Editing Polarization (waves) (section)

=== Jones vector ===
{{Main|Jones vector}}
Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional [[complex number|complex]] vector (the [[Jones calculus|Jones vector]]):

<math display="block"> \mathbf{e} = \begin{bmatrix}
  a_1 e^{i\theta_1} \\
  a_2 e^{i\theta_2}
\end{bmatrix}.</math>

Here {{math|''a''{{sub|1}}}} and {{math|''a''{{sub|2}}}} denote the amplitude of the wave in the two components of the electric field vector, while {{math|''θ''{{sub|1}}}} and {{math|''θ''{{sub|2}}}} represent the phases. The product of a Jones vector with a complex number of unit [[Absolute value|modulus]] gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of [[absolute phase]]. The [[Basis (linear algebra)|basis]] vectors used to represent the Jones vector need not represent linear polarization states (i.e. be [[real numbers|real]]). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero [[inner product]]. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization.