I've built a Michelson interferometer which uses a webcam as screen. This setup shall measure and correct drifts in optical path difference of the two interferometer arms.
I have problems determining the resolution of the setup. So I will describe what I'm doing and my thoughts so far.
The fringe pattern captured by the camera will be Fourier Transformed, in order to get the phase information of the periodic pattern. Now when one axis is moved the pattern moves and this directly translates into the phase that can be gained by the discrete Fourier transform.
Now i want to know what the smallest change in phase is that can be measured with my camera. It has a pixel size of 3 um (micrometer) and a has a output of a 10 bit raw RGB with a S/N of 40 dB. One line has on the camera has a width of 640 pixels.
My approach to calculate the resolution of the setup is the following: I consider the amount of fringes displayed, which is about 20. So I know one fringe has the width of 32 pixels. A change of 2pi in phase needs/translates to a movement of 64 pixels on the camera chip. That means if every pixel only had 1 or 0 as a value the phase could be split in 2pi/64 steps then.
But now every pixel has a 10bit depth resulting in 1024 values instead of two per pixel. Which means - but I'm not sure here - every pixel on its own has a resolution of 2pi/1024 and then the width of a fringe divides this number again through 64, resulting in a resolution of 2pi/(1024*64). This number seem much too low for me and thats's where I'm stuck and couldn't find any good sources that might help me out.
On top of that I think there has to be a more formal way using the Nyquist theorem. Here my considerations were that I take the 3um pixel size of the camera and transform it into the 1/um space. But I'm pretty sure the 3um can only be used if the fringes were exactly that big, which they aren't (such small fringes would make the resolution very bad?). So there must be some scaling factor to the pixel size, but I'm not sure how to determine it.