Diffraction of converging electromagnetic waves

Using the electromagnetic equivalent of the Kirchhoff diffraction integral, we investigate the effect of spherical aberration and defocus on the diffraction of Gaussian, uniform, and centrally obscured beams. We find, among other things, that in high-angular-aperture systems suffering from either spherical aberration or defo- cus the axial intensity distribution is no longer symmetric. Equations are derived for the axial intensity near focus for different beam profiles. Intensity contours in focal and meridional planes are depicted for both ideal and aberrated lenses. It is shown that, contrary to certain previous theories, our theory is valid for both high and low angular aperture systems. In a previous paper' we have given a detailed description of a new electromagnetic diffraction theory based on the vectorial equivalent of the Kirchhoff-Fresnel integral. The diffracted fields were obtained by integration over the (aberrated) wave front. We applied this theory to in- vestigate the effect of spherical aberration on the electro- magnetic field in the focal region of a high-aperture lens. Among other things, we found for an incoming plane wave that the intensity distribution on the axis was no longer symmetric around the peak. A similar feature has re- cently been measured. 2 Our aim in the present paper is twofold. First, we show that the vectorial theory of Richards and Wolf, 3 4 which is valid for high-aperture val- ues, and the paraxial scalar theory of Li and Wolf' are both special cases of our approach. Second, we extend this electromagnetic model to include Gaussian beams, centrally obscured beams, and defocus. Equations are derived for the axial intensity distribution of such sys- tems. It was found for high-aperture lenses with defocus that the displacement theorem (Ref. 6, Chap. 9), which predicts a mere shift of the diffraction pattern, no longer holds. The intensity distribution is now asymmetric and has a lower peak intensity. It is seen in our study that one can clearly distinguish among three types of lens, namely, paraxial, low- aperture, and high-aperture systems, with Fresnel num- bers of order 1, 102, and 104, respectively. A study of different beam profiles, but for a scalar theory in the Fresnel approximation, has been carried out by Mahajan, 7 who compared beams with the same total power. We also mention the research of Mansuripur, who describes a vectorial Fresnel diffraction theory. It should be noted that these Fresnel theories are based on four additional assumptions that are not present in our framework: