Polarization Part #7: Ellipsometry

During our mighty quest on polarization, we saw [»] here that the general state of light is an elliptical one such as shown in Figure 1. We also saw how the [∞] Stokes parameters describe light as degree of polarization, linear polarization amount, linear polarization axis and ellipticity when projected on a [∞] Poincaré sphere. The question we will address in this post is how to measure the exact polarization state of a given input beam. This is the topic covered by ellipsometry.

One of the most popular ellipsometry setups is to use a rotating quarter-wave plate making an angle θ with a linear polarizer. A setup proposal is shown in Figure 2 made of Thorlabs components only. I was however not able to try it yet, so consider it with care if you would like to reproduce it before I do. The complete setup is about 2 k€ and is shown here optimized for a HeNe laser input.

The wave plate is rotated from 0 to 180° and N≥8 measurements are performed with a power meter (choose θ_i=i×180°/N for convenience). From these measurements, it is possible to reconstruct the 4 Stokes parameters describing the input beam. For the rest of the post, I will discuss how the method works and what are the typical performances that can be obtained. The treatment is theoretical only with simulations, experimental work will be done in a separate post.

Figure 3 shows the expected intensity profile for a S=(10,8,0,6) elliptical polarization state as the wave plate rotates, along with 8 data points. Note that the results of Figure 3 does not include sources of errors for now.

To understand how the method works, let’s first dive into a [»] mathematical description of the system itself.

The Jones matrix of the system is

Applied on a polarization state (A,B), the recorded output intensity is then

where

which gives after a few derivations (see appendices)

Introducing the Stokes parameters

we get

or

where we introduced the parameters

These parameters can be extracted from the N equally spaced measurements using a discrete Fourier transform:

such that we can identify the Stokes parameters using the reverse equations

Note that we only need to scan across the [0,ϖ[ range due to the presence of the term sin(2θ). Also, since we have both cos(4θ) and sin(4θ), the Nyquist theorem imposes a strict minimum of 8 measurements to prevent aliasing. In practice, more data points can be used for averaging but, as we will see, more data points do not compensate for all types of noise in the measurement.

Different sources of errors can impact the result, along which only an error in the retardance of the wave plate can be accounted for (within the margin of uncertainty of the measurement). The other sources of errors are: non-zero angle of incidence of the beam on the quarter-wave plate, angular errors on the quarter-wave plate position, angular error on the linear polarizer position and non-infinite maximum achievable extinction ratio of the polarizer, ξ.

Although I was not able to reproduce the math behind it, P. A. Williams in his article in Applied Optics vol. 38 no. 31 (Nov 1999), gives the following correction for quarter-wave plates that are not exactly 90° which allows correction for this parameter within the margin of measurement uncertainty:

From our work on [»] polarizers and [»] quarter-wave plates, we have a good understanding on how to calibrate linear polarizers and wave plates. We will use the typical figures obtained in these posts as a basis for our simulations. I will consider here a linear polarizer oriented with an accuracy of 0.5° using the Brewster-angle method and a 1° accuracy on the retardance value. We might be tempted to use a ~5° accuracy on the quarter-wave plate angle but we can calibrate our system using polarized light produced by the Brewster angle method as a reference of linear polarization state, leading to a total accuracy of 1°. I also used a 0.05° angle of incidence error on the wave plate, a MAER of 10⁴ for the polarizer and a 0.1% noise on the detector read-out.

Under these conditions, the results of Figure 3 shifts towards Figure 4 (note: errors were slightly increased to 2° to yield visually distinguishable curves).

I then ran a Monte-Carlo analysis on 10⁵ samples on normally distributed polarization states, yielding the Poincaré coordinates error histograms of Figure 5. The probable (50%) error on polarization degree, p, was 1.4%. Error on linear polarization state, X, was 1.4°. Error on elliptical polarization state, Y, was 0.1°. At 95% confidence, the errors increased to 6%, 4.1° and 0.4° respectively.

A few notes on the simulations must be discussed. First, increasing the number of samples, N, did not improve the results because the driving factor of the error was the angular error and not the detector noise. Second, neither the MAER at 10⁴ nor the AOI error of the waveplate at 0.05° were found limiting which are reasonable values to use in practice. Similarly, a 1° accuracy on the retardance did not have significant impact on the results as well. Finally, most of the error seems to have come from the quarter-wave plate angle uncertainty although the polarizer angle played some role on S₁ and S₂, but less on S₃.

In conclusion, the analysis performed here through simulations shows that using a relatively inexpensive setup can yield accurate measurements of ellipticity, Y, down to 0.1° probable error, and a relatively satisfactory measurement of polarization axis, down to a 1.4° probable error which directly originates from the calibration of axes. This is to be compared with Thorlab’s own polarimeter which offers 1% accuracy on degree of polarization and 0.25° on X and Y measurements for a price above 6 k€. Efforts should be therefore be put on ensuring a proper reference polarization state to calibrate both the linear polarizer and the quarter-wave plate axes.

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I would like to give a big thanks to Stephen, Lilith, Zach, Michael, James, Karel, Jesse, Kausban, Samy, Sivaraman, Pronto, Sunanda, Benjamin, Tayyab, Themulticaster, Marcel, Anthony, Dennis and Natan who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!

Appendices

The full derivations of the expression are given here-below.

And