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6.3 Single Slit Diffraction

# 6.3 Single Slit Diffraction - Diffraction effects of...

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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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