The Polarization of Skylight: an Example from Nature ͒ Glenn S

The polarization of skylight: An example from nature ͒ Glenn S. Smitha School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250 ͑Received 27 March 2006; accepted 15 September 2006͒ 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 ͑unpolarized͒ 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. ͓DOI: 10.1119/1.2360991͔

I. INTRODUCTION and it contains the points centered on the sun ͑S͒, the scatterer ͑P͒, and the observer ͑M͒. The angles of elevation for ␺ ␺ Polarization is a fundamental property of electromagnetic the sun and scatterer are s and p. radiation ͑light͒ and is discussed at all levels from introduc- The direct light from the sun is natural or unpolarized tory courses in physics to graduate courses in electromagne- light. This light is scattered by the molecules of the air or, tism. The polarization of skylight and its use by insects for alternatively, fluctuations in the density of the air.8 The scat- navigation is a practical example of much interest to stu- tering elements are small compared to all wavelengths of dents. significance, so the scattering of light is by Rayleigh scatter- The polarization of skylight is easily observed by the eye ing. The positions of the elements are random, so the scat- using a simple linear polarizer. Figure 1 shows results for a tering from the various elements is incoherent. Thus, we only clear blue sky: In Fig. 1͑a͒ the polarizer is oriented for maxi- need to consider dipole scattering from one element. The mum transmission, and in Fig. 1͑b͒ it is oriented for mini- scattered light ͑skylight͒ is partially polarized, that is it is mum transmission. equivalent to natural light plus a linearly polarized compo- The first scientific observation of the polarization of sky- nent. The observer views the skylight through a linear polar- light is usually attributed to the French natural philosopher izer with its transmission axis at the angle ␰ to the normal of 1,2 Dominique François Jean Arago in 1809. There are claims the principal plane ͑x axis͒. As the observer rotates the po- that the Vikings knew of this phenomenon nearly 1000 years larizer, he/she sees a maximum and a minimum in the irra- 3,4 ͑␰͒͑ ͒ earlier and used it for navigation. The Vikings supposedly diance It time-average power per unit area , as shown in discovered a naturally occurring dichroic mineral, referred to Fig. 1. as “sunstone,” that served as a linear polarizer. They nor- mally navigated using the position of the sun, but when the sun was obscured by clouds or below the horizon, they could A. Natural light use this device to sense the direction of polarization for the visible portion of the sky. Then, knowing the relation be- To develop a description for the natural or unpolarized ͑ ͒ tween the direction of polarization and the location of the light from the sun, we first consider Fig. 3 a in which the ͑ ͒ sun, they could infer the position of the sun. The claim for electric field of the incident plane wave light is linearly the Vikings’ navigation by polarized skylight has been polarized in the x direction. At the polarizer the electric field disputed.5 is In the popular scientific literature, there are qualitative ex- i ͑ ͒ i ͑ ͒ ͑ ͒ Elin t = Elin,x t xˆ . 1 planations of the polarization of skylight.6,7 The objective of this paper is to go beyond these explanations and to present Throughout the paper we will be interested in the irradi- a simple, analytical model that not only provides a physical ance for a wave, because for optical signals this quantity can explanation for the polarization of skylight, but can also be be measured with a practical detector, in contrast to the elec- used for quantitative calculations that can be compared to tric field, for which there are no detectors available that have measurements. The model brings together material from dif- a response time short enough to resolve the temporal varia- ferent topics in optics: polarization of plane waves, natural tion. The irradiance for a plane wave propagating in the z ͑unpolarized͒ light, and dipole scattering. Unlike other treat- direction is ments, the analysis is done entirely in terms of the time- 1 TD/2 varying field, without resorting to the frequency domain. I = ͵ zˆ · S͑t͒ dt T D t=−TD/2 1 TD/2 1 1 = ͵ ͉E͑t͉͒2dt = ͉͗E͑t͉͒2͘, ͑2͒ II. SIMPLE MODEL T ␨ ␨ D t=−TD/2 0 0 ␨ Figure 2 is a schematic drawing showing the details of an where S is the Poynting vector and 0 is the wave impedance observation made in the “principal plane” or the “sun’s ver- of free space. The time average in Eq. ͑2͒, which is indicated ͗ ͘ tical,” which is the plane that contains the local zenith and by the angle brackets ¯ , is over the time interval TD asso- the center of the sun. This plane is the y-z plane in Fig. 2, ciated with the detector. In an experiment, we take this time

25 Am. J. Phys. 75 ͑1͒, January 2007 http://aapt.org/ajp © 2007 American Association of Physics Teachers 25 Fig. 1. A clear blue sky viewed through a linear polarizer. ͑a͒ The polarizer is oriented for maximum transmission ͑transmission axis, white arrow, is normal to the principal plane͒. ͑b͒ The polarizer is oriented for minimum transmission ͑transmission axis is parallel to the principal plane͒. From the light meter readings that go with these photographs, the degree of linear Ϸ polarization is dl 0.5. interval to be long enough to make the average practically independent of TD, and for mathematical calculations we →ϱ take TD . For the familiar time-harmonic field with an- Fig. 3. The action of an ideal linear polarizer on waves with various states ␻ ͑ ͒ ͑␻ ͒ ͑ ͒ ͑ ͒ ͑ ͒ gular frequency , that is, E t =E0 cos t xˆ, we have I of polarization: a linearly polarized light, b natural light, and c partially ͉ ͉2 ␨ ͑ ͒ polarized light composed of linearly polarized light plus natural light. = E0 /2 0. For the incident electric field in Eq. 1 , the irradiance is 1 1 Ii = ͉͗Ei ͑t͉͒2͘ = ͉͗Ei ͑t͉͒2͘. ͑3͒ lin ␨ lin ␨ lin,x t ͑ ͒ i ͑ ͒ ͑␰͒ ͑ ͒ 0 0 Elin t = Elin,x t cos uˆ , 4 The transmission axis of the ideal linear polarizer in Fig. ͑ ͒ ␰ 1 3 a is at the angle with respect to the x axis. After the It ͑␰͒ = ͉͗Ei ͑t͉͒2͘ cos2 ͑␰͒ = Ii cos2 ͑␰͒, ͑5͒ lin ␨ lin,x lin wave passes through the polarizer, the transmitted electric 0 field and irradiance are where uˆ 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. ͑5͒ 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. 3͑b͒ 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 asso- ciated 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 in- volves time averages, as in Eq. ͑2͒. In Fig. 3͑b͒ the pair of orthogonal axes, x and y, have arbitrary orientation, and the electric field is i ͑ ͒ i ͑ ͒ i ͑ ͒ ͑ ͒ Enat t = Enat,x t xˆ + Enat,y t yˆ . 6 i ͗ i ͑ ͒͘ The component of Enat have zero mean, Enat,x t =0 and ͗ i ͑ ͒͘ Enat,y t =0, and they obey the relations: ͗ i ͑ ͒ i ͑ ␶͒͘ ͗ i ͑ ͒ i ͑ ␶͒͘ ͑␶͒͑͒ Enat,x t Enat,x t − = Enat,y t Enat,y t − = F 7 Fig. 2. Schematic drawing of the observation of the polarization of skylight in the principal plane ͑the plane containing the points S, P, and M͒. and

