is the pseudoinverse of Z All wave-fronts were reconstructed using the first 15 Zernike terms as described in Born and Wolf, 11 which corresponds to 4 th order All wave-front sensing, reconstruction and correction was implemented in MATLAB This is a slow process, but speed was not critical in this application

4 M2 M4 M5 M3 Tx1/2 CCD L Tx2 Tx1 To sample Wave-front sensor PC LD MMDM M1 Figure 1 Schematic of experimental setup M1 to M5, beam-steering mirrors; MMDM, deformable mirror; Tx2, expand by 2 telescope; Tx1, 1:1 telescope for mirror bias compensation; Tx1/2, reduce by 2 telescope; L, lenslet array; LD, laser diode for calibration 3 EXPERIMENTAL A schematic of the experimental setup is shown in Fig 1 The laser system used was a commercial titaniumsapphire system (Clarke MXR, CPA01) based on a CPA It emits pulses of linearly polarized light at a central wavelength of 775nm Although it is possible to vary the pulse duration by varying the distance between the pulse compressor gratings, for this work pulses were 1fs duration at a repetition rate of 1 khz The system is normally used as an ultra-fast materials processing workstation: the laser is directed to the sample via a beam-expanding telescope (x2) and beam-steering mirrors M1, M2 and M3 The pulse energy is controlled by a half-wave plate and polarizing beam splitter placed before mirror M1 For wave-front sensing and correction some additional elements were added to the standard configuration A HS WFS placed directly behind M3 detected the small fraction of the main beam which leaked through this mirror It contains a reducing telescope (x2), an Adaptive Optics Associates standard lenslet array of focal length 19mm, pitch 2µm, and an inexpensive 8-bit CCD camera The relatively long focal length of the lenslet array was chosen to ensure that the spots formed on the CCD had full-width at half-maxima (FWHM) of greater than two CCD pixels, thus ensuring good centroiding accuracy for wave-front sensing The wave-front was corrected using a low-cost 37-channel micro-machined deformable mirror (MMDM) produced by OKO Technologies This mirror was held in a tip-tilt mount to allow independent adjustment of these Zernike terms prior to any wave-front correction An 8-bit control byte (0-2) was fed to each channel to deform the mirror, where the upper limit of 2 was set by the high-voltage power supply used This particular mirror had an additional high-damage threshold multi-layer coating at 775nm which is capable of withstanding the very high laser fluence produced by the system A 1:1 telescope placed in front of the MMDM compensated for the fact that the mirror in its biased position has a slightly spherical shape The MMDM, the 1:1 telescope and beam-steering mirrors M4 and M5, were all mounted on a single aluminium plate which could be easily integrated into the machining stage 31 WFS calibration The reference spot pattern for the WFS was obtained with a collimated CW laser diode operating at 780nm which was inserted into the beam path between mirror M1 and the expanding telescope On addition of M4, wwwokotechcom

5 Act 1 Act 2 Act 3 Act 4 Act 5 Act 6 Act 7 Act 8 Act 9 Act 10 Act 11 Act 12 Act 13 Act 14 Act 15 Act 16 Act 17 Act 18 Act 19 Figure 2 X and y displacements obtained on poking each actuator in turn about the MMDM bias position M5, and the MMDM, this collimated source was also used to calibrate the deformable mirror Starting with the mirror in the biased position, each actuator was poked away from bias in both positive and negative directions, and the resulting WFS slopes for each spot were stored to build up the IFM column by column To ensure good accuracy, 10 frames at each position were stored and the slopes were calculated from the average spot positions The procedure took approximately minutes for 37 actuators and 89 WFS spots The first 19 columns of the IFM are shown in Fig 2 where the x and y displacements recorded by the WFS at each reference coordinate are plotted One can see here that actuators 1 to 7 will have a large effect on the correction, with the actuators in the next ring (8 to 19) being less effective The outermost ring of actuators ( to 37) has very little influence for the 5mm pupil we used in this work, but it was included in the IFM As predicted by theory, the deformation of the mirror as a function of applied voltage was observed to follow a quadratic dependence on voltage, 12 placing the bias position at control byte 1 applied to all channels (1/ 2 2) The mirror can then provide equal correction on both sides of bias as shown in the experimental plot of Fig 3 The maximum deflection of the mirror, and therefore the amount of wave-front correction it can provide in this experiment, is determined by the size of the pupil at the mirror We have measured 5, 2, and 1 µm of total aberration (from CB = 0 to 2) for pupils of 12, 8, and 5 mm respectively For correction of the wave-front of the femtosecond laser the pupil at the mirror is determined by the fact that the laser beam has a Gaussian cross section and the 8-bit camera can only detect a finite portion in the central part With the standard machining set-up, this corresponds to a pupil at the mirror of just 5mm diameter, or to ± 05 µm of available correction about the bias position 32 Performance of the WFS In contrast to, for example, wave-front sensing in astronomy, low light levels were not an issue in this work In fact, the laser was attenuated behind mirror M3 to avoid saturation of the CCD camera The error in determining the centroid positions for each spot was estimated by making repeated measurements of a plane wave-front We found that the standard deviation in spot position was approximately 1/ th of a CCD pixel, or 05 µm The error of wave-front reconstruction was estimated in the same way: a plane wave was detected and reconstructed many times to determine the standard deviation in the wave-front rms This was determined to be 4 nm It agrees very well with the same number calculated by combining the centroiding error and the trace of the matrix Z Z, where Z is the matrix of partial derivatives of the Zernike terms averaged over each lenslet 13

