Optical depth

Optical depth

The optical depth of a material, denoted by ${\displaystyle \tau }$, is defined as:

${\displaystyle \tau =\ln \!\left({\frac {\Phi _{\mathrm {e} }^{\mathrm {i} }}{\Phi _{\mathrm {e} }^{\mathrm {t} }}}\right)=-\ln T}$

where

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

The decadic absorbance ${\displaystyle A}$ is related to optical depth by:

${\displaystyle \tau =A\ln 10\approx 2.3026A}$

Spectral optical depth

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

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

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

where

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

Spectral absorbance relates to spectral optical depth via:

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

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

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:

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

where ${\displaystyle E=\Phi _{\mathrm {e} }^{\mathrm {e} }/\Phi _{\mathrm {e} }^{\mathrm {i} }}$ 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.

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:

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

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;
• ${\displaystyle \hbar }$ is the reduced Planck constant;
• ${\displaystyle \varepsilon _{0}}$ 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 ${\displaystyle \tau '}$ by the airmass factor ${\displaystyle m}$:

${\displaystyle \tau =m\tau '\approx \tau '\sec \theta }$

yielding a total transmittance of:

${\displaystyle T=e^{-\tau }=e^{-m\tau '}}$

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:

${\displaystyle \tau (z)=k_{\text{a}}w_{1}\rho _{0}He^{-z/H}}$

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.