Optics Lens Maker Formula: 5 Clear Steps Explained

What is the optics lens maker formula? For a thin lens in air, it is 1/f = (n − 1)(1/R₁ − 1/R₂). The formula links a lens's focal length f to the refractive index n of its material and the radii of curvature R₁ and R₂ of its two surfaces. It is the fundamental equation lens manufacturers use to design lenses — from eyeglasses to microscope objectives — with a precise focal length.

Here is how to use it in 5 clear steps.

Step 1: What the lens maker formula tells you

The lensmaker's equation is a recipe. Think of it like this: the radii of curvature (R₁ and R₂) define the shape of the mixing bowl. The refractive index (n) is the type of dough. The thickness (d) is how much dough you use. Change any ingredient and the focal length — the finished cake — changes.

The full lensmaker's equation for a thick lens in air is:

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

Where:

• f = focal length of the lens
• n = refractive index of the lens material (relative to the surrounding medium)
• R₁ = radius of curvature of the first surface (closest to the light source)
• R₂ = radius of curvature of the second surface
• d = lens thickness (distance between the two surface vertices along the axis)

The how does a lens work article explains the basics of refraction and focusing that underpin this equation.

Step 2: Apply the thin lens approximation

Most real lenses are thick — they have non-negligible thickness. But when the lens thickness d is much smaller than R₁ and R₂, the thickness term becomes small enough to ignore. This is the thin lens approximation.

It simplifies the equation to:

1/f ≈ (n − 1)(1/R₁ − 1/R₂)

This is the version you will see in most textbooks. It assumes the lens is thin enough that we can treat refraction as happening at a single plane in the centre of the lens.

When is a lens "thin"? There is no hard rule, but a common guideline is: if d is less than about 10% of the smaller radius of curvature, the thin lens approximation introduces less than 1% error for paraxial rays.

Step 3: Get the sign convention right

The equation only works if you assign the correct signs to R₁ and R₂. Here is the standard Cartesian sign convention for lenses in air:

• R₁ > 0 if the first surface is convex (centre of curvature lies to the right of the surface)
• R₁ < 0 if the first surface is concave (centre of curvature lies to the left)
• R₂ > 0 if the second surface is concave (centre of curvature lies to the right)
• R₂ < 0 if the second surface is convex (centre of curvature lies to the left)

For a biconvex lens (the typical magnifying glass shape), R₁ is positive and R₂ is negative. For a biconcave lens, both are negative. For a plano-convex lens, one radius is infinite (meaning 1/R = 0).

The convex lens guide has diagrams of these shapes and their optical effects.

Step 4: Calculate lens power in dioptres

Optical power P is simply the reciprocal of the focal length in metres:

P = 1 / f (with f in metres)

Power is measured in dioptres (D). A converging lens has positive power. A diverging lens has negative power. An optician prescribing −3.00 D glasses is giving you a diverging lens with a focal length of 1 / (−3.00) = −0.333 m (about −33 cm).

Step 5: Work through an example

Let us design a biconvex lens from crown glass with a focal length of 20 cm in air. Crown glass has a refractive index n = 1.52. We want both surfaces to have equal curvature (so R₂ = −R₁).

Using the thin lens formula:

1/f = (n − 1)(1/R₁ − 1/R₂)

1/(0.20) = (1.52 − 1)(1/R₁ − 1/(−R₁))

5.00 = 0.52 × (2/R₁)

5.00 = 1.04 / R₁

R₁ = 1.04 / 5.00 = 0.208 m = 20.8 cm

So each surface should have a radius of curvature of about 20.8 cm. The Snell's law guide covers the refractive indices of common materials if you want to experiment with different glass types.

When to use the thick lens version

The thin lens approximation works well for most everyday lenses. But when the lens is thick relative to its radii — like the human cornea or a thick condenser lens — you need the full version:

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

The extra term (n − 1)d / (n R₁ R₂) corrects for the fact that light actually refracts at two separate surfaces a distance d apart, not at a single plane. For many lenses this correction is small, but for precision applications — intraocular lens implants, high-end camera lenses, microscope objectives — it matters.

The HyperPhysics lensmaker's guide has an interactive calculator that handles both thin and thick lenses. The StatPearls article on the lensmaker's equation covers its clinical applications in ophthalmology. And the Wikipedia page on the lensmaker's formula provides the full derivation and more advanced variants.

Common applications

Lens manufacturers use this formula daily to design:

• Eyeglasses and contact lenses — opticians specify power in dioptres, and the lensmaker's equation tells them the required curvature and material
• Camera lenses — multi-element designs use the equation for each element, then combine powers
• Microscope objectives — short focal lengths require tight radii and high-index glass
• Intraocular lenses (IOLs) — cataract surgeons use adapted lensmaker formulas to calculate the exact IOL power for each patient

A clarification

People sometimes think the lensmaker's equation applies to every lens. It does not. It assumes spherical surfaces, paraxial rays (small angles close to the axis), and a homogeneous material. Modern aspheric lenses, diffractive optics, and gradient-index (GRIN) lenses need more advanced models. The equation is a starting point — an elegant one — but real lens design involves ray tracing software that accounts for aberrations the simple formula cannot handle.