In a beam of monochromatic unpolarized light the electric field vector at a point traces out an ellipse whose size, eccentricity, and orientation are slowly varying functions of time. The statistical properties of the parameters of this ellipse are investigated. It is shown that the quantity S which is defined as twice the product of the principle axes of the ellipse divided by the sum of the squares is uniformly distributed between zero and one. It therefore has median value ½ which corresponds to a ratio of minor to major axis equal to .268. Hence fairly thin ellipses predominate. The square root of the sum of the squares of the semi-major and semi-minor axes, R, is statistically independent of S and has the distribution function (r3/2p4) exp (-r2/2p2) where 2p2 is the average value of R2.
HENRY HURWITZ, JR., "The Statistical Properties of Unpolarized Light," J. Opt. Soc. Am. 35, 525-531 (1945)