Transmittance

This article is about transmission through a volume. For transmission through a surface, see Fresnel equations.
Diagram of Beer-Lambert Law of transmittance of a beam of light as it travels through a cuvette of width l.
Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region[1]). Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.
Transmittance of ruby in optical and near-IR spectra. Note the two broad blue and green absorption bands and one narrow absorption band on the wavelength of 694 nm, which is the wavelength of the ruby laser.

Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.[2]

Internal transmittance refers to energy loss by absorption, whereas (total) transmittance is that due to absorption, scattering, reflection, etc.

Mathematical definitions

Hemispherical transmittance

Hemispherical transmittance of a surface, denoted T, is defined as[3]

T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}},

where

Spectral hemispherical transmittance

Spectral hemispherical transmittance in frequency and spectral hemispherical transmittance in wavelength of a surface, denoted Tν and Tλ respectively, are defined as[3]

T_\nu = \frac{\Phi_{\mathrm{e},\nu}^\mathrm{t}}{\Phi_{\mathrm{e},\nu}^\mathrm{i}},
T_\lambda = \frac{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}{\Phi_{\mathrm{e},\lambda}^\mathrm{i}},

where

Directional transmittance

Directional transmittance of a surface, denoted TΩ, is defined as[3]

T_\Omega = \frac{L_{\mathrm{e},\Omega}^\mathrm{t}}{L_{\mathrm{e},\Omega}^\mathrm{i}},

where

Spectral directional transmittance

Spectral directional transmittance in frequency and spectral directional transmittance in wavelength of a surface, denoted Tν,Ω and Tλ,Ω respectively, are defined as[3]

T_{\nu,\Omega} = \frac{L_{\mathrm{e},\Omega,\nu}^\mathrm{t}}{L_{\mathrm{e},\Omega,\nu}^\mathrm{i}},
T_{\lambda,\Omega} = \frac{L_{\mathrm{e},\Omega,\lambda}^\mathrm{t}}{L_{\mathrm{e},\Omega,\lambda}^\mathrm{i}},

where

Beer–Lambert law

Main article: Beer–Lambert law

By definition, transmittance is related to optical depth and to absorbance as

T = e^{-\tau} = 10^{-A},

where

The Beer–Lambert law states that, for N attenuating species in the material sample,

T = e^{-\sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\mathrm{d}z} = 10^{-\sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\mathrm{d}z},

or equivalently that

\tau = \sum_{i = 1}^N \tau_i = \sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\,\mathrm{d}z,
A = \sum_{i = 1}^N A_i = \sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrm{d}z,

where

Attenuation cross section and molar attenuation coefficient are related by

\varepsilon_i = \frac{\mathrm{N_A}}{\ln{10}}\,\sigma_i,

and number density and amount concentration by

c_i = \frac{n_i}{\mathrm{N_A}},

where NA is the Avogadro constant.

In case of uniform attenuation, these relations become[2]

T = e^{-\sum_{i = 1}^N \sigma_i n_i\ell} = 10^{-\sum_{i = 1}^N \varepsilon_i c_i\ell},

or equivalently

\tau = \sum_{i = 1}^N \sigma_i n_i\ell,
A = \sum_{i = 1}^N \varepsilon_i c_i\ell.

Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

SI radiometry units

SI radiometry units
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy Qe[nb 2] joule J M⋅L2⋅T−2 Energy of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 M⋅L−1⋅T−2 Radiant energy per unit volume.
Radiant flux Φe[nb 2] watt W or J/s M⋅L2⋅T−3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux Φe,ν[nb 3]
or
Φe,λ[nb 4]		 watt per hertz
or
watt per metre		 W/Hz
or
W/m		 M⋅L2⋅T−2
or
M⋅L⋅T−3		 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr M⋅L2⋅T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]		 watt per steradian per hertz
or
watt per steradian per metre		 W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1		 M⋅L2⋅T−2
or
M⋅L⋅T−3		 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 M⋅T−3 Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance Le,Ω,ν[nb 3]
or
Le,Ω,λ[nb 4]		 watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre		 W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3		 M⋅T−2
or
M⋅L−1⋅T−3		 Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Irradiance Ee[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance Ee,ν[nb 3]
or
Ee,λ[nb 4]		 watt per square metre per hertz
or
watt per square metre, per metre		 W⋅m−2⋅Hz−1
or
W/m3		 M⋅T−2
or
M⋅L−1⋅T−3		 Irradiance of a surface per unit frequency or wavelength. The terms spectral flux density or more confusingly "spectral intensity" are also used. Non-SI units of spectral irradiance include Jansky = 10−26 W⋅m−2⋅Hz−1 and solar flux unit (1SFU = 10−22 W⋅m−2⋅Hz−1).
Radiosity Je[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3]
or
Je,λ[nb 4]		 watt per square metre per hertz
or
watt per square metre, per metre		 W⋅m−2⋅Hz−1
or
W/m3		 M⋅T−2
or
M⋅L−1⋅T−3		 Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Radiant exitance Me[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3]
or
Me,λ[nb 4]		 watt per square metre per hertz
or
watt per square metre, per metre		 W⋅m−2⋅Hz−1
or
W/m3		 M⋅T−2
or
M⋅L−1⋅T−3		 Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Radiant exposure He joule per square metre J/m2 M⋅T−2 Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure He,ν[nb 3]
or
He,λ[nb 4]		 joule per square metre per hertz
or
joule per square metre, per metre		 J⋅m−2⋅Hz−1
or
J/m3		 M⋅T−1
or
M⋅L−1⋅T−2		 Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
Hemispherical emissivity ε 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
or
ελ		 1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
or
εΩ,λ		 1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A 1 Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
or
Aλ		 1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
or
AΩ,λ		 1 Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
or
Rλ		 1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
or
RΩ,λ		 1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
or
Tλ		 1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
or
TΩ,λ		 1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ reciprocal metre m−1 L−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
or
μλ		 reciprocal metre m−1 L−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ reciprocal metre m−1 L−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
or
μΩ,λ		 reciprocal metre m−1 L−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also: SI · Radiometry · Photometry
  1. ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
  2. 1 2 3 4 5 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
  3. 1 2 3 4 5 6 7 Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
  4. 1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
  5. 1 2 Directional quantities are denoted with suffix "Ω" (Greek).

See also

References

  1. ↑ "Electronic warfare and radar systems engineering handbook".
  2. 1 2 IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Transmittance".
  3. 1 2 3 4 "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
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