Snell's law

Another more useful form of Snell's Law is the following:

(sin i) / (sin r) = n = C₁ / C₂

Where:

• i = angle of incidence
• r = angle of refraction
• n = refractive index

From this is is plain to see that the refractive index (n), is the ratio of speeds of light in the different media.

n₂u₂ - n₁u₁ = aN

where bold denotes vector quantities; n₁ and n₂ are, respectively, the index of refraction in the medium of incidence and in the medium of refraction; u₁ and u₂ are, respectively, unitary vectors along the direction of propagation in the medium of incidence and the medium of refraction; a is a proportionality constant; and N is a unitary vector normal to the interface between the media.

This only means that the incident and refracted rays are colinear with normal vector N.

That's a lotta' nice formulae you have there...but what does it mean ?

Snell's Law predicts the angle by which electromagnetic radiation refracts when it passes from one material to another. In order words, it predicts how light changes direction when it passes through a lens, for instance. 

The writeups above miss to inform you that the index of refraction for a medium is the ratio of the speed of light in vacuum to the speed of light in the medium. Air at freezing temperature (freezing water, duh!) have n = 1.0003. Warmer air has higher n, and this is why you may see strange mirror effects on hot asphalt roads in the summer; the air has an abnormally high refractive index, and bends the light from cooler air. 

This law was discovered in 1621 by Dutch scientist Willebrord van Roijen Snell (Snellius), but was never published. Renée Descartes (Cartesius) published this result in 1637 without quoting the source. Have you ever heard about such rudeness ? Anyhow, Christiaan Huygens mentioned the discovery made earlier by Snell in his work Traité de la lumière (1690), where he introduces the concept of light as a wave. This results in Snell receiving all due respect from colleagues and the public and all is well, except in France where Snell's Law is called...yes Descarte's Law. 

The story doesn't end here, though. All is well with Snell's Law through the centuries, and it keeps on governing every little refraction in the world; glasses, microscopes, radars etc.

Until now

In the latest issue of Science, scientists from the UCLA San Diego reports to having created a material with a negative index of refraction. This means that the material basically reverses the meaning of Snell's Law. When tested with microwaves of similar frequencies being used in police radars - hint, hint - the reflected microwaves went in the opposite direction of that predicted by Snell's Law. Now, if we only could figure out a use for this outside of the laboratories...

The material is a composite of fiberglass and copper wires, so it does not work on optical frequencies i.e. light. The military is of course looking into what applications - other than speeding - that can come out of this.

Reference: ne.se, britannica, Scientific American

So far no-one has mentioned this neat rule of thumb: when light passes into a denser medium its speed decreases and it "bends" toward the normal. Conversely, when light gains speed it bends away from the normal.

When Snell's Law doesn't work out, total internal reflection happens. For example, say we have a light ray going from a dense medium like water (index of refraction of approximately 1.3) into air (index of refraction of approximately 1.0) at an angle of 55 degrees:

n₁ sin θ₁ = n₂ sin θ₂
1.3 sin 55^o = 1.0 sin θ₂
1.065 = sin θ₂