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The optical depth of a material, denoted by $\tau$, is defined as:^[2]

$\tau =\ln \left(\frac{{\Phi }_e^i}{{\Phi }_e^t}\right)=-\ln T$

where

• ${\Phi }_e^i$ is the radiant flux received (incident) by that material;
• ${\Phi }_e^t$ is the radiant flux transmitted by that material;
• $T$ is the transmittance of that material.

The decadic absorbance $A$ is related to optical depth by:

$\tau =A\ln 10\approx 2.3026A$

The spectral optical depth in frequency (${\tau }_{\nu }$) or in wavelength (${\tau }_{\lambda }$) is given by:^[1]

${\tau }_{\nu }=\ln \left(\frac{{\Phi }_{e,\nu }^i}{{\Phi }_{e,\nu }^t}\right)=-\ln T_{\nu }$

${\tau }_{\lambda }=\ln \left(\frac{{\Phi }_{e,\lambda }^i}{{\Phi }_{e,\lambda }^t}\right)=-\ln T_{\lambda }$

where

• ${\Phi }_{e,\nu }^i$ and ${\Phi }_{e,\nu }^t$ are the incident and transmitted spectral radiant fluxes in frequency;
• $T_{\nu }$ is the spectral transmittance in frequency;
• ${\Phi }_{e,\lambda }^i$ and ${\Phi }_{e,\lambda }^t$ are the incident and transmitted spectral radiant fluxes in wavelength;
• $T_{\lambda }$ is the spectral transmittance in wavelength.

Spectral absorbance relates to spectral optical depth via:

${\tau }_{\nu }=A_{\nu }\ln 10$

${\tau }_{\lambda }=A_{\lambda }\ln 10$

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:

$T=e^{-\tau }$

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

$\text{ATT}=\frac{{\Phi }_e^{att}}{{\Phi }_e^i}=1-e^{-\tau }+E$

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

$\text{ATT}\approx \tau$

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

$\tau ={\int }_0^l\alpha \left(z\right)dz$

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

$\tau =\alpha l$

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

$\tau ={\int }_0^l\sigma n\left(z\right)dz=\sigma N$

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

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

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

where

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

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 $\theta$, the slant optical depth $\tau$ relates to vertical optical depth ${\tau }^{\prime }$ by the airmass factor $m$:

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

yielding a total transmittance of:

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

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 $z$, optical depth in an exponential atmosphere scales as:^[3]

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

giving a sea-level vertical optical depth of:

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

where:

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

For a uniform plane-parallel cloud layer, optical depth can be parameterized as:^[3]

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

where:

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

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

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 $\tau \approx 2/3$, 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.