se of concept of multiconjugate adaptive optics1–3
MCAO� to overcome the limit of the small correctedeld of view �FoV� that is achievable with classicdaptive optics4 has been proposed. In adaptive op-ics �AO� a single reference star is used to measurehe phase delay introduced by the atmospheric col-mn located between the telescope and the star.hase measurements are then used to drive a de-
ormable mirror �DM� to correct the incoming waveront. In MCAO the goal is to reconstruct the three-imensional structure of the turbulence over a FoV ofeveral minutes of arc instead of in the single direc-ion defined by the AO reference. To achieve thisesult a constellation of reference stars is sensed byeveral wave-front sensors �WFSs�, and a few DMsre conjugated to turbulent atmospheric layers at
C. Arcidiacono �[email protected]� is with the Departmentf Astronomy and Space Science, University of Florence, Largonrico Fermi 5, I-50125 Florence, Italy. E. Diolaiti and M. Tordire with the Department of Astronomy, University of Padova,icolo dell’Osservatorio 3, I-35132 Padua, Italy. R. Ragazzoni, J.arinato, and E. Vernet are with the Istituto Nazionale di Astro-sica, Astrophysical Observatory of Arcetri, Largo Enrico Fermi, 5,-50125 Florence, Italy. E. Marchetti is with the European South-rn Observatory, Karl-Schwarzschild-Strasse, 2, D-85748 Garch-ng bei Munchen, Germany.
ifferent altitudes. Each DM corrects the turbu-ence of the conjugated part of the atmosphere bypplying the information retrieved from the WFSeasurements. A subject of some debate is the way
o combine the signals coming from the referencetars or, in other words, the way to perform the wave-ront-sensing operation. Of several possible solu-ions, two approaches are going to be implemented ineveral MCAO instruments in the near future: thetar-oriented �SO� approach and the layer-oriented5,6
LO� approach. In the SO approach every detector isooking at a single reference, in this way allowing ahree-dimensional reconstruction of the distortedave front to be made; the correction is then applied
o a certain number of DMs. In the LO approach theetectors are conjugated to some suitable altitudes,nd they are looking at several stars; the wave frontshus retrieved are then applied to DMs that are op-ically conjugated to the same altitudes as the rela-ive detectors.
It is clear that, apart from possible tricks,7 in theO approach the number of detectors depends on
he number of DMs used in the system rather than onhe number of references, as it does in the SO ap-roach. Another difference is that in the LO ap-roach the light coming from the different referencess normally optically coadded such as to take advan-age of even faint references and to increase theignal-to-noise ratio �SNR� of the detectors. Thismplies the use of a pupil plane WFS, and the pyra-
id sensor8 appears to be a good choice because of itsigh sensitivity.9–11 The LO WFS splits the light of
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ach reference star into four beams by means ofyramids located on an intermediate focal plane.ach of these beams produces a pupil image on aetector �normally a CCD; Fig. 1� that senses theifferent amounts of illumination of the four pupils.he intensity measurements allow one to computehe derivative of the wave front and finally to re-onstruct the phase delay introduced by the per-urbed layers. Of course the aberrations introducedy the layers near the detector’s conjugation altitudere measured with more accuracy than those intro-uced by layers far away, which are smoothed outore as the distance from the plane increases.12
ach detector–DM loop is hardware independentrom the others �even if it is optically related�, andhis property allows one to tune the spatial–temporalarameters, such as the integration time on the WFSnd the subaperture size on the detector �for in-tance, by binning the pixels on the CCD in differentays�, to the statistical characteristics of the conju-ated layers.In all the SO approaches to MCAO the faint natu-
al guide stars �NGSs; magnitude 17 and higher�resent a low SNR, and they are not valid referencesor wave-front sensing. But in LO systems the op-ical coaddition of light allows one to take advantagelso of starlight with a sufficiently high SNR. Inrinciple, the same results can be obtained by use ofwave-front-sensing CCD camera with readout noise
RON� close to zero and the numerical coaddition ofhe various reference signals, even if a larger numberf CCDs is needed in the optical approach becauseach reference is coupled with a phase sensor. How-ver, the numerical approach is strongly dependentn the future availability of CCDs with these char-cteristics.In a LO system the shapes of the footprints on
etectors conjugated to different altitudes depend onhe stars’ positions with respect to the center of theoV and on the conjugation altitude. A metapupil
ig. 1. Left, LO ground layer. The detector collects the beamsoming from three NGSs split into four pupils by the pyramids.or the ground layer all the NGS split footprints overlap perfectly.ight, LO approach for two conjugation planes. There is oneFS, one DM, and one WFC for each loop. Subscript 1, upper
onjugated plane; 2, ground conjugated plane.
t a conjugate altitude is defined by the projection ofhe FoV onto this conjugated plane; in other words, its the projection onto the conjugate plane of the DM.n a ground-conjugated WFS the pupils of differenteferences overlap perfectly on the metapupil be-ause the projection of the FoV, in this case, corre-ponds to the entrance aperture of the telescope.his means that every reference starlight senses thetmosphere on the ground layer independently ofhether its position with respect to the FoV is taken
nto account for correction. In this way it is possibleo consider for this loop a FoV that is larger than thecientific FoV, thus increasing the number of possibleeference stars and of course the sky coverage. Thisind of concept, in which different fields of view areonsidered, is called multiple field of view �MFoV�ayer oriented.13 For example, for a ground-onjugated detector an annular sky region about theorrected FoV is normally considered for the groundFS, and in this way the light of the references in-
ide the central FoV can be used by a detector conju-ated to a higher altitude to drive the correspondingM. In this way, splitting of the light among detec-
ors can be avoided �see Fig. 2 and 3�. To performrror budget analysis in LOST are implemented theossibility to introduce different sources of phase er-or as the misregistration of the couples DM-WFS asn conjugation altitudes as in position with respect ofhe metapupils. To compute the control matrix, theser can choose to use the simulated measurementsf mirror modes with an arbitrary level of SNR, in-tead of the theoretical mirror modes. Otherources, such as those related to chromatism effects,re not taken into account.The LO and MFoV techniques have one consider-
ig. 2. MFoV layout: DM1 and DM2 are the deformable mirrorsonjugated to the ground and to the high layer, respectively. DM1
s driven by the stars lying in the annular 6-arc-min FoV only;M2, by the NGS in the central 2-arc-min FoV.
