# Transmittance

Diagram of Beer-Lambert Law of transmittance of a beam of light as it travels through a cuvette of width l.
Earth's atmospheric transmittance. 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 optics and spectroscopy, transmittance is the fraction of incident light (electromagnetic radiation) at a specified wavelength that passes through a sample.[1][2] The terms visible transmittance (VT) and visible absorptance (VA), which are the respective fractions for the spectrum of light visible radiation, are also used. The natural radiation of the cuvette corresponding to the temperature of the cuvette remains ignored - see radiative transfer equation.

A related terms is absorbance,[3] or absorption factor,[4] which is the fraction of radiation absorbed by a sample at a specified wavelength.

## Beer–Lambert law

In equation form,

$\mathcal{T}_\lambda = {I\over I_{0}} \qquad \mathcal{A}_\lambda = \frac{I_0-I}{I_0}$

where $I_0$ is the intensity of the incident radiation and I is the intensity of the radiation coming out of the sample and $\mathcal{T}_\lambda$ and $\mathcal{A}_\lambda$ are transmittance and absorbance respectively. In these equations, scattering and reflection are considered to be close to zero or otherwise accounted for. The transmittance of a sample is sometimes given as a percentage.

For liquids, transmittance is related to absorbance A (not to be confused with absorptance) as

$A = - \log_{10}\mathcal{T}\ = - \log_{10}\left({I\over I_{0}}\right)$

In the case of gases it is customary to use natural logarithms instead, making absorbance A for gases

$A = - \ln\mathcal{T}\ = - \ln\left({I\over I_{0}}\right)$

From the above equation and the Beer-Lambert law, the transmittance for gases is thus given by

$\mathcal{T} = e^{-\alpha \, x}$,

where $\alpha$ is the attenuation coefficient and $x$ is the path length. For liquids e is replaced by 10.

Note that the term "transmission" refers to the physical process of radiation passing through a sample, whereas transmittance refers to the mathematical quantity.

### Transmittance in the atmosphere

For atmospheric science applications, it is useful to introduce the quantity of optical depth $\tau$. Given a coefficient of extinction within an atmospheric column, $\beta_e$, optical depth is defined to be

$\tau = \int_{z_1}^{z_2} \beta_e(z) \mathrm{d}z$.

With this, transmittance can be given as

$\mathcal{T} = e^{-\tau}$,

or

$\mathcal{T} = e^{-\tau / \mu }$

where, when the plane parallel assumption is invoked, $\mu=|cos(\theta)|$ with $\theta$ the angle of propagation of the ray from the normal of the surface.