## A Image plane

Pupa plane (primary focal plane)

Fig. 2.16 The basic principles of the Lyot coronagraph

Secondary pupil plane

### Secondary pupil plane

Final image plane by a factor of between 10 and 100. Lyot's coronagraph thus allowed observation of the solar corona a few arcminutes from the solar limb.

An elegant description of how Lyot's coronagraph works may be made using Fourier optics and image formation theories. A good introduction to these topics may be found in Goodman, 1996.

From the point of view of the theory of the formation of images, the principle behind Lyot's coronagraph may be described in the way shown in Fig. 2.17:

(a) at the entry pupil plane: the infinite (i.e., nominally plane) wavefront is sampled by the pupil Fig. 2.17 The Airy function (Airy disk): the diffraction pattern produced by a circular diffracting aperture

(b) at the primary image plane (the focus of the main mirror): The image of a single point at an infinite distance (which is known as the Point-Spread Function, PSF) is the square of the modulus of the pupil's Fourier transform. (The Fourier transform is a mathematical function that allows the transformation of a function into another, known as the frequency domain representation of the original function. For instance, a function of time may be transformed into one of frequency. Among the main characteristics of the Fourier transform is the fact that the operation known as a convolution, whereby two functions are manipulated to provide a third, may be simply represented in Fourier space by a single multiplication of the Fourier transform.) This point-spread function may thus be multiplied by the mask's transmission (1-n, where n is the aperture function, corresponding to a mask with an occulting centre. In this image plane we therefore have:

(c) at the secondary pupil plane: the new pupil is once again the Fourier transform of the primary image, in other words: The Fourier transform of the Airy function (i.e., the initial pupil) less the convolution product of the Fourier transform of the Airy function (the initial pupil) with the Fourier transform of the aperture function (another Airy function with a size that is inversely proportional to the size of the mask). It will be recalled that the Airy function is the result of the diffraction of light by a circular aperture (Fig. 2.17). The final transmission of a coronagraph is thus equal to 0 on the optical axis, and there remains only an annulus (of scattered light), which is eliminated by the Lyot stop at the pupil (Fig. 2.18), which may be written as:

where the symbol ® represents the second magnitude convolution product.

This method, with some variants, has been employed in all solar coronagraphs, including those installed on satellites such as SOHO (Koutchmy, 1988), as well as in certain stellar coronagraphs which, when combined with adaptive-optics systems,