Radiant intensity

In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the 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.

Radiant intensity can as well characterize X-ray tube with filtration. X-ray tube output is defined for this purpose.

SI radiometry units

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	watt	W or J/s	M⋅L^2⋅T^−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux	Φe,ν[nb 3] or Φe,λ[nb 4]	watt per hertz or watt per metre	W/Hz or W/m	M⋅L^2⋅T^−2 or M⋅L⋅T^−3	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr^−1⋅m^−2⋅nm^−1.
Radiant intensity	Ie,Ω[nb 5]	watt per steradian	W/sr	M⋅L^2⋅T^−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν[nb 3] or Ie,Ω,λ[nb 4]	watt per steradian per hertz or watt per steradian per metre	W⋅sr^−1⋅Hz^−1 or W⋅sr^−1⋅m^−1	M⋅L^2⋅T^−2 or M⋅L⋅T^−3	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr^−1⋅m^−2⋅nm^−1. This is a directional quantity.
Radiance	Le,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	Le,Ω,ν[nb 3] or Le,Ω,λ[nb 4]	watt per steradian per square metre per hertz or watt per steradian per square metre, per metre	W⋅sr^−1⋅m^−2⋅Hz^−1 or W⋅sr^−1⋅m^−3	M⋅T^−2 or M⋅L^−1⋅T^−3	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".
Irradiance Flux density	Ee[nb 2]	watt per square metre	W/m^2	M⋅T^−3	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance Spectral flux density	Ee,ν[nb 3] or Ee,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m^−2⋅Hz^−1 or W/m^3	M⋅T^−2 or M⋅L^−1⋅T^−3	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 = 10^−26 W⋅m^−2⋅Hz^−1 and solar flux unit (1SFU = 10^−22 W⋅m^−2⋅Hz^−1=10^4Jy).
Radiosity	Je[nb 2]	watt per square metre	W/m^2	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	Je,ν[nb 3] or Je,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m^−2⋅Hz^−1 or W/m^3	M⋅T^−2 or M⋅L^−1⋅T^−3	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".
Radiant exitance	Me[nb 2]	watt per square metre	W/m^2	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	Me,ν[nb 3] or Me,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m^−2⋅Hz^−1 or W/m^3	M⋅T^−2 or M⋅L^−1⋅T^−3	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".
Radiant exposure	He	joule per square metre	J/m^2	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	He,ν[nb 3] or He,λ[nb 4]	joule per square metre per hertz or joule per square metre, per metre	J⋅m^−2⋅Hz^−1 or J/m^3	M⋅T^−1 or M⋅L^−1⋅T^−2	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".
Hemispherical emissivity	ε			1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	εν or ελ			1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ			1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	εΩ,ν or εΩ,λ			1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	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	Aν or Aλ			1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ			1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν or AΩ,λ			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			1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	Rν or Rλ			1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ			1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν or RΩ,λ			1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T			1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	Tν or Tλ			1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ			1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν or TΩ,λ			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	μν or μλ	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	μΩ	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	μΩ,ν or μΩ,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

1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. 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.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).

See also

• Candela
• Luminous intensity

References

1. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
2. de Kraus, John; Marhefka, Ronald J. (2002). Antennas for all applications (3rd ed.). ISBN 0-07-232103-2.

External links

• Radiation: Activity and Intensity NDE/NDT Resource Center