4.3 Advanced Illumination Aperture

As described in Section 2.3.3 the illuminator of a projection printing system is not a simple point source. Spatially extended apertures are often applied to enhance the imaging performance. A schematic configuration providing off-axis rays for the illumination is shown in Figure 2.4. Under the assumption of Köhler illumination described above, two convenient methods exist to model imaging with such advanced illuminators. Both approaches are based on a spatial discretization of the source into discrete point sources. These point sources are mutually incoherent due to the thermal nature of the light source. The statistical independence is of crucial importance for the performance of projection systems as only a local spatial coherence is introduced on the mask. From a simulation point of view the two methods are implemented as follows: Computing the intensity by taking one incident ray at a time, finding its diffraction, and summing the collected field is known as Abbe's method of imaging [11, pp. 418-424]. The second formulation is due to Hopkins [124,125,126], who observed that the integration over the source can be carried out before summing up the diffraction angles accepted by the lens. We have chosen Abbe's method as it calculates the exciting field amplitudes on the wafer--needed for the exposure/bleaching simulation--as an intermediate result. Hopkins' approach directly yields the aerial image intensity. It is thus applied in many simulation programs that use either a simple exposure simulation, e.g., SAMPLE [109], or are exclusively devoted to aerial image simulation, e.g., SPLAT [117]. We will start with a detailed discussion of Abbe's method and then only briefly outline the main formulae of Hopkins' approach for the sake of completeness.