## Dispersion

The velocity of light in a material, and hence the index of refraction of the material, depends on the wavelength of the light.  Since the refractive index depends on the wavelength of the light, light waves with different wavelengths and therefore different colors are refracted through different angles.  This is called dispersion, because white light is dispersed into its component colors while traveling through the material.

The index of refraction n of most materials transparent to visible light increases slightly as the wavelength decreases.
As the index n increases

• the speed of light decreases,
• the wavelength of light in the material decreases,
• the angle of refraction θt decreases for a given θinisinθi = ntsinθt , the light is bent more.

Snell's law combined with a wavelength-dependent index of refraction n explains the dispersive properties of a prism.  The sides of a prism are not parallel and light changes direction when it passes through it.

A ~1% variation in the index of refraction over the entire visible range of electromagnetic radiation still results in a significant change in the direction of the emerging red and blue rays.  Since in general the index of refraction is bigger for shorter wavelengths, blue light bends more than red light.

### Rainbows

A rainbow is produced by dispersion and internal reflection of light in water droplets in the atmosphere.  White light from the sun enters a spherical raindrop.  The different colors are refracted through different angles, reflected off the back of the drop, and then refracted again when they emerge from the drop.  The white light now has been dispersed into its component colors, and the different colors travel in slightly different directions.  You see red light coming from water droplets higher in the sky than violet light.  The other colors are found between these, making a rainbow.  In the figure to the right and below, red light arrives at the eye of the observer from the upper drop and violet light from the lower drop.  Other raindrops yield the other colors.

Rainbows are usually seen as half circles.  From a plane or from a very tall building or mountain, however, one can see a complete circle.

The sunlight comes from a particular direction.  If the angles are just right for refraction, reflection and refraction, the light can be directed towards your eye.

For red light, the angles are just right for drops higher up that for blue light if you are looking towards the center of the rainbow.  The different colors are directed towards your eye by different drops at different heights.  The red rays make a larger angle with the symmetry axis that the blue rays.

This is also true if you rotate the picture about the symmetry axis. As an extreme case, imagine the picture not depicting a vertical but a horizontal plane. The sunlight still comes from the same direction, and the red rays still make a larger angle with the symmetry axis that the blue rays. This is why we see a bow, for any angle less than 90o we rotate about the symmetry axis, different drops can reflect light of a given color at a given angle towards our eyes. Of course, there must be drops, it must be raining in that direction.

Sometimes one can see a double rainbow.  The second, dimmer, band, which is higher in the sky than the first, comes from light reflected twice inside a raindrop.  This reverses the order of the colors in the second band.

 Single rainbow Double rainbow

#### Problem:

At sunset, in which direction are you most likely to see a rainbow?

Solution:

• Reasoning:
The sun should be behind you, so you would most likely see a rainbow looking approximately east.

Sunset rainbows are special.  The sun's rays are nearly horizontal, so the top of the rainbow is high in the sky.  Because of scattering the sunlight at sunset contains more red and less blue hues. Therefore the red bands of the rainbow are emphasized and the blue bands are muted.

Additional information:   The Physics Classroom: Refraction and the Ray Model of Light  Lessons 4