Spectral irradiance is the irradiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The two forms have different dimensions: spectral irradiance of a frequency spectrum is measured in watts per square metre per hertz (W?m-2?Hz-1), while spectral irradiance of a wavelength spectrum is measured in watts per square metre per metre (W?m-3), or more commonly watts per square metre per nanometre (W?m-2?nm-1).

## Mathematical definitions

Irradiance of a surface, denoted Ee ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[2]

${\displaystyle E_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},}$

where

If we want to talk about the radiant flux emitted by a surface, we speak of radiant exitance.

Spectral irradiance in frequency of a surface, denoted Ee,?, is defined as[2]

${\displaystyle E_{\mathrm {e} ,\nu }={\frac {\partial E_{\mathrm {e} }}{\partial \nu }},}$

where ? is the frequency.

Spectral irradiance in wavelength of a surface, denoted Ee,?, is defined as[2]

${\displaystyle E_{\mathrm {e} ,\lambda }={\frac {\partial E_{\mathrm {e} }}{\partial \lambda }},}$

where ? is the wavelength.

## Property

Irradiance of a surface is also, according to the definition of radiant flux, equal to the time-average of the component of the Poynting vector perpendicular to the surface:

${\displaystyle E_{\mathrm {e} }=\langle |\mathbf {S} |\rangle \cos \alpha ,}$

where

• ? o ? is the time-average;
• S is the Poynting vector;
• ? is the angle between a unit vector normal to the surface and S.

For a propagating sinusoidal linearly polarized electromagnetic plane wave, the Poynting vector always points to the direction of propagation while oscillating in magnitude. The irradiance of a surface is then given by[3]

${\displaystyle E_{\mathrm {e} }={\frac {n}{2\mu _{0}\mathrm {c} }}E_{\mathrm {m} }^{2}\cos \alpha ={\frac {n\varepsilon _{0}\mathrm {c} }{2}}E_{\mathrm {m} }^{2}\cos \alpha ,}$

where

This formula assumes that the magnetic susceptibility is negligible, i.e. that ?r ? 1 where ?r is the magnetic permeability of the propagation medium. This assumption is typically valid in transparent media in the optical frequency range.

## Point source

A point source of light produces spherical wavefronts. The irradiance in this case varies inversely with the square of the distance from the source.

${\displaystyle E={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}$

where

• r is the distance;
• P is the radiant power;
• A is the surface area of a sphere of radius r.

For quick approximations, this equation indicates that doubling the distance reduces irradiation to one quarter; or similarly, to double irradiation, reduce the distance to 0.7. When it is not a point source, for real light sources, the irradiance profile may be obtained by the image convolution of a picture of the light source.[4]

## Solar energy

The global irradiance on a horizontal surface on Earth consists of the direct irradiance Ee,dir and diffuse irradiance Ee,diff. On a tilted plane, there is another irradiance component, Ee,refl, which is the component that is reflected from the ground. The average ground reflection is about 20% of the global irradiance. Hence, the irradiance Ee on a tilted plane consists of three components:[5]

${\displaystyle E_{\mathrm {e} }=E_{\mathrm {e} ,\mathrm {dir} }+E_{\mathrm {e} ,\mathrm {diff} }+E_{\mathrm {e} ,\mathrm {refl} }.}$

The integral of solar irradiance over a time period is called "solar exposure" or "insolation".[5][6]

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy density we joule per cubic metre J/m3 M?L-1?T-2 Radiant energy per unit volume.
Radiant flux ?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".
Spectral flux ?e,?[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.
?e,?[nb 4] watt per metre W/m M?L?T-3
Radiant intensity Ie,?[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 Ie,?,?[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.
Ie,?,?[nb 4] watt per steradian per metre W?sr-1?m-1 M?L?T-3
Radiance Le,?[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,?,?[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".
Le,?,?[nb 4] watt per steradian per square metre, per metre W?sr-1?m-3 M?L-1?T-3
Flux density
Ee[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 flux density
Ee,?[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).
Ee,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
Radiosity Je[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 Je,?[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".
Je,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
Radiant exitance Me[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 Me,?[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".
Me,?[nb 4] 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,?[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".
He,?[nb 4] joule per square metre, per metre J/m3 M?L-1?T-2
Hemispherical emissivity ? N/A 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity ??
or
??
N/A 1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity ?? N/A 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity ??,?
or
??,?
N/A 1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A N/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 A?
or
A?
N/A 1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance A? N/A 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance A?,?
or
A?,?
N/A 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 N/A 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance R?
or
R?
N/A 1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance R? N/A 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance R?,?
or
R?,?
N/A 1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T N/A 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance T?
or
T?
N/A 1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance T? N/A 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance T?,?
or
T?,?
N/A 1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient ? 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 ??
or
??
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 ?? 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 ??,?
or
??,?
reciprocal metre m-1 L-1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
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 "?" (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 "?" (Greek).
5. ^ a b Directional quantities are denoted with suffix "?" (Greek).

## References

1. ^ Carroll, Bradley W. (2017-09-07). An introduction to modern astrophysics. p. 60. ISBN 978-1-108-42216-1. OCLC 991641816.
2. ^ a b c "Thermal insulation -- Heat transfer by radiation -- Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved .
3. ^ Griffiths, David J. (1999). Introduction to electrodynamics (3. ed., reprint. with corr. ed.). Upper Saddle River, NJ [u.a.]: Prentice-Hall. ISBN 0-13-805326-X.
4. ^ I. Moreno, P. X. Viveros-Méndez, "Modeling the irradiation pattern of LEDs at short distances," Opt. Express 29, 6845 (2021). https://doi.org/10.1364/OE.419428
5. ^ a b Quaschning, Volker (2003). "Technology fundamentals--The sun as an energy resource". Renewable Energy World. 6 (5): 90-93.
6. ^ Liu, B. Y. H.; Jordan, R. C. (1960). "The interrelationship and characteristic distribution of direct, diffuse and total solar radiation". Solar Energy. 4 (3): 1. Bibcode:1960SoEn....4....1L. doi:10.1016/0038-092X(60)90062-1.