October 1967

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 57, NUMBER 10 OCTOBER 1967

Analysis of Light Fluctuations from Photon Counting Statistics*

GABRIEL BtDARDDepartment of Pkysics and Astronomy, University of Rochiester, Rochester, New York 14627

(Received 9 March 1967)

It is well known that photoelectric measurements yield information about the statistical behavior offluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, fromthe photosensitive surface of the photoelectric detector, upon which the light beam is normally incident,is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help ofsome plausible assumptions, the present analysis provides solutions to the problem of inverting the Poissontransform, thus determining the probability density for the intensity from experimentally obtained photo-counting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typicalexperimental counting distribution. The technique is of particular interest in connection with efforts to under-stand the statistical behavior of optical fields, especially of laser fields.

INDEX HEADINGS: Lasers; Detection; Coherence.

A VERY powerful method used in the investiga-tions of statistical properties of fluctuating light

beams is the photon counting technique, in which thestatistical distribution of photoelectrons emitted withina fixed time interval is measured with a photoelectricdetector upon which the light beam is normally inci-dent.",2 Application of photoelectric measurement tech-niques to a variety of light beams has recently becomethe subject of extensive investigations.' 1 " Wolf and

* Research supported by the Air Force Cambridge ResearchLaboratories.

1 L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); 74,233 (1959); 81, 1104 (1963).

' L. Mandel, in Progress in Optics, Vol. II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.

3 F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics ofQuantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald,Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.

4 C. Freed and H. A. Haus, in Physics of Quantum Electronics,P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-HillBook Co., New York, 1965), p. 715.

6 F. 'r. Arecchi, Phys. Rev. Letters 15, 912 (1965).6 C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).7C. Freed and H. A. Haus, IEEE J. Quantum Electronics

QE-2, 190 (1966).A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650

(1966); see also Phys. Rev. Letters 16, 1169 (1966).9 F. T. Arecchi, A. Bern6, and P. Bulamacchi, Phys. Rev.

Letters 16, 32 (1966).10 F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys.

Rev. Letters 16, 589 (1966).11 W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531

(1966).

Mehta have recently shown'4 that, at least in principle,the complete probability density for the intensity of alight beam can be determined from knowledge of thephotocount distribution associated with the photoelec-tric detection. However, their method can be used onlyin cases in which a closed-form expression for thecounting distribution can be provided. The purpose ofthe present analysis is to propose a new systematic ap-proach to the problem and to provide some techniquesof solution.

FORMULATION OF THE PROBLEM

When a plane wave of quasimonochromatic polarizedlight is incident normally on a photoelectric detector,the probability p(n) that it photoelectrons will be re-leased, in a fixed time interval T, is related to the proba-bility density P(W) for the time-integrated intensityW by the well-known formula" 2l" 5 :

f e]WnpXn)= | expF-WIP(W)dW.

JO !(1)

12 P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).'3 S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R.

Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones,T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).

14 E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).11L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

1201

20 GABRIEL EI}DAR Vm 37

Here

IVt = at I (as

-a is a measure of the quanatun efficiency of the detectorand 1I1) is the intensity of the light (measured inphotons per second) at time /. Assume that the effect ofthe dead time of the counter on the statistical distribu-tion of photoelectrons 10 46 is negligible. The fundamentalEq. (1) then shows that the statistical distributionp(nz) is obtained by averaging a Poisson distributionwith parameter W over the ensemble of the incidentfield. Hence, Eq. (1) can be considered to representp(nt) as a Poisson transform of P(IV), i.e., a linear trans-form with a Poisson kernel, as first pointed out by Wolfand Mehta. 14

The problem of determining the probability densityP(1) from the photocount probabilities p(z) is oftenreferred to as the inversion problem of photocount sta-tistics, since it involves the inversion of the linear trans-form, Eq. (1).

A FORMAL INVERSE RELATION

Consider first the more general linear transform

(/?- W'lf (W)p ('~aWld (2)

where M(n) can be looked at as the it th moment of thefunction f (W) with respect to the weighting functionp(Wi). The principal technique available for the inver-sion of such a transform' involves relating the momentsM(n) to the coefficients of the expansion of f(W) interms of polynomials orthogonal in the domain (a,b)with the weight function p(lfl).

