The polarization of skylight: An example from nature
Glenn S. Smitha兲
School of Electrical and Computer Engineering, Georgia Institute of Technology,
Atlanta, Georgia 30332-0250
共Received 27 March 2006; accepted 15 September 2006兲
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 共unpolarized兲 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. © 2007 American Association of Physics Teachers.
关DOI: 10.1119/1.2360991兴
I. INTRODUCTION
Polarization is a fundamental property of electromagnetic
radiation 共light兲 and is discussed at all levels from introductory courses in physics to graduate courses in electromagnetism. The polarization of skylight and its use by insects for
navigation is a practical example of much interest to students.
The polarization of skylight is easily observed by the eye
using a simple linear polarizer. Figure 1 shows results for a
clear blue sky: In Fig. 1共a兲 the polarizer is oriented for maximum transmission, and in Fig. 1共b兲 it is oriented for minimum transmission.
The first scientific observation of the polarization of skylight is usually attributed to the French natural philosopher
Dominique François Jean Arago in 1809.1,2 There are claims
that the Vikings knew of this phenomenon nearly 1000 years
earlier and used it for navigation.3,4 The Vikings supposedly
discovered a naturally occurring dichroic mineral, referred to
as “sunstone,” that served as a linear polarizer. They normally navigated using the position of the sun, but when the
sun was obscured by clouds or below the horizon, they could
use this device to sense the direction of polarization for the
visible portion of the sky. Then, knowing the relation between the direction of polarization and the location of the
sun, they could infer the position of the sun. The claim for
the Vikings’ navigation by polarized skylight has been
disputed.5
In the popular scientific literature, there are qualitative explanations of the polarization of skylight.6,7 The objective of
this paper is to go beyond these explanations and to present
a simple, analytical model that not only provides a physical
explanation for the polarization of skylight, but can also be
used for quantitative calculations that can be compared to
measurements. The model brings together material from different topics in optics: polarization of plane waves, natural
共unpolarized兲 light, and dipole scattering. Unlike other treatments, the analysis is done entirely in terms of the timevarying field, without resorting to the frequency domain.
Figure 2 is a schematic drawing showing the details of an
observation made in the “principal plane” or the “sun’s vertical,” which is the plane that contains the local zenith and
the center of the sun. This plane is the y-z plane in Fig. 2,
Am. J. Phys. 75 共1兲, January 2007
A. Natural light
To develop a description for the natural or unpolarized
light from the sun, we first consider Fig. 3共a兲 in which the
electric field of the incident plane wave 共light兲 is linearly
polarized in the x direction. At the polarizer the electric field
is
i
i
共t兲 = Elin,x
共t兲x̂.
Elin
http://aapt.org/ajp
共1兲
Throughout the paper we will be interested in the irradiance for a wave, because for optical signals this quantity can
be measured with a practical detector, in contrast to the electric field, for which there are no detectors available that have
a response time short enough to resolve the temporal variation. The irradiance for a plane wave propagating in the z
direction is
I=
=
II. SIMPLE MODEL
25
and it contains the points centered on the sun 共S兲, the scatterer 共P兲, and the observer 共M兲. The angles of elevation for
the sun and scatterer are ␺s and ␺ p.
The direct light from the sun is natural or unpolarized
light. This light is scattered by the molecules of the air or,
alternatively, fluctuations in the density of the air.8 The scattering elements are small compared to all wavelengths of
significance, so the scattering of light is by Rayleigh scattering. The positions of the elements are random, so the scattering from the various elements is incoherent. Thus, we only
need to consider dipole scattering from one element. The
scattered light 共skylight兲 is partially polarized, that is it is
equivalent to natural light plus a linearly polarized component. The observer views the skylight through a linear polarizer with its transmission axis at the angle ␰ to the normal of
the principal plane 共x axis兲. As the observer rotates the polarizer, he/she sees a maximum and a minimum in the irradiance It共␰兲 共time-average power per unit area兲, as shown in
Fig. 1.
1
TD
1
TD
冕
冕
TD/2
ẑ · S共t兲 dt
t=−TD/2
TD/2
t=−TD/2
1
1
兩E共t兲兩2dt = 具兩E共t兲兩2典,
␨0
␨0
共2兲
where S is the Poynting vector and ␨0 is the wave impedance
of free space. The time average in Eq. 共2兲, which is indicated
by the angle brackets 具¯典, is over the time interval TD associated with the detector. In an experiment, we take this time
© 2007 American Association of Physics Teachers
25
Fig. 1. A clear blue sky viewed through a linear polarizer. 共a兲 The polarizer
is oriented for maximum transmission 共transmission axis, white arrow, is
normal to the principal plane兲. 共b兲 The polarizer is oriented for minimum
transmission 共transmission axis is parallel to the principal plane兲. From the
light meter readings that go with these photographs, the degree of linear
polarization is dl ⬇ 0.5.
interval to be long enough to make the average practically
independent of TD, and for mathematical calculations we
take TD → ⬁. For the familiar time-harmonic field with angular frequency ␻, that is, E共t兲 = E0 cos 共␻t兲x̂, we have I
= 兩E0兩2 / 2␨0. For the incident electric field in Eq. 共1兲, the
irradiance is
i
Ilin
=
1
1 i
i
具兩Elin
共t兲兩2典 = 具兩Elin,x
共t兲兩2典.
