Lensmaker's Equation
The lensmaker's equation relates a lens's focal length to its radii of curvature and refractive index.
Essential for optical design and camera lens calculations.
The Formula
The lensmaker's equation (also called the lensmaker's formula) allows optical engineers and physicists to calculate the focal length of a lens from its physical shape and material. It connects three things: the refractive index of the glass, and the radii of curvature of its two surfaces. This formula is fundamental to designing eyeglasses, camera lenses, microscopes, and telescopes.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f | Focal length of the lens | metres or mm |
| n | Refractive index of the lens material | dimensionless |
| R₁ | Radius of curvature of the first surface (light enters) | metres or mm |
| R₂ | Radius of curvature of the second surface (light exits) | metres or mm |
Sign Convention
R is positive if the centre of curvature is to the right of the surface, negative if to the left (using the standard Cartesian sign convention). For a typical biconvex lens: R₁ > 0 and R₂ < 0.
Example 1 — Biconvex Lens
Crown glass lens (n = 1.52), R₁ = +30 cm, R₂ = −30 cm (symmetric biconvex)
1/f = (1.52 − 1) × [1/30 − 1/(−30)] = 0.52 × [1/30 + 1/30]
1/f = 0.52 × (2/30) = 0.52 × 0.0667 = 0.03467
f = 1/0.03467 ≈ 28.8 cm (converging lens)
Example 2 — Plano-Convex Lens
Flint glass (n = 1.70), R₁ = +20 cm, R₂ = ∞ (flat surface)
1/f = (1.70 − 1) × [1/20 − 1/∞] = 0.70 × [0.05 − 0]
f = 1/(0.70 × 0.05) = 1/0.035 ≈ 28.6 cm
Common Refractive Indices
| Material | Refractive Index (n) | Common use |
|---|---|---|
| Crown glass | 1.52 | Standard eyeglasses |
| Flint glass | 1.60–1.70 | Camera lenses, achromats |
| High-index plastic | 1.67–1.74 | Thin eyeglass lenses |
| Sapphire | 1.77 | Watch crystals, specialist optics |
| Diamond | 2.42 | Gemstones, high-power lasers |
When to Use It
- Designing eyeglass lenses from a prescription (dioptre = 1/f in metres)
- Calculating focal lengths for camera and telescope objective lenses
- Selecting lens blanks in optometry manufacturing
- Understanding how lens shape affects focal power
- Comparing converging (f > 0) and diverging (f < 0) lenses