The polarization of skylight: An example from...

The polarization of skylight: An example from natureGlenn S. Smitha�

School of Electrical and Computer Engineering, Georgia Institute of Technology,Atlanta, Georgia 30332-0250

�Received 27 March 2006; accepted 15 September 2006�

A simple analytical model is presented for calculating the major features of the polarization ofskylight over a hemisphere centered on an earthbound observer. The model brings together materialfrom different topics in optics: polarization of plane waves, natural �unpolarized� light, and dipolescattering. Results calculated with the simple model are compared with experimental data. A briefdescription of the ability of insects to sense the polarization of skylight and their use of it fornavigation is given. © 2007 American Association of Physics Teachers.

�DOI: 10.1119/1.2360991�

I. INTRODUCTION

Polarization is a fundamental property of electromagneticradiation �light� and is discussed at all levels from introduc-tory courses in physics to graduate courses in electromagne-tism. The polarization of skylight and its use by insects fornavigation is a practical example of much interest to stu-dents.

The polarization of skylight is easily observed by the eyeusing a simple linear polarizer. Figure 1 shows results for aclear blue sky: In Fig. 1�a� the polarizer is oriented for maxi-mum transmission, and in Fig. 1�b� it is oriented for mini-mum transmission.

The first scientific observation of the polarization of sky-light is usually attributed to the French natural philosopherDominique François Jean Arago in 1809.1,2 There are claimsthat the Vikings knew of this phenomenon nearly 1000 yearsearlier and used it for navigation.3,4 The Vikings supposedlydiscovered a naturally occurring dichroic mineral, referred toas “sunstone,” that served as a linear polarizer. They nor-mally navigated using the position of the sun, but when thesun was obscured by clouds or below the horizon, they coulduse this device to sense the direction of polarization for thevisible portion of the sky. Then, knowing the relation be-tween the direction of polarization and the location of thesun, they could infer the position of the sun. The claim forthe Vikings’ navigation by polarized skylight has beendisputed.5

In the popular scientific literature, there are qualitative ex-planations of the polarization of skylight.6,7 The objective ofthis paper is to go beyond these explanations and to presenta simple, analytical model that not only provides a physicalexplanation for the polarization of skylight, but can also beused for quantitative calculations that can be compared tomeasurements. The model brings together material from dif-ferent topics in optics: polarization of plane waves, natural�unpolarized� light, and dipole scattering. Unlike other treat-ments, the analysis is done entirely in terms of the time-varying field, without resorting to the frequency domain.

II. SIMPLE MODEL

Figure 2 is a schematic drawing showing the details of anobservation made in the “principal plane” or the “sun’s ver-tical,” 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,

25 Am. J. Phys. 75 �1�, January 2007 http://aapt.org/ajp

and it contains the points centered on the sun �S�, the scat-terer �P�, and the observer �M�. The angles of elevation forthe sun and scatterer are �s and �p.

The direct light from the sun is natural or unpolarizedlight. This light is scattered by the molecules of the air or,alternatively, fluctuations in the density of the air.8 The scat-tering elements are small compared to all wavelengths ofsignificance, so the scattering of light is by Rayleigh scatter-ing. The positions of the elements are random, so the scat-tering from the various elements is incoherent. Thus, we onlyneed to consider dipole scattering from one element. Thescattered light �skylight� is partially polarized, that is it isequivalent to natural light plus a linearly polarized compo-nent. The observer views the skylight through a linear polar-izer with its transmission axis at the angle � to the normal ofthe principal plane �x axis�. As the observer rotates the po-larizer, he/she sees a maximum and a minimum in the irra-diance It�� time-average power per unit area�, as shown inFig. 1.

A. Natural light

To develop a description for the natural or unpolarizedlight from the sun, we first consider Fig. 3�a� in which theelectric field of the incident plane wave �light� is linearlypolarized in the x direction. At the polarizer the electric fieldis

Elini �t� = Elin,x

i �t�x̂ . �1�

Throughout the paper we will be interested in the irradi-ance for a wave, because for optical signals this quantity canbe measured with a practical detector, in contrast to the elec-tric field, for which there are no detectors available that havea response time short enough to resolve the temporal varia-tion. The irradiance for a plane wave propagating in the zdirection is

I =1

TD�

t=−TD/2

TD/2

ẑ · S�t� dt

=1

TD�

t=−TD/2

TD/2 1

�0�E�t��2dt =

1

�0��E�t��2� , �2�

where S is the Poynting vector and �0 is the wave impedanceof free space. The time average in Eq. �2�, which is indicatedby the angle brackets �¯�, is over the time interval TD asso-

ciated with the detector. In an experiment, we take this time

25© 2007 American Association of Physics Teachers

interval to be long enough to make the average practicallyindependent of TD, and for mathematical calculations wetake TD→�. For the familiar time-harmonic field with an-gular frequency �, that is, E�t�=E0 cos ��t�x̂, we have I= �E0�2 /2�0. For the incident electric field in Eq. �1�, theirradiance is

Ilini =

1

�0��Elin

i �t��2� =1

�0��Elin,x

i �t��2� . �3�

The transmission axis of the ideal linear polarizer in Fig.3�a� is at the angle � with respect to the x axis. After thewave passes through the polarizer, the transmitted electricfield and irradiance are

Fig. 1. A clear blue sky viewed through a linear polarizer. �a� The polarizeris oriented for maximum transmission �transmission axis, white arrow, isnormal to the principal plane�. �b� The polarizer is oriented for minimumtransmission �transmission axis is parallel to the principal plane�. From thelight meter readings that go with these photographs, the degree of linearpolarization is dl0.5.

