Lens Maker's Equation Calculator
Why lens design (and focal length) matters
Lenses are core components in cameras, microscopes, telescopes, projectors, VR headsets, and many measurement instruments. A lens works by refracting (bending) light at its surfaces; that bending determines where parallel rays come to focus, how large an image appears, and how wide the field of view is. In practical terms, the focal length you compute here is one of the fastest ways to sanity‑check whether a proposed lens shape and material will behave as a converging lens (positive focal length) or a diverging lens (negative focal length).
This calculator uses the common thin‑lens form of the Lens Maker’s equation. It’s widely taught and extremely useful for first‑pass design, homework, and quick verification—but like any simplified model, it relies on assumptions (listed below) and a sign convention that you must apply consistently.
The Lens Maker’s equation (thin lens)
For a thin lens in a uniform surrounding medium (often air), the focal length f is related to refractive index and the radii of curvature of the two surfaces:
Where:
- f = focal length (meters). Positive means converging; negative means diverging.
- n = refractive index of the lens material relative to the surrounding medium. If the lens is in air, you typically use the material’s index in air (e.g., ~1.49–1.52 for many glasses/plastics).
- R1 = radius of curvature of the first surface the light hits (meters).
- R2 = radius of curvature of the second surface (meters).
Sign convention for R1 and R2 (the #1 source of confusion)
Different textbooks and software packages may use different sign conventions. This page follows a common convention that pairs naturally with the thin Lens Maker’s equation written above:
- R1 is positive if the first surface is convex as seen by the incoming light (it “bulges toward” the incoming rays). It is negative if that first surface is concave.
- R2 is positive if the second surface is convex as seen by the outgoing light. It is negative if the second surface is concave as seen by the outgoing light.
Under this convention, a symmetric biconvex lens (typical converging magnifier) often ends up with R1 > 0 and R2 < 0, which makes the bracket term larger and yields a positive focal length.
Interpreting the result
- Positive focal length (f > 0): the lens is converging (can focus parallel rays to a real focal point). Typical of biconvex or plano‑convex shapes with n > 1.
- Negative focal length (f < 0): the lens is diverging (parallel rays spread out as if coming from a virtual focus). Typical of biconcave or plano‑concave shapes.
- Very large |f| (e.g., hundreds of meters): very weak optical power—often caused by large radii (nearly flat surfaces) or n close to 1.
Also note that many optics workflows use optical power in diopters, defined as P = 1/f when f is in meters. If you want power, you can compute it directly from the equation’s left side.
Worked example (step by step)
Suppose you have a thin, symmetric biconvex lens in air with refractive index n = 1.50. Let the first surface have radius R1 = +0.10 m. For a symmetric biconvex lens under the convention above, the second surface typically uses the opposite sign: R2 = −0.10 m.
- Compute n − 1: 1.50 − 1 = 0.50
- Compute the curvature term:
- 1/R1 = 1/0.10 = 10 m−1
- 1/R2 = 1/(−0.10) = −10 m−1
- [1/R1 − 1/R2] = 10 − (−10) = 20 m−1
- Compute 1/f: (0.50)(20) = 10 m−1
- Invert to get f: f = 1/10 = 0.10 m = 10 cm
The positive result indicates a converging lens, which matches expectations for a biconvex lens in air.
Quick comparison table (common shapes and typical signs)
| Lens type (thin, in air) | Typical sign of R₁ | Typical sign of R₂ | Expected sign of f | Notes |
|---|---|---|---|---|
| Biconvex (converging) | + | − | + | Common magnifier; symmetric case often R₁ = −R₂ in magnitude. |
| Plano‑convex (converging) | + (convex side first) or ~∞ (plane) | ~∞ (plane) or − | + | If one side is plane, use a very large radius (approaches infinity). |
| Biconcave (diverging) | − | + | − | Produces negative focal length in air for n > 1. |
| Plano‑concave (diverging) | − or ~∞ | ~∞ or + | − | Sign depends on which side faces incoming light; keep convention consistent. |
Assumptions and limitations (when this calculator may be inaccurate)
- Thin‑lens approximation: lens thickness is assumed small compared with radii of curvature. Thick lenses require additional terms (or a thick‑lens model with principal planes).
- Paraxial (small‑angle) rays: the equation is derived for rays close to the optical axis. At large field angles or with “fast” lenses, aberrations and higher‑order effects become significant.
- Spherical surfaces: the classic form assumes spherical surfaces. Aspheric lenses won’t be represented by a single radius in the same way.
- Uniform surrounding medium: the refractive index n should be relative to the medium around the lens. If the lens is in water, for example, the effective index contrast is lower and the focal length changes.
- Dispersion ignored: refractive index depends on wavelength. If you’re designing for specific wavelengths (e.g., 532 nm vs 1064 nm), use the appropriate n for that wavelength.
- Manufacturing/measurement tolerances: small changes in radii can noticeably change power for strong lenses; treat results as ideal.
Practical tips for using the inputs
- Units: enter radii in meters to get f in meters. (Example: 50 mm = 0.05 m.)
- Plane surface: if a surface is flat, its radius is infinite. In calculators, you can approximate this by entering a very large number (e.g., 1e9 m) so 1/R ≈ 0.
- Sanity checks: if you get an unexpected sign, recheck the sign convention for R₂; that’s the most common mistake.