loading...
running...
How to use
Pick a bench with the presets: Polarizer chain (laser, polarizer, half-wave and quarter-wave plates, analyzer, power meter), Brewster glass (a slab on a rotary stage), Prism spectrum (white lamp, slit, prism and screen) and Dipole antenna (radiation slice). The sliders rotate the real optical parts; the beam ribbon shows the E field (red) and B field (blue) with the polarization state that leaves each element. Try: cross two polarizers and slip a third at 45 degrees between them, make circular light with the quarter-wave plate, find the glass angle where the reflected beam disappears for p-polarized light, and raise the dipole frequency to see the radiated power grow as f4.
The optical table, breadboard, laser, rotary polarizers and wave plates, glass slab and prism on graduated stages, lamp, slit, screen, power meter and the dipole antenna were modelled and baked (PBR textures) in Blender 5.2 as named parts; Three.js r160 moves and rotates them, draws the E and B fields as arrow combs along the exact Jones-calculus state of each beam piece, ray-traces the prism with Sellmeier indices and evaluates the exact dipole field in a fragment shader.
The science
This bench builds the main facts of optical electromagnetic waves from the Jones calculus, the Fresnel equations and the exact field of an oscillating dipole.
Polarization chain. The laser delivers a Jones vector E = (Ex, Ey). A linear polarizer at angle θ is the projector P(θ) = [[cos2θ, sinθ cosθ], [sinθ cosθ, sin2θ]]; a retarder with fast axis θ and retardance δ is R(−θ) diag(e−iδ/2, e+iδ/2) R(θ), with δ = 90 degrees at the 550 nm design wavelength for the quarter-wave plate and 180 degrees for the half-wave plate (retardance scales as 1/λ here; real plates also have dispersion). The power meter reads |E|2. For linearly polarized light the analyzer gives Malus's law I = I0 cos2θ; a quarter-wave plate at 45 degrees to linear light makes circular light (Stokes S3 = ±1), which no analyzer can then extinguish.
Glass at an angle. At an interface the reflected amplitudes are rs = (n1cosθi − n2cosθt)/(n1cosθi + n2cosθt) and rp = (n2cosθi − n1cosθt)/(n2cosθi + n1cosθt) with Snell's law n1sinθi = n2sinθt. Rp vanishes at Brewster's angle tanθB = n2/n1 (56.3 degrees for n = 1.5): the reflected light is then purely s-polarized, which is why polarizing sunglasses remove glare from water and roads. The index of the glass comes from the three-term Sellmeier fit n2 = 1 + Σ Biλ2/(λ2 − Ci) (BK7, SF11 and fused silica coefficients from the published catalogues).
Prism. A ray entering a prism of apex angle A at incidence θi is bent by D = θi + θe − A. Because n depends on wavelength, blue is bent more than red. The deviation is smallest when the ray crosses symmetrically, where n = sin((A + Dmin)/2)/sin(A/2); the page searches the minimum numerically and compares it with that formula. The coloured fan is computed ray by ray with the Sellmeier index of the selected glass.
Dipole antenna. A small oscillating dipole p(t) = p0 cos ωt along z radiates the exact field E = (1/4πε0){k2(n×p)×n eikr/r + [3n(n·p) − p](1/r3 − ik/r2) eikr}, H = (ck2/4π)(n×p)(eikr/r)(1 − 1/(ikr)) (Jackson, eq. 9.18). The coloured slice evaluates this formula for every pixel in the fragment shader, so the near field (1/r3, field lines that close on the dipole), the transition region and the radiation field (1/r, donut pattern sin2θ) all appear in one picture. The radiated power is P = Z0c2k4p02/(12π), which grows as the fourth power of the frequency (why the sky is blue). Antenna length and the ground plane are drawn but not modelled; the field is that of an ideal point dipole in free space.
Numerical checks run in the page. (1) Malus's law through the Jones chain; (2) a quarter-wave plate at 45 degrees gives |S3/S0| = 1; (3) Brewster's angle found by bisection on Rp against arctan n; (4) R + T = 1 for both polarizations over a sweep of angles; (5) the minimum-deviation angle found by search against the symmetric-passage formula; (6) the time-averaged Poynting flux through spheres of radius 0.3 λ and 7 λ against the closed formula for the radiated power; (7) the Sellmeier index of BK7 at the sodium D line, 589.3 nm, against the catalogue value 1.5168.
Simplifications. The beam is drawn as a ribbon whose wavelength is exaggerated (hundreds of nanometres cannot be seen on a bench); intensities are relative; polarizers are ideal projectors; wave plates are zero-order and ignore dispersion of birefringence; glass is lossless and the faces are uncoated; the dipole is a point source in vacuum and the antenna model is purely decorative. The prism spectrum fan is a geometric-optics calculation without diffraction.