# Lensmaker Equation Calculator

Created by Davide Borchia
Last updated: Feb 18, 2023

Many optical systems need carefully manufactured elements: our lensmaker equation calculator will help you with this problem: learn how to calculate the focal length of a lens if you know its shape or to calculate the required parameters to achieve the wanted focal length.

• What is a lens;
• The elements of a lens, with a focus (pun intended) on the radius of curvature;
• The lensmaker formula: how to calculate a lens thickness, focal length, and curvature radii; and
• How to use our lensmaker equation calculator.

## What is a lens?

A lens is an optical device that modifies the path of a light beam crossing it. Lenses can focus light, meaning that parallel rays are concentrated in a single point.

There are two main types of lenses:

• Concave lenses and;
• Convex lenses.

A lens is identified by a small set of parameters:

• The focal length;
• The thickness of the lens;
• The curvature of the lens's surfaces.

Even if we don't think about it often, lenses are pretty much everywhere. If you wear glasses, you are wearing a pair of carefully fabricated lenses that correct your vision. If you are a birdwatcher, your binoculars have a plethora of lenses inside them, and astronomers' telescopes are similarly using lenses to bring the wonders of the cosmos to our eyes.

Lenses are often studied in the thin lens approximation: in this case, we assume that the thickness of the lens is much smaller than the perpendicular size of the device. This approximation simplifies many equations: we can see it in the lensmaker formula's case.

## The lensmaker equation: calculate the focal length of a lens

Knowing the right parameters, you can calculate the focal length of a lens using the lensmaker equation:

$\footnotesize \frac{1}{f} = (n-1)\left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)\cdot d}{n\cdot R_1 \cdot R_2}\right]$

Where:

• $f$ is the **focal length of the lens;
• $n$ is the index of refraction of the lens material;
• $R_1$ and $R_2$ are the radii of curvature of the lens (the first one for the side closer to the light source); and
• $d$ is the thickness of the lens.

Two words about the radius of curvature of a lens in the lensmaker equation. To define it, we need a sign convention: it is necessary to decide if our lens has positive or negative curvature, which in turn helps us decide where to place the center of curvature (the center of the circle that would fit the lens face). We define the vertex as the point where the lens' face intersects the optical axis.

The convention for the radius of curvature is:

• If the face's vertex lies on the left of the center of curvature, we have a positive radius of curvature;
• If the vertex is found on the right of the center of curvature, the radius of curvature will be negative.

## The lensmaker formula in the thin lenses approximation

If you are using the thin lens approximation, you can simplify the lensmaker equation by assuming that the lens thickness is 0.

With a negligible lens thickness, we calculate the focal length with the formula:

$\footnotesize \frac{1}{f} = (n-1)\left[ \frac{1}{R_1} - \frac{1}{R_2} \right]$

## How to use our lensmaker equation calculator

You can use our lensmaker equation calculator in two ways:

• To calculate the focal length of the lens if you know the fabrication parameters; or
• To calculate a missing parameter if you know the focal length.

The calculator uses, by default, the thin lens approximation. If you want to change this, and use the lensmaker equation for a thick lens, click on advanced mode: you will see the variable lens thickness.

🙋 If you want to calculate the lens thickness, you have to lock the variables for the radii of curvature and the refractive index beforehand! Do this by clicking on the right of the input field".

Davide Borchia
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Refractive index
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