Radiant intensity

In

spectrum is taken as a function of frequency or of wavelength. These are directional quantities. The SI unit of radiant intensity is the watt per steradian (W/sr), while that of spectral intensity in frequency is the watt per steradian per hertz (W·sr⁻¹·Hz⁻¹) and that of spectral intensity in wavelength is the watt per steradian per metre (W·sr⁻¹·m⁻¹)—commonly the watt per steradian per nanometre (W·sr⁻¹·nm⁻¹). Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity in branches of physics other than radiometry. In radio-frequency engineering

, radiant intensity is sometimes called radiation intensity.

Mathematical definitions

Radiant intensity

Radiant intensity, denoted I_e,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as [1]

I_{\mathrm {e} ,\Omega }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \Omega }},

where

∂ is the partial derivative symbol;
Φ_e is the radiant flux emitted, reflected, transmitted or received;
Ω is the solid angle.

In general, I_e,Ω is a function of viewing angle θ and potentially

azimuth angle. For the special case of a Lambertian surface, I_e,Ω follows the Lambert's cosine law

I_e,Ω = I₀ cos θ.

When calculating the radiant intensity emitted by a source, Ω refers to the solid angle into which the light is emitted. When calculating radiance received by a detector, Ω refers to the solid angle subtended by the source as viewed from that detector.

Spectral intensity

Spectral intensity in frequency, denoted I_e,Ω,ν, is defined as [1]

I_{\mathrm {e} ,\Omega ,\nu }={\frac {\partial I_{\mathrm {e} ,\Omega }}{\partial \nu }},

where ν is the frequency.

Spectral intensity in wavelength, denoted I_e,Ω,λ, is defined as^[1]

I_{\mathrm {e} ,\Omega ,\lambda }={\frac {\partial I_{\mathrm {e} ,\Omega }}{\partial \lambda }},

where λ is the wavelength.

Radio-frequency engineering

Radiant intensity is used to characterize the emission of radiation by an antenna:^[2]

I_{\mathrm {e} ,\Omega }=E_{\mathrm {e} }(r)\,r^{2},

where

E_e is the irradiance of the antenna;
r is the distance from the antenna.

Unlike power density, radiant intensity does not depend on distance: because radiant intensity is defined as the power through a solid angle, the decreasing power density over distance due to the inverse-square law is offset by the increase in area with distance.

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 confusingly 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 confusingly 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
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
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.
^ ^a ^b ^c ^d ^e Alternative symbols sometimes seen: $W$ or $E$ for radiant energy, $P$ or $F$ for radiant flux, $I$ for irradiance, $W$ for radiant exitance.
^
ν" (Greek letter nu
, not to be confused with a letter "v", indicating a photometric quantity.)