Optical depth

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material.[1] 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.[1]

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, that is the optical depth divided by ln 10.

Mathematical definitions
Optical depth

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

Φ_eⁱ is the radiant flux received by that material;
Φ_e^t is the radiant flux transmitted by that material;
T is the transmittance of that material.

Absorbance is related to optical depth by:

\( \tau =A\ln 10, \)

where A is the absorbance.

Spectral optical depth

Spectral optical depth in frequency and spectral optical depth in wavelength of a material, denoted τν and τλ respectively, are given by:[1]

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

where

Φ_e,ν^t is the spectral radiant flux in frequency transmitted by that material;
Φ_e,νⁱ is the spectral radiant flux in frequency received by that material;
T_ν is the spectral transmittance in frequency of that material;
Φ_e,λ^t is the spectral radiant flux in wavelength transmitted by that material;
Φ_e,λⁱ is the spectral radiant flux in wavelength received by that material;
T_λ is the spectral 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_ν is the spectral absorbance in frequency;
A_λ is the spectral absorbance in wavelength.

Relationship with attenuation

Attenuation
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 its attenuation 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:

\( \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 attenuance of that material;
E = Φ_e^e/Φ_eⁱ is the emittance of that material,

and according to Beer–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

Optical depth of a material is also related to its attenuation 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 at z,

and if α(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 the attenuation cross section of the material, that is its attenuation coefficient divided by its number density:

\( {\displaystyle \tau =\int _{0}^{l}\sigma n(z)\,\mathrm {d} z,} \)

where

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

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

\( \tau =\sigma Nl.

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

\( {\displaystyle \tau _{\nu }={\frac {d^{2}n\nu }{2\mathrm {c} \hbar \varepsilon _{0}\sigma \gamma }},} \)

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;
ε₀ is the vacuum permittivity;
σ the cross section of the beam;
γ the natural linewidth of the transition.

Atmospheric sciences
See also: Beer's law

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 τ = mτ′, where τ′ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. Therefore,

\( T=e^{-\tau }=e^{-m\tau '}. \)

The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols, and gaseous absorption. 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

\( {\displaystyle \tau (z)=k_{a}w_{1}\rho _{0}He^{-z/H}} \) [3]

and it follows that the total atmospheric optical depth is given by

\( {\displaystyle \tau (0)=k_{a}w_{1}\rho _{0}H} \)[3]

In both equations:

k_a 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

\( {\displaystyle \tau =Q_{e}\left[{\frac {9\pi L^{2}HN}{16\rho _{l}^{2}}}\right]^{1/3}} \) [3]

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,

\( {\displaystyle \tau \propto N^{1/3}} \) [3]
Astronomy
Main article: Optical depth (astrophysics)

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.

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.

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Symbol	Notes
Radiant energy	Q_e^{[nb 2]}	joule	J	M⋅L²⋅T⁻²	Energy of electromagnetic radiation.
Radiant energy density	w_e	joule per cubic metre	J/m³	M⋅L⁻¹⋅T⁻²	Radiant energy per unit volume.
Radiant flux	Φ_e^{[nb 2]}	watt	W = J/s	M⋅L²⋅T⁻³	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux	Φ_e,ν^{[nb 3]}	watt per hertz	W/Hz	M⋅L²⋅T⁻²	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm⁻¹.
Spectral flux	Φ_e,λ^{[nb 4]}	watt per metre	W/m	M⋅L⋅T⁻³
Radiant intensity	I_e,Ω^{[nb 5]}	watt per steradian	W/sr	M⋅L²⋅T⁻³	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	I_e,Ω,ν^{[nb 3]}	watt per steradian per hertz	W⋅sr⁻¹⋅Hz⁻¹	M⋅L²⋅T⁻²	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr⁻¹⋅nm⁻¹. This is a directional quantity.
Spectral intensity	I_e,Ω,λ^{[nb 4]}	watt per steradian per metre	W⋅sr⁻¹⋅m⁻¹	M⋅L⋅T⁻³
Radiance	L_e,Ω^{[nb 5]}	watt per steradian per square metre	W⋅sr⁻¹⋅m⁻²	M⋅T⁻³	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance	L_e,Ω,ν^{[nb 3]}	watt per steradian per square metre per hertz	W⋅sr⁻¹⋅m⁻²⋅Hz⁻¹	M⋅T⁻²	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr⁻¹⋅m⁻²⋅nm⁻¹. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Spectral radiance	L_e,Ω,λ^{[nb 4]}	watt per steradian per square metre, per metre	W⋅sr⁻¹⋅m⁻³	M⋅L⁻¹⋅T⁻³
Irradiance Flux density	E_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance Spectral flux density	E_e,ν^{[nb 3]}	watt per square metre per hertz	W⋅m⁻²⋅Hz⁻¹	M⋅T⁻²	Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10⁻²⁶ W⋅m⁻²⋅Hz⁻¹) and solar flux unit (1 sfu = 10⁻²² W⋅m⁻²⋅Hz⁻¹ = 10⁴ Jy).
Spectral irradiance Spectral flux density	E_e,λ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiosity	J_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity	J_e,ν^{[nb 3]}	watt per square metre per hertz	W⋅m⁻²⋅Hz⁻¹	M⋅T⁻²	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m⁻²⋅nm⁻¹. This is sometimes also confusingly called "spectral intensity".
Spectral radiosity	J_e,λ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exitance	M_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance	M_e,ν^{[nb 3]}	watt per square metre per hertz	W⋅m⁻²⋅Hz⁻¹	M⋅T⁻²	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m⁻²⋅nm⁻¹. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Spectral exitance	M_e,λ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exposure	H_e	joule per square metre	J/m²	M⋅T⁻²	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure	H_e,ν^{[nb 3]}	joule per square metre per hertz	J⋅m⁻²⋅Hz⁻¹	M⋅T⁻¹	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m⁻²⋅nm⁻¹. This is sometimes also called "spectral fluence".
Spectral exposure	H_e,λ^{[nb 4]}	joule per square metre, per metre	J/m³	M⋅L⁻¹⋅T⁻²
Hemispherical emissivity	ε	N/A		1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	ε_ν or ε_λ	N/A		1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	ε_Ω	N/A		1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	ε_Ω,ν or ε_Ω,λ	N/A		1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	A	N/A		1	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	A_ν or A_λ	N/A		1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	A_Ω	N/A		1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	A_Ω,ν or A_Ω,λ	N/A		1	Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	R	N/A		1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	R_ν or R_λ	N/A		1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	R_Ω	N/A		1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	R_Ω,ν or R_Ω,λ	N/A		1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T	N/A		1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	T_ν or T_λ	N/A		1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	T_Ω	N/A		1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	T_Ω,ν or T_Ω,λ	N/A		1	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	μ	reciprocal metre	m⁻¹	L⁻¹	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	μ_ν or μ_λ	reciprocal metre	m⁻¹	L⁻¹	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	μ_Ω	reciprocal metre	m⁻¹	L⁻¹	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μ_Ω,ν or μ_Ω,λ	reciprocal metre	m⁻¹	L⁻¹	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also: SI · Radiometry · Photometry

Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).

Directional quantities are denoted with suffix "Ω" (Greek).