Polarization Part #6: Calibrating Waveplates

Now that we can [»] align polarizers to a given axis with controlled precision, it is time to address the calibration of wave plates. I will focus here on quarter-wave plates (QWP) but the theory can be generalized to any type of wave plates.

As we have seen in our post on [»] birefringence and on [»] Jones calculus, a wave plate is characterized by a retardance δ expressed in either angle form or wave form (e.g. a λ/4 wave plate is the same as a 90° wave plate and a ϖ/2 wave plate). When purchasing a wave plate, it will have a nominal retardance (e.g. quarter wave) and an actual retardance that will depend on multiple factors like manufacturing accuracy, used vs. designed wavelength, angle of incidence etc. For precision work, it is therefore important to calibrate your wave plate to know its exact retardance value. This is precisely what we will study in this post, as well as a quantification of the accuracy to which we can calibrate our wave plates – a topic that is often ignored in polarizations textbooks.

When studying [»] birefringence, we saw that rotating a birefringent sample between two crossed polarizers or two parallel polarizers was modifying the light intensity passing through the complete system. Using [»] Jones calculus, we can write our polarizer/wave plate/polarizer system as

where (A,B) is (1,0) for the parallel configuration and (0,1) for the crossed configuration.

By developing the intensity, I=I₀(|A’|²+|B’|²), for both configurations we get

Both configurations show a variation of the intensity with the quadruple of the angle θ, something that we already saw in our [»] birefringence experiment. Mathematically, we can locate extrema’s by looking at the first and second derivatives of the intensity in regards to the angle θ

Two solutions occur leading to either cos(4θ)=1 or cos(4θ)=-1.

At cos(4θ)=1 we get

and at cos(4θ)=-1 we get

Only the second solution is of interest. It shows that the intensity either increases or drops based on the cosine of the retardance value. When this cosine is zero (δ=(2k+1)ϖ/2, quarter-wave plates), the intensity is the same in both configurations. When this cosine is maximum (δ=(2k+1)ϖ, half-wave plates), the difference in intensity is maximum. Measuring the difference in intensity between the two configurations therefore gives an indication of the value of the retardance cosine if we know the intensity I₀.

In practice, we correct this difference by the mean intensity such as to compensate for unknown initial beam intensity losses in the system:

This formula is extremely important as it allows identifying the exact retardance of our wave plate in the range [0,ϖ[.

Experimentally, we will re-use our [»] 3-polarizers system except, this time, we will introduce our wave plate between two polarizers. The setup is shown in Figure 1. The input polarization axis is not important but the output polarizer must be accurately set to parallel or crossed configuration using the strategy explained in our [»] previous post.

The first thing is to align the wave plate to 45°. In theory, you can use either the crossed configuration or the parallel configuration and search respectively for a maximum or a minimum intensity as you rotate the waveplate, but, in practice it is better to use the crossed configuration for reasons that will be clearer below. Once the waveplate is at 45°, you measure the intensity in both configurations and use the formula above to find the exact retardance value.

The experimental procedure is relatively straightforward, but we can spot two main problems:

First, we have shown in our [»] previous post that optimizing for a non-zero transmission yields rather imprecise results. As a consequence, we can expect that the exact angle at which we put the wave plate is not going to be exactly 45°.

Second, we know that our system will never be in perfect crossed or parallel configuration, with the parallel configuration having the least accuracy.

Solving the Jones system for an imperfect θ and output polarization state proved to be extremely complex. I therefore switched to a numerical analysis of the system using Matlab. Through simulations, we learn two important things: (1) half of the error in the polarizer axis will be transferred to the alignment of the wave plate, just like with the [»] 3-polarizers experiment, and, (2) the retardance accuracy does not depend on the exact angle of the waveplate if and only if there is no error in the polarizers crossed and parallel configuration. In practice, any error in the polarizer axis will therefore yield an error of alignment and an error of retardance measurement.

This phenomenon was studied using a Monte-Carlo analysis. I used the assumption developed in my [»] last post that the alignment error is bound to the SNR and considered the alignment error of the crossed and parallel configuration, the alignment error of the waveplate due to (a) the search for the maximum, and, (b) the error introduced by an improper crossed setup, and, finally, the error introduced on the intensity measurement from the SNR of the camera.

Figure 2 shows the numerical analysis for a SNR of 100 and 10⁵ Monte-Carlo samples on a quarter-wave plate. The probable (50%) error on the value of the retardance is 1.30° (λ/277) and the 95% error is 8.65° (λ/42). This is an important result because it is the one obtainable with a 1-pixel reading on the camera.

Second, the 95% error was evaluated for increasing SNR values. The results are shown in Figure 3. They show that a SNR of at least 2,572 is required to achieve a λ/1000 accuracy on the retardance measurement. Using a typical camera of SNR=100, this corresponds to averaging at least 662 pixels or frames. Interestingly, the accuracy seems to follow an exact linear pattern vs. SNR.

Finally, I put the procedure in practice using my Thorlabs WPQ10ME-633 quarter-wave plate with a LED630L source. I measured single-pixel intensities of 232 and 221 leading to an error of -0.96° (λ/375). Using the same images, I averaged over a disk of 10 pixels diameter, yielding the error -0.94° (λ/383). These errors were computed by correcting the QWP expected value for a 630 nm source instead of a 633 nm one. With our averaged SNR, we expect the measurement error to be <λ/500 (95% probability) and Thorlabs claims a <λ/100 error – both of which are consistent with our measured value of ~λ/383.

That concludes the post of today! In the next post, we will glue the polarizer and QWP techniques together to measure any random polarization states in a technique called ellipsometry!

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