HIGH-PRECISION ASTROMETRY WITH A DIFFRACTIVE PUPIL TELESCOPE

Detection and characterization of potentially habitable Earth-mass exoplanets is one of the leading astronomical challenges of our age. Thanks to indirect detection techniques, such as radial velocity, transit photometry, and microlensing, the number of known exoplanets is rapidly growing, providing valuable information on the frequency and diversity of exoplanets. Two approaches—direct planet imaging and host star astrometry—have the potential for revolutionary discoveries by obtaining a complete census of exoplanets around nearby stars and characterizing them:

• 1.  
  Direct imaging of exoplanets with future space telescopes will reveal their atmospheric composition and possibly identify signs of biological activity (Levine et al. 2009).
• 2.  
  Astrometry, by providing a precise measurement of the host star motion on the sky, will yield the planet mass and orbit (Unwin 2005; Shao et al. 2009).

While either technique is suitable to identify nearby planets, both are required for unambiguous characterization of potentially habitable worlds (Shao et al. 2010). It has been so far assumed that coronagraphic imaging/spectroscopic measurement and mass determination require two separate missions. An approach is proposed here that combines the two techniques using a single space telescope in which light is simultaneously fed to a narrow-field coronagraph for exoplanet imaging and spectroscopy and a wide-field astrometric camera imaging a wide annulus around the central field for mass measurement with astrometry.

Astrometric measurement from wide-field images is fundamentally limited, in a perfect system, by photon noise and sampling effects, which are quantified in the Appendix and taken into account in this paper for numerical performance estimates. These fundamental limits are, however, not the focus of this paper, which is aimed at providing a solution to the three main practical challenges to performing precision absolute astrometry of a bright star from a wide-field image when using numerous faint field stars as the astrometric reference.

• 1.  
  Dynamical range. There is a large brightness difference between the central target star and the surrounding field stars, making it difficult for a detector to properly image both.
• 2.  
  Distortions. Slight deformations of the optical surfaces, or the detector focal plane array, introduce astrometric errors. These errors tend to grow larger in amplitude as the field of view (FOV) is increased.
• 3.  
  Detector defects. The geometry and response of pixels are not known to sufficient accuracy to allow high-precision astrometry from a single wide-field image.

The proposed diffractive pupil astrometry optical principle, detailed in Section 2, solves the first two challenges by creating in the wide-field focal plane image diffraction spikes. These spikes are of comparable surface brightness as the field stars (solves the dynamical range challenge) and experience the same distortions as the field stars used as the astrometric reference (solves the distortion challenge).

Detector defects are not calibrated by the diffraction spikes and require averaging of many measurements to be reduced to the desired sub-μas level. Our proposed concept uses a large number (typically hundreds to thousands) of faint (m_V ≈ 14 and fainter) field stars for astrometric referencing and therefore differs from interferometric approaches (Unwin 2005) performing pairwise astrometric measurements using bright stars. The large number of stars used for astrometric referencing averages down by one to two orders of magnitude high-order astrometric errors. As described in Section 3, further averaging is required for sub-μas precision and is achieved by continuously rolling the telescope during observations.

In Section 4, we describe how the astrometric data are acquired and how the measured positions of the field stars are combined toward the final astrometric measurement. Section 5 provides an estimate of the astrometric precision for a medium-sized 1.4 m space telescope with a 0.3 deg² (0.6 deg diameter) FOV.

As shown in Figure 1, the proposed technique uses a conventional wide-field diffraction-limited imaging telescope. The baseline telescope design adopted in this paper uses three mirrors (all conics) to optimize image quality over a wide FOV and is an off-axis system to allow compatibility with all high-performance coronagraph concepts. The central portion of the field is used for coronagraphy and reflected into a coronagraph instrument by a small pick-off mirror in the intermediate focal plane of the system. The rest of the field is imaged by an unfiltered (sensitive from 0.4 μm to 0.8 μm) wide-field diffraction-limited CCD camera at f/40, Nyquist sampled at 0.6 μm. The wide-field off-axis 1.4 m diameter telescope design shown in Figure 1 produces a 0.5 deg × 0.5 deg diffraction-limited wide-field image for astrometric measurement and feeds a coronagraph instrument with a narrow FOV extracted in the intermediate focus. The total on-axis point-spread function (PSF) diameter at the intermediate focus is 6 arcsec: the pick-off mirror must be at least 6 arcsec diameter to avoid vignetting, even for a very small coronagraph FOV. The telescope design was performed with the CodeV software and then coded in a custom C program developed by our team to allow high-precision astrometric computations.

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While the telescope diameter and design adopted are chosen to be cost realistic and inspired from the Pupil mapping Exoplanet Coronagraphic Observatory mission concept study (Guyon et al. 2010), the technique is applicable to other telescope sizes and optical design.

We show in Section 2.2 that, by placing non-reflective dots on the primary mirror (PM), diffraction spikes are created in the wide-field astrometric image to provide a suitable reference (linked to the central star) against which the position of the field stars is accurately measured. We show in Section 2.3 that all astrometric distortions (due, for example, to deformations of the secondary and tertiary mirrors, or deformations of the focal plane array) are by design common to the spikes and the background stars, and a differential astrometric measurement between the diffraction spikes and the field stars is therefore largely immune to large-scale astrometric distortions. Our proposed concept therefore does not require the picometer-level stability (or metrology calibration) on the optics over years that would otherwise be essential for wide-field astrometric imaging (Unwin 2005; Meijer et al. 2009).

