Optics for photographing degree of polarization?

It's relatively common practice in photography to use a polarizing filter to get rid of reflections from non-metallic surfaces. Usually you just put this filter in front of the lens and rotate it manually to take pictures with your desired polarization

Comparing with and without the filter, you would get something like below (from wikipedia).

This is quite useful, but what I'm interested in is if one can make an image that shows the degree of polarization itself. For example, highly polarized light sources like computer monitors would show up bright, which unpolarized light from light bulbs would show up as dark. As a practical use case, since circularly polarized light is really rare in nature, one could use such a camera aetup to send messages with circularly polarized light even in high light pollution areas.

Mathematically, I think this would look like

$$\mathrm{Image} = \frac{I_{\perp}-I_{\parallel}}{I_{\perp}+I_{\parallel}}$$

So my question is: are there some optics that you can attach to a camera to directly give you an image of the degree of polarization? Ideally the optics would be portable, and require only a single exposure of the camera with minimal post-processing.

1. Taking a sequence of pictures with a polarizing wheel for a range of angles. The issue here is that it requires a lot of exposures and manual movement, so the lighting may be inconsistent and/or the scene will change between images.

2. Some sort of electrically controlled polarizer, so images of different polarization could be taken back to back. Issue here is that I'm not finding a way to integrate such a device on a portable camera. A lot of optical polarization switches need really high voltages.

This electronics.se post is relevant, though it's not clear if a solution was found, https://electronics.stackexchange.com/questions/198607/electrically-switching-between-horizontal-and-vertical-light-polarization. I don't really care about financial cost, just if a convenient solution exists that is reasonably attainable

• I think four filters would be enough: two linear polarizers oriented at 90 degrees to each other, a left-circular polarizer, and a right-circular polarizer. – S. McGrew May 19 '19 at 14:55
• @S. McGrew, not sure I follow? – KF Gauss May 19 '19 at 17:00
• Suppose that the light is linearly polarized but at an unknown angle. In that case, two measurements at 90 degrees to each other will tell you the polarization angle. But those will not tell you if the light is circularly polarized- which corresponds 90 degrees relative phase between linear polarizations. If you also measure the right- and left- circular polarizations of the light, it will give you all information needed to calculate the relative phase. Different wavelengths often are polarized differently, so to get full information you need the four measurements at each wavelength. – S. McGrew May 19 '19 at 20:25
• A good place to start is to look up Jones matrix. – S. McGrew May 19 '19 at 20:26
• @S. McGrew, ah I understand. Yes I know you definitely can do this process by taking 4 separate images, but I'm curious if the optics can be setup to do all of this in a single exposure. – KF Gauss May 19 '19 at 23:04

Yes, it is possible to design optics to record the full polarization state of an image (monochromatic) in a single exposure. A very straightforward way would be to substitute for the single lens in a camera, an array of four lenses to form four separate images of the same scene on a single photodetector array. In front of the four lenses you would place a vertical linear, horizontal linear, right circular, and left circular polarizer respectively. There are some variations on that theme, all accomplishing essentially the same thing.

• I should have worded my question better, what I'm looking for is for some optics that can be placed in front of usual off-the-shelf camera lenses to image how strongly polarized things are. I don't necessarily want to know how polarized anything is, just whether it's polarized or not. You're totally right that the most obvious way to do this is by just using four filters in a polarization wheel (as described in my question). I'm just curious if there was some way to skip taking multiple pictures since we don't need all that information, and not just by having 4 different images/lenses/sensors. – KF Gauss May 20 '19 at 6:38
• An electrically switchable polarizer would be equivalent to the wheel and would require four photos. I think you do need all that information, at least to accomplish what you've stated in the question. There are several ways to combine the info from the 4 images into a single exposure: 4 lenses -> 4 reduced images, or else something more complicated that scrambles, overlaps, or interleaves the 4 images in a decipherable way. Instead of 4 lenses/filters, how about a single 4-part optical element that just goes over the lens? That's do-able. – S. McGrew May 20 '19 at 13:14
• Thanks for the reply, wouldn't a single 4-part element just mask the polarization of each quadrant of the image differently (e.g. upper left is H polarized, upper right L polarized etc.)? Or are you suggesting something else? – KF Gauss May 21 '19 at 0:34
• I'm imagining a filter you could screw onto the front of a camera lens. The filter would be divided into quadrants with the different filters, but also each quadrant would include a segment of a weak lens. Each lens segment forms a reduced-size image, offset appropriately from the center of the camera's image sensor. The lens segment could be either a standard refractive lens segment, or could be (much less expensive) a Fresnel lens segment. The result would be four differently-filtered images with slightly reduced resolution, all shot in a single exposure. – S. McGrew May 21 '19 at 3:35
• Thanks for pointing out the article. However, I think the solution given there is equivalent to the one I offered. Their metalens produces four images in essentially the same way as my solution, "simply" using polarization-selective metalenses instead of the segmented lens/filter I suggested. The optical paths are the same in their solution as in mine. – S. McGrew Jul 7 '19 at 13:16

Measuring the polarization of incoming light can indeed by achieved by essentially taking four separate pictures by mechanically rotating filters, or by having four different cameras with distinct optical paths for each polarization.

However, a true single component solution seems to have been just developed by a team of engineers just this past year using a monolithically integrated meta-surface grating, which does all of the optics in a single go. See the paper here: https://science.sciencemag.org/content/365/6448/eaax1839

This is the true solution to my question, as they provide a single optical element that is placed into the camera optical system to get the full Stokes matrix. Once commercialized, this optical component can be easily added to any existing camera, with some caveats regarding working optical wavelength discussed in the paper.

• I don't see how this is qualitatively different to the existing answer. – Emilio Pisanty Jul 7 '19 at 14:22