April 13, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 45April 13, 2005 Physics for Scientists&Engineers 2 2ReviewReview! Imagine light incident on a thin, optically clear medium inair such as a soap bubble! Transmitted light has no phase change! For reflected light• If n1 < n2, the phase of the reflected wave will bechanged by half a wavelength• If n1 > n2 then there will be no phase changeApril 13, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! The criterion for constructive interference of lightincident on a thin, optically clear medium in air such as asoap bubble is! The minimum thickness tmin that will produce constructiveinterference corresponds to! We get the same answer for the destructive interferenceof light passing from air to two clear optical media suchthat nair < n1 < n2 such as the coating on a camera lensm +12!"#$%&'airn= 2t m = 0, m = ±1, m = ±2,...( )tmin=!air4nApril 13, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3)! An interferometer is a device designed to measure lengthsor changes in length using interference of light! An interferometer can measure lengths or changes inlengths to a fraction of the wavelength of light usinginterference fringes! An interferometer can be used to measure the thickness ofa material or the index of refraction of a material byplacing in one of the paths of the interfering light andcounting the change in the number of fringesNmaterial! Nair=2tn"!2t"=2t"n ! 1( )April 13, 2005 Physics for Scientists&Engineers 2 5Diffraction by a Circular OpeningDiffraction by a Circular Opening! We have considered interference through two slits anddiffraction through a single slit! Now we consider diffraction of light through a circularopening! Diffraction through a circular opening relates to observingobjects with telescopes with circular mirrors and camerasthat have circular lens! The resolution of a telescope or camera is limited bydiffraction phenomena! The first diffraction minimum from light with wavelength !passing through a circular opening with diameter d issin!= 1.22"dApril 13, 2005 Physics for Scientists&Engineers 2 6Diffraction by a Circular Opening (2)Diffraction by a Circular Opening (2)! This result is similar to the result from a single slit except for thefactor of 1.22! If one is using a circular lens to observe two distant points objects,such as two stars, whose angular separation is small, diffraction limitsthe ability of the lens to distinguish these two objects! The criterion for being able to separate two point objects is based onthe idea that if the first image is centered on the first diffractionminimum of the second object, the objects are just resolved! This criterion is called Rayleigh’s Criterion and is expressed as! Where "R is the minimum observable angular separation, ! is thewavelength, and d is diameter of the lens!R= sin"11.22#d$%&'()April 13, 2005 Physics for Scientists&Engineers 2 7Examples of ResolutionExamples of ResolutionResolved Barely Resolved Not ResolvedApril 13, 2005 Physics for Scientists&Engineers 2 8Hubble Space TelescopeHubble Space Telescope! The diameter of the Hubble Space Telescopeis 2.4 m. What is the minimum angularresolution of the Hubble Space Telescope?! Using Rayleigh’s Criterion with green lightof wavelength 550 nm we get! which corresponds to the angle subtended by a dime located 64km away! When the Hubble Space Telescope was first launched, flaws werediscovered in the main mirror that limited its ability to resolveimages! A repair mission fixed the mirror so that it now functions at thediffraction limit!R= 1.22550 "10#9 m2.4 m= 2.8 "10#7April 13, 2005 Physics for Scientists&Engineers 2 9Double Slit DiffractionDouble Slit Diffraction! We have discussed the interference pattern produced bytwo slits! For that analysis we assumed that the slits themselveswere very narrow compared with the wavelength of light,a << !! For these narrow slits, the diffraction maxima are verywide and we saw peaks in the intensity that were the sameintensity at all angles! For many sets of real-world double slits, the condition a << !is not met and we observe that not all the interferencefringes have the same intensityApril 13, 2005 Physics for Scientists&Engineers 2 10Double Slit DiffractionDouble Slit Diffraction (2)(2)! With diffraction effects the intensity of the interferencepattern from double slits is given by! If the screen is placed a sufficiently large distance fromthe slits then we can write! On the next slide we calculate the intensity pattern for adouble slit including interference and diffraction assumingL = 2.0 m, a = 5.0!10-6 m, d = 1.0!10-5 m, and " = 550 nmI = Imaxcos2!sin""#$%&'(2 "=)a*sin+ !=)d*sin+!="ay#L and $="dy#LApril 13, 2005 Physics for Scientists&Engineers 2 11Double Slit Diffraction (3)Double Slit Diffraction (3)Pattern for single slitdiffractionPattern for double slitinterferenceApril 13, 2005 Physics for Scientists&Engineers 2 12Real Life Two Slit Diffraction PatternReal Life Two Slit Diffraction PatternDiffraction minimaApril 13, 2005 Physics for Scientists&Engineers 2 13Diffraction GratingsDiffraction Gratings! We have discussed diffraction and interference for a single slit and fortwo slits! Now we will discuss the application of diffraction and interference to asystem of many slits! Putting many slits together forms a device called a diffraction grating! A diffraction grating has a large number of slits, or rulings, placed veryclose together! A diffraction grating can also be constructed using an opaque materialwith grooves rather than actual slits! A diffraction grating produces an intensity pattern that consists ofnarrow bright fringes separated by wide dark areas! This characteristic pattern results from the use of many slits thatproduce destructive interference away from the maximaApril 13, 2005 Physics for Scientists&Engineers 2 14Diffraction GratingsDiffraction Gratings! A portion of a diffraction grating is shown below! In this drawing we see coherent light with wavelengthincident on a series of narrow slits each separated by adistance d! A diffraction pattern is produced on a screen a longdistance L awayApril 13, 2005 Physics for Scientists&Engineers 2 15Diffraction Gratings (2)Diffraction Gratings (2)! We can expand our drawing as we did for the singleslit and double slit to enable our analysis of the pathlength difference for the
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