Transmittance

This article is about transmission through a volume. For transmission through a surface, see Surface 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.

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 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

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

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

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

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

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

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

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

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

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

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

Main article: 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

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

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

Quantity		SI units	Notes
Name	Sym.
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	εν ελ	—	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ	—	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 εΩ,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	Aν Aλ	—	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ	—	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 AΩ,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	Rν Rλ	—	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ	—	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν RΩ,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	Tν Tλ	—	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ	—	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν TΩ,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	μν μλ	m−1	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	μΩ	m−1	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μΩ,ν μΩ,λ	m−1	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

See also

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