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fficient method for the reduction of large pistonrrors in segmented-mirror telescopes

ary Chanan and Agustı Pinto

Phase discontinuity sensing �PDS� is one of two successful approaches to segment phasing that arecurrently in use at the Keck telescopes, but it has only a limited capture range. We describe and presentnumerical simulations of a broadband version of the current �narrowband� PDS algorithm that canextend the capture range from 0.4 to 40 �m. Like the original algorithm, the new broadband PDSalgorithm requires no special-purpose hardware but only a high-resolution area detector operating in the2–3-�m range. The potential application of this algorithm to extremely large telescopes is also dis-cussed. © 2004 Optical Society of America

OCIS codes: 010.7350, 120.5050, 220.1140.

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. Introduction

n a typical generic process in which a physical pa-ameter is adjusted or tuned, the coarse adjustmentends to be relatively easy, with the difficulty increas-ng as one proceeds to finer and finer adjustments.owever, in the case of phasing mirror segments in a

egmented-mirror telescope the situation is reversed.ecause of aliasing, it is the coarse adjustments �i.e.,

hose that correspond to phase errors of many wave-engths� that are the most difficult to make. Oncehe phase errors are reduced below a half-wave �auarter-wave of surface error�, the problem is greatlyimplified.At the Keck Observatory the coarse phasing prob-

em has been solved by means of the so-called broad-and phasing camera system �PCS� algorithm,1hich has a capture range of �30 �m. This full

apture range has proved necessary for optical align-ent not only following the initial installation of seg-ents but also following routine periodic segment

xchanges for aluminization. Fine phasing at theeck Observatory is done with the narrowband PCSlgorithm.2 Both of these algorithms require �theame� special-purpose hardware in the form of an

The authors are with the Department of Physics and Astronomy,niversity of California, Irvine, Irvine, California 92697. G.hanan’s e-mail address is [email protected] 21 November 2003; revised manuscript received 11arch 2004; accepted 12 March 2004.0003-6935�04�163279-08$15.00�0© 2004 Optical Society of America

lignment camera,3 one of which is permanentlyounted on each of the two Keck telescopes.4Phase discontinuity sensing5 �PDS� is an alterna-

ive phasing scheme that is also in use at the Keckbservatory. Unlike the PCS, PDS has the useful

eature that no special-purpose hardware �or evenny supplementary optics� is required; it needs onlyn infrared area detector of suitable image scale.or this purpose we use the Near Infrared Camera6

NIRC� on the Keck 1 telescope. PDS has provedxtremely valuable as a backup to PCS phasing.DS was also used to confirm that the original PCShasing solution was correct. However, to date onlynarrowband PDS algorithm �the analog of narrow-

and PCS� exists: the capture range of narrowbandDS �NPDS� is �400 nm. There is currently noackup to PCS that is capable of capturing segmentsith piston errors of the order of 1 �m, let alone the

ens of micrometers that are required after a segmentxchange. In this paper we explore a broadbandersion of PDS with a capture range similar to that ofroadband PCS.Note that the desire for a backup coarse phasing

ystem for the Keck Observatory is not the sole mo-ivation for the development of a broadband PDSBPDS� algorithm. Many proposed extremely largeelescopes �ELTs� with significantly more segmentshan the Keck telescopes are currently in the plan-ing stages.7–11 Although the baseline conceptualesign of, for example, the California Extremelyarge Telescope12 �CELT� anticipates a PCS-type ap-roach to segment phasing, in view of the likely costnd complexity of the alignment camera that woulde required for this purpose it is worth investigating

1 June 2004 � Vol. 43, No. 16 � APPLIED OPTICS 3279

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lternative phasing schemes in the context of ELTs.dditionally, the backup and verification functions ofuch an alternative scheme are even more critical forLTs than for the Keck telescopes because ultrahigh-ontrast imaging represents a major observationaloal for these proposed telescopes. For contrast ra-ios of 106, the diffraction consequences of even aingle wayward segment �of a total of 1000 segments�an be significant.

In Section 2 of this paper we briefly review theheory and practice of the current NPDS algorithm.n Section 3 we present the relevant theory for BPDS,nd in Section 4 we describe the algorithm in someetail, including practical considerations. Section 5resents results of numerical simulations of BPDSor the Keck telescopes, and in Section 6 we discusshe algorithm in the context of ELTs. Our conclu-ions are summarized in Section 7.

