Abstract

When the sun is near the horizon, a circular band with approximately vertically polarized skylight is formed at 90° from the sun, and this skylight is only weakly reflected from the region of the water surface around the Brewster’s angle (53° from the nadir). Thus, at low solar heights under a clear sky, an extended dark patch is visible on the water surface when one looks toward the north or south quarter perpendicular to the solar vertical. In this work, we study the radiance distribution of this so-called Brewster’s dark patch (BDP) in still water as functions of the solar height and sky conditions. We calculate the pattern of reflectivity R of a water surface for a clear sky and obtain from this idealized situation the shape of the BDP. From three full-sky polarimetric pictures taken about a clear, a partly cloudy, and an overcast sky, we determine the R pattern and compose from that synthetic color pictures showing how the radiance distribution of skylight reflected at the water surface and the BDPs would look under these sky conditions. We also present photographs taken without a linearly polarizing filter about the BDP. Finally, we show a 19th century painting on which a river is seen with a dark region of the water surface, which can be interpreted as an artistic illustration of the BDP.

© 2017 Optical Society of America

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References

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  1. G. P. Können, Polarized Light in Nature (Cambridge University, 1985), pp. 30–34.
  2. D. Pye, Polarised Light in Science and Nature (IOP, 2001).
  3. J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
    [Crossref]
  4. G. Horváth, A. Barta, and R. Hegedüs, “Polarization of the sky,” in Polarized Light and Polarization Vision in Animal Sciences, G. Horváth, ed. (Springer, 2014).
  5. J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
    [Crossref]
  6. M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
    [Crossref]
  7. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, 1988).
  8. P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
    [Crossref]
  9. B. Fougnie, G. Bracco, B. Lafrance, C. Ruffel, O. Hagolle, and C. Tinel, “PARASOL in-flight calibration and performance,” Appl. Opt. 46, 5435–5451 (2007).
    [Crossref]
  10. E. Vuillard, “The Ferryman,” Musee D’Orsay, 1897.
  11. R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
    [Crossref]
  12. R. Hegedüs, S. Åkesson, and G. Horváth, “Anomalous celestial polarization caused by forest fire smoke: why do some insects become visually disoriented under smoky skies?” Appl. Opt. 46, 2717–2726 (2007).
    [Crossref]

2007 (3)

2004 (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

2001 (2)

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
[Crossref]

1994 (1)

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Åkesson, S.

Barta, A.

G. Horváth, A. Barta, and R. Hegedüs, “Polarization of the sky,” in Polarized Light and Polarization Vision in Animal Sciences, G. Horváth, ed. (Springer, 2014).

Berry, M. V.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Bracco, G.

Bréon, F. M.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Bricaud, A.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Buriez, J. C.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Coulson, K. L.

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, 1988).

Dennis, M. R.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Deschamps, P. Y.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Fougnie, B.

Gál, J.

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
[Crossref]

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

Hagolle, O.

Hegedüs, R.

Horváth, G.

R. Hegedüs, S. Åkesson, and G. Horváth, “Anomalous celestial polarization caused by forest fire smoke: why do some insects become visually disoriented under smoky skies?” Appl. Opt. 46, 2717–2726 (2007).
[Crossref]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
[Crossref]

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
[Crossref]

G. Horváth, A. Barta, and R. Hegedüs, “Polarization of the sky,” in Polarized Light and Polarization Vision in Animal Sciences, G. Horváth, ed. (Springer, 2014).

Können, G. P.

G. P. Können, Polarized Light in Nature (Cambridge University, 1985), pp. 30–34.

Lafrance, B.

Lee, R. L.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Leroy, M.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Meyer-Rochow, V. B.

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
[Crossref]

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

Podaire, A.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Pye, D.

D. Pye, Polarised Light in Science and Nature (IOP, 2001).

Ruffel, C.

Seze, G.

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

Tinel, C.

Vuillard, E.

E. Vuillard, “The Ferryman,” Musee D’Orsay, 1897.

Wehner, R.

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Geosci. Remote Sens. (1)

P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
[Crossref]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
[Crossref]

Proc. R. Soc. A (1)

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proc. R. Soc. A 457, 1385–1399 (2001).
[Crossref]

Remote Sens. Environ. (1)

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76, 103–111 (2001).
[Crossref]

Other (5)

G. Horváth, A. Barta, and R. Hegedüs, “Polarization of the sky,” in Polarized Light and Polarization Vision in Animal Sciences, G. Horváth, ed. (Springer, 2014).

G. P. Können, Polarized Light in Nature (Cambridge University, 1985), pp. 30–34.

D. Pye, Polarised Light in Science and Nature (IOP, 2001).

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (Deepak, 1988).

E. Vuillard, “The Ferryman,” Musee D’Orsay, 1897.

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Figures (10)

Fig. 1.
Fig. 1.

