The Digital Camera: Total Internal Reflection

Snell’s Law enables us to predict the way a light ray will refract — or bend — when it enters a new surrounding with a different refractive index. If the new surrounding has a larger refractive index the ray will bend toward the surface normal; if the new surrounding has a smaller refractive index the ray will bend away from the surface normal. For a ray that moves from air into water, for instance, Snell’s Law prescribes the following relationship between the angles for the incident and refracted rays:

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

A ray that enters the water directly in line with the surface normal (0 degrees) will not bend, but a ray that enters the water nearly perpendicular to the surface normal (90 degrees) will bend to about 48.6 degrees. No light ray that enters water from air can travel at an angle greater than 48.6 degrees from the surface normal. This important fact is a consequence of the mathematical form of Snell’s Law:

\sin(\theta_{water}) = \frac{n_{air}}{n_{water}} \sin(\theta_{air}).

When the angle in air is 90 degrees then its sine is 1.0, and, because the ratio of the refractive index in air to the refractive index in water is 0.75, a ray that enters water at 90 degrees will bend so that the sine of its new angle is equal to 0.75, which corresponds to an angle of 48.6 degrees. This angle is called the critical angle for an air-to-water or water-to-air interface. When a light ray travels into a medium with a larger refractive index — air into water, for instance — the critical angle limits the angle at which the ray can travel in the new medium. But when a light ray travels into a medium with a smaller refractive index — water into air, for instance — the critical angle has a different kind of significance.

Snell’s Law for a light ray in water that encounters a water-air boundary is:

\sin(\theta_{air}) = \frac{4}{3} \sin(\theta_{water}),

where I have explicitly written the ratio of the refractive indices in water and air as 4/3. When the sine for the angle in water is less than 3/4 and greater than -3/4, the sine for the angle in air will be between -1 and 1. But when the sine of the angle in water is greater than 3/4, Snell’s Law will require the sine for the angle in air to be larger than one. But the sine of any angle cannot be smaller than -1 or larger than 1, so something must be wrong. When this happens — when the incident angle is greater than the critical angle for a ray traveling into a surrounding with a smaller refractive index — the ray will not enter the new medium and will, instead, reflect completely into the original medium. This phenomenon is called total internal reflection.

Total internal reflection for light rays that travel from water into air.

When a ray encounters the boundary at an angle larger than the critical angle, then the reflection angle is equal to the incident angle, just as it is for standard reflection.

Because of total internal reflection, the region outside of Snell’s Window will be like a mirror for a fish that is looking toward the surface of the water.

Outside of Snell’s window the surface of the water reflects the bottom of the pool.
(Image from the Wikimedia Commons.)

The phenomenon of total internal reflection enables light to stay inside of an optical fiber even when the fiber is bent; however, an optical fiber will not work if you bend it so much that light encounters the fiber’s edge at an angle smaller than the critical angle for the fiber and its cladding.

© 2011 Timothy Schulz

The Digital Camera
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