The Poisson transform, Eq. (1), is a special case ofrelation (2), in which the appropriate orthogonal poly-nomials are the Laguerre polynomials, orthogonal inthe domain (0, rc), with the weight function p(Wh=exp(-W); the moments M(n) are equal to nztp(nt).

Hence, upon formal expansion in terms of the Laguerrepolynomials Ld(W) of the first kind, the probabilitydensity P(TW) can be written in the form

P(TITV)== Ecj.L,_V~). (3)k=O

As shown in the Appendix, it readily follows fromEqs. (1) and (3) that the expansion coefficients ck are re-lated to the discrete set of photocount probabilitiesp (n) in the following way:

Equations (3) and (4) therefore express the inverse ofthe linear transform (1) in the form

A.0 r=k(5)

Furthermore, it can readily be shown that a, formulasimilar to Eq. (4) relates each photocount probabilityp(m) to the expansion coefficients ck;

(6)p w) = E (-1)Z0 Cn.k-o

In fact, Eqs. (4) and (6) express the uniqueness andthe symmetry of the Poisson transformation.

To illustrate these results, let us first consider sometypical photocount distributions. Suppose, for instance,that the experimental photocount distribution is theBose-Einstein distribution

(7)

where (i) denotes the average number of photocountsin the time interval T. Such a photocount distributionwould be obtained from single-mode thermal radiationincident on the photodetector. The expansion coeffi-cients ck, obtained by substitution of Eq. (7) intoEq. (4), are of the form

(8)

The corresponding probability density P(TV), obtainedfrom Eqs. (3) and (8), is given by

(9)

with (IiV)= (X). In deriving Eq. (9), we used the sumrule for the generalized Laguerre polynomials'8 LA;

oa "!Ls(x)L;7fy)zn (xyz)-! (2 (xyz)t)

;=o Przy+a1) (1 -z) 1

Xexp[ Z± (10)

where 'a is the modified Bessel function of order a. Inour case, a=0.

As a second example, suppose that the photocountsare distributed according to the Poisson distribution

(11)

Such a distribution is typical of a single-mode purelycoherent field. In this case, the expansion coeficientsck are

c£= exp(- (f)))Lk((fl)),Cars (- W (r).

r=O(4)

'6 G. B6dard, Proc. Phys. Soc. (London) 90, 131 (1967).17 See, for example, P. M. Morse and H. Feshbach, Methods of

Theoretical Physics (McGraw-Hill Book Co., New York, 1963),p. 947.

(12)

where LA, is the Laguerre polynomial of degree k. Itfollows from Eqs. (12), (3), and (10) tha t P (W;) has the

" See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table ofIntegrals, Series and Products (Academic Press Inc., New YoTk,1965)y p. 1038.

1202 Vol time jJ7

P (11) = 00,/ E (it � + 1 T,-1-1

ck. = E (11) + 1 I-`.

P (W) = (Ilff)) exPE- ff,71(Ul,

P 00 = !) exp (- (n.�).

O 16 ANALYSIS OF LIGHT FLUCTUATIONS 1203

formP(IV) = (1(W- (IV)), (13)

with (W)= (it), and a is the Dirac delta function.As a third example, consider the case in which the

photocounts are governed by the distribution

( 1Th)n nr)(nv)+1)[-(ne) 1Xx T)+ tj, (14)

where L,, is the Laguerre polynomial, as it is the casefor superposed thermal and coherent radiation, whoseintensities are in the ratio (nc)/(nfr).' 9 This last expres-sion represents to a good approximation the photocountdistribution associated with the detection of lightemitted by a laser operating above threshold. 0' 7 Theexpansion coefficients Ck are given by

c,=U11T)+1]-k-1Lk( (n ]. [-1C 5(n0)f exF (nT)+ (

The corresponding P(W), obtained from Eqs. (15),(3), and (10), can be expressed in the form

1 r/W+(,n)\ ,2 (W~n)P (W)= TexPL (1 )IoI ), (16)

where lo is the modified Bessel function of order zero.The present analysis indicates that, whenever a

closed expression is available for p(n), the correspond-ing P(W) can be obtained by application of Eq. (5).In practice, however, experiments provide a finite setof values p(i) and we would like to construct at least agood approximation to the corresponding P(W). Wenow examine some approximate techniques for solvingthis inversion problem.

