Abstract: At short wavelengths, such as the far ultraviolet (FUV), transparent, optically active materials are scarce. Reflection phase retardation by a multilayer thin film can be a good alternative in this wavelength region. We design a multilayer quarterwave retarder by calculating the electric fields at each boundary in the multilayer thin film. Using this method we achieve designs of FUV multilayers which provide high, matched reflectances for both s- and p- polarization states, and at the same time a phase difference between these two states of nearly 90$DGR@. For example, a quarterwave retarder designed at the Lyman-$alpha line ($lambda $EQ 121.6 nm) has 81.05% reflectance for the s-polarization and 81.04% for the p- polarization state. The phase difference between these two polarization states is 90.07$DGR@. For convenience the retarders are designed for 45$DGR angle of incidence, but our design approach can be used for any other angle of incidence. Aluminum and MgF$- 2$/ are used as film materials and an opaque thick film of aluminum as the substrate.!15
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