Lens Physics (2D)

Refraction at two spheres

Each ray is traced exactly: it is intersected with the first sphere, bent by Snell's law n₁ sinθ₁ = n₂ sinθ₂ in vector form, intersected with the second sphere and bent again. No small-angle approximation is used for the drawn rays, so you can see spherical aberration: rays near the edge focus closer to the lens than rays near the axis.

Thick-lens power

The back focal distance measured from the rear vertex is f (1 − (n − 1) t / (n R₁)). The image position is also found independently with a paraxial ray-transfer (ABCD) matrix; the page checks that the traced image and the matrix image agree.

Dispersion

The refractive index follows n(λ) = n_d + k (1/λ² − 1/λ_d²), with k fixed by the Abbe number V = (n_d − 1)/(n_F − n_C). Blue light bends more, so f_F is shorter than f_C by about f/V.

Honest simplifications

• Meridional (2D) rays only: skew rays and coma/astigmatism of off-axis points are not shown, and the object tip is traced exactly while the image arrow uses the paraxial chief ray.
• Surfaces are perfect spheres without coatings; reflection losses and total internal reflection beyond the aperture are ignored (blocked rays are simply not drawn).
• The 3D page uses a thin-lens image formula measured from the lens centre; this page uses the exact thick-lens image, so values differ by a few millimetres.
• The vertical axis is stretched so thin lenses stay readable; the stretch factor is printed on the canvas.

• Load Magnifier: with the object inside the focal length the rays diverge and a virtual, upright image appears (dashed lines).
• Reduce the radius to 60 mm and watch the edge rays cross much nearer than the central ones in the aberration graph.
• Load Flint lens, strong colour and compare the blue and red focal points with the crown glass.
• Move the object past the focal length: the image jumps from virtual to real.