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.
Claude 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.
The Sun
Lyot coronagraph
Observing plane
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3. Lyot drawing of the coronagraph
© Observatoire de Paris — Patrimoine Scientifique de l'Observatoire de Meudon
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4. 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

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.
Claude Aime - Sunspot July 2010

6. On axis point source, no turbulence, perfect instrumentFT FT FT A B C D
(Units are different in pupil and focus planes)
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7. Claude Aime - Sunspot July 2010
Lyot mask:
Alternative not considered here:
Claude Aime - Sunspot July 2010

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

9. Illustration: focal plane
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10. Claude Aime - Sunspot July 2010
Pupil plane
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11. Claude Aime - Sunspot July 2010
Pupil plane
Claude Aime - Sunspot July 2010

12. Claude Aime - Sunspot July 2010
Larger mask
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13. Claude Aime - Sunspot July 2010
Larger mask
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14. Claude Aime - Sunspot July 2010
Lyot Mask, no Lyot stop
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15. Lyot Mask, Lyot stop = aperture (Arago – Poisson – Fresnel spot)
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16. Lyot Mask, Lyot stop = 0.9 aperture (Arago – Poisson –Fresnel spot)
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17. An off-axis point source behind the Lyot mask
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18. An off-axis point source behind the Lyot mask (smaller Lyot stop)
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19. A point source close to the edge of the Lyot mask
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20. Addition in intensity of all contributions
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.
Claude Aime - Sunspot July 2010

22. 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

23. Claude Aime - Sunspot July 2010
The effect similar to the Poisson-Arago spot is well retrieved using the series expansion
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24. 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

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 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

26. 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

27. 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

28. 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

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

31. Claude Aime - Sunspot July 2010
Thank you
Claude Aime - Sunspot July 2010

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

34. Claude Aime - Sunspot July 2010

35. Claude Aime - Sunspot July 2010

36. Claude Aime - Sunspot July 2010