Polarization

·         Introduction

Polarization generally means “orientation”. It comes from the Greek word polos, for the axis of a spinning globe. Wave polarization occurs for vector fields. As we know that light can be described as electromagnetic wave, like radio waves, that propagate via a sinusoidal oscillation of the vectors that are electric fields and magnetic fields. The direction in which the electric field oscillates as it propagates is known as polarization. Generally we expect fields to have three vector components, e.g. (x,y,z), but light waves only have two non-vanishing components: the two that are perpendicular to the direction of the wave as shown in fig 1.

Fig 1: General Light Wave (Electromagnetic Wave)

 The simplest type of polarization of is plane or linear polarization as illustrated in fig 2. For electromagnetic wave travelling through free space, in which the field of the electromagnetic wave oscillates only in a single fixed plane then this type of polarization is known as plane polarization electromagnetic wave or light wave. The vector along which the light travels must also lie in this plane, but this restriction still allows an infinite number of planes of polarization to be defined, each of which describes a separate linear state of polarization (SOP).

Fig 2: Plane Polarized Light-Wave

Electromagnetic waves are the solutions of Maxwell’s equations in a vacuum:

In order to satisfy all four equations, the waves must have the E and B fields transverse to the propagation direction. Thus, if the wave is traveling along the positive z-axis, the electric field can be parallel to the +x-axis and B-field parallel to +y. Half a cycle later, E and B are parallel to –x and –y. Since the fields oscillate back and forth several hundred trillion times per second, we don’t usually know their sense (i.e. +x vs. –x). Polarization of light therefore only refers to direction (e.g., x), not sense.

If the light propagates in the opposite direction, along –z, then the E and B fields are instead respectively parallel to +y and +x. The direction the light travels is determined by the direction of the vector cross product E×B.

For a plane waves travelling along the z-axis of Cartesian space, with wavelength ƛ and frequency f, and can be described as the following ways:

In this expression, ‘cos’ could be equally well replaced by ‘sin’. What is important is the relative sign of the z and t arguments. If z = ct = (ω/k)t, as time advances, the phase of the wave remains constant. This is a plane wave travelling in the positive z-direction at velocity c. Conversely, E (z,t) = E₀ cos (kz + ωt) describes a wave travelling in the negative z-direction. This could equally well be described by E (z,t) = E₀ cos (- kz - ωt). The only criterion is that the sign of the z and t terms are the same for this backward propagation direction.

Fig 3: A wave polarized along the x-direction can equally well be represented by the coherent sum of amplitudes along the xˆ ' and ŷ ' axes 

Here we will ignore the magnetic field B since its value can be immediately inferred from the functional form for E by applying Maxwell’s equations. Thus, it adds no additional degree of freedom to the range of allowable solutions.

Note:

·         We have seen in an EM wave the electric field (E) is the most important and this defines the polarization of the wave.

·         E and B fields are both travelling waves, each oscillating in perpendicular planes to each other and in phase with each other.

An observer looking along Poynting’s Vector (S) would see E and B varying in time.

Fig 4: E and B of light wave varying with time

The E and B fields are both maximal then both zero simultaneously.

·   In general E can be at any position perpendicular to the direction of travel the wave, then B is perpendicular to both.

·   If the EM wave travels though a medium that only allows E to oscillate in one direction that is a polarizing material (e.g. Polaroid). In (and after) the material the E of all of the light is aligned in one direction

This is known as linearly polarized light.

Recall that a vector can be resolved into perpendicular components in any basis. Hence the E field in the wave can be thought of as the sum of two different E fields that are perpendicular.

Fig 5:  The sum of two different E fields those are perpendicular

This means the EM wave can also be considered as being the sum of two in-phase waves with different polarizations.