In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, optical depth or optical thickness is the
natural logarithm of the ratio of incident to ''transmitted''
radiant power
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spec ...
through a material.
Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material.
Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted
spectral radiant power through a material.
Optical depth is
dimensionless, and in particular is not a length, though it is a
monotonically increasing function of
optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.
In
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, a closely related quantity called "
absorbance
Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...
" or "decadic absorbance" is used instead of optical depth: the
common logarithm of the ratio of incident to transmitted radiant power through a material, that is the optical depth divided by ln 10.
Mathematical definitions
Optical depth
Optical depth of a material, denoted
, is given by:
where
*
is the
radiant flux received by that material;
*
is the
radiant flux transmitted by that material;
*
is the
transmittance of that material.
The absorbance
is related to optical depth by:
Spectral optical depth
Spectral optical depth in frequency and spectral optical depth in wavelength of a material, denoted
and
respectively, are given by:
where
*
is the
spectral radiant flux in frequency transmitted by that material;
*
is the spectral radiant flux in frequency received by that material;
*
is the
spectral transmittance in frequency of that material;
*
is the
spectral radiant flux in wavelength transmitted by that material;
*
is the spectral radiant flux in wavelength received by that material;
*
is the
spectral transmittance in wavelength of that material.
Spectral absorbance is related to spectral optical depth by:
where
*
is the spectral absorbance in frequency;
*
is the spectral absorbance in wavelength.
Relationship with attenuation
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 its
attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at var ...
when both the absorbance is much less than 1 and the
emittance of that material (not to be confused with
radiant exitance or
emissivity) is much less than the optical depth:
where
*Φ
et is the radiant power transmitted by that material;
*Φ
eatt is the radiant power attenuated by that material;
*Φ
ei is the radiant power received by that material;
*Φ
ee is the radiant power emitted by that material;
*''T'' = Φ
et/Φ
ei is the transmittance of that material;
*''ATT'' = Φ
eatt/Φ
ei is the attenuation of that material;
*''E'' = Φ
ee/Φ
ei is the emittance of that material,
and according to the
Beer–Lambert law,
so:
Attenuation coefficient
Optical depth of a material is also related to its
attenuation coefficient by:
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 at ''z'',
and if ''α''(''z'') is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes:
Sometimes the relation is given using the
attenuation cross section of the material, that is its attenuation coefficient divided by its
number density:
where
*''σ'' is the attenuation cross section of that material;
*''n''(''z'') is the number density of that material at ''z'',
and if
is uniform along the path, i.e.,
, the relation becomes:
Applications
Atomic physics
In
atomic 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
where
*''d'' is the
transition dipole moment;
*''n'' is the number of atoms;
*''ν'' is the frequency of the beam;
*c is the
speed of light;
*ħ is
Planck's constant;
*ε
0 is the
vacuum permittivity;
*''σ'' the cross section of the beam;
*''γ'' the
natural linewidth of the transition.
Atmospheric sciences
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. The optical depth for a slant path is , where ''τ′'' refers to a vertical path, ''m'' is called the
relative airmass, and for a plane-parallel atmosphere it is determined as 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
Absorption may refer to:
Chemistry and biology
*Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which s ...
. The optical depth of the atmosphere can be measured with a
sun photometer.
The optical depth with respect to the height within the atmosphere is given by
and it follows that the total atmospheric optical depth is given by
In both equations:
* k
a is the absorption coefficient
* w
1 is the mixing ratio
* ρ
0 is the density of air at sea level
* H is the scale height of the atmosphere
* z is the height in question
The optical depth of a plane parallel cloud layer is given by
where:
* Q
e is the extinction efficiency
* L is the
liquid water path
* H is the geometrical thickness
* N is the concentration of droplets
* ρ
l is the density of liquid water
So, with a fixed depth and total liquid water path,
.
Astronomy
In
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, the
photosphere 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.
Note that the optical depth of a given medium will be different for different colors (
wavelengths) of light.
For
planetary 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.
See also
*
Air mass (astronomy)
*
Absorptance
*
Actinometer
*
Aerosol
*
Angstrom exponent
*
Attenuation coefficient
*
Beer–Lambert law
*
Pyranometer
*
Radiative transfer
*
Sun photometer
*
Transparency and translucency
In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable scattering of light. On a macroscopic scale (one in which the dimension ...
References
{{reflist
External links
Optical depth equations
Scattering, absorption and radiative transfer (optics)
Visibility
Spectroscopy