1. Introduction

1.1 Description and outline

The Large Synoptic Survey Telescope (LSST) is a ground-based, wide field-of-view, survey telescope scheduled to be built at Cerro Pachón in Northern Chile and to see first light in 2014 [1,2]. One of the distinguishing features of this telescope is its large étendue, 319 m²deg², nearly two orders of magnitude higher than existing telescopes. The étendue is the product of the collecting area of the primary mirror and the field-of-view area and is a measure of how quickly the telescope can survey the sky to a given depth. The telescope camera will view a 9.6 square degree patch of sky with a 3.2 gigapixel focal plane array with 0.2 arcseconds/pixel sampling. Each exposure is planned to be 15 seconds long and this cadence will enable the LSST to map out the entire sky every three nights. This unprecedented capability will open up many new exciting scientific capabilities.

In the first section of this paper, we review the optical design of the Large Synoptic Survey Telescope and its novel wavefront sensor geometry. In the second section, we review curvature wavefront sensors and the algorithm used to evaluate their use on the LSST. In the third section, we evaluate the expected performance of the wavefront sensors under the postulated conditions of operation and demonstrate that this system will perform to the specifications required to meet the LSST performance goals.

1.2 LSST optical design

The wide field-of-view, 3.5 degrees, of the LSST is enabled by three mirrors in a modified Paul-Baker design. The three mirrors consist of a monolithic primary (8.4 m)/ tertiary (5.0 m) mirror, fabricated from a single honeycomb borosilicate substrate, and a 3.4 m diameter convex secondary mirror. The layout of the telescope is shown below in Fig. 1 . In addition to the three mirrors, there are three lenses used to correct and flatten the focal plane and an array of filters to select the desired wavelength band inside the camera [3–5]. To carry out its science missions, the LSST plans to operate with six different wavelength filter bands: u (330–400 nm), g (402–552 nm), r (552–691 nm), i (691–818 nm), z (818–922 nm) and y (950–1060 nm). The filter set g, r, i, z, and y were based on the Pan-STARRS bandpass filter set [6]. The Pan-STARRS filter set of g, r, and i were in turn based upon the Sloan Digital Sky Survey (SDSS) filters [7].

 

Fig. 1 Layout of the Large Synoptic Survey Telescope (LSST). The LSST contains three mirrors in a modified Paul-Baker configuration, along with three refractive corrector lenses, to enable a reasonably well corrected 3.5 degree field-of-view across the flat focal plane array.

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1.3 LSST wavefront sensing

As the telescope surveys the sky over the course of the night, the mirrors experience aberrations which arise from changing gravitational force vectors and thermal drifts. In order to maintain image quality, these slowly changing effects will be corrected by actively controlling the surface shape of each of the mirrors with actuators. Most large, modern telescopes actively control the shape of their primary mirrors; however LSST will be unique because it will be the first telescope with active control of three mirrors, which is necessary to control off-axis aberrations throughout its large field.

The active optics system for LSST will not correct the atmospheric turbulence, as would an adaptive optics system. Adaptive optics systems operate at much higher temporal frequencies to correct for the quickly changing atmospheric turbulence. The integration time for the LSST wavefront sensors of 15 seconds (the same as for the science detectors) is much longer than the atmospheric coherence time – the amount of time the atmosphere is effectively stationary. Since the phase aberrations caused by the atmosphere are never completely static, adaptive optics systems operate with a time interval short enough so that phase variance of the Kolmogorov atmospheric turbulence between the measurement and the correction is less than some small amount, typically one square radian. For usual conditions, the one square radian time delay is 0.314(r_o/v) ~0.015 seconds (where v = 3 m/s is the wind velocity and r_o = 15 cm is the Fried parameter), several orders of magnitude less than the LSST wavefront sensor integration time.

Because three separate mirror surfaces must be controlled, the phase at each of the mirrors must be tomographically reconstructed using wavefront data at different field angles. For each field angle, a reference star provides an illumination source for a wavefront sensor, located in the focal plane of the camera, to collect the wavefront containing information on the telescope aberrations. Figure 2 conceptually shows the tomography geometry for the LSST using three wavefront sensors collecting data from three field angles. The tomographic problem can be reduced to a matrix problem by assuming a circular Zernike expansion of aberrations at each of the mirror surfaces [8–10].

 

Fig. 2 Tomography geometry for LSST (with the telescope mirrors shown schematically as thin lenses). Wavefront data may be collected from stars at different field angles using multiple wavefront sensors.

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For LSST, it would be ineffective to pick off the beam before the shutters or filter for wavefront sensing because of the fast f number of the system (f/1.23). Therefore, the wavefront sensors will share the space in the focal plane with the science detectors and the minimum number of wavefront sensors needed to solve the tomographic problem and achieve optimal image quality should be used to maximize the available area for science data. A previous paper addressed the issue of tomographic reconstruction of the LSST mirror surfaces given a perfect wavefront sensor [10]. The analysis in this previous paper included an expected residual wavefront error of 200 nm rms (root mean square) from atmospheric turbulence integrated over 15 seconds. In this paper, it was determined that four wavefront sensors in a square geometry would be sufficient to correct the mirror aberrations in the telescope in the presence of the expected residual wavefront error. As a result of this analysis, the baseline configuration of the LSST focal plane array was chosen to contain four wavefront sensors, each with an area equivalent to one science detector, 13 × 13 arcminutes, within the 3.5 degree field-of-view, as shown in Fig. 3a .

 

Fig. 3 a) Layout of the LSST focal plane array. The focal plane of the telescope is primarily populated with CCD arrays performing the telescope surveys, represented by small blue squares depicting 4096 × 4096 pixel sensors (16.8 Mpixels). There are 189 of these science sensors, assembled 3 × 3 into 21 rafts (outlined in red), for a total count of 3.2 Gpixels. There are four wavefront sensors, located in the corners of the array and eight guide sensors adjacent to the wavefront sensors. b) Schematic of the corner raft tower concept, which includes one wavefront sensor and two guider sensor packages.

