Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A253
Tomography approach for multi-objectadaptive optics
Fabrice Vidal,* Eric Gendron, and Gérard Rousset
Laboratoire d’Etudes Spatiales et d’Instrumentation en Astrophysique (LESIA), Observatoire de Paris,CNRS, UPMC, Universite Paris Diderot 5 Place Jules Janssen, 92190 Meudon, France
. INTRODUCTIONne of the key scientific drivers of the future giant tele-
copes of the 30–50 m class is the formation of the earlyniverse, and in particular the way distant galaxies as-embled. These extremely dim and small objects requireoth the large light-collecting power of an Extremelyarge Telescope (ELT) together with the high spatial res-lution brought by adaptive optics (AO). Moreover, thealaxies need to be studied through a statistical ap-roach, which necessitates being able to observe a wealthf galaxies. Dealing with minimum integration times ofhe order of 8 hours for each, the only way to achieve thiss thus to multiplex the observations.
These requirements led in 2004 [1] to the concept ofulti-object adaptive optics, or MOAO, fully inherited
rom the FALCON concept presented in 2001 [2]. MOAOllows the simultaneous observation of objects thatpread in a field as wide as desired—at least as far as theelescope technology allows. It is composed of individualptical trains that split the field. Some compensate theavefront onto small (a few arcsecond) areas of the gal-xies. They are driven by the wavefront information col-ected from other systems: the off-axis WFSs that pickheir signals from either natural or artificial stars acrosshe field. The way the wavefront measurements are com-ined in order to extract the wavefront control is the to-ographic reconstruction.EAGLE [3] (Elt Adaptive optics for GaLaxy Evolution)
s a multi-object integral field spectrograph equipped withOAO and proposed as a future instrument for the Eu-
opean Extremely Large Telescope [4] (E-ELT). It will beble to cover a field of view of 10 arcmin. The top level re-uirements of the instrument call for 20 parallel spectro-copic channels in the near infrared and specify to achieve0% Ensquared Energy (EE) in a resolution element of5mas. Each channel will use one deformable mirrorDM) working in open loop driven by tomographic recon-truction using the wavefront information provided by sixaser guide stars’ (LGSs)’ off-axis WFS and five naturaluide stars (NGSs)’ WFS [5]. EAGLE is a challenging pro-ram and requires a pathfinder called CANARY [6] for anarly on-sky demonstration of the MOAO concept to miti-ate the risk. Hence, CANARY is an open-loop and tomog-aphy experiment that will be installed in 2010 at theilliam Herschel Telescope at La Palma, in the Canary
slands. The development schedule of CANARY is splitnto three stages, starting with three open-loop NGS
FSs and one on-axis truth sensor, then coupling four ad-itional open-loop Rayleigh LGS WFSs, and finally in-luding a woofer–tweeter scheme. The truth sensor islaced in the central (target) direction after the DM andan measure the corrected wavefront.
The work presented in this paper is the approach to besed in CANARY as the tomography reconstruction. Thispproach after an on-sky validation could be extended atlater stage to EAGLE. It was first described by Vidal et
010 Optical Society of America
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A254 J. Opt. Soc. Am. A/Vol. 27, No. 11 /November 2010 Vidal et al.
l. 2008 [7], and we recently obtained laboratory experi-ental results demonstrating the very good performance
f our approach [8].MOAO shares with multi-conjugated AO (MCAO),
round-layer AO (GLAO), and laser tomographic AOLTAO) the problem of the tomographic reconstruction ofhe phase volume above the telescope. Although these ap-roaches differ by the specification of the image quality,he extension of the field of view, and the width of the fieldf optimization, they share the same problem, which is tond the best way to deduce the phase, based on the wave-ront measurements in a number of sky directions.
A particular approach seems to emerge: the minimum-hase-variance approach that can be found in Le Roux etl. 2004 [9] and Gavel 2004 [10]. Even more sophisticated,etit et al. 2006 [11] uses a Kalman-based approach thatses the temporal behavior of the turbulence to minimizecriterion in the spatial and time-optimized domain. In
hese various attempts, a basis for tomographic recon-truction is usually chosen: either Zernike for someBaranec et al. 2006 [12]) or Fourier basis (Poyneer et al.003 [13]). All those attempts are strongly based on thenowledge of the turbulence profile, which is the startingoint of the derivation of the estimators. Not surprisingly,hey all converge toward similar solutions.
