Tuesday, July 19, 2011

Light becomes polarized because of its wave nature.

As electromagnetic radiation (in the visible spectrum), a light wave is composed of both electric and magnetic field components, and is usually represented by a phase vector that encodes information about just the electric field. The phase vector points in the direction of the electric field, and its magnitude denotes the electric field strength. Because the phase vector can be decomposed into orthogonal components that oscillate sinusoidally, light is called a wave, and the phase of the wave at a certain place and time refers to its place along the sine curve.

Examine a planar wave, which travels along the z axis with electric and magnetic fields in the x-y plane. Let E(t,z) be the phase vector with orthogonal components x(t,z) and y(t,z) that are time- and space-dependent. If the x and y field components oscillate with amplitudes Ex and Ey, then

Ex2 + Ey2 = ||E||2 for all t, z
x(z,t) = Excos(kz-ωt1)
y(z,t) = Eycos(kz-ωt2)

where the following are constants:

ω  is the frequency, in units of Hertz (Hz) or radians/second,
k  is the (angular) wavenumber, in units of radians/meter, and
φ1, φ2 are the phases, in units of radians.

Polarization is the phenomenon of light waves having the same spatial orientation of their phase vectors (if it happens in nature), or the restriction of the phase vectors to certain orientations (by experiment). Ordinary sunlight is generally unpolarized because the direction of the individual phase vectors are aligned randomly with each other as they oscillate. By passing unpolarized light through a linear polarizing filter, waves result whose phase vectors only oscillate along a particular axis, say the horizontal (x) axis. One has “filtered out” the vertical (y) electric field components from every wave that passed through the linear polarizer. Thus Ex = E is the amplitude for the whole wave, and

x(t,z) = E cos(kz-ωt)
y(t,z) = 0

If you projected the endpoint of the phase vector onto a cross-section of the travelling wave (a picture called a Lissajous curve), you would see a line - hence the name, linear polarization.

Other polarizations are possible where the direction rather than the magnitude of the phase vector changes. If one takes the horizontally polarized light, tilts it 45 degrees to halfway between horizontal and vertical (like the animation), and then passes it through a quarter wave plate that slows down electric field along the horizontal axis by a quarter phase, one obtains circularly polarized light. The horizontal component of the phase vector now oscillates a quarter phase (2π/4 = π/2) behind the vertical components, resulting in the parametric equations

x(t,z) = (1/√2) E cos(kz-ω(t-π/2))
y(t,z) = (1/√2) E cos(kz-ωt)

which, keeping z constant, represent a circle (the 1/√2 comes from sin 45 deg and cos 45 deg). You can calculate that the phase φis

φ1 = ωπ/2

If the polarizing plate inserts a different phase in the x field component, then you’ll get elliptically polarized light.

How can you tell what polarization light has, or if it’s polarized at all? Unless you have a sensory organ that detects the direction of the electric or magnetic fields around you, the only way change you can see after polarizing a light source is a reduction (usually) in the intensity of the light, which is proportional to the square of the electric field amplitude. For example, the intensity of ordinary sunlight is approximately halved when you put on Polaroid sunglasses.

What happens when you pass light through two linear polarizers? Three linear polarizers?