␨0
␨0
Fig. 3. The action of an ideal linear polarizer on waves with various states
of polarization: 共a兲 linearly polarized light, 共b兲 natural light, and 共c兲 partially
polarized light composed of linearly polarized light plus natural light.
共3兲
i
t
共t兲 = Elin,x
共t兲 cos 共␰兲û,
Elin
The transmission axis of the ideal linear polarizer in Fig.
3共a兲 is at the angle ␰ with respect to the x axis. After the
wave passes through the polarizer, the transmitted electric
field and irradiance are
t
Ilin
共␰兲 =
共4兲
1 i
i
具兩E 共t兲兩2典 cos2 共␰兲 = Ilin
cos2 共␰兲,
␨0 lin,x
共5兲
where û is a unit vector in the direction of the transmission
axis of the polarizer. Here and in the following, we ignore
any time delay common to all field components that is a
result of the wave passing through the polarizer. Notice that
Eq. 共5兲 is just the law of Malus for the action of an ideal
linear polarizer on a linearly polarized wave.9,10
We next consider the case shown in Fig. 3共b兲 in which the
incident wave is natural or unpolarized light. If we could
measure the electric field of natural light, it would produce a
chaotic waveform, similar to the familiar noise voltage associated with electronic circuits. The description of natural
light must be based on statistical quantities that can be measured. The representation for natural light we will use involves time averages, as in Eq. 共2兲. In Fig. 3共b兲 the pair of
orthogonal axes, x and y, have arbitrary orientation, and the
electric field is
i
i
共t兲 = Enat,x
共t兲x̂ + Einat,y共t兲ŷ.
Enat
共6兲
i
i
have zero mean, 具Enat,x
共t兲典 = 0 and
The component of Enat
and they obey the relations:
具Einat,y共t兲典 = 0,
i
i
共t兲Enat,x
共t − ␶兲典 = 具Einat,y共t兲Einat,y共t − ␶兲典 = F共␶兲
具Enat,x
Fig. 2. Schematic drawing of the observation of the polarization of skylight
in the principal plane 共the plane containing the points S, P, and M兲.
26
Am. J. Phys., Vol. 75, No. 1, January 2007
共7兲
and
Glenn S. Smith
26
i
具Enat,x
共t兲Einat,y共t − ␶兲典 = 0.
共8兲
These results are independent of the choice of the origin in
time and to hold for all ␶. We say that the time autocorrelation functions 共7兲 for the x and y components of the field for
natural light are equal, and that the time cross-correlation
function 共8兲 for the x and y components of the field is zero.11
The irradiance for the incident natural light is
i
=
Inat
1
2 i
i
关具兩Enat,x
共t兲兩2典 + 具兩Einat,y共t兲兩2典兴 = 具兩Enat,x
共t兲兩2典, 共9兲
␨0
␨0
where Eq. 共7兲 with ␶ = 0 was used in the last step. After the
wave passes through the linear polarizer, the transmitted
electric field and irradiance are
i
t
共t兲 = 关Enat,x
共t兲 cos 共␰兲 + Einat,y共t兲 sin 共␰兲兴û,
Enat
t
=
Inat
共10兲
1
i
关具兩Enat,x
共t兲兩2典 cos2 共␰兲 + 具兩Einat,y共t兲兩2典 sin2 共␰兲
␨0
i
共t兲Einat,y共t兲典 cos 共␰兲 sin 共␰兲兴
+ 2具Enat,x
=
1 i
1 i
具兩E 共t兲兩2典 = Inat
.
2
␨0 nat,x
共11兲
Equations 共7兲 and 共8兲 with ␶ = 0 were used to obtain the result
t
is seen
in Eq. 共11兲. The irradiance of the transmitted light Inat
i
, no
to be one half of the irradiance of the incident light Inat
matter what the orientation ␰ of the linear polarizer. This
result is an important characteristic of natural light.12
We next consider the case shown in Fig. 3共c兲 in which the
incident wave is the sum of the linearly polarized light in
Fig. 3共a兲 and the natural light in Fig. 3共b兲; that is, the electric
field is Eq. 共1兲 plus Eq. 共6兲. This combination is referred to
as partially polarized light, or, more precisely, as partially,
linearly polarized light.13 We will assume that the linearly
polarized light and natural light are uncorrelated:
i
i
共t兲Enat,x
共t − ␶兲典 = 0,
具Elin,x
i
具Elin,x
共t兲Einat,y共t − ␶兲典 = 0.
共12兲
Fig. 4. Details for the scattering of natural light by an electrically small
element in the atmosphere. The points S, P, and M lie in the principal plane,
and the unit vectors x̂ and x̂⬘ are normal to this plane. The inset shows the
radiation patterns in the principal plane for the two components of the electric dipole moment: px⬘, which is normal to the principal plane, and py⬘,
which is in the principal plane.
obtained by using Eq. 共14兲 to write Eq. 共15兲 in terms of the
transmitted irradiances:
Straightforward calculations, similar to those we have already performed, show that the irradiances for the incident
light and transmitted light are
i
i
i
= Ilin
+ Inat
IPP
=
i
IPP
=
i
Ilin
i .
i
Ilin + Inat
Am. J. Phys., Vol. 75, No. 1, January 2007
t
t
max共IPP
兲 + min共IPP
兲
.
