Abstract

The celestial polarization pattern may be scrambled by refraction at the air–water interface. This polarization pattern was examined in shallow waters with a submersible polarimeter, and it was calculated by using land measurements (“semiempirical predictions”) and models of the skylight polarization. Semiempirically predicted and measured e-vector orientations were significantly similar. Conversely, predicted percent polarization was correlated but lower than measurements. Percent polarization depended on wavelength, where at high sun altitudes maximal percent polarization generally appeared in the UV and red spectral regions. The wavelength dependency of polarization may lead to differential spectral sensitivity in polarization-sensitive animals according to time and type of activity.

© 2006 Optical Society of America

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2004 (2)

N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in sea water: implications for polarization signaling," J. Exp. Biol. 207, 3619-3628 (2004).
[CrossRef] [PubMed]

B. Suhai and G. Horváth, "How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study," J. Opt. Soc. Am. A 21, 1669-1676 (2004).
[CrossRef]

2003 (2)

A. Barta and G. Horváth, "Underwater binocular imaging of aerial objects versus the position of eyes relative to the flat water surface," J. Opt. Soc. Am. A 20, 2370-2377 (2003).
[CrossRef]

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

2002 (1)

2001 (5)

T. W. Cronin and N. Shashar, "The linearly polarized light field in clear, tropical, marine waters: spatial and temporal variation of light intensity, degree of polarization and e-vector angle," J. Exp. Biol. 204, 2461-2467 (2001).
[PubMed]

R. Wehner, "Polarization vision--a uniform sensory capacity?" J. Exp. Biol. 204, 2589-2596 (2001).
[PubMed]

I. Novales Flamarique and H. I. Browman, "Foraging and prey-search behavior of small juvenile rainbow trout (Oncorhynchus mykiss) under polarized light," J. Exp. Biol. 204, 2415-2422 (2001).
[PubMed]

I. Pomozi, G. Horváth, and R. Wehner, "How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation," J. Exp. Biol. 204, 2933-2942 (2001).
[PubMed]

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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

2000 (1)

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

1999 (2)

R. Schwind, "Daphnia pulex swims towards the most strongly polarized light--a response that leads to 'shore flight'," J. Exp. Biol. 202, 3631-3635 (1999).
[PubMed]

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

1998 (2)

N. Shashar, R. T. Hanlon, and A. D. Petz, "Polarization vision helps detect transparent prey," Nature 393, 222-223 (1998).
[CrossRef]

M. Stramska and T. D. Dickey, "Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds," Deep-Sea Res., Part I 45, 1393-1410 (1998).
[CrossRef]

1997 (1)

I. Novales Flamarique and C. W. Hawryshyn, "Is the use of underwater polarized light by fish restricted to crepuscular time periods?" Vision Res. 37, 975-989 (1997).
[CrossRef]

1995 (2)

G. Horváth and D. Varjú, "Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell's window of the flat water surface," Vision Res. 35, 1651-1666 (1995).
[CrossRef] [PubMed]

L. B. Wolff and A. G. Andreou, "Polarization camera sensors," Image Vis. Comput. 13, 497-509 (1995).
[CrossRef]

1993 (1)

D. C. Parkyn and C. W. Hawryshyn, "Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibers," J. Comp. Physiol., A 172, 493-500 (1993).
[CrossRef]

1992 (1)

C. W. Hawryshyn, "Polarization vision in fish," Am. Sci. 80, 479-491 (1992).

1991 (2)

S. M. Goddard and R. B. Forward, "The role of the underwater polarized light pattern, in sun compass navigation of the grass shrimp, Palaemonetes vulgaris," J. Comp. Physiol., A 169, 479-491 (1991).
[CrossRef]

D. A. Ritz, "Polarized-light responses in the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 154, 245-250 (1991).
[CrossRef]

1990 (2)

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

S. M. Goddard and R. B. Forward, "The decay and learning of a y axis orientation behavior: the offshore escape response of the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 142, 137-150 (1990).
[CrossRef]

1989 (1)

S. M. Goddard and R. B. Forward, "The use of celestial cues in the offshore escape response of the shrimp, Palaemonetes vulgaris," Mar. Behav. Physiol. 16, 11-18 (1989).
[CrossRef]

1987 (1)

C. W. Hawryshyn and W. N. McFarland, "Cone photoreceptor mechanisms and the detection of polarized light in fish," J. Comp. Physiol., A 160, 459-465 (1987).
[CrossRef]

1986 (1)

W. N. McFarland, "Light in the sea--correlations with behaviors of fishes and invertebrates," Am. Zool. 26, 389-401 (1986).

