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'Two-mirror and two-lens devices converting an incident plane
wave of a given cross section and intensity distribution into an output
plane wave irradiating at a given target set with prescribed intensity
are required in many applications. Most of the known designs are
restricted to rotationally symmetric mirrors/lenses. In this talk I
will discuss designs with freeform mirrors/lenses, that is, without a
priori assumption of rotational symmetry. Assuming the geometrical
optics approximation, it is shown that the functions describing such
freeform mirrors/lenses satisfy Monge-Amp`ere type partial differential
equations (PDE’s) derived from the basic laws of geometrical optics.
However, because of strong nonlinearities numerical solution of such
PDE’s by known numerical methods is difficult. Fortunately, the above
design problems can also be formulated as problems in calculus of
variations in which instead of solving the nonlinear PDE’s one needs to
find extrema of some Fermat-like functionals taking the role of cost
functions in optimal transportation problems. This approach was
successfully applied in single and two-mirror design problems and
allowed to calculate solutions with high accuracy by methods based on
linear programming. I will describe some of these results and indicate
their extensions to the two-lens design problem.'
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