PHYS 450 Spring semester Lecture 09: Young s Double Slit. Ron Reifenberger Birck Nanotechnology Center Purdue University

Transcription

1 3/3/7 PHYS 45 Spring semester 7 Lecture 9: Young s Double Slit Ron Reifenberger Birck Nanotechnolog Center Purue Universit Lecture 9 pre -96 s coherent light source The Iea of Coherence Ras never cross each other; parallel or raial Perfect wave trains in 3D; neste spherical or plane waves post-96 s coherent light source A perfect point source gives perfectl coherent light while a wie iffuse source gives incoherent light

2 3/3/7 Interference THREE Unerling Assumptions Fresnel s work in 87 combines Hugen s principle of wave propagation an Young s ieas about interference x Coherence requirement is reail achieve b a laser ave spreas out after passing through a narrow slit Phase ifference between two optical paths an etermines intensit at x Conitions that must be satisfie: Monochromatic: Light with a specific wavelength Coherent: Phase ifference between light waves arriving at an location remains constant over time Coherence is a requirement impose on light to prouce interference fringes 3 Young s Double Slit Experiment 83 ( ouble path genre of experiments) Is this what happens? Preiction base on ra nature of light 4

3 3/3/7 Young s EXPERIMENT: A sequence of bright an ark fringes are observe on a viewing screen! ith of slit = b Assume b < Interference Pattern Sie view! Young use sunlight that entere the room through a pinhole in a winow shae 5 Moel to Explain Young s Double Slit Ke Iea: Two coherent light beams interfere after following ifferent paths light source Out of phase b 8 o (or π) Important historicall because it allowe an eas wa to measure the wavelength of light 6 3

4 3/3/7 Recent Histor of -Slits in Phsics In 967, two-source interference using TO separate lasers as light sources In 999, the ouble-slit experiment was emonstrate with Buckball molecules (comprise of 6 carbon atoms, iam=7 nm) In, the ouble-slit experiment with electrons as escribe b Richar Fenman was again carrie out, this time using two slits of 6 nm wie 4 μm long In 3, the ouble-slit experiment was successfull performe with molecules that each comprise 8 atoms (whose total mass was over, atomic mass units) 7 Optical Path Length Difference in Young s Double Slit Geometr a) b) Top view! screen screen near screen far screen c) as istance to screen increases, θ θ an is the limiting conition for parallel ras) two parallel ras OPL= sin() 8 4

5 3/3/7 Top view! Constructive interference when optical path length ifference is an integer slits OPL a ae i if <<, sinθ Θ P m th bright fringe screen a is amplitue of E-fiel at point P from top slit OPL sin phase ifference : OPL sin 9 Intensit at point P i at point P, A a ae sin i i i i I A aae aae a a e e a i i e e a a a cos 4a cos Locations of maximum Intensit maximum when,,,, m or when,, 4,, m sin,,4,,m sin,,4,,m m sin tan Details, etails 3 Separation between maxima (approx): fringe spacing Intensit (au) m m tanm m m tanm max m m m m 4 Exact vs Small angle approximation Two Slits; =m, =638 nm; =5 m exact approximate (in m) 5

6 3/3/7 Constructive/Destructive Interference Two Slits OPL is optical path length ifference between an is resulting phase ifference m= Bright OPL= = Dark OPL=/ = Bright OPL= = m= Complication prior iscussion treats a as a constant an ignores epenence of a on with of single slit (b) ouble slit b Hugens wavelets max = l screen b The observe intensit of fringes is moulate as increases =with of central maximum; relate to with of single slit (see Appenix) See Appenix for more etails 6

7 3/3/7 Apparatus Young s Two Slit Laser Two Slits Reflecting Mirror 3 Measurements Set Pattern Slit ith (b) Slit Separation () Measure C A 4mm 5mm l B A 4mm 5mm l B C 8mm 5mm l, with of central max, with of central max, with of central max 965B PASCO OS-965 slit set: Electroforme Ni foil Dimensions in mm Tolerances: 5 mm Slit Space imension is center-to-center istance between slits 4 7

8 3/3/7 Extension of Young s Interference: Llo s Mirror (834) 5 Review: Tracking the Phase Upon Reflection Must Inclue Possible Phase Change upon Reflection KEY IDEA Keep track of the phase!! air n= thin film n air n= Not to scale Reflection off interface from low n to high n: 8 o () phase change Reflection off interface from high n to low n: no phase change 6 8

