Transmittance

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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.

Video Transmittance

Mathematical definitions

Hemispherical transmittance

Hemispherical transmittance of a surface, denoted T, is defined as

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

where

• ?_e^t is the radiant flux transmitted by that surface;
• ?_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_? and T_? respectively, are defined as

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

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

where

• ?_e,?^t is the spectral radiant flux in frequency transmitted by that surface;
• ?_e,?ⁱ is the spectral radiant flux in frequency received by that surface;
• ?_e,?^t is the spectral radiant flux in wavelength transmitted by that surface;
• ?_e,?ⁱ is the spectral radiant flux in wavelength received by that surface.

Directional transmittance

Directional transmittance of a surface, denoted T_?, is defined as

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

where

• L_e,?^t is the radiance transmitted by that surface;
• L_e,?ⁱ 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_?,? and T_?,? respectively, are defined as

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

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

where

• L_e,?,?^t is the spectral radiance in frequency transmitted by that surface;
• L_e,?,?ⁱ is the spectral radiance received by that surface;
• L_e,?,?^t is the spectral radiance in wavelength transmitted by that surface;
• L_e,?,?ⁱ is the spectral radiance in wavelength received by that surface.

Maps Transmittance

Beer-Lambert law

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

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

where

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

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

${\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\mathrm {d} z}=10^{-\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\mathrm {d} z},}$

or equivalently that

${\displaystyle \tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,}$

${\displaystyle A=\sum _{i=1}^{N}A_{i}=\sum _{i=1}^{N}\varepsilon _{i}\int _{0}^{\ell }c_{i}(z)\,\mathrm {d} z,}$

where

• ?_i is the attenuation cross section of the attenuating specie i in the material sample;
• n_i is the number density of the attenuating specie i in the material sample;
• ?_i is the molar attenuation coefficient of the attenuating specie i in the material sample;
• c_i is the amount concentration of the attenuating specie 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

${\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},}$

and number density and amount concentration by

${\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}$

where N_A is the Avogadro constant.

In case of uniform attenuation, these relations become

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

or equivalently

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

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

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

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SI radiometry units

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See also

• Opacity (optics)

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References

Source of article : Wikipedia