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Curvature wavefront sensors (CWFS) were chosen for LSST because of the simplicity of their implementation and because of their large effective field-of-view. In this article, we investigate the use of curvature wavefront sensors to measure the mirror deformations and describe a new implementation method for curvature sensors. In addition to the active control of the mirror surfaces, the pointing of the telescope will be actively controlled to minimize the effect of tracker errors and two guide sensors will also be located in each of the four corners of the focal plane array, but outside of the target science field of view [11].

A corner raft tower package, holding one wavefront sensor and two guider sensors, as shown in Fig. 3b, will be placed in each of the four corners of the focal plane. The curvature wavefront sensor detectors will use the same type of CCD detectors as the science array. This allows the same infrastructure for data acquisition and control to be used. In addition, the electronics for operating the wavefront sensors and guide sensors are packaged within the volume behind the detectors, similar to the science raft configuration in order to allow the towers to be as similar mechanically and thermally to the science rafts as possible.

As discussed in Ref [10], M1, M2, and M3 can be deformed and moved with five degrees of freedom for each mirror (rotation about the z-axis is excluded). Singular value decomposition is used because there are many possible choices of motion that can achieve an unaberrated wavefront across the field. There are many combinations of motions and deformations that cannot be distinguished, but this is unimportant. All that is important is that it is possible to obtain an unaberrated wavefront across the entire field and that good wavefronts at the wavefront sensor positions guarantee a good wavefront across the entire field. Indeed, when the LSST is tomographically aligned, the solution state may be no closer to the ideal baseline design than the original state, but the solution will give performance across the field that is nearly equal to that of the ideal baseline design.

Because a split detector will be used for the LSST curvature wavefront sensing and because for each curvature wavefront sensor the positions of the two stars that are used will not be known, tip and tilt will not be obtained. Not having the tip and tilt means that the LSST cannot be corrected for distortion. This was a design decision. Even if we had the tip and tilt information, however, we could not use it because five wavefront sensors are required to do the tomographic wavefront alignment with tip and tilt included, and there are only four [10].

2. Curvature wavefront sensors

2.1 Mathematical algorithm

Curvature wavefront sensing was developed as an alternate method of wavefront sensing by François Roddier in 1988 [12]. A curvature wavefront sensor measures the spatial intensity distribution of a star at two positions, one on either side of focus. The difference in intensity between these intra- and extra-focal images is proportional to the Laplacian of the phase of the wavefront, as given by the transport-of-intensity equation (TIE) Eq. (1):

One method for solving the transport-of-intensity equation to find the phase of the wavefront is a modification of the matrix method of Gureyev and Nugent [13], which we have found to be both accurate and fast. There are a variety of other iterative algorithms used to solve the TIE, including successive over relaxation (SOR) and iterative Fast Fourier Transforms (FFT) [14]; however, they may require additional boundary condition data and are potentially slower since they are iterative. There are other algorithms which exist and may offer better performance, but they have not been explored for LSST. The goal of this paper is to show that the curvature technique works for LSST, but it is too early to exclude alternative algorithms which could potentially enhance performance.

The solution for the phase, derived by Gureyev and Nugent in [15] is given by Eq. (2):

The difference between the intra- and extra-focal images is used to form the numerator in the derivative term $\partial I\left(\stackrel{\rightharpoonup }{r}\right)/\partial z$ in F, and the average of the two images is used for the term $I\left(\stackrel{\rightharpoonup }{r}\right)$ found in the matrix M. So as not to be dominated by a huge focus term, the phase reconstruction is done in object space. Thus, the intra- and extra-focal intensity images are relayed to the conjugate locations in object space where the light is collimated, which changes only the coordinates of the images and not their intensity distributions. Therefore, the denominator of the derivative term is the separation$\partial z$of the images in object space.

2.2 Curvature wavefront sensing for LSST

The implementation of the curvature sensor envisioned for the LSST telescope is quite different from what has traditionally been fielded. Traditionally, curvature wavefront sensors have imaged the same star on either side of focus using a membrane mirror that pistons between measurements to change the location of the focus. The curvature sensor implementation envisioned by Chuck Claver for the LSST uses a split detector, one half at an intra-focus location and the other half at an extra-focus location, to image different stars at two nearby field locations. Curvature sensing has also been proposed for the Visible and Infrared Survey Telescope for Astronomy (VISTA) [15,16], although that project utilized a very different design. The geometry of the curvature wavefront sensor configuration for LSST is shown below in Fig. 4 .

 

Fig. 4 a) A three-dimensional view and b) a side view of a schematic of the curvature wavefront sensor (CWFS) located on the focal plane array. The extra- and intra- focus CWFS detectors are displaced by Δz below and above the science focal plane.

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The value of the defocus distance, Δz, needs to be carefully chosen. If the defocus is too small, the wavefront sensor images will be blurred by the atmosphere too much to measure the aberrated wavefront of the telescope. The CWFS sensitivity increases for larger values of defocus, however if the defocus is too much, the light from the reference star will be spread over too many pixels and the noise sources (such as sky background) will degrade the reconstruction. Also, larger amounts of defocus increase the probability that the star images will overlap. We have analyzed these competing effects using simulations to determine that the amount of defocus to effectively balance these effects over changing atmospheric conditions (differing values of r_o), as well as in the different filter bands, should be Δz = 1 mm. This is discussed more in the appendix.

Shack-Hartmann wavefront sensors were also considered for LSST, however, they would be more difficult than curvature sensors to implement. A curvature sensor simply requires two defocused focal planes, while a Shack-Hartmann sensor would require a lens placed below the camera focal plane to reimage the pupil of the telescope onto a lenslet array and a field stop at the wavefront sensor focal plane. This field stop in the Shack-Hartmann sensor limits the field-of-view and would necessitate active tracking of the stars, while tracking is not required for the curvature sensors which have a very large field-of-view as demonstrated below (The LSST does have eight guide star detectors to stabilize the image during each exposure). Each of the intra- and extra-focus detectors in the curvature wavefront sensors has a field of 13 × 6.5 arcminutes, which together as a pair is the same size field as one of the science detectors, as well as the same number of pixels (4096 × 4096). In this large field, there will be many images of defocused stars from which to select for use in the CWFS algorithm. The best images (from the brightest stars without saturated pixels or the ones with the least overlap) can be chosen to optimize the signal-to-noise ratio. Compared to standard curvature wavefront sensors, the sensors envisioned for LSST have the added advantage of no moving parts.

