Polarization Part #3: Birefringence

In my [»] first post in the polarization series, I introduced the concept of decomposition of an electromagnetic wave into two orthogonal components E_x and E_y in the basis (x,y)

such that

I also said that it is sometimes convenient to use a basis (x,y) that corresponds to some particular geometry of a sample under analysis. Now, imagine that the sample produces a phase shift that differs from the x and y components of the field, such that

The wave now starts propagating very oddly as shown in Figure 1. While the initial wave was said to be linearly polarized, this new wave is now said to be elliptically polarized. I will show here-below why we talk about elliptical polarization and an example of physical phenomenon that can produce such polarization state.

The best way to read the results of Figure 1 is to consider the two separate components E_x and E_y (displayed in blue and red respectively in Figure 1) and to accept that the superposition of these two components yields the black path.

Another way to view what is happening to the electromagnetic field is to look at how it behaves in the transverse plane at some position z. By conveniently choosing z=0, we get

which is the polar expression of an elliptical path progressing at an angular speed ω, as shown in Figure 2 (time normalized for better clarity).

The equation of the ellipse being

where δ=φ_y-φ_x is the relative phase retardance between the two components of the wave.

Note that in the absence of an observable relative phase retardance between the two components (δ=kπ with k=0,1,2…), the ellipse becomes a line. Since the former expression (with δ≠kπ) is more general, we conclude that the most general state of polarization is elliptical polarization and that linear polarization is only a very specific case of general polarization states.

You may now wonder why a sample would introduce different phase shifts along different directions. The answer boils down to the generalization of what we said about refractive indices [»] here and [»] here. Briefly summarized, matter slows down electromagnetic waves because the electron clouds oscillate in synchronicity with the input wave but with a phase lag, leading to a delayed (phase shifted) secondary wave that combines with the input wave. The effect results in an overall tiny phase retardance of the input wave by each layer of atoms in the sample lattice and stacking up at each lattice layer. We translate this overall phenomenon as the refractive index of the material being traversed by the electromagnetic wave. Now, even without a full mathematical description of the phenomenon, we can anticipate that if atoms are not stacked similarly along different directions, then the magnitude of the retardance will differ in those two directions. This is exactly what happens and we call this effect birefringence because we observe two different retardance, and so refractive indices, along the two different directions. You will often hear the terms ordinary axis and extraordinary axis to label these two directions but remember it’s only a naming convention that originated from the history of birefringence discovery.

Many crystals have a natural birefringence effect, the most known being calcite, a form of crystalized calcium carbonate, and more specifically Island spar. When viewing an image under natural (unpolarized) light through Island spar, two shifted images appear. I wish I had a sample to illustrate this but a quick search on google will show you plenty of illustrations. The effects comes from the very high birefringence of the crystal (and especially thick crystals) that bends rays differently depending on their polarization state. Since under natural light we have a mixture of different polarization states and that we can always decompose each polarization state in the specific axis of the crystal, we observe two different refractions which we wouldn’t if we had oriented the incoming light polarization along one of the specific axes of the crystal.

Other samples do not have natural birefringence but acquire one when their lattice is deformed such as when applying stress on them. We call this effect stress birefringence and it plays a very important role in optics. This is illustrated in Figure 3 where we visualize stress in a part under stress using a crossed polarizers setup. The area without stress remains black while color stripes appears in the area with the maximum stress. If you pay attention, you will see some residual stress in the non-compressed part (left-hand side of Figure 3) due to the machining process. Crossed polarizer setups are very convenient to spot issues like these after machining or molding transparent parts!

A very good example of stress birefringence can be obtained by pulling a cooking plastic film because it produces stress along the direction of the pull. Although using cooking plastic film works fine, a more repeatable way to address this experiment is by using transparent adhesive tape which is produced by elongating a plastic film in the first place – leaving residual stress in the tape and therefore birefringence. This is precisely what I did in Figure 4 by stacking multiple thicknesses of adhesive tape over a microscopy slide and observing them under a crossed polarizer systems using our fresh new [»] polarized backlight. Note that all the CAD files required to reproduce this experiment can be downloaded [∞] here for free for all our [∞] Premium members.

Not only does the adhesive tape show vivid colors in our test setup, but the colors also fade as we rotate the sample along the optical axis as shown in Figure 5 (color tones logarithmically compressed to better map the different light intensities).

We observe in Figure 4 and Figure 5 that the color depends on the thicknesses of the adhesive tape (number of layers used) and that the light intensity goes to a maximum when the sample is at 45° to the backlight (polarizer) axis and to a minimum when it is aligned with either the backlight (polarizer) axis or its perpendicular axis.

While I will give a full mathematical treatment of polarization and retardance effects in the next post, I can already give the following explanation:

(a) When we rotate the sample, we project different amounts of phase retardances on the two components of the electromagnetic wave. When the phase retardance is along the initial polarization (or its perpendicular), there is no overall effect.

(b) The amount of phase retardance depends on the wavelength following a λ^-3 dependency at first approximation because the refractive index can always be approximated by a Cauchy law (as demonstrated [»] here) in λ^-2 and because phase is given by n×d/λ where n is the refractive index and d the sample thickness.

(c) At 45°, and when the phase difference between the E_x and E_y components approaches 180°, the input linear polarization state is swapped such that a horizontal polarization state becomes a vertical polarization state and vice-versa. This is illustrated in Figure 6 where the initial horizontal polarization state is decomposed into two vectors in the basis E_45° and E_-45°. But because our sample introduces a phase retardance of 180° along the E_45° component (and none along the E_-45° component), the net effect is to flip the E_45° vector (because cos(180°)=-1), yielding a vertical output polarization state.

In consequence, when we orient the sample at 45°, there is usually always one wavelength that will produce the 180° retardance condition, and we switch from a crossed polarizer configuration where no light passes to an aligned polarizer configuration where all the light passes. The wavelength at which this occurs depends on the amount of stress birefringence (i.e. how much the refractive index changes along the stress direction) and the material thickness. Because it drops according to ~λ^-3, we observe relatively narrow chromatic peaks that produces vivid colors on the camera.

Also, if we modify our experimental setup such that both of the polarizer are horizontal (or vertical), we produce the inverted colors because, this time, the sample rotates the polarizer state such that light is blocked. This is illustrated in Figure 7. Compared to Figure 4, the background is now bright because the default behavior is to block light unless its polarization state is rotated – while in the original setup the default behavior is to block the light unless the polarization state is rotated.

This concludes our post on birefringence! In the next post, I will give a full mathematical treatment of polarization and introduces more phase retardance concepts such as wave plates. I however felt important to have some experimental introduction to birefringence before digging into the (more heavy) mathematical theorems. Also, the stress birefringence experiments are very didactic to reproduce with their vivid colors and helps gaining interest in polarization!

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