Fig. 2. Schematic drawing of the observation of the polarization of skylight

in the principal plane �the plane containing the points S, P, and M�.

26 Am. J. Phys., Vol. 75, No. 1, January 2007

Elint �t� = Elin,x

i �t� cos ��û , �4�

Ilint �� =

1

�0��Elin,x

i �t��2� cos2 �� = Ilini cos2 �� , �5�

where û is a unit vector in the direction of the transmissionaxis of the polarizer. Here and in the following, we ignoreany time delay common to all field components that is aresult of the wave passing through the polarizer. Notice thatEq. �5� is just the law of Malus for the action of an ideallinear polarizer on a linearly polarized wave.9,10

We next consider the case shown in Fig. 3�b� in which theincident wave is natural or unpolarized light. If we couldmeasure the electric field of natural light, it would produce achaotic waveform, similar to the familiar noise voltage asso-ciated with electronic circuits. The description of naturallight must be based on statistical quantities that can be mea-sured. The representation for natural light we will use in-volves time averages, as in Eq. �2�. In Fig. 3�b� the pair oforthogonal axes, x and y, have arbitrary orientation, and theelectric field is

Enati �t� = Enat,x

i �t�x̂ + Enat,yi �t�ŷ . �6�

The component of Enati have zero mean, �Enat,x

i �t��=0 and�Enat,y

i �t��=0, and they obey the relations:

�Enat,xi �t�Enat,x

i �t − �� = �Enat,yi �t�Enat,y

i �t − �� = F�� 7�

Fig. 3. The action of an ideal linear polarizer on waves with various statesof polarization: �a� linearly polarized light, �b� natural light, and �c� partiallypolarized light composed of linearly polarized light plus natural light.

and

26Glenn S. Smith

�Enat,xi �t�Enat,y

i �t − �� = 0. �8�

These results are independent of the choice of the origin intime and to hold for all �. We say that the time autocorrela-tion functions �7� for the x and y components of the field fornatural light are equal, and that the time cross-correlationfunction �8� for the x and y components of the field is zero.11

The irradiance for the incident natural light is

Inati =

1

�0��Enat,x

i �t��2� + ��Enat,yi �t��2�� =

2

�0��Enat,x

i �t��2� , �9�

where Eq. �7� with �=0 was used in the last step. After thewave passes through the linear polarizer, the transmittedelectric field and irradiance are

Enatt �t� = �Enat,x

i �t� cos �� + Enat,yi �t� sin ��û , �10�

Inatt =

1

�0��Enat,x

i �t��2� cos2 �� + ��Enat,yi �t��2� sin2 ��

+ 2�Enat,xi �t�Enat,y

i �t�� cos �� sin ��

=1

�0��Enat,x

i �t��2� =1

2Inat

i . �11�

Equations �7� and �8� with �=0 were used to obtain the resultin Eq. �11�. The irradiance of the transmitted light Inat

t is seento be one half of the irradiance of the incident light Inat

i , nomatter what the orientation � of the linear polarizer. Thisresult is an important characteristic of natural light.12

We next consider the case shown in Fig. 3�c� in which theincident wave is the sum of the linearly polarized light inFig. 3�a� and the natural light in Fig. 3�b�; that is, the electricfield is Eq. �1� plus Eq. �6�. This combination is referred toas partially polarized light, or, more precisely, as partially,linearly polarized light.13 We will assume that the linearlypolarized light and natural light are uncorrelated:

�Elin,xi �t�Enat,x

i �t − �� = 0, �Elin,xi �t�Enat,y

i �t − �� = 0.�12�

Straightforward calculations, similar to those we have al-ready performed, show that the irradiances for the incidentlight and transmitted light are

IPPi = Ilin

i + Inati �13�

and

IPPt �� = Ilin

t + Inatt = Ilin

i cos2 �� +1

2Inat

i . �14�

Note that the irradiance of the partially polarized light forboth the incident and the transmitted waves is the sum of theindividual irradiances for the two components �linearly po-larized light and natural light� when treated separately.

The degree of linear polarization of the incident, partiallypolarized light is defined as

dl =irradiance of linearly polarized component

irradiance of total

=Ilin

i

IPPi =

Ilini

Ilini + Inat

i . �15�

A more useful expression for the purposes of measurement is


obtained by using Eq. �14� to write Eq. �15� in terms of thetransmitted irradiances:

dl =IPP

t �� = 0� − IPPt �� = �/2�

IPPt �� = 0� + IPP

t �� = �/2�=

max�IPPt � − min�IPP

t �max�IPP

t � + min�IPPt �

.

�16�

As seen from the right-hand side of Eq. �16�, the degree oflinear polarization can be determined by rotating the linearpolarizer and noting the maximum and minimum values ofthe transmitted irradiance.

