Comparison of LSST and DECam Wavefront Recovery
    Bo Xin
    , Aaron Roodman
    , George Angeli
    , Chuck Claver
    , and Sandrine Thomas
    Large Synoptic Survey Telescope, Tucson AZ, USA
    SLAC National Accelerator Laboratory, Stanford University, Menlo Park CA, USA
    We make a detailed quantitative comparison of the wavefront recovery algorithms between those developed for
    Dark Energy Camera (DECam) and the Large Synoptic Survey Telescope (LSST). Samples used in this study
    include images of out of focus stars collected by the DECam at the Blanco 4-meter telescope and arti?cial
    simulated donut images. The data from DECam include wavefront images collected by the wavefront sensors
    and out-of-focus images where the entire DECam sensor array is used. For simulated images, we have used both
    the forward Fraunhofer di?raction and a LSST-like ZEMAX optical model where the images are convolved with
    Kolmogorov atmosphere. All samples are analyzed with the forward wavefront retrieval algorithm developed for
    DECam and the transport of intensity algorithm for LSST. Good quantitative agreement between results by the
    two implemented algorithms is observed.
    Keywords: LSST, DECam, Wavefront Sensors, Curvature Sensors, Donuts
    The Large Synoptic Survey Telescope (LSST) is a new facility now under construction that will survey ˘ 20000
    square degrees of the southern sky through 6 spectral ?lters (ugrizy) multiple times over a 10-year period.
    The optical system is based on a modi?ed Paul-Baker 3-mirror telescope design having an 8.4m primary, 3.4m
    secondary, and 5.0m tertiary feeding a three-element refractive camera system producing a ?at 9.6 square degree
    ?eld-of-view with an e?ective clear aperture of ˘ 6.5m. LSST will use an active optics system (AOS) to maintain
    alignment and surface ?gure on its three large mirrors. Corrective actions fed to the LSST AOS are determined
    from information derived from 4 curvature wavefront sensors located at the 4 corners of the focal plane. Each
    wavefront sensor is a split detector such that the halves are 2mm on either side of focus.
    The LSST wavefront sensing software is described in detail in Ref.
    . It includes extensions to published
    curvature wavefront sensing algorithms needed to address challenges presented by the LSST, namely the large
    central obscuration, the fast f/1.23 beam, o?-axis pupil distortions, and vignetting at the sensor locations. It
    also takes into account corrections needed for the split sensors and the e?ects from the angular separation of
    di?erent stars providing the intra- and extra-focal images. The underlying baseline algorithms are the iterative
    Fast Fourier Transform (FFT) method by Roddier and Roddier,
    and the series expansion technique by Gureyev
    and Nugent.
    also describes a set of very extensive tests and validations using simulated images. However,
    there is no validation using real data or direct comparison to other wavefront sensing algorithms. In this paper,
    we present a detailed quantitative comparison with the forward modeling algorithm that has been used by Dark
    Energy Camera (DECam), using both data collected by the DECam and simulated images.
    DECam is a 3 square-degree ?eld-of-view imager now in operation on the CTIO Blanco 4-meter telescope.
    It has eight wavefront sensors placed on the same focal plane as the 62 science sensors, and a hexapod to allow
    rapid adjustment of the camera's focus and alignment with respect to the primary mirror. The wavefront sensors
    are divided into four pairs of intra- and extra-focal sensors, located on the edge of the ?eld of view. Each pair
    consists of two 2k by 2k CCD chips, placed out of focus by ? 1.5mm. The wavefront sensing and the AOS on the
    DECam have been proven to operate successfully, maintaining unsupervised control of the focus and alignment.
    Further author information:
    Bo Xin: E-mail:, Telephone: 1 520 318 8352

    Given the similarity between the LSST and DECam, both being suvery telescopes with large ?eld-of-view, and
    both employing area wavefront sensors, DECam can serve as a good test ground for the LSST wavefront sensing
    The DECam wavefront estimation utilizes a forward-modeling non-linear ˜
    The intensity distribution on
    the image plane is modeled as the Fraunhofer integral, where the inputs are the pupil function and the wavefront
    at the exit pupil. The pupil function is a ?xed property of DECam and accounts for DECam's complicated
    obscuration geometry. The wavefront is decomposed into a set of Zernike polynomials, where the coe?cients are
    the free parameters in the ?t. The atmospheric seeing is accounted for using a convolution of the image model
    with a Kolmogorov kernel.