3. CWFS modeling

3.1 Modeling goals

Errors in the recorded wavefront sensor phase propagate through the tomographic wavefront reconstruction and result in errors in the controlled shapes of the telescope mirrors. Bright stars are essential for overcoming the sources that degrade the signal-to-noise ratio in the curvature wavefront sensors for LSST. The fundamental question is: Are there enough field stars of the magnitudes necessary to provide the required wavefront accuracy in each of the four sensors? We have performed extensive modeling of the LSST CWFS to examine the magnitudes of the errors from different sources and verify that there will be enough stars for the wavefront sensors to operate within the required specifications.

The aberrations of the three LSST mirrors, viewed through a particular field angle in the telescope, are described by a wavefront, which we represented by (up to 36) annular Zernike polynomial coefficients. This is the wavefront that we wish to accurately reconstruct during operation by recording images on each side of focus and solving the transport of intensity equation. In order to simulate the wavefront sensor reconstruction, pairs of intra- and extra-focal images were generated using wave optics modeling, implemented in C + + .

3.2 Generation of simulated wavefront sensor images

To generate an image, the combined wavefront phase representing the telescope aberrations and the atmospheric turbulence was Fresnel wave propagated to the defocused image plane. This step was performed twice, at two different wavelengths, and the two resulting images were averaged to approximate the band of wavelengths that forms the real image during operation. This has the effect of averaging out some of the diffraction effects at the edges of the images that are more pronounced in a single frequency simulated image than a real image. Then, the overall brightness of the image was scaled to match the flux from the chosen magnitude star in combination with the telescope transmission and detector quantum efficiency at the chosen wavelength. Other sources of noise were added to the image, including detector read noise (five electrons), one electron/second dark current and sky brightness (described in an upcoming section). Pairs of simulated intra- and extra-focal intensity images, such as in Fig. 5 , were then analyzed to reconstruct the phase.

 

Fig. 5 Simulated intra-focal intensity and extra-focal intensity images used to test the curvature wavefront sensor reconstruction. The defocused images are rings due to the annular pupil shape of LSST and appear speckled due to the simulated atmosphere.

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The difference between the applied annular Zernike coefficients and the best-fit coefficients describe the error in the curvature wavefront sensor reconstruction. The rms reconstruction error was then calculated by summing in quadrature the 4th–36th Zernike coefficients of the wavefront sensor phase error. (Piston, tip and tilt are not measured by the CWFS.)

3.3 Required accuracy and error budget

The image quality error budget for LSST, which will ensure all telescope subsystems will meet performance goals, is currently being refined and has not yet been finalized. The current error budget allocated to wavefront sensor errors is 0.125 arcseconds. This is a root sum square combination of 0.08 arcseconds allocated to atmospheric residuals and 0.10 arcseconds allocated to all other curvature wavefront sensor errors. The atmospheric residuals (time-averaged atmospheric turbulence wavefronts) at the four curvature wavefront sensors result in the telescope tomographic alignment not being perfect. The 0.08 arcsecond number that “corresponds” to good seeing at Cerro Pachón [17] is the no atmosphere FWHM image spot size that results when the telescope is tomographically aligned under good seeing conditions (r ₀ = 17.2 cm at λ = 500 nm, wind = 3.33 meters/sec, turbulence layer at 500 meters altitude). The 0.08 arcsecond number is obtained by ray-tracing the consequently misaligned telescope with no atmosphere over a particular set of field points with particular weightings that has been adopted as a standardized measure of the image quality across the field. We emphasize that the LSST telescope team has not yet decided upon an error budget and that this is simply a reasonably possible error budget.

The image spot FWHM in arcseconds on the sky is obtained by first multiplying the image rms spot radius by a factor of 2.35/sqrt(2) to obtain the FWHM, then multiplying by a factor of 0.02 arcseconds/micron. (The 10 micron pixel spacing corresponds to 0.2 arcseconds on the sky.)

Using the LSST alignment simulator [10], a Monte Carlo simulation was performed to test the effect of wavefront sensor errors in the presence of atmospheric aberrations and realistic mirror misalignment and bending errors. The results are shown in Fig. 6 . All the upper blue diamond-shaped spot sizes were obtained by a three-step process: For each point, first the CWFS wavefront sensor errors were determined. These CWFS errors are due to both the atmospheric residuals for r ₀ = 17.2 cm at λ = 500 nm with a wind of 3.33 meters/second and due to the photon and CCD read noise due to using a dim star (m = 10 to m = 20). Secondly, the LSST was tomographically aligned using the CWFS errors. Thirdly, ray traces were done with no atmosphere to determine the spot sizes at the various field points. The horizontal axisis the amount of added RSS (GQ) wavefront sensor error due to using a faint star (m = 10 to m = 20). The upper blue diamond-shaped data points show the total image spot FWHM for the telescope aligned to the CWFS with errors due to both the atmosphere and due to using a dim star error with m = 10 to m = 20. We emphasize that the atmosphere is always present for all the points plotted in Fig. 6.

 

Fig. 6 Additional curvature wavefront sensor errors due to photon and read noise lead to larger image sizes. Each data point shows the resulting spot size (in the i band) after correcting the telescope for arbitrary mirror deformations and rigid body motions in the presence of a set of wavefront sensor error coefficients and simulated atmosphere. The error bars show the resulting range of image sizes for different realizations of the atmospheres.

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The residual telescope design error of 0.10 arcseconds was subtracted by root difference squared to find the effect of only the additional wavefront reconstruction errors, which were photon and read noise. (The star magnitude was varied from m = 10 to m = 20.) The data is noisy because the particular Zernike aberrations present affect the spot size as well as the amount of rms wavefront error. On average, an image FWHM error budget of < 0.14 arcseconds is achieved for this wavefront sensor configuration with < 200 nm additional wavefront sensor errors combined with the residual atmospheric aberrations for r ₀ = 17.2 cm at λ = 500nm, a wind speed of 3.33 meters/second, and a 15 second exposure. 200 nm is a reasonable goal for the maximum wavefront sensor error because it is similar to the amount of error expected from the residual atmospheric aberrations over a 15 second exposure, as explained in the next section. It is not effective to operate the CWFS at such a high accuracy that would be completely overpowered by atmospheric errors, yet, it would not be ideal to operate with even higher errors than the atmosphere, which would degrade image quality unnecessarily. The curvature wavefront sensor will have a large field-of-view which will allow it to sample many stars over mostly uncorrelated atmosphere. As such, the diagnostic can achieve far better than the 200 nm residual error of the atmosphere by averaging these multiple measurements together.

