${\displaystyle L_{\text{e}}=\frac{\text{d}{I}_{\text{e}}}{\text{d}A}\frac{1}{\cos \alpha }}$

where I_e is radiant intensity, A is area, and α is the angle between the normal to the surface at the specified point and the specified direction

Note 1 to entry: In a practical sense, the definition of radiance can be thought of as dividing a real or imaginary surface into an infinite number of infinitesimally small surfaces which can be considered as point sources, each of which has a specific radiant intensity, I_e, in the specified direction. The radiance of the surface is then the integral of these radiance elements over the whole surface.

The equation in the definition can mathematically be interpreted as a derivative (i.e. a rate of change of radiant intensity with projected area) and could alternatively be rewritten in terms of the average radiant intensity, ${\overline{I}}_{\text{e}}$, as:

${\displaystyle L_{\text{e}}=\underset{A\to \text{0}}{\text{lim}}\frac{{\overline{I}}_{\text{e}}}{A}\frac{\text{1}}{\cos \alpha }}$.

Hence, radiance is often considered as a quotient of averaged quantities; the area, A, should be small enough so that uncertainties due to variations in radiant intensity within that area are negligible; otherwise, the quotient ${\displaystyle {\overline{L}}_{\text{e}}=\frac{{\overline{I}}_{\text{e}}}{A}\frac{\text{1}}{\cos \alpha }}$ gives the average radiance and the specific measurement conditions have to be reported with the result.

Note 2 to entry: For a surface being irradiated, an equivalent formula in terms of irradiance, E_e, and solid angle, Ω, is ${\displaystyle L_{\text{e}}=\frac{\text{d}{E}_{\text{e}}}{\text{d}\Omega }\frac{\text{1}}{\cos \theta }}$, where θ is the angle between the normal to the surface being irradiated and the direction of irradiation. This form is useful when the source has no surface (e.g. the sky, the plasma of a discharge).

Note 3 to entry: An equivalent formula is ${\displaystyle L_{\text{e}}=\frac{\text{d}{\Phi }_{\text{e}}}{\text{d}G}}$, where Φ_e is radiant flux and G is geometric extent.

Note 4 to entry: Radiant flux can be obtained by integrating radiance over projected area, A·cos α, and solid angle, Ω: ${\Phi }_{\text{e}}=\text{}{\displaystyle \iint L_{\text{e}}}\cdot \cos \alpha \text{d}A\text{d}\Omega$.

Note 5 to entry: Since the optical extent, expressed by G·n², where G is geometric extent and n is refractive index, is invariant, the quantity expressed by L_e·n⁻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 6 to entry: The equation in the definition can also be described as a function of radiant flux, Φ_e. In this case, it is mathematically interpreted as a second partial derivative of the radiant flux at a specified point (x, y) in space in a specified direction (ϑ, φ) with respect to projected area, A·cos α, and solid angle, Ω:

${\displaystyle L_{\text{e}}(x,y,\vartheta ,\phi )=\frac{{\partial }^2{\Phi }_{\text{e}}(x,y,\vartheta ,\phi )}{\partial A(x,y)\cdot \cos \alpha \cdot \partial \Omega \left(\vartheta ,\phi \right)}}$

where α is the angle between the normal to that area at the specified point and the specified direction.

Note 7 to entry: The corresponding photometric quantity is "luminance". The corresponding quantity for photons is "photon radiance".

Note 8 to entry: The radiance is expressed in watt per square metre per steradian (W·m⁻²·sr⁻¹).

Note 9 to entry: This entry was numbered 845-01-34 in IEC 60050-845:1987.