Radiant intensity

"Radiation intensity" redirects here. For ionizing radiation, see Radiation flux. For other measures of electromagnetic radiation, see Light intensity (disambiguation).

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<ref name="ISO_9288-1989">"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.</ref>

<math>I_{\mathrm{e},\Omega} = \frac{\partial \Phi_\mathrm{e}}{\partial \Omega},</math>

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<ref name="ISO_9288-1989" />

<math>I_{\mathrm{e},\Omega,\nu} = \frac{\partial I_{\mathrm{e},\Omega}}{\partial \nu},</math>

where ν is the frequency.

Spectral intensity in wavelength, denoted I_e,Ω,λ, is defined as<ref name="ISO_9288-1989" />

<math>I_{\mathrm{e},\Omega,\lambda} = \frac{\partial I_{\mathrm{e},\Omega}}{\partial \lambda},</math>

where λ is the wavelength.

Radio-frequency engineering

Radiant intensity is used to characterize the emission of radiation by an antenna:<ref name="Antennas_for_all_applications">de Kraus, John; Marhefka, Ronald J. (2002). Antennas for all applications (3rd ed.). ISBN 0-07-232103-2.</ref>

<math>I_{\mathrm{e},\Omega} = E_\mathrm{e}(r) \, r^2,</math>

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

Quantity		Unit		Dimension	Notes
Name	Symbol<ref group="nb" name="note-suffix-e">Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.</ref>	Name	Symbol
Radiant energy	Qe<ref group="nb" name="note-alternative-symbol-radiometric">Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.</ref>	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Φe<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	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	Φe,ν<ref group="nb" name="note-suffix-nu">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.)</ref>	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.
	Φe,λ<ref group="nb" name="note-suffix-lambda">Spectral quantities given per unit wavelength are denoted with suffix "λ".</ref>	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Ie,Ω<ref group="nb" name="note-suffix-omega">Directional quantities are denoted with suffix "Ω".</ref>	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	Ie,Ω,ν<ref group="nb" name="note-suffix-nu"></ref>	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.
	Ie,Ω,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Le,Ω<ref group="nb" name="note-suffix-omega"></ref>	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	Le,Ω,ν<ref group="nb" name="note-suffix-nu"></ref>	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".
	Le,Ω,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Irradiance Flux density	Ee<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	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	Ee,ν<ref group="nb" name="note-suffix-nu"></ref>	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).
	Ee,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Je<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	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	Je,ν<ref group="nb" name="note-suffix-nu"></ref>	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".
	Je,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me<ref group="nb" name="note-alternative-symbol-radiometric"></ref>	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	Me,ν<ref group="nb" name="note-suffix-nu"></ref>	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".
	Me,λ<ref group="nb" name="note-suffix-lambda"></ref>	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	He	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	He,ν<ref group="nb" name="note-suffix-nu"></ref>	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".
	He,λ<ref group="nb" name="note-suffix-lambda"></ref>	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: SI Radiometry Photometry

Comparison of photometric and radiometric quantities

See also

• Candela
• Luminous intensity

External links

• Radiation: Activity and Intensity NDE/NDT Resource Center