EXPONENTIAL CURVE FITTING

Let us suppose that the experimental photocount dis-tribution can be fitted by a combination of exponentials,

pM(n) = E aj exp(-4bn).1='1

(17)

distributions. The expansion coefficients Ck are

A1

ck==Z a4EI-exp(-bj)]k.j~-

(18)

The evaluation of the corresponding P(TV) is quitestraightforward and yields

P (W) = E aj exp[bi- {exp (bj) - 1} W]. (19)j-1

In deriving Eq. (19), we made use of Eq. (10). Hence,whenever we can fit the photocount data by Eq. (17),the corresponding P(W) is also given by a combinationof exponentials.

APPROXIMATE CLOSED-FORM SOLUTION

The techniques of inversion, described so far, requirethe direct use of the statistical distribution p(n). Wenow propose a method involving the moments of p(i).In -view of the basic relation (1), the factorial moments

(1,k) dn(n-l)* ... (n-k+l)p(n)n=O

of p(n) and the moments

W=JWkP(W)dW

of the probability density P (W) are equal to each other:

(10 [k] ) = ye. (20)

Hence, knowledge of p(n) allows the determination ofthe moments of P(W); with such information, the in-version problem can be treated as a curve-fitting prob-lem. That is, given a set of moments Pe satisfyingEq. (21), the probability density P(W) whose Poissontransform p(n) fits the experimental photocount datacan be constructed.

A closely related problem is encountered in statistics,where a closed-form expression is often wanted to fitmeasured statistical distributions. It is well known insuch cases that a large variety of distributions, satis-fying some general conditions, can be fitted adequatelyby the Pearson curves, defined by the differentialequation 2 ,12 2

df x-A 1dx ao+ aix+ a2x2-J()

(21)

The coefficients aj and bj, appearing in Eq. (17), arethose which optimize the fit. The Bose-Einstein sta-tistical distribution, Eq. (7), is a special case of such

12 G. Lachs, Phys. Rev. 138, B1012 (1965).20 R. S. Glauber, in Physics of Quantum Electronics, P. L. Kelley,

B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co.,New York, 1965), p. 788.

The method, described extensively in Refs. 21 and 22,is quite straightforward and involves relating the

21 See, for example, M. G. Kendall and A. Stuart, The AdvancedTheory of Statistics (Hafner Publ. Co., New York, 1958), Vol. 1,p. 148.

22 W. P. Elderton, Frequency Curves and Correlation (CambridgeUniv. Press, London, 1938).

October 1967 1203

GABRIEL BEtDARD

parameters A, a0, a1, and a 2 to the moments of the dis-tribution in question.* Even though the inversion problem does not consistof fitting a given experimental distribution, but ratherof constructing its inverse Poisson transform, similartechniques can be applied in a slightly different way.Let P(W) be the required approximation expression forthe actual probability density of IV, and let k(n) be itsPoisson transform,

5(n>= f exp (- W)P(I'V)dW. (22)o lo

First, assume that P(W) obeys the multiparameterdifferential equation

dP N-= [(W-A)/L ajW-11VP()dW P=0

TABLE I. Parameters of the calculated probability densityfor laser below threshold.

(It) A2' AB' A 4 A ao al a2

2.1268 3,5401- 13.3024" 112.1604a 0.2269 0.4835 -1.9118 0.0028

p' are the moments about the mean (1) of ff(W).

p(n). The goodness of the fit will depend mainly on theaccuracy with which the factorial moments (0 1]) canbe determined from the experimental data. A measureof the goodness of the approximate solution could be,for instance, the quantity chi squared, defined in termsof the measured photocount probabilities p(n) asfollows:

X2 ==DIJ jp (k)-P (k) I ,

k (k)(23)

with ao#0. This last equation describes distributionsmore general than the Pearson distributions, defined byEq. (21). Such a generalization is expected to be neces-sary in view of providing a systematic method of in-version for a large variety of probabilities of W en-countered in photocounting statistics.

