Volume133,number1,2 PHYSICSLETTERSA 31 October1988

TWO-FOLD GENERATINGFUNCTION FORGAUSSIAN LIGHTWITH A TWO-PEAKED SPECTRUM

ManmohanSINGH3072/2B/2,St. No. 22, RanjitNagar,NewDelhi110008, India

Received18 March 1988;acceptedfor publication 19August1988Communicatedby A.R. Bishop

By solvingtheFredholmintegralequationof thesecondkind for two non-overlappingintervals,weprovide in thispapertheclosed-formsolutionforthetwo-foldgeneratingfunctionfor gaussianlight havingatwo-peakedspectrumcharacterizingapoly-dispersemedium. The explicit expressionof the second-orderintensitycorrelationis derivedhereto showthe utility of thisgeneratingfunction.

1. Introduction

The studyof macromoleculardynamicshasattractedthe interestof severalresearchersof diverseback-grounds,with pioneeringcontributioncomingfrom Pecorain 1964 [1]. Interestingdetailsonthedevelopmentof this subjectarenow availablein thebooksby BerneandPecora[2] andby Dahneke[3] andsomeof therecentpublications [4—6].However, a systematicandrigorousstudyof the macromoleculardynamics,es-peciallyvia the photoncountingstatistics(PCS),posescertainmathematicaldifficulties, aswerequireaclosed-form solution for the generatingfunction (g.f.) pertainingto the multiple-peakedspectrum,a typical char-acteristicof the polydispersemediumin a laserlight scatteringexperiment.

As a first stej, towardsourunderstandingof the one-fold aspectsof the PCSof atwo-peakedspectrum,wehad shownin ref. [7] for the first time howto arrive atthe uniqueclosedform of the one-fold g.f. usingHadamard’stheorem[8] andcertainboundaryconditions.With the helpof certaindetailsprovidedin refs.[7,9], in the presentpaperwe lay the foundationfor a sensitiveanalysisof the spectral featuresof apoly-dispersemediWnvia the higher-orderPCS,for it is only in the higher-orderstatisticsof anincoherentbeamthatweseeadirectdependenceon thefield correlationaswasshownby Glauber[10]. However,hereweshallconfineour interestto thederivationof thetwo-fold g.f. (section2) anddemonstrateits utility to studytheintensitycorrelations(section3).

The field coftelationof the scatteredlight is givenby Koppel [11] as

(1)

whereG(f’) is *he distributionofthedecayrates.Forthetwo-peakedspectrumconsideredherethehalf-widthsaregivenby [11

r’1=D~k2, f’

2Dtk2+6Dr, (2)

whereD, is thetranslationaldiffusioncoefficient,D. therotationaldiffusioncoefficientandkis themagnitudeof the scatteringwavevector.

For the caseof the scatteredlight havingtwo sharppeaks,the field correlationis givenby

0375-9601/88/S03.50© ElsevierSciencePublishersB.V. 37(North-HollandPhysicsPublishingDivision)

Volume 133, number1,2 PHYSICSLETTERSA 31 October1988

g(~)=J [aio(r_Fi)+a2(r_r2)]e_rT~=gi(ai, T1)+g2(a2,12), (3)

where

g1(a1,T’1)=a1e~1T, g2(a2,r2)=a2e_r2t, a1 +a2=l.

2. Two-fold g.f.

2.1. TheFredhoimdeterminant

To dealwiththetwo-fold PCSof thetwo-peakedspectrumdefinedby eq. (3), weusethefollowing modifiedformof the Fredholmintegralequationofthesecondkind givenin ref. [9] for the caseof gaussian—lorentzianlight,

ti+Ti t2+T2

s~J dt’ g~t—t’ 10! (t’ ) +s2 J dt’ gi t—t’ lO1(t’ ) =2Ø~(t) (or Øj(t)), i~j, (4)

wherethechoiceO!(t) or 01(t) in eq. (4) abovedependson whetherte[t1, t1+T1] or tn[t2, t2+T2]. Weconfineour interestto the caseof two non-overlappingcountingintervals: t1 + T1 ~t2.

