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Chapter 9 I nterf erence andD iff ra ction 9.1 Introduction 453 9:) Interference between Two Coherent Point Sources 454 Coherent sources -t54 Constructive and destructive interference ..55 Interference pattern 455 Xea r field :m d far field -156 "Boundary" between near and far 457 Use of a cOl l\'erging lens 10 ob taiu Iar-Geld interference p3tt em 458 Far-field interference pattern 459 Priucipal maximum 460 Relative phase due to path difference 461 •·.. average.. trn cling W3\"e ,16 1 Photon flux Two-slit interference paU: em -'62 So u rces oscillati ng in phase ...63 Sou rces oscillati ng out of ph ase 463 In terference patt ern near 9 = 0 3 '1&1 Energ)" conserva tion 465 On e plus one equa ls Iour 'l65 On e plus one equals zero ·166 9.3 Interference between Two Independent Sources 466 I. ndepeudent sources and coherence tim e 466 "Incoherence" and interference 467 Brown and Twiss exper iment 468 9.4 How Large Can a "Point" Light Source Be? 470 Classica l point source 470 Simple exten ded source 471 Coherence condition -'72 9.5 Angular \Vidth of a "Beam" of Tra veling w aves An gula r width of a beam is di ffraction limited 4i3 A beam is an interfere nce maximum 4i5 9.6 Diffraction and Huygens' Principle 478 Difference between interference and diffn dion · H8 How an opaque screen works 4i S Shinj- and black opaque screens 4i9 Effect of a hole in an opaque screen 480 Huygeus' pri nciple 48 1 Cnlcujaticn of single·!o lil rliffracliOIl pattern using lluygens' construc tion 482 Sing le-slit diffracti on patt ern 485 Angular width of a diffraction·limited beam 487 Angul ar resolution of the human eye ·l87 Rayleigh 's criterion 488 Nomencla ture: Fraun hofer and Fresne l diffraction 488 Angular width or bea m 4i6 Applir.ntinn: Laser beam IJeTSUS fins Might beam 477 Fourier an alysis of the transverse space de pendence of a coherent sour ce 48 9 Jmportant resul ts of Four ier analysis .4 90 Diffracti on patt ern for two wide slits 491 Diffraction pattern for many identi cal par allel wide slits 492 Multiple-slit int erference pattem 493 Principal maxi ma, central maxim um, white light source 493 Angular width of a principal maximum 495 Transm ission-type diff ract ion grating 496 Diffraction by an opaque obstacle 496 How far downstream does a shadow extend? 497

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Page 1: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

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 -t54Co ns tr uct ive and destructi ve in te rference ..55In te rference pat tern 455Xea r field :m d far field -156

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

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

•·..average.. trn cling W3\"e ,16 1Photon flux ~62

Two-slit interfer ence paU:em -'62

Sources oscillati ng in phase ...63Sou rces oscillati ng out of ph ase 463In te rference patt e rn near 9 = 0 3 '1&1En erg)" conserva tion 465On e plus one equa ls Iour 'l65On 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 467Brown and Twiss exper iment 468

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

C lassica l point source 470Sim ple exten ded source 471Co 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 3A 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 ·H8How an opaque screen works 4i SShinj- and black opa que screens 4i9Effect of a hole in an opaq ue screen 480H uygeus' pri nciple 48 1Cnlcu jatic n of single·!olil rliffrac liOIl patt e rn using l l uygens'

construc tion 482

Sing le-slit diffracti on patt ern 485Angula r width of a diffraction ·limited beam 487Angular reso lut ion of the human eye ·l87Rayleigh 's cr iterion 488Nomencla tu re: Fraun hofer an d Fresne l di ffracti on 488

47~

Angular width or bea m 4i6Applir.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 489Jmpo rt ant resul ts of Four ier ana lysis .490Diffracti on patt ern for tw o wide slits 491Diffraction p attern for man y identi cal par allel wide slits 492Multip le-slit int er ference patt em 493Principal maxi ma, ce ntral maxim um , white ligh t source 493

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

Page 2: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

9,1 In trod uction

Most of our studies so far have been esse ntially one-di me nsional. in thesense that th ere wa s only one pa th by which a wav e emitte d at one placecouJd go to anothe r place. Now we shall conside r situ ations where thereare differ ent possible path s from an emi tte r to a det ect or. Th ese lead towhat are called interf erence or diffraction phenom ena. resulting from con­struc tive and destructiv e superpo sition of waves that have di fferent phaseshifts, 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 esemitte d by two poin t sources having the same frequ e ncy and a co ns tan tph ase relation . Examples are water waves emitt ed by two screwheadsjigg ling the sur face of a pan of water or ligh t e mitt ed by the currents inth 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 bythe 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 rela­non . We find that the interferen ce pa ttern remains constan t only for timeinte rvals of order (av)- I, whe re av is the frequ ency bandwidth of tbesources . 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 apoint so urce , when the sour ce consists of indep endentl y radiating part sand when the de te ctor ave rages over lon g tim e in tervals [Le., long com­pared 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) demon­st 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 spa­tial width D bas an ang ular di vergence (" width") of ord er M ;:::; A/ Dabo 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 athas 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 pat­terns of single and multi ple slits. Th e emphasis is on op tic al and elect ro­magnetic ph en omena. T her e are seve ral hom e expe rime nts involvingdiffr acti on gra tings and various diffr ac tion pa tterns. Fo r th ese expe rimentswe strongly advise the student to ge t a " display lamp" - a light bulb witha 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 experi­me nts use a ile of these as a line source.

9.7 Geo metrica l Optics 498

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

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

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

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

Proble ms and Home Expe riments 519

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

Diverging lens 511Lens power in diopters 5 12Simple magnifier 513Pinhole mag nifier 5 14Do ),ou reallysee th ings upside down ? 514Exercising th e pupils 5 14Telescope 515Microscope 5 15

Thic k spherical or cylindrical lees 516Deviation at a single spherical sur face 516Leeuwenhoek's microscope 517Sco tch llte ret rodi rective reflector 51S

Chapter 9 Interference and Diffraction

Page 3: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

Interference pattern . The patt ern form ed by the various regions of inter­feren ce maxima and minim a is called an in terferen ce patt ern . Even thoughthe waves are traveling waves , th e int erference patt ern is sta tionary in thesense 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 anoscilloscope. In the case of the visible light , we may use our eyes , or ap hotographi c emulsion, or a photomulti plier whose output current we canmeasure. In any case, the detector will experience a total wave that is thelinear superposition of two contr ibutio ns, one fro m each source.

Cons trtlct ive and destrtJct ive intereference. For some locations of thedetector , the arr ival of a wave cre st (or trou gh) from one source is alwaysacco mpanied by the simultaneous ar rival of a crest (or trough) from theother source . Such a location is called a region of constructive -interf erenceor an interference max im um. At other locations the arrival of a crest fromone 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 interferenceminim um. Since (by hypothesis) the two sources maint ain a constant rela­tive phase, a region that is one of constr uctive inter ference at a given timewill always be a region of co nstruct ive int erf er ence, and likewise a regionof 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 thelaw of spec ular reflecti on and Snell's law of refraction from the wavep roperties of light. Th en we co nside r various mirrors, prisms, an d thinlenses.