26 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 26 ͗ i ͑ ͒ i ͑ ␶͒͘ ͑ ͒ Enat,x t Enat,y t − =0. 8 These results are independent of the choice of the origin in time and to hold for all ␶. We say that the time autocorrela- tion functions ͑7͒ for the x and y components of the field for natural light are equal, and that the time cross-correlation function ͑8͒ for the x and y components of the field is zero.11 The irradiance for the incident natural light is 1 2 Ii = ͓͉͗Ei ͑t͉͒2͘ + ͉͗Ei ͑t͉͒2͔͘ = ͉͗Ei ͑t͉͒2͘, ͑9͒ nat ␨ nat,x nat,y ␨ nat,x 0 0 where Eq. ͑7͒ with ␶=0 was used in the last step. After the wave passes through the linear polarizer, the transmitted electric field and irradiance are t ͑ ͒ ͓ i ͑ ͒ ͑␰͒ i ͑ ͒ ͑␰͔͒ ͑ ͒ Enat t = Enat,x t cos + Enat,y t sin uˆ , 10

1 It = ͓͉͗Ei ͑t͉͒2͘ cos2 ͑␰͒ + ͉͗Ei ͑t͉͒2͘ sin2 ͑␰͒ nat ␨ nat,x nat,y 0 ͗ i ͑ ͒ i ͑ ͒͘ ͑␰͒ ͑␰͔͒ +2 Enat,x t Enat,y t cos sin

1 1 = ͉͗Ei ͑t͉͒2͘ = Ii . ͑11͒ ␨ nat,x nat 0 2 Equations ͑7͒ and ͑8͒ with ␶=0 were used to obtain the result ͑ ͒ t in Eq. 11 . The irradiance of the transmitted light Inat is seen i to be one half of the irradiance of the incident light Inat,no 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. 3͑c͒ in which the Fig. 4. Details for the scattering of natural light by an electrically small incident wave is the sum of the linearly polarized light in element in the atmosphere. The points S, P, and M lie in the principal plane, Fig. 3͑a͒ and the natural light in Fig. 3͑b͒; that is, the electric and the unit vectors xˆ and xˆЈ are normal to this plane. The inset shows the field is Eq. ͑1͒ plus Eq. ͑6͒. This combination is referred to radiation patterns in the principal plane for the two components of the elec- as partially polarized light, or, more precisely, as partially, tric dipole moment: pxЈ, which is normal to the principal plane, and pyЈ, which is in the principal plane. linearly polarized light.13 We will assume that the linearly polarized light and natural light are uncorrelated: ͗ i ͑ ͒ i ͑ ␶͒͘ ͗ i ͑ ͒ i ͑ ␶͒͘ Elin,x t Enat,x t − =0, Elin,x t Enat,y t − =0. obtained by using Eq. ͑14͒ to write Eq. ͑15͒ in terms of the ͑12͒ transmitted irradiances: Straightforward calculations, similar to those we have al- It ͑␰ =0͒ − It ͑␰ = ␲/2͒ max͑It ͒ − min͑It ͒ ready performed, show that the irradiances for the incident d = PP PP = PP PP . l t ͑␰ ͒ t ͑␰ ␲ ͒ ͑ t ͒ ͑ t ͒ light and transmitted light are IPP =0 + IPP = /2 max IPP + min IPP i i i ͑ ͒ ͑16͒ IPP = Ilin + Inat 13 ͑ ͒ and As seen from the right-hand side of Eq. 16 , the degree of linear polarization can be determined by rotating the linear 1 polarizer and noting the maximum and minimum values of It ͑␰͒ = It + It = Ii cos2 ͑␰͒ + Ii . ͑14͒ PP lin nat lin 2 nat the transmitted irradiance. 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 ͑linearly po- B. Dipole scattering larized light and natural light͒ when treated separately. The degree of linear polarization of the incident, partially Figure 4 is a schematic drawing of the details of the scat- ͑ ͒ polarized light is defined as tering by an element molecule or density fluctuation in the atmosphere. The incident natural light ͑sunlight͒ is described irradiance of linearly polarized component by Eqs. ͑6͒–͑9͒ with x,y,z replaced by xЈ,yЈ,zЈ. The wave d = l irradiance of total propagates in the direction zЈ. Hence, at the element the in- i i cident electric field and irradiance are Ilin Ilin = = . ͑15͒ i i i i i i E ͑t͒ = E ͑t͒xˆЈ + E ͑t͒yˆЈ ͑17͒ IPP Ilin + Inat nat nat,xЈ nat,yЈ A more useful expression for the purposes of measurement is and

