photon counting on the diffraction pattern

Diffraction of statistically independent photons from a laser source: photon counting on the diffraction pattern E. Panarella National Research Council, Ottawa, Canada K 1A OR6 Received: August 1984

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Follow!itig two previous experiments reported in this journal, we h a ~ ~110\1~ : pesf~3rnic~i another experiment on the photoelectric detection of diffraction patterns caused by statislically independent photons at two light intensities. This time, we counted thc photons at ccluidistant points along a dianleter of the diffraction pattern. It has been found that the patterns did not build u p linearly with light intensity. 111other words, given the samc number of' photons, the dift'raction patterns obtained at two light intensities are not thc same. I t is then demonstrated that such non-linearity of photoelectric detection precludes the possibility of experimentally verifjiing the wave-particle duality concept for single pliotona. Taking ii positive i~ppsoach,it is suggested that perhaps wave phenomena, such as interfere~iceand dif'l'ruction, arc collective phenomena, due t o Inany particles. Some arguments in t'avout. of'this interpretation arc bruugl.ll forward, based o n our experinlental results a n d o n the results of' f-!anbu~.y--Bro~~n and 'T'wiss on the 'clumping effect' for photons. Finally, because of the iniportancc of conf'tonting our experimental results with those obtained on several previous occasions by other investigations. a brief historical study of the latter experiments a n d a critical analysis ol'thcir intcl-pretation arc provided here. Abstract

1 Introduction The theoretical hypothesis that goes under the nat~leof wave-particle duality, so remarkably successful in describing a large nurnber of phenomena and in predicting many others, does not seem to enjoy full and undcbated acceptance among physicists and some of then1 are still perplexed. Although, admittedly, the majority of physicists seems to be able to understand and to accept in toto the dual nature of particles, a few seem to be able to appreciate and acknowledge its descriptive calculation and predictive ability, but fail to clearly understand, o r a t least intuitively accept, why and how a single particle can also have wave nature. We shall not discuss here this philosophical problem because it has been, and continues to be, abundantly debated in the literature. Rather, following the trend of three previous articles published in this Journal,"?" we shall report here the results frorn another experiment done in our laboratory on difl'raclion of statistically independent photons from a laser source. TIIC latest results co11fir111the previous ones and continue to cast some tloubts on the validity of the duality concept, as presently known and accepted. Speculations In Science and Technology, Vol 8, No. 1 , page 35

Rrc:iiise of' tllc In~portztnccof' confronting no\?/ o u r results with those obtained on s r l p1-evioits occ;tsions by other investig;itors, ~ 1 1 1 ~ experimental rcsults will be preceded by a brief historical study of the experiments tipon which the duality hypothesis was built. It will be shown that those results were, of necessity, interpreted in terms of the limited knowledge available a t the time the experiments were performed, knowledge which can be now proven to be in disagreement with present knowledge. Hence, the duality notion f'or a single particle was born o u t of imprecise, if not incorrect knowledge. This does not meall that the wave-particle concept and all its consequent theoretical construction sllould not be retained. Rather, the aim of this work is t o define its limitations. 2 Historical analysis of the experiments and results upon which the waveparticle idea was born The aim of this section is to demonstrate that, historically, the experimental confirmation of the association of a wave t o a single particle was believed t o be obtained from the analysis of experiments that in 110way proved that single particles had wave nature. When these experiments were performed (1909- 1927) the quantutn of corpuscular nature of light had been well established through the fundamental work of Einstein%hich associated a frequency to the energy of a photon in order to explain photoelectric effect phenomena. In 1923 d e B r ~ g l i e , ~ recognising that for light there exists a corpuscular aspect a n d a wave aspect united by the relationship energy = h times frequency (/z = Planck's constant) came naturally to suppose that, for matter a s well, such corpuscular a n d wave aspects should be simultaneously present a n d postulated the relationp = h/h to account for the wavelength of a particle having momentump. It is t o be noted that in the above expressions t'or energy a n d molnentum of a single particle, the frequency and the wavelengths, respectively, are essentially multiplication factors which can be defined, a n d measured, only it' we have a large number of particles disposed on a diff'saction o r interference pattern. In other words, for a single particle, the concepts of frequency a n d wavelength would lose meaning in the absence of other particles. When these concepts were emerging a n d being developed, the idea that a single particle might have wave nature even in the absence of other particles came quite natiirally from the evidence offered by some experiments, initially done by ' r a y l ~ r ,and ~ later by G a n s a n d MiguezF7Zeemai18 a n d Dempster and Batho." 'T'liese investigators had obtained interference o r diffraction patterns at intensities of light so low that it was reasonable to assume that each and every photon recorded on a photographic plate had crossed the interferometer in isolation. Since the interference pattern had been constructed by one photon a t a time in this case, it seemed that each photon was endowed with some sort of wave nature, in the sense of being able to choose o r being uuided in a particular direction a n d of landing on a prescribed point of the P ~nterferencepattern. The detector, namely the photographic detector, was believed a t that time to be a linear device in the sense of being able of recording , more precisely, a number of photons proportional t o every single p l ~ o t o n or,

