Lens Maker Equation Calculator
Work out the focal length of a lens from the glass and the curvature of its two faces, including the effect of thickness.
🔬 What is the Lens Maker Equation?
The lens maker equation, also called the lensmaker's formula, connects the focal length of a lens to the glass it is made of and the shape of its two faces. In its thin-lens form it reads 1/f = (n − 1)(1/R1 − 1/R2), where n is the refractive index and R1 and R2 are the radii of curvature of the first and second surfaces. It is the formula a lens designer uses before any ray tracing: choose a glass, choose the curves, and the focal length follows. This calculator evaluates it, extends it to lenses of real thickness, and runs it backwards to find the curve or the glass you need.
Opticians, camera and telescope builders and physics students all use it. An optician converts a prescription in dioptres into curves for a given plastic. A hobbyist grinding a telescope objective picks the radii that give the right focal length. A student in a lab measures the focal length of a lens and the radius of each face with a spherometer, and then checks the refractive index of the glass. The power of the lens, 1/f in metres, is reported in dioptres, which is the unit on a glasses prescription and the quantity that adds up when thin lenses are placed together.
The most common mistakes come from signs and from assuming the lens is thin. Each radius is signed. With the convention used here, a radius is positive when its center of curvature lies on the side the light is heading toward, so the first face of a biconvex lens is positive and the second is negative. A flat face has an infinite radius, which is why the calculator lets you leave the field blank. For a thick lens the thin formula is only approximate, and the correction term (n − 1) d / (n R1 R2) adds the effect of the glass between the faces. The page shows both values, so you can see when thickness matters and when the thin-lens limit is good enough.
The three modes are Focal length, Design a lens, and Find n. Focal length handles any lens and plots how its power changes with the index. Design a lens returns the radius for a target focal length in equi-convex and plano-convex shapes. Find n estimates the refractive index of an unknown lens. Once you have the focal length, use it with the mirror equation calculator for the matching curved-mirror problem, and the speed of light calculator for how the glass slows light.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Biconvex Magnifier
Thin lens, n = 1.5, R1 = +100 mm, R2 = −100 mm
Example 2 — A Plano-Convex Lens
n = 1.52, R1 = +50 mm, second face flat
Example 3 — A Biconcave Lens
n = 1.5, R1 = −200 mm, R2 = +200 mm
Example 4 — A Thick Lens
n = 1.5, R1 = +50 mm, R2 = −50 mm, thickness d = 10 mm
Example 5 — A Converging Meniscus
n = 1.6, R1 = +40 mm, R2 = +100 mm
Example 6 — Designing an Equi-Convex Lens
Target f = 150 mm in glass with n = 1.52
Example 7 — Finding the Index of an Unknown Lens
A lens with f = 75 mm and radii +50 mm and −50 mm
❓ Frequently Asked Questions
🔗 Related Calculators
What is the lens maker equation?
For a thin lens in air, 1/f = (n - 1)(1/R1 - 1/R2), where n is the refractive index of the glass and R1 and R2 are the radii of curvature of the first and second surface. With n = 1.5 and radii of +100 mm and -100 mm, f = 100 mm.
What is the sign convention for R1 and R2?
A radius is positive when its center of curvature lies on the side the light is going toward, and negative when it lies on the side the light comes from. A biconvex lens therefore has R1 positive and R2 negative. A flat surface has an infinite radius.
What is the thick lens formula?
For a lens of thickness d in air, 1/f = (n - 1)[1/R1 - 1/R2 + (n - 1) d / (n R1 R2)]. The last term corrects for the distance light travels inside the glass. As d tends to zero it vanishes and you recover the thin-lens equation.
How do you find the focal length of a plano-convex lens?
Set the flat radius to infinity, so the equation reduces to 1/f = (n - 1)/R. A plano-convex lens with n = 1.52 and R = 50 mm has f = 50 / 0.52 = 96.15 mm.
How do you calculate the power of a lens in dioptres?
The power is P = 1/f with f in metres. A lens with a focal length of 100 mm = 0.1 m has a power of 10 dioptres. Thin lenses placed close together add their powers.
Is a biconcave lens converging or diverging?
It is diverging, so its focal length is negative. With n = 1.5 and radii of -200 mm and +200 mm the lens maker equation gives f = -200 mm, a power of -5 dioptres.
What radius do I need for a given focal length?
For an equi-convex thin lens R = 2 (n - 1) f. For a plano-convex lens R = (n - 1) f. With n = 1.52 and f = 150 mm an equi-convex lens needs R = 156 mm on each face.
How much does thickness change the focal length?
It depends on the radii. For n = 1.5, R1 = 50 mm, R2 = -50 mm, a 10 mm thickness gives f = 51.72 mm instead of the thin-lens 50 mm, a difference of 3.45 percent. Thick lenses with strong curves need the full formula.
Can the lens maker equation be used for a lens in water?
Only with a change. Replace n by the ratio n(lens) / n(medium) in the equation. This calculator assumes the lens is in air, with an index of 1.
What is a meniscus lens?
A meniscus lens has both surfaces curving the same way, so R1 and R2 have the same sign. A lens with n = 1.6, R1 = 40 mm and R2 = 100 mm is a converging meniscus with f = 111.1 mm, because the first surface curves more strongly.
How do you find the refractive index of a lens from its focal length?
Rearrange the thin-lens form: n = 1 + 1 / [f (1/R1 - 1/R2)]. A lens with f = 75 mm and radii of 50 mm and -50 mm gives n = 1.3333.