This section will introduce some of the basic concepts that are important in understanding the optical behavior of liquid crystals. This is by no means a complete discussion of the topic; it is only intended to be used in the context of liquid crystal optical behavior. Please refer to Jenkins and White for a detailed treatment.
Light can be represented as a transverse electromagnetic wave made up of mutually perpendicular, fluctuating electric and magnetic fields. The left side of the following diagram shows the electric field in the xy plane, the magnetic field in the xz plane and the propagation of the wave in the x direction. The right half shows a line tracing out the electric field vector as it propagates. Traditionally, only the electric field vector is dealt with because the magnetic field component is essentially the same.
This sinusoidally varying electric field can be thought of as a length of rope held by two children at opposite ends. The children begin to displace the ends in such a way that the rope moves in a plane, either up and down, left and right, or at any angle in between.
Ordinary white light is made up of waves that fluctuate at all possible angles. Light is considered to be "linearly polarized" when it contains waves that only fluctuate in one specific plane. It is as if the rope is strung through a picket fence -- the wave can move up and down, but motion is blocked in any other direction. A polarizer is a material that allows only light with a specific angle of vibration to pass through. The direction of fluctuation passed by the polarizer is called the "easy" axis.
If two polarizers are set up in series so that their optical axes are parallel, light passes through both. However, if the axes are set up 90 degrees apart (crossed), the polarized light from the first is extinguished by the second. As the angle rotates from 0 to 90 degrees, the amount of light that is transmitted decreases. This effect is demonstrated in the following diagram. The polarizers are parallel at the top and crossed at the bottom.
Linear polarization is merely a special case of circularly polarized light. Consider two light waves, one polarized in the YZ plane and the other in the XY plane. If the waves reach their maximum and minimum points at the same time (they are in phase), their vector sum leads to one wave, linearly polarized at 45 degrees. This is shown in the following diagram.
Similarly, if the two waves are 180 degrees out of phase, the resultant is linearly polarized at 45 degrees in the opposite sense.
If the two waves are 90 degrees out of phase (one is at an extremum and the other is at zero), the resulting wave is circularly polarized. In effect, the resultant electric field vector from the sum of the components rotates around the origin as the wave propagates. The following diagram shows the sum of the electric field vectors for two such waves.
The most general case is when the phase difference is at an arbitrary angle (not necessarily 90 or 180 degrees.) This is called elliptical polarization because the electric field vector traces out an ellipse (instead of a line or circle as before.)
These concepts can be rather abstract the first time they are presented. The following simulation allows the user to change the phase shift to an arbitrary value to observe the resultant polarization state.