April 8, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 42April 8, 2005 Physics for Scientists&Engineers 2 2ReviewReview! If light waves are traveling from some point, then thephase difference !x can be related to the path differencebetween the two waves! The criterion for constructive interference is given by apath difference !x given by! Destructive interference will take place if the pathdifference !x is a half wavelength plus an integer timesthe wavelength!x = m" m = 0, m = ±1, m = ±2,...( )!x = m +12"#$%&'( m = 0, m = ±1, m = ±2,...( )April 8, 2005 Physics for Scientists&Engineers 2 3Double Slit InterferenceDouble Slit Interference! Our first example of the interference of light is Young’sdouble slit experiment! For this situation we assume that we have coherent light• light with the same wavelength and phase! This light is incident on a pair of slits! Each slit is smaller than the wavelength of light! The slits are separated by a distance d! For each slit we will use a Huygens’ construction and assumeall the light observed passing through each slit is due towavelets emitted from a single point at the center of thatslitApril 8, 2005 Physics for Scientists&Engineers 2 4Double Slit Interference (2)Double Slit Interference (2)! Below we see that spherical wavelets emitted from a pointin the center of the slit! We assume that the slit is much smaller than thewavelength of light so that we can represent the source ofthe wavelets with one pointApril 8, 2005 Physics for Scientists&Engineers 2 5Double Slit Interference (3)Double Slit Interference (3)! Now let’s look at two slits like the one on the previous page! We have coherent light incident from the left and a sourceof spherical wavelets at the center of each slit! We see that the gray dashed lines represent lines alongwhich there is constructive interferenceApril 8, 2005 Physics for Scientists&Engineers 2 6Double Slit Interference (4)Double Slit Interference (4)! If we place a screen to the right of the slits we willobserve an alternating pattern of bright lines and darklines corresponding to constructive and destructiveinterference between the light waves emitted from thetwo slits! To quantify these lines ofconstructive interferencewe expand and simplifythe plot on the previouspageApril 8, 2005 Physics for Scientists&Engineers 2 7Double Slit Interference (5)Double Slit Interference (5)! The two lines r1 and r2 represent the distance from thecenter of slit S1 and slit S2 respectively to a point P on ascreen that is placed a distance L away from the slits! A line drawn from a point midway between the two slits tothe same point on the screen makes an angle " with respectto a line drawn from the slits perpendicular to the screen! The point P on the screenis a distance y above thecenterlineApril 8, 2005 Physics for Scientists&Engineers 2 8Double Slit Interference (6)Double Slit Interference (6)! To further quantify the two slit geometry we expand andsimplify the figure on the previous slide! In this figure we assume that we have placed the screen alarge distance L away from the slits such that the lines r1and r2 are parallel to each other and to the line drawnfrom the center of the two slits to point P! We draw a line from S1perpendicular to r1 and r2making a triangle with sidesd, b, and !x! The quantity !x representsthe path length differencebetween r1 and r2April 8, 2005 Physics for Scientists&Engineers 2 9Double Slit Interference (7)Double Slit Interference (7)! This path length difference will produce different phasesfor light originating from the two slits and illuminating thescreen at point P! The path length difference can be expressed in terms ofthe distance between the slits and the angle at which thelight is observed! For constructive interference this path length differencemust be a multiple of the wavelength of the incident lightsin!="xd or "x = d sin!!x = d sin"= m# m = 0, m = ±1, m = ±2,...( )April 8, 2005 Physics for Scientists&Engineers 2 10Double Slit Interference (8)Double Slit Interference (8)! For destructive interference the path length differencemust be an integer plus a half times the wavelength! A bright fringe on the screen signals constructiveinterference! A dark fringe on the screen signals destructiveinterference! Note that for constructive interference and m = 0, weobtain " = 0, which means that !x = 0 and we have a brightfringe at zero degrees! This bright fringe is called the central maximum!x = d sin"= m +12#$%&'() m = 0, m = ±1, m = ±2,...( )April 8, 2005 Physics for Scientists&Engineers 2 11Order of Interference FringesOrder of Interference Fringes! 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 8, 2005 Physics for Scientists&Engineers 2 12Fringes on a Distance ScreenFringes on a Distance Screen! If the screen is placed a sufficiently large distance fromthe slits, the angle " will be small and we can make theapproximation! We can express constructive interference as! Rearranging this equation gives us the distance of thebright fringes from the central maximum along the screensin!" tan!= y / Ld sin!= dyL= m" m = 0, m = ±1, m = ±2,...( )y =m!Ld m = 0, m = ±1, m = ±2,...( )April 8, 2005 Physics for Scientists&Engineers 2 13Fringes on a Distance Screen (2)Fringes on a Distance Screen (2)! Similarly we can express the distance of the dark fringesfrom the central maximum along the screen as! These formulas allow us to locate the positions of thebright and dark fringes! However, we can also calculate the intensity of the light atany point on the screeny =m +12!"#$%&'Ld m = 0, m = ±1, m = ±2,...( )April 8, 2005 Physics for Scientists&Engineers 2 14Double Slit Intensity on aDouble Slit Intensity on a Distant ScreenDistant Screen! We start our calculation of the intensity of light from adouble slit by assuming that the light emitted at each slit isin phase! The electric field of the light waves can be described by!
View Full Document
Unlocking...