A lens with the bulk taken out
A classical lens thick enough to collimate light from a lamp 0.5 m away would be a ton of glass. Augustin Fresnel's idea (1822) is to keep only the curved surface and drop the glass in between: the lens is cut into concentric rings, each with the same surface slope it had in the thick lens, but stepped back so the total thickness stays a few centimetres. For a ring seen from the lamp at angle φ the ray enters the flat inner face and bends to sin φ = n sin φ′; the outer facet has to turn it parallel to the axis, which gives its tilt tan α = n sin φ′ / (n cos φ′ − 1) with n = 1.51. Beyond about 40° the refractive power runs out, and a ring would lose light at grazing incidence, so the outer part of the lens uses catadioptric prisms: the ray enters a face square to it, is reflected by total internal reflection from the back face (the incidence angle 90° − φ/2 always exceeds the critical angle asin(1/n) = 41.5°) and leaves horizontally. The model exports the exact prism angles from Blender, and the ray paths in the lens are traced from them with the vector law of refraction, with no formula shortcut.
Beam divergence
The lamp is not a point. A source of width w placed at the focus makes a beam of full width about θ ≈ w / f; the bullseye rings give a tight core, the prisms (whose facets are flat and which see the lamp from steeper angles) add vertical spread of about ±1.4° and a wider horizontal pedestal, so the intensity is the sum of boxes, each with its own width. The flux stays constant: when a lens spreads the image of the lamp, the peak intensity falls by the same factor. Peak intensity of an ideal panel is I = τ L A: transmittance τ ≈ 0.8, lamp luminance L and projected lens area A. A single 3rd-order panel has about 0.8 m² of aperture, so a 1000 W halogen lamp (L ≈ 10⁷ cd/m²) gives millions of candela. The facets here are modelled as curved sets of flat strips; real lenses are ground with continuous curves, so real beams are somewhat cleaner.
How far can the light be seen?
Two independent limits. Luminous range: Allard's law E = I T^D / D² with the eye's threshold Et (2×10⁻⁷ lux on a dark night, much higher in twilight and daylight). T is the transmissivity per kilometre, from the meteorological visibility V by T^V = 0.05; the "nominal" T = 0.74 per nautical mile is V = 18.4 km. Solving gives D = (2/a) W((a/2)√(I/Et)) with a = −ln T per km and W the Lambert function, which the page uses to check its numerical root finder. A rotating beam passes a ship so briefly that the eye adds up only part of it: the Blondel–Rey law Ieff = I t / (0.2 s + t) gives the intensity of an equivalent steady light. Geographic range: for a light h₁ and an eye h₂ above the sea, the tangent to a sphere of radius R gives d = √(2Rh₁) + √(2Rh₂) ≈ 3.57(√h₁ + √h₂) km (h in metres). The atmosphere bends light slightly downward, which is modelled as a larger effective radius R/(1−k) with k ≈ 0.13, giving 3.86 km per √m. The published nominal range is the smaller of the two.
Light characters and the rotating carriage
Panels turn with the carriage, so the flash period is the rotation period divided by the number of groups: P = Trot / G, and a group of k adjacent panels gives k quick flashes (Fl(k)). The sweep of the beam across the sea is ω D: at 20 km and one revolution in 20 s the bright spot moves at 6 km/s, which carries no signal and is only a geometric shadow. A real 3rd-order lens assembly weighs about two tonnes. It floats in an annular bath of mercury (density 13,534 kg/m³): the buoyancy supports the weight, and the only friction is the viscous shear of a mercury film a few millimetres thick, τ = η ω r² (2π r d) / g on both walls plus the bottom. Mercury is now rarely used because of its toxicity; the table of bearing options below shows what it replaced (rolling friction τ = μr M g r). Quantities such as the wall gap, rolling-friction coefficients and masses are round model values of the right order, not specifications of a particular lens.
Educational model. Lens facets are idealised, the atmosphere is a single exponential layer, mirages and waves are not modelled, and the lighthouse and ship are original models, not copies of a real station or vessel.