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Feb 1, 2011 - Soc. Pac. 116, 941–951. (2004). 7. M. Le Louarn and N. Hubin, “Wide-field adaptive optics for deep-field spectroscopy in the visible,” Mon. Not.
Laser guide star wavefront sensing for ground-layer adaptive optics on extremely large telescopes Richard M. Clare,* Miska Le Louarn, and Clementine Béchet European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany *Corresponding author: [email protected] Received 31 August 2010; revised 26 November 2010; accepted 28 November 2010; posted 1 December 2010 (Doc. ID 134273); published 27 January 2011

We propose ground-layer adaptive optics (GLAO) to improve the seeing on the 42 m European Extremely Large Telescope. Shack–Hartmann wavefront sensors (WFSs) with laser guide stars (LGSs) will experience significant spot elongation due to off-axis observation. This spot elongation influences the design of the laser launch location, laser power, WFS detector, and centroiding algorithm for LGS GLAO on an extremely large telescope. We show, using end-to-end numerical simulations, that with a noise-weighted matrix-vector-multiply reconstructor, the performance in terms of 50% ensquared energy (EE) of the side and central launch of the lasers is equivalent, the matched filter and weighted center of gravity centroiding algorithms are the most promising, and approximately 10 × 10 undersampled pixels are optimal. Significant improvement in the 50% EE can be observed with a few tens of photons/subaperture/frame, and no significant gain is seen by adding more than 200 photons/subaperture/frame. The LGS GLAO is not particularly sensitive to the sodium profile present in the mesosphere nor to a short-timescale (less than 100 s) evolution of the sodium profile. The performance of LGS GLAO is, however, sensitive to the atmospheric turbulence profile. © 2011 Optical Society of America OCIS codes: 010.1080, 010.7350.

1. Introduction

Currently, a number of extremely large telescopes (ELTs) are in the design and development phases around the world: the Thirty Meter Telescope [1], the Giant Magellan Telescope [2], and the European Extremely Large Telescope (EELT) [3]. Each of these telescopes have planned adaptive optics (AO) [4] systems to overcome the degradatory effect of the turbulent atmosphere. A variety of different types of astronomical AO system exist, in operational or experimental form, in order to meet different science goals [5]. One such AO system is ground-layer AO (GLAO) [6,7], which seeks to provide a modest correction (not diffraction limited, but a reduction in the full width at halfmaximum (FWHM) of 0:1 arcsec or more at optical and near-infrared wavelengths [8]) of the science targets over a large field of view (FOV) (of the order of 0003-6935/11/040473-11$15.00/0 © 2011 Optical Society of America

1–10 arcmin) with one deformable mirror (DM). The principle of GLAO is only to compensate for lowaltitude turbulence (the ground layer), where the majority of atmospheric turbulence lies [6]. While GLAO can operate with natural guide stars (NGSs) for the wavefront sensors (WFS), the scarcity of sufficient bright stars close to the scientific object(s) of interest limits the sky coverage of an NGS GLAO system on an ELT. In order to overcome the sky coverage problem, AO systems generate laser guide star(s) [LGS(s)] [9] from which the WFS measurements can be taken. However, the use of LGS introduces a number of problems for the WFS, including the cone effect, tilt indetermination, and spot elongation. The cone effect [10] arises because the volume of turbulence seen by the LGS WFS is a cone, whereas the corresponding volume for the science object is a cylinder. The cone effect can be reduced on ELTs by using multiple LGSs [5]. The tilt indetermination problem, which is caused by the laser beam being equally deflected 1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS

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on the uplink and downlink paths, can be overcome by using a NGS to measure the tip–tilt modes. For a Shack–Hartmann (SH) WFS, the subapertures that are off axis with respect to the laser launch see an elongated source due to the off-axis observation. This elongation β is given approximately by [11] β¼

cosðζÞbt ; h2

ð1Þ

where ζ is the zenith angle, b is the baseline distance between the launch telescope and a given subaperture, t is the thickness of the sodium layer, and h is the mean height of the sodium layer above the telescope. Therefore, this elongation is more significant for ELTs due to the increased diameter of the telescope and, hence, the off-axis distance b. Example elongation patterns for the EELT with central launching and side launching of the lasers are shown in Figs. 1(a) and 1(b), respectively. The maximum elongation is less for the central launch of the LGS, but the side launch with different launch locations offers a diversity of spot elongations in the different WFS. From a study of the aberrations due to spot elongation, the purpose of this paper is to evaluate a number of design trade-offs relating to the WFS with a SH for LGS GLAO on an ELT. In particular, it is necessary to decide whether to launch the lasers from the side or center of the primary mirror, as shown in Fig. 1. Also, the required LGS photon flux for LGS GLAO needs to be estimated in order to decide the required laser power. Similarly, the required number of detector pixels (and pixel scale) and the acceptable level of read noise need to be calculated to design the LGS WFS detector. The optimal temporal sampling frequency needs to be estimated in order to specify the real-time computer (RTC). The centroiding algorithm also needs to be chosen in order to specify the RTC. As well as the aforementioned design trade-offs, sensitivity analysis of the LGS GLAO system on an ELT with respect to the atmospheric profile, the

