Mathematical Descriptions of Opacity - Background: Unattenuated Wave

Background: Unattenuated Wave

A electromagnetic wave propagating in the +z-direction is conventionally described by the equation: where

E₀ is a vector in the x-y plane, with the units of an electric field (the vector is in general a complex vector, to allow for all possible polarizations and phases),

is the angular frequency of the wave,

k is the angular wavenumber of the wave,

Re indicates real part.

e is Euler's number; see the article Complex exponential for information about how e is raised to complex exponents.

The wavelength is, by definition,

.

For a given frequency, the wavelength of an electromagnetic wave is affected by the material in which it is propagating. The vacuum wavelength (the wavelength that a wave of this frequency would have if it were propagating in vacuum) is

(c is the speed of light in vacuum). In the absence of attenuation, the index of refraction (also called refractive index) is the ratio of these two wavelengths, i.e.,

.

The intensity of the wave is proportional to the square of the amplitude, time-averaged over many oscillations of the wave, which amounts to:

.

Note that this intensity is independent of the location z, a sign that this wave is not attenuating with distance. We define I₀ to equal this constant intensity:

.