Physics 116 Session 27 Diffraction gratings Postulates of Relativity Nov 14, 2011 R. J. Wilkes Email: ph116@u.washington.edu
Announcements • � HW 5 due today!
Lecture Schedule (up to exam 3) Today 3
Diffraction for a circular aperture: resolution Last time: • � Pinholes also show diffraction fringes – � Similar to single slit pattern, but with circular symmetry – � Mathematical form is called the Airy function – � Can just resolve 2 pinholes if their 1st minima overlap: (a): One pinhole (b): Two, just separable (c): Two, not separable! – � Angle to first dark fringe for a pinhole is • � Rayleigh Criterion: resolution for aperture of diameter D Telescope, camera, binoculars and human eye = circular apertures ! Rayleigh criterion lets us estimate resolution limits for optical devices
Examples • � Light with wavelength 676 nm strikes a single slit of width 7.64 microns, and goes to a screen 185 cm away. What is distance in cm of 1 st bright fringe above the central fringe? First bright fringe is about halfway between dark fringes for m=1 and m=2 We can find its angle by using “m=1.5” in the dark-fringe formula: (same as finding location of y dark fringes 1 and 2, and then finding their midpoint) L Tangent of angle = y/L: • � A spy satellite orbits at 160 km altitude. What camera lens diameter is needed to resolve objects of size 30 cm (meaning: distinguish two objects a foot apart)? Assuming light wavelength 550 nm: (~ center of human vision range) 5
2-slits revisited • � Now that we know about diffraction, we can understand details of 2-slit interference patterns • � Each slit’s diffraction pattern modulates the 2-slit interference pattern ! a w w diffraction pattern due to each slit Interference pattern for 2 slits w=0.25mm, a=0.9mm, " =632nm Result:
From 2 to N slits: diffraction gratings • � For N>>2 slits, uniformly spaced, we get an interference pattern that has – � Many sharply defined bright fringes, equally spaced (principal maxima) • � Approximately same intensity for all • � Increasing N sharpens the principal maxima – � In between, many very dim secondary fringes (secondary maxima) • � For very large N, slit mask = “diffraction grating” – � for m=0 all wavelengths have max at q=0 for m>0, maxima at ! � " : can use grating as a spectroscope – � For w=0.25mm, a=0.5mm, " =632nm: N=4 N=8
Diffraction gratings • � N-slit interference produces pattern with fringe spacings dependent on wavelength • � Diffraction grating = thousands of closely spaced slits – � Very sharp fringes build up for each color, with contributions from all slits – � Better than a prism (no light absorption in glass) – � Can use mirror surface with fine-line pattern also: reflection grating Diffraction grating with red + blue wavelengths incident: colors are separated due to different angles for their maxima Rainbow effect looking at white light reflected in a CD 8
Examples typos corrected: • � A laser emits 2 wavelengths, 420nm and 630nm. At what angle and for what order will maxima for both wavelengths coincide, for a grating with N=450 lines/mm? We want same angle to be order number m for one line, and n for the other; both must be integers: ( ) d sin � m = m � 1 d sin � n = n � 2 spacing d (in mm ) = 1/ N lines / mm � m = sin � 1 m � 1 ( ) = sin � 1 m � 1 N 450 lines / mm = 450 � 10 � 6 lines / nm d ( ) ( ) 450 � 10 � 6 nm � 1 ( ) = sin � 1 m 420 nm � = sin � 1 m 0.189 � � � � � � � ( ) ( ) 450 � 10 � 6 nm � 1 ( ) � n = sin � 1 n 630 nm � = sin � 1 n 0.2835 � � � � � � � � � ( ) = n 0.2835 ( ) m = n 0.2835 � � = 1.5 n so we want m 0.189 � � � 0.189 Try n=1, 2… and find first value of n that gives integer m n= 2 gives m=3: so find angle in degrees by putting in m=3 above m = -3 -2 -1 0 +1 +2 +3 n = -2 -1 0 +1 +2 9
“Modern” physics • � Next set of topics: a brief introduction to relativity and quantum theory, atomic and nuclear physics – � Modern physics (term coined in mid-20 th century!) – � As opposed to: “classical physics” (Newton and Maxwell), where • � Time ticks on, independent of physical objects or their motions • � EM radiation consists of waves, not particles • � With fully detailed info on its physical state now, motion of a body can be predicted precisely, into the future • � In 1895, physics was thought to be “almost finished” – � Just a few little problems remained to be settled… (sound familiar?) 1. � How to tweak Maxwell’s equations to make them obey Galilean Relativity? 2. � Why do atoms emit light only at specific wavelengths (“line spectra”) 3. � Explanations for a few peculiar experimental results: “Blackbody radiation” (thermal emission of EM waves) spectrum “Photoelectric effect” (emission of charge when light hits metal surfaces) First, item 1 above: relativity 10
Review: Galileo’s “common sense” relativity (c. 1600) • � Example: Bill and Phil are (at first) both standing still – � Bill fires a gun: bullet’s speed is 1000 m/s relative to gun • � Both agree: bullet speed is 1000 m/s – � Next, Phil rides on train with speed 100 m/s, shoots gun forward • � Phil says v=1000 m/s, but Bill says it is 1100 m/s • � Both are right: describing motion in their reference frames – � Bill agrees he would say 1000 m/s if he were on train – � Phil agrees he would say 1100 m/s if he were on ground – � Both agree bullet “really” moves 1100 m/s (Earth reference) Next: Phil rides on 100 m/s train moving past Bill First: Both are at rest Q: what if Phil fired backward ? P B P B 1000 m/s 1000 m/s 100 m/s Train Earth Earth 11
Oops: A little problem with Maxwell’s equations • � By 1880s, Maxwell’s work was in everyday technology – � Every generator, motor, telegraph, telephone proved him right • � Just like quantum theory today…computer chips, lasers • � Problem: Maxwell equations don’t obey Galilean relativity ! – � Simple example: imagine Phil holds an electric charge • � Both standing still: both see only electric field of charge • � Phil is on moving train, Bill remains at rest: – � Bill sees moving charge = current � magnetic field B – � Phil still sees only electric field E of static charge First: Both are at rest Next: Phil rides on 100 m/s train moving past Bill Both see E field and Phil sees only E field, but Bill sees a B field no B field P + Moving charge = current I B P B + Static charge 100 m/s Train Earth Earth 12
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