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Just what is the limit? To answer that question, consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) [see [link] (a)]. It can be shown that, for a circular aperture of diameter D size 12{D} {} , the first minimum in the diffraction pattern occurs at θ = 1 . 22 λ / D size 12{θ=1 "." "22"λ/D} {} (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The accepted criterion for determining the diffraction limit to resolution based on this angle was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion    for the diffraction limit to resolution states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other . See [link] (b). The first minimum is at an angle of θ = 1 . 22 λ / D size 12{θ=1 "." "22"λ/D} {} , so that two point objects are just resolvable if they are separated by the angle

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

Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
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binding energy per nucleon
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why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
room?
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Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 1

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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