ble advantage in terms of real-time computingower compared with SO systems. In fact, as waslready mentioned, the loops are hardware indepen-ent and the amount of information retrieved by theFSs is essentially much less, i.e., only what is
eeded to drive the corresponding DMs. This re-ults in a need for less CPU time and memory also forimulations related to these systems compared withimulations of systems based on the SO approach.his fact is relevant especially for extremely large
elescopes �ELTs�, �which, however, are not studiedere�, and in fact we emphasize that the completeimulations related to ELTs published so far areomehow related to the LO approach14 or to use of aultigrid solver for the SO approach that is similar to
he LO scheme in terms of atmospheric layer decou-ling.15
The first version of the code, then called a layer-riented simulation tool �LOST�, was developed tontroduce the LO approach. That first version waspgraded16 to analyze the performance of the LOave-front sensor arm of the Multiconjugate Adap-
ive optics Demonstrator17 �MAD� and, finally, of aayer-oriented multiple field of view interferometricystem18 that comprises a Large Binocular TelescopeLBT� Interferometric Camera �LINC� and the Near-R�Visible Adaptive Interferometer for AstronomyNIRVANA� for the LBT.
In this paper, in Section 2 we describe in detail theOST package and then in Section 3 present theain tests done to validate it. Several simulations
esults are presented in Section 4 for two specificases.
. LOST Simulation Package
he LOST code was developed in the Interactive Dataanguage version 5.2, and runs on Unix–Linux orindows platforms. A typical simulation test takes
ew hours and several hundred megabytes of memoryn a PC or a workstation with a 1–2-GHz processor,
ig. 3. Two FoVs, both relative to the ground. The superpositionf the footprints on the nonconjugated plane decreases with dis-ance from the conjugated plane: This shift produces a blur ofrequency lower on the nonconjugated planes. Thus they are seenmoothed on the WFS. This effect depends on the separationngle between the references, and it is larger for large FoVs thanor small FoVs.
ccording to the required precision and the numbernd dimensions of the phase screens used to simulatehe atmosphere.
One defines the system by writing a script file. Inhis way it is possible to set the main parameters ofhe telescope and of the AO system �such as diameter,he central obstruction, wavelengths, number ofMs, and the wave-front sensing and reconstructionethods�. Two WFS noise models are implemented
n the LOST code: the Shack–Hartmann and theyramid wave-front sensors, and in both cases it isossible to set the characteristics of the detector thats used and the information about the geometry of the
FS. The parameters related to the DMs andFSs can be different for each conjugation loop, ex-
loiting the LO approach’s ability to tune these quan-ities according to the statistical features of theonjugated plane. A block diagram of the LOSTode is shown in Fig. 4. It has three main steps:he input definition, the main loop, and the outputomputation. The main loop is composed of a suc-ession of actions executed sequentially and of twoondition points in which the code compares the ac-ual step to the integration timer or to the overallimulation timer. With the LOST code it is possibleo analyze widely different AO systems: fromingle-reference AO by setting only one DM conju-ated to the ground �or to another layer� and to onen-axis NGS, to the more-complex MCAO systemhat defines different DMs at different conjugationltitudes. The code is based on the geometrical pro-ection of the NGSs footprints onto several layers andnto the DMs. The turbulent layers are representeds phase screens, and they are shifted during theimulation according to the wind-speed profile, as-uming a Taylor hypothesis about the turbulence evo-ution. The NGS’s asterism can be set by the user orandomly generated, assuming the distribution de-ned by the star galaxy luminosity function of Bah-all and Soneira19 and according to the galacticatitude and longitude considered. The system’serformance is expressed in terms of evolution of thetrehl ratio �SR� and of the point-spread function
PSF�, both computed for each time evolution step.he PSFs are integrated during the simulation toenerate a long-exposure SR map. The LOST codevaluates these quantities for one or several sky di-ections inside and outside the FoV as well. An op-ical coaddition of the NGS wave front for each loop ismplemented; however, we consider the possibility ofnumerical LO sensor only for the ground loop �using
ommon CCDs or L3 CCDs20�. The physical mean-ng of each simulation section in the block diagramescription is discussed in what follows.
