Question

A quarter-wave plate is made from a piece of calcite with its optic axis oriented parallel to its surface.

(a) Determine the thickness of the plate in terms of the wavelength of the radiation and the two refractive indices.

(b) For incident light which is linearly polarized, determine the angle between the plane of polarization and the optic axis of the crystal in order that the transmitted light be circularly polarized.

(you can assume the equation for elliptically polarized light in the notes but otherwise work from first principles)

Answer 1

Optical paths along the 2 axes of polarization are \(n_xe\) and \(n_ye\) where \(e\) is the thickness of the plate.

A quarter-wave plate must produce a difference in paths of \(\lambda/4\)

Solving for \(e\) is easy.

Answer 2

[ I have this for (a) ]

For some given wavelength of light \(\lambda\), a quarter-wave plate is: a plate of a thickness \(e\), such that the phase difference \(\delta\) (up to an integer \(m\)), between the extraordinary and ordinary rays leaving the plate should be \(\pi/2\).
\begin{align*}
\delta &= 2m\pi+\frac\pi2\\
&& \text{In terms of the path difference $\Delta$}\\
\frac{2\pi}\lambda\Delta
&= 2\pi(m+\tfrac14)\\
\Delta &= (m+\tfrac14)\lambda\\
e|n_x-n_y|
&= (m+\tfrac14)\lambda\\
e &= \frac{m+\tfrac14}{|n_x-n_y|}\lambda
\end{align*}
where the refractive indices are \(n_x\) and \(n_y\) for the extraordinary and ordinary rays respectively.

Considering the thinnest possible quarter-wave plate; \(m=0\):
\[e = \frac{\lambda}{4|n_x-n_y|}\]

Answer 3

Perfect :))
For b) you need to define the properties of a circular polarised light in terms of relative phases and amplitudes between the axes.
When this is done, the answer to the question will be pretty obvious.

Answer 4

I still don't understand b)

Answer 5

A circular polarisation means two perpendicular components of equal amolitudes, with a \(\pi /2\) phase difference.

- For the phase difference, you need a quarterwave plate.
- For the equal components, you need an initial plane-polarised light at 45° to the plates axes.