# Basic Concepts: Polarization of Light

One of my tasks this week, before heading off to the Caribbean for a relaxing vacation, is going to be to find a new pair of polarized sunglasses that aren’t ridiculously ugly. This seems like a decent hook of a physics post, explaining why “polarized” is a selling point for sunglasses, but first, I probably ought to explain what we means when we talk about the polartization of light.

As you know, even if your name isn’t Bob, light can be thought of as an electromagnetic wave. You have an electric field oscillating in space and time, and a magnetic field with it, oscillating at the same frequency. The changing electric field creates a changing magnetic field, which creates a changing electric field, which creates a changing magnetic field, and so on. The fields are perpendicular to each other and to the direction of motion, and support each other as they move through space at the speed of light. The usual graphical representation is shown in the right-hand figure below.

(Image taken from Hyperphysics.)

Now, the electric and magnetic fields are vectors, and as such point in a particular direction in space. We can use the direction of one of the two (by convention, we pick the electric field) to define the polarization of the light field. The actual electric field will oscillate up and down, but at any given instant, it will be pointing in some direction (unless you’re really unlucky, and it’s zero at that instant, in which case wait a bit, and it’ll come back), and we use that axis to define the polraization. If the field is pointing either up or down, we call it vertical polarization, and if it’s pointing either right or left, we call it horizontal polarization.

Of course, electric fields aren’t restricted to pointing exactly along vertical or horizontal axes, but can be at any arbitrary angle to those axes. That’s ok, though, because we can think of any arbitrary angle of polarization as being a combination of horizontally and vertically polarized light, with appropriate amplitude.

This is sort of the reverse process from the bad treasure maps you would make as a kid– if you want to hide something a bit west of north, you can define the position as, say, four steps north followed by three steps west, starting from some landmark. That works out to be five steps away from the landmark in a north-by-northwest sort of direction, but it’s easier to specify in terms of steps along the north-south and east-west axes.

We can do the same thing with electric fields. A field with a magnitude of 5 units at an angle of 37 degrees left of the vertical axis can be broken up into a vertical field with a magnitude of 4 units, and a horizontal field with a magnitude of 3 units. Or, in terms of polarization, we can think of a light field with a polarization angle 37 degrees from the vertical as being 4 units of vertically polarized light, plus 3 units of horizontally polarized light.

Ordinary light from the sun, or a light bulb is unpolarized, a term which is a little deceptive. Individual light waves will, of course, be polarized, but each wave has its own polarization, and they’re not correlated in any way. The polarization of the waves measured at any particular spot will flucuate randomly, and very rapidly, with no preferred direction.

We can make polarized light using special materials in which the transmission of light through the material is different for different orientations of the polarization. A naturally occurring example is calcite, which has a different index of refraction for different polarizations. A propery cut calcite crystal will bend horizontally and vertically polarized light by different amounts, and can be used to separate the two. If you block one beam, you’re left with either horizontally or vertically polarized light.

These days, polarized light is more likely to be made from a Polaroid type material (invented by the same guy responsible for the concept of shaking things like instant-film pictures, who thought up one good name and ran with it). Polaroid sheets are made of polymers that allow one polarization of light to pass through, while strongly absorbing the other.

You can create light of an arbitrary polarization by shining unpolarized light at a polarizer of whatever type you like, arranged to transmit light with its polarization along a particular direction. If you take two polarizers and put one after the other, you find that the amount of light making it through the second varies depending on the angle between the two. If the polarizers are aligned in the same direction, all of the light makes it through the second polarizer. If they’re perpendicular, none of the light makes it through (you get, say, vertical polarization from the first one, which is absorbed by a horizontal polarizer).

For angles in betwee, you pass only the component along the direction of the polarizer. So, for example, if your first polarizer is set up to transmit 5 units of light at 37 degrees to the vertical, a vertical polarizer would pass only the 4 units of vertically polarized light on, and a horizontal polarizer would pass only the 3 units of horizontally polarized light.

We usually think of this in terms of the intensity of the light passed through the polarizer, as a fraction of the initial intensity. Intensity goes like electric field squared, so the vertical polarizer passes 4/5 of the initial electric field, which means 16/25 of the initial intensity. The horizontal polarizer would pass 9/25 of the original intensity.

In general, the amount of light transmitted through a polarizer at an arbitrary angle is given by Malus’s Law, which says that the transmitted intensity is the initial intensity multiplied by the cosine squared of the angle between the polarization of the light and the polarizer. In our simple example, cos(37)=4/5, so everything works out.

This behavior is why polarization is such a useful tool for quantum optics. We can treat the polarization of light as a combination of two polarization states (H and V), and easily prepare any arbitrary combination of those two that we want (there are also devices that allow you to rotate the polarization of light without decreasing the intensity, but I won’t talk about them here). We can then analyze that light using polarizers, and get an excellent realization of the ideal quantum measurement– light either passes through a polarizer or it doesn’t, and the light that makes it through a verical polarizer is perfectly vertically polarized afterwards. This is why so many of the classic experiments on the fundamentals of quantum mechanics have been done with polarized photons.

The discussion above has assumed that we have linearly polarized light, that is, light where the electric field oscillates back and forth along the same axis all the time. This doesn’t have to be the case, though– the horizontal and vertical components of the field are independent, and we can delay one of them relative to the other. If we delay one by one-quarter of an oscillation relative to the other, we get circular polarization, which looks like this:

At some instant, the horizontally polarized wave will be at a maximum, and the vertically polarized wave will be at zero. At that instant, the polarization looks horizontal. A little bit later, though, the horizontally polarized wave has decreased a little, while the vertically polarized wave has started to increase. Now, the light looks like it’s polarized at a slight angle– mostly horizontal, but with a small vertical component. A little while later, the horizontal component has decreased some more, and the vertical component has increased some more, and the angle is bigger. And so on. Eventually, you hit a point where the vertically polarized wave is at a maximum, and the horizontally polarized wave is zero, and you have pure vertical polarization.

If you sit at one point in space, and look at the direction of the electric field, you’ll see it spinning around in a circle like the hand on a clock. It rotates very fast– one complete circle per oscillation of the light wave– and is always changing. We can classify this as either right-hand circular or left-hand circular polarization, depending on the direction of rotation.

Circular polarization can be generated using materials like calcite in which one polarization travels faster than the other (called “birefringent” materials), and cutting them to just the right thickness to produce the necessary delay. A piece of birefringent material of the right thickness to make circular polarization is called a “quarter wave plate,” because it delays one wave by a quarter of an oscillation. Exactly what thickness you need depends on the material you’re using and the wavelength of the light you’re working with. Quarter wave plates tend to be very expensive.

Circular polarization might seem like just a curiousity, but in fact, it’s very important for AMO physics. Cicrularly polarized light carries angular momentum, and thus can be used to selectively produce atoms in particular states. Atoms absorbing right-hand-circular light increase their angular momentum, while atoms absorbing left-hand-circular light decrease their angular momentum, and this allows us to exert fine control over the exact state of an electron inside an atom, which in turn lets us do some amazing experiments.

What does this have to do with ugly sunglasses? Come back tomorrow, and I’ll explain.