Diffraction of statistically independent photons

37

the I~glitintensity. In other words, the appearance ot'a wave plicnonlenon like the interf'erence fringes o n a pliotographic plate, togctlier nit11 tllc aqsumption tliat the detector Mras linear, were sufficient t o p 1 . o c01ic1~1si~eIj ~~ tllat each a n d every plloton was elidowed with a wave na turc. lJnfortunatelg, the a\sumption that the photograpllic d e t e c t o ~wa5 Iinei~r nl;is not corrcct. I n fact, as modern p l i ~ t ~ g ~ . i lt11col.y p h i ~ now points out,'""'"-' a photographic film is a non-linear detector. ?'his is because a pllotograpl~ic grain, wliel? exposed to light, does not become developable unless it absorbs at least three or four photons within a finite short time. This means that a photographic grain acts as a n I\'-fold coincidence countel.. whcrc I< is of' tlic order of' 3, 4 o r 111ore." This c o i ~ ~ c i d e n ccco u ~ i t c rmodel of' photogrl-1p11ic detection 11a\ bcen successful in explaini~igtlie ef'f'rct know11 as 'reciprocitj failure' a t both 11igl1 and low intensity of light,$' ~ I i c r c b ya photographic detector does not respond to just tlie total ~ i u m b e rofphotons,f7'(f'-= photon flux; 7' =. exyocurt: time) incident during timc 7' but, clue to tltc silost time ol' persihtence of the counting meclianism, can record a count only whcn thsce o r more photons impinge 011 tlie grain area within that timc of'pcrsistcncc. With extre~nelylow light flux very few events are recorded and tllc detccto~.s l ~ o ~ ~ s clcarly its non-lil~carity, because the probability of having :it Icr-1st tlircc photons near the grain area hithin the sllort counting time is negl~giblc. Such detccticjn non-linearity has profound cc)nsequetlces. I-lad they k n o w n , a t the t i ~ n ethe ~ v a ~ e - p a r t i cduality le hypothesis was being put i'osn'ard, tliat tilt. experinients which seemed to prove directly and conclusively such I~ypolhrsis for 5ingle particles had never ~.ecordedsingle p l ~ o t o n sbut , bunches or packets ~ s certainly more), the wave-particli~concept cvould of at least t l ~ r e ep h o t o ~ (and c r to a have bcen more logically ascribed t o collectio~isof' particlc.5 ~ ~ i t l i tIi;~n sil-tgle particle:/- In sliort, the widely refkrred to e x p e r i m e ~ ~by?aylorOand ts by Dcmpster and Battle" in n o way provided a proof that single p l i o t o ~ ~have s wave nature. I n order to justify, howcves, why, a t the time of'tlic emcrgencc. ot'tlie waveparticle duality, a photograpliic ti1111was believeti to bc a l i ~ i e adcvicc, ~. we t l ~ i ~ ithat k s u c l ~assumption was the most logical one a t that tinie bccauce previous experiments on tlie photoelectric eSfectl"iad proven tlic linearity of the latter phenomenon. Actually, the very notion of a photon a c a single entity with ~ ~ cdefined ll particle properties was a natural consequcncc oi'thc linearity of the photoelectric effect. And the photographic ef't'ect was bclicvcd to be identical to the photoelectric effect. Unfortunately, it was not realised that the pl~otoelectricef'i'ect ir~vestigateciat that time, with high intensity light sources, such as arcs, sparks ancl mu-cury vapour lamps,L5was in a range so far away from the extreniely low light intensities of the experiments of Taylor" a n d Dcmpsisr and Hatho" tliat nothing could be said about the linearity of the phsnomcnon in this latter ":Act~~:rlly. we belie\,c that such an cxper.irnc11talnon-lincar ct‘l'cct t~.iggcrccltllc r.csc;~l.cllthat ult~ni;~tcly lcil to the plloton counter. nlodcl. .l.Tliis ia indccit ivIi;~t h ; ~ sbee11 suggc5tc~iin Ket's 1-3 \f,llcre the c\pc~,inicnt;~l 1.cs~11tso l > t a i n ~ i~~l , i l h photogl-apliic recordings of'diffraction pattcl.n\ at dil't2retit light ~ ~ l t c u \ i ~clc;l~.ly i e s silo\\ a ~ ~ o ~ i - I i ~ l c01'; r r i t ~ cletcct~c)n.Sucll non-lit~caritycan orlly he intc~.pi.ctcitin tcr.111~ ol'thc :icccptcd ~,llotoy,~,il>llli. tlico~.!. \+llicli ~ n d i c a ~thirt c single i)liotons irrc not ~.cco~.dcd on ;I p l i o t ~ ) g ~ ~ i ~I'il~n. i~liic

