Derivation and Simulation of Higher Numerical Aperture Scalar Aerial Images

The application of scalar diffraction theory to a projection optics system is examined here for somewhat higher numerical aperture conditions upon removing the paraxial, or small angle, approximation that is typically made. A detailed derivation is given that notes the key physical assumptions originally contained in work by others on vector imaging theory. The asymptotic limit of large lenses and focal length sizes to object and image sizes is explicitly carried out, while keeping numerical apertures and magnification fixed. Numerical results of the resulting equations are presented for a variety of imaging conditions, including phase shift masks and modified illumination.