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Refraction, Part 1

Refraction is the name given to the observed phenomenon that light changes direction, or "bends," as it passes the boundary between one medium and another. This is shown to the right, in a general sense.

Here, we see a beam of light traveling through air, until it meets a pool of water. It arrives at some angle to the surface as shown. As it passes through the boundary, going from air into water, it actually slows down. Since even a single ray of light has a finite thickness, the part that enters the water first slows down first, causing the light ray to change direction to a steeper angle in the water.

If we change the angle at which the light enters the water, we find that the angle of the light in the water also changes, such that we see no change at all if the light source is directly overhead so that the entering ray of light is perpendicular to (in mathematical terms normal to) the surface. As we change the entering angle more and more away from the perpendicular, we see that the ray of light in the water has bent more and more away from the direction taken by that ray of light in the air.

To describe this phenomenon mathematically, we will measure the angle of incidence, θi, in the same way we did when discussing reflection. Instead of an angle of reflection, however, this time we have an angle of refraction, θr. Both angles are measured from a line normal to the boundary between the two media. However, unlike the phenomenon of reflection, the angle of refraction is unlikely to be the same as the angle of incidence in the general description of refraction.

The basic Law of Refraction was first formulated by Willebrord Snell in 1621. Consider the diagram to the left. We see here two parallel rays of light in red. They are passing through a boundary between air and water at a measurable angle of incidence, θi. The rays of light in the water remain parallel, and are now leaving the boundary at a measurable angle of refraction, θr.

We now apply a little plane geometry and a small amount of trigonometry. We start by drawing line AB at right angles to the right-hand ray of light, in air. This forms right triangle ABC, such that angle BAC is equal to θi. We also draw line CD at right angles the the left-hand ray of light, in water. This gives us right triangle ACD, whose angle ACD is equal to θr.

If we now measure the lengths of lines BC and AD along the rays of light, we find that these two lengths have a consistent 4:3 relationship. That is, if we divide line BC into four equal parts, line AD will be exactly three of these parts in length. This remains true for any angle of incidence greater than 0. (At θi = 0, θr also becomes 0, and no visible refraction occurs.)

 Mathematically, BC = 4 = 1.33 = n, AD 3

where n is the index of refraction of water.

Using some basic trigonometry, we note that line AC in this figure is the hypoteneuse for both of the right triangles we drew. Therefore,

BC = AC sin θi

and

 AC sin θi = sin θi = 1.33 = n AC sin θr sin θr

All materials through which light can pass have a measurable index of refraction, which is constant for the specific material involved. For water, this index is 1.33.

The other two materials of greatest interest to the technical community are glass and plastic. Although glass can be manufactured with any number of additives to control various physical properties, its basic structure is silicon dioxide. The nominal index of refraction for glass is about 1.50, with higher numbers for specific additives.

Plastics can be manufactured with many different compositions, so there is no basic index of refraction for the general category of plastic.

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