9.2 Interference bet ween Two Coheren t Point Sources

Coh erent sources. The simplest situation involving interference is thatin which there are two identical point sources at different locations, eache mitt ing harmonic traveling waves of the same frequency into an openhomogeneous 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), tbentbe relati ve pbase of the t wo sources (th e differ ence betwee n th eir phaseconsta nts) does not cha nge wit h time and the two sources are said to berelati vely cobere nt , or Simply, coherent. (Even if they have di fferent fre­qu en cies, they are " coherent" if eac h is monochromatic, since th eir rela­tive phase is always completely det ermined .) If eac h source has the samedomin ant frequency and eac h has a finite band width 1 p, then, if the sourcesare " independe nt," the relative phase of th e tw o sources will only remainco nstant over times of the orde r of (ll p)- l. On the othe r hand, two sourcesma y be " locked " in phase with one an othe r becau se they are dri ven by aco mmon driving force . In this case, even though the phase constant ofeach source will drift in an uncon tr ollable manner thr ough a phase of order2'17 in a time (Ilv)- l. where Ilv is the ban dwidth of the common drivingforce, the relative phase will rem ain consta nt. The sources are th en saidto be coherent even though they are not monochromatic.

As an example of two cohe rent sources of waves, consider two rodswhich touch the surface of a body of wate r. If the rods are identicallydriv e n in vertica l oscillations, they p rodu ce sur face-tension wave s on thewat er . Th e relative ph ase of the rods is constant because they are drivenby a common source. As ano the r example of two coherent sources, con­sider two identi cal radio an te nnas dri ven at constant relative phase hy thesame oscillator. Eve n if the oscillator is not per fect ly monochromatic, there lative phase of the two ante nna curre nts remains constant. As an exam­ple of two coherent sources of visib le Light , consider two small holes orparallel slits in an opaque scree n which is illuminated on one side by a dis­tant " point" source of light. C ur rents are indu ced in the edg es of th e slitsby the electric field of the electromagnetic radi ation (light) emitted by tbepoi nt source. Th e two slits are th en said to be coherent sources of light.See Fig. 9.!.

In all these examples we need a " detec tor" that is responsive to thewav es. In the case of the surface-tension wave s in water, we ma y use atin y piece of cork which Boats on the surface and whose vertical disp lace­me nt can be measur ed . In the case of the radi o waves, we may use a

Sec. 9.2

So J--------------------1Slit 1

Slit 2

455

Fig. 9.1 Two coherent sou rces of light.Currents in the edges of slits 1 and 2 aredri cen by in cident waves emitt ed bypain t source So- Th e phase constant ofSo may drif t or change suddenly, but therela tice phas e of the slit CUTretits remai nsconstant.

Page 4: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

456 Interference and Diffraction

an te nnas is turned off and th en on ag ain with a new pha se co nstan t, therela tive ph ase of th e a nte nna cur rents remains unch anged . Simil arl y. ifthe point SOtuCC dri ving th e two slits is turn ed off and on, the slit curr en tsmai ntain constant relative pha se. Th erefore the interference pattern isunc hanged. On th e other hand , if th e point source is moved so as to cha ngethe distance to one sli t by a di fferen t amount from th e distan ce to the o the rslit, th e relati ve phase of th e induced currents will change, and th e locationsof interferen ce maxima an d minima will change. i.e ., the int erfer en ce pat­tern will cha nge. Si milarl y if we insert a delay cable between th e radiooscillator and one of th e ante nnas so as to change th e rela tive phase of theante n na cur rents, th at will change th e interference patt ern in that case.

Sec. 9.2

Source 1 LIP

dl L zp

Source 2

op

457

Fig. 9.2 Far fi euL TIu~ detector at P i,'l

in th e fa r fie ld of the tw o so urce" pro­L;de d L u exceeds LI P by mu ch leu thanone wavelen gth, fo r the co nfigurat ion

shown.

i.e. ,

Near field and f or fi eld. In most of th e cases th at we sha ll co nside r, th edetector is at a dista nce from th e two sources which is large compa red withthe separ atio n of the sources. On e th en says that the detec tor is in the [orfield of th e sources. We usuall y consider th e far field because we ca n makeSimplifying geo metr ical appro xima tions. As far as th e effect of distan ce onwave amplitude is co nce rne d , we ca n th en say that th e two identical sourcesare esse ntially at the sa me distance from th e det ect or. I n this cas e, e achsource will contribu te a tr aveli ng wav e having essentially the sa me ampli­tude as tha t co ntr ih ute d by th e other (p rovided th e sources are identical).

At a give n positi on of the detect or (often ca lled th e field point P), thetim e dependence of th e total wave fun cti on is th erefor e given by super­positio n of two harmoni c oscillatio ns having the same frequen cy and ampli­tud e but having (generally) di fferent ph ase consta nts. Tb e two phase co n­stants (a t a given field point) depend on th e ph ase co nsta nts of the twooscilla ting sources and on the num ber of wavelengths be tween each sourceand the field point. If the distance from tbe field poin t P to one sourceeq uals th at to th e other source or if the)' differ by a whole number ofwa vele ng ths, and if th e sourc es oscilla te in phase. th en P is at an int erf er­ence maximum and th e am plitud e of its harm onic oscilla tion is twi ce theamplitude it would have if eit he r so urce were presen t alone. (If thesources oscillate 180 0 out of ph ase. P is a t an inte rferen ce nod e and haszero ampbtude.) If the distan ce from th e field point P to one sour ceexcee ds th at to the other by t A (plus any w hole nu mber of wavelength s)and if the sour ces osc illa te in ph ase, the n P is at an interf eren ce node andhas zero am plitude. The approxi mation consists in takin g the am plitudesof th e individual contri but ions from th e two sources to be exactly equal. inspite of th e fact s th at they are in ge ne ral at slightly di ffer ent distances fromthe field poin t and that th e amplitudes fall off with dista nce . Th us th eamplitude at an in te rference minimum is ge nerally not exactly zero.

A second importan t simplification tha t can be used in the far field is th eapproximatio n tha t th e direction from source 1 to the field point P ispar alle l to th e d irer-no n fro m source 2 to P. We shall utilize thi s approxi-

mation when we calculate (be low) th e inte rfe rence pattern from two pointsources. \ Ve now give an ap proximate crite rion th at is helpful in dec idin gin a given case whe ther use of the far-field approximation is justified: Wecons ide r a fie ld point P at w hic h the dir ecti on fro m so urce 1 to P IS per­pendicular to the line joining source ] an d source 2. (See Fig. 9.2.) Th efar -field appro ximation is justified provided we ca n take the dir ection fromsource 2 to P to be parallel to that from 1 to P. In thi s case one can assumethat th e relative phase of th e two wave cont ributions at P is essentially thesame as th e relati ve phase of th e two so urces (for th e geo metry of Fig. 9.2).Thi s approximatio n breaks dow n ba dly if th e dista nce Uzp from s~urce 2 toP excee ds th e distance LI P by one half -w avelen gth (o r more), smc e th enth e two wave co ntributions at P d iffer in ph ase hy ]80 deg (or more) whenthe two sources are in phase.

"Boundary" bet ween necr and fa r. Let us define a cru de sort of "bound­ary distance" l-o betw een sources and field poin t , suc h th.at \V.hen. LIP andLop are very large co mpare d with Lo, the far -field approximation IS a goodone. Th us l-c is a rough ho unda ry be tween th e far-field and the near -fieldregi ons. Th e natural choice for the boundary distance Le is a distan ce L,Pat which L2P exceeds LIP by exa ctly one hal f-wav ele ng th. We obtai n anapproximate expression for this approximate boundary as follows: Accord ­ing to Fig . 9.2, we have (exactly)

L z p 2 = L 1 p 2 + cP,

L" Z _ L l p 2 = (L2 P - L ' P)(L 2 P + L,p) = rJ2 .