共14兲
B. Dipole scattering
Figure 4 is a schematic drawing of the details of the scattering by an element 共molecule or density fluctuation兲 in the
atmosphere. The incident natural light 共sunlight兲 is described
by Eqs. 共6兲–共9兲 with x , y , z replaced by x⬘ , y ⬘ , z⬘. The wave
propagates in the direction z⬘. Hence, at the element the incident electric field and irradiance are
共15兲
A more useful expression for the purposes of measurement is
27
t
t
max共IPP
兲 − min共IPP
兲
As seen from the right-hand side of Eq. 共16兲, the degree of
linear polarization can be determined by rotating the linear
polarizer and noting the maximum and minimum values of
the transmitted irradiance.
irradiance of linearly polarized component
irradiance of total
i
Ilin
t
t
IPP
共␰ = 0兲 + IPP
共␰ = ␲/2兲
=
共16兲
Note that the irradiance of the partially polarized light for
both the incident and the transmitted waves is the sum of the
individual irradiances for the two components 共linearly polarized light and natural light兲 when treated separately.
The degree of linear polarization of the incident, partially
polarized light is defined as
dl =
t
t
共␰ = 0兲 − IPP
共␰ = ␲/2兲
IPP
共13兲
and
1 i
t
t
t
i
共␰兲 = Ilin
+ Inat
= Ilin
cos2 共␰兲 + Inat
.
IPP
2
dl =
i
i
i
共t兲 = Enat,x⬘共t兲xˆ⬘ + Enat,y⬘共t兲yˆ⬘
Enat
共17兲
and
Glenn S. Smith
27
i
Inat
=
2 i
具兩E
共t兲典2典.
␨0 nat,x⬘
共18兲
The incident field induces an electric dipole moment in the
electrically small element:
i
i
p共t兲 = px⬘共t兲xˆ⬘ + py⬘共t兲yˆ⬘ = ␣e⑀0关Enat,x⬘共t兲xˆ⬘ + Enat,y⬘共t兲yˆ⬘兴,
共19兲
where ⑀0 is the permittivity of free space, and ␣e is the electric polarizability of the element. Note that the dipole moment has components parallel, py⬘, and perpendicular, px⬘, to
the principal plane. In Eq. 共19兲 we have assumed that the
dipole moment responds instantaneously to the incident electric field 共there is no dispersion兲. This assumption is good for
the molecules of air at optical wavelengths. If the dipole
moment is expressed in terms of the coordinate system of the
observer 共x , y , z兲, we have
pជ 共t兲 =
i
␣e⑀0关Enat,x
共t兲x̂
⬘
+
i
Enat,y⬘共t兲共−
sin 共␪兲ŷ + cos 共␪兲ẑ兲兴,
共20兲
where the angle ␪ is
␪ = ␺ p − ␺s − ␲/2.
共21兲
This dipole moment produces the radiation that is the skylight, and the electric field of this light is14
Esr共r,t兲 =
␮0
兵r̂ ⫻ 关r̂ ⫻ p̈共t − r/c兲兴其,
4␲r
共22兲
where ␮0 is the permeability of free space, r = rr̂ is the radial
vector drawn from the dipole, and the double dot over a
quantity indicates the second derivative with respect to
time.15 If Eq. 共20兲 is inserted into Eq. 共22兲, we obtain the
electric field incident on the polarizer, which is located at a
distance z from the scatterer:
Esr共z,t兲 = −
␣e
i
i
共t − z/c兲x̂ − Ënat,y⬘共t − z/c兲sin共␪兲ŷ兴.
关Ë
4␲c2z nat,x⬘
共23兲
After the wave passes through the linear polarizer, the
transmitted electric field and irradiance are
Et共z,t兲 = −
␣e
i
关Ë
共t − z/c兲cos共␰兲
4␲c2z nat,x⬘
i
− Ënat,y⬘共t − z/c兲sin共␪兲sin共␰兲兴û
共24兲
and
I t共 ␰ 兲 =
␣2e
2 4
16␲ c ␨0z2
i
关具兩Ënat,x⬘共t − z/c兲兩2典cos2共␰兲
i
+ 具兩Ënat,y⬘共t − z/c兲兩2典sin2共␪兲sin2共␰兲
i
i
− 2具Ënat,x⬘共t − z/c兲Ënat,y⬘共t − z/c兲典sin共␪兲cos共␰兲sin共␰兲兴
i
i
= K关具兩Ënat,x⬘共t⬘兲兩2典cos2共␰兲 + 具兩Ënat,y⬘共t⬘兲兩2典sin2共␪兲sin2共␰兲
i
i
− 2具Ënat,x⬘共t⬘兲Ënat,y⬘共t⬘兲典sin共␪兲cos共␰兲sin共␰兲兴.
共25兲
In the last line we have simplified the result by recognizing
that 1 / z changes little in the vicinity of the polarizer, so we
28
Am. J. Phys., Vol. 75, No. 1, January 2007
can replace the factor in front of the brackets by the constant
K, and we have set t⬘ = t − z / c.
