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Chapter 8:
Predicting Scatter from Roughness

Prediction is very difficult, especially if it's about the future. - Niels Bohr

One of the difficulties associated with scatter measurements is the large number of variables on which the scatter distribution depends. In addition to sample parameters such as roughness, bulk defects, and contamination, there are instrumentation-dependent parameters such as polarization, angle of incidence, and wavelength. Separating the various effects is not trivial, although there can be strong economic motivations to do so. The usual approach has been to make scatter measurements under the conditions expected for actual use; that is, to use the polarization, incident angle, and wavelength that are intended for eventual use to make the scatter measurements. Although polarization and incident angle are relatively easy to adjust in most instruments, a huge amount of data would be required to cover all of the combinations that a given sample might encounter in its expected use. In addition, generating scatter data at arbitrary wavelengths is an expensive task. Therefore, there are strong motivations for being able to predict sample scatter and avoid taking the data.

Unfortunately, there has been a tendency to oversimplify some scatter predictions, which has led to poor results and considerable confusion. For example, the relationship TIS = (4πσ/λ)2 can be interpreted as follows: if the rms roughness doubles, the measured TIS will quadruple. This is essentially true (ignoring the high-angle sensitivity limitations of TIS measurements given in Section 1.6) if the increase in σ is produced proportionally over the spatial bandwidth corresponding to the angular collection cone of the scatter instrument (i.e., the PSD doubles over the collection angles). But the equation is easily misinterpreted as indicating that if the illuminating wavelength is doubled, the measured scatter from a given reflector should be reduced by one fourth. Although one could probably guess that the TIS would be reduced in such a case, it would be just a guess because changing the wavelength changes the spatial bandwidth of the measurement. If a grating was being measured, and doubling the wavelength brought the first-order diffraction into the collection optics, the TIS would probably increase!

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