As shown in Figure 2 (left), a regular grid of dark (non-reflective) spots is physically etched/engraved on the front surface of the PM. The dots act as a two-dimensional (2D) diffraction grating: the monochromatic system PSF is an Airy pattern surrounded by a widely spaced grid of fainter Airy patterns. In polychromatic light, the secondary Airy patterns are radially dispersed, producing the long diffraction spikes visible in Figure 2 (right). This PSF appears at the focal plane for each field object displaced so it is centered where its star is imaged, respectively, modulated in brightness by the source magnitude. The spikes from the field stars are therefore very dim, while the spikes from the host star are much brighter. Light in the central part of the field is directed to the coronagraph, therefore suppressing its bright central Airy pattern while passing its polychromatic diffraction pattern (aka spikes) to the focal plane array used for astrometry. When the telescope is pointed at a bright star, these spikes will be superimposed on a background of numerous faint field stars used as the astrometric reference. Precise measurement of the position of the bright central star against this background reference is possible by simultaneously imaging on a diffraction-limited wide-field camera both the spikes and the background of faint reference stars. Our approach uses only the images acquired with the wide-field camera to perform the relative astrometric measurement between the central star and a reference consisting of a large number of background faint stars: the central field that can be directed to the coronagraph instrument does not contribute to the astrometric measurement. The system is thus immune to flexure or non-common path errors between the two optical paths, and there is no need to reference the two paths.

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The spikes provide an adequately bright signal for the host star light, solving the contrast problem between the bright central star and the fainter background field stars used as the astrometric reference. By distributing a few percent of the host star's light over a large number of pixels, the spikes provide a feature that can be imaged without saturation on the same detector as the background stars. A similar magnitude compensation scheme using a grating in front of the telescope has previously been used over small angles for ground-based astrometry of binary stars (Strand 1946), and more recently with a grating in a relay pupil for coronagraphic imaging and astrometry of faint companions with adaptive optics (Sivaramakrishnan & Oppenheimer 2006; Marois et al. 2006; Zimmerman et al. 2010). We note that the schemes proposed by Sivaramakrishnan & Oppenheimer (2006) and Marois et al. (2006) are fundamentally different in goals from our proposed diffractive pupil concept, as (1) they are aimed at performing a relative astrometric measurement between a star and its faint companion in a coronagraph, while our concept is aimed at measuring the absolute position of a star against a reference consisting of other stars in a wide-field image; and (2) they do not allow calibration of wide-field distortions that are introduced by large optical elements (main telescope optics), as this requires the first optical element to be the diffractive element.

In a conventional wide-field telescope, a high-precision astrometric measurement is not possible, as small unknown errors in the shape of the optics create astrometric distortions: for different positions on the sky, the beam footprint on optical elements that are not conjugated to the pupil is different, producing tip-tilt anisoplanatism (aka astrometric distortions). Physical distortions of the focal plane array also contribute to astrometric distortions.

The diffraction spikes created by the diffractive pupil encode all instrumental distortions since the reference pattern (diffraction spikes) is introduced directly on the PM of the telescope. The dots on the PM act as a diffraction grating creating secondary beams that emerge from the PM with slightly different angles and travel through the optical system up to the focal plane. Light from an off-axis star and light from a nearby diffraction spike go through nearly the same path in the optical system (telescope + instrument) and share the same astrometric distortion. The anisoplanatism problem is therefore eliminated in the differential spike/background star astrometric measurement, as illustrated in Figure 3. The proposed scheme also calibrates focal plane array distortions, as they will affect equally the background stars and the diffraction spikes. We note that wave front errors on the PM and telescope pointing errors do not directly produce an astrometric error as they are common to both the diffracted beams and the beam from the astrometric reference stars.

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For the spikes to encode exactly the same astrometric distortions as the background field stars, the telescope design must satisfy the following criteria:

• 1.  
  The dots must cover uniformly the PM; otherwise, changes in PM shape can create a differential motion between the spikes and the background stars. For example, if the dots cover only a zone of the PM, the spikes will move with the average wave front slope over the area of the PM covered with dots, while background stars will move with the overall wave front slope over the whole PM.
• 2.  
  The PM must be the aperture stop for the system, so that aberrations introduced by the PM have no field dependence and therefore produce no astrometric distortions (the diffraction spikes can only calibrate distortions introduced by optics after the plane in which the dots are placed).
• 3.  
  There must not be any chromatic optics between the PM and the wide-field camera detector. Refractive optics have some chromaticity, and the spikes are chromatically elongated (a background star and a spike near it therefore have very different colors and could see different distortions in a system with refractive optics). With refractive elements, the color of background stars would need to be known in order to identify which parts of the spikes should be used for high-precision astrometry.

The concept discussed in this paper fulfills these three requirements. For implementations that do not fulfill these requirements, additional error term(s) would need to be quantified, and the estimated astrometric precision obtained in this paper would therefore not be applicable.

