Polarization Part #5: Aligning Polarizers

Now that we have a good [»] theoretical understanding of polarization, it is time to dig into some more experimental concepts. In this post, I address the alignment of polarizers and, in the next one, the alignment of quarter-wave plates. I have separated the two to discuss each of them in detail, in particular regarding aspects like the precision to which it is possible to perform these alignments. As always, all CAD required to performed the experiments presented in this post are available [∞] here for free to all my Premium members on Patreon.

In my [»] first post on polarization, I briefly discussed linear polarizers as devices that let pass only polarization states parallel to their principal (major) axis. The component perpendicular to this major axis being rejected. Note that this rejection is, in practice, never complete for reasons that depend on the specific type of polarizer being used as well as their quality of manufacturing. The ratio between the rejected and the accepted component is the maximum achievable extinction ratio (MAER) of the polarizer; the extension ratio (ER) being the attenuation of the signal obtained when two polarizers are in parallel vs. crossed configurations.

There are multiple configurations of polarizers setups, and we have already met one [»] here with two polarizers in crossed configuration and in parallel configuration. Other common configurations exist, such as polarizers aligned to 45° but the main concept is always to align the polarizer major axis to a given direction with as much precision as possible. I will cover these three configurations here.

In this post, I will be using mainly dichroic film polarizers such as the Thorlabs LPVISE (economy) and LPVISA (accurate) series. Their MAER are on the order of 10⁴-10⁵ for the LPVISE series and 10⁶-10⁸ for the LPVISA series. Dichroic film polarizers are an extension of the concept we introduced with [»] birefringence. In dichroic film polarizers, the geometry of the material produces a different absorption along two orthogonal directions of the field instead of a phase shift. It is these two different absorptions that produce the polarizing effect. Other types of polarizers exist such as crystal polarizers and wire grid polarizers, but they won’t be discussed here.

Note that the MAER, ξ, is specified by polarizers. When combining multiple polarizers, the maximum achievable extinction ratio becomes

For instance, the MAER of two equivalent polarizers placed in a crossed configuration is therefore ξ/2 and not ξ.

One of the first alignment procedures we must define is how to define a base polarization axis in a system. You can choose any axis you want but, since most optical systems are laid on an optical baseplate or breadboard, it is common to choose a polarization state that is either parallel or perpendicular to the reference plane formed by the baseplate or breadboard.

To achieve this, we can make use of the fact that reflection on an uncoated glass surface is polarization-dependent as demonstrated in [»] this post. When reflecting light at Brewster’s angle (~56° for n=1.5 glass), only light perpendicular to the reference plane is reflected (the light parallel to the reference plane is fully transmitted). Therefore, by sending light that is exclusively parallel to the reference plane, we should observe no reflected light – provided we are exactly at Brewster’s angle, with purely monochromatic light and with a fully polarized collimated input beam. We can then use this principle to tune the orientation of the input polarization state until no light is reflected.

This principle is used in Figure 1 to align a Thorlabs LPVIS polarizer parallel to the reference plane of Thorlab’s cage system structure. The setup comprises a relatively monochromatic illumination made of a parallel beam, a polarizer mounted on a precision rotation stage, and a camera to re-image the input beam pinhole. Side note, most people would use a power meter for the alignment but I like using cameras because they are extremely sensitive with detection thresholds of only a few photons – and are also much cheaper than power meters!

When performing the alignment of the system shown in Figure 1, set the camera exposure to ~100 ms (or any other required exposure time to achieve a mean gray value at 80% saturation of the camera) and tune the polarizer axis until the camera grayscale values histogram shows a minimum as shown in Figure 2. For those who replicate the setup using the [∞] provided CAD, you can use a current source of 25 mA for the LED.

Now that we have a reference axis defined for our polarizer, we can ask ourselves how to align a second polarizer to a crossed configuration relative to this reference polarizer. Here, we can use the setup of Figure 3 using a similar procedure with the grayscale histogram on the camera.

Note that if you would like to substitute this new polarizer as a reference polarization state, such as to have a reference polarization state perpendicular to the system reference plane, you would need to add the precision of the polarizer alignment to the precision at which we aligned the initial polarizer.

It is interesting to question what alignment precision we can get with the setup of Figure 1 and Figure 3. Considering that the alignment error in a crossed configuration is bound to the uncertainty in intensity measurement, we get from Malus Law

But there is an intensity leak induced by the MAER of the system such that

The minimum intensity change observable by the camera is bound to the shot noise and the [»] full-well capacity of the camera of this leaking signal

And therefore

For modern CMOS cameras near saturation the signal-to-noise ratio is around ~100 such that

Under this assumption, the alignment tolerance of two dichroic film polarizers with a MAER of 2×10⁴ is ~0.5°.

Concerning the performance of the Brewster’s angle setup, we have shown in our [»] post on reflection that the coefficients of reflection on a glass are

such that the MAER near Brewster angle is

where δα is the incidence angle error relative to the Brewster angle and n is the refractive index of the glass.

I have spotted 3 different sources of error of incidence angle: (1) mechanical tolerance issues, (2) variation of the index of refraction with wavelength leading to subtle change of Brewster angle, and, (3) collimation errors.

