Abstract

Optical spectra are typically normalized per unit wavelength or per unit photon energy, yielding two different expressions or curves. It is advantageous instead to normalize a spectrum to a constant fractional bandwidth, providing a unique expression independent of whether the bandwidth is in dimensions of wavelength or of photon energy. For the Sun, whereas a per-unit-wavelength spectrum peaks in the green and a per-unit-photon-energy spectrum peaks in the IR, when the proposed normalization is used, the output peaks in the red. This approach applies to any spectral source and provides curves of constant spectral resolving power, as produced by many spectrometers.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), pp. 184–187.
  2. This is the irradiance at the surface of the Sun. To obtain the irradiance intercepting the Earth, this expression must be divided by the square of the ratio of the Earth–Sun distance to the radius of the Sun.
  3. B. H. Soffer, D. K. Lynch, “Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision,” Am. J. Phys. 67, 946–953 (1999).
    [CrossRef]
  4. Although the spectrum is centered in the red, we do not perceive the Sun as being red. This can be explained by the fact that the spectrum covers our visible range and as a result of our physiological ability to redefine perceived colors with reference to the illumination source. See R. Mausfeld, “Color perception: from Grassmann codes to a dual code for object and illumination colors,” in Color Vision Perspectives from Different DisciplinesW. G. K. Backhaus, R. Kliegl, J. S. Werner, eds. (de Gruyter, Berlin, 1998), Chap. 12.
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 333.
  6. Ref. 5, pp. 406–407.

1999 (1)

B. H. Soffer, D. K. Lynch, “Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision,” Am. J. Phys. 67, 946–953 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 333.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), pp. 184–187.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), pp. 184–187.

Lynch, D. K.

B. H. Soffer, D. K. Lynch, “Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision,” Am. J. Phys. 67, 946–953 (1999).
[CrossRef]

Mausfeld, R.

Although the spectrum is centered in the red, we do not perceive the Sun as being red. This can be explained by the fact that the spectrum covers our visible range and as a result of our physiological ability to redefine perceived colors with reference to the illumination source. See R. Mausfeld, “Color perception: from Grassmann codes to a dual code for object and illumination colors,” in Color Vision Perspectives from Different DisciplinesW. G. K. Backhaus, R. Kliegl, J. S. Werner, eds. (de Gruyter, Berlin, 1998), Chap. 12.

Soffer, B. H.

B. H. Soffer, D. K. Lynch, “Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision,” Am. J. Phys. 67, 946–953 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 333.

Am. J. Phys. (1)

B. H. Soffer, D. K. Lynch, “Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision,” Am. J. Phys. 67, 946–953 (1999).
[CrossRef]

Other (5)

Although the spectrum is centered in the red, we do not perceive the Sun as being red. This can be explained by the fact that the spectrum covers our visible range and as a result of our physiological ability to redefine perceived colors with reference to the illumination source. See R. Mausfeld, “Color perception: from Grassmann codes to a dual code for object and illumination colors,” in Color Vision Perspectives from Different DisciplinesW. G. K. Backhaus, R. Kliegl, J. S. Werner, eds. (de Gruyter, Berlin, 1998), Chap. 12.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 333.

Ref. 5, pp. 406–407.

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Butterworth-Heinemann, Oxford, 1980), pp. 184–187.

This is the irradiance at the surface of the Sun. To obtain the irradiance intercepting the Earth, this expression must be divided by the square of the ratio of the Earth–Sun distance to the radius of the Sun.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Normalized spectral output for a 5800 K blackbody source approximating the Sun versus photon energy. The leftmost curve (dJ ω/ℏdω) is intensity per-unit-photon energy from Eq. (1), the rightmost curve (dJ λ/dλ) is intensity per-unit wavelength from Eq. (2) and the central curve J′ is intensity per-unit fractional bandwidth from Eq. (3). The respective photon energies and heights of the peaks are 1.41 eV and 2.81 kW cm-2 eV-1 for dJ ω/ℏdω, 2.48 eV and 8.44 kW cm-2 µm-1 for dJ λ/dλ, and 1.96 eV and 4.72 kW cm-2 for J′.

Fig. 2
Fig. 2

Normalized spectral output for a 5800 K blackbody source approximating the Sun versus wavelength. The rightmost curve (dJ ω/ℏdω) is intensity per-unit-photon energy from Eq. (1), the leftmost curve (dJ λ/dλ) is intensity per-unit wavelength from Eq. (2), and the central curve J′ is intensity per-unit fractional bandwidth from Eq. (3). The respective wavelengths and heights of the peaks are 879 nm and 2.81 kW cm-2 eV-1 for dJ ω/ℏdω, 500 nm and 8.44 kW cm-2 µm-1 for dJ λ/dλ, and 633 nm and 4.72 kW cm-2 for J′.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

dJωdω=14π2c2ω3expω/kT-1=2πc 1λ3exp2πc/kTλ-1,
ωkT=2πckTλ=2.822.
dJλdλ=4π2c21λ5exp2πc/kTλ-1=8π3c3ω5expω/kT-1.
2πckTλ=ωkT=4.965.
J=4π2c2ω4expω/kT-1=4π2c21λ4exp2πc/kTλ-1.
2πckTλ=ωkT=3.921.
Δω/ω=Δλ/λ.
λ/Δλ=|m|N,
λΔλ=tdndλ,

Metrics