Lets briefly summarize what we said about optical rotation
and ellipticity. When a linearly polarized light interacts with matter, it can
be affected by either the refractive index or the extinction coefficient, or
both. Once the electromagnetic wave leaves the material however it can be
described it terms of two arrows representing its left and right circular
polarization components. These arrows can have different amplitudes and/or
different initial (or "absolute") phase. Therefore, all the information
regarding the polarization state of light is contained within those two
quantities - the amplitude and the absolute phase of the components. This is
also true if instead of having the linear polarization represented in terms of
left and right circular polarized waves, we chose to use the Cartesian X and Y
components of the electric field. Thus for a linearly propagating wave along the
Z direction, having an amplitude of E0X along the X
axis and a phase value of dX at the
origin, i.e. at z = t = 0:
(Eq.5.1)
As far as the polarization state is concerned, Eq. 5.1
contains unnecessary information. We know that we only need the amplitude and
the absolute phase, so wouldn't it be nice to somehow separate those two from
the rest? This bright idea occurred to R. Clark Jones (1916-2004) in 1941 and
here is how he did it. From Eq. 5.1 it follows that:
(Eq.5.2)
where "Re" signifies that we mean only the real part of the
complex number within the brackets. Further on:
(Eq.5.3)
And we have done it! In order to simplify the notations, we
can omit the Re as well as the first exponential since that one is constant
(having in mind that they are both still there). Eq. 5.3 becomes then:
(Eq.5.4)
where ẼX is
the complex amplitude of the wave. It is a phasor composed of the
amplitude and the absolute phase. Generally speaking, the set of X and Y complex
amplitudes of a light wave, represented in the from of Eq. 5.4, is called the
Jones vector.
You might have noticed that in
the process of obtaining a succinct representation for the polarization state,
we switched from trigonometric functions to complex numbers. Couldn't we keep it
simple? Couldn't we stay with those beloved trigonometric functions?
Well, as far as I know, there
are three reasons to make use of the complex numbers in optics. The first one
was discussed above - in describing the polarization state of light, it is a way
of separating the physically meaningful quantities of amplitude and absolute
phase from those that are not necessary, namely, the angular frequency and the
wave vector. Second, it simplifies the calculations, for instance, whenever we
have to add up the contributions of several waves having the same frequency but
with different absolute phases. And finally, when solving the wave equation of
light, derived from Maxwell's equations, the general solution involves complex
numbers. In my opinion, this last reason is the least intuitively appealing but
the reasoning behind it goes something like that: "Maxwell's equations are
light. Their solution is complex. Therefore, complex numbers are
the best representation for light."
Next, we shall see how the Jones
vector can be used to construct a general equation for the elliptical
polarization of light in terms of the optical rotation angle and the ellipticity.
