Transmittance

<h2 id="surface-transmittance">Surface Transmittance</h2>
<h3>Hemispherical transmittance</h3>
Hemispherical transmittance of a surface, denoted T, is defined as<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

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

where

<ul><li>Φet is the <a href="/facts/Radiant_flux/ak2d0btz">radiant flux</a> transmitted by that surface into the hemisphere on the opposite side from the incident radiation;</li>
<li>Φei is the radiant flux received by that surface.</li></ul>
Hemispheric transmittance may be calculated as an integral over the directional transmittance described below.

<h3>Spectral hemispherical transmittance</h3>
Spectral hemispherical transmittance in frequency and spectral hemispherical transmittance in wavelength of a surface, denoted Tν and Tλ respectively, are defined as<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

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

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

where

<ul><li>Φe,νt is the <a href="/facts/Radiant_flux/ak2d0btz">spectral radiant flux in frequency</a> transmitted by that surface into the hemisphere on the opposite side from the incident radiation;</li>
<li>Φe,νi is the spectral radiant flux in frequency received by that surface;</li>
<li>Φe,λt is the <a href="/facts/Radiant_flux/ak2d0btz">spectral radiant flux in wavelength</a> transmitted by that surface into the hemisphere on the opposite side from the incident radiation;</li>
<li>Φe,λi is the spectral radiant flux in wavelength received by that surface.</li></ul>
<h3>Directional transmittance</h3>
Directional transmittance of a surface, denoted TΩ, is defined as<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

T
          
            Ω
          
        
        =
        
          
            
              L
              
                
                  e
                
                ,
                Ω
              
              
                
                  t
                
              
            
            
              L
              
                
                  e
                
                ,
                Ω
              
              
                
                  i
                
              
            
          
        
        ,
      
    
    {\displaystyle T_{\Omega }={\frac {L_{\mathrm {e} ,\Omega }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega }^{\mathrm {i} }}},}

where

<ul><li>Le,Ωt is the <a href="/facts/Radiance/5kS1FhpD">radiance</a> transmitted by that surface into the <a href="/facts/Solid_angle/se8IXufg">solid angle</a> Ω;</li>
<li>Le,Ωi is the radiance received by that surface.</li></ul>
<h3>Spectral directional transmittance</h3>
Spectral directional transmittance in frequency and spectral directional transmittance in wavelength of a surface, denoted Tν,Ω and Tλ,Ω respectively, are defined as<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

T
          
            ν
            ,
            Ω
          
        
        =
        
          
            
              L
              
                
                  e
                
                ,
                Ω
                ,
                ν
              
              
                
                  t
                
              
            
            
              L
              
                
                  e
                
                ,
                Ω
                ,
                ν
              
              
                
                  i
                
              
            
          
        
        ,
      
    
    {\displaystyle T_{\nu ,\Omega }={\frac {L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega ,\nu }^{\mathrm {i} }}},}

T
          
            λ
            ,
            Ω
          
        
        =
        
          
            
              L
              
                
                  e
                
                ,
                Ω
                ,
                λ
              
              
                
                  t
                
              
            
            
              L
              
                
                  e
                
                ,
                Ω
                ,
                λ
              
              
                
                  i
                
              
            
          
        
        ,
      
    
    {\displaystyle T_{\lambda ,\Omega }={\frac {L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {t} }}{L_{\mathrm {e} ,\Omega ,\lambda }^{\mathrm {i} }}},}

where

<ul><li>Le,Ω,νt is the <a href="/facts/Radiance/5kS1FhpD">spectral radiance in frequency</a> transmitted by that surface;</li>
<li>Le,Ω,νi is the spectral radiance received by that surface;</li>
<li>Le,Ω,λt is the <a href="/facts/Radiance/5kS1FhpD">spectral radiance in wavelength</a> transmitted by that surface;</li>
<li>Le,Ω,λi is the spectral radiance in wavelength received by that surface.</li></ul>
<h3>Luminous transmittance</h3>
In the field of <a href="/facts/Photometry_(optics)/eo6J7qj3">photometry (optics)</a>, the luminous transmittance of a filter is a measure of the amount of luminous flux or intensity transmitted by an <a href="/facts/Optical_filter/WdCmGqaq">optical filter</a>. It is generally defined in terms of a <a href="/facts/Standard_illuminant/urST5Mz7">standard illuminant</a> (e.g. Illuminant A, Iluminant C, or Illuminant E). The luminous transmittance with respect to the standard illuminant is defined as:

T
          
            l
            u
            m
          
        
        =
        
          
            
              
                ∫
                
                  0
                
                
                  ∞
                
              
              I
              (
              λ
              )
              T
              (
              λ
              )
              V
              (
              λ
              )
              d
              λ
            
            
              
                ∫
                
                  0
                
                
                  ∞
                
              
              I
              (
              λ
              )
              V
              (
              λ
              )
              d
              λ
            
          
        
      
    
    {\displaystyle T_{lum}={\frac {\int _{0}^{\infty }I(\lambda )T(\lambda )V(\lambda )d\lambda }{\int _{0}^{\infty }I(\lambda )V(\lambda )d\lambda }}}

where:

