mathematical descriptions of opacity
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When an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
travels through a medium in which it gets attenuated (this is called an "
opaque Opacity or opaque may refer to: * Impediments to (especially, visible) light: ** Opacities, absorption coefficients ** Opacity (optics), property or degree of blocking the transmission of light * Metaphors derived from literal optics: ** In lingu ...
" or "
attenuating In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable a ...
" medium), it undergoes
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
as described by the
Beer–Lambert law The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied t ...
. 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 The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
; *
penetration depth Penetration depth is a measure of how deep light or any electromagnetic radiation can penetrate into a material. It is defined as the depth at which the intensity of the radiation inside the material falls to 1/e (about 37%) of its original valu ...
and
skin depth Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the con ...
; * complex angular wavenumber and
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
; *
complex refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
; * complex electric permittivity; * AC
conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution **Ionic conductivity (solid state), elec ...
(
susceptance In electrical engineering, susceptance (''B'') is the imaginary part of admittance, where the real part is conductance. The reciprocal of admittance is impedance, where the imaginary part is reactance and the real part is resistance. In SI unit ...
). 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: \mathbf(z, t) = \operatorname \left mathbf_0 e^\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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...
, to allow for all possible polarizations and phases); *''ω'' is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the wave; *''k'' is the
angular wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the wave; *Re indicates
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
; *''e'' is
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
. The
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
is, by definition, \lambda = \frac. 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, where c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in vacuum. In the absence of attenuation, the
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
(also called
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
) is the ratio of these two wavelengths, i.e., n = \frac = \frac. The
intensity Intensity may refer to: In colloquial use *Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, ma ...
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 \left, \mathbf_0 e^\^2 = , \mathbf_0, ^2. 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: I(z) = I_0 \propto , \mathbf_0, ^2.


Complex conjugate ambiguity

Because \operatorname\left mathbf_0 e^\right= \operatorname\left mathbf_0^* e^\right! , either expression can be used interchangeably. MIT OpenCourseWare 6.007 Supplemental Notes
''Sign Conventions in Electromagnetic (EM) Waves''
/ref> 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 In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the comp ...
). 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 In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
, one with a positive imaginary part and one with a negative imaginary part, derived from the two different conventions.For the definition of complex refractive index with a positive imaginary part, se
''Optical Properties of Solids'', by Mark Fox, p. 6
For the definition of complex refractive index with a negative imaginary part, se
''Handbook of infrared optical materials'', by Paul Klocek, p. 588
The two definitions are
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s of each other.


Attenuation coefficient

One way to incorporate attenuation into the mathematical description of the wave is via an
attenuation coefficient The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
:Griffiths, section 9.4.3. \mathbf(z, t) = e^ \operatorname\! \left mathbf_0 e^\right! , where ''α'' is the attenuation coefficient. Then the intensity of the wave satisfies: I(z) \propto \left, e^\mathbf_0 e^\^2 = , \mathbf_0, ^2 e^, i.e. I(z) = I_0 e^. The attenuation coefficient, in turn, is simply related to several other quantities: * absorption coefficient is essentially (but not quite always) synonymous with attenuation coefficient; see
attenuation coefficient The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
for details; * molar absorption coefficient or molar extinction coefficient, also called molar absorptivity, is the attenuation coefficient divided by molarity (and usually multiplied by ln(10), i.e., decadic); see Beer-Lambert law and
molar absorptivity Molar may refer to: *Molar (tooth), a kind of tooth found in mammals *Molar (grape), another name for the Spanish wine grape Listan Negro *Molar (unit), a unit of concentration equal to 1 mole per litre *Molar mass *Molar volume *El Molar, Tarragon ...
for details; * mass attenuation coefficient, also called mass extinction coefficient, is the attenuation coefficient divided by density; see
mass attenuation coefficient The mass attenuation coefficient, or mass narrow beam attenuation coefficient of a material is the attenuation coefficient normalized by the density of the material; that is, the attenuation per unit mass (rather than per unit of distance). Thus, i ...
for details; * absorption cross section and scattering cross section are both quantitatively related to the attenuation coefficient; see
absorption cross section Absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain particle-particle interaction, e.g., scattering, electromagne ...
and
scattering cross section In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle o ...
for details; * The attenuation coefficient is also sometimes called opacity; see
opacity (optics) Opacity is the measure of impenetrability to electromagnetic radiation, electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a transmission mediu ...
.


Penetration depth and skin depth


Penetration depth

A very similar approach uses the
penetration depth Penetration depth is a measure of how deep light or any electromagnetic radiation can penetrate into a material. It is defined as the depth at which the intensity of the radiation inside the material falls to 1/e (about 37%) of its original valu ...
: \begin \mathbf(z, t) &= e^ \operatorname\! \left mathbf_0 e^\right! , \\ I(z) &= I_0 e^, \end where ''δ''pen is the penetration depth.


Skin depth

The
skin depth Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the con ...
is defined so that the wave satisfies:Griffiths, section 9.4.1.Jackson, Section 5.18A \begin \mathbf(z, t) &= e^ \operatorname\! \left mathbf_0 e^\right! , \\ I(z) &= I_0 e^, \end 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 \alpha = 1/\delta_\mathrm = 2/\delta_\mathrm.


