Transmittance

Diagram of Beer-Lambert Law of transmittance of a beam of light as it travels through a cuvette of width l.

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

Φ_e^t is the radiant flux transmitted by that surface;
Φ_eⁱ 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]

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

Φ_e,ν^t is the spectral radiant flux in frequency transmitted by that surface;
Φ_e,νⁱ 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,λⁱ 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]

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

where

L_e,Ω^t is the radiance transmitted by that surface;
L_e,Ωⁱ 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]

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

L_e,Ω,ν^t is the spectral radiance in frequency transmitted by that surface;
L_e,Ω,νⁱ 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,λⁱ 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

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

where

τ is the optical depth;
A is the absorbance.

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

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

\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 N_A 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
v
t
e
Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Dimension	Notes
Radiant energy	$Q e$ ^{[nb 2]}	joule	J	M⋅L²⋅T⁻²	Energy of electromagnetic radiation.
Radiant energy density	$w e$	joule per cubic metre	J/m³	M⋅L⁻¹⋅T⁻²	Radiant energy per unit volume.
Radiant flux	$Φ e$ ^{[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", and called luminosity in Astronomy.
Spectral flux	$Φ e, ν$ ^{[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.
Spectral flux	$Φ e, λ$ ^{[nb 4]}	watt per metre	W/m	M⋅L⋅T⁻³
Radiant intensity	$I e,Ω$ ^{[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	$I e,Ω, ν$ ^{[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.
Spectral intensity	$I e,Ω, λ$ ^{[nb 4]}	watt per steradian per metre	W⋅sr⁻¹⋅m⁻¹	M⋅L⋅T⁻³
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 Specific intensity	$L e,Ω, ν$ ^{[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".
Spectral radiance Specific intensity	$L e,Ω, λ$ ^{[nb 4]}	watt per steradian per square metre, per metre	W⋅sr⁻¹⋅m⁻³	M⋅L⁻¹⋅T⁻³
Irradiance Flux density	$E e$ ^{[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 Spectral flux density	$E e, ν$ ^{[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).
Spectral irradiance Spectral flux density	$E e, λ$ ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiosity	$J e$ ^{[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	$J e, ν$ ^{[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".
Spectral radiosity	$J e, λ$ ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exitance	$M e$ ^{[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	$M e, ν$ ^{[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".
Spectral exitance	$M e, λ$ ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exposure	$H e$	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	$H e, ν$ ^{[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".
Spectral exposure	$H e, λ$ ^{[nb 4]}	joule per square metre, per metre	J/m³	M⋅L⁻¹⋅T⁻²
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.
^ ^a ^b ^c ^d ^e Alternative symbols sometimes seen: $W$ or $E$ for radiant energy, $P$ or $F$ for radiant flux, $I$ for irradiance, $W$ for radiant exitance.
^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)
^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit wavelength are denoted with suffix "λ".
^ ^a ^b Directional quantities are denoted with suffix "Ω".

References

^ "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)
^ ^a ^b IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484 Cite error: The named reference "GoldBook" was defined multiple times with different content (see the help page).
^ ^a ^b ^c ^d "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.

[note-suffix-e-4] Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.

[note-alternative-symbol-radiometric-5] Alternative symbols sometimes seen: $W$ or $E$ for radiant energy, $P$ or $F$ for radiant flux, $I$ for irradiance, $W$ for radiant exitance.

[note-suffix-nu-6] ^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)

[note-suffix-lambda-7] ^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit wavelength are denoted with suffix "λ".

[note-suffix-omega-8] Directional quantities are denoted with suffix "Ω".

[1] "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)

[GoldBook-2] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484 Cite error: The named reference "GoldBook" was defined multiple times with different content (see the help page).

[ISO_9288-1989-3] "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.

[1]

[2]

[3]

[nb 1]

[nb 2]

[nb 3]

[nb 4]

[nb 5]

Mathematical definitions

Hemispherical transmittance

Spectral hemispherical transmittance

Directional transmittance

Spectral directional transmittance

Beer–Lambert law

SI radiometry units

See also

References