How a Fresnel lens works
A glass lens bends light only where the light crosses a surface, and only the angle of the surface at that spot matters. The glass in between does no bending; it only adds weight and absorbs light. Buffon and Condorcet suggested building a lens in steps in the 1700s. Augustin-Jean Fresnel made it work for lighthouses in 1822. He cut the lens into rings, kept each ring's surface angle, and threw the bulk away. The result is a thin plate covered in concentric grooves that focuses almost like the thick lens it came from.
In the side view, each groove has a sloped facet that does the bending and a steep riser that steps back down. Choose how the curved surface is folded: rings of equal pitch (the grooves get deeper toward the rim) or of equal depth (the rings get narrower). Choose the facet shape too: true curves, or flat prisms. Use the magnifier (top right of the side view) to see the grooves up close.
A flat plate made of two materials
Fill the grooves with a second transparent material and the plate becomes flat on both sides, with the same thickness everywhere. At a boundary, light bends according to how different the two refractive indices are. So the buried sawtooth still focuses, but only through the index step Δn = nB − nA. A step of 0.18 (silicone into polycarbonate) needs far steeper facets than acrylic into air (0.49) for the same focal length. If the step is zero, the plate becomes invisible: that's index matching, like a glass rod vanishing in oil. Swap the two materials and the step changes sign, so the same mould makes a diverging lens. Turning a plate around (Flip) keeps its focal length.
The flat faces only shift the rays sideways. An internal boundary also reflects far less than glass to air: about (Δn/2n)², a few hundredths of a percent here, versus about 4% at each acrylic–air surface.
Stacking plates
Thin lenses add their powers: 1/f = 1/f₁ + 1/f₂ − d/(f₁ f₂). Two identical plates touching each other halve the focal length. A converging plate plus its swapped twin cancel, and light comes out (almost) parallel again. Space a +150 mm and a −75 mm plate 75 mm apart so their foci meet, and you have a Galilean telescope. Parallel light goes in, and parallel light half as wide comes out. The status chip under the side view tells you whether the light leaves converging, parallel or diverging, and where the focus is.
Where exactly is the focal point?
The simple lens formula uses only the curvature at the centre, which describes rays close to the axis. That's the paraxial focus, drawn as a hollow diamond. Real rays farther out cross the axis somewhere else (spherical aberration). The page also finds the best focus, where the bundle is narrowest, drawn as a ring. The left plot shows every ray's crossing point against the height it entered at. A flat line means a sharp focus. Ideal facets are solved ring by ring so that every ray heads for the chosen focus, and the line goes flat. With flat facets each ring acts as a little prism sending out a parallel beamlet, so the focus smears to about one ring's width.
Colours
The refractive index changes with wavelength (dispersion), so blue usually focuses closer than red. For a composite plate, what counts is how much the step changes across the spectrum, measured by an effective Abbe number V = Δnd / (ΔnF − ΔnC). A small step between materials of different dispersion can have V ≈ 10, which is far more colourful than any glass (30–70). It can even be negative, when the denser material disperses less. Stack a positive-V plate with a negative-V plate and red and blue meet again: the achromatic stack preset.
Where the light gets lost
A ray bent by a facet drifts sideways as it travels on. Within one groove depth it can hit the riser of the next ring inward. Inside the denser material it hits at a grazing angle and is totally reflected away. Tilting the risers (draft) can't remove this dead zone; it only moves it. Lean a riser parallel to the bent light (auto) and the incoming light hits it instead. Lean it further, or the other way, and the loss grows. A lens asked to bend light too sharply runs into total internal reflection at its outer facets. Fresnel solved that for his lighthouses with outer rings of reflecting prisms.
Mouse: drag to pan, wheel to zoom, double-click to fit. Drag the dashed screen line, the point source and the magnifier box. Keys: F fit, A flowing light, M magnifier.
What is computed and what is approximated
- Exact: every ray is traced with vector Snell's law at every boundary: flat faces, curved facets (intersections refined onto the true curve with Newton's method) and risers. That includes total internal reflection and unpolarised Fresnel reflection, with optional ghost rays.
- Paraxial prediction: an ABCD matrix of the whole stack, using each plate's thickness and its centre curvature.
- Best focus is the least-squares waist of the output rays. For ring lenses each ray is weighted by the ring area it stands for. Rays that hit a riser or were totally reflected count as stray and are left out of the focus.
- Ideal facets are solved for the plate on its own, for parallel light at 587.6 nm, accounting for where the folded boundary sits inside the plate.
- 2D cross-section: exact for a ring lens for rays in a plane through the axis, which covers the side view and the on-axis view-through image. For straight grooves the view-through image treats the direction along the grooves as undeviated.
- Geometric optics only: no diffraction. That's fine while the groove pitch is much larger than the wavelength (pitch ≥ 0.05 mm).
- Materials: typical nd and Abbe values, turned into a Cauchy dispersion fit. Silicone is approximate. No absorption, no polarisation tracking.
- Imperfections are simple random models: a per-ring facet angle error, and surface roughness as a random tilt of the normal at each hit.