Lensmaker's Equation

Equivalent forms

P = (n-1)(1/R_1 - 1/R_2)

1/f = (n-1)(1/R_1 - 1/R_2 + (n-1) t / (n R_1 R_2))

Two curvatures and one index — and you have engineered every microscope, telescope, and pair of glasses ever made.

Symbol	Name	SI unit	Dimension	Range
$n$	Refractive index Refractive index of the lens material (relative to surrounding medium)	dimensionless	1	1 – 2.5
$R_1$	First surface radius Radius of curvature of first surface (positive if convex toward incoming light)	m	L	-2 – 2
$R_2$	Second surface radius Radius of curvature of second surface (sign follows convention)	m	L	-2 – 2
$f$	Focal lengthoutput Focal length of the thin lens	m	L	-2 – 2

Unit systems

Symbol	SI	CGS	Imperial
$R_1$	m	cm	m
$R_2$	m	cm	m
$f$	m	cm	m

Where it holds

Valid for thin lenses (thickness much smaller than radii) and paraxial rays. For thick lenses, the full form with thickness term applies. Breaks down for chromatic aberration since n depends on wavelength.

Discovery

Multiple · 1690

Refined from earlier work by Kepler and Descartes. The modern thin-lens form was used by Huygens and codified during the 18th century as opticians and astronomers needed predictable lens design.

Try this

How does an optician decide the curvature of your eyeglass lens?

A double-convex glass lens (n=1.5) has surface radii R1 = 0.2 m and R2 = -0.2 m. Find its focal length.

Research status: stable

Real-world applications

Eyeglasses: opticians use this to compute curvatures for prescription power.
Camera lens design: optimizing curvatures and indices to balance aberrations.
Telescope objectives: choosing radii for desired focal length given glass type.
Achromatic doublets: combining crown and flint glass with computed curvatures to cancel chromatic aberration.

Common misconceptions

Sign convention is fixed — different textbooks use different sign conventions; the formula form changes accordingly.
Lens power depends only on shape — it also requires the refractive index step from the medium.
Thicker lens means stronger — only the surface curvatures and index matter for thin lenses.

Experimental verification

Verified by direct focal length measurements of lenses with known radii (from spherometer) and known indices (from refractometry).

Derivation

Apply the refraction-at-a-single-surface formula

(n_{1}/s + n_{2}/s' = (n_{2}-n_{1})/R)

at each lens surface and treat the image from the first as the object for the second.

For a thin lens in

air (n_{1} = n_{3} = 1

,

n_{2} = n)

, the result simplifies to

1/f = (n-1)(1/R_{1} - 1/R_{2})

.

Limiting cases

n = 1

⟶

1/f = 0

No index step means no bending — lens does nothing.

R_2 \to \infty

⟶

1/f = (n-1)/R_1

Plano-convex lens — second surface is flat.

R_1 = -R_2

⟶

1/f = 2(n-1)/R_1

Symmetric biconvex lens — twice the power of a plano-convex.

What if…

What if the lens is submerged in water?

Replace n by n_lens/n_water. A glass lens (1.5) in water (1.33) becomes much weaker: effective index step drops from 0.5 to 0.13.

What if R_1 and R_2 have the same sign (meniscus)?

Lens can be converging or diverging depending on the magnitudes — used for corrective vision and field flatteners.

1

Symmetric biconvex glass lens

Given ·

n:: 1.5
R 1:: 0.2
R 2:: -0.2

Find · f

Steps

Apply lensmaker's equation: $1/f = (n-1)(1/R_{1} - 1/R_{2})$
Substitute: $1/f = 0.5 * (1/0.2 - 1/(-0.2))$
Compute: $1/f = 0.5 * (5 + 5) = 5$
$f = 1/5 = 0.2\,\mathrm{m}$

Result ·

1/f = (1.5-1)*(1/0.2 - 1/(-0.2)) = 0.5 * (5 - (-5)) = 0.5 * 10 = 5

, so

f = 0.2\,\mathrm{m}

Thin Lens Equation→Snell's Law→Mirror Equation→