B. Dipole scattering

Figure 4 is a schematic drawing of the details of the scat-tering by an element �molecule or density fluctuation� in theatmosphere. The incident natural light �sunlight� is describedby Eqs. �6�–�9� with x ,y ,z replaced by x� ,y� ,z�. The wavepropagates in the direction z�. Hence, at the element the in-cident electric field and irradiance are

Enati �t� = Enat,x�

i �t�x�̂ + Enat,y�i �t�y�̂ �17�

Fig. 4. Details for the scattering of natural light by an electrically smallelement 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 theradiation patterns in the principal plane for the two components of the elec-tric dipole moment: px�, which is normal to the principal plane, and py�,which is in the principal plane.

and

27Glenn S. Smith

Inati =

2

�0��Enat,x�

i �t��2� . �18�

The incident field induces an electric dipole moment in theelectrically small element:

p�t� = px��t�x�̂ + py��t�y�̂ = �e0�Enat,x�i �t�x�̂ + Enat,y�

i �t�y�̂� ,

�19�

where 0 is the permittivity of free space, and �e is the elec-tric polarizability of the element. Note that the dipole mo-ment has components parallel, py�, and perpendicular, px�, tothe principal plane. In Eq. �19� we have assumed that thedipole moment responds instantaneously to the incident elec-tric field �there is no dispersion�. This assumption is good forthe molecules of air at optical wavelengths. If the dipolemoment is expressed in terms of the coordinate system of theobserver �x ,y ,z�, we have

p��t� = �e0�Enat,x�i �t�x̂ + Enat,y�

i �t��− sin ��ŷ + cos ��ẑ�� ,

�20�

where the angle is

= �p − �s − �/2. �21�

This dipole moment produces the radiation that is the sky-light, and the electric field of this light is14

Esr�r,t� =�0

4�r

r̂ � �r̂ � p̈�t − r/c�� , �22�

where �0 is the permeability of free space, r=rr̂ is the radialvector drawn from the dipole, and the double dot over aquantity indicates the second derivative with respect totime.15 If Eq. �20� is inserted into Eq. �22�, we obtain theelectric field incident on the polarizer, which is located at adistance z from the scatterer:

Esr�z,t� = −�e

4�c2z�Ënat,x�

i �t − z/c�x̂ − Ënat,y�i �t − z/c�sin��ŷ� .

�23�

After the wave passes through the linear polarizer, thetransmitted electric field and irradiance are

Et�z,t� = −�e

4�c2z�Ënat,x�

i �t − z/c�cos��

− Ënat,y�i �t − z/c�sin��sin��û �24�

and

It�� =�e

2

16�2c4�0z2 ��Ënat,x�

i �t − z/c��2�cos2��

+ ��Ënat,y�i �t − z/c��2�sin2��sin2��

− 2�Ënat,x�i �t − z/c�Ënat,y�

i �t − z/c��sin��cos��sin��

= K��Ënat,x�i �t��2�cos2�� + ��Ënat,y�

i �t��2�sin2��sin2��

− 2�Ënat,x�i �t��Ënat,y�

i �t��sin��cos��sin�� . �25�

In the last line we have simplified the result by recognizing

that 1 /z changes little in the vicinity of the polarizer, so we


can replace the factor in front of the brackets by the constantK, and we have set t�= t−z /c.

After performing a series of operations on Eqs. �7� and �8�and setting �=0, we can show that �see Appendix�

��Ënat,x�i �t��2� = ��Ënat,y�

i �t��2�, �Ënat,x�i �t��Ënat,y�

i �t�� = 0,

�26�

so that Eq. �25� can be written as

It�� = K��Ënat,x�i �t��2��cos2 �� + sin2 �� sin2�� . �27�

If we substitute Eq. �21� for the angle and rewrite thetrigonometric terms, we obtain our final result for the irradi-ance seen by the observer:

It�� = K��Ënat,x�i �t��2��sin2��p − �s�cos2��

+ cos2��p − �s�� . �28�

The result Eq. �28� has the same form as our earlier ex-pression for the irradiance of partially, linearly polarizedlight observed with a linear polarizer given by Eq. �14�; bothhave a term that depends on cos2 �� as well as a term that isindependent of �. Thus, the skylight that we observe isequivalent to partially polarized light, which is composed oflinearly polarized light and natural light. It is important torealize that this equivalence applies to the observed irradi-ances as measured by the polarizer and detector; the electricfields for the two types of light could be different and theirradiances the same. We can calculate the degree of linearpolarization for the skylight from Eq. �16�:

dl =It�� = 0� − It�� = �/2�It�� = 0� + It�� = �/2�

=sin2 ��p − �s�

1 + cos2 ��p − �s�. �29�

In summary, we have found skylight to be equivalent to amixture of linearly polarized light and natural light �partiallypolarized light�, with the linearly polarized component nor-mal to the principal plane, and the degree of linear polariza-tion a simple function �Eq. �29�� of the difference in theangles of elevation for the observation point �scatterer� andthe sun, �p−�s. Specifically, the degree of linear polarizationis maximum when the ray from the sun to the scatterer �SP�is orthogonal to the ray from the scatterer to the observer�PM�, then �p−�s=� /2 and dl=1. For other orientations thedegree of polarization is less; the minimum occurs when therays are parallel or antiparallel, then �p−�s=0, � and dl=0.

An examination of the patterns for dipole radiation, shownin the inset of Fig. 4, provides insight into these results. Inthe principal plane, the component of the dipole moment px��viewed end on in Fig. 4� radiates an electric field that isnormal to this plane and independent of �s and �p. The othercomponent of the dipole moment py� radiates an electric fieldthat is in this plane and proportional to �sin�� = �cos��p−�s��. Thus, when we view the element at an angle such that

=0 ��p−�s=� /2�, we only see the component of the elec-tric field that is normal to the principal plane �the x� compo-nent�; hence, the electric field is linearly polarized. For otherangles of observation, we see a mixture of the radiated elec-tric 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 ofskylight, Eq. �29�, is compared with measurements in Fig. 5.