27 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 27 2 can replace the factor in front of the brackets by the constant Ii = ͉͗Ei ͑t͒͘2͘. ͑18͒ nat ␨ nat,xЈ K, and we have set tЈ=t−z/c. 0 After performing a series of operations on Eqs. ͑7͒ and ͑8͒ The incident field induces an electric dipole moment in the and setting ␶=0, we can show that ͑see Appendix͒ electrically small element: ͉͗ ¨ i ͑ Ј͉͒2͘ ͉͗ ¨ i ͑ Ј͉͒2͘ ͗ ¨ i ͑ Ј͒ ¨ i ͑ Ј͒͘ Enat,xЈ t = Enat,yЈ t , Enat,xЈ t Enat,yЈ t =0, p͑t͒ = p ͑t͒xˆЈ + p ͑t͒yˆЈ = ␣ ⑀ ͓Ei ͑t͒xˆЈ + Ei ͑t͒yˆЈ͔, xЈ yЈ e 0 nat,xЈ nat,yЈ ͑26͒ ͑19͒ so that Eq. ͑25͒ can be written as where ⑀ is the permittivity of free space, and ␣ is the elec- 0 e t͑␰͒ ͉͗ ¨ i ͑ Ј͉͒2͓͘ 2 ͑␰͒ 2 ͑␪͒ 2͑␰͔͒ ͑ ͒ tric polarizability of the element. Note that the dipole mo- I = K Enat,xЈ t cos + sin sin . 27 ment has components parallel, p , and perpendicular, p ,to yЈ xЈ ͑ ͒ ␪ the principal plane. In Eq. ͑19͒ we have assumed that the If we substitute Eq. 21 for the angle and rewrite the dipole moment responds instantaneously to the incident elec- trigonometric terms, we obtain our final result for the irradi- tric field ͑there is no dispersion͒. This assumption is good for ance seen by the observer: the molecules of air at optical wavelengths. If the dipole It͑␰͒ = K͉͗E¨ i ͑tЈ͉͒2͓͘sin2͑␺ − ␺ ͒cos2͑␰͒ moment is expressed in terms of the coordinate system of the nat,xЈ p s ͑x y z͒ 2͑␺ ␺ ͔͒ ͑ ͒ observer , , , we have + cos p − s . 28 ជ͑ ͒ ␣ ⑀ ͓ i ͑ ͒ i ͑ ͒͑ ͑␪͒ ͑␪͒ ͔͒ ͑ ͒ p t = e 0 Enat,xЈ t xˆ + Enat,yЈ t − sin yˆ + cos zˆ , The result Eq. 28 has the same form as our earlier ex- ͑ ͒ pression for the irradiance of partially, linearly polarized 20 light observed with a linear polarizer given by Eq. ͑14͒; both 2 where the angle ␪ is have a term that depends on cos ͑␰͒ as well as a term that is independent of ␰. Thus, the skylight that we observe is ␪ ␺ ␺ ␲ ͑ ͒ = p − s − /2. 21 equivalent to partially polarized light, which is composed of This dipole moment produces the radiation that is the sky- linearly polarized light and natural light. It is important to light, and the electric field of this light is14 realize that this equivalence applies to the observed irradiances as measured by the polarizer and detector; the electric ␮ 0 fields for the two types of light could be different and the Esr͑r,t͒ = ͕rˆ ϫ ͓rˆ ϫ p¨͑t − r/c͔͖͒, ͑22͒ 4␲r irradiances the same. We can calculate the degree of linear polarization for the skylight from Eq. ͑16͒: ␮ where 0 is the permeability of free space, r=rrˆ is the radial It͑␰ =0͒ − It͑␰ = ␲/2͒ sin2 ͑␺ − ␺ ͒ vector drawn from the dipole, and the double dot over a d = = p s . ͑29͒ l t͑␰ ͒ t͑␰ ␲ ͒ 2 ͑␺ ␺ ͒ quantity indicates the second derivative with respect to I =0 + I = /2 1 + cos p − s time.15 If Eq. ͑20͒ is inserted into Eq. ͑22͒, we obtain the electric field incident on the polarizer, which is located at a In summary, we have found skylight to be equivalent to a ͑ distance z from the scatterer: mixture of linearly polarized light and natural light partially polarized light͒, with the linearly polarized component nor- ␣ mal to the principal plane, and the degree of linear polariza- Esr͑z,t͒ =− e ͓E¨ i ͑t − z/c͒xˆ − E¨ i ͑t − z/c͒sin͑␪͒yˆ͔. 4␲c2z nat,xЈ nat,yЈ tion a simple function ͓Eq. ͑29͔͒ of the difference in the angles of elevation for the observation point ͑scatterer͒ and ͑23͒ ␺ ␺ the sun, p − s. Specifically, the degree of linear polarization After the wave passes through the linear polarizer, the is maximum when the ray from the sun to the scatterer ͑SP͒ transmitted electric field and irradiance are is orthogonal to the ray from the scatterer to the observer ͑PM͒, then ␺ −␺ =␲/2 and d =1. For other orientations the ␣ p s l Et͑z,t͒ =− e ͓E¨ i ͑t − z/c͒cos͑␰͒ degree of polarization is less; the minimum occurs when the 2 nat,xЈ ␺ ␺ ␲ 4␲c z rays are parallel or antiparallel, then p − s =0, and dl =0. An examination of the patterns for dipole radiation, shown ¨ i ͑ ͒ ͑␪͒ ͑␰͔͒ ͑ ͒ − Enat,yЈ t − z/c sin sin uˆ 24 in the inset of Fig. 4, provides insight into these results. In the principal plane, the component of the dipole moment pxЈ and ͑viewed end on in Fig. 4͒ radiates an electric field that is 2 ␺ ␺ ␣ normal to this plane and independent of s and p. The other It͑␰͒ = e ͓͉͗E¨ i ͑t − z/c͉͒2͘cos2͑␰͒ ␲2 4␨ 2 nat,xЈ component of the dipole moment pyЈ radiates an electric field 16 c 0z ͉ ͑␪͉͒ ͉ ͑␺ that is in this plane and proportional to sin = cos p i ␺ ͉͒ + ͉͗E¨ ͑t − z/c͉͒2͘sin2͑␪͒sin2͑␰͒ − s . Thus, when we view the element at an angle such that nat,yЈ ␪ ͑␺ ␺ ␲ ͒ =0 p − s = /2 , we only see the component of the elec- ͗ ¨ i ͑ ͒ ¨ i ͑ ͒͘ ͑␪͒ ͑␰͒ ͑␰͔͒ ͑ Ј −2 Enat,xЈ t − z/c Enat,yЈ t − z/c sin cos sin tric field that is normal to the principal plane the x compo- nent͒; hence, the electric field is linearly polarized. For other ͓͉͗ ¨ i ͑ Ј͉͒2͘ 2͑␰͒ ͉͗ ¨ i ͑ Ј͉͒2͘ 2͑␪͒ 2͑␰͒ angles of observation, we see a mixture of the radiated elec- = K Enat,xЈ t cos + Enat,yЈ t sin sin tric fields from the two components of the dipole moment; ͗ ¨ i ͑ Ј͒ ¨ i ͑ Ј͒͘ ͑␪͒ ͑␰͒ ͑␰͔͒ ͑ ͒ hence, the electric field is partially polarized. −2 Enat,xЈ t Enat,yЈ t sin cos sin . 25 The simple result for the degree of linear polarization of In the last line we have simplified the result by recognizing skylight, Eq. ͑29͒, is compared with measurements in Fig. 5. that 1/z changes little in the vicinity of the polarizer, so we For these measurements the observer is viewing the zenith

