# Lecture 19

## Physics 126 with Amidei at University of Michigan - Ann Arbor *

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Lecture 19 Interference • Superposition of Linear Waves – Phase, Pathlength Difference – Constructive and Destructive Interference • Superposition of Spherical Waves – Interference Pattern • Two Slit Interference –maxiumumat dsin(θ) = mλ !! • Thin Film Interference – watch out for the fine print! Interference • Recall E, B fields obey superposition principle: they add • If two EM waves overlap, their effects will add: interference • Used by Young (1801) to show that light was a wave • A general feature of all wave phenomena •Facts – Need multiple waves with same wavelength – Sources have definite phase relationship: coherence – Interference arises because waves have a phase difference – The phase difference can arise from a path length difference ΔL • See Superposition Applet – http://www.matter.org.uk/diffraction/geometry/superposition_of_waves_ exercises.htm sin( ) sin( )AtBtω ωφ++ Phases, Wavelengths, Pathlengths, Interference • The phase is where it is in the cycle • Can be measured in units of radians, degrees, wavelength π2π 3π ½λ λ 1½λ source 1 180 0 360 0 540 0 Phases, Wavelengths, Pathlengths, Interference • Add second wave, same λ, overlapping the first • Let source of second wave be farther away by ½λ source 2 • The two disturbances arriving at any point P are “180 0 out of phase” • They cancel in the sum • Destructive interference • Two waves with pathlength difference Δl = ½ λ cancel π2π 3π ½λ λ 1½λ source 1 180 0 360 0 540 0 P ΔL = ½ λ Phases, Wavelengths, Pathlengths, Interference • Add second wave, same λ, overlapping the first • Let source of second wave be farther away by 1½λ • The two disturbances arriving at any point P are “180 0 out of phase” • They cancel in the sum • Destructive interference • Two waves with pathlength difference Δl = (n+½ λ) cancel, n=1,2,… π2π 3π ½λ λ 1½λ source 1 180 0 360 0 540 0 P ΔL = 1½ λ source 2 Phases, Wavelengths, Pathlengths, Interference • Add second wave, same λ, overlapping the first • Let source of second wave be farther away by λ source 2 • The two disturbances arriving at any point P are “in phase” • They add in the sum • Constructive interference • Two waves with pathlength difference Δl = nλ ADD, n=1,2,… π2π 3π ½λ λ 1½λ source 1 180 0 360 0 540 0 P ΔL = λ Flat Representation of Spherical Wave solid lines are “crests” dotted lines are ‘troughs” waves with λ spreading isotropically from source λ P Flat Representation of Spherical Wave solid lines are “crests” dotted lines are ‘troughs” waves with λ spreading isotropically from source λ Q: In units of wavelength, how far is Point P from the source? A. 2 λ B. 4 λ C. 8 λ P Two Overlapping Sources Along this line, peaks overlap peaks and troughs overlap troughs. In phase. Constructive. Add Along this line pathlength from the two sources is equal Along this line, peaks and troughs overlap. Out of phase. Destructive. Cancel Along this line, the pathlength from bottom source is longer by Δl = ½ λ See Two Source Interference Applet http://www.ngsir.netfirms.com/englishhtm/Interference2.htm Two Slit (Young’s) Interference I Minn, BU Physics Project the interference pattern on a screen 0 th (central) constructive maximum 1 st constructive maximum 1 st constructive maximum 1 st destructive minimum 1 st destructive minimum 2 nd destructive minimum 2 nd destructive minimum 2 nd constructive maximum 2 nd constructive maximum See http://www.walter-fendt.de/ph14e/doubleslit.htm Two Slit (Young’s) Interference • Constructive interference when Δl= dsin(θ) = mλ; m th maxima • Destructive interference when Δl= dsin(θ) = (m-½)λ; m th minima Two sources separated by d For point P on screen located by angle θ from the centerline Pathlength difference of the sources is Δl= dsin(θ) Two Slit (Young’s) Interference: Example • Constructive interference when dsin(θ) = mλ m th maxima • Destructive interference when dsin(θ) = (m-½)λ m th minima What is the y position on the screen of the 2 nd bright band from the center? Two Slit (Young’s) Interference: Example • Constructive interference when dsin(θ) = mλ m th maxima • Destructive interference when dsin(θ) = (m-½)λ m th minima What is the y position on the screen of the 2 nd bright band from the center? If we use a smaller wavelength, that y of 2 nd fringe: A. Stays in same place B. Moves toward center C. Moves out away from center Thin Film Interference • http://mysite.verizon.net/vzeoacw1/thinfilm.html Pathlength difference Δl ~ 2 t Simplified concept: Interference when if Δl = 2 t is of order 1 wavelength Thickness = t Thin Film Interference: Fine Print 1) When reflecting from an interface where n is increasing across the boundary, the phase of the wave flips!!! Therefore the reflected wave gets an “effective” Δl = ½ λ at the instant of reflection. Subtract this from the pathlength difference 2t 2) In a dielectric medium the wavelength is reduced by a factor of n λ→λ/n Thin Film Interference: Fine Print 1) When reflecting from an interface where n is increasing across the boundary, the phase of the wave flips!!! Therefore the reflected wave gets an “effective” Δl = ½ λ at the instant of reflection. Subtract this from the pathlength difference 2t 2) Pathlength difference happens in a dielectric medium where the wavelength is reduced by a factor of n: λ→λ/n 1 2 2 1 20,12 2 tm nn tm m n λ λ λ −= =+ = 1 4 t n λ = • Constructive interference when Minimum thickness for constructive interference amidei Microsoft PowerPoint - Lec19_Interference_post.ppt

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