KAHIBARO
Table of Contents
What polarization means
Polarization is a property of transverse waves that describes the direction in which the wave oscillates. In optics, polarization refers to the direction of the electric field oscillation of light.
Light is an electromagnetic wave. As it travels, its electric field and magnetic field both oscillate, and both are perpendicular to the direction of motion. In most optics problems, polarization is described using the electric field, because it is the electric field that most strongly determines how light interacts with matter.
If light travels in the $x$ direction, then its electric field can oscillate in the $y$ direction, in the $z$ direction, or in some changing combination of both. The pattern of this oscillation defines the polarization state.
For a light wave traveling in a given direction, polarization is defined by the behavior of the electric field in the plane perpendicular to propagation.
Unpolarized and polarized light
Many ordinary light sources, such as the Sun or a lamp, emit unpolarized light. This means the electric field direction changes randomly over time among many transverse directions.
Polarized light has a definite pattern of oscillation. The simplest case is linear polarization, where the electric field always oscillates along one fixed line.
A useful comparison is shown below.
| Type of light | Electric field behavior |
|---|---|
| Unpolarized light | Random transverse directions |
| Linearly polarized light | One fixed direction |
| Circularly polarized light | Rotates with constant magnitude |
| Elliptically polarized light | Rotates with changing orientation, tracing an ellipse |
Linear polarization
In linear polarization, the tip of the electric field vector moves back and forth along a straight line in the transverse plane.
For example, if light moves in the $x$ direction and the field is along $y$, one possible wave is
$$
\vec{E}(x,t) = E_0 \cos(kx - \omega t)\,\hat{y}
$$
This is linearly polarized in the $y$ direction.
If the electric field instead points at some angle in the $yz$ plane, the light is still linearly polarized, just along a different axis.
Polarizers
A polarizer is a device that allows only one component of the electric field to pass. If unpolarized light passes through a polarizer, the transmitted light becomes linearly polarized along the polarizer's transmission axis.
A common idealized rule is that a polarizer keeps the component of the electric field parallel to its axis and blocks the perpendicular component.
If unpolarized light of intensity $I_0$ passes through an ideal polarizer, the transmitted intensity is
$$
I = \frac{I_0}{2}
$$
This factor of $\tfrac{1}{2}$ appears because, on average, unpolarized light has equal intensity in all transverse directions.
For ideal unpolarized light passing through one ideal polarizer,
$$
I = \frac{I_0}{2}
$$
and the transmitted light is linearly polarized along the polarizer axis.
Polarization by selective transmission
Many real polarizers work by absorbing one component of the electric field more strongly than the other. The transmitted light then has a preferred oscillation direction.
Suppose linearly polarized light is incident on a polarizer whose axis makes an angle $\theta$ with the light's polarization direction. Only the component parallel to the axis survives. If the incoming electric field amplitude is $E_0$, then the transmitted amplitude is
$$
E = E_0 \cos\theta
$$
Since intensity is proportional to the square of amplitude, the transmitted intensity becomes
$$
I = I_0 \cos^2\theta
$$
This relation is called Malus's law, which is treated separately in the next chapter, but the geometric idea belongs naturally here.
Polarization by reflection
Light can also become partially polarized when it reflects from a nonmetallic surface, such as water or glass. The reflected and transmitted waves do not treat all oscillation directions equally, so one direction can become stronger than the other.
A familiar example is glare from a road or lake. This reflected light is often strongly polarized, which is why polarized sunglasses can reduce glare.
At one special incidence angle, called Brewster's angle, the reflected light is completely linearly polarized. Brewster's angle is discussed in its own chapter, so here it is enough to note that reflection can produce polarization.
Polarization by scattering
When light is scattered by small particles or molecules, the scattered light can become polarized. This happens because the oscillating electric field of the incident wave drives charges in matter, and the reradiated light depends on direction.
This effect helps explain why light from the sky is partially polarized. Sunlight scattered in the atmosphere is often polarized, especially at directions roughly perpendicular to the Sun.
Circular and elliptical polarization
Linear polarization is not the only possible type. If the electric field has two perpendicular components with a phase difference, the direction of the total electric field can rotate as time passes.
Suppose a wave travels in the $x$ direction and has components
$$
E_y = E_{0y}\cos(kx-\omega t)
$$
$$
E_z = E_{0z}\cos(kx-\omega t + \phi)
$$
The resulting polarization depends on the amplitudes $E_{0y}$ and $E_{0z}$ and on the phase difference $\phi$.
If $\phi = 0$ or $\pi$, the result is linear polarization.
If $E_{0y} = E_{0z}$ and $\phi = \pm \frac{\pi}{2}$, the result is circular polarization.
In the general case, the tip of $\vec E$ traces an ellipse, giving elliptical polarization.
Two perpendicular electric field components determine the polarization state.
If their phase difference is
$$
\phi = 0 \text{ or } \pi
$$
the light is linearly polarized.
If
$$
E_{0y} = E_{0z}, \qquad \phi = \pm \frac{\pi}{2}
$$
the light is circularly polarized.
Otherwise, the polarization is generally elliptical.
Why polarization matters
Polarization is important because many materials respond differently to different field directions. This lets us control light in useful ways. Polarization is used in sunglasses, photography, liquid crystal displays, stress analysis, optical communication, and many laboratory instruments.
Polarization also gives information about how light was produced or scattered. By measuring polarization, physicists can learn about surfaces, materials, and atmospheric conditions.
A simple mathematical view
If linearly polarized light has electric field components in two perpendicular directions, we can write
$$
\vec E = E_y \hat y + E_z \hat z
$$
For linear polarization, the ratio $E_z/E_y$ stays fixed in sign and direction during the oscillation. For more general polarization, that ratio changes with time because of phase differences.
This shows that polarization is closely connected to vector components, but here the important point is physical, the transverse electric field has a direction pattern, and that pattern carries useful information.
Key ideas to remember
Polarization can occur only for transverse waves. Since light is a transverse electromagnetic wave, it can be polarized. Unpolarized light has no preferred transverse direction, while polarized light does. Linear polarization means oscillation in one fixed direction. Circular and elliptical polarization arise from combining perpendicular components with phase differences. Polarizers, reflection, and scattering are common ways to produce polarized light.
Essential facts about polarization:
$$
\text{Polarization describes the direction behavior of } \vec E
$$
$$
\text{Only transverse waves can be polarized}
$$
$$
I_{\text{after one ideal polarizer}} = \frac{I_0}{2} \quad \text{for unpolarized light}
$$
$$
I = I_0 \cos^2\theta \quad \text{for linearly polarized light through a polarizer}
$$