For these measurements the observer is viewing the zenith

28Glenn S. Smith

sky ��p=� /2� as the sun rises ��s increases�. Results areshown for two measurements; both were taken at high alti-tude on a clear day. The dots are for results measured �visiblespectrum� at Bocaiuva, Brazil at an altitude of 671 m in1947; the dashed line is for results measured �=0.71 �m�on Mauna Loa in Hawaii at an altitude of 3400 m in1977.2,16 The general trend is predicted by the simple theory,that is, a decrease in the degree of linear polarization as thesun rises. However, the predicted degree of polarization isalways greater than measured. For example, the maximumdegree of linear polarization, which occurs at sunrise, �s=0,is 100% for the theory but only about 84% for the measure-ments. Factors not included in the simple theory cause thisdifference and will be discussed later.

III. DISTRIBUTION FOR POLARIZED SKYLIGHT

From our knowledge of the degree of linear polarization inthe principal plane, Eq. �29�, we can obtain the degree oflinear polarization over the rest of the sky. First, we intro-duce the unit vectors shown in Fig. 6�a�: n̂s points from theobserver to the sun along MS, and n̂p points from the ob-server to the observation point along MP. Because of thegreat distance to the sun, the ray MS in Fig. 6�a� is parallel tothe ray PS in Fig. 2, and both are at the angle of elevation �s.We also observe that n̂p · n̂s=cos ��p−�s�, so Eq. �29� can bewritten as

dl =1 − cos2 ��p − �s�1 + cos2 �� − � �

=1 − �n̂p · n̂s�2

ˆ ˆ 2. �30�

Fig. 5. Degree of linear polarization for light from the zenith sky versus theangle of elevation of the sun, �s. The measured data are from 1947 �Ref. 16�and from 1977 �Ref. 2�.

p s 1 + �np · ns�


From Eq. �30� it is clear that the degree of linear polarizationdepends only on the direction of the observation point rela-tive to the direction of the sun.

Now consider the construction shown in Fig. 6�b�. Theline MP �unit vector n̂p� lies in the principal plane �gray�. Ifthis line is rotated about the line through the sun, that is,about MS �about the unit vector n̂s�, the end of the line tracesout a circle �dotted line�. At every point on this circle, thedegree of linear polarization is the same, because the dotproduct that appears in Eq. �30� is the same. For example, forthe point P� we have n̂p� · n̂s= n̂p · n̂s. At each point on thiscircle, the linearly polarized component of the electric fieldis tangent to the circle.

From these observations we can construct polarization dia-grams for the whole sky.17 Two of these diagrams are shownin Fig. 7 for the case �s=35°. Figure 7�a� shows mainly thesolar half of the sky, and Fig. 7�b� shows mainly the anti-solar half of the sky. The length of a heavy line indicates thedegree of linear polarization, and the line is parallel to thedirection of the linearly polarized component of the electricfield. Note that the circles of constant dl are centered on theline through the sun, MS, and that the maximum �dl=1� oc-curs, as expected, when �p−�s=� /2 �n̂p · n̂s=0�. As the sunmoves across the sky, this pattern for the polarization movesover the hemisphere.

It is convenient to have an analytical description for thepolarization of skylight that applies over the entire hemi-sphere. For this purpose, a parametric expression for a circleof constant dl �the dotted curve in Fig. 6�b�� can be obtained

Fig. 6. �a� Drawing of the principal plane showing the unit vectors n̂s and n̂ppointing from the observer toward the sun and toward the observation point,respectively. �b� Construction for a circle on which the degree of linearpolarization dl is a constant.

in terms of the arc length �. The location of a point on this

29Glenn S. Smith

circle, such as P�, is given by the azimuthal angle relative tothe direction of the sun �az and the angle of elevation �el. Forpoints on the right half of the hemisphere, these angles arerestricted to the ranges 0��az�� and 0��el�� /2; resultson the left half of the hemisphere can be obtained from thoseon the right half by symmetry. When �s is specified, thefollowing parametric equations for �az and �el describe acircle of constant dl:

�az�� =�

2+ tan−1��1 − dl

2dlcos��s�csc��

+ sin��s�cot�� , �31�

�el�� = sin−1�±1 − dl1 + dlsin��s�+ 2dl

1 + dlcos��s�cos�� . �32�

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 hemi-sphere are excluded:

0 � � �� , �1 − dl2dl

tan��s�� 1,cos−1��1 − dl

2dltan��s�� , �1 − dl2dl tan��s�� 1.

�33�

For the lower sign in Eqs. �31�–�33� we use only the valuesfor which

dl � � sin2��s�1 + cos2��s�� , �34�to exclude the cases in which the entire circle for dl lies onthe lower hemisphere. In Eqs. �31�–�33�, the principal valuesof the inverse trigonometric functions are assumed: −� /2�sin−1 � �� /2, 0�cos−1 � ��, and −� /2� tan−1 � �� /2.