Since the PM mask is by design a regular grid containing no low-order aberrations, it does not impact high-contrast coronagraph observations performed by a separate narrow-field instrument (see Figure 4), other than a small loss in throughput: as seen by the coronagraph, the pupil is uniformly gray, with a few percent of the light missing. The effect of the dots is therefore equivalent to a uniform loss in reflectivity in the coating. Small errors in the manufacturing of the dot pattern will, however, introduce low-order amplitude modulations in the pupil and must be kept sufficiently small to be adequately removed by the coronagraph's wave front control system.

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Another potential issue for the coronagraph is the presence of diffraction spikes from other stars in the FOV of the wide-angle camera. Thankfully, the spikes are faint (approximately 1e-8 of the surface brightness of the peak PSF for the baseline concept adopted in this paper) and occupy a small area of the focal plane. In the pessimistic case where a similarly bright star is within the field, the average surface brightness in the coronagraphic FOV is equal to the fractional area covered by the dots divided by the camera FOV (area over which the spikes extend) in units of (λ/D)². With the 1.4 m telescope example considered in this paper, 0.29 deg² (=0.6 deg diameter) FOV, and 1% dot coverage, the average surface brightness in the coronagraphic FOV is at the 2e-11 contrast level, significantly below the expected zodiacal and exozodiacal contrast levels. The effects of spikes can further be reduced by removing from the data set the small fraction of the observations when the spikes are known to move across the central field as the telescope is rolling.

Performing astrometry and coronagraphy with the same telescope is an efficient combination, as both techniques require a stable telescope delivering stable PSFs. A coronagraph mission would observe a small number of bright targets with long exposure times and several visits; this observation mode is also suitable for the astrometric measurement.

Unknown detector imperfections have a large effect on astrometric accuracy and must be taken into consideration in the data acquisition. There are two possible approaches to mitigate this problem.

• 1.  
  Keeping the star(s) on nearly the same pixel position between measurements to perform a differential measurement that is insensitive to static detector imperfections.
• 2.  
  Averaging down detector defects by performing a large number of measurements over different pixels.

The first approach requires a very specific design (Unwin 2005) and would be required to change the detector geometry to follow field stars in an astrometric imaging system. It is otherwise not possible to keep background stars on exactly the same pixels of the detector during the measurement timescale due to the central star's proper motion (which can drag the background stars at approximately 1 arcsec yr⁻¹), its parallax, and the aberration of light. Moreover, using the same limited number of pixels for the astrometric measurement requires very good long-term stability of their response.

If background stars cannot practically be kept on the same pixels, each measurement of the motion of a background star between two epochs compares PSFs falling on different pixels with different unknown characteristics (pixel sensitivity, size, shape, etc.). Many statistically independent measurements are required to average this error term, as the desired sub-μas accuracy corresponds to approximately 10⁻⁵ pixels: with a 1/100th pixel single star single measurement centroiding accuracy, a ≈10³ averaging factor (obtained with 10⁶ uncorrelated measurements) is required to reach sub-μas accuracy. By averaging measurements from a few hundred stars, an averaging factor ≈10 is obtained, and the final measurement is still ≈100 times short of the goal. We therefore achieve the averaging by both combining the position measurements of a large number of background field stars and rolling the telescope along the line of sight to move the background stars' PSFs over a large number of pixels. With a ≈1 rad roll, stars will move over ≈10⁴ pixels, providing the required 100× averaging assuming uncorrelated measurements. As described in Section 3.2, the averaging gain is significantly better than this thanks to a strong roll anticorrelation of the detector errors. The proposed roll geometry is shown in Figure 5.

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The telescope roll is very efficient at reducing the contribution of detector errors to the final astrometric measurement. During a single observation (typically a day), the telescope is slowly rolled around the line of sight to move the background PSFs on the focal plane. On a large format detector, a roll angle of a few tens of degrees is large enough to move the PSF by several thousand pixels, and a 1 rad roll will produce about 10,000 PSF centroid measurements per star.

Unknown flat-field errors produce astrometric measurement errors: a pixel that is more sensitive than assumed will "attract" the measured position, while pixels less sensitive than assumed will "push away" the measured position. Figure 6 shows how this measurement error (red arrows) evolves as the PSF drifts across the detector during telescope roll. The error can be decomposed into a radial component (along the direction to the central star) and an azimuthal component. When integrated over time as the telescope rolls, the residual error is almost entirely radial, as the azimuthal error when the PSF approaches the defective pixel is compensated by the opposite error when the PSF drifts away from the defective pixel. The telescope roll can therefore almost entirely remove flat-field-induced errors along the axis where the astrometric measurement is performed. We note that, as detailed in Section 4, the position measurement in the direction along the spikes is not used, as the spikes' elongation is not suitable for distortion measurement along this axis, and the roll anticorrelation described in this section does not apply.

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The discussion above assumes sensitivity differences between pixels but is also valid for other differences between pixels, such as a pixel with a peculiar color preference, or a pixel with a dead corner. The key advantage of the roll is that it transforms detector-defect-induced astrometric errors into a time-variable error that is strongly anticorrelated on both sides of the defect. These errors therefore disappear in the averaged measurement (this is much better than a decorrelated error that slowly decreases as $\sqrt{N}$).