    For LSST, the degrees of freedom that need to be controlled include those in the two actively supported
    mirror systems and the positioning of the two hexapods. The number of degrees of freedom is signi?cantly greater
    than any currently operating telescopes including DECam. This requires the ability to estimate higher-order
    properties of the aberrated wavefront. The current design is that Zernike coe?cients, Z4 { Z22 in Noll/Mahajan's
    will be measured and used by the AOS. For this reason, LSST has chosen to solve the transport-
    of-intensity equation (TIE) to retrieve the wavefront, rather than a forward-modeling approach. For DECam
    online operations, Zernike coe?cients up to trefoid (Z9, 10) are measured, and only focus (Z4), astigmatism (Z5,
    6), and coma (Z7, 8) are used to control the prime-focus camera's ?ve degrees of freedom. Furthermore, the
    DECam forward-modeling algorithm ?xes the spherical aberration (Z11) to a value predicted by the ZEMAX
    prescription. The Fried parameter of the atmosphere is ?xed at r
    = 0.125m.
    For o?ine analysis, the forward
    ?t includes Z11 and r
    as free parameters. In the comparison study presented in this paper, the DECam results
    come in two versions, one with ?ts to Zernikes up to Z11, the other with additional quadrafoil (Z14, 15). In the
    rest of this paper, the indexing and normalization of the Zernikes always follow the Noll/Mahajan's de?nition.
    We also need to note that this comparison study is actually a comparison between the LSST implementation of
    the baseline algorithms with the iterative wavefront compensation and the DECam forward modeling algorithm.
    In order to make comparison, the LSST software is run in its \paraxial lens" mode, meaning the algorithm's
    LSST speci?c extensions as described in Ref.
    are being bypassed. Because the DECam software only has the
    standard circular Zernikes implemented, results from LSST used for comparison are also in standard Zernikes.
    The obscuration ratio of DECam is 34%. This is less of a problem compared to working with LSST-like images,
    where the obscuration is 61%. Similarly, DECam is a f/2.9 system, while LSST is f/1.23. DECam is 2.2 degree
    wide ?eld, LSST is 3.5 degree. Given DECam's proven success, we expect none of these to lead to signi?cant
    issues to the comparison study.
    This paper is organized as follows. In Section
    we describe the data used in this analysis, including both
    images taken by DECam and simulated data. Section
    then outlines the analysis approach. The results are
    discussed in Section
    . In Section
    we give a summary and what we conclude from this analysis.
    The comparison study is performed by analyzing the same ensembles of donut images using separate code
    implementing each of the two algorithms. Samples used in this study include images of out of focus stars
    collected by the DECam at the Blanco 4-meter telescope and arti?cial simulated donuts. We have looked at
    three types of data collected by the DECam during 2014 { 2015: (1) Wavefront sensor images taken during
    engineering runs, when the telescope was doing large slews throughout the night, with open AOS loop, without
    the look-up-table corrections. We'll later refer to these as the open-loop data. In these data, the wavefront
    aberrations vary more wildly compared to the normal observing runs. (2) Wavefront sensor images taken during
    typical DECam observing nights. (3) Science sensor images taken during engineering runs, where the entire
    camera was moved out of focus. These also do not have the AOS corrections. For the wavefront sensor images,
    including types (1) and (2) above, the images from adjacent CCDs are from both sides of focus at ? 1.5mm. They
    can therefore directly be used by the LSST software. The out-of-focus science images (type (3)) are also from
    both sides of focus, at ? 1.5mm and ? 3.0mm. These were used by DECam to map out the wavefront across
    the focal plane. Both softwares use the cut-out images created by the DECam wavefront image pre-processing
    pipeline. The image stamps are 64 by 64 pixels for ? 1.5mm defocused images and 128 by 128 for ? 3.0mm

    defocused images. The number of DECam exposures that are used in the analysis is 202, 137, and 98 for types
    (1), (2), and (3), respectively. The number of postage stamps are 58,997, 10,688, and 174,472 for the three types
    of data, before any selection that will be discussed in the next section.
