The Lens Maker Equation
The lens maker equation is the fundamental formula that relates the focal length of a thin lens to its refractive index and radii of curvature. Master every form of the equation, understand the derivation, and use our free calculator below.
Use Calculator Now ↓1. The Lens Maker Equation
Standard Form
1/f = (n-1)(1/R₁ - 1/R₂)
f — focal length (meters)
n — refractive index of lens material
R₁ — radius of curvature, first surface (m)
R₂ — radius of curvature, second surface (m)
Sign convention: Convex surface facing light source → positive R. Concave → negative R.
2. Understanding Each Variable
Focal Length
Distance from lens center to focal point. Parallel rays converge (f > 0) or appear to diverge (f < 0).
meters
Refractive Index
Ratio of light speed in vacuum to speed in material. Higher n → stronger bending, shorter f.
dimensionless
First Surface Radius
Radius of first surface (light enters). Convex facing light: +R. Concave: −R.
meters
Second Surface Radius
Radius of second surface (light exits). Same sign convention as R₁.
meters
3. Equation Forms
Standard
1/f = (n-1)(1/R₁ - 1/R₂)
Solve for f
f = 1/[(n-1)(1/R₁ - 1/R₂)]
Solve for n
n = 1 + 1/[f(1/R₁ - 1/R₂)]
Solve for R₁
R₁ = 1/[1/R₂ + 1/((n-1)f)]
Solve for R₂
R₂ = 1/[1/R₁ - 1/((n-1)f)]
4. Thin vs Thick Lens Equation
Thin Lens
1/f = (n-1)(1/R₁ - 1/R₂)
Assumes thickness d ≪ R. Use when lens is thin relative to radii. Valid for most standard optics.
Thick Lens
1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
Includes thickness d. Use when d is significant (rule of thumb: d > R/10).
5. Derivation Overview
The lens maker equation follows from Snell's law applied at each curved surface.
- 1.Snell's Law: n₁ sin θ₁ = n₂ sin θ₂. For paraxial rays: n₁θ₁ ≈ n₂θ₂.
- 2.Single surface refraction: n/s + n'/s' = (n'-n)/R for object distance s and image s'.
- 3.First surface: 1/s₁ + n/s₁' = (n-1)/R₁ (air to glass).
- 4.Second surface: n/s₁' + 1/s₂' = (1-n)/R₂ (glass to air).
- 5.Combine (thin lens): 1/f = (n-1)(1/R₁ - 1/R₂) with focal length f.
6. Worked Examples
Example 1: Biconvex Lens
n = 1.5, R₁ = +0.10 m, R₂ = −0.10 m. Find f.
1/f = (1.5−1)(1/0.10 − 1/(−0.10)) = 0.5 × 20 = 10
f = 0.1 m = 10 cm (converging)
Example 2: Biconcave Lens
n = 1.52, R₁ = −0.12 m, R₂ = +0.18 m. Find f.
1/f = (1.52−1)(1/(−0.12) − 1/0.18) = 0.52 × (−13.89) ≈ −7.22
f ≈ −0.139 m (diverging)
Example 3: Plano-Convex Lens
n = 1.5, R₁ = +0.20 m, R₂ = ∞ (flat). Find f.
1/f = (1.5−1)(1/0.20 − 0) = 0.5 × 5 = 2.5
f = 0.4 m = 40 cm
Lens Maker Equation Calculator
Enter n, R₁, and R₂ to compute focal length instantly.
Lens Maker Formula
Calculate focal length from lens parameters
Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)
Positive for convex, negative for concave
Positive for convex, negative for concave
Frequently Asked Questions
What is the lens maker equation?
The lens maker equation is 1/f = (n-1)(1/R₁ - 1/R₂), relating focal length f to refractive index n and radii R₁, R₂. It is fundamental to optical lens design.
What is the difference between lens maker equation and thin lens equation?
The lens maker equation calculates f from lens geometry. The thin lens equation 1/f = 1/v − 1/u relates f to object distance u and image distance v.
When should I use the thick lens equation?
Use the thick lens form 1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)] when thickness d is significant (e.g. d > R/10).
What are typical refractive index values?
Crown glass ~1.52, flint glass ~1.62, water 1.33, diamond 2.42. Higher n gives shorter focal length.