The data acquired consist, for each observation epoch, of a series of exposures acquired while the telescope is slowly rolling around the line of sight. The individual exposures are kept sufficiently short to avoid smearing of the PSFs at the edge of FOV by more than approximately 1 pixel, but sufficiently long to ensure photon-noise-limited sensitivity. A full sequence of exposures spans several hours to days, while observing epochs are separated by weeks to months. The top left part of Figure 7 illustrates the data acquired for two epochs. The top row shows a sequence of four images acquired at epoch 1 (in an actual observation, the number of individual images would be much greater). The telescope is rolled between each image. While the diffraction spikes are fixed on the detector, the background stars (gray spots labeled 1, 2, and 3 in the top left image of the figure) rotate around the line of sight due to telescope roll. The same sequence of measurements is repeated at epoch 2 (second row).

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The astrometric measurement is performed differentially, between sets of images acquired at different epochs, and must detect a relative motion (translation) between the central star and the field stars. The precise position of the central star is not directly measured; its diffraction spikes are used instead. In the absence of astrometric distortions, the astrometric signal would be a pure field-invariant translation between the two epochs, which would be entirely derived from the measurement of the positions of the field stars on the detector, as described in Section 4.2. As described in Section 4.3, astrometric distortions are measured and corrected for by comparing images of the diffraction spikes between the two epochs.

The displacement of the field stars between the two epochs is measured and shown as small red arrows in the epoch 2 images. The position of each field star on the detector is measured for each exposure by fitting a model of the PSF on a square grid detector. An optimal matched filter taking into account photon noise from both the stellar PSFs and the background (zodiacal light) is assumed for simulations and derivations in this paper, since it yields the best measurement accuracy. The effect of uncalibrated detector defects will be discussed in Section 5. A simpler photocenter algorithm may be used instead but is expected to yield slightly reduced accuracy. We assume in this paper that the position measurement for the background stars is unaffected by the spikes; this approximation is not valid when the star image falls on a spike, which for any given background star occurs for a small fraction of the total observing time.

For the same telescope roll angle (same column in Figure 7), the field stars are located almost (but not exactly) on the same position on the detector at the two epochs. This configuration is largely immune to uncalibrated/unknown PSF shape, as the measurement is differential and the PSF is not expected to change sufficiently over a sub-arcsec field to introduce an astrometric error. Maintaining nearly the same PSF positions between epochs also greatly reduces sensitivity to static astrometric distortions, which cannot be removed from the data (the diffraction spikes are used to measure changes in astrometric distortions but do not measure absolute distortions). The position offset (dx_i, j, dy_i, j), the difference between field star position at two epochs on the focal plane array, is measured for each field star i in the field for each roll angle position j (for which the roll angle is denoted as θ_j).

In the absence of measurement noises, the measured offsets are

$$\begin{equation} dx_{i,\, j} = -dX \cos (\theta _j) - dY \sin (\theta _j) \end{equation} \tag{ 1 }$$

$$\begin{equation} dy_{i,\, j} = -dX \sin (\theta _j) + dY \cos (\theta _j), \end{equation} \tag{ 2 }$$

where (dX, dY) is the astrometric motion of the central star between the two epochs and θ_j is the angle between the sky coordinate system (X, Y) and the camera coordinate system (x, y).

Our proposed calibration of astrometric distortions relies on comparison between diffraction spike images acquired at different epochs and is therefore not sensitive to static astrometric distortions. Only changes in the astrometric distortions between epochs are measured. In the absence of such changes, the diffraction spikes would be kept at exactly the same position on the detector for all observations. Time-variable field distortions are measured by small changes in the location of the spikes and are therefore only sampled along the spikes.

The distortions can only be measured to high accuracy in the direction perpendicular to the spikes due to their strongly elongated shape. The radial component of the spike motion cannot be measured to a good accuracy in the presence of noise. We also note that the highly beneficial anticorrelation effect described in Section 3.2 occurs only on the angular coordinate. For each exposure, only the angular (perpendicular to the spikes) component of the background star's positions therefore contributes to the final astrometric measurement. Each individual astrometric measurement (single telescope roll angle, single background star) is one-dimensional (1D), and the final 2D astrometric measurement is obtained by combining many 1D measurements.

A 2D interpolation is used to build a continuous 2D map of the distortion change (shown in the lower right of Figure 7) between the observation epochs from the measurement of azimuthal distortion along the spikes. This azimuthal distortion map is then removed from the measurements. The final 2D astrometric measurement is obtained by combining all 1D azimuthal measurements with appropriate weighting coefficients (fainter stars where the photon noise contribution is large are given a smaller weight).

This section is aimed at providing an estimate of the astrometric precision of the proposed concept for a baseline design, for which a detailed numerical simulation is performed to quantify the contribution of several key error terms.

A good understanding of how the astrometric error is affected by the brightness of the field stars is essential, as (1) the final astrometric measurement is obtained by combining measurements from background stars covering a large range of brightness and (2) the instrument design and data acquisition may need to be optimized for a relatively narrow range of background star brightness due to technical limitations (detector dynamical range). Our approach is thus to quantify the astrometric information offered by each field star (which is a function of field star brightness) and then to optimally combine the information from all field stars into a single 2D astrometric measurement.

