# Jones vector and matrices

With Jones vectors and matrices one can describe the change in polarization of a EM wave. What is the convention of the reference coordinate system; Is it fixed or does it change whenever the direction of the wave changes?

In other words: I have a +45° linear polarized wave traveling in $+z$ direction towards a metallic mirror where it becomes reflected back in $-z$ direction.

One can now either choose a fixed coordinate system so that the polarization remains +45° also for the back-travelling wave (but with an additional phase of $\pi$). Or one can choose a coordinate system "attached" to the wave so that here the reflected wave again travels in positive z direction but now with a -45° polarization (and an additional phase of $\pi$) in this attached coordinate system.

What is the conventional choice?

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There is no conventional choice, the Jones vector formalism must be attached to

• A local coordinate system dependent on the $\bf{k}$-vector and its direction. The Jones vector is always chosen so that the $\bf{k}$-vector corresponds to the positive $z$-axis. This is the current way that optical professionals and particularly professionals in polarization optics define it (as far as I know), see below for further discussion.
• A right-handed coordinate system.

You do have a choice between the increasing and decreasing phase conventions, which will change signs all over the place, and can make calculations incorrect if you mix them up. Be very careful of the wikipedia article on Jones vectors and matrices, they do not use a consistent phase convention (even though they state that they do at the top).

• Decreasing phase: $({\bf{k}}\cdot {\bf{r}}-\omega t)$
• Increasing phase: $(\omega t-{\bf{k}}\cdot {\bf{r}})$

Since the Jones vector is attached to a (possible changing) local coordinate frame, a good way to do a polarization ray trace is to use a global three dimensional coordinate system. This has been accomplished by Yun, Crabtree, McClain and Chipman in their papers "Three-dimensional polarization ray-tracing calculus": "definition and diattenuation" (Appl. Optics 50 no. 18, pp. 2855-2865 (2011), doi:10.1364/AO.50.002855), and "retardance" (Appl. Optics 50 no. 18, pp. 2866-2874 (2011), doi:10.1364/AO.50.002866).

Take note that because of the inherent rotation of the coordinate system through/from interfaces, the Jones matrices will appear to have retardances that are not physically created, but are just a geometrical artifact of the change of local coordinate system.

Poynting vector vs $\bf{k}$-vector definitions

As user17581 pointed out, some people have defined the Poynting vector as the direction of the $+z$-axis. This makes sense because ${\bf{S}}={\bf{E}}\times{\bf{H}}$ so the electric field is always perpendicular to ${\bf{S}}$ and ${\bf{H}}$. The question is then, why isn't it defined that way?

I believe it is because in a material, like a crystal, there are two or three indices of refraction, in two or three particular directions. If a Jones matrix is described for a particular input angle via the Poynting vector in a crystal then a particular Jones matrix is obtained (i.e., it really doesn't describe the entire space of possible inputs to the crystal). This matrix, however, doesn't really make sense because the underlying material properties are anisotropic. If you, however, define the Jones matrices for each index of refraction (and the associated $\bf{k}$-vectors), and model the propagation as two (or more) rays through the crystal, then recombine them at the end, then you basically have a "basis" of Jones matrices for the crystal, which you do not have to recompute each time.

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Perhaps a better way to ray trace would be to use the Poynting vector definition, with some kind of eigenvalue decomposition for a basis...this could be a good area of research... – daaxix Jan 9 '13 at 21:25
After talking about this with my advisor, Jones vectors and matrices in an anisotropic material don't really make sense, this is why you have to define them per ${\bf{k}}$-vector. – daaxix Jan 10 '13 at 20:01

The conventional choice would be the second one you presented, I would say from intuition and from what I can remember from my bachelor's optics course.

The key is that the cartesian axes used to describe the polarization of a beam of light are chosen in such a way that its Poynting vector points to the $+z$ direction, as a general and simple convention. That tells you 'where to look the polarization from', as per convention the cartesian coordinates must also be right-handed.

Hope it's been useful.

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thanks to both of you! – Dan Jan 9 '13 at 17:27
actually, the $+z$-direction is defined by the ${\bf{k}}$-vector not the Poynting vector. In many media they are the same, but not in crystals for example. – daaxix Jan 9 '13 at 19:22
I thought it was actually the opposite! Never mind though, since optics it's not my field, and I gather you have a certain deal of experience in it (certainly more than I do :-p), so thanks for the fix, daaxix, I'll keep that in mind for the future ;-) – user17581 Jan 9 '13 at 20:19
See my addition to my answer above... – daaxix Jan 9 '13 at 21:14
Thanks a lot for your dedication, I found your explanation quite illustrative :-) – user17581 Jan 9 '13 at 23:16