Figure P 38.69 a is a three-dimensional sketch of a...

Figure $\mathrm{P} 38.69 \mathrm{a}$ is a three-dimensional sketch of a birefringent crystal. The dotted lines illustrate how a thin parallel-faced slab of material could be cut from the larger specimen with the optic axis of the crystal parallel to the faces of the plate. A section cut from the crystal in this manner is known as a retardation plate. When a beam of light is incident on the plate perpendicular to the direction of the optic axis, as shown in Figure $\mathrm{P} 38.69 \mathrm{b}$ , the $\mathrm{O}$ ray and the E ray travel along a single straight line, but with different speeds. (a) Let the thickness of the plate be $d$ and show that the phase difference between the O ray and the E ray is $$ \theta=\frac{2 \pi d}{\lambda}\left|n_{O}-n_{E}\right| $$ where $\lambda$ is the wavelength in air. (b) In a particular case the incident light has a wavelength of 550 $\mathrm{nm}$ . Find the minimum value of $d$ for a quartz plate for which $\theta=\pi / 2$ . Such a plate is called a quarter-wave plate. (Use values of $n_{O}$ and $n_{E}$ from Table $38.1 . )$

Figure $\mathrm{P} 38.69 \mathrm{a}$ is a three-dimensional sketch of a birefringent crystal. The dotted lines illustrate how a thin parallel-faced slab of material could be cut from the larger specimen with the optic axis of the crystal parallel to the faces of the plate. A section cut from the crystal in this manner is known as a retardation plate. When a beam of light is incident on the plate perpendicular to the direction of the optic axis, as shown in Figure $\mathrm{P} 38.69 \mathrm{b}$ , the $\mathrm{O}$ ray and the E ray travel along a single straight line, but with different speeds. (a) Let the thickness of the plate be $d$ and show that the phase difference between the O ray and the E ray is
$$
\theta=\frac{2 \pi d}{\lambda}\left|n_{O}-n_{E}\right|
$$
where $\lambda$ is the wavelength in air. (b) In a particular case the incident light has a wavelength of 550 $\mathrm{nm}$ . Find the minimum value of $d$ for a quartz plate for which $\theta=\pi / 2$ .
Such a plate is called a quarter-wave plate. (Use values of $n_{O}$ and $n_{E}$ from Table $38.1 . )$

Key Concepts

Phase Difference

The phase difference between the two rays in a birefringent material arises because they travel at different speeds and hence accumulate different optical path lengths over the same physical distance. This difference in phase is directly related to the thickness of the material, the wavelength of light, and the difference between the refractive indices of the two rays.

Birefringence

Birefringence is a property of certain materials where the refractive index differs depending on the polarization and propagation direction of light. This leads to the splitting of an incident light beam into two rays that travel at different speeds within the material, each experiencing a different refractive index, commonly referred to as the ordinary and extraordinary rays.

Optic Axis

The optic axis is a specific direction in an anisotropic (birefringent) crystal along which the propagation of light is independent of polarization. When light travels perpendicular to this axis in a correctly cut crystal, it still splits into two rays with different speeds due to the varying interaction with the crystal structure.

Retardation Plate

A retardation plate is a thin, parallel-faced slab of birefringent material that is cut so that its faces are parallel to the optic axis. This arrangement causes a phase difference between the ordinary and extraordinary rays as they traverse the plate because of their different velocities, making such plates useful in controlling the polarization state of light.

Quarter-Wave Plate

A quarter-wave plate is a specific type of retardation plate designed such that the phase difference between the ordinary and extraordinary rays is ?/2. This specific phase shift is used to convert linearly polarized light into circularly polarized light (or vice versa) by introducing a precise delay between the two polarization components.

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