Large telescopes have always been an importantdriver for astrophysical discoveries, leading to thedesign and construction of several 8 to 10m tele-scopes. Practical aspects related to manufacturingand handling limit the size of monolithic mirrorsto ∼8m, forcing the adoption of segmented primarymirrors to reach larger diameters, e.g., for the Kecktelescopes and the Gran Telescopio Canarias. Both ofthem are constituted of 36 hexagonal segments of∼0:9m each, resulting in a total diameter of 10m.The next generation of extremely large telescopes(ELTs), which is currently being designed, will na-turally adopt segmented primary mirrors to reachdiameters from 30 to 42m. The two main telescopesof the ELT era will be the thirty meter telescope [1],
30m in diameter with 492 segments, and theEuropean ELT (E-ELT) [2], 42m with 984 segments.
ELT primary mirrors are subject to various effectsthat can modify the relative position of the segmentssuch as the variation of the gravity vector with re-spect to the telescope, thermal variations, wind,and vibrations. This is why the position of each seg-ment must be actively measured and controlled inpiston, tip, and tilt to reach optical performancesof ∼10nmrms on the wavefront. The strategy cur-rently foreseen is a combination of edge sensors tomonitor the shape of the primary mirror in real-timeand of an optical phasing sensor (OPS) to providezero reference with nanometer precision for thesesensors at regular intervals. Indeed, the accuracyof edge sensors is strongly affected by externalconditions such as mechanical alignment, tempera-ture, and humidity. An absolute reference is thusneeded to recalibrate these sensors and keep low co-phasing errors. Current OPS designs are based onShack–Hartmann wavefront sensors similar in their
principle to the one used in the Keck telescopes [3,4],but several other concepts of sensors have beenstudied.
Requirements faced by these OPSs when cophas-ing a segmented mirror are of two kinds: (1) thecoarse initial phasing of the mirror and (2) the veryfine phasing at high precision. The latter is essentialto reach the highest optical performances when con-straints on the wavefront are exceptionally high, inparticular, for applications such as direct detection ofexoplanets with extreme adaptive optics and corona-graphy on an ELT [5–7], for which a precision of a fewto 30nmrms must be obtained to reach contrasts of10−8 to 10−9. The former is equally important for theinitial phasing of the primary mirror when segmentswill be installed for the first time on the telescope. Itis also foreseen to be faced on a daily basis in the E-ELT for which 2 of the 984 segments will be replacedevery day for recoating. Mechanical alignment ofthe segments will provide a cophasing precision of∼20 μm on the wavefront, leaving the remaining cor-rection down to ∼10nmrms for a dedicated OPS.
It is in this context that the Active Phasing Experi-ment (APE) [8] was designed to test four conceptsof OPS in the laboratory and on the Very LargeTelescope (VLT). The four sensors are a Shack–Hartmann sensor [9], a pyramid sensor [10], a curva-ture sensor [11], and a Zernike phase contrast sensor[12]. The APE bench includes a small (153mm in dia-meter) segmented mirror constituted of 61 segmentscontrollable independently in piston, tip, and tilt[13], dimensioned to reproduce future ELTs in termsof segment size and gaps between segments whenseen from the OPS. The VLT pupil is reimaged onthe active segmented mirror (ASM) in order to createa fake 8m segmented pupil which can be analyzedand corrected by the four sensors. Since the APEis installed on a Nasmyth platform and does notinclude a pupil derotator, the VLT pupil, and in par-ticular the spiders, rotates with respect to the seg-mentation pattern created by the ASM. This canlead to phasing errors when the spiders are in a con-figuration that isolates different areas of the ASM bycovering almost aligned segment borders. This pro-blem is specific to APE and will not take place ina real ELT where the spider will remain fixed withrespect to segmentation. The absolute position ofeach segment in the APE is referenced by an internalmetrology (IM) system [14] that is used to control theASM with a precision better than 5nmrms severaltimes per second. By means of the IM, the ASMcan be set in any configuration of piston, tip, and tiltwithin the limits of �14:4 μm in piston on the wave-front, which is the measurement range of the IM.
One of the sensors in the APE is the Zernike phasecontrast sensor, also called the ZErnike Unit forSegment phasing (ZEUS), was developed by the La-boratoire d’Astrophysique de Marseille in collabora-tion with the European Southern Observatory andthe Instituto de Astrofísica de Canarias. The ZEUSfinds its origins in the Mach–Zehnder (MZ) phasing
sensor concept [15,16], replacing the delicate interfe-rometer setup by a simple phasemask [12]. Similar tothe Zernike phase contrast approach [17], it has beenfound equivalent to the MZ in terms of performancecharacteristics. The phase mask, located in the tele-scope focus, takes the form of a cylindrical depressionmachined into a glass substrate. Its diameter is closeto that of the seeing disk, and its depth corresponds toa phase shift between π=4 and π=2 of the lighttransmitted through the mask compared to thattransmitted through the surrounding substrate.Following the mask, a lens projects an image of thetelescope pupil onto a detector array. When the ob-served star is centered on themask, pupil aberrationsappear as intensity variations on the detector as inthe classical phase contrast method, but since themask is larger than the diffraction spot, low-frequency aberrations, in particular those due to at-mospheric turbulence, are filtered out. Piston errorsbetween segments, containing important high-frequency components, show up as antisymmetric in-tensity variations along segment edges. Enlargingthe mask diameter eliminates more of the atmo-spheric aberrations but reduces at the same timethewidth of the antisymmetric signal,making itmoredifficult to measure. A similar trade-off is also foundformask depth: a deepmask, giving a π=2 phase shift,provides a stronger signal than a shallowermask, butthe signal also contains a larger symmetric compo-nent, which is found to reduce the accuracy of the sig-nal fitting algorithm, particularly for large pistonerrors. To allow for different observing conditions,stellarmagnitudes, observingwavelength, and initialphasing errors, five different phase masks are avail-able in the ZEUS,with a depth of 100 or 175nm, and adiameter of 1.0, 1.5, and 2:0 in: [18]. All the results re-ported here were obtained using the 175nm thickmasks.
