AkCalculators

The Lens-Maker Equation — thin-lens design, sign conventions, and practical guidance

This article explains the thin-lens design equation (often called the lens-maker equation), how to use it for both convex and concave lenses, how to choose and apply sign conventions correctly, and practical considerations for optical design. We present derivations, worked examples and limitations so you can apply the calculator confidently to real lens design tasks.

1. Overview — what the lens-maker equation gives

The thin-lens maker equation (for a lens in air) links the focal length f of a lens to the refractive index n of the lens material and the radii of curvature R₁ and R₂ of the two spherical surfaces:

1 / f = (n − 1) · (1 / R₁ − 1 / R₂)

This formula assumes a thin lens (thickness ≪ radii and focal length) and the surrounding medium is air (n≈1). The lens-maker equation is central to lens design — it tells us how curvature and refractive index determine focal power.

2. Physical meaning of terms

Focal length (f): the distance from the lens' principal plane to its focal point for paraxial (small-angle) rays. Positive f implies a converging lens (typically convex overall), negative implies diverging (typically concave overall).

Refractive index (n): dimensionless ratio describing how much the lens material slows light compared to vacuum. Typical crown glasses are 1.5–1.6; flint glasses are higher (~1.6–1.9).

Radii of curvature (R₁, R₂): radii of the spherical surfaces. Sign conventions matter — a surface bowed toward the incoming light vs away from it has different signs. Correct signs are crucial for correct f.

3. Sign conventions — Cartesian and magnitude modes

There are multiple sign conventions in optics. The two most common are:

• Cartesian convention — the optical axis is the x-axis and light travels left-to-right. The radius R of a surface is positive if the center of curvature is to the right of the surface, negative if to the left. With this convention R₁ pertains to the first surface a ray meets (left surface) and R₂ to the second (right surface).
• Magnitude mode / surface-type — enter radii as positive magnitudes and indicate whether each surface is convex or concave relative to the incoming light; the calculator assigns the sign automatically. This is convenient when you think in terms of surface shapes instead of centers of curvature.

Our calculator supports both modes — select the one you prefer and it will convert your inputs to the correct signed radii for the lens-maker equation.

4. Deriving the thin-lens equation (brief)

The lens-maker equation can be derived by applying Snell's law at each spherical surface and using paraxial approximations (small angles). For a single spherical refracting surface separating media with indices n₁ and n₂, the paraxial imaging relation is

(n₂ / s') − (n₁ / s) = (n₂ − n₁) / R

where s and s' are object and image distances measured from the surface, and R is the radius (with sign). For a thin lens (two refracting surfaces close together) in air (n₁ ≈ 1 outside, n₂ = n inside lens), combining the refractions at both surfaces and simplifying yields the lens-maker equation:

1 / f = (n − 1) (1 / R₁ − 1 / R₂)

A compact physical view: each surface contributes optical power; the lens' total power is the sum of the two surface powers scaled by (n−1).

5. Which variable can you solve for?

The equation is algebraic and easily rearranged to solve for any single unknown when the others are known. Common tasks:

• Compute f given n, R₁, R₂: substitute into 1/f = (n−1)(1/R₁ − 1/R₂).
• Compute n given f, R₁, R₂: rearrange to n = 1 + (1/f) / (1/R₁ − 1/R₂).
• Compute R₁ or R₂ given f and the other parameters: treat 1/R as the algebraic unknown and invert.

6. Convex vs concave examples — worked problems

Example A — Bi-convex lens: Suppose a bi-convex glass lens has equal radii of curvature |R₁| = |R₂| = 100 mm, both convex toward incoming light in magnitude mode, and uses crown glass n = 1.5. Assign signs: if light enters left-to-right, R₁ = +100 mm (center to the right), R₂ = −100 mm (second surface center to left). Then 1/f = (1.5 − 1) (1/100 − 1/(−100)) = 0.5 × (0.01 + 0.01) = 0.5 × 0.02 = 0.01 mm⁻¹ ⇒ f = 100 mm. So a symmetric bi-convex lens with these values focuses at 100 mm.

Example B — Plano-concave lens: Consider a plano-concave lens with R₁ = ∞ (plane), R₂ = 50 mm (concave toward right). In Cartesian mode R₁ = ∞ gives 1/R₁ = 0. For a concave right surface with center to left, R₂ = −50 mm. Using n = 1.5: 1/f = 0.5 × (0 − 1/(−50)) = 0.5 × (0 + 0.02) = 0.01 mm⁻¹ ⇒ f = +100 mm. Note: overall focal sign depends on exact surface arrangement; check signs carefully.

7. Practical notes and limitations

Thin-lens assumption: The lens-maker equation neglects lens thickness. For thick lenses, the separation between surfaces shifts principal planes, and the simple equation is insufficient. Use thick-lens formulas or ray-tracing software for precision designs.

Paraxial approximation: The derivation uses small-angle approximations (sin θ ≈ θ). The formula accurately predicts paraxial focal lengths; off-axis aberrations (coma, astigmatism) are not captured.

Material dispersion: The refractive index n depends on wavelength (dispersion). Design for a specific wavelength (e.g., 550 nm), or use achromatic doublets for broadband imaging.

8. Manufacturing & design considerations

In practice, lens designers combine radii, thickness, glass type, and coatings to meet imaging goals. Glass catalogues provide refractive indices and Abbe numbers (measure of dispersion). Achromatic lens systems pair crown and flint glasses to reduce chromatic aberration.

9. Using the calculator — tips

• Always enter radii and focal length in the same units. The calculator returns focal length in the same units you used.
• If a surface is planar, enter R = 0 or leave blank and use the calculator’s semantics (1/∞ = 0). We treat empty or "0" as infinite radius for that surface.
• When uncertain about signs, use the magnitude mode and mark surfaces convex/concave — that is less error-prone for quick design checks.

10. Summary

The lens-maker equation is a compact, powerful tool for estimating focal length from lens curvature and refractive index. It is essential for early-stage optical design and classroom exercises. Use thin-lens formulas for quick checks and a more advanced model for production or high-precision optics.