How to experimentally find the retardation of a wave plate?

Question

I've been trying to understand how to experimentally verify the retardation of a given wave plate. It is pretty easy to distinguish between HWPs and QWPs but what if the retardation was random (i.e. not limited to being a QWP or HWP). I found one equation online that states that the retardation, $\delta$ can be found using the equation $\cos(\delta/2) = \sqrt{\dfrac{I_{min}}{I_{max}}}$. This equation is used when the waveplate of unknown retardation is placed between two parallel polarizers, and $I_{min}$ is the smallest intensity achieved when rotating the wave plate, and $I_{max}$ the largest. However, I have no clue how this equation was derived and the source I found does not go through the derivation. I'd appreciate if someone could offer more insight into this equation and/or any other ways to verify the retardation of a random wave plate.

Answer 1

The derivation is straight-forward using, e.g., Mueller optical calculus. Assume a unit intensity (ignoring units) unpolarized light source passing through the following optical train: an ideal x-oriented (horizontal) linear polarizer, then a general linear retarder, then another ideal x-oriented linear retarder, and finally a Stokes vector sink, i.e., a light intensity detector. This is depicted in the upper part of the figure below:

The lower part of the figure shows the Mueller optical calculus matrix calculation needed to calculate the transmitted light intensity, I. Note that I is the uppermost component of the Stokes output vector.

The Mueller matrices for the ideal horizontal linear polarizer and the general linear retarder (GLR) are from the wikipedia article on Mueller calculus. These matrices, with trivial notation changes and possible trivial component sign changes in the GLR matrix, are readily found in other sources: see references below.

To find the maximum transmitted light intensity, set $\theta = 0$ degrees and $\delta = 0$ degrees in the GLR matrix. It then reduces to the identity matrix and the calculation of the maximum transmitted light yields a value of 0.5. This calculation is shown in the upper part of the next figure:

To find the minimum transmitted light intensity, let $\theta = 45$ degrees in the GLR matrix, which simplifies considerably, and do the calculation shown in the lower part of the second figure.

References other than wikipedia:

1. W.A. Shurcliff, Polarized Light, Harvard University Press, Cambridge, MA, 1962, Appendix 2.

2. Kliger, D. S.,Lewis, J. W., Randall, C. E., Polarized Light in Optics and Spectroscopy, 1st ed., Academic Press, Boston, 1990, Appendix B II.

As noted above, $\theta = 45$ degrees minimizes the transmitted light intensity. This is easily verified by running the optical calculus simulation program shown in the upper part of the first figure. The program appears to be a block diagram because that is how I programmed it long ago and it is the same program as I used in my answer here.

For the 71 degrees retardance used in the simulation in the first figure, the transmitted light intensity was 0.331392038614 for $\theta = 45$ degrees. Keeping the retardance constant and running the program for $\theta$ angles from 0 to 90 degrees, in increments of 1 degree, yields the following plot:

The minimum light intensity in the plot is 0.331392038614 at $\theta = 45$ degrees. Dividing by 0.5, i.e., the maximum transmitted light intensity, yields 0.6627840772. Taking the square root then yields 0.8141155183, which is the cosine of 35.5 degrees, as expected.