Physics · Ch 6 — Optics
Resolution
Resolution
Diffraction limits the sharpness of any image formed by a finite-sized aperture. Even a perfect optical instrument images a single point source not as a true point, but as a small bright central disc surrounded by faint concentric rings, called an Airy disc, because the aperture itself (such as a lens, or the iris of the eye) diffracts the light passing through it; for a circular aperture of width , the central disc's half-angular spread is given by (the factor 1.22, rather than the plain 1 of the single-slit case, arising specifically from the circular rather than rectangular shape of the aperture). When two point sources are imaged close together, their two Airy-disc patterns can overlap and blur into a single indistinct blob; according to Rayleigh's criterion, the two are said to be 'just resolved' when the central maximum of one image's pattern falls exactly on the first minimum of the other's, corresponding to a minimum resolvable angular separation of radians. This shows that finer resolution (smaller resolvable angle) requires a shorter wavelength and a larger aperture. Resolution is the smallest separation that can still be seen as distinct in an image, and resolving power is defined as the reciprocal of that …
What this figure shows. A circular aperture of width a, images a single point object not as a sharp point but as a small central bright disc surrounded by a series of progressively fainter concentric rings -- the whole pattern together known as an Airy disc. The half-angular spread of the bright central disc, marked theta in the diagram and related to the image-side distance f and the pattern's own radius r0, is given by a sin(theta) = 1.22(lambda), the factor 1.22 arising specifically becau …
What this figure shows. Three side-by-side intensity profiles compare two close point sources' Airy-disc images at decreasing separation. (a) Unresolved: the two central peaks sit so close together their combined intensity profile shows only a single blurred maximum, with no visible dip between them. (b) Just resolved (Rayleigh's criterion): the two peaks are spaced so the central maximum of one image falls exactly on the first minimum of the other, leaving a shallow but clearly visible dip between two still-overlapping peaks. (c) Well resolved: the two peaks sit far enough apart that their profiles barely …
Worked out. The optical telescope at the Vainu Bappu Observatory, Kavalur, has an objective lens of diameter 2.3 m, used with light of wavelength 589 nm. Substituting into theta = 1.22 lambda/a gives theta = 1.22 times 589e-9/2.3 = 321.4e-9 rad, i.e. theta approximately 3.214 times 10^-7 rad (about 0.0011 arcsecond). Comparing this to the unaided human eye's angular resolution of roughly 3 times 10^-4 rad shows the telescope resolves detail nearly a thousand times finer than the eye alone can -- directly illustrating why large-aperture telescopes are built specifically to push thi …