Fig. 1. Example spot elongation patterns for the 42 m EELT for (a) central launch of four lasers from behind the central obscuration and (b) side launch of four lasers from different positions around the edge of the primary mirror. There are only 11 × 11 subapertures here as an illustration. The crosses denote the four laser launch locations in each case. 474

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sodium profile, and the evolution of the sodium profile is undertaken. These design trade-offs and sensitivity analysis to environmental conditions are computed using the ESO end-to-end simulation tool OCTOPUS [12]. The remainder of this paper is structured as follows: in Section 2 the wavefront sensing and wavefront reconstruction algorithms are defined. In Section 3 the LGS GLAO parameters and environmental conditions are explained. The simulation results are presented in Section 4. Finally, conclusions are drawn in Section 5. 2. Mathematical Preliminaries A. Wavefront Sensing

In this paper, we consider four algorithms for estimating the centroid d ¼ ðdx ; dy Þ of the noisy elongated LGS WFS spot intensity Iðx; yÞ: center of gravity (COG) with thresholding, weighted COG (WCOG), correlation (CORR), and matched filter (MF). Rðx; yÞ is the reference image used in the centroiding algorithm. We also investigate the quad cell (QC), a special case of the COG. Comparisons of these algorithms have been published previously in [13,14]. For brevity, the centroiding equations are only given for the x direction; the y direction is analogous. 1. The COG algorithm we use first subtracts a threshold T from the image to mitigate against read noise, and the negative pixels are then zeroed:  I t ðx; yÞ ¼

Iðx; yÞ − T 0

Iðx; yÞ − T > 0 : otherwise

ð2Þ

The thresholded image, I t ðx; yÞ, is then centroided by [14] dx ¼

ΣxIt ðx; yÞ ; ΣI t ðx; yÞ

ð3Þ

where x are the pixel positions in the x axis. Each subaperture image is thresholded by a fraction of the maximum pixel intensity in that subaperture. The absolute maximum pixel intensity changes significantly between subapertures due to the differing spot elongation for different subapertures. This thresholding scheme is the same as the radial thresholding scheme used in [15]. A special case of the COG algorithm is the QC, where, because the LGS images are poorly sampled by the 2 × 2 pixels, the relationship between the centroid and the pixel intensities, I row;col , is given by d x ¼ gx

I 11 þ I 12 − I 21 − I 22 ; I 11 þ I 12 þ I 21 þ I 22

where the QC gains, g ¼ ðgx ; gy Þ, are

ð4Þ

gx ¼

rffiffiffi π θ ; 2 x

ð5Þ

and θ ¼ ðθx ; θy Þ is the spot FWHM in the direction of centroiding. This equation for the QC gains is derived for a Gaussian spot, although we use it for both the Gaussian and the non-Gaussian sodium profiles in this paper. The QC gains are calculated in x and y from the elongation and orientation of each spot, as is done for the Keck LGS system [16]. For both the COG and QC, the reference centroids dref ¼ ðdx;ref ; dy;ref Þ are then subtracted from ðdx ; dy Þ. The reference centroids for the COG and QC are dependent on the sodium profile, and so they evolve as the sodium layer evolves [11]. 2. The second algorithm investigated, WCOG, is a variation on the COG, where the image is weighted by the reference image Rðx; yÞ in order to reduce the effect of detector read noise. We utilize the mean image in each subaperture as the reference image Rðx; yÞ. The centroid equation for WCOG is, therefore, [14] dx ¼ g

ΣxRðx; yÞIðx; yÞ ; ΣRðx; yÞIðx; yÞ

ð6Þ

where g is a gain relating to the relative size of the weighting function and image [14]. This gain is also a function of the structure of the sodium profile [13]. If the weighting function and image have the same FWHM and are symmetric, the gain g ¼ 2. The reference centroids are estimated by taking the WCOG of the reference image with itself: dx;ref

ΣxRðx; yÞRðx; yÞ ; ¼g ΣRðx; yÞRðx; yÞ

ðdx ; dy Þ ¼ ðQT M −1 QÞ−1 QT M −1 ðIðx; yÞ − Rðx; yÞÞ; ð9Þ where Rðx; yÞ is the mean image. M, the noise covariance, is the sum of the shot noise contribution and read noise, σ r : M ¼ diagfRðx; yÞ þ σ 2r g;

