A variation on Snell's Law

The diagram on the left shows a ray of light entered a piece of plastic perpendicular to the side, so it did not change direction when it entered. When it arrives at the next side, because of the way the plastic was designed, the ray is incident on the second side at 45 degrees to the normal to the surface.

When you press '2nd' or 'inv' or 'shift' sin, (depending on what your calculator is marked) you will see "-E-" or "ERROR", or something like that. When a ray leaves a dense material the angle of refraction is larger than the angle of incidence, it could have been predictable that you could have an angle of incidence that is too big -- and that is exactly what happened here.


In this case, the ray literally cannot leave the material through that side - it is reflected back inside the dense material. This is called "total internal reflection". As you probably know, if you have been in this business very long, there is always some reflection from any lens surface; we limit this reflection with anti-reflection coatings. But in the case we have just seen, ALL of the light traveling in this particular direction is reflected, not just a small percentage of it.
















We can find the angles of incidence that will result in total internal reflection. Any angle greater than the critical angle of the material will result in total internal reflection. The critical angle is the angle that results in an angle of refraction of 90 degrees: in other words, the angle where the refracted ray will travel along the surface of the material or parallel to the material. You can find the critical angle of a material using Snell's Law.
What is the critical angle of of a ray in water traveling to air? We know that total internal reflection occurs when traveling from a more dense to less dense material, we also know that the critical angle will be the angle of incidence that creates and angle of refraction of 90 degrees.

This means that if an angle of incidence in water were greater than 48.76 degrees it would result in total internal reflection.

There are a lot of useful examples of total internal reflection. Fiber optics is where fiber can go around corners, or travel for hundreds of miles, and the ray of light that entered at one end will continue inside the fiber until it reaches the other end. The triangle that we started with is the cross section of a prism that is used as an inexpensive mirror inside binoculars and other optical instruments. Think of a light in pool or pond. At night you can see the light shinning on the bottom of the pool this is due to total internal reflection.
I am not going to test you on your ability to solve for the critical angle. I will, however, ask you what happens to a ray of light if it is incident at an angle greater than the critical angle. And I may give you a Snell's Law problem that results in "-ERROR-" on your calculator, and you will need to know what that means! If you are enjoying the subject, however, go ahead and read the pages 24-25 in the text book and do the exercises. If you have any questions about it, give me a call or e-mail me, and I will be happy to help you through the subject!