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Sub-Aperture Piston Phase Diversity for Segmented and Multi-Aperture Systems

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Sub-Aperture Piston Phase Diversity for Segmented and Multi-Aperture Systems Sub-aperture piston phase diversity for segmented and multi-aperture systems Matthew R. Bolcar* and James R. Fienup The Institute of Optics, The University of Rochester, Rochester, New York 14627, USA *Corresponding author: [email protected] Received 2 June 2008; accepted 7 July 2008; posted 27 August 2008 (Doc. ID 96884); published 24 September 2008 Phase diversity is a method of image-based wavefront sensing that simultaneously estimates the un- known phase aberrations of an imaging system along with an image of the object. To perform this es- timation a series of images differing by a known aberration, typically defocus, are used. In this paper we present a new method of introducing the diversity unique to segmented and multi-aperture systems in which individual segments or sub-apertures are pistoned with respect to one another. We compare this new diversity with the conventional focus diversity. © 2009 Optical Society of America OCIS codes: 100.5070, 100.3020, 110.6770, 120.0120. 1. Introduction array. Some form of actuation is necessary to main- Modern ground-based and space-based observatories tain equivalent optical path distances (OPDs) be- are nearing the limits on the size and weight of tween the segments or sub-apertures. In the case monolithic primary mirrors, hindering an increase of segmented systems such as the JWST, actuation in both light collecting efficiency and imaging resolu- is achieved by mounting each hexagonal segment tion. In response, technology is heading in the direc- on a hexapod, allowing for seven degrees of freedom: tion of segmented and multi-aperture systems as the x and y translation, clocking, piston, tip, tilt, next generation of telescopes. Segmented systems and intra-segment radius of curvature. For multi- such as the Keck Observatory in Hawaii, the Thirty aperture systems, a system of optical delay lines Meter Telescope (TMT), and the James Webb Space equalizes the path length between each sub- Telescope (JWST) to be launched after 2013 use an aperture. These optical delay lines are tunable, al- array of actuated hexagonal segments to create a pri- lowing for adjustments of sub-aperture piston phase, mary mirror that is easier to fabricate than a mono- and in some cases tip and tilt. lithic mirror of equivalent size, and, in the case of the It is necessary to know the state of each segment or JWST, capable of being stowed in a launch vehicle sub-aperture to within a small fraction of a wave- and deployed in orbit. Proposed multi-aperture sys- length in order to apply the appropriate corrections tems, such as the Terrestrial Planet Finder Interfe- to the actuators. Given the complexity of modern seg- rometer [1], Keck Interferometer, and MIDAS [2] use mented and multi-aperture systems, laser interfero- an array of afocal telescopes that are interferometri- metry is not a practical method of wavefront sensing; cally combined to achieve a resolution comparable to furthermore, frequent recalibration is likely neces- a primary mirror equivalent to the entire array size. sary due to drift in the mechanical mounting and In either configuration, the segments or sub- actuating systems. Shack–Hartmann wavefront sen- apertures must be aligned to very tight tolerances sors have proved useful for large ground-based sys- in order to achieve the resolution benefit of the entire tems, but generally do not work as well for extended objects or for wavefronts having high complexity. Laser guide stars can provide an isolated point 0003-6935/09/0100A5-08$15.00/0 source for ground-based systems, but do not work © 2009 Optical Society of America 1 January 2009 / Vol. 48, No. 1 / APPLIED OPTICS A5 for space-based systems since they use the properties function (PSF) at wavelength λ, nk is the noise in the of the sodium layer of the upper atmosphere. kth image, and * denotes a convolution. The intensity Image-based wavefront sensing techniques such as PSF is the magnitude squared of the coherent phase retrieval and phase diversity allow for an es- impulse response, timate of the pupil phase to be made directly from 2 images produced by the system and have been iden- sk;λðu; vÞ¼jhk;λðu; vÞj ; ð2Þ tified as enabling technologies for space-based sys- tems, where the aberrations do not change with which in turn is a Fresnel-like transform of the pupil, extreme rapidity. Phase retrieval algorithms require a known object which is usually an unresolved point πD source. Phase diversity