38

E. Panarella

range. 111 other words, besides considering the photographic effect as linear, which we now know is not, they assumed tliat the linearity stemmed from tlie pliotoelectric ei't'ect linearity, wliich they had [tot investigated a t low light ~ntens~ties. Procccding f urtlier with this historical review, it should be said that, even if' the p h o t o g r a p l ~ ~detection, c as we now know, had 1;ot really revealed single photons, still several experiments of photoelectric detection done in subsequent times by Janossy and NarayLhand Pfleegor a n d Mandel,17 t o quote j~tsttwo of them, experiments done with extremely low light i~itensities,seemed to have been able to prove that single photons had wave nature, because incariably a n interference pattern appeared even when the photons were supposed to cross in isolation the interferometer. Therefore, the matter seeriled to be settled anyway. Again, no o11c of' these experiments proved their thesis because invariably they were dolie at a single light intensity, 110matter how low."' Had they done the experiments a t different low light intensities, they would have discoverecl a non-linearity of detection, as we shall prove below where o u r latest experimental results are reported. S u c l ~ nor>-linearity of' photoelectric detection has again profotind conrequelices, just as in the photographic case. This is because, in order to prove concl~~sively that single photon< have wave nature (and this applies just as well to otlicr particles, such a s electrons and neutrons), one needs to have the following two conditions satisfied: An interference pattern S ~ O L I I C Jappear a t a light intensity so low that the assumption that the photons are isolated within the interferometer is a plausible one. The presence of' the i~iterf'crencefringes in this case would prove that we arc dealing with a wave plienonlen, but it would not prove that it is a single-particle phenomenon. 2 11' tlie experintent is repeated now at lower (higher) light intensity atid longer (sl~orter)recording tin~es,sucli tliat the total ~ t ~ ~ mof b ephotons r is tlie same as before, t11e r(it?x. interference pattern should appear. 'I'tii5 woulcl provc that tlte plienornenotl is linear, o r tliat we arc dealirig with a single-particle phenomenon, thus confirming the corpuscular nature of light. 111conclusion, tlie ref'erence t1i;lt is always 111adeto t11e experiments done i n t l ~ epast a \ proot' tliat single photons h a \ e wavc nature is not entirely appropriate, because all those experiments satisfied only the first of the above two conditions and therefore the wave pattern obtained could riot be u~iecjuivocallyascribed to single photons. 111Section 5 below we shall take up this n?atteteagain and put f'O~.~'iird sonie arguments, based on the results of o u r experiments and ot' those of' the c1assic;ll Hanbury- Brown and Twiss" experinicnt on p1i~)to11 c ~ ) ~ - r e l a t i o ~in i s , Favour of an explanation that interference and diffraction are collective, rather than single-particle, plicnoiiiena. In other words, while i t wlll be confirmed that collections of particles ha\/c wave nature, as proven by all cxperin~entscarried out either in 1

Diffraction of statis tically independent photons

39

the past o r in the presetzt. it will be argued that single particles eitlieldon't 1l;lc.e wave nature o r , if they h a w i t , it has not been demonstrated as yet.

