JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Spectrum of Detected Light: A Particle Model

SHERMAN KARP

Douglas Aircraft Comnpany, Inc., Missile & Space Systems Division, Santa Monica, California 90406

(Received 24 October 1964)

The power-spectrum density of light at the output of a photodetector is determined using a particle model.

The model is the same as that introduced by Rice in determining the spectrum of shot noise. A time-varying

Poisson statistic is used to represent the intensity variations. The conditions are shown under which the

intensity variations of the detected light are representative of the intensity variations of the light in both

time and frequency. An intrinsic signal-to-noise ratio of photon to photoelectron conversion is defined.

Spectrum alternations due to delay are also determined. Finally, the mixing of light from two or more light

sources is considered and the power-spectrum density determined.

INTRODUCTION

CLASSICALLY, photon emissions (from coherentlight) are considered statistically independent,

obeying nonvarying Poisson statistics.1 In an experi-ment by Hanbury Brown and Twiss,2 a deviation fromclassical counting (due to intensity variations) wasobserved. This has stimulated interest in obtaining theautocorrelation function and power-spectrum densityof the detected light.1 In this paper an attempt is madeto obtain the power-spectrum density while preservingthe identity of the particle nature of light and to showwhen this is representative of the light itself (the auto-correlation function is the inverse Fourier transformof the power-spectrum density and has not beencomputed).

PHOTO DETECTOR

We can consider the quantum efficiency of a photo-detector n as the probability of emitting a photoelectronafter a photon enters (this assumes no saturation). Weneed to know the requirements for making the statisticsof the output photoelectrons "reasonably similar" tothe statistics of the input photons. Consider a timeinterval AT. Suppose K photons arrive. The probabilitythat j(j1 K) photoelectrons will be emitted is

P(j)= ( ),qj(1-7q)K-j.

The average number of photoelectrons emitted isbK

E[j] = Z jP (j) =

j=o

with variance

K K

Var[j]=7 j 2 P(j)-a jP(j)]2 ==,q(1-71)K.j=o j=o

(1)

the binomial distribution (or to approximate it by anormal distribution) and find the probability that j isin some narrow range. A more informative alternativewould be to define a signal-to-noise ratio as the ratio ofthe square of the mean to the variance, or

(4)

If K is the average number of photons that can arrivein the interval AT (this is a function of which intervalis chosen), then the probability of K arriving is P(K),3

(5)

Therefore, the expected signal-to-noise ratio is

E[(S/N)K]= q]Di/(I-rt)]E[K]= =E1/(1--n)]K. (6)

K can be written as n5(t)AT, where n5 (t) is the photonrate as a function of time and the signal-to-noise ratiocan be written as [i1 /(l-7)]n5 (t)AT. If we were onlyinterested in determining the average statistics, thenclearly we would make AT as large as possible. However,we also want the photon-electron rate to "follow" thephoton rate as the latter varies. Therefore, AT cannotbe considered larger than the variations of n5(t) whencomputing the signal-to-noise ratio, or these variationswill be lost even though the average statistics will becorrect. The variations of n8 (t) are related to its band-width B, hence AT must be of the form

AT= 1/kB

(to satisfy the Nyquist criterion, k= 2). Looked atanother way, consider iit (t) existing on the internal

(2) (- T/2, T/2) and bandlimited to B. This function canbe represented as a point in orthogonal space of 2BTdimensions4' 5 (samples)-2BT degrees of freedom. If thesamples are spaced evenly in time, AT seconds apart,

,^\ each sample satisfies(I5)

This can be interpreted in one of two ways. The mostrigorous way would be to set a confidence interval for

1 L. Mandel, Progress in Optics, edited by E. Wolf (John Wiley& Sons, Inc., New York, 1963).

2 R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London)A242, 300 (1957).

Hence2BAT= 1.

AT= 2B-'.

3 S. 0. Rice, Bell System Tech. J. 23, 282 (1944).4 R. M. Fano, Transmission of Information (John Wiley &

Sons, Inc., and Technology Press, New York, 1961).6 G. Sansone, Ortliogonal Functions (Interscience, John Wiley &

Sons, Inc., New York, 1959).

686

JUNE 1965VOLUME 55, NUMBER 6

(SIN) K = 77-TCI (I - 77) -

P(K)=E(10K1K!]e-j?.

