Mathematical descriptions of opacity

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 Beer-Lambert 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:

• Absorption coefficient,
• Penetration depth and Skin depth,
• Propagation constant, attenuation constant, phase constant, and complex wavenumber,
• Complex refractive index and extinction coefficient,
• Complex dielectric constant,
• AC conductivity.

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.

Background: Unattenuated wave

Main article: Electromagnetic wave equation

A electromagnetic wave propagating in the +z-direction is conventionally described by the equation: $\mathbf {E} (z,t)=\mathrm {Re} (\mathbf {E} _{0}e^{i(kz-\omega t)})$ 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),

$\omega$ 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, in this case, is

$\lambda ={\frac {2\pi }{k}}$

while the vacuum wavelength (the wavelength which a wave of this frequency would have if it were in vacuum) is

$\lambda _{0}={\frac {2\pi c}{\omega }}$.

The index of refraction (also called refractive index) is the ratio of these two, i.e.

$n={\frac {\lambda _{0}}{\lambda }}={\frac {ck}{\omega }}$.

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

$I(z)\propto |\mathbf {E} _{0}e^{i(kz-\omega t)}|^{2}=|\mathbf {E} _{0}|^{2}$.

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:

$I(z)=I_{0}\propto |\mathbf {E} _{0}|^{2}$.

Absorption coefficient

Main articles: Absorption coefficient and Beer-Lambert law

One way to incorporate attenuation into the mathematical description of the wave is via an absorption coefficient:

$\mathbf {E} (z,t)=e^{-\alpha _{abs}z/2}\mathrm {Re} (\mathbf {E} _{0}e^{i(kz-\omega t)})$

where $\alpha _{abs}$ is the absorption coefficient. The intensity in this case satisfies:

$I(z)\propto |e^{-\alpha _{abs}z/2}\mathbf {E} _{0}e^{i(kz-\omega t)}|^{2}=|\mathbf {E} _{0}|^{2}e^{-\alpha _{abs}z}$

i.e.,

$I(z)=I_{0}e^{-\alpha _{abs}z}$

(See the articles Beer-Lambert law and molar absorptivity for how the absorption coefficient relates to the molar extinction coefficient (also called molar absorptivity) and to the mass extinction coefficient. See the article Attenuation coefficient for how the absorption coefficient relates to the attenuation coefficient and mass attenuation coefficient. See the article Absorption cross section for how the absorption coefficient relates to the absorption cross section.)

Penetration depth, skin depth

Main articles: Penetration depth and Skin depth

A very similar approach uses the penetration depth:

$\mathbf {E} (z,t)=e^{-z/(2\delta _{pen})}\mathrm {Re} (\mathbf {E} _{0}e^{i(kz-\omega t)})$

$I(z)=I_{0}e^{-z/\delta _{pen}}$

where $\delta _{pen}$ is the penetration depth.

The skin depth $\delta _{skin}$ is defined so that the wave satisfies:

$\mathbf {E} (z,t)=e^{-z/\delta _{skin}}\mathrm {Re} (\mathbf {E} _{0}e^{i(kz-\omega t)})$

$I(z)=I_{0}e^{-2z/\delta _{skin}}$

where $\delta _{skin}$ is the skin depth.

Physically, the penetration depth is the distance which the wave can travel before its intensity reduces by a factor of $1/e\approx 0.37$. 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

$\alpha _{abs}=1/\delta _{pen}=2/\delta _{skin}$

Complex wavenumber, propagation constant

Main article: Propagation constant

Another way to incorporate attenuation is to use essentially the original expression:

$\mathbf {E} (z,t)=\mathrm {Re} (\mathbf {E} _{0}e^{i({\tilde {k}}z-\omega t)})$

but with a complex wavenumber (as indicated by writing it as ${\tilde {k}}$ instead of k). Then the intensity of the wave satisfies:

$I(z)\propto |\mathbf {E} _{0}e^{i({\tilde {k}}z-\omega t)}|^{2}$

i.e.,

$I(z)=I_{0}e^{-2z\mathrm {Im} ({\tilde {k}})}$

Therefore, comparing this to the absorption coefficient approach,

$\mathrm {Im} ({\tilde {k}})=\alpha _{abs}/2$,     $\mathrm {Re} ({\tilde {k}})=k$

(k is the standard (real) angular wavenumber, as used in any of the previous formulations.)

A closely-related approach, especially common in the theory of transmission lines, uses the propagation constant:

$\mathbf {E} (z,t)=\mathrm {Re} (\mathbf {E} _{0}e^{-\gamma z+i\omega t})$

$I(z)=I_{0}e^{-2z\mathrm {Re} (\gamma )}$

where $\gamma$ is the propagation constant. (The exponential here has the time-dependence $+i\omega t$, not $-i\omega t$; this is the more common convention for AC current, see impedence.)

Comparing the two equations, the propagation constant and complex wavenumber are related by:

$\gamma ^{*}=-i{\tilde {k}}$

(where the * denotes complex conjugation), or more specifically:

$\mathrm {Re} (\gamma )=\mathrm {Im} ({\tilde {k}})=\alpha _{abs}/2$

(This, the real part of the propagation constant, is also called the attenuation constant, sometimes denoted $\alpha$.)

$\mathrm {Im} (\gamma )=\mathrm {Re} ({\tilde {k}})=k$

(This, the imaginary part of the propagation constant, is also called the phase constant, sometimes denoted $\beta$.)