After performing a series of operations on Eqs. 共7兲 and 共8兲
and setting ␶ = 0, we can show that 共see Appendix兲
i
i
具兩Ënat,x⬘共t⬘兲兩2典 = 具兩Ënat,y⬘共t⬘兲兩2典,
i
i
具Ënat,x⬘共t⬘兲Ënat,y⬘共t⬘兲典 = 0,
共26兲
so that Eq. 共25兲 can be written as
i
It共␰兲 = K具兩Ënat,x⬘共t⬘兲兩2典关cos2 共␰兲 + sin2 共␪兲 sin2共␰兲兴.
共27兲
If we substitute Eq. 共21兲 for the angle ␪ and rewrite the
trigonometric terms, we obtain our final result for the irradiance seen by the observer:
i
It共␰兲 = K具兩Ënat,x⬘共t⬘兲兩2典关sin2共␺ p − ␺s兲cos2共␰兲
+ cos2共␺ p − ␺s兲兴.
共28兲
The result Eq. 共28兲 has the same form as our earlier expression for the irradiance of partially, linearly polarized
light observed with a linear polarizer given by Eq. 共14兲; both
have a term that depends on cos2 共␰兲 as well as a term that is
independent of ␰. Thus, the skylight that we observe is
equivalent to partially polarized light, which is composed of
linearly polarized light and natural light. It is important to
realize that this equivalence applies to the observed irradiances as measured by the polarizer and detector; the electric
fields for the two types of light could be different and the
irradiances the same. We can calculate the degree of linear
polarization for the skylight from Eq. 共16兲:
dl =
sin2 共␺ p − ␺s兲
It共␰ = 0兲 − It共␰ = ␲/2兲
=
.
It共␰ = 0兲 + It共␰ = ␲/2兲 1 + cos2 共␺ p − ␺s兲
共29兲
In summary, we have found skylight to be equivalent to a
mixture of linearly polarized light and natural light 共partially
polarized light兲, with the linearly polarized component normal to the principal plane, and the degree of linear polarization a simple function 关Eq. 共29兲兴 of the difference in the
angles of elevation for the observation point 共scatterer兲 and
the sun, ␺ p − ␺s. Specifically, the degree of linear polarization
is maximum when the ray from the sun to the scatterer 共SP兲
is orthogonal to the ray from the scatterer to the observer
共PM兲, then ␺ p − ␺s = ␲ / 2 and dl = 1. For other orientations the
degree of polarization is less; the minimum occurs when the
rays are parallel or antiparallel, then ␺ p − ␺s = 0, ␲ and dl = 0.
An examination of the patterns for dipole radiation, shown
in the inset of Fig. 4, provides insight into these results. In
the principal plane, the component of the dipole moment px⬘
共viewed end on in Fig. 4兲 radiates an electric field that is
normal to this plane and independent of ␺s and ␺ p. The other
component of the dipole moment py⬘ radiates an electric field
that is in this plane and proportional to 兩sin共␪兲 兩 = 兩cos共␺ p
− ␺s兲兩. Thus, when we view the element at an angle such that
␪ = 0 共␺ p − ␺s = ␲ / 2兲, we only see the component of the electric field that is normal to the principal plane 共the x⬘ component兲; hence, the electric field is linearly polarized. For other
angles of observation, we see a mixture of the radiated electric fields from the two components of the dipole moment;
hence, the electric field is partially polarized.
The simple result for the degree of linear polarization of
skylight, Eq. 共29兲, is compared with measurements in Fig. 5.
For these measurements the observer is viewing the zenith
Glenn S. Smith
28
Fig. 5. Degree of linear polarization for light from the zenith sky versus the
angle of elevation of the sun, ␺s. The measured data are from 1947 共Ref. 16兲
and from 1977 共Ref. 2兲.
sky 共␺ p = ␲ / 2兲 as the sun rises 共␺s increases兲. Results are
shown for two measurements; both were taken at high altitude on a clear day. The dots are for results measured 共visible
spectrum兲 at Bocaiuva, Brazil at an altitude of 671 m in
1947; the dashed line is for results measured 共␭ = 0.71 ␮m兲
on Mauna Loa in Hawaii at an altitude of 3400 m in
1977.2,16 The general trend is predicted by the simple theory,
that is, a decrease in the degree of linear polarization as the
sun rises. However, the predicted degree of polarization is
always greater than measured. For example, the maximum
degree of linear polarization, which occurs at sunrise, ␺s = 0,
is 100% for the theory but only about 84% for the measurements. Factors not included in the simple theory cause this
difference and will be discussed later.
III. DISTRIBUTION FOR POLARIZED SKYLIGHT
From our knowledge of the degree of linear polarization in
the principal plane, Eq. 共29兲, we can obtain the degree of
linear polarization over the rest of the sky. First, we introduce the unit vectors shown in Fig. 6共a兲: n̂s points from the
observer to the sun along MS, and n̂ p points from the observer to the observation point along MP. Because of the
great distance to the sun, the ray MS in Fig. 6共a兲 is parallel to
the ray PS in Fig. 2, and both are at the angle of elevation ␺s.
We also observe that n̂ p · n̂s = cos 共␺ p − ␺s兲, so Eq. 共29兲 can be
written as
dl =
29
1 − cos2 共␺ p − ␺s兲 1 − 共n̂ p · n̂s兲2
=
.
