Total Internal Reflection

We begin our examination of Total Internal Reflection (T.I.R.) by noting that when light travels from a more optically dense material [larger index of refraction] to a less dense material the angle of refraction is larger than the incident angle. The concept of optical density relates to the difficulty that light has travelling through the medium. Index of refraction is thus a measure of this quantity. There are numerous cases where a larger optical density is accompanied by a smaller mass density.

Because the refracted angle is always larger than the incident angle, it is possible for the refracted angle to reach 90° before the incident angle reaches 90°. If the light were to refract out of the more dense medium, it would then run along the surface. Larger angles would then yield situations which would force the sine function to be larger than 1.00, which is mathematically impossible.

When the incident angle reaches the condition whereby the refracted ray would bend to an angle of 90°, it is called the CRITICAL ANGLE. The critical angle obeys the following equation:


For light going from water to air, the critical angle is calculated from the equation above:

And so the critical angle for the air-water interface is 48.6°.

What does this answer mean?

  1. Any angle between the light ray and the normal less than 48.6° inside the water will yield a refracted light ray that will at least partially emerge from the water.
  2. Any angle between the light ray and the normal more than 48.6° inside the water will not yield any refracted rays emerging from the water. All of the light rays will stay under the water surface.
  3. 48.6° is the angle where the phenomenon of Total Internal Reflection (T.I.R.) just begins.

Now, we don't have the full story, yet. It seems that we're forgetting a very important part of the picture, namely what has been happening at the underside of the water all along -- namely, reflection. In our diagram we are now showing a Reflected Ray in addition to the Refracted Ray.

This reflected ray changes in intensity as we vary the angle of incidence. At small incident angles (almost perpendicular to the surface) the reflected ray is weak and the refracted ray is strong. As the angle increases, though, the reflected ray becomes stronger and the refracted ray weakens. What actually happens at the critical angle, then?

As the angle approaches the critical angle, the percentage that refracts out reduces to 0% while the amount that reflects increases to 100%.



Perhaps the most important application of Total Internal Reflection is fiber optics. In fiber optics, light passes through a material with a fairly high index of refraction, typically glass or plastic. This material is surrounded by material with a lower index of refraction. If the light strikes the sides at a large enough angle, it totally reflects and stays inside. If the material is flexible, light can be "bent" around corners while keeping all of it inside the fiber. "Fiber" comes from the fact that most applications use material that is very thin, about the size of a hair.


One application is communications where the phone companies are competing to put in as much fiber as they can. It turns out that using light as the carrier of sound signals makes them less succeptible to interference from electrical noise and it takes less space than a conventional copper cable.

A second application is in medicine. Using fiber optics, physicians are able to look inside the body with very little invasive effect. The fiber the doctors insert is surrounded by additional fibers that carry light down to the end. In this scheme, the lens is formed right onto the end of the optical fiber.


A further application is in binoculars and stereo microscopes. A pair of 45-45-90 prisms are arranged so that the incoming light totally reflects off the inside surface of the prisms. This increases the optical path while keeping the size of the instrument relatively compact. And the alignment of the mirrors stays constant.

There's much more to this seemingly innocent device. In practice, the two prisms not only make the light's path longer, but they also invert the image seen if one looks through them. Ask your teacher to show you this. The question then becomes, why?

Uploaded 1/2001