Photon noise on field stars sets a fundamental limit to the astrometric accuracy, and selecting the few brightest stars in the field may therefore appear preferable, rather than optimizing the instrument for a large number of fainter stars. Uncalibrated distortions and systematic errors will, however, require using many stars toward the measurement to average down non-photon noise errors. This is illustrated in Figure 8, which shows for a particularly simple example (fixed astrometric distortion residual equal to 1 μas) how the two error terms (photon noise and non-photon noise) evolve as a function of the field stars' stellar magnitude, assuming that only stars within 0.5 mag of a nominal stellar magnitude are selected. In an actual flight system, a wider range of brightness can be selected—the narrower range is only adopted here to identify which stars contribute the most to the final accuracy. In this simple example, it is assumed that the 1 μas non-photon noise error per star is uncorrelated between stars. Selecting stars around m_V = 19 then offers the best compromise between photon noise and distortions/systematic errors, yielding a single-axis astrometric measurement accuracy below 0.1 μas in a 1 day observation for the 1 deg² (0.6 deg diameter) galactic pole field considered. With reduced systematic errors, brighter stars should be selected, even if they are less numerous, and the overall astrometric accuracy would be better. To estimate the final astrometric measurement error of our concept, it is therefore necessary to quantify measurement noise as a function of field star brightness and optimally select and combine field stars according to their apparent luminosities: this analysis is performed in this section using a numerical model that includes dominant sources of noise.

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When optical design and numerical simulations are required, we chose to adopt a medium-sized 1.4 m space telescope with a 0.3 deg² (0.6 deg diameter) FOV, although the technique could be applied to larger telescopes to offer higher astrometric accuracy. A numerical tool was developed to quantify the measurement accuracy and identify the main terms in the error budget. Section 5.2 identifies possible sources of astrometric measurement error and, when possible, discusses their likely amplitude. These errors guide the numerical simulation, described in Section 5.3, which is designed to quantify the main error terms and produce an overall error budget. Table 1 summarizes the error terms discussed and lists which ones are to be included in the numerical model.

5.2.1. Photon Noise and Related Effects

5.2.2. Fundamental Astronomical Effects

5.2.3. Detector Terms (Static)

5.2.4. Detector Terms (Time Variable)

5.2.5. Other Detector Terms

A numerical simulation was developed to quantify the astrometric accuracy of the system. The top part of Figure 9 gives an overview of the numerical simulation, and a more detailed step-by-step description is given in the bottom part of the same figure. The relevant sources of noise identified in Section 5.2 are included in the model, and the detailed block diagram of the numerical simulation shows where and how they are inserted.

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The model input (telescope, instrument, and detector models) is first established. As detailed in the bottom left block of Figure 9, this includes a model of the focal plane detector array distortion (static and dynamic), a model of the optical surface errors on mirrors M2 and M3 (static and dynamic), and variations in detector flat field. The main model parameters are also chosen, such as telescope diameter and pupil mask geometry. The stellar field is computed in coordinates relative to the central star and is therefore translated between observation epochs according to proper motion and parallax (the actual astrometric signature of a planet is much smaller than proper motion and parallax and therefore has no impact on the noise and astrometric measurement accuracy).

The data simulation is performed using the model inputs and a model of the stellar field to be observed. As shown in the "data simulation" block in the bottom of Figure 9, this step relies on ray tracing to transform the optical surface errors into astrometric distortions and applies these distortions (along with other defects such as detector flat-field variations) to a simulated image of the diffraction spikes. Several sources of errors are also included to compute the set of measured background star positions, consisting of a 2D centroid in detector pixel units for each star, at each roll angle.

The simulated data consist of two simulated diffraction spike images (one with no source of noise or distortion, and one with distortions and noise included) and noisy sets of measured background star positions (one set for each observation epoch).

The data analysis (bottom right box in the bottom of Figure 9) is performed by differentiating the two diffraction spike images to first estimate the noise residual on the astrometric distortion measurement. This step includes a 2D interpolation of a signal that is only present along the 1D spikes to create a full continuous 2D map of angular distortion variations between two observation epochs. The residual distortion error is applied to the noisy pixel coordinate centroiding measurements previously computed. These sets of pixel coordinates are roll averaged and combined into a 2D astrometric measurement. This whole process is repeated for each observation epoch with different noise realizations.

The main parameters of the baseline system are listed in Table 2. The simulation is performed over a 0.1 deg radius field to keep the image size sufficiently small (16k pixels on a side, 1 GB size per image in single precision) for fast computations. The results obtained with the small 0.03 deg² (= 0.1 deg radius) field are then scaled to larger fields assuming that the final measurement error scales as the inverse square root of the FOV, provided that the surface brightness of the diffraction spikes is kept constant (this last requirement implies that scaling small FOV results to larger FOV also requires the fraction of the PM area covered by dots to scale linearly as the FOV).