Our approach is similar to the approaches of most ofhe above-mentioned authors by using covariance matri-es in the minimum-phase-variance reconstructor. Buthat we discover is that we are able directly from the realeasurements of the wavefront sensors to identify all the
arameters relevant to defining these covariance matri-es. Therefore we propose a procedure in two steps: first,he recording of a set of open-loop wavefront measure-ents for the identification of the turbulent and instru-ental parameters (what we call Learn) and second, the
omputation of the optimal reconstructor defined by thesearameters (what we call Apply). In Section 2, we derivehe proposed tomography approach, linking the WFSs inhe different directions and measuring directly the re-uired covariance matrices needed for the reconstructor.ection 3 presents the theoretical expressions of the tur-ulence spatial covariance of wavefront slopes, allowingne to derive any turbulent covariance matrix betweenwo wavefront sensors. In Section 4, we discuss the con-ergence issue on the measured covariance matrices, weropose the fitting of the data based on the theoreticallope covariance using a reduced number of turbulencearameters, and we present the computation of a fullyodelized reconstructor.
. TOMOGRAPHYne of the main limitations of AO is the anisoplanatismngle [14]. A guide star is required to sense the wavefront.t may be either the object itself (if bright enough) or aearby star. The angular distance between the object andhe guide star needs to be sufficiently short, or the AOerformance will decrease as the guide star gets fartherway. This limiting angle is called anisoplanatism angle.ecause of this problem, the sky coverage is usually lowhatever the size of the telescope. To solve the sky cover-ge problem, tomography is a method that uses the stars
urrounding the target object even beyond the anisoplan-tism angle to sense the wavefront. These stars may beither NGSs or LGSs. Although they are in off-axis direc-ions they give lots of turbulence information. Tomogra-hy’s goal is to predict how the turbulence is in the on-xis direction (i.e., where the target object is).Several tomography algorithms are currently under
est in various laboratories. Ammons et al. [15] use thepherical waves algorithm in the MOAO case for a 10 melescope, and Costille et al. [16] uses the Kalman-basedpproach in the MCAO scheme, but no approach has beenemonstrated in a real on-sky instrument.Early on-sky experiments have been carried out by
agazzoni et al. [17] and Velur et al. [18], which aim toemonstrate the validity of the tomographic analysis. Ansterism of three off-axis stars with a fourth one at theenter was used, and the relations between the off-axisavefronts and the on-axis one were exhibited. The firstulti-WFS on-sky demonstrator is MAD [19], which usedleast-squares approach to demonstrate a MCAO correc-
ion. In 2010, CANARY will demonstrate the feasibility ofhe tomography principle in the MOAO case by recon-tructing the wavefront and compensating it in the on-xis direction with one DM working in open loop.Our concern is to develop a robust tomography algo-
ithm that takes into account not only the specificities ofhe atmosphere along with the geometry of the tomogra-hic problem but also the real physical and optical char-cteristics of the instrument. This work has been alreadyresented in Vidal et al. 2008 [7] and Vidal et al. 2009 [8].
. Classical Approachn AO the optimal approach [20] consists in minimizinghe residual phase variance from a measurement vector� related to the turbulence phase �� . This vector mayome from different wavefront sensors looking at a NGSr a LGS in various directions after a concatenation of theff-axis measurements. In all these papers authors as-ume a linear behavior of the WFS in terms of the phase.e will also assume the same in our paper, whether theFS measurements are linear or have been linearized.
his leads us to consider a linear approach to reconstructhe wavefront.
We define R as the reconstruction matrix from thelope’s measurements to the phase. This matrix will beptimal when there is no more correlation between the er-or and the set of measurements. This assumption takeshe form
���� − Rm� �m� t� = 0, �1�
here � � denotes an ensemble average. The minimumean square error (MMSE) solution takes the form
R = ��� m� t��m� m� t�−1. �2�
This approach is presented by authors to solve theroblem by modeling all the parameters involved in theomographic process. These parameters, such as DM,
FS configurations, and turbulence profile, are pure geo-etric a priori injected on the model. The main difficulty
s to estimate these parameters as close as possible to theeal instrument. For the MOAO case the situation is even
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Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A255
orse since calibration is not obvious because of the vari-us optical trains in parallel.
For any instrument working in a typical MOAOcheme, one should have a mirror model as close as pos-ible to the reality, plus a model of the WFSs, and—erhaps the most difficult one—a model of the optical re-ationship among all the elements: registration of WFSsetween each other and WFSs between DMs.It seems to us that this assumption is too challenging.
n classical AO, the problem of establishing the links be-ween the WFS and the DM has been solved by measur-ng the interaction matrix [21]. This calibration step ofhe interaction matrix allows the rubbing out of any de-iation between the model and the reality. The whole con-rol of the loop is based on this calibration.
