Transmittance

In optical physics, 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.

Luminous transmittance

In the field of photometry (optics), the luminous transmittance of a filter is a measure of the amount of luminous flux or intensity transmitted by an optical filter. It is generally defined in terms of a standard illuminant (e.g. Illuminant A, Iluminant C, or Illuminant E). The luminous transmittance with respect to the standard illuminant is defined as:

T_{lum}={\frac {\int _{0}^{\infty }I(\lambda )T(\lambda )V(\lambda )d\lambda }{\int _{0}^{\infty }I(\lambda )V(\lambda )d\lambda }}

where:

$I(\lambda )$ is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).
$T(\lambda )$ is the spectral transmittance of the filter
$V(\lambda )$ is the luminous efficiency function

The luminous transmittance is independent of the magnitude of the flux or intensity of the standard illuminant used to measure it, and is a dimensionless quantity.

Beer–Lambert law

By definition, internal 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 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

\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^[4]

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.

Other radiometric coefficients

Radiometry coefficients
v
t
e
Quantity		SI units	Notes
Name	Sym.	SI units	Notes
Hemispherical emissivity	$ε$	—	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	$ε ν$ $ε λ$	—	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	$ε Ω$	—	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	$ε Ω, ν$ $ε Ω, λ$	—	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	$A$	—	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	$A ν$ $A λ$	—	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	$A Ω$	—	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	$A Ω, ν$ $A Ω, λ$	—	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	$R$	—	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	$R ν$ $R λ$	—	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	$R Ω$	—	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	$R Ω, ν$ $R Ω, λ$	—	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	$T$	—	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	$T ν$ $T λ$	—	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	$T Ω$	—	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	$T Ω,ν$ $T Ω, λ$	—	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	$μ$	m⁻¹	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	$μ ν$ $μ λ$	m⁻¹	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	$μ Ω$	m⁻¹	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	$μ Ω, ν$ $μ Ω, λ$	m⁻¹	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

References

^ "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001.{{cite web}}: CS1 maint: unfit URL (link)
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484
^ ^a ^b ^c ^d "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626

[1] "Electronic warfare and radar systems engineering handbook". Archived from the original on September 13, 2001.{{cite web}}: CS1 maint: unfit URL (link)

[GoldBook-2] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Transmittance". doi:10.1351/goldbook.T06484

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

[GoldBook2-4] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626

[1]

[2]

[3]

[4]

Mathematical definitions

Hemispherical transmittance

Spectral hemispherical transmittance

Directional transmittance

Spectral directional transmittance

Luminous transmittance

Beer–Lambert law

Other radiometric coefficients

See also

References