Lensmaker's Equation Calculator

Introduction to the Lensmaker Equation

The lensmaker equation connects a lens's signed surface curvatures and refractive index with the focal length it produces. Given the refractive index and the signed radii of both surfaces, you can estimate focal length; if one of those quantities is unknown, the relationship can be rearranged to find it. This calculator performs that lens-design calculation in the browser: enter four values, leave one field blank, and compute the remaining value.

This lensmaker calculator is useful for checking an optics exercise, comparing candidate lens shapes, or seeing when center thickness changes a focal-length estimate. It includes the center thickness $d$, so it covers both the usual thin-lens case and the thick-lens expression used when thickness cannot be ignored. Setting thickness to zero removes the thickness correction.

Lensmaker results depend on sign convention as much as on the numerical radii. The entries for $R_1$ and $R_2$ are signed radii of curvature rather than unsigned distances. Reversing a radius sign can change a converging result into a diverging one, so the calculator can represent biconvex, plano-convex, meniscus, biconcave, and other lens forms when the signs are entered consistently.

How to Use the Lensmaker Calculator

To solve a lensmaker-equation problem here, choose the quantity you need and leave exactly that one field empty. Enter the other four values and press Compute. If multiple fields are blank, or none is blank, the page requests exactly one unknown because a single lensmaker calculation otherwise has no unique target.

Each lensmaker input has a distinct optical role. The refractive index $n$ describes how strongly the lens material bends light relative to its surrounding medium. The radius $R_1$ belongs to the first surface met by incoming light, while $R_2$ belongs to the second. The thickness $d$ is the axial distance between lens vertices, and focal length $f$ is the paraxial distance from the lens to its focal point.

Consistent units are essential for a lensmaker calculation. The page does not convert lengths, so use the same unit for $R_1$, $R_2$, and $d$; the computed focal length will use that unit too. Meters and millimeters are both suitable, but mixing them within one calculation produces an incorrect result.

The lensmaker sign convention on this page is the standard introductory-optics convention. A radius is positive when its center of curvature is to the right of the surface as incoming light travels, and negative when that center is to the left. Thus many converging lenses have $R_1>0$ and , while a biconcave diverging lens often reverses those signs. Sketching the lens and its centers of curvature is a reliable check before entering values.

After a lensmaker value is submitted, the result message identifies the computed variable and the script places that value in the formerly blank field. You can then change one surface or the material index and calculate again to compare focal lengths or required curvatures.

Lensmaker Equation Formula

The calculator uses the thick-lens lensmaker equation below. Its notation combines focal length with refractive index, two signed surface radii, and center thickness.

Formula: 1 / f = n - 1 1 / R_1 - 1 / R_2 + (n - 1 d) / (n R_1 R_2)

$\frac{1}{f}=n-1\left(\frac{1}{R_1}-\frac{1}{R_2}+\frac{n-1d}{n{R}_1{R}_2}\right)$

In this lensmaker relationship, reciprocal focal length depends on refractive-index contrast, the curvature of each surface, and a center-thickness correction for a non-thin lens. Greater curvature generally gives a shorter focal length in magnitude. For a converging shape with unchanged radii, a higher refractive index also generally increases optical power and shortens focal length.

For a thin lens, where center thickness is negligible, the lensmaker equation reduces to:

Formula: 1 / f = n - 1 1 / R_1 - 1 / R_2

$\frac{1}{f}=n-1\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

This thin-lens expression is commonly introduced first because it isolates the main effect of material index and surface curvature. Physical lenses are not infinitely thin, however, so the full form is better suited to lenses whose center thickness is not small compared with their radii.

The calculator can also solve the lensmaker equation for $R_1$, $R_2$, $d$, or $n$. The radius and thickness cases use algebraic rearrangement. When refractive index is the blank field, the script applies Newton-Raphson iteration because the thick-lens expression is nonlinear in $n$.

Lensmaker Equation Worked Example

Consider a biconvex glass lens with refractive index $n=1.52$, first radius $R_1=0.1$ m, and second radius $R_2=-0.1$ m. If its center thickness is $d=0$, the thin-lens result is approximately $f\approx 0.096$ m. Under paraxial conditions, parallel incoming rays focus about 9.6 cm from the lens.

For the same lensmaker inputs except a 5 mm center thickness, or 0.005 m, focal length becomes about 0.097 m. This slightly longer focal length follows from the signed radii: a biconvex lens has radii of opposite signs, making their product negative, so the Gullstrand thickness term reduces the optical power. The difference is small in this example, but thickness can matter in precision optical work.

The calculator also supports reverse lensmaker design estimates. If focal length and material are known but one surface radius remains undecided, leave that radius blank to obtain a starting curvature. Such a result is a first-pass paraxial estimate; practical design also considers aberrations and manufacturing limits.

A negative focal length from the lensmaker equation denotes a diverging rather than converging lens in this convention. It is not automatically an error: the selected signed curvatures and refractive index may correctly describe a lens that spreads parallel rays apart, as many concave shapes do.

Lensmaker Equation Assumptions and Limitations

The lensmaker equation is a paraxial model of lens behavior. It assumes rays remain near the optical axis and form small angles with it, allowing a compact focal-length relationship. Rays far from the axis or at large angles can form images that differ from this prediction.

A lensmaker focal-length result is not an image-quality assessment. A lens with the predicted focal length can still show spherical aberration, coma, astigmatism, field curvature, distortion, or chromatic aberration. Those effects require information beyond paraxial optical power and are commonly examined with ray tracing or multi-element optical design tools.

The refractive index in this lensmaker form is relative to the surrounding medium. In air, ordinary glass index values can be used directly. In water or another fluid, the relevant value changes. For example, glass with absolute index 1.5 in water of index 1.33 has a relative index of approximately $\frac{1.5}{1.33}\approx 1.13$, which gives a more realistic underwater calculation.

Lensmaker numerical edge cases are possible. A zero radius makes the equation undefined because it represents infinite curvature. Some combinations can produce an extremely long or short focal length or no physically useful configuration. When solving numerically for refractive index, unusual inputs can also converge slowly or return a mathematically valid value that is implausible for ordinary optical materials.

Within those limits, this lensmaker calculator is a useful first-order design and learning aid. It shows how surface curvature, thickness, and refractive index interact: steepening a surface usually strengthens a lens, increasing index lets the same shape bend light more, and increasing thickness introduces the thick-lens correction.

Interpreting Lensmaker Equation Results

A focal length returned by the lensmaker calculator carries a sign as well as a magnitude. Positive focal length generally identifies a converging lens and negative focal length a diverging lens. A returned radius sign identifies the side of its surface on which the center of curvature lies. If the unknown was refractive index, compare the result with known optical materials to judge whether it is plausible; common crown glasses are near 1.5, while high-index glasses can be larger.

When checking a lensmaker exercise, estimate the trend before computing. A strongly curved glass lens should not have an exceptionally long focal length unless its surface powers nearly cancel. The same lens placed in water is usually weaker than it is in air, and a thick lens can differ noticeably from the thin-lens estimate. These expectations help expose sign and unit errors.

Lensmaker Equation Questions and Answers