The Lens Maker's Equation

The thin-lens equation $1/f = 1/d_o + 1/d_i$ tells you where the image goes once you know the focal length. But where does the focal length come from? How does a piece of curved glass end up with $f = 10$ cm rather than $f = 20$ cm?

The answer is the lens maker's equation, which connects the focal length to the physical properties of the lens — the curvature of its two surfaces and the refractive index of the glass:

$\dfrac{1}{f} = (n - 1)\left[\dfrac{1}{R_{1}} - \dfrac{1}{R_{2}}\right]$

Here $n$ is the refractive index of the lens material, $R_{1}$ is the radius of curvature of the front surface (the one the light hits first), and $R_{2}$ is the radius of curvature of the back surface.

Sign convention for radii

The sign convention for $R$ is critical and frequently confusing:

• A surface is convex (center of curvature on the far side) → $R > 0$
• A surface is concave (center of curvature on the near side) → $R < 0$
• A flat surface → $R = \infty$ (or $1/R = 0$)

For our problem: $R_{1} = +20$ cm (front surface is convex — it bulges toward the incoming light). $R_{2} = -30$ cm (back surface is concave — it curves away from the exiting light, also bulging toward the incoming light). This is a biconvex lens where both surfaces curve outward.

Solving the problem

$\dfrac{1}{f} = (1.50 - 1)\left[\dfrac{1}{20} - \dfrac{1}{-30}\right] = 0.50 \times \left[\dfrac{1}{20} + \dfrac{1}{30}\right]$

$= 0.50 \times \left[\dfrac{3}{60} + \dfrac{2}{60}\right] = 0.50 \times \dfrac{5}{60} = 0.50 \times 0.08\overline{3} = 0.041\overline{6}$

$f = \dfrac{1}{0.041\overline{6}} = 24 \text{ cm}$

The focal length is 24 cm, positive (converging lens).

Understanding each factor

$(n - 1)$: This is the lens's "refractive power" relative to the surrounding medium (air, $n_{\text{air}} = 1$). The higher the refractive index, the more strongly the lens bends light. Diamond ($n = 2.42$) has $n - 1 = 1.42$, more than twice that of glass ($n - 1 = 0.50$), which is why diamond lenses (if practical) would have very short focal lengths.

$1/R_{1}$: The curvature of the front surface. Smaller $R$ means more sharply curved, means more bending power. A flat surface ($R = \infty$) contributes zero.

$1/R_{2}$: The curvature of the back surface, but with the opposite sign convention. Notice the minus sign in $[1/R_{1} - 1/R_{2}]$: for a biconvex lens where both surfaces add converging power, $R_{1} > 0$ and $R_{2} < 0$, so $-1/R_{2}$ becomes positive, and both terms add.

Lens shapes and their properties

Different combinations of $R_{1}$ and $R_{2}$ give different lens shapes:

• Biconvex ($R_{1} > 0$, $R_{2} < 0$): Both surfaces converge. Strongest converging lens for given radii.
• Plano-convex ($R_{1} > 0$, $R_{2} = \infty$): Flat on one side. Used in many optical instruments.
• Meniscus converging ($R_{1} > 0$, $R_{2} > 0$, with $R_{1} < R_{2}$): Both surfaces curve the same way, but the front is more strongly curved. Still converging overall. Eyeglass lenses are often meniscus.
• Biconcave ($R_{1} < 0$, $R_{2} > 0$): Both surfaces diverge. Gives $f < 0$ (diverging lens).
• Plano-concave ($R_{1} = \infty$, $R_{2} > 0$): Flat on one side, concave on the other. Diverging.
• Meniscus diverging ($R_{1} > 0$, $R_{2} > 0$, with $R_{1} > R_{2}$): Looks like a meniscus but diverges.

The power of a lens (diopters)

Optometrists don't talk about focal length — they use power, measured in diopters (D):

$P = \dfrac{1}{f \text{ (in meters)}}$

For our lens: $P = 1/0.24 \approx 4.17$ D. A strong reading glass might be +3 D; a powerful magnifier might be +10 D.

Diopters are additive: placing two thin lenses in contact gives a combined power $P_{\text{total}} = P_{1} + P_{2}$. This is much more convenient than combining focal lengths (which involves $1/f_{\text{total}} = 1/f_{1} + 1/f_{2}$).

Worked examples with different shapes

Plano-convex, $R_{1} = 15$ cm, $R_{2} = \infty$, $n = 1.50$: $1/f = 0.50 \times [1/15 - 0] = 0.50/15 = 1/30$ $f = 30$ cm. Only one surface does the bending.

Symmetric biconvex, $R_{1} = 20$ cm, $R_{2} = -20$ cm, $n = 1.50$: $1/f = 0.50 \times [1/20 + 1/20] = 0.50 \times 1/10 = 1/20$ $f = 20$ cm. Both surfaces contribute equally.

Biconcave, $R_{1} = -25$ cm, $R_{2} = 25$ cm, $n = 1.50$: $1/f = 0.50 \times [-1/25 - 1/25] = 0.50 \times (-2/25) = -1/25$ $f = -25$ cm. Diverging — negative focal length.

Common mistakes

• Sign errors on $R_{2}$. Remember that in $1/R_{1} - 1/R_{2}$, the minus is already baked in. If the back surface is concave ($R_{2} < 0$), then $-1/R_{2}$ is positive, adding to the converging power. Don't accidentally apply a second negative.
• Using radius of curvature as focal length. $R$ and $f$ are different. For a plano-convex lens with $R_{1} = R$, $f = R/(n-1) = 2R$ (for $n = 1.5$). They're only equal in special cases.
• Forgetting this equation is for thin lenses in air. The full version for a lens of thickness $d$ in a medium of refractive index $n_m$ is more complex. The thin-lens-in-air version works when the lens is thin and surrounded by air on both sides.

Try it in the visualization

Drag the $R_{1}$ and $R_{2}$ sliders and watch the lens shape morph from biconvex through plano-convex to biconcave. The focal length updates live. Switch the glass material (crown glass, flint glass, diamond) to see how $n$ affects $f$. Toggle the ray diagram to see parallel rays converging (or diverging) at the computed focal point.