With more than 205 extra-solar planets having been detected to date, interest in extra-solar planet science has exploded. All of these detections have been of planets much larger than Earth (most larger than Jupiter), a fact due to the limitations of the indirect methods being used (stellar wobble, transits, and gravitational lensing). While upcoming space missions—e.g., Convection Rotation and planetary Transits (CoRoT); Kepler; and the Space Interferometry Mission—will have the ability to detect some Earth-size terrestrial planets, they will not be able to characterize these planets nor will they perform an exhaustive search. For that, direct imaging is necessary. NASA plans two missions with the goal of directly detecting Earth-like planets: the Terrestrial Planet Finder-Interferometer (TPF-I) and the Terrestrial Planet Finder-Coronagraph (TPF-C). Each works in a different wavelength range and provides complementary scientific information.

Detecting Earth-size planets is difficult for two reasons. First, the contrast ratio between the planet- and the starlight is less than 10^{−10}; this is far greater contrast than provided by the Airy function of a reasonably sized telescope. For TPF-C, modifications of the point spread function (PSF) will achieve this contrast via some form of coronagraphy. One such approach being given serious consideration is the shaped-pupil coronagraph^{1}. The shaped pupil is a binary apodization in the entrance pupil of the telescope that produces a PSF with regions of extremely high contrast. The center frame of Figure 1 shows the image of a star (the PSF) when using the shaped pupil in the left frame, for example.

**Figure 1.** Left: binary-shaped pupil. Center, right: ideal image of a star showing a log scale of the contrast. Right: image of a star in the presence of system aberrations in log scale. λ is the wavelength of light, and D is the diameter of the telescope's primary mirror. Colorbars in the center and right frames show the log of the contrast between a point in the image and the central peak.

Second, achieving high contrast arises from system aberrations that produce a halo of light in the high-contrast region of the coronagraph. Both amplitude aberrations (caused by non-uniformity of reflectivity across the optical surfaces) and phase aberrations (caused by deformations in the shape of the optical surfaces) scatter light throughout the system and create a degraded image. The right frame in Figure 1 shows the degradation of a shaped pupil PSF in the presence of system aberrations. This scattered halo is often called ‘speckle’.

Additional blurring, as seen in the top right frame of Figure 2 arises from considering a broadband source: multiple wavelengths will result in speckle at slightly different locations in the image plane. This blurring further complicates the task of managing the speckle interference. The goal of TPF-C is to observe in bands no smaller than 10% bandwidth; narrower bands will result in either many multiple channels, increasing cost and complexity, or dramatically increased mission time as each narrow band is observed. A narrow band also comes at the expense of system throughput, further increasing observation time.

The only solution to the speckle interference is to correct the aberrated wavefront produced by the errors in the system. This is accomplished via adaptive optics (AO). The goal is to create a ‘dark hole’ in a region of the image plane close to the star where we search for planets.^{2} By dark hole we mean a predefined search space in the image plane where the AO system creates the needed contrast. Only a single deformable mirror (DM) is necessary to perform this correction in broadband light. The first algorithm for doing so is called ‘speckle nulling’. This algorithm works by forming a sinusoidal ripple on the DM with a frequency and amplitude corresponding to the location and intensity of the brightest pixel in the image plane.^{3} While speckle nulling has been proven in the laboratory, it requires a large number of iterations when speckles are close to one another. In this paper we present a new approach based on minimizing the energy across the entire search region at once, dramatically reducing the time needed for correction. In the remainder of this note we describe our implementation of such an algorithm.

**Figure 2.** Top left: aberrated image in monochromatic light. Top right: aberrated image in broadband light with 10% bandwidth about 600nm. Bottom left: reimaged pupil plane intensity pattern in monochromatic light. Bottom right: reimaged pupil plane intensity pattern in broadband light with 10% bandwidth. Note that while using broadband light has the effect of smearing the intensity pattern in the image plane, that effect is almost completely undone in the reimaged pupil plane.

**Wavefront correction** The wavefront correction algorithm, designed to achieve high contrast in minimum time, consists of two stages: a reconstruction stage, where the complex wavefront is estimated from multiple measurements in the reimaged pupil plane; and a correction stage, where those measurements are used to determine the DM setting. The correction stage in this algorithm is based on a method developed by Bordé et al.^{4} The main difference in this application of their correction method is that we do not assume a first-order expansion of the aberrations. Moreover, the method developed by Bordé describes an open-loop correction method for a Lyot coronagraph while the correction method described here is a general closed-loop method for many classes of high-contrast imaging systems. While we present a complete, closed-loop implementation, it is useful to recognize that these two stages are independent in the sense that the wavefront reconstruction could be used with any correction algorithm and vice versa.

We begin by describing the correction algorithm as it sets the requirements on the reconstruction stage. Then we describe the ‘Peek-a-boo’ algorithm for forming the needed complex field estimation.

**Energy minimization-based correction** Let the effect of the optical system be modeled as the linear transformation, *C*, between the electric field at the DM plane, *E*_{0}, and the electric field at the science camera plane, *E*_{f}, where the measurements will take place,

For the system in Figure 3, , where denotes the Fourier transform, *S* is the shaped-pupil mask, and *M* is the image plane mask. Light is incident on the DM and then reflected to the shaped pupil, which is in a plane conjugate to the DM (relay optics are not shown in the figure). Once passing through the shaped pupil, the light is focused to an image. At the image plane, we place a binary mask that passes only the light in the region where we would like to correct and search for a planet. This light is then reformed into a reimaged pupil where the measurements are taken. The field at the DM can be modeled as:

where *A* is the unaberrated, ideal, electric field (including the effect of the shaped pupil), λ is the wavelength, α and β are the amplitude and phase aberrations, respectively, and ψ is the DM surface height.

