Polarization Part #8: Circular Birefringence and Circular Dichroism

Now that we have a better understanding of [»] generalized polarization, we can also address the specific case of circular birefringence. In this post, I will present the general theory behind it as well as a simple experiment that you can reproduce at home that highlights its most important characteristic. In the next post, I will present a more advanced metrology instrument based on what we will learn today. So be sure you understand this post before moving to the next one.

Before I presented the theory of generalized polarization, I also briefly discussed about [»] birefringence – a natural phenomenon occurring when a sample has two different atomic stacking directions, leading to two different indices of refraction along the different axis of the sample. I gave a qualitative description of the phenomenon based on an analysis of how each layer of atoms introduce a slight phase shift in the outcoming wave. The more layers we stack, the more phase shift accumulate, which is ultimately translated into the refractive index of the material as an apparent speed reduction of the electromagnetic wave in the material. The important take-away was that having different stacking density along different directions produces different indices of refractions and effects on polarized light.

When studying [»] generalized polarization, we saw that the most general state of light is in the form of an elliptical helix, not just plain plane waves. Although we did not mention it in our post, you probably also know that helixes can wrap in either clockwise or counter-clockwise direction as shown in Figure 1. Circular birefringence is a specific type of birefringence that affects clockwise helixes differently than counter-clockwise ones. Although the maths behind it are more complex, we can give a similar qualitative appreciation as for the standard, linear, case: materials that have a clockwise-related atoms stacking will introduce a different phase shift to clockwise electromagnetic waves than to counter-clockwise ones, therefore introducing a different index of refraction for them.

This is particularly interesting because many substances have such a clockwise/counter-clockwise nature, leading to a method that allows discriminating between the two forms they can take. Such substances are called chiral and, rather than naming them from a clockwise/counter-clockwise aspect, the names right-form (D, dextrogyre), and left-form (L, levogyre) are universally accepted.

When placed between two crossed-polarizers, such as in the setup developed [»] here, the two different types of crystal are resolved in some specific cases. This is illustrated in Figure 2 with sodium chlorate (NaClO₃) crystals where one form turns blue and the other one turns to a pale red. Just like we met dichroism in the linear birefringence case, this change of color due to circular birefringence is called circular dichroism.

Thicker crystals also have more vivid colors as illustrated in Figure 3. This indicates that the effect is proportional to the thickness of the traversed material.

Although out of the 235 crystalline forms 65 are chiral, only the ones belonging to the cubic symmetry group can be observed using the crossed polarizer like in Figure 2. This is because cubic crystals are isotropic along the different axis while other crystal groups are anisotropic and also induces standard, linear, birefringence into the mix. Sodium chlorate belongs to the P2₃1 point group space and has the crystalline structure shown in Figure 4. The helical nature of sodium chlorate crystals clearly pops-up with crystal existing in either the D and L form. It is important to understand that it is not the sodium chlorate itself that is chiral, neither the cubic lattice, but how the different atoms stack up in the lattice.

Sodium chlorate is a specific case of an achiral molecule (a molecule that does not have L/D form on its own) that crystalize under a chiral structure. This type of chiral crystals will spontaneously produce a 50/50 mix of L and D crystals, unless specific conditions are imposed during crystallization. When a D (or L) crystal is isolated, redissolved and recrystallized, it will reappear as either a D or L form with a 50/50 probability ratio as well.

Some molecules are however chiral by nature, such as sugars, and will form crystals of the same handedness (L/D) as their molecule. Literature mentions that less than 10% of chiral molecules will spontaneously resolve as L and D chiral crystals, the rest forming achiral crystals that mixes both of the forms of the molecules and called a racemate. Pasteur originally discovered this phenomenon with sodium ammonium tartrate crystals which undergo spontaneous resolution upon crystallization. Sodium ammonium tartrate, however, crystalizes in the orthorhombic system, making it unsuitable for the polarizer approach, and Pasteur had to resolve the handedness of the crystals by eye which is, by personal experience, not an easy task.

Finally, some molecules are achiral in solution because their bonds rotate freely in solution, but “freeze” into a given chiral conformation upon crystallization.

If you would like to dig deeper into chirality and chirality symmetry breaking, you can have a glance at [∞] this old report I wrote as an undergraduate student. The report details experimental procedures on how to produce sodium chlorate crystals but also reviews in more detail the concept of chirality and the different types of chiral crystals.

In this post, I would like to discuss in more details the results of Figure 2 and, more specifically, why one form of crystal turns red and the other turns blue. Note that we cannot associate the colors red/blue with either L or D form because rotating the analyzing polarizer will change red into blue and vice-versa as we will see.

Let’s study what happens when a linearly polarized electromagnetic wave passes through a chiral crystal. For the sake of simplicity, I will choose a polarization state E=(1,0) although this is by no means limiting (think about the [»] canonical representation of a wave). Such a polarization state can be decomposed into two circular waves of opposing directions as shown in Figure 5 for different time values t₀, t₁, t₂ etc.

