radiance (in a given direction, at a given point of a real or imaginary surface) [*L*_{e}]

quantity defined by the equation:

where

d*Φ*_{e} is the radiant flux transmitted by an elementary beam passing through the given point and propagating in the solid angle, d*Ω*, containing the given direction;

d*A* is the area of a section of that beam containing the given point;

*Θ* is the angle between the normal to that section and the direction of the beam

Unit: W·m^{-2}·sr^{-1}

NOTE 1 The above equation does not represent a derivative (i.e. a rate of change of flux with solid angle or area) but rather the quotient of an element of flux by an element of solid angle and an element of area. In strict mathematical terms the definition could be written:

In practical measurements, *A *and *Ω *should be small enough that variations in *Φ*_{v} do not affect the result. Otherwise, the ratio _{} gives the average radiance and the exact measurement conditions must be specified.

NOTE 2 In the following notes the symbols for the quantities are without subscripts because the formulae are also valid for the terms “luminance” and “photon radiance”.

NOTE 3 For an area, d*A*, of the surface of a source, since the intensity, d*I*, of d*A* in the given direction is _{}, then an equivalent formula is _{}, a form mostly used in illuminating engineering.

NOTE 4 For an area, d*A*, of a surface receiving the beam, since the irradiance or illuminance, d*E*, produced by the beam on d*A* is _{}, then an equivalent formula is _{}, a form useful when the source has no surface (e.g. the sky, the plasma of a discharge).

NOTE 5 Making use of the geometric extent, d*G*, of the elementary beam, since d*G* = d*A* cos*Θ* d*Ω*, then an equivalent formula is _{}.

NOTE 6 Since the optical extent, *G**n*^{2}, (see NOTE to “geometric extent”) is invariant, then the quantity *Ln*^{-2} is also invariant along the path of the beam if the losses by absorption, reflection and diffusion are taken as 0. That quantity is called “basic radiance”.

NOTE 7 The relation between d*Φ* and *L* given in the formulae above is sometimes called the “basic law of radiometry and photometry”:

with the notation given here and at “geometric extent”.

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