Representation of Polarized Radiation

Light, or electromagnetic radiation, is a transverse wave (also called a vector wave). That is, the vibrations are in a plane transverse, or perpendicular, to the direction of propagation. At the same time light is also a particle, with a `particle' of light referred to as a photon but for the purposes of monochromatic radiative transfer, only the wave properties of light need be considered. However, it will prove convenient to use the concept of photons in describing multiple scattering. The wavelike properties of electromagnetic radiation are governed by Maxwell's equations, a set of four simultaneous differential equations which connect the basic electromagnetic quantities. As is true for the complete description of any vector wave, four properties must be specified. These are: wave amplitude, wavelength, phase, and polarization.

The basic quantities associated with an electromagnetic wave are the mutually perpendicular electric and magnetic field vectors, $\v{\cal{E}}$ and $\v{\cal{H}}$, respectively. The flow of energy resulting from the propagation of an electromagnetic wave is in the direction given by $\v{\cal{E}} \times \v{\cal{H}}$. For historical reasons, the electric field vector is used when representing an electromagnetic wave with the implicit assumption that there is an associated magnetic field vector.

Consider an electromagnetic wave of circular frequency $\omega$ propagating in the positive $\hat{z}$-direction. The electric field vector can then be resolved into two mutually perpendicular components which are parallel and perpendicular to some reference plane, $\v{\cal E}={\cal E}_{\parallel} \hat{e}_{\parallel} + {\cal E}_{\perp} \hat{e}_{\perp}$, such that,

 

$$\cal{E}_{\parallel} a_{\parallel} e^{i(wt-kz+\delta_{\parallel})}$$ = (3.1)

$$\cal{E}_{\perp} a_{\perp} e^{i(wt-kz+\delta_{\perp})}$$ = (3.2)

where $i=\sqrt{-1}$, $k=2\pi/\lambda$ is the wavenumber, $\lambda$ is the wavelength, $a_{\parallel}$ and $a_{\perp}$ are the amplitudes of the parallel and perpendicular components, respectively, and $\delta_{\parallel}-\delta_{\perp}$ is the phase difference between these two components. Radiation as described by equation (2.2) represents a simple wave. That is, a monochromatic, plane, elliptically polarized wave solution of Maxwell's equations. This representation includes the four wave properties listed above. The state of polarization of a wave is described by $\delta_{\parallel}-\delta_{\perp}$ and the magnitudes of $a_{\parallel}$ and $a_{\perp}$. For $\sin(\delta_{\parallel}-\delta_{\perp})>0$, the polarization is said to be right-handed while for $\sin(\delta_{\parallel}-\delta_{\perp})<0$, the polarization if said to be left-handed. This refers to the direction of rotation (from the fingers) if the thumb represent the direction of propagation. Special cases of elliptical polarization include linear ($\delta=0$) and circular ( $\delta=\pi/2$ and $a_{\parallel}= a_{\perp}$).

Measured light (coherent sources notwithstanding) is the `net' effect of a great many individual waves, and not all necessarily possessing the same polarization or wavelength. Over a measurement time of one second, more than a million individual waves may be collected by a detector. As such, the measured quantity will be the superposition of a great many simple waves and so it becomes necessary to introduce a time average. The possible time-averaged polarization states are: completely polarized, partially polarized, and completely unpolarized. Completely polarized light implies that all constituent simple waves individually possess identical polarization states. Completely unpolarized, or randomly polarized, radiation implies that there is no relationship between the polarization states of the individual simple waves. Partially polarized light is a mixture of the two.