We have searched for a kind of equivalent calibrationor the MOAO concept that could take into account thealibrations of the exact registration between elements asell as possible flaws of the optical system. The open-loopperation prevents measuring directly any interactionatrix between the off-axis sensors and the DM. Even in
he case where it could be possible, one should be able toalibrate for pupil misregistration and sensitivity dispari-ies that will have an impact on the tomographic recon-truction.
We present hereafter a new tomography sub-optimalpproach based on the measurements, that does not needny model a priori, and that takes into account the cali-ration specificities in MOAO.
. New Two-Step Approach
. Instrumental Deviationse think that it is necessary to calibrate for the devia-
ions between the instrument model and the reality. Inarticular, the WFS may suffer from a misregistration ofhe pupil, affected by at least six main parameters: x, y, zshifts and conjugation height), rotation �, and magnifica-ion G, plus, possibly, distortion. The sensitivity of the
FSs may be calibrated too. The optical relationship be-ween DMs and WFSs as well as the sensitivity of all thenfluence functions of the DM should be characterized. Ife consider a system like EAGLE where pick-off mirrors
end the light toward DMs located around the focal plane,e can see that the pupil is rotated on the DM by anmount that depends on the position of the pick-off mirrorn the focal plane. The registration between WFSs andMs is affected by a rotation that will depend on the tar-et position: we think it is highly desirable to calibrate fort, rather than to rely on a model, and to monitor it duringbservation. Despite of the efforts put in the alignment ofhe system, no model will be reliable enough to stick tohe real system. We will call instrumental deviations thensemble of these optomechanical flaws that sum up to-ether and that need to be calibrated.
. Transformation of WFS Measurementse initially planned to propose to compensate the instru-ental deviations by applying a linear transformation on
he WFS measurements. The issue is to measure the in-trumental deviations. We found a solution that consistsn feeding all the WFSs with the same wavefront. In prin-
iple, for a given wavefront in the pupil plane, all theFS measurements should be the same. This common
upil-plane wavefront could, for example, be produced byhe telescope DM. In EAGLE, this mirror is M4 mirror ofhe E-ELT.
We define m� as the slope’s measurement vector. Usingne Shack–Hartmann WFS, m� has the following dimen-ion: [2�number of subapertures] (in x and y directions).ctually, we think that there could exist a matrix C that
ransforms, at a given time t, the measurements of a WFSex: m� 1) into the other �m� 2� and can be written as
m� 2�t� = C . m� 1�t�, �3�
here the matrix C has the dimension [2�number ofubapertures of WFS#2, 2�number of subapertures ofFS#1]. It can be retrieved by sensing with both WFSs
he same set of different wavefronts. The correspondinget of measurements are appended to M1 and M2 matri-es, leading to
M2 = C . M1. �4�
Matrices M1 and M2 have, respectively, dimensions [2number of subapertures of WFS#1, number of measure-ents] and [2�number of subapertures of WFS#2, num-
er of measurements]. They are of rectangular shape andannot be inverted directly. We solve the matrix C byinimizing the following quantity:
�2 = �CM1 − M2�2. �5�
eveloping the expression gives
�2 = �i
�j��
kcikm1kj
− m2ij�2�6�
ith the first index of coefficients of M1 and M2 referringo the spatial position of pupil sample and the second oneeferring to time (or draw number). In order to minimize2, we differentiate with respect to the coefficients of C,nd we obtain
C�M1M1t� = M2M1
t. �7�
When the matrix M1M1t can be inverted, we can write
he matrix C as
C = �M2M1t��M1M1
t�−1. �8�
For the same wavefront seen by both WFSs, C acts as ahange of basis matrix between measurements of sensorsand 2. If there is no deviation between the 2 WFSs, C
educes the identity. Getting the measurements m1� andsing Eq. (3) we are able to predict how the measure-ents of sensor 2 �m2� � should be compensated for the de-
iations between the two sensors. Let us note that, in or-er to be able to invert the matrix �M1M1
t�, one shouldrovide a series of linearly independent measurementshat span the full vector space of the sensor. We will comeack to this particular point later on; we assume for theoment that we have a way to span this whole space.Thanks to this calibration procedure, all the measure-ents of the WFSs become normalized in terms of shifts,
ensitivity, rotation…. This calibration procedure may besed for all the WFSs in order to feed any tomographicrocedure with normalized data.
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A256 J. Opt. Soc. Am. A/Vol. 27, No. 11 /November 2010 Vidal et al.
Now, we will see how this method can be pushed a stepurther.
