Adaptive optics sky coverage modeling for extremely large ... · Adaptive optics sky coverage...

Adaptive optics sky coverage modeling for extremelylarge telescopes

Richard M. Clare, Brent L. Ellerbroek, Glen Herriot, and Jean-Pierre Véran

A Monte Carlo sky coverage model for laser guide star adaptive optics systems was proposed by Clare andEllerbroek [J. Opt. Soc. Am. A 23, 418 (2006)]. We refine the model to include (i) natural guide star (NGS)statistics using published star count models, (ii) noise on the NGS measurements, (iii) the effect oftelescope wind shake, (iv) a model for how the Strehl and hence NGS wavefront sensor measurementnoise varies across the field, (v) the focus error due to imperfectly tracking the range to the sodium layer,(vi) the mechanical bandwidths of the tip–tilt (TT) stage and deformable mirror actuators, and (vii)temporal filtering of the NGS measurements to balance errors due to noise and servo lag. From thismodel, we are able to generate a TT error budget for the Thirty Meter Telescope facility narrow-fieldinfrared adaptive optics system (NFIRAOS) and perform several design trade studies. With the currentNFIRAOS design, the median TT error at the galactic pole with median seeing is calculated to be 65 nmor 1.8 mas rms. © 2006 Optical Society of America

OCIS codes: 010.1080, 010.7350.

1. Introduction

Several extremely large telescopes (ELTs) of primarymirror diameters 20–100 m are currently in the plan-ning and design stages, such as the Thirty MeterTelescope (TMT),1 the European ELT,2 the GiantMagellan Telescope (GMT),3 and the Euro50 project.4Successful utilization of the unprecedented resolu-tion of these ELTs is contingent upon the correction ofatmospheric turbulence with adaptive optics (AO).5

Current AO systems6 employ a laser guide star(LGS) to provide wavefront sensor (WFS) measure-ments of the instantaneous wavefront aberrationswithout relying upon the availability of a bright nat-ural guide star (NGS). However, because the laserjitters and is deflected on both the upward and down-ward paths through the atmosphere, the tip–tilt (TT)modes of the atmospheric aberration cannot be de-termined from the LGS. Consequently, NGS WFS(s)are also required to estimate these modes. The skycoverage problem is the probability of finding suffi-

ciently bright NGS(s) within the isoplanatic patch ofthe science object that will allow for the accurateestimation of the TT modes.

Multiconjugate adaptive optics (MCAO), wherethere are multiple deformable mirrors (DMs) conju-gate to different altitudes in the atmosphere, havebeen proposed to overcome the cone effect and provideimaging over a wider field of view (FOV). MultipleLGSs are used to tomographically reconstruct atmo-spheric turbulence in three dimensions and determinethe commands for the multiple deformable mirrors. Analternative AO configuration for tomography andwide-field correction is multiobject adaptive optics(MOAO), where the DMs apply independent correc-tions to objects within the FOV.

An additional fundamental effect limiting LGS AOsystems is tilt anisoplanatism, which arises when asingle TT NGS is viewed off axis with respect to thescience object. For LGS MCAO, tilt anisoplanatismwill degrade the uniformity of turbulence compensa-tion over an extended field and may become the dom-inant wavefront error term. A number of differentapproaches to overcome tilt anisoplanatism havebeen proposed: (1) using multiple TT NGSs,7 (2) usinga NGS that measures TT and focus,8 and (3) usingLGSs at different altitudes7–9 (i.e., a combination ofRayleigh and sodium LGS). For option (1), Ellerbroekand Rigaut7 state that only three TT NGSs are nec-essary, and we use three NGS WFSs for this option.Because the altitude of the sodium layer is constantlychanging,10 and it is not possible to disentangle at-

R. M. Clare ([email protected]) and B. L. Ellerbroek arewith the Thirty Meter Telescope Project, California Institute ofTechnology, 1200 E. California Boulevard, Mail Code 102-8, Pas-adena, California 91125. G. Herriot and J.-P. Véran are with theHerzberg Institute of Astrophysics, 5071 West Saanich Road, Vic-toria, British Columbia, Canada.

Received 12 May 2006; revised 4 August 2006; accepted 11 Au-gust 2006; posted 11 August 2006 (Doc. ID 70899).

0003-6935/06/358964-15$15.00/0© 2006 Optical Society of America

8964 APPLIED OPTICS � Vol. 45, No. 35 � 10 December 2006

mospheric focus aberrations from these altitude vari-ations with the LGS measurements, we modify option(1) to be two TT sensors and a TT-focus sensor. In thispaper, we evaluate the performance of these threeoptions.

In Ref. 11, a method for producing Monte Carlo skycoverage simulations over random NGS constella-tions is presented. The essence of this method is topropagate the turbulence phase screens at each alti-tude, which are represented as a Zernike basis sum,to the aperture using geometric optics. This modelaccounts for the cone effect for the finite height of theLGSs as well as the anisoplanatism caused by theguide stars being off axis with respect to the scienceobject. The expected wavefront error is then calculatedusing a minimum variance estimator from these trans-formation matrices and the statistical properties of theatmosphere.

In this paper, we continue with the methodology setout in Ref. 11 but include a number of practical con-siderations in order to generate a TT error budget forthe narrow-field infrared adaptive optics system(NFIRAOS) of the TMT.12 These upgrades include (i)guide star statistics using the Bahcall–Soneira andSpagna models; (ii) background, photon, and readoutnoise on the NGS measurements; (iii) a telescope wind-shake model; (iv) a model for how the Strehl, and hencethe NGS WFS measurement noise, varies across thefield in the IR; (v) the error due to imperfectly trackingthe range to the sodium layer; (vi) the mechanicalbandwidths of the NFIRAOS TT stage and DM actua-tors; and (vii) temporal filtering of the NGS measure-ments to balance the errors due to wind shake, noise,servo lag, and sodium altitude tracking. We employthis updated sky coverage model to perform a numberof design trades for the NFIRAOS. These includewhether to use optical (V band) or IR (J band) starsand sensors for the NGS WFS. Second, we find theoptimal patrol field diameter required to find sufficientNGS. Third, we use this methodology to evaluatethe different methods for correcting the tilt aniso-planatism described previously. We also compare theperformance of a quad-cell Shack–Hartmann (SH) de-tector with a matched-filter or noise-weighted least-squares approach for measuring the TT from the NGS.

The theory of the sky coverage simulator, includingthe transformation matrices, control algorithm, andminimum variance reconstructors, is presented inSection 2. The computational details of the sky cov-erage simulator, in particular, the calculation of thenoise and atmospheric covariance matrices, are out-lined in Section 3. Sky coverage Monte Carlo simu-lation results for NFIRAOS are presented in Section4. Conclusions are drawn in Section 5.

2. Theory

A. Background and Notation

We consider the atmosphere to consist of Nl discretelayers of turbulence. We model the phase screen atthe ith atmospheric layer, ��i; r, ��, as a finite sum of

N orders of Zernike polynomials13:

��i; r, �� n�0

N

�m�0

n

anm�i�Znm�r, ��, (1)

where Znm is the mth Zernike polynomial of radialorder n, and anm�i� are the coefficients of the corre-sponding Zernike polynomials at the ith layer.

