Optical bench

Paraxial thin lenses in air, light travelling toward increasing x. Positions, object height, focal lengths and clear diameter are in centimetres. Positive focal length converges; negative diverges. The object must be before every lens. Lenses are processed by position, preserving their row identities; lenses at the same position are ideal contact lenses.

Propagation uses [[1,d],[0,1]], a lens [[1,0],[−1/f,1]]. From the object plane to immediately after a lens the matrix is [[A,B],[C,D]]. A finite conjugate is at distance−B/D with total magnification 1/D. D=0 means collimated output at that stage; later lenses may focus it. An intermediate image in a later lens plane is valid. Per-lens conjugates are predictions before later elements act, not necessarily images physically formed between lenses.

The displayed fan samples rays aimed across the first lens aperture. A ray stops if a later aperture blocks it. An analytic slope interval determines whether any rays can pass and adds a representative transmitted ray when needed. Geometric image predictions ignore blocking; if all rays are blocked no image arrow is shown. Dotted lines are straight backward extensions of final rays for a virtual image. Grey dots mark focal points.

The diagram uses independent x/y scales, so apparent angles are exaggerated; the numeric ray coordinates are unchanged. Extremely distant/tall images can lie outside the finite view while their values remain available. Move elements by dragging or exact fields. Three presets illustrate a single image, a collimated intermediate focused by another lens, and a relay. Ideal monochromatic geometry only: no lens thickness, aberrations, diffraction or real intensity integration. Large slopes exceed the paraxial approximation. Valid settings survive reload.