    Tests using simulated images have the advantage of being able to better cover the parameter space. We have
    used two sets of simulated images. In the ?rst set, the forward-modeling code utilizing the forward Fraunhofer
    di?raction intergral, which is part of the DECam algorithm, is used to generate the synthetic donut images. The
    wavelength is 700nm. The second set of test images are generated using a simple ZEMAX paraxial lens model,
    with f/1.23 and central obscuration of 61%. The parameters are the same as the LSST optical system, but the
    model doesn't have the fast-beam non-linear e?ects and the o?-axis distortions. Geometric Image Analysis is
    performed at 1mm above and below the focus. Both sets of images are then convolved with Kolmogorv PSF
    kernels representing the atmospheric seeing. For each Zernike among Z4 { Z15, intra- and extra-focal images are
    simulated for magnitudes of 0.05, 0.10, 0.25, 0.50, 1.00, 1.50, and 2.00 waves, with wavelength of 770nm.
    The DECam wavefront sensing algorithm only requires one donut image at a time, in order to ?t for the wavefront.
    For the TIE-based LSST algorithm, both intra- and extra-focal images are required. Ideally, the TIE requires
    that the intra- and extra-focal images be taken with the same optics state and at the same ?eld position. With
    the area sensors, these conditions can not be satis?ed. We need to minimize the di?erence between the intra-
    and extra-focal images, in terms of their ?eld positions and how much the telescope state might have changed
    between them if they are from separate exposures. Another consideration is that if we match an intra-focal
    image with good Signal-to-Noise Ratio (SNR) with an extra-focal image with bad SNR, and vice versa, we will
    not get a good wavefront measurement. Therefore, care needs to be taken when pairing images from di?erent
    On the wavefront sensor images from each exposure, we typically have tens of donut images on each of the
    eight sensors. Because the wavefront varies across the focal plane, we only use images from adjacent CCDs to
    form the intra- and extra-focal pairs. Saturated images are excluded from further use. DECam chips saturate
    at around 40,000 counts, except that FS4
    (ampli?er with column number IX >1024) and FN2 (ampli?er with
    IX <1024) saturate at much lower - around 5000 counts. We then remove the background by means of a
    least-square ?t to the intensity outside of the signal region using a 2D linear model. The signal region is de?ned
    as an annulus large enough to cover the aberrated signals. The SNR is then calculated for each donut image
    based on this ?t. To make sure the input images are of good quality, donut images with SNR less than 20 are
    not used. Next, all the donuts on each wavefront sensor are ranked using SNR from high to low. Intra- and
    extra-focal pairs are then formed by taking the ?rst image from the intra-focal sensor and the ?rst image from
    the extra-focal sensor, the second image from the intra-focal sensor and the second image from the extra-focal
    sensor, and so on, until the list on either sensor is exhausted. As an example, Figure
    (left) shows one pair of
    intra- and extra-focal images on the wavefront sensors with the telescope in the open-loop mode. With 15?m
    DECam pixels, the donuts are roughly 40 pixels in diameter. The SNR distributions on the pairs of wavefront
    sensors for the same exposure is shown in Figure
    (right). The number of image pairs that pass the above
    selection criteria are 6,284, 1,344, and 29,427 for data types (1), (2), and (3).
    After running the LSST wavefront sensing software on each of these image pairs, the recovered wavefronts,
    in terms of Zernike coe?cients, are averaged over the same pair of wavefront sensors. These are then compared
    to the results from the DECam algorithm. For each pair of images used by the LSST algorithm, there are two
    sets of Zernike coe?cents from the DECam algorithm, each from one image. The Z4 values are approximately
    ? 8.7?, where the wavelength ?=700nm. To compare to results from the LSST software, we average each Zernike
    over the adjacent sensors, to get one set of Zernikes for each sensor pair. Bad ?ts are excluded by requiring that
    the RMS of Z5 { Z10 be less than 3?m and the magnitude of measured Z4 on each individual image be between
    5?m and 15?m.
    For the out-of-focus DECam images, we follow the same procedure as described above for the wavefront
    images, except the pairing. After ranking the donut images from each sensor based SNR from high to low, the
    The naming of the DECam sensors can be found in, for example, Figure 1 of Ref.