3 E:xperirnental results Bcforc reporting ~ L I Slatest results of an experlnicnt with \tatistlcally independent laser photons in \4jhich we counted the photons at cquidlst~int points along tlie vertical diameter ot'the diffraction pattern, wc woi~ldlike to review brief y tlie results ot' Ref'. 3 in order to provide new a ~ , g u m c t ~int sJ'avo~!r of' the initial tentative interpretation which was oi'l'cred tlierc.

Some oscilloscope records 01' diffraction pa ttcrns f'rotii \ t ; tist ~ icaIIy independent photons under dif'ferent conditions of light ilitc'n\~tyllave bcen reported.' Let us recall these experimental results. With a photon t'lux ot' 1.90 x 10'" photons/sec witliln the interfel.ometcr (the average photon \eparatiorl ~n this case is less than t11c length ol the intcrt'crometer), a clear drf'i'~.act~ol~ p:1tterti appCars(Figure l a ) which i \ coniposetl o i ' a n intensity ~naxiniums\~l.l.oundcd by hinges o r subsidiary maxima. When the photon flux is reduced by 21 f actor of' -- 48 to -4 X lOQphotons/scc, such that the average photon separation 15 greater tl1a11 the length of tlie interJ'eromcter, tile i'~'lngesot'si~bs~diat-y maxima seem to disappear (I'igure 1b), cfcspitc a n overall signir l a l~iplil'ication 01' -- 44 1 in g o ~ n gfrom Figure la to I b. I n Figi~rcI wc 11a~cno^ adtlcci anotlicr oscilloscope record (Fig~11.eIc) obtaineci \$ ith a niuch 1o~vc.1 11g11tJ'l\ix of' -7 X 10' pliotons/\cc, i.e. a plloton i l u r lo\z~erbv a i i ~ c t o rof' -273 than tlic tnitiirl one. All these records wcre obtained with a high-gain pl~otomultiplic~1~y scanning the vertical diameter of'the dii'f't.act~onpattern at 21 con5tarit velocltj of I cm/sec, the orifice in front of the photomultiplier having a diametcr of 5.08 X 10-kcm. For clarity, beside each oscilloscope record, we sllow the \ame dif't'r;ictio~~ pattern drawn with a thin Ilne passlng in tllc nliddlc of the basclinc or in the middle of tlie broadelzed trace. In Ref. 3 we said that the abseticc of the subsidiary maxlnia on tlic dil'iractiotl p a t t e ~ nof Figure l b was unjustif'ied. We ~ l o u l dlike to SLIP POI-^ this clalln as follo\vs.

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1 Figure la shows that, with a photon llux of' 1.90 X 10"' photon\/scc within tlle interfero~neterthe diffraction pattern (Airy pattern) is clearly dcf~tiedand is c o n ~ p o r e dof a central peak surrounded by two subsidiary rnaxin~a.'l'he central peak has an a ~ ~ l p l i t u dof'e -- 10.2 divisions (by 'division' we I I I C ~ I I I , of' course, the separation between two consecutive solid horimntal Ilncs on tlie pliotographic grid), as derived from the knowledge that tlic original centl.al peak of 4 divisions (see Figure 2a of Ref. 3) has bcen amplil'ied by a factor of 2.55 by increasing the photoinultiplier voltage from 2,000 to 2,200 V. According to classical optics,"the first subsidiary maximuni on the dit'i'l.action pattern should have a n anlplitude equal to 0.0175 times the central peak alnplitude, i.e.

