A simple analytical model is presented for calculating the major features of the polarization of skylight over a hemisphere centered on an earthbound observer. The model brings together material from different topics in optics: polarization of plane waves, natural (unpolarized) light, and dipole scattering. Results calculated with the simple model are compared with experimental data. A brief description of the ability of insects to sense the polarization of skylight and their use of it for navigation is given.

1.
D. F. J.
Arago
,
Oeuvres Complètes de François Arago
(
Gide
, Paris,
1858
), Vol.
7
, pp.
394
395
.
2.
K. L.
Coulson
,
Polarization and Intensity of Light in the Atmosphere
(
Deepak
, Hampton, VA,
1988
), p.
2
.
3.
H.
LaFay
, “
The Vikings
,”
Natl. Geogr.
37
,
492
541
(
1970
).
4.
C. P.
Können
,
Polarized Light in Nature
(
Cambridge U. P.
, Cambridge,
1985
), p.
30
.
5.
C.
Roslund
and
C.
Beckman
, “
Disputing Viking navigation by polarized light
,”
Appl. Opt.
33
,
4754
4755
(
1994
).
6.
J.
Walker
, “
The Amateur Scientist: More about polarizers and how to use them, particularly for studying polarized skylight
,”
Sci. Am.
238
(
1
),
132
136
(
1978
).
7.
D. K.
Lynch
and
W.
Livingston
,
Color and Light in Nature
, 2nd ed. (
Cambridge U. P.
, Cambridge,
2001
), pp.
26
27
.
8.
The two viewpoints as to the cause of the scattering that gives rise to the color and polarization of skylight were first introduced by Lord Rayleigh (molecules) and A. Einstein (density fluctuations): Lord Rayleigh, “
On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue sky
,”
Philos. Mag.
47
,
375
384
(
1899
).
A.
Einstein
, “
Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes
,”
Ann. Phys.
33
,
1275
1298
(
1910
).
English translation, “
Theory of the opalescence of homogeneous liquids and mixtures of liquids in the vicinity of the critical state
” in
Colloid Chemistry: Theoretical and Applied
, edited by
J.
Alexander
(
Chemical Catalog Company
, New York,
1926
), Vol.
1
, pp.
323
339
.
9.
E. T.
Malus
, “
Sur une propriété des forces répulsives qui agissent sur la lumiére
,”
Mémoires de physique et de chimie de la Société D’Arcueil
2
,
254
267
(
1809
).
English translation, “
On a property of the repulsive forces, that act on light
,”
A Journal of Natural Philosophy, Chemistry, and the Arts (Nicholson’s Journal)
30
,
161
168
(
1811
).
10.
W. A.
Shurcliff
,
Polarized Light: Production and Use
(
Harvard U. P.
, Cambridge, MA,
1966
), p.
39
.
11.
P. Z.
Peebles
, Jr.
,
Probability, Random Variables and Random Signal Principles
, 4th ed. (
McGraw-Hill
, New York,
2001
).
12.
In Eqs. (6) and (8) we have postulated the properties for the electric field of natural light. Then we used these properties to predict the action of the linear polarizer on this light. Historically, the opposite was true: The action of polarizers on natural light was used to infer the properties of the light. See, for example,
G. G.
Stokes
, “
On the composition and resolution of streams of polarized light from different sources
,”
Trans. Cambridge Philos. Soc.
9
,
399
416
(
1852
); and
E.
Wolf
, “
Coherence properties of partially polarized electromagnetic radiation
,”
Nuovo Cimento
13
,
1165
1181
(
1959
).
13.
We have discussed partially polarized light that is formed by adding linearly polarized light to natural light. Partially polarized light can be formed in other ways. For example, partially polarized light that is quasi-monochromatic is formed by adding elliptically polarized light to natural light. The composition of partially polarized, quasi-monochromatic light can be determined by performing a series of measurements with a linear polarizer and a linear retarder. The details are given in
M.
Born
and
E.
Wolf
,
Principles of Optics
, 7th ed. (
Cambridge U. P.
, Cambridge,
1999
), Chap. 10;
L.
Mandel
and
E.
Wolf
,
Optical Coherence and Quantum Optics
(
Cambridge U. P.
, Cambridge,
1995
), Chap. 6.
14.
The scattering element (molecule) is in random motion with the speed vc. We have assumed that this motion does not change the statistical properties of the field, and we have used the expression for the radiated field of a stationary dipole:
G. S.
Smith
,
An Introduction to Classical Electromagnetic Radiation
(
Cambridge U. P.
, Cambridge,
1997
), pp.
452
465
.
15.

The superscript sr is used to indicate that this is the “scattered radiated” field: the part of the scattered field that behaves as 1r.