ð7Þ

where, here again, the gain g ¼ 2 if Rðx; yÞ is symmetric. No thresholding of Iðx; yÞ is performed for WCOG. In this paper, as per [13], we assume the sodium profiles are symmetric, even when the simulated profiles are not symmetric, and use a gain g ¼ 2. 3. The CORR algorithm that we employ in this paper is defined in [14,17]. The cross correlation, Cðx; yÞ, of the noisy image Iðx; yÞ and the reference image Rðx; yÞ are given by Cðx; yÞ ¼ F −1 f½F fRðx; yÞg F fIðx; yÞgg;

rithm on Cðx; yÞ with an adaptable threshold depending on the noise level. The disadvantage of parabola fitting is that only a few pixels around the peak are used, and, consequently, any information contained in the wings of Cðx; yÞ is ignored [14]. For CORR, the FT product of Eq. (8) can be zero padded to increase the resolution of Cðx; yÞ. Initial simulations showed that an oversampling factor of 4 was sufficient; there was little improvement in the error variance in oversampling by more than this, while the real-time computation is slower, the larger the oversampling. Using the COG algorithm to find the maximum of Cðx; yÞ has the additional problem of setting the optimum threshold. We employ radial thresholding [15] on Cðx; yÞ, and we optimize the radial threshold for each photon level. No thresholding is done to the images Iðx; yÞ. The CORR algorithm gives the centroid relative to the reference image, so the reference centroid is not subtracted. The CORR is the most computationally intensive of the four algorithms due to the FFTs, so it is the least appealing from the perspective of the design of the RTC. 4. The MF, or noise-weighted least-squares algorithm, is defined in [18]. The centroid is given by

ð8Þ

where F is the Fourier transform (FT). In the simulation code, the Fourier transforms are computed using the fast FT (FFT) algorithm. As with WCOG, we use the mean image in each subaperture as the reference image Rðx; yÞ. The estimate of the centroid is then given by the position of the maximum of Cðx; yÞ. A number of possibilities exist to find the position of the maximum of Cðx; yÞ, such as parabola and Gaussian fitting [17], but we follow the example of [14] and use the COG algo-

ð10Þ

where diag indicates that the values are placed on the diagonal of matrix M. Q is the interaction matrix for the MF approach, and it is defined by  Q¼

 Qx

Qy

 ¼

∂Rðx;yÞ ∂x0

∂Rðx;yÞ ∂y0

 :

ð11Þ

The partial derivatives of Rðx; yÞ with respect to shifts of ðx0 ; y0 Þ can be found by expressing the shifted reference LGS image R0 ðx; yÞ by use of the Fourier shift theorem: R0 ðx; y; x0 ; y0 Þ ¼ F −1 fF fRðx; yÞg expð−j2πðux0 þ vy0 ÞÞg;

ð12Þ

where ðu; vÞ are the coordinates in the Fourier plane. The partial derivatives of the LGS images in the x direction evaluated at ðx0 ; y0 Þ ¼ ð0; 0Þ are  ∂R  ¼ F −1 f−j2πuF fRðx; yÞgg: ∂x0 ðx0 ;y0 Þ¼ð0;0Þ

ð13Þ

The derivatives of the reference image, Rðx; yÞ, with respect to the shifts in x and y are calculated with the 1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS

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use of the FT. It should be noted that the real-time computation for the MF does not require a FT operation, unlike the CORR. The MF, like the CORR, gives centroids relative to the reference image, so reference centroids do not need to be subtracted. B.

Reconstruction

We employ the noise-weighted maximum a posteriori (MAP) reconstruction method introduced in [19], whereby the spot elongation of the SH WFS is explicitly included in the reconstructor. This method was adapted to GLAO in [20], and it is here used with a matrix-vector-multiply (MVM) reconstructor. A noise-weighted MAP algorithm has previously been used for tomographic AO in [21]. The method is recapped briefly here. First, the wavefront slopes, dk ¼ ðdkx ; dky Þ, in the kth WFS are linearly related to the wavefront w: dk ¼ Sk w þ nk ;

ð14Þ

where Sk is the interaction matrix of the kth guide star to WFS, nk is measurement noise in the kth WFS. The estimate of the wavefront w in the DM space from the vector of measurements d is given by ^ ¼ Gd; w

ð15Þ

where the command matrix G for a MVM MAP reconstructor is defined by [21] G ¼ ðST N −1 S þ W −1 Þ−1 ST N −1 ;

ð16Þ

where N ¼ hnnT i is the noise covariance matrix, S is the interaction matrix, and W ¼ hwwT i is the covariance of the turbulent wavefront. We assume von Kármán turbulence to calculate the wavefront covariance W. S, the interaction matrix, is the concatenation of the L individual WFS interaction matrices Sk. d is the concatenation of the L individual sets of WFS measurements dk . The controller used in this paper is a simple integrator defined by aj ¼ aj−1 þ γGdj ;

ð17Þ

where aj is the DM command at time step j, aj−1 is the DM command at time step j − 1, γ is the integrator loop gain, and Gdj is the estimate of the residual wavefront in DM space at time step j. N, the noise covariance matrix, is a block diagonal matrix of the form 0 B B B N¼B B B @

N1

0 476

..

0 .

Nk

..

.

1 C C C C; C C A

ð18Þ

NL

APPLIED OPTICS / Vol. 50, No. 4 / 1 February 2011

where N k is the noise covariance of the kth WFS, where there are a total of L WFSs. For the LGS, because the SH spots are nonorthogonally oriented with respect to the pixel axes of the detector, the measurement noise between x and y is correlated. These correlations and the effect of the elongation, given by ðβx ; βy Þ in each subaperture, is included as a prior in the inverse of the noise covariance matrix ½N k −1 of the kth WFS [20,21]: k −1

½N 

  1 θ2 1 þ α2 β2y =θ2 −α2 βx βy =θ2 ; ¼ 2 2 σ n θ þ α2 ðβ2x þ β2y Þ −α2 βx βy =θ2 1 þ α2 β2x =θ2 ð19Þ

where α is a weighting term on the elongation, σ 2n is the noise variance in the nonelongated direction, and θ is the FWHM of the nonelongated point-spread function (PSF). The weighting term, α, needs to be optimized numerically and is dependent on the laser launch location. The case of α ¼ 0 yields a diagonal noise covariance matrix with equal entries on the diagonal. The calculation of σ 2n and optimization of α are discussed in [21]. 3. GLAO and Environmental Parameter Definition

In this section we define the GLAO system and the relevant environmental parameters for the simulations that are presented in Section 4. The pertinent parameters for LGS GLAO are listed in Table 1. The four LGSs are on a ring with a diameter of 6:0 arcmin. The optimal number and position of the LGSs are not investigated in this paper. The launch location of the lasers is designated to be either side or central launch. In the case of central launch, all the lasers are projected from on axis, i.e., at coordinates ð0; 0Þ in the pupil plane. For the side launch, four laser launch telescopes at ð16:26; 16:26Þ m are used (23:0 m off axis, so 1:5 m from the edge of the primary mirror), with one laser beacon per launch telescope. The laser launch locations and the spot elongation geometry are shown in Fig. 1. Unless otherwise stated, the default launch position is the side launch. The EELT is a 42 m primary mirror telescope, with a central obscuration of diameter 12 m. The four Table 1.

GLAO System Parameters

Parameter

Value

Number of LGS (L) LGS asterism diameter Telescope diameter Central obscuration (diameter) WFS Integrator gain (γ) Zenith angle (ζ) Nonelongated spot FWHM (θ) Pixels per subaperture Pixel scale Frame rate Outer scale

4 6:0 arcmin 42 m 12 m 60 × 60 (SH) 0.2 0° 1:1 arcsec 10 × 10 1:0 arcsec=pixel 500 Hz 25 m

Table 2.

Good, Median, and Bad Atmospheric Profiles Used in This Paper

Layer

Height (m)

% of C2N Good

% of C2N Median

% of C2N Bad

Wind Speed (m=s)

