Polarization

Polarization of light is the phenomenon in which the electric field oscillations of a light wave are restricted to a single direction or plane, perpendicular to the direction of propagation.

At Brewster’s angle the reflected and refracted rays are perpendicular, so \(i_p+r=90^\circ\Rightarrow r=90^\circ-i_p\). By Snell’s law:

For glass with \(\mu=1.5\):

Refractive index \(\mu=1.5\) and angle of incidence \(i_p=56.3^\circ\).

Plane-polarized light is light in which the electric vector oscillations are strictly confined to a single plane (the plane of vibration) containing the direction of wave propagation.

When unpolarized light is incident on a transparent dielectric surface (glass or water), the reflected beam is generally partially polarized. At a particular angle of incidence, the Brewster angle \(i_p\), the reflected light becomes completely plane-polarized.

At this angle the reflected and refracted rays are at right angles (\(90^\circ\)) to each other. The vibrations in the plane of incidence are completely transmitted into the medium, and only the vibrations perpendicular to the plane of incidence are reflected. The refracted beam is partially polarized.

Polaroids are thin synthetic sheet polarizers containing long-chain molecules (for example polyvinyl alcohol impregnated with iodine) oriented parallel to each other. They absorb light vibrations parallel to the molecular chains (dichroism) and transmit the vibrations perpendicular to them, along a single direction called the transmission axis.

The intensity is reduced to half:

Unpolarized light has equal components of vibration along all directions. The first Polaroid transmits only the component along its axis and absorbs the perpendicular component.

The transmitted intensity becomes zero (\(I_2=0\)). The light leaving the first Polaroid is plane-polarized along its axis. The second Polaroid (the analyser) at \(90^\circ\) has no component of this vibration along its axis, because \(\cos90^\circ=0\).

As the second Polaroid is rotated from \(90^\circ\) to \(0^\circ\), the transmitted intensity increases steadily from zero to a maximum, following Malus’s law:

Significance: this continuous variation of intensity shows that light is a transverse wave. A longitudinal wave (such as sound) would show no change in intensity on rotation.

Brewster’s law: when unpolarized light is incident on a transparent surface at the polarizing angle \(i_p\), the tangent of the polarizing angle is equal to the refractive index \(\mu\) of the medium:

Derivation: at Brewster’s angle the reflected and refracted rays are perpendicular. From the geometry at the point of incidence:

Applying Snell’s law:

The reflected light is produced by the electric dipoles of the refracting medium, which are made to oscillate by the refracted wave. The oscillations are perpendicular to the refracted ray. When the reflected and refracted rays are at \(90^\circ\), the dipoles that oscillate in the plane of incidence point exactly along the direction of the reflected ray.

An oscillating dipole cannot radiate energy along its own line of oscillation. So no light vibrating in the plane of incidence is reflected. The reflected light contains only the vibrations perpendicular to the plane of incidence, and is therefore completely plane-polarized.

Brewster’s angle \(i_p\) is the particular angle of incidence at which the light reflected from a transparent boundary is completely plane-polarized.

Significance: it gives a way to produce plane-polarized light without absorption losses, and it can be used to find the refractive index of a material.

At Brewster’s angle the reflected and refracted rays are perpendicular (\(90^\circ\)). The dipoles excited in the second medium oscillate along the direction of the reflected ray and cannot radiate in that direction. Consequently the vibrations in the plane of incidence are not reflected (they are transmitted), and the reflected beam contains only the perpendicular vibrations.

By the condition for Brewster’s angle, the angle between the reflected and refracted rays is exactly \(90^\circ\).

Let the refractive indices of the denser and rarer media be \(\mu_d\) and \(\mu_r\). The reflected and refracted rays are perpendicular, and by the law of reflection \(r=i\):

By Snell’s law at the boundary:

The critical angle \(C\) for total internal reflection from the denser to the rarer medium satisfies \(\sin C=\dfrac{\mu_r}{\mu_d}\), so

Plane-polarized light is light in which the electric field vector oscillates in only one plane perpendicular to the direction of propagation. Unpolarized light has electric field oscillations in random orientations, in all directions perpendicular to the direction of propagation.

Light is an electromagnetic wave with transverse electric field oscillations. At Brewster’s angle the angle of refraction satisfies \(i+r’=90^\circ\). The direction of the reflected ray is then parallel to the oscillations of the atomic dipoles (excited by the refracted wave) that correspond to the electric field in the plane of incidence. Since an oscillating dipole cannot radiate along its line of motion, none of the vibrations in the plane of incidence is reflected. The reflected ray is completely polarized, with vibrations perpendicular to the plane of incidence.

Polarization is the restriction of the electric field vibrations of a light wave to a single direction in a plane perpendicular to the direction of propagation.

Significance: it is the experimental proof that light waves are transverse and not longitudinal, because longitudinal waves have no directional asymmetry around the axis of propagation.

Brewster’s law states that the tangent of the polarizing angle \(i_p\) of a medium equals its refractive index:

When light is incident at \(i_p\), the reflected ray is completely plane-polarized, and the reflected and refracted rays make an angle of \(90^\circ\) with each other.

Polaroids are synthetic sheet polarizers made of long-chain polymer molecules aligned parallel during manufacture. When unpolarized light passes through, the component of the electric field parallel to the molecular chains is absorbed and the perpendicular component passes through, producing plane-polarized light.