  Direct detection of extrasolar planets will require imaging systems capable
of unprecedented contrast. Apodized pupils provide an attractive way to achieve
such contrast but they are difficult, perhaps impossible, to manufacture to the
required tolerance and they absorb about 90% of the light in order to create
the apodization, which of course lengthens the exposure times needed for planet
detection. A recently proposed alternative is to use two mirrors to accomplish
the apodization. With such a system, no light is lost. In this paper, we
provide a careful mathematical analysis, using one dimensional mirrors, of the
on-axis and off-axis performance of such a two-mirror apodization system. There
appear to be advantages and disadvantages to this approach. In addition to not
losing any light, we show that the nonuniformity of the apodization implies an
extra magnification of off-axis sources and thereby makes it possible to build
a real system with about half the aperture that one would otherwise require or,
equivalently, resolve planets at about half the angular separation as one can
achieve with standard apodization. More specifically, ignoring pointing error
and stellar disk size, a planet at $1.7 \lambda/D$ ought to be at the edge of
detectability. However, we show that the non-zero size of a stellar disk pushes
the threshold for high-contrast so that a planet must be at least $2.5
\lambda/D$ from its star to be detectable. The off-axis analysis of
two-dimensional mirrors is left for future study.
