Lensmaker's Equation

The lensmaker's equation calculates focal length from lens curvature and refractive index.
Essential for optics and lens design.

The Formula

1/f = (n − 1) × [1/R₁ − 1/R₂]

The lensmaker's equation relates the focal length of a thin lens to its refractive index and the radii of curvature of its two surfaces.

Symbol	Meaning
f	Focal length of the lens (measured in meters, m, or centimeters, cm)
n	Refractive index of the lens material (dimensionless — glass is typically 1.5 to 1.9)
R₁	Radius of curvature of the first lens surface (positive if center of curvature is to the right)
R₂	Radius of curvature of the second lens surface (positive if center of curvature is to the right)

Using the standard sign convention (light travels left to right):

R is positive if the center of curvature is on the transmission side (right side)
R is negative if the center of curvature is on the incoming side (left side)
For a biconvex lens: R₁ is positive, R₂ is negative
For a biconcave lens: R₁ is negative, R₂ is positive
For a flat surface: R = infinity, so 1/R = 0

A biconvex crown glass lens (n = 1.52) has R₁ = +20 cm and R₂ = −20 cm. What is its focal length?

Apply the formula: 1/f = (1.52 − 1) × [1/20 − 1/(−20)]

1/f = 0.52 × [0.05 + 0.05] = 0.52 × 0.10

1/f = 0.052

f = 19.2 cm (converging lens)

A plano-convex lens (n = 1.50) has one flat side and one curved side with R = 30 cm. What is its focal length?

R₁ = +30 cm (curved side), R₂ = ∞ (flat side), so 1/R₂ = 0

1/f = (1.50 − 1) × [1/30 − 0] = 0.50 × 0.0333

1/f = 0.01667

f = 60 cm

Use the lensmaker's equation when designing or analyzing thin lenses.

Determining the focal length of a lens from its physical properties
Designing lenses with specific focal lengths for cameras, microscopes, or telescopes
Understanding how lens material (refractive index) affects focusing power
Comparing lens designs with different curvatures and materials
Note: This formula applies to thin lenses. For thick lenses, additional terms account for lens thickness.