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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