In general, the sky is polarized tangential to a circle centered in the sun and maximum polarization is found at ninety degrees from it. Therefore, with the sun close to the zenith the sky will be polarized horizontally along the entire horizon. On the other hand, when the sun is setting West, the sky will be maximally polarized along the meridian and thus vertically at the horizon due North and South. Towards the zenith just after sunset (or before sunrise) the degree of polarization of the sky can reach its maximum of about 75% in very clear days. Don't forget to test these facts using your polarized sunglasses next time you sunbathe on a tropical beach (a vacation-friendly non-strenuous activity).

The polarization pattern of the sky can be a little more complicated than the simplest (single scattering) theory would predict. In fact, Arago himself discovered that the sky opposite to the sun is not completely unpolarized as one could expect but somewhat vertically polarized (up to 20-30% at sunset: this is sometimes given the perplexing name of negative polarization). The point where the polarization switches from vertical to horizontal going from the horizon to the zenith is called the Arago neutral point. The reason for this is that when the sun is close to (or below) the horizon its light competes with the twilight sky itself as an illumination source, or in other words, the sky at the horizon is illuminated by the rest of the sky. Two other neutral points (much more difficult to detect) are rumored to be above and below the sun (the Babinet and Brewster neutral points) [2] .

With the sun close to the zenith the sky is polarized mostly horizontally.

At sunset the polarization along the horizon is mostly vertical.

Why scatter can polarize light?

Scatter happens because a photon excites an electron that absorbs its energy and vibrates, and this vibration re-radiates like an antenna a new photon in a random direction. The direction of the electron vibration is the same as the direction of the electric field of the incident photon. Conversely, the radiated photon has the electric field aligned with the direction of the electron vibration.

Light is a transversal wave what means that the electric field "vibrates" perpendicularly to the direction of the beam. If the incident light is unpolarized, the electric field vibrates in every direction in a plane perpendicular to the beam. The electron of the scattering molecule will also vibrate confined to that plane. But if the plane is seen from the side, the vibrations appear to be confined to a line, as shown in this animation. Therefore, light scattered backwards (or forward) remains unpolarized, while light scattered at 90 degrees becomes linearly polarized (in intermediate directions it is partially polarized).

Now, it is important to realize that if a photon is scattered multiple times instead of just once before it reaches the observer, its polarization becomes random.  The reason is that each scattering event is in a random 3D direction and therefore the final polarization also becomes random.

Why can a polarized filter make the sky bluer?

If the air were completely transparent, we would be able to see the stars during the day (nice!) but there would be no visible "sky" (not so nice). This is the same reason why you have to blow smoke on a laser beam to see it from the side. The sky tends to look sky-blue because the shorter (blue) wavelengths are more likely to be scattered than the longer (red) ones by the much smaller air molecules ("Rayleigh scattering"). Contrary to the Little Red Riding Hood tale, being dressed blue puts the photon at higher risk of interception before reaching home.

But if your are looking through a lot of air (i.e. at the horizon) or when there are other larger particles (e.g. pollution) that are much more efficient scatterers than the air molecules, longer (redder) wavelengths are also likely to be scattered towards you. At the same time, there is a higher chance that blue light is scattered  a second time on its way to your eyes, limiting the increase of its contribution. The end result is that the sky will loose some of it color saturation and become more whitish.

 Therefore, skylight colors that reach your eye have different pedigree. The blue lineage is more likely to have been scattered multiple times than does the red part of the spectrum. As multiple scatterings destroy the polarization of light, blue light is less polarized than other colors in skylight.

Thus, by aligning a polarizer so it blocks the polarization of the sky its blue color is enhanced (but darkened).  Photographers frequently use polarized filters on their cameras to capture skies that look more saturated than to the unaided eye. 

This fisheye photograph (hi-res) of the full sky at sunset was kindly provided by Forrest Mims of electronics authoring fame.  He took it from his South-Central Texas Observatory using a polarizing filter aligned East-West. The roughly North-South line of dark blue sky shows where polarization was at its maximum. As the title of that country-western song not yet written says: "the skies of Texas are bluest for the cowboy with sunglasses and no blues to sing."

Mims will grant use of this picture for noncommercial use by students and individuals so long as acknowledgment is given. If you are interested in computers, you should read his first-hand account of the history of Altair, the first commercial personal computer, at his site www.forrestmims.org.

[2] G. P. Können, Polarized light in Nature, Cambridge University Press, 1985 (original in Dutch, 1980). Highly recommended.