But , for th e case of interest, Lzr and L 1P are near ly eq ual. to one anothe rand bo th are esse ntially eq ual to Lo, since L 2P exceeds Lv by tAo

d2 = (Lz p - L,p)(L-ll' + Lv ) :::: (! A)(Lo + Lo)·

Thus for a rou gh criterion we can say that far-field app roxim atio ns arejustified for field points P mu ch farth er from th e sources than a dista nce Lo

Page 5: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

458 Interieren ce and Diffra ct ion Sec. 9.2 459

satisfying the relati on

, .

,I

. ,

Fig. 9.4 Two point so urces emittingwaves whi ch are detected at a distantfiel d poi nt P.

.'

'.

Large

distance

, t.

. ' .~T 5, : "!: .:

~:·: ,: s~~; i, _",. •if:: d' ~ir! ~

..... '1'

Far-field interference pattern. In Fig. 9.4 we show two point sourcesemitting elect romagnetic waves that are detected at a distan t field point P.We are only going to look at the inter ferenc e patt ern in the plane contain­ing the tw o sour ces an d th e field point P. Our resul ts will also apply totwo " line" sources (consisting of slits, in the case of light), or to tworadio ante nnas, or to sur face waves in water.

'i .

parallel. But thi s sa me len s (with acco m modat ion mu scles relaxed ) willfocus any parallel rays on the retina, 'whether or not the y arise from a " dis­tant poi nt source. " The focusing act ion of th e lensis shown in Fig. 9,3 .It tur ns out (as we shall show in Sec . 9.7) that although the actu al distancefrom source 1 to P (in Fig, 9,3) is less than that from source 2 to P, thenumber of wavelengths is the same. Tha t is possible beca use the pat h from51 to P has a larger amou nt of path length in the lens, where the wave­length is sho rte r tha n in air , The point P is " effectively" infinitely far away,in the sense that the paraUel rays sho wn leaving sources 1 and 2 reach thedetection point P after traversing the same num ber of wavelen gth s. Thusthe point P is at an interferen ce maximum (assuming sources 1 and 2 oscil­late in phase) just as it would be if th e entire region had constant index ofrefra ction and P were infinitely far to the right.

F rom now on we shall assu me that P is in the far field of sources 1 and2, either because P is actually very far from the sources or because we areusing a lens and P is "e ffectively" ve ry far from the sources .

(1)

p,IIIIIIIII

,I

S2

Use of a conve rging lens /0 obtain far-fi eld interference patt ern. Youwill study experimentally th e two-slit interfe rence pat tern for visible light.(See Home Exp. 9.18.) Th e tw o cohe rent sources are produced as in Fig.9.1. A typical slit separation is ! mm. Let us calculate how far down­strea m from th e slits the field point must be in order to be in the far fieldof the double slit. Using Eq . (I ) with A = 5000 A and d = t mm, we get

r _ _ '!?. _ (0.5 X 10- ' em)' _ . 0YJ - A - 5.0 X 10- ' em -" em.

Th us one should be per hap s lOLa ;::: 5 meters from the slit to be in the farfield. That is inconvenient and un necess ary ; here is how we ca n get a far­field patt ern with the double slit held right in fron t of your detector: Thedetec tor is yow' eye, whic h consists essentially of a ph otosensitive surfa ce(the retina ) and a lens. (We shan study lenses in Sec . 9.7.) The lens has avariable focal length that is varied by cha nging th e tension in the accomo­da tio n muscles of th e eye . Wh en you look at a dist an t object, these mus­cles are relaxed (for a norm al eye); the lens is then shaped so that rays froma dj s:~n t p~~nt source ~ tr iking different parts of the lens surface are broughtto a focus at the retina. (If the refractive power of the lens is either toostro ng or too weak, the rays will not focus at the retina, and the dista nt ob­ject will appear blurr ed.) Since the source is distan t, these rays are almost

Fig. 9.3 Con verging lens. Parallel raysfrom sou rces 1 and 2 are f ocu sed atpoint P provided the two sou rces oscil­late wi th the sam e phos e constant. Thedista nce f rom the cen ter of the lens tothe foc al poin t P is called the focal len gthf, f or a lens whose thi ck ness is sma llcom pared IL'1t h f

Page 6: Chapter 9 Interference and Diffractionhep.ucsb.edu/courses/ph25_08/waves01.pdfChapter 9 Interference and Diffraction 9.1 Introduction 453 9:) Interference between Two Coherent Point

460 Interference and Diffraction Sec. 9.2 461

(8)

(7)

~ep :::: wtl - wtS

=k(r, - r, )

=kid sin 9)

2 d sin 0= ·"-A- '

wli = (o.,(t - ~) = wI - krl

Wt'2 =w(t-~) = cd - h z.

where d sin 0 is the path difference as indica ted in Fig. 9.4. All the variouslines of Eq . (8) are equivalent mathem ati cally, but they correspond to dif ­ferent mental pictu res, each of which should be learned independently.Th us, in the first line , we thin k abo ut di ffere n t emissio n times; in th e lastline , we thi nk abo ut the fact tha t the ph ase difference is 2'1T tim es th e num­ber of wavele ng ths of pat h diff e ren c e; in th e second an d third lin es, wethink ahou t the nu mbe r of radians of phase per un it distan ce (the wavenumber k) tim es the pa th difference. In addition to Cicp as given b y Eq .(8), there is of course the phase difference '1'1 - '1'2 of the oscillat ions of thetw o sources.

The total field £ at P is the superposition of E, an d £, :

£(r,O,t) = £ , + £ 2

= A(r) cos (wt , + '1'1) + A(r) cos (W1'2 + '1" )

= A(r) cos (wt + '1'1 - !crl) + A(r) cos (wI + '1'2 - !cr2). (9)

Relative phase due to path difference. Because of the fact th at the pathdiff eren ce ra - r 1 depen ds on the angle 8, the relative phase of the twowaves at P depe nds on 8. I t is just thi s va riat ion of the relat ive phase withangle tha t giv es rise to the int erferen ce pa tte rn. T his relative phase dueto path difference is important , so we brive it a nam e, D.ep:

The emissio n times 11and t2of th e radi ation d e tecte d at the la ter t ime t

ar e given by

(2)

(3)

£ I(t) = _ qij, (t l)rl c2

_ w2q yo cos (wt1+ '1'1)- T I C2

Prin cipal maxi mu m. \ Vh en the distan ces '1and '2from sources 1 and 2 tothe field point P arc large co mpared with the sepa ration d, then t he tworays along the lines of sigh t from the two sources to point P are near ly par­allel, bo th being at essentially the same angle 8 to the z axis as shown in thefigure. In that case, the path differe nce ra - r, is essentially equal tod sin 8. Therefore, if the tw o sou.rces oscillate in phase, P lies in a regionof co nst ructive inte rference when d sin 8 = O. ±A, ± 2A, et c. The inter­feren ce maximum at 8 = 0 is called the principa l or zeroth-order ma x im um.Th e first maximum on eit her side, wher e d sin (j is ±A, is called the first­order m aximum, etc. The regions of destru ctive interferen ce, where thetotal wave is always zero, arc called nodes. Th ey occ ur at angles wherethe path difference d sin 0 is :tt A, :ttA, etc .

