Millisecond exoplanet imaging: II. regression equations and technical discussion.

The leading difficulty in achieving the contrast necessary to directly image exoplanets and associated structures (e.g., protoplanetary disks) at wavelengths ranging from the visible to the infrared is quasi-static speckles (QSSs). QSSs are hard to distinguish from planets at the necessary level of precision to achieve high contrast. QSSs are the result of hardware aberrations that are not compensated for by the adaptive optics (AO) system; these aberrations are called non-common path aberrations (NCPAs). In 2013, Frazin showed how simultaneous millisecond telemetry from the wavefront sensor (WFS) and a science camera behind a stellar coronagraph can be used as input into a regression scheme that simultaneously and self-consistently estimates NCPAs and the sought-after image of the planetary system (exoplanet image). When run in a closed-loop configuration, the WFS measures the corrected wavefront, called the AO residual (AOR) wavefront. The physical principle underlying the regression method is rather simple: when an image is formed at the science camera, the AOR modules both the speckles arising from NCPAs as well as the planetary image. Therefore, the AOR can be used as a probe to estimate NCPA and the exoplanet image via regression techniques. The regression approach is made more difficult by the fact that the AOR is not exactly known since it can be estimated only from the WFS telemetry. The simulations in the Part I paper provide results on the joint regression on NCPAs and the exoplanet image from three different methods, called ideal, naïve, and bias-corrected estimators. The ideal estimator is not physically realizable (it is useful as a benchmark for simulation studies), but the other two are. The ideal estimator uses true AOR values (available in simulation studies), but it treats the noise in focal plane images via standard linearized regression. Naïve regression uses the same regression equations as the ideal estimator, except that it substitutes the estimated values of the AOR for true AOR values in the regression formulas, which can result in problematic biases (however, Part I provides an example in which the naïve estimate makes a useful estimate of NCPAs). The bias-corrected estimator treats the errors in AOR estimates, but it requires the probability distribution that governs the errors in AOR estimates. This paper provides the regression equations for ideal, naïve, and bias-corrected estimators, as well as a supporting technical discussion.

Millisecond exoplanet imaging: I. method and simulation results.

One of the top priorities in observational astronomy is the direct imaging and characterization of extrasolar planets (exoplanets) and planetary systems. Direct images of rocky exoplanets are of particular interest in the search for life beyond the Earth, but they tend to be rather challenging targets since they are orders-of-magnitude dimmer than their host stars and are separated by small angular distances that are comparable to the classical λ/D diffraction limit, even for the coming generation of 30 m class telescopes. Current and planned efforts for ground-based direct imaging of exoplanets combine high-order adaptive optics (AO) with a stellar coronagraph observing at wavelengths ranging from the visible to the mid-IR. The primary barrier to achieving high contrast with current direct imaging methods is quasi-static speckles, caused largely by non-common path aberrations (NCPAs) in the coronagraph optical train. Recent work has demonstrated that millisecond imaging, which effectively "freezes" the atmosphere's turbulent phase screens, should allow the wavefront sensor (WFS) telemetry to be used as a probe of the optical system to measure NCPAs. Starting with a realistic model of a telescope with an AO system and a stellar coronagraph, this paper provides simulations of several closely related regression models that take advantage of millisecond telemetry from the WFS and coronagraph's science camera. The simplest regression model, called the naïve estimator, does not treat the noise and other sources of information loss in the WFS. Despite its flaws, in one of the simulations presented herein, the naïve estimator provides a useful estimate of an NCPA of ∼0.5 radian RMS (≈λ/13), with an accuracy of ∼0.06 radian RMS in 1 min of simulated sky time on a magnitude 8 star. The bias-corrected estimator generalizes the regression model to account for the noise and information loss in the WFS. A simulation of the bias-corrected estimator with 4 min of sky time included an NCPA of ∼0.05 radian RMS (≈λ/130) and an extended exoplanet scene. The joint regression of the bias-corrected estimator simultaneously achieved an NCPA estimate with an accuracy of ∼5×10-3 radian RMS and an estimate of the exoplanet scene that was free of the self-subtraction artifacts typically associated with differential imaging. The 5σ contrast achieved by imaging of the exoplanet scene was ∼1.7×10-4 at a distance of 3λ/D from the star and ∼2.1×10-5 at 10λ/D. These contrast values are comparable to the very best on-sky results obtained from multi-wavelength observations that employ both angular differential imaging (ADI) and spectral differential imaging (SDI). This comparable performance is despite the fact that our simulations are quasi-monochromatic, which makes SDI impossible, nor do they have diurnal field rotation, which makes ADI impossible. The error covariance matrix of the joint regression shows substantial correlations in the exoplanet and NCPA estimation errors, indicating that exoplanet intensity and NCPA need to be estimated self-consistently to achieve high contrast.

