# Transmittance

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Effectiveness of a material in transmitting radiant energy

This article is about several kinds of transmission of electromagnetic radiation into and through substances. For the reduction of transmittance by scattering, see [Scattering](/source/Scattering).

Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region[1]). Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.

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](/source/Ruby_laser).

Electromagnetic radiation can be affected in several ways by the medium in which it propagates. It can be [scattered](/source/Scattering), [absorbed](/source/Absorption_(electromagnetic_radiation)), and [reflected and refracted](/source/Fresnel_equations) 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](/source/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](/source/Transmission_coefficient).

## Surface transmittance

### Hemispherical transmittance

**Hemispherical transmittance** of a surface, denoted *T*, is defined as[2]

- 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](/source/Radiant_flux) *transmitted* by that surface into the hemisphere on the opposite side from the incident radiation;

- Φei 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[2]

- 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](/source/Radiant_flux) *transmitted* by that surface into the hemisphere on the opposite side from the incident radiation;

- Φe,νi is the spectral radiant flux in frequency received by that surface;

- Φe,λt is the [spectral radiant flux in wavelength](/source/Radiant_flux) *transmitted* by that surface into the hemisphere on the opposite side from the incident radiation;

- Φe,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[2]

- 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

- *L*e,Ωt is the [radiance](/source/Radiance) *transmitted* by that surface into the [solid angle](/source/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[2]

- 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

- *L*e,Ω,νt is the [spectral radiance in frequency](/source/Radiance) *transmitted* by that surface;

- *L*e,Ω,νi is the spectral radiance received by that surface;

- *L*e,Ω,λt is the [spectral radiance in wavelength](/source/Radiance) *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)](/source/Photometry_(optics)), the luminous transmittance of a filter is a measure of the amount of luminous flux or intensity transmitted by an [optical filter](/source/Optical_filter). It is generally defined in terms of a [standard illuminant](/source/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](/source/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](/source/Dimensionless_quantity).

## Internal transmittance

### Optical depth

By definition, internal transmittance is related to [optical depth](/source/Optical_depth) and to [absorbance](/source/Absorbance) as

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

where

- *τ* is the optical depth;

- *A* is the absorbance.

### Beer–Lambert law

Main article: [Beer–Lambert law](/source/Beer%E2%80%93Lambert_law)

The [Beer–Lambert law](/source/Beer%E2%80%93Lambert_law) 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

- *σ**i* is the [attenuation cross section](/source/Cross_section_(physics)) of the attenuating species *i* in the material sample;

- *n**i* is the [number density](/source/Number_density) of the attenuating species *i* in the material sample;

- *ε**i* is the [molar attenuation coefficient](/source/Molar_attenuation_coefficient) of the attenuating species *i* in the material sample;

- *c**i* is the [amount concentration](/source/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](/source/Avogadro_constant).

In case of *uniform* attenuation, these relations become[3]

- 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](/source/Atmospheric_science) applications and [radiation shielding](/source/Radiation_shielding) theory for instance.

## Other radiometric coefficients

Radiometry coefficients v t e Quantity SI units Notes Name Sym. Hemispherical emissivity ε —N/a Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface. Spectral hemispherical emissivity εν ελ —N/a Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. Directional emissivity εΩ —N/a Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. Spectral directional emissivity εΩ,ν εΩ,λ —N/a Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface. Hemispherical absorptance A —N/a Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance". Spectral hemispherical absorptance Aν Aλ —N/a Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance". Directional absorptance AΩ —N/a Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance". Spectral directional absorptance AΩ,ν AΩ,λ —N/a Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance". Hemispherical reflectance R —N/a Radiant flux reflected by a surface, divided by that received by that surface. Spectral hemispherical reflectance Rν Rλ —N/a Spectral flux reflected by a surface, divided by that received by that surface. Directional reflectance RΩ —N/a Radiance reflected by a surface, divided by that received by that surface. Spectral directional reflectance RΩ,ν RΩ,λ —N/a Spectral radiance reflected by a surface, divided by that received by that surface. Hemispherical transmittance T —N/a Radiant flux transmitted by a surface, divided by that received by that surface. Spectral hemispherical transmittance Tν Tλ —N/a Spectral flux transmitted by a surface, divided by that received by that surface. Directional transmittance TΩ —N/a Radiance transmitted by a surface, divided by that received by that surface. Spectral directional transmittance TΩ,ν TΩ,λ —N/a Spectral radiance transmitted by a surface, divided by that received by that surface. Hemispherical attenuation coefficient μ m−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. Spectral hemispherical attenuation coefficient μν μλ m−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume. Directional attenuation coefficient μΩ m−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume. Spectral directional attenuation coefficient μΩ,ν μΩ,λ m−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

## See also

- [Opacity (optics)](/source/Opacity_(optics))

- [Photometry (optics)](/source/Photometry_(optics))

- [Radiometry](/source/Radiometry)

## References

1. **[^](#cite_ref-1)** ["Electronic warfare and radar systems engineering handbook"](https://web.archive.org/web/20010913091738/http://ewhdbks.mugu.navy.mil/EO-IR.htm#transmission). Archived from the original on September 13, 2001.

1. ^ [***a***](#cite_ref-ISO_9288-1989_2-0) [***b***](#cite_ref-ISO_9288-1989_2-1) [***c***](#cite_ref-ISO_9288-1989_2-2) [***d***](#cite_ref-ISO_9288-1989_2-3) ["Thermal insulation — Heat transfer by radiation — Vocabulary"](http://www.iso.org/iso/homehttps://www.iso.org/standard/82088.html/store/catalogue_tc/catalogue_detail.htm?csnumber=16943). *ISO 9288:2022*. [ISO](/source/International_Organization_for_Standardization) catalogue. August 1, 2022. Retrieved February 12, 2025.

1. **[^](#cite_ref-GoldBook2_3-0)** [IUPAC](/source/International_Union_of_Pure_and_Applied_Chemistry), *[Compendium of Chemical Terminology](/source/IUPAC_books#Gold_Book)*, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "[Beer–Lambert law](https://goldbook.iupac.org/terms/view/B00626.html)". [doi](/source/Doi_(identifier)):[10.1351/goldbook.B00626](https://doi.org/10.1351%2Fgoldbook.B00626)

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Adapted from the Wikipedia article [Transmittance](https://en.wikipedia.org/wiki/Transmittance) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Transmittance?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.