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HARVARD PHYS 11b - Lecture 22

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Physics 11bLecture #22Diffraction and PolarizationS&J Chapter 38What We Did Last Time Studied interference >2 waves overlap Æ Amplitudes add upÆ Intensity = (amplitude)2does not add up Two-body interference Intensity depends on angle Young’s experiment demonstrated wave-ness of light Thin-film interference Reflectivity of thin film depends on thickness, plus hard/soft-ness of the two boundaries Soap bubbles Newton’s rings 1/4-wavelength anti-reflective coatings20sincosdIIπθλ=Today’s Goals Interference of light from many sources Î Diffraction Diffraction grating Fraunhofer diffraction: what happens when light goes through a not-too-narrow slit Diffraction limit of optical devices’ resolution Polarization of light Direction of E field (perpendicular to propagation) Polarizing filters Polarization of reflected lightN-Body Interference It’s easy to extend 2-body interference to n bodies Result is interesting There are practical applications Difference between paths fromneighboring sources is Add up E from all sourcesθdsindθsindθ1xnx1(1)sinmxxmdθ=+ −{}01011cos( )cos ( 1) sinnmmnmEEkxtEkx m d tωθω===−=+−−⎡⎤⎣⎦∑∑Calculating this sum is possible but difficultN-Body Interference The total E turns out to be Intensity goes with ∝ E2as usual is 1/2 of the phase difference between neighbors I is a periodic function (period 2π) of φ At φ= 0 (i.e. θ= 0) it goes to()1sin202sin2sin( )cossin( )nkndxxkdEE k tθθω+=−22sin2220022sin2sin ( )sinsin ( ) sinkndkdnIE Eθθφφ∝=sin2kdθφ≡θdsindθ1xnx0sinlimsinnnφφφ→=N-Body Interference Calculate intensity relative to θ= 0 Peaks nearφ= 0, π, 2π, …gets narrower as n increases With large enough n,waves vanish except forvery sharp peaks at First peak at220sin()sinnIEφθφ⎛⎞∝⎜⎟⎝⎠2() sin(0) sinInInθφφ⎛⎞=⎜⎟⎝⎠Large n0IIx3π2ππ0sin2kdmθφπ==sinmdλθ=3π2ππsin dθλ=220(0)IEn∝φDiffraction Grating Consider a plate with fine stripes e.g., a transparent film with regularscratches Gaps between scratches transmit light Like Young’s experiment, with manymany many slits Light is “diffracted” at angle θonly if Trivial solution: θ= 0 for any wavelength First non-trivial solution: m = 1 Æ Light goes to different θdepending on λ Can measure light intensity as a function of wavelengthlightsinmdλθ=θsin dθλ=Wide Slit Now we make a wide slit on the wall We know what happens:a shadow shaped just like the slit We also know that we getspreading wavesfrom a narrow slit Observer behind the wall seeslight with a narrow slit, but notwith a wide slit Does this make sense? What happens if the width ofthe slit is “in between”?LightLightWide Slit Huygens’ principle tells us what to do Imagine a lot of wave sourcesin the slit Waves spread from each source Add up the amplitude This is n-body interferencewith big n We know the answer for any n Make n Æ infinity while keeping nd = a = width of the slit2() sin(0) sinInInθφφ⎛⎞=⎜⎟⎝⎠dasin2kdθφ≡Wide Slit Since , Take the limit of n Æ infinity Let’s define Intensity is (this)2 This is known as the Fraunhofer diffractionsin2sin2sinsinsin sinkakannnnθθφφ=sin sin22kd kanθθφ≡=sin2kanθφ=nd a=sin sin sin222sin sin sin222sin sin sinlim limsinka ka kaka ka kannnnnnθθθθθθ→∞ →∞==22() sin(0)IIθαα=sin sin2ka aθπθαλ≡=sinαα=Fraunhofer Diffraction Diffraction gives a bright bandsurrounded by dimmer sidebands Dark bands occur at Width of the central band is If the slit is wide (a >> λ) Î ∆θsmall, i.e. no spread If the slit is narrow (a << λ) Î ∆θlarge, i.e. spherical waves22() sin0(0)IIθαα==darksinmaλθ=sinaπθαλ≡sinaλθ=sinaλθ=−sin 2aλθ=sin 2aλθ=−aθλ∆=Optical Devices Optical devices use lenses to collect light from object Telescopes, microscopes, cameras, human eyes Lenses have finite aperture It’s like passing light througha hole Æ Diffraction Light spreads out according to This blurs the imagesinaλθ=Optical Resolution Consider two points of an object Light from these points end up in two points But they are spread out due to diffraction by If the angle θbetween two points is θ < θd,the points cannot be distinguished (or resolved) in the image We can consider θdas the ultimate resolution It’s fundamental – due to the wave nature of light It’s determined by the aperture a of the devicesindaλθ=dθθAiry Disc Fraunhofer diffraction was calculated for a slit Lenses are usually round We need the diffraction formula for a round hole Calculation tedious, but not fundamentally different Solution known as Airy Disclightθaθsin 1.22daλθ=dθDiffraction Limit Resolution of an optical device with aperture a is Called the diffraction limit or Rayleigh criterion It can be worse, of course Example: 10-inch telescope Eyes good for 10-3rad If magnification > 400, you start to see the blur No point in making >500 power 10-inch telescope Common wisdom: 50x per inchsin 1.22aλθθ≈>96550 10 m1.22 2.6 10 rad0.254m−−×=×Large Telescopes The larger, the better – It’s that simple Better resolution + light collection Keck + KeckII (10 m) on Mauna Kea, Hawaii Location chosen to minimize the disturbancedue to air density fluctuation Index refraction of air limitsall telescopes on EarthPolarization For a given k, there are two possible directions of E E is in the x- or y-plane Direction of E is calledthe polarization They are 2 independentsolutions of the wave eqns Linear combination ofthese solutions give youall the possible directionsof EEBEBLinear Combinations Looking up from downstream (+x) Can make any anglefrom the horizontaland vertical wavesEBEBHorizontally polarized Vertically polarizedEBAddArbitrary directionPolarizing Filters A polarizing filter passes only one polarization e.g. vertically polarizing filter blockshorizontally-polarized light Analogy: transverse waves on a stringrunning through a narrow slit Widely used in photography and sunglassesPolarization of Reflected Light Reflected light is (partially) polarized E vector tends to lie parallel to the surface Polarizing filters can cut reflection


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HARVARD PHYS 11b - Lecture 22

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