How an Optician Knows Exactly Which Lens You Need

Spectacles, camera lenses and the lens in your eye all bend light at curved surfaces. Starting from refraction at one spherical surface, we can derive the formulas that let an optician choose a lens of exactly the right power — and see how a prism bends and spreads light.

This part covers refraction at a spherical surface, the lens maker's and thin lens formulas, the power of lens combinations, and refraction through a prism.

**For light passing from a medium of index $n_1$ into one of index $n_2$ across a spherical surface of radius $R$, an object at distance $u$ forms an image at distance $v$ given by $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$.

Derivation outline, for small angles:**

- A ray from object O meets the surface at N and refracts towards image I; C is the centre of curvature and M the pole
- The angles with the axis are $\alpha \approx \frac{MN}{OM}$, $\beta \approx \frac{MN}{MC}$ and $\gamma \approx \frac{MN}{MI}$
- From the triangles, $i=\alpha +\beta$ and $r=\beta -\gamma$
- Snell's law for small angles gives $n_{1}i=n_{2}r$
- Substituting and applying the sign convention gives the result

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

Worked example. An object in air is $40$ cm from a convex glass surface with $n_2=1.5$ and $R=+10$ cm, so $u=-40$ cm:

$$\frac{1.5}{v}=\frac{1.5-1}{10}-\frac{1}{40}=0.025\Rightarrow v=+60\text{ cm}$$

A real image forms $60$ cm inside the glass.

An everyday example. A glass paperweight dome makes the picture beneath it look larger, because its curved top refracts light like a single spherical surface.

The substance. The same relation works for light travelling either way — just swap $n_1$ and $n_2$ and keep the sign convention.

**Applying the spherical-surface relation to both faces of a thin lens gives the lens maker's formula $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$, and with object and image distances it gives the thin lens formula $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$, with magnification $m=\frac{v}{u}$.

Derivation.** The first surface forms an image at $v_1$ with $\frac{n}{v_1}-\frac{1}{u}=\frac{n-1}{R_1}$. That image is the object for the second surface: $\frac{1}{v}-\frac{n}{v_1}=\frac{1-n}{R_2}$. Adding the two:

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

Worked example 1 — lens maker's formula. A biconvex lens has $R_1=+20$ cm, $R_2=-20$ cm and $n=1.5$:

$$\frac{1}{f}=0.5\left(\frac{1}{20}+\frac{1}{20}\right)=\frac{1}{20}\Rightarrow f=20\text{ cm}$$

Worked example 2 — locating the image. An object sits $30$ cm in front of this lens, so $u=-30$ cm:

$$\frac{1}{v}=\frac{1}{20}-\frac{1}{30}=\frac{1}{60}\Rightarrow v=+60\text{ cm},m=\frac{v}{u}=-2$$

A real, inverted image twice the size forms $60$ cm behind the lens.

An everyday example. A magnifying glass held close to the small print on a medicine strip is a convex lens closer than its focal length, forming an enlarged virtual image.

The substance. A lens's focal length depends on the surrounding medium — a glass lens is much weaker in water.

**The power of a lens is $P=\frac{1}{f}$ with $f$ in metres, measured in dioptres, and thin lenses in contact combine as $\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}$, so their powers simply add, $P=P_1+P_2$.

Power:

- A convex, converging lens has positive power
- A concave, diverging lens has negative** power
- $1$ dioptre $=1$ m${}^{-1}$

Combination. The image from the first lens acts as the object for the second; with no gap between them,

$$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2},P=P_1+P_2,m=m_1{m}_2$$

Worked example. A convex lens of focal length $25$ cm touches a concave lens of focal length $50$ cm:

$$P_1=\frac{1}{0.25}=+4\text{ D},P_2=\frac{1}{-0.5}=-2\text{ D},P=+2\text{ D},F=\frac{1}{2}=0.5\text{ m}$$

An everyday example. **A spectacle prescription of $-1.5$ D** means a diverging lens of focal length about $0.67$ m, used to correct short sight.

The substance. Powers add directly only for thin lenses in contact — once the lenses are separated, the gap must be taken into account.

**A ray through a prism bends at both faces, with $r_1+r_2=A$ and deviation $\delta =i+e-A$, and at minimum deviation, when the ray passes symmetrically, the refractive index is $n=\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$.

Geometry of the prism:**

$$r_1+r_2=A,\delta =i+e-A$$

where $i$ and $e$ are the angles of incidence and emergence.

Minimum deviation. As $i$ increases, $\delta$ first falls and then rises. At the minimum, $i=e$ and $r_1=r_2=\frac{A}{2}$, so $i=\frac{A+{\delta }_m}{2}$ and

$$n=\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$$

Worked example. An equilateral glass prism with $A=60^{\circ }$ gives a minimum deviation of $38^{\circ }$:

$$n=\frac{\sin 49^{\circ }}{\sin 30^{\circ }}=\frac{0.755}{0.5}\approx 1.51$$

Thin prism. For a small angle $A$, $\delta =(n-1)A$, so a $5^{\circ }$ prism with $n=1.5$ deviates light by $2.5^{\circ }$.

An everyday example. Sunlight through a glass prism hanging in a window spreads into a band of colours, because each colour has a slightly different refractive index and so a different deviation.

The substance. Violet light deviates most and red least, because glass has a higher refractive index for violet.