# Physics:Transmittance

Short description: Effectiveness of a material in transmitting radiant energy
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.

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.[2]

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[3]

$\displaystyle{ T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}}, }$

where

• Φet is the radiant flux transmitted 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[3]

$\displaystyle{ T_\nu = \frac{\Phi_{\mathrm{e},\nu}^\mathrm{t}}{\Phi_{\mathrm{e},\nu}^\mathrm{i}}, }$
$\displaystyle{ T_\lambda = \frac{\Phi_{\mathrm{e},\lambda}^\mathrm{t}}{\Phi_{\mathrm{e},\lambda}^\mathrm{i}}, }$

where

### Directional transmittance

Directional transmittance of a surface, denoted TΩ, is defined as[3]

$\displaystyle{ T_\Omega = \frac{L_{\mathrm{e},\Omega}^\mathrm{t}}{L_{\mathrm{e},\Omega}^\mathrm{i}}, }$

where

• Le,Ωt is the radiance transmitted by that surface;

### 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[3]

$\displaystyle{ T_{\nu,\Omega} = \frac{L_{\mathrm{e},\Omega,\nu}^\mathrm{t}}{L_{\mathrm{e},\Omega,\nu}^\mathrm{i}}, }$
$\displaystyle{ T_{\lambda,\Omega} = \frac{L_{\mathrm{e},\Omega,\lambda}^\mathrm{t}}{L_{\mathrm{e},\Omega,\lambda}^\mathrm{i}}, }$

where

## Beer–Lambert law

Main page: Physics:Beer–Lambert law

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

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

where

• τ is the optical depth;
• A is the absorbance.

The Beer–Lambert law states that, for N attenuating species in the material sample,

$\displaystyle{ T = e^{-\sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\mathrm{d}z} = 10^{-\sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\mathrm{d}z}, }$

or equivalently that

$\displaystyle{ \tau = \sum_{i = 1}^N \tau_i = \sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\,\mathrm{d}z, }$
$\displaystyle{ A = \sum_{i = 1}^N A_i = \sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrm{d}z, }$

where

Attenuation cross section and molar attenuation coefficient are related by

$\displaystyle{ \varepsilon_i = \frac{\mathrm{N_A}}{\ln{10}}\,\sigma_i, }$

and number density and amount concentration by

$\displaystyle{ c_i = \frac{n_i}{\mathrm{N_A}}, }$

where NA is the Avogadro constant.

In case of uniform attenuation, these relations become[4]

$\displaystyle{ T = e^{-\sum_{i = 1}^N \sigma_i n_i\ell} = 10^{-\sum_{i = 1}^N \varepsilon_i c_i\ell}, }$

or equivalently

$\displaystyle{ \tau = \sum_{i = 1}^N \sigma_i n_i\ell, }$
$\displaystyle{ A = \sum_{i = 1}^N \varepsilon_i c_i\ell. }$

Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.