For each WFS in the AO system, the phasescreens, represented by a vector of Zernike coeffi-cients at each atmospheric layer, a, are projected tothe aperture plane by a transformation matrix, T,such that

b � Ta, (2)

where b is the vector of the Zernike coefficients of thewavefront measured by the WFS, and

a � �a�1� · · · a�i� · · · a�Nl��T. (3)

The T matrix takes into account the off-axis effect ofthe NGS or LGS, as well as the cone effect due to thefinite height of the LGS, and also includes a summa-tion over all of the atmospheric layers. The exact formof the T matrix is defined in Ref. 11. The wavefrontmodes as seen by the LGS, bL (both sodium, and whenapplicable, Rayleigh), are given by

bL � TLa, (4)

where the L subscript in this paper refers to the LGS.In this paper, the LGS measurements are assumed tobe noiseless and instantaneous since, we are primar-ily concerned with the dynamic errors with respect tothe NGS, and only the lower-order component of theTT-removed LGS measurements are considered (typ-ically Zernike radial orders 2–6). The effects of noiseon the LGS for the TMT are addressed in Ref. 14.

Similarly, the wavefront modes seen from the NGS,bn, are given by

bn � Tna� � n�, (5)

where the n subscript refers to the NGS, and thevector n� is the additive noise on the NGS WFS mea-surements as discussed further in Subsection 3.D.The prime �� notation on a and n is to show that theNGS measurements, bn, are temporally filtered by acontroller Hol�s� to minimize the combined effect ofthe measurement noise and the servo lag.

The Zernike coefficients, bs, of the wavefronts pro-jected into the aperture plane for sources in the di-rections of the science evaluation points are given by

bs � Tsa, (6)

10 December 2006 � Vol. 45, No. 35 � APPLIED OPTICS 8965

where the s subscript refers to the science field.The Ns science evaluation points we consider forNFIRAOS are defined in Section 4.

Finally, the DMs provide a correction to the wave-fronts associated with the science evaluation direc-tions that is given by

bm � TmPma, (7)

where a is the estimate of the Zernike coefficients ateach layer; the m subscript refers to mirrors through-out this paper; and Pm, the optimal fit of the Zernikemodes to the DM to compensate for the estimatedscience phase profiles, is defined in Ref. 11 by

Pm � �TmTWTm��1Tm

TWTs. (8)

Here, W is a block diagonal weighting matrix of Ns

blocks whose elements are wkI, where I is an identitymatrix of dimensions equal to the number of Zernikemodes computed for each science wavefront. If theentire science field is considered, wk � 1�Ns for all kpoints. If only the on-axis science point is considered,w1 � 1 and wk � 0, otherwise. Similar mirror trans-formation matrices can be derived for the correctionof the NGS modes, Tm,n, and the correction of the LGSmodes, Tm,L.

B. Control Algorithm Overview

The control algorithm we employ in this paper is atwo-step process. We first estimate the wavefrontZernike coefficients at each layer, a, and the associatedNGS measurements, bn, from the LGS measure-ments, bL. The NGS WFS measurements are esti-mated by first performing atmospheric tomographyto estimate the full 3D turbulence profile from theLGS WFS measurements and then projecting the vol-ume turbulence estimate into wavefront estimates inthe direction of each NGS. Of course, the overall TTmode of each NGS measurement will be poorly esti-mated due to the LGS position uncertainty. But rea-sonable estimates of the focus and astigmatismmodes (as well as the tilt anisoplanatism modes) canbe obtained and can be incorporated into the wave-front reconstruction process. We then estimate theresidual uncertainty in a, ares, from the new informa-tion provided by the actual NGS measurements, bn. Itmay be shown that this two-step approach is equiv-alent to an integrated one-step minimum varianceestimator, but the two-step approach is a more effi-cient approach to Monte Carlo simulations of theNGS constellations. The control model is shown inFig. 1 and is described in more detail here.

First, the Zernike coefficients at each of the atmo-spheric layers are estimated from the LGS measure-ments, bL, using

aL � ELbL � ELTLa, (9)

where EL is a minimum variance estimator as de-scribed in Subsection 2.C. The residual error in thisestimate is given by

ares � a � aL, (10)

and from Fig. 1 the estimate of the residual error(computed from the NGS measurements) is related tothe final estimate a by

ares � a� aL. (11)

Note that although Eq. (9) indicates that the LGS WFSmeasurement vector bL is measured in an open loopwithout the corrections applied by the DMs, NFIRAOSwill actually generate pseudo-open-loop measure-ments by combining closed-loop measurements withthe knowledge of DM actuator commands, as illus-trated in Fig. 1.

Next, the residual uncertainty, ares, is estimatedbased upon the new information provided by the tem-porally filtered NGS WFS measurement. Temporalfiltering is applied to balance the estimation errorsdue to NGS noise, servo lag, sodium-layer altitudeuncertainty, and wind shake. The filtered NGS WFSmeasurement is given by

bn �Hol�s�

1 � Hol�s��Tna � n�

� Hcl�s��Tna � n�� Tn�Hcl�s�a� � Hcl�s�n� Tna� � n�, (12)

where Hcl�s� is the closed-loop temporal filter, and s isthe Laplace coordinate. The new information pro-vided by this measurement is the component not pre-dicted by the LGS WFS measurement,

�bn � bn � bn � bn � EnbL, (13)

where En is the minimum variance estimator for bn asdescribed further in Subsection 2.C. Combining Eqs.(12) and (13) yields

�bn � Tna� � n� � EnTLa. (14)

Finally, the estimate of the residual uncertainty aresis given by the expression

ares � Eres�bn, (15)

where Eres is a minimum variance estimator as de-scribed further in Subsection 2.C. The derivations ofEL, En, and Eres, and the associated formulas for eval-


uating their performance, are described in the follow-ing subsections.

C. Minimum Variance Estimators

The minimum variance estimator, EL, of the atmo-spheric modes, a, from the LGS measurements bL isgiven by11

aL � �abLT��bLbL

T�†bL � ELTLa, (16)

where

EL � �aaT�TLT�TL�aaT�TL

T�†, (17)

and † is the pseudoinverse operator. Similarly, theminimum variance estimator, En, of the NGS mea-

surements, bn, from the LGS measurements, bL, isgiven by

bn � �bnbLT��bLbL

T�†bL � EnTLa, (18)

where

En � Tn�a�aT�TLT�TL�aaT�TL

T�†. (19)

Equations (16) and (18) may be combined as

aL

bn�� a

bnbL

T��bLbLT�†bL. (20)

The mean-squared difference between the atmo-spheric modes, a, and the estimated atmosphericmodes, aL, and the NGS measurements, bn, and theestimate of the NGS measurements, bn is

Fig. 1. AO control system model used in this paper. Inside the dotted box is the control algorithm. See the text for the matrix defini-tions.


The four terms of this block matrix are the covariancematrices needed to define and evaluate the second-stage minimal variance estimator Eres. The compu-tation of the new terms appearing in Eq. (21) aresummarized below.

