# Color and light

#### Prerequisites

Light isn't just "light"; it comes in an extraordinary variety of types. One of the characteristics of light that is very obvious is color. Because animals use light as a major source of information about the world they live in, and because the chemistry of life interacts with light in useful and interesting ways that distinguish colors, color communicates large amounts of useful information; and many animals and plants use color as ways of both hiding and displaying themselves.

But what is color? In our different models of light it is described in different ways.

## Color in the ray model

The first person who figured out how color works was Isaac Newton. He noted that white light put through a triangular block of glass broke into different colors. If run back through the prism, the colors would reassemble into white, but small bits of the color (red or blue) by themselves could not be broken up further. Since he believed that light was made up of very fast moving particles, he conjectured that light was a mixture of different particles of different colors.

Newton was rather lucky that glass has the properties it does. It turns out (from a complex combination of chemistry and quantum mechanics) that the speed of light (and therefore the index of refraction ) in glass is a smooth function of frequency (color in the wave model) — almost a straight line for the range of frequencies of visible light. In general, transparent materials can have indices of refraction that vary wildly as a function of frequency.

## Color in the wave model

In the wave model color corresponds to the frequency of the light (or equivalently, the wavelength). In the wave model, three key properties are responsible for what makes color useful, two physical , the other mathematical.

1. Superposition — Electric fields don't interact with each other. The fields from different sources just add. The result may be an enhancement or a cancellation, but in some sense we can think of the underlying fields as still there, even when they cancel. This tells us we can break up an electric field in different ways if we choose.
2. Fourier's theorem — There is a mathematical principle that says (loosely) almost any function can be written as a sum of sinusoidal functions oscillating at different frequencies. (The "almost" is because sinusoidal functions have trouble adding up to produce sudden changes. You're OK as long as everything is pretty smooth.
3. Interaction of light with matter — Light is emitted and scattered from moving electric charge. Since matter is bound together in stable situations by forces, it has lots of natural ways to oscillate and resonate: normal modes. Because the matter is naturally oscillating at particular frequencies, it emits and absorbs light at those frequencies.

These ideas combine to tell us that whatever light signal we get, we can express it as a sum of light having different frequencies. When we plot the amount of each frequency that is present, the result is called a spectrum (plural: spectra). The spectrum of a complex light signal (indeed, of any time series signal) can give a lot of information. When we make the next step -- the photon model -- we'll see that light spectra can give us a lot of information about molecular structures.

## Color in the photon model

Einstein's photon model (and our current picture of light in quantum field theory) adds that light can only interact with matter in packets and relates the wave properties of the light, its frequency ($f$) and wavelength ($\lambda$), to its particle properties, its energy ($E$) and momentum ($p$):

$$E = hf \quad \quad p = h/\lambda.$$ Since light can only be absorbed or emitted in these packets, the way different colors of light interact with matter tells us about the spaces between the allowed excitation states of atoms and molecules. Doing a spectral analysis of light emitted or absorbed by something can give us a lot of information about it. Stokes (famous for his study of viscosity) was the first to show that hemoglobin was the molecule responsible for carrying oxygen in the blood using spectral analysis. (See the figure at the right, from his original paper.) It's spectral analysis that permits us to figure out the composition of stars.

How animals perceive light and color is a fascinating are of zoology. Look at the follow-on for an introduction to color perception.

Joe Redish 5/12/12

Article 705