. Linking Off-Axis WFSs and DMOAO implies necessarily an open-loop operation of theM, which means that we cannot measure directly the re-
ation between each off-axis sensor and the DM. We pro-ose then to add another wavefront sensor, on axis, afterhe DM. The setup is equivalent to a classical closed-loopO scheme: it provides a way to calibrate a classical in-
eraction matrix. Of course, this on-axis sensor will notse the flux from the scientific target: it is used only foralibration purposes.
Provided that we can feed the on-axis sensor with a suf-ciently bright source, we will then transform all theeasurements from each off-axis sensors into a measure-ent registered to the on-axis one: for this, we use the
trategy we described in Subsection 2.B.2: we use a per-urbation in a common pupil plane and replace M2 with
central in Eq. (4), and Mi becomes the measurement ofhe ith off-axis sensor:
Mcentral = Ci . Mi, �9�
hich leads to
Ci = �McentralMit��MiMi
t�−1. �10�
The principle of the registration calibration procedures summarized Fig. 1 for three off-axis WFSs. We nowresent the overall calibration and compensation proce-ure having one on-axis WFS behind a DM for calibration
ig. 1. In steps 1 and 3 we show the same wavefront (in the pupre the common basis removed from instrumental deviations. Thmay be any tomographic algorithm that gives the on-axis slope
urposes and three off-axis WFSs for turbulence mea-urements. There are three steps.
The goal of the first step (denoted 1 in Fig. 1) is to beble to tomographically blend, without any deviationroblems in the data, any tomographic model on theransformed measurement of the off-axis WFS becausehey are expressed in a common basis, free from instru-ental deviations. This common basis is related to theM in an unambiguous way thanks to the interaction ma-
rix done on the on-axis sensor (denoted 3 Fig. 1).In this scheme the two calibration procedures can be
erformed before observation by showing to the three off-xis sensors and the on-axis one a common perturbationn the pupil plane. The advantage is to free the tomogra-hic model from a modelization of the mirror, but still oneas to deliver the prediction of the on-axis measurementsdenoted 2 in Fig. 1).
Again, we will push the method a step further and seehat it can lead to a full tomographic solution.
. Toward Another Tomographic Algorithmubsections 2.B.2 and 2.B.3 both introduced a method toalibrate all the deviations of the system. The computa-ions of the matrices Eqs. (9) and (10) were done thanks toset of common wavefront perturbations in the pupil that
ould, as an example, be generated by mirror M4 of the-ELT.Now let us consider measurements made on turbulent
avefronts with a thick atmosphere. We rewrite Eq. (9)
ach sensor as well as an on-axis one. The on-axis measurementsaction matrix also gives the voltage to apply to the mirror. Stepphase).
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Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A257
ut now replace the Mi matrix with a concatenation of allhe off-axis sensors Moffaxis. We are looking for a new ma-rix W:
Mcentral = W . Moffaxis. �11�
This equation. links the off-axis measurements to then-axis sensor, and the solution is
W = �McentralMoffaxist��MoffaxisMoffaxis
t�−1. �12�
When the altitude of the wavefront perturbation is inhe pupil plane, then W can be seen as a change of basisatrix from the off-axis sensors to the on-axis one (matrix). Now when the perturbation is lying out of the pupillane, it is no longer common for each wavefront sensor,nd then the matrix W becomes a tomographic recon-tructor in the on-axis measurement basis. The idea is tose the turbulence seen by all the sensors (off- and on-xis) to compute the tomographic reconstructor W. Ofourse this can be done only when the on-axis sensor isointed toward a bright star (in a calibration procedure).Note that unlike in the classical approach, the recon-
tructor W is able to reconstruct only the on-axis slopesnstead of the phase. The missing step, which is the phaseeconstruction from the estimated on-axis slopes, is en-ured by the calibrated interaction matrix, as describedubsections 2.B.2 and 2.B.3. When the number of drawshat constitute the matrix Moffaxis tends toward infinity,hen the matrix �MoffaxisMoffaxis
t� tends toward the covari-nce matrix of the off-axis sensors slopes. We call this ma-rix COffOff. The same remark applies to the matrixMonaxisMoffaxis
t�, which tends toward the covariance ma-rix between the off-axis sensor slopes and the on-axisensor slopes (called COnOff).
Only the COnOff matrix contains the target direction in-ormation (called central). The tomographic reconstructor
can be computed thanks to the covariance matrices ofll the sensors. Equation (12) can be written as
W = COnoff � COffOff−1. �13�
One recognize here the same expression as Eq. (2): theMSE approach.Notice that this estimator:
• incorporates all the knowledge about the optimalompensation of the system deviations, as presented inubsections 2.B.2 and 2.B.3,• retrieves only the on-axis slopes instead of the on-
xis phase: this is inherent to the way it has beeneasured.
