Extending scalar aerial image calculations to higher numerical apertures

The usual formula for the scalar aerial image of an isolated object due to a projection lens system has been generalized beyond the paraxial approximation in an attempt to extend scalar diffraction theory to include numerical aperture (NA) values up to about 0.6. Beyond this regime, or certainly beyond NA=0.7, polarization effects need to be included, thereby demanding a full vector treatment and invalidating the present scalar formulation. A key point to the present scalar result without the paraxial approximation is the predicted functional dependence of the aerial image on magnification as NA increases. A second key point is that the usual scaling of λ/NA for the object dimensions and λ/NA2 for defocus become invalid for high NA systems. Numerical results of illustrative test cases are shown.