Theoretical treatment of the ZEUS and fine co-phasing performance within the single-wavelengthcapture range with the ZEUS has been studied inSurdej et al. [19] using observations performed on theAPE bench at the VLT. Here, we present the resultsobtained during the same observing runs using aclosed-loop multiwavelength phasing scheme, whichallows reaching a capture range of several micro-meters. We first describe our phasing scheme inSection 2. Then in Section 3, we present the two ob-serving runs of February and April 2009 with the dif-ferent configurations that have been tested. InSections 4 and 5, we present the results obtainedon-sky when phasing at a single wavelength in open-and closed-loop with large piston errors, respectively,and when using our multiwavelength scheme to es-timate and correct these errors. In Section 6, we ex-trapolate our results to elaborate a phasing strategyfor an actual ELT, and finally in Section 7, we brieflycompare our results with a Shack–Hartmann-typesensor.
In this Section, we present a multiwavelength phas-ing scheme which allows reaching a capture range ofseveral micrometers in piston. Before generalizing toa complete segmented mirror, we consider the idealsituation of two segments having one border in com-mon that we wish to cophase. The two segments haveabsolute piston values of p1 and p2, respectively, gen-erating a phase difference Δϕ ¼ 2π
λ ðp2 − p1Þ ¼ 2πλ Δp
at their common border; Δp is called the edge piston.Unless specified otherwise, from now on, all valueswill be given in units of λ or in nanometers on thewavefront.
A. Capture Range in an Open Loop
The ZEUS normalized signal is constituted of twoparts: an antisymmetric part, which is equivalent tothe signal obtained with an MZ interferometer [16],and a symmetric part, which is specific to the ZEUSsignal. The amplitude of the antisymmetric part isproportional to the sine of the phase difference be-tween the two segments
Sasym ¼ A sinðΔϕÞ; ð1Þ
whereA is a calibration coefficient, which depends onthe phase mask physical properties (thickness anddiameter) and the observing conditions (seeing).The phasing algorithm developed for the ZEUS al-lows to retrieve the value of Sasym through signal fit-ting [19], an thus to determine the value of Δp with
Δp ¼ λ2π Δϕ ¼ λ
2π arcsin�Sasym
A
�: ð2Þ
Since Sasym is a periodic function of Δϕ, the rangeof measurable edge pistons is therefore limited to�λ=4 on the wavefront when operating in an openloop. In the following Sections, we are going to seehow this very narrow capture range can be extendedby the use of a closed loop and several wavelengths.
As mentioned by Surdej et al., the symmetric partof the ZEUS signal Ssym, which is proportional tocosðΔϕÞ, could be used to extend the capture rangeto �λ=2. However, on-sky results have shown thatthe value retrieved for Ssym through signal fittingis not reliable, and thus cannot be used to removeambiguity on the edge piston estimation, limitingthe effective capture range to �λ=4. We note that theamplitude of the symmetric part strongly depends onmask thickness, so the use of a thinner phase maskgives a cleaner, more easily exploitable signal.
B. Closed Loop Phasing
Another possibility to extend the capture range to�λ=2 is to perform closed-loop operation of theOPS. The main idea behind this concept is that foran edge piston Δp ∈ ½λ=4…λ=2�, the correction cal-culated assuming Δp ∈ ½0…λ=4� will necessarily beoriented toward a decrease of the edge piston. By per-forming successive measurements and corrections,
the edge piston will eventually be brought into theopen-loop capture range, and then to zero. This pro-cedure is illustrated in Fig. 1, where an edge pistonΔp0 ¼ 0:41λ is first estimated to 0:09λ, leading to anew edge piston Δp1 ¼ ð0:41 − 0:09Þλ ¼ 0:32λ whichis still in ½λ=4…λ=2�. The following estimation of0:018λ leads to a third edge piston Δp2 ¼ 0:14λ,which is now within ½0…λ=4�, leading to a final cor-rection that removes the remaining edge piston.
This closed-loop process can be applied for edgesteps within ½−λ=2…λ=2� but also around any multi-ple of λ, i.e., in ½λðn − 1=2Þ…λðnþ 1=2Þ�, with n as aninteger. In that case, the closed-loop process willbring the edge piston Δp toward nλ.
C. Dual-Wavelength Phasing
If the edge piston Δp is larger than λ=2, the single-wavelength capture range is not sufficient for phasingthe two segments with a zero phase error but only tothe closest integer multiple of λ, leaving an ambiguityof nλ. The only way to increase the capture range is tousemultiplewavelengths tomeasure and remove thisambiguity. Two-wavelength interferometry, and inparticular how to choose the right wavelengths, hasalready been extensively described in literature [20]with several improvements [21,22]. However, thesemethods require accurate measurements at two dis-tinctwavelengths.Aswill be shown inSubsection4.A,the ZEUSmeasurement accuracy in an open-loop on-sky is rather poor, requiring the use of a differentmethod. The multiwavelength scheme that wasadopted for the ZEUS [18] uses closed-loop phasingat two separate wavelengths λ0 and λ1 ([23]), in orderto determine and remove the ambiguity on the edgepiston.
Phasing is first performed at λ0, until convergenceis reached. The edge piston is thenΔp0 ¼ nλ0, with nas an integer. The wavelength is switched to λ1, sothat a new edge piston Δp1 ¼ nλ1 appears:
Closed-loop phasing
-3λ/2 -λ/2 -λ/4 0 λ/4 λ/2 3λ/2Edge piston (λ)
-1.0
-0.5
0.0
0.5
1.0
Nor
mal
ized
sig
nal
0
1
2
3
3 2 1Corrections
After correction 3After correction 2After correction 1Initial position
Fig. 1. (Color online) Illustration of closed-loop phasing for an in-itial edge piston Δp ¼ 0:41λ, i.e., outside of the open-loop capturerange. Successive estimations and corrections will always beoriented toward a decrease of the edge piston.
where Δλ ¼ λ0 − λ1. While Δp0 and Δp1 are un-known, we assume that the difference between them,ΔΠ, is known. The ambiguity n can then be easilydetermined from Eq. (3):
n ¼ ΔΠΔλ : ð4Þ
In order for the ambiguity n to remain identical forthe two wavelengths, Eq. (4) is valid only as long as
jΔΠj <�λ2; ð5Þ
with �λ ¼ ðλ0 þ λ1Þ=2. With the approximation thatΔp≃ n�λ, Eqs. (4) and (5) lead to the following condi-tion of validity:
2jΔpj≲�λ2Δλ : ð6Þ
For simplicity, we note Λ ¼ �λ2=Δλ the syntheticwavelength, which is a function of only the twowavelengths λ0 and λ1. The new multiwavelengthcapture range in a closed loop is then equal to �Λ=2.