. Generation of Atmospheric Layers
s in most AO simulation packages, in the LOST codeach atmospheric phase screen is computed by aourier-transform technique in which the subhar-onic frequencies21 are added to match the Kolmog-
rov or the von Karman power spectrum �the latter isharacterized by outer scale L �. In Fourier space a
0
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andom value 0–2� of the phase is associated withach frequency �and, iteratively, with each screen�.inally, we normalize each phase screen to have, onverage, the correct amount of phase variance in theupil aperture according to the theoretical predic-ion.22,23 In the Kolmogorov case our proceduresenerate the phase screens by using the power spec-rum function, PS�r0, k�, defined by the Wiener rela-ion
PS�r0, k� � 0.023r0�5�3k�11�3, (1)
here r0 is the Fried parameter and k is the spatialrequency. The von Karman power spectrum is de-ned by
PS�r0, k� � 0.023r0�5�3�k2 � k0
2��11�6, (2)
here k0 is the frequency that corresponds to theuter scale L0. We normalize the arrays by dividinghe phase screens by the square root of the averageariance. We compute this value by averaging thehase’s standard deviation relative to a square-gridelescope pupil over the phase screen. Then eachcreen is multiplied by a factor that is the theoreti-ally predicted22,23 phase variance over the pupil, ��
2.or the Kolmogorov spectrum, ��
2 is expressed by
� 2 � 1.03�D�r �5�3, (3)
ig. 4. Block diagram of the LOST code. The input parametersrovided by filling a script text with the parameter values. This scodes, or any other information about the telescope or the adaptivain MCAO loop all the constant quantities, such as the control mere the main loop starts with the definition i � 1: First the wa
he PSFs are computed for each direction in which the user wantycles of the loop. The NGSs’ wave fronts are combined in the Leasurements are computed and integrated at each loop. The in
oop. The phase noise that is due to the WFS is then added to thpplied to the DMs at the same frequency as that used for the Weal-time computer and to other possible sources taken into accounFSs. At each iteration the phase screens are shifted according tohen the last iteration, i � n, is reached the integrated PSFs, th
nd the residual wave-front error, are saved in the output files.
� 0
hereas, in the von Karman spectrum, ��2 is defined
y
��2 � 0.09�L0�r0�
5�3, (4)
here r0 is related to the Kolmogorov value by aore-complex relation.23 We compute the phase
creens by normalizing for the theoretical phase vari-nce and using a fixed spectrum; then the averagemplitude of each phase screen is determined.The physical dimensions of the layers generated
epend on the pixel size used. This size is defined ashe ratio of the dimensions of the pupil diameterxpressed by the user in pixel units and in meters.he lower limit for the pixel size should be the small-st r0 taken into account for the various phasecreens.
. Loop
he basic clock is defined by phase screen evolutionnterval �t. This user-defined parameter sets theower temporal step of the simulation. Every otheremporal parameter, such as the integration time ofhe wave-front sensors and the delay applied to theM correction, must be an integer multiple of thisumber. One simulates the atmospheric evolutiony shifting each phase screen by a displacement
�s � v �t, (5)
define the characteristics of the atmosphere and the system arendicates to the code whether the atmosphere, the stars, the mirrortem has to be computed or loaded from existing files. Before the, are computed and the variables such as the PSF, are initialized.onts of the NGSs are retrieved as seen by the telescope and thenetrieve the SR. These PSFs are saved and integrated along thede for each conjugation altitude. In this way the LO noise-freetion covers fk cycles, where the subscript k refers to the relevantmeasurements, and the correction is computed. Corrections are
with the delay dk that is due to the operations performed by thehe integration time values fk are usually different for the differentind speed and direction, and finally iteration counter i is updated.
evolution, and other required data, such as the mode coefficients
thatript ie sysatrixve frs to rO motegrae LOFS,t. Tthe w
here vwind, i is the speed vector of the ith layer.his shift is performed in an accurate way through
he combination of a rigid shift of the screen matrixnd a linear interpolation. To preserve the statisti-al layer characteristics we apply the shift at eachtep by starting from the original layer array ratherhen by reiterating the shift of the previously shiftedcreen. Otherwise, this procedure would introducesmearing if it were applied several times.The control algorithm is a pure integrator, but
ther filters can be implemented. For every tempo-al step the code computes the measured wave frontf each NGS by coadding the portions of the phasecreens illuminated by the guide stars and subtract-ng the previous computed DM correction:
WFi�x� � j�1
nlayer
Lj�x � hji� � k�1
nDM
DMk�x � hki�,
(6)
here Lj is the jth layer, DMk is the kth DM, x is aosition vector on the pupil, i are the off-axis coor-inates of the ith star, hk is the conjugation altitudef the kth DM, and hj is the altitude of the jth layer.n each layer we define an x, y coordinate systemith the origin in the center of the FoV to identify the
ff-axis angle and the position of each star. Now were able to focus on the desired layer altitude accord-ng to the LO scheme. We superpose the NGS waveronts, considering the positions of the footprints athe conjugation altitudes. Each star’s wave front iseighted according to the related NGS intensities as
hown in Eq. �7� below. The final result is an arrayhat contains the phase measurements of the layerst the conjugation altitude:
M �i�1
nstar
WFi� xi, yi�Ii
j�1
nstar
Ij
, (7)
here M is the array that represents the weightedum of the NGS wave fronts as seen focused on theonjugation altitude, WFi is the ith NGS wave front,nd Ij is the jth value of star intensity �in linearnits�. Of course, one must superpose WFi by tak-
ng into account the position of the footprint at theonjugation altitude.
The integration time of the WFS has to be largerhan an evolution time step of the atmosphere toimulate the blurring of the wave-front informationhat is due to the finite exposure time of the WFS.
These layer measurements are then used to com-ute the correction, and, after a number of stepsquivalent to the time delay, the DMs are subtractedrom the wave fronts of the NGSs. In a closed loophe reconstructed phase refers to the residual turbu-ence, and the corresponding correction is actually aifferential correction.The incoming phases combined in LO mode focused
o the conjugation altitude are sampled with a reso-ution that is much better than that applied to the
FS measurements. To achieve the spatial sam-ling used, we spatially average the measurements toet a measurement relative to each WFS pixel �seeig. 5�. In the real system these WFS pixels can beead while they are assembled in squares to minimizeFS noise. In the simulator we also consider this
ossibility by averaging the phase values of a screenf pixels in a square that comprises the assembledixels and finally associating them to the relativeoise �see Subsection 2.C below�.Finally, features of the MFoV approach are imple-ented. In particular, we consider that the NGSs
hat lie in the external annular FoV are used to drivehe ground layer wave-front sensor, whereas thetars of the central FoV are used to drive the higherMs �see also Ref. 24�.
