Lenses

How Lighthouses Work 3 ended on a limit: a lamp six times as bright reached only about 1.4 times as far, so more lamps are a poor way to make a light reach further. The better way is to stop wasting the light a lamp already makes. A flame sends light in every direction and only a thin slice of it travels toward a ship. This chapter is about the glass and metal that bend the rest into that slice: first the reflector, then the lens, then the stepped lens of Augustin-Jean Fresnel that put an end to the reflectors.

Reflectors: Catching the Light That Goes the Wrong Way

A bare flame lights the sky and the sea equally and the sky does not need it. The first practical fix was a mirror. The lighthouse article says the Argand hollow-wick lamp and the parabolic reflector were introduced in the late 18th century, and a steady flame made focusing practical in the first place. It also says Hutchinson's 1777 catoptric system was the first practical optical system, focusing light into a concentrated beam. A parabolic mirror sends the light from a small flame at its focus out as a parallel beam. That is general geometry, not a claim from a source.

But a reflector wastes most of the light. The two Wikipedia articles give figures that do not quite line up:

The 20% and the "about half" are not quite the same quantity (focused into the beam, against reflected at all), so I treat them as a source conflict that I cannot resolve, and I have not found how either was measured. The lighthouse article also says the first Fresnel lens at Cordouan "increased the luminosity of the lighthouse lamp by a factor of four". The 85% and the 20% differ by 4.25 times. What does that do to the range?

The columns are the source, its share of the light, its intensity in candela (the 1,000 cd for the 20% reflector is an assumption), its nominal range in nautical miles at 10 nm visibility, and the range as a multiple of the reflector's. The last line is 0.85 divided by 0.20. The lens at 4,250 cd has 4.25 times the intensity but only 1.32 times the range, which is the inverse square law and the loss in the air, as in How Lighthouses Work 3. Even so, a third more range is worth having, and the lens did it with the same lamp.

From Buffon to Fresnel

The idea of the stepped lens is older than Fresnel. A lens bends light only at its two surfaces; the glass between them does no bending, adds weight and absorbs light. That is general optics, not a quote. A stepped lens keeps the surfaces and removes the glass in the middle. The timeline from the Fresnel lens article:

A source conflict: the article dates Condorcet's proposal to 1790 but says secondary sources give 1773 or 1788, so I do not rely on any of them. The article also says Fresnel died on 14 July 1827, aged 39, having designed the catadioptric prism but not seen a full-sized version. The lighthouse article gives the same Cordouan year (1823) and the same visibility, more than 20 miles (32 km). Buffon, Condorcet, Rogers and Brewster all had the idea; Fresnel is the one who made it work.

Lenses, Prisms and Why They Are Lighter

A lighthouse lens is built from rings. A ring near the middle refracts: the light goes in one side, bends and comes out as a beam. The article calls these dioptric parts: refraction only. For light leaving the flame at a steep angle, the article describes catadioptric prisms: each does two refractions and one total internal reflection, which it says captures more oblique light and avoids the loss from a silvered mirror. The light that a reflector wastes sideways is gathered into the beam.

The size of a lens is given by its order. According to the lighthouse article, a first-order lens is the largest, most powerful and most expensive of the usual sizes, and coastal lighthouses generally use first, second or third order. The Fresnel lens article lists these focal lengths. The numbering differs between its "modern" and "Fresnel" orders, so I use the modern ones.

The article lists more orders than shown, down to an 8th order of 70 to 75 mm focal length; I have kept the ones used below. It gives one weight: a first-order assembly about 2.54 m tall weighing about 1.5 tonnes (3,300 lb). Why does focal length matter? The flame is not a point. A flame of radius r at a focal distance f spreads the beam by a half-angle whose tangent is r divided by f, so a long focal length gives a narrow beam, and a narrow beam is a bright one. Here is that model for one 10 cd burner and a 10 mm flame radius (both assumed), with 85% of the light collected (the source's share for a lens), at the focal lengths of six orders (700 to 750 mm shown as 725, my midpoint):

The columns are the order, the focal length, the full width of the beam in degrees, the peak intensity and the nominal range at 10 nm visibility. The same burner in the hyper-radial lens makes a beam 0.86 degrees wide of about 601,000 cd and in a sixth-order lens a beam 7.63 degrees wide of about 7,700 cd. The light is the same; only the width of the beam changes. A large lens does not add light, it keeps it in a narrow beam. The model is a simplification: it treats the lens as a round lens with the flame at its focus and ignores the faults a real lens has. That is why a 10 cd flame comes out at hundreds of thousands of candela; a real light is not this tidy, and the figures show the trend, not any real lens.

Why a stepped lens is lighter

A solid lens of focal length f and glass index n needs surfaces of radius about (n − 1) × f. The thickness it adds in the middle (the sag) grows as the square of the aperture and gets worse as the focal length shortens. A stepped lens has the same optics in a flat panel of roughly constant thickness. The code compares the glass of a solid plano-convex lens with a flat panel 20 mm thick. The index, density, panel thickness and 5 mm edge are assumptions; the 920 mm focal length is the first-order figure from the table.

The columns are the diameter, the thickness of glass added in the middle, the mass of the solid lens, the mass of the stepped panel and the ratio of the two. At 0.2 m across, the stepped panel is heavier (1.6 kg against 0.8 kg for the solid one) because 20 mm is more than the solid lens needs at that size. Only above a diameter of about 0.33 m (the break-even found in Exercise 3) does stepping win: at 0.8 m the solid lens has a 233 mm bulge and weighs 169 kg against 25 kg, 6.7 times heavier, and the ratio keeps growing with size. The saving is largest exactly where a lighthouse needs it, in a lens a metre or more across. The thick middle of such a lens would also absorb light, the problem the article says Rogers's lenses had. These numbers come from the model: real panels have prisms and frames and weigh more than 20 mm of flat glass.