1981 (1)

J. Janssen, "Searching for zooplankton just outside Snell's window," Limnol. Oceanogr. 26, 1168-1171 (1981).
[CrossRef]

1973 (2)

R. B. Forward and T. H. Waterman, "Evidence for e-vector and light intensity pattern discrimination by the teleost Demogenys," J. Comp. Physiol. 87, 189-202 (1973).
[CrossRef]

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

1972 (1)

R. B. Forward, K. W. Horch, and T. H. Waterman, "Visual orientation at the water surface by the teleost Zenarchopterus," Biol. Bull. 143, 112-126 (1972).
[CrossRef]

1970 (4)

T. H. Waterman and R. B. Forward, "Field evidence for polarized light sensitivity in the fish Zenarchopterus," Nature 228, 85-87 (1970).
[CrossRef] [PubMed]

R. L. Snyder and J. Dera, "Wave-induced light-field fluctuations in the sea," J. Opt. Soc. Am. 60, 1072-1079 (1970).
[CrossRef]

V. A. Timofeyeva, "The degree of polarization of light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 6, 513-522 (1970).

O. Munk, "On the occurrence and significance of horizontal band-shaped retinal areas in teleosts," Vidensk. Medd. Dan. Naturhist. Foren. 133, 85-120 (1970).

1969 (1)

V. A. Timofeyeva, "Plane of vibrations of polarized light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 5, 1049-1057 (1969).

1965 (2)

C. Groot, "On the orientation of young sockeye salmon (Oncorhynchus nerka) during their seaward migration out of lakes," Behaviour (Suppl.) 14, 1-198 (1965).

E. J. Denton and J. A. C. Nicol, "Polarization of light reflected from the silvery exterior of the bleak Alburnus alburnus," J. Mar. Biol. Assoc. U.K. 45, 705-709 (1965).
[CrossRef]

1958 (2)

A. Ivanoff and T. H. Waterman, "Elliptical polarization of submarine illumination," J. Mar. Res. 16, 255-282 (1958).

A. Ivanoff and T. H. Waterman, "Factors, mainly depth and wavelength, affecting the degree of underwater light polarization," J. Mar. Res. 16, 283-307 (1958).

1957 (1)

1954 (1)

T. H. Waterman, "Polarization patterns in submarine illumination," Science 120, 927-932 (1954).
[CrossRef] [PubMed]

Allison, W. D.

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

Andreou, A. G.

L. B. Wolff and A. G. Andreou, "Polarization camera sensors," Image Vis. Comput. 13, 497-509 (1995).
[CrossRef]

Arnold, M. G.

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

Barta, A.

Bernath, B.

Boal, J. G.

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

Bowering, D.

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

Browman, H. I.

I. Novales Flamarique and H. I. Browman, "Foraging and prey-search behavior of small juvenile rainbow trout (Oncorhynchus mykiss) under polarized light," J. Exp. Biol. 204, 2415-2422 (2001).
[PubMed]

Cole, R. L.

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

Collett, E.

E. Collett, Polarized Light--Fundamentals and Applications (Marcel Dekker, 1994).

Coulson, K. L.

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

Cronin, T. W.

N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in sea water: implications for polarization signaling," J. Exp. Biol. 207, 3619-3628 (2004).
[CrossRef] [PubMed]

T. W. Cronin and N. Shashar, "The linearly polarized light field in clear, tropical, marine waters: spatial and temporal variation of light intensity, degree of polarization and e-vector angle," J. Exp. Biol. 204, 2461-2467 (2001).
[PubMed]

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

Denton, E. J.

E. J. Denton and J. A. C. Nicol, "Polarization of light reflected from the silvery exterior of the bleak Alburnus alburnus," J. Mar. Biol. Assoc. U.K. 45, 705-709 (1965).
[CrossRef]

Dera, J.

Dickey, T. D.

M. Stramska and T. D. Dickey, "Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds," Deep-Sea Res., Part I 45, 1393-1410 (1998).
[CrossRef]

Egan, W. G.

W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, 1985).

Flamarique, I. Novales

I. Novales Flamarique and H. I. Browman, "Foraging and prey-search behavior of small juvenile rainbow trout (Oncorhynchus mykiss) under polarized light," J. Exp. Biol. 204, 2415-2422 (2001).
[PubMed]

I. Novales Flamarique and C. W. Hawryshyn, "Is the use of underwater polarized light by fish restricted to crepuscular time periods?" Vision Res. 37, 975-989 (1997).
[CrossRef]

Forward, R. B.

