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Applied Optics

Applied Optics


  • Vol. 49, Iss. 7 — Mar. 1, 2010
  • pp: 1116–1130

Correct equations and common approximations for calculating Rayleigh scatter in pure gases and mixtures and evaluation of differences

Wynn L. Eberhard  »View Author Affiliations

Applied Optics, Vol. 49, Issue 7, pp. 1116-1130 (2010)

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Equations for Rayleigh scattering in a mixture of gases are derived and compared to frequent approxi mations in the literature. The traditional Rayleigh scattering equation as modified by King for scatter from a pure gas is correct, whereas another version sometimes appearing in modern literature is erroneous. Use of a mixture’s refractive index, which is equivalent to assuming the isotropic molecular polarizabilities of the component gases are identical, is an approximation. Another common approximation is using only number-density weighting of the King factors. Approximation errors can be large when the major components of a mixture have disparate optical properties. Fortunately, the errors for Earth’s air are much smaller and comparable to errors from other sources.

OCIS Codes
(010.1320) Atmospheric and oceanic optics : Atmospheric transmittance
(280.3640) Remote sensing and sensors : Lidar
(290.1310) Scattering : Atmospheric scattering
(290.5870) Scattering : Scattering, Rayleigh

ToC Category:

Original Manuscript: December 1, 2009
Manuscript Accepted: January 14, 2010
Published: February 23, 2010

Wynn L. Eberhard, "Correct equations and common approximations for calculating Rayleigh scatter in pure gases and mixtures and evaluation of differences," Appl. Opt. 49, 1116-1130 (2010)