**Figure 3**. In this system, light is incident onto the DM. It is then reflected to the shaped pupil and focused to an image. At the image plane, we place a mask that passes only the light in the region we want to correct. This light is then brought to a second pupil plane where the measurements are taken.

Using the the first two terms in the Taylor expansion for and Equations (1) and (2), the electric field at the science camera plane can be approximated as:

where and we assume the cross term is negligible since the coronagraph operator creates extremely high contrast in the search region by design. The total energy in the dark zone is given by:

where the inner product is defined as and the asterisk represents the complex conjugate.

Letting the DM have *N*_{dm}× N_{dm} actuators, and using an influence function model for its surface, ψ can be written as:

where *a*_{k,l} is the *kl*^{th} coefficient and is the DM's influence function, that is, the surface height for a single activated actuator, centered at the location of the *kl*^{th} actuator.

The problem is now reduced to finding the appropriate coefficients, *a*_{k,l}, for the DM that minimizes the energy in the dark hole. This is done by setting , or:

To solve for the coefficients from the minimization condition in Equation (6), we need to estimate the complex, valued electric field in the CCD plane (the reimaged pupil plane in Figure 3) of an uncorrected, aberrated system given by . With such an estimate, Equation (6) reduces to a simple matrix problem that can be solved for the coefficients using a pseudo-inverse equation.

Why minimize the energy to achieve a dark hole? We recognize that given a perfect DM, that is, a DM with a surface that can be controlled to any arbitrary shape with arbitrary accuracy, and given a deterministic model of the system, the intensity in the dark hole could be reduced to zero through a direct inversion. Since that is more contrast than is necessary, particularly given the other interfering light in the discovery region (such as exo-zodiacal emissions from the host solar system), one might consider other minimizations, such as voltages on the actuators, under a desired contrast constraint. We are studying such approaches. Nevertheless, not only is the DM not perfect, but our model of the DM is only approximate, and our first-order expansion of the energy limits what can be achieved at any given step. The minimum-energy approach we use has the advantage of reducing to a quadratic equation that can easily and quickly be solved and, under our various approximations, results in a contrast slightly better than we desire.

**The ‘peek-a-boo’ estimation method** As we described above, solving Equation (6) for the actuator commands requires knowledge of the complex electric field in the pupil plane. Unfortunately, the camera measurement provides only the intensity, or magnitude squared, of the electric field at the image. Determining the full complex field requires at least three different intensity measurements to provide enough diversity to extract the real and imaginary parts and the sign. This diversity must also be known *a priori* so that it can be subtracted from the calculation. Our proposed ‘Peek-a-boo’ algorithm uses three intensity measurements in the re-imaged pupil plane, thus allowing reconstruction of the complex electric field. The first measurement is taken by placing a mask in the image plane that allows only the light in the correction region to pass through. The second measurement is taken by using the same mask with a pinhole of known size added to the center. The third and final measurement is taken by using the same image plane mask again, but with a π/2-phase shift added to the pinhole (quarter-wave plate). The pinhole provides the necessary diversity, and its effect can be mathematically accounted for because the solution for the field propagating through a pinhole is well known. Once these three measurements (*I*_{1}, *I*_{2}, and *I*_{3} respectively) have been taken, can be reconstructed via:

where *h* is the diameter of the circular pinhole.

**Simulation results** In Figure 4, we present simulation results of the full estimation and correction algorithm in a 10% bandwdth about 600nm. Convergence to the desired contrast is achieved after only five iterations, where each consists of taking the three measurements for the estimation described above and performing the optimal correction algorithm. The algorithm is iterative because higher order terms neglected in the first-order Taylor expansion of the DM surface limit the accuracy at each step. As the bandwidth increases beyond 8%, uncorrected light leaks in through the image plane mask due to the smearing of speckles in broadband light. To overcome this, we decreased the size of the image plane mask with each iteration, letting through a slightly smaller region.

**Figure 4**.Top left: the aberrated image with a 10% bandwidth about 600nm. Moving to the right and then the second row: the first five iterations of the estimation and correction algorithm. By the fifth iteration, the dark zone on the right-hand side of the figure contains a dark hole with the necessary contrast of below 10^{10}.

**Conclusion and future work**

The energy minimization-based correction algorithm and the Peek-a-boo estimation algorithm demonstrate how a single DM can be used to create a dark hole, a region in the image plane with the contrast necessary to directly detect extra-solar Earth-like planets. The simulations shown in Figure 4 have overcome the problems of system aberrations and image blurring caused by broadband measurements. This work is the first simulation of broadband wavefront sensing and control at levels required by TPF-C.

The technique is applicable over at least a 10% bandpass around 600nm. We will study the effects of noise sources (e.g. shot noise), detector linearity, drift, mask imperfections, and DM calibration and response errors to determine the true robustness of this approach. We also will attempt to employ this approach on coronagraph testbeds at Princeton University and Jet Propulsion Laboratory in the near future.

Amir Give'on

Infrared Processing and Analysis Center

California Institute of Technology

Pasadena, CA

Jason Kay, Jeremy Kasdin

Mechanical and Aerospace Engineering

Princeton University

Princeton, NJ

Stuart Shaklan

Jet Propulsion Laboratory

Pasadena, CA