16.
R. A.
Richardson
and
E. O.
Hulburt
, “
Sky-brightness measurements near Bocaiuva, Brazil
,”
J. Geophys. Res.
54
,
215
227
(
1949
).
17.
R.
Wehner
and
S.
Rossel
, “
The bee’s celestial compass – A case study in behavioural neurobiology
,” in
Experimental Behavioral Ecology and Sociobiology, In Memoriam Karl von Frisch 1886–1982
, edited by
B.
Hölldobler
and
M.
Lindauer
(
Sinauer Associates
, Sunderland, MA,
1985
). pp.
11
53
.
18.
K. J.
Voss
and
Y.
Liu
, “
Polarized radiance distribution measurement of skylight. I. System description and characterization
,”
Appl. Opt.
36
,
6083
6094
(
1997
).
19.
Y.
Liu
and
K. J.
Voss
, “
Polarized radiance distribution measurement of skylight. II. Experiment and data
,”
Appl. Opt.
36
,
8753
8764
(
1997
).
20.
Selected Papers on Scattering in the Atmosphere
, edited by
C. F.
Bohren
(
SPIE
, Bellingham, WA,
1989
), pp.
261
326
.
21.
W.
Haidinger
, “
Uber das directe Erkennen des polarisirten Lichts und der Lage der Polarisationsebene
” (On the direct recognition of polarized light and the polarization plane),
Ann. Phys. Chem.
63
,
29
39
(
1844
).
22.
M. G. J.
Minnaert
,
Light and Color in the Outdoors
(
Springer-Verlag
, New York,
1993
), pp.
276
278
.
23.
D.
Auerbach
, “
Optical polarization without tools
,”
Eur. J. Phys.
21
,
13
17
(
2000
).
24.
A. P.
Ovcharenko
and
V. D.
Yegorenkov
, “
Teaching students to observe Haidinger brushes
,”
Eur. J. Phys.
23
,
123
125
(
2002
).
25.
T.
Labhart
and
E. P.
Meyer
, “
Detection of polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye
,”
Microsc. Res. Tech.
47
,
368
379
(
1999
).
26.
K.
von Frisch
, “
Gelöste und ungelöste Rätsel der Bienensprache
,”
Naturwiss.
35
,
38
43
(
1948
).
27.
K.
von Frisch
, “
Die Polarisation des Himmelslichtes als orientierender Faktor bei den Tänzen der Bienen
,”
Experientia
5
,
142
148
(
1949
).
28.
K.
von Frisch
, “
Die Sonne als Kompass im Leben der Bienen
,”
Experientia
6
,
210
221
(
1950
).
29.
K.
von Frisch
,
The Dance Language and Orientation of Bees
(
Harvard U. P.
, Cambridge, MA,
1967
).
30.
J. L.
Gould
and
C. G.
Gould
,
The Honey Bee
(
Scientific American Library
, New York,
1988
).
31.
F. G.
Barth
,
Insect and Flower
(
Princeton U. P.
, Princeton, NJ,
1991
).
32.
R. H.
Schinz
, “
Structural specialization in the dorsal retina of the bee, Apis mellifera
,”
Cell Tissue Res.
162
,
23
34
(
1975
).
33.
T.
Labhart
, “
Specialized photoreceptors at the dorsal rim of the honeybee’s compound eye: polarizational and angular sensitivity
,”
J. Comp. Physiol.
141
,
19
30
(
1980
).
34.
R.
Wehner
and
S.
Strasser
, “
The POL area of the honey bee’s eye: Behavioural evidence
,”
Physiol. Entomol.
10
,
337
349
(
1985
).
35.
E. P.
Meyer
and
T.
Labhart
, “
Pore canals in the cornea of a functionally specialized area of the honey bee’s compound eye
,”
Cell Tissue Res.
216
,
491
501
(
1981
).
36.
R.
Menzel
and
A. W.
Snyder
, “
Introduction to photoreceptor optics—an overview
,” in
Photorecpetor Optics
, edited by
R.
Menzel
and
A. W.
Snyder
(
Springer-Verlag
, New York,
1975
), pp.
1
13
.
37.
R.
Wehner
, “
Polarized-light navigation by insects
,”
Sci. Am.
235
(
7
),
106
115
(
1976
).
38.
S.
Rossel
and
R.
Wehner
, “
The bee’s map of the e-vector pattern in the sky
,”
Proc. Natl. Acad. Sci. U.S.A.
79
,
4451
4455
(
1982
).
39.
S.
Rossel
and
R.
Wehner
, “
How bees analyse the polarization patterns in the sky, experiments and model
,”
J. Comp. Physiol., A
154
,
607
615
(
1984
).
40.
S.
Rossel
and
R.
Wehner
, “
Polarization vision in bees
,”
Nature (London)
323
,
128
131
(
1986
).
41.
K.
Kirschfeld
, “
Navigation and compass orientation by insects according to the polarization pattern of the sky
,”
Z. Naturforsch. C
43c
,
467
469
(
1988
).
42.
K.
Kirschfeld
, “
The role of the dorsal rim ommatidia in the bee’s eye
,”
Z. Naturforsch. C
43c
,
621
623
(
1988
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.