Being able to compute the tomographic reconstructor Wn the on-axis measurement basis, we use Eq. (11) to es-imate the on-axis slopes that our on-axis sensor shouldee. Then we compute the voltage to be applied to the mir-or using an inverse of the measured interaction matrixomputed using a singular value decomposition (SVD).
. THEORETICAL EXPRESSION OF THEOVARIANCES OF WAVEFRONT SLOPEShe covariance matrices can be determined thanks to theavefront slopes measured by the WFS [Eq. (12)]. How-
ver, in the following sections we want to compare theroperties of these matrices with theoretical expectations.or that purpose, we derive the theoretical expression of
he covariance of slopes.
. Correlation of Slopes: Case of a Single WFShe local wavefront slope measured by the subaperture ofShack–Hartmann (or any slope sensor) is given by av-
raging the phase gradient over this subaperture cen-ered at the distance r�:
sx�r�� =1
S ��
�x�u� ���u� − r��du2, �14�
hich is the correlation of �� /�x�u� � with ��u� �, a functionqual to 1 within a subaperture centered around the ori-in distance r� and 0 elsewhere.
Now the covariance between two slopes from subaper-ures separated by a distance r� along x direction is
�sx�0� �sx�r��� = 1
S2 ��
�x�u� ���u� �du � ��
�x�v����v� − r��dv� ,
�15�
�sx�0� �sx�r��� =1
S2 ��
�x�u� �
��
�x�v��� � ��u� ���v� − r��du�dv� .
�16�
oddier 1981 [22] gives the expression leading to thehase structure function D� by
��
�x�u� �
��
�x�v��� = −
1
2
�2D�
�x2 �v� − u� �. �17�
y replacing we have
�sx�0� �sx�r��� = −1
2
1
S2 �2D�
�x2 �v� − u� � � ��u� ���v� − r��du�dv� ,
�18�
�sx�0� �sx�r��� = −1
2
1
S2
�2D�
�x2 �r�� � ��r�� � ��− r��, �19�
here � denotes the product of convolution.Assuming that the subapertures are symmetrical, this
xpression in the Fourier domain [F�f� denotes Fourierransform of the function f]will give
F��sxsx�� � F� �2D�
�x2 � � F��� � F��*�. �20�
ith the Kolmogorov hypothesis and an infinite outercale, the phase power spectral density is the Fourierransform of the structure function D�:
���k� � = �0.023/r05/3��k� �−11/3. �21�
By considering that derivatives translate in the Fourieromain into frequency multiplications, we can write
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A258 J. Opt. Soc. Am. A/Vol. 27, No. 11 /November 2010 Vidal et al.
here kx is the conjugate Fourier variable of x and �̃F���. We can write the final expression of the covari-nce between slopes along x of two subapertures sepa-ated by r�. In the Fourier domain, this leads to
F��sxsx�� � kx2 � r0
−5/3�k� �−11/3 � �̃�k� � � �̃*�k� �. �23�
The relations are the same for crossed correlations be-ween slopes along the x and y directions, just replacingx by ky, the conjugate variable of y. Of course, we canossibly rewrite the expression of the structure functiono take into account a finite outer scale L0 to model a Vonarman spectrum. Figure 2 shows an example of a cova-
iance map between x slopes of a Shack–Hartmann com-uted with Eq. (23).The slope covariance spectrum in x exhibits a larger ex-
ension along the kx axis than along the ky one. Conse-uently, the corresponding covariance map shows a corre-ation of x slopes that is higher along the y axis thanlong the x one, as expected [23].Equation (23) allows us to compute the covariance term
etween two subapertures of one single WFS, and there-ore its full covariance map (Figs. 2–4). In the case of a 3Durbulence, the final covariance map will be computed ashe sum of individual covariance maps, each correspond-ng to a single layer. Those single-layer covariance mapsdd up together because layers are independent.
. Generalization to Several WFSse present in this section how to compute the covarianceatrix between two different WFSs. A single-layer cova-
iance matrix is computed as a collection of slope covari-nces, computed for all possible couples of subaperturesrojected onto the considered altitude layer along the di-ection of observation. Although not difficult, it may re-uire a tricky shuffling of subaperture position data. It is
ig. 2. Left: Fourier domain covariance map between the x slopef x slopes in the real domain sampled for a 14�14 subaperture=50 cm).
0
uilt first by computing, for each of its coefficients, theeparation vector between the pair of considered subaper-ures. We have proceeded as follows.
