MSU PHY 184 - Physics for Scientists & Engineers 2

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April 10, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 43April 10, 2005 Physics for Scientists&Engineers 2 2ReviewReview! Coherent light with wavelength ! incident on two narrowslits separated by a distance d produces an interferencepattern on a screen located a large distance L from the slits! The position of the bright fringes from the center line isgiven by! The position of the dark fringes from the center line isgiven byy =m!Ld m = 0, m = ±1, m = ±2,...( )y =m +12!"#$%&'Ld m = 0, m = ±1, m = ±2,...( )April 10, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! The integer m is called the order of the fringe! The order has a different meaning for bright fringes andfor dark fringes! For constructive interference•m = 1 would give us the angle of the first order bright fringe•m = 2 would give us the second order fringe, etc.! For destructive interference•m = 0 would give us the angle of the first order dark fringe•m = 1 would give us the second order fringe, etc.! For both bright and dark fringes, the first order fringe isthe one closest to the central maximumApril 10, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3)! Coherent light with wavelength ! incident on two narrow slits separatedby a distance d produces an interference pattern on a screen located alarge distance L from the slits with intensity given byI = 4 Imaxcos2!dy"L#$%&'(April 10, 2005 Physics for Scientists&Engineers 2 5DiffractionDiffraction! Any wave passing through an opening that is about the samesize as the wavelength of the wave will experiencediffraction! The same applies to light waves! Diffraction means that the wave will spread out on theother side of the opening rather than having the openingcast a sharp shadow! In addition, if light passes through a narrow slit, it willproduce an interference pattern called a diffractionpattern! Light passing a sharp edge will also exhibit a diffractionpatternApril 10, 2005 Physics for Scientists&Engineers 2 6Diffraction (2)Diffraction (2)! Huygens’ principle can describe this spreading out and wecan use a Huygens’ construction to quantify the diffractionphenomenon! For example, to the right weshow coherent light incident onan opening with dimensioncomparable to the wavelengthof the light! Rather than casting a sharpshadow, light spreads out on theother side of the openingApril 10, 2005 Physics for Scientists&Engineers 2 7Diffraction (3)Diffraction (3)! We can describe this spreading outby using a Huygens’ constructionand assuming that sphericalwavelets are emitted at severalpoints inside the opening! The resulting light waves on theright side of the opening canundergo interference and produce acharacteristic diffraction pattern! Light waves can also encounterbarriers such as the one shownApril 10, 2005 Physics for Scientists&Engineers 2 8Diffraction (4)Diffraction (4)! In this case, the light far from the edge of thebarrier continues to travel in a straight line! The light near the edge of the barrier seems tobend around the barrier is described by thesources of wavelets near the edge! Diffraction phenomena cannot be described bygeometric optics. Often the resolution of anoptical instrument can be limited by diffractioneffects rather than geometric effectsApril 10, 2005 Physics for Scientists&Engineers 2 9Single Slit DiffractionSingle Slit Diffraction! We start our quantitative description of diffraction bystudying the diffraction of light through a single slit ofwidth a that is comparable to the wavelength of light thatis passing through the slit! We will approach the calculation using a Huygens’construction! We assume that the light passing through the single slit isdescribed by spherical wavelets emitted from adistribution of points located in the slit! The light emitted from these points will superimpose andinterfere based on the path length difference for eachwavelet at each positionApril 10, 2005 Physics for Scientists&Engineers 2 10Single Slit Diffraction (2)Single Slit Diffraction (2)! At a distant screen we will observe an intensitypattern characteristic of diffraction! This intensity pattern will consist of bright anddark fringes! For the case of two-slit interference, we wereable to work out the equations for the brightfringes based on constructive interference! For diffraction we will analyze the dark fringes asdestructive interference because constructiveinterference is beyond the scope of this bookApril 10, 2005 Physics for Scientists&Engineers 2 11Single Slit Diffraction (3)Single Slit Diffraction (3)! To study the interference let’s expand and simplify ourprevious figure for single slit diffraction! We assume coherentlight with wavelength !incident on a slit withwidth a that producesan interference patternon a screen a distanceL away! We employ a simplifyingmethod of analyzing pairs of light waves emitted frompoints in the slitApril 10, 2005 Physics for Scientists&Engineers 2 12Single Slit Diffraction (4)Single Slit Diffraction (4)! We start with light emitted from the top edge of the slitand from the center of the slit as shown! To analyze the pathdifference we show anexpanded version of ourfigure to the right! Here we assume that thepoint P on the screen is farenough away that the raysr1 and r2 are parallel andmake an angle ! with thecentral axisApril 10, 2005 Physics for Scientists&Engineers 2 13Single Slit Diffraction (5)Single Slit Diffraction (5)! Therefore the path length difference for these two rays is! The criterion for the first dark fringe is! Although we chose one ray originating from the top edge ofthe slit and one from the middle of the slit to locate thefirst dark fringe, we could have used any two rays thatoriginated a/2 apart inside the slitsin!="xa / 2 or "x =a sin!2!x =a sin"2=#2April 10, 2005 Physics for Scientists&Engineers 2 14Single Slit Diffraction (6)Single Slit Diffraction (6)! Now let’s consider four rays instead of two! Here we choose a ray from the top edge of the slit andthree more rays originating from points spaced a/4 apartApril 10, 2005 Physics for Scientists&Engineers 2 15Single Slit Diffraction (7)Single Slit Diffraction (7)! We can expand the previousdrawing to represent the caseof the screen being far away asshown to the right! The path length differencebetween the pairs of rays(r1,r2), (r2,r3), (r3,r4) is given by! The dark fringe is given bysin!="xa / 4 or "x


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MSU PHY 184 - Physics for Scientists & Engineers 2

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