We now deriv e a n exp res sio n for th e total elec t ric field a t P under theassu mption that bo th sources u nder go th e same harm onic " mo tion,U exceptthat they may ha ve differ en t phase constants. We shalJ use a ment al pic­tur e for the sources of two oscillating point cha rges . \ Ve co nside r a Singlepolariza tion co mponent , w hich we can tak e to be one or th e other of thetwo ind ependent di recti on s transverse to the line of sight from the sourc esto P. vVe need not speci fy the polarizati on , because the results ob tainedhold indepe ndent ly for either (or any other) polarization ; for example, left­handed or right-han ded ci rcular polari zati on ). However, for co ncre te nesswe conside r the linear polari za tion co mpo nent alon g y, wh ere y is pe rp en.dicular to the plane of Fig. 9.4. Th en th e motions of the point charg es Jand 2 have y co mpo ne nts

YI(t) = yocos (wt + '1'1),

Y2(t) = yo cos (wt + '1'2).

Th e field poin t P is locat ed at the an gle 0 give n by Fig. 9.4 and at a dista ncer, where we take l' to be the averag e of r t a nd T2 (i.e ., we pu t the origin ofcoordina tes halfwa y be tween th e two sources). The radiation field EI(t) atfield point P due to the ea rlier retarded motion y , (t , ) is given by

Th e radi ation field £2(t) du e to Y2(tZ) is given by an analogous expression.In th e far-field approximati on , we take r, and ra bo th essentially eq ual toth e ave rag e distanc e T:

r == !ir, + r2),

£,(t) = A(r) cos (wt , + '1',),

£2(t) = A(r) cos (w1'2 + '1'2),

A(r) == w2qyo .

1C2

(4)

(5)

(6)

"A ve rage" t raveli ng wa ve. Rather th an exp ress E as a superposition oftwo outgo ing sphe rica l t ra veling waves from sour ces 1 an d 2, we canexp ress it as a sing le " average" outgo ing sp herical tr aveling wave with anam plitu de that is modulat ed as a fun cti on of th e pr op aga tion di recti on 8and wit h a ph as e co nstant that is th e average of the phase co nstants fPl andq!2 of the two sour ces. To sho w thi s, we use the trigonomet ric identities

cos a + cos b = cos [t (a + b) + 1(a - b)] + cos [1(a + b) - 1(a - b))

= 2 cos 1(a + b) cos t (a - b),

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4 62 Interference and Diffract.ion Sec. 9.2 463

with

a = wt + tJll - krl>

b = wi + '/'2 - krz.

PllUton fl ux. The photon 11m at th e field poin t P is prop ortional to thetime-averaged ene rgy Hux ( 5). If we have only the sing le polarizationcomponent along 5' that we have been considering, the energy Hux is givenby

Fig. 9.5 Intensi t y of superposition f romuco sources osc illa ting in ph ase. Thesep aration d is large co mpared with A.

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In Fig. 9.5 we plot this angula r distribution in the region near e = 0 underthe assumption that the sources are separated by many wavele ngths (d~ A),so tha t 1(8) goes th rough m an y maxima and mi nima while 8 is still rathersmall. This enables us to make a diagram in which we show severalmaxi ma and minima in the same small region (near 8 = 0).

Sources oscillat ing III phase. If <PI and CP2 are equal, the angular depend­e nce of the two-slit (or two-paint-source) pa ttern is

1(0) = I maa cosz t tl<p

o [ d Sill 8]= I max cos- 7T- -A- .

(10)

(12)

(13)

(ll)

Then

!( a + b) = wt + !(<pt + '/'2) - k · !( r, + rz)

= wt + C{}av - kr,

!( a - b) = !(<PI - <pz) - ~ k( r J - rz)

= t (<P l - <Pz) + ttl<p .

Then Eq. (9) beco mes

£(r,8,t) = (2A(r) cos ft{<pt - '/'2) + t -"<pll cos (wt + <P.,. - kr)

=A(r,O) cos (wt + <Pay - kr),

with amplitude A(r,O) given by

A(r,O) = 2A(r) cos [!(<pt - <pz) + t tl<p] ,

tl<p = k(rz _ r, ) = 277 d s~n 0

( 5) = -.£.. ( EZ), (14)4"

withE = 5'£ (r,O,t). (15)

Then( EZ) = ( [A(r,O) cos (wt + <Pay - kr)]2)

= tAZ (r,O), (16)wi th

AZ(r,O) = {2A(r) cos ft (<pt - '/'2) + t tl<pJj2. (17)

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Two-slit interfer ence pa tte rn. Let us hold r fixed and look at th e variationof th e photon flux with angle O. According to Eq s. (14) th rough (17), wehave [calling the photon flux 1(0)]

1(0) = I maa cosz f!( <pt - <pz) + t tl<p]. (18)

Acco rding to Eq . (18), the intensit y varies as the squared cosine of halfthe relative phase, where the relative phase is pa rtly that of the oscillatingsources and par tly that du e to the depend en ce of the path differen ce onangle.

Sources oscillating aut of phase. If <P, an d <pz differ in ph ase by :!: 1r,

then half their phase di fferen ce is ±t77 , so th at Eq. (18) gives

1(8) = 1m ax sinz t f), <p

1 . 2 "d sin 0= max sm --A--·

I II Fig. 9.6 we pint Eq. (20) near 8 = 0 for the case where d is manywa velen gths, so that several maxim a of 1(0) occ ur near 8 = O.

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464 Interference and Diffraction Sec. 9.2 465

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Le t us ca ll th e co rres po nding spatial separ ation b etw een success ive maximaby the na me %0 . According to eithe r Fig. 9.5 or 9.6, for 0 nea r 0 deg, 1'0

is th e distance L tim es 00 :

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with!'>cp = 2" d sin O/ A. (27)

Th us th e energy flux when bo th sources are turned on is. the product ofthe angula r modulation factor 2 cos' [t (cp, - cp, ) + t !'>cp] times the sum ofth e fluxes th at would be produced by each source acting by itself . Ifth ere are man y maxima a nd minima be tw een 0 = 0 ° and 0 =.360 °, theangular modulatio n factor will be zero as often as it is :?O.and \V1U h~v~ anaverage value of unity. i n order to prod uce many maxima and muu rna,th e t wo sources must be man y wavelen gth s apart. Thus we see tha t thetotal energy emitt ed (in the plan e of th e figur es we have shown) is just thesum of wha t the sourc es would give indi viduall y, pr ovid ed th e two sourcesar e man y wavelengths apart . That seems reasonabl e.

which is inde pen de nt of O. Simila rly, if only source 2 is tur ned on, thephoton flux is proporti onal to

( E, '> = t A' (r). (25)

When both sourc es are tu rned on, the ph oton flux is propo rtional (with thesame proportionality constant as ab ove) to

( E'> = «(E, + E, F >=tA' (r,O)= t · (2A(r) cos [Hcp , - cp,) + t !'>cplF

= A' (r) ' 2 cos ' [Hcp, - 'l'2) + t !'>cp].