. Addition of Phase Noise
he LOST code implements the procedures to simu-ate the error in phase measurements relative to theH WFS and the pyramid. In the latter case in theomputation of the NGS photon flux the splitting ofight among the various loops is taken into accountin the MFoV case this is done only for the centraloV reference stars�. To simulate the WFS re-ponse we take into account all the sources of noise:oissonian distribution of photons, RON, dark cur-
ig. 5. Results of the LO procedure described in the text. Top,igh-altitude WFS measurement: the footprints of three NGSso not perfectly overlap because of their different positions in theoV. Metapupils �the circles that define the projection of the FoVn the conjugated planes� are also shown. The real measure-ents are at the right, where the effects of the WFS spatial sam-
ling �7 � 7 and 8 � 8, respectively, for high and ground WFSs� areisible. Bottom, measurements of the ground WFS. Here theupil footprints and the metapupil overlap perfectly. For the twomages we apply the same linear scale to show that the dimensionf the highest metapupil is larger than the ground state metapupil.
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ent, and sky background. The LOST code com-utes the error that is due to the WFS directly inerms of phase for all the subapertures that composehe metapupil. In the SH case the LOST code ap-lies the noise propagation equations in Ref. 25,hich associates the noise relative to the intensityeasurements with the noise relative to the mea-
ured phases. The noise that is due to each consid-red source is given in terms of variance in aubaperture, ��
2, expressed in square radians:
��,photon2 �
�2
21
nph�NT
ND�2
, (8)
��,RON�dark2 �
�2
3�e
2
nph2 �NS
2
ND�2
, (9)
��,sky2 �
�2
3nbg
nph2 �NS
ND�2
, (10)
here nph is the number of photons detected in theubaperture, NS
2 is the total number of WFS pixelser subaperture, and NT is the image’s FWHM.his FWHM is equal approximately to the ratio �r0,here is the wave-front-sensing wavelength, if r0 is
arger than the dimension of the subaperture, d,hereas ND is the FWHM in pixels of the diffractionattern of the subaperture, ND � �d. nbg is theumber of photons detected in the subaperture thatome from the sky background. �e is the root meanquare, in photoelectrons, that is due to RON andark counts:
�e � ��e,RON2 � Ne,dark�
1�2, (11)
here �e, RON is the root mean square that is due tohe RON �equal to the RON value� and Ne, dark are theark counts. Applying Eq. �11�, we obtain an arrayf variance values �for each subaperture� of the sameimensions of the array that represent the phaseeasurements. This variance array permits the
omputation of random phase noise maps: By mul-iplying this array point by point by a random arrayf unitary variances, one can retrieve the noise array.n array of random numbers is generated for eachhase measurement. This noise is added algebra-cally to the measurements given by the LO proce-ure before the measurements are used for the fitith the Zernike polynomials. In the WFSs conju-ated to the high-altitude layers the footprints of thetars do not completely overlap, and illumination ofhe subapertures depends on the positions and mag-itudes of the references. This dependence affectshe phase noise variance array, which is not uniformn that it has valleys where the footprints overlap andeaks onto which only the light of a few faint NGSs isrojected. The model of the pyramid WFS in theimulator is based on geometrical approximationsnd on the WFS noise theory developed for the SHensor. We assume, in an operative sense, that eachH subaperture corresponds to the four WFS pixelssed to sample the image of the four pupils reimaged
y the pyramid. Under this assumption the pyra-id is quite equivalent to a quad-cell SH sensor: the
our pixels of the quad cell correspond to the sameixels on the four pupils of the pyramid. These fourixels, one for each pupil, permit measurement of thehase in a single subaperture of the main pupil �seeig. 6�. The LOST code uses a pure integrator con-
rol law, and the wave-front reconstructor is not noiseeighted. The code applies the same gain to all theirror modes computed, relative to each DM. To cut
he noisier modes, the user has to set a high conditionumber �the standard value applied is 10� for the cutf the eigenvalues in the computation of the inverse ofhe interaction matrix �see Subsection 2.D below�.n this way is possible to manage measurements inhe proper way when the SNR is low.
To simulate the effect of a pyramid-based WFS weonsider the SH procedure described above, assum-ng a modulation of the pyramid that always gives ainear response of this device for every degree of cor-ection. We model the coefficients NS, ND, and NT toe consistent with the characteristics of the pyramidFS, and we substitute these values into Eq. �8�–
10�.Several studies9,26 predicted a gain in terms of lim-
ting magnitude of the pyramid with respect to theH sensor; anyway, we chose a conservative ap-roach in which this advantage is not considered.rom the comparison with the quad cell, the numberf pixels NS
2 used to sense the wave front in eachubaperture is 4. ND in the SH sensor is the dimen-ion in pixels of the PSF pattern of a diffraction-imited spot for a subaperture d. Instead, the LOSTode uses for a pyramid the linear dimension of theuad cell itself, expressed in pixels �ND � 2�. Theyramid is a pupil plane WFS, and there is no NGSpot �as there is in the SH sensor� on the 4 pixels thatompose the quad cell, but they are completely illu-inated by a portion of the pupil image �except those
ig. 6. Analogy between the quad-cell SH �left� and the pyramidFS �right�. D, dimension of the metapupil; d, dimension of the
ubaperture. In both cases, the light collected by a single subap-rture is measured by four pixels: the quad cell in the SH casend the four corresponding pixels on the four reimaged pupils inhe pyramid case. In the SH sensor we measure shift � of thepot, whereas in the pyramid we compute the intensities of the fourixels to retrieve the wave-front derivatives.
