# What ways are there to measure the local polarization of a laser beam?

Measuring the polarization of a laser beam is a simple enough task if the polarization is the same everywhere. You can even buy commercial polarimeters.

How do you go about it if the light beam has different polarizations in different parts of the transverse plane? One example is a radially polarized beam. More generally, is there a good technique for sampling the local polarization (which might be linear, elliptical, or circular, anywhere on the Poincaré sphere) in one transverse plane?

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The usual way linear polarisation is measured is by shining polarised light onto a polarising filter, rotating that filter and then using Malus' law to fit the data to a $I_0 cos^2(\theta_{beam} - \theta_{polariser})$ shape. By finding the angular position of the intensity peak we can infer the angle of polarisation of the incoming beam.

Now, assume we shine a beam of a certain spread in the transverse plane, having different polarisations everywhere. If we shine this beam onto a polariser, we will get a pattern of intensities $I(x, y) = I_0 cos^2(\theta_{beam}(x, y) - \theta_{polariser})$. We can find the values of $\theta_{beam}(x, y)$ as follows:

1. Calibration -- rotate $\theta_{polariser}$ and calculate $\max_{x, y} (\theta_{beam}(x, y))$, the peak intensity. The maximum of the peak intensity over all values of $\theta_{polariser}$ will give you $I_0$.
2. Fix $\theta_{polariser}$ to the value that gives the maximum peak intensity. Now you know that you have aligned your polariser with one of the polarisations present in the beam. Then, the ratio $I(x,y) / I_0$ will give you $cos^2(\theta_{beam}(x, y) - \theta_{polariser})$, from which $\theta_{beam}(x, y)$ can be inferred.

This method works for polarisation that is constant in time. For other types of polarisation, you can always take a snapshot over a short period of time that has roughly constant polarisation. Or, if the polarisation is circular, you can use a half waveplate to convert it to linear.

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Nice comprehensive answer! This is about the same as the best idea I had come up with myself so far. I was hoping there was a simpler way, but this isn't so difficult - just tedious ;-) –  ptomato Nov 5 '10 at 0:11

Ernst Mach has once designed an experiment which nicely illustrates linear polarization using a glass cone. Polarized light falls on the cone from the top at Brewster angle. In case of unpolarized light the reflected light has symmetric distribution while with linearly polarized light two dark strips occur in the plane of polarization. It is demonstration rather than measurement but I thought it could be of interest (see the image below).

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That's pretty clever. I expect with a good detector arrangement and some data reduction you could get a pretty good result out of that, and at a high sampling frequency too. –  Colin K Apr 4 '12 at 19:26

Maybe raster scanning the beam in the polarimeters. Or using the same techniques used to measure a single polarization beam but with a camera as a detector.

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Polarization gradients are used extensively in laser cooling and trapping experiments. There is no reason why one cannot use some parameter of an optical trap to obtain local polarization gradients.

Also, you might want to take a look at the Magneto-Optic Kerr effect.

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