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 r ...
, optical depth or optical thickness is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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 spe ...
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
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, 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 science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, 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
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
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
Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is t ...
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 variable att ...
when both the absorbance is much less than 1 and the
emittance of that material (not to be confused with
radiant exitance
In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on wheth ...
or
emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
) 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
The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied t ...
,
so:
Attenuation coefficient
Optical depth of a material is also related to its
attenuation coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
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
The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
:
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
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, 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
The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the tra ...
;
*''n'' is the number of atoms;
*''ν'' is the frequency of the beam;
*c is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
;
*ħ is
Planck's constant;
*ε
0 is the
vacuum permittivity;
*''σ'' the cross section of the beam;
*''γ'' the
natural linewidth
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identi ...
of the transition.
Atmospheric sciences
In
atmospheric sciences
Atmospheric science is the study of the Earth's atmosphere and its various inner-working physical processes. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather forecasting. Climatology is the study of ...
, 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
The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "highe ...
corresponding to the given path. Therefore,
The optical depth of the atmosphere can be divided into several components, ascribed to
Rayleigh scattering
Rayleigh scattering ( ), named after the 19th-century British physicist Lord Rayleigh (John William Strutt), is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the ...
,
aerosols
An aerosol is a suspension of fine solid particles or liquid droplets in air or another gas. Aerosols can be natural or anthropogenic. Examples of natural aerosols are fog or mist, dust, forest exudates, and geyser steam. Examples of anthrop ...
, and gaseous
absorption. The optical depth of the atmosphere can be measured with a
sun photometer
A sun photometer is a type of photometer conceived in such a way that it points at the sun.
Recent sun photometers are automated instruments incorporating a sun-tracking unit, an appropriate optical
system, a electromagnetic spectrum , spectrally ...
.
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 Liquid water path - in units of g/m2 is a measure of the total amount of liquid water present between two points in the atmosphere.
LWP is an important quantity in understanding radiative transfer in the atmosphere. It is defined as the integral o ...
* 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 astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, the
photosphere
The photosphere is a star's outer shell from which light is radiated.
The term itself is derived from Ancient Greek roots, φῶς, φωτός/''phos, photos'' meaning "light" and σφαῖρα/''sphaira'' meaning "sphere", in reference to it ...
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 (
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s) of light.
For
planetary rings
A ring system is a disc or ring, orbiting an astronomical object, that is composed of solid material such as dust and moonlets, and is a common component of satellite systems around giant planets. A ring system around a planet is also known as ...
, 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)
In astronomy, air mass or airmass is a measure of the amount of air along the line of sight when observing a star or other celestial source from below Earth's atmosphere ( Green 1992). It is formulated as the integral of air density along the lig ...
*
Absorptance Absorptance of the surface of a material is its effectiveness in absorbing radiant energy. It is the ratio of the absorbed to the incident radiant power. This should not be confused with absorbance and absorption coefficient.
Mathematical definitio ...
*
Actinometer
Actinometers are instruments used to measure the heating power of radiation. They are used in meteorology to measure solar radiation as pyranometers, pyrheliometers and net radiometers.
An actinometer is a chemical system or physical device which ...
*
Aerosol
An aerosol is a suspension (chemistry), suspension of fine solid particles or liquid Drop (liquid), droplets in air or another gas. Aerosols can be natural or Human impact on the environment, anthropogenic. Examples of natural aerosols are fog o ...
*
Angstrom exponent The Angstrom exponentGregory L. Schuster, Oleg Dubovik and Brent N. Holben (2006): "Angstrom exponent and bimodal aerosol size distributions". ''Journal of Geophysical Research: Atmospheres'', volume 111, issue D7, article D07207, pages 1-14. Itaru ...
*
Attenuation coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
*
Beer–Lambert law
The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied t ...
*
Pyranometer A pyranometer is a type of actinometer used for measuring solar irradiance on a planar surface and it is designed to measure the solar radiation flux density (W/m2) from the hemisphere above within a wavelength range 0.3 μm to 3 μm. The name pyra ...
*
Radiative transfer
Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative trans ...
*
Sun photometer
A sun photometer is a type of photometer conceived in such a way that it points at the sun.
Recent sun photometers are automated instruments incorporating a sun-tracking unit, an appropriate optical
system, a electromagnetic spectrum , spectrally ...
*
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 dimensions ...
References
{{reflist
External links
Optical depth equations
Scattering, absorption and radiative transfer (optics)
Visibility
Spectroscopy