The covariance of the NGS measurements is foundfrom substituting Eq. (5) into �bnbn

T�:

�bnbnT� � Tn�a�a�T�Tn

T � CN, (22)

where CN � �n�n�T� is the covariance of the temporallyfiltered noise and is discussed in Subsection 3.D. Next,the covariance of the NGS measurements, bn, with theatmospheric turbulence, a, is similarly given by

�bnaT� � Tn�a�aT�. (23)

The covariance of the LGS measurements, bL, withthe NGS measurements, bn, is found by substitutingEqs. (4) and (5) into �bnbL

T�:

�bnbLT� � Tn�a�aT�TL

T. (24)

The covariance matrix of the atmospheric modes,C� � �aaT�, is a block diagonal matrix with the ithblock representing the covariance of the ith layer ofthe atmosphere:

C� � C��1� 0

Ì

C��i�Ì

0 C��Nl��, (25)

where there are i � 1, . . . , Nl layers of turbulence,and the covariance of the ith layer is given by

C��i� � �a�i�a�i�T� � ��i�C, (26)

��i� is the strength of layer i, and C is a normalizedcovariance matrix for a single phase screen with unitstrength. The covariance of the filtered atmosphericmodes with the nonfiltered atmospheric modes, C�� a�aT�, is a similarly defined block-diagonal matrixwhere the filtered covariance of the ith layer is

C��i� � �a�i��a�i�T� � �a�i�a�i��T�. (27)

The covariance of the filtered atmospheric modes,C�� a�a�T�, is again a block-diagonal matrix wherethe filtered covariance of the ith layer is

C��i� � �a�i��a�i��T�. (28)

The calculation of C��i� and C��i� is discussed inSubsection 3.E.

By applying Eqs. (22)–(25), (27), and (28) to Eq. (21)for the covariances of the measurements and Eqs.(17) and (19) for EL and En, we obtain

The residual error in estimating ares � a � aL from�bn � bn � bn is given by the minimum varianceestimator:

ares � �ares�bnT��bn�bn

T��1�bn � Eres�bn, (30)

where the estimator, Eres, is given by substitution ofthe terms in Eq. (29) in Eq. (30),

Eres � BT�D � CN��1. (31)

Finally, the mean-squared difference between theresidual atmospheric modes, ares, and the estimate ofthe residual atmospheric modes, ares, is given by

��ares � ares��ares � ares�T� � A � EresB � BTEresT

� Eres�D � CN�EresT (32)

�A � BT�D � CN��1B. (33)

� ares

�bn ares

�bn T��a � aL

bn � bna � aL

bn � bnT�

��abnabn

T��abnbL

T��bLbLT�†�bLabn

T�� aaT� � �abL

T��bLbLT�†�bLaT�

�bnaT� � �bnbL

T��bLbLT�†�bLaT�

�abnT� � �abL

T��bLbLT�†�bLbn

T��bnbn

T� � �bnbLT��bLbL

T�†�bLbnT��. (21)

�a � aL

bn � bna � aL

bn � bnT�� C� � ELTLC� C��Tn

T � C�TLTEn

T

TnC�� EnTLC� TnC� � TnT � EnTLC��Tn

T � CN��A BT

B D � CN�. (29)


Subtracting Eq. (11) from Eq. (10) shows that

a � a � ares � ares, (34)

so that Eq. (32) actually describes the overall estima-tion error in the two-step estimation algorithm:

��a�a��a�a�T� � A � BT�D � CN��1B. (35)

It is insightful to deconstruct the estimation errorcalculated in Eq. (35) into individual errors for noise,tilt anisoplanatism, and servo lag. The mean-squaredcontribution from the NGS measurement noise to theestimate of the atmospheric modes, �N, is given fromthe noise-covariance-dependent term in Eq. (32):

�N � ��Eresn��Eresn��T� � �EresCNEresT�. (36)

Next, the mean-squared error due to tilt aniso-planatism, �a, can be found by replacing the filteredatmospheric covariance terms (C�� and C��) of B andD of Eq. (29) with the unfiltered atmospheric covari-ance matrix �C�� such that

Ba � TnC� � EnTLC�, (37)

Da � TnC�TnT � EnTLC�Tn

T. (38)

This substitution directly assumes that the NGSWFS measurements are noise free and are not tem-porally filtered. The mean-squared difference be-tween the atmospheric modes and the estimate fromthese ideal measurements, aa, is then given by sub-stituting Eqs. (37) and (38) for Ba and Da into Eq. (32)and ignoring the noise covariance term:

�a � ��a � aa��a � aa�T�� A � EresBa � Ba

TEresT � EresDaEres

T. (39)

Finally, the mean-squared error due to servo lag, �l

is then defined as the remaining error, namely,

�l � ��a�a��a�a�T� � �N � �a

� �A � Eres�B � �BTEresT � Eres�DEres

T, (40)

where

�A � 0, (41)

�B � Tn�C�� C��, (42)

�D � Tn�C� � �C��TnT � EnTL�C�� C��Tn

T,(43)

such that A � �A � A, Ba � �B � B, and Da � �D� D.

D. Residual Phase Variance Formulas

The estimation error covariance matrix ��a � a��a� a�T� defines the error in estimating the atmospheric

turbulence profile not the wavefront error for thescience instrument. In Ref. 11, we derive the mean-squared wavefront error �rad2� for a MOAO system as

MOAO2 � Tr�Ts

TWTs��a�a��a�a�T��, (44)

where Tr is the trace of the matrix. If we substituteEq. (35) for ��a � a��a � a�T� in Eq. (44), we obtain

MOAO2 � Tr�Ts

TWTs�A � BT�D � CN��1B��. (45)

For a MCAO system, the residual phase error�rad2� is the sum of the error in estimating the wave-front and the error in fitting the estimated modes tothe DMs:

MCAO2 � est

2 � fit2. (46)

The fitting and estimation errors are, respectively,11

fit2 � Tr��Ts � TmPm�TW�Ts � TmPm�C��, (47)

est2 � Tr��TmPm�TW�TmPm��a�a��a�a�T��. (48)

If we substitute Eq. (35) for ��a � a��a � a�T� in Eq.(48), we obtain the estimation error for MCAO:

est2 � Tr��TmPm�TW�TmPm��A � BT�D � CN��1B��.

(49)

The wavefront errors due to noise, tilt anisoplanatism,and servo lag, respectively, can similarly be found bysubstituting Eqs. (36), (39), and (40) into Eq. (44) forMOAO and Eq. (48) for MCAO, respectively.

For both MOAO and MCAO modes, the wavefronterror, �, in nanometers of the phase is often moreconvenient. The conversion from 2 �rad2� to �m� issimply

�m� ��2 �rad2�

2��e, (50)

where �e is the r0 evaluation wavelength.

3. Computational Details

A. Wavefront Reconstruction Error due to Noise for aLow-Order Shack–Hartmann Wavefront Sensor

In this paper, we compare the noise performance of aquad-cell SH WFS15 against a more complex sensorestimating the displacements of subaperture NGSimages using a matched-filter16 (or noise-weightedleast-squares) approach. In this subsection, we cal-culate the covariance matrix of estimated Zernikecoefficients for the SH WFS. We consider the cases ofthe NGS WFS measuring the first radial order ofZernike polynomials (i.e., tip and tilt) with a single


subaperture and the NGS WFS measuring the firsttwo orders of Zernike polynomials [i.e., tip, tilt, focus,and the astigmatism modes (TTFA)], which requiresa 2 � 2 array of subapertures. For the former ap-proach, the number of Zernike modes measured, Nz,is 2, and for the latter approach, Nz � 5. The SH WFSactually measures wavefront gradients that must bereconstructed (or scaled) into these low-order wave-front modes. For the ideal gradient WFS model, thewavefront slopes in the x and y directions for the kthNGS, sx�k�, and sy�k�, are given by

sx�k� � Gxb�k� � �x�k�, (51)

sy�k� � Gyb�k� � �y�k�, (52)

where Gx and Gy are the interaction matrices of theslopes to the Zernike polynomials, b�k� are the coef-ficents of the Zernike polynomials for the kth NGS,and �x and �y are the additive noise on the measure-ments in the x and y directions. The variance of theone-axis noise for the kth NGS, �

2�k�, is given by

�2�k� � � �x�k� 2� � � �y�k� 2�. (53)

The statistics of the �x�k� and �y�k� will depend uponthe NGS signal level, the WFS read noise, and theparticulars of the processing algorithm used to esti-mate the subaperture TT from the measured inten-sities.