Using Eqs, (24) and (25) , we write this in th e form

( £2) = {( E,' > + ( E,2>J 2 cos ' [t("" - CP2) + t !'>cp], (26)

One plus one equals fo ur. However , conside r th e case wher e the twosources are very close togeth er . Let th em be at a separation d that is verymuch less than one wavelength . 1f th e sources are in ph ase, Eq s. (26) and

(27) give

( £2> ::=2[(E, 2> + ( E,2) ]. (28)

Thus, instead of ha ving an a mou nt of energy th at is th e sum of wh at thetw o sources give indi vid ually, we get twice th at amount. ~iS ~ay .see~st range. Doesn't it violate energy conservation? No. Th e imp licatio n lS

Energy co nservation. If sourc e 2 is turn ed off, the elect ric field a t I' isgiven by source 1 only:

E = E, = A(r) cos (wt + cp , - kr,). (23)

The photon flux is then propor tio nal to

( E,') = N (r)(cos' (WI + cp, - kr,»=t A'(r),

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Int erjerence pat tern near fJ = 0°. Wh en you look at a line source oflight wit h a double slit, you can not usually tell exactly where 0 = 0 occurs .Thus Fi gs. 9.5 and 9.6 co ntain more in formation th an is usually available(at least in the home expe riments). Th e imp ort ant inform ati on is theangula r int er val be t wee n successive maxi ma or the correspo nding spa tialint er val on a detecti ng screen (which may be your reti na, for example).Succe ssive maxima in Figs. 9.5 and 9.6 co rrespond to an increase in pa thdi fferen ce of one wa velength . i.e., to an increase of d sin eby an amo unt A.For 0 near 0 deg, we can use the small-angle approximati on sin 0 z O.Th en th e angular int erva l bet ween su ccessive maxima is Ai d radians. Letus call thi s angular interval 80 :

Fig. 9.6 Intensity of su perposi tion fromtwo sources osci lla ting 180 deg out ofph ase.

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we see tha t tcoh is 2 71/ j, w , i.e., leob is (6 11) - 1. For time intervals smallerth an ( j, v) - l, we ca n thi nk of the relat ive phase of the two sources asrem ainin g esse ntially cons tant. (There ca n be man y oscillatio ns in such atim e interval, because we assume vo(~v) - l is large.)

by a n opaq ue jacket which is pierced by two pin holes or slits. Each pin ­hole is illuminated by differ ent ato ms of the gas. Alternatively, we mayhave two pinh oles or slits in an opaque piece of materi al set in front of anordinary light bulb , (In order to have a reasonably small band of frequen­cies, we could put a red gelatin filt er over th e slits.)

We shall suppose th at the freq uency bandwid th 6v is small co m paredwith the do minant frequency Va. Th en there are many oscilla tio ns at fre­qu ency eo during a t ime inte rval of lengt h (.l V)-1 Th e tim e interval(.~ V)- I is th e cohe re nce tim e, tcob; it is th e tim e interval requi red for fre­qu ency components at extre mes of th e frequ ency band to get out of phaseby abo ut 2". Thus if t, ob is defined by

4 6 6 Interjerence and Diffraction

th at each source emits twice as much ene rgy whe n the othe r source is sit­tin g on top of it (and oscillating in phase) as it does when oscillati ng by it­self. How ca n that be? \ Ve have prescri bed the motion of each sourceby Eqs. (2), ind ependent of th e se para tion d. The ene rgy output doub lesnot because the motion of eithe r source changes bu t be cause the impedanceexpe rienced by eac h source has doub led ! Wh y is tha t? It is because theres istiv e dr ag force exerted on th e elec t rons in on e ante nna by the radioate d field (taking th e two radio antennas as an example) is du e not just toth e field being e mitte d by tha t an ten na; it is that force plus th e force dueto the field being em itte d by th e othe r antenna. Since th e curre nts are inphase (by h ypo thesis) an d since th e an tennas arc in ve ry close proximity.the net d rag force exerted on the elec tro ns in one an tenna is twice what itwould be if the other ante nna wer e not presen t. Th e power supply musttherefore push twice as hard to main tain the pr escribed veloc ity, and th uswe ge t twice the work don e by th e power supply. Since thi s holds foreac~ ~nten na, we have acco unted for th e tw ofold increase in total energyermssron.

Sec. 9.3

j,w tcob ;:::; 217,

467

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"Incoherence" and int erference. Let us conside r only the situ ation inwhich th e se paratio n d between the two sources is large co mpared wi ththe wavelen gth A. Th en the interfe re nce pa tt ern looks like Fig. 9.5 attim es wh en the relati ve ph ase of the tw o sources happ ens to be zero . Itlooks like F ig. 9.6 at times whe n th e relative phase happens to be 180 deg.For relati ve ph ases be t ween 0 and 180 deg, th e int erfer en ce patt ern liesbetween those shown in Figs, 9.5 and 9.6.

If th e detector is one which requires a long time to detect th e int ensityat a given posit ion , such as th e eye (which has a reso lution time of abo ut"').~ sec ), then th e plot of time-averaged int ensit y versus 0 will sho w noodependence, hecause during a tim e long compared with (6V) - 1 the inter­fer en ce pa ttern will ha ve taken on all appearances bet ween th e ext rem esgiven by Figs. 9.5 and 9.6, and every value of d sin 0 will have exper ience dthe same ti me-averaged in tensity. One then says that th e two pointsources are "incoherent ." The tim e-averaged ene rgy flux (the photon flux)is then just th e sum of the fluxes one woul d get for either source by itself.The interference patt ern is " washed out" because of the long time averageduring th e measuri ng pro cess . Thi s fact is expressed algebrai cally bynoti ng tha t Eq. (26), Sec. 9.2, gives ( £2) ::::: ( £ ,') + ( £ , ') , independentof 0, provided th at th e relati ve phase <p, - '1" takes on all possib le valueswith roughly equal amoun ts of tim e spe nt in each small interv al of relat ivephase be tween zero and 2". Th at follows from

One plus o ne eq uals ze ro. If the sourc es oscillate 180 deg out of phasea nd if you then supe rpose one an tenna almo st on top of the othe r, you getalmost zero for the tot al wav e a mplitu de . I n th e limi t th at th e antennasare on top of one a nothe r, th e output is zero , accordin g to Eq . (20). Th epower supply does no work, and no energy is radi ated . Th e field emitt edby one ante nna pushes on the elect rons in th e othe r an tenn a in such a wayas to help the osci llato r. I n the lim it of zero se para t ion of th e an ten nas,th e electrons in th e two an te n nas dr ive one another with no help from theosci lla tor. We th en have a " clos ed" system with ene rgy going out of oneantenna int o the othe r and back again. The ant ennas are then just part ofth e reson ant circuit of th e oscillator, and th e power supply need only re­pl eni sh th e losses du e to th e resistance of th e antenn as. T he rad iat ionre sista nce- the charac teristic imped ance-s-has gone to zero.

9.3 Int erference between Tw o I ndependent Sou rces

Indep enden t sources and coherence time. Suppose th at ea ch of twosources has domina nt angula r frequen cy Wo and bandwidth 6w . Supp osefurther that th e sources are ind ependent. Th at is, they are nol dri ven by~ co mmon dr iving force . Th e n the re is nothi ng that keeps th em exactlym phase. I n th e case of t wo radio an tennas , th at would mea n th at eachan tenna is dri ven by a separate osci llato r an d powe r supply . In the caseof sources of visible light, it means we hav e two independe nt sources withdifferent atoms contributing to eac h source. For example, we may have amercur y vapo r lamp consisting of a gas discharge in a glass tube surrounded

(cos ' [!('/'l - '/'2) + t 6<p]) = t,

for fixed 6'1' and for '1', - '/'2 uniform ly distributed from zero to 2".

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468 Interf erence and Diffract ion See. 9.3 469

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Now let us find th e ave rage of 1112 when the separat ion X2 - Xl is thatbetween an " instantaneous" in ter fer en ce maximum an d th e neighboringmini mu m, i.e ., whe n it is half of xo, where Xo (as shown in Fig. 9.7) is these pa ration between successive maxima of the instan tan eous tw o-slit inter­fe ren ce pattern. (It is given by Eq . (22). Sec. 9.2 .) If x, - X, = txo.th en at an insta nt when PM 1 happe ns to ha ve curr ent a, PM 2 has (ac­cording to Fig. 9.7) current c. Wh en PM I has h, PM 2 has d. etc. Th us.for th e time ave rage over the four representa tive curre nts a, b, c, d forPM 1, we have

We see that (1112)av is three tim es larger when r a - X I is ze ro than whenit is halI of the separat ion between successive maxim a of the instan tan eouspattern. Th us we see tha t a plot of {/112)av ver sus X2 - Xl will determineth e relati ve phase t. <p = 2." d sin OI A.

