The Lensmaker's Equation

The lensmaker's equation connects a lens's focal length to its physical properties: the refractive index of the glass and the radii of curvature of its two surfaces.

The Equation

$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Where:

• $n$ is the refractive index of the lens material
• $R_1$ is the radius of curvature of the first surface (positive if center of curvature is to the right)
• $R_2$ is the radius of curvature of the second surface (positive if center of curvature is to the right)
• $f$ is the focal length

Sign Convention for Radii

• If a surface is convex toward incoming light: $R > 0$
• If a surface is concave toward incoming light: $R < 0$

For a standard biconvex lens: $R_1 > 0$ and $R_2 < 0$.

Example: A biconvex glass lens ($n = 1.5$) with $R_1 = 20\,\text{cm}$ and $R_2 = -20\,\text{cm}$:

$\frac{1}{f} = (1.5 - 1)\left(\frac{1}{20} - \frac{1}{-20}\right) = 0.5 \times 0.1 = 0.05$ $f = 20\,\text{cm}$

Two-Lens System

When two thin lenses are separated by a distance $d$, the combined focal length is:

$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$

When the lenses are in contact ($d = 0$):

$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$

Your Task

Implement the lensmaker's equation and the two-lens combination formula.