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Chapter 7:
Poincaré Sphere Representation of Polarized Light

7.1 Introduction

The Poincaré sphere representation was conceived by the French physicist Henri Poincaré in 1892. It is a simple and extremely useful geometrical representation of various polarization states and their evolution through a birefringent medium. We will see that this representation makes the formulation as well as the explanation of a difficult problem very easy due to the graphical depiction of the actions of the polarizers and birefringent media involved. According to this representation, one represents various polarization states, polarizers, and birefringent media by specific points on the surface of a sphere (unit radius) in terms of the longitude and latitude, as discussed in the following section.

7.2 Various Polarization States

The most general polarization state is an elliptically polarized state, which is described by the following three parameters:

(i) The orientation θ of the major axis of the polarization ellipse, which is measured with respect to some fixed direction. Let us select this direction as the horizontal direction (x axis) in the (x-y) plane, transverse to the direction of propagation, taken as the z axis (see Fig. 7.1).

(ii) The ellipticity of the ellipse representing the SOP, which is measured in terms of an angle ε = tan-1(b/a), where b and a represent the semi-minor and semi-major axess, respectively. Obviously, a and b are both positive quantities, and ab; thus 0 ≤ ε ≤ π/4.

(iii) The sense of rotation of the electric field with time is specified by assigning a positive or negative sign to ε; the positive sign is for right rotations, and the negative sign for left rotations. Here, the right/left rotations are the clockwise/counterclockwise rotations, as seen by an observer looking into the source of the light.

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