Polarization & Bows ~ A fogbow showing an extraordinary number of inner supernumeraries imaged using a special polarizing technique by Chris Cox.   Locale: The Greenland Ice Cap ICECAPS Atmospheric Observatory, Summit Station. Date: July 15, 2012 during the recent Greenland warm event that for several hours on July 11-12 saw temperatures warming up to around melting point.   ©Chris Cox
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View a fogbow or rainbow through a polarizing filter. Part is bright with enhanced contrast while other parts almost disappear. Rotate the filter and the bright areas move along the curve of the bow. Rainbows are strongly polarized, fogbows are less so but still enough to show significant effects.

When unpolarized light strikes water, part is reflected and a part enters the liquid. Both rays are partially polarized. Waves with their electric vector parallel to the surface are more strongly reflected. Waves whose electric vector is perpendicular are more strongly refracted. The degree of their polarization depends on the angle of incidence and at a certain angle, the Brewster angle, the reflected and refracted beams are 100% polarized.

Fogbow and rainbow light has undergone two refractions and an internal reflection and is therefore polarized. At any point the polarization directions are radial and tangential to the bow’s curve.

Chris Cox relied on this polarization to capture unusually clear supernumerary structure. He took three images with the polarizer turned 45° between each and then composited them.

Supernumeraries show peculiar polarization effects. As the filter is rotated they shift in position. When the polarizer's electric vector is tangential to the bow the supernumeraries occupy the position of the dark gaps between the fringes seen when the filter is rotated 90°. The changing position arises because supernumeraries are produced by interference between two rays having the same deflection angle. For one polarisation component, one of these two rays is phase shifted during reflection inside the droplet.

For more about polarization, download Gunther Konnen's book.
An unpolarized ray strikes a water surface. The ray can be regarded as having two components. One (shown red) has its electric vector always parallel to the surface. The second (shown blue) has its electric vector at right angles.

The reflected ray is partially polarized. The parallel (red coloured) component is stronger than the other.

The refracted ray is weaker in the parallel component.
A rainbow or fogbow ray. Partial polarization occurs at the two refractions and at the internal reflection.