Abstract

The first likely photographic observation of the tertiary rainbow caused by sunlight in the open air is reported and analyzed. Whereas primary and secondary rainbows are rather common and easily seen phenomena in atmospheric optics, the tertiary rainbow appears in the sunward side of the sky and is thus largely masked by forward scattered light. Up to now, only a few visual reports and no reliable photographs of the tertiary rainbow are known. Evidence of a third-order rainbow has been obtained by using image processing techniques on a digital photograph that contains no obvious indication of such a rainbow. To rule out any misinterpretation of artifacts, we carefully calibrated the image in order to compare the observed bow’s angular position and dispersion with those predicted by theory.

© 2011 Optical Society of America

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References

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  1. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
    [CrossRef]
  2. R. L. Lee, Jr., “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
    [CrossRef]
  3. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  4. Y. Ayatsuka, “Ungewöhnliche Regenbögen,” Meteoros 9, 128–136 (2006).
  5. L. Cowley, “Twinned rainbows,” http://www.atoptics.co.uk/rainbows/twin1.htm.
  6. L. Cowley, “Reflection rainbows,” http://www.atoptics.co.uk/rainbows/reflect.htm.
  7. R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.
  8. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular p. 243.
    [CrossRef]
  9. Named after a hypothetical zero-order rainbow, i.e., a rainbow without any internal reflection. This possibility has caused some rumors in history. However, due to the monotonic character of the deflection function of zero-order scattering, a zero-order rainbow does not exist . Nonetheless in the Sun facing hemisphere, zero-order scattered light is dominating.
  10. R. L. Lee, Jr. and P. Laven, “Visibility of natural tertiary rainbows,” Appl. Opt. 50, F152–F161.
    [PubMed]
  11. This estimation is somehow difficult because of the intensity singularity at the angle of extreme deviation for the tertiary rainbow. Such singularities are typical features for caustics within geometric optics calculations.
  12. P. Laven, 9 Russells Crescent, Horley, Surrey, RH6 7DJ, United Kingdom (personal communication, 2011).
  13. C. B. Boyer, “The tertiary rainbow: An historical account,” ISIS (1913-) / Isis 49, 141–154 (1958).
    [CrossRef]
  14. T. C. Lewis, “A tertiary rainbow,” Nature 32, 523–626 (1885).
    [CrossRef]
  15. W. R. Corliss, Rare Halos, Mirages, Anomalous Rainbows and Related Electromagnetic Phenomena (1984), pp. 30–31.
  16. R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Bellingham, 2001), pp. 290–295.
  17. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).
  18. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–432 (1976).
    [CrossRef]
  19. F. Billet, “Mémoirs sur les dix-neuf premiers arcs-en-ciel de l’eau,” Annales Scientifiques de l’Ecole Normale Supérieure 5, 67–109(1868).
  20. R. L. Lee, Jr. and P. Laven, “Visibility of natural tertiary rainbows,” Appl. Opt. 50, F152–F161.
    [PubMed]
  21. Arbeitskreis Meteore e.V. http://www.meteoros.de/indexe.htm.
  22. M. Großmann, “Regenbogen 3. Ordnung 15.05.2011,” http://www.meteoros.de/php/viewtopic.php?t=8463.
  23. M. Großmann, “Natural tertiary rainbow 3rd order,” http://atoptics.wordpress.com/2011/06/01/rainbow-3th-order-3/.
  24. as quoted from by Cowley in http://www.atoptics.co.uk/rainbows/ord34.htm.
  25. Our method is based on the well-known transformation routines from one angular coordinate system into another, e.g., from declination and right ascension into elevation and azimuth for astronomical purposes. Here, we have to deal with four such coordinate systems, having their individual poles at (1) the center of the star field image, (2) the zenith, (3) the center of the rainbow image, and (4) the Sun.
  26. A. Haußmann, “Die rechnerische Bestimmung von Winkeldistanzen und Positionen durch die Vermessung von Fotos,” http://www.meteoros.de/download/haussmann/NLC-Haussmann.zip.
  27. W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 203–210.
  28. C. Mauer, “Measurement of the spectral response of digital cameras with a set of interference filters,” Diploma thesis (University of Applied Sciences Cologne, 2009).
  29. M. Menat, “Atmospheric phenomena before and during sunset,” Appl. Opt. 19, 3458–3468 (1980).
    [CrossRef] [PubMed]
  30. International Association for the Properties of Water and Steam, “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure,” (1997), http://www.iapws.org/relguide/rindex.pdf.
  31. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular pp. 243–245.
    [CrossRef]
  32. R. L. Lee, Jr. and P. Laven, “Visibility of natural tertiary rainbows,” Appl. Opt. 50, F152–F161.
    [PubMed]
  33. P. Laven, “Mieplot Software,” http://www.philiplaven.com/mieplot.htm.
  34. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
    [CrossRef]
  35. M. Theusner, “3rd and 4th order rainbows,” http://atoptics.wordpress.com/2011/06/12/3rd-and-4th-order-rainbows/.
  36. W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 200–201.
  37. M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133.
    [PubMed]

