Lensmaker's Equation Calculator
Enter the refractive index, front and back radii of curvature, and (optionally) the centre thickness — this calculator returns the focal length in millimetres and the optical power in diopters using the lensmaker's equation.
Focal length f
100 mm (10 cm, converging)
- Optical power (diopters, 1/m)
- 10
- 1/f (per mm)
- 0.01
- Lens shape
- biconvex
- Focal length (metres)
- 0.1
Uses the lensmaker's equation 1/f = (n − 1)·[1/R₁ − 1/R₂ + (n − 1)·d / (n·R₁·R₂)] with the Cartesian sign convention (R > 0 if the centre of curvature lies on the far side). Setting d = 0 recovers the thin-lens form.
How to use this calculator
Enter the refractive index n of the lens material (crown glass ≈ 1.5). Enter the two radii of curvature R₁ (front) and R₂ (back) in millimetres, following the Cartesian sign convention: R is positive when the centre of curvature lies on the far side of the surface. For a symmetric biconvex lens, R₁ > 0 and R₂ < 0. Enter 1e9 (a billion millimetres) to represent a flat surface. Leave centre thickness d at 0 for the thin-lens form or set a positive value to include the thick-lens correction.
How the calculation works
The lensmaker's equation relates the focal length f of a lens in air to the refractive index n, the two radii of curvature R₁ and R₂, and the centre thickness d: 1/f = (n − 1) · [1/R₁ − 1/R₂ + (n − 1)·d / (n·R₁·R₂)]. It follows from applying the single-surface refraction equation twice (at the front then the back surface) in the paraxial limit. When d is zero the thick term vanishes and the classic thin-lens form 1/f = (n − 1)·(1/R₁ − 1/R₂) is recovered. The optical power in diopters (D) is the reciprocal of the focal length in metres — for f in millimetres, D = 1000 / f_mm. A positive focal length identifies a converging lens; a negative focal length a diverging lens.
Worked example
A biconvex crown-glass lens with n = 1.5 and radii R₁ = +100 mm, R₂ = −100 mm (thin lens, d = 0). Plugging in: 1/f = (1.5 − 1) × (1/100 − 1/(−100)) = 0.5 × (0.01 + 0.01) = 0.01 mm⁻¹. Inverting gives f = 100 mm (10 cm). Optical power D = 1000 / 100 = 10 diopters. This matches Hecht Optics example 5.11 style problems for a symmetric biconvex lens. If the same lens has a 5 mm centre thickness, the thick-lens correction adds (0.5·5)/(1.5·100·(−100)) = −1/6000 mm⁻¹, changing 1/f to 0.01 − 0.000167 ≈ 0.009833, so f ≈ 101.7 mm — a small but non-trivial shift for high-precision optics.
Frequently asked questions
What is the lensmaker's equation used for?
It relates the focal length of a thin or thick lens to the refractive index of the glass and the curvature of the two surfaces. Lens designers use it to pick radii that will give a target focal length; opticians use it to check a prescription; physics students use it to link ray optics to refraction. It assumes paraxial rays (small angles relative to the optical axis) and monochromatic light.
What sign convention should I use for R₁ and R₂?
This calculator uses the Cartesian sign convention (the one in Hecht Optics and most modern physics texts): light travels left-to-right, and a radius is positive when the centre of curvature lies on the far (right) side of the surface. For a symmetric biconvex lens R₁ > 0 and R₂ < 0. For a symmetric biconcave lens R₁ < 0 and R₂ > 0. For a flat surface use R = ∞ — the calculator accepts 1e9 (a billion millimetres) as effectively flat.
When is the thin-lens approximation good enough?
When the centre thickness d is small compared to the focal length and the radii — a common rule of thumb is d < 0.1·|f| for better than about 1% accuracy. For thick camera lenses, ophthalmic lenses in high prescriptions, and telescope objectives, include the thickness term. Set d > 0 in this calculator to switch on the correction.
How do I get optical power in diopters?
Optical power P = 1/f when f is in metres. This calculator returns diopters directly. If the focal length is 100 mm (0.1 m), the power is 10.0 diopters. Prescription eyewear is specified in diopters (e.g. a −3.5 D lens is diverging with f ≈ −286 mm).
Does the equation work for lenses immersed in water or oil?
The standard form assumes the lens is in air (n_medium ≈ 1). If the surrounding medium has refractive index n_m, replace (n − 1) with (n/n_m − 1) throughout. A lens immersed in water has a longer focal length than the same lens in air because the refractive-index contrast is smaller.
Why does the same glass lens have different focal lengths for red and blue light?
Refractive index depends on wavelength (dispersion), so n in the equation is slightly higher for blue light than for red. That gives a shorter focal length for blue and a longer one for red — the chromatic-aberration effect that achromatic doublets are designed to cancel. Use the n value at the wavelength you care about (typically the sodium d-line, 589.3 nm, for visible-light optics).