The multiwavelength phasing is illustrated inFig. 2. In practice, ΔΠ is determined by performingphasing of the segments at λ1 while recording thesuccessive corrections that are applied on individualsegments until convergence is reached at λ1. Ambigu-ity is then determined with Eq. (4) for every segmentand finally removed.
Contrary to other methods ([21,22]) which requireto carefully choose the wavelengths, the choice here
is mainly driven by the difference Δλ, over which thecapture range depends. Two close wavelengths willprovide a wide capture range but will be less precisefor small piston errors because the piston differencebetween the two phased positions will be small, andthus more sensitive to errors. On the contrary, morewidely separated wavelengths will provide a shortercapture range, but a greater precision for smallpiston errors.
D. Generalization and Practical Implementation
The closed-loop and multiwavelength schemes fromSubsections 2.B and 2.C have been illustrated in thecase of two segments having one border in common.However, these schemes can be generalized to a com-plete segmented mirror with hexagonal geometry. Asa matter of fact, the edge piston values at the bordersof each segment are related to the segment piston ofeach segment by a system of linear equations. Thesegment piston defined with respect to a referencesegment which has a piston p ¼ 0 by definition.The determination of the edge piston at all segmentborders then allows deducing the piston of each seg-ment by the resolution of a set of linear equationsusing singular value decomposition (SVD).
Another important aspect, which has been over-looked in the previous Sections, is the measurementnoise. In the presence of noise, the OPS will never seea completely phased mirror, i.e., the peak-to-valley(PtV) and rms values of the measured piston errorswill never be zero. It is then necessary to define a con-vergence criterion, which tells if the mirror is phasedfrom the point of view of the OPS. This criterion hasbeen defined as thresholds TPtV and Trms on the PtVand RMS values of the measured piston errors: forthe mirror to be considered as phased, the PtV andrms must both be lower than their respective thresh-old. Their exact value is not critical, they mustsimply be tight enough to phase the mirror with suf-ficient accuracy, but loose enough to prevent oscillat-ing around the phased configuration because ofmeasurement noise. The optimal values depend onseveral parameters such as observing conditions,wavelength, and phase mask properties. In practice,we used previous results obtained in similar condi-tions to choose appropriate values, which were ingeneral around ∼20nm for Trms and ∼50nm forTPtV. Further study and observations would be re-quired to define an automatic adjustment procedurefor these values.
The calibrations for the multiwavelength schemeleading to the normalized image are identical to thatof the single-wavelength scheme [19]: a dark frameand a reference image taken without the phasemask. For the latter, it is sufficient to off centerthe mask by a few arcseconds to eliminate its influ-ence on the signal at the segment edges. It is neces-sary to acquire such an image in each filter used forthe multiwavelength scheme.
Multi-wavelength phasing
1200 1300 1400 1500 1600 1700 1800 1900
Edge piston (nm)
-1.0
-0.5
0.0
0.5
1.0
Nor
mal
ized
sig
nal
nλ0nλ1
∆Π
0
1
2
3
4
5
6
λ1 = 650 nmλ0 = 750 nmλ1 = 650 nmλ0 = 750 nm
Fig. 2. Multiwavelength phasing scheme illustrated with λ0 ¼750nm and λ1 ¼ 650nm. Phasing is first performed at λ0 for theedge piston to converge to nλ0, then the wavelength is switchedto λ1 and phasing is performed again until convergence. TheΔΠ piston difference between the two phased positions is directlyrelated to the ambiguity n and the two wavelengths (see text fordetails).
The ZEUS was tested on-sky during a total of sixnights from December 2008 to April 2009. A smallpart of two half-nights (∼6h in total) was dedicatedto study multiwavelength phasing of large piston er-rors in February and April 2009, respectively . Differ-ent random configurations were applied on the ASMranging from 600 to 8000nm PtV on the wavefront,outside of the single-wavelength capture range of theZEUS. All these observations were performed onbright stars with magnitudes comprised betweenV ¼ 4:0 and V ¼ 5:0 to avoid being in a noise-limitedphasing regime. Investigations of performance in thephoton starving regime have been presented bySurdej et al.
In February 2009, the 600, 1200, and 2000nm PtVrandom configurations were tested with narrow-band filters at λ0 ¼ 750nm (Δλ ¼ 40nm) and λ1 ¼650nm (Δλ ¼ 40nm), producing a multiwavelengthcapture range of �4:9 μm. Observing conditions wereaverage, with seeing varying between 0.7 and 1:2 in:.The diameter of the phase mask used for the obser-vations was 1:5 in:, and its thickness was 175nm.
In April 2009, the 2000, 6000, 7000, and 8000nmPtV random configurations were tested at λ0 ¼800nm (Δλ ¼ 40nm) and λ1 ¼ 750nm, producing acapture range of �12:0 μm. Observing conditionswere good, with a seeing disk around 0:7 in:, leadingto the use of a smallermask of 1:0 in:with a thicknessof 175nm.
4. Single Wavelength Results
In this Section, we present the results obtained bothin the open and closed loop at a single wavelength.We focus on the analysis of the results at the levelof segment borders, where the normalized signal ismeasured and converted into an edge piston.
