# Question regarding the three polarizer paradox

Staff Emeritus
Why are you guys dragging QM into this? Is the OP's difficulty that this isn't complicated enough?

vanhees71
Gold Member
I've also no clue, why the quantum issue came up at all. I've also no clue, why this is called a paradox, but maybe that's because I've learnt classical electrodynamics and optics quite a time ago already.

As @Dale stresses, all this is completely understandable with classical electrodynamics and there's nothing mysterious nor paradoxical.

Without going into the microscopic details just consider the most simple polarizer, a polaroid foil, is a device which lets an electromagnetic wave polarized in one direction (say the $x$-direction) completely through and absorbs one polarized in the direction perpendicular to it (say the $y$ direction, and take the $z$ direction as the direction the wave travels, i.e., the direction of $\vec{k}$).

Now an arbitrary electromagnetic wave with $\vec{k}=k \vec{e}_z$ can be written as
$$\vec{E}(t,\vec{x})=(E_{0x} \vec{e}_x + E_{0y} \vec{e}_y) \exp(\mathrm{i} k z -\mathrm{i} \omega t),$$
where the physical field is of course just the real part, but the complex notation makes life much easier particularly in this polarization business.

So we can describe the polarizer oriented such that it lets $x$-polarized waves through and absorbs $y$-polarized waves completely by the linear (projection) operator.
$$\vec{E}'(t,\vec{x})=\hat{P}_x \vec{E}(t,\vec{x})=E_{0x} \vec{e}_x \exp(\ldots).$$
The intensity $I$ is given (up to an uninteresting constant) by
$$I=\langle \vec{E}^* \cdot \vec{E} \rangle_{T}=\vec{E}_0^{*} \vec{E}_0.$$
Now if you have linearly polarized waves we have $E_{0x},E_{0y} \in \mathbb{R}$ and you can write
$$\vec{E}_0=E_0 (\cos \varphi \vec{e}_x + \sin \varphi \vec{e}_y).$$
The intensity (before the polaroid foil) thus is
$$I=E_0^2=E_{0x}^2+E_{0y}^2.$$
and after
$$I'=E_{0x}^2=E_0^2 \cos^2 \varphi,$$
which is Malus's Law for linearly polarized light.

Now it's easy to show that for a polaroid with it's direction oriented with an angle $\vartheta$ to the $x$ axis acts as the projection operator
$$\hat{P}_{\vartheta}= \vec{e}_{\vartheta} \vec{e}_{\vartheta}^{\text{T}}=\begin{pmatrix} \cos \vartheta \\ \sin \vartheta \end{pmatrix} (\cos \vartheta,\sin \vartheta),$$
i.e., for the amplitude after the polaroid you have
$$\vec{E}_0'=\vec{e}_{\vartheta} (\vec{e}_{\vartheta} \cdot \vec{E}_0).$$
Now it's clear that with two filters in perpendicular relative orientation (say one in $x$ and one in $y$ direction) you get
$$\vec{E}_0'=\hat{P}_{0} \hat{P}_{\pi/2} \hat{E}_0 = \vec{e}_x (\vec{e}_x \cdot \vec{e}_y) \vec{e}_y \cdot \vec{E}_0=0.$$
If you now have first one in $x$ then one in $\pi/4$ and then one in $\pi/2$ orientation you have
$$\vec{E}_0' = \hat{P}_{\pi/2} \hat{P}_{\pi/4} \hat{P}_{0} = \vec{e}_y (\vec{e}_y \cdot \vec{e}_{\pi/4}) (\vec{e}_{\pi/4} \cdot \vec{e}_x) E_{0x}=\frac{1}{2} \vec{e}_y E_{0x}.$$
The intensity after the filter thus is
$$I'=\frac{1}{4} |E_{0x}|^2.$$
Where is here a paradox? Where do I need photons?

Ibix
I've also no clue, why this is called a paradox
If you think as a wave as something either passing through the polaroid or not, it's slightly surprising. I've certainly seen polaroids illustrated as a gate with a plastic sinusoid slid through it (which will only go one way, and can't be rotated by a polaroid). That's a wrong picture, of course.

I think a better picture is to think of the polaroid as an array of antennae that are driven by the incident wave to emit a new wave. Then it's easier to believe that the intermediate polariser could re-emit radiation with components that wouldn't be completely absorbed by the third polariser.

sophiecentaur
Gold Member
@Dale I knew you knew that!!!

anorlunda
Mentor
I've also no clue, why the quantum issue came up at all.
My bad. I thought it was a QM-only thing.

sophiecentaur
Gold Member
My bad. I thought it was a QM-only thing.
You don't use QM to describe what happens in your VP VHF boat antenna. RF electromagnetism make so much more sense when people want some 'understanding'. You can 'feel' the currents flowing and the result of tilting an antenna.

Dale
Mentor
I think a better picture is to think of the polaroid as an array of antennae that are driven by the incident wave to emit a new wave.
I had never heard that before. Very Huygens-like.

Ibix
I had never heard that before. Very Huygens-like.
Really? I thought it was a fairly standard explanation of how wire grid (and similar) polarisers work, an anisotropic relative of the usual explanation of why EM waves don't penetrate metals. The charges can't move very far across the grid.

vanhees71
Gold Member
Usually the treatment is using "macroscopic electrodynamics", i.e., Maxwell's equations for $\vec{E}$, $\vec{B}$ and $\vec{D}$ and $\vec{H}$ with consitutive equations motivated from linear-response theory (in very advanced special treatments the latter derived from quantum many-body theory). Then of course the microscopic picture that the electromagnetic field is due to the superposition of the incoming (external) em. field and the em. field from the motion of the charges in the material.

For a treatment in the latter spirit see the very nice textbook by Schwartz:

M. Schwartz, Principles of Electrodynamics, Dover Publications (1972)

It's a bit like a book with the didactical intentions of Purcell in the Berkeley physics book done right, though there are still some rough edges in there (e.g., the use of the concept of "relativistic mass").

Demystifier
2018 Award
Argh! That's also a paradox now. Well, I've to reformulate next week's problem set for my students to make them solve a paradox rather than some projection operator applications ;-)).
Well, it's always more fun to solve a straightforward problem when it is presented as a "paradox". At least to me.

sophiecentaur
Gold Member
Well, it's always more fun to solve a straightforward problem when it is presented as a "paradox". At least to me.
That's probably true. It can bring out your argumentative side and makes you more determined to disprove the paradox. Everyone loves a Devil's Advocate.

Andy Resnick