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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.

** Next:** 4.3.1 Abbe's Method
**Up:** 4. Aerial Image Simulation
** Previous:** 4.2.4 In-Lens Filters
*Heinrich Kirchauer, Institute for Microelectronics, TU Vienna *

1998-04-17