. Review of Narrowband Phase Discontinuity Sensing

he current NPDS algorithm5 exploits the fact thator monochromatic light of wavelength � the intensityf the difference between inside- and outside-of-focusmages in the neighborhood of an intersegment edgearies as sin�2k��, where k � 2�� is the wave num-er and � is the edge height difference between thewo segments in question. The problem of extract-ng � from the so-called difference image is linearizedy restriction of the algorithm to sufficiently small �,n practice ��8 � ��8, so the capture range is400 nm for the operating wavelength of � � 3.3 �m.his is adequate for monthly tune-ups of the segmenthasing but is 2 orders of magnitude too small toapture segments after a segment exchange, as, forxample, after realuminizing. The piston accuracys 40 nm, quoted here—like the capture range—at theurface, not at the wave front.NPDS is subject to two conditions that effectively

estrict it to infrared wavelengths. These conditionsre the following: �a� The scale of diffraction effectsassociated with the primary-mirror segments� in themage plane must be small compared with radius a of

segment mapped onto the out-of-focus image plane,.e.,

f ��a al�f, (1)

here f is the effective focal length and l is the ex-rafocal distance. �b� The segment diffraction effectsust be well resolved on the detector:

��a �� p�206265, (2)

here p is the pixel size �image scale� in arcseconds.more thorough discussion of the restrictions on

DS can be found in the original paper.5Figure 1 shows a numerically generated out-of-

ocus image of a star for the Keck telescope, at aavelength of 3.3 �m, with one segment pistoned out

f the page by ��8. The resultant diffraction effectsre well localized and correspond to the position ofhe segment in the primary mirror.

280 APPLIED OPTICS � Vol. 43, No. 16 � 1 June 2004

As noted above, the NPDS algorithm operates onhe difference image:

S � I� � I, (3)

hich is the pixel-by-pixel difference between inside-f-focus �I�� and outside-of-focus �I� stellar images,ith the outside-of-focus image rotated by 180°, such

hat a given ray from the primary mirror hits theame nominal pixel location on the detector in bothmages. Because the diffraction effects in questionxtend somewhat beyond the corresponding locationf the segment in the pupil �as is clear from Fig. 1� thelgorithm does not converge in a single step butather is iterative, with full convergence typically re-uiring four or five iterations.

. Theory of Broadband Phase Discontinuity Sensing

t is straightforward to extend greatly the captureange of a monochromatic wavelength phase mea-urement technique, such as NPDS, by makingeasurements at two or more discrete wavelengths

nd constructing the so-called synthetic wave-ength,13 which is much larger than any of the in-ividual wavelengths. However, we have electedot to apply this approach to PDS. The reason ishat synthetic wavelength algorithms are highlyonlinear; thus, if the measurement uncertaintiesre only modestly larger than expected or the initialhase errors slightly exceed the formal captureange, the resultant phase errors can be huge, es-entially because the solution can unexpectedlyump to the wrong branch of the relevant multiple-alued curve.14 We prefer instead a white-light, orroadband, algorithm. This approach exploits theact that the diffraction effects associated with largehase errors are washed out as the bandwidth is

ig. 1. Numerically generated inside-of-focus image of a starormed by the Keck telescope at a monochromatic wavelength of.3 �m. The upper segment in the middle ring has been pistonedut of the page by ��8. The diffraction pattern has an obviousorrespondence with the position of the mispistoned segment in therimary mirror.

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ncreased and the image becomes incoherent.uch an approach has been used successfully in thehack–Hartmann-type broadband PCS phasing al-orithm1 at the Keck Observatory. If it is pre-ented with initial phase errors that are beyond theapture range, then—rather than coming up with arong answer �as a synthetic wavelength algorithmill�—a broadband algorithm will realize that

omething is wrong and the data can be flaggedppropriately. This is a significant advantage inhe phasing of highly segmented-mirror telescopes,n which one would much rather omit some ques-ionable measurements �the problem is in any caseighly overdetermined� than contaminate the over-ll precision phasing solution with a few measure-ents that are grossly wrong.To acquire some insight into the way in which the

roadband PDS algorithm works, consider a se-uence of difference images for a segmented mirrorelescope such as the Keck, for which the pistonrrors for an isolated segment or two are given by� ��n�2 � 1�8� �, where n � 0, 1, 2, . . . and � �for all other segments. For truly monochromatic

ight, all difference images in this sequence willhow identical maximal signals Smax�� S��8�,ut, for a finite bandpass �, the difference imageill gradually disappear for sufficiently large �.

pecifically, we have S Smax for � �� lc, where lcs the coherence length of the wavelength filter:

lc ��2

2 �. (4)

e take � to be the FWHM of the filter transmissionurve; this gives consistent results �see below� for aaussian or square transmission curve. Figure 2

hows the expected washing out of the signal for rep-esentative simulated difference images for such aequence for the Keck geometry and a filter with �0 �.31 �m and � � 0.063 �m �Gaussian transmissionurve�, so that lc � 87 �m.

These considerations can be made more precise byeans of an analysis similar to that used in the der-

vation of the broadband PCS signal. We begin withhe basic expression that relates the optical field athe aperture of an optical system, described by a pupilunction P�x, y� and a focal length f, to the optical field

at a defocus distance l f

��, k� � �� P� x, y�exp�2ik��exp�ikl�2

4f 2�� exp�ik� � ��dxdy, (5)

ig. 2. Numerically generated difference images for the Keck telescope with two mispistoned segments, through a filter with � � 3.3 �mnd � � 0.063 �m. The piston error of the upper and lower segments �positive and negative values, respectively, are as follows: �a��8, �b� ��5� � ��8�, �c� ��10� � ��8�, �d� ��15� � ��8�, �e� ��20� � ��8�, and �f � ��25� � ��8�. The fading of the difference image with


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here ��x, y� is the position vector in the aperturelane, ��, �� is the position vector in the defocusedmage plane, and ��x, y� is the telescope surface errorincluding segment piston and higher-order aberra-ions�. A constant optical field �plane wave� incidentpon the aperture has been assumed.Image intensity I�� is then equal to ��, k��2.

he effects of seeing in the long exposure limit can bencluded by means of a simple convolution. We ob-ain the broadband image by adding together severalonochromatic images that have been properlyeighted by the spectrum G�k�:

�I�� G�k��, k��2dk. (6)

inally the difference image is obtained by use of Eq.3�.

Following an analysis similar to that which haseen employed elsewhere,5 we have that, in theeighborhood of an edge,

S��, k� � M�� k�S��8, 0�, (7)

here S is the intensity of the difference image and Ms the modulation, which is the Fourier transform ofhe spectrum of the incident light. The first argu-ent of S is the piston error; the second represents

he filter FWHM in k-space. Note that M dependsn the quantities � and k through the product � knly. The quantity S��8, 0� is understood to bevaluated at the center of the bandwidth. Equation7� holds point by point �or pixel by pixel� in theeighborhood of an edge; that is, there are implicitubscripts �i, j� that indicate the pixel locations on Sbut not on M� on both sides of this equation. For anssumed Gaussian bandpass in k we have

M�� k� � exp�2�k2�2�sin 2k�, (8)

here �k and lc are related by

lc�k ��

�8 ln 2� 1.334 (9)

nd �8 ln 2�k is the FWHM of the Gaussian band-ass. For a square bandpass we have

M�� k� �sin�� k�

� ksin 2k�. (10)

e now want to extract the pixel-independent,iston-sensitive quantity M�� k� for a given edge.o do this, we multiply both sides of Eq. �7� by S��8,�, make the �i, j� dependence explicit, sum over i and, and rearrange to obtain

c��, k� � � Sij��, k�Sij��8, 0�

� Sij2��8, 0�

� M�� k�, (11)

here the expression in the middle, which we abbre-iate as c��, k�, is the correlation coefficient of theifference image and S��8, 0�, where the latter


uantity is obtained from a numerically generatedemplate. The sums are taken over all the pixels inbox by which we formally define the neighborhood

f an edge. We condition the data by normalizinghe inside- and outside-of-focus images before sub-racting to form the difference image, and the meanalues of S��, k� and S��8, 0� are forced to zeroefore the summation. An example of a templatemage consisting of a template for all edges is shownn Fig. 3. The correlation coefficient c��, k� repre-ents a robust estimate of the modulation on theight-hand side of Eq. �11�. A plot of the right-handide of Eq. �11� versus � is shown for a filter with � �.2 �m and � � 0.59 �m in Fig. 4.At this point the problem of determining the un-

