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The most important configuration of the illuminator in projection printing
systems is due to Köhler [11, pp. 524526].
In this method a converging lens commonly
called condenser is positioned in front of the light source, whereby
the light source lies in the focal plane of the lens
(cf. Section 2.3.2). As illustrated in
Figure 4.5 the rays from each source point then emerge
from the condenser as parallel beams. This arrangement has the advantage
that irregularities in the brightness distribution on the source do not
cause irregularities in the intensity of the object illumination. Thus
the quality of image composition severely depends on a careful and accurate
setup of the Köhler arrangement. For the mathematical description we
assume an ideal setup, so that the emerging beams from one source point
formafter passage through the condensera plane wave.
In complex phasor notation we get for the disturbance on the object

(4.35) 
with amplitude A_{s} and normalized wavevector

(4.36) 
Figure 4.5:
Köhler illumination of an
object. The Köhler arrangement provides a plane wave illumination
of the mask. The quality of the image composition severely depends on
an accurate setup.

In photolithography the object to be imaged is the photomask, whereby the
pattern is either exclusively stored in the amplitude transmittance or in
both amplitude and phase. Technological aspects of binary and
phaseshifting masks were already discussed in Section 2.4.
For aerial image simulation the mask is modeled to be infinitesimally thin
with ideal transitions of the transmission characteristic, i.e., the mask
thickness
d_{ø} vanishes as shown in
Figure 4.5. Depending
on the mask type, the socalled transmission function t(x, y) is realvalued
(zero or one) or complexvalued with module less than one. Similar
to the phase transformation of the lens in (4.19)
the field disturbance U_{s}(x, y) incident on the mask is multiplied with
the transmission function t(x, y) to obtain the object field immediately
behind the mask, i.e.,

(4.37) 
This relation holds for the same reasons as discussed in connection with
the Kirchhoff boundary conditions below (4.6) on
page .
For Köhler illumination we get with (4.38)
for the object amplitude

(4.38) 
Now we make a fundamental assumption that is used throughout all aerial
image simulators: The photomask is periodic in both lateral
directions x and y with periods a/M and b/M, respectively, i.e.,

(4.39) 
The object periods are chosen as a/M and b/M, so that the image periods
are a and b for a M : 1projection system (cf. (4.37)).
A direct consequence of this assumption is the possibility to represent
the transmission function t(x, y) by a Fourier series, i.e.,

(4.40) 
whereby the Fourier coefficients are given by

(4.41) 
Combining the two equations (4.41)
and (4.43) yields for the object amplitude

(4.42) 
whereby
k_{0} = 2/
has been inserted for the wavenumber. The object
coordinates have to be scaled by the magnification M resulting in
(cf. (4.35))

(4.43) 
Next we restrict the source point locations to a rectangular grid.
Hence, only homogeneous plane waves from certain discrete orientations
have to be considered further. We choose the spacing of the grid in the
source plane in such a way that the lateral components of
the normalized wavevector
s_{pq} equal

(4.44) 
The reason for this discretization will be explained later.
Throughout aerial image simulation, it is more or less
irrelevant although the resulting formulae are more compact. However,
for an efficient exposure/bleaching simulation this specific choice
is crucial as will be explained in Chapter 6.
Insertion of the discrete wavevectors (4.47)
into (4.46) yields
for the scaled object amplitude due to one source point (p, q).
Because of the periodicity of
U_{o}^{pq}( x/M,  y/M) its Fourier transform
consists of a train of impulses occurring at the harmonically related
frequencies n/a and m/b and writes to

(4.45) 
Now we can easily determine the image spectrum from (4.37).
Employing the equivalence property of the deltafunction^{k}

(4.46) 
Consequently, we obtain a Fourier series representation for the image such as

(4.47) 
whereby the Fourier coefficients are given by
As can be seen the discrete pupil function P(n, m) is related to the real
physical pupil function
(x, y) by
P(n, m) = M(d_{i}n/a,d_{i}m/b) and writes
with (4.22) to

(4.49) 
The intensity
I_{i}^{pq}(x, y) due to the illumination of the mask
by one source point (p, q) is simply the squared module of the amplitude
divided by twotimes the freespace resistance ,^{l}
i.e.,

(4.50) 
Footnotes
 ... deltafunction^{k}
 The equivalence
or filtering property of the delta function (x) states that
f (x) = h(x)(x  x^{}) equals
f (x) = h(x^{})(x  x^{}).
 ...,^{l}
 The freespace
resistance has a value of
=
377 .
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Up: 4.1 Principles of Fourier
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Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417