    Figure 1. Left: An intra-focal image (top) on wavefront sensor FN1 and an extra-focal image (bottom) on wavefront
    sensor FN2. Right: SNR distributions on the pairs of wavefront sensors. The plot titles show the sensor names and the
    number of image pairs where both images have SNR > 20. These examples are with the telescope in the open-loop mode.
    image pairs are formed by taking one image from an intra-focal exposure and the other image from an extra-focal
    exposure. The intra- and extra-focal images are always from the same sensor, but di?erent exposures.
    The procedure is even simpler for the simulated images. The images are already paired up, and there is no
    background removal needed.
    Another note we need to make here is the orthogonality issue we mentioned already in Section
    . In order to
    make a valid comparison, both algorithms need to use the same number of Zernike coe?cients when estimating or
    ?tting the wavefront. In the LSST algorithm con?guration ?le, an option is provided to the user to use standard
    Zernikes, regardless of the obscuration of the system. However, we ?nd that the agreements between the LSST
    and DECam algorithms are generally improved if we still use annular Zernikes in the LSST algorithm, then post-
    process the results by reconstructing a wavefront map from the annular Zernike coe?cients then decomposing
    it into standard Zernikes. This is consistent with the fact that mathematically the TIE solvers in the LSST
    algorithm relies more on the orthogonality of the basis set. We use this post-processor to get the LSST results
    used in this comparison.
    For the results shown in the next section, the LSST results are obtained using the series expansion algorithm
    as the baseline. As we discussed in Ref.
    , the two baseline algorithms produce very similar results. The series
    expansion algorithm is faster.
    4. RESULTS
    4.1 Comparison Using DECam Wavefront Sensor Data
    shows the comparison of Zernike coe?cients using one exposure from the open-loop wavefront images.
    Each plot is for one pair of wavefront sensors. For each wavefront sensor pair, the thin red dashed lines are
    the Zernike coe?cients measured by the LSST algorithm for individual image pairs. The thick red solid lines
    are the average of all the LSST measurements. The thin blue (green) dashed lines are the DECam solutions
    without (with) quadrafoil for individual donut images. The thick blue (green) solid lines are the DECam averages
    without (with) quadrafoil. In general, the agreement on the averaged solutions between the algorithms is very

    Figure 2. Comparison of Zernike coe?cients using one exposure from the open-loop wavefront images.
    good. Looking at the results from individual images, or image pairs in case of the LSST solutions, both algorithms
    are robust and give consistent results.
    The comparison by Zernike between the two algorithms is shown in Figure
    . The DECam results are from
    the ?ts without quadrafoil. Very good agreement is seen for astigmatism, coma, and trefoil. Some deviations
    are seen on the focus term. Meanwhile the spherical aberration (Z11) coming out of the DECam algorithm
    has a very small spread. Apparently there is cross-talk between the focus and spherical terms in one or both
    algorithms. Focus and spherical aberration are both rotationally symmetric, and includes radial terms up to ˆ
    and ˆ
    , respectively, where ˆ is the radial coordinate on the pupil. The di?culty in separating spherical from
    focus arises because of the limited number of pixels along the radial direction coupled with noise. The fact that
    the wavefront is di?erent on adjacent wavefront CCDs also contributes to the di?erences between the results.
    Results we have discussed so far are with the telescope in the open-loop mode. Next we will show some
    results from running both algorithms on wavefront sensor images during typical DECam observing runs. The
    scatter plots look similar to those in Figure
    , except that the aberrations are smaller and the range of variation
    of the aberrations are smaller. We have also looked at how the wavefront aberrations measured on the four
    pairs of wavefront sensors vary as a function of time throughout a typical DECam observing night. Figure
    shows the time history of 45
    astigmatism (Z5), as measured by both algorithms. Figure
    is the coma-y (Z7)
    time history for the same night. The DECam wavefront is very stable over time. Generally good agreement
    is observed between the two wavefront sensing algorithms, in terms of both the absolute magnitude and the
    exposure-to-exposure variations. The RMS di?erence between the measurements made by the two algorithms
    is approximately in the range of 40 { 70 nm. This is true for all the other measured Zernikes. A missing data
    point on these plots means for a paticular exposure there is no pair of stars that could pass our saturation and
    SNR selections, which we described in Section

    Figure 3. Scatter plots of DECam results vs. LSST by Zernike. The data used are wavefront sensor images with the
    telescope in open-loop mode. The red diagonal lines with slope of unit are for reference.