Figure 8 presents contour plots for the degree of linearpolarization when the elevation angle of the sun is �s=44.7°. These are polar graphs in which the radial variable is�el �0° at the outer edge and 90° at the center� and the an-gular variable is �az. The results in Fig. 8�a� are from thesimple theory, Eqs. �31�–�34�, and those in Fig. 8�b� are mea-sured data. The measurements were made at the wavelength

=0.439 �m on a very clear day during February 1996 at

18,19
the University of Miami in Miami, FL. The measured

results are not shown near the horizon ��el�10° –15° �where they are irregular. The simple theory and the measure-ments show the same general structure for the polarization,particularly the direction of the linearly polarized componentof the field, which is parallel to a contour. However, themaximum degree of linear polarization is 100% for thetheory but only 50%–60% for the measurements. In bothcases, the maximum, as expected, occurs when �az180°and �el90°−�s45°.

The difference between the theory and measurements canbe attributed to several factors not included in the simpletheory that decrease the linearly polarized component of thelight: the anisotropic polarizability of the air molecules, mul-tiple scattering of light between air molecules, scattering oflight from aerosol particles and dust in the atmosphere, andsunlight reflected from the clouds and the ground. Some ofthese factors are more significant in the urban environmentof Miami than at the high altitude sites for the measurementsshown in Fig. 5. This difference is probably why the mea-

Fig. 7. Polarization diagram for the entire sky when �s=35°. The length ofthe heavy line indicates the degree of linear polarization dl; the line isparallel to the direction of the linearly polarized component of the electricfield. Results are shown for two orientations: �a� mainly the solar half of thesky and �b� mainly the anti-solar half of the sky.

sured maximum dl0.5 in Fig. 8�b� is significantly lower

30Glenn S. Smith

than the measured maximum dl0.84 in Fig. 5. These addi-tional factors introduce other interesting effects into the mea-surements that are not predicted by the simple theory. Forexample, there are points where dl=0 other than those at�p−�s=0 and 180° that are not predicted by the simpletheory. A comprehensive discussion of these effects is givenin Refs. 2 and 20.

IV. INSECT NAVIGATION BY POLARIZEDSKYLIGHT

Under very favorable conditions, human beings can detectthe presence of polarized light through a faint pattern knownas Haidinger’s brush.21 Recognizing and interpreting thispattern takes practice, and it plays no known role in ourfunctioning. The reader is referred to Refs. 22–24 for detailsof this interesting phenomenon. The situation is quite differ-ent for many insects, because they can readily detect thepolarization of light �specifically skylight� and make use of itin various ways.25 The insects that detect and use the polar-ization of light include honey bees, ants, crickets, flies, andbeetles.

The first conclusive evidence for the use of polarized sky-

Fig. 8. Contour plots for the degree of linear polarization dl for �s=44.7°.�a� Simple theory. �b� Measured data at =0.439 �m �Refs. 18 and 19�. Theposition of the sun is shown by the small symbol.

light for orientation by insects was obtained by Karl Ritter


von Frisch for honey bees in 1948.26–29 Since then, the honeybee has been studied intensely in this regard, and we willrestrict our discussion to this insect. Figure 9 is a schematicdrawing of one of von Frisch’s experiments.

When foraging for food �nectar and pollen�, worker beesuse the location of the sun to determine direction, that is,they determine direction with respect to the sun much as wedetermine direction with respect to magnetic north using acompass.29–31 On returning to the hive, she �all worker beesare female� informs other workers of the location of the food.When the food is at a distance of about 100 m or greater, shecommunicates this information through the tail-waggingdance, which is shown in Fig. 9�a� for a dance on the hori-zontal comb of a hive with the sun visible. The dance has astraight portion that is continually repeated after circling tothe right or left. Along the straight portion, the bee wagglesher body, hence the name for the dance. The distance to thefood is encoded in the characteristics of the dance, such as itstempo, and the direction to the food is indicated by the di-rection of the straight portion of the dance relative to thedirection of the sun, the solar azimuth angle �az in Fig. 9�a�.

Von Frisch noticed that bees could communicate the loca-tion of the food through the tail-wagging dance even whenhe blocked their view of the sun at the hive, as indicated inFig. 9�b� by the gray area. This observation held as long as

Fig. 9. Schematic drawings showing von Frisch’s experiment demonstratingthe honey bee’s orientation by polarized light. Each figure shows the bee’stail-wagging dance on a horizontal comb of the hive. �a� During the dance,the sun is visible to the bee. �b� During the dance, a patch of clear blue skyis visible to the bee, and the view of the sun is blocked �indicated by grayarea�. �c� Same as in �b� with the skylight passing through a linear polarizer,and the transmission axis of the polarizer û aligned with the linearly polar-ized component of the electric field of the skylight. �d� Same as in �c� withthe transmission axis of the linear polarizer rotated.

he left a patch of clear blue sky visible to the bees. He

31Glenn S. Smith

surmised that the bees were using the polarization of theskylight for orientation; possibly they could determine thelocation of the hidden sun by knowing the relation betweenthe pattern of polarization for skylight and the position of thesun. To test this hypothesis, he passed the skylight visible tothe bees through a linear polarizer. When the transmissionaxis of the polarizer û was aligned so as to pass the linearlypolarized component of the electric field of the skylight, as inFig. 9�c�, the bee’s dance was unaltered from that in Fig.9�b�; that is, it still pointed in the direction of the food.However, when the polarizer was rotated to change the di-rection of the transmitted electric field, as in Fig. 9�d�, thedirection of the bee’s dance changed, so that it no longerpointed in the direction of the food.

