62 The intensity of a beam of light is defined as the power per
unit area incident on a perpendicular surface. Suppose that a beam
of light in a medium with index of refractionnreaches the surface of
the medium, with air on the outside. Its incident angle with respect
to the normal isθ. (All angles are in radians.) Only a fraction
f of the energy is transmitted, the rest being reflected. Because
of this, we might expect that the transmitted ray would always be
less intense than the incident one. But because the transmitted ray
is refracted, it becomes narrower, causing an additional change in
intensity by a factorg >1. The product of these factorsI=fgcan
be greater than one. The purpose of this problem is to estimate the
maximum amount of intensification.
We will use the small-angle approximationθ 1 freely, in order to
make the math tractable. In our previous studies of waves, we have
only studied the factorf in the one-dimensional case whereθ =
- The generalization toθ 6 = 0 is rather complicated and depends
on the polarization, but for unpolarized light, we can use Schlick’s
approximation,
f(θ) =f(0)(1−cosθ)^5 ,where the value off atθ= 0 is found as in problem 17 on p. 395.
(a) Using small-angle approximations, obtain an expression forgof
the formg≈1 +Pθ^2 , and find the constantP. .Answer, p. 1065
(b) Find an expression forI that includes the two leading-order
terms inθ. We will call this expressionI 2. Obtain a simple expres-
sion for the angle at whichI 2 is maximized. As a check on your
work, you should find that forn= 1.3,θ= 63◦. (Trial-and-error
maximization ofIgives 60◦.)
(c) Find an expression for the maximum value ofI 2. You should
find that forn= 1.3, the maximum intensification is 31%.
63 In an experiment to measure the unknown index of refraction
nof a liquid, you send a laser beam from air into a tank filled with
the liquid. Letφbe the angle of the beam relative to the normal
while in the air, and letθbe the angle in the liquid. You can setφto
any value you like by aiming the laser from an appropriate direction,
and you measureθas a result. We wish to plan such an experiment
so as to minimize the error dnin the result of the experiment, for
a fixed error dθin the measurement of the angle in the liquid. We
assume that there is no significant contribution to the error from
uncertainty in the index of refraction of air (which is very close to
- or from the angleφ. Find dnin terms of dθ, and determine the
optimal conditions. .Solution, p. 1051
Problems 843