We consider the coordinates in the pupil of a subaper-ure number i of the WFS number n as �xin ,yin�. The sub-pertures will project onto the turbulent layer number lt an altitude hl in the direction �n ,n�, and their coor-inates will become xin+nhl and yin+nhl. The separa-ion vector between subapertures i and j of two WFSs nnd m projected on an altitude layer will then be
�xin − xjm
yin − yjm� + hl�n − m
n − m�. �24�
Expression (24) is the sum of two terms: the right-handne accounts for the global separation vector between thewo WFSs, while the left-hand one accounts for the par-icular separation of the couple of subapertures in the pu-il.Now, the right-hand term depends on neither i nor j;
.e., it is not related to subapertures but just to the twoartmann shifts. We take it into account by computingirectly a “shifted map” in the Fourier domain using thequation
��sxnsxm�� � kx2�̃n�k� ��̃m
* �k� �
�r0−5/3�k� �−11/3e−2i�hl�kx�n−m�+ky�n−m��. �25�
ote that when the separation between the two WFSs isero ��n ,n�= �0,0�� the maximum correlation is in theenter of the map. The same effects appears when theayer is on the ground �hl=0�.
An example of a covariance map between two Shack–artmanns for two different altitude layers is shown inig. 5. An example of computation of the covariance ma-
rix in y from the covariance map is presented Fig. 6. It iso be noted that many couples of subapertures have theame separation vector (shown by the black arrow in Fig.
hack–Hartmann computed from Eq. (23). Right: covariance map4dssp ,+14dssp , �, where dssp is the pitch of the subapertures (size
s of a S�r= �−1
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Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A259
): consequently, they will have the same covariancealue. This explains the symmetry observed in the cova-iance matrix (white arrows). An example of theoreticalomputation of the covariance matrix among three differ-nt Shack–Hartmann WFSs is shown in Fig. 7.
In the above theoretical derivation, we assume isotropyf the turbulence, which is a rather good approximationhen considering the small angle between the WFS direc-
ions. To manage possible non-stationarity of the turbu-ence, it will require regular measurement of the covari-nce matrices to update the identified parameters andherefore the reconstructor.
. EXAMPLES OF TOMOGRAPHYROCEDURES IN OPEN LOOPe have presented in Subection 2.B.4 how to retrieve the
omographic reconstructor directly from the open-loop on-
ig. 4. Left: Fourier domain covariance map between the x and yeal domain sampled for a 14�14 subaperture �r= �−14d ,+14d
ig. 3. Left: Fourier domain covariance map between the y slopomain sampled for a 14�14 subaperture �r= �−14dssp ,+14dssp �,
ssp ssp
ky measurements of the instrument. In Section 3 we de-cribed how to link theoretically the measured covari-nces of wavefront slopes to the turbulence profile andther geometric parameters [Eq. (25)].
Now, three strategies can be followed, each describedy Subsections 4.A, 4.B, and 4.C.
. Learn and Apply (L&A) Algorithmhe solution of the tomographic problem is given by Eq.
13), but we still have to solve two main points to bemplemented in a real system.
First, the tomographic reconstructor can be computedrom the on-sky measured covariance matrices withoutny model at all but only when an infinite time sequencerom both on-axis and off-axis WFSs is provided. Ofourse, in the real world, we are limited by the statisticalonvergence of the estimated covariance matrices.
s of a Shack–Hartmann. Right: Covariance map of x slopes in thehere d is the pitch of the subapertures (size r =50 cm).
Shack–Hartmann. Right: covariance map of y slopes in the realdssp is the pitch of the subapertures (size r0=50 cm).
slope � w
es of awhere
ssp 0
Fds
Fa
A260 J. Opt. Soc. Am. A/Vol. 27, No. 11 /November 2010 Vidal et al.
ig. 6. Top left: Arrows represent all the subapertures pairs with the same shift in the pupil of +10 in the x direction and +2 in the yirection with a 14�14 Shack–Hartmann. Top right: covariance map for the y slopes. Bottom right: associated covariance matrix of ylopes (148 subapertures).
ig. 5. Example of covariance map of y slopes between two Shack–Hartmanns with an atmospheric layer at twodifferent altitudes (0 mnd 5000 m) for an angle =2’ between two guide stars.
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Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A261
Second, the two covariance matrices COnOff and COffOffre needed to compute the tomographic reconstructor.ff-axis slopes are always provided so that we can easily
ompute the COffOff matrix at any time. However, during aeal observation run, the on-axis slopes, provided by aentral WFS, are usually not available to compute theOnOff matrix. We can compute only half a tomographic
econstructor. To overcome these two problems we have tontroduce a Kolmogorov (or possibly Von Karman) atmo-pheric model a priori.