As in theone-foldcaseof atwo-peakedspectrum[7], wefind on differentiationof eq. (4) thatthefunctionsO)~( t) satisfy the following differentialequation,

d40L~(t)/dt4—Ad2O2(t)/dt2+BOL~(t)=0, (5)

with

A=f’~+fl—2s12*~(a1f’1+a21’2), B=r112[f11’2 —2s12~(a1F2+a2f’1)], ~=1/A. (6)

A simplesolution of eq. (5) is given by

0L~(t)=a)j~e”’+b~,~e”~t, (7)

wherep,, are theroots of the equation

p~1—Ap,~1+B=0 (8)

andaregiven by

±P~.=±[A± (A2~~4B)U2]l/2/~/~ (9)

where i= 1, 2 andj= 3, 4. Substitutingeq. (7) into eq. (4) andperformingthe integrationswe get,(i) when t

1~<t~<t1+T1,

s1a1 e~’’~{a!(I’~+p~)—‘ [e~” +m):_e(rI +rn)ti ] +b! (I’~—p,)—1 [e(

11 _p~)t_e(rI—~n] }+s

1a2e~’~{a)(F2 +p,) —~[e2 t_e(F2+P)tI] +b! (1’2 —Ps) [e’2/~~)l_eU’2_~)t1 ] }

+S~aienht{a)(F1 —p~)’[e_1_Pt_e_1_~~~~h+T)] +bJ(F1 +p~)’[e_1+t_e_~’1+~)~h+T~1}

+s1a2e’~’{a)(F2—p~)’[e~ (f2 ~‘~‘~e (r2_~~)(ti+ri)]+b! (F2 +p~)’ [e_(T2+P)t_e_~’2+P1~1+T~ ]}+s2aiel’1t{a](Fi —p3)—~ [e_~~_Pt2_e__1~1)~2+T2)] +b

1 (F1 +p1)’ ~

+s2a2eT2{ai(F2 —pd) 1 [e_~2_PJ)t2_e_(T2_~~J~12+T2)] +b1 (F

2 +p1) ‘ ~ (12+T2) ] }=)~[a)e”

t+b!e”], (10)

38

Volume133, number1,2 PHYSICSLETTERSA 31 October1988

(ii) whent2~<t~<t2+T2,

s1 a~e’”{a) (F1 +p~)— I [e~” +p,)(Ii +Ti _e(~IPP’),’] +b~(F1 —ps) — I ~ —p,)(ti +1,) ~ ] }+s~a2e~~~2t{a!(F2 +p~)—‘ [e2+~~~ (II+ Ti) — e21)hI J + b! (F2 p~)— [e~’2_.~1) (Ii +Ti) _e(J2_~0~~] }+S2cx1 e’’~{a1(Fi+p~)— I [e~” +Pi)t_e(rI+PJ)t2] +b1(F~—.p,)’ ~

+s2cr2e2{a

1(F2+p~) ~[e~2J)1_e~’2+ui)t2] +bI (F2—p3)’ [e’2~~J)t_eU~2&i2]}

+s2a1e’~’{a1(F~—pa) I[e_~_Pi)t_e~~’Pi)~2+T2)] +bI (F~+p~) ~[e(F+Pi)1_ePit2+T2)J}

+s2a2e2{a~(F

2—p1)—‘ [e 2PJ)t_e(I2P)(12+T2)] +b~(F2+p~) ‘ [e 2+Pj)t_e~(F2+Pjxt2+T2) j}

=t[aie1~Jt+bie_1~it]. (11)

Now on collectingthecoefficientsof e~T12tin eqs.(10) and(11), we get for thenon-singularsolutionthefollowing Fre~1holmdeterminant(FD):

—E1(p1) —F1(—p1) —E2(p1) —F,(—p1) E1(p1)11(p1) 0 E2(p,)I2(p~) 0

—E1(—p1) —F~(p1) —E2(—p1) —F2(p~) E1(—p1)I~(—p1) 0 E2(—p1)I2(—p~) 0

—E1(p2) —F~(—p,) —E2(p,) —F2(—p2) E1(p2)11(p2) 0 E2(p2)12(p2) 0

—E1(—p2) —F1(p,) —E,(—p2) —F2(p2) E~(—p2)I1(—p2) 0 E2(—p2)12(—p2) 0

o H1(p1)J1(p3) 0 H2(p3)J2(p3) —E~(p3) —F1(—p3) —E2(p3) —F2(—p3)

o Hi(—p3)J1(—p3) 0 H2(—p3)J2(—p3) —E1(—p3) —F1(p3) —E2(—p3) —F2(p3)

o H1(p4)J~(p4 0 H2(p4)J2(p4) —E1(p4) —F1(—p4) —E2(p4) —F2(—p4)

o H~(—p4)J1(—p4) 0 H2(—p4)J2(—p4) —E1(—p4) —F~(p4) —E2(—p4) —F2(p4)