<ul><li>
 
 
 
 I
 (
 λ
 )
 
 
 {\displaystyle I(\lambda )}
 
 is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).</li>
<li>
 
 
 
 T
 (
 λ
 )
 
 
 {\displaystyle T(\lambda )}
 
 is the spectral transmittance of the filter</li>
<li>
 
 
 
 V
 (
 λ
 )
 
 
 {\displaystyle V(\lambda )}
 
 is the <a href="/facts/Luminous_efficiency_function/7H2G30dE">luminous efficiency function</a></li></ul>
The luminous transmittance is independent of the magnitude of the flux or intensity of the standard illuminant used to measure it, and is a <a href="/facts/Dimensionless_quantity/TINrpQIq">dimensionless quantity</a>.

<h2 id="internal-transmittance">Internal Transmittance</h2>
<h3>Optical Depth</h3>
By definition, internal transmittance is related to <a href="/facts/Optical_depth/XQwtIdFH">optical depth</a> and to <a href="/facts/Absorbance/Rnv6VMX1">absorbance</a> as

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

where

<ul><li>τ is the optical depth;</li>
<li>A is the absorbance.</li></ul>
<h3>Beer–Lambert law</h3>
Main article: <a href="/facts/Beer%25E2%2580%2593Lambert_law/nl64rI1C">Beer–Lambert law</a>
The <a href="/facts/Beer%25E2%2580%2593Lambert_law/nl64rI1C">Beer–Lambert law</a> states that, for N attenuating species in the material sample,

τ
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          τ
          
            i
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          σ
          
            i
          
        
        
          ∫
          
            0
          
          
            ℓ
          
        
        
          n
          
            i
          
        
        (
        z
        )
        
        
          d
        
        z
        ,
      
    
    {\displaystyle \tau =\sum _{i=1}^{N}\tau _{i}=\sum _{i=1}^{N}\sigma _{i}\int _{0}^{\ell }n_{i}(z)\,\mathrm {d} z,}

A
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          A
          
            i
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          ε
          
            i
          
        
        
          ∫
          
            0
          
          
            ℓ
          
        
        
          c
          
            i
          
        
        (
        z
        )
        
        
          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

<ul><li>σi is the <a href="/facts/Cross_section_(physics)/lo014h2D">attenuation cross section</a> of the attenuating species i in the material sample;</li>
<li>ni is the <a href="/facts/Number_density/qtyDnLhz">number density</a> of the attenuating species i in the material sample;</li>
<li>εi is the <a href="/facts/Molar_attenuation_coefficient/TwglQEAW">molar attenuation coefficient</a> of the attenuating species i in the material sample;</li>
<li>ci is the <a href="/facts/Amount_concentration/P2WL5b9q">amount concentration</a> of the attenuating species i in the material sample;</li>
<li>ℓ is the path length of the beam of light through the material sample.</li></ul>
Attenuation cross section and molar attenuation coefficient are related by

ε
          
            i
          
        
        =
        
          
            
              
                N
                
                  A
                
              
            
            
              ln
              ⁡
              
                10
              
            
          
        
        
        
          σ
          
            i
          
        
        ,
      
    
    {\displaystyle \varepsilon _{i}={\frac {\mathrm {N_{A}} }{\ln {10}}}\,\sigma _{i},}

and number density and amount concentration by

c
          
            i
          
        
        =
        
          
            
              n
              
                i
              
            
            
              
                N
                
                  A
                
              
            
          
        
        ,
      
    
    {\displaystyle c_{i}={\frac {n_{i}}{\mathrm {N_{A}} }},}

where NA is the <a href="/facts/Avogadro_constant/SwajCq22">Avogadro constant</a>.
In case of uniform attenuation, these relations become<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

τ
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          σ
          
            i
          
        
        
          n
          
            i
          
        
        ℓ
        ,
      
    
    {\displaystyle \tau =\sum _{i=1}^{N}\sigma _{i}n_{i}\ell ,}

A
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          ε
          
            i
          
        
        
          c
          
            i
          
        
        ℓ
        .
      
    
    {\displaystyle A=\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell .}

Cases of non-uniform attenuation occur in <a href="/facts/Atmospheric_science/HKjTA3F4">atmospheric science</a> applications and <a href="/facts/Radiation_shielding/5LaNNgXe">radiation shielding</a> theory for instance.

