chapter 9 interference and chapter 9 interference and diffraction 9.1 introduction 453 9:)...

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  • Chapter 9

    Interference and Diffraction

    9.1 Int roduction 453

    9:) Interfe rence between Tw o Cohe re n t Point Sour ces 454

    Co herent sources -t54 Co ns tr uct ive and destructi ve in te rference ..55 In te rference pat tern 455 Xea r field :m d far field -156

    " Bou ndary" bet ween near and fa r 457 Use of a cOll\'erging lens 10 ob taiu Iar -Geld int e rfe rence

    p3ttem 458 Far-field inte rference pa tte rn 459 Priucipal maximum 460 Relative phase due to path differenc e 461

    •·..average.. trn cling W3\"e ,16 1 Photon flux ~62 Two-slit interfer ence paU:em -'62

    Sources oscillati ng in phase ...63 Sou rces oscillati ng out of ph ase 463 In te rference patt e rn near 9 = 0 3 '1&1 En erg)" conserva tion 465 On e plus one equa ls Iour 'l65 On e plus one equa ls zero ·166

    9.3 I nt erference between Two Independent Sources 466

    I.n de pe udent sources a nd co he rence tim e 466 " Incoherence" and interference 467 Brown and Twiss exper iment 468

    9.4 How Large Can a " Poin t" Ligh t Source Be? 470

    C lassica l point source 470 Sim ple exten ded source 471 Co herence condition -'72

    9.5 Ang ular \Vidth of a "Beam" of Tra veling w aves

    An gula r width of a beam is di ffractio n limited 4i 3 A beam is an inte rfere nce maximum 4i5

    9.6 Diffr acti on and Huygens' Principle 478

    Differ ence be tween int erf er en ce and di ffn dion ·H8 How an opaque screen works 4i S Shinj- and black opa que screens 4i9 Effect of a hole in an opaq ue screen 480 H uygeus' pri nciple 48 1 Cnlcu jatic n of single·!olil rliffrac liOIl patt e rn using l l uygens'

    construc tion 482 Sing le-slit diffracti on patt ern 485 Angula r width of a diffraction ·limited beam 487 Angular reso lut ion of the human eye ·l87 Rayleigh 's cr iterion 488 Nomencla tu re: Fraun hofer an d Fresne l di ffracti on 488

    47~

    Angular width or bea m 4i6 Applir.ntinn: Laser beam IJeTSUS finsMight beam 477

    Fourier an alysis of the t ransverse spa ce dependen ce of a co he ren t

    sour ce 489 Jmpo rt ant resul ts of Four ier ana lysis .490 Diffracti on patt ern for tw o wide slits 491 Diffraction p attern for man y identi cal par allel wide slits 492 Multip le-slit int er ference patt em 493 Principal maxi ma, ce ntral maxim um , white ligh t source 493

    Angu la r wid th of a princi pal maximum 495 Transm ission-type diff ract ion gra ting 496 Diffrac tion by an opaq ue obstacle 496 How far downstr eam does a shadow exte nd? 497

  • 9,1 In trod uction

    Most of our studies so far have been esse ntially one-di me nsional. in the sense that th ere wa s only one pa th by which a wav e emitte d at one place couJd go to anothe r place. Now we shall conside r situ ations where there are differ ent possible path s from an emi tte r to a det ect or. Th ese lead to what are called interf erence or diffraction phenom ena. resulting from con- struc tive and destructiv e superpo sition of waves that have di fferent phase shifts, depending on the path taken .

    I n Sec . 9.2 we conside r the supe rpos itio n at a detect or of the wav es emitte d by two poin t sources having the same frequ e ncy and a co ns tan t ph ase relation . Examples are water waves emitt ed by two screwheads jigg ling the sur face of a pan of water or ligh t e mitt ed by the currents in th e edges of two slits which are illuminated by a line or po in t source (Home Exp . 9.18) or sound waves emitte d by two loudspeakers driven by the same audio oscillator.

    In Sec. 9.3 we conside r int erfer ence between two " independent" sources, i.e ., so urces whose phases are not co nstr ained to maintain a definit e rela- non . We find that the interferen ce pa ttern remains constan t only for time inte rvals of order (av)- I, whe re av is the frequ ency bandwidth of tbe sources . Never the less, by a sufficie ntly fas t mea surement one can deter - mine th e interferen ce patt er n.

    In Sec. 9.4 we find how lar ge a sour ce can be an d still be hav e like a point so urce , when the sour ce consists of indep endentl y radiating part s and when the de te ctor ave rages over lon g tim e in tervals [Le., long com- pared with (Llp)- l ). Th e result can be verified in an easy hom e exp erime nt (Home Exp. 9.20). Another home expe rime nt (Home Exp. 9.21) demon- st rates the coherence of a Lloyd 's mirror.

    I n Sec. 9.5 we give a crude deri vati on of th e result th at a bea m of spa- tial width D bas an ang ular di vergence (" width") of ord er M ;:::; A/ D abo ut th e do minan t dir ecti on of travel. Th is fact is ma the ma tica lly rela ted (by the theory of Fourier analysis) to the fact that a pulse of time width at has a frequ ency width of order (a t)- l.

