Spectral radiance

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Template:Short description 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. The SI unit of spectral radiance in frequency is the watt per steradian per square metre per hertz (Template:Nobreak) and that of spectral radiance in wavelength is the watt per steradian per square metre per metre (Template:Nobreak)—commonly the watt per steradian per square metre per nanometre (Template:Nobreak). The microflick is also used to measure spectral radiance in some fields.[1][2]

Spectral radiance gives a full radiometric description of the field of classical electromagnetic radiation of any kind, including thermal radiation and light. It is conceptually distinct from the descriptions in explicit terms of Maxwellian electromagnetic fields or of photon distribution. It refers to material physics as distinct from psychophysics.

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. The concept refers to an area, projected from the element of source area into a plane at right angles to the line of propagation, and to an element of solid angle subtended by the detector at the element of source area.[3][4][5][6][7][8][9]

The term brightness is also sometimes used for this concept.[3][10] The SI system states that the word brightness should not be so used, but should instead refer only to psychophysics.

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

Definition

The specific (radiative) intensity is a quantity that describes the rate of radiative transfer of energy at Template:Math, a point of space with coordinates Template:Math, at time Template:Mvar. It is a scalar-valued function of four variables, customarily[3][4][5][11][12][13] written as I(𝐱,t;𝐫1,ν) where: Template:Unbulleted list Template:Math is defined to be such that a virtual source area, Template:Math, containing the point Template:Math , is an apparent emitter of a small but finite amount of energy Template:Math transported by radiation of frequencies Template:Math in a small time duration Template:Math , where dE=I(𝐱,t;𝐫1,ν)cos⁡(θ1)dA1dΩ1dνdt, and where Template:Math is the angle between the line of propagation Template:Math and the normal Template:Math to Template:Math; the effective destination of Template:Math is a finite small area Template:Math , containing the point Template:Math , that defines a finite small solid angle Template:Math about Template:Math in the direction of Template:Math. The cosine accounts for the projection of the source area Template:Math into a plane at right angles to the line of propagation indicated by Template:Math.

The use of the differential notation for areas Template:Math indicates they are very small compared to Template:Math, the square of the magnitude of vector Template:Math, and thus the solid angles Template:Math are also small.

There is no radiation that is attributed to Template:Math itself as its source, because Template:Math is a geometrical point with no magnitude. A finite area is needed to emit a finite amount of light.

Invariance

For propagation of light in a vacuum, the definition of specific (radiative) intensity implicitly allows for the inverse square law of radiative propagation.[12][14] The concept of specific (radiative) intensity of a source at the point Template:Math presumes that the destination detector at the point Template:Math has optical devices (telescopic lenses and so forth) that can resolve the details of the source area Template:Math. 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 refers to the area Template:Math of the detecting surface.

This may be understood by looking at the diagram. The factor Template:Math has the effect of converting the effective emitting area Template:Math into a virtual projected area Template:Math at right angles to the vector Template:Math from source to detector. The solid angle Template:Math also has the effect of converting the detecting area Template:Math into a virtual projected area Template:Math at right angles to the vector Template:Math , so that Template:Math . Substituting this for Template:Math in the above expression for the collected energy Template:Math, one finds Template:Math: when the emitting and detecting areas and angles Template:Math and Template:Math, Template:Math and Template:Math, are held constant, the collected energy Template:Math is inversely proportional to the square of the distance Template:Math between them, with invariant Template:Math .

This may be expressed also by the statement that Template:Math is invariant with respect to the length Template:Mvar of Template:Math ; 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 Template:Mvar of the ray Template:Math.[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]

Reciprocity

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.

Étendue and reciprocity

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, Template:Math , of the pencil of light which "connects" the two surface elements Template:Math and Template:Math is defined as d2G=dA1cos⁡(θ1)dΩ1=dA1dA2cos⁡(θ1)cos⁡(θ2)r2=dA2cos⁡(θ2)dΩ2.

This can help understand the geometrical aspects of the Stokes-Helmholtz reversion-reciprocity principle.