A. Open-Loop Performance
The ZEUS shows poor open-loop performance mainlydue to large fitting errors in the determination ofSasym. Although the ZEUS capture range is intrinsi-cally limited to �λ=4 when fitting only the antisym-metric part of the signal, it is interesting to evaluatethe precision with which the signal is fitted in theopen loop since it will impact on the convergencespeed in the closed loop. Figure 3 shows a typical ex-ample for a 2000nm PtV configuration of the ASM atλ0 ¼ 750nm. Iteration 0 represents the open-loopmeasurement on this configuration. The data pointsclearly follow a sinusoidal calibration curve, as ex-pected from Eq. (1), but for edge pistons outside of�λ=6, the error can be as high as 50%. Within thisrange, the antisymmetric part Sasym dominates,providing good accuracy on the signal fitting, whileoutside this range, the symmetric part Ssym be-comes more important, and the signal fitting is lessaccurate.
The loss of open-loop accuracy for edge pistons out-side of the quasilinear part of the sine calibrationcurve is illustrated on Fig. 4, which shows the stan-dard deviation of the error on the determination ofSasym in bins of λ0=8 for the same ASM configurationand iterations as Fig. 3. At iteration 0, all bins arepopulated with 15 to 20 points (ensuring reliable sta-tistics), and we see that the fitting error clearly in-creases for edge pistons outside of �λ=8, thusproviding accurate measurements only close to zero.This poor open-loop performance, even within thesingle-wavelength capture range, requires the useof closed-loop phasing to bring all edge pistons closeto zero.
B. Closed-Loop Performance
Although open-loop performance is poor, all edge pis-tons finally converge toward integer multiples of λ0in the closed loop. Surdej et al. have shown thatconvergence is reached with good accuracy for edge
-2000 -1000 0 1000 2000Edge piston (nm)
-2
-1
0
1
2
Sas
ym (
norm
aliz
ed u
nit)
-2λ0 -λ0 0 λ0 2λ0
Iteration 10Iteration 5Iteration 0
-400 -200 0 200 400Edge piston modulo λ0/2 (nm)
-2
-1
0
1
2-λ0/2 -λ0/4 0 λ0/4 λ0/2
(a) (b)
Fig. 3. (Color online) ZEUS antisymmetric part of the signal, Sasym, as a function of the edge piston on the wavefront given by the IM, (a)and the same measurements folded within �λ0=2, (b) for the 2000nm PtV random configuration on the ASM at λ0 ¼ 750nm. Measure-ments are given for three iterations during closed-loop phasing. Iteration 0 corresponds to the initial unphased configuration. The ASM ispartially phased at iteration 5, and completely phased at iteration 10 according to the ZEUS. Borders covered by the VLT pupil and spidershave been removed (see text for details).
pistons within a single-wavelength capture range,but we now demonstrate that the same result is ob-tained for edge pistons around larger integer multi-ples of λ0. We see in Fig. 3 that all edge pistons arewithin �λ0=4 around a multiple of λ0 after five itera-tions, and within �λ0=10 after 10 iterations, whenthe ASM is considered to be phased for the ZEUS.
In Fig. 4, we see that although the error is largeand far from zero, all edge pistons finally converge inthe closed loop after 10 iterations because the smallerror in the linear part of the calibration curve allowsto retain edge pistons in that area once they are closeenough to the zero phase error. The same behavior isobserved for all other configurations tested on-sky: atthe last iteration in single wavelength, most edge pis-tons are within �λ0=10 around a multiple of λ0, andthe error on Sasym is below 0.2. In the next Section,we will see how this result translates into the deter-mination of the piston error of the segments.
5. Multiwavelength Results
As we have seen from the previous Section, edge pis-tons converge in a closed loop toward an integer mul-tiple of the wavelength. In this Section, we are goingto see how it leads to the phasing of the ASM at in-dependent wavelengths and how it allows the deter-mination of the piston ambiguity for each segment.From now on, we will analyze the results at the levelof the segments, i.e., after solving the linear systemof equations with SVD that converts the edge pistonsinto estimated segment piston errors with respect toan arbitrary zero position. These estimates are fedback to the ASM, leading, eventually, to convergenceto a phased state, at least in a single-wavelengthsense. The absolute piston of the segments is mea-sured by the IM, allowing us to follow the phasingprocess iteration by iteration and to see the exactposition of each individual segment.
A. Convergence at Two Wavelengths
The convergence of all edge pistons toward integermultiples of λ can equivalently be seen as the conver-gence of the piston error of each segment toward aninteger multiple of λ. This process is illustrated inFig. 5, which shows the piston error of each segmentsof the ASM for the 24 iterations that have beenperformed for the configuration previously shown.This allows us to follow the evolution of individualsegments or groups of segments. For all ASM config-urations that were tested on-sky, the followingprogression is observed.
1. Convergence at λ0 from a totally random con-figuration is somewhat chaotic; segments with simi-lar piston errors at the first iteration can follow verydifferent paths (e.g., segments 7 and 48 in Fig. 5).
2. After a certain number of iterations (the exactnumber depends on observing conditions and thresh-olds TPtV and Trms) convergence is reached at λ0, andthe wavelength is switched to λ1.
3. The loop is closed at λ1 until convergence isreached again. This process takes much less itera-tions since the mirror is already in an ordered state.Usually all segments converging at λ0 will similarlyconverge at λ1.
Convergence was reached for most segments in allASM configurations tested on-sky, at the exception ofa few cases that will be detailed in Subsection 5.D.Table 1 summarizes all the important phasing infor-mation in the closed loop at λ0 and λ1. In particular,we see that 10 to 30 iterations are necessary to reachphasing at λ0, while only 4 to 11 iterations are neces-sary at λ1. In most cases, all segments converge at
-400 -200 0 200 400
Edge piston modulo λ/2 (nm WF)
0.0
0.2
0.4
0.6
0.8
1.0
Sas
ym e
rror
(no
rmal
ized
uni
t)
-λ0/2 -λ0/4 0 λ0/4 λ0/2
Iteration 0Iteration 5Iteration 10
Fig. 4. (Color online) Standard deviation of the error on the de-termination of Sasym in bins of λ0=8 for the same ASM configurationand iterations as Fig. 3. Only bins with more than 5 availablepoints have been represented, but most bins have at least 10points, thus providing reliable statistics.
