  We search for the best possible transmission for an external occulter
coronagraph that is dedicated to the direct observation of terrestrial
exoplanets. We show that better observation conditions are obtained when the
flux in the focal plane is minimized in the zone in which the exoplanet is
observed, instead of the total flux received by the telescope. We describe the
transmission of the occulter as a sum of basis functions. For each element of
the basis, we numerically computed the Fresnel diffraction at the aperture of
the telescope and the complex amplitude at its focus. The basis functions are
circular disks that are linearly apodized over a few centimeters (truncated
cones). We complemented the numerical calculation of the Fresnel diffraction
for these functions by a comparison with pure circular discs (cylinder) for
which an analytical expression, based on a decomposition in Lommel series, is
available. The technique of deriving the optimal transmission for a given
spectral bandwidth is a classical regularized quadratic minimization of
intensities, but linear optimizations can be used as well. Minimizing the
integrated intensity on the aperture of the telescope or for selected regions
of the focal plane leads to slightly different transmissions for the occulter.
For the focal plane optimization, the resulting residual intensity is
concentrated behind the geometrical image of the occulter, in a blind region
for the observation of an exoplanet, and the level of background residual
starlight becomes very low outside this image. Finally, we provide a tolerance
analysis for the alignment of the occulter to the telescope which also favors
the focal plane optimization.This means that telescope offsets of a few
decimeters do not strongly reduce the efficiency of the occulter.
