Optical depth
In
In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided by loge(10), because of the different logarithm bases used.
Mathematical definitions
Optical depth
Optical depth of a material, denoted , is given by:[2]
- 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:[1]
- 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:
- 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
- Φet is the radiant power transmitted by that material;
- Φeatt is the radiant power attenuated by that material;
- Φei is the radiant power received by that material;
- Φee is the radiant power emitted by that material;
- T = Φet/Φei is the transmittance of that material;
- ATT = Φeatt/Φei is the attenuation of that material;
- E = Φee/Φei is the emittance of that material,
and according to the Beer–Lambert law,
Attenuation coefficient
Optical depth of a material is also related to its attenuation coefficient by:
- 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:
- 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
- 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 the reduced Planck constant;
- ε0 is the vacuum permittivity;
- σ the cross section of the beam;
- γ the natural linewidthof the transition.
Atmospheric sciences
In
The optical depth with respect to the height within the atmosphere is given by[3]
In both equations:
- ka is the absorption coefficient
- w1 is the mixing ratio
- ρ0 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[3]
- Qe is the extinction efficiency
- L is the liquid water path
- H is the geometrical thickness
- N is the concentration of droplets
- ρl is the density of liquid water
So, with a fixed depth and total liquid water path, .[3]
Astronomy
In astronomy, 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.[citation needed][clarification needed]
Note that the optical depth of a given medium will be different for different colors (wavelengths) of light.
For
See also
- Air mass (astronomy)
- Absorptance
- Actinometer
- Aerosol
- Angstrom exponent
- Attenuation coefficient
- Beer–Lambert law
- Pyranometer
- Radiative transfer
- Sun photometer
- Transparency and translucency