Validity of the Hopkins approximation in simulations of hyper-NA (NA>1) line-space structures for an attenuated PSM mask

In our previous work [1] we have shown that as the NA of a lithographic projection system increases some of the simulation assumptions that are traditionally made - such as the so-called Hopkins assumption, i.e. the assumption that diffraction at the mask is independent of the angle of incidence of the illuminating light waves - break down, at least in some cases. Reliable simulation results will then only be obtained if this Hopkins assumption is eliminated, i.e. when the diffraction effect is re-evaluated for each incident direction. The differences in the results between two such simulations, one using the Hopkins approach, the other with this assumption removed, have been demonstrated to be very significant in some case, but today there is no clear understanding when the removal of the Hopkins assumption is essential: a systematic study is not available. As simulations without the Hopkins approximation are significantly more time consuming than simulations done unde r the Hopkins assumption, a better understanding of which model can or must be used under which circumstances, would be of significant practical importance. The aim of this paper is to provide such a more systematic study for the case of 6% attenuated PSM with line/space structures targeting at a 45 nm resist linewidth for a variety of pitches, for a NA = 1.2 water immersion system. Standard lithographic metrics such as process windows will be used to compare the two simulation approaches. All this work will be done taking the mask topography and optical material parameters into account. As the polarization state of the mask-illumination will also greatly affect the imaging quality at hyper-NA, we will compare the results for different polarization states and illumination modes.