By passing the skylight visible to the bees through band-pass filters, von Frisch and others determined that beesmainly use the ultraviolet �UV� portion of the spectrum for

Fig. 10. Elements in the bee’s detection of polarized light. �a� The compounends of the ommatidia visible on the surface of the eye. The specialized ommis shown in black. �b� Longitudinal and transverse cross sections for one ofsingle visual cell from �b�, showing the details for the microvilli �Ref. 37�.showing the three UV sensitive visual cells �A, B, and C� in gray, and the

the orientation by polarized light. It was also shown that it is


the direction of the linearly polarized component of the elec-tric field that is most important; the degree of linear polar-ization need only be greater than about 10%.

Since von Frisch’s pioneering behavioral research, therehas been substantial effort devoted to identifying the physi-ology of the bee’s eye responsible for sensing the polariza-tion of light. Bees have compound eyes that are made frommany individual sensing units called ommatidia. The inset inFig. 10�a� shows the ends of the ommatidia visible on thesurface of the bee’s eye. The eye of a worker bee containsabout 5000 ommatidia. The ommatidia used for orientationwith polarized light in the UV are believed to be specializedones �about 150� located at the upper rim of the eye, that is,in the dorsal rim area.25,32–34 Figure 10�b� is a schematicdrawing showing the longitudinal and transverse cross sec-tions for one of these specialized ommatidia. It consists ofnine long, nearly straight cells of equal length that are evi-

of the worker bee composed of about 5000 ommatidia. The inset shows theinvolved in the detection of polarized light are in the dorsal rim area, whichecialized ommatidia composed of nine visual cells �Refs. 25 and 29�. �c� An expanded view of the transverse cross section of the ommatidium in �b�onal orientation of the microvilli in the rhabdom �Ref. 25�.

d eyeatidiathe sp�d� A

orthog

dent in the transverse cross section. These cells are fused at

32Glenn S. Smith

the center of the ommatidium to form the rhabdom. Lightenters the ommatidium through the lens, which in these spe-cialized ommatidia is covered with pore canals that increasethe visual field.35 The light accepted by the lens and crystal-line cone is guided down the rhabdom, much as light is in anoptical fiber.

Figure 10�c� is a schematic drawing for one of the ninevisual cells shown in Fig. 10�b�. The rhabdomere is the por-tion of a cell that contributes to the rhabdom. It is made up ofmany small protrusions called microvilli that are perpendicu-lar to the optical axis of the ommatidium.36,37 There are di-polar pigment molecules in the membrane of the microvilli,and the axes of these molecules are aligned with the mi-crovilli. As indicated in Fig. 10�c�, light with its electric fieldparallel to the microvilli is absorbed more by these mol-ecules than light with its electric field perpendicular to themicrovilli. This feature makes the cell sensitive to the direc-tion of the linearly polarized component of light. The mea-sured response �electrical signal� of these specialized cells isas much as ten times greater when the electric field is parallelto the microvilli than when it is perpendicular to the mi-crovilli.

Three of the nine cells in a specialized ommatidium aresensitive to UV and take part in the orientation by polarizedlight; they are shaded gray in the transverse cross sectionshown in Fig. 10�d�.25,32–34 The microvilli of one of thesecells �marked C� are perpendicular to those in the other twocells �marked A and B�. Thus, in a single ommatidium thereare sensors for electric fields in two orthogonal directions.Presumably, electrical signals enter the bee’s nervous systemfrom an ommatidium that indicate the relative magnitudes ofthe electric fields in these two directions.

With the physiology of the bee’s eye responsible for sens-ing polarized light established, the question that remains ishow do bees use polarization for orientation and navigation?Different theories have been proposed, and we will give abrief sketch of one by Rossel and Wehner, which is welldescribed in the literature including its limitations.17,38–42 Inthis theory bees mainly make use of the polarization of sky-light in the anti-solar half of the sky, that is, for 90° ��az�270° and 0° ��el�90° in Fig. 7. This observation makessense, because when �s�0°, the skylight from the anti-solarhalf of the sky is more highly polarized than that from thesolar half �compare Fig. 7�a� with Fig. 7�b��. Also, the lightfrom the solar half of the sky is composed of skylight plusdirect sunlight, and the latter is unpolarized.

Rossel and Wehner assume that the special ommatidia inthe dorsal rim area of the bee’s eyes are arranged so that theymatch some gross features of the polarization of the anti-solar sky when the bee faces the anti-solar meridian ��az=180° �. Specifically, for an ommatidium pointing in the so-lar azimuth direction �az, the axis of polarization �the axisfor one of the two sets of microvilli in Fig. 10�d�� is alignedwith a mean representation of the electric field of the sky-light from that direction.

When the bee views a patch of blue sky, she rotates abouther vertical body axis. The signal the bee receives from thearray of specialized ommatidia in the dorsal rim area changesduring the rotation, and it is maximum when the bee is ap-proximately facing the anti-solar meridian. With this proce-dure the bee effectively determines the location of the sun�position in azimuth� from the polarized light it receives

from a patch of blue sky.


Recall that in each of the specialized ommatidium thereare actually two orthogonal sensors of polarization �sets ofmicrovilli� �Fig. 10�d��. Thus, when the response from oneset of sensors is maximum, the response from the other set ofsensors is minimum. The bee may use the contrast betweenthe signals from the two sets of sensors to enhance the accu-racy for its orientation; that is, the bee may orient to maxi-mize the difference in the signals from the two orthogonalsets of sensors.

