# Condenser aberrations in Köhler illumination

## Abstract

This paper presents a general analysis of condenser aberrations in projection systems that employ Köhler illumination. We first analyze condenser aberrations in terms of Hopkins’ theory of partially coherent image formation, and then discuss practical consequences for lithography. The intensity in an aerial image depends jointly on the object, the imaging lens, and the mode of illumination. The angular distribution of illumination isequivalent to the so-called effective source, and is determined by the actual source and the illumination optics. In order that image formation be the same for all points on the mask, the relationship between the entrance pupil of the imaging lens and the effective source should be the same for all points in the field. Condenser aberrations (including defocus) cause the effective source to vary with position in the field. Thus, it is appropriate to refer to the “local effective source”. Changes with field position in the effective source cause variations in the imaging of a particular pattern. One phenomenon, which is well known, is that of lateral image shift with wafer defocus, caused by a lateral shift in the effective source due to condenser aberrations. This is usually analyzed geometrically, by considering a tilt, relative to the principal ray, of the illumination direction at each mask point. According to this geometrical analysis, the image translates linearly with defocus, whereas the more accurate analysis employing the shifted effective source shows a shift that is different and that is nonlinear. There are joint effects involving aberrations of both the imaging lens and condenser. For example, if the the imaging lens has coma, the image suffers a radial expansion that can be compensated by wafer and condenser defocus. In general, the plane of optimum image placement can be displaced axially from the plane of optimum resolution and the paraxial focal plane. Condenser aberration effects are entirely accounted for with the standard formulations for partially coherent imaging, so long as the correct effective source is used. Thus, these effects can be modeled with any partially coherent imaging program that permits an arbitrary effective source. © 1988 SPIE.