The Lensmaker's Equation: Turning Two Curved Surfaces Into a Focal Length

The equation and what every symbol means

For a thin lens in air, the lensmaker's equation is:

• 1/f = (n − 1) · (1/R₁ − 1/R₂)

Here f is the focal length (metres), n is the refractive index of the lens glass, and R₁ and R₂ are the radii of curvature of the first and second surfaces the light meets. The quantity 1/f is the optical power P, measured in dioptres (1 D = 1 m⁻¹): a +5 D lens has f = +0.20 m. Opticians add powers, not focal lengths, precisely because the (n−1)(1/R₁−1/R₂) form is additive.

The genius of the equation is its factorisation. The material contributes (n − 1) — only the contrast between glass and surroundings matters, not n alone. The geometry contributes (1/R₁ − 1/R₂), the lens's total curvature. Steeper curves (small R) and higher index both increase power. A perfectly flat slab has R₁ = R₂ = ∞, so 1/f = 0 and it does not focus — it only shifts the image sideways.

Sign convention: the part everyone gets wrong

The equation is useless without a consistent sign rule, and this is where most errors creep in. In the standard convention, light travels left to right, and a radius is positive if its centre of curvature lies to the right (on the outgoing side of that surface), negative if to the left. Equivalently: a surface that bulges toward the incoming light gets a positive R₁; a surface that bulges away gets a negative R₂.

• An equiconvex lens: R₁ > 0 and R₂ < 0, so (1/R₁ − 1/R₂) is doubly positive → strong converging power.
• A biconcave lens: R₁ < 0 and R₂ > 0 → the bracket is negative → f < 0, a diverging lens.
• A plano surface: R = ∞, so its 1/R term is zero and simply drops out.

A flat piece of glass has infinite radius, and 1/∞ = 0 — that is why a plano-convex lens has exactly half the power of the equiconvex lens made from the same tool. Get a single sign wrong and a converging lens becomes a diverging one, so designers memorise the rule as "positive R means the centre of curvature is downstream."

Where it comes from: two refractions in a row

The equation is not a fit to data; it derives cleanly from applying Snell's law in the paraxial (small-angle) approximation at each surface. For a single spherical refracting surface separating media n₁ and n₂ with radius R, the object–image relation is:

• Step 1: n₂/v − n₁/u = (n₂ − n₁)/R at surface 1 (media 1 → glass n).
• Step 2: apply the same relation at surface 2 (media n → media 1), taking the image from step 1 as the object for step 2.
• Step 3: assume the lens is thin — the separation d between surfaces is negligible, so the intermediate image distance carries through unchanged.
• Step 4: add the two equations. The intermediate image distance cancels, leaving 1/v − 1/u = (n − 1)(1/R₁ − 1/R₂).

Setting the object at infinity (u = ∞) makes v = f by definition, and out drops the lensmaker's equation. The whole derivation rides on sin θ ≈ θ; rays far from the axis violate it, which is exactly the origin of spherical aberration. The equation gives you the ideal Gaussian focal length that all the real rays are trying, and failing, to hit.

The medium matters: (n − 1) becomes (n/n_m − 1)

The plain (n − 1) factor secretly assumes the lens sits in a vacuum or air (n ≈ 1.0003). Immerse the lens in a medium of index n_m and the correct factor is (n/n_m − 1), because refraction depends only on the ratio of indices across each surface. This has dramatic consequences:

• An equiconvex crown-glass lens (n = 1.52) with f = 96 mm in air balloons to f ≈ 350 mm in water (n_m = 1.33), because the glass–water contrast (1.52/1.33 − 1 = 0.14) is far weaker than the glass–air contrast (0.52).
• This is why your eyes cannot focus underwater: the cornea, which does two-thirds of the eye's focusing in air, loses almost all its power once water replaces the air on its front surface. A diving mask restores an air gap and your focus with it.
• Oil-immersion microscope objectives exploit the same effect in reverse, using n_m ≈ 1.515 oil to match the glass and eliminate a refraction, capturing steeper rays for higher resolution.

The fish that see sharply underwater solve it with nearly spherical, ultra-high-index lenses (n up to ~1.55 at the centre, graded) to claw back power lost to the small index contrast with water.

Worked design and thick-lens corrections

Suppose you need a +5.00 D spectacle lens (f = 200 mm) in polycarbonate, n = 1.586. Pick a plano-convex form (R₂ = ∞): 1/f = (0.586)(1/R₁), so R₁ = 0.586 × 200 mm = 117 mm. Grinding one face to a 117 mm radius delivers the prescription — no ray tracing needed. Switch to high-index n = 1.74 glass and the same power needs only R₁ = 0.74 × 200 = 148 mm: a flatter, thinner, lighter lens, which is precisely why premium spectacle lenses advertise their index.

Real lenses have thickness, and for that the thick-lens version restores the term dropped in Step 3:

• 1/f = (n − 1) [ 1/R₁ − 1/R₂ + (n − 1)d / (n R₁ R₂) ]

where d is the centre thickness. For a d = 5 mm, n = 1.52 equiconvex lens with R = ±100 mm, the correction term shifts f by about +0.9%, moving f from 96.2 mm to roughly 97 mm — small but the difference between a lens that meets spec and one that does not for a precision instrument. As d → 0 the extra term vanishes and the thin-lens form returns.

Chromatic dispersion: n is not one number

The equation hides a subtlety in the letter n: refractive index depends on wavelength (dispersion), so a single lens has a different focal length for every colour. For crown glass, n ranges from about 1.514 at 656 nm (red, Fraunhofer C line) to 1.523 at 486 nm (blue, F line). Plug both into the lensmaker's equation and the blue focal length is roughly 1.7% shorter than the red — blue focuses closer to the lens. This is chromatic aberration, the coloured fringing on cheap optics.

The spread is quantified by the Abbe number V = (n_d − 1)/(n_F − n_C), typically ~59 for crown and ~36 for dense flint glass; higher V means less dispersion. Because the lensmaker's equation is linear in (n − 1), you can cancel the colour error by cementing a positive crown element to a negative flint element (an achromatic doublet), choosing curvatures so the red and blue focal lengths coincide while the net power stays positive. That trick, refined from the 1750s onward, is why every serious camera lens is a stack of elements, not a single piece of glass.

The same crown-glass blank (n = 1.52) gives wildly different focal lengths depending on how the two faces are curved. Radii in mm, positive toward outgoing light.

Lens shape	R₁ (mm)	R₂ (mm)	f in air (mm)	Optical power (D)
Equiconvex (converging)	+100	−100	+96.2	+10.4
Plano-convex	+100	∞ (flat)	+192	+5.2
Symmetric biconcave (diverging)	−100	+100	−96.2	−10.4
Positive meniscus	+50	+100	+192	+5.2
Same equiconvex, but in water (n_m=1.33)	+100	−100	+350	+2.9