Optical depth

Relationship with attenuation

Attenuation

Optical depth quantifies the attenuation of transmitted radiant power in a medium. Attenuation can occur via absorption, scattering, reflection, and other physical processes.

According to the Beer–Lambert law, the transmitted flux decays exponentially with optical depth:

${\displaystyle T=e^{-\tau }}$

The fractional attenuation (${\displaystyle \text{ATT}}$) is given by:

${\displaystyle \text{ATT}=\frac{{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{a}\mathrm{t}\mathrm{t}}}{{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}}=1-e^{-\tau }+E}$

where ${\displaystyle E={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{e}}/{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}}$ is the relative emittance of the medium. For optically thin media (${\displaystyle \tau \ll 1}$) with negligible emission (${\displaystyle E\ll \tau }$), attenuation reduces approximately to the optical depth itself:

${\displaystyle \text{ATT}\approx \tau }$

Attenuation coefficient

The optical depth along a path of length ${\displaystyle l}$ is related to the local attenuation coefficient ${\displaystyle \alpha (z)}$ by:

${\displaystyle \tau ={\int }_0^l\alpha (z)\mathrm{d}z}$

If ${\displaystyle \alpha }$ is uniform along the path, the relation simplifies to:

${\displaystyle \tau =\alpha l}$

In terms of the attenuation cross-section ${\displaystyle \sigma }$ per particle and number density ${\displaystyle n(z)}$:

${\displaystyle \tau ={\int }_0^l\sigma n(z)\mathrm{d}z=\sigma N}$

where ${\displaystyle N={\int }_0^{l}n(z)\mathrm{d}z}$ is the column density along the line of sight.

Applications

Atomic physics

In atomic physics, the spectral optical depth of a cloud of resonant two-level atoms can be determined from quantum-mechanical dipole transitions:

${\displaystyle {\tau }_{\nu }=\frac{d^{2}n\nu }{2c\hslash {\epsilon }_{0}A\gamma }\mathcal{L}(\nu )}$

where

• ${\displaystyle d}$ is the transition dipole moment;
• ${\displaystyle n}$ is the total number of atoms;
• ${\displaystyle \nu }$ is the frequency of the incident radiation;
• ${\displaystyle c}$ is the speed of light;
• ${\displaystyle \hslash }$ is the reduced Planck constant;
• ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity;
• ${\displaystyle A}$ is the cross-sectional area of the beam;
• ${\displaystyle \gamma }$ is the natural linewidth of the atomic transition;
• ${\displaystyle \mathcal{L}(\nu )}$ is the normalized spectral line profile shape.

Atmospheric sciences

In atmospheric sciences, optical depth often refers to a vertical path extending from Earth's surface to outer space. For a slanted line of sight at zenith angle ${\displaystyle \theta }$, the slant optical depth ${\displaystyle \tau }$ relates to vertical optical depth ${\displaystyle {\tau }^{\prime }}$ by the airmass factor ${\displaystyle m}$:

${\displaystyle \tau =m{\tau }^{\prime }\approx {\tau }^{\prime }\sec \theta }$

yielding a total transmittance of:

${\displaystyle T=e^{-\tau }=e^{-m{\tau }^{\prime }}}$

The total atmospheric optical depth encompasses contributions from Rayleigh scattering, aerosol optical depth (AOD), and trace gas absorption, routinely measured using Sun photometers.

With altitude ${\displaystyle z}$, optical depth in an exponential atmosphere scales as:

${\displaystyle \tau (z)=k_{\text{a}}{w}_1{\rho }_{0}H{e}^{-z/H}}$

giving a sea-level vertical optical depth of:

${\displaystyle \tau (0)=k_{\text{a}}{w}_1{\rho }_{0}H}$

where:

• ${\displaystyle k_{\text{a}}}$ is the absorption coefficient;
• ${\displaystyle w_1}$ is the species mixing ratio;
• ${\displaystyle {\rho }_0}$ is the air density at sea level;
• ${\displaystyle H}$ is the atmospheric scale height;
• ${\displaystyle z}$ is the altitude above sea level.

For a uniform plane-parallel cloud layer, optical depth can be parameterized as:

${\displaystyle \tau =Q_{\text{e}}{\left[\frac{9\pi L^{2}HN}{16{\rho }_{\text{l}}^2}\right]}^{1/3}}$

where:

• ${\displaystyle Q_{\text{e}}}$ is the extinction efficiency;
• ${\displaystyle L}$ is the liquid water path;
• ${\displaystyle H}$ is the geometrical thickness of the cloud;
• ${\displaystyle N}$ is the droplet number concentration;
• ${\displaystyle {\rho }_{\text{l}}}$ is the density of liquid water.

Thus, for a constant cloud thickness and total liquid water content, optical depth scales with droplet concentration as ${\displaystyle \tau \propto N^{1/3}}$.

Astronomy

In astronomy, optical depth governs radiation escape from stellar atmospheres, planetary rings, and nebulae. In stellar physics, the photosphere is defined as the atmospheric depth where ${\displaystyle \tau \approx 2/3}$, representing the mean surface from which thermal photons escape into space.

For planetary rings, optical depth measures the fraction of light blocked by ring material during stellar occultations.