S. M. Goddard and R. B. Forward, "The role of the underwater polarized light pattern, in sun compass navigation of the grass shrimp, Palaemonetes vulgaris," J. Comp. Physiol., A 169, 479-491 (1991).
[CrossRef]

S. M. Goddard and R. B. Forward, "The decay and learning of a y axis orientation behavior: the offshore escape response of the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 142, 137-150 (1990).
[CrossRef]

S. M. Goddard and R. B. Forward, "The use of celestial cues in the offshore escape response of the shrimp, Palaemonetes vulgaris," Mar. Behav. Physiol. 16, 11-18 (1989).
[CrossRef]

R. B. Forward and T. H. Waterman, "Evidence for e-vector and light intensity pattern discrimination by the teleost Demogenys," J. Comp. Physiol. 87, 189-202 (1973).
[CrossRef]

R. B. Forward, K. W. Horch, and T. H. Waterman, "Visual orientation at the water surface by the teleost Zenarchopterus," Biol. Bull. 143, 112-126 (1972).
[CrossRef]

T. H. Waterman and R. B. Forward, "Field evidence for polarized light sensitivity in the fish Zenarchopterus," Nature 228, 85-87 (1970).
[CrossRef] [PubMed]

Gál, J.

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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

Gensler, P.

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

Goddard, S. M.

S. M. Goddard and R. B. Forward, "The role of the underwater polarized light pattern, in sun compass navigation of the grass shrimp, Palaemonetes vulgaris," J. Comp. Physiol., A 169, 479-491 (1991).
[CrossRef]

S. M. Goddard and R. B. Forward, "The decay and learning of a y axis orientation behavior: the offshore escape response of the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 142, 137-150 (1990).
[CrossRef]

S. M. Goddard and R. B. Forward, "The use of celestial cues in the offshore escape response of the shrimp, Palaemonetes vulgaris," Mar. Behav. Physiol. 16, 11-18 (1989).
[CrossRef]

Goldsmith, T. H.

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

Groot, C.

C. Groot, "On the orientation of young sockeye salmon (Oncorhynchus nerka) during their seaward migration out of lakes," Behaviour (Suppl.) 14, 1-198 (1965).

Hagen, R.

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

Haimberger, T. J.

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

Hanlon, R. T.

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

N. Shashar, R. T. Hanlon, and A. D. Petz, "Polarization vision helps detect transparent prey," Nature 393, 222-223 (1998).
[CrossRef]

Hawryshyn, C. W.

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

I. Novales Flamarique and C. W. Hawryshyn, "Is the use of underwater polarized light by fish restricted to crepuscular time periods?" Vision Res. 37, 975-989 (1997).
[CrossRef]

D. C. Parkyn and C. W. Hawryshyn, "Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibers," J. Comp. Physiol., A 172, 493-500 (1993).
[CrossRef]

C. W. Hawryshyn, "Polarization vision in fish," Am. Sci. 80, 479-491 (1992).

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

C. W. Hawryshyn and W. N. McFarland, "Cone photoreceptor mechanisms and the detection of polarized light in fish," J. Comp. Physiol., A 160, 459-465 (1987).
[CrossRef]

Horch, K. W.

R. B. Forward, K. W. Horch, and T. H. Waterman, "Visual orientation at the water surface by the teleost Zenarchopterus," Biol. Bull. 143, 112-126 (1972).
[CrossRef]

Horváth, G.

B. Suhai and G. Horváth, "How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study," J. Opt. Soc. Am. A 21, 1669-1676 (2004).
[CrossRef]

A. Barta and G. Horváth, "Underwater binocular imaging of aerial objects versus the position of eyes relative to the flat water surface," J. Opt. Soc. Am. A 20, 2370-2377 (2003).
[CrossRef]

G. Horváth, B. Bernath, B. Suhai, A. Barta, and R. Wehner, "First observation of the fourth neutral point in the atmosphere," J. Opt. Soc. Am. A 19, 2085-2099 (2002).
[CrossRef]

I. Pomozi, G. Horváth, and R. Wehner, "How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation," J. Exp. Biol. 204, 2933-2942 (2001).
[PubMed]

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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

G. Horváth and D. Varjú, "Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell's window of the flat water surface," Vision Res. 35, 1651-1666 (1995).
[CrossRef] [PubMed]

G. Horváth and D. Varjú, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer-Verlag, 2003).

Hyde, D.

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

Ivanoff, A.

A. Ivanoff and T. H. Waterman, "Elliptical polarization of submarine illumination," J. Mar. Res. 16, 255-282 (1958).

A. Ivanoff and T. H. Waterman, "Factors, mainly depth and wavelength, affecting the degree of underwater light polarization," J. Mar. Res. 16, 283-307 (1958).

A. Ivanoff, "Polarization measurements in the sea," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 151-175.

Janssen, J.

J. Janssen, "Searching for zooplankton just outside Snell's window," Limnol. Oceanogr. 26, 1168-1171 (1981).
[CrossRef]

Jerlov, N. G.

N. G. Jerlov, "Significant relationships between optical properties of the sea," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 77-94.

Jerlov, N. J.

N. J. Jerlov, Optical Oceanography, Vol. 5of Elsevier Oceanography Series (Elsevier, 1968).
[CrossRef]

Kleerekoper, H.

