Telescopes and optics

Telescope. Enter the aperture D and focal length F (mm or inches — the switch at the top), the eyepiece’s focal length (always in mm) and its apparent field, and any Barlow (2 for a 2×) or reducer (0.63). You get:

Camera. For a camera at the telescope’s focus: the image scale 206.265 × pixel size (µm) ÷ F (mm) in arcseconds per pixel, the sensor’s field of view, and how many pixels span the seeing disc. Two to three pixels across the seeing’s full width at half maximum is well sampled (Nyquist); fewer is under-sampled (square stars), more over-sampled (wasted signal). For planetary "lucky" imaging the target is the diffraction limit instead: f-ratio ≈ 2 × pixel ÷ λ.

Binoculars. For M × D: exit pupil D ÷ M, the twilight factor √(M × D) (detail in dim light), relative brightness (exit pupil²), the field at 1,000 m and 1,000 yd from the true field, and the apparent field by ISO 14132-1 (tan ½AFOV = M tan ½TFOV) as well as the older M × TFOV.

Ray bench. A light source on the left and thin lenses and mirrors on the axis. Drag the object arrow (sideways for distance, up and down for height) and the elements; or edit them in the list, add more, and pick a preset — converging and diverging lenses, a magnifier, concave and convex mirrors, Keplerian and Galilean telescopes, a compound microscope and a reflector. Tick At infinity for a star-like object at an angle. The three principal rays are drawn for the first element — red parallel to the axis then through the focal point, green through the centre, blue through the near focal point then parallel — plus two more rays; every ray is traced on through all the elements. Each image is drawn as an arrow (solid real, dashed virtual), with its distance, whether it is real or virtual, upright or inverted, and the magnification, step by step when there are several elements. ⛶ Full screen (or F) gives the bench the whole screen.

Lensmaker. The focal length of a lens from its refractive index and surface radii, with an optional thickness (thick-lens form, with the back and front focal distances) and the surrounding medium (try water).

Refraction. Snell’s law n₁ sin θ₁ = n₂ sin θ₂ at a flat surface. Drag the incoming ray in the picture. Going into a less dense medium beyond the critical angle arcsin(n₂/n₁), all the light reflects (total internal reflection, how optical fibres and prism binoculars work). Also the Brewster angle and the Fresnel reflectance for s- and p-polarised light.

Thin lenses and mirrors use the "real is positive" Gaussian form 1/f = 1/d_o + 1/d_i with m = −d_i/d_o: f > 0 for converging lenses and concave mirrors (f = R/2), d_i > 0 for a real image. A virtual object (negative d_o) arises when light already converging meets the next element. The ray trace is paraxial: a ray at height y with slope u leaves a thin element with slope u − y/f, and a mirror also reverses it. The lensmaker’s equation uses the Cartesian sign convention for radii (positive when the centre of curvature is to the right). The Dawes limit is W. R. Dawes’s 1867 empirical 4.56″ ÷ D(inches); the Rayleigh criterion is the first zero of the Airy pattern.

The tests check these against textbook values: a 200 mm f/5 with a 25 mm eyepiece gives 40× and a 5 mm exit pupil; Dawes for 100 mm is 1.16″; a 3.76 µm pixel at 1,000 mm is 0.776″; f = 10, d_o = 30 gives d_i = 15 and m = −0.5; a biconvex lens with R = ±20 mm and n = 1.5 has f = 20 mm; glass to air has a 41.81° critical angle and 4% reflection at normal incidence; and two-lens systems step by step.

Related: physics (thin lens calculator, the eye in dioptres), astrophotography (500 and NPF rules, star trails), depth of field, field of view and diffraction, sun and moon, exposure, spectroscopy.

Share link copies a link with all the tabs' settings (the ray bench is packed into ?s=). Links can be written by hand; add the tab after #: telescope, camera, binoculars, bench, lensmaker or refraction. Lengths are in mm.