Thin Lenses & the Lens Maker's Formula
Two curved surfaces close together: how their curvatures set a lens's focal length.
Lesson at a glanceShow the key points
- A thin lens: transparent material between two spherical surfaces, thin compared with object distances; principal axis through C₁ and C₂; optical centre P.
- Parallel rays converge to the second focus F₂ (converging/convex lens) or seem to diverge from it (diverging/concave lens). The first focus F₁: object position giving parallel emergent rays.
- Sign convention: optical centre as origin; f > 0 for converging, f < 0 for diverging lenses.
- Two refractions give 1/v − 1/u = (μ₂/μ₁ − 1)(1/R₁ − 1/R₂) (18.14–18.15); in air, the lens maker's formula 1/f = (μ − 1)(1/R₁ − 1/R₂) (18.16).
- Lens formula: 1/v − 1/u = 1/f (18.17).
A lens is just two spherical surfaces close together. Put the last lesson’s formula to work twice and out comes the rule every lens designer uses.
Thin lenses
A lens is a piece of transparent material bounded by two spherical surfaces (or one spherical and one flat). It is thin if its thickness is small compared with the object and image distances. Its two surfaces have centres C₁ and C₂ and radii R₁ and R₂. The line through C₁ and C₂ is the principal axis, and the point P of the lens on it is the optical centre.
The foci
A narrow beam parallel to the axis, near the optical centre, is refracted twice and emerges either converging to a point F₂ or diverging as if from F₂. The lens is then a converging (convex) or diverging (concave) lens.
If the medium is the same on both sides, the two focal lengths are equal. “Focus” and “focal length” will mean F₂ and PF₂.
Sign convention
As before: optical centre at the origin, principal axis as the x-axis, positive along the incident light; u, v, f, R₁, R₂ are x-coordinates; heights up are positive. With light from the left, f is positive for a converging lens and negative for a diverging lens.
| Lens (light from the left) | R₁ | R₂ | f |
|---|---|---|---|
| Biconvex | + | − | + |
| Biconcave | − | + | − |
| Plano-convex, curved face first | + | ∞ | + |
| Converging meniscus (both centres to the right) | + (smaller) | + | + |
The lens maker’s formula
Let the lens (index μ₂) sit in a medium of index μ₁. At the first surface, light goes from μ₁ into μ₂ and forms an image at v₁. Since the lens is thin, take the origin at P for both surfaces:
That image is the object for the second surface, where light goes from μ₂ back into μ₁:
Adding the two eliminates v₁:
For u → ∞, v = f. In air (μ₁ = 1, μ₂ = μ):
Combining the two equations gives the lens formula:
A biconvex lens has both radii 30 cm and refractive index 1.6. Find its focal length.
- 1Thin lens: two spherical surfaces; principal axis through C₁C₂; optical centre P.
- 2Parallel rays converge to (or seem to diverge from) F₂; f > 0 for converging, f < 0 for diverging lenses.
- 3Lens maker's formula in air: 1/f = (μ − 1)(1/R₁ − 1/R₂).
- 4Lens formula: 1/v − 1/u = 1/f.
Check yourself
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