Appendix A: Contribution of the aberrations to the residual energy on the axis of the AIC

We calculate here the mean intensity behind AIC with an OPD error:

$$\begin{eqnarray*}I(\vec{\rho}) &=& RT<\vert{\rm TF} [P( {\bf r} ) \exp \left( i... ...{\bf r}) \right) \exp \left( id \right) ]_{\vec{\rho}}\vert^2 >. \end{eqnarray*}$$

We approximate $\exp\left(ia\right)\approx 1+ia$ because the fluctuations of the turbulent phase are small because the first aberrations are corrected by the AO. The fluctuations of the OPD are also small (about $\frac{\lambda}{15}$ in IR).

$$I(\vec{\rho})\approx RT<\vert i\mathrm{TF}[P(\vec{r})(\phi(\vec{r})-\phi(-\vec{r})-d)]_{\vec{\rho}}\vert^2>.$$

We develop the phase errors in Zernike polynomials (Noll 1976, Paper I). The even Zernike polynomials are removed because of the rotation of the pupil in one arm and the sum is only done on odd radial polynomials.

$$I(\vec{\rho})\approx RT<\vert\mathrm{TF}[2i\sum_{{\rm odd},i=... ...fty}{a_{i} Z_{i}(\vec{r})}-idP(\vec{r})]_{\vec{\rho}}\vert^2>.$$

Remembering that $P(\vec{r})=Z_{1}(\vec{r})$ and calling $\mathrm{TF}[Z_i(\vec{r})]_{\vec{\rho}}=\hat{Z_i}(\vec{\rho})$

$$I(\vec{\rho})\approx RT<\vert 2i\sum_{\mathrm{odd},i=J}^{\inf... ...i} \hat{Z}_{i}(\vec{\rho})}-id\hat{Z}_{1}(\vec{\rho})\vert^2>.$$

As
$$\begin{align*}\hat{Z}_{i}(\vec{\rho})&=\hat{Z}_{nk}(\rho,\theta)\\ &=\pi\sqrt{n... ...^{(n-k)/2} i^{-k} \frac{2J_{n+1}(2\pi\rho)}{2\pi\rho}\exp(ik\theta) \end{align*}$$
depends on i^-k the term $2i\sum_{\mathrm{odd},i=J}^{\infty}{a_{i} \hat{Z}_{i}(\vec{\rho})}$ is real while the term $id\hat{Z}_{1}(\vec{\rho})$ is imaginary.

Then the cross values are null and we obtain:

$$I(\vec{\rho}) \approx RT \left[<\vert 2\sum_{\mathrm{odd},i=J... ...ho})}\vert^2>+<d^2\vert\hat{Z}_{1}(\vec{\rho})\vert^2>\right].$$

The first term is the same as the one found in Paper I and gives no light contribution on the axis of the AIC. The second term is then the term that explains the residual peak found on the axis of the AIC.

Acknowledgements

We are grateful to the Observatoire de Haute Provence staff and the Office Nationale d'Étude et de Recherche Aérospatiale staff. Part of this work has been performed using the computing facilities provided by the program "Simulations Interactives et Visualisation en Astronomie et Mécanique (SIVAM)''.