How do you measure small change in the angle of polarisation of a light beam? What is the smallest measurable angle?
With the right experimental setups, nanoradian rotations are measurable.
Polarizers, optical bridges and Sagnac interferometers for nanoradian polarization rotation measurements Alistair Rowe, Indira Zhaksylykova, Guillaume Dilasser, Yves Lassailly, Jacques Peretti
The ability to measure nanoradian polarization rotations, $\theta_f$,, in the photon shot noise limit is investigated for partially crossed polarizers (PCP), a static Sagnac interferometer and an optical bridge, each of which can in principal be used in this limit with near equivalent figures-of-merit (FOM). In practice a bridge to PCP/Sagnac source noise rejection ratio of $1/4 \theta^2_f$ enables the bridge to operate in the photon shot noise limit even at high light intensities. The superior performance of the bridge is illustrated via the measurement of a 3 nrad rotation arising from an axial magnetic field of 0.9 nT applied to a terbium gallium garnet.
Briefly, they're comparing three methods:
(TGG creates the test rotation) The top is the traditional Partially Crossed Polarizer (PCP) approach. The sensitivity comes from having them partially crossed.
The center one is a Sagnac interferometer. Those are normally used to sense global rotations, like a gyroscope. But an optical rotation can be sensed too. Light going in the two directions around the path is rotated in opposite ways, affecting the end-point interference.
The third approach is called a "Polarising Bridge". The Partial Beam Splitter (PBS) allows the two detectors to look at the X and Y coordinates of the rotated beam, allowing a better comparison (less noise) than the PCP approach, at least in theory.
The experimental result is a comparison of measuring a 3 nanoradian ($3 \times 10^-9$ radian, less than a thousandth of a second of arc) rotation of visible light. That’s the effect due to 25mm of material in a 0.9nT magnetic field, about 1/10,000 of Earth’s field.