1 + cos2 共␺ p − ␺s兲 1 + 共n̂ p · n̂s兲2
Am. J. Phys., Vol. 75, No. 1, January 2007
共30兲
Fig. 6. 共a兲 Drawing of the principal plane showing the unit vectors n̂s and n̂ p
pointing from the observer toward the sun and toward the observation point,
respectively. 共b兲 Construction for a circle on which the degree of linear
polarization dl is a constant.
From Eq. 共30兲 it is clear that the degree of linear polarization
depends only on the direction of the observation point relative to the direction of the sun.
Now consider the construction shown in Fig. 6共b兲. The
line MP 共unit vector n̂ p兲 lies in the principal plane 共gray兲. If
this line is rotated about the line through the sun, that is,
about MS 共about the unit vector n̂s兲, the end of the line traces
out a circle 共dotted line兲. At every point on this circle, the
degree of linear polarization is the same, because the dot
product that appears in Eq. 共30兲 is the same. For example, for
the point P⬘ we have n̂ p⬘ · n̂s = n̂ p · n̂s. At each point on this
circle, the linearly polarized component of the electric field
is tangent to the circle.
From these observations we can construct polarization diagrams for the whole sky.17 Two of these diagrams are shown
in Fig. 7 for the case ␺s = 35°. Figure 7共a兲 shows mainly the
solar half of the sky, and Fig. 7共b兲 shows mainly the antisolar half of the sky. The length of a heavy line indicates the
degree of linear polarization, and the line is parallel to the
direction of the linearly polarized component of the electric
field. Note that the circles of constant dl are centered on the
line through the sun, MS, and that the maximum 共dl = 1兲 occurs, as expected, when ␺ p − ␺s = ␲ / 2 共n̂ p · n̂s = 0兲. As the sun
moves across the sky, this pattern for the polarization moves
over the hemisphere.
It is convenient to have an analytical description for the
polarization of skylight that applies over the entire hemisphere. For this purpose, a parametric expression for a circle
of constant dl 关the dotted curve in Fig. 6共b兲兴 can be obtained
in terms of the arc length ␨. The location of a point on this
Glenn S. Smith
29
circle, such as P⬘, is given by the azimuthal angle relative to
the direction of the sun ␣az and the angle of elevation ␤el. For
points on the right half of the hemisphere, these angles are
restricted to the ranges 0 ⱕ ␣az ⱕ ␲ and 0 ⱕ ␤el ⱕ ␲ / 2; results
on the left half of the hemisphere can be obtained from those
on the right half by symmetry. When ␺s is specified, the
following parametric equations for ␣az and ␤el describe a
circle of constant dl:
␣az共␨兲 =
冋冑
册
␲
+ tan−1 ⫿
2
1 − dl
cos共␺s兲csc共␨兲
2dl
+ sin共␺s兲cot共␨兲 ,
冋冑
␤el共␨兲 = sin−1 ±
+
冑
1 − dl
sin共␺s兲
1 + dl
共31兲
册
2dl
cos共␺s兲cos共␨兲 .
1 + dl
共32兲
For some values of dl, there are two separate curves,
hence, the two signs in these equations. The parameter ␨
must be constrained to ensure that points on the lower hemisphere are excluded:
0ⱕ␨ⱕ
冦
␲,
−1
cos
冋冑
⫿
冏冑
册 冏冑
1 − dl
tan共␺s兲 ,
2dl
冏
冏
1 − dl
tan共␺s兲 ⱖ 1,
2dl
冋
册
sin2共␺s兲
,
1 + cos2共␺s兲
共34兲
to exclude the cases in which the entire circle for dl lies on
the lower hemisphere. In Eqs. 共31兲–共33兲, the principal values
of the inverse trigonometric functions are assumed: −␲ / 2
ⱕ sin−1 共 兲 ⱕ ␲ / 2, 0 ⱕ cos−1 共 兲 ⱕ ␲, and −␲ / 2 ⱕ tan−1 共 兲
ⱕ ␲ / 2.
Figure 8 presents contour plots for the degree of linear
polarization when the elevation angle of the sun is ␺s
= 44.7°. These are polar graphs in which the radial variable is
␤el 共0° at the outer edge and 90° at the center兲 and the angular variable is ␣az. The results in Fig. 8共a兲 are from the
simple theory, Eqs. 共31兲–共34兲, and those in Fig. 8共b兲 are measured data. The measurements were made at the wavelength
␭ = 0.439 ␮m on a very clear day during February 1996 at
the University of Miami in Miami, FL.18,19 The measured
30
Am. J. Phys., Vol. 75, No. 1, January 2007
共33兲
1 − dl
tan共␺s兲 ⬍ 1.
2dl
For the lower sign in Eqs. 共31兲–共33兲 we use only the values
for which
dl ⬎
Fig. 7. Polarization diagram for the entire sky when ␺s = 35°. The length of
the heavy line indicates the degree of linear polarization dl; the line is
parallel to the direction of the linearly polarized component of the electric
field. Results are shown for two orientations: 共a兲 mainly the solar half of the
sky and 共b兲 mainly the anti-solar half of the sky.
results are not shown near the horizon 共␤el ⱕ 10° – 15° 兲
where they are irregular. The simple theory and the measurements show the same general structure for the polarization,
particularly the direction of the linearly polarized component
of the field, which is parallel to a contour. However, the
maximum degree of linear polarization is 100% for the
theory but only 50%–60% for the measurements. In both
cases, the maximum, as expected, occurs when ␣az ⬇ 180°
and ␤el ⬇ 90° −␺s ⬇ 45°.