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0.0 175 X 10.2 = 0. I78 division

E. Panarella

TIME SCALE . 0 . 5 sec/div

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( i ~ [ / ~ ~ ( ~ c/ ~, tlilot t(~~ o/1t(iit1~(/ ~,t~ 111 ( a ) it,ilif (1 p l ~ o t o ~ ? it.^ q/' I . 90 X 1 0 ' " Figurc 1 I / I ( J i~c~gul~11. pho/o11\./s(~, i.\ 1 1 0 1 /)~.('.\e~.\'~~il (11i(1t1lil l ~ t ~ ~ i . ~ ~ l , iko f i . 1101 i ~ l(7/)/1(~(1i, g ~ ~ \ 11.11~11 IIIC lj,~rhi,//il.\.1.s ~.o(/l/(,(~illo 4 X lliS~~hoton.s/,\c~c, ( b ) cc~lrll o 7 X 10' l)l?oto~r.~/.rr~c. ( c )tk~.c,t~ilc~ rl~c.firt~l I / I N / I / I ( ~(~/~?,t~lifi'~,(tliot/ O/'I/IP ~ / ( ~ l c > (.~\ ;t I ~o Y/ ~~ /i(/.s c I ~ ~I)wI? ~ I I ~ , I . ~ ( / , S441-/01(/, ~Y/ f ; ~ ) / i ?( 2 1 ) 10 ( b ) , t111(/ 221iOy/?)/(l fi.o/u ( 3 )1 0 (c) 11'IiIlc1 1 1 ~ /I:/II ~ I I I ~ \il.i9 ~ I I r ~ , c , ~ r1oi1,11 ll 011/\. 4STf0/ik1111(/ 273-/0/ri, I . L J S / I ~ J ~ , ~ I ~ ~ L J ~ ' 7

As Figure la shows, this is indeed so and the first subsidiary m a x i ~ ~ ~isu m clearly scen. Actually even the second subsidiary maxirnum is revealed, wllose amplitude is:'"

0.0042 X 10.2 = 0.042 division

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Diffraction of statistically independent photons

Now, if the same experiment is repeated with reduced light intensity, the f'ringes (or subsidiary maxima) should be seen again psovidcd tlic :rn~yliI'ication is sut'f'iciently Iiigli to yield a fringe amplitucic of 0. 178 division o l higlicl-. Af'ter rcdt~cingthere1'c)r.e the light intensity within the inte~.Scsomctc~b> a iircto~01' -- 48 to 4 X lo4 pliotons/sec, we obtained Figuse l b wllicli sl~o\v\only t l ~ c centsal peak. N o sign of fringes or s~tbsldlarymaxima is present in thi\ picturc. In ordcr to see ii' the 1'ringeh had to be expected, MY f'issf c;tlculatcd tlie anlplitude of' the central peak of' Figure 1b as follows: Original peak amplitude (see Figurc 2b in Iiet'. 3) - 2.5 divisions Al~iplif'icationdue to increase ol' photon~ultiplies\~oltage 4.41 Oscilloscope amplii'ica tion = 4 Ilence, the centsal peak of Figure 1b has amplltudc:

2.5

X

4.41 X 4 = 44.1 divisions

arid tlie amplitude ot' tlie first subsidiasy maximum

01'

f7igul-e

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should be:

0.0 175 X 44.1 -- 0.77 divisions i,e. larger than in Figure la, where it was detected. C o n s c q ~ ~ c n t liyt ,s e c l ~ tliirt ~s the expected fringes (lid not appcar and tlie co~lclusioiithat tlic f'ri~igr\did not exist is justified by this scsult.