nown edge height � has been reduced to determiningn unknown horizontal offset in the curve in Fig. 4.o make this determination we measure the correla-ion coefficient c for a variety of values of the edgeeight that differ from one another by knownmounts, where the telescope’s active control sys-em15 is used to increment the edge height in cali-rated steps. We then determine the unknownriginal edge height by fitting Eq. �11� to �. Al-hough the normalization of the right-hand side ofq. �11� is in principle known, we simultaneously fit

o the overall normalization as well, because the pres-nce of noise in the measurements will systematicallyeduce the correlation coefficient below its theoreticalalue.Thus far we have discussed the measurement pro-

ess in the context of a single edge, but one can mea-ure all intersegment edges simultaneously in anfficient way by stepping the primary mirror throughsequence of configurations, as is done in the PCS

ig. 3. Numerically generated template for � � 3.3 �m. Eachectangle measures 1.5 arc sec by 1.2 arc sec and is centered on onef the intersegment edges.

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roadband algorithm.16 As with the PCS, here wese an 11-step sequence. Singular value decompo-ition17,18 is then used to determine the 36-segmentiston corrections that will minimize the edge heightsn a least-squares sense.

. Details of the Algorithm

n this section we first consider those details of thePDS algorithm that are specific to BPDS; then weonsider those that are common to both the broad-and and the narrowband algorithms.

. Algorithm Details Specific to Broadband Phaseiscontinuity Sensing

. Filtersor these simulations we have assumed filter prop-rties that are identical to those of filters that areurrently available in the NIRC6 on the Keck 1 tele-cope. This ensures that the filter properties areealistic and also provides for continuity between theimulations and future experiments. Properties ofhe two filters considered are listed in Table 1. Notehat the designations “broadband” and “narrowband”haracterize the width of the filter not in absoluteerms but rather relative to the quantity �2�2�max,here �max is the largest piston error with which thelgorithm is expected to contend. Thus the same.31–0.063-�m filter is used—under different

ig. 4. Modulation of the difference image in the neighborhood ofn intersegment edge for a filter with � � 2.2 �m and � � 0.59m. This theoretical quantity should be equal to the experimen-

ally determined correlation coefficient defined in Eq. �11�. Thelled diamonds show the typical sampling of the modulation curvey the BPDS algorithm. These data points were determined byeans of the Monte Carlo technique described in the text, without

eference to the equation for the plotted curve. The points do notand exactly on the theoretical curve because the bandwidth wasot small compared with the central wavelength �as assumed inhe derivation of the theoretical expression�. The numericallyenerated curve used in the data analysis does not suffer from thisiscrepancy.

ircumstances—for both the broadband and the nar-owband algorithms.

. Capture Rangehe capture range of the BPDS algorithm is clearly ofhe order of coherence length lc, but we require aore-precise working definition. We also need to

istinguish between the capture range for edges andhe capture range for segment pistons, because theaximum value of the edge height may be as much as

wice the maximum piston error for a given mirroronfiguration. Empirically we have found it usefulo characterize the segment piston capture range aslc�2.

. Step Sequences noted above, we have used here an 11-step mea-urement sequence similar to that used in the PCS.owever, in BPDS—unlike in a PCS—the precise

hoice of the increment between steps is important;or example, we would not want a sequence in whichhe measurements could accidentally all coincideith zero crossings in the modulation curve of Fig. 4.e have found empirically that a useful edge-step

equence is given by n s, where n takes on the 11nteger values from 5 to 5 and

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here m is an integer such that the 11 steps span theull capture range, which is roughly �lc�2.

. Box Sizen principle, the narrowband and the broadband al-orithms should use the same size for the boxes overhich the diffraction pattern for a given edge is de-ned. However, for historical reasons the originalPDS algorithm used a segment-based, rather thann edge-based, approach. The edge-based BPDS al-orithm described here uses boxes that are 1.5 arcec � 1.2 arc sec, centered on the intersegment edges,ith the long axis oriented parallel to the edge. Fig-re 3 shows the boxes for the 84 Keck edges, includ-

ng the template diffraction pattern S��8, 0� for eachox.