6 15 1 Bias CB = 1 05 PV (µ m) Control Byte x 10 4 Figure 3 V squared dependence of wave aberration produced by the MMDM These values were measured with the WFS using a pupil of approximately 8 mm diameter at the mirror x Figure 4 Example wave-fronts measured with interferometer (left-hand side) and HS WFS (right-hand side) The interferometer measurement is shown in an 8mm diameter pupil, while the wave-front measured by the WFS is plotted in a normalized pupil corresponding to 8 mm This particular comparison represented the worst discrepancy in PV aberration (-25%) between the two methods in a series of measurements of different wave-fronts The ability of the WFS to faithfully reproduce any measured wave-front was tested by comparing results obtained with it to those obtained with a FISBA OPTIK µphase r HR digital interferometer This comparison was performed at 6328 nm which is the wavelength of operation for the interferometer An example of a measured wave-front is shown in Fig 4, for a specific voltage pattern applied to the MMDM actuators In order to compare directly the measured peak-to-valley (PV) and root-mean-square (rms) wave-front aberrations in each system, a measurement mask of 8 mm diameter was applied to all FISBA data, thus ensuring that the aberrations were calculated over the same pupil We found agreement to within 25 %, worst case, for all comparisons made wwwfisbacom

7 Act 1 Act 2 Act 3 Act 4 Act 5 Mode 1 Mode 2 Mode 3 Mode 4 Act 6 Act 7 Act 8 Act 9 Act 10 Mode 5 Mode 6 Mode 7 Mode 8 Act 11 Act 12 Act 13 Act 14 Act 15 Mode 9 Mode 10 Mode 11 Mode 12 Act 16 Act 17 Act 18 Act 19 Mode 13 Mode 14 (a) (b) Figure 5 (a) Reconstructed wavefronts corresponding to the influence functions of the first 19 MMDM actuators (b) Mirror modes obtained after application of SVD to the IFM of Zernike coefficients Note that that mode 15, corresponding to piston, is not shown here 41 Influence functions of mirror 4 RESULTS In section 31 we described acquisition of the IFM by poking of each mirror actuator in turn The columns of this matrix consist of wave-front slopes It is also possible to construct a type of IFM where each column contains the 15 Zernike coefficients required to reconstruct the wave-front obtained on poking one actuator The dimensions of this IFM are 15 rows x 19 cols, the number of Zernike terms and the number of actuators considered Using Singular Value Decomposition (SVD) it is possible to extract from this IFM the set of surfaces that the mirror can reproduce exactly - the spatial modes of the mirror as given by the columns of matrix U in Eq 3 These modes correspond to non-zero singular values in the IFM decomposition Any surface composed of a linear combination of these modes will be produced exactly on application of the appropriate control voltages to the mirror electrodes The mirror influence functions for the inner 19 MMDM actuators and the accompanying mirror modes from SVD are plotted in Fig 5 42 Procedure for correction Fig 6 shows a typical HS spot pattern obtained from the femtosecond laser and including the centroid search boxes and pupil aperture The centroids were ordered from top to bottom and from left to right before calculation of wave-front slopes Prior to the implementation of any wave-front correction loop, the wave-front obtained from the laser was compared to the reference wave-front in order to reduce the difference between the two, while keeping the MMDM in the biased position This was done by careful manual adjustment of the tip-tilt mirror mount that held the MMDM so that the Z 2 and Z 3 terms in the Zernike expansion were less than the other low-order terms In this way we could reduce the rms difference between the two wave-fronts without using any mirror stroke Before correction, the largest terms in the expansion were generally defocus and the two astigmatism components, as seen for example in Fig 9(b) Due to the Gaussian nature of the laser beam, careful adjustment of the laser intensity was required to ensure that the center of the profile did not saturate the CCD camera, while the outer spots were still bright enough to be detected The process of wave-front correction applied in the work was quite simple At the beginning of the script the IFM was read in and SVD was used to calculate the LSQ control matrix as described in section 2 A spot pattern from the laser was stored and used to determine the finding pupil for the WFS centroids: a mask obtained from this image was used to define a region of interest on the CCD camera The reference HS image produced by the collimated beam was read

