A light wave (or, any other electromagnetic wave) consists of two components, the electric and the magnetic, which are mutually perpendicular. One of these, the electric component, is known as the plane of oscillation for that wave.

Light emitted by an atom in a single event has a fixed plane of oscillation. However, light waves emitted by other such atoms micht not have the same plane. Hence. light emitted by the substance as a whole, consists of different light waves, each with its different plane of oscillation, i.e. a random change in the direction of the light keeps occuring. Such light is known as unpolarized, and is emitted by sources like the Sun, bulbs, candles etc.

Let us consider light travelling in a direction perpendicular to the plane of the screen. Two axes, Y and Z are selected on the screen, and all the electric fields are resolved on these two. Let us consider one light wave whose components are Ey and Ez. The fact that the resultant electric field changes its direction randomly with time, can be mathematically phrased by saying that Ey and Ez have a phase difference of δ, which changes randomly. Hence,
     Ey = E₁ sin(ωt - kx + δ)
     Ez = E₂ sin(ωt - kx)
The angle which this resultant electric field makes with the Y axis can be expressed as,
     tan θ = Ez / Ey = (E₁ sin(ωt - kx + δ)) / (E₂ sin(ωt - kx))
Since there is a random chage in δ with respect to time, so is there a random change in θ. Hence, the light is unpolarized.

If δ = 0,
     tan θ = E₂ / E₁ = constant.
Therefore, θ is a constant and the light is linearly polarised.

If δ = π,
     tan θ = -E₂ / E₁ = constant.
Therefore, θ is a constant and the light is again linearly polarized.

If δ = π/2,
     tan θ = ((E₂ sin(ωt - kx) / (E₁ sin(ωt - kx + π/2)))
Now, if E₂ = E₁,
     tan θ = tan(ωt - kx)
Therefore,
     θ = ωt - kx
This means that the angle θ, increases uniformly with time. The resultant electric field therefore is
     E² = Ey² + Ez² = E₁² cos² (ωt - kx) + E₂² sin² (ωt - kx)
          = E₁²
Therefore, the magnitude of the resultant electric field remains constant, but as θ increases uniformly with time, the tip of the electric field goes in a circle with uniform angular speed, ω. This light will then be called circularly polarized light.

If δ = π/2, and E₂ is not equal to E₁, the magnitude of the electric field will change periodically. Hence, its tip will trace out an ellipse, due to which the light will then be called elliptically polarized light.

Several methods can be employed to polarize light in the required direction. The instrument which is made use of for this purpose is known as a polarizer. Polarizing sheets, commercially known as polaroids, consist of long-chained molecules, such as hydrocarbons, which are aligned in one direction, and embedded in plastic sheets. When light is sent through these sheets, only the electric components which are parallel to one particular direction are transmitted, whereas the rest are absorbed by the polarizer. That direction is known as the polarizing direction or the transmission axis of the polarizer. Thus the emergent light is linearly polarized, whose direction can be controlled by changing the orientation of the polarizer.

Intensity of transmitted light

It can be shown that the intensity of the electric field of transmitted light is half of that of the incident light.
     I = (1/2)I₀
Now, if linearly polarised light is made to pass through another polarizing sheet, known as the analyzer (refer to diagram), whose transmission axis might not be parallel to plane of oscillation of the polarized light, θ being the angle between the E vector of the light and transmission axis of the analyzer, the transmitted component will be
     Ey = E cos θ
But since the intensity is directly proportional to the square of amplitude,
     I = I₀ cos² θ (law of Malus)
Hence, if unpolarized light is made to pass through two polarizing sheets, one after the other, the intensity of the transmitted light depends on the difference between their polarizing directions. If the polarizing axes are parallel, the analyser does not bring about any change in the polarised light. In other cases, the analyzer, further reduces the intensity, the maximum reduction being when θ = 90°.

Polarization can also be brought about by reflection, refraction and scattering.

Since the laws of reflection say that the incident ray, reflected ray and the normal at the point of incidence, all lie in the same plane, the electric components which are parallel to the plane of reflection will be most strongly reflected as compared to the other planes. Hence, the reflected light will have electric compenonts concentrated in one direction, and hence will be linearly polarized.

If the angle of incidence of a light ray incident on a surface capable of both refraction and reflection, i, is such that
     tan i = μ
the reflected ray will be completely polarized whereas the refracted ray will never be completely polarized. This is known as Brewster's law and the corresponding angle of incidence is called Brewster's angle. When unpolarized light is scattered by a dust particle, the scattered light is partially polarised. A particular phenomenon, birefringence, occurs when light enters a medium like iceland spar. Waves moving in one plane are bent by different amounts than those at right angles to them, thus producing two images.

Polarization is widely used today, in many fields like the development of polarizing sunglasses, transmission of radio signals and liquid crystal display panels (in calculators, laptops etc.). Many substances that are normally colourless become brightly coloured when viewed under polarised light. Human eyes cannnot distingiush between polarized and unpolarized light, but insects such as bees can, besides distinguishing between them, also determine the angle of polarization.

Optical Activity

If a polarized wavetrain, whose plane of vibration is parallel to the transmission axis of the quartz plane it is incident upon, is rotated through a certain angle while passing through the quartz, the quartz is said to be optically active. A right-handed optically active quartz plane would rotate an incident polarized wavetrain by 90° towards the right (looking in the direction of the source), whereas a left-handed plane would rotate it by 90° towards the left. Both these planes however, have a similar arrrangement of molecules. Certain glucose solutions can also be used as optically active planes.

One area in which optical activity is made use of is in liquid crystal display panels, which consist of a liquid crystal placed between two polarising sheets whose transmission axes are perpendicular to wach other. When the display is off, the plane of vibration of polarized light coming from the first polarizer is rotated by the liquid crystal so as to allow it to pass through the other polarizer as well. Hence light reaches the screen, and is reflected back from it. However, when the display is switched on, electric field reduces the optical activity of certain areas of the liquid crystal due to which a dark area is formed on the screen.

Interference of polarized light This occurs when a crushed cellophane paper, placed between two polarizers shows changes in colour when the angle between transmission axes of the two polarizers is changed by rotating the analyzer, due to interference of polarized light.