Optimization of a one dimensional hypertelescope for a direct imaging in astronomy

doi:10.1016/j.ejor.2008.02.002

European Journal of Operational Research

Volume 195, Issue 2, 1 June 2009, Pages 519-527

https://doi.org/10.1016/j.ejor.2008.02.002 Get rights and content

Abstract

We describe an application of nonlinear optimization in interferometric optical astronomy. The aim is to find the relative positions of the output pupils and the modulus of the beams through each pupil of a linear array of telescopes in order to design an instrument capable of imaging exoplanets. The problem is modelized under the form of a semi-infinite nonlinear minimization problem. The model problem is transformed by a simple discretization into a minimization problem with a finite number of constraints, then it is solved by using a minimization solver. Numerical experiments are reported.

Introduction

The new generation of high resolution optical imaging system for astronomy is based on the linkage of a telescope array in order to reach micro or nanoradian resolution [6]. The principle of the image restoration uses an indirect method. The interferometric signals give partial information on the spectrum of the object spatial distribution. A numerical image reconstruction is necessary to post-process the data. The usage of this technique never allows to select the light of one pixel in the observed field, for example in order to achieve a direct spectral analysis. The hypertelescope concepts proposed by Labeyrie [4], Vakili et al. [9] and Reynaud–Delage [8] (see also the recent book [5]), answer this problem. By using a specific conditioning of the beam coming from the telescopes, it is possible to perform a direct imaging, thanks to an accurate equalisation of the optical paths and a pupil densification, see Fig. 1.

The first operation, the so called cophasing, has to be achieved with a sub-micrometric accuracy. Overall, in this process, coherent properties of the beams have to be preserved taking care of polarization and dispersion differential effects. The pupil densification is a specific step of the hypertelescope design. The input pupil mapping is homothetically reduced and the resulting beams are expended in order to maximize the densified pupil coverage. The goal of such instruments is to design an optical instrument with a strong dynamic in the frame of a high resolution imaging. For example, the angular separation between a star and an exoplanet is expected to be in the range of nanoradian and the ratio of their intensities can be less than $10^{- 6}$ . Of course, this tremendous result will be reached in a limited field and the number of pixels remains low as reported in [4], [8], [9].

In a general way, the optimization of optical instrument is addressed in various domains such as microscopy, see for example [7].

The purpose of this paper is to find the modulus of each beam and the relative positions of the output pupils to obtain high resolution and dynamic of the image. In this preliminary study, we will consider a linear array of telescopes corresponding to pupils arranged along a straight line. Applied optimization to the design of an optical instrument capable to image exoplanets has been already investigated by Kasdin et al. [3] and Vanderbei [10]. As in Vanderbei approach, we propose to formulate the design problem as a semi-infinite minimization problem. After discretization, the model problem is transformed into a constrained nonlinear minimization problem. Then, the problem is solved by using a nonlinear optimization solver.

The paper is organized as follows. In Section 2, we first present our general experimental setup, then we state the particular case of a linear array of telescopes in Section 3. The optimization model is presented in Section 4. A starting point strategy for the optimization solver is discussed in Section 5, then numerical experiments are presented in Section 6.

Section snippets

Densified pupil and point spread function

This section is focused on the optical field description in the densified pupil and to derive the corresponding point spread function.

The optical field of a monochromatic plane wave through a pupil is characterised by a modulus and a phase. We assume that all beams have the same phase (cophasing assumption). It follows that the optical field of n pupils centered at $(u_{k}, v_{k})$ , $k = 1, \dots, n$ and with the same diameter d is given by $g (u, v) = \sum_{k = 1}^{n} a_{k} 1_{B_{k}} (u, v),$ where $a_{k}$ is the modulus of beam $k and 1_{B_{k}}$ is the

Linear array of telescopes

In this first paper, we consider the particular case of a linear array of telescopes, that is $v_{k} = 0$ for all $k = 1, \dots, n$ in (1). Taking (2) into account, the normalized PSF becomes $Ψ (x, y) = \frac{1}{s^{2}} | \hat{h} (x, y) |^{2} {|\sum_{k = 1}^{n} a_{k} e^{- \frac{2 i π}{λ f} {xu}_{k}}|}^{2} .$ In the right-hand side of the above formula, the term which contains the optimization parameters $a_{k} and u_{k}$ does not depend on y. It follows that in our study we can then set y to any arbitrary value. Let us set $y = 0$ . By introducing the polar coordinates $u = r \cos θ and v = r \sin θ$ , the value of $\hat{h}$

Optimization model

In the particular case of regularly spaced pupils, the second term in the right-hand side of (3) can be interpreted as a truncated Fourier series of some periodic function. This function can be seen as a size function to obtain a given PSF. This technique is used for the design of antenna array in telecommunication network where such a function is called an apodization function. In this specific case the parameters $a_{k}$ are then the Fourier coefficients of the apodization function (see e.g., [2]

Starting point strategy

A difficulty when solving problem (6) is to avoid to be trapped by a locally optimal point which is not a global minimizer, a common situation which is hard to deal within nonlinear optimization. To cope with this difficulty we considered to solve several occurrences of the same problem but with a random starting point for the solver. We solved an instance of problem (6) with eight pupils ( $m = 4$ ), $d = 1$ , $α_{\min} = 0.25 and α_{\max} = 0.75$ . The starting points were chosen following a uniform distribution such

Numerical experiments

We first analyse the effect of the choice of the CLF interval on an optimal configuration. Let us define a third measure of the quality of a given configuration. Let $ρ \in [0, 1]$ be such that $Ψ (ρ) = \min \{α \in [0, 1] : Ψ (α) = \frac{1}{2}\} .$ This is the width of the PSF curve at half height of the main central lobe. The number of resels is the ratio $R = \frac{α_{\max} - α_{\min}}{ρ} .$ This is the number of resolved elements in the clean field of view interval.

Let us observe the variations of the three parameters, dynamic (D), number of resels (R)

Conclusion

Hypertelescopes appear to be good candidates for the next generation of high resolution and high dynamic imaging device for astronomy. The optimization of the point spread function is a crucial step to maximize the efficiency of these imaging systems. This paper has demonstrated the potential of nonlinear optimization technology to adjust the densified pupil configuration in order to address specific observation program (i.e. resolution, dynamic). As a preliminary approach, this work has been

Acknowledgement

This work is supported by the Centre National d’Étude Spatiale (CNES).

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O.R. ApplicationsOptimization of a one dimensional hypertelescope for a direct imaging in astronomy

Abstract

Introduction

Section snippets

Densified pupil and point spread function

Linear array of telescopes

Optimization model

Starting point strategy

Numerical experiments

Conclusion

Acknowledgement

AMPL A Modeling Language for Mathematical Programming

Optics

Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs

The Astrophysical Journal

Resolved imaging of extra-solar planets with future 10–100 km optical interferometric arrays

Astronomy & Astrophysics

An Introduction to Optical Stellar Interferometry

O.R. Applications
Optimization of a one dimensional hypertelescope for a direct imaging in astronomy