Mathematical descriptions of opacity

When an electromagnetic wave travels through a medium in which it gets absorbed (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 absorbed. 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.

Background: Unattenuated wave

Main article: Electromagnetic wave equation

A electromagnetic wave propagating in the +z-direction is conventionally described by the equation: $\mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})$ 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),

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; see the article Complex exponential for information about how e is raised to complex exponents.

The wavelength is, by definition,

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

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

$\lambda_0 = \frac{2\pi c}{\omega}$

(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.,

$n = \frac{\lambda_0}{\lambda} = \frac{ck}{\omega}$.

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

$I(z) \propto |\mathbf{E}_0 e^{i(k z - \omega t)}|^2 = |\mathbf{E}_0|^2$.

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:

$I(z) = I_0 \propto |\mathbf{E}_0|^2$.

Complex conjugate ambiguity

Because

$\mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)}) = \mathrm{Re} (\mathbf{E}_0^* e^{-i(k z - \omega t)}),$

either expression can be used interchangeably. 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.^[1] The two definitions are complex conjugates of each other.

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:^[2]

$\mathbf{E}(z,t) = e^{-\alpha_{abs} z / 2} \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})$

where α_abs is the absorption coefficient. The intensity in this case satisfies:

$I(z) \propto |e^{-\alpha_{abs} z/2}\mathbf{E}_0 e^{i(k z - \omega t)}|^2 = |\mathbf{E}_0|^2 e^{-\alpha_{abs} z}$

i.e.,

$I(z) = I_0 e^{-\alpha_{abs} z}$

The absorption coefficient, in turn, is simply related to several other quantities:

• Attenuation coefficient is essentially (but not quite always) synonymous with absorption coefficient; see attenuation coefficient for details.
• Molar absorption coefficient or Molar extinction coefficient, also called molar absorptivity, is the absorption 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 absorption coefficient divided by density; see mass attenuation coefficient for details.
• Absorption cross section and scattering cross section are both quantitatively related to the absorption coefficient (or attenuation coefficient); see absorption cross section and scattering cross section for details.
• The absorption coefficient is also sometimes called opacity; see opacity (optics).

Penetration depth, skin depth

Main articles: Penetration depth and Skin depth

A very similar approach uses the penetration depth:^[3]

$\mathbf{E}(z,t) = e^{-z / (2 \delta_{pen})} \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})$

$I(z) = I_0 e^{-z/\delta_{pen}}$

where δ_pen is the penetration depth.

The skin depth δ_skin is defined so that the wave satisfies:^[4][5]

$\mathbf{E}(z,t) = e^{-z / \delta_{skin} } \mathrm{Re} (\mathbf{E}_0 e^{i(k z - \omega t)})$

$I(z) = I_0 e^{-2z/\delta_{skin}}$

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 $1/e \approx 0.37$. 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

α_abs = 1 / δ_pen = 2 / δ_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).^[4][6] 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,^[2]

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

(k is the standard (real) angular wavenumber, as used in any of the previous formulations.) In accordance with the ambiguity noted above, some authors use the complex conjugate definition, $\mathrm{Im}(\tilde{k}) = -\alpha_{abs}/2.$^[7]

A closely related approach, especially common in the theory of transmission lines, uses the propagation constant:^[8][9]

$\mathbf{E}(z,t) = \mathrm{Re} (\mathbf{E}_0 e^{-\gamma z + i \omega t})$

I(z) = I₀e ^{− 2zRe(γ)}

where γ is the propagation constant.

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 quantity is also called the attenuation constant,^[7][10] sometimes denoted α.)

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

(This quantity is also called the phase constant, sometimes denoted β.)^[10]

Unfortunately, the notation is not always consistent. For example, $\tilde{k}$ is sometimes called "propagation constant" instead of γ, which swaps the real and imaginary parts.^[11]

Complex refractive index, extinction coefficient

Main article: Refractive index

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 $\mathrm{Im}(\tilde{n}) = \frac{c \alpha_{abs}}{2\omega}=\frac{\lambda_0 \alpha_{abs}}{4\pi}$.

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

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 $\tilde{n}$.^[1][12]

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 μ 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:^[2]

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

Squaring both sides and using the results of the previous section gives:^[6]

$\mathrm{Re}(\tilde{\epsilon}/\epsilon_0) = \frac{c^2}{(\omega^2)(\mu/\mu_0)}(k^2-\frac{\alpha_{abs}^2}{4})$

$\mathrm{Im}(\tilde{\epsilon}/\epsilon_0) = \frac{c^2}{(\omega^2)(\mu/\mu_0)}(k\alpha_{abs})$

(this is in SI; in cgs, drop the $\epsilon_0$ and μ₀).

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: Electrical conductivity

Another way to incorporate attenuation is through the conductivity, as follows.^[13]

One of the equations governing electromagnetic wave propagation 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 σ 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. 
• 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).

1. ^ ^{a b} For the definition of complex refractive index with a positive imaginary part, see Optical Properties of Solids, by Mark Fox, p. 6. For the definition of complex refractive index with a negative imaginary part, see Handbook of infrared optical materials, by Paul Klocek, p. 588.
2. ^ ^{a b c} Griffiths, section 9.4.3.
3. ^ IUPAC Compendium of Chemical Terminology
4. ^ ^{a b} Griffiths, section 9.4.1.
5. ^ Jackson, Section 5.18A
6. ^ ^{a b} Jackson, Section 7.5.B
7. ^ ^{a b} Integrated Photonics: Fundamentals, by Ginés Lifante, p.35
8. ^ "Propagation constant", in ATIS Telecom Glossary 2007
9. ^ Advances in imaging and electron physics, Volume 92, by P. W. Hawkes and B. Kazan, p.93
10. ^ ^{a b} Electric Power Transmission and Distribution, by S. Sivanagaraju, p.132
11. ^ See, for example, Encyclopedia of laser physics and technology
12. ^ Pankove, pp. 87-89
13. ^ Jackson, section 7.5C