t the border of the metapupil�. So for a pyramid wean assume that the dimension in pixels of the illumi-ated portion of the quad cell is the side of the quad cell
tself. If we refer to the variance of a single subaper-ure, Eq. �9� becomes, for dark current and RON,
��,RON�dark2 �
4�2
3 � �e
nph�2
. (12)
he NT parameter is present only in the Eq. �8� �rel-tive to the photon noise� and its value is computed atvery step of the loop on the instantaneous PSF byse of the Fourier transform. One finds the skyackground noise by substituting the parametersound for the pyramid WFS:
��,sky2 �
�2
3nbg
nph2 . (13)
. Zonal and Modal Reconstruction and Correction
he user can choose either a zonal or a modal methodith which to compute the shapes of the DMs. Theeasurements computed with the simulated phase
ensors are the input data of the reconstruction-orrection procedure. These are expressed in arraysf phase values, where the number of arrays is equal tohe number of WFSs. There is a one-to-one corre-pondence between the elements of the arrays and theeasurements that come from the assembled pixels
which we also call subapertures�. But not all theubaperture measurements are used in the reconstruc-ion procedure. Several subapertures can be onlyartially illuminated by the stars’ footprints or not atll. The code uses in the reconstruction only thoseubapertures illuminated at least for a user-definedraction �the standard is 10%�, even if the computationf the DM surfaces is performed over all the metapu-ils.In a zonal reconstruction the DM is computed by
ig. 7. Left, twin primary mirrors geometry of the LBT. The pecondary mirror and is only 8.2 m. Right, diffraction-limited Pomputed with the LOST.
inear spline interpolation of the phase noise mea-urements retrieved by the WFS. But usually onenalyzes the performance by taking into accountodal reconstruction. In modal reconstruction theeasurements are fitted with a user-defined number
f Zernike polynomials or set of modes �such asarhunen–Loeve or user-defined modes or, for in-
tance, the measured mirror modes�. This fit is per-ormed by inverse matrix operation: for each loop le solve the system that relates the LO phase noiseeasurements array Ml to DM modes Zl and obtain
s a result the coefficient vector cl relative to eachirror mode considered:
cl � Il�Ml, (14)
here I� is the inverse of the interaction matrix com-uted by singular-value decomposition. As in theeal LO systems this matrix is computed once forach NGS asterism. We assume that the DMs areble to produce the modes Z used to fit the measure-ents, and we compute the DMs as linear combina-
ions of Z. For each loop we take into account a gainoefficient, gl, such that the mirror mode coefficientn�1,l applied to each DMl is defined by
an�1,l � an,l � glcn�1,l, (15)
here the subscripts n and n � 1 refer to two consec-tive steps of the same loop. The maximum numberf modes that is useful for reconstructing the phaseeasurements is related to the number of subaper-
ures considered. One selects the modes by keepingnly the polynomials with associated singular valuesarger than a condition number �usually defined as the0% of the largest eigenvalue� to ensure greater sta-ility of the correction-rejecting unstable modes.
ry mirror diameter is 8.4 m. The entrance pupil is due to thethe interferometric case in the K band as seen by the LBT and
rimaSF in
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. Interferometry
e implemented interferometric features to simulatehe LBT telescope with two independent 8-m-classelescopes on the same mount �Fig. 7�. The two sys-ems run in parallel, using the same parameters butach looking to its own portion of the same layers.e collected the two wave fronts of each test star insingle array �according to the specific dimension of
he LBT� and used them to compute the interfero-etric PSF of the stars by performing a Fourier
ransform. As in the single-channel case, after aumber of iterations defined by the user the long-xposure PSFs start to be integrated as the PSFs thatere computed for all the steps of the simulation are
ollected. These instantaneous PSFs are used alsoo compute the instantaneous interferometric SR �de-ned as the ratio of the central value of the computedSF to the PSF in the diffraction-limited case�.owever, more interferometric properties will be ob-
ained by development of the code specifically to en-ure more-accurate and -realistic results, includingetailed behavior of the fringe tracker27 �an issuehat, however, is not discussed in this paper�. TheOST code at present assumes that the piston termetween the two arms is corrected by the system,xcept for small, random errors.
. Simulation Outputs
or every temporal step the residual wave fronts aresed for computing the instantaneous PSF in all theky directions defined by the user. The SR is eval-ated as the ratio between the center of these PSFsnd the same value relative to the diffraction-limitedase for each iteration. We coadd the PSFs step bytep for each direction to retrieve, at the end of theimulation, the long-exposure PSFs relative to theemporal interval over which the coaddition was sety the user. Finally, the code saves the instanta-eous SR data and the integrated PSFs and computeshe long-exposure SR map over the FoV by linearlynterpolating the SR value of each sky direction. Inhis way the user is able to compute the SR and theSF for the directions previously set.
. Validation of the Code
. Comparison with Theory
e performed several tests to check the validity ofhe LOST simulations. The phase screens simu-ated the same properties predicted theoretically for
Table 1. SR for Two Kinds of Noise: Ph
MR WFS Sampling Gain RON
14.83 10 � 10 0.683 5.015.81 14 � 14 0.400 0.0
aComputed for the parameters in Ref. 31. The SR is compared inhe same. The error values are the standard deviations of the inshe simulation.
bRef. 31.
olmogorov turbulence as well as the isoplanaticatch size angle and the temporal error fit that wereheoretically predicted.28 We also checked the rela-ionship between phase variance �� and the wave-ength28 and whether the distribution of the residual� with respect to the Zernike fitted modes fitted theistribution predicted.22 We tested the influence oftting error in the zonal case and the influence of theumber of Zernike polynomials used in the modalorrection on achievable performance. In all theseases we took into account an on-axis single-referenceystem without considering the effects of noiseaused by the WFS on the phase measurements.e repeated each test several times with different
andomly generated phase screens to compare theistribution of the LOST code results with the theo-etical expectations. All the tests performed yieldedesults that were in agreement with the theory: Theifference between the LOST results and the theo-etical values was less than 10%, as was also reportedn Refs. 29 and 30.