. Mitigating the Statistical Convergence Problemsing the theoretical expression given by Eq. (25), we cant the on-sky measured covariance matrices with severalarameters:
1. for the turbulence strength: essentially r0 for eachayer and the outer scale L0,
2. for the geometric configuration: altitude h for eachayer and pointing directions for each WFS �i ,i�,
3. and possibly, for registration parameters:i ,yi ,zi ,�i ,Gi (see Subsection 2.B.1). However, we did notry to determine these parameters in the approach de-ailed hereafter. We think that they could be more effi-iently retrieved by a specific calibration procedure basedn the one described in Subsections 2.B.2 and 2.B.3.
Input of the algorithm is a set of turbulence slopeseasured by all the WFSs (including the on-axis one if
vailable). We typically use 1000 to 10,000 slopes re-orded in open loop. We use these data to compute theeasured covariance matrix C and C . In
ig. 7. Left: Covariance matrix CoffOff between three 14�14 Shand 10,000 m of relative strength 0.5,0.35, and 0.15. An 8 m telehe WFS 2 versus WFS 1 covariance map.
OffOffraw OnOffraw
rder to retrieve the above-listed parameters, we mini-ize the distance � between the model covariance matrix
We have used a Levenberg–Marquardt fitting algo-ithm [24] to perform the minimization of �. The numberf layers is not retrieved by the algorithm but is chosen byhe user before the fitting procedure. The input guess ofhe turbulence parameters, h ,r0�h� ,L0, can be either pro-ided by any turbulence profile or defined by the user toypical values. The known asterism configuration is alsosed for the geometrical guess. The convergence stopshen � becomes lower than the value defined by the user.At present this method allows us to retrieve from the
aw data the turbulence profile and geometric configura-ion with only a few seconds of recorded data. We call thistep Learn.
Our proposed approach ends with the computation ofhe tomographic reconstructor W. Using the determinedurbulence and geometric parameters we compute theOffOff and COnOff matrices, and using Eq. (13) we obtain. We call this step Apply.The Learn & Apply (L&A) procedure has the advantage
o measure, on-sky, the tomographic reconstructor with-ut using any external a priori from the turbulence pro-le. The computed tomographic reconstructor properlyts the instrument behavior. Notice that because the typi-al variation scale of the atmospheric parameters (seeing,
, C2) is few minutes or more, the retrieval algorithm
rtmann subapertures. Conditions: three layers placed at 0, 3000,s simulated. Right: equivalent covariance map with a close-up of
ck–Hascope i
0 n
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oes not need to be implemented in the real-time com-uter (RTC) but only as a monitoring tool in the AO su-ervisor computer.
. Without Data from the Central Sensorhe COnOff matrix cannot be measured on-sky. However,e can still minimize the first term of Eq. (26), which al-
ows us to retrieve with good confidence the turbulencend geometric parameters for the off-axis WFS. Neverthe-ess it is not enough to compute the COnOff matrix, be-ause the measurement of the on-axis relative direction ishe last parameter missing. Assuming that the metrologyf the instrument provides us this knowledge, we are ableo compute COnOff and therefore the tomographic recon-tructor W.
Let us underscore that the error on this direction is notritical until it is kept lower than the isoplanatism angle.ote that to calibrate the registration parameters linked
o the central WFS, a dedicated measurement run can beerformed on chosen guide stars before the observationun.
. Numerical Simulationn example of retrieved parameters for a one-layer con-guration and three off-axis NGS WFSs is presented inable 1. Figure 8 shows the raw (left) and fitted (right)OffOff covariance matrix. We used the open-source inter-reted programming language called Yorick [25] with thedaptive Optics simulation tool called YAO [26] (Yorickdaptive Optics) developed by F.Rigaut to perform ourimulations. The simulated telescope is a 8 m. The threeff-axis WFSs and the on-axis one have 14�14 subaper-ures with a typical size of 50 cm. There are 1616 pixels per subaperture. Diffractive WFS are simu-
ated. One layer is placed at 8000 m altitude.The relative error of the retrieved parameters with the
&A method is 6% on altitude, 5% in the WFS positions,nd 10% on r0. L0 is usually difficult to retrieve. In a realxperiment, we will have to test different numbers of lay-rs in order to find the best fit of the measured covarianceatrices. In the case of noisy slopes, we are able to fit the
oise covariance matrix from the COffOff taking advantagef the redundancy of the data. This matrix is also usefulo regularize the tomographic reconstructor.