. Computation Timehe BPDS algorithm requires nonnegligible compu-ation time to generate the required templates as inig. 3 �30 s on a Pentium IV 2.4-GHz machine� as wells the theoretical curves as in Fig. 4 �10 h�. How-ver, these need to be generated only once for each

Table 1. Properties of NIRC Filters Suitable for BPDS

Filter

CentralWavelength

��m�FWHM

��m� Profile

CoherenceLength��m�

PAH 3.31 0.063 Gaussian 87.0KW 2.25 0.59 Square 4.26


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lter for all time. The calculations required forurve fitting �once per edge per difference image� andor the singular value decomposition �once per itera-ion� total less than 60 s per iteration. BPDS simu-ations, however, can be very time-consuming,equiring 2 h per iteration for a filter with � � 0.6m, because the integral in Eq. �6� is approximatedy a sum in which the number of terms increasesith �. The last two named times are dominatedy the time required for generating the desired dif-raction patterns by means of fast Fourier transformsFFTs�.

. Algorithm Details Common to Broadband andarrowband Phase Discontinuity Sensing

e can obtain useful estimated operational parame-ers for BPDS from the following parameters, whichre currently used for NPDS: The NPDS algorithms used on the NIRC on the Keck 1 telescope. TheIRC detector is a 256 � 256 InSb array with pixels

hat are 0.15 arc sec on a side. The computationalFT array that we used to generate the templates is024 � 1024. At the f�25 focus of the Keck 1 tele-cope the extrafocal distances are �0.8 m ��4 mm ofecondary motion�. At the operating wavelength of.3 �m the atmospheric coherence diameter is com-arable to the 1.8-m segment diameter, so the effectsf seeing on the diffraction patterns are modest.ypical integration times are 30 s on source and 30 sff source. Images are background subtracted, andad pixels are removed and replaced by local aver-ges. Illumination is provided by stars of V magni-ude 3–4, typically of spectral type G or K, within 30°f the zenith. A set of �12 such stars is adequate toover the entire sky.

. Numerical Simulations

o predict the performance of the BPDS algorithm weave run a large number of numerical simulations.hese simulations closely parallel the way in which

he algorithm would be implemented in practice, ex-ept that the images are generated numerically byeans of a FFT.In the simulations, segment aberrations at the

0-nm level �rms surface� were defined by means ofnd- and 4th-order Zernike polynomials. These areomparable to the actual segment aberrations at theeck telescopes. A set of piston errors within the3-�m capture range for the 3.31-�m polycyclic aro-atic hydrocarbon �PAH� filter �Table 1� was then

rawn from a Gaussian distribution. Random seg-ent tip and tilt errors that are typical for the Keck

elescopes were then introduced �0.018 arc sec rms19

n each dimension�. Inside- and outside-of-focus im-ge pairs were generated numerically from Eq. �5�,valuated by means of a FFT, and from Eq. �6� forach of the standard 11 configurations of the primaryirror as defined by Eq. �12� with s � 17.80 �m.he effects of seeing were included in the long expo-ure limit. Small centration and scale errors wereeliberately imposed on each simulated exposure andhen backed out by use of an automatic centration


nd scaling algorithm. Finally the images were nor-alized and subtracted to produce the difference im-

ges. Correlation coefficients were determined forach difference image and each edge by use of a tem-late similar to the one shown in Fig. 3. We deter-ined the best-fit edge heights by curve fitting as

escribed above, and these edge heights were thenonverted to 36 best-fit piston values by means ofingular value decomposition. We corrected theriginal piston errors by subtracting off the lattereasured values, the rms piston error was recom-

uted, and the procedure was then iterated severalimes.

Results of a typical trial are given in Table 2. Therst iteration reduced the rms piston error from 27.5o 0.84 �m. The next iteration led to little improve-ent, indicating that the accuracy of the PAH filter is1 �m.Once the rms piston error has been reduced to �1

m, further improvement can be effected by switch-ng to a second filter with a larger bandpass andence a smaller coherence length. In particular, athis point we switched to the K-wide �KW� filter cen-ered at 2.25 �m with � � 0.59 �m, corresponding toc � 4.29 �m. The step size decreased �in proportiono the coherence length� to 0.86 �m, and the accuracymproved dramatically to 0.07 �m, limited by the sizef the assumed segment aberrations and segmentip–tilt.

Figure 5 shows the piston error as a function ofteration number for 10 simulations with aberratedegments as described above. Again the filter band-idth was changed �from 0.06 to 0.6 �m� for the thirdnd fourth iterations. Also included are the resultsor one simulation with aberration-free segmentswith no tip–tilt errors�. For aberration-free seg-ents the convergence is essentially perfect; other-ise the piston errors converge to the level of the

egment aberrations.