3.4 Atmospheric simulation

The higher temporal frequencies of atmospheric turbulence for LSST will mostly average out over the integration time. However, observations at the Gemini telescope at Cerro Pachón have indicated that for good seeing conditions there will be a residual atmospheric turbulence level of somewhat more than 200 nm for a 15 second integration time. The residual atmospheric turbulence adds additional phase errors that affect the overall solution for the deformation and rigid body motions of the three telescope mirrors. This source of error was addressed in our previous paper [10], and the aim of the current paper was to study errors in the curvature wavefront reconstruction itself. However, the atmospheric turbulence cannot be ignored because it affects the images in the CWFS detector used for the reconstruction. In order to simulate the image blur and speckle-like patterns (Fig. 5) caused by the atmosphere, an atmospheric phase screen generated using Kolmogorov statistics was added to the phase representing the telescope aberrations at the pupil plane. This is the combined phase that was Fresnel propagated to the two planes of the curvature wavefront sensor to form the instantaneous intra- and extra-focus images. We had no a priori analysis demonstrating that the chosen CWFS algorithm would even be able to successfully use speckle-like images for phase reconstruction for the specific defocus value chosen. However, the atmospherically-distorted images did not cause the reconstruction to break down; the reconstructed phase was the sum of the phase aberration representing the telescope and the atmospheric turbulence, as expected for a successful reconstruction.

To isolate the CWFS phase reconstruction errors from the atmospheric turbulence phase, this process was repeated using the negative of the atmospheric phase screen. Each intra- and extra-focal image was an average of the image formed using the original atmosphere and the image formed using the oppositely signed phase atmosphere. Since we plan to use split CWFS detectors, each intra- and extra-focal image will use a different star (whose light travels through a unique atmosphere) as the source, and different atmospheric phase screens were used for each defocused image. The result of reconstructions done using images formed in the manner was the applied telescope aberration and some small amount of CWFS errors.

Finally, the effect of wind was modeled by shifting the generated atmospheric phase screen a number of times (typically five) to represent an integration time of 15 seconds. At each shift, the phase aberration representing the telescope was combined with the instantaneous atmospheric turbulence as described previously and propagated to the image plane. All of the instantaneous intensity images comprising the fifteen second integration time were averaged together. For the results shown in this paper, the Kolmogorov turbulence was generated using a Fried parameter of r ₀ = 17.2 cm (at λ = 500 nm) and a wind speed of 3.33 m/s (or 50 meter total shift over the 15 second integration time) to represent the average residual atmospheric aberrations of 200 nm over 15 seconds at Cerro Pachón.

In summary, this is what is done in the curvature wavefront sensor simulation: For each position of the frozen phase screen that is carried by the wind and for each wavelength in the discretized spectral band, intensity images are first computed using wave optics for the positive phase screen. Then the intensity images are computed using wave optics for the oppositely signed phase screen. All these intensity images are then averaged together as would happen in the real world and the reconstruction is then done to obtain the aberration coefficients. It is true that averaging with positive and negative phase screens does lead to zero mean turbulence (that’s the idea), or, in other terminology, zero atmospheric residuals, but the speckles remain in all the images. The idea was to determine the errors that are caused by the speckles with no atmospheric residuals. These errors turn out to be very small, typically only several nanometers even for the u band.

High-resolution atmospheric phase screens, which are required to simulate the atmosphere well, result in high-resolution CWFS images (typically 1024 × 1024). Considering that one defocused image is an average of 20 individual images (2 wavelengths × 2 phase and oppositely signed phase screens × 5 instantaneous atmosphere positions to simulate wind), this requires significant computing power. After these high-resolution intra-and extra-focal images are created, they are binned into fewer pixels (163 × 163) to simulate the image size in the detector. (For an f/1.25 beam, the image of a point source, defocused by 1 mm is about 0.8 mm in diameter, or about 80 10-μm pixels.) These CCD-resolution images are saved and used as the baseline images for finding the effect of adding varying amounts of other sources of error in the detector, such as electron or photon noise.

We have also done extensive CWFS reconstructions where we do not use the oppositely signed phase screen and we do indeed recover the residual atmospheric turbulence wavefronts. We can also use the oppositely signed phase screen and add in aberrations so that the aberrations are not zero and we have done this too. The reason that we sometimes use the oppositely signed phase screens are so that we can separate the effects due to the residual atmospheric turbulence and the effects due to there being speckles. Using positive and oppositely signed phase screens also lets us easily add a selected spectrum of aberrations. We have done extensive simulations in which we add a spectrum of thirty-six aberrations to images with turbulence for both positive and oppositely signed phase screens and accurately reconstruct these aberrations (except for piston and tip and tilt). Of course, when we use only positive phase screens and do the CWFS reconstruction we get the known atmospheric residuals. (They are known because we fit annular Zernikes to the phase screen annuli for the curvature wavefront sensors). And if we use only positive phase screens and add aberrations, the curvature wavefront reconstruction gives the atmospheric residuals plus the added aberrations.

3.5 Sky brightness

The night sky is never perfectly dark. Artificial light pollution is a common source of light in the night sky in urban areas, but will not be significant in the remote area of Chile where LSST will operate. Other sources of light in the night sky are airglow, Zodiacal light, Rayleigh and Mie scattered moonlight, scattered light from other sources, and diffuse light from galaxies and faint stars. In order to collect useful science data in the shorter wavelength bands, LSST will generally use these filters on the darker nights when the Rayleigh and Mie scattered moonlight is minimized. Sources of photon noise in the images can be troublesome for CWFS's because as a star goes out of focus, it becomes dimmer, while an out-of-focus sky is still as bright as when in-focus. For source stars of decreasing stellar magnitude (brighter stars), relative to a constant sky background value, the errors in the reconstructed wavefront decrease, as shown in Fig. 7 .