The difference between the theory and measurements can
be attributed to several factors not included in the simple
theory that decrease the linearly polarized component of the
light: the anisotropic polarizability of the air molecules, multiple scattering of light between air molecules, scattering of
light from aerosol particles and dust in the atmosphere, and
sunlight reflected from the clouds and the ground. Some of
these factors are more significant in the urban environment
of Miami than at the high altitude sites for the measurements
shown in Fig. 5. This difference is probably why the measured maximum dl ⬇ 0.5 in Fig. 8共b兲 is significantly lower
Glenn S. Smith
30
Fig. 8. Contour plots for the degree of linear polarization dl for ␺s = 44.7°.
共a兲 Simple theory. 共b兲 Measured data at ␭ = 0.439 ␮m 共Refs. 18 and 19兲. The
position of the sun is shown by the small symbol.
than the measured maximum dl ⬇ 0.84 in Fig. 5. These additional factors introduce other interesting effects into the measurements that are not predicted by the simple theory. For
example, there are points where dl = 0 other than those at
␺ p − ␺s = 0 and 180° that are not predicted by the simple
theory. A comprehensive discussion of these effects is given
in Refs. 2 and 20.
IV. INSECT NAVIGATION BY POLARIZED
SKYLIGHT
Under very favorable conditions, human beings can detect
the presence of polarized light through a faint pattern known
as Haidinger’s brush.21 Recognizing and interpreting this
pattern takes practice, and it plays no known role in our
functioning. The reader is referred to Refs. 22–24 for details
of this interesting phenomenon. The situation is quite different for many insects, because they can readily detect the
polarization of light 共specifically skylight兲 and make use of it
in various ways.25 The insects that detect and use the polarization of light include honey bees, ants, crickets, flies, and
beetles.
The first conclusive evidence for the use of polarized skylight for orientation by insects was obtained by Karl Ritter
31
Am. J. Phys., Vol. 75, No. 1, January 2007
Fig. 9. Schematic drawings showing von Frisch’s experiment demonstrating
the honey bee’s orientation by polarized light. Each figure shows the bee’s
tail-wagging dance on a horizontal comb of the hive. 共a兲 During the dance,
the sun is visible to the bee. 共b兲 During the dance, a patch of clear blue sky
is visible to the bee, and the view of the sun is blocked 共indicated by gray
area兲. 共c兲 Same as in 共b兲 with the skylight passing through a linear polarizer,
and the transmission axis of the polarizer û aligned with the linearly polarized component of the electric field of the skylight. 共d兲 Same as in 共c兲 with
the transmission axis of the linear polarizer rotated.
von Frisch for honey bees in 1948.26–29 Since then, the honey
bee has been studied intensely in this regard, and we will
restrict our discussion to this insect. Figure 9 is a schematic
drawing of one of von Frisch’s experiments.
When foraging for food 共nectar and pollen兲, worker bees
use the location of the sun to determine direction, that is,
they determine direction with respect to the sun much as we
determine direction with respect to magnetic north using a
compass.29–31 On returning to the hive, she 共all worker bees
are female兲 informs other workers of the location of the food.
When the food is at a distance of about 100 m or greater, she
communicates this information through the tail-wagging
dance, which is shown in Fig. 9共a兲 for a dance on the horizontal comb of a hive with the sun visible. The dance has a
straight portion that is continually repeated after circling to
the right or left. Along the straight portion, the bee waggles
her body, hence the name for the dance. The distance to the
food is encoded in the characteristics of the dance, such as its
tempo, and the direction to the food is indicated by the direction of the straight portion of the dance relative to the
direction of the sun, the solar azimuth angle ␣az in Fig. 9共a兲.
Von Frisch noticed that bees could communicate the location of the food through the tail-wagging dance even when
he blocked their view of the sun at the hive, as indicated in
Fig. 9共b兲 by the gray area. This observation held as long as
he left a patch of clear blue sky visible to the bees. He
Glenn S. Smith
31
Fig. 10. Elements in the bee’s detection of polarized light. 共a兲 The compound eye of the worker bee composed of about 5000 ommatidia. The inset shows the
ends of the ommatidia visible on the surface of the eye. The specialized ommatidia involved in the detection of polarized light are in the dorsal rim area, which
is shown in black. 共b兲 Longitudinal and transverse cross sections for one of the specialized ommatidia composed of nine visual cells 共Refs. 25 and 29兲. 共c兲 A
single visual cell from 共b兲, showing the details for the microvilli 共Ref. 37兲. 共d兲 An expanded view of the transverse cross section of the ommatidium in 共b兲
showing the three UV sensitive visual cells 共A, B, and C兲 in gray, and the orthogonal orientation of the microvilli in the rhabdom 共Ref. 25兲.
surmised that the bees were using the polarization of the
skylight for orientation; possibly they could determine the
location of the hidden sun by knowing the relation between
the pattern of polarization for skylight and the position of the
sun. To test this hypothesis, he passed the skylight visible to
the bees through a linear polarizer. When the transmission
axis of the polarizer û was aligned so as to pass the linearly
polarized component of the electric field of the skylight, as in
Fig. 9共c兲, the bee’s dance was unaltered from that in Fig.