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

Loesy, G. S.

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

Lythgoe, J. N.

J. N. Lythgoe, The Ecology of Vision (Oxford U. Press, 1979).

Marshall, N. J.

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

Matis, J. H.

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

McAlary, F. A.

W. N. McFarland, C. M. Wahl, T. H. Suchanek, and F. A. McAlary, "The behavior of animals around twilight with emphasis on coral reef communities," in Adaptive Mechanisms in Ecology of Vision, S.N.Archer, ed. (Academic, 1999), pp. 583-628.

McFarland, W. N.

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

C. W. Hawryshyn and W. N. McFarland, "Cone photoreceptor mechanisms and the detection of polarized light in fish," J. Comp. Physiol., A 160, 459-465 (1987).
[CrossRef]

W. N. McFarland, "Light in the sea--correlations with behaviors of fishes and invertebrates," Am. Zool. 26, 389-401 (1986).

W. N. McFarland, "The visual world of coral reef fishes," in The Ecology of Fishes on Coral Reefs (Academic, 1991), pp. 16-37.

W. N. McFarland, C. M. Wahl, T. H. Suchanek, and F. A. McAlary, "The behavior of animals around twilight with emphasis on coral reef communities," in Adaptive Mechanisms in Ecology of Vision, S.N.Archer, ed. (Academic, 1999), pp. 583-628.

Meyer-Rochow, V. B.

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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

Moyer, H. D.

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

Munk, O.

O. Munk, "On the occurrence and significance of horizontal band-shaped retinal areas in teleosts," Vidensk. Medd. Dan. Naturhist. Foren. 133, 85-120 (1970).

Nicol, J. A. C.

E. J. Denton and J. A. C. Nicol, "Polarization of light reflected from the silvery exterior of the bleak Alburnus alburnus," J. Mar. Biol. Assoc. U.K. 45, 705-709 (1965).
[CrossRef]

Parkyn, D. C.

D. C. Parkyn and C. W. Hawryshyn, "Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibers," J. Comp. Physiol., A 172, 493-500 (1993).
[CrossRef]

Petz, A. D.

N. Shashar, R. T. Hanlon, and A. D. Petz, "Polarization vision helps detect transparent prey," Nature 393, 222-223 (1998).
[CrossRef]

Pomozi, I.

I. Pomozi, G. Horváth, and R. Wehner, "How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation," J. Exp. Biol. 204, 2933-2942 (2001).
[PubMed]

Ritz, D. A.

D. A. Ritz, "Polarized-light responses in the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 154, 245-250 (1991).
[CrossRef]

Sabbah, S.

N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in sea water: implications for polarization signaling," J. Exp. Biol. 207, 3619-3628 (2004).
[CrossRef] [PubMed]

Schenck, H.

Schwind, R.

R. Schwind, "Daphnia pulex swims towards the most strongly polarized light--a response that leads to 'shore flight'," J. Exp. Biol. 202, 3631-3635 (1999).
[PubMed]

Shashar, N.

N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in sea water: implications for polarization signaling," J. Exp. Biol. 207, 3619-3628 (2004).
[CrossRef] [PubMed]

T. W. Cronin and N. Shashar, "The linearly polarized light field in clear, tropical, marine waters: spatial and temporal variation of light intensity, degree of polarization and e-vector angle," J. Exp. Biol. 204, 2461-2467 (2001).
[PubMed]

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

N. Shashar, R. T. Hanlon, and A. D. Petz, "Polarization vision helps detect transparent prey," Nature 393, 222-223 (1998).
[CrossRef]

Snyder, R. L.

Stramska, M.

M. Stramska and T. D. Dickey, "Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds," Deep-Sea Res., Part I 45, 1393-1410 (1998).
[CrossRef]

Suchanek, T. H.

W. N. McFarland, C. M. Wahl, T. H. Suchanek, and F. A. McAlary, "The behavior of animals around twilight with emphasis on coral reef communities," in Adaptive Mechanisms in Ecology of Vision, S.N.Archer, ed. (Academic, 1999), pp. 583-628.

Suhai, B.

Timms, A. M.

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

Timofeyeva, V. A.

V. A. Timofeyeva, "The degree of polarization of light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 6, 513-522 (1970).

V. A. Timofeyeva, "Plane of vibrations of polarized light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 5, 1049-1057 (1969).

V. A. Timofeyeva, "Optics of turbid waters (results of laboratory studies)," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 177-219.

Varjú, D.

G. Horváth and D. Varjú, "Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell's window of the flat water surface," Vision Res. 35, 1651-1666 (1995).
[CrossRef] [PubMed]

G. Horváth and D. Varjú, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer-Verlag, 2003).

Wahl, C. M.

