The mesospheric sodium layer is located at a meanaltitude of hNa � 90 km and has a mean thickness of�Na � 10 km. As a result, a sodium laser guide star(LGS) will have perspective elongation, and a Shack–Hartmann wavefront sensor (SH-WFS) subapertureimage of such a laser beacon will be elongated. Thedegree of elongation, �Na, increases approximatelyproportionally to the distance between the subaper-ture and the laser launch telescope (LLT), the thick-ness of the layer, and decreases proportionally to theinverse of the square of the profile mean altitude:�Na � rSA�Na�hNa
2 . For the Thirty Meter Telescope1
(TMT), the LLT will be located behind the secondarymirror of the telescope, producing radially elongatedLGS subaperture focal-plane spots. hNa, �Na, and thedetailed structure of the sodium profile PNa�h� allevolve significantly on time scales of seconds tominutes. For edge subapertures of the TMT �rSA �14.5 m�, the average angular size of the sodiumlayer along the radial direction, is of the order of�Na � 3 arc sec � 3 �seeing, which is at least three
times larger than the seeing-limited angular size�seeing � �WFS�r0��WFS� of the transverse laser beaconintensity pattern at the laser focus on the sodiumlayer.
Here we analyze the impact of these radially elon-gated and temporally varying LGS spots on the mea-surement accuracy, the associated wavefront error,and the linear dynamic range of the standard cen-troid algorithm and of a more refined noise-optimalmatched filter spot position estimation algorithm forthe TMT facility adaptive optics (AO) system. Theresults presented are based on a contiguous set of 88lidar sodium profile measurements with temporaland spatial resolutions of 72 s and 24 m, respective-ly.2 For square subapertures of size equal to dSA
� 0.5 m at the primary mirror and integration timesof the order of 1 ms, which correspond to the TMTbaseline AO system design, a 17 W cw sodium laser isanticipated to provide a mean photon return, yieldingof the order of N � 103 photodetected electrons persensing subaperture per integration time.3 This levelof signal is assumed throughout the paper and is therequirement currently imposed upon the TMT LGSfacility.
It is found that the rms spot position estimationerror due to noise is significantly increased at theedge of the TMT aperture due to the impact of LGSelongation, but the effect can be reduced with noise-optimal matched filter processing. This is particu-larly true when CCD readout noise is nonzero. The
The authors are with the Thirty Meter Telescope Project Office,1200 East California Boulevard, Mail Code 102-8, Pasadena, Cal-ifornia 91125. L. Gilles’s e-mail address is [email protected].
Received 21 November 2005; accepted 10 April 2006; posted 17April 2006 (Doc. ID 65931).
wavefront error for the TMT baseline AO system em-ploying 16 � 4 CCD arrays per subaperture is of theorder of 32 nm in the absence of read noise and 45 nmwith 5 electrons rms read noise per pixel per readfor the matched filter algorithm. The additionalroot-sum-square (rss) wavefront error for a centroidalgorithm is of the order of 14 and 55 nm, respec-tively.
In terms of linear dynamic range, the centroid algo-rithm provides 2–3 times more dynamic range thanthe matched filter, but the effect is expected to be smallsince (i) the null point for each LGS WFS subaperturemay be calibrated to account for noncommon pathwavefront aberrations without dynamic range degra-dation, and (ii) the time-varying residual tip–tilt sub-aperture wavefront aberrations due to atmosphericturbulence is expected to be smaller than the dynamicranges in question.
The paper is organized as follows: Section 2 pro-vides an overview of the LGS SH-WFS subaperturespot model. Subsections 2.B and 2.C present the cen-troid and the matched filter spot position estimationalgorithms, respectively. Sample numerical resultsare presented in Section 3. Finally, Section 4 con-cludes the study.