1 2 3 4 5 6 7 8 9

47 140 281 562 1125 2250 4500 9000 18,000

53.28 1.45 3.50 9.57 10.83 4.37 6.58 3.71 6.71

52.24 2.60 4.44 11.60 9.89 2.95 5.98 4.30 6.00

48.91 4.25 7.26 13.91 9.56 1.87 5.92 4.11 4.21

15.0 13.0 14.0 10.0 9.0 15.0 25.0 40.0 21.0

LGS WFS are SH, with a square array of 60 × 60 subapertures. The integrator loop gain, γ, is kept constant (at 0.2) for all photon fluxes simulated. This value was optimized for the high flux (500 photons/subaperture/ frame) case, so performance at lower fluxes is somewhat pessimistic. The number of photons/ subaperture/frame refers to the number of photodetection events on the detector (i.e., the WFS quantum efficiency is not included here) for an LGS WFS spot that is not truncated. For more elongated subapertures, a percentage of these photons is lost. The number of photons/subaperture/frame is per LGS, not total. For all the simulations in this paper, the zenith angle is assumed to be 0°, the worst case for spot elongation. The nonelongated LGS spot size (the median short-term nonelongated FWHM LGS spot size as observed on a WFS), θ, is assumed to be 1:10 arcsec in this paper, to take into account uplink and downlink propagation and laser launch telescope aberrations [22]. Unless otherwise stated, the default detector used is the NGS detector (NGSD) of 10 × 10 pixels [23] with 1:0 arcsec=pixel and with a noise of 4 e=pixel=readout, and the temporal sampling frequency of the LGS WFS is 500 Hz. However, the number of pixels, pixel scale, read noise, and sampling frequency are all subject to design trade-offs in Section 4. For LGS GLAO on the EELT, the positions of the LGSs are fixed with respect to the sky. So when the pupil rotates to track the science target, the LGS WFS images will rotate on the LGS WFS detector. This rotation does affect the low-order “LGS aberrations,” as have been measured in Ref. [11] on the sidelaunched Keck II LGS AO system. The reference images for the centroiding algorithms will need to be updated (as an average from the LGS WFS detector images) as the LGS images rotate. However, the limiting factor for the update rate for these images is the evolution of the sodium layer (a few seconds), not the rotation of the LGS WFS images on the detector. Additionally, the command matrix G will have to be recomputed offline to take into account the exact orientation and elongation as the LGS WFS images rotate on the detector. The time scale of this update is much larger than the exposures studied in this paper.

Three atmospheric C2n profiles have been obtained using MASS/DIMM and SLODAR profilers at Paranal during the calendar year 2007 [24], and they are tabulated in Table 2. The three profiles are referred to as median seeing (s ¼ 0:8 arcsec, r0 ¼ 12:9 cm, τ0 ¼ 2:5 ms), good seeing (s ¼ 0:6 arcsec, r0 ¼ 17:2 cm, τ0 ¼ 3:3 ms), and bad seeing (s ¼ 0:8 arcsec, r0 ¼ 9:4 cm, τ0 ¼ 1:8 ms). Unless otherwise stated, the default profile used for the simulations is the median profile. The distribution of the strength of the layers is not significantly different [they all have approximately 50% of C2n at the lowest (47 m) layer], and the wind speeds at each layer are identical for all three profiles. The outer scale in all three cases is 25:0 m. The four normalized sodium profiles that are simulated in this paper are shown in Fig. 2. All the profiles are centered at an altitude of 90:0 km above the telescope. The default profile used in the simulations is Fig. 2(a), which is the average of 88 consecutive lidar frames (sampled 72 s apart) from the University of Western Ontario (UWO) lidar [25]. This profile has been used by several studies [11,18,21] as a representative sodium profile for AO simulations. The second

Fig. 2. Normalized sodium profiles: (a) UWO lidar mean profile, (b) Gaussian of 11:4 km FWHM, (c) UBCt1000, and (d) UBCt2000. 1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS

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profile is a Gaussian of FWHM of 11:4 km, as shown in Fig. 2(b), which is a common approximation used for AO simulations. The remaining two profiles in Figs. 2(c) and 2(d) are taken from the University of British Columbia (UBC) lidar study [26]. Both profiles are from the night of 4 August 2008, the night chosen in the University of Victoria laboratory experiment on wavefront sensing [13]. This night was chosen by the authors of [13] because of the high number of sporadic events occurring during the night. As with the experiment, the sodium profiles (which are taken every 1 s) are averaged over 20 frames to reduce the photon noise on individual frames. Figure 2(c) is taken from averaging the 20 frames starting at t ¼ 1000 s after UCT ¼ 10 h and is referred to as UBCt1000, and the second profile (bottom right) is taken from averaging the 20 frames starting at t ¼ 2000 s after UCT ¼ 10 hr and is referred to as UBCt2000. The LGS WFS subaperture images are generated for the lidar sodium profiles using the code developed in [11]. The OCTOPUS end-to-end simulation code used in this paper has previously been checked against an independent analytical simulation tool in Ref. [27]. The two codes simulated NGS and LGS GLAO systems for 8 and 32 m diameter telescopes using the same atmospheric statistics. In the high flux case (the analytic code could not simulate WFS noise), the field-averaged FWHM in the K band of the two codes agree to within 0.01 to 0:02 arcsec (or 5%–10% of the estimated FWHM). The difference was thought to be due to additional terms, such as servo lag in the end-to-end simulations that could not be reproduced in the analytic code. This is a similar level of agreement between analytic and end-to-end simulations as found by Andersen et al. in [8]. 4. Simulation Results

