Transmittance Explained

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.

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^[1]

T = (\Phi ^{t} _{e})/(\Phi ^{i} _{e}) , 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} = (\Phi ^{t} _{e ,\nu})/(\Phi ^{i} _{e ,\nu}) , T _{λ} = (\Phi ^{t} _{e , λ})/(\Phi ^{i} _{e , λ}) , 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} = (L ^{t} _{e ,\Omega})/(L ^{i} _{e ,\Omega}) , 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} = (L ^{t} _{e ,\Omega,\nu})/(L ^{i} _{e ,\Omega,\nu}) , T _{λ,\Omega} = (L ^{t} _{e ,\Omega, λ})/(L ^{i} _{e ,\Omega, λ}) , 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} = (\int ^{inf ty} _{0} I( λ )T( λ )V( λ )d λ)/(\int ^{inf ty} _{0} I( λ )V( λ )d λ)

where:

I( λ ) is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).T( λ ) is the spectral transmittance of the filterV( λ ) 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

See 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 ^{N} _{i =1} \sigma _{i} \int ^{\ell} _{0} n _{i(z)dz}} = 10 ^{- \sum ^{N} _{i =1} \varepsilon _{i} \int ^{\ell} _{0} c _{i(z)dz}} , or equivalently that\tau = \sum ^{N} _{i =1} \tau _{i} = \sum ^{N} _{i =1} \sigma _{i} \int ^{\ell} _{0} n _{i(z)dz,} A = \sum ^{N} _{i =1} A _{i} = \sum ^{N} _{i =1} \varepsilon _{i} \int ^{\ell} _{0} c _{i(z)dz,} 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} = (N _{A})/(ln { 10) }\,\sigma_i,and number density and amount concentration byc _{i} = (n _{i})/(N _{A}) , where N_A is the Avogadro constant.

In case of uniform attenuation, these relations become

T = e ^{- \sum ^{N} _{i =1} \sigma _{i} n _{i\ell}} = 10 ^{- \sum ^{N} _{i =1} \varepsilon _{i} c _{i\ell}} , or equivalently\tau = \sum ^{N} _{i =1} \sigma _{i} n _{i\ell,} A = \sum ^{N} _{i =1} \varepsilon _{i} c _{i\ell.}

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

See also

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

Notes and References

1. Web site: Thermal insulation — Heat transfer by radiation — Physical quantities and definitions. ISO 9288:1989. ISO catalogue. 1989. 2015-03-15.