For other uses, see Radiance (disambiguation).

## Description

Radiance is useful because it indicates how much of the power emitted, reflected, transmitted or received by a surface will be received by an optical system looking at that surface from some angle of view. In this case, the solid angle of interest is the solid angle subtended by the optical system's entrance pupil. Since the eye is an optical system, radiance and its cousin luminance are good indicators of how bright an object will appear. For this reason, radiance and luminance are both sometimes called "brightness". This usage is now discouraged (see the article Brightness for a discussion). The nonstandard usage of "brightness" for "radiance" persists in some fields, notably laser physics.

The radiance divided by the index of refraction squared is invariant in geometric optics. This means that for an ideal optical system in air, the radiance at the output is the same as the input radiance. This is sometimes called conservation of radiance. For real, passive, optical systems, the output radiance is at most equal to the input, unless the index of refraction changes. As an example, if you form a demagnified image with a lens, the optical power is concentrated into a smaller area, so the irradiance is higher at the image. The light at the image plane, however, fills a larger solid angle so the radiance comes out to be the same assuming there is no loss at the lens.

## Mathematical definitions

Radiance of a surface, denoted Le,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is given by:[3]

$L_{\mathrm{e},\Omega} = \frac{\partial^2 \Phi_\mathrm{e}}{\partial \Omega\, \partial A \cos \theta},$

where

• ∂ is the partial derivative symbol;
• Ω is the solid angle;
• A cos θ is the projected area.

In general Le,Ω is a function of viewing direction, depending on θ through cos θ and azimuth angle through ∂Φe/∂Ω. For the special case of a Lambertian surface, 2Φe/(∂ΩA) is proportional to cos θ, and Le,Ω is isotropic (independent of viewing direction).

When calculating the radiance emitted by a source, A refers to an area on the surface of the source, and Ω to the solid angle into which the light is emitted. When calculating radiance received by a detector, A refers to an area on the surface of the detector and Ω to the solid angle subtended by the source as viewed from that detector. When radiance is conserved, as discussed above, the radiance emitted by a source is the same as that received by a detector observing it.

Spectral radiance in frequency of a surface, denoted Le,Ω,ν, is given by:[3]

$L_{\mathrm{e},\Omega,\nu} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \nu},$

where ν is the frequency.

Spectral radiance in wavelength of a surface, denoted Le,Ω,λ, is given by:[3]

$L_{\mathrm{e},\Omega,\lambda} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \lambda},$

where λ is the wavelength.

Radiance of a surface is related to étendue by:

$L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e}}{\partial G},$

where

• n is the refractive index in which that surface is immersed;
• G is the étendue of the light beam.

As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore, basic radiance defined as:[4]

$L_{\mathrm{e},\Omega}^* = \frac{L_{\mathrm{e},\Omega}}{n^2}$

is also conserved. In real systems, the étendue may increase (for example due to scattering) or the radiant flux may decrease (for example due to absorption) and, therefore, basic radiance may decrease. However, étendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase.

## References

1. ^ Palmer, James M. "The SI system and SI units for Radiometry and photometry". Archived from the original on August 2, 2012.
2. ^ Rowlett, Russ. "How Many? A Dictionary of Units of Measurement". Retrieved 10 August 2012.
3. ^ a b c "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.
4. ^ William Ross McCluney, Introduction to Radiometry and Photometry, Artech House, Boston, MA, 1994 [ISBN 978-0890066782]