Forget the wave/particle thing. It will only confuse you. In the vast majority of cases, light behaves like a wave. Photons are not particles in the sense that most non-physicists would interpret that word, and it is complicated to explain the differences.
The wavey nature of light shows up in interference and diffraction experiments. It also limits the capabilities of conventional microscopes and telescopes (in different ways). This even applies to CD/DVD/... technologies, where shorter wavelength lasers are needed to read the ever-shrinking markings. On the opposite scale, you have radio waves. These have very long wavelengths, which keeps them from being easily blocked by buildings or terrain.
Anyway, the thing that is "waving" to produce light is the electromagnetic field. This is an abstract thing that basically represents a potential to interact with charged particles (in a well-defined way). It may be represented by two vectors at each point in space - the electric and magnetic fields. These vectors are what are usually drawn in intro books.
If you have a very simple situation where the electric field is oscillating in a single plane, then the wave is said to be (completely) polarized. Usual light isn't (strongly) polarized, but is instead formed by a sum of waves oscillating in all different directions. Polarizing sunglasses filter out all of the waves oscillating outside of a particular plane. The reason that this is useful is that light becomes partially polarized when it is reflected. So the glasses are designed to filter out light bouncing off the ground. This is only effective at filtering out mirror-like surfaces, which is exactly what you'd want in a lens.
Also, don't think of waves too literally as sinusoids. Almost no real-life waves are ever pure sines. The reason that things are introduced that way is that it is simple to find the properties of waves with that shape. It also happens that arbitrary waves may be built up by summing together a number of different sine waves with different frequencies and amplitudes (called the Fourier transform). This is only possible because the equations governing the electromagnetic field are linear (at least outside of certain materials).
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