 

Fig. 7 Pairs of curvature WFS images for different magnitude stars are used to reconstruct the applied phase (shown in the upper right). Each of the intra- and extra-focal wavefront sensor CCD images includes the same electronic and sky background noise, which is more apparent for the dimmer source stars (on the left, with larger values of stellar magnitude). The images of dim stars are background limited which causes the applied phase to be poorly reconstructed.

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The detailed results of the LSST Operations Simulator describe a possible cadence of observations over a ten-year period, including the locations on the sky observed each night, which filters were used and other observing conditions, such as sky background and seeing [2]. For purposes of CWFS analysis, the relevant result of this simulation is the expected range of sky brightness for each filter band. Sample histograms showing the expected number of visits versus sky brightness for the u and i bands are shown below in Fig. 8 . The numerical value of the magnitude of the sky background describes the brightness of a star with the same magnitude, with the light spread over one square arcsecond. (These histograms are for an early run of the LSST operations simulator and include some artifacts, such as the spike near magnitude 17 in the i band plot.)

 

Fig. 8 Histograms of observations during a ten-year LSST survey period vs. magnitude of sky brightness in two different filter bands. The magnitudes in the graph are the brightness values per square arcsecond in the wavelength band of interest (Left image: u band, Right image: i band). Courtesy of Kem Cook.

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The observation cadence of the LSST, which is driven by its various science goals, determines the distribution of sky brightness. The main bump where the sky is darkest is what is important. Although the other bumps are “real,” no importance should be attached to them. The sky brightness depends of course upon the time of night, where in the sky one observes, and, if above the horizon, where the moon is in the sky and what its phase is. This figure was created using a particular cadence and a different cadence might result in a figure without the other smaller bumps or with the other bumps in different places.

3.6 Wavefront sensor error vs. stellar magnitude

Reconstructions were performed using pairs of defocused images for a range of stellar magnitudes and sky backgrounds for all of the filter bands. (Although we will use a split curvature wavefront sensor and each source star will be a different magnitude, the simulation here assumed that the same magnitude star for each image in a pair.) Fig. 9 shows the reconstruction error vs. stellar magnitude for the mean expected sky brightness for each band, based on the current LSST cadence simulation.

 

Fig. 9 Phase reconstruction error versus star magnitude for each of the six wavelength bands in LSST. The amount of sky brightness noise added to the images used for the reconstructions corresponds to the mean expected level in each band (values in parentheses) over a 10-year period. Brighter stars (lower stellar magnitude values) have only small amounts of curvature wavefront sensor reconstruction errors. All magnitudes (stellar and background) are specified at the appropriate filter band wavelength.

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The CWFS reconstruction using simulated stars of magnitudes < 10 is very good (the errors follow the asymptote seen on graph at ~10 nm rms), but stars this bright will not be used for curvature sensing. This is because they saturate the CWFS detector and are not very plentiful. The number of stars that exist at each larger magnitude grows geometrically. Most of the stars used for curvature sensing will be in the magnitude range of 12 < m < 17, depending on the wavelength band, and the following section addresses the availability of stars in this range.

3.7 Distribution of stars

Finding suitable natural stars bright enough for wavefront sensing is a common problem for adaptive optics systems. However, this will not be an issue for the LSST curvature sensors, which have a much wider field of view that enables more stars to be found. The question of whether two sufficiently bright stars will always be available for wavefront sensing is best answered by looking at the region of the sky with the fewest stars, the galactic pole. Figure 10 shows the cumulative probability of finding a star of some magnitude or brighter at the north galactic pole in each half of the four CWFS detectors.Thus, the total probability shown is the probability that at least one star is found in a single intra- or extra-focal detector, of area 88 square arcminutes, raised to the eighth power. If instead there were four detectors of areas twice as large (i.e. for a pistoning CWFS), then it would be much easier to find stars for wavefront sensing, in effect shifting the probability curves in Fig. 10 to the left by about two magnitudes. Because stars are different colors, the magnitude describing the stellar brightness varies depending on the wavelength band of the observation. The stellar magnitudes used in Fig. 10 (and throughout the paper) are those expected at the wavelength band of interest for each curve. The model used for star statistics matches the data from SDSS and this is described in detail in the appendix.

 

Fig. 10 Probability of finding a star brighter than magnitude m, in the six wavelength bands, at the north galactic pole in each of eight 88 sq. arcminute detectors. More stars are found near the red end of the spectrum (y) than the blue end (u) due to the abundance of low-mass, red dwarf type stars.

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3.8 Probability of achieving desired wavefront sensor performance

It is essential that there is a very high probability for the curvature wavefront sensor reconstruction error to be less than 200 nm. The cumulative probability of achieving a given rms reconstruction error, shown in Fig. 11 , is found by combining the rms CWFS error vs. stellar magnitude (Fig. 9) and the stellar magnitude vs. star probability (Fig. 10).

 

Fig. 11 Probability vs. CWFS Error at the north galactic pole, the region of the sky with the fewest stars. The probability curves are similar for each of the middle bands (g, r, i) and there are easily enough stars available in this region to fill each of the eight detector halves for curvature wavefront sensor operate well below the 200nm error goal. However, ~5% of fields in the y band and ~15% of fields in the u band will not meet the 200 nm accuracy requirement.

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Even in the most sparsely populated area of the sky, there are enough stars of the magnitudes necessary to provide the required wavefront accuracy in each of the four corner wavefront sensors for each of the middle bands: g, r, i and z. However, at high galactic latitudes, ~5% of fields in the y band and ~15% of fields in the u band will not meet the 200 nm accuracy requirement. In the y band, the lower probability is due primarily to the fact this filter will be used when there is significant sky background, and in the u band, the lower probability is due to the absence of bright field stars. In these cases, if the wavefront sensor errors are too large, the control system can delay mirror adjustments until the next pointing. There will be little degradation in optical performance because the active optics system corrects slowly changing errors. Also, these curves assume only one pair of star images are used in the reconstruction, but performance may be improved by averaging the results from multiple stars that may be available and this is certainly the case in the y band.