Efficient, nonlinear phase estimation with the nonmodulated pyramid wavefront sensor.

The sensitivity of the pyramid wavefront sensor (PyWFS) has made it a popular choice for astronomical adaptive optics (AAO) systems. The PyWFS is at its most sensitive when it is used without modulation of the input beam. In nonmodulated mode, the device is highly nonlinear. Hence, all PyWFS implementations on current AAO systems employ modulation to make the device more linear. The upcoming era of 30-m class telescopes and the demand for ultra-precise wavefront control stemming from science objectives that include direct imaging of exoplanets make using the PyWFS without modulation desirable. This article argues that nonlinear estimation based on Newton's method for nonlinear optimization can be useful for mitigating the effects of nonlinearity in the nonmodulated PyWFS. The proposed approach requires all optical modeling to be pre-computed, which has the advantage of avoiding real-time simulations of beam propagation. Further, the required real-time calculations are amenable to massively parallel computation. Numerical experiments simulate a PyWFS with faces sloped 3.7° to the horizontal, operating at a wavelength of 0.85 µm, and with an index of refraction of 1.45. A singular value analysis shows that the common practice of calculating two "slope" images from the four PyWFS pupil images discards critical information and is unsuitable for the nonmodulated PyWFS simulated here. Instead, this article advocates estimators that use the raw pixel values not only from the four geometrical images of the pupil, but from surrounding pixels as well. The simulations indicate that nonlinear estimation can be effective when the Strehl ratio of the input beam is greater than 0.3, and the improvement relative to linear estimation tends to increase at larger Strehl ratios. At Strehl ratios less than about 0.5, the performances of both the nonlinear and linear estimators are relatively insensitive to noise since they are dominated by nonlinearity error.

Statistical framework for the utilization of simultaneous pupil plane and focal plane telemetry for exoplanet imaging. I. Accounting for aberrations in multiple planes.

A new generation of telescopes with mirror diameters of 20 m or more, called extremely large telescopes (ELTs), has the potential to provide unprecedented imaging and spectroscopy of exoplanetary systems, if the difficulties in achieving the extremely high dynamic range required to differentiate the planetary signal from the star can be overcome to a sufficient degree. Fully utilizing the potential of ELTs for exoplanet imaging will likely require simultaneous and self-consistent determination of both the planetary image and the unknown aberrations in multiple planes of the optical system, using statistical inference based on the wavefront sensor and science camera data streams. This approach promises to overcome the most important systematic errors inherent in the various schemes based on differential imaging, such as angular differential imaging and spectral differential imaging. This paper is the first in a series on this subject, in which a formalism is established for the exoplanet imaging problem, setting the stage for the statistical inference methods to follow in the future. Every effort has been made to be rigorous and complete, so that validity of approximations to be made later can be assessed. Here, the polarimetric image is expressed in terms of aberrations in the various planes of a polarizing telescope with an adaptive optics system. Further, it is shown that current methods that utilize focal plane sensing to correct the speckle field, e.g., electric field conjugation, rely on the tacit assumption that aberrations on multiple optical surfaces can be represented as aberration on a single optical surface, ultimately limiting their potential effectiveness for ground-based astronomy.

