Transmittance

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]

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.

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

• 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]

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

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

where

• 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]

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

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]

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

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

where

• 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, internal transmittance is related to optical depth and to absorbance as

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

where

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

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

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

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

• σ_i is the attenuation cross section of the attenuating species i in the material sample;
• n_i is the number density of the attenuating species i in the material sample;
• ε_i is the molar attenuation coefficient of the attenuating species i in the material sample;
• c_i is the amount concentration of the attenuating species 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

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

and number density and amount concentration by

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

where N_A 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.

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⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	watt	W = 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	Φe,ν[nb 3]	watt per hertz	W/Hz	M⋅L^2⋅T^−2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm^−1.
	Φe,λ[nb 4]	watt per metre	W/m	M⋅L⋅T^−3
Radiant intensity	Ie,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	Ie,Ω,ν[nb 3]	watt per steradian per hertz	W⋅sr^−1⋅Hz^−1	M⋅L^2⋅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,Ω,λ[nb 4]	watt per steradian per metre	W⋅sr^−1⋅m^−1	M⋅L⋅T^−3
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]	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,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	Ee[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 Spectral flux density	Ee,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 = 10^4 Jy).
	Ee,λ[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiosity	Je[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	Je,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".
	Je,λ[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiant exitance	Me[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	Me,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".
	Me,λ[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiant exposure	He	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	He,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".
	He,λ[nb 4]	joule per square metre, per metre	J/m^3	M⋅L^−1⋅T^−2
Hemispherical emissivity	ε	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	εν  or ελ	N/A		1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ	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	εΩ,ν  or εΩ,λ	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	Aν  or Aλ	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	AΩ	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	AΩ,ν  or AΩ,λ	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	Rν  or Rλ	N/A		1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ	N/A		1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν  or RΩ,λ	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	Tν  or Tλ	N/A		1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ	N/A		1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν  or TΩ,λ	N/A		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. 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 "ν" (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 "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).

See also

• Opacity (optics)

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. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
4. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626