Tables of Planck’s Radiation and Photon Functions*

Lowan, Arnold N.; Blanch, Gertrude

doi:10.1364/josa.30.000070

Cited by 45 publications

(9 citation statements)

References 2 publications

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“…25,26 The calculation of this integral has a long and distinguished history 27 with a range of approximations being purposed for it in the past. [28][29][30][31][32][33][34] By application of the polylogarithm function it is now possible to express Eq. (7) in closed form.…”

Section: Radiometric Casementioning

confidence: 99%

“…(33) dates back to work first performed in 1931 61 and has a long history of being tabulated. 28,[62][63][64][65][66][67] Here we have written this fractional function for the very first time in closed form in terms of polylogarithms. 68 A value for the thermal contrast equal to three in the actinometric case plays an equally important role to the value of four found for the radiometric case.…”

Section: Actinometric Casementioning

confidence: 99%

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Exact expressions for thermal contrast detected with thermal and quantum detectors

Stewart

Johnson

2014

SPIE Proceedings

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The detected thermal contrast is a recently defined figure of merit introduced to describe the overall performance of a detector detecting radiation from a thermal source. We examine the detected thermal contrast for the case where the target emissivity can be assumed to be a function of the temperature and independent of the wavelength within a narrow wavelength interval of interest. Exact expressions are developed to evaluate the thermal contrast detected by both thermal and quantum detectors for focal-plane radiation detecting instruments. Expressions for the thermal contrast of a blackbody, an intrinsic radiative quantity of a body independent of the detection process, and simplified expressions for the detected thermal contrast for target emissivities which are well approximated by the grey body approximation are also given. It is found the contribution in the detected thermal contrast consists of two terms. The first results from changes occurring in the emissivity of a target with temperature while the second results from purely radiative processes. The size of the detected thermal contrast is found to be similar for the two detector types within typical infrared wavelength intervals of interest, contradicting a result previously reported in the literature. The exact results are presented in terms of a polylogarithmic formulation of the problem and extend a number of approximation schemes that have been proposed and developed in the past.

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Section: Radiometric Casementioning

confidence: 99%

Section: Actinometric Casementioning

confidence: 99%

Exact expressions for thermal contrast detected with thermal and quantum detectors

Stewart

Johnson

2014

SPIE Proceedings

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“…41 Only recently was it formally shown the integral could not be evaluated in closed form in terms of any of the elementary transcendental functions k of mathematics. 42 While a number of approximations for its accurate numerical evaluation have been developed, 32,33,[43][44][45][46][47] what is not widely known is if one writes the spectral radiant exitance in terms of a polylogarithm, using known properties for the function the integral can be readily evaluated in closed form in terms of this special function.…”

Section: Radiometric Casementioning

confidence: 99%

“…43 An alternative form leading to the fractional function of the first kind for the actinometric case can also be defined. In terms of an integrand consisting of a product between wavelength and the spectral radiant exitance, it can be shown that 36…”

Section: Actinometric Casementioning

confidence: 99%

Analysis of the relative merits of the 3-5 μm and the 8-12 μm spectral bands using detected thermal contrast

Stewart

2015

SPIE Proceedings

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The two most important atmospheric transmission bands in the infrared occur at 3-5 µm and 8-12 µm respectively. For a given infrared detector a common question that continues to be asked is, of the two spectral bands, which, if any, gives the better performance? While seemly an innocent enough question, the literature attests it has not been without controversy. Conflicting and often contradictory results have been given that tend to reflect the predilections of the proponent rather than any sort of measured consideration based on technical factors likely to affect performance. In this study an analysis designed to assess the relative merits of infrared detectors operating in the 3-5 µm and 8-12 µm spectral bands based on the recently defined figure of merit known as the detected thermal contrast is undertaken. The detected thermal contrast attempts to describe the overall performance of the sequence of events from the initial emission of thermal radiation at the surface of a target to the final measurable output signal seen in the detecting instrument. Under ideal limiting conditions typical of those found for many industrial and scientific applications, by considering targets whose spectral emissivities vary as a function of both wavelength and temperature, exact expressions based on the recently introduced polylogarithmic formulation of the problem are developed for both thermal and quantum detectors. It is found the 3-5 µm waveband for either detector type gives better performance while differences between the two types of detectors is not as significant as one might initially expect. The work not only extends upon a number of approximate schemes that have been proposed and developed in the past where target emissivities as a function of the temperature have been used, it also challenges a number of previously reported results.

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“…(8) -the fractional function of the first kind for the actinometric case -a fact not recognised by either author. While not as frequently used as the corresponding radiometric case, initial study of the function dates back to the work of Zanstra in 1931 [4] and has a long history of being tabulated [10][11][12][13][14][15][16]. Other relatively simple examples in the infrared where the blackbody fractional function of the first kind for the actinometric case makes an appearance is in the calculation of the effective radiance for an infrared detector [17] or in the calculation of the detected thermal contrast [18] for a quantum detector [19][20][21][22].…”

Section: From Vu Jádè To Dèjá Vumentioning

confidence: 99%

A note on the general class of blackbody fractional functions

Stewart

2014

Infrared Physics & Technology

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Tables of Planck’s Radiation and Photon Functions*

Cited by 45 publications

References 2 publications

Exact expressions for thermal contrast detected with thermal and quantum detectors

Exact expressions for thermal contrast detected with thermal and quantum detectors

Analysis of the relative merits of the 3-5 μm and the 8-12 μm spectral bands using detected thermal contrast

A note on the general class of blackbody fractional functions

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