Singularity Theory Yields Elegant Description of Polarized Sunlight

Daniel S. Burgess

Scientists have long been intrigued by the polarization of light in the daytime sky. In 1863, Sir David Brewster noted the similarity of the phenomenon with that produced by refracting crystals. Less than a decade later, Lord John W.S. Rayleigh proposed that the polarization was the result of scattering by molecules in the atmosphere, leading to Subrahmanyan Chandrasekhar's description in the mid-20th century of the polarization in terms of multiple scattering effects.

Singularity theory offers a "beautiful" approach to explaining the polarization patterns observed in the sunlit sky. Courtesy of Sir Michael V. Berry.

Despite the explanatory power and predictive success of such theories, they lacked a certain elegance in that they involved multiple, complicated formulas. Now researchers at the University of Bristol in the UK and the US Naval Academy in Annapolis, Md., have employed geometric singularity theory to describe the pattern and the intensity of the polarization of sunlight in terms of the contours of elliptical integrals. They note that the work replaces pages of formulas and calculations with a single, simple solution.

The researchers explained singularity theory in terms of the topography of a terrain. A few exceptional places, such as the tops of hills and the lowest depths of valleys, give order to the arrangement of contour lines that describe the place. A knowledge of the hilltops and the valleys, therefore, offers a global understanding of the landscape.

Mark R. Dennis, now at the University of Southampton in the UK, and his colleagues, Sir Michael V. Berry and Raymond L. Lee Jr., looked to similar exceptional places in the pattern of polarization of sunlight that give it order. They realized that the four observable unpolarized points in the sky -- above and below the position of the sun, and above and below the position opposite the sun -- are just such "fingerprint" singularities that define the foci of a family of confocal ellipses.

To test their theory, the scientists compared the intensity patterns it generated with those observed in polarization photographs taken with a digital camera and fish-eye lens. They found them in good agreement, especially farther from the singularities, with 90 percent of the angular deviations from the observed polarization directions being less than 8°. Dennis noted that this is comparable to the agreement for multiple-scattering theory.

Although multiple-scattering theory can account for several additional atmospheric effects that influence the polarization patterns in the sunlit sky, such as haze and surface reflection, the elegance of singularity theory wins the beauty contest that is used to decide among otherwise similarly successful, competing scientific theories.

"Part of the aesthetic beauty of a scientific result is its simplicity and the variety of different ways it can be approached," Dennis said, citing the response of Paul A.M. Dirac when asked how he had found his equation for the quantum mechanical electron: " 'I found it beautiful.' "