Polarisation Optics - Demo of Radial Birefringence in Yogurt Pots

Some round plastic food package lids exhibit well-defined linear birefringence caused by their manufacture: plastic injection into a thin mould from a central point. Viewed between crossed polarisers we see a clear dark cross oriented with the two polarisers. The pattern is independent of rotation of the lid. We put this down to the existence of two values of refractive index at each point in the clear plastic, one for the light electric field oriented radially at that point, one tangential. Such containers provide a simple image, explained via rotationally symmetric stresses frozen in at manufacture.

Laptop or PC monitor with LCD screen (not OLED).

Small pieces of polariser, big enough to look through. e.g. sunglasses.

Circular transparent plastic lids (we use Fage brand yogurt pot lids).

The internet abounds with photos of stressed plastic objects viewed between crossed polarisers. Plastic rulers, Cellophane wrapping, plastic bags and CD cases all show beautiful coloured patterns caused by the spatially varying birefringence. However, other that saying "it's down to stresses in the plastic", in the complexity it's hard to explain what is going on. Here we provide a demonstration providing a simpler image, one that is easier to explain via the manufacturing process.

Bring up a blank white page in your word-processor or Notepad editor and expand it to fill the whole screen. Look at the blank screen through the polariser and rotate it for minimum intensity. Now hold the plastic lid between the screen and the second polariser.

That's it! You should see some coloured areas and, most strikingly, a dark cross. If you rotate the plastic the pattern should remain largely unchanged. The cross remains aligned with the polarisers' axes.

To understand what's going on we have to look at the lid and think about how it is made. If there is a rough point in the center of the lid, that is probably where hot liquid plastic was injected into a mould. The mould consists of two pieces of metal with a gap between them that will define the lid thickness and overall shape.

As liquid plastic flows radially outward from the injection point there will be some molecular alignment. Perhaps the plastic is also compressed in the radial direction and stretched tangentially. As the plastic cools, the alignment will be frozen in as stress, stress which manifests itself as differences in refractive index. Clearly, from the symmetry of the moulding process, the stress distribution must have rotational symmetry. Therefore we have just two refractive index values at each point, one in the radial direction (n_r) and one tangential (n_t). Around any fixed radius these values remain constant. However, each radius could have different values. At any point on the disc the plastic behaves like a uniaxially birefringent crystal, whereby the two special directions are radial and tangential. Our yogurt pot lids are labelled PP (polypropylene), which generally shows quite low levels of birefringence compared to some other polymers.

Let's assume that the light that comes from the LCD screen (first polariser) is linearly polarised with its electric field vector oriented vertically and the second polariser blocks this polarisation. The electric field entering the plastic lid will be aligned with a vertical radial line in the lid. Because it is so aligned, this field does not "see" any birefringence - it only sees one refractive index n_r. The light exits the plastic with state of polarisation unchanged, still linear and still oriented vertically, so it's absorbed by the crossed second polariser as before and we see a dark up-down radial line.

The same is true along a horizontal radius of the lid. Here the input vertically oriented electric vector sees only the tangential refractive index n_t, propagates unchanged and we see the second, left-right dark line.

For all other parts of the plastic lid the input electric vector will see both radial and tangential refractive indices. The light there is effectively separated into two waves travelling at different speeds in the plastic, proportional to 1/n_r and 1/n_t. The output state of polarisation is determined by the phase shift between the two waves. It could be linear, circular or in general elliptical and will vary with position. The second polariser then absorbs some of the light, depending on the state of polarisation and hence on the phase shift. This phase shift is inversely proportional to the light wavelength, so after the second polariser the intensity will also vary with wavelength, giving the visible colours.

Circularly symmetric food containers provide an easily obtained alternative to the usual plastic rulers and CD cases for demonstrating stress patterns by viewing between crossed polarisers. The simple geometry and resultant image also allow for a simple explanation based on stresses frozen into the plastic during the injection moulding process.