Note that conflicting terminologies are in use: In particular, the parameter ${\tilde {k}}$ is sometimes called "propagation constant".

Complex refractive index, extinction coefficient

Main articles: Refractive index and Extinction coefficient

Recall that in nonattenuating media, the refractive index and wavenumber are related by:

$n={\frac {ck}{\omega }}$

A complex refractive index can therefore be defined in terms of the complex wavenumber defined above:

${\tilde {n}}={\frac {c{\tilde {k}}}{\omega }}$.

In other words, the wave is required to satisfy

$\mathbf {E} (z,t)=\mathrm {Re} (\mathbf {E} _{0}e^{i\omega (({\tilde {n}}z/c)-t)})$.

Comparing to the preceding section, we have

$\mathrm {Re} ({\tilde {n}})={\frac {ck}{\omega }}$, and $\mathrm {Im} ({\tilde {n}})={\frac {c\alpha _{abs}}{2\omega }}$.

The real part of ${\tilde {n}}$ is often (ambiguously) called simply the refractive index. The imaginary part is called the extinction coefficient.

An alternate, equally-valid convention is to write

$\mathbf {E} (z,t)=\mathrm {Re} (\mathbf {E} _{0}e^{-i\omega (({\tilde {n}}z/c)-t)})$

where the time-dependence is $+i\omega t$ instead of $-i\omega t$. Under this convention, the real part of ${\tilde {n}}$ would be unchanged, but the imaginary part would change signs; in other words, the extinction coefficient would be negative the imaginary part of ${\tilde {n}}$.

Complex permittivity

Main article: Complex permittivity

In nonattenuating media, the permittivity and refractive index are related by:

$n=c{\sqrt {\mu \epsilon }}$ (SI),     $n={\sqrt {\mu \epsilon }}$ (cgs)

where $\mu$ is the permeability and $\epsilon$ is the permittivity. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex permittivity:

${\tilde {n}}=c{\sqrt {\mu {\tilde {\epsilon }}}}$ (SI),     ${\tilde {n}}={\sqrt {\mu {\tilde {\epsilon }}}}$ (cgs).

Squaring both sides and using the results of the previous section gives:

$\mathrm {Re} ({\tilde {\epsilon }}/\epsilon _{0})={\frac {c^{2}}{(\omega ^{2})(\mu /\mu _{0})}}(k^{2}-{\frac {\alpha _{abs}^{2}}{4}})$

$\mathrm {Im} ({\tilde {\epsilon }}/\epsilon _{0})={\frac {c^{2}}{(\omega ^{2})(\mu /\mu _{0})}}(k\alpha _{abs})$

(this is in SI; in cgs, drop the $\epsilon _{0}$ and $\mu _{0}$).

This approach is also called the complex dielectric constant; the dielectric constant is synonymous with $\epsilon /\epsilon _{0}$ in SI, or simply $\epsilon$ in cgs.

AC conductivity

Main article: Conductivity

Another way to incorporate attenuation is through the conductivity, as follows.

Part of the equation governing electromagnetic wave propegation is the Maxwell-Ampere law:

$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {d\mathbf {D} }{dt}}$ (SI)     $\nabla \times \mathbf {H} ={\frac {4\pi }{c}}\mathbf {J} +{\frac {1}{c}}{\frac {d\mathbf {D} }{dt}}$ (cgs)

where D is the displacement field. Plugging in Ohm's law and the definition of (real) permittivity

$\nabla \times \mathbf {H} =\sigma \mathbf {E} +\epsilon {\frac {d\mathbf {E} }{dt}}$ (SI)     $\nabla \times \mathbf {H} ={\frac {4\pi \sigma }{c}}\mathbf {E} +{\frac {\epsilon }{c}}{\frac {d\mathbf {E} }{dt}}$ (cgs)

where $\sigma$ is the (real, but frequency-dependent) conductivity, called AC conductivity. With sinusoidal time dependence on all quantities, i.e. $\mathbf {H} =\mathrm {Re} (\mathbf {H} _{0}e^{-i\omega t})$ and $\mathbf {E} =\mathrm {Re} (\mathbf {E} _{0}e^{-i\omega t})$, the result is

$\nabla \times \mathbf {H} _{0}=-i\omega \mathbf {E} _{0}(\epsilon +i{\frac {\sigma }{\omega }})$ (SI)     $\nabla \times \mathbf {H} _{0}={\frac {-i\omega }{c}}\mathbf {E} _{0}(\epsilon +i{\frac {4\pi \sigma }{\omega }})$ (cgs)

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,

${\tilde {\epsilon }}=\epsilon +i{\frac {\sigma }{\omega }}$ (SI)     ${\tilde {\epsilon }}=\epsilon +i{\frac {4\pi \sigma }{\omega }}$ (cgs).

Comparing to the previous section, the AC conductivity satisfies

$\sigma ={\frac {k\alpha _{abs}c^{2}}{(\omega )(\mu /\mu _{0})}}$ (SI)     $\sigma ={\frac {k\alpha _{abs}c^{2}}{4\pi \omega \mu }}$ (cgs).

References and footnotes

• Jackson, John David (1999). Classical Electrodynamics (3rd ed. ed.). New York: Wiley. ISBN 0-471-30932-X. {{cite book}}: |edition= has extra text (help)
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• J. I. Pankove, Optical Processes in Semiconductors, Dover Publications Inc. New York (1971).