A popular classroom optics demonstration consists of placing two plastic sheet polarizers together, rotating one with respect to the other. Against a light source we see the view changing from bright to dark with a 90° rotation of one polarizer. However, it is not often pointed out that extinguishing the light in the "crossed state" is only effective when light passes perpendicularly through the polarizers; skew rays are not extinguished to the same degree. This Instructable shows how to demonstrate the effect of skew light passing through crossed polarizers.

Supplies

Bright LED with a wide light divergence

Battery as power supply and current limiting resistors

Two linear polarizer sheets, about 25x25 mm or larger

Diffuser sheet such as tracing paper or frosted plastic drawing film

Miscellaneous pieces of cardboard as spacers.

Step 1: Background

Plastic sheet polarizers are often introduced in school light and optics classes. The demonstration usually consists of placing two polarizers together, rotating one with respect to the other. Against a light source we see the view changing from bright to dark with a 90° rotation of one polarizer. This is explained by saying that the polarizers have one direction which is “easy” for the light’s electric field to vary, and a perpendicular "hard" direction. They are linear polarizers, allowing light to pass with a single transverse electric field direction. When the polarizer easy directions are at 90° to each other, very little light passes – the polarizers are “crossed”.

A similar variation of light transmission can be seen with a single polarizer sheet rotated in front of an LCD monitor or TV screen (not OLED). These light sources are already linearly polarized because of a plastic polarizer covering the whole screen area.

However, it is not often pointed out that extinguishing the light in the crossed state is only effective when light passes perpendicularly through the polarizers; skew rays are not extinguished to the same degree. This also explains the reduced contrast with TVs viewed from the sides. Demonstrating this effect in the classroom takes a little more effort.

Step 2: What We Need

First we need two pieces of polarizer sheet, at least 25x25 mm. Amazon, eBay and others sell these. Just get unmounted sheets, not the rings designed for attaching to camera lenses. Sheets 100 mm square or A4 size are most economical and can be cut with scissors. Remember to remove any protective film after cutting.

Next, the light source. You can of course just tilt the polarizers so that light passes through at a skew angle, but this does not give a total picture. A nice way to demonstrate the skew effects is to use a light source with a wide divergence. In this way we see a range of transmission angles simultaneously.

So, torches, sunlight, laser pointers, narrow beam LEDs are not useful. We need to use a small, high brightness, unpolarized light source with a high divergence. Many types of surface mount LEDs with a flat top are suitable. I used some white, high-power LEDs removed from a light fitting. They feature a semi-spherical silicone dome over the small LED chip. As the chip is at its center, the sphere has no focussing power, but simply allows light to escape with a wide divergence. White and single-colour LEDs perform similarly.

With just the light source and two polarizers you can still only look in one direction at a time. To make the whole divergence of light visible at once we use a diffuser. This directs light which has passed through to our eyes at many skew angles simultaneously. It can be tracing paper, frosted plastic drawing film or ground glass. You can even make your own from clear plastic or glass and abrasion with fine silicon carbide paper. The whole apparatus is assembled as a sandwich.

Step 3: Powering the LED

You will also need a power supply for the LED. A battery 3 V to 6 V is fine, along with a current limiting resistor. Start with a resistor value which guarantees a safe current, say 1000 ohm. Measure the voltage across the LED and the battery voltage using a multimeter. The LED voltage will not change much with current, so you can now choose a new resistor to power the LED near its maximum current. The resistor value you need is just (V_battery – V_LED)/current-you-want.

For example, with 6 V and 1000 ohm my white LED voltage was 2.66 V. I thought 20 mA would be fine, so chose a resistor of (6.0 - 2.66)/0.02 = 167 ohm; 160 ohm gave a current of 19 mA.

Step 4: Assembly

Rotate the two polarizers into the crossed position while looking at a bright light source like a table lamp (not the sun!). If they were cut as squares from a larger sheet, then when crossed they will also overlap almost perfectly. When it looks as dark as possible, fix them together with two small pieces of adhesive tape on the edges. Recheck the null, as it is quite critical and the polarizers may have shifted during taping. Using small pieces of cardboard or plastic foamboard as spacers, place the two-polarizer sandwich on the LED.

Now apply power to the LED. Looking directly down on the LED the light should be well extinguished, at skew angles less so. However, without the diffuser you will only be able to see light that has gone through the polarizers at one angle. So, place the diffuser on top of the polarizers. You should now see the symmetric cross pattern of transmitted light. In directions aligned with the easy and hard axes of the polarizers the extinction is good. In all other directions there is light leakage. Using a white LED you may even be able to discern a blue tinge at the very center. Extinction performance of these polarizers is optimum for red light but drops off for blue wavelengths.

Step 5: What Do You See?

This is the kind of pattern seen looking down on the diffuser sheet.

Step 6: Explanation

The imperfect extinction of light at skew angles is not down to any imperfection in the polarizers, but to purely geometrical effects. Light hitting the first polarizer sets up an oscillating electric field along the easy axis (Y) of the polarizer (thick brown line). This is then resolved on a plane perpendicular to the direction of propagation (k), shaded in light green. So, looking back along the propagation direction the easy-axis electric vector and the propagating electric vector will line up on top of each other. This is true for all directions of propagation.

When this light then hits the second polarizer, the transmitted intensity depends on the angle between the electric vector (brown) and the second polarizer easy direction. We can see this by looking vertically down on the polarizer.

If the propagation direction is in the XZ- or YZ-plane as shown, referred to the polarizer easy direction, the electric vector is precisely perpendicular to the second easy direction - almost perfect extinction.

Step 7: Skew Light

However, for propagation skew to the XZ and YZ planes the electric vector is rotated with respect to the second easy axis, so perfect extinction is impossible – there is light leakage.

So, the next time someone says that two polarizers can be used to block light you can say

"Well, yes, but it depends on the angles".