# Irradiance

In radiometry, **irradiance** is the radiant flux (power) *received* by a *surface* per unit area. The SI unit of irradiance is the watt per square metre (W/m^{2}). The CGS unit erg per square centimetre per second (erg·cm^{−2}·s^{−1}) is often used in astronomy. Irradiance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

**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

Irradiance of a surface, denoted *E*_{e} ("e" for "energetic", to avoid confusion with photometric quantities), is defined as^{[1]}

where

- ∂ is the partial derivative symbol;
- Φ
_{e}is the radiant flux received; *A*is the area.

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

### Spectral irradiance

Spectral irradiance in frequency of a surface, denoted *E*_{e,ν}, is defined as^{[1]}

where *ν* is the frequency.

Spectral irradiance in wavelength of a surface, denoted *E*_{e,λ}, is defined as^{[1]}

where *λ* 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:

where

- < • > is the time-average;
**S**is the Poynting vector;*α*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^{[2]}

where

*E*_{m}is the amplitude of the wave's electric field;*n*is the refractive index of the medium of propagation;- c is the speed of light in vacuum;
- μ
_{0}is the vacuum permeability; - ε
_{0}is the vacuum permittivity.

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

## Solar energy

The global irradiance on a horizontal surface on Earth consists of the direct irradiance *E*_{e,dir} and diffuse irradiance *E*_{e,diff}. On a tilted plane, there is another irradiance component, *E*_{e,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 *E*_{e} on a tilted plane consists of three components:^{[3]}

The integral of solar irradiance over a time period is called "solar exposure" or "insolation".^{[3]}^{[4]}

## SI radiometry units

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol^{[nb 1]} |
Name | Symbol | Symbol | ||||

Radiant energy | Q_{e}^{[nb 2]} |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}^{[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 | Φ_{e,ν}^{[nb 3]}or Φ _{e,λ}^{[nb 4]} |
watt per hertzorwatt per metre |
W/HzorW/m |
M⋅L^{2}⋅T^{−2}orM⋅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 | I_{e,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 | I_{e,Ω,ν}^{[nb 3]}or I_{e,Ω,λ}^{[nb 4]} |
watt per steradian per hertzorwatt per steradian per metre |
W⋅sr^{−1}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−1} |
M⋅L^{2}⋅T^{−2}orM⋅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 | 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 | L_{e,Ω,ν}^{[nb 3]}or L_{e,Ω,λ}^{[nb 4]} |
watt per steradian per square metre per hertzorwatt per steradian per square metre, per metre |
W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−3} |
M⋅T^{−2}orM⋅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 |
E_{e}^{[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 |
E_{e,ν}^{[nb 3]}or E_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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^{4}Jy). | |||

Radiosity | J_{e}^{[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 | J_{e,ν}^{[nb 3]}or J_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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 | M_{e}^{[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 | M_{e,ν}^{[nb 3]}or M_{e,λ}^{[nb 4]} |
watt per square metre per hertzorwatt per square metre, per metre |
W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} |
M⋅T^{−2}orM⋅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 | H_{e} |
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 | H_{e,ν}^{[nb 3]}or H_{e,λ}^{[nb 4]} |
joule per square metre per hertzorjoule per square metre, per metre |
J⋅m^{−2}⋅Hz^{−1}orJ/m ^{3} |
M⋅T^{−1}orM⋅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 | ε_{ν}or ε_{λ} |
1 | Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | |||||

Directional emissivity | ε_{Ω} |
1 | Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. | |||||

Spectral directional emissivity | ε_{Ω,ν}or ε_{Ω,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 | A_{ν}or A_{λ} |
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_{Ω} |
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_{Ω,ν}or A_{Ω,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 | R_{ν}or R_{λ} |
1 | Spectral flux reflected by a surface, divided by that received by that surface. | |||||

Directional reflectance | R_{Ω} |
1 | Radiance reflected by a surface, divided by that received by that surface. | |||||

Spectral directional reflectance | R_{Ω,ν}or R_{Ω,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 | T_{ν}or T_{λ} |
1 | Spectral flux transmitted by a surface, divided by that received by that surface. | |||||

Directional transmittance | T_{Ω} |
1 | Radiance transmitted by a surface, divided by that received by that surface. | |||||

Spectral directional transmittance | T_{Ω,ν}or T_{Ω,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 | μ_{ν}or μ_{λ} |
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 | μ_{Ω} |
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 | μ_{Ω,ν}or μ_{Ω,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 |

- ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- 1 2 3 4 5 Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - 1 2 3 4 5 6 7 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.
- 1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- 1 2 Directional quantities are denoted with suffix "Ω" (Greek).

## See also

- Illuminance
- Spectral flux density
- Albedo
- Fluence
- Insolation
- Light diffusion
- PI curve (photosynthesis-irradiance curve)
- Solar azimuth angle
- Solar irradiance
- Solar noon
- Stefan–Boltzmann law

## References

- 1 2 3 "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions".
*ISO 9288:1989*. ISO catalogue. 1989. Retrieved 2015-03-15. - ↑ 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. - 1 2 Quaschning, Volker (2003). "Technology fundamentals—The sun as an energy resource".
*Renewable Energy World*.**6**(5): 90–93. - ↑ 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.