We are developing high performance mid-infrared (especially 30-40μm wavelength regions) multilayer interference
filters with mechanical strength and robustness for thermal cycling toward cryogenic infrared astronomical missions.
Multilayer interference filters enable us to design a wide variety of spectral response by controlling refractive index and
thickness of each layer. However, in mid- and far-infrared (MIR/FIR) regions, there are a few optical materials so that
we can only use limited refractive index values to design filters, which makes difficult to realize high performance
filters. It is also difficult to deposit thick layers required for MIR/FIR multilayer filters. Furthermore, deposition of two
materials, which have different coefficients of thermal expansion, makes filters fragile for thermal cycling. To clear these
problems, we introduce sub-wavelength structures (SWS) for controlling the refractive index. Then, only one material is
necessary for fabricating filters, which enables us to fabricate filters with mechanical strength and robustness for thermal
cycling. According to the effective medium approximation (EMA) theory, the refractive index of randomly mixing
materials in sub-wavelength scale is controllable by changing the ratio of mixing materials. However, it is not clear that
EMA can be applied to such simple SWS, periodic cylindrical holes on a bulk material, which is easily fabricated by
photolithography. In order to verify the controllability of refractive index by simple SWS, we have fabricated simple
SWS on a silicon substrate and measured its transmittance. Comparing measured transmittance with theoretical
transmittance calculated by EMA, we confirm that EMA can be applied to simple SWS fabricated by photolithography.