Transmittance

Hemispherical transmittance of a surface, denoted T, is defined as{{cite web |date=August 1, 2022 |year= |title=Thermal insulation — Heat transfer by radiation — Vocabulary |url=http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 |access-date=February 12, 2025 |work=ISO 9288:2022 |publisher=ISO catalogue}}

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

where

• Φ_e^t is the radiant flux transmitted by that surface into the hemisphere on the opposite side from the incident radiation;
• Φ_eⁱ is the radiant flux received by that surface.

Hemispheric transmittance may be calculated as an integral over the directional transmittance described below.

Spectral hemispherical transmittance in frequency and spectral hemispherical transmittance in wavelength of a surface, denoted T_ν and T_λ respectively, are defined as

:$T\_\backslash nu\; =\; \backslash frac\{\backslash Phi\_\{\backslash mathrm\{e\},\backslash nu\}^\backslash mathrm\{t\}\}\{\backslash Phi\_\{\backslash mathrm\{e\},\backslash nu\}^\backslash mathrm\{i\}\},$

:$T\_\backslash lambda\; =\; \backslash frac\{\backslash Phi\_\{\backslash mathrm\{e\},\backslash lambda\}^\backslash mathrm\{t\}\}\{\backslash Phi\_\{\backslash mathrm\{e\},\backslash lambda\}^\backslash mathrm\{i\}\},$

where

• Φ_e,ν^t is the spectral radiant flux in frequency transmitted by that surface into the hemisphere on the opposite side from the incident radiation;
• 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 into the hemisphere on the opposite side from the incident radiation;
• Φ_e,λⁱ is the spectral radiant flux in wavelength received by that surface.

Directional transmittance of a surface, denoted T_Ω, is defined as

:$T\_\backslash Omega\; =\; \backslash frac\{L\_\{\backslash mathrm\{e\},\backslash Omega\}^\backslash mathrm\{t\}\}\{L\_\{\backslash mathrm\{e\},\backslash Omega\}^\backslash mathrm\{i\}\},$

where

• L_e,Ω^t is the radiance transmitted by that surface into the solid angle Ω;
• L_e,Ωⁱ is the radiance received by that surface.

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

:$T\_\{\backslash nu,\backslash Omega\}\; =\; \backslash frac\{L\_\{\backslash mathrm\{e\},\backslash Omega,\backslash nu\}^\backslash mathrm\{t\}\}\{L\_\{\backslash mathrm\{e\},\backslash Omega,\backslash nu\}^\backslash mathrm\{i\}\},$

:$T\_\{\backslash lambda,\backslash Omega\}\; =\; \backslash frac\{L\_\{\backslash mathrm\{e\},\backslash Omega,\backslash lambda\}^\backslash mathrm\{t\}\}\{L\_\{\backslash mathrm\{e\},\backslash Omega,\backslash lambda\}^\backslash mathrm\{i\}\},$

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.

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:

:$T\_\{lum\}\; =\; \backslash frac\{\backslash int\_0^\backslash infty\; I(\backslash lambda)T(\backslash lambda)V(\backslash lambda)d\backslash lambda\}\{\backslash int\_0^\backslash infty\; I(\backslash lambda)V(\backslash lambda)d\backslash lambda\}$

where:

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

By definition, internal transmittance is related to optical depth and to absorbance as

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

where

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

{{main|Beer–Lambert law}}

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

:$\backslash tau\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash tau\_i\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash sigma\_i\; \backslash int\_0^\backslash ell\; n\_i(z)\backslash ,\backslash mathrm\{d\}z,$

:$A\; =\; \backslash sum\_\{i\; =\; 1\}^N\; A\_i\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash varepsilon\_i\; \backslash int\_0^\backslash ell\; c\_i(z)\backslash ,\backslash mathrm\{d\}z,$

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

:$\backslash varepsilon\_i\; =\; \backslash frac\{\backslash mathrm\{N\_A\}\}\{\backslash ln\{10\}\}\backslash ,\backslash sigma\_i,$

and number density and amount concentration by

:$c\_i\; =\; \backslash frac\{n\_i\}\{\backslash mathrm\{N\_A\}\},$

where N_A is the Avogadro constant.

In case of uniform attenuation, these relations become{{GoldBookRef|title=Beer–Lambert law|file=B00626|accessdate=2015-03-15}}

:$\backslash tau\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash sigma\_i\; n\_i\backslash ell,$

:$A\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash varepsilon\_i\; c\_i\backslash ell.$

Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

{{Radiometry coefficients}}

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

{{reflist}}

Category:Physical quantities

Category:Radiometry

Category:Spectroscopy