The model is not a true end-to-end simulation, as it was designed to be as simple as possible while accurately quantifying the relevant sources of errors identified in Table 1 and discussed in Section 5.2. The numerical model does not produce complete simulated images as would be seen by the detector: diffraction spike images are computed without field stars (top left part of the lower chart in Figure 9), and the distortions are measured from these diffraction spike images alone. Errors on the measurement of the position of field stars are treated separately and added to the errors due to the imperfect calibration of distortions by the spikes. This simplification to the numerical simulation is made possible by the non-overlap between sources of errors affecting distortion calibration with the spikes (error terms N5, N16, N17, N25, and N28) and the errors affecting the measurement of the field stars' locations in pixel coordinates (error terms N1, N2, N3, N4, N6, N7, N8, N12, N13, N14, N15, and N25).

Key steps of the numerical simulation are shown in Figure 10. The figure shows static and dynamic distortion maps computed with realistic assumptions about mirror shapes, deformations, and focal plane array shape and stability. A power spectral density measured on previously manufactured mirrors of similar size, curvature, and asphericity was kindly provided by L-3 Tinsley and used for this model. The mirror deformations between observations are assumed to be 40 pm per surface, which is similar to what is experienced on mirrors of this size in optical manufacturing laboratories with 0.25°C temperature control (with mK-level temperature control on orbit, the deformations would likely be much smaller). It is assumed that the true pixel location differs from expectation by 0.2%, with most of this difference at spatial frequencies higher than can be sampled by the diffraction spikes. We note that this is a conservative assumption and that such geometry errors could be calibrated prior to launch. The focal plane array is assumed to have uncalibrated temperature inhomogeneities varying by 20 mK between observations, and the coefficient of thermal expansion of silicon is assumed to convert these temperature differences into deformations of the focal plane. With these assumptions, a ray-trace model of the optical design is used to simulate both the static distortion and its change (dynamic) between observations, as shown by images (a) and (b) in Figure 10. A pair of simulated images of the central bright star as seen by the wide-field camera can then be computed: one with the static distortion only, and one with the static and dynamic distortion terms (images c and d). Differentiating the two images produces the measured angular distortion signal (e), which only contains a high-S/N measurement along the diffraction spikes (f). This signal is then interpolated into a 2D map of measured distortion change between the two observations (g), which is subtracted from the true distortion change to produce the residual error in distortion (h). The telescope roll averages this residual, and the resulting map (i) shows, for each possible field star position in the field, the error due to uncalibrated changes in distortion.

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Our numerical model makes a number of conservative assumptions, often motivated by the desire to keep the numerical simulation simple.

• 1.  
  Galactic pole pointing is assumed, resulting in a low density of background field stars. Only stars are used for astrometric referencing: extragalactic objects, which outnumber stars at m_V = 20 and fainter, are not considered and would help with astrometric referencing at high galactic latitude.
• 2.  
  We assume no calibration of static errors of the system prior to launch. The amplitude of optical and detector errors adopted in our model corresponds to currently available hardware, but we assume no knowledge of these errors. This is a highly conservative assumption, as ground testing can measure these errors, which can then be taken into account in the data analysis. For example, optical surfaces of M2 and M3 would likely be available from the manufacturer, providing a good estimate of static optical distortions. Detector errors (flat field and regularity of pixel positions) can also be measured prior to launch.
• 3.  
  We assume no on-orbit calibration of error, and no attempt to correlate errors with system variables (such as spacecraft Sun angle, target star stellar type) is made. We assume that each observation is analyzed with no knowledge of other observations. This is a very conservative assumption, as static distortions can be partially calibrated by dithered observations (see, for example, Bellini et al. 2011) and changes in the optical system such as optics deformation are likely to be either gradual (can be measured using the previous and next observations) or correlated to identifiable system variables (for example, spacecraft Sun angle) and can be at least partially compensated for by analyzing globally all measurements, including measurements on other targets. We also note that dominant error sources (for example, due to temperature changes in the telescope structure or intermittent detector effects), even if they cannot be predicted, modeled, or linked to a physical process, will produce recognizable signatures in the data, which can be identified by processing all data acquired during the mission and removed from the data prior to the final astrometric measurement. This is made possible by the large redundancy in the measurement, where the positions of many stars, over many locations of the focal plane, are used to compute a few variables (planet orbital parameters and mass(es)). Past and future astrometric missions rely on such techniques to reduce astrometric errors, but for simplicity, we have chosen to not do this, as it is difficult to quantify how much this global analysis of the measurements improves performance without detailed simulations of the disturbances in the telescope and instrument.

Our simple model also makes a number of optimistic assumptions.