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  1. V. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. K. N. Liou, An Introduction to Atmospheric Radiation, Second Edition (Academic, 2002).
  3. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean. Technol. 16, 1854-1861 (1999). [CrossRef]
  4. C. Tomasi, V. Vitale, B. Petkov, A. Lupi, and A. Cacciari, “Improved algorithm for calculations of Rayleigh-scattering optical depth in standard atmospheres,” Appl. Opt. 44, 3320-3341(2005). [CrossRef] [PubMed]
  5. B. W. Forgan, “General method for calibrating sun photometers,” Appl. Opt. 33, 4841-4850 (1994). [CrossRef] [PubMed]
  6. V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004). [CrossRef]
  7. F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482-489 (1972). [CrossRef]
  8. J. Biele, G. Beyerle, and G. Baumgarten, “Polarization lidar: corrections of instrumental effects,” Opt. Express 7, 427-434(2000). [CrossRef] [PubMed]
  9. E. J. McCartney, Optics of the Atmosphere (Wiley, 1976).
  10. J. A. Sutton and J. F. Driscoll, “Rayleigh scattering cross sections of combustion species at 266, 355, and 532 nm for thermometry applications,” Opt. Lett. 29, 2620-2622 (2004). [CrossRef] [PubMed]
  11. S. Kampmann, A. Leipertz, K. Döbbeling, J. Haumann, and T. Sattelmayer, “Two-dimensional temperature measurements in a technical combustor with laser Rayleigh scattering,” Appl. Opt. 32, 6167-6172 (1993). [CrossRef] [PubMed]
  12. Lord Rayleigh, “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107-120, 274-279 (1871).
  13. Lord Rayleigh, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375-384(1899).
  14. R. M. Measures, Laser Remote Sensing (Krieger, 1992).
  15. R. B. Miles, W. R. Lempert, and J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33-R51 (2001). [CrossRef]
  16. D. A. Long, Raman Spectroscopy (McGraw-Hill, 1977).
  17. L. V. King, “On the complex anisotropic molecule in relation to the dispersion and scattering of light,” Proc. R. Soc. London Ser. A 104, 333-357 (1923). [CrossRef]
  18. Lord Rayleigh, “On the scattering of light by a cloud of similar small particles of any shape and oriented at random,” Philos. Mag. 35, 373-381 (1918).
  19. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  20. A. T. Young, “Rayleigh scattering,” Appl. Opt. 20, 533-535 (1981). [CrossRef] [PubMed]
  21. A. T. Young, “ On the Rayleigh-scattering optical depth of the atmosphere,” J. Appl. Meteorol. 20, 328-330 (1981). [CrossRef]
  22. A. T. Young, “Revised depolarization corrections for atmospheric extinction,” Appl. Opt. 19, 3427-3428 (1980). [CrossRef] [PubMed]
  23. A. O. Langford, R. Schofield, J. S. Daniel, R. W. Portmann, M. L. Melamed, H. L. Miller, E. G. Dutton, and S. Solomon, “On the variability of the Ring effect in the near ultraviolet: understanding the role of aerosols and multiple scattering,” Atmos. Chem. Phys. 7, 575-586 (2007). [CrossRef]
  24. D. R. Bates, “Rayleigh scattering by air,” Planet. Space Sci. 32, 785-790 (1984). [CrossRef]
  25. A. Bucholtz, “Rayleigh-scattering calculations for the terrestrial atmosphere,” Appl. Opt. 34, 2765-2773 (1995). [CrossRef] [PubMed]
  26. T. J. Greytak and G. B. Benedek, “Spectrum of light scattered from thermal fluctuations in gases,” Phys. Rev. Lett. 17, 179-182 (1966). [CrossRef]
  27. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006). [CrossRef]
  28. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  29. Lord Rayleigh, “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag. 10, 73-78 (1880).
  30. J. S. Marshall and W. Hitschfeld, “Interpretation of the fluctuating echo from randomly distributed scatterers: Part I,” Can. J. Phys. 31, 194 (1953). [CrossRef]
  31. A. E. Siegmann, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350-1356 (1966). [CrossRef]
  32. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak, 1988).
  33. H. Naus and W. Ubachs, “Experimental verification of Rayleigh scattering cross sections,” Opt. Lett. 25, 347-349(2000). [CrossRef]
  34. D. A. Whiteman, “Examination of the traditional Raman lidar technique. I. Evaluating the temperature-dependent lidar equations,” Appl. Opt. 42, 2571-2592 (2003). [CrossRef] [PubMed]
  35. M. Sneep and W. Ubachs, “Direct measurement of the Rayleigh scattering cross section in various gases,” J. Quant. Spectrosc. Radiat. Transfer 92, 293-310 (2005). [CrossRef]
  36. D. Ityaksov, H. Linnartz, and W. Ubachs, “Deep-UV Rayleigh scattering of N2, CH4 and SF6,” Mol. Phys. 106, 2471-2479(2008). [CrossRef]
  37. S. S. Srivastava, N. K. Vyas, J. Rai, and B. Karikeyan, “On the different approaches of Rayleigh optical depth determination,” Adv. Space Res. 44, 1058-1066 (2009). [CrossRef]
  38. R. Penndorf, “Tables of the refractive index for standard air and the Rayleigh scattering coefficient for the spectral region between 0.2 and 20.0 μ and their application to atmospheric optics,” J. Opt. Soc. Am. 47, 176-182(1957). [CrossRef]
