Transmittance

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File:Ruby transmittance.svg

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">{{#invoke:citation/CS1|citation |CitationClass=web }}</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

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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>Template:GoldBookRef</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.

See also

• Opacity (optics)
• Photometry (optics)
• Radiometry