T he crux of the tech nique of Brown and Twiss is th at in the pro d uct ' 1 ' 2

eac h cur rent is only averag ed over tim es of orde r 10- 8 sec, an d during thisli me the curr ents are essentially co nstant. The average ( 1112 ) over a t imein te rval of min ut es is just wha t th ey wo uld ge t by ave ragi ng ove r seve raldozen co herence times. say over 10- 6 sec. (Th ey average over muchlonge r ti mes so as to average out ph oto multip lier noise and for othe r ex­perimen tal reaso ns.) The prod uct ( II}( I,} , 011 th e other hand , is in de­pendent of xi - X2 , beca use each photomul ti plier has sampled the en ti reint er fer ence pattern during th e tim e of averaging. Th e essential thi ng isto find ou t which separa tio ns Xl - 1'2 corre spond to th e situation th at [ 1 islarge when / 2 is la rge and is small when h is small (as whe n X l - 1'2 is zero)and which separatio ns correspo nd to I I small when 12 is large and vice versa.

Fig. 9. i h It ens it y venus .r (It a given" instant" of d uration le' 3 than (!1vr i .

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(I , I ' )n = t( ee + bd + eo + db)

= t( O' 1 + ~ . t + I ' °+ t ·t )=t

Tt is cl ear that th ere are no «int rinsically" incoh eren t so urces. " Inco­herence" is merely the result of a measurem ent process which- throwsa way inform ati on that is available in th e int erf er en ce patt ern if one hast~~ tech.niqu e to look at ti mes co mparable to or sho rte r th an (.6. ,,)- 1. ForVISible ligh t , th e co herence times are of order 10- 9 to 10- 8 sec (for asource c.onsisting of indep enden tly radiating ato ms in a gas-discha rge tube).so that It ta kes some exp erime ntal inge nuity to measure th e in ter ferencepatt ern before it changes. Nevert heless. it has been done in a very beauti­ful exp eri ment by R. Brown and R. Twiss. t

Brown an d Tw iss experime nt. Th e meth od by which Brow n and Twisse ffect ively " read" the int erfer en ce pattern in a tim e less th an 10- 8 sec is

as fell ows. On e uses t wo ph ot omultiplie rs a t different valu es of .r (as de­fined in Figs. 9.5 an d 9.6) and with a vari able separati on Xl _ X2. Th eoutpu t curr ent of one phot omu lti plier, [ 1. is multiplied by th at of the other,12 , in a fast circu it which ca n follow current fluctuati on.s tha t occur intimes ~f order 10- 8 sec. (In othe r words, the fast circuit ry has lOO-~1c

~and:\'ldth . ) Th e produ ct / 112 is determined " instan taneously," Le., in atim e In ~e rval at the 10- 8 sec level, bu t then th e average of th is pr od uct.( 1I I 2 ) , 15 taken over a long time in terval of man y minutes. Th e separa tionX l - Xz of the two photomultipli ers is varied , and the time average of thel: rod u ct of curr ents is taken at eac h separa tio n. Fi nally, one plots thetim e-averaged produ ct versus Xl - X2. Now , the instan tan eous cur rent ineach photomul tip lier is proport ional to the flux of light ene rgy , i.e. , to 1(0)a.t that p bot o mul ttp lier. First let us co nside r th e case whe re the separa­~Ion Xl - :\"2 15 zero, so th at each ph otomultiplier is subject to th e sameInstan taneous light aux. Let us perform a very cru de average of thep roduct of the two cur re nts. Let us say tha t 1(0) only tak es o n th e four~alues ind ica ted by u, b, c, and d in F ig . 9. 7. Let us call th e corr espond­m g curr ents by th e nam es a, b, c. and d and give th em units in which weshall have 0 =0, b =1.e = 1, and d = t . For one-fourth of the " instants"(o f durati on about 10- 8 sec ), PM 1 (photomultiplier I) has cur rent I I co r­r~sponding ~pproxj mately to a. At th e same times, / 2 is also eq ua l to a,sm ce PM 2 IS a t th e same place as PM 1. On e-fourth of th e tim e, eachhas curr ent correspo nding to b, one- fourth of the tim e to c, and one-fourthof the tim e to d as the in terference pattern shifts . Th us the tim e avera geof the .pro~u~t of the two curre nts for X2 = XI is give n (in our cru deap proxi mation) by

(II I, ).v =t(aa + bb + ee + dd)

= t( O. °+ t . t + 1· 1 + t .t ) = t ·, R. Hanbury Brown and R. O. Twiss, "The Question o( Co rrelation between Photons in C0­

herem Light Rays: ' Nature 178, 1447 (1956). For a more recent expe rimen t using lasers. see~.~~~egor and L. Mand el. "Interference of Independent Phot on Beams: ' PhlJ3. Rev. 159, 1084

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470 Interf erence ami Diffra ct ion Sec. 9.4 471

Fi g, 9.8 (a) Sourccs } and 2 arc drivenby a single poi nt sou rce an d maintainco nstant relati ve pha se. Th ey are co­

heren t. (b) Sources 1 and 2 are driv enby different sets of ind ependently radi­ating atoms. For measur em ent t imeslong com pared wi th (tH.)-I, th ey areincoherent.

In ter ms of ph otons, one finds that th e probabi lity for detect ing a pho­ton in photo mult iplier 2 is larger th an average when ph otomultiplier 1 has"r ecentl y" (wi thin 10- 8 sec) det ect ed a photon, pro vided one has X l = X2 ;

it is smaller than average if one has Xl - Xz = !xo. To put it very crudelyand " semlclasslcally," if one has (for example) a wave th at co rresponds inint ensity to abo ut 100 photons inter fering with another wave th at also cor­res po nds to abou t 100 ph otons, th en if th e wave trains happen to overlapin space, their supe rpos itio n ca n give a tot al int ensity tha t corres po nds to400 photons (completely const ruc tive interf erence) or to zero intensity(completely destru ctive inter fer ence). This is expe rimentally distinguishable(by th e techniqu e of Brown and Twiss) from a situation where the wave ­trains never ove rlap and where one th erefore always has approximate ly100 + 100 ;:::; 200 photons. It is obvious from this way of expressing itthat th e experiment is helped by having a n intense ligh t source (to increasethe chance of an overla p be tween th e wavetrains of two photons) and byhaving ph otons with narr ow ba ndw id th s [be cause the length of the wave­t rain is essentia lly c ti mes the mean decay time T (that is, c/ t. v), and ~ longwa vetrain mean s a higher chance for overlap].

9,4 How Large Can a "Point" Li ght So u rce Be?

In Fig. 9.1 we showed how one can obta in two cohe re nt sources of light(two sourc es whose relative phase remains constant) by irr adiatin g twoslits in a n opaque screen with ra diati on from a " point" source of ligh t. Onth e othe r hand, if th e source is so broad that one slit is illuminated mostlyby one set of at oms and the other hy anothe r ind ep enden t se t, then thet wo slits are co mpletely incoher en t, i.e ., thei.r phases are uncorr elat ed [formeasurem en t tim es th at are long compare d with (LiV)- l ]. Th ese twoextre mes are illustr ated in Fig. 9.8.

