Overview

• The polarization state of light describes which direction the electric field is oscillating and is general drawn as a polarization ellipse
• The Poincaré sphere is a useful tool for visually describing polarization states.
• The Stokes vector is a four-element vector that fully defines the polarization state of a beam of light.

Polarized Light

Light is an electromagnetic wave governed by Maxwell equations and the mathematics that describe it get complicated quickly. But it is possible to break it down into relatively easy-to-understand properties. Essentially, light has three basic properties: brightness, color, and polarization state.

The intensity or brightness is related to the amplitude of the wave



The spectral content or color is related to the wavelength or oscillation frequency of the wave



The polarization state is related to the direction that the wave is oscillating, and there are wide range of possibilities.



Most people can see the brightness and color of light, so these properties tend to be easily understood. But human eyes are (almost) completely insensitive to polarization, making it is much more difficult to get an intuitive understanding of polarization.

Many people’s only experience with polarization is through polarized sunglasses. Most of the light found in nature is unpolarized, however light reflected from horizontal surfaces tends to become horizontally polarized. Therefore, the glare from your car dashboard or from the road or from your favorite fishing spot tends to be horizontally polarized. By wearing sunglasses that block horizontal polarization but allow vertical polarization to pass through, we can affectively remove the glare from these horizontal surfaces.



Since we cannot directly observe polarization, we end up relying on mathematics and other visualization tools to help our understanding. There are some excellent textbooks for people interested in learning the mathematics of the electric-field wave equation and how it relates to Jones vectors for describing polarization at a fundamental level. But it is possible to gain some useful intuition without such detailed mathematics.

We begin by defining the polarization ellipse. Consider a fixed point in space as a wave of light travels in the z-direction and plot the location of the electric field over time. We find that the electric field could trace out a line, a circle, or an arbitrary ellipse. The example shown below illustrates the electric field tracing out an ellipse with about 50% ellipticity.


We define the polarization ellipse to be the trajectory of the electric field in the XY plane as viewed looking into the beam (light is traveling in the +z direction), as shown below:



From this ellipse, we define the orientation, θ, as the angle of the major axis. The ellipticity, ε, will be +1 for right-hand circular polarization, 0 for linear polarization, and -1 for left-hand circular. Positive ellipticity indicates right-handed polarization (CW rotation when looking into the beam) and negative ellipticity indicates left-handed polarization (CCW rotation when looking into the beam).

There are a continuum of possible polarization ellipses, and we can organize some examples in a grid as shown below:



And we find that an extremely convenient map of all polarization ellipses can be projected onto the surface of a sphere. Every possible polarization state is then uniquely defined by a single longitude and latitude coordinate. This is the famous Poincaré sphere.



It is easy to understand what the polarization ellipse would look like for any point on the sphere. In the example below, the green dot represents horizontal linear polarization, and the blue dot represents right handed circular polarization. The red dot is a right-handed (clockwise rotation) elliptical polarization state with ellipticity approximately 50% and an orientation of about 40°.



In the example below, we have color-coded polarization states on the Poincaré sphere by wavelength. The same data series is also shown as a series of ellipse drawings.



Polarization researchers much prefer the Poincaré sphere representation over drawing lots of little circles and lines and ellipses.

While the Poincaré sphere provides an excellent way to graphically display polarization states, ultimately we are still going to need to do some mathematics. We begin by defining a typical XYZ coordinate system to describe points on the Poincaré sphere. We label the axes S1, S2 and S3 as shown below:



The coordinate (S1, S2, S3) defines any polarization state on or inside the Poincaré sphere. We also need to define a fourth value, S0, to be the intensity of the beam, and we set the diameter of the sphere to be S0.

These four values make up the Stokes vector:

S = (S­₀, S₁, S₂, S₃)

This Stokes vector completely describes the polarization state of a beam of light.

Admittedly, the Stokes vector coordinate system is pretty strange. It is not a simple orthogonal coordinate system like we learned in Geometry class. Instead, S0 tells how much light there is, while S1 tells us whether that light tends to more polarized in the 0° orientation (positive S1 values) or the 90° orientation (negative S1 values). Similarly, S2 and S3 tell us whether the light is polarized in the +45° vs. -45° orientations or polarized in the right-hand circular vs. left-hand circular states, respectively.

The basis states for this strange coordinate system are shown below:



And there are a wide range of parameters that can be calculated from the Stokes vector such as the degree of polarization, orientation of the polarization state, and ellipticity of the polarization state. These calculations are shown below:



But the key point is that the Stokes vector contains four numbers that fully describe the polarization state of light. This will be critical in understanding how light interacts with different materials via the Mueller matrix.