28 Am. J. Phys., Vol. 75, No. 1, January 2007 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 Ref. 16 and from 1977 ͑Ref. 2͒.

͑␺ ␲ ͒ ͑␺ ͒ sky p = /2 as the sun rises s increases . Results are ͑ ͒ shown for two measurements; both were taken at high alti- Fig. 6. a Drawing of the principal plane showing the unit vectors nˆ s and nˆ p tude on a clear day. The dots are for results measured ͑visible pointing from the observer toward the sun and toward the observation point, spectrum͒ at Bocaiuva, Brazil at an altitude of 671 m in respectively. ͑b͒ Construction for a circle on which the degree of linear d 1947; the dashed line is for results measured ͑␭=0.71 ␮m͒ polarization l is a constant. 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 From Eq. ͑30͒ it is clear that the degree of linear polarization sun rises. However, the predicted degree of polarization is depends only on the direction of the observation point rela- always greater than measured. For example, the maximum tive to the direction of the sun. ͑ ͒ degree of linear polarization, which occurs at sunrise, ␺ =0, Now consider the construction shown in Fig. 6 b . The s ͑ nˆ ͒ ͑ ͒ is 100% for the theory but only about 84% for the measure- line MP unit vector p lies in the principal plane gray .If this line is rotated about the line through the sun, that is, ments. Factors not included in the simple theory cause this about MS ͑about the unit vector nˆ ͒, the end of the line traces difference and will be discussed later. s out a circle ͑dotted line͒. At every point on this circle, the degree of linear polarization is the same, because the dot product that appears in Eq. ͑30͒ is the same. For example, for the point PЈ we have nˆ pЈ·nˆ s =nˆ p ·nˆ s. At each point on this circle, the linearly polarized component of the electric ﬁeld III. DISTRIBUTION FOR POLARIZED SKYLIGHT is tangent to the circle. From these observations we can construct polarization dia- 17 From our knowledge of the degree of linear polarization in grams for the whole sky. Two of these diagrams are shown ␺ ͑ ͒ the principal plane, Eq. ͑29͒, we can obtain the degree of in Fig. 7 for the case s =35°. Figure 7 a shows mainly the linear polarization over the rest of the sky. First, we intro- solar half of the sky, and Fig. 7͑b͒ shows mainly the anti- duce the unit vectors shown in Fig. 6͑a͒: nˆ points from the solar half of the sky. The length of a heavy line indicates the s degree of linear polarization, and the line is parallel to the observer to the sun along MS, and nˆ points from the ob- p direction of the linearly polarized component of the electric server to the observation point along MP. Because of the ͑ ͒ ﬁeld. Note that the circles of constant dl are centered on the great distance to the sun, the ray MS in Fig. 6 a is parallel to line through the sun, MS, and that the maximum ͑d =1͒ oc- ␺ l the ray PS in Fig. 2, and both are at the angle of elevation s. curs, as expected, when ␺ −␺ =␲/2 ͑nˆ ·nˆ =0͒. As the sun ͑␺ ␺ ͒ ͑ ͒ p s p s We also observe that nˆ p ·nˆ s =cos p − s ,soEq. 29 can be moves across the sky, this pattern for the polarization moves written as 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 2 ͑␺ ␺ ͒ ͑ ͒2 1 − cos p − s 1− nˆ p · nˆ s ͓ ͑ ͔͒ d = = . ͑30͒ of constant dl the dotted curve in Fig. 6 b can be obtained l 2 ͑␺ ␺ ͒ ͑ ͒2 ␨ 1 + cos p − s 1+ nˆ p · nˆ s in terms of the arc length . The location of a point on this

29 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 29 circle, such as PЈ, 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 Յ␣ Յ␲ Յ␤ Յ␲ restricted to the ranges 0 az and 0 el /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:

␲ 1−d ␣ ͑␨͒ −1ͫϯͱ l ͑␺ ͒ ͑␨͒ az = + tan cos s csc 2 2dl

͑␺ ͒ ͑␨͒ͬ ͑ ͒ + sin s cot , 31

1−d ␤ ͑␨͒ −1ͫ ͱ l ͑␺ ͒ el = sin ± sin s 1+dl 2d ͱ l ͑␺ ͒ ͑␨͒ͬ ͑ ͒ + cos s cos . 32 1+dl ␺ For some values of dl, there are two separate curves, Fig. 7. Polarization diagram for the entire sky when s =35°. The length of hence, the two signs in these equations. The parameter ␨ the heavy line indicates the degree of linear polarization dl;thelineis must be constrained to ensure that points on the lower hemi- parallel to the direction of the linearly polarized component of the electric ﬁeld. Results are shown for two orientations: ͑a͒ mainly the solar half of the sphere are excluded: sky and ͑b͒ mainly the anti-solar half of the sky.

