Jones calculus

The Jones formalism describes linear optical images taking into account the polarization. The light is represented as a plane electromagnetic wave having a complex two-dimensional Jones vector, the amplitude of the wave. The images are represented by Jones matrices. The formalism was named after R. Clark Jones, of this representation introduced in 1941. The Jones formalism is particularly suitable for analysis of optical systems in which a light beam is passed through a cascade of optical components.

Mathematical Description

In complex notation, the elongation of a monochromatic plane wave in a Cartesian coordinate system, the spatial and temporal dependence

The circular wave number and the angular frequency, respectively, and is chosen as the propagation direction of the axis. The Jones vector of this wave is then

That is, the explicit spatial and time dependence of the amplitude is suppressed in the description of the shaft. Can be the effect of an optical component to the optical fiber to describe the effect of a complex- 2 × 2 matrix of the Jones vector, if the element does not have any non-linear properties,

Going through the light beam, a system of optical elements with Jones matrices, so can the overall effect of the optical system by a Jones matrix

Describe (if multiple reflections between the individual components do not play a role ). The intrinsic polarization of an optical system corresponding to the eigenvectors of its Jones matrix. The Jones vector is only suitable for the description of completely polarized light, and accordingly can only optical components that have no depolarizing properties, are characterized by Jones matrices. Are depolarization effects are important, must be resorted to elaborate Stokes formalism.

Jones matrices can describe, for example, linear polarization or circular polarizations (rotation of the polarization plane ) and retardation plates. In the plate section, for example a direction of polarization is delayed with respect to the perpendicular to a quarter wavelength. With circular polarization and deceleration, the amount of total amplitude does not change, and the unitary matrices, it is used ( complex conjugate, and T represents the transpose of the matrix) and. For linear polarization, the amount of the total amplitude may change, the associated matrices are not unitary.

In H -position

In V- position

In 45 ° position

In -45 ° position

According to the usual manner of speaking in the optics, denote "H" and horizontal and "V" such as the vertical orientation in the x and y direction. If it does not depend on the interference with other beams, a common ( complex ) phase prefactor can be excluded, and the matrices are frequently indicated that the first diagonal point is real.

Rotated Component

An optical component with respect to its optical axis by the angle θ rotated so is the Jones matrix for the rotated component, M ( θ ). Said matrix obtained from the matrix M to the non-twisted part by the following transformation:

Transition to quantum mechanics

One can understand the pure x - and y- polarization as pure orthonormal basis and display them in Bra- Ket notation, as indicated in the table above. A polarization filter can be conceived, for example, then the quantum mechanical operator, an own state of the system (pure x -or y- polarization) is projected ( collapse of the wave function). The corresponding projector would be for an x - polarization filter: The eigenvalue is then the proportion of the incident light, which has the corresponding polarization. The observable is the polarization in the x- direction. Analogously, the stated above construct filter for circularly polarized light.

In the Bra- Ket - representation can also be a base change easily perform. The change of basis matrix that has transferred from the x / y basis in the representation by superposition of oppositely circularly polarized waves following form.

Such considerations give a clear reference to the otherwise rather abstract formalism of quantum mechanics.