2006 (1)

Y. Ayatsuka, “Ungewöhnliche Regenbögen,” Meteoros 9, 128–136 (2006).

2002 (3)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular p. 243.
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular pp. 243–245.
[CrossRef]

1998 (2)

1986 (1)

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

1983 (1)

1980 (1)

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–432 (1976).
[CrossRef]

1958 (1)

C. B. Boyer, “The tertiary rainbow: An historical account,” ISIS (1913-) / Isis 49, 141–154 (1958).
[CrossRef]

1885 (1)

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523–626 (1885).
[CrossRef]

1868 (1)

F. Billet, “Mémoirs sur les dix-neuf premiers arcs-en-ciel de l’eau,” Annales Scientifiques de l’Ecole Normale Supérieure 5, 67–109(1868).

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular p. 243.
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular pp. 243–245.
[CrossRef]

Ayatsuka, Y.

Y. Ayatsuka, “Ungewöhnliche Regenbögen,” Meteoros 9, 128–136 (2006).

Billet, F.

F. Billet, “Mémoirs sur les dix-neuf premiers arcs-en-ciel de l’eau,” Annales Scientifiques de l’Ecole Normale Supérieure 5, 67–109(1868).

Boyer, C. B.

C. B. Boyer, “The tertiary rainbow: An historical account,” ISIS (1913-) / Isis 49, 141–154 (1958).
[CrossRef]

Corliss, W. R.

W. R. Corliss, Rare Halos, Mirages, Anomalous Rainbows and Related Electromagnetic Phenomena (1984), pp. 30–31.

Cowley, L.

L. Cowley, “Twinned rainbows,” http://www.atoptics.co.uk/rainbows/twin1.htm.

L. Cowley, “Reflection rainbows,” http://www.atoptics.co.uk/rainbows/reflect.htm.

Fraser, A. B.

A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Bellingham, 2001), pp. 290–295.

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.

Großmann,

M. Großmann, “Regenbogen 3. Ordnung 15.05.2011,” http://www.meteoros.de/php/viewtopic.php?t=8463.

Großmann, M.

M. Großmann, “Natural tertiary rainbow 3rd order,” http://atoptics.wordpress.com/2011/06/01/rainbow-3th-order-3/.

Haußmann, A.

A. Haußmann, “Die rechnerische Bestimmung von Winkeldistanzen und Positionen durch die Vermessung von Fotos,” http://www.meteoros.de/download/haussmann/NLC-Haussmann.zip.

Langley, D. S.

Laven, P.

Lee, R. L.

Lewis, T. C.

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523–626 (1885).
[CrossRef]

Marston, P. L.

Mauer, C.

C. Mauer, “Measurement of the spectral response of digital cameras with a set of interference filters,” Diploma thesis (University of Applied Sciences Cologne, 2009).

Menat, M.

Moilanen, J.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 203–210.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 200–201.

Pedgley, D. E.

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Tape, W.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 203–210.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 200–201.

Theusner, M.

M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133.
[PubMed]

M. Theusner, “3rd and 4th order rainbows,” http://atoptics.wordpress.com/2011/06/12/3rd-and-4th-order-rainbows/.

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–432 (1976).
[CrossRef]

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–432 (1976).
[CrossRef]

Annales Scientifiques de l’Ecole Normale Supérieure (1)

F. Billet, “Mémoirs sur les dix-neuf premiers arcs-en-ciel de l’eau,” Annales Scientifiques de l’Ecole Normale Supérieure 5, 67–109(1868).

Appl. Opt. (7)

ISIS (1913-) / Isis (1)

C. B. Boyer, “The tertiary rainbow: An historical account,” ISIS (1913-) / Isis 49, 141–154 (1958).
[CrossRef]

J. Opt. Soc. Am. (1)

Meteoros (1)

Y. Ayatsuka, “Ungewöhnliche Regenbögen,” Meteoros 9, 128–136 (2006).

Nature (1)

T. C. Lewis, “A tertiary rainbow,” Nature 32, 523–626 (1885).
[CrossRef]

Phys. Rep. (3)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular pp. 243–245.
[CrossRef]

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002), see in particular p. 243.
[CrossRef]

Weather (1)

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Other (20)

International Association for the Properties of Water and Steam, “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure,” (1997), http://www.iapws.org/relguide/rindex.pdf.

W. R. Corliss, Rare Halos, Mirages, Anomalous Rainbows and Related Electromagnetic Phenomena (1984), pp. 30–31.

R. L. Lee, Jr. and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Bellingham, 2001), pp. 290–295.

M. Theusner, “3rd and 4th order rainbows,” http://atoptics.wordpress.com/2011/06/12/3rd-and-4th-order-rainbows/.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 200–201.

P. Laven, “Mieplot Software,” http://www.philiplaven.com/mieplot.htm.

Named after a hypothetical zero-order rainbow, i.e., a rainbow without any internal reflection. This possibility has caused some rumors in history. However, due to the monotonic character of the deflection function of zero-order scattering, a zero-order rainbow does not exist . Nonetheless in the Sun facing hemisphere, zero-order scattered light is dominating.