ClaSS'ica l poi nt source. Th e closest we ca n ge t to havin g a p oint sourceis to have a single ato m. Acc ordi ng to the classical pict ure. this atome mits elec t ro mag ne tic waves in all di recti on s an d dri ves th e slit curr entsin Fig. 9.8a with equal ph ase. (Th e quantum th eory gives effectiv ely thesa me result. ) A practi cal sourc e of light will have a huge number ofradiating ato ms. If they we re all sitti ng at exactly th e sam e point, wewould have a point source. (It would be eve n more like a classical pointsource th an is a single real ato m.) But in any practic al sour ce the atomsare in a region of finite dimensions. How lar ge can a light source be andst ill be an "effect ive" point source (meaning th at th e slit curre nts in thedoub le slit irrad iated by the " point" source maintain constant relati veph ase)?

Sim ple exte nded source, Let us co nside r a very simple source tha t is nota point source. It co nsists of three independen t point sources a. b, and c,each with the sa me dom inant frequency, ban dw id th, and ave rage inten­sity , arra yed as sho wn in Fig. 9.9. Suppose we start out with only pointso urc e a tu rned on . Then slits 1 and 2 are d riven at co nsta n t re lativephase (which ha ppens to be zer o for our figure ) and are cohe rent over anytime int erval. Next tur n on bo th sources a and c. Source c is a lightsource with the same frequen cy and bandwid th as source a bu t not cor­relat ed in ph ase with sourc e a. Th us c and a do not maintain co nstantrelati ve phase over times long co mpar ed with (.6. 11)- 1. Nevert h eless, therelative ph ase of slit.s 1 and 2 rem ain s zero for all tim e. becau se source cdri ves th e slit currents with zero relative phase just as does source a. Th eslit currents may be regarded as a supe rposition of th e cur rents ind uced byth e t wo sources, and if each source contribution gives zero relative ph asebet ween the slit curr ents, so does th e superp osition. Th us we concl udeth at we can extend the po int source along th e line connec ti ng a and cwithout spoiling th e co herence of slits 1 and 2.

Now co nsider the situation where both sourc es a an d b are tur ne d on(with c turned off). Sources a and b are indepen den t source s having thesa me domin an t freque ncy and bandwidth and the sam e average int ensity.D uri ng any time inte rval sho rt co mpare d with (t.V)- l , th e ampli tu de andphase co nstan t of each sourc e remain constant. Suppose that fo r a giveninstant {an instant means a ti me inter val short co mpared with th e coher­ence time (t. v)- l but long enough to have at least one complete fast oscu­lation , so th at we ca n tell what th e amplitudes an d phases are] it so happensthat th e amp lit ud e of b is very small co mpared with th at of a . Th en to agood approximation th e two slits are irr ad iated only by a , and th e slit cur ­re nts th er efore have ze ro relati ve phase. Now let us wait a t ime longco mpared with the cohere nce tim e of sourc es a and b and lo ok again .

Fig. 9.9 Coherence. Slits 1 a nd 2 aredriVeTl by the th ree in depe nde n t sources0, b, ond c. Afust l ite three so urces becoalesced to a single point f or diu I and2 to be coherent?

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4 72 Interference and Diffractian Sec. 9.5 473

Coherence condition. Th e "extended source" consisting of points a, b,and c therefore acts like an effective point source provided that it satisfiesthe coherence condition,

where Xo is the spatial se paration between successive maxima and is givenaccnrding to Eq. (22), Sec. 9.2, by

AXo = L d . (34)

where one or the other of these fonn s may be most appropriate dependin gon what par am eters are expe rimen tally variable. [Ynu can verify Eq. (37)in an easy hnme experiment (see Home Exp . 9.20). In that experiment Lis the variable.] The easiest way to rememb er the coherence co ndition isin the form

D~ U (35)d'i.e .,

d~ r; , (36)

i.e .,

L::>dD (37)A '

which says that the product of the two transverse widths d and D must besmall compared with the product of the two longitu di nal lengths L and A.

If the source co nsists of a huge number of point s between a and b, sothat the sour ce has width d, Eq . (38) app lies to the entire source if it appliesto the extreme points a and b (i.e., point sources close r together than d arecohe rent if th ose d apart are). Similarly, when we (la te r) consider seve ralor many slits in a screen instead of just two, the coherence co ndition Eq.(38) can be a pplied to the entire array of slits with D taken as the separa­tion of the outermost slits.

A ngular width of a beam is diffraction limited. Now co mes the interest­ing and very important question: Can one, by very careful design, make abeamof waves that is just like a «cross-sectional segme nt" of a plane wave,in the sense that all the waves are traveling in exactly the same direc tion,so that one has a perfectly parallel beam that will co ntinue foreve r withthe same width? No. No matter how small the point source at the focus ofa per fect par abola, tb e radiation in the beam will not be perfectly parallel.If th e " domi na nt" directi on is along z and the spatial widt h of th e beam(at a given value of z, say at the reflector) is D. then there will be an angu­lar distribution of propagation directions with a "full width at half maxi­mum intensity" of about AID. (We will show this be low.) Similarly if wehave a perfe ctly plane wave from a distant point source falling on a hole

9.5 .4 ngular lVidth of a "Beam" of Travelin g lVaves

A "be am" of traveling waves is a pattern of waves traveling in a givendir ection and havi ng a finite lateral width. A flashlight beam of visiblelight and a radar beam of microwa ves can each be made by pu tting asmall source of elec tromagnetic radiation at the focal point of a parabolicreflector. The small source drives the electrons in the metallic surface ofthe reflector with just the proper phase relations so that the reflected radia­tion from all poin ts of the surface interferes constructively along the direc­tion nf the bea m. Anot her way to get a light beam is to reflect light frnma small or distant scurce (the sun, for example) nff a small plane mirror.Alternatively , we can use a hole in an opaque scree n instead of a mirror.LJ the source is sufficiently far away and sufficie ntly small, the radiationincident on the mirror (or bole) can be approximated as a plane wave-Le.,one with all the radiation traveling in exac tly the same direction. Then themir ror reflects " part nf the plane wave." Similarly in the case of the smallsource at the focus of a parabolic mirror, if the source is sufficiently smalland the mirror is a perfect paraboloid, the beam is (to a certain appr oxima­tion) like a "segment of a plane wave," consisting of radiation all travelingin the same direction. AU these considerations hold equ ally well for sound.waves or wate r waves.

(38)IdD~U,1

Sup pose that this time it so happens that the amp litud es of the oscillationsof a and b are practical ly equal. In this case , the screen with the slits isirradiated by the two-source interference pattern that we have seen inFigs. 9.5, 9.6, and 9.7. Th e locatinns of the maxima and minima dependon the relative phase o f sources a and b. The question of interest iswhether or not the two slits 1 and 2 are still driven with zero relativeph ase. We know that the amplitude of the interf erence patt ern changessign when we go from one interference maximum to the next. [Accordingto Eq . (13), Sec. 9.2, th e a mplitude A(r,O) is proportinnal to th e cosine of~<Pl - '/>2) + -nd sin OjA. It thus changes sign whe n d sin 0 increasesby an amount 'A. as betw een success ive interference maxima .] We see thatboth slits are dri ven at zero relative pbase most of the time only if tbeyare se parated by much less than the separation :1"0 betwe en succes siveinterference maxima of the two-source interference pattern. (Even whenthe slits are closely spaced. it may happen that a ze ro of the two-sourcepattern irradiating them faUs be twee n the two slits, in which case they aredriven 180 deg out of phase. However, this happens a smaller and smallerfraction of the time as th e slits get closer together.) Thu s we need

D ~xo, (33)

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474 Interference and Diffract ion Sec. 9.5 475

(.

la)

etc.

