Derivation of Lens Maker Formula: 1/f = (n − 1)(1/R₁ − 1/R₂)

Q: What is the lens maker formula?

It is 1/f = (n − 1)(1/R₁ − 1/R₂), giving the focal length of a thin lens from its refractive index and the radii of its two faces.

Q: What are the assumptions in the derivation of lens maker formula?

The lens is thin, so its thickness is ignored; the aperture is small; the object is a point on the axis; and the rays make small angles with the axis.

Q: Why is it called the lens maker's formula?

Because it tells the person making the lens what radii to grind on each face to get the focal length they need from a given glass.

Q: What happens to the focal length of a lens in water?

It increases. Glass and water have closer refractive indices than glass and air, so n₂₁ − 1 shrinks. A glass lens with f = 20 cm in air has f ≈ 78 cm in water.

Q: What is the sign of R for a plane surface?

A plane surface has an infinite radius, so 1/R = 0 for that face, whichever side it is on.

The derivation of lens maker formula leads to $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ , which gives the focal length of a thin lens from the refractive index n of its material and the radii of curvature R₁ and R₂ of its two faces. It is called the lens maker's equation because it tells a lens grinder what curves to cut to get a chosen focal length.

The proof applies the formula for refraction at one curved surface twice, once at each face, and adds the results. It is a regular long answer in Class 12 board papers. Signs follow the New Cartesian convention: with light travelling left to right, a radius whose centre of curvature lies to the right of the surface is positive. For a biconvex lens that makes R₁ positive and R₂ negative.

Quick facts

Formula
$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
In another medium
Replace n by $n_{21} = \frac{n_{\text{lens}}}{n_{\text{medium}}}$
Biconvex, equal radii R
$f = \frac{R}{2(n - 1)}$ ; equals R when n = 1.5
Plano-convex
$f = \frac{R}{n - 1}$ ; equals 2R when n = 1.5
Assumptions
Thin lens, small aperture, rays close to the axis
Unit of f
metre (m)
Course
Class 12 ray optics, JEE, NEET, AP Physics 2 (as a concept)

Starting point: refraction at one spherical surface

For a ray passing from a medium of index $n_1$ into a medium of index $n_2$ across a spherical surface of radius R, an object at distance u forms an image at distance v given by

$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$

This result is itself derived from Snell's law with small angles, and most syllabi let you quote it in the lens maker proof. Every distance is measured from the pole of that surface with the New Cartesian signs.

Derivation of lens maker formula, class 12 steps

Take a thin lens of index $n_2$ in a medium of index $n_1$ . Its first face has radius $R_1$ and its second face $R_2$ . A point object O sits on the axis at distance u.

Step 1, first surface. Light goes from $n_1$ into $n_2$ . Ignoring the second face for now, it would form an image I₁ at distance $v_1$ :

$\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1}$

Step 2, second surface. Before the rays reach I₁ they meet the second face, so I₁ acts as a virtual object for it. The lens is thin, so the distance of I₁ from the second face is still $v_1$ . Light now goes from $n_2$ back into $n_1$ and forms the final image at v:

$\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2}$

Step 3, add. The $\frac{n_2}{v_1}$ terms cancel:

$\frac{n_1}{v} - \frac{n_1}{u} = (n_2 - n_1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Step 4, divide by n₁ and write $n_{21} = \frac{n_2}{n_1}$ :

$\frac{1}{v} - \frac{1}{u} = (n_{21} - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Step 5, use the definition of focus. An object at infinity ( $u = -\infty$ ) forms its image at the focus ( $v = f$ ):

$\frac{1}{f} = (n_{21} - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

In air $n_1 = 1$ , so $n_{21} = n$ and the familiar form appears. Comparing Step 4 with Step 5 also hands you the thin lens equation for free; it is worked through on the formula of lens page.

Examiners look for the virtual object argument in Step 2 and the thin lens assumption that lets you reuse v₁.

Worked examples

Example 1: an equiconvex lens

A biconvex lens has both radii equal to 20 cm ; the glass has n = 1.5. Find its focal length.

Signs: R₁ = +20 cm, R₂ = −20 cm.

$\frac{1}{f} = 0.5\left(\frac{1}{20} + \frac{1}{20}\right) = 0.5 \times 0.1 = 0.05\ \text{cm}^{-1}$

f = +20 cm, or 0.20 m in SI units.

Example 2: a plano-convex lens

A plano-convex lens of glass n = 1.5 has a curved face of radius 15 cm. Light enters through the flat face. Find f.

R₁ = ∞ (flat face first), R₂ = −15 cm.

$\frac{1}{f} = 0.5\left(0 + \frac{1}{15}\right) \Rightarrow f = +30\ \text{cm}$

Example 3: a biconcave lens

A biconcave lens of index 1.6 has radii of 30 cm on both faces. Find f.

R₁ = −30 cm, R₂ = +30 cm.

$\frac{1}{f} = 0.6\left(-\frac{1}{30} - \frac{1}{30}\right) = -0.04\ \text{cm}^{-1} \Rightarrow f = -25\ \text{cm}$

Negative f: a diverging lens, as expected.

Example 4: the same lens under water

The lens from Example 1 (f = 20 cm in air) is placed in water of index 1.33. Find its new focal length.

$n_{21} = \frac{1.5}{1.33} = 1.128, \qquad \frac{f_w}{f_a} = \frac{n - 1}{n_{21} - 1} = \frac{0.5}{0.128} = 3.91$

f in water ≈ 3.91 × 20 ≈ 78 cm. The lens is much weaker because the glass and water bend light by similar amounts.

What the formula tells you

Higher n means shorter f. Dense flint glass makes stronger lenses than crown glass of the same shape.
Sharper curves (smaller R) mean shorter f.
If the lens index equals the surrounding index, $n_{21} = 1$ and the lens vanishes optically: f becomes infinite.
If the lens is less dense than its surroundings (an air bubble in water), $n_{21} < 1$ and a convex shape diverges light.
Turning a lens round swaps R₁ and R₂ and changes both signs, so f is unchanged.

The reciprocal of f is the power, so the same formula gives the power of a lens in dioptres when the radii are in metres. The refractive index itself comes from the index of refraction formula. Browse every optics page from the waves and optics hub.

Sign slips in lens maker problems

Writing R₂ as positive for a biconvex lens. With light from the left, the centre of the second face lies on the left, so R₂ is negative.
Forgetting n − 1. The factor is (n − 1), not n.
Using the index of the glass when the lens is in a liquid. Use the relative index.
Mixing cm and m when you want dioptres. Convert the radii to metres first.

Frequently asked questions

What is the lens maker formula?

It is 1/f = (n − 1)(1/R₁ − 1/R₂), giving the focal length of a thin lens from its refractive index and the radii of its two faces.

What are the assumptions in the derivation of lens maker formula?

The lens is thin, so its thickness is ignored; the aperture is small; the object is a point on the axis; and the rays make small angles with the axis.

Why is it called the lens maker's formula?

Because it tells the person making the lens what radii to grind on each face to get the focal length they need from a given glass.

What happens to the focal length of a lens in water?

It increases. Glass and water have closer refractive indices than glass and air, so n₂₁ − 1 shrinks. A glass lens with f = 20 cm in air has f ≈ 78 cm in water.

What is the sign of R for a plane surface?

A plane surface has an infinite radius, so 1/R = 0 for that face, whichever side it is on.