Next, determine the parameters A and a1 (j=O,1, - * ,N) from the moments of the experimental p(n) inthe following way. Rewrite Eq. (23) in the form

iv d-PEL ajW-,]-= (W-A )P(V),

5=0 dw(24)

multiply each side of Eq. (24) by W", and integrate overthe range (0, oc) to obtain the set of simultaneous linearequations

N

L (kA i)aj(nlk+i-J )j (-,rk+IJ))5=0

-A (a011 )-aoP(0)Sko= 0, (25)

(k=0,1,. -*).

Here, 6kD is the Kronecker delta and the factorialmoment (nf[lt) satisfies Eq. (20). In deriving Eq. (25),we assumed that the probability density of W is not toosingular at the origin and vanishes sufficiently rapidlyat infinity. The quantity P(0) is the value of P(W) atthe origin and is obtained from the experimental databy application of Eq. (5). The set of linear equations(25) can then be solved for the parameters A and a1,(j= 0, 1, *- -,).

Upon substitution of these parameters into Eq. (23),we can determine P(W) by straightforward integration.At this point, owing to the uniqueness of the Poissontransform, we can take P(W) to be a good approxima-tion to the actual probability density P(W) of W, if itsassociated transform (in), calculated from Eq. (22),fits adequately the experimental photocount distribution

(26)

here a is the total number of samples constituting theexperimental photocount distribution p (n).

We should mention at this point that the set of equa-tions (25) can be solved even if the quantity P(0) can-not be evaluated with reasonable accuracy. Recallingthat .P(0) appears only in the first equation, corre-sponding to kl= 0, all we need to do is to consider the setof equations k= 1, 2, - - *, M instead of the set k= 0, 1, 2,M-1, when P(0) cannot be evaluated. Such a pro-cedure will, however, require knowledge of still-higher-order moments of p (it).

APPLICATION

We now illustrate this last method by analyzing someexperimental results. We consider one of the countingdistributions obtained experimentally by Freed andHaus7 with a He-Ne laser (6328 A) operating below thethreshold of oscillation. The counting time interval wasT= 10-0 sec, the bandwidth tA260 Hz, and the totalnumber N of samples was 15 991. The experimentaldata allow determination of at most the first four fac-torial moments of p(n) with good accuracy. The set oflinear equations (25) can then be solved for the parame-ters A, ao, a1, and a2 : In the present case, the quantityP(0), evaluated by use of Eq. (5), is approximatelyzero. The values of these parameters appear in Table Ialong with the first few moments of P(IV).

The corresponding P(W), evaluated from Eq. (23),can be expressed in the form

P(W)=Z~eaexp(-Z)/ar(,ya), O<Z<co (27)=0 Z<O,

with the reduced variable

A=Z (TT7- (W) + 2'T) ,

where

a= 2 b62'/43,a=A2'Y-y-

1204 V'olun-e 57

ANALYSIS OF LIGHT FLUCTUATIONS

TABLE II. Number of samples S(n) vs photoelectron count nfor laser below threshold.

S Soe(n Se(fl-)b S.(fl)O

0 4329 4344 43181 3887 3897 38992 2729 2676 26773 1727 1759 17634 1128 1149 11555 739 751 7566 490 490 4957 325 320 3248 223 209 2129 154 136 13910 87 89 9111 53 58 5912 38 38 3913 26 25 2614 23 16 1715 13 11 11

a S(n) is the number of observed samples (after Freed and Haus7).bSc () is the calculated uamber of samples using the present approxi-

mate method.CS. (n) is the calculated number of samples using the model of Freed and

Hia Is.1

and I' is the Ith moment about the mean of the densityP(W). Substitution of Eq. (27) into Eq. (22) yields thePoisson transform

expQ-y)

ait P(.YesI+r1) y-T

X:I ( (28)r-\ Byaf ya) (yf+l

withy= (TV)-A2"y.

In the present case, the values of the parameters -yand a are, respectively, 0.5322 and 0.0054. We thenevaluate 7 (n) with the help of Eq. (28) and compare itwith the experimental p(n). Table II provides such acomparison through the quantities S`== 91p(n) andSO-=Zfp(n), where DIh==15 991 is the total number ofsamples. The agreement between S, and So is seen to bequite satisfactory. We conclude that P(W), given byEq. (27), is a good approximation to the actual proba-bility density P(W) for the light intensity. The densityP(W) has been evaluated and is shown in Fig. 1.

