W9_parallel_resonance.eps

380 Week 11: Light

Variational principles prove to be of great use in more advanced physics, as nature

appears to be intrinsically “economical” and choose extremal paths, usually ones that

minimize a quantity called theaction. Newton’s laws themselves can be derived in a

generalized form from a suitable variational principle of a quantity called the “action”,

and this proves to be a useful way to derive and understand partsof quantum theory as

well!

Total Internal Reflection, Critical Angle

θ

reflected

θc θc

r = π/2

incident

(does not escape medium)

refracted ray

Figure 148: Light travelling from a denser medium to a lighter one is totally internally reflected if
θi≥θc= sin(nn^12 ), corresponding to an angle of refraction ofπ/2, where the refracted rayfails to
escape the medium.

If a ray is travelling from a denser medium to a lighter one, one quickly observes a curious

thing. Since the ray is bentawayfrom the normal, there exist angles for which Snell’s

law has no solution!

In fact, it is easy to identify an angle of incidence such that the angleof refraction isθr=

π/2. If we assume thatn 2 > n 1 and we are going from mediumn 2 (the heavier/denser)

to mediumn 1 (the lighter/less dense):

n 2 sin(θ 2 ) =n 2 sin(θc) =n 1 sin(π/2) =n 1 (932)

or

θc= sin−^1

(

n 1

n 2

)

(933)

If we increaseθ 2 > θc, we make the left hand side of Snell’s lawbigger thann 1 but we

cannot find any angleθrfor which sin(θr)>1!. We conclude that at all anglesθcand

greater the rayfails to escape the medium!

Since it is not absorbed by the interface, and is not transmitted intomediumn 1 , the

only place the energy in this ray can go is into thereflectedray. The ray is thustotally

internally reflected.

Total internal reflection is extremely useful in our modern society. It is the basis offiber

opticswhere (laser) light signals are “trapped” inside a “light pipe” that transmits the

light down the fiber and around sufficiently gentle bends without allowing the light to

escape through the sides of the optical fibers that have an index of refraction greater

than that of the surrounding air or other media.