Appendix A: Analytical expressions of the residual pupil amplitudes for cosine apodizers

The residual amplitude left in the entrance aperture by the coronagraphic process (Eq. (10)) depends on the function $F\left(x\right)$ (Eq. (11)) which describes the diffraction pattern of the coronagraphic mask.

• For a raw unapodized aperture $P(x)=\Pi(x/L)$, one obtains (Aime et al. 2001):
  
   

  $$F\left(x\right)=\frac{1}{\pi} \left( {\rm Si} \left[ \pi a\left(x... ...\right) \right] -{\rm Si}\left[\pi a \left(x-\frac{L}{2}\right)\right]\right) .$$ (A.1)

  
  

• For the partial cosine apodized aperture $\Pi\left(\frac{x}{L}\right) \cos\left(\frac{\pi x}{b L}\right)$ we have:
  

  $$F\left(x\right)= \frac{-1}{2\pi} \left(\frac{\sin(\pi x)}{b L} \left(-{\rm Ci}[A]+{\rm Ci}[B]+{\rm Ci}[C]-{\rm Ci}[D]\right) \right.$$

  $$\left. + \frac{\cos(\pi x)}{b L}\left(-{\rm Si}[A]-{\rm Si}[B] +{\rm Si}[C]+{\rm Si}[D]\right)\right)$$ (A.2)

  
  
  with:
  
  $A=\pi(a-\frac{1}{b L})(x-\frac{L}{2})$,
  $B=\pi(a+\frac{1}{b L})(x-\frac{L}{2})$,
  $C=\pi(a-\frac{1}{b L})(x+\frac{L}{2})$,
  $D=\pi(a+\frac{1}{b L})(x+\frac{L}{2})$.
  
  
• For the partial cosine squared apodization: $\Pi\left(\frac{x}{L}\right)\cos^2\left(\frac{\pi x}{b L}\right)$, we obtain:
  

  $$F\left(x\right)= \frac{-1}{4\pi} \left(\frac{\sin(2\pi x)}{b L}\left(-{\rm Ci}[A]+{\rm Ci}[B]+{\rm Ci}[C]-{\rm Ci}[D]\right)\right.$$

  $$\left.+ \frac{\cos(2\pi x)}{b L}\left({\rm Si}[A]+{\rm Si}[B]+{\r... ... \left({\rm Si}[a \pi (\frac{L}{2} - x)]+{\rm Si}[a \pi (\frac{L}{2}+x)]\right)$$ (A.3)

  
  
  with:
  
  $A=\pi(a-\frac{2}{b L})(x-\frac{L}{2})$,
  $B=\pi(a+\frac{2}{b L})(x-\frac{L}{2})$,
  $C=\pi(a-\frac{2}{b L})(x+\frac{L}{2})$,
  $D=\pi(a+\frac{2}{b L})(x+\frac{L}{2})$.
  
  

In these expressions, we have used the sine integral ${\rm Si}(x)$ and cosine integral ${\rm Ci}(x)$functions defined by:

  

$${\rm Si}(x)=\int_0^x \frac{\sin t}{t}{\rm d}t,$$ (A.4)

$${\rm Ci}(x)=-\int_{\vert x\vert}^\infty \frac{\cos t}{t}{\rm d}t.$$ (A.5)