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Unformatted text preview: Diffraction effects of apertures 6.2 Interference and diffraction Huygen’s Principle Single slit diffraction Circular diffraction The diffraction limit W hen light is pas sed though a n a perture the light bea m is smeared out. Although this ef fec t is small for lar ge apertures . It li mi ts the i ma gi ng properties of the li ght waves. Hyugen’s Principle All points on a wavefront act as point sources of spherically propagating “wavelets” that travel at the speed of light appropriate to the medium. At a short time Dt later, the new wavefront is the unique surface tangent to all the forward propagating wavelets. Huygen’s principle explains diffraction around a barrier. large ho le plane wa ve spherical wave sma ll ho le Single slit diffraction Single slit diffraction W aves from 5 sources propagati ng at a ngle q Find the co ndi tion for des tructive i nterference. Divide the slit into 2 halves From Hyugen’s pri nciple si ngle sli t diffrac tion res ults from wave le ts from a large number of sources The i nterference from thes e wa ves deter mine the i nte nsity prof i le of the wave For each source in the botto m ha lf there is a source in the top ha lf a distance of a/2 a way the two sources interfere des tructive ly under the conditio n a l sin q = 2 2
The i nterference bet wee n the top ha lf cancels the a mpli tude from the botto m ha lf. No light 1 Single slit diffraction Now divide the slit up into 4 sections Agai n each source wi ll have a not her source at a distance of a/4 a wa y. The conditio n for des tr uctive interference Single slit diffraction The general condition for destructive interference for single slit diffraction a sin q = ml
m= +1, +2, ......... Minima a l sin q = 4 2 a sin q = 2l q a/4 sinq This rel ation is sim ilar to the rel ation for twoslit interference but be careful to note the differenc es. Intensities for single slit diffraction This ana lysis is for the cas e where the screen is far from the s li t L>>a, Fraunhof er diffrac tion. Phas or additio n to ge t a mplitudes Eq Square the a mplitude to get the i nte nsity. b N points N1 segments c Intensity Elec tric field a mpli tude E s in(f / 2) q = E o f/2
2 Intensity µ Ampli tude a é s in( f / 2) ù S q = So ê ú ë f/2 û
where 2 f = 2p
central ma xi mum mi ni mum genera l poi nt a sin q l Single slit diffraction The width of the diffraction increases as the slit bec om es sm aller Example For singl e slit diffraction (a>>λ) find the am plitude of the first sec ondary m axim um in term s of the intensity of the central m axim um So. Assum e that the peak lies between the first two m inim a. 2 Two slit interference pattern Diffraction Limit Light is s meared o ut when pas sed thro ugh an aperture This s mearing out li mits the resolutio n for optical syste ms. d a d a W hat is the ratio of d/a? Diffraction Limit single slit T wo objec ts we ll res olve d The separation bet wee n peaks is greater tha n the wi dth of the peak Rayleigh Criteria Single slit diffraction The Rayl eigh criteri on is that the peak of one object is at the position of the first m inimum of the other object. a qmin T wo objec ts just res olved The separation bet wee n peaks is just e qual to the wi dth of the peak Raylei gh criteria Pos ition of the firs t mi ni mum s in qm in =
For small a ngles l a qm in = l a Resolution W ell res olve d Rayleigh Criterion circular diffraction Most optical syst em s use a circular aperture e.g. cam era lenses, telescopes, eye. The Rayl eigh criteri on for a circular aperture is Just res olved Raylei gh criteria qmin = 1.22l D D is the diameter of the circular aperture Not res olved Yo u cannot te ll if there are two peaks pres ent D qmin 3 Resolution of two images by a lens resolved Diffraction Limit • The diffraction lim it determ ines the m inim um spot size to which parallel light can be focused. • The diffraction lim it lim its the sm allest objects that can be resolved by an optical system . just resolved not resolved The resolution of im ages is lim ited by the diffraction pattern. Diffraction Lim ited spot size A camera has an f/1.4 lens, meaning the ratio of focal length to lens diameter is 1.4. Find the radius to which the lens can focus parallel light with wavelength of 580 nm. D Microscope q f f = 1 4 . D q D f r If we use the same le ns as in a micros cope the n the Raylei gh criterion li mits the size of the s ma lles t o bjec t that can be i ma ged by the micros cope. qmin = 1.22l r » D f r = (1 . 2 2 ) l f D = (1.22)(580)(1.4) = 1000nm = 1mm 1 Diffraction limit of vision. Light enters the hum an eye through an iris with an aperture with a diam eter of about 4mm. In the diffraction lim it how far away can you see two objects separat ed by 1mm (use l =500nm ) r =1mm GeoEye 1 Ima gi ng Sate lli te 1.1 m dia meter mirror Res olutio n 0 .41 m Altitude of 684 km Diffraction limit (λ=550nm) qm in = y l = 1.22 L D y L D θ in m y = qmin L 1 22 L 1. 2( 50 x10 -9 m 684 x10 m ) . l 2 5 )( 3 = = 0. 1m 4 D 1. m 1 1.22l r = qm in » D L L = rD 10 -3 ( 4 x10 -3 ) = = 6.5m 1 .22l 1.22(500 x10 -9 ) Satellite picture of Sea W orld 4 Optical CD focused l aser beam Diffraction limits the resolution of light microscopes 106 m The am ount of inform ation that can be encoded is lim ited by the diam eter of the diffractionlim ited spot. To get to smaller spot size use shorter wavelengths of light • A DVD hold more information than a CD. DVDs need a shorter wavelength laser to focus on the sm aller bumps. • Light m icroscopes can’t resolve m olecular dimensions. – Xrays diffraction with shorter wavelengths can. (but there are no xray microscopes) – Electron microscopes can (electrons are waves too). 5 ...
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