Transmittance

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
:$T = \frac,$
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
:$T_\nu = \frac,$
:$T_\lambda = \frac,$
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
:$T_\Omega = \frac,$
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
:$T_ = \frac,$
:$T_ = \frac,$
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.

Beer–Lambert law

By definition, internal transmittance is related to optical depth and to absorbance as
:$T = e^ = 10^,$
where
*''τ'' is the optical depth;
*''A'' is the absorbance.

The Beer–Lambert law states that, for ''N'' attenuating species in the material sample,
:$T = e^ = 10^,$
or equivalently that
:$\tau = \sum_^N \tau_i = \sum_^N \sigma_i \int_0^\ell n_i(z)\,\mathrmz,$
:$A = \sum_^N A_i = \sum_^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrmz,$
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
:$\varepsilon_i = \frac\,\sigma_i,$
and number density and amount concentration by
:$c_i = \frac,$
where N_A is the Avogadro constant.

In case of ''uniform'' attenuation, these relations become
:$T = e^ = 10^,$
or equivalently
:$\tau = \sum_^N \sigma_i n_i\ell,$
:$A = \sum_^N \varepsilon_i c_i\ell.$

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

Other radiometric coefficients

See also

*Opacity (optics)

References

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Physical quantities
Radiometry
Spectroscopy