2. Laser Guide Star Shack–Hartmann WavefrontSensor Spot Model and Spot PositionEstimation Algorithms
A. Spot Model
The LGS SH-WFS subaperture spot model developedfor this study was inspired by a similar model proposedby Ellerbroek.4,5 The continuous subaperture averagespot will be denoted i��x, �y�, where �x and �y denotethe angular position coordinates in the subaperturefocal plane along the horizontal and vertical direc-tions. In the angular frequency domain, the subaper-ture spot spectrum will be denoted i�ux, uy�, where ux
and uy denote the angular frequency coordinatesalong the horizontal and vertical directions. Iso-planatic conditions are assumed to approximatelyhold, so that i��x, �y� can be modeled as the convo-lution of the subaperture point-spread function, de-noted PSFSA��x, �y�, with the beacon object, denotedibeacon��x, �y�:
i��x, �y� � PSFSA��x, �y� � ibeacon��x, �y�. (1)
The subaperture PSF is modeled like a short-exposure Kolmogorov turbulence degraded PSF, andthe beacon object as the convolution of the laser beamtransverse cross section at the laser focus on the so-dium layer with a geometric image of the sodiumprofile, denoted iNa��x, �y�, modeling the depth of thesodium layer. Invoking reciprocity, the laser beamtransverse cross section at the laser focus on the so-dium laser is modeled as the LLT aperture PSF re-flected about the origin, denoted PSFLLT ��x, �y�. As forthe sensing subaperture, the LLT PSF is modeledlike a short-exposure Kolmogorov turbulence de-graded PSF. Thus we have
The beacon object is proportional to signal level N,and it is normalized such that the integral of thesubaperture spot over an infinite focal plane is equalto N:
���
�
d�x���
�
d�yi��x, �y� � N. (3)
Invoking the convolution theorem, the angular fre-quency spectrum of the subaperture spot is expressedas a product of the respective spectra:
i�ux, uy� � OTFSA�ux, uy�ibeacon�ux, uy�, (4)
ibeacon�ux, uy� � OTF*LLT�ux, uy�iNa�ux, uy�. (5)
The sensing subaperture and LLT aperture PSFs aremodeled like the inverse Fourier transform of therespective short-exposure Kolmogorov turbulence de-graded OTFs:
where USA�LLT� denotes the subaperture (LLT aper-ture) field amplitude, DOPDSA�LLT�
is the piston tip–tiltremoved Kolmogorov optical path difference (OPD)structure function, �x, y� is the spatial coordinate ofa point in the subaperture (LLT aperture), andOTFSA�LLT�
DL denotes the diffraction limited OTFs.DOPDSA�LLT�
is not a shift-invariant function and mustbe evaluated numerically as described in Appen-dix A. USA�LLT� is expressed as follows:
ref� denotes the angular coordinates ofthe subaperture focal plane null point (which will benonzero to account for WFS noncommon path aber-rations), and ��x
in, �yin� denotes the input subaperture
Zernike tilt, which is estimated by the centroid andmatched filter spot position estimation algorithmsdetailed in Subsections 2.B and 2.C. The 1�e2 laserbeam intensity diameter is equal to 2 2�laser. The useof short-exposure Kolmogorov turbulence degradedOTFs is based on the assumption that (i) the LGSSH-WFS and laser pointing loop operate in closedloop and are perfectly tip–tilt compensated, and (ii)dSA and dLLT are small so that the Kolmogorov statis-tics approximately hold.
Finally, expressing the Cartesian coordinates of agiven subaperture in terms of its polar coordinates,xSA � rSA cos��SA�, ySA � rSA sin��SA�, it is convenientto introduce locally rotated angular coordinates���, ���, related to ��x, �y� by a rotation:
Note that the � axis points thus from the LLT to thesubaperture of interest, and the � axis is orthogonalto that direction (see Fig. 1). These two directions willbe referred to hereafter as radial and azimuthal. Thegeometrical image of the sodium profile PNa�h� at adistance rSA from the LLT is then modeled as follows:
iNa��x, �y� �1
rSA ����PNa�h���� hNa�, (16)
h���� � h hLGS
2 ��
rSA, (17)
h � hLGS � hNa, (18)
where �� and �� are given by Eq. (14), hNa is thecentroid of the sodium profile, and hLGS denotes theLGS SH-WFS focus altitude.