Collimated beam

For the present purposes, the light from a star can be treated as a practically collimated beam, but apart from this, a collimated beam is rarely if ever found in nature, though artificially produced beams can be very nearly collimated. For some purposes the rays of the sun can be considered as practically collimated, because the sun subtends an angle of only 32′ of arc.[17] The specific (radiative) intensity is suitable for the description of an uncollimated radiative field. 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]

Rays

Specific (radiative) intensity is built on the idea of a pencil of rays of light.[19][20][21]

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, the energy does not in general propagate at right angles to the wavefronts.[22][23]

Alternative approaches

The specific (radiative) intensity is a radiometric concept. 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.

Another way to describe the radiative field is in terms of the Maxwell electromagnetic field, which includes the concept of the wavefront. The rays of the radiometric and photon concepts are along the time-averaged Poynting vector of the Maxwell field.[26] In an anisotropic medium, the rays are not in general perpendicular to the wavefront.[22][23]

References

Template:Reflist

  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ 3.0 3.1 3.2 Planck, M. (1914) The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, pages 13-15.
  4. ↑ 4.0 4.1 Chandrasekhar, S. (1950). Radiative Transfer, Oxford University Press, Oxford, pages 1-2.
  5. ↑ 5.0 5.1 5.2 Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York Template:ISBN., pages 311-312.
  6. ↑ Goody, R.M., Yung, Y.L. (1989). Atmospheric Radiation: Theoretical Basis, 2nd edition, Oxford University Press, Oxford, New York, 1989, Template:ISBN, page 16.
  7. ↑ Liou, K.N. (2002). An Introduction of Atmospheric Radiation, second edition, Academic Press, Amsterdam, Template:ISBN, page 4.
  8. ↑ Hapke, B. (1993). Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, Template:ISBN, page 64.
  9. ↑ Rybicki, G.B., Lightman, A.P. (1979/2004). Radiative Processes in Astrophysics, reprint, John Wiley & Sons, New York, Template:ISBN, page 3.
  10. ↑ Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, Template:ISBN, page 194.
  11. ↑ Kondratyev, K.Y. (1969). Radiation in the Atmosphere, Academic Press, New York, page 10.
  12. ↑ 12.0 12.1 12.2 Mihalas, D. (1978). Stellar Atmospheres, 2nd edition, Freeman, San Francisco, Template:ISBN, pages 2-5.
  13. ↑ Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, Template:ISBN, pages 194-199.
  14. ↑ 14.0 14.1 Rybicki, G.B., Lightman, A.P. (1979). Radiative Processes in Astrophysics, John Wiley & Sons, New York, Template:ISBN, pages 7-8.
  15. ↑ Bohren, C.F., Clothiaux, E.E. (2006). Fundamentals of Atmospheric Radiation, Wiley-VCH, Weinheim, Template:ISBN, pages 191-192.
  16. ↑ Planck, M. (1914). The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, page 35.
  17. ↑ Goody, R.M., Yung, Y.L. (1989). Atmospheric Radiation: Theoretical Basis, 2nd edition, Oxford University Press, Oxford, New York, 1989, Template:ISBN, page 18.
  18. ↑ Hapke, B. (1993). Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, Template:ISBN, see pages 12 and 64.
  19. ↑ Planck, M. (1914). The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, Chapter 1.
  20. ↑ Levi, L. (1968). Applied Optics: A Guide to Optical System Design, 2 volumes, Wiley, New York, volume 1, pages 119-121.
  21. ↑ Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, Template:ISBN, pages 116-125.
  22. ↑ 22.0 22.1 Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, Template:ISBN, pages 792-795.
  23. ↑ 23.0 23.1 Hecht, E., Zajac, A. (1974). Optics, Addison-Wesley, Reading MA, page 235.
  24. ↑ Mihalas, D. (1978). Stellar Atmospheres, 2nd edition, Freeman, San Francisco, Template:ISBN, page 10.
  25. ↑ Lamb, W.E., Jr (1995). Anti-photon, Applied Physics, B60: 77-84.[1]
  26. ↑ Mihalas, D. (1978). Stellar Atmospheres, 2nd edition, Freeman, San Francisco, Template:ISBN, page 11.
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