2000 nm PtV - Internal Metrology
0 5 10 15 20 25Iteration
-1000
-500
0
500
1000
Pis
ton
(nm
)
-λ0
-λ1
0
λ1
λ0
7 48 750 nm650 nm
Fig. 5. (Color online) Convergence of all segments using our mul-tiwavelength scheme for the random 2000nm PtV configuration.At the end of iteration 14, the nλ ambiguity for each segment isdetermined and removed, in theory bringing all segments withinthe single-wavelength capture range. Segments 7 and 48 havebeen highlighted because they start with similar piston errorsbut follow very different paths to reach phasing at λ0.
both wavelengths within λ=6 or better toward an in-teger of λ, translating into 10 to 30nmrms phasingprecision on the whole mirror. The final phasingperformance is strongly related to the observingconditions, and in particular to the stability of theseeing, but these numbers are comparable to the re-sults obtained in the closed loop around zero [19]. Alarger sample of measurements would be necessaryto draw final conclusions on the influence of seeingvariations.
B. Ambiguity Determination
We now demonstrate that the accurate closed-loopconvergence at both λ0 and λ1 allows us to determineand remove thenλambiguity on thepiston error of thesegments to bring themwithin the single-wavelengthcapture range. The ambiguity is determined usingEq. (4) and then removed for each segment. The deter-mination is considered successful for a segment if itspiston error is within the single-wavelength capturerange around zero after the ambiguity has been re-moved. The results obtained on-sky are presentedin Fig. 6, which shows the proportion of segmentswithin a single-wavelength capture range aroundzero, �λ0=2, and larger multiples of �λ0 (includingsegments outside of the multiwavelength capturerange).
The essential result is that for six of the sevenASM configurations tested on-sky, 100% of the seg-ments are in ½−3λ=2…3λ=2� after ambiguity removal,and at least 80% of them are in ½−λ=2…λ=2�, which isthe single-wavelength capture range. For the8000nm PtV configurations, two segments are com-pletely lost because ambiguity determination wasfaulty, and two segments are within capture rangearoundmultiples of λ0 larger than 2. Using the multi-wavelength scheme again can of course allow tocapture these last two segments.
The proportion of segments inside the single-wavelength capture range after ambiguity has re-moved decreases for configurations with large initialpiston errors, at the benefit of segments that end upclose to�λ0. This is expected since the determination
of the ambiguity is quite prone to small errors: thevalue determined from Eq. (4) is rounded to the clo-sest integer, so small convergence errors at either λ0or λ1 can potentially propagate to produce an error of�1 on n. However, only large convergence errors orsegments that did not converge at all could produceerrors larger than 1.
Finally, we expect from these results that the num-ber of lost segments will increase for configurationswith larger PtV. In particular, configurations withlarge piston errors are more sensitive to convergenceproblems that can result in the loss of segments. Con-figurations having segments close to�Λ=2 or close tothe coherence length of the filters will certainly endup with a large proportion of segments either with avery large piston error or within capture range oflarge multiples of λ0. The way to overcome thislimitation is to use filters with central wavelengthscloser together that will offer a much larger capturerange. However, this leads to larger uncertainty inthe determination of the ambiguity n (Eq. (4)), hencethe piston error determination. A second couple of fil-ters with wavelengths more widely separated wouldthen be necessary to recover full phasing precisiononce all segments are within a few λ around the zeroposition.
C. Systematic Effects
In these results, a systematic effect has been re-moved from the data to provide accurate values. Onthe ASM, the central segment cannot be actively con-trolled and is maintained in a fixed position to serveas the zero reference for measuring the piston errorof all other segments. While this reference is usableby the OPS in a laboratory where there is no tele-scope pupil covering the ASM, it becomes useless
within �λi=2.cNumber of segments outside of nλi � λi=6 for which phasing is
considered to have failed.
Segments position after ambiguity determination
600 1200 2000Feb.
2000Apr.
6000 7000 8000
PtV (nm)
0
20
40
60
80
100
Pro
port
ion
of s
egm
ents
(%
)
Within capture range around:0±λ0±nλ0 (n ≥ 2)
Fig. 6. Proportion of segments brought within the single-wavelength capture range around 0 (light gray),�λ0 (intermediategray) or larger multiples of �λ0 ðn ≥ 2Þ (dark gray) after determi-nation and subtraction of the nλ ambiguity using our multiwave-length scheme for the different ASM random configurations testedon-sky.
on the VLT where the central reference segmentremains hidden behind the secondary mirror.
However, the IM always sees this reference seg-ment and can thus provide a piston error measure-ment with respect to the central segment. This hasno impact on edge pistons measurements at allborders, which gives a relative positioning informa-tion between segments. But when converting theedge piston information into piston error measure-ments through SVD, the lack of absolute referenceintroduces a variable offset in all segment pistons de-termined by the OPS. This changing reference can bereferred to as a “floating” reference.
This effect is just an artifact introduced by the IM,and it has no impact on the final single-wavelengthphasing performance or on the nλ ambiguity determi-nation: the ZEUS phases all segments together, sim-ply ignoring the central one, which is the referencefor the IM. This is why the floating reference hasbeen estimated and removed from the data pre-sented in Fig. 5 and Table 1. The estimation is per-formed at the last iteration of each wavelength usingthe segments phasing close to zero (zero being de-fined by the IM). The systematic offset of this groupof segments has been measured and removed fromall previous iterations at the same wavelength.
D. Convergence Problems
Although inmost cases all segments converge towardan integer multiple of λ0 or λ1, there are two notableexceptions visible in Table 1: the 600nm PtV config-uration at both wavelengths, and the 8000nm PtVconfiguration at λ0, for which a few segments failedto converge. These segments have been highlightedin Fig. 7 on the science images obtained for the lastiteration at λ0. This failure mainly affects segmentswhich are partly covered by the telescope pupil, andwhere the number of usable borders is generally lim-ited to two or three instead of six, leaving a possibleambiguity on the determination of the piston error ofthe segment. The spiders can also cover some of theborders and distort the edge piston measurement.Moreover, adjacent segments are influenced by themisbehavior of their neighbor, resulting in groups
of segments that fail to converge within the �λ=6interval.