• 1.  
  A 100% duty cycle is assumed for the measurement, and the detector readout is assumed to be instantaneous.
• 2.  
  Several detector effects and errors are not included, such as cosmic-ray hits and detector blooming of saturated stars in the field. Detector readout noise is not included in the simulation, and it is therefore assumed that the exposure time is chosen to operate in the photon-noise-limited regime for the field star brightness range offering the best astrometric signal. Charge diffusion and inter-pixel capacitance are not taken into account and can contribute to the detector spatial modulation transfer function, which we assume here is dominated by the pixel size and geometry.
• 3.  
  As described earlier in this section, we compute images of the diffraction spikes without field stars for calibration of astrometric distortions. This assumes that the data analysis can perfectly separate diffraction spikes from field stars. While this assumption is optimistic, we note that discarding parts of each frame where field stars and spikes overlap would result in a very small loss of sensitivity, as field stars are sparse on the focal plane array.
• 4.  
  We have assumed that the diffraction spikes are fixed on the focal plane array between observing epochs and that the distortion change between epochs is measured by comparing the intensity over the same pixels. This approximation implies that static uncalibrated errors in the detector flat field (term N12) have no effect on the distortion measurement and that variations in pixel sensitivities over time (term N16) are the dominant non-photon noise error term. Figure 10 (image (a)) does indicate that variations in optical surfaces will produce 10⁻³-level variations in the location of the spikes, and the focal plane detector array distortions (excluding translation and focus) could be stable to a few times 10⁻³ pixels between epochs (a 100 mK change in temperature non-uniformity on a detector array with 25,000 pixels across the field radius will produce a shift under 10⁻³ pixels at the edge of the field). The assumption that spikes are static on the detector therefore seems reasonable at the few that 10⁻³ pixel level, assuming that the telescope pointing is driven to position the diffraction spikes on a fixed location, which removes the first-order effect of optical alignment errors. We note that even if the spikes were moved between epochs by a pixel or more, pixelation errors at the 1/100 pixel level per λ/D long spike segment (corresponding to percent-level uncalibrated pixel responses) would be averaged down to approximately 0.1 μas thanks to the number of spikes (≈100) and their length (≈10, 000 resolution elements).

Misalignments between the optical elements of the telescopes are not considered in the error budget but are discussed separately in Section 5.6, where tolerances on the rigid body motions of optics are estimated.

Finally, we do not consider in our model variations of the astrometric distortions during a single observation. Our error budget only considers variation between different observations of the same source. This simplification should not affect the final astrometric measurement, as variations during a single measurement are averaged through the telescope roll, even if this averaging is, in the case of field stars, both spatial and temporal due to the roll of the telescope.

Rigid body motions of the telescope optics create variations in the instrumental distortion, affecting the position of both the diffraction spikes and the field stars on the focal plane detector array.

The effect of rigid body motions of optical elements on diffraction spike motion is quantified in Table 5, and the morphology of these distortions is shown in Figure 12. For each alignment error (rotation or translation of an individual component), high-accuracy ray tracing was used to compute astrometric distortion variation, with ≈3e6 rays per source and 1000 sources across the FOV. For each source, ray tracing was performed with and without the alignment error, and the difference between the locations of the two images in the focal plane constitutes the astrometric distortion variation. The average value of this distortion across the field (global pointing offset) is first removed, as the telescope would be pointed to locate the spikes on the same pixels between observations. The global roll term (rotating the spikes' pattern) is not removed, as it is assumed that the focal plane array cannot be rotated against the telescope PM. The residual distortion is then measured in the angular coordinate only (the spikes are radially elongated, and a radial shift of their positions does not have an effect on the measurement). The values given in the second column of Table 5 are, for each mode, the angular distortion variation (rms averaged across the field) in detector pixel per unit of rigid body motion. For example, a 1 mm translation of M2 along the x-axis will move the diffraction spikes by 0.502 pixel rms in the focal plane. We have assumed for this table the baseline design (1.4 m telescope, 44 mas pixel size) used in this paper. The third column gives for each alignment mode the amount of rigid body motion required to produce a 0.1 pixel rms motion of the spikes, assuming that only a single misalignment mode is present.

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As shown in Figure 12, the distortion variations induced by alignment errors are of low order and are therefore well sampled by the motion of the diffraction spikes they produce. To compute a 2D map of angular distortions from images of the spikes acquired at different epochs, the data analysis proposed in this paper performs a geometrical interpolation between the spikes. This approach is well suited when no a priori information is available about the nature of the distortions, and it is used in the numerical simulation.

Astrometric distortions induced by alignment errors consist of a limited number of modes (shown in Figure 12) and are likely to be the dominant source of astrometric distortion variation. Table 5 shows that a 10 μm motion of M2 or M3 produces a 1/100th pixel variation in astrometric distortion, which is approximately 10× larger than the dynamic distortion anticipated from changes of the mirrors' optical figure and detector deformations (image (a) in Figure 10) combined.

A more appropriate approach is to first fit and subtract the known alignment-induced distortion modes from the measured diffraction spike motion and then perform the 2D interpolation on the residual spike motion. There are 16 alignment modes: with the PM serving as a reference, there are 5 modes for M2 and M3 each (3 translations and 2 rotations; the rotation around the conic surface vertex is not included due to rotational symmetry of the surfaces) and 6 modes for the detector plane. Since this is a small number of modes to fit compared to the large number of measurements (number of pixels illuminated by the spikes), this modal subtraction should not otherwise affect the measurement.

