• Source: Transmittance

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 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
=



Φ


e




t




Φ


e




i





,


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


where

Φet is the radiant flux transmitted by that surface;
Φei 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 Tν and Tλ respectively, are defined as





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

Φe,νt is the spectral radiant flux in frequency transmitted by that surface;
Φe,νi is the spectral radiant flux in frequency received by that surface;
Φe,λt is the spectral radiant flux in wavelength transmitted by that surface;
Φe,λi is the spectral radiant flux in wavelength received by that surface.




= Directional transmittance

=
Directional transmittance of a surface, denoted TΩ, is defined as





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

Le,Ωt is the radiance transmitted by that surface;
Le,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 Tν,Ω and Tλ,Ω respectively, are defined as





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

Le,Ω,νt is the spectral radiance in frequency transmitted by that surface;
Le,Ω,νi is the spectral radiance received by that surface;
Le,Ω,λt is the spectral radiance in wavelength transmitted by that surface;
Le,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:





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:




I
(
λ
)


{\displaystyle I(\lambda )}

is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).




T
(
λ
)


{\displaystyle T(\lambda )}

is the spectral transmittance of the filter




V
(
λ
)


{\displaystyle V(\lambda )}

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.




Beer–Lambert law



By definition, internal transmittance is related to optical depth and to absorbance as




T
=

e

−
τ


=

10

−
A


,


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


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

−

∑

i
=
1


N



σ

i



∫

0


ℓ



n

i


(
z
)

d

z


=

10

−

∑

i
=
1


N



ε

i



∫

0


ℓ



c

i


(
z
)

d

z


,


{\displaystyle 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},}


or equivalently that




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

σi is the attenuation cross section of the attenuating species i in the material sample;
ni is the number density of the attenuating species i in the material sample;
εi is the molar attenuation coefficient of the attenuating species i in the material sample;
ci 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





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 Avogadro constant.
In case of uniform attenuation, these relations become




T
=

e

−

∑

i
=
1


N



σ

i



n

i


ℓ


=

10

−

∑

i
=
1


N



ε

i



c

i


ℓ


,


{\displaystyle T=e^{-\sum _{i=1}^{N}\sigma _{i}n_{i}\ell }=10^{-\sum _{i=1}^{N}\varepsilon _{i}c_{i}\ell },}


or equivalently




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 atmospheric science applications and radiation shielding theory for instance.




Other radiometric coefficients






See also


Opacity (optics)
Photometry (optics)
Radiometry




References

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