[Click here for a PDF of this post with nicer formatting]

Another problem from [1].

# Problem

The general case is represented by the Jones vector

Show that this represents elliptically polarized light in which the major axis of the ellipse makes an angle

with the axis.

# Solution

Prior to attempting the problem as stated, let’s explore the algebra of a parametric representation of an ellipse, rotated at an angle as in figure (1). The equation of the ellipse in the rotated coordinates is

which is easily seen to have the required form

We’d like to express and in the “fixed” frame. Consider figure (2) where our coordinate conventions are illustrated. With

and we find

so that the equation of the ellipse can be stated as

or

Observing that

we have, with a Jones vector representation of our rotated ellipse

Since we can absorb a constant phase factor into our argument, we can write this as

This has the required form once we make the identifications

What isn’t obvious is that we can do this for any , , and . Portions of this problem I tried in Mathematica starting from the elliptic equation derived in section 8.1.3 of [2]. I’d used Mathematica since on paper I found the rotation angle that eliminated the cross terms to always be 45 degrees, but this turns out to have been because I’d first used a change of variables that scaled the equation. Here’s the whole procedure without any such scaling to arrive at the desired result for this problem. Our starting point is the Jones specified field, again as above I’ve using

We need our cosine angle addition formula

Using this and writing we have

Subtracting from we have

Squaring this and using , and 1.2.17 we have

which expands and simplifies to

which is an equation of a rotated ellipse as desired. Let’s figure out the angle of rotation required to kill the cross terms. Writing , and rotating our primed coordinate frame by degrees

we have

To kill off the cross term we require

or

This yields 1.1.2 as desired. We also end up with expressions for our major and minor axis lengths, which are respectively for

which completes the task of determining the geometry of the elliptic parameterization we see results from the general Jones vector description.

# References

[1] G.R. Fowles. *Introduction to modern optics*. Dover Pubns, 1989.

[2] E. Hecht. *Optics*. 1998.