9共b兲; that is, it still pointed in the direction of the food.
However, when the polarizer was rotated to change the direction of the transmitted electric field, as in Fig. 9共d兲, the
direction of the bee’s dance changed, so that it no longer
pointed in the direction of the food.
By passing the skylight visible to the bees through bandpass filters, von Frisch and others determined that bees
mainly use the ultraviolet 共UV兲 portion of the spectrum for
the orientation by polarized light. It was also shown that it is
32
Am. J. Phys., Vol. 75, No. 1, January 2007
the direction of the linearly polarized component of the electric field that is most important; the degree of linear polarization need only be greater than about 10%.
Since von Frisch’s pioneering behavioral research, there
has been substantial effort devoted to identifying the physiology of the bee’s eye responsible for sensing the polarization of light. Bees have compound eyes that are made from
many individual sensing units called ommatidia. The inset in
Fig. 10共a兲 shows the ends of the ommatidia visible on the
surface of the bee’s eye. The eye of a worker bee contains
about 5000 ommatidia. The ommatidia used for orientation
with polarized light in the UV are believed to be specialized
ones 共about 150兲 located at the upper rim of the eye, that is,
in the dorsal rim area.25,32–34 Figure 10共b兲 is a schematic
drawing showing the longitudinal and transverse cross sections for one of these specialized ommatidia. It consists of
nine long, nearly straight cells of equal length that are evident in the transverse cross section. These cells are fused at
Glenn S. Smith
32
the center of the ommatidium to form the rhabdom. Light
enters the ommatidium through the lens, which in these specialized ommatidia is covered with pore canals that increase
the visual field.35 The light accepted by the lens and crystalline cone is guided down the rhabdom, much as light is in an
optical fiber.
Figure 10共c兲 is a schematic drawing for one of the nine
visual cells shown in Fig. 10共b兲. The rhabdomere is the portion of a cell that contributes to the rhabdom. It is made up of
many small protrusions called microvilli that are perpendicular to the optical axis of the ommatidium.36,37 There are dipolar pigment molecules in the membrane of the microvilli,
and the axes of these molecules are aligned with the microvilli. As indicated in Fig. 10共c兲, light with its electric field
parallel to the microvilli is absorbed more by these molecules than light with its electric field perpendicular to the
microvilli. This feature makes the cell sensitive to the direction of the linearly polarized component of light. The measured response 共electrical signal兲 of these specialized cells is
as much as ten times greater when the electric field is parallel
to the microvilli than when it is perpendicular to the microvilli.
Three of the nine cells in a specialized ommatidium are
sensitive to UV and take part in the orientation by polarized
light; they are shaded gray in the transverse cross section
shown in Fig. 10共d兲.25,32–34 The microvilli of one of these
cells 共marked C兲 are perpendicular to those in the other two
cells 共marked A and B兲. Thus, in a single ommatidium there
are sensors for electric fields in two orthogonal directions.
Presumably, electrical signals enter the bee’s nervous system
from an ommatidium that indicate the relative magnitudes of
the electric fields in these two directions.
With the physiology of the bee’s eye responsible for sensing polarized light established, the question that remains is
how do bees use polarization for orientation and navigation?
Different theories have been proposed, and we will give a
brief sketch of one by Rossel and Wehner, which is well
described in the literature including its limitations.17,38–42 In
this theory bees mainly make use of the polarization of skylight in the anti-solar half of the sky, that is, for 90° ⱕ ␣az
ⱕ 270° and 0 ° ⱕ ␤el ⱕ 90° in Fig. 7. This observation makes
sense, because when ␺s ⬎ 0°, the skylight from the anti-solar
half of the sky is more highly polarized than that from the
solar half 关compare Fig. 7共a兲 with Fig. 7共b兲兴. Also, the light
from the solar half of the sky is composed of skylight plus
direct sunlight, and the latter is unpolarized.
Rossel and Wehner assume that the special ommatidia in
the dorsal rim area of the bee’s eyes are arranged so that they
match some gross features of the polarization of the antisolar sky when the bee faces the anti-solar meridian 共␣az
= 180° 兲. Specifically, for an ommatidium pointing in the solar azimuth direction ␣az, the axis of polarization 关the axis
for one of the two sets of microvilli in Fig. 10共d兲兴 is aligned
with a mean representation of the electric field of the skylight from that direction.
When the bee views a patch of blue sky, she rotates about
her vertical body axis. The signal the bee receives from the
array of specialized ommatidia in the dorsal rim area changes
during the rotation, and it is maximum when the bee is approximately facing the anti-solar meridian. With this procedure the bee effectively determines the location of the sun
共position in azimuth兲 from the polarized light it receives
from a patch of blue sky.
33
Am. J. Phys., Vol. 75, No. 1, January 2007
Recall that in each of the specialized ommatidium there
are actually two orthogonal sensors of polarization 共sets of
microvilli兲 关Fig. 10共d兲兴. Thus, when the response from one
set of sensors is maximum, the response from the other set of
sensors is minimum. The bee may use the contrast between
the signals from the two sets of sensors to enhance the accuracy for its orientation; that is, the bee may orient to maximize the difference in the signals from the two orthogonal
sets of sensors.
