JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Statistical properties of ensembles of classical wave packets

Sherman Karp

Naval Electronics Laboratory Center, San Diego, California 92152(Received 22 August 1974)

Over the past two decades there has been a renewed interest in the properties of light and itsinteractions with matter. In particular, the phenomenon of mixing on the surface of a quantumdetector, first performed by Forrester and Gudmundsen, has had significant impact on the treatmentof the interaction of light with matter. Mandel and Wolf have interpreted optical mixing of twofields in terms of wave-packet interactions. The wave packets of the fields were allowed to add in aclassical manner, without any interaction between photons. In this paper, we consider the extensionto an ensemble of statistical wave packets. This is accomplished by considering the wave-packetensemble as a generalized shot-noise process, and generating statistical properties of the ensemble.The first-order probability density is obtained for the envelope of an ensemble in which each memberhas uniform and independently distributed phase. This is then used to compute the resultingphotoelectron-counting distribution. The results can be shown to be somewhat different thanwould arise from a gaussian field, and a statistical test is derived to compare the two.

Index Headings: Fluctuations of light; Detection.

In order to explain the phenomenon of mixing on the sur-face of a quantum detector, using quantum mechanics,Mandel and Wolf' employed a procedure that, in essence,is to allow the wave packets associated with photons oftwo different frequencies to add in a classical manner,without any interaction. We will address three pointsthat arise in connection with this approach. First, if wecan add two wave packets as though they behaved classi-cally, we should formalize this to include statistical en-sembles of wave packets with phase. Second, if thisprocedure is valid for wave packets of different frequen-cies, it should be valid for wave packets that have thesame frequency. Finally, if wave packets must betreated classically, we should consider the ramificationsof their being classical.

We will consider an ensemble of wave packets as ashot-noise process. The theory associated with thistechnique was first proposed by Rice, 2 who used it torepresent discrete current distributions. It was ex-tended to a narrow-band-noise process with randomphase by Furutsu and Ishida. 3 Use of a wave-packet en-semble to represent a known narrow-band signal was in-troduced by Karp, Gagliardi, and Reed.4

In all these approaches, first- and second-order sta-tistics were computed, as well as the spectral proper-ties. This is accomplished by using the general expres-sion for the characteristic function computed by Rice2

and inverting it for specific conditions. We will consid-er a wave-packet ensemble of the form

(1)

where k, ti, ai, and pi are all random parameters. Byextending the results of Furutsu and Ishida, we willcompute the envelope distribution of the process s(t)when 4I i is uniformly distributed, and relate it to theRayleigh density that would occur for a gaussian process.We will assume such a field to be incident on a quantumdetector and compute both the counting distribution andits characteristic function. We will derive a test thatuses the resulting photoelectron count to compare the

computed field with a gaussian field.

I. FIRST-ORDER DENSITY OF s(t)

Using the basic approach of Rice, and standard tech-niques for Cartesian to cylindrical-coordinate conver-sion, Furutsu and Ishida showed that the probability den-sity of the envelope process s(t), when Oi is uniformlydistributed, can be expressed as

(2)Pit (R) = f (XR)J0 (XR)f(X, T) dX,0

where

f(X, T) = exp [kr Jf dx f da p%(x)Pa(a)

X{Jo(Xr(t- x; a))- 1-] - (3)

In this expression, Rt in boldface letters represents therandom variable that describes the envelope process atany time t; R denotes the outcome. Similarly, x is arandom variable that describes the time distribution ofthe wave packet in the interval (0, T) and a allows forshape variations of r(t). The assumption underlying thissolution is that the random variable k (kEk) can be takento be a Poisson random variable in the interval (0, T),described by

P() =- exp[- T]k!

(4)

where kT is the expected value. Thus 1FT is the average

number of wave packets in the (0, T) interval. The gen-erality of this assumption is described elsewhere, 4-6

but it is a common and accurate assumption for prob-lems of this type. The cumulative distribution of theenvelope was also computed to be3

FRt(R) = PRt (R) dR'= R J,(XR)f(G, T) dX . (5)

Let us first consider the conditions under which thecentral-limit theorem applies. This can easily be seento be when

421

APRIL 1975VOLUME 65, NUMBER 4

k

s(t) = R,1i=1 r(t- xi; ai)eJ(wit+0i1

SHERMAN KARP

Jo(xr[ (t- x); a]) L-1 - X(Xr[(t- x); a])2 X

for then

f(XT) = exp[- TTEX2],

where

E = | dx f dapx(x)P,(a) (r2 [(t - x); a]),0 a

and

PR, (R) = Rf rXJ0 (XR) exp[- FTEX 2] dX

(6)

(7)

= R E) [-R2/4)TE], (8)

which is the Rayleigh density with parameter 2 kTE, thepower in each orthogonal component of s(t). This re-sult now gives us a little insight into how the generalsolution should be obtained. Thus, if we expandf(X, T)as

f(, T) = exp [Z ! )2X

where

Vi = kTfT dx f da Px(x)Pa(a) r2 [(t- x); a]1I0 a L J

then

f(AX, T) = ep[ - X2krE+ i (- 1)2

(9)

we obtain our main result,

R FR 2 1[ R2 I E E C- L,, /pit,(R) = REexpLjE (En4E (17)

where Lj(x) is the Laguerre polynomial. We can alsowrite this as

t [ 2 ]=E (TTE)' (2)to show the dependence on lTE.