9 3/3/7 Source S (near grazing incience) OPL Virtual Source S (smmetric to mirror plane) Llo s Mirror Relevant Geometr phase shift D OPL tan D ❶ a ❷ ae i (834) Viewing Screen P Front Surface Mirror hen light from two paths are in phase, a bright fringe appears on viewing screen hen light from two paths are out of phase, a ark fringe appears on viewing screen OPL sin phase ifference : sin D 7 Multiple Fringes D Viewing Screen m- Source S (near grazing incience) Virtual Source S (smmetric to mirror plane) m- m m+ Front Surface Mirror Grazing Beam m m+ 8 9

10 3/3/7 Intensit i sin Aaae where i i i i I A aae aae a a e e a i i e e a a a cos 4a cos 3 Separation between maxima (approx) Details, etails D m m 3 D m m 3D D D m m m m Locations of maximum intensit maximum when,,,, m or when,, 4,, m,,4,,m D D D,, 4,, D 3 5,,,, m D m,, 4,, m m The ratio D/ is ifficult to measure accuratel 9 Apparatus Llo s Mirror Laser Ajustable Vertical Slit Llo s Mirror path of laser beam Llo s Mirror Lens (f~ +5 cm) Reflecting Mirror Lens Reflecting Mirror

11 3/3/7 Fringes from Llo s Mirror Proceure: ~ inches Lens Imperfections Calibrate D/ using known HeNe laser line NO changes in optical alignment allows wavelength calibration of other laser lines =6 nm ~7 inches D m m Know, measure l, calibrate D/ Available HeNe gas lasers: re (638 nm) orange (6 nm) ellow (594 nm) green (543 nm) =638 nm Switch in new laser, D ' m m Measure l, use calibrate value for D/, calculate Comparison between Young an Llo Young Llo central feature bright ark, separation between slits basicall fixe ajustable Primar use Measure wavelength of light Prouces interference patterns over large areas

12 3/3/7 Up Next Interferometers 3 Appenix: The Intensit through a Single Slit I E tot I=(E +E ) NOT I= E +E Let b now equal the with of slit b hat happens when b is comparable to? i N θ Δ(OPL) i E o E coskrt N Eo Ei coskrti N E o sin OPL i i i Sum over all i 4

13 3/3/7 In the limit of infinitel man point sources within one slit, the sum can be converte to an integral over position of all points in the slit: Let the center of the slit efine = Here, the with of the slit equals b It follows that varies from b/ to +b/ For the E-fiel amplitue we have b E Eo sin sin coskrt bsin For the time average intensit of E, we have (ω=π/t =πf) E b b o E E cos kr t kr t b b b sin sin E b sin T time o Eo sin cos b b b b Eo sin kr t sin sin kr t sin b sin cos kr t t T t 5 T E T time o cos b sin sin E b sin cos kr t t sin T T 4 t kr t kr t sin kr T kr T sin kr kr T 4 T 4 sin kr kr 4 sin krkr 4T sin krcos4 krsin 4 sin kr 4T sin kr4 sin kr 4 T kr t t orking it out - Single Slit t T b sin sin I E E time o b sin b sin sin I o b sin Note that I above is the same as a which appears in slies 9-6 3

14 3/3/7 b sin sin b I I o when sin sin b sin b sin,, 3, m m integer b sin,, 3, m sin,, 3, m b b b b minima equall space in sin( ) m sin, sin,, sin m b b b hen oes I()=? Single Slit m=- m=+ m=+ b b zero zero b zero zero m m 7 phase= Appenix: Defining the phase of a wave phase x = x = π λ x x (x) = Asinπ λ x phase x = x = π λ x x Δ = x-x = π -π λ λ x π π x = x -x = Δx πm m = integer λ λ x E E x 8 4

15 3/3/7 Ke Iea: Aing in-phase an out-of-phase signals The phase of the sine wave is measure in raians a) x x Asin o π Asin o π Δφ λ λ In Phase SUM The superposition of two waves b) E (N/m) position (m) Out of Phase Δφ =,π, 4π, SUM= E (N/m) - Δφ = π, 3π, position (m) Principle of Superposition 9 Example Monochromatic light of unknown wavelength passes through two narrow slits that are separate b =5 m A viewing screen is place m behin the ouble slits The 4 th bright fringe observe on the screen is locate 8 cm from the central bright fringe hat is the wavelength of the light? sin θ =mλ constructive λ sinθ' = m+ λ = m+ estructive where m =,±,±,±3, θ = cm Y=8 cm Viewing screen sinθ =mλ or mλ Is <<? -6 8 cm 5 m 4λ cm -6 5 m 8 cm λ = 4 cm -6 5 =375-7 =565 m=565nm 3 5