We have assumed, as discussed in Section 5.3.1, that the diffraction spikes fall on the same pixels for all observations and that uncalibrated static detector inter- and intra-pixel sensitivity errors do not contribute to errors in the spike position measurement. As noted in Section 5.3.1, if the spikes fall on different pixels between observations, we may assume that percent-level errors in the detector flat field at high spatial frequency will produce a pixelation centroid error of approximately 1/100th of a pixel for each λ/D long segment of a spike, and the overall resulting astrometric error would be at about 0.1 μas thanks to the averaging of many such segments. For this error term to be well below 0.1 μas, as assumed in our error budget, the spikes have to fall on the same pixels for all observations.

Table 5 shows that the positions of optical elements need to be stable at the 100 μm level (translation)/10 arcsec level (rotation) for the spikes to stay on the same pixel location on the detector between observations, at the 1/10th of a pixel level. With the tolerances given in the table, a quadratic sum of the distortions would produce a 0.35 pixel rms displacement of the spikes on the focal plane array. This amount of displacement is sufficiently small to allow the same pixels to be used for the spike location measurement between observations, therefore removing the effect of unknown pixel-to-pixel sensitivity variations on the spike position measurement accuracy—as assumed in our error budget. Unknown intra-pixel sensitivity variations will, however, still impact the measurement with the 0.35 pixel rms spike motion, but this second-order effect will be significantly smaller than the first-order pixel-to-pixel flat-field errors and should therefore not contribute to astrometric errors at the 0.1 μas level.

With an ideal telescope (no wave front aberration, no central obstruction), the accuracy with which a monochromatic PSF (exact Airy function) can be localized in 2D is, in the photon noise regime,

$$\begin{equation} \sigma _{2{\rm D}} [\lambda /D] = \sqrt{2}/(\pi \sqrt{N_{{\rm ph}}}) [\lambda /D] = 0.450/\sqrt{N_{{\rm ph}}} [\lambda /D], \end{equation} \tag{ A1 }$$

where brackets indicate the unit, D is the telescope diameter, λ is the wavelength, and N_ph is the total number of photons available for the measurement. If the PSF position is measured along one axis only, the measurement error is

$$\begin{equation} \sigma _{1{\rm D}} [\lambda /D] = 1/(\pi \sqrt{N_{{\rm ph}}}) [\lambda /D] = 0.318/\sqrt{N_{{\rm ph}}} [\lambda /D]. \end{equation} \tag{ A2 }$$

These measurement accuracies are obtained with an optimal matched filter that optimally weights each pixel according to S/N. Numerically these equations can be verified by starting from the PSF (noted PSF(x, y)) and computing

$$\begin{eqnarray} (1/\sigma _x)^2 &=& \sum _{x,y} (1/\sigma _x(x,y))^2\nonumber\\ &=& \sum _{x,y} \left(\left(\frac{\delta {\rm PSF}(x,y)}{\delta x}\right)^2 \bigg/{\rm PSF}(x,y) \right), \end{eqnarray} \tag{ A3 }$$

with σ_1D = σ_x = σ_y and $\sigma _{2{\rm D}} = \sqrt{2} \,{\times}\, \sigma _{1{\rm D}}$. Equation (A3) shows for each pixel the signal (equal to δPSF(x, y)/δx) and the noise (equal to $\sqrt{{\rm PSF}(x,y)}$ in the photon noise regime considered here).

Equations (A1) and (A2) assume a monochromatic PSF, but a real polychromatic PSF is not as sharp and will lead to a lower astrometric accuracy for the same total number of photons (as shown in Figure 13). This performance loss is, however, modest, especially compared to the sensitivity gain offered by including more flux as the spectral bandwidth increases. Figure 13 shows that with a 50% spectral bandwidth, the astrometric error is 38% larger than it would be in monochromatic light with the same number of photons.

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The astrometric error induced by photon noise is well described by the following analytical fit, derived empirically from the points in Figure 14, and also shown in the figure:

$$\begin{equation} \sigma _{1{\rm D}} [\lambda /D] \,{=}\, N_{{\rm ph}}^{1/2} \,{\times}\, (1/\pi \,{+}\, 0.1 \,{\times}\, {\it Sampling}\ {\it Factor}^{-1.2}) [\lambda /D], \end{equation} \tag{ A4 }$$

where SamplingFactor is 0.5 times the linear number of pixels per λ/D and is therefore equal to 1 for Nyquist sampling (SamplingFactor = 1.5 means 3 pixels per λ/D). Figure 14 shows that astrometric accuracy in the photon noise regime continues to improve as the sampling increases beyond the Nyquist limit. Provided that a good model of the PSF exists, the astrometric accuracy remains quite good below Nyquist sampling: at 0.4× Nyquist (pixel linear size >λ/D), the 2D astrometric accuracy is still below $1/\sqrt{N_{{\rm ph}}} \lambda /D$. Below 0.4 Nyquist, the astrometric accuracy is a function of the exact PSF location, with optimal sensitivity when the PSF falls on the corner between pixels.

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The combined effect of finite detector sampling and polychromaticity on astrometric accuracy is shown in Figure 15 in the absence of a background (no zodiacal light). For this figure, the flux is assumed to be constant in units of photon per wavelength bin from 0.52 to 0.78 μm and decreases continuously outside this band, with the 50% level at 0.47 and 0.83 μm and the 0% level at 0.4 and 0.9 μm. For the sampling and astrometric accuracy units, λ = 0.6 μm is assumed.

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