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Jones Calculus

The Jones matrix calculus is a matrix formulation of polarized light that consists of 2 × 1 Jones vectors to describe the field components and 2 × 2 Jones matrices to describe polarizing components. While a 2 × 2 formulation is "simpler" than the Mueller matrix formulation the Jones formulation is limited to treating only completely polarized light; it cannot describe unpolarized or partially polarized light. The Jones formulation is used when treating interference phenomena or in problems where field amplitudes must be superposed. A polarized beam propagating through a polarizing element is shown below.

polarizing_element

The 2 × 1 Jones column matrix or vector for the field is

equation_1

where E0x and E0y are the amplitudes, δx and δy are the phases, and i = √-1. The components Ex and Ey are complex quantities. An important operation in the Jones calculus is to determine the intensity I:

equation_2

The row matrix is the complex transpose † of the column matrix, so I can be written formally as

equation_3

It is customary to normalize I to 1.

The Jones vectors for the degenerate polarization states are:

equation_4

The Jones vectors are orthonormal and satisfy the relation Ei† · Ej = δij, where δij(I = j ,1, Ij,0) is the Kronecker delta.

The superposition of two orthogonal Jones vectors leads to another Jones vector. For example,

equation_5

which, aside from the normalizing factor of 1/√2 , is L+45P light. Similarly, the superposition of RCP and LCP yields

equation_6

which, again, aside from the normalizing factor is seen to be LHP light. Finally, in its most general form, LHP and LVP light are

equation_7

Superposing ELHP and ELVP yields

equation_8

This shows that two orthogonal oscillations of arbitrary amplitude and phase can yield elliptically polarized light.

A polarizing element is represented by a 2 × 2 Jones matrix

equation_9

It is related to the 2 × 1 output and input Jones vectors by E' = J · E. For a linear polarizer the Jones matrix is

equation_10

For an ideal linear horizontal and linear vertical polarizer the Jones matrices take the form, respectively,

equation_11

The Jones matrices for a wave plate (E0x = E0y = 1) with a phase shift of φ/2 along the x-axis (fast) and φ/2 along the y-axis (slow) are (i = √-1 )

equation_13

The Jones matrices for a QWP φ = π/2 and HWP φ = π are, respectively,

equation_14

For an incident beam that is L-45P the output beam from a QWP aside from a normalizing factor is

equation_15

which is the Jones vector for RCP light. Finally, the Jones matrix for a rotator is

equation_16

For a rotated polarizing element the Jones matrix is given by

equation_17

The Jones matrix for a rotated ideal LHP is

equation_18

Similarly, the Jones matrix for a rotated wave plate is

equation_19

For a HWP φ = π the matrix reduces to

equation_20

The matrix is almost identical to the matrix for a rotator except that the presence of the negative sign with cosθ rather than with sinθ along with the factor of 2 shows that the matrix is a pseudo-rotator; a rotating HWP reverses the polarization ellipse and doubles the rotation angle.

An application of the Jones matrix calculus is to determine the intensity of an output beam when a rotating polarizer is placed between two crossed polarizers.

rotating_polarizer

The Jones vector for the output beam is E' = J · E and the Jones matrix for the three polarizer configuration is

equation_21

For input LHP light the intensity of the output beam is I = E'† · E' = [1 - cos(4θ)]/8.

intensity_output_beam

An important optical device is an optical isolator.

optical_isolator

The Jones matrix equation and its expansion is

equation_22

Thus, no light is returned to the optical source and the circular polarizer acts as an ideal optical isolator.

Citation:

E. Collett, Field Guide to Polarization, SPIE Press, Bellingham, WA (2005).



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