Snell's Law

There are at least three ways of deriving this law of refraction, or how light bends when it travels from one material to another. The easiest method is indicated in the two pictures below. The light is traveling ^ to the wave crests, and when it enters the lower material and slows down, it still must travel ^ to the crests. The sketch on the right is a magnification of the one on the left. It shows two triangles with a common hypotenuse H. Play around with this and show that (1/v₁)sinq ₁ = (1/v₂)sinq ₂, Then multiply both sides by the speed of light in a vacuum, c, and define n = c/v, so
n₁sinq ₁=n₂sinq ₂.

Another method is to invoke the principle of least time. Light gets to its destination in the least time. In the sketch below, if you were going to travel from A to B by running to the line, then swimming the rest of the way, you would not go in a straight line, because you can run faster than you can swim, so a better strategy would be to run a little farther, and cut down on the swimming distance. Light is smart enough to behave the same way: it chooses the path of least time. This is a calculus problem: find total time t as a function of x, where x is the distance to where the slant line from A intersects the horizontal line. Then dt/dx = 0 at minimum t, and from this you can derive Snell's law.

t = (D₁²+x²)^1/2/v₁ + (D₂² + [L-x]²)^1/2/v₂ . The derivative puts those hypotenuses in the denominator, and we readily get 1/(v₁sinq₁)=1/(v₂sinq₂), which reduces to Snell’s law.

Another method is to work with field equations at the surface. Look this up in a book on electromagnetism for upper undergrads or grad students.

You might get the impression that refraction is for transparent materials only. Wrong- light penetrates anything to some extent. Take a thin enough slice of any opaque material and some light will go through it.

Now beam your way back to the main waves junk.

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