. Shack–Hartmann Test
e simulate the SH WFS by using Eqs. �8�–�10� toompute the phase noise associated with WFS mea-urements. We validated our procedures by com-aring the LOST code results with the cross-checkesults published in Ref. 31. Here the results ofhree simulation codes are presented and their per-ormance compared. The three results refer to apecific system in specific atmospheric conditions andse the same simulation parameters. We followedhe same strategy to check our code. In several sim-lations under the same conditions we found results
n good agreement �see Table 1 and Fig. 8�.
. Pyramid Test
e cross checked our model for the noise of the pyr-mid WFS, comparing our simulations with similarimulations performed by with the numerical codeAOS.32 The MAD system was simulated with bothOST and CAOS packages.24 For MCAO the WFSensor was a pyramid, and no modulation was con-idered. If we assume an optimistic value of r0 �.18 m at the V band �0.5 �m�, the two codes giveimilar results, with a peak SR of approximately 50%nd an average SR of 38% computed on the guide starthe SR includes tilt�. For details of the system pa-ameter that we used, see the Subsection 4.A below.he CAOS simulation package uses an end-to-end
Noise and RON, and Photon Noise Onlya
Integration Time�s� Rigaut SRb LOST SR
0.011 0.5 0.46 � 0.020.004 0.5 0.53 � 0.02
mns 6 and 7 to that given in Ref. 31, and the two are approximatelyaneous SR contribution to the long-exposure SR computed during
odel of a pyramid WFS, whereas the LOST codeses the relations derived for the SH WFS. How-ver, our simulations agree with these simulationsecause the case considered belongs to the brightegime of integrated magnitude where the pyramidain9,26 with respect to the SH sensor is negligible.e also compare our results with those obtained by
edrigo et al.14 and with those that are relative to theAD. Again, the results of the two codes are in good
greement, as listed in Table 2.
. Applications
he LOST software was developed to simulate a ge-eric layer-oriented wave-front sensor system. Inhat follows, we present four different results rela-
ive to the MAD and NIRVANA MCAO LO systems.
ig. 8. Dashed curve, data of Rigaut et al.31; LOST results areepresented by the dotted curve. Both curves refer to the Geminielescope and the classic AO case with one DM conjugated to theround layer, r0,V � 0.25, r0,H � 0.90, �e of 5 e� per pixel per frame,0 � 10 sampling, and a zonal DM reconstruction. � � 0.22�m� bandwidth; overall quantum efficiency, 0.5; scientific andave-front sensing wavelengths, respectively, sc � 1.6 ��m Hand� and WFS � 0.7 ��m R band�; diameter, 7.9 m; and centralbstruction, 1.2 m. Data relative to our simulations were ob-ained with 1-s exposure. Curves were obtained after optimiza-ion of the two free-parameter gain and integration times of the
FS. Error bars, standard deviations of the instantaneous SRontribution to the long-exposure SR computed during the simu-ations.
Table 2. Comparison of LOST MAD Simulations and the SimulationDescribed in Ref. 14a
aThe same eight-star asterism over a 2-arc-sec FoV is assumedn both cases, as is an identical atmospheric model characterizedy an overall r0 � 0.83 m at the K band and the Kolmogorov powerpectrum.
n both cases we considered a seven-layer atmo-phere with the typical conditions measured at Cerroaranal �Chile�, characterized by an average seeingalue at the V band of 0.73 arc sec, equivalent to anverall r0,V � 0.14 m. In both cases the results areomputed at the K band, � 2.2 �m, with a corre-ponding r0,K of 0.83 m. The MAD simulations areerformed with an existing set of phase screens �Ta-le 3�, whereas the atmosphere relative to the NIR-ANA system is computed by the LOST code.To measure the value of the SR over the 2-arc min
oV, for the MAD we use a constellation of sky direc-ions, whereas for NIRVANA we consider a simplequare grid �Fig. 9�.
. LO Wave-Front Sensor for MAD
he MAD module will be mounted at the visitorasmyth focus of UT3, an 8-m-diameter telescope of
he Very Large Telescope. The layer-oriented wave-ront sensor of the instrument is currently installedt the Astrophysical Observatory of Arcetri, the SHFS, the main optical train, and a real-time com-
uter are located at the European Southern Obser-atory at Garching, Germany, and the CAMCAOnfrared camera is located in Lisbon �see also Ref. 33�.
The capabilities of the MAD layer-oriented wave-ront sensor �for specifications used see Tables 4 and� can be analyzed numerically with good accuracy byhe LOST code. Two sets of DM modes are used, oneor each conjugation altitude; each set contains the 59losest modes to the 59 first Zernike polynomials thatDM can generate. In the following simulations a
enith angle of 30° was assumed.The values of the integration time used in the sim-
lations are selected by an optimization test. Byunning an automatic interactive data language pro-edure, we use all the possible combinations of inte-ration times of the two WFSs �ground and high� inhort simulations of 1 s in which only the SR on theenter of the field is computed. In this way the bestalues for these parameters are retrieved for bothases considered below. Also, the gains used in theimulations are chosen through short simulations; inarticular, we considered a grid of gain values �from.4 to 0.9, with 0.1 spacing between consecutiveoops� applied to the loops by use of the best integra-ion times.
