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Optical depth

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Physics concept

For other uses, seeOptical depth (astrophysics).

Aerosol Optical Depth (AOD) at 830 nm measured with the same LED sun photometer from 1990 to 2016 at Geronimo Creek Observatory, Texas. Measurements made at or near solar noon when the Sun is not obstructed by clouds. Peaks indicate smoke, dust and smog. Saharan dust events are measured each summer.

Inphysics,optical depth oroptical thickness is thenatural logarithm of the ratio of incident totransmittedradiant power through a material.Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material.Spectral optical depth orspectral optical thickness is the natural logarithm of the ratio of incident to transmittedspectral radiant power through a material.^[1] Optical depth isdimensionless, and in particular is not a length, though it is a monotonically increasing function ofoptical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.^[1]

Inchemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: thecommon logarithm of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided bylog_e(10), because of the different logarithm bases used.

Mathematical definitions

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Optical depth

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The optical depth of a material, denoted ${\textstyle \tau }$ , is given by:^[2] $\tau =\ln \!\left({\frac {\Phi _{\mathrm {e} }^{\mathrm {i} }}{\Phi _{\mathrm {e} }^{\mathrm {t} }}}\right)=-\ln T$ where

${\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }}$ is theradiant flux received by that material;
${\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }}$ is theradiant flux transmitted by that material;
${\textstyle T}$ is thetransmittance of that material.

The absorbance ${\textstyle A}$ is related to optical depth by: $\tau =A\ln {10}$

Spectral optical depth

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The spectral optical depth in frequency (denoted $\tau _{\nu }$ ) or in wavelength ( $\tau _{\lambda }$ ) of a material is given by:^[1] $\tau _{\nu }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}}\right)=-\ln T_{\nu }$ $\tau _{\lambda }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)=-\ln T_{\lambda },$ where

$\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }$ is thespectral radiant flux in frequency transmitted by that material;
$\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }$ is the spectral radiant flux in frequency received by that material;
$T_{\nu }$ is thespectral transmittance in frequency of that material;
$\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }$ is thespectral radiant flux in wavelength transmitted by that material;
$\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }$ is the spectral radiant flux in wavelength received by that material;
$T_{\lambda }$ is thespectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by: $\tau _{\nu }=A_{\nu }\ln 10,$ $\tau _{\lambda }=A_{\lambda }\ln 10,$ where

$A_{\nu }$ is the spectral absorbance in frequency;
$A_{\lambda }$ is the spectral absorbance in wavelength.

Relationship with attenuation

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Attenuation

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Main article:Attenuation

Optical depth measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by absorption, but also reflection, scattering, and other physical processes. Optical depth of a material is approximately equal to itsattenuation when both the absorbance is much less than 1 and the emittance of that material (not to be confused withradiant exitance oremissivity) is much less than the optical depth: $\Phi _{\mathrm {e} }^{\mathrm {t} }+\Phi _{\mathrm {e} }^{\mathrm {att} }=\Phi _{\mathrm {e} }^{\mathrm {i} }+\Phi _{\mathrm {e} }^{\mathrm {e} },$ $T+ATT=1+E,$ where

Φ_e^t is the radiant power transmitted by that material;
Φ_e^att is the radiant power attenuated by that material;
Φ_eⁱ is the radiant power received by that material;
Φ_e^e is the radiant power emitted by that material;
T = Φ_e^t/Φ_eⁱ is the transmittance of that material;
ATT = Φ_e^att/Φ_eⁱ is the attenuation of that material;
E = Φ_e^e/Φ_eⁱ is the emittance of that material,

and according to theBeer–Lambert law, $T=e^{-\tau },$ so: $ATT=1-e^{-\tau }+E\approx \tau +E\approx \tau ,\quad {\text{if}}\ \tau \ll 1\ {\text{and}}\ E\ll \tau .$

Attenuation coefficient

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Optical depth of a material is also related to itsattenuation coefficient by: $\tau =\int _{0}^{l}\alpha (z)\,\mathrm {d} z,$ where

l is the thickness of that material through which the light travels;
α(z) is the attenuation coefficient or Napierian attenuation coefficient of that material atz,

and ifα(z) is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: $\tau =\alpha l$

Sometimes the relation is given using theattenuation cross section of the material, that is its attenuation coefficient divided by itsnumber density: $\tau =\int _{0}^{l}\sigma n(z)\,\mathrm {d} z,$ where

σ is the attenuation cross section of that material;
n(z) is the number density of that material atz,

and if $n {\displaystyle n}$ is uniform along the path, i.e., $n(z)\equiv N$ , the relation becomes: $\tau =\sigma Nl$

Applications

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Atomic physics

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Inatomic physics, the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by $\tau _{\nu }={\frac {d^{2}n\nu }{2\mathrm {c} \hbar \varepsilon _{0}\sigma \gamma }}$ where

d is thetransition dipole moment;
n is the number of atoms;
ν is the frequency of the beam;
c is thespeed of light;
ħ is thereduced Planck constant;
ε₀ is thevacuum permittivity;
σ is the cross section of the beam;
γ is thenatural linewidth of the transition.

Atmospheric sciences

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Astronomy

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Main article:Optical depth (astrophysics)

Inastronomy, thephotosphere of a star is defined as the surface where its optical depth is 2/3. This means that each photon emitted at the photosphere suffers an average of less than one scattering before it reaches the observer. At the temperature at optical depth 2/3, the energy emitted by the star (the original derivation is for the Sun) matches the observed total energy emitted.^{[citation needed]}^{[clarification needed]}

Note that the optical depth of a given medium will be different for different colors (wavelengths) of light.

Forplanetary rings, the optical depth is the (negative logarithm of the) proportion of light blocked by the ring when it lies between the source and the observer. This is usually obtained by observation of stellar occultations.

Mars dust storm – optical depth tau – May to September 2018
(Mars Climate Sounder;Mars Reconnaissance Orbiter)
(1:38; animation; 30 October 2018;file description)

References

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^^a ^b ^cIUPAC,Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Absorbance".doi:10.1351/goldbook.A00028
^Christopher Robert Kitchin (1987).Stars, Nebulae and the Interstellar Medium: Observational Physics and Astrophysics.CRC Press.
^^a ^b ^c ^dPetty, Grant W. (2006).A first course in atmospheric radiation. Sundog Pub.ISBN 9780972903318.OCLC 932561283.