Since 1995, more than 300 planets have been discovered orbiting stars other than our sun. This has led to increased interest in extrasolar planetary science among both the public and scientists. While the vast majority of these objects were detected using indirect techniques such as radial-velocity measurements and planetary transits, direct imaging additionally enables spectral characterization.1
Several research groups are developing instrumentation to directly detect thermal emission from ‘hot Jupiters’ forming in orbit around nearby young stars. However, the pathway from Jovian-planet imagers to an Earth-like planet observatory is extremely challenging. First, the planet-to-starlight contrast ratio is less than 10−10. The point-spread function (PSF) of a typical circular telescope, known as the ‘Airy function,’ does not provide the contrast required to see terrestrial extrasolar planets. Imaging thus requires a coronagraphic device (e.g., a shaped-pupil coronagraph, the performance of which we are currently testing) to artificially create high contrast at the Earth-like planet's expected location. This type of instrument modifies a telescope's entrance pupil to produce a PSF that provides the requisite contrast in the image plane.2
Unfortunately, system aberrations cause PSF distortion, even with a perfect coronagraph, thus reducing the contrast by several orders of magnitude. Errors in the reflectivity and surface shape across the system's optical surfaces cause amplitude and phase aberrations that scatter light into the high-contrast planet-detection region in the image plane. Figure 1 shows an example of a shaped pupil, the corresponding PSF (with two discovery regions on either side of the core), and the aberrated PSF.
Figure 1. (a) Shaped pupil. (b) Corresponding point-spread function (PSF). (c) PSF affected by system aberrations. The color bars represent logarithmic contrast ratios. λ/D: Wavelength/aperture diameter.
The technique of adaptive optics aims to compensate for these aberrations using one or more deformable mirrors (DMs). Using actuation behind a reflective surface, the shape of a DM can be adjusted to any desired surface profile, thus enabling phase changes of the light after reflection. The proper choice of DM surface shape can be used to reestablish the desired contrast in a given region of the image plane, commonly referred to as the dark hole (DH). Using a number of estimation and control algorithms, we managed to successfully use one DM to create a DH in one half of the image plane with a contrast in excess of 10−9 at the Jet Propulsion Laboratory's High Contrast Imaging Testbed.3 Because the DM only corrects phase, we can only reject the combined effects of amplitude and phase errors over half the image plane with a single DM.4 It was recently shown that symmetric DH creation, on both sides of the image plane with double the ‘discovery space,’ is achievable by employing two DMs in series.5 Here, we report on laboratory results using two sequential DMs (see Figure 2).
Figure 2. Optical layout at the High-Contrast Imaging Laboratory at Princeton University. DM1 and DM2: Deformable mirrors. OAP1 and OAP2: Off-axis paraboloids.
Figure 3. (top) Aberrated and (middle) corrected images resulting from the stroke-minimization symmetric dark-hole (DH) experiment using two DMs in series. (bottom) Graph showing contrast as a function of iteration in the two DHs individually and combined.
Various approaches have been taken to formulate a high-contrast wavefront-correction algorithm, including minimization of the total energy6 and conjugation of the complex field in the DH.7 Here, we use a stroke-minimizing algorithm,8 where A(x, y) is the amplitude mask in the coronograph's pupil plane and the aberrations in the field are described by the function 1+g(x, y). We assume that a DM is located in the pupil plane and that Fresnel propagation between the two DMs can be accounted for. With ψ(x, y) defined as the phase on the surface of the DM (in radians), we can write the electric field in the pupil plane as
where we have dropped the dependence on x and y, assumed that the DM's surface height is small, and ignored the cross term between the aberration (g) and the DM surface. is a linear operator that takes the electric field in the pupil/DM plane to the plane of interest (often the combination of a series of Fourier transforms and mask multiplications), so that the electric field in the image plane can be approximated by
where we have assumed that the system's PSF is negligibly small in the DH. We represent the DM surface by the sum of basis functions, fk(x, y), i.e.,
and define the vector of basis-function strengths, X= [a1, a2, …, aN]. Since the field in the DH is approximately linear in X, the intensity can be written as a quadratic function.8
The correction algorithm seeks to minimize the stroke of the DM actuators (i.e., the basis-function coefficients) provided that the intensity stays below a given contrast. We find the optimal value of X by taking the partial derivative with respect to X and normalizing the result to zero. Given an estimate of the electric field in the DH, the algorithm can be used to determine the ideal DM surfaces needed to achieve the desired contrast. We obtain this field estimate using a pairwise estimation algorithm with DM diversity.7
In our first experimental results, the corrected image displays a contrast of two orders of magnitude better than the initial, aberrated image. In 60 iterations, the contrast inside the DHs improved from 2.5×10−4 to 3×10−6. Figure 3 shows the aberrated image prior to correction and after 60 iterations of the stroke-minimization correction algorithm, respectively. In addition, the figure shows a graph of contrast versus iteration.
High-contrast imaging aimed at finding Earth-like planets orbiting other stars requires a sophisticated adaptive-optics system with multiple DMs acting in series to compensate for phase and amplitude errors. We have demonstrated a wavefront-correction algorithm that can be used to determine the optimal DM surfaces required to achieve high contrast in the image plane. The limiting factors in our laboratory experiments are the wavefront estimation and the DM model. We are currently developing algorithms that can better estimate the complex electric field in the image plane, even in the face of DM model errors.
Jason Kay, N. Jeremy Kasdin, Tyler Groff, Michael McElwain
High-Contrast Imaging Laboratory
Jason Kay received his bachelor's degrees in optics and applied mathematics in 2004 from the University of Rochester. He is currently in his fifth year of graduate work in the Mechanical and Aerospace Engineering Department. His research is in the field of wavefront estimation and correction and system modeling. He is a recipient of the National Defense Science and Engineering Graduate Fellowship and has a Science, Mathematics And Research for Transformation (SMART) scholarship through the Department of Defense.
N. Jeremy Kasdin received his BSE degree from Princeton University in 1985 and his PhD in Aeronautics and Astronautics from Stanford University in 1991. His research interests are in the areas of space optics, spacecraft design, and control and astrodynamics. He is currently an associate professor in the Mechanical and Aerospace Engineering Department and the principal investigator of the Princeton Terrestrial Planet Finder project.
Tyler Groff received his bachelor's degrees in mechanical engineering and astrophysics in 2007 from Tufts University. He is currently in his second year of graduate study in the Mechanical and Aerospace Engineering Department. His research focuses on coronagraph design and wavefront control.
Michael McElwain is currently the Henry Norris Russell postdoctoral fellow in the Department of Astrophysical Sciences. He earned his PhD in astrophysics from the University of California at Los Angeles in 2007. His research focuses on instrumentation, observational techniques, and scientific analysis for the direct detection of extrasolar planets. He works on ground- and space-based technology development to improve wavefront estimation and control using adaptive optics.
Jet Propulsion Laboratory
Laurent Pueyo is currently a NASA postdoctoral fellow. He received his PhD in mechanical and aerospace engineering from Princeton University in 2008. His research focuses on the design and development of high-contrast imaging technologies for direct detection of exoplanets. He was the recipient of the 2004 Daniel and Florence Guggenheim and the 2007 Harari Fellowships.
1. C. Marois, B. Macintosh, T. Barman, B. Zuckerman, I. Song, J. Patience, D. Lafreniere, R. Doyon, Direct imaging of multiple planets orbiting the star HR 8799, Science 322, no. 5906, pp. 1348-1352, 2008. doi:10.1126/science.1166585