  39. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527-610 (1974). [CrossRef]
  40. R. R. Rudder and D. R. Bach, “Rayleigh scattering of ruby-laser light by neutral gases,” J. Opt. Soc. Am. 58, 1260-1266(1968). [CrossRef]
  41. F. T. Gucker, S. Basu, A. A. Pulido, and G. Chiu, “Intensity and polarization of light scattered by some permanent gases and vapors,” J. Chem. Phys. 50, 2526-2535 (1969). [CrossRef]
  42. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  43. H. R. Gordon, J. W. Brown, and R. H. Evans, “Exact Rayleigh scattering calculations for use with the Nimbus-7 Coastal Zone Color Scanner,” Appl. Opt. 27, 862-871 (1988). [CrossRef] [PubMed]
  44. P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. 29, 1897-1900 (1990). [CrossRef] [PubMed]
  45. K. V. Chance and R. J. D. Spurr, “Ring effect studies: Rayleigh scattering, including molecular parameters for rotational Raman scattering, and the Fraunhofer spectrum,” Appl. Opt. 36, 5224-5230 (1997). [CrossRef] [PubMed]
  46. D. V. Hoyt, “A redetermination of the Rayleigh optical depth and its application to selected solar radiation problems,” J. Appl. Meteorol. 16, 432-436 (1977). [CrossRef]
  47. Shardanand and A. D. Prasad-Rao, “Absolute Rayleigh scattering cross section of gases and freons of stratospheric interest in the visible and ultraviolet regions,” NASA Tech. Note TN D-8442 (1977).
  48. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  49. W. F. Murphy, “The Rayleigh depolarization ratio and rotational Raman spectrum of water vapor and the polarizability components for the water molecule,” J. Chem. Phys. 67, 5877-5882 (1977). [CrossRef]
  50. G. R. Alms, A. K. Burnham, and W. H. Flygare, “Measurement of the dispersion in polarizability anisotropies,” J. Chem. Phys. 63, 3321-3326 (1975). [CrossRef]
  51. N. J. Bridge and A. D. Buckingham, “The polarization of laser light scattered by gases,” Proc. R. Soc. London Ser. A 295, 334-349 (1966). [CrossRef]
  52. R. L. Rowell, G. M. Aval, and J. J. Barrett, “Rayleigh-Raman depolarization of laser light scattered by gases,” J. Chem. Phys. 54, 1960-1964 (1971). [CrossRef]
  53. J. Oddershede and E. N. Svendsen, “Dynamic polarizabilities and Raman intensities of CO, N2, HCl and Cl2,” Chem. Phys. 64, 359-369 (1982). [CrossRef]
  54. W. L. Eberhard, “Comment: On the different approaches of Rayleigh optical depth determination,” Adv. Space Res. (to be published).
  55. R. D. Sharma, “Contribution of the polarizability anisotropy to Rayleigh scattering,” J. Geophys. Res. 112, A05306 (2007). [CrossRef]
  56. J. Joiner, P. K. Bhartia, R. P. Cebula, E. Hilsenrath, R. D. McPeters, and H. Park, “Rotational Raman scattering (Ring effect) in satellite backscatter ultraviolet measurements,” Appl. Opt. 34, 4513-4525 (1995). [CrossRef] [PubMed]
  57. C. E. Sioris, W. F. J. Evans, R. L. Gattinger, I. C. McDade, D. A. Degenstein, and E. J. Llewellyn, “Ground-based Ring-effect measurements with the OSIRIS development model,” Can. J. Phys. 80, 483-491 (2002). [CrossRef]
  58. C. J. F. Böttcher, Theory of Electric Polarization (Elsevier, 1952).
  59. C. J. F. Böttcher, O. C. van Belle, P. Bordewijk, and A. Rip, Theory of Electric Polarization (Elsevier, 1973), Vol. I.
  60. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  61. H. A. Lorentz, “Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. 245, 641-664 (1880). [CrossRef]
  62. L. V. Lorenz, “Ueber die Refractionsconstante,” Ann. Phys. 247, 70-103 (1880). [CrossRef]
  63. D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704-709 (1982). [CrossRef]
  64. J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition,” Appl. Opt. 6, 51-59(1967). [CrossRef] [PubMed]
  65. E. R. Peck and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am. 56, 1059-1063 (1966). [CrossRef]
  66. C. R. Mansfield and E. R. Peck, “Dispersion of helium,” J. Opt. Soc. Am. 59, 199-204 (1969). [CrossRef]
  67. A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “Interferometric determination of the refractive index of carbon dioxide in the ultraviolet region,” Opt. Commun. 9, 432-434 (1973). [CrossRef]
  68. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566-1573 (1996). [CrossRef] [PubMed]
  69. K. P. Birch and M. J. Downs, “Correction to the updated Edlén equation for the refractive index of air,” Metrologia 30, 155-162 (1993). [CrossRef]
  70. E. R. Peck and B. N. Khanna, “Dispersion of nitrogen,” J. Opt. Soc. Am. 56, 1059-1063 (1966). [CrossRef]
  71. P. L. Smith, M. C. E. Huber, and W. H. Parkinson, “Refractivities of H2, He, O2, CO, and Kr for 168 ≤ λ ≤ 288 nm,” Phys. Rev. A 13, 1422-1434 (1976). [CrossRef]
  72. G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edléns formulae,” Metrologia 35, 133-139 (1998). [CrossRef]

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