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2 It' one looks at Figure la, and more specii'ically a t t h y l'igu1.c on tlie right, one observes that thc 1'11-stsubsidiary maximunl lias a n a m p l i t ~ ~ cof' i c 250 m\'. Nou,ifone reduces tlic liglit intensity by a i'dctor of' -48, ttic : i m p l i t ~ ~ oi'5~1ch dc subcidiary maxiiiium should bc reduced accorciingly to -- 5 11iV. I'liis signal amplitude is sul'ficiently high and a clear uptl~arddispl;tce~iientol'tlic basclinc in 1-igurc l b at tlic position of the fringes should have occur rcd. On tlie other hand, tlie absence of the t'singes cannot bc j~lstii'iccl (311 tlic g r o ~ i ~ that, \ d sincc the pliotom~tltiplie1--gives very short pulses, thcy d o not overlap at the positionof' the fringes \vl~entllc light intensity is weak, t l ~ u s precluding the trace to elevate from tht; baseline.':' In fact, at the position 13 o n the central peak at the same heigli t as point A of the fringe (sec I - : I ~ L II bI -) ~the liglit intensity is just as wcak. Although in R the trace doe(; not elevate from the baseline by as much a s it sllould, namely - 5 m V , still an upward displacetnent takes place by 1 mV, wl~ereasnotliing of this happens on tlic fringcs. In c c ~ ~ ~ c l u s iito nseems , that the absence of f'ringes is due to non-linearity 01' detection a t very low light intensity, and this assumption will ~.cccicca confirmation fro111 the experiment t o bc reported below.

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3 A n indication that the results might be departing f'sorn tlic prcdlctions of t17a11eo p t ~ c sand tending towards tlie predictions of geometrical o p t ~ c \i, pro\ ided by Figures 1b and lc which show a sudden disconti~il~it> of' Itglit intensity a t points rvhlch are closer to the optlcr-11nxii oi'the sy\tcrn a \ tlic light intensity goes dourn. 111other urords, although the ccntr.'11 1-singe \eeni\ to bc

r n ~ t ~ ~ l t ~ t litl~zals, l l ~ n g '11 the I~gltti ~ i t e n i ~ t i\cL Cs Iiavc ~ n ~ c s.At ~ e a taezx~dth d, ot 13.5 Iiim, L lie light ~ntt.n\itydictr~butionprcicntj ii budder^ discontinuit> closer to t17c gcomcti-icril axis of' tlic \y\tem a \ the I~glittntenjity goes down.

Photon counting at equidistant poi~ltsalong a vertical diameter of the dijji.uction I)(/ T 1Ol.17

Let us IIOW report tlie results fro111 our latest experiment. The experimental ;ipparatus I S basically the one described in Kei'. 3 (see Figurr 2). A 5-I~ILVcw E M , , , , mode Spcctra-Physics Mod. 135 I-Ic-Ne laser was used as a source of light. The laser einitted a Gausrian be;tm of radius = 0.35 mi11 a t l/~,"points. The peak light intensity in the ccntr;il part of tlie beam was: = 2/',,/71.~' = 2.56 W/cm2

(P,,

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5 x 10'' W)

The light intensity prof'ilc was smoothcd out by means of' a pinhole of ciia~neterd - 5.08 .X lo-' cn1 positionetl at the centre oi'the beam, a t the point of' ~iiaxiniuiiilight intensity. The resultant emerging bright central disc o f t h e Airy pattern was coliimatecf by means of a simple double-convex lens locatect a t a distance from the pinhole equal to tlie lens focal 1ength.f'- 30crn. 'T'iie intensity of' light :it the centre 01' the Airy pattern resulted in:?"

