Lensmaker Equation Calculator

What is the lensmaker’s equation?

The lensmaker’s equation computes the focal length (f) of a thin lens in air from the material’s refractive index (n) and the radii of curvature of its two surfaces (R1, R2): 1/f = (n−1)·(1/R1 − 1/R2). Example: n = 1.52, R1 = 100 mm, R2 = −100 mm (symmetric biconvex) → 1/f = 0.52·(0.01 + 0.01) = 0.0104 → f ≈ 96.15 mm.

What is the R1/R2 sign convention in this calculator?

This calculator’s default is the Cartesian convention: light travels left to right, and R is positive when the center of curvature is on the incoming-light side (right). Hence a biconvex lens is written R1 > 0, R2 < 0; a biconcave lens R1 < 0, R2 > 0. The f sign follows: positive f = converging, negative f = diverging.

What is the “real-is-positive” convention and how does it differ?

In the real-is-positive convention, R is positive when the surface is convex toward the incoming light, so a biconvex lens is written R1 > 0, R2 > 0. The only difference from Cartesian is the R2 sign (R2 is flipped); R1 and the formula stay the same. The toggle in this calculator flips the R2 sign automatically so the physical lens shape is unchanged.

What does R = infinity (plano) mean?

Infinite R means a flat (plano) surface, so 1/R = 0 and that term drops out of the equation. Example: a plano-convex lens with n = 1.52, R1 = 50 mm, R2 = ∞ → 1/f = 0.52/50 = 0.0104 → f ≈ 96.15 mm, the same as a biconvex with R = 100 mm because half the radius on one surface gives the same power.

What happens if R1 = R2?

If both radii are equal in size and sign (e.g. R1 = R2 = 100 mm), then 1/R1 − 1/R2 = 0, so the lens power is zero and f is infinite — the lens behaves like a flat glass window that does not focus light. The calculator shows ∞ for this case.

Can the refractive index n be less than or equal to 1?

Not for a lens in air — lens material is always optically denser than air (n > 1). If n = 1, the (n−1) factor becomes zero and the lens has no power. The calculator rejects n ≤ 1 with an error message.

How does the lensmaker’s equation differ from the thin-lens equation 1/f = 1/s + 1/s′?

The lensmaker’s equation determines f from the lens geometry and material (n, R1, R2) — used when designing/making lenses. The thin-lens equation 1/f = 1/s + 1/s′ uses a known f to find the image position (s′) from the object position (s) — used when working with a finished lens.

How can a meniscus lens be positive or negative?

A meniscus has one convex and one concave side; its sign is set by the steeper curvature. Cartesian example with n = 1.52: R1 = 100 mm, R2 = 200 mm (convex side steeper) → f ≈ +384.62 mm (positive); reversed R1 = 200 mm, R2 = 100 mm (concave side steeper) → f ≈ −384.62 mm (negative).

Does this calculator support thick lenses?

Yes — enter the center thickness (d) and the calculator uses the thick form: 1/f = (n−1)·[1/R1 − 1/R2 + (n−1)·d/(n·R1·R2)]. Example: n = 1.52, R1 = 100 mm, R2 = −100 mm, d = 10 mm → f ≈ 97.83 mm (not 96.15 mm under the thin approximation). For d = 0 the result is identical to the thin form.

I need contact lens calculations, not lens design — is this the right tool?

No — contact lenses use optometric vertex distance correction (power compensation for the lens-to-eye distance), not the lensmaker’s equation. Use the Contact Lens Correction calculator (/tools/contact-lens-vertex/) for that; this tool only computes lens-geometry f, n, R1, R2.