View source for Transmittance

{{short description|Effectiveness of a material in transmitting radiant energy}}
{{About|transmission through a ''volume''|transmission through a ''surface''|Surface transmittance}}
<!-- Deleted image removed: [[File:Beer lambert1.png|thumb|240px|Diagram of Beer-Lambert Law of transmittance of a beam of light as it travels through a [[cuvette]] of width ''l''.]] -->
[[File:Atmosfaerisk spredning.png|thumb|Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region<ref>{{cite web|url=http://ewhdbks.mugu.navy.mil/EO-IR.htm#transmission |title=Electronic warfare and radar systems engineering handbook |url-status=unfit |archive-url=https://web.archive.org/web/20010913091738/http://ewhdbks.mugu.navy.mil/EO-IR.htm#transmission |archive-date=September 13, 2001 }}</ref>). Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.]]
[[File:Ruby transmittance.svg|thumb|240px|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]].]]

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 (physics)|power]] that is transmitted through a sample, in contrast to the [[transmission coefficient]], which is the ratio of the transmitted to incident [[electric field]].<ref name=GoldBook>{{GoldBookRef|title=Transmittance|file=T06484|accessdate=2015-03-15}}</ref>

'''Internal transmittance''' refers to energy loss by [[Absorption (electromagnetic radiation)|absorption]], whereas (total) transmittance is that due to absorption, [[scattering]], [[Reflection (physics)|reflection]], etc.

==Mathematical definitions==
===Hemispherical transmittance===
'''Hemispherical transmittance''' of a surface, denoted ''T'', is defined as<ref name="ISO_9288-1989">{{cite web|url=http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943|title=Thermal insulation — Heat transfer by radiation — Physical quantities and definitions|work=ISO 9288:1989|publisher=[[International Organization for Standardization|ISO]] catalogue|year=1989|access-date=2015-03-15}}</ref>
:<math>T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}},</math>
where
*Φ<sub>e</sub><sup>t</sup> is the [[radiant flux]] ''transmitted'' by that surface;
*Φ<sub>e</sub><sup>i</sup> 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''<sub>ν</sub> and ''T''<sub>λ</sub> respectively, are defined as<ref name="ISO_9288-1989" />
:<math>T_\nu = \frac{\Phi_{\mathrm{e},\nu}^\mathrm{t}}{\Phi_{\mathrm{e},\nu}^\mathrm{i}},</math>
:<math>T_\lambda = \frac{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}{\Phi_{\mathrm{e},\lambda}^\mathrm{i}},</math>
where
*Φ<sub>e,ν</sub><sup>t</sup> is the [[Radiant flux|spectral radiant flux in frequency]] ''transmitted'' by that surface;
*Φ<sub>e,ν</sub><sup>i</sup> is the spectral radiant flux in frequency received by that surface;
*Φ<sub>e,λ</sub><sup>t</sup> is the [[Radiant flux|spectral radiant flux in wavelength]] ''transmitted'' by that surface;
*Φ<sub>e,λ</sub><sup>i</sup> is the spectral radiant flux in wavelength received by that surface.

===Directional transmittance===
'''Directional transmittance''' of a surface, denoted ''T''<sub>Ω</sub>, is defined as<ref name="ISO_9288-1989" />
:<math>T_\Omega = \frac{L_{\mathrm{e},\Omega}^\mathrm{t}}{L_{\mathrm{e},\Omega}^\mathrm{i}},</math>
where
*''L''<sub>e,Ω</sub><sup>t</sup> is the [[radiance]] ''transmitted'' by that surface;
*''L''<sub>e,Ω</sub><sup>i</sup> 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''<sub>ν,Ω</sub> and ''T''<sub>λ,Ω</sub> respectively, are defined as<ref name="ISO_9288-1989" />
:<math>T_{\nu,\Omega} = \frac{L_{\mathrm{e},\Omega,\nu}^\mathrm{t}}{L_{\mathrm{e},\Omega,\nu}^\mathrm{i}},</math>
:<math>T_{\lambda,\Omega} = \frac{L_{\mathrm{e},\Omega,\lambda}^\mathrm{t}}{L_{\mathrm{e},\Omega,\lambda}^\mathrm{i}},</math>
where
*''L''<sub>e,Ω,ν</sub><sup>t</sup> is the [[Radiance|spectral radiance in frequency]] ''transmitted'' by that surface;
*''L''<sub>e,Ω,ν</sub><sup>i</sup> is the spectral radiance received by that surface;
*''L''<sub>e,Ω,λ</sub><sup>t</sup> is the [[Radiance|spectral radiance in wavelength]] ''transmitted'' by that surface;
*''L''<sub>e,Ω,λ</sub><sup>i</sup> is the spectral radiance in wavelength received by that surface.

==Beer–Lambert law==
{{main|Beer–Lambert law}}
By definition, internal transmittance is related to [[optical depth]] and to [[absorbance]] as
:<math>T = e^{-\tau} = 10^{-A},</math>
where
*''τ'' is the optical depth;
*''A'' is the absorbance.

The [[Beer–Lambert law]] states that, for ''N'' attenuating species in the material sample,
:<math>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},</math>
or equivalently that
:<math>\tau = \sum_{i = 1}^N \tau_i = \sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\,\mathrm{d}z,</math>
:<math>A = \sum_{i = 1}^N A_i = \sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrm{d}z,</math>
where
*''σ''<sub>''i''</sub> is the [[Cross section (physics)|attenuation cross section]] of the attenuating species ''i'' in the material sample;
*''n''<sub>''i''</sub> is the [[number density]] of the attenuating species ''i'' in the material sample;
*''ε''<sub>''i''</sub> is the [[molar attenuation coefficient]] of the attenuating species ''i'' in the material sample;
*''c''<sub>''i''</sub> 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
:<math>\varepsilon_i = \frac{\mathrm{N_A}}{\ln{10}}\,\sigma_i,</math>
and number density and amount concentration by
:<math>c_i = \frac{n_i}{\mathrm{N_A}},</math>
where N<sub>A</sub> is the [[Avogadro constant]].

In case of ''uniform'' attenuation, these relations become<ref name=GoldBook2>{{GoldBookRef|title=Beer–Lambert law|file=B00626|accessdate=2015-03-15}}</ref>
:<math>T = e^{-\sum_{i = 1}^N \sigma_i n_i\ell} = 10^{-\sum_{i = 1}^N \varepsilon_i c_i\ell},</math>
or equivalently
:<math>\tau = \sum_{i = 1}^N \sigma_i n_i\ell,</math>
:<math>A = \sum_{i = 1}^N \varepsilon_i c_i\ell.</math>

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

==Other radiometric coefficients==
{{Radiometry coefficients}}

==See also==
*[[Opacity (optics)]]
*[[Photometry (optics)]]
*[[Radiometry]]

==References==
{{reflist}}

[[Category:Physical quantities]]
[[Category:Radiometry]]
[[Category:Spectroscopy]]