The sensitivity of the ith subaperture to the jthZernike polynomial in the x, Gx�i, j�, and y directions,Gy�i, j� is given by

Gx�i, j� �

�e

2�sai

�Zj�x, y��x dxdy

�sai

1dxdy

, (54)

Gy�i, j� �

�e

2�sai

�Zj�x, y��y dxdy

�sai

1dxdy

, (55)

where Zj is the jth Zernike polynomial using Noll’sordering approach,13 �e is the evaluation wavelength,and the integration domain sai is the ith subaperture.By converting to polar coordinates and using Noll’sdefinitions for the derivatives of the Zernikes,13 it canbe shown that for a single fully illuminated subaper-ture

Gx �2�e

D�1 0�, (56)

Gy �2�e

D�0 1�, (57)

where D is the telescope diameter. Similarly for the2 � 2 fully illuminated subapertures, Gx and Gy aregiven by

Gx �2�e

D

1 0

8

�3

4�63

4�63

1 0 �8

�3

4�63

�4�63

1 0 �8

�3�

4�63

�4�63

1 08

�3�

4�63

4�63

, (58)

Gy �2�e

D

0 1

8

�3

4�63

�4�63

0 18

�3�

4�63

�4�63

0 1 �8

�3�

4�63

4�63

0 1 �8

�3

4�63

4�63

. (59)

The noise-weighted least-squares estimate of theZernike modes, bk, from the wavefront slopes is

bk ��GxT Gy

T��N�1Gx

Gy��1

�GxT Gy

T��N�1sx

sy,(60)

where �N is the noise covariance matrix in the slopespace.

The covariance matrix of the phase estimation er-ror due to the noise of the kth NGS �b

2�k� is given by

�b2�k� � �

2�k��GxT Gy

T�Gx

Gy��1

, (61)

where for 1 � 1 subapertures

��GxT Gy

T�Gx

Gy��1

� D2�e

2�1 00 1�, (62)


and for 2 � 2 subapertures

��GxT Gy

T�Gx

Gy��1

� D4�e

2

1 0 0 0 00 1 0 0 0

0 032

1280 0

0 0 032

640

0 0 0 032

64

.

(63)

The noise covariance matrix, CN, is a diagonal ma-trix and is found by expanding Eq. (91) for multipleNGS:

CN � ��fs� �b

2�1� 0Ì

�b2�k�

Ì

0 �b2�NNGS�

�, (64)

where the matrices �b2�k� are diagonal for the case

where first- or second-radial-order Zernikes are re-constructed from a SH WFS with 1 � 1 or 2 � 2subapertures, respectively, and � is the noise gain.

B. Tip–Tilt Measurement Noise for a Quadrant Detector

Here, we describe a model for the noise equivalentangle, �, assuming a NGS SH WFS with quad-celldetectors.17 The rms one-axis TT measurement erroron each subaperture (alternatively, the noise equiv-alent angle) is given by

� ��B

SNR, (65)

where �B is the effective spot size of the subapertureNGS image, and SNR is the signal-to-noise ratio of asingle subaperture. For a quadrant detector, the SNRis given by

SNR �Np

�Np � 4Nb � 4e2, (66)

where Np is the number of photodetection events persubaperture, Nb is the number of background photo-detection events per subaperture, and e is the rmsdetector read noise per pixel.

In the optical (V band), we assume that the NGSimages are effectively seeing limited, and the effec-tive spot size is given by17

�B ��

4r0�0

�

exp��3.44�5�3�d�

, (67)

where � is the imaging wavelength, and r0 is the Friedparameter.18 The integral�0

� exp��3.44�5�3�d� is com-puted numerically, with a value of 0.4258.

In the IR (J band), we assume that the NGS imagescontain a diffraction-limited core, and the effectivespot size is given by15

�B �3��Nsa

16D , (68)

where Nsa is the total number of subapertures for theNGS WFS.

The number of photons per subaperture is given by17

Np �z10�m�2.5�tsAT

Nsa, (69)

where z is the intensity of a 0 magnitude star, m isthe magnitude of the NGS, � is the end-to-end effi-ciency of the optics and WFS detectors, ts is the inte-gration time, and AT is the area of the telescope pupil�R2�. The value of z, and hence Np, is scaled by theStrehl ratio for J-band sensing with diffraction-limited pixels, since it is assumed that the size of thisquad cell is matched to the diffraction-limited core.

Finally, the number of background photodetectionevents per pixel in each subaperture is given by17

Nb �zb�tsATw2

Nsa, (70)

where zb is the background intensity per square arcsecond in the plane of the telescope, and w is theangular subtense of the square pixel.

C. Subaperture Tip–Tilt Estimation Using a MatchedFilter

In this subsection, we calculate the variance of theunfiltered noise �b

2 for the matched-filter (or noise-weighted least-squares) approach. This processing al-gorithm is applicable if the SH WFS forms actualimages of the NGS in each subaperture. The NGSimage, I, is simply a shifted version of the normal-ized image I0 times the number of photodetectionevents, Np:

I�r� � NpI0�r � ��. (71)

The number of photodetection events, Np, is calcu-lated with Eq. (69).

Equivalently, the NGS image is given by

I � Np��1{��I0�exp��j2��}, (72)

where � � ��x, �y�, and � and ��1 are the Fourier andinverse Fourier transforms. The partial derivatives ofthe NGS images in the x and y directions evaluated at� � 0 are


�I��x

��0

� Np��1{�j2�x��I0�}, (73)

�I��y

��0

� Np��1{�j2�y��I0�}. (74)

The interaction matrix for the matched-filter ap-proach, Q, is

Q � �Qx Qy� � � �I��x

�I��y

�. (75)

The matched-filter algorithm for estimating the tilt onthe ith NGS image, �i, from the ith NGS image Ii is

�i � �QTCN�1Q��1QTCN

�1�Ii � I0�, (76)

where CN, the noise covariance, is the sum of theshot noise contribution I, background Nb, and readnoise e:

CN � I � Nb � e2. (77)

Note that the noise covariance does not include theso-called speckle noise due to the time-varying per-formance of the AO system and the resulting vari-ability in the NGS WFS point-spread function. Thebackground noise, Nb, is calculated using Eq. (70).The variance of the unfiltered measurement noise foran on-axis NGS is given by

�b2�0� � �QTCN

�1Q��1. (78)

The x and y coordinates of the noise will have unequalvariances if the NGS images are asymmetrical. If thelocation of the NGS in the FOV is at an angle of � tothe x axis, the noise covariance matrix may be com-puted using

�b2�� T��b

2�0�T�T, (79)

where the rotation matrix, T�, is of the standard form:

T� � � cos � sin �

�sin � cos ��. (80)

D. Natural Guide Star Wavefront Sensor MeasurementNoise Filtering

The atmospheric Zernike coefficients measured bythe NGS WFS are given by Eq. (5), which is repeatedhere

bn � Tna� � n�, (81)

where n� is the temporally filtered additive noise onthe NGS WFS measurements. As mentioned previ-ously, the NGS measurements are temporally filteredby the open-loop transfer function Hol�s�. The noise

transfer function, Hn�s�, is defined by

Hn�s� �Hcl�s�

HWFS�s�, (82)

where HWFS(s) is the sample-and-hold WFS transferfunction and is given by19

HWFS�s� �fs

s�1 � exp�sfs�. (83)

The filtered noise signal with respect to time, n��t�,is the convolution of the noise impulse response, hn�t�,with the unfiltered noise signal, n�t�:

n��t� � n�t� � hn�t�. (84)

Using the convolution theorem in Eq. (84) yields,

N��f � � N�f �Hn�f�, (85)

where f is the frequency domain coordinate. We takethe inverse Fourier transform of Eq. (85) to get thefiltered noise signal as a function of the unfilterednoise spectrum:

n��t� ��

�

N�f �exp�j2ft�Hn�f �df, (86)

and j � ��1. The ensemble mean-squared average ofthe filtered noise is given by multiplying Eq. (86) byits complex conjugate and taking the ensemble:

� n��t� 2� ��

�

� N�f � 2� Hn�f � 2df. (87)

The noise spectrum N�f � is only nonzero between�fs�2 and fs�2, where fs is the sampling frequency ofthe NGS WFS. If we also assume that � N�f � 2� isconstant with respect to f within this passband, then

� n��t� 2� � � N�f � 2��fs�2

fs�2

Hn�f � 2df. (88)

The WFS sample time, ts, is defined by

ts �1fs

. (89)

The unfiltered noise spectrum is given by

� N�f � 2� � ts�b2 �

�b2

fs, (90)

where �b2 is the variance of the unfiltered measure-

ment noise on a single measurement as derived in


Subsection 3.A for a quad cell and Subsection 3.C forthe matched filter. Substituting Eq. (90) into Eq. (88)gives

� n��t� 2� ��b

2

fs�

�fs�2

fs�2

Hn�f � 2df � �b2��fs�, (91)

where the noise gain, ��fs�, is given by

��fs� �1fs�

�fs�2

fs�2

Hn�f � 2df. (92)

The noise gain due to the temporal filtering is thus afunction of the sampling frequency of the NGS WFS.

E. Covariance of Temporally Filtered Turbulence

Noll defines the covariance of the atmosphere interms of the Zernikes, a, as13

�anman*m*T� � �0.046��R�r0�5�3��n � 1��n* � 1��1�2

� ��1��n�n*�2n��2�mm*Inn*, (93)

where in Ref. 11, we show that for a finite-outer scale,L0, the term Inn* is given by

Inn* ��0

�

��1Jn�1�2��Jn*�1�2��

��2 � RL0

2�11�6 d�, (94)

where Jn is an nth-order Bessel function of the firstkind, n and m are the radial and azimuthal orders ofthe Zernikes, respectively, R is the telescope radius,and � is the coordinate in the Fourier space. Equation(94) was derived by including the von Karman spec-trum in Noll’s original definition of Inn. Similar for-mulas are derived here for the temporally filteredcovariances �a�aT� and �a�a�T� assuming one particu-lar wind-speed model again using power spectraltechniques.

The closed-loop transfer function, Hcl�s�, of the tem-poral filter is given by

Hcl�s� �Hol�s�

1 � Hol�s�. (95)

In Ref. 20, it is shown that for a wind profile with aknown wind speed and a random wind direction uni-formly distributed between 0 and 2, the crossstatistics between the spectra of the filtered and un-filtered phase profiles is given by

��i; ��*�i; �� i; �� 2�h��; i�, (96)

where � ��i; �� 2� is the power spectrum of the ithlayer of the unfiltered atmosphere, and

h��; i� ��0

2

Hcl�jv�i�� cos��d�, (97)

where we have made the substitution s � jv�i�� cos �,and v�i� is the wind speed at the ith atmosphericlayer. Similarly, the power spectrum of the tempo-rally filtered atmosphere, � ��i; �� 2�, is

� ��i; �� 2� � � ��i; �� 2�h��; i�, (98)

where

h��; i� ��0

2

Hcl�jv�i�� cos�� 2d�. (99)

Note that the integrals in Eqs. (97) and (99) arisefrom the particular form of the wind model assumedhere, i.e., a constant wind speed and a random winddirection.

The cross covariance between the filtered and un-filtered Zernike modes �anm��i�an*m*�i�T� is now foundby using Eq. (96) in a derivation otherwise identicalto Noll’s development of Eq. (93),

�anm��i�an*m*�i�T� � �0.046��R�r0�5�3

� ��n � 1��n* � 1��1�2

��1��n�n*�2n��2�mm*Inn*��i�, (100)

where the value of Inn*��i� for the ith layer is given by

Inn*��i� ��0

�2sec2��tan��

Jn�1�tan��Jn*�1�tan��

�tan��2 2

� RL02�11�6

� h��tan��v�i�2R ; i�d�, (101)

and we have made the substitution tan�� 2� for� in Eqs. (94) and (97). Similarly, the filtered modecovariance is found by using Eq. (98) in Eq. (93):

�anm��i�an*m*��i�T� � �0.046��R�r0�5�3

� ��n � 1��n* � 1��1�2

��1��n�n*�2n��2�mm*Inn* � �i�,(102)

where

Inn*��i� ��0

�2sec2��tan��

Jn�1�tan��Jn*�1�tan��

�tan��2 2

� RL02�11�6

� h � �tan��v�i�2R ; i�d�. (103)


In this paper, the cross covariance and filtered covari-ance of the atmosphere, �a�aT� and �a�a�T�, are foundby integrating Eqs. (101) and (103) numerically, andsubstituting Inn*��i� and Inn*��i� into Eqs. (100) and(102), respectively. These integrals are numericallydifficult when the filter Hcl is optimized to providegood or very good correction.

F. Star Generation

In the V band, the NGSs are generated using theBahcall–Soneira star model.21 This produces stars upto a magnitude of 30. In the J band, the NGSs aregenerated using the Spagna model,22 which producesstars with magnitudes up to 22 at the north galacticpole (NGP) and 19 for other galactic latitudes. Thereare more stars at each magnitude in the J band thanin the V band with these models. For example, for a2 arc min diameter patrol field at the NGP, the me-dian expected magnitude in J is 16.5 and in V is 18.3.

G. Natural Guide Star Sharpening

The NGSs are expected to be partially corrected(sharpened) by the AO system in the J band. TheStrehl of each NGS is a function of its position in thefield. Strehl ratios were generated in the Liner AOSimulator (LAOS)14 at five points along the x axis andat four points along a line at 30° to the x axis, al-though as shown in Fig. 2, there is little angulardependence on the Strehl, and the angular depen-dence is ignored. This calculation of the Strehl doesinclude the higher-order errors from the LGS such asspot elongation. A cubic fit was made to these ninepoints as shown in Fig. 2. The Strehl on axis in the Vband was calculated using LAOS to be 0.02, which issignificantly less than in the J band justifying theassumption of seeing-limited images in the V band.

