Radiant exposure

In radiometry, radiant exposure or fluence is the radiant energy received by a surface per unit area, or equivalently the irradiance of a surface, integrated over time of irradiation, and spectral exposure is the radiant exposure per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant exposure is the joule per square metre (J/m²), while that of spectral exposure in frequency is the joule per square metre per hertz (J⋅m⁻²⋅Hz⁻¹) and that of spectral exposure in wavelength is the joule per square metre per metre (J/m³)—commonly the joule per square metre per nanometre (J⋅m⁻²⋅nm⁻¹).

Mathematical definitions

Radiant exposure

Radiant exposure of a surface, denoted H_e ("e" for "energetic", to avoid confusion with photometric quantities), is defined as¹ $H_{\mathrm {e} }={\frac {\partial Q_{\mathrm {e} }}{\partial A}}=\int _{0}^{T}E_{\mathrm {e} }(t)\,\mathrm {d} t,$ where

∂ is the partial derivative symbol;
Q_e is the radiant energy;
A is the area;
T is the duration of irradiation;
E_e is the irradiance.

Spectral exposure

Spectral exposure in frequency of a surface, denoted H_e,ν, is defined as¹ $H_{\mathrm {e} ,\nu }={\frac {\partial H_{\mathrm {e} }}{\partial \nu }},$ where ν is the frequency.

Spectral exposure in wavelength of a surface, denoted H_e,λ, is defined as¹ $H_{\mathrm {e} ,\lambda }={\frac {\partial H_{\mathrm {e} }}{\partial \lambda }},$ where λ is the wavelength.

SI radiometry units

SI radiometry units
v
t
e
Quantity		Unit		Dimension	Notes
Name	Symbol^{nb 1}	Name	Symbol	Dimension	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", and called luminosity in astronomy.
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 called "intensity".
Spectral radiance Specific intensity	$L e,Ω, ν$ ^{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 called "spectral intensity".
Spectral radiance Specific intensity	$L e,Ω, λ$ ^{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 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 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 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 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 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 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
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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.)
Spectral quantities given per unit wavelength are denoted with suffix "λ".
Directional quantities are denoted with suffix "Ω".

References

IUPAC, Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Radiant exposure". doi:10.1351/goldbook.R05042

[note-suffix-e-2] Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.

[note-alternative-symbol-radiometric-3] Alternative symbols sometimes seen: $W$ or $E$ for radiant energy, $P$ or $F$ for radiant flux, $I$ for irradiance, $W$ for radiant exitance.

[note-suffix-nu-4] Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)

[note-suffix-lambda-5] Spectral quantities given per unit wavelength are denoted with suffix "λ".

[note-suffix-omega-6] Directional quantities are denoted with suffix "Ω".

[GoldBook-1] IUPAC, Compendium of Chemical Terminology, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "Radiant exposure". doi:10.1351/goldbook.R05042

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Mathematical definitions