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When a ray of ordinary light is passed on the surface of the water the reflected light will be completely polarized( vibrations in one plane).

My question is what will be plane of vibration in the partially polarized light that undergoes refraction? How many planes of vibration will be there? Deep explanation focusing on the planes of vibration of the partially polarized light would be appreciated.

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Only when the incidence angle is the Brewster's angle is reflected light completely polarized. – Siyuan Ren Aug 12 '12 at 8:52

The reflected light is not completely polarised. It's certainly true that $s$ polarised light (i.e. light polarised in the plane of the surface) is more strongly reflected, but except at the Brewsters angle $p$ polarised light is reflected as well.

The plane of polarisation of $s$ and $p$ polarised light is not changed by reflection/transmission, however a mixture of polarisations will be changed because the $s$ to $p$ ratio of the reflected and transmitted light will not be the same as the incident light.

The reflection and transmission of the light can be calculated using the Fresnel equations. The link I've given is to the Wikipedia article Fresnel equations and this is an excellent starting point if you want a thorough explanation of the subject. To go much farther you'll need a good book on Optics. My favourite is "Optics" by Hecht and Zajac, though I think that can be hard to find these days.

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Hecht isn't hard to find. It's still one of the standards. – Colin K Aug 19 '12 at 15:57
@ColinK ...although Alfred Zajac somehow dropped off along the way - don't know what happenned there. – WetSavannaAnimal aka Rod Vance Aug 28 '15 at 11:47

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