Physics 116 Session 24 Interference in gratings and thin films Nov 8, 2011 R. J. Wilkes Email: ph116@u.washington.edu
• � 2 clickers have quiz data logged, but no registration: • � 649314 • � 614235 If one of these is yours, see me pls Exam 2 results • � Avg 84 • � Std dev 15 • � Median 88 • � Grades should be on Webassign gradebook later today
Lecture Schedule (up to exam 3) Today 3
Interference: reflected waves • � Phase shift on reflection – � Recall behavior of wave on a rope: • � If end of rope is rigidly secured, reflected wave gets inverted (180 deg phase change) • � If end of rope can move up and down freely, no inversion – � Similar behavior for reflection of EM waves if reflecting surface is on a medium with higher index of refraction (slower light propagation speed) • � SO phase flip going from air to water, but no flip from water to air • � Phase shift means interference can occur between incident and reflected waves C=180deg out of phase with A A B’ n’ 2 = 1.0 n 1 = 1.0 n’ 1 = 1.5 n 2 = 1.5 C’ = in phase with A’ C’ B A’ 10
Interference in a thin layer of air • � Air gap between flat glass plates – � Make a layer of air between 2 glass plates by inserting a shim (fine wire, or hair) at both ends • � Thickness of air layer is constant: t = T shim • � Ray reflected from top surface (air to glass): phase flip • � Ray reflected from 2 nd surface (glass to air): no flip (Ray 1) • � Ray reflected from 3 rd surface (air to glass): phase flip again (Ray 2) • � Suppose angle of incidence is nearly vertical – � so neglect the angle shown in drawing: Ray 2 travels 2t further than ray 1 • � Rays 1 and 2 show destructive or constructive interference if 2t is some multiple of � wavelength ( ) � 2 � � 1 = 2 t = m � / 2 m = 0,2,4 … m = 1,3,5 … Constr : Destr : T shim Not to scale! If there were no phase flip for ray 2, would be 2 1 Glass is much Constructive if m= even (so full wavelength difference) thicker than air Destructive if m= odd (so half-wavelength difference) layer Phase flip of ray 2 in effect adds another half-wavelength: � 2 � � 1 = 2 t + � � = m � � � � � � � 2 t � + 1 � = m t � � � � � 2 , � � � � � � 2 2 2 if : m = 2,4,6 … m = 1,3,5 … Constr Destr : 11
Interference in reflected waves • � Example: wedge of air between flat glass plates – � Make a wedge of air between 2 glass plates by inserting a shim (fine wire, or hair) in one end • � Thickness of air layer is proportional to x: t(x) = T shim x T shim x=0 x=1 – � We see fringes as t increases: dark fringe when m is odd, bright when m is even � 2 � � 1 = 2 t + � � = m � � � � � � � 2 t � + 1 � = m � if : m = 2,4,6 … m = 1,3,5 … � � � � 2 , Constr Destr : � � � � � � 2 2 2 • � Requirements: – � Illuminate with monochromatic light, or fringes will be smeared in a rainbow! – � Plates have to be very flat (“optical quality”) or ripples will make interference pattern impossible to see • � Flat to within � wavelength: this is a very sensitive flatness test! – � Light source and observer are “distant” www.oldham-optical.co.uk • � Meaning: ~ parallel light rays 12
Newton’s rings • � Air gap between a convex lens and a flat glass plate – � Another air layer of variable thickness – � This setup is used to test lenses – again, we can immediately see flaws at the scale of � wavelength R • � If you enjoy geometry and algebra, here’s how to determine radius r of the m th ring, for a given lens radius of curvature R and wavelength of light r t • � Usually need a microscope to see rings for a typical lens 13
Thin films • � Reverse the air gaps we’ve just discussed – � Make a thin film of some substance with n>n air • � Soap bubble (water layer, held together by surface tension) • � Oil slick on a puddle of water • � Film of plastic material specifically designed for coating optical parts – � Now ray 1 has a phase flip, and ray 2 does not 2 1 – � Otherwise, exactly the same analysis – except • � Path difference occurs in medium with n>1 • � We have to take into account the wavelength in the medium (but we measure wavelengths in air t (n air ~ n vacuum =1.0) � n = � air n , effective � 1 = � air (just the phase flip) , � 2 = 2 t (path length in film) 2 � � = � 2 � � 1 , � measure � � in units of wavelength to get phase relations � � � � � � interference if � 2 � 1 � � 2 t 1 � = m � � � = � � � � � � � � � n � air � n � � � � � � 2 2 � � � � 2 t � 1 � = m 2 nt � 1 � = m � � if : m = 0,2,4,6 … m = � 1, + 1, 3,5 … 2 , 2 , Constr Destr : � � � n � air � � � � 2 2 • � Notice: if t is nearly 0, we still get destructive interference due to the phase flip for ray 1, so we need to add that case to our list – that’s why the -1 is in there 14
Antireflection coatings • � Application of thin films: arrange for destructive interference of reflected rays – � Then film = non-reflective coating for lenses, glasses, etc • � Example: glass (n=1.5 coated with magnesium fluoride (n=1.38) What is the thickness of MgF layer needed? – � Now both ray 1 and ray 2 have a phase flip, but ray 2 also has path 2t – � Choose wavelength in center of human-visible range: 555 nm – � For destructive interference, we want m=1 (m=-1 means t=0) � MgF = � air 1.38 , effective � 1 = � air (just the phase flip) , � 2 = 2 t + � n (path length in MgF film plus phase flip) 2 2 � � = � 2 � � 1 , � measure � � in units of wavelength to get phase relations 2 1 � � � � � � interference if � 2 � 1 � � 2 t + 1 1 � = m � � � = � � � � � � � � � � n � � � air � � � n � 2 2 2 � � 2 nt � = m if : m = 0,2,4,6 … m = � 1, + 1, 3,5 … t 2 , Constr Destr : MgF � � air � � glass � air � air � � t = 1 2 n = 1 � = 555 nm 4(1.38) = 100.5 nm so � � � 2 4 n 15
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