Transmittance

{{short description|Effectiveness of a material in transmitting radiant energy}} <!-- 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''.]] --> {{About|several kinds of transmission of electromagnetic radiation into and through substances|the reduction of transmittance by scattering|Scattering}}

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.]]

Electromagnetic radiation can be affected in several ways by the medium in which it propagates. It can be scattered, absorbed, and reflected and refracted at discontinuities in the medium. This page is an overview of the last 3. The '''transmittance''' of a material and any surfaces is its effectiveness in transmitting radiant energy; the fraction of the initial (incident) radiation which propagates to a location of interest (often an observation location). This may be described by the transmission coefficient.

==Surface transmittance== ===Hemispherical transmittance=== '''Hemispherical transmittance''' of a surface, denoted ''T'', is defined as<ref name="ISO_9288-1989">{{cite web |date=August 1, 2022 |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}}</ref> :<math>T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}},</math> 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=== '''Spectral hemispherical transmittance in frequency''' and '''spectral hemispherical transmittance in wavelength''' of a surface, denoted ''T''_ν and ''T''_λ 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_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=== '''Directional transmittance''' of a surface, denoted ''T''_Ω, 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''_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=== '''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<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''_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:

:<math>T_{lum} = \frac{\int_0^\infty I(\lambda)T(\lambda)V(\lambda)d\lambda}{\int_0^\infty I(\lambda)V(\lambda)d\lambda}</math>

where: *<math>I(\lambda)</math> is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude). *<math>T(\lambda)</math> is the spectral transmittance of the filter *<math>V(\lambda)</math> 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.

== Internal transmittance ==

=== Optical depth === 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.

=== Beer–Lambert law === {{main|Beer–Lambert law}}

The Beer–Lambert law states that, for ''N'' attenuating species in the material sample, :<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''_''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 :<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_A 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>\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