where A = n &/4 is the pinhole area a n d P, -= IpA. A second pinhole of diameter t l - 5.08 X I O w ' cni \\as po\~tionecfat the ccntse of tlie A ~ r yd ~ s cS. ~ n c the c light ~ntcnritya c 1 . o ~this ~ pinholc was c\\entinlly cc~n\t:int.tlie photon i'lux entering tlic pinhole resulted in being 1.90 X 10"' pl.totons/scc. I'11e dil'l'sacted light out of this second pinhole wa\ then rccollimatcd by me:lns of :i si~lipledouble-convcx lens located ; i t a distance from the pinliole equal to the lens focal length f - 202111 and thc light detection occurred along tlic vertical ciianietcr of' tlic diffraction pattern by n ~ e a n sof a high gain was pro\w,ied photo~nultiplier,For goocl \pace resolution. the photorn~llt~plier cvith a s~iiltllor~i'lceof' 5.08 X 10-'cm diameter drilled on its front cover. 'The s ~ g n a lfrom the photomultiplier was then a~i~plifiect and sent to a counter. More in detail. the detection cvstem consisted of a t'ourtccn-stage, f1;itI'iiccplate IXCA photomultipl~cr.tGpe 7265, Iiau~ng;i m~11tialk;rliphotodiodc ([C'sj)Na-KSb)with S-20 reymnsc. I'lie photo~nultiplicrcurrent nnlplif'icat~on was 2 x 10'. I llc tube was oper:ited ;it 2,050 V, i.e. below tlie rn;txitnum pcrmisr~blcvoltago of 2,400 V, in order to reduce the dark currellt count t'roni t l i e r n ~ i o n ~erni\iion c anti to increase tlie signal-to-noisc ratio." In order to 1'~1rtlierrcduce rlic cla1.k current. the p h o t o m ~ ~ l t i p l iwas e ~ . cooled with a blanket ot'clry icc to - 15 C'. L iglit unifornrity occr the photocathotfc;ll-ca w a i acliicvrd by ~nxcrting dl1 fuscr w1t1111-1the pllotom~~ltiplicr c a w , right behind the entrance orifice. Great cnrc was also taken to reduce any stray light e n t c s ~ n g tlic plioto~uultipl~cr ;.in4 this z v a s achivvcd by enclosing the entire appirlatr~s containirig the 1;1sc1-and related optics \+ithin ;I black box, so that only a jlliall opening was :iv;~il;tblctor tlie Iascr bean1 to get out o f p ~ n l i o l zNo. 2. As to the

Diffraction of statistically independent photons

~*e\icluallight fronl the laser ci15cliai.ge tube going t h r o u g l ~the p ~ n h o l e ~t , was cut o i ~ allnost t cotilplctuly by placing in I'ront o t ' t l ~ ep l ~ o t o n ~ ~ ~ l t i pa l 11tgIiier g t l6,328 l A ilild pii\\ ttlter hav~ligt ~ s n r n i \ s ~ o84'';) n a t the l:i\c~.u ~ : i v c I ~ ~ ~01. i f t:tllii,g d o w n to 0.03(,r 'it 5.540 A. Finally, the entire experiment W;I\ c;lrs~cdout in ;r \tnall, ~ 7 1 n d o ~ v l e(lark \ s , room c o ~ i ~ p l e t eshieldeci ly t'ronn any external light. The euperimc~ltc o ~ ~ s i s t ci dl l ~ n o c i n gthe j>hotoliiultipller by t'cluidiitat~t jtepscot' 5/1000 of' a n inch (- 1.27 x 1 0 - cnt) and ~irrcstingit a t e ~ t c hstep just f'or the ti141e rcqui~.cd 1'C)i. I _ ) L I ~ coil~lting. ~ 111e coilt~tillgwas done wit11 a Tcnnelec 54GP Scaler ancl 531 A Timer, the iignal fr-0111 the photomultiplier h ~ i \ ~ ~been r r g amplified by a t'rtctor of' 10 through an amplifier liavi~igi n p l ~ t rcs~st:incc 1000 o h ~ i l s . The coirnti~igtime wit5 chosen ~.iitliershort, 2 X 10-I scc and 2 {ec for the two c\perirncnts that we 1.a11,respectively, becauw this ol'f'ere(l sonic clistinct a d \ , ~ n t a g eover \ lorig coutlt~ngtIIne5. 1'11isavoids p r o b l c ~ i o~ ts' p l ~ o t o m u i t i ~ ~ l i e r f'rt t ~ g u earxi dccrea\c ot' \en%itivity ," anci the dark count call be greatlj reduced ~ ~ t t a11 l r iiPPl.Opl.ltitC c11oice 01' sllort coiltrting ti111~'. The e\perinicntal rcsirlts arc reported in 1:igilr.e 3. 'The solid circles rep~escnt t l ~ cccjt~ritsobtai~lcci whc11 the p l ~ o t o n lluu wittiin the intcrt'c~roiiictc!, i.c. (the :iver:ige p l ~ o t o ~ l between pit11101~\ I L 1 ~ l ( l2. wa\ 1.90 X 10'" pl~otc)t~s/sec \ C ~ ) ; ~ I . ~ ~ ~is I C 1.57 ) I I c111,1.c. I I I U C ~ IIC