This study addresses the impacts of the sodiumlayer structure and structural variability but doesnot address the impacts of a temporal lag on refocus-
ing the laser to the correct mean sodium layer alti-tude, hence we assume hereafter that h � hLGS �hNa � 0. In the angular frequency domain, the sodiumprofile image spectrum is given by the following ex-pression:
iNa�ux, uy� �1
hLGS2 PNaf �
xSAux ySAuy
hLGS2 �
� exp�j2�fhLGS�. (19)
Higher-order effects generated by the 3D LGS thatare not included in the model are speckle noise anddepth of focus.
Pixel intensities averaged over the Poisson photonarrival statistics and over the normally distributedreadout noise are obtained by integrating the contin-uous LGS SH-WFS subaperture spot i��x, �y� overeach CCD pixel bin B�k���x, �y�. In vector notation, wehave
I�avg ����
�
d�x���
�
d�yi��x, �y�B� ��x, �y� (20)
����
�
dux���
�
duyi�ux, uy� �B*�ux, uy�, (21)
where the last equality follows from the fact that theFourier transform is a unitary transformation. Pixelbins are modeled as square boxes of angular subtense�pix, with radial and azimuthal coordinate vectors ���
and ��� in the locally rotated frame, blurred by aGaussian response function modeling charge diffu-sion:
Fig. 1. Illustration of subaperture focal-plane radial geometryCCD arrays.
� Npix� pixels is expressed as the sum of the individual
pixel bins B�k���x, �y�. Integrating the subaperture spotover the total field of view (FOV) of the detector arrayyields the average signal level multiplied by a leak-age factor:
1�TI�avg ����
�
d�x���
�
d�yi��x, �y�1�TB� ��x, �y� � �N,
(26)
where � � 1 is the leakage factor, i.e., the energy lossfactor due to photons falling outside the CCD array.The subaperture signal-to-noise ratio (SNR) is thenequal to
SNR �1�TI�avg
Tr�C���
�N
�N Npix� Npix
� �e2. (27)
B. Centroid Algorithm
The centroid algorithm has been extensively used incombination with 2 � 2 pixel arrays known as quad-rant detectors or quad cells. A detailed analysis of thealgorithm’s noise properties was presented in thiscontext by Tyler and Fried6 in their seminal 1982paper. This material is briefly reviewed below withinthe framework of an arbitrary CCD array geometrywith Npix
� � Npix� pixels and an arbitrary subaperture
focal plane null point.The centroid spot position estimate is given by the
following expressions:
��(�)in � �� �(�)
T (�I� � I�0avg), (28)
I�0avg � I�avg���in � 0��, (29)
�� ���� ���(�)
B
1�TI�0avg
������, (30)
� �1�TI�0
avg
1�TI�, (31)
��(�)B
1�TI�0avg
�d���(�)T ��avgI�avg � I�0
avg�d�����
in ���(�)
in �0��1
�1
g� �(�)T ������
, (32)
g� ���� ��I�avg
���(�)in �
��(�)in �0
. (33)
�����B in Eq. (30) and g� ���� in Eq. (33) denote, respec-
tively, the centroid gain and the radial and azimuthalslopes of the average pixel intensity transfer curvesat null. The latter can be estimated in practice bycontinually dithering the laser beacon on the sky andcan be updated on slow time scales of a few seconds.
