ISEGORIA / MATH ENCYCLOPEDIA
Optics: rays, waves, and colour
Lenses, interference, and polarization.
Before you begin: Waves and geometry
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Thin lens
Drag the object arrow along the axis, or drag the focal point F′. Three principal rays are traced from the tip of the object, each bent by the thin-lens rule, and a faint fan of other rays shows that every ray from that point meets again at the image. Move the object inside the focal length and the outgoing rays diverge: the image becomes virtual, found by extending them backwards. The graph below follows the image distance as the object moves.
Worked example. With f = 2 and d_o = 5, d_i = 10/3 ≈ 3.33 and m ≈ −0.67: beyond 2f the image is real, inverted and reduced.
Watch out. The thin-lens model ignores thickness and aberration. Distances are measured from the lens; d_i < 0 means a virtual image on the object side, and a diverging lens has f < 0.
What happens at d_o=f?
The image distance diverges: rays leave parallel.
2. Slit interference
Light of one wavelength passes through N narrow slits. The upper picture adds the waves from every slit and shows the time-averaged intensity: bright bands leave the slits along the directions where the path difference is a whole number of wavelengths. The graph shows the exact far-field intensity, the product of the multi-slit factor and the single-slit envelope, and the strip above it is what a distant screen would show. Change the wavelength, the slit spacing, the slit width and the number of slits.
Worked example. With λ = 550 nm and d = 2.5 µm, sin θ₁ = 0.22, so the first bright order is at about 12.7°.
Watch out. The graph is the far-field (Fraunhofer) pattern and uses no small-angle approximation. Close to the slits, in the upper picture, the bright bands are hyperbolae that only become straight far away.
What widens the fringe spacing, and what do extra slits change?
A longer wavelength or a smaller slit spacing spreads the orders apart. More slits leave the bright directions where they are but make each one narrower and brighter relative to the background.
3. Polarization
Unpolarised light passes a vertical polariser and then an analyser at angle θ. Drag the analyser's handle. The field that survives the first filter is projected onto the analyser axis, so its amplitude is multiplied by cos θ and the intensity by cos²θ, which is Malus’s law. Then insert a third filter halfway between them: crossed filters that block everything start passing light again.
Worked example. Crossed polarisers transmit no light in the ideal model; with a third filter at 45° between them, I = I₀/4.
Watch out. I₀ is the intensity after the first polariser, half of the unpolarised input. Real filters have finite extinction and wavelength dependence.
Why does intensity depend on angle twice?
The electric-field amplitude projects once, then intensity squares amplitude.