Concerning (1), the setup of Figure 1 has fairly low accuracy in the positioning of the glass interface and I evaluate the associated error to roughly ~1°, leading to a MAER of 600 at n=1.5 which is pretty bad. Alone, this would translate into a positioning tolerance of 1.2° - clearly something we could improve on. By using a precision rotation stage and a carefully machined baseplate, I evaluate that we can get this incidence angle error down to 0.1° or better (mainly the flatness of the baseplate), such that we can reduce the angle tolerance to 0.5°. At this point, the MAER of the reference polarizer becomes the limiting element.

Concerning (2), the change of index of N-BK7 from 630 nm to 650 nm is 1.5152 to 1.5145 with respective Brewster angles of 56.58° and 56.56° leading to a MAER of 3×10⁴ - an amount negligible compared to MAER due to mechanical tolerances and somewhat comparable to the MAER of an economy dichroic film polarizer. However, this is true only for narrow bandwidth LEDs and is not applicable to broadband sources.

Concerning (3), it is important to select a pinhole that is small enough compared to the focal length being used to ensure proper collimation, as opening in the beam will also change the incidence angle. In the setup Figure 1, I used a 150 µm pinhole combined with 50 mm achromatic lens, leading to an incidence angle error of (at most) 0.1° although we get a fan of rays of different angles. This is small compared to the angular error done due to mechanical tolerances, but not negligible if you try the improved version – case in which you might want to use a smaller pinhole size.

In conclusion, setting two polarizers in crossed configuration is achievable within 0.5° or better depending on the performances of the polarizers, and aligning the polarizer parallel to the reference plane should be possible in the range of 1° without much issues. It is important to bear these orders of magnitude in mind when evaluating system requirements! Also, remember that these orders of magnitude were computed under the assertion that the MAER is the limiting factor in alignment precision.

These values are also given for a typical CMOS sensor and only one pixel which corresponds to the procedure of Figure 2. By writing a custom software to average multiple pixels over multiple frames, we can improve the SNR as √N where N is the total number of pixels used (this is valid until we reach the [»] quantization threshold of the camera ADC). But since the alignment precision improves with the square root of the SNR, 10⁴ pixels are required to improve the alignment by a factor of 10 – something that is already difficult to achieve. The size of the pinhole and the magnification of the imaging systems therefore play an important role when averaging but this kind of improvements go beyond the scope of this post.

Also, in Figure 3 we aligned a crossed configuration setup by searching for the minimum transmission and our alignment predictions are all based under the assumption of seeking for a minimum. In theory, we could align a parallel configuration system by searching for a maximum transmission, but we would run into an experimental issue here. This time the intensity varies as

such that the base intensity is much higher than with the minimum optimization we met previously.

This leads to

which is to be compared with the formula for the minimum where we also have a √ξ in the denominator.

With ξ=10⁴, a typical value for economy polarizers, aligning at the maximum is therefore 10 times less accurate than aligning the same polarizer at the minimum. At ξ=10⁸, for precision polarizers, things are even worse with a 100-fold difference. Again, this is an important consideration to bear in mind when aligning polarizers.

If you need to align two polarizers in a parallel configuration, I would therefore recommend putting the precision rotation mount on the side and aligning the polarizer in a crossed configuration. Assuming 50 µm accuracy in the squareness of the 30 mm precision rotation mount, putting the cage on the side would equate to an additional ~0.2° of precision loss. Alternatively, it is possible to rotate a polarizer using the dial of the rotation mount, or, better, to use a motorized mount. The error made using Thorlabs CRM1T mount this way is 1° (2° marks) and 5’ (via vernier scale) on the CRM1PT. The CRM1PT, despite its higher price compared to the CRM1T, is therefore a valuable tool for polarization optics but we have seen that, unless you are using very high quality components and setup, the overall order of magnitude of alignment tolerance is in the range of 0.5-1°. Last but not least, one extremely annoying disadvantage of the low-cost CRM1T mount is its tendency to move as you tighten the rotation lock screw! In conclusion, if you can afford the CRM1PT mounts, they will make your life easier and you will be good to go for future investments in polarization optics.

Finally, I would like to address the case of aligning a polarizer at 45° to the reference plane which is useful in some experimental setups that we will cover later. In this [»] former post we have seen that putting a third polarizer at an angle θ between two crossed polarizers yielded an intensity proportional to cos²(θ)×sin²(θ) which is maximum at 45°. This concept is put in action in the setup of Figure 4. Note that this system suffers from the same shot noise problem as we previously discussed, except that here it is not possible to use the cage rotation trick.

Also, the formula cos²(θ)×sin²(θ) applies to a system with perfectly aligned crossed polarizer. When the crossed polarizer has a small angular error ε the equation becomes

which admit a maximum at

Such that we align the polarizer at 45° plus or minus half of the error made on the alignment of the crossed polarizer – to which you also need to add the error made when identifying the maximum of the curve. You therefore need to balance the cost vs. benefits of this alignment strategy compared to using a precision rotation mount to shift a crossed configuration by 45°. Nonetheless, I find the setup of Figure 4 very interesting in its concept.

This concludes today’s post! In my next post, I will cover precise alignment of quarter-wave plates before moving to ellipsometry :) It doesn’t look like, but you should now be close to being an expert in polarization!

Do not hesitate to share your thoughts on the [∞] community board to let me know if you enjoyed this post!

I would like to give a big thanks to Alex, Stephen, Lilith, James, Zach, Michael, Karel, Jesse, Kausban, Samy, Sivaraman, Shaun, Sunanda, Benjamin, Themulticaster, Tayyab, Pronto, Marcel, Dennis, Natan, Anthony and Wein 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!