Snell or Fresnel: the influence of material index on hyper-NA lithography

As immersion lithography is extended to ever increasing resolution, the resulting propagation angles in the materials involved become closer to grazing than to normal incidence. Classical laws of refraction and reflection cannot be used with either assumption however, as a collection of angles may exist across the entire range. Fresnel reflection at these angles becomes large enough that small disparities in refractive indices at material interfaces may lead to adverse effects. As an example, when water is used at numerical apertures approaching its refractive index, reflection effects are greater than the constraints imposed by refraction or absorption. This will limit the maximum NA value allowed by any given material to values sufficiently lower than its refractive index. Additionally, we have grown accustom to expanding the application of the Snell-Descartes Law to materials with low absorption, assuming that the contribution of the imaginary component of the refractive index is negligible. This is not the case for photoresists, fluids, or glasses, which can not strictly be considered as non-absorbing media. We have expanded the Snell-Descartes Law for absorbing media, with some interesting consequences. We will show that there is no real limit on the numerical aperture into a material, so long as its extinction coefficient is not zero. The relationship that lithographers have been using recently where NA< min(nglass, nfluid, nresist) will be shown to be inadequate and imaging at numerical apertures up to 1.85 will be demonstrated using materials with significantly lower (real) refractive index values. Keyword list: Hyper-NA, reflection, refraction, Snell's Law, Fresnel, immersion lithography