Chapter 36. a λ 2 2. (minima-dark fringes) Diffraction and the Wave Theory of Light. Diffraction by a Single Slit: Locating the Minima, Cont'd

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1 Chpter 36 Diffrction In Chpter 35, we sw how light bes pssing through ifferent slits cn interfere with ech other n how be fter pssing through single slit flres-iffrcts- in Young's experient. Diffrction through single slit or pst either nrrow obstcle or n ege prouces rich interference ptterns. The physics of iffrction plys n iportnt role in ny scientific n engineering fiels. In this chpter we explin iffrction using the wve nture of light n iscuss severl pplictions of iffrction in science n technology. Diffrction n the Wve Theory of Diffrction Pttern fro single nrrow slit. Fresnel Bright Spot. Centrl xiu Sie or seconry xi These ptterns cnnot be expline using geoetricl optics (Ch. 34)! Bright spot Diffrction by Single Slit: Locting the Mini When the pth length ifference between rys r n r is /, the two rys will be out of phse when they rech P on the screen, resulting in estructive interference t P. The pth length ifference is the istnce fro the strting point of r t the center of the slit to point b. For D>>, the pth length ifference between rys r n r is (/) sin. Diffrction by Single Slit: Locting the Mini, Cont' Repet previous nlysis for pirs of rys, ech seprte by verticl istnce of / t the slit. Setting pth length ifference to / for ech pir of rys, we obtin the first rk fringes t: sin = sin = (first iniu) For secon iniu, ivie slit into 4 zones of equl withs /4 (seprtion between pirs of rys). Destructive interference occurs when the pth length ifference for ech pir is /. sin = sin = (secon iniu) 4 Diviing the slit into incresingly lrger even nubers of zones, we cn fin higher orer ini: sin =, for =,,3K (ini-rk fringes) Fig Fig To obtin the loctions of the ini, the slit ws eqully ivie into N zones, ech with with x. Ech zone cts s source of Huygens wvelets. Now these zones cn be superipose t the screen to obtin the intensity s function of, the ngle to the centrl xis. To fin the net electric fiel E (intensity α E ) t point P on the screen, we nee the phse reltionships ong the wvelets rriving fro ifferent zones: Fig Intensity in Single-Slit Diffrction, Qulittively phse π pth length = ifference ifference N=8 = 0 sll φ = π ( xsin ) st in. st sie x. Fig Intensity in Single-Slit Diffrction, Quntittively Here we will show tht the intensity t the screen ue to single slit is: sinα I ( ) = I (36-5) π where α = φ = sin (36-6) In Eq. 36-5, ini occur when: α = π, for =,,3K If we put this into Eq we fin: π π = sin, for =,,3K or sin =, for =,,3 K (ini-rk fringes)

2 Proof of Eqs n 36-6 Distnt point source, e,g., str Diffrction by Circulr Aperture lens sin =. (st in.- circ. perture) sin =. (st in.- single slit) Ige is not point, s expecte fro geoetricl optics! Diffrction is responsible for this ige pttern Resolvbility Ryleigh s Criterion: two point sources re brely resolvble if their ngulr seprtion R results in the centrl xiu of the iffrction pttern of one source s ige is centere on the first iniu of the iffrction pttern of the other source s ige. Diffrction by Double Slit Double slit experient escribe in Ch. 35 where ssue tht the slit with <<. Wht if this is not the cse? Two vnishingly nrrow slits << Single slit ~ Fig Two Single slits ~ Fig R = sin.. (Ryleigh's criterion) Diffrction Grtings Device with N slits (rulings) cn be use to nipulte light, such s seprte ifferent wvelengths of light tht re contine in single be. How oes iffrction grting ffect onochrotic light? With of Lines The bility of the iffrction grting to resolve (seprte) ifferent wvlength epens on the with of the lines (xi) Fig Fig Fig sin = for = 0,, K (xi-lines) Fig If we ivie slit into infinitesilly wie zones x, the rc of the phsors pproches the rc of circle. The length of the rc is E. φ is the ifference in phse between the infinitesil vectors t the left n right ens of the rc. φ is lso the ngle between the rii rke R. E The sh line bisecting f fors two tringles, where: sin φ = E R In rin esure: φ = R E Solving the previous equtions for E one obtins: E = sin φ φ The intensity t the screen is therefore: I ( ) E sinα = I ( ) = I I E φ is relte to the pth length ifference cross the entire slit: π φ = ( sin ) 36-7 Fig R sll π β = sin sinα I ( ) = I ( cos β ) (ouble slit) π 36-9 α = sin

