When an electromagnetic wave travels through a medium in which it gets attenuated (this is called an "opaque" or "attenuating" medium), it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:
Note that in many of these cases there are multiple, conflicting definitions and conventions in common use. This article is not necessarily comprehensive or universal.
An 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);
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
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:
either expression can be used interchangeably.[1] Generally, physicists and chemists use the convention on the left (with e−iωt), while electrical engineers use the convention on the right (with e+iωt, for example see electrical impedance). The distinction is irrelevant for an unattenuated wave, but becomes relevant in some cases below. For example, there are two definitions of complex refractive index, one with a positive imaginary part and one with a negative imaginary part, derived from the two different conventions.[2] The two definitions are complex conjugates of each other.
One way to incorporate attenuation into the mathematical description of the wave is via an attenuation coefficient:[3]
where α is the attenuation coefficient.
Then the intensity of the wave satisfies:
i.e.
The attenuation coefficient, in turn, is simply related to several other quantities:
absorption coefficient is essentially (but not quite always) synonymous with attenuation coefficient; see attenuation coefficient for details;
molar absorption coefficient or molar extinction coefficient, also called molar absorptivity, is the attenuation coefficient divided by molarity (and usually multiplied by ln(10), i.e., decadic); see Beer-Lambert law and molar absorptivity for details;
mass attenuation coefficient, also called mass extinction coefficient, is the attenuation coefficient divided by density; see mass attenuation coefficient for details;
Physically, the penetration depth is the distance which the wave can travel before its intensity reduces by a factor of 1/e ≈ 0.37. The skin depth is the distance which the wave can travel before its amplitude reduces by that same factor.
The absorption coefficient is related to the penetration depth and skin depth by
Complex angular wavenumber and propagation constant[edit]
This quantity is also called the phase constant, sometimes denoted β.[11]
Unfortunately, the notation is not always consistent. For example, is sometimes called "propagation constant" instead of γ, which swaps the real and imaginary parts.[12]
In accordance with the ambiguity noted above, some authors use the complex conjugate definition, where the (still positive) extinction coefficient is minus the imaginary part of .[2][13]
where σ is the (real, but frequency-dependent) electrical conductivity, called ACconductivity.
With sinusoidal time dependence on all quantities, i.e.
the result is
If the current were not included explicitly (through Ohm's law), but only implicitly (through a complex permittivity), the quantity in parentheses would be simply the complex electric permittivity. Therefore,
Comparing to the previous section, the AC conductivity satisfies