Abstract
We consider a partial differential equation (PDE) approach to
numerically solve the reflector antenna problem by solving an optimal
transport problem on the unit sphere with cost function $c(x,y) = - 2\log || {x -
y} ||$. At each point on the sphere, we
replace the surface PDE with a generalized Monge–Ampère type equation
posed on the local tangent plane. We then use a provably convergent
finite difference scheme to approximate the solution and construct the
reflector. The method is easily adapted to take into account highly
nonsmooth data and solutions, which makes it particularly well adapted
to real-world optics problems. Computational examples demonstrate the
success of this method in computing reflectors for a range of
challenging problems including discontinuous intensities and
intensities supported on complicated geometries.
© 2021 Optical Society of
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