(A) Geometry of the polarization optics of the Brewster’s dark patch (BDP) visible on the water surface toward the north or south quarter when the sun is on or near the horizon. Double-headed arrows represent the horizontally polarized light coming from the solar–antisolar vertical of the celestial hemisphere. From the north–south vertical, being perpendicular to the solar–antisolar vertical, vertically polarized skylight originates. (B) Calculation of the horizontal distance d=mh of the center of BDP from an observer viewing at the water surface from height h with an angle of view θBrewster=arctan(m)=53° relative to the vertical, where m=1.33 is the refractive index of water.

Fig. 2.
Fig. 2.

Pattern of reflectivity R of the water surface calculated for clear skies with solar heights hS = (A) 0°, (B) 18°, (C) 36°, (D) 54°, (E) 72°, and (F) 90°. The position of the mirror image of the sun is represented by a dot. The Brewster’s point with minimum reflectivity Rmin represented by + is considered as the center of the BDPs (the two central pear-shaped black patches, if they exist). In (A)–(D) the two central pear-shaped black patches represent regions with R2%, corresponding with the BDPs, and the alternating white and black annular outer regions represent 2%<R3%,,19%<R20%, while the outermost white annular zone has 20%<R100%.

Fig. 3.
Fig. 3.

Continuous curve with its vertical scale on the left side: Area percentage b of the two BDPs of the water surface relative to the whole downward pointing hemisphere filled by the water surface as a function of the solar height angle hS computed for clear skies, the polarization patterns of which are described by the model of Berry et al. [6]. Dotted curve with its vertical scale on the right side: Minimum Rmin of reflectivity of the water surface versus hS computed for clear skies.

Fig. 4.
Fig. 4.

(A) Color photograph of a clear sky at solar height hS=0° taken with a 180° field-of-view fisheye lens. (B) Synthetic photograph of the mirror image of the sky in (A) with 50 times brightening, as computed from the polarimetric full-sky data simultaneously measured with picture (A). (C) Computed pattern of reflectivity R of the water surface under the sky in (A). The dark areas are the BDPs. In (A)–(C) the position of the (mirror) sun is marked with a red dot. (D), (E) The regions of the sky and the sky-mirroring water surface in windows 1 and 2 of (A) and (B), represented as rectangular photographs, showing the view an observer (painter) would see when looking perpendicular to the solar vertical. Their horizontal and vertical fields of view are 90° and 72°, respectively. The picture was taken by Gábor Horváth at Göd, Hungary (47°70N, 19°15E) on 7 August 2012 at 20:15 (UT+2h).

Fig. 5.
Fig. 5.

As in Fig. 4 for a partly cloudy sky at solar height hS=26° on 1 May 2012 at 16:38 (UT+2h). The dark areas are the BDPs.

Fig. 6.
Fig. 6.

As in Fig. 4 for an overcast sky at solar height hS=58° on 15 April 2012 at 14:30 (UT+2h). The BDPs do not show up in this diagram.

Fig. 7.
Fig. 7.

Taj Mahal and reflecting pool looking to north at sunset. The nearer water does not reflect the blue sky, although it reflects the building very clearly. This proves that the darkening could not have been produced by a polarizing filter on the camera. This photograph originates from promotional material produced in 1995 by the travel company Abercrombie & Kent.

Fig. 8.
Fig. 8.

Black-and-white picture of the Lake Hyllsjon near Havnmantorp and Lessibo, southern Sweden. The horizontal field of view of the camera was facing north. This photograph was taken by David Pye early on the morning on 4 June 2011 without a polarizing filter.

Fig. 9.
Fig. 9.

180° field-of-view photographs taken about the sky and five different lakes through a horizontal polarizer (A), (C), (E), (G), (I) and without such a filter (B), (D), (F), (H), (J) at Göd, Hungary (47°70N, 19°15E) on 15 July 2017 at 20:30 (UT+2h) when the horizontal optical axis of the camera saw toward west (A)–(D), south (E)–(H), and north (I)–(J). In (B) and (D), there is no BDP, while in (F), (H), and (J) it can be seen on the water surface.

Fig. 10.
Fig. 10.

Edouard Vuillard (1897): The Ferryman (Musee D’Orsay, Paris [10]). The dark patch at the water surface is most likely BDP. For detailed discussion, see text.

Equations (6)

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Sskylight=Iskylight[1,p·cos(2α),p·cos(2α),0],
M__=12(tanθsinθ+)2(cos2θ+cos2θ+cos2θcos2θ+00cos2θcos2θ+cos2θ+cos2θ+00002cosθcosθ+00002cosθcosθ+),θ=θiθr,θ+=θi+θr,
θr=arcsin(sinθim),
S_reflected=M__S_skylight,
Ireflected=S0refl,preflected=S1refl2+S2refl2+S3refl2S0refl2S1refl2+S2refl2S0refl2,
S3refl0,αreflected=12arctan(S2reflS1refl),

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