8 HS image Figure 6 A typical HS spot pattern showing the pupil and search areas for each spot in and all reference centroids were ordered and stored A loop counter was started, the mirror was placed in the biased position, and several frames of HS images were obtained and averaged to find the femtosecond laser centroids These were ordered and matched to the reference coordinates to find the wave-front slopes At this stage the laser wave-front was reconstructed and the applied control bytes, Zernike coefficients, and wave-front PV and rms were stored The LSQ control matrix was then applied to the wave-front slopes to calculate the correction to the actuator control bytes The new control vector was applied to the MMDM and the loop was executed again On completing the loop, the set of control bytes which resulted in the lowest value of rms error was applied to the mirror to perform the correction 43 Quality of correction In this demonstration experiment a single value for the loop gain was applied at every iteration This value was chosen by examining the convergence of the wave-front rms towards a stable value The gain needs to be small enough that the rms value does not oscillate, but large enough to force convergence in a reasonable number of iterations If oscillation occurs the correction control bytes tend to get clipped at their upper and lower limits, thereby reducing the effectiveness of the next loop calculation An example of running the Matlab script for 12 iterations using different gain values is shown in Fig 7 In general the gain used in this work was fixed at g = 004 In the SVD described in section 2, modes which have small sensitivities λ i correspond to large system gains λ 1 i in the control matrix These modes are extremely sensitive to noise, and when included in the correction loop, they can cause it to diverge The advantage of SVD is that modes associated with large gains can be filtered out by setting their singular values equal to zero in the pseudoinverse 14 The ratio of the largest to the smallest singular value is called the condition factor of the matrix We measured condition factors of for λ 1 /λ 37 and just 10 for λ 1 /λ 19 for the experimental IFM of wave-front slopes measured in section 31 The effect of the number of modes on the quality of wave-front correction is shown in Fig 8 With just a few modes the correction is quite poor, as it is for or more modes This trend is most likely a consequence of the small beam size at the MMDM Recall from Fig 5, that the effect of the inner 7 actuators is quite strong, while the effect of those in the next ring (8 to 19) is reduced The outermost actuators ( to 37) have very little effect in the IFM and therefore on the control matrix Measurement of the IFM, in a separate simulation, using a 10 mm diameter beam routinely returned condition factors λ 1 /λ 37 of about and no real drop off in wave-front correction quality was observed on discarding the smaller singular values from the control matrix 44 Some typical results The results of a typical run of the correction loop are shown in Figures 9 for loop gain of 004 and on retaining just the first 7 singular values in the SVD Note that Zernike coefficients 2 (y tilt) and 3 (x-tilt) are smaller than

9 G=002 G=004 G= Wave front rms (nm) G=006 G=008 G= Iteration Figure 7 Residual wave-front rms error as a function of loop gain Final wavefront rms (nm) Measured stdev in rms is 4nm Number of singular values Figure 8 Best rms achieved over 10 iterations as a function of the number of singular values retained in the control matrix Loop gain is g = 004

10 Start Minimum 004 rms(nm) 45 µ m (a) Iteration 004 (b) Zernike coefficient Figure 9 (a)wave-front rms obtained on running the correction loop 10 times with g = 004 and 7 singular values (b) Zernike coefficients of the reconstructed starting wave-front and that obtained on iteration 9 Start Minimum microns Figure 10 Reconstructed wave-fronts corresponding to iterations 1 and 9 of Fig 9 the other low-order terms (4 is defocus, 5 and 6 are astigmatism) at the start of the process to ensure that mirror stroke is not wasted on correcting these terms The wave-fronts corresponding to these Zernike expansions are shown in Fig 10 Over many runs the initial wave-fronts are always similar (providing tilt has been minimized) as the mirror always starts in the biased position The final (corrected) wave-fronts are also very similar with the largest terms in the Zernike expansion usually being the two astigmatism terms We also found that the actuator voltages required to produce the best wave-front on each run of the correction script were quite similar This would suggest that the aberrations being corrected are static, and that there is no real requirement for a high-speed correction loop in this particular application

