Appendix A: Formalism for the coronagraphy of circular apertures

In Eq. (6), the two dimensional convolution product between the 2 radial functions $f(\vec{r})=\Psi_{\rm A}(\vec{r})$ and $g(\vec{r})=\frac{a~{J_1}(\pi ~a~\vec{r})}{2~\vec{r}}$ can be simplified, using the properties of FT and HT. The Fourier Transform (FT) of the convolution product is a simple product between the FTs of each functions:

$$\mathcal{F}\left(f(\vec{r})~\ast~g(\vec{r})\right)=\widehat{f}(\vec{\rho})\times\widehat{g} (\vec{\rho}),$$ (A.1)

where $\vec{\rho}$ denotes the radial vector in the Fourier space and $\rho$ its modulus. Since these functions are radial functions, a radial cut (denoted $\widehat{f}(\rho)$) and $\widehat{f}(\rho)$ can be computed by a Hankel Transform (HT). Using Eq. (2), we have, for $\rho \geq 0$:

$$\widehat{f}(\rho)=\int _{0}^{\infty}\Pi\left(\frac{r}{D}\right)\Phi (r)~{{J}_0}(2~\pi ~\rho ~r)~2~\pi ~r~{\rm d}r$$ (A.2)

$$\widehat{g}(\rho)=\Pi\left(\frac{\rho}{a}\right)\cdot$$ (A.3)

The radial cut of the FT of the convolution product is then:

$$\left[\mathcal{F}(f(\vec{r})~\ast~g(\vec{r}))\right]_{\rm r... ...frac{D}{2}}\Phi (r)~{{J}_0}(2~\pi ~\rho ~r)~2~\pi ~r~{\rm d}r.$$ (A.4)

Taking again the HT of this radial expression (reminding that (HT[HT]=Id)), we come back to the expression of the radial cut of the convolution product:

 

$$\left[f(\vec{r})\ast g(\vec{r})\right]_{\rm r.c.}= \int _{\rho = ... ...~\rho ~\xi )~2~\pi ~\xi ~{\rm d}\xi {{J}_0}(2~\pi ~\rho ~r)~2~\pi ~r ~{\rm d}r,$$ (A.5)

that can be written, with the kernel notation:

$$\left[f(\vec{r})~\ast~g(\vec{r})\right]_{\rm r.c.}={\left( 2... ... _{0}^{\frac{D}{2}}\xi ~\Phi (\xi )~{{K}_0}(\xi ,r)~{\rm d}\xi$$ (A.6)

where the Kernel:

$${{K}_0}(\xi ,r) = \int _{0}^{\frac{a}{2}}\rho ~{{J}_0}(2~\pi ~\rho ~\xi )~{{J}_0}(2~\pi ~\rho ~r)~{\rm d}\rho,$$ (A.7)

was already given by Frieden (1971) in his presentation of circular prolate functions properties (see Appendix B)

An analytical expression of the Kernel can be obtained with the help of Mathematica software (Wolfram 1999):

$${{K}_0}(\xi ,r) = \frac{a~r~{{J}_0}(a~\pi ~\xi )~{{J}_1}(a~... ...{{J}_1}(a~\pi ~\xi )}{4~\pi ~\left( r^2 - {\xi }^2 \right) },$$ (A.8)

which limit for $r=\xi$ equals:

$${{K}_0}(\xi ,\xi ) = \frac{1}{8}~a^2~\left( {{{J}_0}(a~\pi ~\xi )}^2 + {{{J}_1}(a~\pi ~\xi )}^2 \right),$$ (A.9)

for $\xi=r$.
Theses analytical expressions of the Kernel make it possible to use the integral of Eq. (A.6) for the direct computation of the convolution product, with better ratio precision / computation time.