Lens Maker Equation Calculator

Lens Maker Equation Calculator using the thick-lens formula to get the focal length and lens power from R₁, R₂, d and n, or the front radius, back radius or index for any target f.

Lens Maker Equation Calculator With Steps

The Lens Maker Equation Calculator works the equation in four directions, picked from the Solve For list. Each answer comes with a step list that shows the formula, the values used and the result, so you can follow the full solution.

Four result cards follow the main answer. They show the lens power in diopters with the lens shape, the shape factor with equal-radii and best-form options, and the back focal length with edge thickness. The last card gives the working f-number, the color focus spread and the Airy spot size.

The Advanced panel, titled Medium, Size and Glass, sets the surrounding medium index n₀, the lens diameter and the glass’s Abbe number. Enter 0 for a flat surface, and set d = 0 for the thin-lens answer. The page runs in any phone browser, so it works as a lens maker equation calculator app without an install.

Lens Maker’s Formula

The lensmaker’s equation links the focal length of a lens to its surface radii and the index of its glass. For a thin lens in air it takes its most familiar form, the one most textbooks and the Lens Maker Equation Calculator start from.

$$\frac{1}{f} = (n – 1)\left(\frac{1}{R_1} – \frac{1}{R_2}\right)$$

Here f is the focal length, n the refractive index of the lens, R₁ the radius of the surface light meets first and R₂ the radius of the second surface. A lens with n = 1.5 and two 200 mm radii, R₁ = +200 mm and R₂ = −200 mm, has f = 200 mm.

The Class 12 NCERT textbook writes it as 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂), equation 9.21 in its Ray Optics chapter. There n₂₁ = n₂/n₁ is the lens index relative to the medium around it, which equals n when the lens sits in air.

Lens power is the inverse of focal length in meters, so P = 1/f gives diopters (D), with 1 D = 1 m⁻¹. A 200 mm lens has a power of +5 D. In a medium other than air the calculator reports P = n₀/f, the form used for vision optics.

Thick Lens Formula

Real lenses have a center thickness d, and the full equation adds a term for it. The medium index n₀ enters as well, so the same formula covers a lens in air, water or oil.

$$\frac{1}{f} = \frac{n – n_0}{n_0}\left[\frac{1}{R_1} – \frac{1}{R_2} + \frac{(n – n_0)\,d}{n R_1 R_2}\right]$$

The default lens in the Lens Maker Equation Calculator is N-BK7 glass, with n = 1.5168 at 587.6 nm. With R₁ = 60 mm, R₂ = −120 mm and d = 6 mm, the thick-lens focal length is 78.29 mm and the power is +12.77 D. Dropping the thickness gives 77.40 mm, a 1.1% gap that grows with thicker, more curved lenses.

Focal length is measured from the principal planes, which sit inside a thick lens. On a bench you measure from the glass, so the third card gives the back focal length, 75.62 mm here, and the front focal length, 76.95 mm.

$$BFL = f\left(1 – \frac{(n – n_0)\,d}{n R_1}\right)$$

How to Decide R1 and R2

The calculator uses the Cartesian sign convention, with light traveling left to right. A radius is positive when the surface’s center of curvature lies to the right of the surface, and negative when it lies to the left. NCERT uses the same rule, measuring distances along the direction of incident light as positive.

For a biconvex lens that makes R₁ positive and R₂ negative. Some sources, including the RP Photonics lens article, count every convex surface as positive and write the formula with a plus sign. Both give the same focal length, as long as the signs match the formula.

The radius of curvature is the radius of the sphere a lens surface is cut from. A flat surface has an infinite radius and no curvature, so enter 0 for it in the calculator. The six lens shapes follow from the two signs.

The two broad types of lens are converging, with positive f, and diverging, with negative f. A double concave lens of N-BK7 with R₁ = −50 mm, R₂ = +50 mm and d = 2 mm gives f = −48.05 mm, a power of −20.81 D. Its thin-lens value is −48.37 mm.

Lens Formula vs Lens Maker Formula

The lens maker formula finds f from the shape and glass of a lens. The lens formula then uses that f to locate an image, which is why the two are often confused.

$$\frac{1}{v} – \frac{1}{u} = \frac{1}{f}$$

Here u is the object distance and v the image distance, both measured from the lens with the same sign convention. Rearranged, 1/v = 1/f + 1/u. An object 30 cm in front of a 20 cm lens has u = −30 cm, so 1/v = 1/20 − 1/30 = 1/60. The real image forms 60 cm behind the lens, with magnification v/u = −2.

The minus sign in the lens formula comes from the sign convention, since a real object always sits on the negative side. For a convex lens, v turns negative only when the object is inside the focal length, giving a virtual, upright image on the same side as the object.

Use the lens maker formula, or the Lens Maker Equation Calculator, when you design or check a lens from its radii and material. Use the lens formula when the lens is already known and you need the image position or size.

Lens Maker’s Formula Derivation

The derivation applies refraction at a single spherical surface twice. For a surface of radius R between media n₁ and n₂, paraxial rays obey this relation.

$$\frac{n_2}{v} – \frac{n_1}{u} = \frac{n_2 – n_1}{R}$$

The first surface forms an image inside the glass, and that image is the object for the second surface. For a thin lens the two distances inside the glass cancel when the two equations are added. Setting the object at infinity then leaves v = f and the lens maker’s formula.

Keeping the glass thickness instead of letting it shrink to zero adds the d term of the thick-lens version. The name comes from its use: lens makers choose the radii to grind for a needed focal length.