algorithms do not require the h u; v i k u2 v2 k;λð Þ¼exp λB ð þ Þ object to be known and will estimate the object in ad- ZZ k dition to the system phase. πAk 2 2 × Pk;λðx; yÞ exp i ðx þ y Þ Phase diversity was first proposed as a method of λBk wavefront sensing by Gonsalves [3] and later devel- 2π oped by Paxman and Fienup for multi-aperture sys- × exp −i ðxu þ yvÞ dxdy; ð3Þ tems [4]. Paxman, Schulz, and Fienup formulated λBk phase diversity from a statistical model that better captured the affect of noise on the algorithms [5]. where ðx; yÞ are the pupil plane coordinates and Further developments by Paxman and Seldin com- Ak, Bk, and Dk are the elements of the ABCD ray- bined the method of phase diversity with speckle transfer matrix that relates the pupil plane to the imaging and introduced a broadband model of phase image plane [10] for the kth diversity image. The gen- diversity incorporating a gray-world model of the ob- eralized pupil, Pk;λðx; yÞ, is given by a sum over the ject [6] that was later tested by Bolcar and Fienup sub-aperture functions, [7]. In this paper we continue the development of phase diversity by utilizing the unique architecture XQ of segmented and multi-aperture systems to intro- Pk;λðx; yÞ¼ Pq;k;λðx; yÞ duce sub-aperture piston phase as the diversity func- q¼1 tion, as an alternative to focus diversity. This XQ technique of sub-aperture piston phase diversity P x; y (SAPPD) was first reported in [8]. We include here ¼ j qð Þj q 1 the effects of a broadband object and algorithm, as ¼ 2π well as compare the performance of sub-aperture div × exp i ½Wqðx; yÞþW ðx; yÞ ; ð4Þ piston diversity and focus diversity with respect to λ q;k object reconstruction error. SAPPD can be useful for Fourier transform ima- where Q is the number of sub-apertures or segments, ging spectroscopy with a multi-aperture system [9] Wq is the unknown contribution of the phase on sub- where the sub-apertures are purposely pistoned to aperture q in terms of optical path delay (OPD), and collect spectral data. Also, SAPPD can be useful as div Wq;k is the known diversity contribution to the phase a risk reduction method on systems where the focus on sub-aperture q in terms of OPD. diversity mechanism fails or is disabled. The unknown OPD, Wqðx; yÞ, can be parameterized In Section 2 we review the phase diversity concept, in a number of ways. The most straightforward meth- incorporating the statistical techniques of [5] and the od is as a pixel-by-pixel phase map, broadband techniques of [6,7] and introducing the X implementation of sub-aperture piston phase. In W x; y α δ x − mΔx; y − nΔy ; Section 3 we describe digital simulations that com- qð Þ¼ q;m;n ð Þ ð5Þ m;n pare SAPPD to conventional focus diversity. In Section 4 we present the results and in Section 5 α we summarize and conclude the paper. where q;m;n is the value of the OPD at pixel index ðm; nÞ in the qth sub-aperture, ðΔx; ΔyÞ are the pixel spacings, and δ is a delta function. Another popular 2. Statement of Problem method is to estimate the phase by an expansion over basis functions: A. Image Model The detected images are modeled as XJ X Wqðx; yÞ¼ αq;jZq;jðx; yÞ; ð6Þ j 1 dkðu; vÞ¼ f λðu; vÞsk;λðu; vÞþnkðu; vÞ; ð1Þ ¼ λ where J is the number of terms in the expansion where dk is the kth detected image, ðu; vÞ are the im- and Zq;jðx; yÞ is the jth basis function defined over age plane coordinates, f λ are the object pixel values the qth sub-aperture. Zernike-like polynomials are at wavelength λ, sk;λ is the kth intensity point spread commonly chosen for the expansion since each A6 APPLIED OPTICS / Vol. 48, No. 1 / 1 January 2009 polynomial represents a balanced optical aberration. f λðu; vÞ¼Φλf ðu; vÞ; ð8Þ In either representation, the phase is parameterized in terms of the vector of coefficients, α. where Φλ is a spectral coefficient at wavelength λ. The goal of phase diversity is to estimate the phase Under this assumption, the reduced Gaussian metric parameters, α, and the object pixels, f λðu; vÞ, at each becomes wavelength λ, from the set of detected images X XK fdkðu; vÞg. 2 LRG½fdkðu; vÞg; α¼ jDkj B. Nonlinear Optimization f u;f v∈χ k¼1 In [5], the phase diversity problem is formulated as a P P 2 K D Φ SÃ α nonlinear optimization in which the appropriate j¼1 j λ j;λð Þ − λ ; likelihood function is maximized for a given noise P P 2 ð9Þ mechanism. We follow the same treatment here K Φ S α m¼1 λ m;λð Þ and maximize the Gaussian log-likelihood function λ in the Fourier domain, given by where χ is the set of pixels where the denominator D XK X does not equal zero and the dependence of j and Sj;λ on ðf u; f vÞ has been left out for brevity.
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