Use of Kramers-Kronig relations to extract the conductivity of high-Tcsuperconductors from optical data

In principle the conductivity of the cuprate superconductors can be obtained from reflectivity measurements using the Kramers-Kronig-transform technique. However, at low temperatures and for frequencies below ∼300 ${\mathrm{cm}}^{\mathrm{-}1}$ the reflectivities of materials such as ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ are close to unity. Uncertainty in the precise signal level corresponding to unity reflectivity and a lack of knowledge of the reflectivity below the lowest measured frequency cause this method to become unreliable. To address this problem we have used a bolometric technique and a resonant technique to obtain accurate submillimeter and microwave data for the residual losses in epitaxial thin films of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ at low temperatures. The Kramers-Kronig analysis of our data is in good agreement with results from fitting our data to simple weakly coupled grain and two-fluid models for the a-b plane conductivity. However, below 450 ${\mathrm{cm}}^{\mathrm{-}1}$ it is in disagreement with some published results of other workers obtained from Kramers-Kronig analysis of reflectivity data. To understand this discrepancy we analyze how the conductivity determined by the Kramers-Kronig-transform technique depends on some commonly used low-frequency extrapolations of reflectivity data.