Dual-shaped offset synthesis using parameter optimization of the solutions to the first-order geometrical optics partial differential equations

When integrating the geometrical-optics-derived nonlinear partial differential equations (PDEs) for offset dual-shaped reflector synthesis from a prescribed outer projected perimeter inwards, computational problems arise in the central region of the reflectors. A new parameter of the PDEs has been introduced, the radius, rho /sub 0/, of the inner projected rim of the main reflector when the inner value of theta = theta /sub 0/ for the subreflector is prescribed. A poorly selected inner radius parameter rho /sub 0/ can preclude the obtaining of useful solutions to the first-order PDEs by commonly used methods. Two complementary methods for choosing the parameter rho /sub 0/ and other parameters have been used. The first method utilizes the initial values found from an approximate solution to the PDEs. The second method utilizes an optimization algorithm which searches for the parameters of the PDEs which minimize a prescribed object function. Such an object function may measure the deviation of the projected outer perimeter from a circular shape (or some other desired shape).<>