Numerical Simulation of Binary Lenses by Direct Solution of the Helmholtz Equation

Binary diffractive optics have become an important class of structures for use in optical interconnects and coupling into fibers1. Such structures have been successfully designed in the past using the techniques of Fourier analysis. These techniques are entirely adequate for near-paraxial optics such as occur with high-F-number lenses. The design of fast (F<1.5) lenses such as those required for coupling light into optical fibers, however, requires the use of more accurate techniques, most of which are based upon modal expansions2,3. In this paper we describe a new finite-difference method for modeling small-feature-size binary optical structures that involves the direct solution of the scalar Helmholtz in the vicinity of the structure, followed (in the case of lenses) by wide-angle beam propagation to predict the size of the focal spot. In this approach, the structure is described entirely by by its resulting dielectric function, allowing the treatment of aperiodic binary structures of arbitrary complexity. All reflections together with effects due to sub-wavelength feature sizes are automatically included. Polarization effects are treated approximately owing to the use of semivectorial boundary conditions at all dielectric interfaces. This technique has been shown previously4 to correctly model the behavior of first-and second-order diffraction gratings etched into waveguides. (The present method differs slightly from that described in reference 4 in that the solution method is direct matrix inversion instead of the iterative method described therein.) It involves no expansions or simplifying assumptions (aside from vectorial considerations) and is quite fast, requiring only a few minutes of workstation runtime.