In this section, we present the end-to-end simulation results for the GLAO system and environmental conditions presented in Section 3. The performance metric used for all the simulations in this paper is the linear size (1 dimension or box width) of the pixel required for 50% ensquared energy (EE) in the K band (2:2 μm). Each data point is run over 500 time steps (1:0 s at 500 Hz). These short time exposures are due to the slow speed of calculating the large WFS FOVs required when considering elongated LGS WFS spots. We have checked for one simulation point— the default set of parameters and with 500 photons/ subaperture/frame—that the 50% EE does indeed remain stable. After 500 iterations (1:0 s), the 50% EE is 0:234 arcsec; after 1500 iterations (3:0 s), the 50% EE is 0:236 arcsec; and after 3000 iterations (6:0 s), the 50% EE is 0:238 arcsec. The 50% EE in the K band versus the radial position off axis of 25 PSF stars arranged regularly (5 × 5 grid) over a 5 arcmin square are plotted in Fig. 3 without GLAO correction and with GLAO correction with 500 photons/subaperture/frame and with side launch. The GLAO correction produces the best cor478

APPLIED OPTICS / Vol. 50, No. 4 / 1 February 2011

Fig. 3. 50% EE in the K band versus the radial position of the PSF star in the field without correction (circles) and with correction with 500 photons/subaperture/frame for side launch (crosses).

rection on-axis, although the 50% EE at the edge of the field is only 12% higher than on-axis. Given the consistent correction across the field and the fact that we do not perform trade studies on the number or position of the LGS, we compute only the 50% EE in the K band on axis for the remainder of the simulations, in order to reduce the computation from calculating multiple PSFs. It is not possible to include all physical effects in these end-to-end simulations for time and computational reasons. First, the Rayleigh fratricide effect arising from one LGS WFS looking through the Rayleigh plume of another has been excluded. This phenomenon is expected to degrade the correction of the central launch more than the side launch [28]. Second, the majority of simulations here assume that the sodium profile is static and, hence, the reference images/centroids/gains for the various WFS algorithms are known exactly. Therefore, this neglects the LGS aberrations seen at the Keck LGS system [11] and on the University of Victoria LGS WFS test bed [13]. The telescope spiders, which can mask out some columns/rows of subapertures and create distinct islands of subapertures, are also not considered. Finally, the inclusion of the NGS, which is required to overcome the tilt indetermination problem, is a limiting factor in the duration of the simulations. Consequently, we have assumed that the tilt can be measured from the four LGS instead. We have checked that this approximation does not affect the 50% EE significantly. The 50% EE for the default set of parameters and with 500 photons/subaperture/frame with a bright NGS (10,000 photons/subaperture/frame) is 0:237 arcsec. The 50% EE with the same parameters but no NGS and instead using the tilt measured from the LGS is 0.230. While the tilt indetermination is not included in the simulations, the cone effect is still included. This approximation does not in any way affect the trade-offs we then make on the LGS WFS

centroiding algorithm, LGS launch location, or LGS WFS detector FOV, which are the focus of this paper. A.

Design Trade-Offs

The first step is to optimize the weighting term α for both central and side launch of the lasers. This optimization is made at a signal level of 100 photons/ subaperture/frame and gives an optimal α value of 0.4 for the side launch and 0.3 for the central launch. These values are consistent with those found in [20,21] for GLAO and laser tomography AO systems, respectively, for the EELT. In Fig. 4, the 50% EE in the K band on axis is plotted for the optimal values of α for both the side and the central launch. With noise-optimal weighting, the performance of central launch and side launch is almost the same, with the central launch marginally better at lower flux and the side launch at higher flux. In all the following simulations, the side launch is assumed for the baseline, and while the performance with respect to photon flux is similar, the central launch configuration will suffer more from Rayleigh backscatter than the side launch [28]. The 50% EE in the K band on axis is plotted versus the frame rate of the LGS WFS in Fig. 5. Three different values of constant detected laser photon flux are plotted: 50 k photons/subaperture/ second, 100 k photons/subaperture/second and 1 M photons/subaperture/second. For all cases, the exposure time is kept constant at 1:0 s, meaning that the different sampling frequencies require a different number of loop iterations. At a sampling frequency of 100 Hz, the EE is dominated by the temporal error rather than the photon noise, so all three signal levels have almost the same EE. The three curves separate at higher sampling frequencies due to the different photon noise. There is some (small) improvement in going to sampling rates greater than 500 Hz at the higher flux levels, but 500 Hz is used

Fig. 4. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame with noise-optimal weighting for the central (dashed curve) and side (solid curve) launch. The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere.

