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.
An electromagnetic wave propagating in the +z-direction is conventionally described by the equation:
For a given frequency, the wavelength of an electromagnetic wave is affected by the material in which it is propagating.
Note that this intensity is independent of the location z, a sign that this wave is not attenuating with distance.
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.
The attenuation coefficient, in turn, is simply related to several other quantities: A very similar approach uses the penetration depth:[4]
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.
Another way to incorporate attenuation is to use the complex angular wavenumber:[5][7]
In accordance with the ambiguity noted above, some authors use the complex conjugate definition:[8]
A closely related approach, especially common in the theory of transmission lines, uses the propagation constant:[9][10]
Comparing the two equations, the propagation constant and the complex angular wavenumber are related by:
This quantity is also called the attenuation constant,[8][11] sometimes denoted α.
This quantity is also called the phase constant, sometimes denoted β.
is sometimes called "propagation constant" instead of γ, which swaps the real and imaginary parts.
[12] Recall that in nonattenuating media, the refractive index and angular wavenumber are related by:
This quantity is often (ambiguously) called simply the refractive index.
This quantity is called the extinction coefficient and denoted κ.
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] In nonattenuating media, the electric permittivity and refractive index are related by:
where ε is the complex electric permittivity of the medium.
Squaring both sides and using the results of the previous section gives:[7]
Another way to incorporate attenuation is through the electric conductivity, as follows.
[14] One of the equations governing electromagnetic wave propagation is the Maxwell-Ampere law:
Plugging in Ohm's law and the definition of (real) permittivity
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.
Comparing to the previous section, the AC conductivity satisfies