1−d ␲ ͯͱ l ͑␺ ͒ͯ Ն , tan s 1, 2dl 0 Յ ␨ Յ ͑33͒ Ά 1−dl 1−dl −1ͫϯͱ ͑␺ ͒ͬ ͯͱ ͑␺ ͒ͯ Ͻ cos tan s , tan s 1. 2dl 2dl

͑ ͒ ͑ ͒ ͑␤ Յ ͒ For the lower sign in Eqs. 31 – 33 we use only the values results are not shown near the horizon el 10° –15° for which where they are irregular. The simple theory and the measurements show the same general structure for the polarization, sin2͑␺ ͒ d Ͼ ͫ s ͬ, ͑34͒ particularly the direction of the linearly polarized component l 2͑␺ ͒ 1 + cos s of the field, which is parallel to a contour. However, the maximum degree of linear polarization is 100% for the to exclude the cases in which the entire circle for dl lies on theory but only 50%–60% for the measurements. In both ͑ ͒ ͑ ͒ ␣ Ϸ the lower hemisphere. In Eqs. 31 – 33 , the principal values cases, the maximum, as expected, occurs when az 180° ␲ ␤ Ϸ ␺ Ϸ of the inverse trigonometric functions are assumed: − /2 and el 90°− s 45°. Յsin−1 ͑͒Յ␲/2, 0Յcos−1 ͑͒Յ␲, and −␲/2Յtan−1 ͑͒ The difference between the theory and measurements can Յ␲/2. be attributed to several factors not included in the simple Figure 8 presents contour plots for the degree of linear theory that decrease the linearly polarized component of the ␺ polarization when the elevation angle of the sun is s light: the anisotropic polarizability of the air molecules, mul- =44.7°. These are polar graphs in which the radial variable is tiple scattering of light between air molecules, scattering of ␤ ͑ ͒ el 0° at the outer edge and 90° at the center and the an- light from aerosol particles and dust in the atmosphere, and ␣ ͑ ͒ gular variable is az. The results in Fig. 8 a are from the sunlight reflected from the clouds and the ground. Some of simple theory, Eqs. ͑31͒–͑34͒, and those in Fig. 8͑b͒ are mea- these factors are more significant in the urban environment sured data. The measurements were made at the wavelength of Miami than at the high altitude sites for the measurements ␭=0.439 ␮m on a very clear day during February 1996 at shown in Fig. 5. This difference is probably why the mea- 18,19 Ϸ ͑ ͒ the University of Miami in Miami, FL. The measured sured maximum dl 0.5 in Fig. 8 b is significantly lower

30 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 30 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. ͑a͒ During the dance, the sun is visible to the bee. ͑b͒ During the dance, a patch of clear blue sky is visible to the bee, and the view of the sun is blocked ͑indicated by gray area͒. ͑c͒ Same as in ͑b͒ with the skylight passing through a linear polarizer, and the transmission axis of the polarizer uˆ aligned with the linearly polarized component of the electric field of the skylight. ͑d͒ Same as in ͑c͒ with ␺ the transmission axis of the linear polarizer rotated. Fig. 8. Contour plots for the degree of linear polarization dl for s =44.7°. ͑a͒ Simple theory. ͑b͒ Measured data at ␭=0.439 ␮m ͑Refs. 18 and 19͒.The position of the sun is shown by the small symbol. von Frisch for honey bees in 1948.26–29 Since then, the honey bee has been studied intensely in this regard, and we will Ϸ than the measured maximum dl 0.84 in Fig. 5. These addi- restrict our discussion to this insect. Figure 9 is a schematic tional factors introduce other interesting effects into the mea- drawing of one of von Frisch’s experiments. surements that are not predicted by the simple theory. For When foraging for food ͑nectar and pollen͒, worker bees example, there are points where dl =0 other than those at use the location of the sun to determine direction, that is, ␺ ␺ p − s =0 and 180° that are not predicted by the simple they determine direction with respect to the sun much as we theory. A comprehensive discussion of these effects is given determine direction with respect to magnetic north using a in Refs. 2 and 20. compass.29–31 On returning to the hive, she ͑all worker bees are female͒ informs other workers of the location of the food. IV. INSECT NAVIGATION BY POLARIZED When the food is at a distance of about 100 m or greater, she SKYLIGHT communicates this information through the tail-wagging dance, which is shown in Fig. 9͑a͒ for a dance on the hori- Under very favorable conditions, human beings can detect zontal comb of a hive with the sun visible. The dance has a the presence of polarized light through a faint pattern known straight portion that is continually repeated after circling to as Haidinger’s brush.21 Recognizing and interpreting this the right or left. Along the straight portion, the bee waggles pattern takes practice, and it plays no known role in our her body, hence the name for the dance. The distance to the functioning. The reader is referred to Refs. 22–24 for details food is encoded in the characteristics of the dance, such as its of this interesting phenomenon. The situation is quite differ- tempo, and the direction to the food is indicated by the di- ent for many insects, because they can readily detect the rection of the straight portion of the dance relative to the ͑ ͒ ␣ ͑ ͒ polarization of light specifically skylight and make use of it direction of the sun, the solar azimuth angle az in Fig. 9 a . in various ways.25 The insects that detect and use the polar- Von Frisch noticed that bees could communicate the loca- ization of light include honey bees, ants, crickets, flies, and tion of the food through the tail-wagging dance even when beetles. he blocked their view of the sun at the hive, as indicated in The first conclusive evidence for the use of polarized sky- Fig. 9͑b͒ by the gray area. This observation held as long as light for orientation by insects was obtained by Karl Ritter he left a patch of clear blue sky visible to the bees. He

