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.

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
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
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.
^ ^a ^b ^c ^d ^e ^f ^g 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.)
^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit wavelength are denoted with suffix "λ".
^ ^a ^b Directional quantities are denoted with suffix "Ω".

Comparison of photometric and radiometric quantities

References

^ ^a ^b ^c "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
^ 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

[note-suffix-e-3] 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-4] 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-5] ^ ^a ^b ^c ^d ^e ^f ^g 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-6] ^ ^a ^b ^c ^d ^e ^f ^g Spectral quantities given per unit wavelength are denoted with suffix "λ".

[note-suffix-omega-7] Directional quantities are denoted with suffix "Ω".

[ISO_9288-1989-1] "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.

[Antennas_for_all_applications-2] Kraus, John; Marhefka, Ronald J. (2002). Antennas for all applications (3rd ed.). ISBN 0-07-232103-2.

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Mathematical definitions