W. N. McFarland, C. M. Wahl, T. H. Suchanek, and F. A. McAlary, "The behavior of animals around twilight with emphasis on coral reef communities," in Adaptive Mechanisms in Ecology of Vision, S.N.Archer, ed. (Academic, 1999), pp. 583-628.

Waterman, T. H.

R. B. Forward and T. H. Waterman, "Evidence for e-vector and light intensity pattern discrimination by the teleost Demogenys," J. Comp. Physiol. 87, 189-202 (1973).
[CrossRef]

R. B. Forward, K. W. Horch, and T. H. Waterman, "Visual orientation at the water surface by the teleost Zenarchopterus," Biol. Bull. 143, 112-126 (1972).
[CrossRef]

T. H. Waterman and R. B. Forward, "Field evidence for polarized light sensitivity in the fish Zenarchopterus," Nature 228, 85-87 (1970).
[CrossRef] [PubMed]

A. Ivanoff and T. H. Waterman, "Elliptical polarization of submarine illumination," J. Mar. Res. 16, 255-282 (1958).

A. Ivanoff and T. H. Waterman, "Factors, mainly depth and wavelength, affecting the degree of underwater light polarization," J. Mar. Res. 16, 283-307 (1958).

T. H. Waterman, "Polarization patterns in submarine illumination," Science 120, 927-932 (1954).
[CrossRef] [PubMed]

T. H. Waterman, "Polarization sensitivity," in Comparative Physiology and Evolution of Vision in Invertebrates. B: Invertebrate Visual Centers and Behavior I, H.Autrum, ed. (Springer-Verlag, 1981), pp. 281-469.

Wehner, R.

G. Horváth, B. Bernath, B. Suhai, A. Barta, and R. Wehner, "First observation of the fourth neutral point in the atmosphere," J. Opt. Soc. Am. A 19, 2085-2099 (2002).
[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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

I. Pomozi, G. Horváth, and R. Wehner, "How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation," J. Exp. Biol. 204, 2933-2942 (2001).
[PubMed]

R. Wehner, "Polarization vision--a uniform sensory capacity?" J. Exp. Biol. 204, 2589-2596 (2001).
[PubMed]

Wolff, L. B.

L. B. Wolff and A. G. Andreou, "Polarization camera sensors," Image Vis. Comput. 13, 497-509 (1995).
[CrossRef]

Am. Sci. (1)

C. W. Hawryshyn, "Polarization vision in fish," Am. Sci. 80, 479-491 (1992).

Am. Zool. (1)

W. N. McFarland, "Light in the sea--correlations with behaviors of fishes and invertebrates," Am. Zool. 26, 389-401 (1986).

Behaviour (1)

C. Groot, "On the orientation of young sockeye salmon (Oncorhynchus nerka) during their seaward migration out of lakes," Behaviour (Suppl.) 14, 1-198 (1965).

Biol. Bull. (1)

R. B. Forward, K. W. Horch, and T. H. Waterman, "Visual orientation at the water surface by the teleost Zenarchopterus," Biol. Bull. 143, 112-126 (1972).
[CrossRef]

Deep-Sea Res., Part I (1)

M. Stramska and T. D. Dickey, "Short-term variability of the underwater light field in the oligotrophic ocean in response to surface waves and clouds," Deep-Sea Res., Part I 45, 1393-1410 (1998).
[CrossRef]

Image Vis. Comput. (1)

L. B. Wolff and A. G. Andreou, "Polarization camera sensors," Image Vis. Comput. 13, 497-509 (1995).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (2)

V. A. Timofeyeva, "The degree of polarization of light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 6, 513-522 (1970).

V. A. Timofeyeva, "Plane of vibrations of polarized light in turbid media," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 5, 1049-1057 (1969).

J. Comp. Physiol. (2)

R. B. Forward and T. H. Waterman, "Evidence for e-vector and light intensity pattern discrimination by the teleost Demogenys," J. Comp. Physiol. 87, 189-202 (1973).
[CrossRef]

H. Kleerekoper, J. H. Matis, A. M. Timms, and P. Gensler, "Locomotor response of the goldfish to polarized light and its e-vector," J. Comp. Physiol. 86, 27-36 (1973).
[CrossRef]

J. Comp. Physiol., A (5)

C. W. Hawryshyn, M. G. Arnold, D. Bowering, and R. L. Cole, "Spatial orientation of rainbow trout to plane-polarized light: the ontogeny of e-vector discrimination and spectral characteristics," J. Comp. Physiol., A 166, 565-574 (1990).
[CrossRef]

C. W. Hawryshyn and W. N. McFarland, "Cone photoreceptor mechanisms and the detection of polarized light in fish," J. Comp. Physiol., A 160, 459-465 (1987).
[CrossRef]

D. C. Parkyn and C. W. Hawryshyn, "Polarized light sensitivity in rainbow trout (Oncorhynchus mykiss): characterization from multiunit ganglion cell responses in the optic nerve fibers," J. Comp. Physiol., A 172, 493-500 (1993).
[CrossRef]

S. M. Goddard and R. B. Forward, "The role of the underwater polarized light pattern, in sun compass navigation of the grass shrimp, Palaemonetes vulgaris," J. Comp. Physiol., A 169, 479-491 (1991).
[CrossRef]

C. W. Hawryshyn, H. D. Moyer, W. D. Allison, T. J. Haimberger, and W. N. McFarland, "Multidimensional polarization sensitivity in damselfishes," J. Comp. Physiol., A 189, 213-220 (2003).