31 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 31 Fig. 10. Elements in the bee’s detection of polarized light. ͑a͒ 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. ͑b͒ Longitudinal and transverse cross sections for one of the specialized ommatidia composed of nine visual cells ͑Refs. 25 and 29͒. ͑c͒ A single visual cell from ͑b͒, showing the details for the microvilli ͑Ref. 37͒. ͑d͒ An expanded view of the transverse cross section of the ommatidium in ͑b͒ showing the three UV sensitive visual cells ͑A, B, and C͒ in gray, and the orthogonal orientation of the microvilli in the rhabdom ͑Ref. 25͒. surmised that the bees were using the polarization of the the direction of the linearly polarized component of the elec- skylight for orientation; possibly they could determine the tric field that is most important; the degree of linear polar- location of the hidden sun by knowing the relation between ization need only be greater than about 10%. the pattern of polarization for skylight and the position of the Since von Frisch’s pioneering behavioral research, there sun. To test this hypothesis, he passed the skylight visible to has been substantial effort devoted to identifying the physi- the bees through a linear polarizer. When the transmission ology of the bee’s eye responsible for sensing the polariza- axis of the polarizer uˆ was aligned so as to pass the linearly tion of light. Bees have compound eyes that are made from polarized component of the electric field of the skylight, as in many individual sensing units called ommatidia. The inset in Fig. 9͑c͒, the bee’s dance was unaltered from that in Fig. Fig. 10͑a͒ shows the ends of the ommatidia visible on the 9͑b͒; that is, it still pointed in the direction of the food. surface of the bee’s eye. The eye of a worker bee contains However, when the polarizer was rotated to change the di- about 5000 ommatidia. The ommatidia used for orientation rection of the transmitted electric field, as in Fig. 9͑d͒, the with polarized light in the UV are believed to be specialized direction of the bee’s dance changed, so that it no longer ones ͑about 150͒ located at the upper rim of the eye, that is, pointed in the direction of the food. in the dorsal rim area.25,32–34 Figure 10͑b͒ is a schematic By passing the skylight visible to the bees through band- drawing showing the longitudinal and transverse cross sec- pass filters, von Frisch and others determined that bees tions for one of these specialized ommatidia. It consists of mainly use the ultraviolet ͑UV͒ portion of the spectrum for nine long, nearly straight cells of equal length that are evi- the orientation by polarized light. It was also shown that it is dent in the transverse cross section. These cells are fused at

32 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 32 the center of the ommatidium to form the rhabdom. Light Recall that in each of the specialized ommatidium there enters the ommatidium through the lens, which in these spe- are actually two orthogonal sensors of polarization ͑sets of cialized ommatidia is covered with pore canals that increase microvilli͓͒Fig. 10͑d͔͒. Thus, when the response from one the visual field.35 The light accepted by the lens and crystal- set of sensors is maximum, the response from the other set of line cone is guided down the rhabdom, much as light is in an sensors is minimum. The bee may use the contrast between optical fiber. the signals from the two sets of sensors to enhance the accu- Figure 10͑c͒ is a schematic drawing for one of the nine racy for its orientation; that is, the bee may orient to maxi- visual cells shown in Fig. 10͑b͒. The rhabdomere is the por- mize the difference in the signals from the two orthogonal tion of a cell that contributes to the rhabdom. It is made up of sets of sensors. many small protrusions called microvilli that are perpendicular to the optical axis of the ommatidium.36,37 There are di- polar pigment molecules in the membrane of the microvilli, V. CONCLUSION and the axes of these molecules are aligned with the mi- ͑ ͒ crovilli. As indicated in Fig. 10 c , light with its electric field We have used the material in this paper to supplement a parallel to the microvilli is absorbed more by these mol- conventional treatment of the polarization of plane electro- ecules than light with its electric field perpendicular to the magnetic waves. In addition to a classroom presentation, microvilli. This feature makes the cell sensitive to the direc- each student is given a simple linear polarizer ͑laminated tion of the linearly polarized component of light. The mea- film polarizer, Edmund Optics, NT38-396͒. They are asked ͑ ͒ sured response electrical signal of these specialized cells is to make a qualitative observation of the polarization of sky- as much as ten times greater when the electric field is parallel light using the hand-held polarizer, and to see if their assess- to the microvilli than when it is perpendicular to the mi- ment corresponds to Fig. 7. Readings from the current re- crovilli. search on insects’ use of polarized skylight are assigned or Three of the nine cells in a specialized ommatidium are suggested. Students are often surprised to find that they can sensitive to UV and take part in the orientation by polarized readily understand the current research with only the addi- light; they are shaded gray in the transverse cross section tional knowledge of polarized light that they have received in 25,32–34 shown in Fig. 10͑d͒. The microvilli of one of these this presentation. The general feeling of students is that this cells ͑marked C͒ are perpendicular to those in the other two material on the polarization of skylight, particularly that per- cells ͑marked A and B͒. Thus, in a single ommatidium there taining to navigation by honey bees, makes what they be- are sensors for electric fields in two orthogonal directions. lieve to be a rather ordinary topic much more exciting. 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 sens- ACKNOWLEDGMENTS ing polarized light established, the question that remains is how do bees use polarization for orientation and navigation? The author would like to thank R. Todd Lee for assistance Different theories have been proposed, and we will give a with the photographs shown in Fig. 1, and a reviewer for brief sketch of one by Rossel and Wehner, which is well making useful suggestions that were included in the paper. described in the literature including its limitations.17,38–42 In The author is grateful for the support provided by the John this theory bees mainly make use of the polarization of sky- Pippin Chair in Electromagnetics that furthered this study. Յ␣ This paper is dedicated to the memory of an exceptional light in the anti-solar half of the sky, that is, for 90° az Յ Յ␤ Յ teacher, Ronold W. P. King ͑1905–2006͒, Professor of Ap- 270° and 0° el 90° in Fig. 7. This observation makes ␺ Ͼ plied Physics at Harvard University. 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 ͓compare Fig. 7͑a͒ with Fig. 7͑b͔͒. Also, the light from the solar half of the sky is composed of skylight plus APPENDIX: CORRELATION FUNCTIONS direct sunlight, and the latter is unpolarized. FOR THE SECOND DERIVATIVE OF THE Rossel and Wehner assume that the special ommatidia in ELECTRIC FIELD OF NATURAL LIGHT the dorsal rim area of the bee’s eyes are arranged so that they match some gross features of the polarization of the anti- The argument for the first of the relations in Eq. ͑26͒ be- ͑␣ solar sky when the bee faces the anti-solar meridian az gins by differentiating the left-hand side of Eq. ͑7͒ with re- =180°͒. Specifically, for an ommatidium pointing in the so- spect to ␶, then using integration by parts to obtain: ␣ ͓ lar azimuth direction az, the axis of polarization the axis 1 TD/2 dE ͑tЈ͒ for one of the two sets of microvilli in Fig. 10͑d͔͒ is aligned ͵ ͑ ͒ͫ x ͬ Ex t − dt with a mean representation of the electric field of the sky- T dtЈ ␶ D t=−TD/2 t− light from that direction. When the bee views a patch of blue sky, she rotates about 1 1 TD/2 dE ͑t͒ ͓ ͑ ͒ ͑ ␶͔͒TD/2 ͵ x her vertical body axis. The signal the bee receives from the =− Ex t Ex t − + T −TD/2 T dt array of specialized ommatidia in the dorsal rim area changes D D t=−TD/2 during the rotation, and it is maximum when the bee is ap- ϫE ͑t − ␶͒ dt. ͑A1͒ proximately facing the anti-solar meridian. With this proce- x →ϱ dure the bee effectively determines the location of the sun For TD we assume that the first term is negligible. We ͑position in azimuth͒ from the polarized light it receives differentiate the second term with respect to ␶ and integrate from a patch of blue sky. by parts:

33 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 33 1 TD/2 dE ͑t͒ dE ͑tЈ͒ partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165– ͵ x ͫ− x ͬ dt 1181 ͑1959͒. 13 T dt dtЈ ␶ We have discussed partially polarized light that is formed by adding D t=−TD/2 t− linearly polarized light to natural light. Partially polarized light can be formed in other ways. For example, partially polarized light that is quasi- 1 dE ͑t͒ TD/2 1 TD/2 d2E ͑t͒ =− ͫ x E ͑t − ␶͒ͬ + ͵ x monochromatic is formed by adding elliptically polarized light to natural x 2 light. The composition of partially polarized, quasi-monochromatic light TD dt −T /2 TD t=−T /2 dt D D can be determined by performing a series of measurements with a linear ϫ ͑ ␶͒ ͑ ͒ polarizer and a linear retarder. The details are given in M. Born and E. Ex t − dt. A2 Wolf, Principles of Optics, 7th ed. ͑Cambridge U. P., Cambridge, 1999͒, →ϱ As before, for TD we assume that the first term is neg- Chap. 10; L. Mandel and E. Wolf, Optical Coherence and Quantum Op- ligible, and we differentiate the second term twice with re- tics ͑Cambridge U. P., Cambridge, 1995͒,Chap.6. 14 spect to ␶ to obtain The scattering element ͑molecule͒ is in random motion with the speed vc. We have assumed that this motion does not change the statistical 1 TD/2 d2E ͑t͒ d2E ͑tЈ͒ properties of the field, and we have used the expression for the radiated ͵ x ͫ x ͬ ͉͗ ¨ i ͑ ␶͉͒2͘ 2 2 dt = Enat,x t − . field of a stationary dipole: G. S. Smith, An Introduction to Classical T dt dtЈ ␶ ͑ ͒ D t=−TD/2 t− Electromagnetic Radiation Cambridge U. P., Cambridge, 1997 ,pp. 452–465. ͑A3͒ 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. If the same operations are applied to the right-hand side of 16 Eq. ͑7͒ and the results are equated, we have R. A. Richardson and E. O. Hulburt, “Sky-brightness measurements near Bocaiuva, Brazil,” J. Geophys. Res. 54, 215–227 ͑1949͒. 17 ͉͗ ¨ i ͑ ␶͉͒2͘ ͉͗ ¨ i ͑ ␶͉͒2͘ ͑ ͒ R. Wehner and S. Rossel, “The bee’s celestial compass – A case study in Enat,x t − = Enat,y t − , A4 behavioural neurobiology,” in Experimental Behavioral Ecology and So- ␶ → ciobiology, In Memoriam Karl von Frisch 1886–1982, edited by B. Höll- which for =0 and x,y xЈ,yЈ is the first relation in Eq. ͑ ͒ ͑ ͒ ͑ ͒ dobler and M. Lindauer Sinauer Associates, Sunderland, MA, 1985 . pp. 26 . The argument for the second relation in Eq. 26 is 11–53. obtained in the same way. 18 K. J. Voss and Y. Liu, “Polarized radiance distribution measurement of skylight. I. System description and characterization,” Appl. Opt. 36, 6083–6094 ͑1997͒. 19 Y. Liu and K. J. Voss, “Polarized radiance distribution measurement of ͒ a Electronic mail: [email protected] skylight. II. Experiment and data,” Appl. Opt. 36, 8753–8764 ͑1997͒. 1 D. F. J. Arago, Oeuvres Complètes de François Arago ͑Gide, Paris, 20 Selected Papers on Scattering in the Atmosphere, edited by C. F. Bohren 1858͒, Vol. 7, pp. 394–395. ͑SPIE, Bellingham, WA, 1989͒, pp. 261–326. 2 K. L. Coulson, Polarization and Intensity of Light in the Atmosphere 21 W. Haidinger, “Uber das directe Erkennen des polarisirten Lichts und der ͑Deepak, Hampton, VA, 1988͒,p.2. Lage der Polarisationsebene” ͑On the direct recognition of polarized light 3 H. LaFay, “The Vikings,” Natl. Geogr. 37, 492–541 ͑1970͒. and the polarization plane͒, Ann. Phys. Chem. 63, 29–39 ͑1844͒. 4 C. P. Können, Polarized Light in Nature ͑Cambridge U. P., Cambridge, 22 M. G. J. Minnaert, Light and Color in the Outdoors ͑Springer-Verlag, 1985͒,p.30. New York, 1993͒, pp. 276–278. 5 C. Roslund and C. Beckman, “Disputing Viking navigation by polarized 23 D. Auerbach, “Optical polarization without tools,” Eur. J. Phys. 21, light,” Appl. Opt. 33, 4754–4755 ͑1994͒. 13–17 ͑2000͒. 6 J. Walker, “The Amateur Scientist: More about polarizers and how to use 24 A. P. Ovcharenko and V. D. Yegorenkov, “Teaching students to observe them, particularly for studying polarized skylight,” Sci. Am. 238͑1͒, Haidinger brushes,” Eur. J. Phys. 23, 123–125 ͑2002͒. 132–136 ͑1978͒. 25 T. Labhart and E. P. Meyer, “Detection of polarized skylight in insects: a 7 D. K. Lynch and W. Livingston, Color and Light in Nature, 2nd ed. survey of ommatidial specializations in the dorsal rim area of the com- ͑Cambridge U. P., Cambridge, 2001͒, pp. 26–27. pound eye,” Microsc. Res. Tech. 47, 368–379 ͑1999͒. 