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}}

[[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]].]]

Electromagnetic radiation can be affected in several ways by the medium in which it propagates. It can be [[Scattering|scattered]], [[Absorption (electromagnetic radiation)|absorbed]], and [[Fresnel equations|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=[[International Organization for Standardization|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 [[Radiant flux|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 [[Radiant flux|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 [[Radiance|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 [[Radiance|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 [[Cross section (physics)|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]]