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**An analytic approach to the Lyot coronagraph**

1. Illustrative numerical examples for the response of a Lyot coronagraph to point sources 2. Outline of the analytical approach based on a Zernike decomposition (due to André Ferrari), and first results for a resolved source. Claude Aime - Sunspot July 2010

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**The diffraction halo of the Sun at the output of a Lyot coronagraph**

Each point of the solar disc produces its own diffraction pattern in the image plane through the coronagraph. The observed diffraction halo is the sum of all contributions. The Sun Lyot coronagraph Observing plane Claude Aime - Sunspot July 2010

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**Lyot drawing of the coronagraph**

© Observatoire de Paris — Patrimoine Scientifique de l'Observatoire de Meudon Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

The 4 planes. A B C D Pupil plane Focal plane Pupil plane Focal plane MASK STOP Claude Aime - Sunspot July 2010

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**An analytic approach to the Lyot coronagraph**

1. Illustrative numerical examples for the response of the Lyot coronagraph to point sources 2. Outline of the analytical approach based on a Zernike decomposition (due to André Ferrari), and first results for a source of large angular diameter. Claude Aime - Sunspot July 2010

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**On axis point source, no turbulence, perfect instrument**

FT FT FT A B C D (Units are different in pupil and focus planes) Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Lyot mask: Alternative not considered here: Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Lyot mask + Lyot stop Residual image A few l/D D or <D Claude Aime - Sunspot July 2010

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**Illustration: focal plane**

Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Pupil plane Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Pupil plane Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Larger mask Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Larger mask Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Lyot Mask, no Lyot stop Claude Aime - Sunspot July 2010

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**Lyot Mask, Lyot stop = aperture (Arago – Poisson – Fresnel spot)**

Claude Aime - Sunspot July 2010

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**Lyot Mask, Lyot stop = 0.9 aperture (Arago – Poisson –Fresnel spot)**

Claude Aime - Sunspot July 2010

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**An off-axis point source behind the Lyot mask**

Claude Aime - Sunspot July 2010

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**An off-axis point source behind the Lyot mask (smaller Lyot stop)**

Claude Aime - Sunspot July 2010

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**A point source close to the edge of the Lyot mask**

Claude Aime - Sunspot July 2010

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**Addition in intensity of all contributions**

Claude Aime - Sunspot July 2010

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**An analytic approach to the Lyot coronagraph**

1. Illustrative numerical examples for the response of the Lyot coronagraph to point sources 2. Outline of the analytical approach based on a Zernike decomposition (due to André Ferrari), and first results for a source of large angular diameter. Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Outline of the analytic approach (see Ferrari 2007, Ferrari et al 2010) Starting point: decompose the waves on a Zernike base where r and q are the polar coordinates, and are the Zernike radial polynomials, m < n, same parity (otherwise = 0) For a point source in the direction a in units of l/D, the wavefront writes: Then use the properties of Fourier transform of Zernike polynomials: where r and f are the conjugate variable to r and q . Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

The effect similar to the Poisson-Arago spot is well retrieved using the series expansion Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

The integrated intensity in plane D (and C) takes the form of (intricate) infinite series with Claude Aime - Sunspot July 2010

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**Convergence and limitations**

The series converges with a reasonable number of terms for a star of small angular diameter (a fraction of or a few l/D), but not for the solar case, for which the diameter is thousands of l/D. The expression in plane D assumes that the Lyot stop is exactly the size of the entrance aperture (no analytic expression for a different size) This strong limitation for the solar case is acceptable for the stellar case since (prolate) apodized aperture will be used rather than clear aperture. NUMERICAL ILLUSTRATIONS => Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Radial cut of the intensity in plane C, inside the pupil image, for a Lyot mask of diameter 12 l/D Stars of different angular diameters “diffraction ring” Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Radius of the source in units of resolution Lyot mask of radius: Focal plane in units of resolution Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Radius of the source in units of resolution Lyot mask of radius: Focal plane in units of resolution Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Radius of the mask in units of resolution Source angular diameter Claude Aime - Sunspot July 2010

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**Pro et contra of the approach**

(+) Exact calculation of the propagation through the coronagraph. (+) Approach can be very general (for exoplanet). (-) The result is given by slowly converging series: difficult to apply to the solar case (not yet realistic). (-) The computation is fully analytic only for a Lyot stop equal to the aperture (OK if an apodized aperture is used – not presented here) Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

Thank you Claude Aime - Sunspot July 2010

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**Clear vs apodized (Sonine, s=1) aperture**

Claude Aime - Sunspot July 2010

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**Claude Aime - Sunspot July 2010**

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**Claude Aime - Sunspot July 2010**

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**Claude Aime - Sunspot July 2010**

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**Claude Aime - Sunspot July 2010**

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