(12)where

Ek(pIJ) =ak(Fk +p1~)(F~—p~1)exp[ (Fk—pjJ)tI21,

Fk(pIJ)=ak(Fk +p~~)(F~—p~1)exp[ (F,., —p,,~)(t~,2+ T~,2)],

Ik(P1)=exp[ (Fk—pI) T1 1—1, Hk(pJ) =ak(Fk—pI)(F~.—p~),

Jk(pJ)=1—e*p[—(Fk+pJ)T2], k,k’=l,2(k�k’), i=l,2, j=3,4. (13)For ~=0, we Obtainfrom eqs.(12) and(13)

~ (14)

2.2. Thenon-analyticityoftheFD

If wego arounda completecircle in thecomplexi~-plane,wenoticethat thisFD (eq. (12)) doesnotreturnto its original valueandit doesundergoa signchangeas [7]

PI~P2 and p3-p4.

Also since the order of thep~in the rows of the FD (eq. (12)) canbe any oneof the 2X 4! possibleper-mutations,we geta signchangein the FD foreachexchangeamongthep or p~.Lastly, noticingthefactthat

39

Volume 133, number1,2 PHYSICSLETTERSA 31 October1988

the indexing of the countingintervalsis completelyarbitrary,the following exchangesare possible,

Pi4-~±pj.

Thus if we now divide or multiply the FD in eq. (12) by a factorPIP2P3P4,wewill notgetany sign flippingsin it. Further,if weallow for the following possibilities,

P1P2 and p~=p~, (15)

the FD (eq. (12)) will becomezeroat four points in thecomplexi~-p1ane.To counterthis,weneedto divideit by thefactor (p~_p~)(p~—p~).But in ordertosatisfythe physicsof the PCSit canbe easilydeducedfromthe discussionin ref. [7] that we needto divide the FD (eq. (12)) by the factorPi P2P3 P4 (p~~ ) ~

p~)2andthusobtainthe following uniqueanalytic function,say P(~):

P(~)=D(i~)/p1p2p3p4(p~—p~)

2(p~—p~)2. (16)

It is well known that the two-fold g.f. ofthe gaussianlight canbeexpressedasthefollowing infinite product,

Q(s1,s2)=fl [l+~<I) ]—‘, (17)

k

wherethe~‘k are the eigenvaluesof eq. (4) and <I> is the meancount rate.Determiningthe order [8] of theentirefunction P(c5) to be ~ (from eqs.(9) and(12)) andapplyingHadamard’stheorem[8], wecanwrite

P(i~)=P(0)fl (l—~/~), (18)k

whereP(0) is a constant.On comparingeqs.(17) and(18) weget the following form of the two-fold g.f.,

(19)

whereP(0) andP( <i>) canbe easilyobtainedfrom eq. (16) by putting ~=0 and ~= <I> respectively.

3. Two-fold correlation

Havingobtainedtheclosedform of thetwo-foldg.f. for the two-peakedspectrumgivenby eqs. (12)—(14),(16), (19) and (20), wenow give anexampleof the two-fold PCSof this spectrum,namelythe correlationfunction.

The unnormalizedcorrelationfunctionin the two-fold statisticsis given by

~ n1n2P(n1,n2)=<n1n2>, (20)ni ..O fl20

where n1 and n2 are the numberof photocountsregisteredin the intervals [t1, t~+ T~]and [t2, t2+ T2] re-spectivelyandP(n1, n2) representsthejoint probabilityof thesephotocounts.Thejoint probabilityP(n1, n2)is relatedto the two-fold g.f. in the following way:

~ (1__si)Pu(l_s2)n2P(ni,n2). (21)Pit 0 ?120

The two-fold correlationfunction is given by

40

Volume 133,number1,2 PHYSICSLETTERSA 31 October1988

C(2~=<ntn2>=8~1~52~ , (22)8s1t952 si=s~=0

andthenormalizedcorrelationfunctionby

ct2~(r)=<fin

2 >/<n1 > <n2>. (23)

