Foundations — Aberrations — chromatic, spherical (concepts)

This page is the toolbox. Before you can watch a lens fail, you must know exactly what each letter in $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ means, what it looks like on a drawing, and why the topic cannot live without it. We build them in order — each one leans only on the ones above it.

0. What is a "point image"? (the thing that goes wrong)

Look at the figure. On the left, a perfect lens: every ray leaving the point on the left is bent so they all cross at one place on the right. On the right, a real lens: the rays miss each other — they cross at slightly different spots, so instead of a dot you get a fuzzy smear.

1. The ray and its angle $\theta$

Why do we need $\theta$ at all? Because bending depends on the angle of arrival. A ray that grazes the glass nearly flat (small $\theta$) barely bends; a ray that slams in steeply (large $\theta$) bends hard. In the figure, the near-axis ray in blue makes a tiny $\theta$; the rim ray in pink makes a big $\theta$. That single difference is the entire seed of spherical aberration — so $\theta$ earns its place first.

2. Ray height $h$ and "paraxial"

$h$ and $\theta$ are two views of the same fact: the higher up a ray enters (bigger $h$), the steeper it must bend to reach the focus (bigger $\theta$). For a lens of radius $R$, roughly $\theta$ grows with $h$.

3. Why $\sin \theta$, and why we chop it: the small-angle approximation

Light bending obeys Snell's law, which contains $\sin \theta$ — the sine of the angle. Sine is the tool that converts an angle into a ratio of side lengths on a right triangle (opposite over hypotenuse). We need it because refraction physically depends on that ratio, not on the raw angle.

But $\sin \theta$ is awkward to solve equations with. So physicists use its Taylor series — an exact way to write a curvy function as an endless sum of simple powers:

$\sin \theta =\theta -\frac{{\theta }^3}{6}+\frac{{\theta }^5}{120}-\dots$

The figure plots the true curve $\sin \theta$ (white) against the straight line $\theta$ (blue). Near $\theta =0$ they lie on top of each other — the approximation is superb. But as $\theta$ grows, the white curve droops below the line: the gap is exactly the $-{\theta }^3/6$ term we threw away.

4. Refractive index $n$

The figure is a prism splitting white light. The blue ray bends hardest (largest $n$), red least (smallest $n$). This single picture is the birth of chromatic aberration: if a lens bends blue harder than red, blue must come to focus sooner.

5. Radii $R_1$, $R_2$ and the shape factor $K$

Why split them? Because chromatic aberration changes only $n$ while $K$ stays fixed — this separation is what lets us differentiate later.

6. Focal length $f$ and the lensmaker's equation

Why $n-1$ and not $n$? Because bending is relative to the surrounding air ($n\approx 1$). A lens submerged in a medium of its own index wouldn't bend light at all — the "$-1$" measures the contrast.

7. Power $P$ — the same thing, flipped

That additivity is exactly why the Achromatic Doublet works — you glue two lenses and add their powers, arranging the colour errors to cancel while the total power survives.

8. Dispersive power $\omega$ and the Fraunhofer lines

To fix chromatic aberration we need one number for "how much a glass spreads colours." That number is $\omega$.

The prerequisite map

Read it top-down: the left branch (ray, height, sine cut) feeds spherical aberration; the middle/right branch ($n$, colour, $\omega$) plus the lensmaker's equation feeds chromatic aberration and its cure. Both pour into the parent topic.

Equipment checklist

Cover the right side and answer each. If any stumps you, reread that section above.

Once every item above is second nature, you are ready to watch both lies break in the parent note: Aberrations — Chromatic & Spherical.