asinkLkysintdywhereListhedistancefromthetopoftheslittothedetectorClearlythisisafunctionperiodicintimewithangularfrequencyWewishtoputitintotheformytamplitudesinphasetOnceitsinth ID: 298687
Download Pdf The PPT/PDF document "IntensityinSingle-slitDi raction.Awaveof..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Intheseterms,y(t)==mnAei(kL�!t)+Aei(kL+ �!t)o==mnAei(kL+ =2� =2�!t)+Aei(kL+ =2+ =2�!t)o==mnAei(kL+ =2�!t)[e�i =2+e+i =2]o==mnAei(kL+ =2�!t)[2cos( =2)]o=2Acos( =2)sin(kL+ =2�!t):Intermsoftheformabove,[amplitude]=2Acos( =2):Theintensityofthissignalisproportionaltotheamplitudesquared.Ifwedenetheintensityat=0tobeIm(\Intensityatthemiddle"or,asitturnsout,\Intensityatthemaximum"),thenintensity=Imcos2 2where =2d sin:(2)IntensityinSingle-slitDi raction:Setup.Awaveofwavelengthpassesthroughasingleslitofwidtha: PlacethedetectoralongdistanceLfromthetopoftheslit(\Fraunhoferlimit").ThenthedetectorisadistanceL+zsinalongthepathshowninthegure.Hencethewavesignalatthedetectorduetothepathshownisy(x;t)=[amplitude]sin(k(L+zsin)�!t):Becausethiswaveisthewaveduetoaninnitesimalwindowofwidthdz,theamplitudewillbeverysmall.Wewrite[amplitude]=Adz a2 (wherethedivisionbyainsuresthat\[amplitude]"hastheproperdimensions).Thetotalwavesignalreceivedbythedetectoristhesignalintegratedoverallpossiblepaths,fromz=0toz=a:y(t)=Za0A asin(kL+kzsin�!t)dz:Clearly,thisisafunctionperiodicintimewithangularfrequency!.Wewishtoputitintotheformy(t)=[amplitude]sin([phase]�!t):Onceit'sinthisform,theintensityisproportionalto[amplitude]2.IntensityinSingle-slitDi raction:Math.Tomakethisalgebraeasier,weuseEuler'srelationeit=cost+isint:Intheseterms,y(t)==mZa0A aei(kL�!t)eikzsindz==mA aei(kL�!t)Za0eikzsindz:ButZa0eikzsindz=1 iksineikzsinaz=0=1 iksin�eikasin�1sowedene =1 2kasinandndy(t)==mA 2i ei(kL�!t)�ei2 �1==mA 2i ei(kL+ �!t)�e+i �e�i ==mA 2i ei(kL+ �!t)(2isin )==mAsin ei(kL+ �!t)=Asin sin(kL+ �!t):(3)Thisexpressionisinthedesiredform.Becauseintensityisproportionaltoamplitudesquared,intensity=Imsin 2where =a sin:(4)3