Mathematically,

with

and

such that

which is indeed a linear polarization state aligned with X of amplitude E₀.

When traversing the chiral medium, due to the preferential handedness nature of the crystal, the left-handed component sees a phase shift -φ and the right-handed component a phase shift of +φ:

and

which then combines as

which is nothing but

This new polarization state is still linearly polarized but with an angle φ relative to the X axis. Put differently, the chiral medium made the polarization state rotate by an angle φ.

In generalized form we get

This is illustrated in Figure 6 which shows, in a more graphical manner, the rotation of polarization axis when compared to the one Figure 5.

Alternatively, we can make the same analysis in a more compact form using Jones vectors,

and

which is also the Jones vector of a linear polarization state making an angle φ to the x axis.

The three methods here-above all describe the same phenomenon but using different approaches – pick the one that speaks the most to your brain configuration. They all show that when linearly polarized light passes through a chiral medium, it rotates the plane of polarization of the incoming wave. Advanced EM theory, which explains the origin of the +φ and -φ for light handedness, also shows that the effect is independent on the specific orientation of the sample to the light. Indeed, in the experiment of Figure 2, we can rotate the sample by any amount, and neither the colors nor the light intensities change. This is very different from the results obtained with standard, linear, birefringence which showed to be sensitive to the sample orientation to the incoming beam.

Linear and circular birefringence are two different physical effects but they both share a common qualitative analysis. The similitudes however vanish more when careful EM analysis is performed – so keep in mind that the two effect are only remotely connected.

The property of a solution to rotate plane-polarized light quickly attracted attention of experimental physicists and chemists who studied the topic extensively and yielded many experimental techniques. The earlier works were entirely empirical ones, as the explanation of the phenomenon only came much later.

Empirical physicists called the amount by which a given sample rotates plane-polarized light the optical activity of the sample. More specifically, Biot’s law states that the angle by which the polarization plane is rotated, φ, is proportional to the optical activity of the sample, α, the concentration of the sample (in case of dilutions), c, and the length of the optical path, L,

where the optical activity is always specified for a given wavelength, λ, and temperature, T.

When the specific optical activity of a given substance is known, the here-above formula allows finding the concentration of the solution by measuring the amount by which plane polarized light is rotated when passing through an experimental cell of given length. Due to the low values usually reached, a typical cell length, L, measures 10 cm although any other lengths can be used. Most of the tables of optical activities are therefore tabulated with units degrees per decimeter per g/cm³ of concentration, deg° dm^-1 g^-1 cm³. This non-SI choice is directly connected to the history of the empirical work performed by Biot’s and others. Indeed, work on optical activity predates the concept of mole and matter quantity by several decades!

One question remains to be answered regarding the results of Figure 2, which is the reason for the different colors.

When studying linear birefringence, we saw that the phase shift effect was related to wavelength by a factor of λ^-3. Here, the effect also changes with wavelength but with a factor of λ^-2

For practical work, only one oscillator is often satisfactory.

For the sake of completeness, despite it is not required to explain the red/blue colors, temperature affects the poles positions and weights B_i in a much more complex way but can always be linearized on a small domain

A workable model over a small temperature range can therefore be approximated as

where A gives the amplitude of the rotation, λ₀ is the position of the oscillator pole, and γ is the temperature sensitivity.

Let’s now focus on the color change aspects.

The intensity collected by our instrument is given by [»] Malus law of two crossed polarizers

where θ₀ is the angle between the reference and analyzer polarizers and φ is given by the optical activity.

Since the optical activity depends on wavelength, the intensity also depends on wavelength according to

where the sign of A depends on the crystal handedness, represented here by the ± symbol (temperature effects neglected here).

In consequences, there is a specific wavelength at which the intensity cancels

All other wavelengths induce a non-null intensity with a quadratic dependence to wavelength. Since the phase-shift is inverted depending on the crystal handedness, we can set an angle that will cancel the blue-green region of the spectra for one crystal type and the red-green region of the spectra for the second crystal type. This also explains why we can swap the colors by selecting the opposite value of θ₀. In practice, angles larger than this threshold must be used.

Note that when the polarizers are perfectly crossed, such that θ₀=90°, the expression becomes symmetrical and no distinction occurs anymore between the L and D crystals because

Figure 7 shows simulated crystals for conditions A=0.015 °/nm² and λ₀=200 nm under five different polarizer angles. The simulation perfectly reproduces the experimental observations of Figure 2. Note that although the case is fictional, it is representative of most clear solutions and crystals which admit a pole in the UV region and can therefore be seen as a relatively general representation of the circular dichroism phenomenon. The factor A determines how far away from the perfectly crossed position we must place our polarizer to observe the effect.

This concludes today’s post! I did not anticipate to make such a detailed experimental post at first but I hope you enjoyed the post :) Do not hesitate to share your thoughts on the [∞] community board to let me know!

Now that we have a good, empirical, understanding of circular birefringence, it is time to move to a more sophisticated setup in our following post.

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!