V. CONCLUSION
We have used the material in this paper to supplement a
conventional treatment of the polarization of plane electromagnetic waves. In addition to a classroom presentation,
each student is given a simple linear polarizer 共laminated
film polarizer, Edmund Optics, NT38-396兲. They are asked
to make a qualitative observation of the polarization of skylight using the hand-held polarizer, and to see if their assessment corresponds to Fig. 7. Readings from the current research on insects’ use of polarized skylight are assigned or
suggested. Students are often surprised to find that they can
readily understand the current research with only the additional knowledge of polarized light that they have received in
this presentation. The general feeling of students is that this
material on the polarization of skylight, particularly that pertaining to navigation by honey bees, makes what they believe to be a rather ordinary topic much more exciting.
ACKNOWLEDGMENTS
The author would like to thank R. Todd Lee for assistance
with the photographs shown in Fig. 1, and a reviewer for
making useful suggestions that were included in the paper.
The author is grateful for the support provided by the John
Pippin Chair in Electromagnetics that furthered this study.
This paper is dedicated to the memory of an exceptional
teacher, Ronold W. P. King 共1905–2006兲, Professor of Applied Physics at Harvard University.
APPENDIX: CORRELATION FUNCTIONS
FOR THE SECOND DERIVATIVE OF THE
ELECTRIC FIELD OF NATURAL LIGHT
The argument for the first of the relations in Eq. 共26兲 begins by differentiating the left-hand side of Eq. 共7兲 with respect to ␶, then using integration by parts to obtain:
1
TD
冕
TD/2
t=−TD/2
=−
冋
Ex共t兲 −
dEx共t⬘兲
dt⬘
册
dt
t−␶
1
1
TD/2
关Ex共t兲Ex共t − ␶兲兴−T
+
D/2
TD
TD
冕
TD/2
t=−TD/2
dEx共t兲
dt
⫻Ex共t − ␶兲 dt.
共A1兲
For TD → ⬁ we assume that the first term is negligible. We
differentiate the second term with respect to ␶ and integrate
by parts:
Glenn S. Smith
33
1
TD
冕
TD/2
t=−TD/2
=−
冋
dEx共t兲
dEx共t⬘兲
−
dt
dt⬘
冋
1 dEx共t兲
Ex共t − ␶兲
TD
dt
册
册
dt
t−␶
TD/2
+
−TD/2
1
TD
冕
TD/2
t=−TD/2
⫻Ex共t − ␶兲 dt.
d2Ex共t兲
dt2
共A2兲
As before, for TD → ⬁ we assume that the first term is negligible, and we differentiate the second term twice with respect to ␶ to obtain
1
TD
冕
TD/2
t=−TD/2
冋
d2Ex共t兲 d2Ex共t⬘兲
dt2
dt⬘2
册
t−␶
i
dt = 具兩Ënat,x
共t − ␶兲兩2典.
共A3兲
If the same operations are applied to the right-hand side of
Eq. 共7兲 and the results are equated, we have
i
共t − ␶兲兩2典 = 具兩Ëinat,y共t − ␶兲兩2典,
具兩Ënat,x
共A4兲
which for ␶ = 0 and x , y → x⬘ , y ⬘ is the first relation in Eq.
共26兲. The argument for the second relation in Eq. 共26兲 is
obtained in the same way.
a兲
Electronic mail: glenn.smith@ece.gatech.edu
D. F. J. Arago, Oeuvres Complètes de François Arago 共Gide, Paris,
1858兲, Vol. 7, pp. 394–395.
2
K. L. Coulson, Polarization and Intensity of Light in the Atmosphere
共Deepak, Hampton, VA, 1988兲, p. 2.
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H. LaFay, “The Vikings,” Natl. Geogr. 37, 492–541 共1970兲.
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C. P. Können, Polarized Light in Nature 共Cambridge U. P., Cambridge,
1985兲, p. 30.
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J. Walker, “The Amateur Scientist: More about polarizers and how to use
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D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed.
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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
共molecules兲 and A. Einstein 共density fluctuations兲: 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,
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In Eqs. 共6兲 and 共8兲 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
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1
34
Am. J. Phys., Vol. 75, No. 1, January 2007
partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–
1181 共1959兲.
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 quasimonochromatic 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. 共Cambridge U. P., Cambridge, 1999兲,
Chap. 10; L. Mandel and E. Wolf, Optical Coherence and Quantum Optics 共Cambridge U. P., Cambridge, 1995兲, Chap. 6.
14
The scattering element 共molecule兲 is in random motion with the speed
v c. 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 共Cambridge U. P., Cambridge, 1997兲, 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 1 / r.
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11–53.
18
K. J. Voss and Y. Liu, “Polarized radiance distribution measurement of
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19
Y. Liu and K. J. Voss, “Polarized radiance distribution measurement of
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20
Selected Papers on Scattering in the Atmosphere, edited by C. F. Bohren
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W. Haidinger, “Uber das directe Erkennen des polarisirten Lichts und der
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M. G. J. Minnaert, Light and Color in the Outdoors 共Springer-Verlag,
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D. Auerbach, “Optical polarization without tools,” Eur. J. Phys. 21,
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A. P. Ovcharenko and V. D. Yegorenkov, “Teaching students to observe
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Glenn S. Smith
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35