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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.

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Presentation on theme: "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."— Presentation transcript:

1 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. 1Claude Aime - Sunspot July 2010

2 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. Claude Aime - Sunspot July The SunLyot coronagraphObserving plane

3 Lyot drawing of the coronagraph © Observatoire de Paris — Patrimoine Scientifique de l'Observatoire de Meudon 3Claude Aime - Sunspot July 2010

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

5 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. 5Claude Aime - Sunspot July 2010

6 On axis point source, no turbulence, perfect instrument Claude Aime - Sunspot July FT A B C D (Units are different in pupil and focus planes)

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

8 Lyot mask + Lyot stop Claude Aime - Sunspot July A few  /D D or

9 Illustration: focal plane Claude Aime - Sunspot July

10 Pupil plane Claude Aime - Sunspot July

11 Pupil plane Claude Aime - Sunspot July

12 Larger mask Claude Aime - Sunspot July

13 Larger mask Claude Aime - Sunspot July

14 Lyot Mask, no Lyot stop 14 Claude Aime - Sunspot July 2010

15 Lyot Mask, Lyot stop = aperture (Arago – Poisson – Fresnel spot) 15 Claude Aime - Sunspot July 2010

16 Lyot Mask, Lyot stop = 0.9 aperture (Arago – Poisson –Fresnel spot) 16 Claude Aime - Sunspot July 2010

17 An off-axis point source behind the Lyot mask 17 Claude Aime - Sunspot July 2010

18 An off-axis point source behind the Lyot mask (smaller Lyot stop) 18 Claude Aime - Sunspot July 2010

19 A point source close to the edge of the Lyot mask 19 Claude Aime - Sunspot July 2010

20 Addition in intensity of all contributions 20 Claude Aime - Sunspot July 2010

21 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. 21Claude Aime - Sunspot July 2010

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

23 The effect similar to the Poisson-Arago spot is well retrieved using the series expansion 23Claude Aime - Sunspot July 2010

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

25 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 /D), but not for the solar case, for which the diameter is thousands of /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

26 Radial cut of the intensity in plane C, inside the pupil image, for a Lyot mask of diameter 12 /D 26Claude Aime - Sunspot July 2010 Stars of different angular diameters “diffraction ring”

27 Focal plane in units of resolution Radius of the source in units of resolution Lyot mask of radius: 27Claude Aime - Sunspot July 2010

28 Focal plane in units of resolution Radius of the source in units of resolution Lyot mask of radius: 28Claude Aime - Sunspot July 2010

29 Source angular diameter Radius of the mask in units of resolution 29Claude Aime - Sunspot July 2010

30 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) 30Claude Aime - Sunspot July 2010

31 Thank you Claude Aime - Sunspot July

32 Clear vs apodized (Sonine, s=1) aperture 32 Claude Aime - Sunspot July 2010

33 33

34 Claude Aime - Sunspot July

35 Claude Aime - Sunspot July

36 Claude Aime - Sunspot July


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