Tool · Polarization

Jones Calculus Calculator

What comes out of a stack of polarizers and wave plates? Choose the input polarization and up to four elements, and the calculator multiplies their Jones matrices to give the transmitted intensity, the output state, its Stokes parameters and the polarization ellipse. Background: Jones vectors, polarization states, and wave plates.

Input light

Azimuth ψ (°)

Ellipticity angle χ (°), −45 to 45

χ = 0 is linear light and χ = ±45° circular, positive for right-handed (clockwise as seen facing the light); tan χ is the ratio of the minor to the major axis of the ellipse. Angles are measured from the horizontal x axis towards y, as seen facing the oncoming light.

Elements, in the order the light meets them

Element 1

Type

Axis angle, or rotation for a rotator (°)

Retardance δ (°), retarder only

Element 2

Type

Axis angle, or rotation for a rotator (°)

Retardance δ (°), retarder only

Element 3

Type

Axis angle, or rotation for a rotator (°)

Retardance δ (°), retarder only

Element 4

Type

Axis angle, or rotation for a rotator (°)

Retardance δ (°), retarder only

For wave plates and retarders the angle is that of the fast axis. Polarizers are ideal: they pass the component along their axis and block the other completely.

Presets

Readouts

Polarization ellipse, input and output

inputoutput

Rotate one element

element

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The matrix products are compared with results worked out by hand, every retarder matrix is checked to be unitary, and the output of an arbitrary chain is checked to be fully polarized.

Check	Expected	Computed	Tolerance

The expected values follow the Jones matrices in Hecht, Optics, and Goldstein, Polarized Light, evaluated by hand for the stated cases. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

Fully polarized light is described by its Jones vector, the complex amplitudes of the horizontal and vertical field components, with the field taken as $R e {(E_{x}, E_{y}) e^{i (k z - ω t)}}$ . Each element is a 2 × 2 matrix, and a chain of elements acts as the product of their matrices, the first element on the right: $E_{o u t} = M_{4} M_{3} M_{2} M_{1} E_{i n}$ . An element whose axis is at angle $θ$ from horizontal is $R (- θ) M_{0} R (θ)$ , with

R (θ) = (\begin{array}{cc} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{array}), M_{0}^{p o l} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}), M_{0}^{r e t} = (\begin{array}{cc} 1 & 0 \\ 0 & e^{i δ} \end{array})

for a linear polarizer and for a retarder with its fast axis along $θ$ ; a half-wave plate has $δ = π$ and a quarter-wave plate $δ = π / 2$ . A rotator, such as an optically active crystal or a Faraday rotator, turns any linear polarization by $ρ$ toward $+ y$ . Its matrix, $(\begin{array}{cc} \cos ρ & - \sin ρ \\ \sin ρ & \cos ρ \end{array}) = R (- ρ)$ , has no fixed linear axis (its eigenvectors are the two circular states), so it is not of the $R (- θ) M_{0} R (θ)$ form.

The matrices differ in kind. A polarizer’s matrix is singular: its determinant is zero, it projects every input onto its axis, and the component along the other eigenvector, with eigenvalue 0, is lost, which is why a polarizer can block light. Retarders and rotators are unitary, $M^{†} M = I$ , so they change the polarization state but conserve the intensity; the validation table checks this for four retarders at arbitrary angles and retardances. The output is summarized by the Stokes parameters $S_{0} = ∣ E_{x} ∣^{2} + ∣ E_{y} ∣^{2}$ , $S_{1} = ∣ E_{x} ∣^{2} - ∣ E_{y} ∣^{2}$ , $S_{2} = 2 R e (E_{x}^{*} E_{y})$ and $S_{3} = - 2 I m (E_{x}^{*} E_{y})$ , from which the azimuth is $ψ = \frac{1}{2} atan2 (S_{2}, S_{1})$ (the arctangent of $S_{2} / S_{1}$ , with the quadrant taken from the signs of $S_{1}$ and $S_{2}$ ) and the ellipticity angle $χ = \frac{1}{2} \arcsin (S_{3} / S_{0})$ . With this sign, $S_{3} > 0$ is right-handed light, turning clockwise as seen facing the oncoming beam, as in Hecht. The elements are ideal: polarizers have infinite extinction, and reflection losses and the wavelength dependence of real wave plates are left out. Partially polarized light needs the Stokes and Mueller description instead.

Worked example

Horizontal light meets polarizers at 0°, 45° and 90°. The first passes everything, the second passes cos² 45° = 0.5, and the third passes half of that again, so 0.25 of the light gets through, although the first and last polarizers alone, crossed, would pass nothing. A half-wave plate at 22.5° instead turns the horizontal light to 45° with no loss, and a quarter-wave plate at 45° makes it right circular, with χ = 45°.

References: R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488 (1941). E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 8. D. H. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).