<h2 id="other-radiometric-coefficients">Other radiometric coefficients</h2>
Radiometry coefficients<ul><li>v</li><li>t</li><li>e</li></ul><table><tbody><tr><th colspan="2">Quantity</th><th rowspan="2">SI units</th><th rowspan="2">Notes</th></tr><tr><th>Name</th><th>Sym.</th></tr><tr><td><a href="/facts/Emissivity/LUxfMRRI">Hemispherical emissivity</a></td><td>ε</td><td>—</td><td>Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.</td></tr><tr><td><a href="/facts/Emissivity/LUxfMRRI">Spectral hemispherical emissivity</a></td><td>εν ελ</td><td>—</td><td>Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.</td></tr><tr><td><a href="/facts/Emissivity/LUxfMRRI">Directional emissivity</a></td><td>εΩ</td><td>—</td><td>Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.</td></tr><tr><td><a href="/facts/Emissivity/LUxfMRRI">Spectral directional emissivity</a></td><td>εΩ,ν εΩ,λ</td><td>—</td><td>Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.</td></tr><tr><td><a href="/facts/Absorptance/n6C9pnDR">Hemispherical absorptance</a></td><td>A</td><td>—</td><td>Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "<a href="/facts/Absorbance/Rnv6VMX1">absorbance</a>".</td></tr><tr><td><a href="/facts/Absorptance/n6C9pnDR">Spectral hemispherical absorptance</a></td><td>AνAλ</td><td>—</td><td>Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "<a href="/facts/Absorbance/Rnv6VMX1">spectral absorbance</a>".</td></tr><tr><td><a href="/facts/Absorptance/n6C9pnDR">Directional absorptance</a></td><td>AΩ</td><td>—</td><td>Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "<a href="/facts/Absorbance/Rnv6VMX1">absorbance</a>".</td></tr><tr><td><a href="/facts/Absorptance/n6C9pnDR">Spectral directional absorptance</a></td><td>AΩ,ν AΩ,λ</td><td>—</td><td>Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "<a href="/facts/Absorbance/Rnv6VMX1">spectral absorbance</a>".</td></tr><tr><td><a href="/facts/Reflectance/TT4sBwIw">Hemispherical reflectance</a></td><td>R</td><td>—</td><td>Radiant flux reflected by a surface, divided by that received by that surface.</td></tr><tr><td><a href="/facts/Reflectance/TT4sBwIw">Spectral hemispherical reflectance</a></td><td>RνRλ</td><td>—</td><td>Spectral flux reflected by a surface, divided by that received by that surface.</td></tr><tr><td><a href="/facts/Reflectance/TT4sBwIw">Directional reflectance</a></td><td>RΩ</td><td>—</td><td>Radiance reflected by a surface, divided by that received by that surface.</td></tr><tr><td><a href="/facts/Reflectance/TT4sBwIw">Spectral directional reflectance</a></td><td>RΩ,νRΩ,λ</td><td>—</td><td>Spectral radiance reflected by a surface, divided by that received by that surface.</td></tr><tr><td>Hemispherical transmittance</td><td>T</td><td>—</td><td>Radiant flux transmitted by a surface, divided by that received by that surface.</td></tr><tr><td>Spectral hemispherical transmittance</td><td>TνTλ</td><td>—</td><td>Spectral flux transmitted by a surface, divided by that received by that surface.</td></tr><tr><td>Directional transmittance</td><td>TΩ</td><td>—</td><td>Radiance transmitted by a surface, divided by that received by that surface.</td></tr><tr><td>Spectral directional transmittance</td><td>TΩ,νTΩ,λ</td><td>—</td><td>Spectral radiance transmitted by a surface, divided by that received by that surface.</td></tr><tr><td><a href="/facts/Attenuation_coefficient/ppCIsSGw">Hemispherical attenuation coefficient</a></td><td>μ</td><td>m−1</td><td>Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.</td></tr><tr><td><a href="/facts/Attenuation_coefficient/ppCIsSGw">Spectral hemispherical attenuation coefficient</a></td><td>μνμλ</td><td>m−1</td><td>Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.</td></tr><tr><td><a href="/facts/Attenuation_coefficient/ppCIsSGw">Directional attenuation coefficient</a></td><td>μΩ</td><td>m−1</td><td>Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.</td></tr><tr><td><a href="/facts/Attenuation_coefficient/ppCIsSGw">Spectral directional attenuation coefficient</a></td><td>μΩ,νμΩ,λ</td><td>m−1</td><td>Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Opacity_(optics)/l90q0sAV">Opacity (optics)</a></li>
<li><a href="/facts/Photometry_(optics)/eo6J7qj3">Photometry (optics)</a></li>
<li><a href="/facts/Radiometry/w9j9PXSB">Radiometry</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Thermal insulation — Heat transfer by radiation — Vocabulary". ISO 9288:2022. ISO catalogue. August 1, 2022. Retrieved February 12, 2025. <a href="http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943" target="_blank">http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Thermal insulation — Heat transfer by radiation — Vocabulary". ISO 9288:2022. ISO catalogue. August 1, 2022. Retrieved February 12, 2025. <a href="http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943" target="_blank">http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Thermal insulation — Heat transfer by radiation — Vocabulary". ISO 9288:2022. ISO catalogue. August 1, 2022. Retrieved February 12, 2025. <a href="http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943" target="_blank">http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Thermal insulation — Heat transfer by radiation — Vocabulary". ISO 9288:2022. ISO catalogue. August 1, 2022. Retrieved February 12, 2025. <a href="http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943" target="_blank">http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Beer–Lambert law". doi:10.1351/goldbook.B00626 <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" target="_blank">/wiki/International_Union_of_Pure_and_Applied_Chemistry</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

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