J. Exp. Biol. (6)

I. Pomozi, G. Horváth, and R. Wehner, "How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation," J. Exp. Biol. 204, 2933-2942 (2001).
[PubMed]

R. Schwind, "Daphnia pulex swims towards the most strongly polarized light--a response that leads to 'shore flight'," J. Exp. Biol. 202, 3631-3635 (1999).
[PubMed]

I. Novales Flamarique and H. I. Browman, "Foraging and prey-search behavior of small juvenile rainbow trout (Oncorhynchus mykiss) under polarized light," J. Exp. Biol. 204, 2415-2422 (2001).
[PubMed]

N. Shashar, S. Sabbah, and T. W. Cronin, "Transmission of linearly polarized light in sea water: implications for polarization signaling," J. Exp. Biol. 207, 3619-3628 (2004).
[CrossRef] [PubMed]

T. W. Cronin and N. Shashar, "The linearly polarized light field in clear, tropical, marine waters: spatial and temporal variation of light intensity, degree of polarization and e-vector angle," J. Exp. Biol. 204, 2461-2467 (2001).
[PubMed]

R. Wehner, "Polarization vision--a uniform sensory capacity?" J. Exp. Biol. 204, 2589-2596 (2001).
[PubMed]

J. Exp. Mar. Biol. Ecol. (2)

D. A. Ritz, "Polarized-light responses in the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 154, 245-250 (1991).
[CrossRef]

S. M. Goddard and R. B. Forward, "The decay and learning of a y axis orientation behavior: the offshore escape response of the shrimp Palaemonetes vulgaris (Say)," J. Exp. Mar. Biol. Ecol. 142, 137-150 (1990).
[CrossRef]

J. Fish Biol. (1)

G. S. Loesy, T. W. Cronin, T. H. Goldsmith, D. Hyde, N. J. Marshall, and W. N. McFarland, "The UV visual world of fishes: a review," J. Fish Biol. 54, 921-943 (1999).
[CrossRef]

J. Mar. Biol. Assoc. U.K. (1)

E. J. Denton and J. A. C. Nicol, "Polarization of light reflected from the silvery exterior of the bleak Alburnus alburnus," J. Mar. Biol. Assoc. U.K. 45, 705-709 (1965).
[CrossRef]

J. Mar. Res. (2)

A. Ivanoff and T. H. Waterman, "Elliptical polarization of submarine illumination," J. Mar. Res. 16, 255-282 (1958).

A. Ivanoff and T. H. Waterman, "Factors, mainly depth and wavelength, affecting the degree of underwater light polarization," J. Mar. Res. 16, 283-307 (1958).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Limnol. Oceanogr. (1)

J. Janssen, "Searching for zooplankton just outside Snell's window," Limnol. Oceanogr. 26, 1168-1171 (1981).
[CrossRef]

Mar. Behav. Physiol. (1)

S. M. Goddard and R. B. Forward, "The use of celestial cues in the offshore escape response of the shrimp, Palaemonetes vulgaris," Mar. Behav. Physiol. 16, 11-18 (1989).
[CrossRef]

Nature (2)

N. Shashar, R. T. Hanlon, and A. D. Petz, "Polarization vision helps detect transparent prey," Nature 393, 222-223 (1998).
[CrossRef]

T. H. Waterman and R. B. Forward, "Field evidence for polarized light sensitivity in the fish Zenarchopterus," Nature 228, 85-87 (1970).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. 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. London, Ser. A 457, 1385-1399 (2001).
[CrossRef]

Science (1)

T. H. Waterman, "Polarization patterns in submarine illumination," Science 120, 927-932 (1954).
[CrossRef] [PubMed]

Vidensk. Medd. Dan. Naturhist. Foren. (1)

O. Munk, "On the occurrence and significance of horizontal band-shaped retinal areas in teleosts," Vidensk. Medd. Dan. Naturhist. Foren. 133, 85-120 (1970).