It should be pointed out that a spot position es-timation bias occurs in the radial direction if thecentroid algorithm is not updated rapidly enough totrack the variations in the shape of the sodiumprofile. Mathematically, this bias is expressed asfollows:
�bias�t; t � � �� �(�)T �t����t; t �I�0
avg�t �� I�0
avg�t��, (34)
��t; t � �1�TI�0
avg�t�1�TI�0
avg�t �. (35)
Note that if I�0avg�t � is simply proportional to
I�0avg�t�, the spot position estimation bias is equal to
zero. A rough estimate of the telescope full-aperturewavefront error due to a nonzero bias can be obtainedby reconstructing the wavefront at the subapertureresolution and summing up radially the biases foreach subaperture from the LLT to the edge of theaperture. In integral notation, this is expressed asfollows:
�bias�r� ��0
r
dr��bias�r��. (36)
The piston removed and piston-focus removed wave-fronts can then be computed by using the usual for-mulas:
where Z1(4) denote the Zernike piston and focusmodes. The rms error due to the biases is finallyexpressed as
�1(4)2 �
�0
2�
d��0
D�2
rdr��bias�4� �r��2
�D2�4. (40)
Due to photon and readout noise, the centroid es-timate in Eq. (28) is a random variable, whose vari-ance at the null point, known as the centroid noisepropagation, is equal to the following expression:
���(�)
2 � ��(�)B
1�TI�0avg�2
var����(�)T ��I�0
avg ��� � I�0avg��,
(41)
� �1
1 �� 1 � �, (42)
� �1�T�
1�TI�0avg
. (43)
After a little algebra, the following expressions areobtained:
���(�)
2 � ���(�)B �2 �����
SNR2���in � 0��
q�(�)2
SNR2���in � 0��
� 2q�(�)
2
1�TI�0avg�, (44)
q���� � ���(�)T
I�0avg
1�TI�0avg
, (45)
����� �Tr����������(�)
T Cmod�Tr�Cmod�
��k ���(�)
(k) �2�I0avg�k� �e
2��k �I�0
avg�k� �e2�
, (46)
Cmod � C����in � 0��. (47)
Note that for a quadrant detector, ����� is simply equalto a quarter of the pixel area subtense.
C. Matched Filter Algorithm
We define a matched filter algorithm by the followingnoise-weighted least-squares optimization problem:
���in, ��
in, N� � arg min���in, ��
in, N�J���
in,��in, N�, (48)
J���in, ��
in, N� � y�TCmod�1 y�, (49)
y� � I� � I�0avg g� ���
in g����in
I�0
avg
N N�, (50)
Cmod � C����in�0, ��
in � 0, N � 0�, (51)
where g� ���� are given by Eq. (33).The solution for ��
in and ��in is given by the following
expressions:
��(�)in � �� �(�)
T �I� � I�0avg�, (52)
�� � � ���2 Cmod
�1 �g� � � �I�0avg�, (53)
� �g� �
TCmod�1 I�0
avg
I�0avgTCmod
�1 I�0avg
, (54)
�� � � ���
2 Cmod�1 g��. (55)
The matched filter noise propagation coefficients areexpressed as follows:
���2 �
1
g� �TCmod
�1 �g� � � �I�0avg�
, (56)
���
2 �1
g��TCmod
�1 g��
. (57)
Equations (52)–(57) follow from the following symme-tries of vectors g� �, g��, and I�0
avg when displayed asNpix
� � Npix� arrays:
(i) Array �g� �� is symmetric along the � direction,i.e., its rows are identical.
(ii) Array �g��� is antisymmetric along the � direc-tion, i.e., g��
T1� � 0. In particular, g��Tg� � � 0.
(iii) Array �I�0avg� is symmetric along the � direc-
tion, i.e., its rows are identical. In particular,g��
TCmod�1 I�0
avg � g��TCmod
�1 g� � � 0.
Finally, as for the centroid algorithm, a spot posi-tion estimation bias occurs in the radial direction ifthe matched filter algorithm is not updated rapidly
enough to track the variations in the shape ofthe sodium profile. Note that the parameter � inEq. (53) ensures that �bias � 0 and N � N if I� �
�N N��NI�0avg.
3. Simulation Results
Figure 2 displays the mean sodium profile obtainedby averaging and centering 88 contiguous frames oflidar measurements with a spatial resolution equal to24 m (Ref. 2) as well as a sample sodium profileframe.
