When an electromagnetic wave travels through a medium in which it get absorbed (this is called an "opaque" or "attenuating" medium), as described by the BeerLambert law, there are a wide array of mathematical descriptions of the parameters involved in the propagation and attenuation of the wave. 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 in that respect.
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A electromagnetic wave propagating in the +zdirection is conventionally described by the equation: where
The wavelength, in this case, is
while the vacuum wavelength (the wavelength which a wave of this frequency would have if it were in vacuum) is
The index of refraction (also called refractive index) is the ratio of these two, i.e.
The intensity of the wave is proportional to the square of the amplitude, timeaveraged 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 I_{0} to equal this constant intensity:
One way to incorporate attenuation into the mathematical description of the wave is via an absorption coefficient:^{[1]}
where α_{abs} is the absorption coefficient. The intensity in this case satisfies:
i.e.,
The absorption coefficient, in turn, is simply related to another of other quantities:
A very similar approach uses the penetration depth:^{[2]}
where δ_{pen} is the penetration depth.
The skin depth δ_{skin} is defined so that the wave satisfies:^{[3]}^{[4]}
where δ_{skin} is the skin depth.
Physically, the penetration depth is the distance which the wave can travel before its intensity reduces by a factor of . 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
Another way to incorporate attenuation is to use essentially the original expression:
but with a complex wavenumber (as indicated by writing it as instead of k).^{[3]}^{[5]} Then the intensity of the wave satisfies:
i.e.,
Therefore, comparing this to the absorption coefficient approach,^{[1]}
(k is the standard (real) angular wavenumber, as used in any of the previous formulations.)
A closelyrelated approach, especially common in the theory of transmission lines, uses the propagation constant:^{[6]}
where γ is the propagation constant. (The exponential here has the timedependence + iωt, not − iωt; this is the more common convention for alternating current, see electrical impedance.)
Comparing the two equations, the propagation constant and complex wavenumber are related by:
(where the * denotes complex conjugation), or more specifically:
(This, the real part of the propagation constant, is also called the attenuation constant, sometimes denoted α.)
(This, the imaginary part of the propagation constant, is also called the phase constant, sometimes denoted β.)
Note that conflicting terminologies are in use: In particular, the parameter is sometimes called "propagation constant".^{[7]}
Recall that in nonattenuating media, the refractive index and wavenumber are related by:
A complex refractive index can therefore be defined in terms of the complex wavenumber defined above:
In other words, the wave is required to satisfy
Comparing to the preceding section, we have
The real part of is often (ambiguously) called simply the refractive index. The imaginary part is called the extinction coefficient.
An alternate, equallyvalid convention^{[8]} is to write
where the timedependence is + iωt instead of − iωt. Under this convention, the real part of would be unchanged, but the imaginary part would change signs; in other words, the extinction coefficient would be negative the imaginary part of .
In nonattenuating media, the permittivity and refractive index are related by:
where μ is the permeability and ε is the permittivity. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex permittivity:^{[1]}
Squaring both sides and using the results of the previous section gives:^{[5]}
(this is in SI; in cgs, drop the ε_{0} and μ_{0}).
This approach is also called the complex dielectric constant; the dielectric constant is synonymous with ε / ε_{0} in SI, or simply ε in cgs.
Another way to incorporate attenuation is through the conductivity, as follows.^{[9]}
One of the equations governing electromagnetic wave propagation is the MaxwellAmpere law:
where D is the displacement field. Plugging in Ohm's law and the definition of (real) permittivity
where σ is the (real, but frequencydependent) conductivity, called AC conductivity. With sinusoidal time dependence on all quantities, i.e. and , the result is
If the current J was not included explicitly (through Ohm's law), but only implicitly (through a complex permittivity), the quantity in parentheses would be simply the complex permittivity. Therefore,
Comparing to the previous section, the AC conductivity satisfies
