Geometry of general Jones vector (problem 2.8)
Posted by peeterjoot on August 9, 2012
Another problem from .
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.
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
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 . 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
To kill off the cross term we require
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.
 G.R. Fowles. Introduction to modern optics. Dover Pubns, 1989.
 E. Hecht. Optics. 1998.