II. PHOTOELECTRON COUNTING

If we were to develop a statistical test to determinewhether the radiation from a purely random thermalsource (blackbody) came from the probability densityp,(R) or a gaussian density (Rayleigh envelope), wewould construct a likelihood function9 and compare it toan appropriate threshold. Clearly, the likelihood ratioA(R), the ratio of the two probability densities, becomes

(18)A(R) = Z nEfL (R 2TE

Generally, tests of this nature are performed at high-er frequencies, where the particulate nature of radiation

(10) is more pronounced. Considering wide-band detectors,Mandel and Wolf' showed that the photoelectron-countdistribution in a time interval less than the inversebandwidth of the radiation can be expressed as

(11)

and

f(Ax, T) = exp[- X2kTE][J +(Z)+ (Z)+* ] * (12)

If we now collect terms with the same power in X2, andset

6 = (W) i>2

=0, j=1,

then

f(A, T) =exp [- WT EA2] [+ 2! ) +(4) + (2Xs;A+ *--]

=exp[-)TEX 2 ] Cnx 2n, C0 =1, C1 =0,n=O

with7

Cn =, .. ! 6n i=nn n ***n

Now, using the relationships 8

0Oj xI exp[- a2x 2]J"(fx) dx

1Tr((v+ g + 1)/2) ( 1.+v+l 1 -;2 av+I+lr(v \ 2 ; v+1;

and

j(a; b; z)= e'jF(b-a;b;-z)

(b). Lbb ze

(13)

(14)

Pk(k) Ptp(R)dR() T exp[ aR2])

Inserting Eq. (2) yields

Pk(k) = Xf(A, T) exp[- X 2/4a]L,(X 2/4 a) dX .

Inserting Eq. (17) yields

Pk(k) 1+ 4UTTE (1 4a1fTE\

X:( Cng,2Fl(-n;k+1;1;+4e-E'n= OTY + 4c4dTEI

2F1(a; b; c; x) =] (a)(b)n x"

where

(a)0=1(a+) * * * (a+n- 1)ao =I1 .

(19)

(20)

(21)

(22)

Because 2F1(0; b; c; x)=1, we see that the n=0 term hasa Bose-Einstein distribution, i. e.,

___1_ 4 4 ak- EE n=PkA) = 1+ 4U)FE(1 + 4 ak-TE/

(15)

(23)

corresponding to the Rayleigh portion of the envelopedistribution. The moment-generation function,

(24)(s)= E pk(k)e,h=o

(16) can also be obtained. Inserting the counting distributionin Eq. (21) yields

422 Vol. 6 5

STATISTICAL PROPERTIES OF WAVE PACKETS

Notic 1 + 4 aWE(1 - es) En!C c(1 +4 kE(l 's()

n=O (25)

Notice again that the n= 0 term corresponds to a Bose-Einstein counting distribution or

(s) -(0III. STATISTICAL TESTS

The most-efficient way to test between the count dis-tribution computed here and a Bose-Einstein distributionwould be to calculate the likelihood function and to com-

pare it to an appropriate threshold. This ratio can beshown to be

A(k) = (Cn )n 2F1 (; k+1; 1; 1+24akE) )

but requires a knowledge of the wave-packet envelope

r(t).

Having the characteristic function, however, allowsus to set up another test, less efficient, but one that re-quires no knowledge of r(t). First, compute the firsttwo moments. Because C0 = 1, and Cl = 0, the first mo-

ment is the same as for the Bose-Einstein distribution,and must be so because it is proportional to the averageenergy. The n = 2 term yields an additional contribution

to the second moment of the Bose-Einstein distribution,m2 . Hence the variance is

ma2_m 2= 4 a)TE(l + 4 aFkTE) + 2(4!) 2Y2 (28)

where

ml = 4 akTE (29)

is the average photoelectron count. In the expressionfor m2 - Mil, 4 kaTE represents the quantizing effect ofthe detector. Recall that

y2 kTfO p,(x) dx{ Pa(a) da [x), (30)

where 2(4a)2y2 represents the contribution due to dis-creteness in the envelope. Using the Schwarz inequality

[f H(y)G(y) dy] < f [H(Y)]2 dy f [G(Y)]

2 dy

and setting

G = [ P1 (x)p.(a)] (31)

H= [px(x)pa(a)]112 4 2 [(t- x); a]

we find that

y2 2 kTE2 (32)

and m2 - m 2 can be written as

mi2 - i=4alTE (1+ hf E) + (4 alTE)2,

where aj is the quantum efficiency.

a = 7T/hf,

the distance, D, from the source. Consequently,

2kTE- 1D 2 ,

which means that either

kT 1/D2, E = const,

or

E- 1/D 2, kT = const .

