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 scat- terer ͑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 mea- sured. 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 .