Fig. 5. 50% EE in the K band on axis versus the temporal sampling frequency (Hz) of the LGS WFS for a constant photon flux of 50 k photons/subaperture/second (solid curve), 100 k photons/ subaperture/second (dashed curve), and 1 M photons/subaperture/second (dashed–dotted curve). The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere.

by default in all of the simulations in this paper. While Fig. 5 shows the optimum sampling frequency is 500 Hz for three different flux levels, it should be noted that this is for only one wind speed profile (the median), and the optimum sampling frequency may change in better or worse wind conditions. The baseline read noise level σ r for the NGSD WFS detector under study for the EELT is four electrons/ pixel/frame [23]. In Fig. 6 we plot the 50% EE in the K band on axis for this read noise level, as well as 1 e, 2 e, and 3 e. At lower laser flux (less than 100 photons/subaperture/frame), there is a significant improvement in 50% EE in going from the baseline 4 e to a lower read noise level. However, at flux levels higher than 100 photons/subaperture/frame, there is no noticeable gain achieved in 50% EE in lower noise

Fig. 6. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for σ r ¼ 1 1, 2, 3, and 4 electrons of read noise/pixel/frame. The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere. 1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS

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detectors. There is a clear trade-off between a lower read noise detector or more laser photon flux. In Fig. 7, the 50% EE in the K band on axis for the COG, WCOG, MF, and CORR centroiding algorithms are plotted versus the number of photons/ subaperture/frame for the side launch of the LGS GLAO. The performance of the WCOG and MF algorithms is almost identical and clearly preferable to both the COG and the CORR algorithms, particularly at lower flux levels. One possible reason for this is that the radial thresholds employed for the COG and CORR may not be optimal; that is, not all subapertures should be thresholded with the same percentage threshold. It is not surprising that the COG algorithm gives the largest 50% EE: it is the only one that does not make use of the known elongation. The WCOG and MF algorithms do not need any threshold optimization, unlike the CORR and COG routines, and this is a further advantage of the WCOG and MF algorithms. The MF algorithm is used in the remainder of the simulations, although the WCOG is equally appealing. Figure 8 shows the gain in 50% EE in the K band on axis compared to the uncorrected atmosphere versus LGS flux for six combinations of pixel scale, FOV, and number of pixels of the LGS WFS detector. The pixel scales simulated here are between Nyquist (two pixels/FWHM) and half-Nyquist (one pixel/FWHM). Sampling of less than one pixel/FWHM in [13] was found to be insufficient and consequently is not explored here. The absolute 50% EE values were not plotted here because the uncorrected 50% EE values varied slightly for different FOVs, due to the different phase screen resolution used in each case. The determining factors in this trade-off are the increased read noise from more pixels, the reduction in photons due to truncation from a smaller FOV, biases introduced by truncating the elongated LGS WFS image, and errors associated with under-

Fig. 7. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for the COG (dotted curve), WCOG (dashed curve), MF (solid curve), and CORR (dashed– dotted curve) WFS algorithms. The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere. 480

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Fig. 8. Gain in 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for different combinations of detector pixels and pixel scales.

sampling the nonelongated direction of the spot. For ease of computation, the Gaussian sodium profile Fig. 2(b) was assumed for all six combinations. Clearly the worst choice is 6 × 6 pixels with 0:4 arcsec=pixel, which does not properly accommodate the spot elongation. The best choice is the 10 × 10 pixels with 1:0 arcsec=pixel, which is almost undersampled by a factor of 2 in the nonelongated direction. Increasing the number of pixels with this pixel scale to 14 × 14 marginally diminishes the 50% EE gain, due to the increased read noise. The three cases with 0.7 and 0:5 arcsec=pixel all yield worse returns than the 1:0 arcsec=pixel cases. The default of 10 × 10 pixels with 1:0 arcsec=pixel is used for the rest of the simulations in this paper. We note in Fig. 7 that, at a flux level of 20 photons, the 50% EE for the COG and CORR algorithms is larger than the 50% EE for the uncorrected atmosphere.

Fig. 9. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for the QC (dashed curve) and the 10 × 10 pixel detector with MF (solid curve). The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere.

Fig. 10. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for the good (dashed curve), median (solid curve), and bad (dashed–dotted curve) atmospheric profiles. The horizontal lines are the 50% EE in the K band on axis for the uncorrected atmosphere for the good (dashed curve), median (solid curve), and bad (dashed–dotted curve) atmospheric profiles.

Similarly, in Fig. 8, at 20 photons, the gain in 50% EE compared to the uncorrected atmosphere for some of the LGS WFS detector FOV combinations is less than 1. Thus, in both low flux cases, the GLAO system has increased the 50% EE, instead of reducing it. This problem arises because the integrator gain of 0.2 was optimized for the higher flux case (500 photons) and applied in all the simulations. Obviously, an integrator gain of 0 would mean the 50% EE in these cases would be the same as 50% of the atmosphere. In Fig. 9, the performance of the QC is compared with that of the 10 × 10 pixel array with the MF. The FOV of the QC is 10:0 × 10:0 arcsec, the same as the multiple pixel detector. The QC is assumed to have a

Fig. 11. 50% EE in the K band on axis versus the number of detected photons/subaperture/frame for the four different sodium profiles. The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere.

