Reason for the convention about polarization states

Question

I'd like to know if there is a special reason for limiting convention of polarization state to waves that can be split in just two components of equal frequency.

Answer 1

Galmeida,

I think you are thinking of a Jones vector polarization formalism, which works for plane waves in a homogeneous, linear, anisotropic medium (like air, amorphous glass, or vacuum). The reasons they are defined this way are the following

1. Any electric field can classically be decomposed into a superposition of plane waves via a Fourier transform. So we can think about manipulating plane wave components, which correspond to single frequencies.
2. In homogeneous, linear, anisotropic media, the propagation direction of a plane wave is the $k$-vector, which is perpendicular to the electric and magnetic field directions. If we only consider the electric field, then it's vector will trace some shape in the plane perpendicular to the $k$-vector. Since the shape traced is in a plane, we only need two numbers (say x and y) to the electric field vector and any given time/spatial point. So the reason that we define the single frequency is because we think about plane waves usually, and the reason that the Jones formalism only uses two components is because in a lot of materials that we care about the electric field vector is always perpendicular to the $k$-vector.

This can be generalized (and has been) to a full 3-d vector and spectral (all wavelengths) polarization description. See Emil Wolf and his work on Coherence and Polarization or Goodman's Statistical Optics for more details...