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Re: [Phys-L] reflection scaling law --> theoretical physics exercise



Dear Colleagues: I am looking for a CD of Physics Quizzes that the late Ted Vittitoe made up several years ago. They are no longer available from Ted and I wanted to know if anyone had the CD and was not using it anymore or could part with it. If you have this CD of Physics Quizzes and would be willing to sell it or part with it, I would be willing to pay shipping costs. Thanks!Frank CangePhysics Teacher

Date: Fri, 11 Apr 2014 09:42:21 -0700
From: jsd@av8n.com
To: Phys-L@Phys-L.org
Subject: [Phys-L] reflection scaling law --> theoretical physics exercise

Executive summary: Suppose there is a small index step, such
that n = n1/n2 = 1 + ε for some small ε. It turns out that
the amount of reflection scales like ε squared. For any angle
you choose, the amount of reflection at that angle exhibits
the same ε^2 scaling.

This ε^2 scaling is a famous result, worth remembering.
Perhaps more importantly, you don't have to remember it,
because it is a simple exercise in theoretical physics
to show that the result *must* hold.

1) Fire up ye olde spreadsheet and plot the Fresnel equations
for the coefficient of reflection, as a function of angle, for
some small ε, such as ε = 0.001.

On the same axes, plot the curve for some other value of ε,
such as ε = 0.0001. Observe that the curves are different,
depending on ε.

Copy the graph. On the new graph, plot the same thing, except
scale each curve, that is, multiply each curve by ε^2, using
the curve's own value of ε^2. Observe that the curves are now
the same, independent of ε. Seriously, you should probably
plot one of them with a dotted line, so that it doesn't entirely
cover up the other curve.

We say that there is a /universality/ property here. There is
a universal curve for ε^2 Rp(ε, θ), independent of ε, when ε
is small. Change one of the ε values to ε = 1.01 and observe
that this is "almost" small enough to exhibit universal behavior.
Observe that ε = 1.1 is in the ballpark, but not particularly
close.

2) Show mathematically that this is guaranteed to happen, in
the small-ε limit.

Hint: Write the Fresnel equation as a Taylor series in powers
of ε. Show that the zeroth-order term is zero, the first-order
term is zero, and the second-order term is nonzero.

Depending on how you write the Fresnel equations, you may
need to invoke Snell's equation and/or the trig identity
sin^2 + cos^2 = 1. I call that the "Pythagorean" trig identity,
for obvious reasons.

3) Look at the structure of the Fresnel equations. Convince
yourself that the scaling law *must* be true, as a consequence
of the structure of the equations.

It's a somewhat complicated equation, but the ε-dependence is
really simple. Any competent theoretical physicist should be
able to look at this and immediately notice that the reflection
*must* scale like ε^2 for small ε.

Yesterday I looked at this and didn't notice the scaling law,
which is humiliating. In retrospect I can do it without
writing anything down. There are about four steps in the
analysis, but each is so simple that you can do it in your
head, just looking at the structure of the Fresnel equation,
plus Snell's law and/or trig identities.

Viewed as a function of ε, the Fresnel equation is a rational
function. It's even simpler as a function of n, and dn/dε=1,
so it comes to the same thing. We're looking for zeros, and
the denominator is uniformly positive, so the only thing that
matters is the numerator. You should be able to differentiate
a polynomial in your head. The coefficients are complicated
functions of θ, but it's still just a polynomial in n.

=============

While we're in the neighborhood, let me mention the concept
of /logarithmic derivative/. In reflection/refraction problems
like this, neither the absolute index nor the change thereof
matter nearly so much as the /relative/ change in the index.
In other words, the change that matters is Δ(ln(n)). Note
that my ε is precisely this: ε = Δ(ln(n)).

Logarithmic derivatives show up all over the place in physics.

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