H. Wind Shake

Wind at ground-based telescopes generates TT wave-front errors that are nonnegligible. In this paper, thewind shake is modeled as being statistically indepen-dent from atmospheric turbulence, so the total TTerror is the sum of the contributions of the two effects.

Initial simulations showed that the wind-shaketerm was the dominant term in the TT error budgetfor NFIRAOS and that the tip–tilt stage (TTS) alonewas insufficient to correct for the TT induced from thewind shake. Instead, we use both the TTS and theDM to correct for the wind shake (woofer–tweetercontrol). Additionally, we control the TTS with a pro-portional integral controller such that there are threecontrol paths: the DM, the TTS with a single integra-tor, and the TTS with two integrators. Each path hasa separate gain, which is dependent on the samplingfrequency. The woofer–tweeter control of the DM andTTS for NFIRAOS is described in Ref. 19.

The level of uncorrected wind shake assumed forthis study is 25 mas, which was calculated from anintegrated modeling run of the telescope for a medianwind profile.19,23 The TTS is assumed to have a 20Hz mechanical bandwidth. The residual telescopewind-shake TT jitter after correction by the TTSand DM, res

2, for this model is shown in Fig. 3. Thediscontinuous nature of this curve is due to theconstraints chosen to optimize the gains of the con-troller (see Ref. 19).

I. Sodium-Layer Range Estimation Error

Experimental results have shown that the mean alti-tude of the sodium layer can vary by several meters persecond.10 This temporal variation of the sodium layerresults in a focus error, which cannot be determinedfrom the sodium LGS WFS, because it cannot bedisentangled from the atmospheric focus aberration.Therefore one of the NGSs is required to measure thefocus term, and the sampling rate of the TTFA NGSWFS determines the focus error from the sodium-layeraltitude variations. So, although the purpose of thispaper is to estimate TT errors and not higher-ordererrors such as this focus error, we also include thiserror when we optimize the sampling frequency of theNGS WFS.

The calculation of the error in tracking the meanaltitude of the sodium layer is presented in Ref. 24.This focus error calculation is made by extrapolatinga power spectrum of height variations from lidar data

Fig. 2. Strehl ratio in the J band as a function of the position ofthe NGS in the field. The squares represent the Strehls generatedusing LAOS, and the curve is a best-fit cubic approximation.

Fig. 3. (Color online) Residual telescope wind-shake tilt jitterafter the correction as a function of the sampling frequency of theNGS WFS.


and applying the rejection transfer function for usingelectronic offsets, which will correct for the focus ofthe LGS WFSs in real time. The residual wavefrontfocus error, Na, is plotted versus the NGS WFS sam-pling frequency in Fig. 4 for the median observedsodium altitude variations.

J. Sampling Frequency Optimization

The overall wavefront error, 2 �rad2�, is the sum ofthe atmospheric error atm

2, which is computed witheither Eq. (45) for the MOAO mode or Eq. (46) for theMCAO mode; the residual telescope wind-shake jit-ter, res

2; and the sodium layer tracking error, Na2;

2�fs� � atm2�fs� � res

2�fs� � Na2�fs�. (104)

All the error terms in Eq. (104) are functions of thesampling rate fs. We therefore find the optimum sam-pling frequency, fs*, for each NGS constellation with

fs* � arg minfs

�atm2�fs� � res

2�fs� � Na2�fs��. (105)

4. Narrow-Field Infrared Adaptive Optics SystemSimulations

In this section, the sky coverage simulations for theAO system NFIRAOS at the NGP (latitude � 0°,longitude � 90°), which represents the worst case forsky coverage, are presented. The turbulence andwind velocity profile used in these simulations aretabulated in Table 1. This profile is generated frommeasurements obtained at Cerro Pachon.25 The otheratmospheric and telescope parameters are tabulatedin Table 2.

The wavefront errors are evaluated on axis fora single conjugate AO system using Eq. (104).NFIRAOS, in fact, has a 10 arc sec square field in thebaseline design and a 30 arc sec square field in theupgrade path, although overall TT performance forthese fields is typically within 4% of the on-axis case.

We investigate both optical (V band) and IR (Jband) sensing for NFIRAOS. We assume the pixels in

the J band are twice diffraction limited (i.e., w in theJ band is ��D rads � 0.0086 arc sec) and seeing lim-ited in the V band �w � 0.5 arc sec�. The levels of theread noise, �e, considered are 0, 5, 10, and 15 elec-trons per pixel per readout. Although the first sixZernike orders are considered in the problem, onlythe errors arising in the TT terms are evaluated.

Using the J band for the NGS WFS passband yieldsa conservative bound on the expected sky coverage ofan IR TT WFS, since using a wider J � H passbandimproves TT WFS performance significantly on dimstars. We have not included these results in this pa-per, since the modeling becomes more complex whenAO system performance and the NGS irradiance varysignificantly across the wider WFS passband.

The sampling rate of the NGS WFS, fs, is optimizedfor each NGS constellation. The allowable range of

Fig. 4. (Color online) Estimated focus wavefront error due tovariations in the mean altitude of the sodium layer as a function ofthe sampling frequency of the NGS WFS.

Table 1. Six-Layer Turbulence Profile Typical of Cerro Pachon(Ref. 25)a

Layeri

h(i)(m) �(i)

v(i)(ms1)

1 0 0.6523 52 2577 0.1723 133 5155 0.0551 204 7732 0.0248 305 12,887 0.0736 206 15,464 0.0219 10

aThe profile shows the height h(i) of each layer, the relativeturbulence strength, �(i), and wind speed, v(i).

Table 2. System and Atmospheric Parameters

Parameter Symbol Value

Telescope diameter D 30 mOuter scale L0 30 mPixel subtense (V band) w 0.5 arc secPixel subtense (J band) w ��D radsFried’s parameter r0 0.15 mr0 evaluation wavelength �e 0.5 mHeight of sodium LGS H 90 kmHeight of Rayleigh LGS HR 20 kmDM conjugate altitudes hm 0, 12 kmAO order of correction — 60 � 60Zernike radial order N 6End-to-end efficiency of

opticsa� 0.4

Background intensityb

(V band)zb 37.6 photons m�2 arc sec�2 s�1

Background intensityb

(J band)zb 1385 photons m�2 arc sec�2 s�1

Intensity of m � 0 starb

(V band)z 9.71 � 109 photons m�2 s�1

Intensity of m � 0 starb

(J band)z 5.52 � 109 photons m�2 s�1

Imaging wavelength(V band)

� 0.5 m

Imaging wavelength(J band)

� 1.25 m

aReference 17.bReference 26.


sampling frequencies is 10–1000 Hz. A single simu-lation of one NGS constellation takes of the order of2 s. The majority of this time is spent in optimizingthe sampling frequency. The wavefront error, 2, foreach element in the simulation space is computed for500 NGS constellations. By reseeding the randomnumber generator, every option in the simulationspace is simulated over the same set of 500 NGSconstellations.

The baseline NFIRAOS LGS asterism is shown inFig. 5 and consists of six LGSs: one LGS is on axis,and the remaining five are equally spaced on a ringof a diameter of 70 arc sec. The Rayleigh LGS, whenused, is on axis and at a range of 20 km. The sciencefield consists of 49 points arranged in a square grid ofa linear dimension of 10 arc sec in the baseline designand 30 arc sec in the upgrade path.