3.9 Overlapping stars

While it is essential that there are a sufficient number of bright stars for the CWFS to operate at low noise levels, if there are too many stars, it will be difficult to choose a star whose image is not overlapped by a neighboring star. A defocused image of a pair of closely neighboring stars with overlapping edges causes additional errors in the CWFS reconstruction. Two stars will overlap if they are not at least ~16 arcseconds (80 pixel diameter × 0.2 arcseconds/pixel) apart.

The quality of the phase reconstruction depends on both the amount of overlap and the magnitude of the crowding star. The errors due to the overlap shown in Fig. 12 (50% pupil shift) are higher than for other amounts of pupil shifting, so this represents the worst case. (Stars that are separated by smaller angles will cause small errors in the low order aberrations and stars separated by larger angles only have a small area that is actually overlapping.) For this amount of overlap, the crowding star needs to be five magnitudes dimmer for the phase reconstruction to have an error very much less than 200 nm. Also, stars with only very slight overlaps can possibly use even brighter stars (Δm > 3, perhaps).To see if this is likely to be a problem, we examine the star field in the most crowded area of the sky in the wavelength band with the most stars. Figure 13 shows a typical star field in y band at the galactic equator, with the stars color-coded by magnitude.

 

Fig. 12 A crowding star (shifted by 50% of the diameter) in one of the intra- or extra- focal detectors causes errors in the phase reconstruction. a) The applied phase representing the telescope aberrations (a combination of the first 36 annular Zernike coefficients), b) the intra-focal image with a neighboring star, shown here one magnitude dimmer, causing crowding c) the extra-focal image, with no crowding, and d) the reconstructed phase and CWFS error for a variety of different crowding star magnitudes. As the fraction of the intensity in the crowding star goes lower, the reconstruction improves.

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Fig. 13 A typical star field in y band at galactic equator seen by one pair of intra- and extra-focal CWFS detectors (with |Δz| = 1 mm defocus). The field shown, which is 176 square arcminutes (4096 × 4096 pixels), represents the most crowded area of the sky in the band with the most stars (reddest band) and there are still usable stars wavefront sensing.

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Two stars are needed over this area, one for each intra- and extra-focal image. Although there is only one star with magnitude m < 13 that is completely isolated from all other stars (with m < 18), there are 20 stars of magnitude m < 13, crowded by a second star with relative intensity of less than 1% (Δm > 5). This is the smallest difference in magnitude that causes negligible reconstruction errors. Identifying bright stars with no or negligible crowding to use for curvature wavefront sensing is not a trivial task. Usable stars can possibly be located in the image by examining the second moments of the individual defocused stellar images or they can be identified using a star catalog. While galaxies are common and not included in the defocused star field above, they are not expected to cause significant reconstruction errors because they are dimmer than the sky background when out of focus. Since this analysis demonstrates that even the most crowded fields still have usable stars for wavefront sensing, there is no need for more complicated deconvolution algorithms that could allow star images with crowding to be used.

3.10 Other issues

Compared to traditional curvature wavefront sensors that use images of the same star on either side of focus, the split-detector curvature wavefront sensors envisioned for LSST require some additional steps in the reconstruction algorithm to account for the fact that different stars are used. These steps include registering the two images, normalizing their total intensities and correcting for the vignetting differences between the two stars. All of these issues are discussed in further detail in the appendix.

4. Summary

This paper describes the methodology for measuring the telescope aberrations using a set of curvature wavefront sensors located in the four corners of the LSST camera focal plane. We simulated curvature wavefront sensor images and performed phase reconstructions to demonstrate that this system will meet the specifications necessary for the LSST performance goals. Sufficient numbers of reference stars required for the wavefront sensor operation were shown to be statistically available for any LSST wavelength band and zenith pointing angle. A comprehensive analysis of many sources of phase reconstruction errors was performed and the performance goals were achieved without any averaging to decrease sources of noise. By averaging the results (the individual Zernike aberration coefficients that describe the wavefront) of many different pairs of defocused stellar images (preferably with some sort of weighting algorithm on the star brightness), the errors from noise or residual atmospheric aberrations would be even further reduced.

Appendix A: Additional CWFS reconstruction steps and issues

The appendix describes some of the additional steps in the CWFS reconstruction algorithm necessary because the intra- and extra-focal images come from two different stars, including registration, intensity normalization and vignetting.

A.1 Registration and intensity normalization

In a traditional CWFS, images of the same star are collected on either side of focus so the registration between the two images is known. However since for LSST, the curvature wavefront sensor focal plane is split, each intra- and extra- focal image will be extracted separately from the total star field in the CWFS detector and the registration will be unknown. Before combining the two images to make the average image $I\left(\stackrel{\rightharpoonup }{r}\right)$ or difference image $\partial I\left(\stackrel{\rightharpoonup }{r}\right)/\partial z$ used to solve the transport of intensity equation, they need to be properly aligned, or registered. Initially, the registration can be found by computing the intensity centroid or an area centroid. However, using this initial registration will give non-zero tilt coefficients and since the curvature wavefront sensor is not able to measure tilts (which appear only as displacements in the image plane) the tilt values should be forced to vanish. This can be done by displacing one of the images by a small amount iteratively until the tilt coefficients vanish in the phase reconstruction solution.

Also, since the stars will be different magnitudes, the intensities of the images will need to be normalized. This step involves subtracting the background from each image, which is found in an annular region outside of the defocused pupil region. Background subtraction has the capacity to introduce errors, especially for dim stars that are only slightly brighter than the sky background. Another example where errors will be introduced is when there is a neighboring star, not overlapping the chosen image, but still close enough to be in the surrounding area that is averaged to find the background, though there are many well-known techniques for removing sky using annuli containing stars.

The photocenters that are used to do the initial registration are different than the effective photocenters for the curvature wavefront sensor reconstruction. This is because the photocenters used to do the initial registration are based upon clipping the images at some point on their histograms. This essentially draws contours around the two images. There are no holes. (This is based upon an algorithm that has also been used to do phase unwrapping for phase-shifting interferometry). The initial photocenters are then based upon the intensities in these two areas. The curvature wavefront reconstruction, however, is based upon two circular areas that are larger than these two contoured areas, but are centered upon their photocenters initially.