This estimation is somehow difficult because of the intensity singularity at the angle of extreme deviation for the tertiary rainbow. Such singularities are typical features for caustics within geometric optics calculations.

P. Laven, 9 Russells Crescent, Horley, Surrey, RH6 7DJ, United Kingdom (personal communication, 2011).

L. Cowley, “Twinned rainbows,” http://www.atoptics.co.uk/rainbows/twin1.htm.

L. Cowley, “Reflection rainbows,” http://www.atoptics.co.uk/rainbows/reflect.htm.

R. Greenler, Rainbows, Halos, and Glories (Cambridge, 1980), pp. 6–7.

Arbeitskreis Meteore e.V. http://www.meteoros.de/indexe.htm.

M. Großmann, “Regenbogen 3. Ordnung 15.05.2011,” http://www.meteoros.de/php/viewtopic.php?t=8463.

M. Großmann, “Natural tertiary rainbow 3rd order,” http://atoptics.wordpress.com/2011/06/01/rainbow-3th-order-3/.

as quoted from by Cowley in http://www.atoptics.co.uk/rainbows/ord34.htm.

Our method is based on the well-known transformation routines from one angular coordinate system into another, e.g., from declination and right ascension into elevation and azimuth for astronomical purposes. Here, we have to deal with four such coordinate systems, having their individual poles at (1) the center of the star field image, (2) the zenith, (3) the center of the rainbow image, and (4) the Sun.

A. Haußmann, “Die rechnerische Bestimmung von Winkeldistanzen und Positionen durch die Vermessung von Fotos,” http://www.meteoros.de/download/haussmann/NLC-Haussmann.zip.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2005), pp. 203–210.

C. Mauer, “Measurement of the spectral response of digital cameras with a set of interference filters,” Diploma thesis (University of Applied Sciences Cologne, 2009).

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Figures (6)

Fig. 1
Fig. 1

Distribution of the scattered intensity per solid angle from a water sphere in air as function of the scattering angle, according to geometric optics. Within this ray tracing calculation, up to five internal reflections are considered. The calculation has been done for three different wavelengths and both polarizations have been summed up. Numbers indicate the rainbow order. For comparison, the contribution from the tertiary rainbow is included.

Fig. 2
Fig. 2

(a) Original image from May 15, 2011, 18 00 UTC, Kämpfelbach, Germany. Two reference positions (A and B) for image orientation are indicated. (b) Processed version after contrast expansion and unsharp masking, showing a rainbow like pattern next to the image center, marked by the arrows. The solar elevation is 8.1 ° . Our analysis (Section 3A) shows that the center of the picture points to 18.7 ° elevation and 252.1 ° azimuth, with an anticlockwise rotation angle of the image with respect to the vertical of 2.6 ° .

Fig. 3
Fig. 3

Star field image used for calibration of the rainbow image. Elevation and azimuth of the positions A and B can be calculated using stars as references. These positions can be identified in the original image [Fig. 2a], taken from approximately the same position, thus serving as references to locate the recorded bow at the celestial sphere (picture taken on June 02, 2011, 21:22 UTC).

Fig. 4
Fig. 4

(a) Sample points (white circles) at the outermost red rim of the recorded bow. The x, y coordinate system for the determination of pixel coordinates of these sample points is indicated by white arrows. (b) Lines of equal angular distance from the Sun: 41.5 ° (red, solid), 40.6 ° (green, dashed), 39.3 ° (blue, dotted), corresponding to the Descartes angles for 600 nm , 530 nm , and 460 nm . Comparison with Fig. 2b reveals the coincidence of these circles with the rainbow.

Fig. 5
Fig. 5

Sector area (white) for quantitative analysis of intensity, covering solar distances from 29 ° to 59 ° along a 40 ° section of the clock angle.

Fig. 6
Fig. 6

Brightness data of the red, green, and blue channel from the arc segment area marked in Fig. 5 as taken (a) from the original image and (b) after subtraction of an individual polynomial background for each color channel. Dashed vertical lines mark the Descartes angles for the tertiary rainbow. The data were read out at 0.01 ° intervals of angular distance and have subsequently been smoothed by a moving average of 0.4 ° width [solid lines in (b)]. While barely visible in (a), each curve shows a pronounced maximum in (b). These maxima are observed at 41.1 ° (red), 39.9 ° (green), and 38.5 ° (blue) distance from the Sun, respectively. This corresponds to a shift of 0.4 ° 0.8 ° in sunward direction with respect to their individual Descartes angles. This shift is in qualitative agreement with corrections obtained from wave optics.

Tables (2)

Tables Icon

Table 1 Pixel Coordinates, Celestial Coordinates and Solar Distance for the Sample Points from Fig. 4a a

Tables Icon

Table 2 Peak and Centroid Wavelengths of the Canon 450D CMOS Sensor Response Spectra with Correction for the Evening Solar Spectrum and Final Choice for Descartes Angle Calculation

Equations (1)

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R = 3633 . 7 tan ϑ 262 . 45 tan 2 ϑ .

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