2.6. Brief theory of polarization microscopy with variable retarders

A specimen is considered to have retardance magnitude R and slow axis azimuth ϕ distributed in the Cartesian coordinate x-y plane, termed the specimen plane. Using the instrument described above, retardance values up to π radians (273.5 nm for a 547 nm wavelength source) and azimuth values from 0° to 180° can be measured. Birefringence maps are calculated from intensity values of images collected under different voltage settings applied to liquid crystal variable retarders in the polarization state generator [42]. This approach is rapid (∼seconds) and requires no moving parts during acquisition.

Elliptical polarization states are generated by a pair of LCVRs (LCC1423-A, Thorlabs) which alter linearly polarized light passing through a fixed polarizer. The slow axes of LC-A and LC-B are aligned at 45° and 0° with respect to the principal axis of the linear polarizer. A Jones vector represents light polarization after passing through the two retarders is:

where α and β refer to the retardance of LC-A and LC-B, respectively, at given voltage settings.

The final intensity, transmitted through the specimen, quarter waveplate, linear analyzer, and recorded at the camera, I(α, β, x, y), can then be derived as:

where I_max(x, y) is the illumination intensity distribution, τ(x, y) is an isotropic attenuation factor due to the specimen, and I_min(x, y) is specimen background illumination. Γ<!-- Γ -->(x,y) and θ<!-- θ -->(x,y) are the desired retardance and azimuth maps, respectively.