Polarization Part #9: Maintaining Polarization States

With our [»] theoretical understanding of polarization, we can address real polarization issues that occur in many optical setups. More specifically, I show here how we can fix our systems to better maintain polarization states.

A typical example of such polarization issue is how reflection on optical surfaces (mirrors, prism, beamsplitter etc.) make our beam elliptic and reduce its maximum achievable extinction ratio. This is illustrated in Figure 1 where a beam reflecting on a beamsplitter degrades the extinction ratio. Although this depends on the alignment of the system, I measured an extinction of 10³ with the Thorlabs BS013 beamsplitter and LPVISE100-A polarizers and 10⁴ without – a tenfold difference!

The [»] Jones matrix of the beamsplitter is

where r_s and r_p are complex reflection coefficients (more on this later).

When linearly-polarized light along an axis θ falls on the beamsplitter, the output polarization state becomes

and the [»] ellipticity between the two components, S₃, becomes

This equation is important because it shows that, for linearly polarized light, the ellipticity of the resulting beam is fixed by the phase difference between the two reflection coefficients modulated by the sine of the angle at which the beam strikes the surface. It is then null when the beam is aligned with s (θ=0°) or p (θ=90°) and maximum when the beam is oriented 45° relative to s and p. It is also obviously null if there is no phase difference between the s and p reflection coefficients.

Also, recall that any Jones vector can be decomposed into a canonical form (a,jb) where a and b are real numbers and that the maximum achievable extinction ratio, ξ, is given by

which also states, using a different approach, that the extinction ratio is finite as soon as the reflection coefficients have imaginary parts and θ≠kϖ/2.

The only way to counter this phenomenon is to alter the phase difference between the two components of the Jones vector by introducing a [»] quarter-wave plate (QWP) in the system. This is illustrated in Figure 2 which experimentally restored the extinction ratio of 10⁴, thereby cancelling all unwanted effects of the beamsplitter.

Indeed, when oriented along the axis of the canonical form, it comes

which is a linear polarization state since a and b are both real numbers.

In practice you don’t have to evaluate this angle – you just rotate the QWP until the remaining ellipticity disappears. But with the equation above, you have the explanation on why it always works.

You might now ask why reflection coefficients have imaginary parts – an excellent question in my opinion! I will cover here 4 important cases:

Reflection on metallic mirrors have complex reflection coefficients because the index of refraction of metal is complex. This boils down to the sea of electrons in a metal that dampens the secondary waves of the matter-light interaction; this damping motion is represented by an imaginary part in the [»] oscillator model. Furthermore, since the [»] Fresnel reflection coefficients involve the index of refraction of the material, this makes the reflection coefficients themselves complex.

In practice, reflection on metallic mirrors is the largest source of undesired ellipticity upon reflection in an optical system and you can expect fairly large amounts of linear polarization losses when using metallic mirrors.

Reflection on coated surfaces also have complex reflection coefficients but for a different reason. Coated surfaces use thin-film interferences where multiple reflected waves acquire different phase shifts. This is represented by a phase term that differs for the r_s and r_p coefficients directly.

Although most people will ignore treating coated surface as a source of ellipticity, our beamsplitter experiment clearly shows that it is non-negligible for high-performance systems.

Total internal reflection prisms also have complex reflection coefficients, but for yet another reason. Above the critical refraction angle, the normal component of the wavevector in the second medium becomes imaginary, even though the refractive indices themselves remain real. This gives rise to an evanescent field whose amplitude decays exponentially away from the interface. Applying the Fresnel boundary conditions then leads to complex reflection coefficients for both r_s and r_p. Importantly, their magnitudes remain equal to unity, but they acquire different phase shifts.

This happened to me more than 10 years ago when working on an ESA subsystem. The experiment was conceived around a TIR prism to avoid the problems introduced by metallic mirrors and coated surfaces, but failed to preclude the ellipticity introduced by the TIR process itself! This ultimately limited the performance of the instrument and required extremely cautious alignment of the laser beam to achieve the mandatory performances.

Finally, any intrinsic, residual or stress birefringence can also introduce a phase difference between the two components of your Jones vector. Some materials like quartz have enormous intrinsic birefringence but remember that standard optical glasses have typical remaining ∆n between 10^-6~10^-7 (see [∞] Schott datasheet of N-BK7 as an example) due to their manufacturing process and stress birefringence can easily reach the same levels under normal clamping. While these effects are low, they accumulate throughout the system and also limit the maximum performances you can reach.

Therefore, if you need to maintain a high-quality linear polarization state in your system, remember to always align your beam with either the s or p direction of your reflecting surface and foresee a quarter-wave plate to compensate for any remaining ellipticity introduced by imperfect alignments and birefringence effects.

I would like to give a big thanks to Stephen, Lilith, Zach, Michael, Karel, Jesse, Samy, Kausban, Sivaraman, Pronto, Benjamin, Sunanda, 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!