In Table III, the probability density .P(W) pre-dicted by the present analysis is compared with theprobability density

PM1(T/F)= (Ilyer)) expE-(W-(nb>)1(nII)I, > (l)=0, W< (Itb< ,

(29)

proposed by Freed and Haus7 ; the values of theparameters appearing in Eq. (29) are (nT)=)1.8955 and(fn)=0.2460. The essential difference between P(W)and PPM(W), as shown by Table III, is that .P(W), pre-dicted by the present analysis, starts at zero, rises to itsmaximum value, and decays exponentially, while

P 1w)

0 2 4 6 8 10

FIG. 1. Probability density for the intensity of a single-modelaser operating below the threshold of oscillation, correspondingto a typical photocount distribution.

PM1 (W) is a shifted exponentially decaying distributioncurve. Referring again to Table II, we see that the cal-culated quantities SC(n) and Sm(n) represent, equallywell, reasonable fits to the experimental data So(n),taking into account the statistical fluctuations of So(n)due to the finite number of observed samples. Thequantity K2, defined by Eq. (26) and evaluated withthe data of Table 1,1 is 6.2 for the present analysis com-pared to 5.8 for the model of Freed and Haus. However,we should stress at this point that, unlike in the modelof Freed and Haus, the present probability densityP(W) has been constructed directly from the experi-mental photocount distribution without invoking anyparticular statistical model to describe the behavior ofthe fluctuating light beam.

TABLE III. Comparison of the probability density P(W) pre-dicted by the present analysis and the probability density Pm (W)proposed by Freed and Haus' in connection with the photocountdata under consideration.

W P (W) PM1(w)

0.242 0 00.244 0.5218 00.246 0.5226 0.52750.250 0.5227 0.52640.300 0.5119 0.51270.400 0.4868 0.48630.500 0.4622 0.46141.000 0.3553 0.35442.000 0.2092 0.20914.000 0.0723 0.07286.000 0.0249 0.02538.000 0.0086 0.0088

10.000 0.0030 0.0031

October 1967 12905

1206 GABRIEL BEDARD

CONCLUSION

Inversion of a given photocount distribution, meas-ured for counting time intervals shorter than the co-herence time, yields the probability density P(I) forthe light intensity I. At such counting time intervals,however, the effect of the dead-time of the counter isnot negligible, unless the experiments are performed atlow intensity, corresponding to a small average numberof photocounts in the counting time interval.1 0"6

The approximate solution in terms of the multi-parameter expression (23) has interesting properties.First, it is always positive, as should be expected of aprobability density. Furthermore, as demonstrated by atypical example, only a small number of parametersmight be needed to obtain a good approximate solution.This is very useful, considering that, in practice, onlythe first few factorial moments of the experimentalphotocount distribution can be evaluated with reasona-ble accuracy. This limitation, however, is not too criti-cal. We could make use of unreliable values of thehigher-order factorial moments to determine a largernumber of parameters, by regarding them as initialvalues in an optimization procedure leading to the best-fit solution. Finally, the inversion is performed withoutinvoking any particular statistical model to describethe field.

ACKNOWLEDGMENTS

I am indebted to Professor Emil Wolf for stimulatingdiscussions and critical comments in connection withthe present work. I also wish to acknowledge the co-operation of C. Freed and H. A. Haus, who kindly pro-

vided the experimental photocount data used to illus-trate the theory.

APPENDIX: INVERSE OF THE POISSONTRANSFORM IN TERMS OF LAGUERRE

POLYNOMIALS

Let us formally expand P (W) in the form

00P(W) = E CkLk(W),

L=0(Al)

where Lk (W) is the Laguerre polynomial of degree k,

k (k (-WV)-0 r!

(A2)

satisfying the orthonormality relation

I exp (- W)L n(W)L,1 (W) = Armsao

(A3)

It follows from (Al) and (A3) that the expansion co-efficients Ck are given by

c= j exp(- W)P(FV)Lk(W)dW. (A4)

From Eqs. (1), (A2), and (A4), the expansion co-efficients are found to be given by the formula

11 kC = E (- l)r p (r).

r=O(A5)

Kenneth N. Ogle, rillyer Medalist, and Clarence H. Graham,chairman, Tillyer Medal committee, at Columbus meeting.

1206 Volume 57