. Extension to Larger Segmented Telescopes

n this section we present some general consider-tions regarding the scaling of the PDS algorithmsNPDS and BPDS scale similarly� to larger tele-copes. Detailed numerical simulations and analy-is will be deferred to another paper; here we simplystablish that such extensions are feasible.A fundamental relation of the discrete Fourier

ransform in the context of aperture and imagelanes states that

AB � N�, (13)

Table 2. Convergence of the �Simulated� BPDS Algorithm

IterationInitial rmsPiston ��m�

Residual rmsPiston ��m�

CaptureRange ��m� Filter

1 27.5 0.84 �43.5 PAH2 0.84 0.80 �43.5 PAH3 0.80 0.07 �2.1 KW4 0.07 0.07 �2.1 KW

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here the computational array used for the FFT �inimulations and in template generation� is N � N.his value is not to be confused with the size in pixelsf the detector array used for the observations. Here

is the size of the computational aperture array ineters, B is the size of the computational image ar-

ay in radians, and � is the wavelength. Note that Aust be somewhat larger than diameter D of the

elescope �we take A � 1.5 D�, and B must be simi-arly larger than the diameter of the defocused image.

e define sampling � of the segmented mirror to be� B��n, where n is the total number of segments

n the telescope. Based on our Keck BPDS simula-ions and our empirical results for NPDS,5 a value of� 3.3 � 105 rad �7 arc sec� is sufficiently large �butay not be necessary� for good convergence. Com-

ining the various relations above gives

N � 9.25 � 105�an��, (14)

here the last relation uses D � 1.85a�n and a ishe side length �circumscribed radius� of the hexago-al segments.For ELTs, correspondingly large computational ar-

ays may be required. For example, for the CELT12

n � 1080 segments, a � 0.5 m, � � 3.3 �m�, Eq. �14�rounded up to the nearest power of 2� gives N �6,384. This number is very large, but two pointshould be made: �1� This N applies only to the com-utational array, not to the detector size, which ineneral will be much smaller. �2� This numberhould be sufficient to ensure good convergence of thelgorithm, but this may not be necessary. Furtherumerical simulations should resolve the latteroint.

ig. 5. Convergence of the BPDS algorithm for 10 typical simu-ations with aberrated segments and one with aberration-free seg-

ents. The filter bandwidth for iterations 1 and 2 was 0.06 �m,or which the limiting rms piston error is �1 �m. The filterandwidth for iterations 3 and 4 was 0.6 �m, for which the limitingms piston error is the size of the segment aberrations, or essen-ially zero in the absence of aberrations.

As for the required size of the detector, 1024 � 1024nfrared arrays are currently available. These put5–20 pixels across a CELT segment, which may al-eady be sufficient. In the worst case, a 2048 � 2048rray would be required. It is true that, for ex-remely large telescopes, the extrafocal images haveo be extremely far out of focus to produce the re-uired resolution, corresponding to image diametersf perhaps as much as 2 or 3 arc min. Although thisight require reimaging optics for the appropriate

mage scale to be produced, it is unlikely that thesearge extrafocal distances would lead to insufficientignals because—for a given value of � and a giventar—the number of photons per square arcsecondepends only on the segment size, not on the totalumber of segments.

. Conclusions

e have described a broadband generalization of ourxisting narrowband phase discontinuity sensing al-orithm, which extends the capture range from 0.4 to0 �m. Numerical simulations show that thisPDS algorithm should converge within a few itera-

ions to the point where the residual piston errors areomparable to whatever higher-order segment aber-ations happen to be present. Although the simula-ions were presented in the context of the Keckelescopes, scaling arguments imply that it should betraightforward to adapt BPDS to extremely largeelescopes that have many times more segments thanhe Keck telescopes. Whereas Shack–Hartmann-ype phasing �PCS�—or another approach—mayrove to be the phasing method of choice for ELTs,he fact that BPDS requires no special-purpose hard-are should at a minimum make it valuable forackup and verification purposes.

This research has been supported in part by theational Science Foundation Science and Technologyenter for Adaptive Optics, managed by the Univer-ity of California at Santa Cruz under cooperativegreement AST-9876783. A. Pinto thanks the De-artment of Universities, Research and the Informa-ion Society for the Fulbright–Generalitat deatalunya �Spain� grant that he received, which en-bled him to take part in this research.

eferences1. G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson,

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efficient method for the reduction of large piston errors in segmented-mirror telescopes

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