Polarization (Brit. polarisation) is a property of waves that describes the orientation of their oscillations. For transverse waves, it describes the orientation of the oscillations in the plane perpendicular to the wave's direction of travel. Longitudinal waves such as sound waves in liquids and gases do not exhibit polarization, because for these waves the direction of oscillation is by definition along the direction of travel. Some media can carry waves with both transverse and longitudinal oscillations. Such waves do have polarization.

Polarization is used in areas of science and technology dealing with wave propagation, such as optics, seismology, and telecommunications. For electromagnetic waves such as light, the polarization is described by specifying the direction of the wave's electric field. According to the Maxwell equations, the direction of the magnetic field is uniquely determined for a specific electric field distribution and polarization. For transverse sound waves in a solid, the polarization is associated with the direction of the shear stress in the plane perpendicular to the propagation direction.

Basics: plane waves

The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation of most light waves (a plane wave is a wave with infinitely long and wide wavefronts). For plane waves the transverse condition requires that the electric and magnetic field be perpendicular to the direction of propagation and to each other. Conventionally, when considering polarization, the electric field vector is described and the magnetic field is ignored since it is perpendicular to the electric field and proportional to it. The electric field vector of a plane wave may be arbitrarily divided into two perpendicular components labeled x and y (with z indicating the direction of travel). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner in time, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase, that is they may not reach their maxima and minima at the same time. Mathematically, the electric field of a plane wave can be written as,

$\backslash vec\{E\}(\backslash vec\{r\},t)\; =\; Real\; (A\_\{x\}\; ,\; A\_\{y\}\backslash cdot\; e^\{i\backslash phi\}\; ,\; 0)\; e^\{i(k\; z\; -\; \backslash omega\; t)\}$

or alternatively,

$\backslash vec\{E\}(\backslash vec\{r\},t)\; =\; (A\_\{x\}\backslash cdot\; Cos(k\; z\; -\; \backslash omega\; t)\; ,\; A\_\{y\}\backslash cdot\; Cos(k\; z\; -\; \backslash omega\; t\; +\; \backslash phi)\; ,\; 0)$

where $A\_\{x\}$ and $A\_\{y\}$ are the amplitudes of the x and y directions and $\backslash phi$ is the relative phase between the two components The shape traced out in a fixed plane by the electric vector as such a plane wave passes over it (a Lissajous figure) is a description of the polarization state. The following figures show some examples of the evolution of the electric field vector (blue), with time(the vertical axes), at a particular point in space, along with its x and y components (red/left and green/right), and the path traced by the tip of the vector in the plane (purple): The same evolution would occur when looking at the electric field at a particular time while evolving the point in space, along the direction opposite to propagation.