  Cooling down test masses to cryogenic temperatures is a way to reduce the
thermal noise of gravitational wave detectors. Crystalline materials are
considered the most promising materials for fabricating cryogenic test masses
and their coatings because of their excellent thermal and optical properties at
low temperatures. However, birefringence owing to local impurities and
inhomogeneities in the crystal can degrade the performance of the detector. The
birefringence measurement or mapping over a two-dimensional area is thus
important. This study describes a method for fast birefringence measurements of
a large sample by simply combining a series of transmission wavefront error
measurements using linearly polarised light with Fizeau interferometers. Using
this method, the birefringence inhomogeneity information of KAGRA's two input
test masses with a diameter of 22 cm was fully reconstructed. The birefringence
information was then used to calculate the transverse beam shapes of the light
fields in orthogonal polarisation directions when passing through the
substrate. It was possible to find a calculated beam shape consistent with
in-situ measurements using the KAGRA interferometer. This technique is crucial
for birefringence characterisation of test masses in future detectors, where
even larger sizes are used.
