# Transmittance

*volume*. For transmission through a

*surface*, see Fresnel equations.

**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.

## Contents

## Mathematical definitions

### Hemispherical transmittance

**Hemispherical transmittance** of a surface, denoted *T*, is defined as^{[3]}

where

- Φ
_{e}^{t}is the radiant flux*transmitted*by that surface; - Φ
_{e}^{i}is the radiant flux received by that surface.

### Spectral hemispherical transmittance

**Spectral hemispherical transmittance in frequency** and **spectral hemispherical transmittance in wavelength** of a surface, denoted *T*_{ν} and *T*_{λ} respectively, are defined as^{[3]}

where

- Φ
_{e,ν}^{t}is the spectral radiant flux in frequency*transmitted*by that surface; - Φ
_{e,ν}^{i}is the spectral radiant flux in frequency received by that surface; - Φ
_{e,λ}^{t}is the spectral radiant flux in wavelength*transmitted*by that surface; - Φ
_{e,λ}^{i}is the spectral radiant flux in wavelength received by that surface.

### Directional transmittance

**Directional transmittance** of a surface, denoted *T*_{Ω}, is defined as^{[3]}

where

*L*_{e,Ω}^{t}is the radiance*transmitted*by that surface;*L*_{e,Ω}^{i}is the radiance received by that surface.

### Spectral directional transmittance

**Spectral directional transmittance in frequency** and **spectral directional transmittance in wavelength** of a surface, denoted *T*_{ν,Ω} and *T*_{λ,Ω} respectively, are defined as^{[3]}

where

*L*_{e,Ω,ν}^{t}is the spectral radiance in frequency*transmitted*by that surface;*L*_{e,Ω,ν}^{i}is the spectral radiance received by that surface;*L*_{e,Ω,λ}^{t}is the spectral radiance in wavelength*transmitted*by that surface;*L*_{e,Ω,λ}^{i}is the spectral radiance in wavelength received by that surface.

## Beer–Lambert law

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

where

*τ*is the optical depth;*A*is the absorbance.

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

or equivalently that

where

*σ*_{i}is the attenuation cross section of the attenuating specie*i*in the material sample;*n*_{i}is the number density of the attenuating specie*i*in the material sample;*ε*_{i}is the molar attenuation coefficient of the attenuating specie*i*in the material sample;*c*_{i}is the amount concentration of the attenuating specie*i*in the material sample;*ℓ*is the path length of the beam of light through the material sample.

Attenuation cross section and molar attenuation coefficient are related by

and number density and amount concentration by

where N_{A} is the Avogadro constant.

In case of *uniform* attenuation, these relations become^{[2]}

or equivalently

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 | Q_{e}^{[nb 2]} |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}^{[nb 2]} |
watt | W or J/s |
M⋅L^{2}⋅T^{−3} |
Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||

Spectral flux | Φ_{e,ν}^{[nb 3]}or Φ _{e,λ}^{[nb 4]} |
watt per hertzorwatt per metre |
W/HzorW/m |
M⋅L^{2}⋅T^{−2}orM⋅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 | I_{e,Ω}^{[nb 5]} |
watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. |
|||

Spectral intensity | I_{e,Ω,ν}^{[nb 3]}or I_{e,Ω,λ}^{[nb 4]} |
watt per steradian per hertzorwatt per steradian per metre |
W⋅sr^{−1}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−1} |
M⋅L^{2}⋅T^{−2}orM⋅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 | L_{e,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 | L_{e,Ω,ν}^{[nb 3]}or L_{e,Ω,λ}^{[nb 4]} |
watt per steradian per square metre per hertzorwatt per steradian per square metre, per metre |
W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−3} |
M⋅T^{−2}orM⋅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 | E_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". |
|||

Spectral irradiance | E_{e,ν}^{[nb 3]}or E_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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 | J_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
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 | J_{e,ν}^{[nb 3]}or J_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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 | M_{e}^{[nb 2]} |
watt per square metre | W/m^{2} |
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 | M_{e,ν}^{[nb 3]}or M_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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 | H_{e} |
joule per square metre | J/m^{2} |
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 | H_{e,ν}^{[nb 3]}or H_{e,λ}^{[nb 4]} |
joule per square metre per hertzorjoule per square metre, per metre |
J⋅m^{−2}⋅Hz^{−1}orJ/m ^{3} |
M⋅T^{−1}orM⋅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 | ε_{ν}or ε_{λ} |
1 | Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. |
|||||

Directional emissivity | ε_{Ω} |
1 | Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. |
|||||

Spectral directional emissivity | ε_{Ω,ν}or ε_{Ω,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 | A_{ν}or A_{λ} |
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_{Ω} |
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_{Ω,ν}or A_{Ω,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 | R_{ν}or R_{λ} |
1 | Spectral flux reflected by a surface, divided by that received by that surface. |
|||||

Directional reflectance | R_{Ω} |
1 | Radiance reflected by a surface, divided by that received by that surface. |
|||||

Spectral directional reflectance | R_{Ω,ν}or R_{Ω,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 |
T_{ν}or T_{λ} |
1 | Spectral flux transmitted by a surface, divided by that received by that surface. |
|||||

Directional transmittance |
T_{Ω} |
1 | Radiance transmitted by a surface, divided by that received by that surface. |
|||||

Spectral directional transmittance |
T_{Ω,ν}or T_{Ω,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 | μ_{ν}or μ_{λ} |
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 | μ_{Ω} |
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 | μ_{Ω,ν}or μ_{Ω,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 |

- ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}Alternative symbols sometimes seen:*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - ↑
^{3.0}^{3.1}^{3.2}^{3.3}^{3.4}^{3.5}^{3.6}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. - ↑
^{4.0}^{4.1}^{4.2}^{4.3}^{4.4}^{4.5}^{4.6}Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek). - ↑
^{5.0}^{5.1}Directional quantities are denoted with suffix "Ω" (Greek).

## See also

## References

- ↑ "Electronic warfare and radar systems engineering handbook".
- ↑
^{2.0}^{2.1}IUPAC,*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". Cite error: Invalid`<ref>`

tag; name "GoldBook" defined multiple times with different content - ↑
^{3.0}^{3.1}^{3.2}^{3.3}"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions".*ISO 9288:1989*. ISO catalogue. 1989. Retrieved 2015-03-15.