2 Equations for coronagraphy with circular apodized apertures

The general formalism to describe a coronagraphic experiment with an apodized aperture has been given by Aime et al. (2002). They showed that a complete extinction of the starlight could be obtained with a coronagraph implemented at the focus of an apodized telescope. Denoting P(x,y) the aperture transmission function ( P(x,y)=1 inside the aperture, and 0 outside), and $\Phi(x,y)$ an apodization function ( $0~\leq~\Phi(x,y)~\leq~1$), the total extinction is obtained if the following equation is satisfied:

$${P}(x,y) \Phi(x,y) \ast \frac{1}{\lambda^2 f^2}\; \widehat{{... ...},\frac{y}{\lambda f}\right)=\frac{1}{\varepsilon}\;\Phi(x,y),$$ (1)

where $\lambda$ is the wavelength, f the focal length of the telescope and M(x,y) a function that describes the mask shape, equal to 1 inside the coronagraphic mask and 0 outside.

With a rectangular aperture, the problem is simplified because this two dimensional equation separates into the product of two symmetric one dimensional equations. The apodization function solution is equal to 1 at the center of the rectangular aperture and decreases towards the edges as the product of two linear prolate functions aligned along the length and the width of the rectangle.

With a circular aperture, the problem is much more complicated since we cannot separate anymore the solution into two linear conditions in x and y. Fortunately, the radial properties of the aperture and mask shape make the solution possible: a two dimensional radial function $\Phi(\vec{r})$ can be represented by its radial cut $\Phi(r)$, where $r=\sqrt{x^2+y^2}$ is the modulus of the position vector $\vec{r}=(x,y)$. For the sake of simplicity, we will use the same notation for these two functions, the interpretation being straightforward for the alert reader, depending on the context. In particular, the computations developed using Fourier transforms in the rectangular separable case, are here solved by means of Hankel transforms.

We consider a circular aperture of diameter D apodized by a radial apodization function $\Phi(r)$: $0~\leq~\Phi(r)~\leq~1$. We consider the four successive planes denoted A (telescope pupil), B (telescope focus, coronagraphic mask), C (relay pupil plane, Lyot Stop) and D (final focus plane)

The complex amplitude at the entrance apodized pupil plane is proportional to:

$$\Psi_{\rm A} (\vec{r})=\Pi\left(\frac{r}{D}\right)~ \Phi(\vec{r})$$ (2)

where $\Pi(r)=1$ for $\vert r\vert\leq1/2$ and 0 otherwise.

The wave propagation between each planes (A, B, C, D) writes as a scaled Fourier Transform (FT) and we assume that the optical layout is properly designed to eliminate the quadratic phase terms associated with the propagation of the waves (Goodman 1996). The FT of a radial function is also a radial function, which radial cut (for $r\geq 0$) can be expressed using the Hankel Transform (HT):

$$\widehat{\Psi}_{\rm A}(r)=\int_0^\infty 2~ \pi~ \rho ~ \Psi_{\rm A}(\rho)~ {J_0}(2~\pi ~\rho~ r)~ {\rm d}\rho.$$ (3)

In plane B, the coronagraphic mask is a disk of diameter d proportional to the size of the diffraction pattern of the circular apodized aperture. At the wavelength $\lambda$, the mask size is:

$$d=a~\lambda ~ f,$$ (4)

where a is a parameter that expresses the diameter of the mask in units of 1/D and f is the telescope focal length. The wave amplitude after the mask writes:

$$\Psi_{\rm B}(\vec{r})=\frac{1}{\imath \lambda f}\;\widehat{... ...t)\times\left[1-\epsilon ~ \Pi\left(\frac{r}{d}\right)\right],$$ (5)

where the symbol $\; \widehat{ }\;$ denotes the HT. We do not consider in this paper other mask functions for which the mask transmission itself is also apodized (Watson et al. 1991). The parameter $\epsilon$ has only two relevant values (Aime et al. 2001, 2002): $\epsilon=1$ for an opaque mask (Lyot's coronagraphy) and $\epsilon=2$for a $\pi$ Phase Mask (PM) (Roddier and Roddier's coronagraphy).

An optical system is then used to obtain an image of the telescope aperture in plane C. Here also, the complex amplitude of the wave is written using a scaled HT of Eq. (5). For simplicity, we assume that the focal lengths of the successive optical systems are identical (if not, an appropriate change of variables leads to a similar result). In the relay pupil plane C, we denote by $\Psi^-_{\rm C}(\vec{r})$ the amplitude before the application of the diaphragm (Lyot Stop):

$${\Psi^-_{\rm C}}(\vec{r}) = {\Psi_{\rm A}}(\vec{r}) - \epsilon ~{{\Psi }_{\rm A}}(r)\ast \frac{a~{J_1}(\pi ~a~r)}{2~r}\cdot$$ (6)

Note that the convolution product $\ast$ is a two-dimensional convolution product over the variables x and y. However, this convolution of two radial functions can be simplified using the properties of the Hankel Transform, as detailed in Appendix A, to obtain for the radial cut of the wave amplitude:

$${\Psi^-_{\rm C}}(r) = \Pi \left(\frac{r}{D}\right)\Phi (r) -... ...{0}^{\frac{D}{2}}\xi ~\Phi (\xi )~{{K}_0}(\xi ,r)~ {\rm d}\xi.$$ (7)

A diaphragm (called Lyot Stop) exactly equal to the entrance pupil shape is then set in plane C, to remove the light diffracted outside the aperture:

$${\Psi^+_{\rm C}}(r) = \left[ \Phi(r) - \epsilon ~{\left( 2~\... ...,r)~{\rm d}\xi\right]% \times \Pi\left(\frac{r}{D}\right)\cdot$$ (8)

The kernel ${K_0}(\xi ,r)$ expression (given in Eq. (A.8) and Eq. (A.9)) permits a numerical study of circular coronagraphy, for a given aperture transmission, using a unique one-dimensional integration that can be computed easily. This method is more interesting and precise than a fully numerical approach (avoiding the sampling problem of both the pupil and the mask that imposes the use of large images), with competitive computing time.

Note that the kernel ${K_0}(\xi ,r)$ is analog to the sine cardinal kernel for the rectangular aperture coronagraphy (Aime et al. 2002).