Vision Res. (3)

G. Horváth and D. Varjú, "Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell's window of the flat water surface," Vision Res. 35, 1651-1666 (1995).
[CrossRef] [PubMed]

I. Novales Flamarique and C. W. Hawryshyn, "Is the use of underwater polarized light by fish restricted to crepuscular time periods?" Vision Res. 37, 975-989 (1997).
[CrossRef]

N. Shashar, R. Hagen, J. G. Boal, and R. T. Hanlon, "Cuttlefish use polarization sensitivity in predation on silvery fish," Vision Res. 40, 71-75 (2000).
[CrossRef] [PubMed]

Other (12)

N. G. Jerlov, "Significant relationships between optical properties of the sea," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 77-94.

N. J. Jerlov, Optical Oceanography, Vol. 5of Elsevier Oceanography Series (Elsevier, 1968).
[CrossRef]

G. Horváth and D. Varjú, Polarized Light in Animal Vision: Polarization Patterns in Nature (Springer-Verlag, 2003).

J. N. Lythgoe, The Ecology of Vision (Oxford U. Press, 1979).

T. H. Waterman, "Polarization sensitivity," in Comparative Physiology and Evolution of Vision in Invertebrates. B: Invertebrate Visual Centers and Behavior I, H.Autrum, ed. (Springer-Verlag, 1981), pp. 281-469.

W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, 1985).

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

E. Collett, Polarized Light--Fundamentals and Applications (Marcel Dekker, 1994).

A. Ivanoff, "Polarization measurements in the sea," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 151-175.

V. A. Timofeyeva, "Optics of turbid waters (results of laboratory studies)," in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 177-219.

W. N. McFarland, "The visual world of coral reef fishes," in The Ecology of Fishes on Coral Reefs (Academic, 1991), pp. 16-37.

W. N. McFarland, C. M. Wahl, T. H. Suchanek, and F. A. McAlary, "The behavior of animals around twilight with emphasis on coral reef communities," in Adaptive Mechanisms in Ecology of Vision, S.N.Archer, ed. (Academic, 1999), pp. 583-628.

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Figures (6)

Fig. 1
Fig. 1

Definition of light directionality and measurements in the underwater setup. The solar zenith angle ( θ i ) —the vertical angle between the zenith and the sun as viewed from outside the water—ranges from 0°, when the sun is at the zenith, to 90°, when the sun is at the horizon, to angles greater than 90°, when the sun is below the horizon; the vertical angle between the refracted light beam and the zenith ( θ r ) ranges from 0°, when the sun is at the zenith, to 48.6°, when the sun is at the horizon, to larger angles, when the sun is below the horizon; the detector zenith angle ( θ p ; also referred to as the viewing zenith angle)—the vertical angle between the zenith and the detector—ranges from 0°, when the detector is facing the zenith, to 90°, when the detector is facing horizontally, to 180°, when the detector is pointing toward the nadir; the solar and detector azimuth angles φ i and φ p are the horizontal angles between the north direction and the sun or the detector, respectively (the azimuth angles are measured clockwise from the north when looking downward); and the relative direction ( φ ) is the horizontal angle between the two vertical planes containing the sun and the detector.

Fig. 2
Fig. 2

Celestial polarization patterns measured by full-sky imaging polarimetry (denoted “Sky”) and predicted sky patterns viewed through Snell’s window of a flat water surface calculated with the use of these celestial patterns (denoted “Snell’s window”) at sun altitudes θ S = 0 ° , 31°, and 64°. The center and the perimeter of the celestial patterns are the zenith and the horizon, respectively, while, in Snell’s window, the center and the perimeter of patterns due to refraction of light represent the zenith and Snell’s window boundary, respectively. (a), (g), (m) 180° field-of-view color photographs of the clear sky; (d), (j), and (p) 97.5° field-of-view color pictures of the sky visible through Snell’s window. For each sun altitude, the celestial and Snell’s window patterns of the percent polarization d and e-vector orientation α (measured clockwise from the local meridian) in the green ( 550 nm ) region of the spectrum are presented. In the measured d and α patterns, the overexposed celestial regions are shaded by red and white, respectively, and the radial black bar in the measured patterns is the wire of the sun occulter (a small black disk). In the α patterns, black bars and white rectangles represent the measured and predicted d and α at four detector positions (coinciding with the centers of the bars). The alignment of the bars and rectangles shows the orientation of polarization, while their width corresponds to four ranges of d, shown in the inset at the bottom.

Fig. 3
Fig. 3

Predicted celestial e-vector orientation patterns viewed from water through Snell’s window of a flat water surface at sun altitudes θ S = 0 ° , 31°, and 64°. Patterns are calculated from the Fresnel theory of refraction using measurements at (a), (e), (i) 450, (b), (f), (j) 550, and (c), (g), (k) 650 nm and (d), (h), (l) Rayleigh celestial patterns. Due to refraction of light, the center and the perimeter of the circular patterns are the zenith and the boundary of Snell’s window, respectively, and the apparent sun altitudes are θ S = 41.26 ° , 49.86°, and 70.43°. Black bars and white rectangles represent the measured and predicted percent polarization and e-vector orientation at several detector positions (coinciding with the centers of the bars). The alignment of the bars and rectangles shows the orientation of polarization, while their width corresponds to their percent polarization range.