Figure 3 displays the Nyquist sampled short-exposure subaperture and LLT aperture PSFs. Thesubaperture size was taken equal to dSA � 0.5 m(order 60 � 60 wavefront sensor), the LLT diameterequal to dLLT � dSA � 0.5 m, and the 1�e2 Gaussianlaser beam intensity diameter equal to 0.6dLLT� 0.3 m. The Fried parameter was chosen equal tor0��0 � 500 nm� � 0.15 m. Note that theLLT Strehl ratio (SR) is in excellent agreement withMaréchal’s approximation, SR��WFS� � exp���2�,where �2 � 0.134�dLLT�r0��WFS��5�3 is the piston tip–tilt removed Kolmogorov phase variance in radianssquared. PSFs were computed in the Fourier domainby using 64 � 64 FFT grids.
Figure 4 displays the Nyquist sampled normalizedaverage beacon radial and the azimuthal cross sec-tions as seen from a subaperture 1 and 14.5 m awayfrom the LLT, together with the total subaperture
spot obtained by convolving the beacon with theshort-exposure subaperture PSF. The FWHM of theradial and azimuthal cross sections of the edge sub-
Fig. 2. Left panel, mean sodium profile obtained by averagingand centering 88 contiguous frames of lidar measurements of thesodium layer with spatial resolution equal to 24 m (Ref. 2). Rightpanel, sample sodium profile frame.
Fig. 3. Left panel, Nyquist sampled subaperture short-exposurePSF. Right panel, Nyquist sampled LLT aperture short-exposurePSF. The subaperture size was taken equal to dSA � 0.5 m (order60 � 60 wavefront sensor), the LLT diameter equal to dLLT �
dSA � 0.5 m, and the 1�e2 Gaussian laser beam diameter equal to0.6 dLLT � 0.3 m. These quantities were computed in the Fourierdomain by using a 32 � 32 subaperture grid embedded into a64 � 64 FFT grid. The Fried parameter is r0��0 � 500 nm� �
0.15 m and the turbulence outer scale is infinite.
Fig. 4. Left panels, Nyquist sampled normalized average beaconradial and azimuthal cross sections as seen from a subaperture 1and 14.5 m away from the LLT. Right panels, total subaperturespot obtained by convolving the beacon with the short-exposuresubaperture PSF at a distance of 1 and 14.5 m.
Fig. 5. Radial and azimuthal photon and readout noise propaga-tion levels associated with the matched filter (left panels) andcentroid spot (right panels) position estimators, as a function of thesubaperture-to-LLT separation. These curves are for the mediansodium profile displayed in Fig. 2. The beacon brightness has beenscaled to provide a mean signal level equal to N � 103 photode-tected electrons per subaperture per integration time, and thecases of �e � 0 (top) and �e � 5 (bottom) electrons rms readout noiseare compared for a 16 � 4 subaperture focal-plane CCD pixel arraywith �pix � 0.5 arc sec pixel subtense and �blur � �pix�4 pixel blur-ring due to charge diffusion. The corresponding SNRs are of theorder of 31 and 19, respectively. Blue and red curves refer to thenull point set, respectively, at the origin (center) of the subaperturefocal plane and at a null position shifted by half a pixel in both theradial and azimuthal directions (as might be the case with samplenoncommon path wavefront errors). Such a null point offset has noimpact on the noise properties of the algorithms.
aperture spot is of the order of 3.5 and 0.8 arc sec,respectively.
