Formula & Calculator
Lensmaker's Equation
Calculates a lens's focal length from its refractive index and the curvature of its two surfaces.
Interpretation
1/f = (n−1)(1/R₁ − 1/R₂). Focal length of a thin lens in air, from radii of curvature and refractive index. Used in lens design.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f | Focal length | m |
| n | Lens refractive index | |
| R1 | Radius of curvature of first surface | m |
| R2 | Radius of curvature of second surface | m |
What it means
The lensmaker’s equation relates the focal length of a thin lens to its refractive index (n) and the radii of curvature of its surfaces (R₁ and R₂). The sign convention is important. This equation is used by optical designers to calculate lens curvatures for desired focal lengths. It is fundamental for designing camera lenses, eyeglasses, microscope objectives, and many other optical systems. Understanding this equation is essential for lens manufacturing and optical engineering. It assumes the lens is in air and is thin; for thick lenses, a more complex form is used.
Worked example
Lensmaker's Equation – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| n | 1.5 |
| R1 (cm) | 20 |
| R2 (cm) | −20 |
| Parameter | Value |
|---|---|
| n | 1.5 |
| f (cm) | 30 |
| R2 | ∞ |
Common mistakes
- Lensmaker’s equation: 1/f = (n−1)·(1/R₁ − 1/R₂) – for a thin lens in air.
- Sign convention for radii: R₁ is the radius of curvature of the first surface (light incidence side). R₂ of the second. Positive if the centre of curvature is on the outgoing side (for light).
- n: Refractive index of the lens material relative to air – must be > 1.
- Units: R₁, R₂, and f in the same units (e.g., metres).
- Assumes: Thin lens in air – for lenses in other media, adjust (n_lens/n_medium − 1).
Applications
The lensmaker's equation, 1/f = (n−1)(1/R₁ − 1/R₂), relates the focal length of a thin lens to its refractive index (n) and the radii of curvature (R₁, R₂) of its two surfaces. This is used by optical engineers to design lenses with specific focal lengths, to select materials, and to optimise lens shapes for aberration control. It is essential in designing lenses for cameras, microscopes, telescopes, and eyeglasses. By choosing appropriate radii and glass types, engineers can achieve the desired optical properties. This equation also underpins the design of complex lens systems. Understanding it is fundamental for anyone involved in lens manufacturing or optical system design.
- Optical lens design and manufacturing
- Selection of glass materials for desired focal lengths
- Design of achromatic and apochromatic lens systems
- Camera lens and microscope objective design
- Educational understanding of lens curvature effects
Frequently Asked Questions
It calculates the focal length of a lens based on its refractive index and the radii of curvature of its two surfaces: 1/f = (n−1)(1/R₁ − 1/R₂).
n = refractive index of the lens material.
R₁ = radius of curvature of the first surface (positive if convex toward incident light).
R₂ = radius of curvature of the second surface (positive if convex toward emerging light).
f = focal length.
Using the Cartesian sign convention: R is positive if the surface is convex toward the incoming light (for the first surface) and convex toward the outgoing light (for the second surface). This can be tricky.
It is derived by applying Snell's Law at each surface, using the paraxial approximation, and combining the two surface powers.
Power P = 1/f (in diopters). The Lensmaker's equation gives P = (n−1)(1/R₁ − 1/R₂).
A lens has n = 1.5, R₁ = 10 cm, R₂ = −15 cm (second surface concave). Then 1/f = (1.5−1)(1/10 − 1/(−15)) = 0.5(0.1 + 0.0667) = 0.5×0.1667 = 0.0833 → f = 12 cm.
- Using inconsistent sign conventions for R₁ and R₂.
- Forgetting that the formula is for a lens in air; if the surrounding medium is different, n must be relative.
- Neglecting lens thickness (thin lens assumption).
For a plano‑convex lens, one radius is infinite, so 1/R = 0, simplifying the equation to 1/f = (n−1)/R for the curved surface.
A higher refractive index increases the power for the same radii, giving a shorter focal length.
It allows designers to choose radii and materials to achieve a desired focal length while controlling aberrations.