Jones Calculus - Jones Vectors

Jones Vectors

The Jones vector describes the polarization of light.

The x and y components of the complex amplitude of the electric field of light travel along z-direction, and, are represented as

\begin{pmatrix} E_x(t) \\ E_y(t)\end{pmatrix} =E_{0} \begin{pmatrix} E_{0x} e^{i(kz- \omega t+\phi_x)} \\ E_{0y} e^{i(kz- \omega t+\phi_y)} \end{pmatrix} =E_{0}e^{i(kz- \omega t)} \begin{pmatrix} E_{0x} e^{i\phi_x} \\ E_{0y} e^{i\phi_y} \end{pmatrix} .

Here is the Jones vector ( is the imaginary unit with ). Thus, the Jones vector represents (relative) amplitude and (relative) phase of electric field in x and y directions.

The sum of the squares of the absolute values of the two components of Jones vectors is proportional to the intensity of light. It is common to normalize it to 1 at the starting point of calculation for simplification. It is also common to constrain the first component of the Jones vectors to be a real number. This discards the phase information needed for calculation of interference with other beams. Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is, which is used by Hecht. In this definition, increase in (or ) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of indicates retardation by (or 90 degree) compared to 1 . Collett uses the opposite definition . The reader should be wary when consulting references on Jones calculus.

The following table gives the 6 common examples of normalized Jones vectors.

Polarization Corresponding Jones vector Typical ket Notation
Linear polarized in the x-direction
Typically called 'Horizontal'
Linear polarized in the y-direction
Typically called 'Vertical'
Linear polarized at 45° from the x-axis
Typically called 'Diagonal' L+45
Linear polarized at −45° from the x-axis
Typically called 'Anti-Diagonal' L-45
Right Hand Circular Polarized
Typically called RCP or RHCP
Left Hand Circular Polarized
Typically called LCP or LHCP

When applied to the Poincaré sphere (also known as the Bloch sphere), the basis kets ( and ) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign = and = . These assignments are arbitrary. Opposing pairs are

  • and
  • and
  • and

The ket is a general vector that points to any place on the surface. Any point not in the table above and not on the circle that passes through is collectively known as elliptical polarization.

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