Elliptical polarization as a function of the
optical rotation and the ellipticity
Let's start with the representation of the elliptical
polarization in the coordinate frame of its own long and short axes. We shall
call these the X' and Y' coordinates. Furthermore, let us chose an amplitude A =
1 for our polarization and, to simplify things further, an absolute phase equal
to
zero (see Fig. 5).
Fig.
5 Elliptically polarized light of amplitude A=1 in the X'Y' coordinate
system of its long and short axis.
Pay attention to the X' and Y' components of the polarization (blue arrow).
You can see that both follow a sinusoidal type motion. Furthermore, when the X'
component is at its maximum, the Y' is zero and vice-versa. This it to say that
they have a p/2 phase difference or, in other words,
one is a cosine and the other is a sine. Therefore:
(Eq.6.1)
where the ± sign indicates
the direction of rotation of the polarization vector and is different
depending on whether we look at the polarization from the point of view of
an incoming wave or an outgoing wave along Z. Furthermore, the amplitudes of both components are
proportional to q and for A = 1 we have:
(Eq.6.2)
We can now switch to the Jones vector:
(Eq.6.3)
From this notation it is very easy to obtain the general expression for the
elliptical polarization in the XY coordinates. First, we shall multiply equation
6.3 by A in order to change the amplitude from 1. Then, a multiplication by exp(id)
changes the phase from 0 to d. Finally, we can
multiply Eq. 6.3 by a rotation matrix R(±f)
and revert from the X'Y' coordinates to the XY ones.
(Eq.6.4)
Once again the ± in R(±f)
indicates that the rotation by f
can be effectuated in two possible directions and
there is a rotation matrix associated with each and them. There is an easy way
to know which rotation matrix to use. Let's take the example in Fig. 6.
Fig.
6 Elliptically polarized light rotated by f in the XY coordinate
system.
We wish to go from the X'Y'
coordinate system to the XY one. Let's then examine what happens to the unit
vectors of X' and Y' in the XY referential. First, in the XY coordinate system,
X' becomes (cosf,-sinf).
Then we see that Y' becomes (sinf,cosf).
Therefore we can write:
(Eq.6.5)
Which gives the expression for R(+f).
Similarly, you can find that for R(-f):
(Eq.6.6)
In the case illustrated in Fig. 6, Eq.6.4 becomes:
(Eq.6.7)
We now have achieved our goal - here is the equation for the elliptical
polarization in terms of the optical rotation and the ellipticity.
Since we are only interested in the optical rotation f and
the ellipticity q, we can consider the ratio EY/EX.
Let's call this ratio the complex polarization. We can designate it with
the letter K and it then follows that:
(Eq.6.8)
But what exactly is this complex polarization? Let us
separate its expression in Eq. 6.8 into a real and imaginary part.
(Eq.6.9)
We see then that, for small angles of f
and q, the real and imaginary parts of the
complex polarization correspond to the optical rotation and the ellipticity
respectively.
(Eq.6.10)
Note again the meaning of the sign of the ellipticity! It is
positive if we look at the wave as it comes towards us along the Z axis and
negative in the opposite conventions that we have adapted so far (see Fig.1 and
Fig.5).