Figure 5 shows the radial and azimuthal rms noisepropagation levels for the matched filter and centroidspot position estimators, as a function of subaperture-to-LLT separation. These curves are for the mediansodium profile displayed in Fig. 2. The beacon bright-ness has been scaled to provide a mean signal level ofN � 103 photodetected electrons per subaperture perintegration time, which is the requirement currentlyimposed upon the TMT LGS facility, and the casesof �e � 0 and �e � 5 electrons rms readout noise arecompared for a 16 � 4 subaperture focal-plane CCDpixel array with �pix � 0.5 arc sec pixel subtense and�blur � �pix�4 pixel blurring modeling charge diffusion.The corresponding SNRs are of the order of 31 and 19,respectively. It is seen that the spot position estima-tion error due to noise is significantly increased at theedge of the TMT aperture due to the impact of LGSelongation, but the effect can be reduced through theuse of the noise-optimal matched filter. This is par-ticularly true when the detector readout noise is non-zero. It is also seen that shifting the null point fromthe origin (center) of the subaperture focal plane tohalf a pixel in both the radial and the azimuthaldirections (as might be the case with sample noncom-mon path wavefront errors) only marginally degradesthe noise properties of the algorithms. These resultsare summarized for a central and an edge subaper-
ture in Table 1. The wavefront error due to the noisehas been computed for the TMT facility AO systemand is displayed in Table 2. The system consists of 5LGSs in a 35 arc sec radius and 1 LGS on axis, order60 � 60 sensing and correction (0.5 m subaperturesand 0.5 m actuator pitch), 16 � 4 CCD arrays persubaperture, 1 tip–tilt focus natural guide star WFSon axis, and two deformable mirrors conjugate toground and 12 km, respectively. Wavefront control isdone by using a double-pole integrator with a gain of0.5 operating in pseudo-open loop.7 The error budgethas been computed by subtracting in quadraturenoise-free from noisy closed-loop Monte Carlo simu-lation results by using the same mean sodium profileas in Fig. 5. Noise-free simulations were run with aminimum variance wavefront reconstructor incorpo-rating 15 mas subaperture regularization noise. Thewavefront error is of the order of 32 nm in the ab-sence of read noise and 45 nm with 5 electrons rmsread noise per pixel per read for the matched filter
Fig. 6. Average spot position estimation error curves, ����� �
�����in , for a central and an edge subaperture as a function of input tilt
level when the null point of the subaperture focal plane is at theorigin. The curves for the matched filter algorithm (left panels) arefor a mean signal level of 1000 photodetected electrons per subap-erture (0.5 and 14.5 m, top and bottom, respectively, for left andright panels) and per integration time and a read noise of either 0or 5 e rms. For the centroid algorithm (right panels), the curves areindependent of signal and read noise levels since the algorithmdoes not incorporate statistical prior information.
Table 1. Radial, Azimuthal, and rss Photon and Readout Noise Propagation Levelsa
aThe levels are read at rSA � 0.5 m and rSA � 14.5 m for the centroid and matched filter spot position estimators operating on the sameCCD array and mean signal level as in Fig. 5.
Table 2. Wavefront Error for the TMT Facility AO Systema
Spot PositionEstimationAlgorithm
Read Noise perPixel per Read
(electrons)
Wavefront Error(nm) due to
LGS WFS Noise
On Axis10 arc sec
FOV Average
Centroid 0 35 325 71 67
Matched Filter 0 32 305 45 42
aThe error is due to LGS WFS noise for the same signal level,subaperture, CCD geometry, and mean sodium profile as in Fig. 5.The error budget was obtained by subtracting in quadrature noise-free from noisy closed-loop Monte Carlo simulation results. Noise-free simulations were run with a minimum variance wavefrontreconstructor incorporating 15 mas subaperture regularizationnoise.
algorithm. The additional rss wavefront error for acentroid algorithm is of the order of 14 and 55 nm,respectively.
