Left of the page is the lens, right is the sea: a lamp at the focus of a ring lens sends a near-parallel beam out; whether a ship sees it depends on the lamp's brightness, the air, and the curve of the Earth.
The science
Fresnel lens. Concentric prism rings give each zone the slope of a thick lens, so a lamp at the focus f leaves in nearly parallel light. Rays follow the vector Snell law n1 sin t1 = n2 sin t2 through glass n = 1.51 (total internal reflection is checked).
Beam. The horizontal beam width is the lamp size over the focal length, theta = w / L, widened by prism aberration. Peak intensity I = tau L_lamp A (luminance times projected lens area).
Flash. One panel sweeps the observer in t = theta/omega; the Blondel-Rey law gives the effective intensity I_e = I t/(0.2 + t).
Allard. E = I T^D / D^2; the luminous range is where E equals the eye threshold. The closed form uses the Lambert W function.
Horizon. Over a sphere of radius R/(1-k) the geographic range is 3.57 (sqrt h1 + sqrt h2)/sqrt(1-k) km (heights in m). The nominal range is the smaller of the two.
Honest limits. Meridian section only (the 3D page traces the full prism geometry); one panel and a uniform atmosphere; the eye threshold is a fit to sun elevation; standard refraction k is a constant. Not a navigation tool; always use official charts.