Complex angular wavenumber and propagation constant


Complex angular wavenumber

Another way to incorporate attenuation is to use the complex angular wavenumber:Jackson, Section 7.5.B \mathbf(z, t) = \operatorname\! \left mathbf_0 e^\right! , where ''k'' is the complex angular wavenumber. Then the intensity of the wave satisfies: I(z) \propto \left, \mathbf_0 e^\^2 = , \mathbf_0, ^2 e^, i.e. I(z) = I_0 e^. Therefore, comparing this to the absorption coefficient approach, \begin \operatorname(\underline) &= k, \\ \operatorname(\underline) &= \alpha/2. \end In accordance with the ambiguity noted above, some authors use the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
definition: \begin \operatorname(\underline) &= k, \\ \operatorname(\underline) &= -\alpha/2. \end


Propagation constant

A closely related approach, especially common in the theory of
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s, uses the
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
: \mathbf(z, t) = \operatorname\! \left mathbf_0 e^\right! , where ''γ'' is the propagation constant. Then the intensity of the wave satisfies: I(z) \propto \left, \mathbf_0 e^\^2 = , \mathbf_0, ^2 e^, i.e. I(z) = I_0 e^. Comparing the two equations, the propagation constant and the complex angular wavenumber are related by: \gamma = i\underline^*, where the * denotes complex conjugation. \operatorname(\gamma) = \operatorname(\underline) = \alpha/2. This quantity is also called the
attenuation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, sometimes denoted ''α''. \operatorname(\gamma) = \operatorname(\underline) = k. This quantity is also called the
phase constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, sometimes denoted ''β''. Unfortunately, the notation is not always consistent. For example, \underline is sometimes called "propagation constant" instead of ''γ'', which swaps the real and imaginary parts.


Complex refractive index

Recall that in nonattenuating media, the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
and angular wavenumber are related by: n = \frac = \frac, where * ''n'' is the refractive index of the medium; * c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in vacuum; * ''v'' is the speed of light in the medium. A complex refractive index can therefore be defined in terms of the complex angular wavenumber defined above: \underline = \frac. where ''n'' is the refractive index of the medium. In other words, the wave is required to satisfy \mathbf(z, t) = \operatorname\! \left mathbf_0 e^\right! . Then the intensity of the wave satisfies: I(z) \propto \left, \mathbf_0 e^\^2 = , \mathbf_0, ^2 e^, i.e. I(z) = I_0 e^. Comparing to the preceding section, we have \operatorname(\underline) = \frac. This quantity is often (ambiguously) called simply the ''refractive index''. \operatorname(\underline) = \frac=\frac. 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 \underline.Pankove, pp. 87–89


Complex electric permittivity

In nonattenuating media, the
electric permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
and
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
are related by: n = \mathrm\sqrt\quad \text,\qquad n = \sqrt\quad \text, where * ''μ'' is the
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
of the medium; * ''ε'' is the
electric permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the medium. * "SI" refers to the SI system of units, while "cgs" refers to Gaussian-cgs units. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex electric permittivity: \underline = \mathrm\sqrt\quad \text,\qquad \underline = \sqrt\quad \text, where ''ε'' is the complex electric permittivity of the medium. Squaring both sides and using the results of the previous section gives: \begin \operatorname(\underline) &= \frac\! \left(k^2 - \frac\right)\quad \text, \quad & \operatorname(\underline) &= \frac\! \left(k^2 - \frac\right)\quad \text, \\ \operatorname(\underline) &= \frack\alpha\quad \text, & \operatorname(\underline) &= \frack\alpha\quad \text. \end


AC conductivity

Another way to incorporate attenuation is through the electric conductivity, as follows.Jackson, section 7.5C One of the equations governing electromagnetic wave propagation is the Maxwell-Ampere law: \nabla \times \mathbf = \mathbf + \frac\quad \text,\qquad \nabla \times \mathbf = \frac \mathbf + \frac\frac\quad \text, where \mathbf is the displacement field. Plugging in
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
and the definition of (real)
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
\nabla \times \mathbf = \sigma \mathbf + \varepsilon \frac\quad \text,\qquad \nabla \times \mathbf = \frac \mathbf + \frac\frac\quad \text, where ''σ'' is the (real, but frequency-dependent) electrical conductivity, called AC
conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution **Ionic conductivity (solid state), elec ...
. With sinusoidal time dependence on all quantities, i.e. \begin \mathbf &= \operatorname\! \left mathbf_0 e^\right! ,\\ \mathbf &= \operatorname\! \left mathbf_0 e^\right! , \end the result is \nabla \times \mathbf_0 = -i\omega\mathbf_0 \! \left(\varepsilon + i\frac\right)\quad \text,\qquad \nabla \times \mathbf_0 = \frac \mathbf_0 \! \left(\varepsilon + i\frac\right)\quad \text. If the current \mathbf 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. Therefore, \underline = \varepsilon + i\frac\quad \text,\qquad \underline = \varepsilon + i\frac\quad \text. Comparing to the previous section, the AC conductivity satisfies \sigma = \frac\quad \text,\qquad \sigma = \frac\quad \text.


Notes


References

* * * {{cite book, author=J. I. Pankove, title=Optical Processes in Semiconductors, publisher=Dover Publications Inc. , location=New York , year=1971 Electromagnetic radiation Scattering, absorption and radiative transfer (optics)