Math 2.0

Optics, electromagnetic theory Geometrical optics Proposition

Lensmaker's equation

Builds on Paraxial approximationRefractive indexSnell's law

Statement

Let a lens be made of a homogeneous material of refractive index nn and be surrounded by air of index 11. Let its two surfaces be spherical with radii of curvature R1R_1 (the surface met first by the light) and R2R_2. A radius is positive when its centre of curvature lies beyond the surface in the direction of travel of the light. Let dd be the thickness of the lens on its axis. Then, in the paraxial approximation, the focal length ff of the lens satisfies 1f=(n−1)[1R1−1R2+(n−1)dnR1R2].\frac{1}{f} = (n - 1)\left[\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n - 1) d}{n R_1 R_2}\right]. For a thin lens, with dd negligible, this becomes 1f=(n−1)(1R1−1R2).\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right).

Remarks

A single spherical surface of radius RR between indices n1n_1 and n2n_2 has paraxial power (n2−n1)/R(n_2 - n_1)/R; this follows from the paraxial form of Snell's law. The lens is two such surfaces with a gap dd of glass between them, and combining the two refractions gives the formula. With ray transfer matrices, 1/f1/f is minus the CC entry of the product of the three matrices (surface, thickness dd with reduced angle, surface).

Example: a symmetric biconvex lens with n=1.5n = 1.5 and R1=20R_1 = 20 cm, R2=−20R_2 = -20 cm. The thin-lens formula gives 1/f=0.5⋅(1/20+1/20)=1/201/f = 0.5 \cdot (1/20 + 1/20) = 1/20, so f=20f = 20 cm.

For a lens in a medium of index nmn_m, replace n−1n - 1 by (n−nm)/nm(n - n_m)/n_m in the thin-lens formula. A glass lens that converges in air therefore converges less strongly in water.

Going deeper

For a student

The equation reduces the design of a simple lens to two radii and an index. Because nn depends on the wavelength, ff does too, which is chromatic aberration; achromatic doublets combine two glasses to cancel it to first order. For a thick lens, the focal length is measured from the principal planes, and the full formula above is needed.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Calculus of variations, concept, no evidence levelCalculus of variationsconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptNormal vector, concept, no evidence levelNormal vectorconceptParaxial approximation, concept, no evidence levelParaxial approximationconceptRefractive index, concept, no evidence levelRefractive indexconceptFermat's principle, statement, no evidence levelFermat's principlestatementSnell's law, statement, no evidence levelSnell's lawstatementLensmaker's equation, statement, no evidence levelLensmaker's equationstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Euler–Lagrange equation concept
  3. Geodesic concept
  4. Normal vector concept
  5. Paraxial approximation concept
  6. Refractive index concept
  7. Fermat's principle statement
  8. Snell's law statement
  9. Lensmaker's equation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lens, Lensmaker's equation
About this page
TagTHR3PN9PKN
URIm2://THR3PN9PKN
Kindstatement, proposition
WritingW0 stub
MSC78A05
StatusThis copy of the site has no build data. Run m2 status THR3PN9PKN.
AboutRefractive index
UsesParaxial approximation, Refractive index, Snell's law
Changed3 November 2026

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