Transmittance

This article is about transmission through avolume.For transmission through asurface,seeSurface transmittance.

Inoptical physics,transmittanceof the surface of a material is its effectiveness in transmittingradiant energy.It is the fraction of incident electromagneticpowerthat is transmitted through a sample, in contrast to thetransmission coefficient,which is the ratio of the transmitted to incidentelectric field.

Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region). 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 theruby laser.

Internal transmittancerefers to energy loss byabsorption,whereas (total) transmittance is that due to absorption,scattering,reflection,etc.

Mathematical definitions

Hemispherical transmittance

Hemispherical transmittanceof a surface, denotedT,is defined as

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

where

• Φ_e^tis theradiant fluxtransmittedby that surface;
• Φ_eⁱis the radiant flux received by that surface.

Spectral hemispherical transmittance

Spectral hemispherical transmittance in frequencyandspectral hemispherical transmittance in wavelengthof a surface, denotedT_νandT_λrespectively, are defined as

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

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

where

• Φ_e,ν^tis thespectral radiant flux in frequencytransmittedby that surface;
• Φ_e,νⁱis the spectral radiant flux in frequency received by that surface;
• Φ_e,λ^tis thespectral radiant flux in wavelengthtransmittedby that surface;
• Φ_e,λⁱis the spectral radiant flux in wavelength received by that surface.

Directional transmittance

Directional transmittanceof a surface, denotedT_Ω,is defined as

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

where

• L_e,Ω^tis theradiancetransmittedby that surface;
• L_e,Ωⁱis the radiance received by that surface.

Spectral directional transmittance

Spectral directional transmittance in frequencyandspectral directional transmittance in wavelengthof a surface, denotedT_ν,ΩandT_λ,Ωrespectively, are defined as

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

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

where

• L_e,Ω,ν^tis thespectral radiance in frequencytransmittedby that surface;
• L_e,Ω,νⁱis the spectral radiance received by that surface;
• L_e,Ω,λ^tis thespectral radiance in wavelengthtransmittedby that surface;
• L_e,Ω,λⁱis the spectral radiance in wavelength received by that surface.

Luminous transmittance

In the field ofphotometry (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 astandard 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 theluminous 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

Main article:Beer–Lambert law

By definition, internal transmittance is related tooptical depthand toabsorbanceas

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

where

• τis the optical depth;
• Ais the absorbance.

TheBeer–Lambert lawstates that, forNattenuating 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_iis theattenuation cross sectionof the attenuating speciesiin the material sample;
• n_iis thenumber densityof the attenuating speciesiin the material sample;
• ε_iis themolar attenuation coefficientof the attenuating speciesiin the material sample;
• c_iis theamount concentrationof the attenuating speciesiin the material sample;
• ℓis 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_Ais theAvogadro constant.

In case ofuniformattenuation, these relations become

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

or equivalently

$\tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell,$

$A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell.$

Cases ofnon-uniformattenuation occur inatmospheric scienceapplications andradiation shieldingtheory for instance.

Other radiometric coefficients

Radiometry coefficients

Quantity		SI units	Notes
Name	Sym.
Hemispherical emissivity	ε	—	Radiant exitance of asurface,divided by that of ablack bodyat the same temperature as that surface.
Spectral hemispherical emissivity	εν ελ	—	Spectral exitance of asurface,divided by that of ablack bodyat the same temperature as that surface.
Directional emissivity	εΩ	—	Radianceemittedby asurface,divided by that emitted by ablack bodyat the same temperature as that surface.
Spectral directional emissivity	εΩ,ν εΩ,λ	—	Spectral radianceemittedby asurface,divided by that of ablack bodyat the same temperature as that surface.
Hemispherical absorptance	A	—	Radiant fluxabsorbedby asurface,divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	Aν Aλ	—	Spectral fluxabsorbedby asurface,divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ	—	Radianceabsorbedby asurface,divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν AΩ,λ	—	Spectral radianceabsorbedby asurface,divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	R	—	Radiant fluxreflectedby asurface,divided by that received by that surface.
Spectral hemispherical reflectance	Rν Rλ	—	Spectral fluxreflectedby asurface,divided by that received by that surface.
Directional reflectance	RΩ	—	Radiancereflectedby asurface,divided by that received by that surface.
Spectral directional reflectance	RΩ,ν RΩ,λ	—	Spectral radiancereflectedby asurface,divided by that received by that surface.
Hemispherical transmittance	T	—	Radiant fluxtransmittedby asurface,divided by that received by that surface.
Spectral hemispherical transmittance	Tν Tλ	—	Spectral fluxtransmittedby asurface,divided by that received by that surface.
Directional transmittance	TΩ	—	Radiancetransmittedby asurface,divided by that received by that surface.
Spectral directional transmittance	TΩ,ν TΩ,λ	—	Spectral radiancetransmittedby asurface,divided by that received by that surface.
Hemispherical attenuation coefficient	μ	m−1	Radiant fluxabsorbedandscatteredby avolumeper unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	μν μλ	m−1	Spectral radiant fluxabsorbedandscatteredby avolumeper unit length, divided by that received by that volume.
Directional attenuation coefficient	μΩ	m−1	Radianceabsorbedandscatteredby avolumeper unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μΩ,ν μΩ,λ	m−1	Spectral radianceabsorbedandscatteredby avolumeper unit length, divided by that received by that volume.

See also

• Opacity (optics)
• Photometry (optics)
• Radiometry