The Statistical Properties of Unpolarized Light
JOSA, Vol. 35, Issue 8, pp. 525-531 (1945)
http://dx.doi.org/10.1364/JOSA.35.000525
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Abstract
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 <i>S</i> 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 <i>½</i> 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, <i>R</i>, is statistically independent of <i>S</i> and has the distribution function (<i>r</i><sup>3</sup>/2<i>p</i><sup>4</sup>) exp (-<i>r</i><sup>2</sup>/2<i>p</i><sup>2</sup>) where 2<i>p</i><sup>2</sup> is the average value of <i>R</i><sup>2</sup>.
Citation
HENRY HURWITZ, JR., "The Statistical Properties of Unpolarized Light," J. Opt. Soc. Am. 35, 525-531 (1945)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-35-8-525
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References
- We assume that the electric field is constant over the part of the cross section of the beam under consideration.
- H. Hurwitz and M. Kac, Ann. Math. Stat. 15, 173 (1944); S. O. Rice, Bell Sys. Tech. J. 23, 282 (1944), 24, 46 (1945). Some insight as to why the quantities in Eq. (5) are Gaussianly distributed can be had from the following considerations. Let[Equation].From Eq.4.[Equation]If the t_{i}’s are randomly distributed the quantities a_{k} and b_{k} can be regarded as the two components of a vector which is the sum of N two-dimensional vectors of constant length but random direction. Hence from the familiar results of the problem of "random walk" we can immediately conclude that a_{k} and b_{k} are Gaussianly distributed in the limit that N becomes infinite.
- The assumption that E_{x} and E_{y} are statistically independent can be justified in terms of the somewhat more physical assumption that the statistical properties of an intense beam of unpolarized light cannot be changed by passing the light through a wave plate. By the use of a wave plate of suitable thickness, one field component can be retarded with respect to the other by any desired amount. It is evident that no possible type of correlation between E_{x} and E_{y} could remain invariant under this general class of transformations.
- M. Kac and H. Steinhaus, Studia Mathematica 6, 89 (1936).
- L. Brillouin, Die Quantenstatistik (Julius Springer, Berlin, 1931), p. 111, 177.
- M. Born, Optik (Julius Springer, Berlin, 1933), p. 23,
- Private communication.
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