Focal Length in Water and Other Media

A lens bends light because its index differs from its surroundings. The smaller that difference, the weaker the lens, so the same glass loses most of its power under water.

The values use the default N-BK7 lens, with water and ethanol from the standard index list and glycerol at n20/D 1.474 from Sigma-Aldrich. In glycerol the lens is almost index-matched, so its power nearly vanishes.

A lens with an index below its medium reverses its behavior. An air bubble in water is a convex shape that diverges light, and the calculator flags that case. The Index of Refraction Calculator finds n for other liquids and glasses.

Shape Factor and Best Form

Many shapes give the same focal length, from a symmetric biconvex lens to a meniscus. The Coddington shape factor q = (R₂ + R₁)/(R₂ − R₁) describes this bending. It is 0 for a symmetric lens and +1 for a plano-convex lens with its curved side first.

Bending does not change the focal length, but it changes spherical aberration. For a distant object the least aberration comes from q = 2(n² − 1)/(n + 2), which gives 0.714 at n = 1.5 in the University of Arizona aberrations lab. For N-BK7 it gives 0.740.

The default lens has q = 0.333. For the same 78.29 mm focal length, the second card lists an equal-radii biconvex lens at ±79.88 mm and a best-form lens at R₁ = 46.25 mm and R₂ = −309.1 mm. A plano-convex lens comes close to best form when its curved side faces the collimated beam, and turning it around raises spherical aberration sharply.

Edge Thickness and Lens Speed

Each curved surface sags below the plane of its vertex across the lens diameter. Over a 25.4 mm diameter, the default lens sags 1.359 mm at the front and 0.674 mm at the back, leaving an edge 3.967 mm thick.

When the surfaces would cross before the rim, the Lens Maker Equation Calculator warns that edge thickness is below zero and gives the minimum center thickness. At d = 0 it reports the thinnest glass you could actually grind, since no real lens has zero thickness.

The fourth card turns focal length and diameter into the working f-number, N = f/D. The default lens is f/3.08, and its diffraction-limited Airy spot is 4.42 µm across at 587.6 nm. A diverging lens shows its full divergence angle in place of the Airy spot. The Angular Resolution Calculator works from the same Airy pattern.

Index, Glass and Wavelength

Glass index changes with wavelength, so a lens has a slightly different focal length for each color. Glass catalogs quote n_d at 587.6 nm, and N-BK7 is listed at 1.5168 with an Abbe number of 64.17 on refractiveindex.info.

For a single lens the spread between blue and red focus is about f ÷ V. That is 1.220 mm for the 78.29 mm default lens, and the fourth card shows it as the color focus spread. It always uses the focal length in air, since V is a property of the glass alone.

Enter the Abbe number of your glass under Advanced; the default 64.17 is N-BK7. A flint glass with a lower V spreads the colors further, which is why achromatic doublets pair a crown and a flint element. When only one wavelength matters, such as a laser line, enter the index at that wavelength instead.

The refractive index mode of the Lens Maker Equation Calculator works backward from a measured focal length. A lens with R₁ = 60 mm, R₂ = −120 mm and d = 6 mm that focuses at 100 mm in air must be made of glass with n = 1.4039.

Lens and Aperture Questions

These answers cover the questions people search most often next to a lens maker result. Each uses the same sign convention as the Lens Maker Equation Calculator, so the numbers can be checked in the tool.

What is the formula for the lensmaker’s equation?

For a thin lens in air it is 1/f = (n − 1)(1/R₁ − 1/R₂). For a thick lens, add (n − 1)d/(nR₁R₂) inside the bracket, and replace n − 1 with (n − n₀)/n₀ for a lens in another medium.

For a double concave lens the same formula applies with R₁ negative and R₂ positive. The bracket then turns negative, giving a negative focal length, which the Lens Maker Equation Calculator labels as a diverging lens.

How do I use the lens maker’s equation?

Assign signs to both radii, look up the glass index at your wavelength, and enter the center thickness. The equation returns f in the same length unit as the radii, and 1/f in meters gives the power in diopters.

To design a lens instead, fix the focal length you need and solve for one radius. In the calculator, a target of 100 mm with R₂ = −120 mm and d = 6 mm in N-BK7 needs R₁ = 89.23 mm.

What does an f/2.8 lens mean?

The f-number is focal length divided by aperture diameter. An f/2.8 lens with a 50 mm focal length has an aperture about 17.9 mm across, and a smaller f-number means a wider opening.

Light gathering scales with the inverse square of the f-number. An f/2 lens collects four times as much light as an f/4 lens, so f/2 is a wide aperture, not a low one. The Aperture Area Calculator converts any f-number into an opening area.

Is f/1.8 enough for low light?

An f/1.8 lens gathers (2.8 ÷ 1.8)² = 2.42 times the light of an f/2.8 lens, a gain of about 1.3 stops. That is often enough for indoor and evening shots at moderate ISO.

The trade-off is depth of field. A wider aperture keeps less of the scene sharp, which matters for close subjects and group photos where every face needs to stay in focus.

Why is it called a lens?

The word comes from Latin lens, meaning lentil, and entered English in the 1690s. A biconvex lens has the same double-convex shape as a lentil seed, according to Etymonline.

For eyeglass lenses specifically, the Diopter Calculator converts between power and focal length, and the Contact Lens Vertex Calculator moves a prescription from glasses to the eye.