. Using Covariance Matrices in a Classical Scheme:pply Onlyhe previous section presented the principle of the “L&A”lgorithm. The algorithm uses the instrument data to re-rieve the parameters needed to compute the two covari-nce matrices and finally the tomographic reconstructor.
Table 1. Parameters Retrieved with 10,000 Slopesa
Parameters r0 Altitude L0 WFS Positions
Initial 10 cm 8000 m 50 m (30�, 15�, 25�)Retrieved 11 cm 7500 m 75 m (28.5�, 14�, 23.8�)
Relative error 10% 6% 50% 5%
aThe number of layers is not deduced by the algorithm. No calibration parametersnjected.
Although the advantage of the L&A method is to be freerom an external turbulence profiler (and consequently toe autonomous), we can still use this external a priori toompute the two covariance matrices and finally the to-ographic reconstructor. The external profile given by
ny Cn2�h� profiler can be used as an input to compute the
lopes’ covariance matrix in the tomographic reconstruc-or.
In this case the algorithm “applies only” the given pa-ameters. It is closer to a classical approach: the turbu-ence profiler measurement is deduced thanks to an exter-al system and is used as an input to a model in charge ofhe on-axis prediction. Such algorithm has to modelizehe entire system extremely precisely. We think that itan be very difficult to implement such an approach forOAO since the WFS and the DM work in open loop and
re on separate optical trains.
. Retrieving the Turbulence Profile from Instrumentata: Learn Only
n this strategy, we use the open-loop slopes to retrieve, inarticular, the turbulence profile. This output may besed in any wide-field or tomographic AO system. Doingo, we transform the instrument into a SLODAR [27]. Theain differences are the use of independent WFS devices
o measure slopes and wider angular separations betweentars and the requirement to manage by dedicated cali-ration the instrumental deviations. The sensitivity in al-itude of our algorithm evaluated by numerical simula-ion is around 500 m with 50 cm subapertures and three0� off-axis guide stars, such as for an OA system at anm telescope. We can estimate the sensitivity provided
y a MOAO system on a 42 m telescope such as EAGLE3,5], considering 50 cm subapertures and three 2.5� off-xis guide stars: it will be of the order of 100 m! This isxactly what is required in terms of resolution to mini-ize the tomographic reconstruction error in such a sys-
em [28]. Taking into account that very wide separationsf the guide stars are envisioned for MOAO, its set ofFSs is potentially the best turbulence profiler deliver-
ng the highest resolution in altitude. The L&A methodoes not intend to determine the absolute turbulence pro-le but only what is required by the instrument in termsf the tomographic problem.
. CONCLUSIONe propose a new approach for tomography in MOAO
alled Learn & Apply (L&A). This method allows us to fithe tomographic reconstructor on the turbulence condi-ions directly from the instrument measurements andlso takes into account the registration between the off-xis WFS and the on-axis DM.We show that this tomographic reconstructor is the
roduct of two covariance matrices. We propose a calibra-ion method for these two covariance matrices on sky us-ng bright stars for the off-axis and on-axis WFSs. We al-ernatively propose to use knowledge of the targetirection in order to be free from the on-axis measure-ents during the real observation and monitor all the pa-
ameters using only the off-axis WFS measurements.
spfittstei
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Vidal et al. Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A A263
We demonstrate by numerical simulation that with aequence of 10,000 open-loop slopes we are able to com-ute the real shape of the tomographic reconstructor bytting tomographic parameters to the raw covariance ma-rices. Then with these parameters, we build the MOAOomographic reconstructor. We find that in the WFS mea-urements we are able to fit the full turbulence profile tohe required resolution, typically 500 m for 50 cm subap-rtures and three 30� off-axis guide stars. The instruments similar to a high-resolution SLODAR.
The next step is to test the monitoring of the param-ters using only the off-axis data in the laboratory. Welso plan to test temporal aspects in open-loop and tomog-aphy operations using modal gain optimization. The&A approach will be tested soon on sky with the CA-ARY instrument, and we want to translate this algo-
ithm for application to the MOAO instrument for the-ELT (EAGLE).
CKNOWLEDGMENTShis work was supported by the French research agencygence Nationale de la Recherche (ANR)—programAUI, Région Ile de France, and was a part of theAGLE phase A study and the CANARY program.AGLE and CANARY are two projects involving Labora-
oire d’Astrophysique de Marseille (LAM), Laboratoire’Etudes Spatiales et d’Instrumentation en AstronomieLESIA), Galaxies Etoiles Physique et InstrumentationGEPI), and Office National d’Etudes et Recherches Aero-patiales (ONERA) in France, and Astronomy Technologyenter (UK-ATC) and Durham University in UK.
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-axis S
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