entrance pupil apodization function was computed using the iterative algorithm described in this section, is shown in Figure \[fig:APCMLCprinciple\].

The APCMLC is described here analytically for monochromatic light using notations shown in Figure \[fig:APCMLCprinciple\]. The entrance pupil shape is defined by the real function $P(\mathbf{r})$, with $\mathbf{r}$ the 2-D position vector in the pupil plane, and $P(\mathbf{r})=1$ for points within the pupil and $P(\mathbf{r})=0$ outside of the pupil. The apodizer function $Apo(\mathbf{r})$ is applied to the pupil, yielding the following complex amplitude in plane $A$:

$$\Psi_A(\mathbf{r}) = Apo(\mathbf{r}) P(\mathbf{r})$$

The $\mathbf{r}$-dependence is dropped in subsequent equations. The iterative algorithm previously described is used to numerically compute the apodization function $Apo(\mathbf{r})$, which will converge to a pupil function $\psi_a$ which is the eigenvector of the “truncate (by P), Fourier Transform, truncate (by $|\mathbf{r}|<a$), and inverse Fourier Transform” operator, with eigenvalue equal to the scaling factor $\Lambda_a$ used in step (4) of the iterative algorithm given previously.

$$\label{equ:La}
( \psi_a P ) \otimes \widetilde{M_a} = \Lambda_a \psi_a$$

where $\otimes$ is the convolution operator, $M_a$ is is equal to 1 within a disk of diameter $a$ and is equal to 0 outside it, and $\widetilde{M_a}$ is the Fourier Transform of $M_a$. In the 