Figures 6 and 7 display the average spot positionestimation error, ����� � �����
in , for a central and an edgesubaperture as a function of input subaperture tiltlevel when the null point is set, respectively, at theorigin of the subaperture focal plane and at half apixel in both the radial and the azimuthal directions.The curves for the matched filter algorithm are for amean signal level of 1000 photodetected electrons persubaperture and per integration time and a readnoise of either 0 or 5 electrons rms. For the centroidalgorithm, the curves are independent of signal andread noise levels since the algorithm does not incor-porate statistical prior information. The linear dy-namic range of the matched filter algorithm isapproximately from �100 to 100 mas, i.e., approx-imately from ����2 to ���2, where �� denotes therms angle of arrival fluctuations over a subapertureof size dSA, i.e., �� � PVOPD�dSA with PVOPD ����2��4�a2
2�1�2 and �a22� � 0.448�dSA�r0�5�3. Note that
these are open-loop results for a single LGS WFSsubaperture. The resulting wavefront error for aclosed-loop AO system is expected to be small andwill be evaluated by using a full wave optics MonteCarlo simulation. The centroid algorithm provides2–3 times more dynamic range, but the effect isexpected to be small. Indeed, shifting the null pointto half a pixel in both the radial and the azimuthaldirections has no impact on these average spot po-sition estimation error curves. The null point foreach LGS WFS subaperture may thus be calibratedto account for noncommon path wavefront aberra-tions without dynamic range degradation. More-over, the time-varying residual subaperturewavefront tip–tilt aberrations due to atmosphericturbulence will be smaller than the dynamic rangesquoted above.
Finally, Fig. 8 illustrates sample average spot po-sition estimation error curves for the centroid and
matched filter algorithms with 72 s (i.e., 1 frame)update latency. The azimuthal curves are identicalfor all pairs of profiles for the centroid algorithm as aconsequence of the symmetry properties of the algo-rithm. The rms bias is of the order of 10 mas atrSA � 14.5 m for both algorithms. The full-aperturewavefront error corresponding to this rms bias asgiven by Eq. (40) is approximately equal to 92 nm.Most of this wavefront error is a focus error. Indeed,the focus removed wavefront error is approximatelyequal to 12 nm only. Here again, a full wave opticsMonte Carlo simulation is required to quantify moreprecisely the associated wavefront error, and this willbe the subject of a future publication.
4. Conclusion
Sodium LGS SH-WFS spot elongation is a significantchallenge for future extremely large telescopes suchas the TMT. The LGS angular spot size along theelongation direction at the edge of the TMT exceedsthree times the angular size of the seeing-limitedtransverse laser beacon intensity at the laser focus onthe sodium layer. Possible approaches to defeat thiseffect include (i) radial-format CCDs8 combined witha noise-optimal spot position estimation algorithm4,5
and (ii) dynamic refocusing. We discussedthe first approach. By using a contiguous set of lidarmeasurements of the sodium profile, the performanceof a standard centroid and a more refined noise-optimal matched filter spot position estimation algo-rithm were analyzed and compared for a nominalmean signal level equal to 1000 photodetected elec-trons per subaperture per integration time, as afunction of subaperture to laser launch telescope sep-
Fig. 7. Same as Fig. 6 but when the null point of the subaperturefocal plane is at half a pixel in both radial and azimuthal direc-tions.
Fig. 8. Average spot position estimation error curves, �����
� �����in , for a central and an edge subaperture (0.5 and 14.5 m, left
and right panels, respectively, for 1-frame latency) for the cen-troid (top panels) and matched filter (bottom) algorithms as afunction of input tilt level when the null point of the subaperturefocal plane is at the origin and the algorithms have 72 s (i.e., 1frame) update latency. Different curves correspond to the 87 dif-ferent pairs of contiguous sodium profile frames. Azimuthal curvesare identical for all pairs of profiles for the centroid algorithm as aconsequence of the symmetry properties of the algorithm.
aration distance and CCD pixel readout noise. Bothalgorithms were compared in terms of their averagespot position estimation error due to noise, theirwavefront error budget for the TMT facility adaptiveoptics system, their linear dynamic range, and theirbias when detuned from the current sodium profile.