V. CONCLUSION

We have used the material in this paper to supplement aconventional treatment of the polarization of plane electro-magnetic waves. In addition to a classroom presentation,each student is given a simple linear polarizer �laminatedfilm polarizer, Edmund Optics, NT38-396�. They are askedto make a qualitative observation of the polarization of sky-light using the hand-held polarizer, and to see if their assess-ment corresponds to Fig. 7. Readings from the current re-search on insects’ use of polarized skylight are assigned orsuggested. Students are often surprised to find that they canreadily understand the current research with only the addi-tional knowledge of polarized light that they have received inthis presentation. The general feeling of students is that thismaterial on the polarization of skylight, particularly that per-taining to navigation by honey bees, makes what they be-lieve to be a rather ordinary topic much more exciting.

ACKNOWLEDGMENTS

The author would like to thank R. Todd Lee for assistancewith the photographs shown in Fig. 1, and a reviewer formaking useful suggestions that were included in the paper.The author is grateful for the support provided by the JohnPippin Chair in Electromagnetics that furthered this study.This paper is dedicated to the memory of an exceptionalteacher, Ronold W. P. King �1905–2006�, Professor of Ap-plied Physics at Harvard University.

APPENDIX: CORRELATION FUNCTIONSFOR THE SECOND DERIVATIVE OF THEELECTRIC FIELD OF NATURAL LIGHT

The argument for the first of the relations in Eq. �26� be-gins by differentiating the left-hand side of Eq. �7� with re-spect to �, then using integration by parts to obtain:

1

TD�

t=−TD/2

TD/2

Ex�t��− dEx�t��dt� �t−� dt=−

1

TD�Ex�t�Ex�t − ��−TD/2

TD/2 +1

TD�

t=−TD/2

TD/2 dEx�t�dt

�Ex�t − �� dt . �A1�

For TD→� we assume that the first term is negligible. Wedifferentiate the second term with respect to � and integrate

by parts:

33Glenn S. Smith

1

TD�

t=−TD/2

TD/2 dEx�t�dt

�− dEx�t��dt�

�t−�

dt

=−1

TD�dEx�t�

dtEx�t − ��

−TD/2

TD/2

+1

TD�

t=−TD/2

TD/2 d2Ex�t�dt2

�Ex�t − �� dt . �A2�

As before, for TD→� we assume that the first term is neg-ligible, and we differentiate the second term twice with re-spect to � to obtain

1

TD�

t=−TD/2

TD/2 d2Ex�t�dt2

�d2Ex�t��dt�2

�t−�

dt = ��Ënat,xi �t − ��2� .

�A3�

If the same operations are applied to the right-hand side ofEq. �7� and the results are equated, we have

��Ënat,xi �t − ��2� = ��Ënat,y

i �t − ��2� , �A4�

which for �=0 and x ,y→x� ,y� is the first relation in Eq.�26�. The argument for the second relation in Eq. �26� isobtained in the same way.

a�Electronic mail: [email protected]. F. J. Arago, Oeuvres Complètes de François Arago �Gide, Paris,1858�, Vol. 7, pp. 394–395.

2K. L. Coulson, Polarization and Intensity of Light in the Atmosphere�Deepak, Hampton, VA, 1988�, p. 2.

3H. LaFay, “The Vikings,” Natl. Geogr. 37, 492–541 �1970�.4C. P. Können, Polarized Light in Nature �Cambridge U. P., Cambridge,1985�, p. 30.

5C. Roslund and C. Beckman, “Disputing Viking navigation by polarizedlight,” Appl. Opt. 33, 4754–4755 �1994�.

6J. Walker, “The Amateur Scientist: More about polarizers and how to usethem, particularly for studying polarized skylight,” Sci. Am. 238�1�,132–136 �1978�.

7D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed.�Cambridge U. P., Cambridge, 2001�, pp. 26–27.

8The two viewpoints as to the cause of the scattering that gives rise to thecolor and polarization of skylight were first introduced by Lord Rayleigh�molecules� and A. Einstein �density fluctuations�: Lord Rayleigh, “Onthe transmission of light through an atmosphere containing small par-ticles in suspension, and on the origin of the blue sky,” Philos. Mag. 47,375–384 �1899�. A. Einstein, “Theorie der Opaleszenz von homogenenFlüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischenZustandes,” Ann. Phys. 33, 1275–1298 �1910�. English translation,“Theory of the opalescence of homogeneous liquids and mixtures of liq-uids in the vicinity of the critical state” in Colloid Chemistry: Theoreticaland Applied, edited by J. Alexander �Chemical Catalog Company, NewYork, 1926�, Vol. 1, pp. 323–339.

9E. T. Malus, “Sur une propriété des forces répulsives qui agissent sur lalumiére,” Mémoires de physique et de chimie de la Société D’Arcueil 2,254–267 �1809�. English translation, “On a property of the repulsiveforces, that act on light,” A Journal of Natural Philosophy, Chemistry, andthe Arts �Nicholson’s Journal� 30, 161–168 �1811�.

10W. 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 SignalPrinciples, 4th ed. �McGraw-Hill, New York, 2001�.

12In Eqs. �6� and �8� we have postulated the properties for the electric fieldof natural light. Then we used these properties to predict the action of thelinear polarizer on this light. Historically, the opposite was true: Theaction of polarizers on natural light was used to infer the properties of thelight. See, for example, G. G. Stokes, “On the composition and resolutionof 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�.

