# How do you measure small change in the angle of polarisation of a light beam? What is the smallest measurable angle?

Say you have a beam or pulse of light with known polarisation which then gets perturbed by something which causes that polarisation to rotate slightly, how could you measure this and what would be the smallest measurable change in angle? I'm aware of beam splitters but they have a limit on angle resolution. Is there anything better?

• Depends a bit. If the rotation is due to magnetization (Faraday effect, Kerr effect), then one can modulate the magnetic field and measure really small angles. If that is not possible, there is the problem of the effect of strains.
– user137289
Commented Jun 4, 2018 at 20:50
• Couldn't you just pass the (linearly?) polarized light through a linear polarizer and measure the transmitted intensity as a function of angle? By fitting the (polarizer angle, transmitted intensity) data points to a $\cos^2(\theta)$ function (Malus' law), you should be able to determine the polarization angle of the light beam both before and after it is perturbed very accurately (fraction of a degree with the linear polarizer mounted on a manually operated rotational stage?).
– user93237
Commented Jun 4, 2018 at 21:39
• @SamuelWeir in theory this works out great, in practice not so much unless you do long experiments and the rotation stays for a prolonged time. This is due to small fluctuations in power over time that would cause measurement errors. Some of the sources are, mechanical vibrations, temperature fluctuations, grid power fluctuations, laser fluctuations (especially if it's not super expensive), fabrication tolerances in the measurement tools, etc. Commented Jun 4, 2018 at 22:39
• @BobvandeVoort - I think that the factors you mention all depend on the details of the experimental setup, including the particular light source, whether a quality optical table is used, and whether the measurement is performed in a professional lab or in someone's garage, as well as the required collection times. But I do now see the basic problem that the $\cos^2(\theta)$ function is extremely flat-topped around $\theta$=0, and that the intensity would be reduced by only about 1 part in 1000 in going from 0˚ to 2˚. That makes it difficult to collect data points near $\theta$=0.
– user93237
Commented Jun 4, 2018 at 23:27
• Somewhat related: Usefulness of half-shade in Laurent half-shade polarimeter. Commented Jun 5, 2018 at 8:28

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.

• Can you or anyone explain in a nutshell the key idea(s) or key principle(s) involved in enabling the measurement of nano radian rotations with this apparatus? It isn't immediately clear from the abstract or the figures or the dense mathematics in the paper what's going on.
– user93237
Commented Jun 5, 2018 at 5:58
• @SamuelWeir Added a bit more. Was that in the right direction? Commented Jun 5, 2018 at 6:13
• So what you are saying is it requires extremely specialized equipment to even get to the order of 10^-9 radians? What is the extreme limit that these methods can be pushed to? Commented Jun 5, 2018 at 22:14
• @LUPHYS It’s not all that specialized; any of those can be built in an optical table with care to well enough to do tenths of micro-radians. And I’m not sure “even 10^{-9} radians” captures how small that really is. That’s measuring the rotation due to a 0.9nT field, about 1/10,000 of Earth’s field. Commented Jun 5, 2018 at 22:45
• @Bob Jacobsen I know it's very small, but for the purposes I want to know for it really isn't. I'm talking about induced polarisation due to the presence of axions. Tiny tiny amounts of change. Thanks for your help anyway. Commented Jun 7, 2018 at 7:09

You could use a slab of transparent material with a known index of refraction such that the specular reflection corresponds to Brewster's angle for the polarization perpendicular to the desired polarization. That is, in the ideal case, the laser pulse is incident on the surface with p-polarization, and the specular reflection is zero. See the picture below. Your laser pulse should be polarized parallel to the plane of incidence. If everything is aligned correctly, there will be no reflected ray.

This way, you can measure the variation in polarization with a light meter placed to intercept the reflected light. Deviations from zero power are much easier to measure than deviations from maximum power.

I have not done the math to figure out the smallest polarization angle defect this could measure.

Image from wikipedia article.

• "Deviations from zero power are much easier to measure than deviations from maximum power." - Is there any other advantage to this Brewster's angle setup? If it's simply a matter of devising a system which shows deviations from zero power due to a slight change in the polarization of the incoming light, couldn't one simply introduce a linear polarizer oriented at 90˚ with respect to the incoming polarized light beam and then measure any changes in the transmitted light intensity through the linear polarizer?
– user93237
Commented Jun 5, 2018 at 2:09
• @SamuelWeir A polarizer at 90 degrees would not allow the beam to be used for its original purpose, whereas the Brewster's angle arrangement is purely transmissive in the ideal. Depending on the power of the laser, 100-percent absorption might be a bad idea as well. Commented Jun 5, 2018 at 2:24
• Deviations from zero power go as the cosine of the rotation. Changes around mid-power (45 degrees) go like the sine, so change fastest with small angle changes. If you have a stable detector & repeatable signals to measure, sine-like is better. For discovery of an unknown, cosine-like might be better. Commented Jun 5, 2018 at 14:26