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Spectral radiance

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In radiometry, specific radiative intensity or spectral radiance is a fundamental quantity that fully describes the classical electromagnetic radiation field. It is a description based on radiometry rather than Maxwellian electromagnetic fields or photon distribution.

The concept is used for some practical measurements, when it is often called spectral radiance. Also, the theory of radiative transfer uses the concept of specific radiative intensity to give a radiometric description of a continuous field of radiation.

Definition

The geometry for the definition of specific radiative intensity. Note the potential in the geometry for laws of reciprocity.

The specific radiative intensity is a radiometric concept concerned with the propagation of energy rather than with the wavefronts. Specific radiative intensity at P1, a point of space x at t, has a scalar-value function of:

I (x, t ; r1, ν)

Where:

ν denotes frequency. As such, specific radiative intensity is somewhat related to the notion of intensity in terms of the photon distribution function. Specific radiative intensity has SI units W·sr·m·Hz.
r1 denotes a unit vector with the direction and sense of the geometrical vector r, from
the source point P1, to
the detection point P2.

There is no radiation that is attributed to P1 itself as its source. The single points P1 and P2 are connected with their respective areas, dA1 and dA2, to make the geometry explicit.

The virtual source, dA1, containing the point P1 is an apparent emitter of a small but finite amount of energy dE equal to:

dE = I (x, t ; r1, ν) (dA1 cos θ1 dΩ1) dν dt ,

where θ1 is the angle between r and the normal P1N1 to dA1, of which the effective destination is a finite small area dA2 around P2 that defines a finite small solid angle dΩ1. The element dE is the energy transported by radiation of frequencies (ν, ν + dν) across a surface element dA1, that contains the point P1, into a solid angle dΩ1 around r1. Included is classical electromagnetic radiation of any kind, including thermal radiation and light, carrying radiant energy, in the direction and sense of r1 with frequency ν.

The use of the differential notation for areas dAi indicates they are very small compared to r, the square of the magnitude of vector r, and thus the angles dΩi are also small.

Reciprocity

The definition of specific radiative intensity implicitly allows for the inverse square law of radiative propagation. 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. Then the specific radiative intensity of the source is independent of the distance from source to detector; it is a property of the source alone. This is because it is defined per unit solid angle, the definition of which includes the area of the effective aperture dA2 of the detecting surface. This may be expressed also by the statement that I (x, t ; r1, ν) is invariant with respect to the length r of r: provided the optical devices have adequate resolution, and that the transmitting medium is perfectly transparent, as for example a vacuum, no matter how far it is from source to detector, the specific radiative intensity of the source is unaffected.

Étendue

The term étendue is used to focus attention specifically on the geometrical aspects. The reciprocal character of étendue is indicated in the article about it. Étendue is defined as a second differential. In the notation of the present article, the second differential of the étendue, dG , of the pencil of light which "connects" the two surface elements dA1 and dA2 is defined as

dG = dA1 cos θ1 dΩ1 = d A 1   d A 2   cos ⁡ θ 1   cos ⁡ θ 2 r 2 {\displaystyle {\frac {{\mbox{d}}A_{1}\ \,{\mbox{d}}A_{2}\ \cos {\theta _{1}}\ \cos {\theta _{2}}}{r^{2}}}} = dA2 cos θ2 dΩ2.

This can help understand the Fresnel-Stokes-Helmholtz-Stewart reversion-reciprocity principle, which was stated in part by Stokes (1849) and with reference to polarization on page 169 of Helmholtz's Handbuch der physiologischen Optik of 1856 as cited by Planck.

Alternative approaches

Specific radiative intensity is built on the idea of a pencil of rays of light.

A similar notion is the intensity in terms of the photon distribution function, which uses the metaphor 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 are in the direction of the time-averaged Poynting vector. In an optically isotropic medium, the rays are normals to the wavefronts, but in an optically anisotropic crystalline medium, they are in general at angles to those normals. That is to say, in an optically anisotropic crystal, in general the energy does not propagate at right angles to the wavefronts.

According to Paltridge and Platt (1976), there are various nomenclatures and general approaches to electromagnetic radiative transfer. An alternative term for the present concept of specific radiative intensity is spectral radiance. These two terms are purely physically oriented without concern for the psychophysical aspect referred to by the term luminance.

See also

References

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