For a given field, a single call of the Bahcall–Soneira or Spagna models can generate more NGSsthan there are NGS WFSs in the NFIRAOS designoptions. If there are more NGSs in the field than NGSWFSs, the wavefront error for every combination ofNGSs is evaluated, and the NGS combination thatproduces the smallest wavefront error is chosen.

The median TT error versus the patrol band diam-eter is shown in Fig. 6. When the patrol diameter isless than or equal to 40 arc sec, the median casecorresponds to zero NGSs in the field and the open-loop TT error of 718 nm. The lowest wavefront erroris effectively obtained with a 2 arc min diameter pa-trol field; stars further away from the science fieldeffectively have too little partial correction and suffertoo much anisoplanatism to be useful. We use thisdiameter for the remainder of the simulations.

In this section, the TT errors are reported innanometer rms. It is possible to convert the re-ported errors from nanometers to tilt jitter in mil-liarc seconds by

�mas� �4 � 1000 � 180 � 60 � 60

D�m�. (106)

For a 30 m diameter telescope, 1 mas of TT jitter cor-responds to 36.4 nm of TT error.

The median TT errors for the three WFS options inthe J and V bands are displayed in Table 3 for 0, 5, 10,and 15 electrons of read noise per pixel. The firstthing we note from Table 3 is that the results usingseeing-limited stars in the V band are, in all cases,significantly worse than using partially compensatedstars in the J band in terms of �nm� and, in fact, failto reach the current requirements for the NFIRAOSsky coverage. We therefore eliminate using the Vband stars and sensors and concentrate on the J bandfor the remainder of this paper.

There is a clear hierarchy in the performance of thethree WFS architectures with the auxiliary RayleighLGS the best at all read noise levels, followed by theTTFA and the two TT NGS WFS option, and last, thesingle TTFA NGS WFS. This introduces a cost versusperformance trade-off in the system design with theRayleigh LGS being nontrivially more complex thanthe other two options.

The cumulative density function (CDF), Pr� � ��versus � �nm�, is shown for the three WFS options inthe J band in Fig. 7, with ten electrons of read noise,which is the baseline value. We see from Fig. 7 thatthe auxiliary Rayleigh LGS option always provides alower TT error than the one TTFA � two TT NGS

Fig. 5. FOV for the NFIRAOS showing the sodium LGS (Œ), the2 arc min diameter patrol field for the NGS (dotted circle), a ran-dom constellation of four NGSs (�), and the science evaluationpoints (�) for the 30 arc sec square science field.

Fig. 6. Median TT error (nm) over 500 NGS constellations versusthe patrol field diameter (arc sec) for two TT NGSs and one TTFANGS in the J band.

Table 3. Median TT Errors (nm) for the NFIRAOS for the Three NGSWFS Options with SH Sensors Using Quad-Cell Detectors

Band WFS Option

TT Error(nm)

�e � 0 �e � 5 �e � 10 �e � 15

V band One TTFA � two TTs 371 374 377 386One TTFA 395 398 409 426One TT � Rayleigh 387 388 391 396

J band One TTFA � two TTs 43 54 65 76One TTFA 67 82 103 128One TT � Rayleigh 37 39 42 45


WFSs option, which always produces a lower TT er-ror than one TTFA NGS WFS.

In Table 4, the median TT error is broken downinto wind shake, servo lag, tilt anisoplanatism, andnoise on the NGS WFS measurements for the threeWFS options with ten electrons of read noise perpixel. Here, the median TT error is the error termaveraged over the 20 middle NGS constellationssorted on the total TT error. The median samplingrate is 90 Hz for the two TT NGS WFSs � one TTFANGS WFS option, 90 Hz for the one TTFA NGS WFS,and 140 Hz for the auxiliary Rayleigh option. TheNGS WFS noise is currently the largest term in theerror budget for all options, followed by tilt aniso-planatism, wind shake, and servo lag. For the two TTNGS WFSs � one TTFA NGS WFS and one TTFANGS WFS options, the median sodium tracking erroris 19 nm and 17 nm for the auxiliary Rayleigh LGSoption. For the single TTFA NGS WFS, the medianmagnitude of the TTFA star is 16.3, at a medianoffset of 43 arc sec off axis. For the auxiliary RayleighLGS, the median magnitude is 16.1 at a median offsetof 44 arc sec off axis. For the two TT NGS WFSs � one

TTFA NGS WFS option, the median magnitude of theTTFA star is 16.9 at 48 arc sec off axis, the median ofthe brighter of the two TT stars is at a magnitude of19.0 at 34 arc sec off axis, and the median dimmer TTstar is at a magnitude of 20.1 at 34 arc sec.

Finally, we compare the performance of the SHWFS quad cell with a matched filter for three TT NGSWFSs. We also investigate different pixel widths inconjunction with the matched-filter approach: ��2D,��D, and 3��2D. The median total TT errors for thematched filter and also for the quad-cell as a compar-ison are displayed in Table 5. The matched-filter ap-proach produces lower TT errors than the quad cellfor all pixel sizes and read-noise levels investigated.The optimal pixel size is ��2D for 0 electrons of readnoise per pixel and ��D for 5, 10, and 15 electrons ofread noise. The improvement with the matched filteris not due to the noise term alone. Although thematched-filter approach produces less noise, this canallow for a faster sample rate to reduce wind shakeand servo lag and can allow for a NGS constellationwith less tilt anisoplanatism to be chosen. Using thematched-filter algorithm is contingent upon improve-ments in IR technology to allow for large, low noise,IR detector arrays.

5. Conclusions

In this paper, we have presented the modeling of skycoverage for an ELT, in particular, the TMT facilityAO system NFIRAOS and have hence generated a TTerror budget. From the simulations presented in Sec-tion 4, we conclude that IR detectors are preferable tooptical detectors for the NGS WFS, which is mainlydue to the expected partial correction in the IR, andalso to the higher stellar densities in the IR comparedto the optical. We find that a 2 arc min diameterpatrol field for finding NGS is sufficient.

At least one NGS WFS is required to measure thefocus from the NGS in order to track variations in thesodium-layer altitude, and we find that an additionaltwo TT NGS WFSs significantly improves the TTerror for the NFIRAOS. The best TT estimate, how-ever, is gained by using a Rayleigh LGS in conjunc-tion with a TT NGS and the sodium LGS asterism.We have discarded this option for now, due to theoptical complexity of using the LGS at two differentaltitudes.

Simulation results also indicate a significantimprovement in using a matched-filter approach toestimating TT from the NGS rather than with a

Fig. 7. CDF �Pr� � �� versus � �nm�� from 500 different NGSconstellations in the J band with ten electrons of read noise for theNFIRAOS for one TTFA NGS WFS and two TT NGS WFSs (solidcurve), a single TTFA NGS WFS (dashed curve), and a single TTNGS WFS used in conjunction with a Rayleigh LGS (dotted curve).

Table 4. Median TT Errors (nm) for the NFIRAOS Broken Down intoWind Shake, Noise, Tilt Anisoplanatism, and Servo Lag for the Three,

WFS Options in the J Banda

Error Source

Median TT Error(nm)

Option 1 Option 2 Option 3

Wind shake 26 26 21Servo lag 15 15 14Tilt anisoplanatism 37 54 21NGS WFS noise 44 84 26Total 65 104 42

aOption 1 is one TTFA NGS WFS � two TT NGS WFSs, option2 is one TTFA NGS WFS, and option 3 is one TT NGS WFS � oneRayleigh LGS. There are ten electrons of read noise per pixel.

