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θ = 1 . 22 λ D , size 12{θ=1 "." "22" { {λ} over {D} } } {}

where λ size 12{λ} {} is the wavelength of light (or other electromagnetic radiation) and D size 12{D} {} is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression, θ size 12{θ} {} has units of radians.

Part a of the figure shows a graph of intensity versus theta. The curve has a central maximum at theta equals zero and its first minima occur at plus one point two two lambda over D and minus one point two two lambda over D. Farther from the central peak, several small peaks occur, but they are much much smaller than the central maximum. Part b of the figure shows a drawing in which two light bulbs, labeled object one and object two, appear in the foreground positioned next to each other. Two rays of light, one from each light bulb, pass through a pinhole aperture and continue on to strike a screen that is farther back in the drawing. On the screen is an x y plot of the two resulting intensity patterns. Because the rays cross in the pinhole, the ray from the left light bulb makes the right-hand intensity pattern, and vice versa. The angle between the rays coming from the light bulbs is labeled theta min. Each ray hits the screen at the central maximum of the intensity pattern that corresponds to the object from which the ray came. The central maximum of object one is at the same position as the first minimum of object two, and vice versa.
(a) Graph of intensity of the diffraction pattern for a circular aperture. Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.

Connections: limits to knowledge

All attempts to observe the size and shape of objects are limited by the wavelength of the probe. Even the small wavelength of light prohibits exact precision. When extremely small wavelength probes as with an electron microscope are used, the system is disturbed, still limiting our knowledge, much as making an electrical measurement alters a circuit. Heisenberg’s uncertainty principle asserts that this limit is fundamental and inescapable, as we shall see in quantum mechanics.

Calculating diffraction limits of the hubble space telescope

The primary mirror of the orbiting Hubble Space Telescope has a diameter of 2.40 m. Being in orbit, this telescope avoids the degrading effects of atmospheric distortion on its resolution. (a) What is the angle between two just-resolvable point light sources (perhaps two stars)? Assume an average light wavelength of 550 nm. (b) If these two stars are at the 2 million light year distance of the Andromeda galaxy, how close together can they be and still be resolved? (A light year, or ly, is the distance light travels in 1 year.)

Strategy

The Rayleigh criterion stated in the equation θ = 1 . 22 λ D size 12{θ=1 "." "22" { {λ} over {D} } } {} gives the smallest possible angle θ size 12{θ} {} between point sources, or the best obtainable resolution. Once this angle is found, the distance between stars can be calculated, since we are given how far away they are.

Solution for (a)

The Rayleigh criterion for the minimum resolvable angle is

θ = 1 . 22 λ D . size 12{θ=1 "." "22" { {λ} over {D} } } {}

Entering known values gives

θ = 1 . 22 550 × 10 9 m 2 . 40 m =  2.80 × 10 7 rad.

Solution for (b)

The distance s size 12{s} {} between two objects a distance r size 12{r} {} away and separated by an angle θ size 12{θ} {} is s = size 12{s=rθ} {} .

Substituting known values gives

s = ( 2.0 × 10 6 ly ) ( 2.80 × 10 −7 rad ) = 0 . 56 ly.

Discussion

The angle found in part (a) is extraordinarily small (less than 1/50,000 of a degree), because the primary mirror is so large compared with the wavelength of light. As noticed, diffraction effects are most noticeable when light interacts with objects having sizes on the order of the wavelength of light. However, the effect is still there, and there is a diffraction limit to what is observable. The actual resolution of the Hubble Telescope is not quite as good as that found here. As with all instruments, there are other effects, such as non-uniformities in mirrors or aberrations in lenses that further limit resolution. However, [link] gives an indication of the extent of the detail observable with the Hubble because of its size and quality and especially because it is above the Earth’s atmosphere.

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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