Apr. 1975

It is generally accepted that Eq. (35) is correct. Thatis, energy is transferred from the field to the detectorvia discrete units of energy, each having an energy hf.It is the density of these photons which decreases withdistance from the source. For this case we would have2.ET=hf and

ma - M' = 4akkTE[1 +471 + (4aak E)2 . (37)

On the other hand, if Eq. (36) were correct, that is if thedetector extracts energy from the resultant sum of sev-eral wave packets, then TT is fixed at the source and

2ET- hf /) ) .srource)

(38)

where abs is the solid angle subtended by the molecular-absorption cross section from the source and nsource isthe solid angle into which the source radiates. We couldcompare the two forms by computing the statistic

(m2 - 2ml2- Ml)/Ml - (39)

If the correct distribution was Bose-Einstein, Eq. (39)would be zero. If the correct distribution was the one

derived in this paper, Eq. (21), then the statistic in Eq.(39) would be equal to 471, if the condition in Eq. (35)

were correct, and equal to 477(nabs/nsource) if the condi-tion in Eq. (36) were correct.

IV. DISCUSSION

Although it is clear that the model presented here isunsubstantiated, the validity of Eq. (36) would lead to aninteresting interpretation of a photon. This interpreta-tion would be that of an interference effect that existsbecause of the overlap of a multitude of wave packets.This effect would be localized to a cross-sectional areaequal to the square of the wavelength and would exist fora duration comparable to the inverse bandwidth of a wavepacket. Because the transient irradiance would be pro-portional to the square of the number of overlappingwave packets, sufficient energy would exist in a smallarea to interact with the molecules that constitute thedetector. Also, being an interference effect, it wouldhave zero rest mass. The results of a two-slit experi-ment would also be satisfied because each wave packetwould diffract through both slits. The main differencewould be that the transient energy would not be quantized

in units of hf. That quantization would occur in the de-tection process. Instead, discreteness would occur inthe radiation field, due to the overlap of wave packets.This discreteness would be characterized by y2.

Recall that 2)FTE is the power in each of the quadra-ture components of the process Rt. Thus Eq. (29) showsthat the quantum detector is detecting the energy in thefield. This energy falls off inversely as the square of

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(35)

(36)

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SHERMAN KARP

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(1960).4S. Karp, R. M. Gagliardi and I. S. Reed, Proc. IEEE 56, 1704

(1968).5S. Karp and R. M. Gagliardi, IEEE Trans. IT-16, 142 (1970).6P. Frost, thesis (Stanford University, 1968). University Mi-

crofilm order No. 69-221.

7M. G. Kendall and A. Stuart, The Advanced Theory of Statis-tics, Vol. 1 (Griffin, London, 1963).

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Technical Council

ROBERT V. POLE (Chairman), IBM Corporation, Mon-terey and Cottle Roads, San Jose, California 95 193

ANTHONY J. DEMARIA (Vice Chainnan), United Air-craft Research Laboratories, East Hartford, Connecticut06108

The chairmen of the Technical Groups comprise (exofficio) the Technical Council.

Aeronautics and Space Optics-LLOYD G. MUNDIE,Rand Corporation, 1700 Main Street, Santa Monica, Cal-ifornia 90406

Atmospheric Optics-GILBERT N. PLASS, Department ofPhysics, Texas A & M University, College Station, Texas77843

Color-S. LEON GUTH, Department of Psychology, Indi-ana University, Bloomington, Indiana 47401

Information Processing, Holography, & Coherence-ADAMIKOZMA, Environmental Research Institute of Michigan,P. 0. Box 618, Ann Arbor, Michigan 48107

Lasers and Electro-Optics-WILLIAM B. BRIDGES, HughesResearch Laboratories, 3011 Malibu Canyon Road,Malibu, California 90265

Lens Design-THOMAS I. HARRIS, Optical Research As-sociates, 1774 North Sierra Madre Villa Avenue, Pasa-dena, California 91107

Optical Fabrication and Testing-FRANK COOKE, 66Summer Street, North Brookfield, Massachusetts 01535

Optical Materials-IRVING H. MALITSON, A-25 I PhysicsBuilding, National Bureau of Standards, Washington,D. C. 20234

Radiometry and Photometry-BRUCE W. STEINER, B-2 13Metrology Building, National Bureau of Standards, Wash-ington, D. C. 20234

Raman-JAMES E. GRIFFITHS, Bell Laboratories, MurrayHill, New Jersey 07974

Spectroscopy-JACK SUGAR, A-167 Physics Building, Na-tional Bureau of Standards, Washington, D. C. 20234

Thin Films & Interferometry-KENNETH M. BAIRD, Ap-plied Physics Division, National Research Council, Otta-wa KIA OSl, Canada

Vision-GERALD WESTHEIMER, Physiology-AnatomyDept., 2575 LSB, University of California, Berkeley,California 94720

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