Similarly, the Jones matrix for a rotated wave plate is The Jones matrix for a rotated ideal LHP is Finally, the Jones matrix for a rotator isįor a rotated polarizing element the Jones matrix is given by The Jones matrices for a QWP φ = π/2 and HWP φ = π are, respectively,įor an incident beam that is L-45P the output beam from a QWP aside from a normalizing factor is The Jones matrices for a wave plate ( E 0 x = E 0 y = 1) with a phase shift of φ/2 along the x-axis (fast) and φ/2 along the y-axis (slow) are ( i = √-1 ) For a linear polarizer the Jones matrix isįor an ideal linear horizontal and linear vertical polarizer the Jones matrices take the form, respectively, It is related to the 2 × 1 output and input Jones vectors by E' = J This shows that two orthogonal oscillations of arbitrary amplitude and phase can yield elliptically polarized light.Ī polarizing element is represented by a 2 × 2 Jones matrix Finally, in its most general form, LHP and LVP light are Which, again, aside from the normalizing factor is seen to be LHP light. Similarly, the superposition of RCP and LCP yields Which, aside from the normalizing factor of 1/√2, is L+45P light. The superposition of two orthogonal Jones vectors leads to another Jones vector. E j = δ ij, where δ ij( I = j ,1, I ≠ j,0) is the Kronecker delta.The Jones vectors are orthonormal and satisfy the relation E i† The Jones vectors for the degenerate polarization states are: The row matrix is the complex transpose † of the column matrix, so I can be written formally as An important operation in the Jones calculus is to determine the intensity I: The components E x and E y are complex quantities. Where E 0 x and E 0 y are the amplitudes, δ x and δ y are the phases, and i = √-1. The 2 × 1 Jones column matrix or vector for the field is A polarized beam propagating through a polarizing element is shown below. ![]() ![]() The Jones formulation is used when treating interference phenomena or in problems where field amplitudes must be superposed. 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 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.
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