For the 600nm PtV configuration [Fig. 7(a)], seg-ments 48 and 58 remained stuck close to −λ0=4, whilesegment 47 was influenced by segment 48 and con-verged only within �λ0=5 around an integer multipleof λ0. For the 8000nm PtV configuration [Fig. 7(b)], agroup of four adjacent segments failed to converge atλ0: two of them are covered by the pupil and haveonly three borders available for analysis, and the spi-der is very close to one of these borders. These seg-ments are highlighted in Fig. 8, and we can see thatsegments 34, 56, and 58 remained in between two in-teger multiples of λ0, while segment 33 convergedclose to −4λ0. Segment 54 is different, as it startedto diverge at the very first iteration. However, it isinteresting to see that all these segments finally con-verged after the wavelength was switched to λ1. Wecan assume that starting from a partially phasedASM and using another wavelength, the uncertain-ties remaining for these segments were cleared,allowing a final convergence.
The main problem with these misbehaving seg-ments comes when the nλ ambiguity is determinedand removed. Since they did not converge properly atone or both of the wavelengths, their value of n canpotentially be wrong, sending them outside of themultiwavelength capture range. This is exactly whathappened for segments 54 and 58 in the 8000nm PtVconfiguration: the sign of ΔΠ, and thus of n, was er-roneous, and they were sent outside of �Λ=2. For the600nm PtV configuration, since the piston errors arealready close to zero, the error on n cannot be largerthan 1, which explains why in the end all segmentsare still within �3λ=2.
E. Summary and Limitations
In the previous Sections, we have presented variousresults for the multiwavelength scheme of the ZEUS.For clarity, different information have been summar-
Fig. 7. Segments that failed phasing for the (a) 600 and(b) 8000nm PtV random ASM configurations. Images correspondto the last iteration at λ0 when the ASM is considered to be phasedfrom the point of view of the ZEUS.
8000 nm PtV - Internal Metrology
0 5 10 15 20 25 30Iteration
-6000
-4000
-2000
0
2000
4000
Pis
ton
(nm
)
-7λ0
-6λ0
-5λ0
-4λ0
-3λ0
-2λ0
-λ0
0
λ0
2λ0
3λ0
4λ0
5λ0
33
34
54
56
58
750 nm800 nm750 nm800 nm
Fig. 8. (Color online) Convergence of all segments using our mul-tiwavelength scheme for the random 8000nm PtV configuration.
ized in Table 2. They are divided in three categories:the starting configuration, the state after the ambi-guity has been determined and removed, and thestate when the phasing procedure was ended. Thephasing procedure was generally stopped beforereaching a new phased state, so the final PtV andrms values are not necessarily reflecting the ulti-mate result, but it gives a general view of the numberof phased, recoverable, and lost segments.
With the use of the multiwavelength scheme, thepiston error of most segments is largely reduced, ascan be seen for example in Fig. 8. The initial coarsephasing would then be followed by another applica-tion of the scheme for a finer phasing with a secondpair of wavelengths offering a less extended capturerange. Some of the segments have been either lost(only two in the configuration with largest piston er-rors) or did not converge properly. The latter are,however, recoverable since their piston error still lieswithin the multiwavelength capture range.
There are two main limitations inherent to theAPE system limiting the performance of the ZEUS.The first is that there is no pupil derotator, whichmeans that the spiders are rotating with respect tothe segmentation pattern, creating a difference be-tween the images being analyzed and the calibrationimages taken at the beginning of the phasing proce-dure. The second is the convergence problems ofsome segments described in Subsection 5.D. Itmostly concern segments covered by the telescopepupil edge (or their close neighbors) for which the lin-ear system between the edge pistons and the seg-ment piston error is less overdetermined. To betterrepresent a real segmented telescope, the ASMshould have been smaller than the VLT pupil.
6. Phasing Strategy for an ELT
These results can now be analyzed in terms of phas-ing strategy for a future ELT. We have seen from pre-vious Sections that the phasing of large piston errorswith our multiwavelength scheme is reliable up to7000nm PtV, and that it could certainly be mademore reliable for even larger piston values using fil-
ters closer together in wavelength. With a couple offilters at 750 and 725nm, the capture range would be�Λ=2 ¼ �10:9 μm. This capture range would be suf-ficient to phase an ELT primary mirror, which hasbeen mechanically phased with a precision of∼20 μm, or to recapture a segment that has beenchanged and reinstalled with equivalent precision.
For an actual ELT, a three-step phasing strategywould appear optimal with a ZEUS-like sensor. First,coarse phasing using a very large capture range inorder to remove the largest piston errors should beimplemented using filters with wavelengths closetogether (e.g., 750 and 725nm). Since a correspond-ingly small bandwidth, typically 10nm, would benecessary to offer sufficiently long coherence length(∼50 μm), a bright star (V ≲ 4) would be requiredfor this step. A second step, using more widely spacedwavelengths (e.g., 750 and 650nm) would bring allsegments within the single-wavelength capturerange. While correspondingly larger filter band-widths can be used, a bright star would still be re-quired to limit exposure time. Final phasing is nowdone using the single-wavelength regime. Here, awide-band filter can be used, allowing the use ofmuch fainter stars (V ≲ 10).
Considering the results presented here and bySurdej et al., phasing of an ELT would require a totalof 80 to 100 closed-loop iterations depending onobserving conditions in the case of a completely un-phased primary mirror. Considering 10 to 20 s ex-posures, this would require between 15 and 35 minof exposure time. Including target acquisition(∼10 min), calibrations (∼5 min), and computingtime (∼20 s=iteration), this leads to a final estimateof less than 1h 30 min for initial phasing of an ELT.For an already partially phasedmirror, only the thirdstep would probably be necessary, consequentlyreducing the amount of phasing time. Phasing of asingle segment with respect to fixed neighbors wasnot studied, but we believe convergence would bemuch faster in this case.
Finally, it is important to underline the potentialproblem of the spiders. In APEs, the absence of a
Table 2. Summary of the Multiwavelength Results
Start Ambiguity Removed End of Phasing Procedure
PtV rms PtV rms Timeb PtV rms Timeb
(nm) (nm) (nm) (nm) Na (min) (nm) (nm) Na (min) Phasedc Recoverabled Loste
600 163 834 172 25 15 113 25 40 24 61 01200 339 821 119 29 18 735 94 33 20 60 1 02000 533 643 153 15 10 763 100 25 15 60 1 02000 533 1093 265 18 8 1867 272 27 11 58 3 06000 1668 1388 318 20 13 895 278 30 16 53 8 07000 1698 1243 265 36 14 1544 196 45 18 57 4 08000 2429 18802 1938 22 10 19676 1980 28 12 22 37 2aNumber of iterations since the beginning of the phasing procedure.bTotal time since the beginning of the phasing procedure.cSegments in the monowavelength capture range (�λ=2).dSegments within of the multiwavelength capture range (�Λ=2), i.e., still recoverable.eSegments outside of the multiwavelength capture range, i.e., unrecoverable.
pupil derotator makes the spider move with respectto the segmentation pattern, sometimes coveringseveral segment borders. The worst configuration oc-curs when the spiders are almost aligned with bor-ders, thus isolating different parts of the mirrorfrom the point of view of the OPS. The upper rightspider of Fig. 7(a) would be in such a configurationif it was rotated ∼5° clockwise. Inversely, the bestconfiguration occurs when the spiders are almostorthogonal to segment borders similarly to the lowerleft spider of Fig. 7(b). Although it will still preventgood edge piston measurement for these borders, theparts of the mirror lying on each side of the spiderwill not be isolated from one another because the seg-ments covered have borders on each side of the spi-der. In order to avoid potential phasing problems, thedesign of an ELT needs to take into account the sizeand position of the spiders to avoid as much aspossible configurations where different parts of themirror are isolated.
7. Comparison with a Shack–Hartmann Sensor
In this Section, we briefly compare the ZEUS with aShack–Hartmann-type sensors. We base this discus-sion on the results published by Chanan et al. [3,4] onthe phasing at the Keck telescopes with their “broad-band algorithm”.
Contrary to a Shack–Hartmann-type sensor, theZEUS does not require any particular alignment be-tween the sensor and the telescope pupil. It is, there-fore, mostly insensitive to alignment variations oroptical distortion. Although the success of the Keckphasing sensor proves that such effects are perfectlymanageable on a 10m class telescope, they can beexpected to have a larger impact on the phasing per-formance for larger telescopes. Also, if continuousphasing is implemented, requiring the use of a guidestar whose position in the field varies, pupil distor-tion may not be constant during the observations.The only constraint for the ZEUS is the alignmentof the phase mask in the focal plane of the telescope,which needs to be better than 0:1 in: on-sky (between5% and 10% of the phase mask diameter). Such analignment could be easily reached with the use ofa tip–tilt corrector.
It has been shown that the ZEUS phasing accuracyis better than 10nmrms with stars brighter thanV≃5 [19], which is similar to the accuracy obtainedat Keck with the narrow-band phasing algorithm [4].When it comes to phasing using a faint star, theZEUS has been shown to reach a precision of 15nmrms for stars of magnitude V ≃ 13. To the best of ourknowledge, measurements in similar conditions havenot been reported for any other sensor, but this is acriterion for comparison that could potentially be dis-criminating. Although the ability to phase with faintstars is not necessary for reducing large piston er-rors, it is certainly important if a continuous phasingstrategy is foreseen. For the phasing of large pistonerrors, it seems that the ZEUS is very similar to theKeck Shack–Hartmann sensor in terms of perfor-
mance and time required for phasing. Chananet al. report reducing 30 μm errors in approximately2h. As explained in Section 6, such errors could bereduced with the ZEUS in less than 1h 30 min (in-cluding calibration) using our multiwavelengthscheme.
8. Conclusions
Cophasing of segmented primary mirrors is requiredto reach high wavefront quality of future ELTs.Given the large number of segments, is it necessaryto be able to phase segments with piston errors ofseveral micrometers and to reach a phasing precisionof a few nanometers. Moreover, this phasing must beexecuted simultaneously for all segments. In thiswork, we have demonstrated on-sky the use of theZernike phase contrast sensor for phasing of largepiston errors using a multiwavelength scheme. Per-forming closed-loop phasing at two close wave-lengths, it is possible to determine and remove thenλ ambiguity on the piston of each segments.
Although the open-loop performance is poor due tolarge errors in the signal fitting procedure outside ofλ=4 around each integer multiples of λ, the good pre-cision of the fitting close to zero, where the antisym-metric part of the signal dominates, allows us toreach convergence in the closed loop after 10 to 30iterations at the first wavelength, and in 3 to 11 atthe second wavelength. In most cases, all segmentsphase within �λ=6 around an integer multiple of λ,so that the nλ ambiguity can be estimated accurately.We have demonstrated that for all the ASM config-urations tested on-sky, at least 90% of the segmentsare within �3λ=2 after ambiguity estimation, and forall configurations except the one with the largestPtV, 80% of the segments are inside �λ=2, i.e., insidethe single-wavelength capture range around zero.Problems directly related to the APE experiment,such as the lack of a derotator, are identified aspossible reasons for segments failing to convergeproperly.
We have also proposed a phasing strategy for anELT with a ZEUS-like sensor. Using different cou-ples of filters, very large piston errors could even-tually be reduced in less than 1h 30 min. We havealso underlined important problems related to thespider position with respect to the segmentation pat-tern, and we advocate a configuration where the spi-ders are perpendicular to segment borders. Such aconfiguration would avoid isolating different partsof the mirror, resulting in independently phasedareas. Finally, we conclude that the ZEUS offers si-milar performances to a Shack–Hartmann-type sen-sor, but it certainly is much more robust in the sensethat it does not require any particular alignmentwith the pupil of the telescope. While this is not pro-blematic in the case of a 10m telescope, this wouldcertainly be a clear advantage in the 30 to 42mpupil.
