Polarization

Polarization is a phenomenon peculiar to transverse waves.  Longitudinal waves such as sound cannot be polarized.  Light and other electromagnetic waves are transverse waves made up of mutually perpendicular, fluctuating electric and magnetic fields.  In the diagram below an EM wave is propagating in the x-direction, the electric field oscillates in the xy-plane, and the magnetic field oscillates in the xz-plane.  A line traces out the electric field vector as the wave propagates.

image

An unpolarized electromagnetic wave traveling in the x-direction is a superposition of many waves.  For each of these waves the electric field vector is perpendicular to the x-axis, but the angle it makes with the y-axis is different for different waves.  For unpolarized light traveling in the x-direction Ey and Ez are randomly varying on a timescale that is much shorter than that needed for observation.

image
Unpolarized light: Natural light is, in general, unpolarized.

For a linearly polarized electromagnetic wave traveling in the x-direction, the angle the electric field makes with the y-axis is unique.

imageAn ideal polarizer is a material that passes only EM waves for which the electric field vector is parallel to its transmission axis.  The electric field is a vector and can be written in terms of the components parallel and perpendicular to the polarizer's transmission axis. 
E
= Eparallel + Eperpendicular
An ideal polarizer passes Eparallel and absorbs Eperpendicular.

If E0 is the incident field vector and the angle between E0 and the transmission axis is θ, then the magnitude of transmitted field vector is E0 cosθ and its direction is the direction of the transmission axis.  The intensity I of an electromagnetic wave is proportional to the square of the magnitude of the electric field vector.  We therefore have

Itransmitted = I0 cos2θ.

This is called the law of Malus.  If θ = 90o the transmitted intensity is zero.

image
The lines indicate the direction of the transmission axis.

Problem:

imageA beam of unpolarized light of intensity I0 passes through a series of ideal polarizing filters with their transmission axis turned to various angles, as shown in the figure.
(a)  What is the light intensity (in terms of I0) in regions A, B, and C?
(b)  If we remove the middle filter, what will be the light intensity at point C?

Solution:
When unpolarized light passes through a polarizer, the intensity is reduced by a factor of ½.  The transmitted light is polarized along the axis of the polarizer.
When polarized light of intensity I0 is incident on a polarizer, the transmitted intensity is given by I = I0cos2θ, where θ is the angle between the polarization direction of the incident light and the axis of the filter.  For the second polarizer θ = 30o.  For the third polarizer θ = 90o - 30o = 60o.
We then have that:
(a) In region A the intensity is I0/2 and the light is polarized along the vertical direction.
In region B the intensity is (I0/2)cos230o, = 0.375 I0, and the light is polarized along the axis of the second polarizer.
in region C the intensity is (0.375 I0)cos260o = 0.0938 I0 and the light is horizontally polarized.
(b) If we remove the middle filter, for the last filter we now have that θ = 90o.  Thus I = 0.
It is important to visualize the fact that adding the middle filter increases the transmitted intensity!
This "paradoxical" effect is a signature of wave phenomena in general.

The most common method of producing polarized light is to use polaroid material, made from chains of organic molecules, which are anisotropic in shape.  Light transmitted is linearly polarized perpendicular to the direction of the chains.  The transmission axis is perpendicular to the chains.

A polarizer produces linearly polarized light.  It is often convenient to orient the transmission axis of a polarizer vertically or horizontally to produce light with vertical or horizontal linear polarization.

image
Vertical and horizontal polarization

Polarization by reflection:

imageWhen unpolarized light is incident on a boundary between two dielectric surfaces, for example on an air-water boundary, then the reflected and transmitted components are partially polarized.  The reflected wave is 100% linearly polarized when the incident angle is equal to an angle called the Brewster angle.
For water this angle is is ~53o with respect to the normal or 37o with respect to the water surface.
For are considerable angular range around the Brewster angle the reflected light is highly polarized in the horizontal direction.

imageWhen the sun is at a low angle in the sky, the sunlight reflecting off the surface of water is nearly 100% horizontally polarized because the angle of incidence is close to the Brewster angle.  Glare-reducing sunglasses are coated with a polarizer with a vertical transmission axis and therefore block the reflected light.

Link:  Polarization of light (Youtube)
Link:  Polarization and reflection (Youtube)