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Specific radiative intensity or spectral radiance is a fundamental quantity that fully describes the classical electromagnetic radiative field. It gives a radiometric description as distinct from a Maxwellian electromagnetic field description and from a photon distribution description. The theory of radiative transfer uses the concept of specific radiative intensity to give a radiometric description of a continuous field of radiation. Also the concept is used for some practical measurements, when it is often called spectral radiance.

Definition

The classical electromagnetic radiative field at a point P₁ of space x and instant of time t can be described in radiometric terms by a quantity I (x, t ; r₁, ν), the specific radiative intensity. I (x, t ; r₁, ν) is a scalar-valued function of its arguments x, t, r₁, and ν, where 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.

It refers to electromagnetic radiation of any kind, including thermal radiation and light, carrying radiant energy, in the direction and sense of r₁ with frequency ν, and is such that the element dE of 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, is

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 . There is no radiation that is attributed to a single point as its source. The single points P₁ and P₂ are there to make the geometry explicit. The virtual source containing the point P₁ is a finite small area dA₁ which is the apparent emitter of a small but finite amount of energy dE of which the effective destination is a finite small area dA₂ around P₂ that defines a finite small solid angle dΩ₁. It is implicit in the use of the differential notation that the vector r has a square magnitude r large in comparison with the areas dA_i and solid angles dΩ_i.

Specific radiative intensity has SI units W·sr·m·Hz.

Alternative approaches

As noted for example by Paltridge and Platt (1976), there are diversities of nomenclature and general approach to electromagnetic radiative transfer, according to different disciplines and temperaments. Some approaches are more practical and oriented to authoritative consensus of committees and others are more oriented towards coherence and consistency in the logical development of physical theory. An alternative term, with authoritative support, for the present concept of specific radiative intensity is spectral radiance. These two are purely physically oriented without concern for the psychophysical aspect referred to by the term luminance.

Energetic and geometrical aspects

The notion of specific radiative intensity has geometrical and energetic aspects, which are distinct.

The inverse square law and Lambert's cosine law are geometrical aspects of the definition of specific radiative intensity.

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.

The geometry also takes into account Lambert's cosine law.

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 = ${\displaystyle {\frac {{\mbox{d}}A_{1}\ \,{\mbox{d}}A_{2}\ \cos {\theta _{1}}\ \cos {\theta _{2}}}{r^{2}}}}$ = 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.

Optical ray aspect

Specific radiative intensity is a radiometric concept, but it is closely related to the notion of intensity in terms of the photon distribution function.. The radiometric concept is built on the idea of a pencil of rays of light, while the photon concept 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. The specific radiative intensity is concerned with the propagation of energy rather than with the wavefronts.

References

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