1. What is a Quarter Wave Plate

A Quarter wave plate (QWP) is an optical element that can be used to convert a linearly polarized light into a circularly polarized light. Linearly polarized light has electric field in one direction while a circularly polarized light has two perpendicular electric field components such that the net electric field moves in a circle. Depending on how that rotation of electric field vector happens as the wave propagates, it can be left circularly polarized (LCP) or right circularly polarized (RCP). The orientation of the the wave plate with respect to the electric field of the linearly polarized light decides the sense of rotation of resultant circularly polarized light.

2. How does it work?

QWP works by delaying the propagation along one axis (also called as the slow axis) in comparison to a perpendicular axis (also called as the fast axis). QWP is made of materials that have difference refractive indices along two direction - also known as birefringence. The direction with a smaller refractive index leads to less phase acquired while the direction with the greater refractiev index leads to more phase acquired. Assume the d is the thickness of the QWP along the direction of propagation.

\[ \mathrm{OPL}_1 = n_1 d \]

\[ \mathrm{OPL}_2 = n_2 d \]

\[ \Delta\mathrm{OPL} = (n_2-n_1)d \]

\[ \Delta\phi = \frac{2\pi}{\lambda_0}(n_2-n_1)d \]

For the QWP we want the phase difference to be pi/2, which corresponds to quarter wave, and hence the name.

\[ |\Delta\phi| = \frac{\pi}{2} \]

\[ |\Delta\mathrm{OPL}| = \frac{\lambda_0}{4} \]

This makes the thickness of the QWP to be:

\[ d = \frac{\lambda_0}{4|n_2-n_1|} \]

Thus with this thickness of the QWP and at the design wavelenght (for which the OPL diff is quarter wave or the phase difference is pi/2), a appropriately aligned linearly polarized light into a circularly polarized light.

3. Orientation of plate's fast axis wrt linear polarized light

The QWP fast axis has to be 45 or 135 degrees from the electric field orientation of the linear polarized light. Only then would we end up with equal magnitudes of electric field components along, say x and y - which is a requirement for circularly polarized light. Any other angle would lead to elliptical polarized light.

4. Materials, Design criteria, Order

Materials that are used are transparent crystals with birefriengent properties at the wavelenghts of interest. Some examples could be Calcite, Quartz, MgF2.

Design wavelengths, and design angle of incidences can be narrow.

Since the phase difference can also be multipes of the minimum required - these are further divided into orders. The case dicussed above is zero order QWP. The others are called Multiple Order QWP.