Appendix A
Denoting the subaperture (LLT aperture) piston tip–tilt modal matrix as MSA�LLT�, the projector onto theorthogonal complement to its range space is given bythe following expressions:
PSA�LLT� � I � MSA�LLT�MSA(LLT)† , (A1)
MSA(LLT)† � �MSA(LLT)
T USA�LLT�MSA�LLT���1
� MSA(LLT)T USA�LLT�. (A2)
Piston tip–tilt removed subaperture (LLT aperture)OPDs are then given by
OPD→
SA�LLT� � PSA�LLT�OPD→
SA�LLT�. (A3)
The discrete structure function associated with theseOPDs is given by the following expression:
�DOPDSA�LLT��kl� ��OPDSA(LLT)
(k) � OPDSA(LLT)(l) �2� (A4)
��COPDSA�LLT��kk �COPDSA�LLT��ll
� 2�COPDSA�LLT��kl, (A5)
where the covariance matrix elements are given by
�COPDSA�LLT��kl� �
12 �PSA�LLT�DOPDSA�LLT�
PSA(LLT)T �kl
,
(A6)
and DOPDSA�LLT�denotes the Kolmogorov shift-invariant
OPD structure function matrix, whose elements areequal to
�DOPDSA�LLT��kl� ��OPDSA(LLT)
(k) �OPDSA(LLT)(l) �2� (A7)
�6.88�x��k� � x��l��
r0��0� �5�3 �0
2��2
. (A8)
The 2D correlation integral defining the short-exposure OTFs given in Eq. (7) is then computed forall spatial shifts ��ux, �uy� from Eq. (A4).
The authors thank the Purple Crow Lidar team fromthe University of Western Ontaria, Canada, for mak-ing their sodium layer measurements available to us.The authors also thank Glen Herriot and Jean-PierreVéran from the Herzberg Institute of Astronomy, Can-ada, for fruitful discussions. The authors acknowledgethe support of the TMT partner institutions includingthe Association of Canadian Universities for Researchin Astronomy (ACURA), the Association of Universi-ties for Research in Astronomy (AURA), the CaliforniaInstitute of Technology, and the University of Califor-nia. This work was also supported by the CanadaFoundation for Innovation, the Gordon and BettyMoore Foundation, the National Optical AstronomyObservatory, which is operated by AURA under coop-erative agreement with the National Science Founda-tion, the Ontario Ministry of Research and Innovation,and the National Research Council of Canada.
References and Notes1. B. L. Ellerbroek, M. Britton, R. Dekany, D. Gavel, G. Herriot, B.
Macintosh, and J. Stoesz, “Adaptive Optics for the Thirty MeterTelescope,” in Astronomical Adaptive Optics Systems and Ap-plications II, R. K. Tyson and M. Lloyd-Hart, eds., Proc. SPIE5903, 20–31 (2005).
2. P. S. Argall, R. J. Sica, O. Vassiliev, and M. M. Mwangi, “Lidarmeasurements taken with a large-aperture liquid mirror:sodium resonance-fluorescence system,” Appl. Opt. 39, 2393–2399 (2000).
3. C. d’Orgeville, F. Rigaut, and B. L. Ellerbroek, “LGS AO photonreturn simulations and laser requirements for the GeminiLGS AO program,” Gemini Observatory preprint #55, availableonline at www.gemini.edu/documentation/webdocs/preprints/gpre55.pdf.
4. B. L. Ellerbroek and G. M. Cochran, “Wave optics propagationcode for multiconjugate adaptive optics,” in Adaptive OpticsSystems and Technology II, R. K. Tyson, D. Donaccini, and M. C.Roggemann, eds., Proc. SPIE 4494, 104–120 (2002).
5. B. L. Ellerbroek, “Wavefront reconstruction algorithms andsimulation results for multiconjugate adaptive optics on gianttelescopes,” in Second Backaskog Workshop on Extremely LargeTelescopes, A. L. Ardeberg and T. Andersen, eds., Proc. SPIE5382, 478–489 (2004).
6. G. A. Tyler and D. L. Fried, “Image-position error associatedwith a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
7. L. Gilles, “Closed-loop stability and performance analysis ofleast-squares and minimum-variance control algorithms formulticonjugate adaptive optics,” Appl. Opt. 44, 993–1002(2005).
8. J. W. Beletic, “Follow the yellow-orange rabbit: a CCD opti-mized for wavefront sensing a pulsed sodium laser guide star,”in Optical and Infrared Detectors for Astronomy, J. D. Garnettand J. W. Beletic, eds., Proc. SPIE 5499, 302–309 (2004).
