Polarization

Fields are written Re{J e^i(kz − ωt)^}, and you look towards the source (the light comes out of the screen at you). Right-handed circular light then turns clockwise; its Jones vector is (1, −i)/√2, its S₃ is positive and it sits at the north pole of the Poincaré sphere. This is the convention of Born & Wolf and Hecht’s Optics; engineers (IEEE) call the same light left-handed. Angles run from horizontal (x) towards vertical (y). A retarder with its fast axis at θ and retardance Γ is R(−θ) diag(e^−iΓ/2, e^iΓ/2) R(θ); a polarizer at θ is [[cos²θ, cos θ sin θ], [cos θ sin θ, sin²θ]].

⛶ Full screen (or F; Esc to leave) gives the current tab’s picture the whole screen; ◫ Panels hides the controls there. Tabs can be linked with #ellipse, #train, #malus, #poincare, #fresnel, #calcite, #waveplate, #colours, #photoelastic and #applications.

Ellipse. Set the orientation ψ and ellipticity angle χ (or pick H, V, D, A, R, L) and watch the field vector trace the ellipse; beside it Ex and Ey against time. You get the Jones vector, the amplitudes and phase difference, the Stokes parameters and the axial ratio tan χ.

Jones train. Light goes in on the left (unpolarised or any state) and passes polarizers, quarter- and half-wave plates, general retarders and rotators. Drag an element’s round handle to turn it (it snaps to 15° steps), drag the element sideways to reorder, tap it to edit it. Under each element: the polarisation after it and the intensity left; at the end the output Jones vector, Stokes parameters and the total Jones matrix. Unpolarised light is handled exactly as the incoherent sum of H and V halves. Try the presets: Malus’s law, the three-polarizer paradox, a quarter-wave plate turning linear into circular light, a half-wave plate rotating it by 2θ, and a circular polarizer that blocks its own reflection.

Malus. I = I₀ cos²θ; the three-polarizer paradox ⅛ sin²2θ (12.5% at 45° from nothing); and N polarizers stepping to 90° passing ½ cos^2N(90°/N), approaching 50%.

Poincaré sphere. The state as a point (S₁, S₂, S₃): longitude 2ψ, latitude 2χ. Drag the point to change the light, drag elsewhere to turn the sphere. Choose a retarder (fast axis and Γ): it rotates the sphere by Γ about the axis at longitude 2θ on the equator; the arc shows the path, Apply it moves the state.

Fresnel equations. For any n₁ → n₂: Rs, Rp and their average against angle, the amplitude coefficients, transmittances, and the phase change on reflection. Brewster’s angle arctan(n₂/n₁) (56.31° for glass, n = 1.5), the critical angle, total internal reflection, the largest s–p phase difference 2 atan((1 − n²)/2n), and the angle for a 45° difference (the Fresnel rhomb).

Calcite. A calcite rhomb on printed text shows two images; turn the crystal and the extraordinary image circles the ordinary one; look through a polarizer and they fade in turn. The walk-off angle tan ρ = (n_o²/n_e² − 1) tan θ / (1 + (n_o²/n_e²) tan²θ) is 6.2° for a natural rhomb (θ = 44.6°).

Waveplates. Thickness d = (m + Γ) λ/|Δn| for quartz, MgF₂, sapphire, calcite or rutile, and a graph of how the retardance of zero- and multi-order plates drifts with wavelength.

Interference colours. A birefringent plate at 45° between crossed (or parallel) polarizers passes sin²(πR/λ) of each wavelength; summing over daylight gives the colour, from first-order grey through the 550 nm "sensitive tint" to the pale higher orders. The Michel-Lévy chart plots those colours against retardation with lines of equal birefringence and the 30 µm thin-section line; pick a mineral or tap the chart.

Photoelasticity. A 50 mm polycarbonate disc squeezed across a diameter, using the exact elastic stress field (Frocht), between crossed polarizers: coloured isochromatic fringes, one per wavelength of retardation C t (σ₁ − σ₂), plus dark isoclinics in a plane polariscope (turn it to sweep them). At the centre σ₁ − σ₂ = 8P/πDt.

Applications. Glare off water and polarised sunglasses (the glare left is Rp/(Rs + Rp), none at Brewster’s angle, a 36.9° sun); a twisted-nematic LCD pixel computed with its Jones matrix; and 3-D cinema glasses, where linear polarisation leaks tan² of your head tilt and circular does not (but leaks a little at colours away from the quarter-wave plate’s design wavelength).

Brewster’s angle for n = 1.5 (56.31°) and 4% reflection at normal incidence; Rp = 0 at Brewster and |r| = 1 in total internal reflection; the Fresnel rhomb’s 45.9° largest phase difference for n = 1.51 (Hecht §8.6); a quarter-wave plate at 45° turning H into right-circular light and a half-wave plate at θ turning linear light by 2θ; Malus’s cos²θ and the three-polarizer ⅛ sin²2θ; S₁² + S₂² + S₃² = S₀² and Jones ↔ Stokes ↔ Poincaré round trips; a retarder as a rotation of the sphere; calcite’s 6.2° walk-off; a zero-order quartz quarter-wave plate at 633 nm (17.4 µm); black at zero retardation and first-order red near 550 nm; the Gooch–Tarry condition; and the disc’s centre stress 8P/πDt.

Related: wave optics lab (interference, diffraction, thin films, ripple tank), telescopes and ray optics (Snell’s law, total internal reflection), spectroscopy, matrices, complex numbers, physics calculators.

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