Spectral radiance

In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength.

It is conceptually distinct from the descriptions in explicit terms of Maxwellian electromagnetic fields or of photon distribution.

For the concept of specific intensity, the line of propagation of radiation lies in a semi-transparent medium which varies continuously in its optical properties.

[3][10] The SI system states that the word brightness should not be so used, but should instead refer only to psychophysics.

The specific (radiative) intensity is a quantity that describes the rate of radiative transfer of energy at P1, a point of space with coordinates x, at time t. It is a scalar-valued function of four variables, customarily[3][4][5][11][12][13] written as

and where θ1 is the angle between the line of propagation r and the normal P1N1 to dA1; the effective destination of dE is a finite small area dA2 , containing the point P2 , that defines a finite small solid angle dΩ1 about P1 in the direction of r. The cosine accounts for the projection of the source area dA1 into a plane at right angles to the line of propagation indicated by r. The use of the differential notation for areas dAi indicates they are very small compared to r2, the square of the magnitude of vector r, and thus the solid angles dΩi are also small.

[12][14] The concept of specific (radiative) intensity of a source at the point P1 presumes that the destination detector at the point P2 has optical devices (telescopic lenses and so forth) that can resolve the details of the source area dA1.

This is because it is defined per unit solid angle, the definition of which refers to the area dA2 of the detecting surface.

Substituting this for dΩ1 in the above expression for the collected energy dE, one finds dE = I (x, t ; r1, ν) cos θ1 dA1 cos θ2 dA2 dν dt / r2: when the emitting and detecting areas and angles dA1 and dA2, θ1 and θ2, are held constant, the collected energy dE is inversely proportional to the square of the distance r between them, with invariant I (x, t ; r1, ν) .

This may be expressed also by the statement that I (x, t ; r1, ν) is invariant with respect to the length r of r ; that is to say, provided the optical devices have adequate resolution, and that the transmitting medium is perfectly transparent, as for example a vacuum, then the specific intensity of the source is unaffected by the length r of the ray r.[12][14][15] For the propagation of light in a transparent medium with a non-unit non-uniform refractive index, the invariant quantity along a ray is the specific intensity divided by the square of the absolute refractive index.

[16] For the propagation of light in a semi-transparent medium, specific intensity is not invariant along a ray, because of absorption and emission.

Nevertheless, the Stokes-Helmholtz reversion-reciprocity principle applies, because absorption and emission are the same for both senses of a given direction at a point in a stationary medium.

The term étendue is used to focus attention specifically on the geometrical aspects.

In the notation of the present article, the second differential of the étendue, d2G , of the pencil of light which "connects" the two surface elements dA1 and dA2 is defined as

The integrals of specific (radiative) intensity with respect to solid angle, used for the definition of spectral flux density, are singular for exactly collimated beams, or may be viewed as Dirac delta functions.

Therefore, the specific (radiative) intensity is unsuitable for the description of a collimated beam, while spectral flux density is suitable for that purpose.

[18] Specific (radiative) intensity is built on the idea of a pencil of rays of light.

That is to say, in an optically anisotropic crystal, the energy does not in general propagate at right angles to the wavefronts.

Related to it is the intensity in terms of the photon distribution function,[5][24] which uses the metaphor[25] of a particle of light that traces the path of a ray.

The idea common to the photon and the radiometric concepts is that the energy travels along rays.

The rays of the radiometric and photon concepts are along the time-averaged Poynting vector of the Maxwell field.