3 With of Lines, cont In this course, soun wve is roughly efine s ny longituinl wve (prticles oving long the irection of wve propgtion). Grting Spectroscope Seprtes ifferent wvelengths (colors) of light into istinct iffrction lines N sin =, sin = (hlf with of centrl line) N Fig Fig. 36- = (hlf with of line t ) N cos Fig. 36- Opticlly Vrible Grphics Grtings ebee in evice sen out hunres or even thousns of iffrction orers to prouce virtul iges tht vry with viewing ngle. Coplicte to esign n extreely ifficult to counterfeit, so kes n excellent security grphic. Grtings: Dispersion n Resolving Power Dispersion: the ngulr spreing of ifferent wvelengths by grting D = (ispersion efine) D = (ispersion of grting) (36-30) cos Resolving Power vg R = (resolving power efine) R = N (resolving power of grting) (36-3) Angulr position of xi Proof of Eq Differentil of first eqution (wht chnge in ngle oes chnge in wvelength prouce?) For sll ngles sin = ( cos ) = n ( cos ) = = cos ( ) Ryleigh's criterion for hlf-with to resolve two lines Proof of Eq Substituting for in clcultion on previous slie = N cos = N R = = N Fig

4 Dispersion n Resolving Power Copre In this course, soun wve is roughly efine s ny longituinl wve (prticles oving long the irection of wve propgtion). X-Ry Diffrction X-rys re electrognetic rition with wvelength ~ Å = 0-0 (visible light ~5.5x0-7 ) Tble 36- Grting N (n) D ( o /µ) R X-ry genertion A o B o C o Dt re for = 589 n n = Fig Fig X-ry wvelengths to short to be resolve by stnr opticl grting ( )( 0. n) = sin = sin = n X-Ry Diffrction, cont X-Ry Diffrction, cont Diffrction of x-rys by crystl: spcing of jcent crystl plnes on the orer of 0. n three-iensionl iffrction grting with iffrction xi long ngles where reflections fro ifferent plnes interfere constructively sin = for = 0,, K (Brgg's lw) Fig interplnr spcing is relteto the unit cell iension 0 5 = or = = Not only cn crystls be use to seprte ifferent x-ry wvelengths, but x-rys in turn cn be use to stuy crystls, for exple eterine the type crystl orering n 0 Fig hitt In Young's ouble-slit experient, the slit seprtion is ouble. This results in: A. n increse in fringe intensity B. ecrese in fringe intensity C. hlving of the wvelength D. hlving of the fringe spcing E. oubling of the fringe spcing hitt A light wve with n electric Fiel plitue of E 0 n phse constnt of zero is to be cobine with one of the following wves: wve A hs n plitue of E 0 n phse constnt of zero wve B hs n plitue of E 0 n phse constnt of π wve C hs n plitue of E 0 n phse constnt of zero wve D hs n plitue of E 0 n phse constnt of π wve E hs n plitue of 3E 0 n phse constnt of π Which of these cobintions prouces the gretest intensity? 3 4 4

5 The light wves represente by the three rys shown in the igr ll hve the se frequency. 4.7 wvelengths fit into lyer, 3. wvelengths fit into lyer, n 5.3 wvelengths fit into lyer 3. Rnk the lyers ccoring to the spees of the wves, lest to gretest. A.,, 3 B.,, 3 C. 3,, D. 3,, E., 3, Lyer Lyer Lyer 3 5 In Young s ouble-slit experient, light of wvelength 500 n illuintes two slits tht re seprte by. The seprtion between jcent bright fringes on screen 5 fro the slits is: A. 0.0 c B. 0.5 c C c D..0 c E. none of the bove 6 5