Transmittance
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Transmittance
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]

${\displaystyle T={\frac {\Phi _{\mathrm {e} }^{\mathrm {t} }}{\Phi _{\mathrm {e} }^{\mathrm {i} }}},}$

where

• ?et is the radiant flux transmitted by that surface;

### Spectral hemispherical transmittance

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

${\displaystyle T_{\nu }={\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}},}$
${\displaystyle T_{\lambda }={\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}},}$

where

### Directional transmittance

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

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

where

• Le,?t is the radiance transmitted by that surface;

### Spectral directional transmittance

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

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

where

## Beer-Lambert law

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

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

where

• ? is the optical depth;
• A is the absorbance.

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

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z},}$

or equivalently that

${\displaystyle \tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,}$
${\displaystyle A=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\,\mathrm {d} z,}$

where

Attenuation cross section and molar attenuation coefficient are related by

${\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},}$

and number density and amount concentration by

${\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}$

where NA is the Avogadro constant.

In case of uniform attenuation, these relations become[4]

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}n_{i}\ell }=10^{-\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell },}$

or equivalently

${\displaystyle \tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell ,}$
${\displaystyle A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell .}$

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

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy density we joule per cubic metre J/m3 M?L-1?T-2 Radiant energy per unit volume.
Radiant flux ?e[nb 2] watt W = 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 ?e,?[nb 3] watt per hertz W/Hz M?L2?T-2 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W?nm-1.
?e,?[nb 4] watt per metre W/m M?L?T-3
Radiant intensity Ie,?[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,?,?[nb 3] watt per steradian per hertz W?sr-1?Hz-1 M?L2?T-2 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W?sr-1?nm-1. This is a directional quantity.
Ie,?,?[nb 4] watt per steradian per metre W?sr-1?m-1 M?L?T-3
Radiance Le,?[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,?,?[nb 3] watt per steradian per square metre per hertz W?sr-1?m-2?Hz-1 M?T-2 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".
Le,?,?[nb 4] watt per steradian per square metre, per metre W?sr-1?m-3 M?L-1?T-3
Flux density
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 flux density
Ee,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-2 Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10-26 W?m-2?Hz-1) and solar flux unit (1 sfu = 10-22 W?m-2?Hz-1 = 104 Jy).
Ee,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
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,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-2 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".
Je,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
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,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-2 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".
Me,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
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,?[nb 3] joule per square metre per hertz J?m-2?Hz-1 M?T-1 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".
He,?[nb 4] joule per square metre, per metre J/m3 M?L-1?T-2
Hemispherical emissivity ? N/A 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity ??
or
??
N/A 1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity ?? N/A 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity ??,?
or
??,?
N/A 1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A N/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 A?
or
A?
N/A 1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance A? N/A 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance A?,?
or
A?,?
N/A 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 N/A 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance R?
or
R?
N/A 1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance R? N/A 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance R?,?
or
R?,?
N/A 1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T N/A 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance T?
or
T?
N/A 1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance T? N/A 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance T?,?
or
T?,?
N/A 1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient ? 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 ??
or
??
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 ?? 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 ??,?
or
??,?
reciprocal metre m-1 L-1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
1. ^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. Spectral quantities given per unit frequency are denoted with suffix "?" (Greek)--not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
4. Spectral quantities given per unit wavelength are denoted with suffix "?" (Greek).
5. ^ a b Directional quantities are denoted with suffix "?" (Greek).

## References

1. ^ "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001.CS1 maint: unfit URL (link)
2. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Transmittance". doi:10.1351/goldbook.T06484
3. ^ a b c d "Thermal insulation -- Heat transfer by radiation -- Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved .
4. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Beer-Lambert law". doi:10.1351/goldbook.B00626