DI

IV

t 1etc. L

Ib)

! etc.

-,etc.

Ie)

Every one of th e four ske tches in Fig. 9.10 ca n be taken to rep resen teither wate r wa ves, sound wave s, or electro magn etic wa ves (visible ligh t ofwa veleng th 5 X 10- 5 e rn, for exa mple, or microw aves of wa vele ng th 10cm, for anothe r example).

of width D (or a mirror of width D), the ang ula r wid th of the t ransmi tt edbea m is a bout X/ D. Th e angula r widt h can only be zer o if D is infinite(or jf A is ze ro). Th e angula r width of the bea m is said to be diffra c tionlimit ed. I n Fig. 9.10 we show some examples of beams. Note that if theorigi nal wi dth of th e beam is D, and if eve ry a tt em p t is m ad e to mak e th ebeam as perfectl y parallel as possible, th e wid th \V after th e beam has tr av­eled a large dist ance L is approximately the original width D plus L timesthe angula r full wid th A/ D. For large eno ugh L , '\\ ' C can neglect the origi­nal width D. Th us we have

(39)

(40)

I~o ~ fJ · 1IV~ LfJ .Beam width :

Angular full width:

A beam is an interference max-imum. Here is a crud e der ivati on of Eq.(39). (I n Sec. 9.6 we shall give an exac t deriva tio n.) Th e result is inde­pendent of the kind of wav es and is independent of how th e waves areproduced . We might as well take the simplest source, which is p robablya plane radi ator as sho wn in Fig. 9.10d . For sound wav es, this can be anoscillating piston in f ree air. Fo r elect roma gnetic wa ves, it can be an oscil­lating she et of cha rge of finit e exte nt. for exa mp le a plan e an tenna arra y.In any case, the entire rad iator is cohe ren t. Th at is, all th e " moving parts"move ill phase with one anot her. (If that is not the case, the an gularspre ad will be larger than tha t given by Eq. (39). I n the limit of an inco­her ent radiat or, there is no bea m a t all .) In th e dominant di rection of th ebeam , a field point sufficiently far fro m the radi ator is esse ntially equidis­tant from all pa rts of the radiator. Th erefore waves from all pa rts of theradiat or add with th e same relative phase and we have a const ruc t ive int er­ference maximum. 'That is what defines the domina nt direction of thebeam . (If one vari es th e re lative ph ase over th e su rfac e of the ra d iator ,one can "s teer" the beam in a directi on that is not norm al to th e sur faceof the radiator. Th at is exac tly wha t happens in Fig. 9. IOc, whe re differentparts of t he mir ror tilt ed at 45 deg to the in cid ent plan e wave are drive nwith differ en t ph ase by th e incident wave, so th at th e region of maximumconstruct ive interference-the dir ection of the reflect ed bea m- is notnormal to the mir ror but instead satisfies th e law of "s pecular reflection .")

made by plan e wave incident On hole inopaque screen- (e) Beam made by planewav e incident on plan e mirr or. (d)Beam .:mitted by plane radiator with allparts oscillating in phase.

Fig. 9.1 0 Diffraction, Beam of u;idthD has angula r width =::.>../ 0 and spreadsby an a mount \V::::: LAID in tra velinga dist an ce L (a) Beam made by pointsource and parabolic mirr or. (b) Beam

(d)

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476 Int erference and Diff raction Sec. 9.5 477

A ng ular lddth of beam. At a distant field point that is not quite in thedirection of the beam, one does not have completely constructive interfer­ence. To see where we get the first zero in the interference pattern, let usdivide the radiator into two halves, a top and a bottom. Then we approxi­mate the radiator by two coherent point (or line) so urces, one at themiddle of the top half and one at the midd le of th e bottom half. Thesesources have lateral separation t D. Their first interference zero (the firstzero on either side of the principal maximum along the beam direction)occurs for a path-length difference of half a wave length , i.e., when (~D) sin 0is .p... For small angles we take sin 0 :::: O. and thus we get

Half angular width to first zero :::=: -5. (41)

with contributions cliffering in phase between successive quarters by 'TT)but rather whe n we have three thirds of the radiator with adjacent thirdsdiffering in phase by 1T from one another. Two of the thirds cancel oneanother, but the thir d third remains. Thus the amplitude for the first sub­sidiary maxim um is smaller than that of the principal maximum by at leasta factor of ! (it is ac tua lly smaller by more than th at beca use oI phase dif­feren ces within the one-third co ntribution that is left). we see that thesubsidiary maxirna have small amplitude co mpared with the ce ntral maxi­mum that gives the " beam" direction. when we study the exact patt ern ,we wiu find tha t the half angula r wid th to the first zero is equal to the fullanguJar width at about half maximum intensity, which is how we have de­fined the angular wid th of the beam in Eq . (39). Th us we have der ivedEq. (39), roughly. (Th e exac t result is given in F ig. 9.14, Sec. 9.6.)

This is shown in Fig. 9. 11.Application: Laser beam versus flashlight beam

Suppose you have a diffraction-limited laser beam of diameter D = 2 nun,wi th wavel engt h 6000 A. Ho w much does the beam diameter increase ina distan ce of 50 ft? The ang ular spread of th e beam is

Where does the next maximum occ ur? If point s 1 and 2 of Fig. 9. 11really we re point (or line) source s, the next maxim um wo uld occur whenth e path length from sourc e 2 to the field po int exceeded th at from source1 b)' on e wa velength. In deed th e to p half and bottom half are in ph asethen, but they each contribute zero! Th at is becau se if you divide the topand botto m halves themselves into halves. so that the entire radiator isdivided into fou r quart ers, th en th e contr ibution of the first quar ter is ISOdeg out of phase with that of the seco nd and cance ls it ; th at of th e thirdis 180 deg out of phase with that of the fourth and cancels it. Thus the firstsubsidiary maximum actually comes not when we have two halves withcontributions differing in phase by 2 1T (since then 'we have four quarters

Clr < f M :::; (0.5)(3 X 10-') ;::;: 1.5 X 10-' em.

Such a small filament is hard to make.

If we want to ohtain a diffraction-limited (rather than filament- size-limit ed)flashlight beam that starts at a 2-mm wi d th , then we want I:!J. () due to thefilame nt to be less than the diffracti on widt h . which is about 3 X 10- ' radaccording to our calculation above. For a typical penl ight , the filament isabout 0 .5 em from the lens; i.e. , f ::::: 0.5 em. Thus the filament must havetransverse dimension Lix given by

Th e angular spread tim es the dista nce L = 50 ft :::; 1500 em gives a spatialsprea d of W :::; (1500)(3 X 10- ' ) ;::;: 0.5 em = 5 mm. (This ca n be nicelydemonstrated in the classroom with a laser.) If you have a "penlight" typeof flashlight with a beam of dia meter 2 mm at the flashlight form ed by a"point" filament at the focus of a lens, how small wo uld the filament haveto be for th e flashli ght bea m to be diffraction -limited? If the filam en t isnot a point, then different parts of the filament give "independe nt" beams.The angular spread due to the finite size of the filament turns out to beapproximate ly th e width of the filam ent divided by th e focal length f :

n firfiu ;::;: y .

3 X 10- ' rad.fi n _ ~ _ 6 X 10- 5 em

v - D - 0.2---------~------ ­5,

s~ ~ _

\

\ L- I D sin 8.-; 2\ \

rI D

t

Fig. 9.11 Plane radiat or. Source 1 r(,.11­resents the contributions f rom the tophalf , source 2 those f rom the bottomhalf