13We have discussed partially polarized light that is formed by addinglinearly polarized light to natural light. Partially polarized light can beformed in other ways. For example, partially polarized light that is quasi-monochromatic is formed by adding elliptically polarized light to naturallight. The composition of partially polarized, quasi-monochromatic lightcan be determined by performing a series of measurements with a linearpolarizer 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 Op-tics �Cambridge U. P., Cambridge, 1995�, Chap. 6.

14The scattering element �molecule� is in random motion with the speedv�c. We have assumed that this motion does not change the statisticalproperties of the field, and we have used the expression for the radiatedfield of a stationary dipole: G. S. Smith, An Introduction to ClassicalElectromagnetic Radiation �Cambridge U. P., Cambridge, 1997�, pp.452–465.

15The superscript sr is used to indicate that this is the “scattered radiated”field: the part of the scattered field that behaves as 1/r.

16R. A. Richardson and E. O. Hulburt, “Sky-brightness measurements nearBocaiuva, Brazil,” J. Geophys. Res. 54, 215–227 �1949�.

17R. Wehner and S. Rossel, “The bee’s celestial compass – A case study inbehavioural neurobiology,” in Experimental Behavioral Ecology and So-ciobiology, In Memoriam Karl von Frisch 1886–1982, edited by B. Höll-dobler and M. Lindauer �Sinauer Associates, Sunderland, MA, 1985�. pp.11–53.

18K. J. Voss and Y. Liu, “Polarized radiance distribution measurement ofskylight. I. System description and characterization,” Appl. Opt. 36,6083–6094 �1997�.

19Y. Liu and K. J. Voss, “Polarized radiance distribution measurement ofskylight. II. Experiment and data,” Appl. Opt. 36, 8753–8764 �1997�.

20Selected Papers on Scattering in the Atmosphere, edited by C. F. Bohren�SPIE, Bellingham, WA, 1989�, pp. 261–326.

21W. Haidinger, “Uber das directe Erkennen des polarisirten Lichts und derLage der Polarisationsebene” �On the direct recognition of polarized lightand the polarization plane�, Ann. Phys. Chem. 63, 29–39 �1844�.

22M. G. J. Minnaert, Light and Color in the Outdoors �Springer-Verlag,New York, 1993�, pp. 276–278.

23D. Auerbach, “Optical polarization without tools,” Eur. J. Phys. 21,13–17 �2000�.

24A. P. Ovcharenko and V. D. Yegorenkov, “Teaching students to observeHaidinger brushes,” Eur. J. Phys. 23, 123–125 �2002�.

25T. Labhart and E. P. Meyer, “Detection of polarized skylight in insects: asurvey of ommatidial specializations in the dorsal rim area of the com-pound eye,” Microsc. Res. Tech. 47, 368–379 �1999�.

26K. von Frisch, “Gelöste und ungelöste Rätsel der Bienensprache,” Natur-wiss. 35, 38–43 �1948�.

27K. von Frisch, “Die Polarisation des Himmelslichtes als orientierenderFaktor bei den Tänzen der Bienen,” Experientia 5, 142–148 �1949�.

28K. von Frisch, “Die Sonne als Kompass im Leben der Bienen,” Experi-entia 6, 210–221 �1950�.

29K. von Frisch, The Dance Language and Orientation of Bees �Harvard U.P., Cambridge, MA, 1967�.

30J. L. Gould and C. G. Gould, The Honey Bee �Scientific American Li-brary, New York, 1988�.

31F. G. Barth, Insect and Flower �Princeton U. P., Princeton, NJ, 1991�.32R. H. Schinz, “Structural specialization in the dorsal retina of the bee,

Apis mellifera,” Cell Tissue Res. 162, 23–34 �1975�.33T. Labhart, “Specialized photoreceptors at the dorsal rim of the honey-

bee’s compound eye: polarizational and angular sensitivity,” J. Comp.Physiol. 141, 19–30 �1980�.

34R. Wehner and S. Strasser, “The POL area of the honey bee’s eye: Be-havioural evidence,” Physiol. Entomol. 10, 337–349 �1985�.

35E. P. Meyer and T. Labhart, “Pore canals in the cornea of a functionallyspecialized area of the honey bee’s compound eye,” Cell Tissue Res.216, 491–501 �1981�.

36R. Menzel and A. W. Snyder, “Introduction to photoreceptor optics—anoverview,” in Photorecpetor Optics, edited by R. Menzel and A. W.Snyder �Springer-Verlag, New York, 1975�, pp. 1–13.

37R. Wehner, “Polarized-light navigation by insects,” Sci. Am. 235�7�,106–115 �1976�.

38S. 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�.

34Glenn S. Smith

39S. Rossel and R. Wehner, “How bees analyse the polarization patterns inthe sky, experiments and model,” J. Comp. Physiol., A 154, 607–615�1984�.

40S. Rossel and R. Wehner, “Polarization vision in bees,” Nature �London�323, 128–131 �1986�.


41K. Kirschfeld, “Navigation and compass orientation by insects accordingto the polarization pattern of the sky,” Z. Naturforsch. C 43c, 467–469�1988�.

42K. Kirschfeld, “The role of the dorsal rim ommatidia in the bee’s eye,” Z.Naturforsch. C 43c, 621–623 �1988�.

35Glenn S. Smith

The polarization of skylight: An example from...

Documents

Transcript of The polarization of skylight: An example from...