Fig. 4
Fig. 4

Measured versus predicted (a) percent polarization and (b) e-vector orientation of skylight viewed from water through Snell’s window. Solid lines, predicted polarization using measured celestial patterns at 450 and 650 nm (solid circles, n = 29 ) and 550 nm (open circles, n = 15 ); dashed lines, predicted polarization using the Rayleigh celestial patterns (compared only with the measured values at 550 nm ); dotted lines, identity ( y = x ) . In all cases, p < 0.001 . Polarization values within Snell’s window, predicted in both manners, did not vary from each other (Wilcoxon, n = 15 , p > 0.6 and p > 0.3 for d and α values, respectively).

Fig. 5
Fig. 5

Changes in percent polarization according to wavelength within Snell’s window. (a) Gray and black circles represent individual measurements serving as examples of the four main “wavelength dependency patterns,” classified as type 1 (small black circles), type 2 (large gray circles), type 3 (large black circles), and type 4 (small gray circles). Dependency types were defined by the relative relationships of the percent polarization values at given wavelengths; type 1 ( d 350 < d 450 > d 700 , similar to a downward-opening parabola), type 2 (decreasing with wavelength, d 350 > d 500 > d 700 ), type 3 ( d 350 > d 500 < d 700 , similar to an upward-opening parabola), and type 4 ( d 350 < d 500 < d 700 , increasing with wavelength). (b) Dependency types, appearing as circles, at different combinations of the sun’s altitude and relative direction. Measurements that did not fit any of these dependency patterns were classified as type 0 and are depicted by open squares. Each point represents a single measurement [such as the ones presented in (a)]. Traces originate from measurements taken with the sensor fixed at one of five relative directions (0°, 45°, 90°, 135°, and 180° away from the sunrise meridian), while the sun changes altitude during the day’s measurements.

Fig. 6
Fig. 6

Distribution of differences between wavelengths in (a) percent polarization ( Δ d d av ) and (b) e-vector orientation ( Δ α ) as a function of the sun altitude and relative direction. Δ d is defined as the maximal difference in the percent polarization between any examined wavelengths, and d av as the percent polarization averaged over the eight examined wavelengths ( 350 700 nm , with a 50 nm interval). Δ α is defined as the maximal difference in the e-vector orientation between wavelengths. In (a) and (b), dots represent individual measurements taken at a certain combination of sun altitude and relative direction, and surfaces stand for an extrapolation (distance-weighted least squares) calculated from the measurements. Sun altitude is measured off the horizon, while the relative direction corresponds to the horizontal angular distance between the directions of the line of sight and the sun. (c) Median Δ d d av (with the first and third quartiles) as a function of the wavelength at which the maximal percent polarization was attained. The proportion of measurements in which the maximal percent polarization was attained at the indicated wavelength is provided above each bar.

Equations (17)

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θ = ( 1 2 ) arctan ( I 0 + I 90 2 I 45 I 90 I 0 ) ,
Then if ( I 90 < I 0 ) [ if ( I 45 < I 0 ) θ = θ + 90 ° else θ = θ 90 ° ] .
if ( θ > 90 ° ) α = θ 90 ° else α = θ + 90 ° .
I = I 0 + I 90 ,
d = 100 ( I 0 I 90 ) 2 + ( 2 I 45 I 90 I 0 ) 2 I 0 + I 90 .
β SW = arcsin ( 1 n ) = arcsin ( 1 1.33 ) = 48.75 °
β * = arccos ( cos β n ) .
S = ( I , Q , U , V ) ,
Q = I d cos ( 2 α ) cos ( 2 ϵ )
U = I d sin ( 2 α ) cos ( 2 ϵ )
V = I d sin ( 2 ϵ )
M = sin ( 2 θ i ) sin ( 2 θ r ) 2 sin 2 ( θ i + θ r ) cos 2 ( θ i θ r ) × [ cos 2 ( θ i θ r ) + 1 cos 2 ( θ i θ r ) 1 0 0 cos 2 ( θ i θ r ) 1 cos 2 ( θ i θ r ) + 1 0 0 0 0 2 cos ( θ i θ r ) 0 0 0 0 2 cos ( θ i θ r ) ] ,
S t = M S i .
I Rayleigh ( γ ) = I max 2 ( 1 + cos 2 γ ) ,
d Rayleigh ( γ ) = d max 1 cos 2 γ 1 + cos 2 γ ,
d = Q 2 + U 2 I ,
α = 1 2 arctan ( U Q ) .

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