Lens Combinations & Aberrations

A single lens can magnify a few times. But the microscopes and telescopes that revolutionized science use two or more lenses working together, achieving magnifications no single lens could deliver. The principle is simple and elegant: the image produced by the first lens becomes the object for the second lens. Chain enough lenses together and you can image bacteria, atoms, or distant galaxies.

This section also covers the two big imperfections of real lenses (called aberrations) and how lens designers correct for them. Both topics are AAMC content-list items.

Two-Lens Systems

The recipe for any two-lens problem is short:

1. Ignore the second lens for now. Use the thin-lens equation with lens 1 to find $d_{i1}$ and $m_1$.
2. The image from lens 1 becomes the object for lens 2. Object distance for lens 2: $d_{o2} = (\text{lens separation}) - d_{i1}$.
3. Apply the thin-lens equation again to lens 2 to find $d_{i2}$ and $m_2$.
4. Total magnification: $m_{total} = m_1 \times m_2$.

A Worked Example

Two converging lenses are 30 cm apart. Lens 1 has $f_1 = 10$ cm, Lens 2 has $f_2 = 15$ cm. Object is 20 cm in front of Lens 1.

Step 1 — Lens 1. $1/10 = 1/20 + 1/d_{i1} \Rightarrow d_{i1} = 20$ cm. $m_1 = -20/20 = -1$. Image: 20 cm behind lens 1, real, inverted, same size.

Step 2 — Object for Lens 2. Image from lens 1 is 20 cm behind lens 1; the lenses are 30 cm apart, so the image sits 10 cm in front of lens 2 → $d_{o2} = 10$ cm.

Step 3 — Lens 2. $1/15 = 1/10 + 1/d_{i2} \Rightarrow 1/d_{i2} = 1/15 - 1/10 = -1/30 \Rightarrow d_{i2} = -30$ cm. $m_2 = -(-30)/10 = +3$. Image: virtual ($d_{i2} < 0$), upright relative to its object, 3× magnified.

Step 4 — Total. $m_{total} = (-1)(+3) = -3$. Final image is inverted relative to the original object and 3× its size.

Lenses in Contact

When two thin lenses are placed directly against each other (separation = 0), you can skip the multi-step process. Their powers in diopters simply add:

Chromatic Aberration

Real lenses suffer from imperfections called aberrations. The first one: chromatic aberration, caused by dispersion (§8.6). The index of refraction depends on wavelength, so different colors focus at slightly different points.

• Violet light (higher $n$) bends more → focuses closer to the lens.
• Red light (lower $n$) bends less → focuses farther from the lens.
• The result: colored halos or fringes around the edges of an image, especially near contrast boundaries.

Correction: an achromatic doublet pairs a converging lens (made of “crown glass”) with a diverging lens (made of “flint glass”) of different dispersive properties. The two lenses’ chromatic effects largely cancel, producing a much sharper image. This is what’s inside high-quality cameras, microscopes, and telescopes.

Spherical Aberration

The second imperfection: spherical aberration. The outer edges of a spherical lens (or mirror) focus light at a shorter distance than the center. Rays hitting near the rim converge sooner than rays hitting near the optical axis. The result: a fuzzy image rather than a sharp one.

Correction: Use a lens with a non-spherical (aspheric) surface — engineering perfection but expensive. Or place an aperture (a small opening) in front of the lens to block the problematic outer rays. Cameras do this every time you “stop down” the aperture (smaller f-number) for a sharper image.