Mathematical descriptions of opacity
Encyclopedia
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:
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.
where
The wavelength
is, by definition,
.
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
(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.,
.
The intensity
of the wave is proportional to the square of the amplitude, time-averaged 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 I0 to equal this constant intensity:
.
either expression can be used interchangeably. Generally, physicists and chemists use the convention on the left (with
), while electrical engineers use the convention on the right (with 
, 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. The two definitions are complex conjugate
s of each other.
where
is the absorption coefficient. The intensity in this case satisfies:
i.e.,
The absorption coefficient, in turn, is simply related to several other quantities:
:
where
is the penetration depth.
The skin depth
is defined so that the wave satisfies:
where
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
but with a complex wavenumber
(as indicated by writing it as
instead of k). Then the intensity of the wave satisfies:
i.e.,
Therefore, comparing this to the absorption coefficient approach,
, 
(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,
A closely related approach, especially common in the theory of transmission line
s, uses the propagation constant
:
where
is the propagation constant.
Comparing the two equations, the propagation constant and complex wavenumber are related by:
(where the * denotes complex conjugation), or more specifically:
(This quantity is also called the attenuation constant, sometimes denoted
.)
(This quantity is also called the phase constant, sometimes denoted
.)
Unfortunately, the notation is not always consistent. For example,
is sometimes called "propagation constant" instead of 
, which swaps the real and imaginary parts.
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
, and 
.
The real part of
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
.
and refractive index
are related by:
(SI
),
(cgs
)
where
is the permeability and 
is the permittivity
. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex permittivity:
(SI
),
(cgs
).
Squaring both sides and using the results of the previous section gives:
(this is in SI; in cgs, drop the
and 
).
This approach is also called the complex dielectric constant; the dielectric constant
is synonymous with
in SI, or simply 
in cgs.
One of the equations governing electromagnetic wave propagation is the Maxwell-Ampere law
:
(SI) 
(cgs)
where D is the displacement field
. Plugging in Ohm's law
and the definition of (real) permittivity
(SI) 
(cgs)
where
is the (real, but frequency-dependent) conductivity, called AC
conductivity. With sinusoidal time dependence on all quantities, i.e.
and 
, the result is
(SI) 
(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,
(SI) 
(cgs).
Comparing to the previous section, the AC conductivity satisfies
(SI) 
(cgs).
Opacity (optics)
Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc...
" 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 depthPenetration depthPenetration 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 of its original value at the surface.When electromagnetic radiation is incident on...
and Skin depth, - Propagation constantPropagation constantThe propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
, attenuation constant, phase constant, and complex wavenumberWavenumberIn the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
, - Complex refractive indexRefractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
and extinction coefficient, - Complex dielectric constantDielectric constantThe relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
, - ACAlternating currentIn alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
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
A electromagnetic wave propagating in the +z-direction is conventionally described by the equation: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),
is the angular frequencyAngular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the wave, - k is the angular wavenumber of the wave,
- Re indicates real part.
- e is Euler's numberE (mathematical constant)The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
; see the article Complex exponential for information about how e is raised to complex exponents.
The wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
is, by definition,
.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
(c is the speed of light in vacuum
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
). In the absence of attenuation, the index of refraction (also called refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
) is the ratio of these two wavelengths, i.e.,
.The intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
of the wave is proportional to the square of the amplitude, time-averaged 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 I0 to equal this constant intensity:
.Complex conjugate ambiguity
Becauseeither expression can be used interchangeably. Generally, physicists and chemists use the convention on the left (with
), while electrical engineers use the convention on the right (with , for example see electrical impedanceElectrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
). 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
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
, one with a positive imaginary part and one with a negative imaginary part, derived from the two different conventions. The two definitions are complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
s of each other.
Absorption coefficient
One way to incorporate attenuation into the mathematical description of the wave is via an absorption coefficient:where
is the absorption coefficient. The intensity in this case satisfies:i.e.,
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 coefficientAttenuation coefficientThe attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" as it passes through the medium, and a small...
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 lawBeer-Lambert lawIn optics, the Beer–Lambert law, also known as Beer's law or the Lambert–Beer law or the Beer–Lambert–Bouguer law relates the absorption of light to the properties of the material through which the light is travelling.-Equations:The law states that there is a logarithmic dependence between the...
and molar absorptivityMolar absorptivityThe molar absorption coefficient, molar extinction coefficient, or molar absorptivity, is a measurement of how strongly a chemical species absorbs light at a given wavelength...
for details. - Mass attenuation coefficient, also called mass extinction coefficient, is the absorption coefficient divided by density; see mass attenuation coefficientMass attenuation coefficientThe mass attenuation coefficient is a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per unit mass...
for details. - Absorption cross section and scattering cross section are both quantitatively related to the absorption coefficient (or attenuation coefficient); see absorption cross sectionAbsorption cross sectionAbsorption 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, electromagnetic absorption, etc...
and scattering cross section for details. - The absorption coefficient is also sometimes called opacity; see opacity (optics)Opacity (optics)Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc...
.
Penetration depth, skin depth
A very similar approach uses the penetration depthPenetration 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 of its original value at the surface.When electromagnetic radiation is incident on...
:
where
is the penetration depth.The skin depth
is defined so that the wave satisfies:where
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
Complex wavenumber, propagation constant
Another way to incorporate attenuation is to use essentially the original expression:but with a complex wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
(as indicated by writing it as
instead of k). Then the intensity of the wave satisfies:i.e.,
Therefore, comparing this to the absorption coefficient approach,
, (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,
A closely related approach, especially common in the theory of transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s, uses the propagation constant
Propagation constant
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
:
where
is the propagation constant.Comparing the two equations, the propagation constant and complex wavenumber are related by:
(where the * denotes complex conjugation), or more specifically:
(This quantity is also called the attenuation constant, sometimes denoted
.)(This quantity is also called the phase constant, sometimes denoted
.)Unfortunately, the notation is not always consistent. For example,
is sometimes called "propagation constant" instead of , which swaps the real and imaginary parts.Complex refractive index, extinction coefficient
Recall that in nonattenuating media, the refractive indexRefractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
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
, and .The real part of
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
.Complex permittivity
In nonattenuating media, the permittivityPermittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
and refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
are related by:
(SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
),
(cgsGaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...
)
where
is the permeability and is the permittivityPermittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
. In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex permittivity:
(SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
),
(cgsGaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units...
).
Squaring both sides and using the results of the previous section gives:
(this is in SI; in cgs, drop the
and ).This approach is also called the complex dielectric constant; the dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
is synonymous with
in SI, or simply in cgs.AC conductivity
Another way to incorporate attenuation is through the conductivity, as follows.One of the equations governing electromagnetic wave propagation is the Maxwell-Ampere law
Ampère's law
In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...
:
(SI) (cgs)where D is the displacement field
Displacement field
Displacement field may refer to:*Displacement field *Electric displacement field...
. Plugging in Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
and the definition of (real) permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
(SI) (cgs)where
is the (real, but frequency-dependent) conductivity, called ACAlternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
conductivity. With sinusoidal time dependence on all quantities, i.e.
and , the result is (SI) (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,
(SI) (cgs).Comparing to the previous section, the AC conductivity satisfies
(SI) (cgs).