8 The two viewpoints as to the cause of the scattering that gives rise to the 26 K. von Frisch, “Gelöste und ungelöste Rätsel der Bienensprache,” Natur- color and polarization of skylight were first introduced by Lord Rayleigh wiss. 35, 38–43 ͑1948͒. ͑molecules͒ and A. Einstein ͑density fluctuations͒: Lord Rayleigh, “On 27 K. von Frisch, “Die Polarisation des Himmelslichtes als orientierender the transmission of light through an atmosphere containing small par- Faktor bei den Tänzen der Bienen,” Experientia 5, 142–148 ͑1949͒. ticles in suspension, and on the origin of the blue sky,” Philos. Mag. 47, 28 K. von Frisch, “Die Sonne als Kompass im Leben der Bienen,” Experi- 375–384 ͑1899͒. A. Einstein, “Theorie der Opaleszenz von homogenen entia 6, 210–221 ͑1950͒. Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen 29 K. von Frisch, The Dance Language and Orientation of Bees ͑Harvard U. Zustandes,” Ann. Phys. 33, 1275–1298 ͑1910͒. English translation, P., Cambridge, MA, 1967͒. “Theory of the opalescence of homogeneous liquids and mixtures of liq- 30 J. L. Gould and C. G. Gould, The Honey Bee ͑Scientific American Li- uids in the vicinity of the critical state” in Colloid Chemistry: Theoretical brary, New York, 1988͒. and Applied, edited by J. Alexander ͑Chemical Catalog Company, New 31 F. G. Barth, Insect and Flower ͑Princeton U. P., Princeton, NJ, 1991͒. York, 1926͒, Vol. 1, pp. 323–339. 32 R. H. Schinz, “Structural specialization in the dorsal retina of the bee, 9 E. T. Malus, “Sur une propriété des forces répulsives qui agissent sur la Apis mellifera,” Cell Tissue Res. 162, 23–34 ͑1975͒. lumiére,” Mémoires de physique et de chimie de la Société D’Arcueil 2, 33 T. Labhart, “Specialized photoreceptors at the dorsal rim of the honey- 254–267 ͑1809͒. English translation, “On a property of the repulsive bee’s compound eye: polarizational and angular sensitivity,” J. Comp. forces, that act on light,” A Journal of Natural Philosophy, Chemistry, and Physiol. 141, 19–30 ͑1980͒. the Arts ͑Nicholson’s Journal͒ 30, 161–168 ͑1811͒. 34 R. Wehner and S. Strasser, “The POL area of the honey bee’s eye: Be- 10 W. A. Shurcliff, Polarized Light: Production and Use ͑Harvard U. P., havioural evidence,” Physiol. Entomol. 10, 337–349 ͑1985͒. Cambridge, MA, 1966͒,p.39. 35 E. P. Meyer and T. Labhart, “Pore canals in the cornea of a functionally 11 P. Z. Peebles, Jr., Probability, Random Variables and Random Signal specialized area of the honey bee’s compound eye,” Cell Tissue Res. Principles, 4th ed. ͑McGraw-Hill, New York, 2001͒. 216, 491–501 ͑1981͒. 12 In Eqs. ͑6͒ and ͑8͒ we have postulated the properties for the electric field 36 R. Menzel and A. W. Snyder, “Introduction to photoreceptor optics—an of natural light. Then we used these properties to predict the action of the overview,” in Photorecpetor Optics, edited by R. Menzel and A. W. linear polarizer on this light. Historically, the opposite was true: The Snyder ͑Springer-Verlag, New York, 1975͒, pp. 1–13. action of polarizers on natural light was used to infer the properties of the 37 R. Wehner, “Polarized-light navigation by insects,” Sci. Am. 235͑7͒, light. See, for example, G. G. Stokes, “On the composition and resolution 106–115 ͑1976͒. of streams of polarized light from different sources,” Trans. Cambridge 38 S. Rossel and R. Wehner, “The bee’s map of the e-vector pattern in the Philos. Soc. 9, 399–416 ͑1852͒; and E. Wolf, “Coherence properties of sky,” Proc. Natl. Acad. Sci. U.S.A. 79, 4451–4455 ͑1982͒.

34 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 34 39 S. Rossel and R. Wehner, “How bees analyse the polarization patterns in 41 K. Kirschfeld, “Navigation and compass orientation by insects according the sky, experiments and model,” J. Comp. Physiol., A 154, 607–615 to the polarization pattern of the sky,” Z. Naturforsch. C 43c,467–469 ͑1984͒. ͑1988͒. 40 S. Rossel and R. Wehner, “Polarization vision in bees,” Nature ͑London͒ 42 K. Kirschfeld, “The role of the dorsal rim ommatidia in the bee’s eye,” Z. 323, 128–131 ͑1986͒. Naturforsch. C 43c, 621–623 ͑1988͒.

35 Am. J. Phys., Vol. 75, No. 1, January 2007 Glenn S. Smith 35