Polarizer = angle sensor

Let’s talk about angle sensors. Basically, angle sensors are needed in the machine tool industry for CNC machines. In electronics, this is always a sore subject, as it involves moving parts. And, as you know, everything that moves, all this, first of all, breaks!

Variable resistors, resolvers, encoders, accelerometers, gyroscopes, high-precision GNSS receivers are classic methods for measuring angles. I did a comparative analysis of known angle measurement technologies in this registry

https://docs.google.com/spreadsheets/d/188FBrvaL_DRcebT3SAtTdFdYGmmIdF6op_9DPNZDvps/edit#gid=0

But there is another witty and speculative way to measure angles: using polarizers.

Everyone knows the physical effect of polarization. This is when they take two polarizing filters from the camera, put them on top of each other and turn them. As a result, the light either passes or does not pass through the plate sandwich.

this is how it looks

The inverse problem can be solved in the same way. In mathematics there is always a direct and inverse problem. Given the level of illumination, calculate the angle of rotation between the planes of polarization. This suggests that on the basis of this effect it is possible to design an electro-optical angle sensor.

There is the law of Malus (1810). This is a physical law expressing the dependence of the intensity of linearly polarized light after it has passed through a polarizer on the angle phi between the polarization planes of the incident light and the polarizer.

Io is the intensity of the light incident on the polarizer, I is the intensity of the light leaving the polarizer, k is the transparency coefficient of the polarizer, phi is the angle between the allowed direction of the polarizer and the polarization direction of the input light.

The angle itself is expressed from this formula phi

If you assemble such a stand

then the lighting graph from the corner will look like this

But here’s the problem. With an illumination of 0.4, 4 different angles are possible: 50.7 Deg; 129.2 Deg; 230.7 Deg; 309.2 Deg. And how to understand that the true angle is, for example, 50.7 degrees and the remaining 3 dimensions are fake. These 4 numbers are simply the result of solving the trigonometric equation (3)

It’s not great. How to weed out 3 extra solutions and leave one true one?

In technology, there is a principle of adding new dimensions. You can also do it here. One more pair of polarizers can be added and one more measurement can be provided to reduce ambiguity.

It turns out such a ratio of angles and sensor readings

This is something you can work with.

For example, when the sensor Ir1 sees the value 0.8 Lx and Ir2=0.092 lx, then the program thinks that the angle is

The situation is similar when the true angle is 150%. Electronics will show 2 equally likely options 150 Deg and 330 Deg. It is worth noting that these two solutions differ by 180 degrees.

It turns out that by adding one polarization sensor shifted by +45%, we expanded the measurement range from 0…90 Deg to 0…180 Deg. But this is still not enough. This is of course no good. It would be strange if the plane’s heading sensor showed that we were flying either north or south. For a full-fledged angle sensor, it is necessary to unambiguously measure the range from 0 to 360 degrees.

How to be?

We all remember from school the formula cosine to cosine

In connection with (4), formula (1) can be rewritten in the form (5)

The deuce in front of phi suggests that the device needs a reducer. Therefore, the solution to the ambiguity of the angle measurement can be a gearbox on gears, which reduces the angular velocity of rotation of the polarizers by 2 times.

In other words, for one revolution of the angle sensor, the polarizer will only make half a revolution. Then the graphs of readings of photosensors 1 and 2 from the angle of the shaft phi will be as follows

There was a certainty. Great.
You can assemble just such an electronic filling for processing sensor readings.

The task of the firmware is to constantly and continuously numerically solve the system of equations (6) and issue a solution to the street (in UART)

where phi is the angle of the angle sensor shaft. The true angle of rotation of the polarizers psi is related to phi by the formula (7)

Benefits of measuring angle with polarizers

++1 no angle limit. You can measure the angle of the wheels.

++2 the angular velocity of rotation of internal parts is 2 times less than the angular velocity of rotation of the angle sensor shaft.

++3 There is no need to look for zero, as it happens in incremental encoders and flywheels of internal combustion engines. I immediately turned it on and you can immediately calculate the angle for any position of the shaft.

++4 there is no need for micrometric parts, like the same incremental encoders, where several thousand holes are made per revolution.

++5 there is nothing here that would have more dimensions than in resolvers.

++6 This sensor does not need gravity, similar to how inclinometers work on accelerometers

++7 Polarizers are a very affordable component. You can at least pick it out of an old monitor. Yes, and you don’t need much. Only to be wider than the laser beam in cross section.

Disadvantages of Angle Measurement with Polarizers

–1 The complexity of the design. We need a microcontroller that will interrogate the light sensors and recalculate the values ​​into angles.

–2 need 2 precision light source

–3 need 2 hermetic dark rooms to eliminate errors in photosensors

–4 Structural isolation from dust and excess light is needed. Ideally a vacuum chamber.

-5 you need two absolutely identical light sensors. We’ll have to work as a matchmaker to find a pair of identical sensors in the box.

Conclusion

As you can see, the measurement of angles with polarizing plates in theory is a completely possible way to automatically measure angles, and hence angular velocities. However, the implementation of such a sensor requires advanced manufacturing capabilities and a serious design.

Surprisingly, in 213 years since the discovery of the Malus law, not a single office has released its own version of the angle sensor based on this physical effect. Apparently this is not easy.

I doubt that such an angle sensor on polarizers will actually be implemented somewhere in hardware. Too many places of probable breakdown and failure. Those who really need a high-quality angle sensor usually put inductosin, and this more than suits them.
https://www.youtube.com/watch?v=3m8UZ1ML8W0&t=13s

Nevertheless, as for me, the technology of measuring angles with polarizers is suitable for university laboratory work in physics or information converters. There it is just customary to take measurements, analyze graphs and calculate errors. Or just for fun. They make solenoid motors.

Links
http://latex.codecogs.com/eqneditor/editor.php

https://pythonru.com/biblioteki/pyplot-uroki

https://ru.wikipedia.org/wiki/Trigonometric_functions

https://www.wolframalpha.com/

https://www.youtube.com/watch?v=L3Hbr_ObTn0
https://www.youtube.com/watch?v=wR-WPocc128&t=17s

https://www.youtube.com/watch?v=Y5MGDltGxCY&t=418s
https://www.youtube.com/watch?v=dD1ZsXRMKMg
https://www.youtube.com/watch?v=TtVwALGK72I
https://www.youtube.com/watch?v=Zzj-8PvQHuU

https://docs.google.com/spreadsheets/d/188FBrvaL_DRcebT3SAtTdFdYGmmIdF6op_9DPNZDvps/edit#gid=0

https://www.youtube.com/watch?v=3m8UZ1ML8W0&t=13s