Optical systems: focal length, ray diagram and spherical aberration

The Optical system cell traces light through lenses and mirrors in your browser. Paraxial values (focal lengths, focal points, principal planes, the image) come from ABCD ray-transfer matrices; real rays are traced exactly with Snell's law to show spherical aberration. The same engine powers the Lens & Mirror Ray Tracer tool, so the cell and the tool page give identical numbers.

1. Open the Lens bench

Choose Notebook → Templates… and open Lens bench, then press Run all. The notebook holds five optical systems: a plano-convex lens both ways round, two thin lenses, a thick lens imaging an object and a cemented achromat.

2. Describe a system

Open the cell Plano-convex lens, curved side first. The field Elements (one per line, in the order the light meets them) contains:

thick R1=51.68 R2=inf t=5 n=N-BK7

Each line is one element, lengths in millimetres:

• lens f=100 — an ideal thin lens.
• gap 20 — the distance to the next element.
• surface R=50 n=1.5168 — a refracting surface into index n (R=inf is flat).
• thick R1=50 R2=-50 t=6 n=N-BK7 — a thick lens in the current medium.
• mirror R=-200 — a concave mirror facing the light.

A radius is positive when its centre of curvature lies to the right. Glass names (N-BK7, N-BAK4, N-SK16, F2, N-SF5, N-SF11, fused silica) fill in the index and Abbe number; # starts a comment.

Below the list set Object distance (mm, or inf), the Object height (mm) (finite objects) or Field angle (°) (objects at infinity), the Aperture radius at the first element (mm) and the number of Real rays.

3. Read the result

The summary gives the effective focal length (100 mm) and the back focal distance (96.7036 mm). The key figures add the front focal distance and the f-number.

The Ray diagram shows the real rays above and below the axis, the dashed paraxial chief ray, and the focal points F, F′ and principal planes H, H′ on the axis. The plot next to it shows the spherical aberration: how far each ray height misses the paraxial focus.

Open All result details for the tables Cardinal points (along the light), Surfaces and paraxial ray heights (marginal and chief ray height at every surface — the lens must be at least that large), Real rays and spherical aberration and, for glasses with an Abbe number, Chromatic focal shift (Abbe-number estimate).

4. Compare the two orientations

The cell Same lens, flat side first is a variant of the first one, so a comparison table shows both side by side. The focal length stays at 100 mm, but the marginal ray now misses the focus by about 4.4 mm instead of 1.1 mm: face the curved side towards parallel light.

5. Plot the spherical aberration

Under the first cell, Continue with offers Plot the spherical aberration (fit LSA = a·h² + b·h⁴). It adds a Curve fit cell with the ray heights and aberrations. Run it: the coefficient a is the third-order spherical aberration, b the fifth-order rest, and the plot shows how quickly the error grows towards the edge of the lens.

Image an object at twice the focal length makes a variant with the object at 2f, where a simple lens forms an inverted image of the same size.

6. Try your own system

Use Add cell and pick Optical system. Two thin lenses 50 mm apart (lens f=100, gap 50, lens f=100) give 66.6667 mm, as 1/f = 1/f₁ + 1/f₂ − d/(f₁f₂) predicts. The Cemented achromat N-BAK4 / N-SF5 cell shows a colour focus shift of about 0.13 mm instead of the 1.5 mm of the single lens.

The model is two-dimensional and monochromatic (plus the Abbe estimate): no skew rays, field aberration maps, lens-edge vignetting or diffraction.