## Minimum Deviation by a Prism

Introduction

Ray tracing through a prism is nothing conceptually new -- it is just an application of the rules of reflection and refraction. There can be surprises, though. We will look at the ray transmitted through a triangular prism, ignoring internal reflections. The surprise is that "most" transmitted rays are deviated through roughly the same angle, a bit less than 40 degrees in our example, irrespective of the angle of incidence of the ray on the prism! This is not a violation of Snell's Law, as it might at first seem, but rather a subtle consequence of it.

Here is a simulation of a ray striking a triangular prism. We will not follow the internal reflections, but rather just look at the transmitted ray, if there is one. Of course if the ray is totally internally reflected, there is no transmitted ray. You can use the mouse to drag any corner of the prism and thus change the angle of incidence. You will see that there is a minimum angle of deviation, about 37.2 degrees. If you drag the prism either direction from orientation that gives this minimum deviation, you find that the deviation is quite insensitive to the change. That is just a familiar fact of calculus: at a minimum, the derivative is zero. And that is just the property mentioned above: "most" rays are deviated at about that angle. Note the symmetrical position of the minimum deviation ray.

The angle of minimum deviation is responsible for some meteorological phemomena, like halos and sundogs , produced by deviation of sunlight in the hexagonal prisms of ice crystals in the air. Reflection from raindrops -- another exercise in ray-tracing -- shows a minimum deviation angle: that's the rainbow! In all cases you can imagine as a first approximation that the light is deviated through just one special angle, the angle of minimum deviation.

The minimum deviation D in a prism occurs when the entering angle and the exiting angle are the same, a particularly symmetrical configuration. Applying Snell's Law at the interfaces you can derive the following relationship:

n=sin[(D+a)/2]/sin(a/2)

where n is the relative index of refraction of the prism, and a is the angle between the two relevant prism faces (60 degrees in our example). Using this relationship, you could figure out what index of refraction was assumed for this simulation. You could also use this method to measure the index of refraction of real materials.