Polarization Optics & Jones Vector Back
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Polarization Optics

Polarization Optics & Jones Vector Calculator

Set input polarization and cascade up to 4 optical elements (polarizer, QWP, HWP, retarder). Output Jones vector, Stokes parameters, and Poincaré sphere updated in real time.

Input Polarization
Polarization angle θ [°]
Ellipticity χ [°]
-45°=L circular / 0°=linear / +45°=R circular
Phase δ [°]
Optical Elements (max 4)
Stokes Parameters
|Ex| output amplitude
|Ey| output amplitude
Degree of polarization
Ellipticity χ [°]
Azimuth ψ [°]

Theory

Jones vector from polarization parameters:

$$\mathbf{J} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} \cos\chi\cos\theta - i\sin\chi\sin\theta \\ \cos\chi\sin\theta + i\sin\chi\cos\theta \end{pmatrix}$$

Jones matrix (QWP, fast axis at angle α):

$$M_{QWP}(\alpha) = R(-\alpha)\begin{pmatrix}1 & 0 \\ 0 & e^{i\pi/2}\end{pmatrix}R(\alpha)$$

Stokes parameters:

$$S_0 = |E_x|^2+|E_y|^2,\quad S_1 = |E_x|^2-|E_y|^2$$ $$S_2 = 2\operatorname{Re}(E_xE_y^*),\quad S_3 = -2\operatorname{Im}(E_xE_y^*)$$

Degree of polarization: $\mathrm{DOP} = \sqrt{S_1^2+S_2^2+S_3^2}/S_0$

Applications: LCD liquid-crystal cell design / Fiber-optic polarization mode dispersion / LiDAR polarimetric scattering / Photoelastic stress measurement / Quantum-key-distribution qubit operations / Solar cell anti-reflection coating.