Formula and Method for the Lens Maker's Equation
The lensmaker's equation relates a lens's focal length f to its shape and material: the refractive index n of the lens material, the radii of curvature R₁ and R₂ of its two surfaces, and its center thickness d. The full (thick-lens) form is 1/f = (n − 1)[1/R₁ − 1/R₂ + (n − 1)d / (nR₁R₂)]. When the lens is thin compared to its radii of curvature (d → 0), the thickness term drops out and the equation reduces to the familiar thin-lens form 1/f = (n − 1)(1/R₁ − 1/R₂). This calculator computes both the full-equation focal length and the thin-lens approximation, plus the corresponding optical power in diopters.
How the calculation works
Enter the refractive index of the lens material, the radius of curvature of each surface, the center thickness, and the length unit those radii and thickness are measured in. The calculator converts all lengths to meters, evaluates the bracketed term [1/R₁ − 1/R₂ + (n − 1)d/(nR₁R₂)], multiplies it by (n − 1) to get 1/f, then inverts that to find the focal length f. Optical power is simply P = 1/f expressed in diopters (with f in meters). For comparison, the thin-lens focal length is calculated the same way but with the thickness term left out.
Sign convention for radii of curvature
This calculator uses the standard optics convention: light travels left to right through the lens. A radius of curvature is positive if the surface's center of curvature lies to the right of (after) that surface, and negative if it lies to the left (before) it. For a common biconvex lens, the front surface bulges toward the incoming light (R₁ > 0) and the back surface bulges away from it (R₂ < 0) — hence the calculator's defaults of R₁ = 100 mm and R₂ = −100 mm. A flat (planar) surface has an infinite radius of curvature and should be modeled as a very large number, not zero — entering R₁ or R₂ = 0 makes the equation undefined.
Common mistakes and practical notes
- Mixing up the surface order: R₁ is always the surface light hits first; swapping R₁ and R₂ flips their signs and changes the computed focal length.
- Entering R = 0 for a flat surface: a flat surface has an infinite radius, not zero — use a very large number, or use the thin-lens form and treat that term as negligible.
- Ignoring thickness for a thick lens: the thin-lens formula is only an approximation — for magnifying glasses, camera lenses, and other lenses with d comparable to R₁ or R₂, use the full equation for an accurate focal length.
- Forgetting the surrounding medium: n here is the lens material's refractive index relative to the surrounding medium (usually air, n ≈ 1.00). If the lens is submerged in water or another medium, use the relative refractive index between the two materials instead.