# 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 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 optics and spectroscopy, transmittance is the fraction of incident light (electromagnetic radiation) at a specified wavelength that passes through a sample.[2][3] 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,[4] or absorption factor,[5] which is the fraction of radiation absorbed by a sample at a specified wavelength.

## Definition

In equation form,

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

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 $\mathcal{\tau}_\lambda$ are transmittance and optical depth 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.

Transmittance is related to optical depth τ (not to be confused with absorbance nor absorptance) as

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

If one want to express optical depth in decibels:

$\tau/dB= - 10 \log_{10}\mathcal{T} = 10 \log_{10}\left({I_0 \over I}\right)$

From the above equation the transmittance is thus given by

$\mathcal{T} = e^{-\tau} = 10^{-\tau/dB}$

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

In plane geometry:

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

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.

## Beer-Lambert law

Main article: Beer–Lambert law

In case of uniform attenuation optical depth is simply:

$\tau = \Sigma l(x,y,z) = N \sigma l(x,y,z)$

where Σ is the attenuation coefficient, N is the medium concentration, σ the total cross section[disambiguation needed] and $l$ the geometrical path length, so the transmittance is:

$\mathcal T = e^{-\Sigma l(x,y,z)} = e^{N \sigma l(x,y,z)}$

In the general nonuniform case the optical depth is an integral quantity in position $\vec r = (x,y,z)$:

$\mathcal T = e^{-\int_0^l(\vec r) \Sigma(\vec r') dl'(\vec r')}$

for example if there is a strong temperature or pressure nonuniformity in a material, the concentration is nonuniform but the cross section is uniform, so:

$\mathcal T = e^{-\sigma \int_0^l(\vec r) N(\vec r') dl'(\vec r')} = [C(\vec r)]^\sigma$

This is the case of atmospheric science applications and also of radiation shielding theory.

## Beer's law

Another equation that can be useful in solving for $\tau$ is the following:[6]

$A = 2 - \log(\tau)$

Where $A$ is a measure of absorbance. By manipulating the equation you can generate the more direct form of the equation:

$\tau = 10^{2-A}$