We analyzed two real star asterisms on a 2-arc min
Table 3. Atmospheric Parameters Used in the Simulationsa
aFor each layer an outer scale of 20 m is assumed. r0 wasomputed for the K band.
FsmlwmwTitssfinguiS
tbT0w
B
TmiTwfitaTtt
Fegd
a 0.4 �
oV with six and eight stars, respectively. The six-tar asterism right ascension was centered in 16 h 25in 17.7 s and dec � � 40°39�53� of galactic latitude5.5° with V � 9.85 integrated magnitude �Fig. 10�,hereas the eight stars were centered in RA 10 h 3550.1 s and dec �58°12�50� of galactic latitude 0°
ith V � 5.03 integrated magnitude �Fig. 11�.hese are bright stars, and to obtain more-
nteresting ones we shifted the integrated magnitudeo the 14th. Assuming a maximum difference intar brightness of 3.5 magnitude and a minimumeparation of stars of 20 arc sec, the probability ofnding a suitable star asterism brighter than mag-itude 14 integrated on a 2-arc-min FoV is 70% for aalactic plane and 3% for the galactic poles.34 Fig-res 10 and 11 summarize the results. Even if the
ntegrated magnitude of the two cases is equal, theR is more nearly uniform for the six stars than for
ig. 9. Left, each cross indicates a direction where the SR is compuach cross indicates a direction where the SR is computed in the NIenerated by the LOST code. In both cases the two circles haveirection.
Table 4. Common Simulation P
D �m�FoV
�arc min� science
��m� WFS
��m�� WFS
��m�
8 2 2.2 0.55 0.4
aThe pixel size of the phase screens is always 0.07 m�pixel, withflux of 5.36 � 1011 photons�s at � 0.55 �m wavelength �� �
Table 5. Common Simulation Par
DM Altitude �km� Gain WFS SamplingIntegratTime �m
1 0 0.6 8 � 8 5.02 8.5 0.6 7 � 7 10.0
he eight stars because the difference in brightnessetween the references is larger in the latter one:he average SRs over the 2-arc-min FoV are 0.21 and.17 for the six and the eight stars, respectively,hereas the peak SRs are 0.29 and 0.28.
. NIRVANA
he LBT has two D � 8.4 m primary mirrorsounted upon a common altitude-azimuth mount-
ng, with a center-to-center distance of 14.4 m.35
he two channels are symmetrical and are suppliedith twin adaptive secondary units.36 Whereas therst-generation instruments37–40 are close to the in-egration phase, the second-generation instrumentsre in the design phase, as is LINC–NIRVANA.18
he LINC module will combine the two LBT channelso use the natural 22.8-m baseline given by the edge-o-edge distance of the two primary mirrors. To
n the MAD case; these coordinates are defined by the user. Right,NA case; these directions are positioned on a square grid and werend 2-arc-min diameters. The origin corresponds to the on-axis
eters Considered for the MADa
MR,sky
ExposureTime �s�
OverallQuantumEfficiency
DelayTime �ms�
20.0 1.0 0.2 2.5
iameter of 112 pixels. The overall quantum efficiency, 0.2, givesm bandwidth� and for a star of magnitude V � 0.
chieve this result, one needs an high level of wave-ront correction. In a first phase the LINC instru-
ent will work by using the single-reference AOorrection provided by the secondary units only. Insecond phase the correction will be enlarged to a
quare field of view of 20 arc sec � 20 arc sec. Tochieve this goal requires a uniform correction, which
ig. 10. Six-star asterism for the MAD case. Left, reference guiong-exposure SR map in the K band. The guide stars’ positions-arc-min diameters. The origin corresponds to the on-axis direc
ig. 11. Eight-star asterism for the MAD case. Right, positionsap in the K band. In both figures the two circles have 1- and 2
Table 6. LINC–NIRVANA Common S
D �m� FoV science
��m� WFS
��m�� WFS
��m�
8.2 2� and 6� 1.2 and 2.2 0.75 0.5
aIn the interferometric mode we consider the same parameters fhe DMs we use the Zernike polynomials, assuming that DMs are
s supplied by the MCAO NIRVANA instrument.he adaptive secondary units will perform theround-layer correction; NIRVANA will provide aigh-level correction over the 2-arc-min FoV with theddition of another DM per arm. Finally, the cor-ected wave fronts will be combined with the pistonorrection provided by the fringe tracking,27 enabling
rs’ positions and magnitudes are indicated by diamonds. Right,arked by triangles. In both figures the two circles have 1- and
magnitudes of the reference guide stars. Left, long-exposure SRmin diameters. The origin corresponds to the on-axis direction.
tion Parameters of Both Channelsa
R,sky
ExposureTime �s�
OverallQuantumEfficiency Delay Time
0.0 1.0 0.3 2 � integration time
th arms. In this case instead of the measured �mirror� modes ofto reproduce every Zernike polynomial.
