How this instrument works
A lens bends light because it has two curved surfaces, and the lensmaker's equation is simply the sum of their two contributions. At each surface, light bends by an amount set by the jump in refractive index — from air, roughly n = 1, into the glass, or back out again — and by how sharply that particular surface curves, R1 or R2. The (n − 1) factor is the excess bending the glass supplies beyond doing nothing: for a flat pane, both radii are infinite, 1/R1 and 1/R2 vanish, and 1/f goes to zero, meaning f is infinite and parallel light stays parallel, exactly as a window behaves.
The equation comes from applying the single-refracting-surface formula twice — once where light enters the element, once where it leaves — then letting the element's thickness shrink toward zero, the thin-lens limit. That is why only two curvatures and one index appear: the front face contributes a bending power of (n − 1)/R1, the back face contributes (n − 1)/(−R2), and the two add. Each radius carries a sign under the convention this instrument uses: positive when that surface's center of curvature lies on the side the light is heading toward, negative when the center sits back on the side the light came from.
The thin-lens assumption is also the equation's boundary. Real elements have thickness, and once that thickness stops being negligible next to R1, R2, or f, the plain formula drifts from what a bench measurement shows, and a fuller thick-lens version — with an added term carrying the element's center thickness and index — takes over. The formula is also monochromatic: n itself depends on wavelength, so a piece ground to land exactly on 50 mm at yellow light focuses red and blue slightly differently, the chromatic aberration that camera and eyepiece designers correct with multi-element stacks rather than a single curve.
- Enter the Refractive index of the lens material, n — about 1.5 for ordinary optical crown glass, higher for flint glass or many plastics.
- Set Radius of curvature, surface 1 (R1) for the face the light meets first; positive if its center of curvature lies downstream, in the direction light travels.
- Set Radius of curvature, surface 2 (R2) for the second face; for a symmetric biconvex shape this matches R1 in size but flips sign.
- Read the Focal length result, f, and switch its unit between millimetres, centimetres, and metres as the job needs.
Worked example — a 50 mm biconvex lens in n = 1.5 glass
Take a symmetric biconvex lens ground from ordinary crown glass, n = 1.5, with both faces curved to a 50 mm radius: R1 = 50 mm and R2 = −50 mm, negative because the second surface curves the opposite way under the sign convention. The formula gives 1/f = (1.5 − 1)(1/0.05 − 1/−0.05) = 0.5 × (20 − (−20)) = 0.5 × 40 = 20 per metre, so f = 1/20 = 0.05 m — exactly 50 mm.
This particular geometry — equal curvature on both faces, opposite in sign — is the shape of an ordinary magnifying glass or loupe, chosen because it is the simplest symmetric form to grind and polish to a matching pair of surfaces. An optical shop runs this exact calculation before touching a blank, confirming that a 50 mm/50 mm curve pair in stock crown glass lands the focal length at the specified 50 mm rather than finding out the curves were wrong after the piece is already finished.
Questions
What does a negative radius of curvature mean?
It marks which way a surface bulges relative to the direction light travels. Under the convention this instrument uses, a radius is positive when its center of curvature lies on the side the light is heading toward, and negative when the center sits back on the entry side. In the worked example's symmetric biconvex lens, the first face bulges toward the light (R1 = +50 mm) and the second bulges away from it (R2 = −50 mm) — both faces are convex, but the sign flips because the direction changes.
Why is the bending term (n − 1) and not just n?
Refraction only bends light by however much a material's index differs from whatever the ray was already travelling through — usually air, at n ≈ 1. Glass at n = 1.5 surrounded by air bends light by the excess, 0.5, at each curved face. That is also why a slab of glass ground into two flat faces, infinite radii both, has no focusing power at all: (n − 1) gets multiplied by zero curvature, however large the index is.
Does this equation account for the lens's thickness?
No — it assumes a thin lens, where the gap between the two surfaces is small next to R1, R2, and f. That holds well for most eyeglass and magnifier elements. A genuinely thick lens, such as a short, stubby camera element, needs the expanded thick-lens formula, which adds a term for the center thickness divided by the index; dropping that term for a thick piece introduces a real, measurable error in the focal length.
How much does refractive index change the focal length?
Substantially, for the same curvatures. Keep R1 = 50 mm and R2 = −50 mm but swap ordinary crown glass, n = 1.5, f = 50 mm, for a denser glass at n = 1.6, and the focal length shortens to 41.67 mm. A higher index bends light more sharply per unit of surface curvature, which is why designers reach for denser glass, or a high-index plastic, when they need more optical power without carving a tighter, harder-to-manufacture curve.
What focal length results from a flat first surface?
Flattening the first surface to an effectively infinite radius turns the lens plano-convex — all the bending happens at the one curved face left. With R1 essentially infinite, n = 1.5, and R2 = −50 mm, the focal length doubles to 100 mm compared with the 50 mm symmetric biconvex case, because a single surface is now doing the work two surfaces previously shared.