The Digital Camera: Snell’s Law of Refraction

In about 140 AD the Greek scientist Claudius Ptolemy compared the direction light traveled in air with the direction it traveled after passing into water, and he recorded his observations in a table that looked something like this:

Ptolemy's measurements for the refraction of light from air into water.

Although Ptolemy presented his results as measurements, historians of science believe he used a mathematical rule to compute the angles. Nevertheless, the drawing below uses some of Ptolemy’s measurements to show the way light rays bend when they enter water from air.

Illustration of some of Ptolemy's measurements.

Using a laser pointer, an aquarium, and a good protractor you could observe the way light bends for any angle between 0 and 90 degrees, and, if you were meticulous about your measurements, you could make a plot that looks like this:

Air-into-water refraction for all angles between 0 and 90 degrees.

Instead of using measurements, though, I made this plot by using a mathematical equation that was derived in 1621 by a Dutch astronomer and mathematician named Willebrord Snellius. The Persian mathematician Abu Said al-Ala Ibn Sahl discovered the same equation in about 984, but the mathematical relationship I used to make this plot is widely referred to as Snell’s Law in respect of Snellius’s discovery.

Based on Ptolemy’s measurements and my graph, we can see that light bends away from the water’s surface when it enters from air. This observation is general: light bends away from the surface if the new surrounding has a larger refractive index; and light bends toward the surface if the new surrounding has a smaller refractive index. We can visualize this phenomenon with a picture like this:

A general representation of refraction at a boundary.

If the refractive index for the second medium (represented as n2 in the figure) is larger than the refractive index for the first medium (n1 in the figure), then the angle of refraction will be smaller than the angle of incidence. If n2 is smaller than n1, then the angle of refraction will be larger than the angle of incidence. Snell’s equation gives us the exact amount of bending, but we can gain insight by first using a simple approximation to Snell’s equation:

\theta_2 = \frac{n_1}{n_2} \theta_1.

According to this approximation, if the refractive index of the second medium is 4/3 times larger than the refractive index of the first medium (as it is for air and water), then the refracted angle is three-fourths the incident angle. Here is a plot that uses this approximation for our example of light entering water from air:

A simple linear approximation to Snell's Law.

For incidence angles smaller than about 30 degrees the simple linear approximation looks pretty good; however, the approximation is poor for larger angles, and even at smaller angles the approximation will likely produce unacceptable results for precision work like the design of a camera lens. In these situations, optical engineers use the more precise form of Snell’s formula:

n_2 \sin \theta_2 = n_1 \sin \theta_1,

where sin θ is the sine of the angle θ.

The sine of an angle is a number between -1 and 1. For angles between -30 and 30, the sine of the angle is roughly the angle divided by 60 degrees, but for other angles this is not true. The following table shows the relationship between several angles and their sine:

The relationship between some angles and their sine.

For negative angles — those that deviate to the opposite side of the normal direction — the sine is negative. The sine for -20 degrees, for instance, is -0.342.

Another common way to represent the relationship between an angle and its sine is with the graph like this:

A graphical plot of the sine as a function of the angle.

Regardless of whether you use a table, graph, calculator, or computer to determine the sine of an angle, this trigonometric function is crucial for predicting the behavior of light rays that pass from one surrounding (such as air) into another (such as glass or water).

To use Snell’s law to determine the angle in water for a light ray that enters from air at an angle of 50 degrees, for instance, we first note that the sine of the angle in water must satisfy

\sin \theta_2 = \frac{n_1}{n_2} \sin \theta_1.

The refractive index for air (the first medium) is 1, the refractive index for water (the second medium) is 1.33, and the sine of 50 degrees is 0.766. This means that the sine of the angle in water will be equal to 0.766/1.33 = 0.575, which corresponds to an angle of 35.1 degrees. Snell’s law can determine the effects of refraction at all possible angles at which a ray can enter water from air, and if we compare Ptolemy’s original ’measurements’ with Snell’s predictions, we find good agreement for small angles but slight deviation at the larger ones:

Comparison of Ptolemy’s measurements and the prediction from Snell’s Law.

For boundaries with irregular shapes, we measure the incidence and refraction angles relative to the surface normal at the position the ray crosses the boundary, just like we did for reflection.

Refraction of light according at an irregular boundary.

A wonderful thing happens when light rays from a point source pass through a spherical surface:

The focusing property of a spherical surface.

The rays from the source bend and come together at roughly the same location behind the surface. This refracting property of spherical surfaces is the secret behind the magic of a camera’s lens.

Before we transition our discussion to the lens, we’ll use two upcoming posts to learn about two important optical phenomena: total internal reflection; and the pinhole camera.

© 2011 Timothy Schulz

The Digital Camera
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