    I n Sec . 9.6 we use I1uyge ns' co nst ructio n to find the int erferen ce pat- terns of single and multi ple slits. Th e emphasis is on op tic al and elect ro- magnetic ph en omena. T her e are seve ral hom e expe rime nts involving diffr acti on gra tings and various diffr ac tion pa tterns. Fo r th ese expe riments we strongly advise the student to ge t a " display lamp" - a light bulb with a cle ar glass envelope and a single str aigh t filam ent about 3 inches long (about 40 ce n ts in most groc ery or ha rd ware stores). Most of the experi- me nts use a ile of these as a line source.

    9.7 Geo metrica l Optics 498

    Spec ular reflection -199 Nonspec ular reflect ion from a reguJar 3rra~' 500 Image of a point source in a mlr rc r -c-virt ual source

    and real source 500 Refraction-SocII's law-Fermat' s p rinciple 50 1 EJlipsoida l mirro r 504 Concave parabolic mirro r 50-1 Concave spherical mirro r 505 Sphe rical abe rra tion 506 Devi at ion of a light ra)' nt ncar-norm al inciden ce on a thin glass

    pri sm 506 Color dispers ion of prism 50i Focusing of paraxial ligh t Tap b)' a thin lens 507 tecessary conditi on for a focus 50S

    Lens-make r's fonnula 50S Focal plan e 509 Real poin t image of a poin t object 509

    Proble ms and Home Expe riments 519

    Thin -lens form ula 509 Lateral magnificati on 510 Co nverging lens 5 10 Virtu al ima ge 5 11

    Diverging lens 511 Lens power in diopters 5 12 Simple magnifier 513 Pinhole mag nifier 5 14 Do ),ou reallysee th ings upside down ? 514 Exercising th e pupils 5 14 Telescope 515 Microscope 5 15 Thic k spherical or cylindrical lees 516 Deviation at a single spherical sur face 516 Leeuwenhoek's microscope 517 Sco tch llte ret rodi rective reflector 51S

    Chapter 9 Interference and Diffraction

  • Interference pattern . The patt ern form ed by the various regions of inter- feren ce maxima and minim a is called an in terferen ce patt ern . Even though the waves are traveling waves , th e int erference patt ern is sta tionary in the sense just menti oned . Notice that even if the oscillator that drives the two

    d etect or consisting of a receiving ant enna, a tun ed resonant circuit, and an oscilloscope. In the case of the visible light , we may use our eyes , or a p hotographi c emulsion, or a photomulti plier whose output current we can measure. In any case, the detector will experience a total wave that is the linear superposition of two contr ibutio ns, one fro m each source.

    Cons trtlct ive and destrtJct ive intereference. For some locations of the detector , the arr ival of a wave cre st (or trou gh) from one source is always acco mpanied by the simultaneous ar rival of a crest (or trough) from the other source . Such a location is called a region of constructive -interf erence or an interference max im um. At other locations the arrival of a crest from one source is always acco mpanied by the arrival of a trough from the other, a nd we then have a region of destructio e int erference or an interference minim um. Since (by hypothesis) the two sources maint ain a constant rela- tive phase, a region that is one of constr uctive inter ference at a given time will always be a region of co nstruct ive int erf er ence, and likewise a region of destru cti ve interfere nce at a given time will remain one for all time.

    454 Interf eren ce arid Diffraction

    In Sec. 9.7 we study so-called " geometrical" optics. \Ve first derive the law of spec ular reflecti on and Snell's law of refraction from the wave p roperties of light. Th en we co nside r various mirrors, prisms, an d thin lenses.

    9.2 Interference bet ween Two Coheren t Point Sources

    Coh erent sources. The simplest situation involving interference is that in which there are two identical point sources at different locations, each e mitt ing harmonic traveling waves of the same frequency into an open homogeneous medium. If eac h source bas a perfectly de finite freq uency (rather th an a dominan t frequency and a finite frequency band width), tben tbe relati ve pbase of the t wo sources (th e differ ence betwee n th eir phase consta nts) does not cha nge wit h time and the two sources are said to be relati vely cobere nt , or Simply, coherent. (Even if they have di fferent fre- qu en cies, they are " coherent" if eac h is monochromatic, since th eir rela- tive phase is always completely det ermined .) If eac h source has the same domin ant frequency and eac h has a finite band width 1 p, then, if the sources are " independe nt," the relative phase of th e tw o sources will only remain co nstant over times of the orde r of (ll p)- l. On the othe r hand, two sources ma y be " locked " in phase with one an othe r becau se they are dri ven by a co mmon driving force . In this case, even though the phase constant of each source will drift in an uncon tr ollable manner thr ough a phase of order 2'17 in a time (Ilv)- l. where Ilv is the ban dwidth of the common driving force, the relative phase will rem ain consta nt. T