The APE experiment was part of the ELT DesignStudy and has been supported by the EuropeanCommission, within Framework Programme 6,
contract 011863. Mechanical design andmanufactur-ing of the ZEUS sensor were provided by the Institu-to de Astrofísica de Canarias (IAC). The EuropeanSouthern Observatory (ESO) provided instrumentcontrol, most of the signal analysis software, and wasused for sensor integration and exploitation withinthe APE. Laboratoire d’Astrophysique de Marseille(LAM) maintained overall responsibility for the de-velopment of the ZEUS sensor and provided concep-tual design, optical design, phase masks, andalignment and validation in the laboratory, as wellas part of the signal analysis software.
A. Vigan acknowledges support from a Science andTechnology Facilities Council grant (ST/H002707/1).We are grateful to Patrick Lanzoni, Marc Ferrari,and Maud Langlois at LAM for their various contri-butions to the ZEUS instrument. We also wish tothank Marcos Reyes and the opto-machanical teamat IAC, as well as Isabelle Surdej, Natalia Yaitskova,Frédéric Gonte, and Frédéric Derie at ESO. Moregenerally we thank all the people from the APEexperiment who have contributed to these results.
References1. J. Nelson and G. H. Sanders, “The status of the Thirty Meter
Telescope project,” Proc. SPIE 7012, 70121A (2008).2. R. Gilmozzi and J. Spyromilio, “The 42m European ELT:
status,” Proc. SPIE 7012, 701219 (2008).3. G. Chanan,M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast,
and D. Kirkman, “Phasing the mirror segments of the Kecktelescopes: the broadband phasing algorithm,” Appl. Opt.37, 140–155 (1998).
4. G. Chanan, C. Ohara, and M. Troy, “Phasing the mirrorsegments of the Keck telescopes II: the narrow-band phasingalgorithm,” Appl. Opt. 39, 4706–4714 (2000).
5. C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan,“Fundamental limitations on earth-like planet detection withextremely large telescopes,” Astron. Astrophys. 447, 397–403(2006).
6. P. Martinez, A. Boccaletti, M. Kasper, C. Cavarroc, N.Yaitskova, T. Fusco, and C. Vérinaud, “Comparison of corona-graphs for high-contrast imaging in the context of extremelylarge telescopes,” Astron. Astrophys. 492, 289–300 (2008).
7. N. Yaitskova, “Adaptive optics correction of segment aberra-tion,” J. Opt. Soc. Am. A 26, 59–71 (2008).
8. F. Gonte, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P.Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L.Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E.Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment:
preliminary results and prospects,” Proc. SPIE 7012,70120Z (2008).
9. R. Mazzoleni, F. Gonté, I. Surdej, C. Araujo, R. Brast, F. Derie,P. Duhoux, C. Dupuy, C. Frank, R. Karban, L. Noethe, and N.Yaitskova, “Design and performances of the Shack-Hartmannsensor within the Active Phasing Experiment,” Proc. SPIE7012, 70123A (2008).
10. E. Pinna, S. Esposito, A. Puglisi, F. Pieralli, R. M. Myers, L.Busoni, A. Tozzi, and P. Stefanini, “Phase ambiguity solutionwith the Pyramid Phasing Sensor,” Proc. SPIE 6267,62672Y (2006).
11. L. Montoya-Martínez, M. Reyes, A. Schumacher, and E.Hernández, “DIPSI: the diffraction image phase sensinginstrument for APE,” Proc. SPIE 6267, 626732 (2006).
12. K. Dohlen, “Phase masks in astronomy: from the Mach-Zehnder interferometer to coronagraphs,” EAS PublicationsSeries 12, 33–44 (2004).
13. C. Dupuy, F. Gonté, and C. Frank, “ASM: a scaled down activesegmented mirror for the Active Phasing Experiment,” Proc.SPIE 7012, 70123B (2008).
14. R. Wilhelm, B. Luong, A. Courteville, S. Estival, and F. Gonté,“Optical phasing of a segmented mirror with sub-nanometerprecision: experimental results of the APE internal metrol-ogy,” Proc. SPIE 7012, 701212 (2008).
15. L. Montoya, “Applications de l’interféromètre de Mach-Zehnder au cophasage des grands télescopes segmentés,”Ph.D. dissertation (Université de Provence, 2004).
16. N. Yaitskova, K. Dohlen, P. Dierickx, and L. Montoya,“Mach-Zehnder interferometer for piston and tip-tilt sensingin segmented telescopes: theory and analytical treatment,” J.Opt. Soc. Am. A 22, 1093–1105 (2005).
17. F. Zernike, “Diffraction theory of the knife-edge test andits improved form, the phase-contrast method,” Mon. Not.R. Astron. Soc. 94, 377–384 (1934).
18. K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L.Montoya, E. Hernandez, M. Reyes, I. Surdej, andN. Yaitskova,“ZEUS: a cophasing sensor based on the Zernike phase con-trast method,” Proc. SPIE 6267, 626734 (2006).
19. I. Surdej, N. Yaitskova, and F. Gonte, “On-sky performance ofthe Zernike phase contrast sensor for the phasing of segmen-ted telescopes,” Appl. Opt. 49, 4052–4062 (2010).
20. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12,2071–2074 (1973).
21. M. G. Lofdahl and H. Eriksson, “Algorithm for resolving 2piambiguities in interferometric measurements by use ofmultiple wavelengths,” Opt. Eng. 40, 984–990 (2001).
22. K. Houairi and F. Cassaing, “Two-wavelength inter-ferometry: extended range and accurate optical path differ-ence analytical estimator,” J. Opt. Soc. Am. A 26, 2503–2511(2009).
23. S. Esposito, INAF-Osservatorio Astrofisico di Arcetri(personal communication, 2006).