Fig. 12. 50% EE in the K band versus the number of detected photons/subaperture/frame for different levels of staleness of the MF (in seconds). The solid horizontal line is the 50% EE in the K band on axis of the uncorrected atmosphere.

readout noise of 1 e=pixel=frame, in comparison with the 4 e=pixel=frame for the 10 × 10 pixel detector. The QC is superior in performance at flux levels less than 100 photons/subaperture/frame, but above this, the 10 × 10 pixel detector with MF yields a lower 50% EE in the K band on axis. While the simulation results in Fig. 9 are promising for the use of the QC, there are a number of practical problems that need to be overcome. The QC is sensitive to static aberrations, which will have the effect of driving the spot off null and hence reducing the dynamic range. In [13], the QC was found to be more susceptible to the evolution of the sodium layer than other centroiding algorithms. The optical design of a large FOV QC to account for spot elongation is also a challenge. B. Environmental Conditions

We now investigate the performance of the LGS GLAO with respect to the environmental conditions, namely, atmospheric profile, sodium profile, and variations in the sodium profile. In Fig. 10, the 50% EE in the K band on axis is plotted versus the number of photons/subaperture/frame for the good, median, and bad atmospheric profiles in Table 2. The performance of the LGS GLAO is strongly dependent on the atmospheric profile assumed. In fact, without any correction, the good profile already provides better performance than correcting the bad profile with 500 photons. For all three profiles, a considerable improvement over the uncorrected 50% EE in the K band on axis is possible with LGS GLAO. At 500 photons, the reduction in the 50% EE over the uncorrected atmosphere is 2.1 times for the good profile, 1.9 times for the median profile, and 1.6 times for the bad profile. Note that while approximately a 2 times reduction in 50% EE is possible for all three atmospheric profiles with LGS GLAO on the EELT, all three profiles do have approximately 50% of C2n at the ground layer. 1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS

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The performance of the LGS GLAO versus photon flux is shown in Fig. 11 for the four sodium profiles plotted in Fig. 2. The three profiles derived from means of lidar measurements all perform almost equally and better than the Gaussian of the FWHM 11:4 km. There are two possible reasons for this: first, all three lidar profiles are narrower in terms of equivalent FWHM than the Gaussian sodium profile and, second, the MF algorithm utilizes the structure of the sodium profile, and there is more structure in the lidar profiles than the Gaussian sodium profile. In all the simulations presented previously, the sodium layer is assumed to be static, and hence the MF (or CORR or WCOG reference images) are known exactly. However, the sodium layer is evolving, as can be seen in lidar time series such as in [26], and, consequently, there will be a lag in updating the reference and hence an additional wavefront error. In order to estimate the effect of the sodium profile evolution on LGS GLAO, we generate the MFs with the profile UBCT1000 and then generate the noisy LGS images with profiles with delays of 10, 100, 200, 500, and 1000 s from the UBC lidar set [26]. The 50% EE in the K band on axis versus LGS photon flux is shown in Fig. 12 with MFs that have these different levels of temporal staleness. There is no noticeable difference in the 50% EE for staleness of less than 200 s. 5. Conclusions

We have performed end-to-end simulations of a LGS GLAO system for an ELT, and we have shown that, under a range of turbulence conditions, the 50% EE can be improved of the order of 2 times. Also, we have seen that the central and side launch of the laser give similar performance if the elongation is taken into account in the reconstructor. The MF and WCOG algorithms perform better than the COG or CORR, and the optimum temporal sampling frequency is approximately 500 Hz. The optimum sampling and FOV of the LGS WFS detector are approximately 10 × 10 pixels of scale 1:0 arcsec=pixel, although only a small loss in performance is seen with a QC. In the default configuration, there is no significant improvement in performance with more than 200 photons/subaperture/frame. The performance of the LGS GLAO is very dependent on the assumed atmospheric turbulence profile, but not on the sodium profile. The evolution of the sodium profile is only significant for LGS GLAO on time scales of hundreds of seconds. The authors gratefully acknowledge the funding of this work by the European Community through Framework Program 6 (ELT Design Study, contract 11863) and Framework Program 7 (Preparation for ELT Construction, contract 211257). References 1. J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008). 2. M. Johns, “Progress on the GMT,” Proc. SPIE 7012, 70121B (2008). 482

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