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:
attenuation coefficient; penetration depth and skin depth; complex angular wavenumber and propagation constant; complex refractive index; complex electric permittivity; AC conductivity (susceptance). 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.
Background: unattenuated wave
Description An electromagnetic wave propagating in the +z-direction is conventionally described by the equation:
E ( z , t ) = Re [ E 0 e i ( k z − ω t ) ] , {\displaystyle \mathbf {E} (z,t)=\operatorname {Re} \left[\mathbf {E} _{0}e^{i(kz-\omega t)}\right]\!,}
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); ω is the angular frequency of the wave; k is the angular wavenumber of the wave; Re indicates real part; e is Euler's number. The wavelength is, by definition,
λ = 2 π k . {\displaystyle \lambda ={\frac {2\pi }{k}}.}
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
λ 0 = 2 π c ω , {\displaystyle \lambda _{0}={\frac {2\pi \mathrm {c} }{\omega }},}
where 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.,
n = λ 0 λ = c k ω . {\displaystyle n={\frac {\lambda _{0}}{\lambda }}={\frac {\mathrm {c} k}{\omega }}.}
The intensity of the wave is proportional to the square of the amplitude, time-averaged over many oscillations of the wave, which amounts to:
I ( z ) ∝ | E 0 e i ( k z − ω t ) | 2 = | E 0 | 2 . {\displaystyle I(z)\propto \left|\mathbf {E} _{0}e^{i(kz-\omega t)}\right|^{2}=|\mathbf {E} _{0}|^{2}.}
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:
I ( z ) = I 0 ∝ | E 0 | 2 . {\displaystyle I(z)=I_{0}\propto |\mathbf {E} _{0}|^{2}.}
Complex conjugate ambiguity Because
Re [ E 0 e i ( k z − ω t ) ] = Re [ E 0 ∗ e − i ( k z − ω t ) ] , {\displaystyle \operatorname {Re} \left[\mathbf {E} _{0}e^{i(kz-\omega t)}\right]=\operatorname {Re} \left[\mathbf {E} _{0}^{*}e^{-i(kz-\omega t)}\right]\!,}
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