# Optical depth

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Optical depth, or optical thickness is a measure of transparency, and is defined as the negative logarithm of the fraction of radiation (e.g., light) that is not scattered or absorbed on a path. The optical depth is a measure of the proportion of radiation absorbed or scattered along a path through a partially transparent medium; the optical depth to a close-by object is zero; as the distance to the object increases, so does the optical depth.

## Contents

### Formulations

The optical depth expresses the quantity of light removed from a beam by scattering or absorption during its path through a medium. If I0 is the intensity of radiation at the source and I is the observed intensity after a given path, then optical depth τ is defined by the following equation:[1]

### Calculation from fundamental principles

In atomic physics, the optical depth of a cloud of atoms can be calculated from the quantum mechanical properties of the atoms. It is given by

$\tau = \frac{d^2 \nu N} {2 c \hbar \epsilon_0 A \gamma},$

where d denotes the transition dipole moment, γ the natural linewidth of the transition, ν the frequency, N the number of atoms, and A the cross-section of the beam.

### Atmospheric Science

In atmospheric sciences, one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. Since τ refers to a vertical path, the optical depth for a slant path is τ' = mτ, where m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = 1 / cosθ, where θ is the zenith angle corresponding to the given path. Therefore

The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols, and gaseous absorption. The optical depth of the atmosphere can be measured with a sun photometer. See also: Beer's law.