Table 5. Median TT Errors (nm) for the NFIRAOS with Three TT NGSWFSs for the Quad-Cell and Matched-Filter Approaches

DetectionMethod

PixelSubtense

(rads)

Median TT Error(nm)

�e � 0 �e � 5 �e � 10 �e � 15

Quad cell ��D 49 65 80 93Matched filter ��2D 31 53 70 79Matched filter ��D 32 46 56 66Matched filter 3��2D 33 46 57 68


quad-cell Shack–Hartmann. IR detector developmentwill be necessary to implement this option.

Although this model is largely complete, furtherrefinements will continue to be made when more ac-curate telescope wind-shake and sodium-layer alti-tude tracking models are available and when furtherturbulence data is available from the proposed TMTsites.

The authors acknowledge the support of the TMTpartner institutions. They are the Association ofCanadian Universities for Research in Astronomy(ACURA), the Association of Universities for Re-search in Astronomy (AURA), the California Insti-tute of Technology, and the University of California.This work was also supported by the Canada Foun-dation for Innovation; the Gordon and Betty MooreFoundation; the National Optical Astronomy Obser-vatory, which is operated by AURA under cooperativeagreement with the National Science Foundation; theOntario Ministry of Research and Innovation; theNational Research Council of Canada; and the Na-tional Science Foundation Science and TechnologyCenter for Adaptive Optics, managed by the Univer-sity of California at Santa Cruz under cooperativeagreement AST-9876783.

References1. J. Nelson and G. H. Sanders, “TMT status report,” in Ground-

Based and Airborne Telescopes, L. M. Stepp, ed., Proc. SPIE6267, 745–761 (2006).

2. I. Hook, G. Dalton, and R. Gilmozzi, “Scientific requirementsfor a European ELT,” in Ground-Based and Airborne Tele-scopes, L. M. Stepp, ed., Proc. SPIE 6267, 726–734 (2006).

3. M. W. Johns, “The Giant Magellan Telescope (GMT),” inGround-Based and Airborne Telescopes, L. M. Stepp, ed., Proc.SPIE 6267, 762–776 (2006).

4. T. E. Andersen, A. Ardeberg, J. Beckers, M. Browne, A.Enmark, P. Knutsson, and M. Owner-Petersen, “From Euro50toward a European ELT,” in Ground-Based and Airborne Tele-scopes, L. M. Stepp, ed., Proc. SPIE 6267, 716–725 (2006).

5. H. W. Babcock, “The possibility of compensating astronomicalseeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).

6. D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Advance-ments in Adaptive Optics, Proc. SPIE 5490 (2004).

7. B. L. Ellerbroek and F. R. Rigaut, “Methods for correcting tiltanisoplanatism in laser-guide-star-based multiconjugate adap-tive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).

8. M. Le Louarn and M. Tallon, “Analysis of modes and behaviorof a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A19, 912–925 (2002).

9. B. Femenia, “Tip–tilt reconstruction with a single dim naturalguide star in multiconjugate adaptive optics with laser guidestars,” J. Opt. Soc. Am. A 22, 2719–2729 (2005).

10. D. J. Butler, R. I. Davies, H. Fews, R. M. Redfern, N. Ageorges,W. K. Hackenberg, R.-R. Rohloff, S. Rabien, T. Ott, and S.Hippler, “Sodium layer monitoring at Calar Alto by LIDAR,” inAdaptive Optical Systems Technology, P. L. Wizinowich, ed.,Proc. SPIE 4007, 358–367 (2000).

11. R. M. Clare and B. L. Ellerbroek, “Sky coverage estimates foradaptive optics systems from computations in Zernike space,”J. Opt. Soc. Am. A 23, 418–426 (2006).

12. G. Herriot, P. Hickson, B. L. Ellerbroek, D. A. Anderson, T.Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C.Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, “NFIRAOS:TMT narrow field near-infrared facility adaptive optics,” inAdvances in Adaptive Optics II, B. L. Ellerbroek and D. B.Calia, eds., Proc. SPIE 6272, 228–239 (2006).

13. R. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. A 66, 207–211 (1976).

14. L. Gilles, B. L. Ellerbroek, and J.-P. Véran, “Laser guide starmulti-conjugate adaptive optics performance of the ThirtyMeter Telescope with elongated beacons and matched filter-ing,” in Advances in Adaptive Optics II, B. L. Ellerbroek andD. B. Calia, eds., Proc. SPIE 6272, 1025–1032 (2006).

15. G. A. Tyler and D. L. Fried, “Image-position error associatedwith a quadrant detector,” J. Opt. Soc. Am. 72, 804–808(1982).

16. W. L. Melvin, “A STAP overview,” IEEE Aerosp. Electron.Syst. Mag. 19, 19–35 (2004).

17. B. L. Ellerbroek and D. W. Tyler, “Adaptive optics sky coveragecalculations for the Gemini-North telescope,” Publ. Astron.Soc. Pac. 110, 165–185 (1998).

18. D. L. Fried, “Optical resolution through a randomly inhomog-enous medium for very long and very short exposures,” J. Opt.Soc. Am. A 56, 1376–1379 (1966).

19. J.-P. Véran and G. Herriot, “Woofer-tweeter tip-tilt control forNFIRAOS for TMT,” in Advances in Adaptive Optics II, B. L.Ellerbroek and D. B. Calia, eds., Proc. SPIE 6272, 555–562(2006).

20. B. L. Ellerbroek, “Linear systems modeling of adaptive opticsin the spatial-frequency domain,” J. Opt. Soc. Am. A 22, 310–322 (2005).

21. J. N. Bahcall and R. M. Soneira, “The universe at faint mag-nitudes. I. Models for the galaxy and the predicted starcounts,” Astrophys. J. Suppl. 44, 73–110 (1980).

22. A. Spagna, “Guide star requirements for NGST: deep NIRstarcounts and guide star catalogs,” Space Telescope ScienceInstitute Document NGST-R-OO13B (Space Telescope ScienceInstitute, Baltimore, Md., 2001).

23. D. Kerley, S. Roberts, J. Dunn, N. Stretch, M. Smith, S. Sun,J. Pazder, and J. Fitzsimmons, “Validation and verification ofintegrated model simulations of a thirty meter telescope,”in Optical Modeling and Performance Predictions II, M. A.Kahan, ed., Proc. SPIE 5867, 220–231 (2005).

24. G. Herriot, P. Hickson, B. L. Ellerbroek, J.-P. Véran, C.-Y. She,R. M. Clare, and D. Looze, “Focus errors from tracking sodiumlayer altitude variations with laser guide star adaptive opticsfor the Thirty Meter Telescope,” in Advances in AdaptiveOptics II, B. L. Ellerbroek and D. B. Calia, eds., Proc. SPIE6272, 486–495 (2006).

25. J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin,E. Masciadri, L. Sanchez, and A. Ziad, “1998 Gemini site test-ing campaign: Cerro Pachon and Cerro Tololo,” Gemini Docu-ment RTP-AO-G0094 (Gemini Observatory, Hilo, Hawaii,2000).

26. P. Puxley, “Observing constraints—sky background,” GeminiObservatory, October 2002, http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocSkyBackground.html.


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