Table of Contents
What Is the Lensmaker's Equation?
The lensmaker's equation relates the focal length of a lens to its physical properties: the refractive index of the lens material and the radii of curvature of its two surfaces. This equation is fundamental in optical design, allowing engineers to calculate the focusing power of lenses from their geometry. It applies to both converging (positive) and diverging (negative) lenses.
For thin lenses where thickness is negligible compared to the radii of curvature, the simplified thin lens form is used. For thick lenses, an additional term accounts for the finite path length of light through the lens material, which shifts the focal length slightly.
The Formula
Where f is the focal length, n is the lens refractive index, R1 and R2 are the radii of curvature of the two surfaces, and d is the lens thickness. For a thin lens (d=0), the last term vanishes.
Sign Conventions
| Surface | Convex (center left) | Concave (center right) | Flat |
|---|---|---|---|
| R1 (front) | Positive (+) | Negative (-) | Infinity |
| R2 (back) | Negative (-) | Positive (+) | Infinity |
- A biconvex lens: R1 > 0, R2 < 0 (converging)
- A biconcave lens: R1 < 0, R2 > 0 (diverging)
- Plano-convex: R1 > 0, R2 = infinity
Frequently Asked Questions
What is optical power?
Optical power is the reciprocal of focal length (P = 1/f) measured in diopters (D) when f is in meters. Positive power means converging, negative means diverging. Eyeglass prescriptions are written in diopters.
When should I use the thick lens equation?
Use the thick lens form when the lens thickness is more than about 10% of the smaller radius of curvature. For most thin eyeglass lenses and camera lenses modeled individually, the thin lens approximation is adequate. Thick condensers and ball lenses require the full equation.
Does the surrounding medium matter?
Yes. The standard lensmaker's equation assumes the lens is in air (n=1). If the lens is immersed in another medium (like water, n=1.33), replace (n-1) with (n/n_medium - 1) throughout the equation.