Lens Maker's Formula

A thin lens is two spherical refracting surfaces close together. Applying the single-surface formula at each surface and combining gives the lens maker's formula, which lets us design a lens of a chosen focal length.

Concept

A thin lens of material refractive index $n$ surrounded by air has two surfaces with radii $R_1$ (first surface met by light) and $R_2$ (second). With the Cartesian sign convention,

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

For a biconvex lens, $R_1 > 0$ and $R_2 < 0$, giving $f > 0$ (converging). For a biconcave lens, $R_1 < 0$ and $R_2 > 0$, giving $f < 0$ (diverging).

Derivation

Step 1: First surface. Object at $u$, image at $v_1$ in glass:

$\frac{n}{v_1} - \frac{1}{u} = \frac{n - 1}{R_1}$

Step 2: Second surface. The image $I_1$ from step 1 acts as the (virtual or real) object for the second surface. The light now goes from glass (index $n$) to air (index $1$), so

$\frac{1}{v} - \frac{n}{v_1} = \frac{1 - n}{R_2}$

Add the two equations. The $n/v_1$ terms cancel:

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

For a parallel beam ($u \to -\infty$), $v = f$, so the right-hand side equals $1/f$:

$\boxed{\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)}$

Power. $P = 1/f$ (with $f$ in metres). Unit: dioptre (D).

Worked Example

A biconvex lens has $R_1 = 20$ cm, $R_2 = -30$ cm, $n = 1.5$. Find $f$.

$\frac{1}{f} = (1.5 - 1)\left(\frac{1}{20} - \frac{1}{-30}\right) = 0.5\left(\frac{1}{20} + \frac{1}{30}\right)$

$\frac{1}{f} = 0.5 \times \frac{5}{60} = \frac{5}{120} = \frac{1}{24}\text{ cm}^{-1}$

$f = +24 \text{ cm (converging)}$

Power: $P = 1/0.24 \approx +4.17$ D.

Common Confusions

• Apply the sign convention to $R_1$ and $R_2$ before substituting. Both being positive does not mean both surfaces are convex.
• The formula assumes the lens is thin (thickness negligible compared to $|R_1|, |R_2|, |u|, |v|$).
• If the lens is immersed in a medium of index $n_m$, replace $n$ by $n/n_m$: $\frac{1}{f} = \left(\frac{n}{n_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ A glass lens in water has a longer focal length than in air.

Key Takeaways

• $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ for a thin lens in air.
• Derived by applying the single-surface formula twice and adding.
• Power $P = 1/f$ in dioptres when $f$ is in metres.
• Focal length depends on $n$ (so on wavelength: chromatic aberration) and on the surrounding medium.