Information content of the near field: three-dimensional samples.

We present an analysis of the accuracy and information content of three-dimensional reconstructions of the dielectric susceptibility of a sample from noisy, near-field holographic measurements, such as those made in scanning probe microscopy. Holographic measurements are related to the dielectric susceptibility via a linear operator within the accuracy of the first Born approximation. The maximum-likelihood reconstruction of the dielectric susceptibility is expressed as a linear combination of basis functions determined by singular value decomposition of the weighted measurement operator. Maximum a posteriori estimates based on prior information are also discussed. Semianalytical expressions are given for the likely error due to measurement noise in the basis function coefficients, resulting in effective resolution limits in all three dimensions. These results are illustrated by numerical examples.

Tomographic imaging of dynamic objects with the ensemble Kalman filter.

We address the image formation of a dynamic object from projections by formulating it as a state estimation problem. The problem is solved with the ensemble Kalman filter (EnKF), a Monte Carlo algorithm that is computationally tractable when the state dimension is large. In this paper, we first rigorously address the convergence of the EnKF. Then, the effectiveness of the EnKF is demonstrated in a numerical experiment where a highly variable object is reconstructed from its projections, an imaging modality not yet explored with the EnKF. The results show that the EnKF can yield estimates of almost equal quality as the optimal Kalman filter but at a fraction of the computational effort. Further experiments explore the rate of convergence of the EnKF, its performance relative to an idealized particle filter, and implications of modeling the system dynamics as a random walk.

On Advanced Estimation Techniques for Exoplanet Detection and Characterization Using Ground-based Coronagraphs.

The direct imaging of planets around nearby stars is exceedingly difficult. Only about 14 exoplanets have been imaged to date that have masses less than 13 times that of Jupiter. The next generation of planet-finding coronagraphs, including VLT-SPHERE, the Gemini Planet Imager, Palomar P1640, and Subaru HiCIAO have predicted contrast performance of roughly a thousand times less than would be needed to detect Earth-like planets. In this paper we review the state of the art in exoplanet imaging, most notably the method of Locally Optimized Combination of Images (LOCI), and we investigate the potential of improving the detectability of faint exoplanets through the use of advanced statistical methods based on the concepts of the ideal observer and the Hotelling observer. We propose a formal comparison of techniques using a blind data challenge with an evaluation of performance using the Receiver Operating Characteristic (ROC) and Localization ROC (LROC) curves. We place particular emphasis on the understanding and modeling of realistic sources of measurement noise in ground-based AO-corrected coronagraphs. The work reported in this paper is the result of interactions between the co-authors during a week-long workshop on exoplanet imaging that was held in Squaw Valley, California, in March of 2012.

Information content of the near field: two-dimensional samples.

Limits on the effective resolution of many optical near-field experiments are investigated. The results are applicable to variants of total-internal-reflection microscopy (TIRM), photon-scanning-tunneling microscopy (PSTM), and near-field-scanning-optical microscopy (NSOM) in which the sample is weakly scattering and the direction of illumination may be controlled. Analytical expressions for the variance of the estimate of the complex susceptibility of an unknown two-dimensional object as a function of spatial frequency are obtained for Gaussian and Poisson noise models, and a model-independent measure is examined. The results are used to explore the transition from near-zone to far-zone detection. It is demonstrated that the information content of the measurements made at a distance of even one wavelength away from the sample is already not much different from the information content of the far field.

Computational lens for the near field.

A method is presented to reconstruct the structure of a scattering object from data acquired with a photon scanning tunneling microscope. The data may be understood to form a Gabor type near-field hologram and are obtained at a distance from the sample where the field is defocused and normally uninterpretable. Object structure is obtained by the solution of the inverse scattering problem within the accuracy of a perturbative, two-dimensional model of the object.