Quarter wave plate

Figure 16.12: A quarter wave plate

Consider a birefringent crystal of thickness $h$ with its optic axis along the $y$ direction as shown in Figure 16.12. Consider light that passes through the crystal as shown in the figure. If we decompose the incident light into two perpendicular polarizations along the $x$ and $y$ axes respectively, these two polarizations will traverse different optical path lengths through the crystal. The optical path length is $n_o h$ for the $x$ component and $n_e h$ for the $y$ component respectively. The crystal is called a quarter wave plate if this difference in the optical path lengths is $\lambda/4$, ie.

$$\mid n_e -n_o \mid h = \frac{\lambda}{4}$$ (16.7)

and it is called a half wave plate if,

$$\mid n_e -n_o \mid h = \frac{\lambda}{2} \,.$$ (16.8)

A quarter wave plate can be used to convert linearly polarized light to circularly polarized light as shown in the right Figure 16.12. The incident light should be at $45^{\circ}$ to the optic axis. The incident wave can be expressed as

$$\vec{E}(z,t)=E_0 (\hat{i}+\hat{j}) \cos(\omega t - k z)$$ (16.9)

The quarter wave plate introduces a phase difference between the $x$ and $y$ polarizations. An optical path difference of $\lambda/4$ corresponds to a phase difference of $90^{\circ}$. The wave that comes out of the quarter wave plate can be expressed as

$$\vec{E}(z,t)=E_0 [\hat{i} \cos(\omega t - k z) +\hat{j} \cos(\omega t - k z+90^{\circ})]$$ (16.10)

which is circularly polarized light.

The output is elliptically polarized if the angle between the incident linear polarization and the optic axis is different from $45^{\circ}$. The quarter wave plate can also be used to convert circularly polarized or elliptically polarized light to linearly polarized light. In contrast a half wave plate rotates the plane of polarization as shown in the left of the Figure 16.13.