1 Lyot Filter
Here we consider a Lyot (Pronounced like “Leo”) filter, which can be used to select a wavelength band from an image. The filter consists of multiple stages. The first stage consists of crossed polarizers separated by a liquid crystal layer, that can be tuned by adjusting a voltge to have a particular optical path difference (OPD) between x and y polarizations, that is rotated by 45 degrees to the coordinate system of the polarizers. The second stage is similar but with a different optical path length and with the polarizations reversed from the first stage. For some wavelength, the OPD in the liquid crystal will be an odd multiple of half that wavelength, so that light will be completely transmitted through the (assumed perfect) crossed polarizers. For other wavelengths, the transmission will be less. Because the two liquidcrystal layers have different thicknesses the wavelength behavior is different.
a. Write the Jones matrices for Px, Py, and the waveplates in terms of OPD and wavelength. Then generate the matrix for each stage.
b. Calculate the transmission, T1 of the first stage for x–polarized input and T2 for the second stage with y–polarized input. Plot these for wavelengths from 400 to 800 nm. Use the following optical path differences for the two stages: OPD1 = 600nm × 1.5 and OPD1 = 600nm × 3.5
c. Calculate the transmission of the whole system for x–polarized input, and include on the same plot.
d. How would your results in part c be different if the input were randomly polarized.
2 T/R
Switch Consider the T/R switch discussed in class. Assume that the matrix for the target is the identity matrix.
a. Write the Jones matrices for the components and for the complete T/R switch from source to detector.
b. If the input is perfectly polarized in the x direction, find the output. What is the efficiency of the switch?
c. Do the same calculation using coherency matrices.
d. Assume the input is 80% polarized in the x direction and 20% in y. What is the output coherency matrix and what is the efficiency.
e. Repeat for unpolarized input.
3 Waveplate Orientation
It’s said that people often don’t know or care which axis of a waveplate is which. Prove this by repeating the solution with the QWP in the opposite orientation. That is, exchange the eigenvectors of the QWP matrix.