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MICROLITHOGRAPHY

MICROLITHOGRAPHY 9.13

9.2.3 Partial Coherence
Although we have completely described the behavior of a simple ideal imaging system, we must add
one more complication before describing the operation of a projection system for lithography. So far,
we have assumed that the mask is illuminated by spatially coherent light. Coherent illumination
means simply that the light striking the mask arrives from only one direction. We have further
assumed that the coherent illumination on the mask is normally incident. The result was a diffrac-
tion pattern that was centered at the entrance to the objective lens. What would happen if we changed
the direction of the illumination so that the light struck the mask at some angle q′? The effect is sim-
ply to shift the position of the diffraction pattern with respect to the lens aperture (in terms of spa-
tial frequency, the amount shifted is sinq′/l). Recalling that only the portion of the diffraction pattern
passing through the lens aperture is used to form the image, it is quite apparent that this shift in the
position of the diffraction pattern can have a profound effect on the resulting image.
If the illumination of the mask is composed of light coming from a range of angles rather than
just one angle, the illumination is called partially coherent. If one angle of illumination causes a shift
in the diffraction pattern, a range of angles will cause a range of shifts, resulting in broadened dif-
fraction orders. One can characterize the range of angles used for the illumination in several ways,
but the most common is the partial coherence factor s (also called the degree of partial coherence,
the pupil filling function, or just the partial coherence). The partial coherence is defined as the sine
of the half-angle of the illumination cone divided by the objective lens numerical aperture. It is thus
a measure of the angular range of the illumination relative to the angular acceptance of the lens.
Finally, if the range of angles striking the mask extends from −90 to 90° (i.e., all possible angles),
the illumination is said to be incoherent.
The extended source method can be used to calculate partially coherent images. In this method,
the full source is divided into individual point sources. Each point source is coherent and results in
an aerial image calculated using the diffraction pattern appropriately shifted in the pupil for that
source point. Two point sources from the extended source, however, do not interact coherently with
each other. Thus, the contributions of these two sources must be added to each other incoherently
(i.e., the intensities of the resulting images are added together). The full aerial image is determined
by calculating the coherent aerial image from each point on the source, and then integrating the
image intensity over the source.

9.2.4 Aberrations and Defocus
Aberrations can be defined as the deviation of the real behavior of an imaging system from its ideal
behavior (the ideal behavior was described earlier using Fourier optics as diffraction limited imag-
ing). Aberrations are inherent in the behavior of all lens systems and come from three basic
sources—defects of construction, defects of use, and defects of design. Defects of construction
include rough or inaccurate lens surfaces, inhomogeneous glass, incorrect lens thicknesses or spac-
ings, and tilted or decentered lens elements. Defects of use include use of the wrong illumination or
tilt of the lens system with respect to the optical axis of the imaging system. Also, changes in the
environmental conditions during use, such as the temperature of the lens or the barometric pressure
of the air, result in defects of use. Defects of design may be a bit of a misnomer, since the aberra-
tions of a lens design are not mistakenly designed into the lens, but rather were not designed out of
the lens. All lenses have aberrated behavior since the Fourier optics behavior of a single lens element
is only approximately true. It is the job of a lens designer to combine elements of different shapes
and properties so that the aberrations of each individual lens element tend to cancel in the sum of all
of the elements, giving a lens system with only a small residual amount of aberrations. It is impos-
sible to design a lens system with absolutely no aberrations.
Mathematically, aberrations are described as wavefront deviations, the difference in phase (or
path length) of the actual wavefront emerging from the lens compared to the ideal wavefront as pre-
dicted from Fourier optics. This phase difference is a function of the position within the lens pupil,
most conveniently described in polar coordinates. This wavefront deviation is, in general, quite com-
plicated, so the mathematical form used to describe it is also quite complicated. The most common

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