Annular Zernike polynomials, not circular Zernike polynomials, are used for the curvature wavefront reconstructions. It is true that due to vignetting these are not the exact orthogonal functions on the pupils, but they are pretty close.

A.2 Vignetting

The curvature wavefront sensor algorithm depends on intensity variations between the intra- and extra-focal images. Ideally, the only source of intensity variation for the LSST curvature sensor images would be the wavefront aberrations. Unfortunately, the wavefront sensors are located near the edge of the field-of-view, where the vignetting is the largest and the unique split-detector implementation of curvature sensing for LSST causes each intra- and extra-focal image, located at different field positions to be vignetted differently. If the algorithm does not account for the difference in vignetting between the two images, there will be significant errors in the phase reconstruction. Vignetted pupils for the range of field angles covered by the wavefront sensors in LSST are shown in the Fig. 14 .

 

Fig. 14 The amount of vignetting of the pupil increases toward the edge of the wavefront sensor field. The pupils are shown for the following field angels: a) θ = 1.45° (inside corner of CWFS), b) θ = 1.62° (CWFS area center), c) θ = 1.75° (edge of 3.5° diameter FOV), d) θ = 1.78° (outside corner of CWFS).

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The nominal LSST field-of-view is 3.5 degrees diameter. Vignetting due to the primary and secondary mirrors gradually increases but remains small until vignetting from the camera body optics commences at about 1.75 degrees radius. This vignetting increases very rapidly with further field angle because the spot sizes are small and are at the edges of the L1, L2, filter, and L3 optics, and therefore a further small change in the field angle consequently causes a large change in the vignetting.

Fortunately, potential errors caused by the differences in vignetting can be calibrated out. Since the location of the stars in the image plane is known, the amount of vignetting can be calculated, as well as the effect on the reconstructed Zernike coefficients. The procedure is similar to propagating the applied telescope phase aberration at the pupil plane to the defocused image plane to form the images for CWFS algorithm. However, instead of applying a phase aberration to a uniform intensity distribution in the annular pupil to form the wavefront, a uniform phase with an intensity distribution representing the amount of vignetting for the location of the star in the image plane is applied in the pupil (Fig. 14). This wavefront is then propagated to the defocused image plane for each intra- and extra- focal image and the regular CWFS algorithm is used to find the annular Zernike aberration coefficients. Using these coefficients as correction factors exactly calibrates out the errors that come from vignetted images.

Since the vignetting will never be exactly known due to changing errors in the telescope mirror positions or the limited knowledge of the image location in the detector, we need to quantify the errors due to the amount of vignetting being different from what we think. To model this, we can do reconstructions with pairs of images with an error in the field angle of one of the images (Fig. 15 ). Figure 15 shows that as long as the assumed vignetting is close to the actual vignetting, then the errors will be low. For example, if the position of the star is unknown by an amount of 0.3 arcminutes (or 18 arcseconds, about the diameter of the defocused image), then the errors due to the vignetting will be only 10 nm.

 

Fig. 15 Reconstruction errors due to vignetting of star images. When there is no correction at all for vignetting (blue dashed curve), the errors will be exceeding large. When there is correction for vignetting, but the amount of vignetting is incorrect (pink solid curve), the errors are manageable.

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The stars in Fig. 15 were fixed at 1.45° and 1.75°, but the correction was done assuming an error in the position of the star at 1.45°. The x-axis is the error in the assumed angular position of this star. The sensitivity of the error in the vignetting correction caused by a radial error in the star’s position on the CCD is approximately independent of its position until vignetting by the camera optics starts occurring, which only happens very near the edge of the field. Thus only a small corner section of the 13’ x 13’ split curvature wavefront sensor area should not be used.

A.3 Spider

Obscurations from the spider structure that holds the camera can also affect the reconstruction errors. The effect from the obscuration depends on the location of the star in the field and can be calibrated out in the same way as the vignetting described above. However, there will be a similar error due to any unmodeled shifts in the spider structures.

A.4 Defocus

Small phase variations in the atmosphere due to turbulence cause light to diffract over an angle $\lambda /r_0$ which blurs the image. It is important that the size of this blur at the defocused image plane is small compared to the blur from the aberrations to be measured $r_0\Delta z/f$ [10]. This condition is described by Eq. 4:

(4)$$\frac{\lambda \left(f-\Delta z\right)}{r_0}<<\frac{r_0\Delta z}{f}$$

where λ is the wavelength of the light, f is the focal length and r ₀ is the atmospheric correlation length, Fried’s parameter. Smaller values of r ₀ describe more turbulent atmospheric conditions and thus larger image blurring. To continue to operate the curvature sensors with the same amount of error as the atmosphere worsens, the defocus distance needs to increase.

Figure 16 shows the case where no noise sources are included and the errors continually decrease for larger defocus distances. However, other effects, such as the limited brightness of the star compared to background noise sources counteract this effect. For increasing defocus, the images take up more and more pixels, the signal becomes lost in the noise and the curvature wavefront sensor errors increase.

 

Fig. 16 Large defocus distances enable better curvature wavefront sensor reconstructions (ignoring sources of electron and photon noise and crowding). For small defocus distances, the defocused image covers too few pixels to achieve accurate reconstructions. The defocused image of a point source covers about 80 × 80 pixels in this system when the defocus distance is ± 1 mm. (This plot shows the rms errors for reconstructing 36 annular Zernike coefficients with a power law distribution, using simulated defocused images in the i band.)

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If each entire LSST CWFS detector could piston to different focus positions, the optimal value of defocus could be selected to minimize the noise, based on the current atmospheric conditions and noise due to the combination of star brightness and sky background. In addition, the errors from other sources (registration, background subtraction, intensity normalization, crowding) would decrease. Also, since only one star is needed to obtain both intra- and extra-focal images and the available area to find the star is doubled, there would be a much higher probability of finding bright stars for wavefront sensing. However, these advantages are not enough to counter the increased difficulty and risk of using a pistoning detector, and a split detector with fixed values of +1 mm and −1 mm was chosen because it is a much simpler solution that is expected to achieve the performance specifications almost all of the time.