Rewritingóq. (19) as

Q(s1,s2)=A(0)/A(s1,s2) (24)

andusingBodewig’srelation [12] for convertingthedifferentiationof adeterminantA to thatof a matrix d,

DA—=ATr(d’Dd), (25)

whereTr stai~dsfor thetraceoperation,wecannow easilyobtainthe followingexpressionfor thesecond-orderunnormalizedcorrelationfunction,

Ct2~(r)={Tr(.~)Tr(d

2) +Tr(.~d2)}iS~=~2=O~ (26)

whered~ i~’~Od/ös1And d2~d~8d/ôs2.

We finally obtain the following expressionfor thenormalizedsecond-ordercorrelationfunction:

It~T’.iI

I I I I Fig. 1. Thebehaviouroftheexcesscorrelation [c(2Xr)—1] for

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.8 0.9 1.0 the two-peakedspectrumcharacterizingthe polydisperseme-t dium asafunctionoftheparametersa

1,2, f’~,2, andT.

41

Volume 133, number1,2 PHYSICSLETTERSA 31 October1988

c~2~(t)=l+[g~(al,FI)sinh2(FlTIFlT)+~(a2,F2)sinh2(F2r/F2T)

+2g~(a~,F1)g2(a2,F2)(sinh[(f~+F~T/2])2] (27)

whereg1 (aI, F1) andg2’( a2,F2) aregivenbyeq.(3). Thefactorslike sinh

2 IF?’)2 accountfor the arbitrarysampletimes.Thus eq. (27) providesusanexactanalysisof the second-ordercorrelationfunction for a two-peakedspectrumcharacterizingthepolydispersemedium.Now asregardstheexperimentalsituation,wenoticethat though“nano-secondcorrelators”areyet to bethe in-things,effortsare on to constructcheapcorrelatorsgiving us reasonableresults,thoughonly for small sampletimes (—.~~.ts) [13,14].

In fig. 1 wegive thebehaviourof the excesscorrelation(= C~2~1) fora two-peakedspectrum.We observethefollowing importantfeaturesfrom fig. 1:

(i) Foranequalmixtureofthetwo peaks,thecorrelationdecayisslowerthanin thecaseofunequalmixtureof the two peaks.

(ii) Fortwo peaksof equalmixturebuthaving greaterhalf-widths, thecorrelationdecaysfasterthan in thecaseof two peakshavingsmallerhalf-widths.This implies that the scatteredfield correlatesfor a short timewhenmacromoleculeshavelargediffusion coefficients.

Acknowledgement

The authoris pleasedto acknowledgethe keeninterestandthe financialhelp of Mr. JoginderSingh. Theauthoris alsothankful to Mr. NarainderSingh Bhau for his help in the computationalwork.

References

[I ] R. Pecora,J.Chem. Phys.40 (1964) 1604.[2] B.J. BerneandR. Pecora,Dynamiclight scatteringwith applicationsto chemistry,biology andphysics (Wiley—Interscience,New

York, 1976).[3] B. Dahneke,ed.,Measurementof suspendedparticleby quasielasticlight scattering(Wiley, NewYork, 1983).[4] J.S.HwangandH.Z. Cummins,J. Chem.Phys.77 (1982)616.[5] S.R.AragonandR. Pecora,J. Chem.Phys.82 (1985)5346.[6] A.K. Livesey,P. Licinio andM. Dolaye,J. Chem.Phys.84 (1986)5102.[7] M. Singh,Phys.Lett. A 126 (1988)463.[8] E. Hille, Analytic function theory,Vol. 2 (Blaisdell, Waltham, 1962)ch. 14.[9] M. Singh,Opt. Acta 31 (1984)1293.

[10] R.J.Glauber,Phys.Rev. 131 (1963) 2766.[11] D.E. Koppel,J. Chem.Phys.57 (1972)4814.[12] E. Bodewig,Matrix calculus(North-Holland,Amsterdam,1959)p. 41.[13] 0. Glatteretal., Rev. Sci. Instrum.58 (1987) 350.[14] H.5. Dhadwal,B. ChuandR.Xu, Rev. Sci. Instrum.58 (1987)1445.

42