Spatial dispersion of crystals as a critical problem for deep UV lithography

Abstract

The birefringence effect caused by spatial dispersion in crystals has been known for more than a hundred years, but until recently it was studied only from the viewpoint of theoretical physics. This article discusses the mathematical description of the effect as well as problems associated with the presence of the effect in modern lithographic optical systems and methods of solving these problems.

Search for Hermite-Gauss mode rotation in cholesteric liquid crystals

“…This is well known in crystal optics: Within the dipole approximation, cubic crystals should be isotropic, however, many show birefringence due to spatial dispersion. This effect has been predicted by Lorentz in 1878, firstly discovered in [32], and became relevant in UV lithography today [33]. It is not clear yet if such structure could allow for OAM-sensitive scattering.…”

On the theory of light propagation in crystalline dielectrics

“…comprising the optical properties of a dielectric crystal. Assuming spatial dispersion is a weak effect, which assumption applies for many transparent media, the Taylor expansion of q, ab T e w ( ) ( ) around q=0 then provides (yet in an implicit manner) access to various optical phenomena featuring the propagation of light in dielectric crystals [3], for instance chromatic dispersion and birefringence, rotary power (natural optical activity) and also the (weak) effects of a spatial-dispersion-induced birefringence, the latter being a critical problem for the design of lens elements made from crystalline materials like CaF 2 and BaF 2 widely used in optical lithograpy systems in the ultraviolet [18,19]. Further we summarize in section 4.6, see table 1, figure 10 and also figure 11, to what large extend our theory of the dielectric tensor for crystalline dielectrics agrees with measurements over a wide range of optical frequencies for a series of well known crystalline materials, including for example Bi 12 TiO 20 and also Bi 12 SiO 20 , both crystals featuring a large number of basis atoms (M=66) in the unit cell, thus demonstrating the utility of our approach.…”

“…But for crystals with cubic symmetry the number of independent components of that tensor reduces substantially: for symmetry group T and T h there exist four independent components, for symmetry group T d , O h and O there exist three independent components and for isotropic systems that number reduces to two [3]. The (weak) effects of a dispersioninduced-birefringence indeed give as a matter of fact reason for concern regarding the image quality of dielectric lenses made from CaF 2 and BaF 2 , a topic of prime importance designing modern lithographic optical systems in the ultraviolet [18,19].…”

Cholesteric polymers and the orbital angular momentum of light

“…This effect has been predicted by Lorentz in 1878, firstly discovered 34 in 1971, and it is shown to have important implications in UV lithography today. 35 It is not clear yet if this could allow for OAM interaction, however, our results make this appear improbable.…”