    4.2 Comparison Using DECam Defocused Focal Plane Data
    Comparison is also made using out-of-focus focal plane images taken during the engineering runs. These are also
    without the AOS corrections. As the result, the wavefront aberrations can also be a bit far from zero. Figure
    shows the scatter plots comparing the two algorithms, for each Zernike separately. Looking at the Z4 plot in
    , all image pairs show large focus term in the wavefront. In turn, this may have negatively impacted
    performance of the wavefront sensing algorithms. The LSST algorithm relies on the the intra- and extra-focal
    image planes stay well within the geometric limit.
    It has long been observed that a large non-zero focus term
    a?ects the TIE-based algorithm's ability to recover the other wavefront Zernikes. Coma is a term easily a?ected,
    because it doesn't alter the boundary of the intensity distribution on the image. As the result, the TIE-based
    algorithm breaks down at relatively low magnitude of coma. Figure
    shows good agreement between the two
    algorithms, except focus and coma. Other factors that might contributed to the di?erences between the results
    from the two algorithms include the temporal and spatial separations between the intra- and extra-focal images.
    Because the out-of-focus sensor images were taken at about 30 minutes apart, the underlying wavefront may
    have drifted to some degree. If the wavefront variation within a CCD chip is large, forming image pairs solely
    based on SNR without taking into account the location on the CCD may also introduce bias.
    Figures 4 and 5 in Ref.
    show how the DECam wavefront vary with coordinates on the focal plane. It is
    seen that the wavefront variation within single CCD chips is more pronounced at the edge of the focal plane.
    Therefore, if the agreement between the two algorithms can be improved by excluding the sensors on the outside
    of the focal plane in the above analysis, that would be evidence that the wavefront variation within chips is the
    major contributor to the di?erences in the results. However, the level of agreement between the two algorithms
    barely changes after we exclude the sensors on the outside of the focal plane. We have also looked at the defocused
    focal plane images where the camera is moved by ? 3.0mm. If this leads to better agreement between the results,
    we would have proven that it is the geometric limit that is a?ecting the performance of the LSST algorithm. No
    signi?cant di?erence is observed between the ? 1.5mm and ? 3.0mm results. The exact source of the di?erence
    between results by the two algorithms, speci?cally on coma as shown in Figure
    , is currently unknown.
    Another note to make here is the di?erence between pistoning the entire camera and the focal plane only.
    The LSST algorithm requires that the only change between the intra- and extra-focal images is the position
    of the focal plane. Moving the entire camera out of focus means the lenses are also moved. In principle, this

    Figure 4. Variations of 45
    astigmatism (Z5) through a typical DECam observing night, as measured by the DECam
    (red) and LSST (blue) algorithms. The four panels are for the four pairs of wavefront sensors. The RMS di?erences
    between the measurements by the two algorithms are shown in the upper right corner of each panel.
    violates the requirements of the TIE solver. In practice, due to the low correcting power of the DECam corrector,
    which does little more than changing the f-number of the converging beam, the TIE still can be used, with a
    \e?ective" image o?set. This is con?rmed by creating intra- and extra-focal images with a ZEMAX DECam
    model by moving the focal plane and the entire camera separately, then running the images through the LSST
    algorithm. For DECam, the e?ective image o?set is 1.2 times the real camera o?set.
    4.3 Comparison Using Simulated Data
    Tests using simulated data is more straightforward, because they are not subject to the real-world e?ects like
    those discussed above. However, the simulated data su?er from certain artifacts, such as the approximations
    made in propagating the light and the e?ects of the atmosphere.
    shows the comparison between the two algorithms using the donut images simulated by the DECam
    forward-modeling code. Note that in this test the DECam algorithm has the advantage of having the same
    optical model in both the simulated data and its forward ?ts. As we discussed in Section
    , the image pairs all
    have single-Zernike aberrations, with various magnitudes. The tests are done for Z4 { Z15. The missing DECam
    measurements are because the ?t results didn't satisfy RMS(Z5{Z10)<3?m and 20?m< j Z4 j <40?m.
    Both algorithms perform well on astigmatism, trefoil, and quadrafoil.
    For focus and spherical aberration, the comparisons between the input wavefront and measurements by both
    algorithms do not look good. The true input wavefront to the simulations are single-term circular Zernikes,
    while the LSST algorithm uses a ?nite number of annular Zernikes to recover the wavefront. Taking this into
    account, the di?erence in the two sets of results on focus and spherical doesn't necessarily mean that there is
    more signi?cant cross-talk between focus and spherical aberration in the LSST algorithm. The DECam algorithm
    diverges when the spherical aberration gets larger than about 0.5?, where ?=700nm. The DECam algorithm
    also fails with large second astigmatism (Z12, 13). Take note that Z11 is not a free parameter in DECam online
    operations, and Z12 and Z13 are not free parameters in DECam online or o?ine ?ts. Due to the geometric limit,
    the LSST algorithm's ability to recover coma starts to break down at around 1?, where ?=700nm.

    Figure 5. Variations of coma-y (Z7) through a typical DECam observing night, as measured by the DECam (red) and
    LSST (blue) algorithms. The four panels are for the four pairs of wavefront sensors. The RMS di?erences between the
    measurements by the two algorithms are shown in the upper right corner of each panel.
    The same kind of comparison using the images simulated with the LSST-like paraxial ZEMAX model is
    shown in Figure
    . The ZEMAX model is independent of the LSST algorithm. For LSST results, some data
    points are missing due to a warning-?ag that we are close to or outside of the geometric limit. Again, good
    agreement between the two algorithms is seen. For coma and spherical aberration, the LSST algorithm runs into
    geometric limit at around 700 { 800nm.
    We have demonstrated good agreement between the TIE-based wavefront sensing algorithm used by LSST and
    the forward-modeling algorithm by DECam. The data used in this analysis include DECam images from both
    the wavefront sensors and out-of-focus focal plane and simulated images. The correction algorithms developed
    by LSST targeting fast f-number and wide-?eld systems are not tested here.
    There appears to be some cross-talk between focus and spherical terms in one or both algorithms. Addi-
    tionally, the LSST algorithm breaks down when certain aberrations, such as coma and spherical aberration, get
    large due to the geometric limit. However, a ? 1.5mm image o?set for a f/2.9 system is equivalent to a ? 0.6mm
    o?set for a f/1.23 system. The nominal wavefront sensor image o?set for LSST is ? 2mm. With 10?m pixels,
    the LSST donuts will be about 165 pixels in diameter. We expect the geometric limit not an problem for LSST
    operation. The better pupil sampling and the use of the annular Zernikes on the annular pupil will both help
    reduce the cross-talk between Zernike modes with same azimuthal frequency.
    Lastly, as we have seen on DECam, during normal observing, the wavefront aberrations measured by the
    wavefront sensors are small. For LSST, at the beginning of each night, the aberrations may be relatively large.
    As long as the wavefront retrieval algorithm doesn't break down, we expect the AOS will quickly pull the system
    back into convergence, so that it doesn't need to work anywhere close to the geometric limit.

    Figure 6. Scatter plots of DECam results vs. LSST by Zernike. The data used are out-of-focus science sensor data with
    the camera hexapod pistoned by ? 1.5mm. The red diagonal lines with slope of unit are for reference.
    We thank Paul Schechter for his critical comments on our work and helpful discussions. We also thank the LSST
    internal reviewer, Kevin Reil, for helping us improve the paper.
    This material is based upon work supported in part by the National Science Foundation through Cooperative
    Support Agreement (CSA) Award No. AST-1227061 under Governing Cooperative Agreement 1258333 managed
    by the Association of Universities for Research in Astronomy (AURA), and the Department of Energy under
    Contract No. DEAC02-76SF00515 with the SLAC National Accelerator Laboratory. Additional LSST funding
    comes from private donations, grants to universities, and in-kind support from LSSTC Institutional Members.
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    Figure 7. Comparison of wavefront recovery results using images simulated with the DECam forward-modeling code.
    The blue diagonal lines with slope of unit are for reference.
    Figure 8. Comparions of wavefront recovery results using ZEMAX-simulated images. The blue diagonal lines with slope
    of unit are for reference.
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