11 5 CONCLUSIONS In summary, we have demonstrated extra-cavity wave-front correction of a high-power femtosecond laser using an inexpensive micro-machined deformable mirror The wave-front was sensed with a Hartmann-Shack wave-front sensor which uses stray light exiting from the system Typical pulse energies at the MMDM were µj, a factor of approximately 15 lower than the number used when specifying the MMDM multi-layer coating In this work we did not simultaneously perform any laser machining, although the coating was designed to withstand the higher pulse energies that are generally required to do this The accuracy of the wave-front sensor was quantified by comparing its performance to that of a commercial interferometer measuring the same wave-fronts Initial experiments show a % reduction in the femtosecond wave-front rms when compared to a plane wave generated by a collimated source at the same wavelength This is despite the fact that the beam size at the deformable mirror covers only half of the mirror actuators, resulting in limited mirror stroke and therefore limited correction capability The relatively modest correction we demonstrate can be attributed to the fact that only the central part of the laser beam is being detected by the WFS, and this is not highly aberrated to start with From typical pre and post correction wave-front rms values, we can infer Strehl ratios of about 076 and 094 respectively The high Strehl of the corrected wave-front offers the potential to achieve a diffraction-limited spot on focusing the beam The speed of the correction loop is quite slow; it takes approximately 1s for each iteration of the loop, although this does include averaging over CCD frames and wave-front reconstruction However, we find that the final voltage pattern required to correct the laser beam is similar on each run through the loop This would suggest that the aberrations being corrected are essentially static, and that there is no requirement to make the correction loop any faster The performance of the loop evaluated as a function of the loop gain and of the number of singular values (mirror modes) retained in the matrix decomposition In this way we were able to identify the optimum loop gain and number of modes for this particular geometry ACKNOWLEDGMENTS This research was supported by Science Foundation Ireland under Grant No SFI/01/PI2/B039C The NCLA would like to acknowledge the support of the Higher Education Authority under the Programme for Research in Third Level Institutes (1999) and Enterprise Ireland under the Technology Development Fund (03) REFERENCES 1 D Strikland and G Mourou, Compression of amplified chirped optical pulses, Opt Commun 56, pp , P Maine, D Strikland, P Bado, M Pessot, and G Mourou, Gereration of ultrahigh peak power pulses by chirped pulse amplification, IEEE J Quant Electron 24, pp 398 3, S Ranc, G Chériaux, S Ferré, J-P Rousseau, and J-P Chambert, Importance of spatial quality of intense femtosecond pulses, Appl Phys B, pp S181 S187, 00 4 W Lubeigt, G Valentine, J Girkin, E Bente, and D Burns, Active transverse mode control and optimisation of an all-solid-state laser using an intracavity adaptive-optic mirror, Opt Exp 10, pp 5 555, 02 5 E Zeek, R Bartels, M M Murnane, H C Kapteyn, S Backus, and G Vdovin, Adaptive pulse compression for transform-limited 15-fs high-energy pulse generation, Opt Lett 25, pp , 00 6 K Ohno, T Tanabe, and F Kannari, Adaptive pulse shaping of phase and amplitude of an amplified femtosecond pulse laser by direct reference to frequency-resolved optical gating traces, J Opt Soc Am B 19, pp , 02 7 J C Chanteloup, H Baldis, A M G Mourou, and B L J P Huignard, Nearly diffraction-limited laser focal spot obtained by use of an optically addressed light valve in an adaptive-optics loop, Opt Lett 23, pp , F Druon, G Chériaux, J Faure, J Nees, M Nantel, A Maksimchuk, G Mourou, J C Chanteloup, and G Vdovin, Wave-front correction of femtosecond terawatt lasers by deformable mirrors, Opt Lett 23, pp , 1998

12 9 W H Press, S A Teukolosky, W T Vetterling, and B P Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, B Wattellier, J Fuchs, J-P Zou, J-C Chanteloup, H Bandulet, P Michel, C Labaune, S Depierreux, A Kudryashov, and A Aleksandrov, Generation of a single hot spot by use of a deformable mirror and study of its propagation in an underdense plasma, J Opt Soc Am B, pp , M Born and E Wolf, Principles of Optics, Cambridge University Press, New York, G V Vdovin, Adaptive Mirror Micromachined in Silicon PhD thesis, Delft University of Technology, L Diaz Santana Haro, Wavefront sensing in the Human Eye with a Shack-Hartmann Sensor PhD thesis, Imperial College of Science, Technology and Medicine, C Paterson, I Munro, and J C Dainty, A low cost adaptive optics system using a membrane mirror, Opt Exp 6, pp , 00