Appendix B: Effect of wind speed

For a nominal atmosphere of r ₀ = 15 cm at λ = 500 nm, we looked at the effect of wind speed on the image spot size after correction from the CWFS signals. This value of r ₀ is midway between good and typical seeing. One must note that the method used to estimate the overall average spot size can have a significant effect on the results as can be shown in the difference between the purple and green curve in Fig. 17 .

 

Fig. 17 The upper two curves are the average image spot FWHM values for two sets of field points: the four CWFS field point positions and the 24 field point positions defined by using four rings (0.461°, 1.005°, 1.432°, 1.688°) and six spokes. The innermost and outermost rings have half the weighting of the two center rings. The lower two curves have the as-designed telescope 0.14 arcsecond FWHM removed by Gaussian quadrature. Hence the lower two curves give the added GQ image spot FWHM due to the atmosphere.

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Wind reduces both the residuals and the speckle. The speckle can be quite pronounced if there is no wind, especially in the u band, as can be seen from Fig. 18 .

 

Fig. 18 left) u band, no wind, r ₀ = 15 cm at λ = 500 nm, z = + 1 mm., right) u band, wind = 3 m/s, r ₀ = 15 cm at λ = 500 nm, z = + 1 mm.

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Even when the speckle is quite pronounced, the errors caused by the speckle are surprising small, even for reconstruction using 36 annular Zernikes. The two images in Fig. 18 were generated using a normal phase screen followed by the reversed phase screen. The reconstruction errors for Fig. 18 (left, right) were 4 and 3 nm, respectively.

Appendix C: Star counts and star fields

A star distribution model is used that is based upon the tables in Mihalas and Binney [18]. Their Table 4-7 on page 227 gives the star densities in the solar neighborhood by spectral type (O B A F G K M) and by absolute visual magnitude M_V. The scale heights by spectral type and classification are given in their Table 4-16. The authors also give the local interstellar absorption coefficient, its wavelength dependence, and the absorption layer characteristic thickness. A de Vacouleurs spheroid [19] with a density equal to 1/1000 of the density in the sun’s local neighborhood is also used. For simplicity, the distribution by spectral type and by absolute magnitude in the spheroidal population was assumed to be the same as the disk population even though the spheroidal population is known to be older and more evolved. This assumption is probably responsible for predicting a greater number of faint stars in the u band at the north galactic pole (NGP) than are actually observed by SDSS (See Fig. 18). Otherwise good agreement was obtained between the SDSS star counts and the star counts from the model based upon the tables in Mihalas and Binney [18]. The star distribution model also used an in plane scale height of 7.5 kiloparsecs for the disk density so that the disk star density scales as exp(-ρ/7.5) where ρ is the distance in kiloparsecs from the galactic polar axis. But this scale height does not affect the star counts at the NGP.

Here’s how the star counts are obtained: Given a wavelength and a limiting magnitude, for each absolute magnitude the limiting distance is found. This limiting distance takes into account the interstellar absorption along the path. Then the cumulative star count per unit solid angle for that absolute magnitude bin is found by integrating the star density for that absolute magnitude bin along the observation direction to the limiting distance with a weighting factor of r ², where r is the distance along the observation direction. The total cumulative star count per unit solid angle is then found by summing over the absolute magnitude bins. The total differential star count is obtained by taking the difference of two total cumulative star counts.

The star counts have been compared to SDSS star counts [20] and the star counts in Allen’s Astrophysical Quantities [21] in Table 1 . The star counts at the NGP are the most important because this is the direction in which the star counts are least.

Table 1. Comparison of the star densities at the NGP in the visual band between Allen’s Astrophysical Quantities [21] and the model based upon Mihalas and Binney [18]. The model star counts are given for the disk only and also for the two-component disk and spheroidal [19] populations.

View Table

Figure 19 shows that the cumulative NGP star count comparison between SDSS and the model. The u g r i z curves are from SDSS data [20] and the _u _g _r _i _z _y curves are from the model based upon Mihalas and Binney [18]. The same colors are used for the same corresponding curves. Also, the same symbols are used for corresponding curves except the symbols are open for the u g r i z curves and filled for the _u _g _r _i _z _y curves. For fainter stars, the comparison is quite good. The greatest difference is for the u band. The model based upon the Mihalas and Binney [18] tables includes both interstellar absorption and the spheroidal component.

 

Fig. 19 Comparison of the cumulative star counts per square degree at the NGP in the LSST u g r i z bands between NGPcountsSDSSWeistrop.dat [20] and the corresponding star counts obtained from the model based upon the tables in Mihalas and Binney [18]. The spheroidal component is included in the model. The underscored variables are for the model. The _y curve from the model is also shown. SDSS is saturated at the bright end, so the curves were extended using Weistrop’s counts [22] as tabulated by Bahcall and Soneira [23]. These were for the V band. The extensions for the other bands were done by scaling the V band curve. The measured SDSS counts are used for u > 17.10, g > 15.60, r > 15, i > 14.80 and z > 14.60.

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For generating simulated star fields, it is always assumed that the angular extent of the simulated star field is small. Then for each apparent magnitude bin the number of stars in that apparent magnitude bin and in the solid angle of the star field is determined. These stars are then placed randomly in the simulated star field.

Acknowledgments

LSST is a public-private partnership. Funding for design and development activity comes from the National Science Foundation, private donations, grants to universities, and in-kind support at Department of Energy laboratories and other LSSTC Institutional Members. This work is supported by in part the National Science Foundation under Scientific Program Order No. 9 (AST-0551161) and Scientific Program Order No. 1 (AST-0244680) through Cooperative Agreement AST-0132798. Portions of this work are supported by the U.S. Department of Energy under contract DE-AC02-76SF00515 with the Stanford Linear Accelerator Center, contract DE-AC02-98CH10886 with Brookhaven National Laboratory, and with the Lawrence Livermore National Laboratory under the auspices of the U.S. Department of Energy in part under Contract W-7405-Eng-48 and in part under Contract DE-AC52-07NA27344.

Additional funding comes from private donations, grants to universities, and in-kind support at Department of Energy laboratories and other LSSTC Institutional Members.

The authors would like to acknowledge discussions with Tim Gureyev, Chuck Claver and Carmen Carrano.

This document was prepared as an account of work sponsored by an agency of the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

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