de staare m
and
imula
M
2
or boable
tm
cMwdotwtmmLTppAthSvafiitsgu
dtmtpotMa
sethiscuats
tracswsbmt m
lga
FaF6
1
ircb
he large-FoV interferometric focal station to beounted with the other scientific instruments.The NIRVANA module is composed of two MCAO
hannels that provide high layer correction. TheCAO system is driven by natural stars that lieithin a 6-arc-min FoV. The NIRVANA module18 isesigned to use an optical or a numerical coadditionf the NGS light following the LO approach. In par-icular, to extend the FoV, and thus the sky coverage,hen one is looking for the references, the MFoV
echnique is applied.13 The capabilities of an instru-ent such as LINC–NIRVANA can be analyzed nu-erically with a good degree of accuracy with theOST code �the system parameters used are listed inable 6�. The values of the spatial–temporal sam-ling of the WFS used in the simulations were com-uted through an automatic optimization procedure:
grid of the possible combinations of integrationimes and sampling of the two WFSs �ground andigh� was used for short simulations of 0.3 s; only theR on the NGS was computed. In this way the bestalues for these parameters were retrieved for eachsterism, with the average SR assumed to be thegure of merit to be optimized. Also, the gains used
n the simulation were evaluated by short simula-ions in which a grid of values �0.05–1.2, with 0.05pacing� was checked by application of the best inte-ration times retrieved. The conjugation altitudessed for the high DM were selected by a similar test.We recall here that in the MFoV the ground loop is
riven by use of references in a FoV that is largerhan the corrected FoV. The secondary adaptiveirrors are controlled by 672 electromagnetic actua-
ors at a 1-kHz rate that in the first phase of theroject will perform single-reference AO correctionnly. But when the NIRVANA module is placed onhe LBT it will perform the ground layer loop. TheCAO system is designed according to the MFoV
pproach, as described in Subsection 2.B, with the
Table 7. AO System Parameter
DMAltitude
�km� GainsIntegrationTime �ms�
1 0.1 0.55 8.02 6 0.45 6.0
aThe binning factors used are 2 � 2 for the ground layer DM an2.
Table 8. Long-Exposure Data of the Cases Taken into Account forNIRVANAa
CaseMaxSR
AverageSR
rmsSR
Peak toValley
Single channel 0.21 0.13 0.03 0.30Interferometric 0.13 0.12 0.002 0.004
aA differential piston error with standard deviation �piston � �4s considered for the interferometric case. The results listed hereefer to the 2-arc-min FoV for the single-channel case and to theentral 20 arc sec � 20 arc sec for the interferometric case. Inoth cases Magnitude R integrated on a 2-arc-min FoV was 14.18.
tars in the external annular 6-arc-min FoV as ref-rences for the ground layer wave-front sensor andhe stars in the 2-arc-min FoV as references for theigh WFS. According to the optical design of the
nstrument, the NGS in the annular 6-arc-min FoVees only the ground layer DM and does not use theorrections performed by the high DM. In the sim-lation code we define the telescope by taking intoccount the diameter of the primary mirror seen byhe secondary and the 0.89-m-diameter central ob-truction.We take into account the minimum separation be-
ween the NGSs in the ground and in the high FoVs,espectively, of 25 and 20 arc sec. These separationsre due to the dimensions occupied by the mechanicalonstraint. In the case we consider in the followingimulations the central wavelength used for theave-front sensing was the R band, whereas the re-
ults presented refer to the K band. We analyzedoth the single channel and the interferometricodes. In the latter case we assumed that the pis-
on correction residual has a rms value of p�4, wherep is the scientific wavelength �K band�, and a zeroean.The star asterism studied in this simulation be-
ongs to the faint range of magnitude for the lowalactic latitudes. In fact, the probability of findingn asterism, in view of the LINC–NIRVANA con-
ig. 12. Reference guide star positions and their magnitudes rel-tive to the NIRVANA cases. The inner circle has a 2-arc-minoV, with integrated R magnitude of 14.2; the larger circle has a-arc-min FoV, with total integrated magnitude of 13.6.
d in the NIRVANA Simulationsa
RON �rms� DarkZernikeModes
.5 e��pixels�frame 500 e��pixels�s 609
.5 e��pixels�frame 500 e��pixels�s 324
4 for the high DM.41,42 In both cases the WFS sampling is 12 �
traints on the minimum separation between theGSs, that is brighter than the 14th integrated mag-itude on the 6-arc-min FoV is more than 90%. Thisumber decreases to 35% for the North galacticole.34 Figure 12 shows the results for both cases.or the single arm the full 2-arc-min corrected FoV ishown, where a significant SR of 0.21 was reached.n this case the correction shows a peak close to theenter of the field because of the nonuniform NGSonfiguration: The correction is driven mainly byhe brightest stars close to the on-axis direction.ut this feature improves performance from the in-
erferometric point of view, because for this option weonsidered a scientific camera centered on the on-axisirection with a FoV of 20 arc sec � 20 arc sec �Fig.3�. In this field the correction is quite uniform, andncluding also the piston-term error and the long-xposure SR �in this case the interferometric SR� is.12–0.13 �Table 8�.
. Conclusions
n this paper we have described the LOST simulationool for multiconjugate layer-oriented systems. Wealidated all the main issues of the code by present-ng the tests performed. These checks have shownhat even if the LOST code does not compute theystem performance in an end-to-end way, it givesesults very similar to other end-to-end simulationodes �such as CAOS�. This code can be a useful toolith which to evaluate the performance of the LO
ystems and to determine the system’s parameters ashe spatial and temporal samples that optimize theR. Both the LO MAD and LINC–NIRVANA sys-ems were analyzed by the LOST code, allowing us tostimate the instruments’ performance and to opti-ize the AO parameters for the median atmosphereodels considered. The authors will provide the
ource files for the code in response to an e-mail mes-age from the reader.
ig. 13. Long-exposure SR maps obtained for a 2-arc-min FoV forase. For both figures the same set of phase screens and the sam
We thank LINC–NIRVANA preliminary design re-iew board members A. Glindemann, G. Herriot, N.ubin, P. Lena, and F. Rigaut, to E. Fedrigo for use-
ul discussions about the SH and pyramid WFS noiseodels, and to M. Le Louarn for the phase screensed in the MAD.
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