Lens Maker Equation Calculator
Enter the two radii of curvature and the refractive index of the lens material to compute focal length and optical power. Switch to thick-lens mode to add the lens thickness, or pick a common material to auto-fill its refractive index. The result panel shows the full worked calculation and a chart of how focal length varies with R1.
Formula
Worked example
A biconvex crown-glass lens (n = 1.52) with R1 = +50 mm, R2 = -50 mm, thickness d = 5 mm: (n-1) = 0.52; 1/R1 - 1/R2 = 1/50 - 1/(-50) = 0.04 mm^-1; thick-lens term = 0.52 x 5 / (1.52 x 50 x (-50)) = -0.000068 mm^-1; 1/f = 0.52 x (0.04 - 0.000068) = 0.020765 mm^-1; f = 48.2 mm.
The lens maker equation
The lens maker equation (also written lensmaker's equation) links the focal length of a thin or thick lens to the geometry of its two curved surfaces and the refractive index of the material. In its full thick-lens form it reads: 1/f = (n-1) * [1/R1 - 1/R2 + (n-1)*d/(n*R1*R2)]. For a thin lens, where the thickness d is negligible, the last term drops out and the formula simplifies to 1/f = (n-1) * (1/R1 - 1/R2). Positive focal lengths indicate converging lenses that bring parallel light to a real focus. Negative focal lengths indicate diverging lenses that spread light as if from a virtual focal point.
Sign convention for radii of curvature
This calculator uses the standard Cartesian sign convention. R1 is the radius of the front surface (the one the light hits first): it is positive when the centre of curvature is to the right of the surface (convex toward incoming light) and negative when it is to the left (concave toward incoming light). R2 is the radius of the rear surface with the same rule. For a classic biconvex converging lens: R1 > 0, R2 < 0. For a biconcave diverging lens: R1 < 0, R2 > 0. A plano-convex lens has one flat surface: represent it with a very large radius (for example 1e9) to approximate infinity. The sign of the computed focal length tells you immediately whether the lens converges or diverges light.
Optical power and diopters
Optical power (P) is the reciprocal of focal length measured in metres: P = 1/f. It is expressed in diopters (D), which are inverse metres. A lens with f = 50 mm = 0.05 m has P = 1/0.05 = +20 D. Eyeglass prescriptions are written in diopters: a +2 D lens corrects farsightedness (hyperopia) and a -3 D lens corrects nearsightedness (myopia). When two thin lenses are placed in contact, their powers simply add: P_total = P1 + P2. This additive property makes diopters convenient for optometry and optical system design.
Thin lens vs. thick lens
The thin-lens approximation is valid when the lens thickness is much smaller than both radii of curvature. Most introductory optics problems use this form because the algebra is simpler. However, real camera and microscope lenses have significant thickness, and the thin-lens formula can produce errors of several percent or more. The thick-lens version of the lensmaker's equation adds a correction term that accounts for the fact that the two refractions happen at surfaces that are spatially separated by d. Switching this calculator to "thick lens" mode unlocks the thickness field and applies the full formula automatically.
Practical applications
Eyeglass lenses, camera objectives, microscope objectives, telescope eyepieces, and contact lenses are all designed using the lensmaker's equation. An optician uses the formula in reverse: starting from a required focal length (the patient's prescription) and choosing a lens material (its refractive index), they calculate the radii of curvature that must be ground onto the glass or polished into the polymer blank. Material choice matters because a higher refractive index allows the same focal length to be achieved with shallower (flatter) curves, producing thinner and lighter lenses. This is why high-index plastic lenses (n approximately 1.6 to 1.74) have become popular for strong prescriptions.
Refractive indices of common optical materials
| Material | Refractive index (n) | Typical use |
|---|---|---|
| Crown glass | 1.52 | Camera lenses, eyeglasses |
| Flint glass | 1.62 | Achromatic doublets, prisms |
| Borosilicate glass | 1.47 | Lab glassware, telescope mirrors |
| Fused silica (quartz) | 1.46 | UV optics, fiber optics |
| Acrylic (PMMA) | 1.49 | Plastic lenses, displays |
| Polycarbonate | 1.586 | Safety glasses, eyewear |
| Sapphire | 1.77 | Watch crystals, high-durability windows |
| Water (20 degrees C) | 1.33 | Aquatic optics, immersion objectives |
| Air (standard) | 1.000293 | Reference medium |
Values at visible wavelengths (~589 nm, sodium D line). Refractive index varies with wavelength (dispersion).
Frequently asked questions
What does a negative focal length mean?
A negative focal length means the lens is diverging: it spreads parallel light outward so that the rays appear to come from a virtual focus on the same side as the incoming light. Biconcave lenses and plano-concave lenses typically have negative focal lengths. In eyewear, negative-power lenses correct myopia (nearsightedness).
What sign convention does this calculator use?
This calculator uses the Cartesian sign convention. Distances measured to the right of a surface are positive; distances to the left are negative. For a biconvex lens: R1 is positive (the front surface curves away from the incoming light) and R2 is negative (the rear surface curves back toward the incoming light). For a biconcave lens the signs are reversed.
How do I enter a flat (plano) surface?
A perfectly flat surface has an infinite radius of curvature. Enter a very large number such as 1000000 (or 1e6) mm to approximate it. For example, a plano-convex lens with R1 = 50 mm and R2 = flat would use R2 = 1000000 mm.
What is optical power in diopters?
Optical power P = 1/f, where f is the focal length in metres. The unit is diopters (D) or inverse metres (m^-1). A converging lens with f = 100 mm = 0.1 m has P = +10 D. Eyeglass prescriptions list the power in diopters directly: +2.50 D means a 400 mm focal-length converging lens.
When should I use thick-lens mode?
Use thick-lens mode whenever the lens center thickness is not negligible compared with the radii of curvature. A common rule of thumb is: if d/R is more than about 0.05 (5%), the thin-lens approximation introduces noticeable error. Camera lenses, wide-angle optics, and strong eyeglass lenses all benefit from the thick-lens correction.
Why does refractive index matter for focal length?
A higher refractive index means the material bends light more per unit of surface curvature. Consequently, a lens made from high-index glass (n = 1.7) achieves the same focal length with shallower, flatter surfaces than a lens made from crown glass (n = 1.52). Flatter surfaces are easier to manufacture, produce thinner and lighter lenses, and reduce some optical aberrations.
Can I use this calculator for contact lenses?
Yes, but with caveats. Contact lenses sit on the eye, so the relevant refractive index is that of the lens material in contact with the tear film (n approximately 1.336), not air. The lensmaker's equation still applies, but the surrounding medium index (assumed to be 1 here) would need adjustment for a fully accurate calculation in non-air media. For a quick estimate of the required curvatures, the standard formula gives a good starting point.