June 1965 SPECTRUM OF DETECTED LIGHT: A PARTICLE MODEL

We can now define an average intrinsic signal-to-noiseratio [v/(1-E)](ii/kB) [iS is the average of n8 (l)]that can be measured at the photodetector output,which is a measure of how well the time-dependentphotoelectron statistics represent the time-dependentphoton statistics. The limit on B imposed by the photo-detector is determined by the time required for photonto photoelectron conversion [when the average delayis of the same order as the variations in n8(t), there isno way to differentiate between the two-see examplelater in text]. However, since this time is "significantlyless than 10-10 sec,"6 bandwidths up to 1010 cps shouldbe easily reproduced.

MODEL

If each photon produces a photoelectron and eachphotoelectron produces a current ef(t) where e= 1.6X 10-'1 C and f(t) is the impulse response of the photo-detector, then the total current is

(7)

where the tm1's are statistically distributed in time, withrate n,(t). Consider the interval (-T, T) with anaverage number of events K. Let Kp(t)dt be the prob-ability than an event occurs in the interval (t, t+dt)with

J p(t)dt= 1-T

[if there are no variations in that), p(t)=2T' andKp(t)dt=ftfdt]. Rice3 has shown that if

S(w) = f I(t)e-"'t di,

then

TAP Ia ; - OUTPUT

IN PUT --

CkP(t) dt)/

TAP |(I -0

FIG. 1. Model of a photodetector.

Finally, the power-spectrum density of I(t), cb(w), is

b (co) lim ( S (co)I 2/2T)av= e2

ni I h (c) I 2

T- n:eb()+qe(9] 11

where 3(co) is an impulse function, K/2T is ii,, and thepower-spectrum density of variation e (t) is7

''(co)= lim [ f e(t)e-jtdt 2T. (12)

It is apparent that if 7 were equal to one, the statisticsof the photoelectrons would be an exact representationof the statistics of the photons. However, since this isnot the case, it is of interest to determine what thedifferences are. We have already shown that out-put signal as a function of time is degraded { S/N=[r/(1-n7)]nis,/2B}. Therefore, the only remainingquestion is how the spectrum is affected. A model ofthe physical process in a photodetector is shown inFig. 1. Photons enter at an average rate in Arandom switch selects tap (1) with probability q andtap (2) with probability (1-7). The output of tap (2)is discarded. The probability of having an event-in theinterval (1, t+dt) at the output of tap (1) is Kp(t)dtintersection - or,

i7Kp (t)dt.

(I S(w) 12)av=e 2KI OR(I ) 2[)1±2 p(I)e wtd11 2]

where

h/(w) = X f (t)e-ij tdt.

Without loss of generality, we can represent p(t) as

p(t) = [1+e(t)]/2T,such that

We now have

T

liT

(8)

(9)

e(t)d1= 0.

(IS(w) )av =e2 K h(w) 12 (1+ (K/2T)

| f [1+e(t)]ej-stdt } 2T). (10)

6 A. T. Forrester, R. A. Gudmundsen, and P. D. Johnson, Phys.Rev. 99, 1961 (1955).

(13)

Finally, the pulses go through a filter h(co) whoseimpulse response is f (t). Therefore, the power-spectrumdensity has the same form as in the previous problemwith 1s, replaced by cane or

b~a,)=e~rJh(w) 2( ( (14)

As the sample qi - 0, the term niis,,2wa(c)+ +e(w)}<<

and the result approaches. the "photon" noise,

e27r;, I h (co) 12.

[A more refined version of the photodetector (Fig. 1)would include a random delay D denoted by the dashedbox.]

Judged as a communications receiver, the photo-detector has a noise figure8 equal to 1/7. Unwanted

I If (l) is a statistical variation, an additional average over theensemble of e(t) is required to obtain this result. However, nodistinction is made here between P(w) and ('(w))av.

8 W. B. Davenport, Jr., and W. L. Root, An Introduction to theTheory of Random Signals and Noise (McGraw-Hill Book Co.,Inc., New York, 1958).

687

I (1) = 57' ef (I - 11.) I

SHERMAN KARP

light (noise) can be described in a similar manner. Infact, if the additive noise it(l) (with average rate it,) isPoisson distributed,

y(t) ==I ()+n(t),

=0, (T) = ElT([I ))+1I (+ )II (YOt (12)])( v

= 0 ,(r) + .(Tr) +2 fz(t) 1(1)

= Op, (T)+c0.(Sr) +2e 2 (qii.) Oflf 8) ,

(b, (X) = -1)(w) + ( (w) +47rei~i,6nn (co).or

(15)

(16)

Ignoring the dc terms we have

1y(o,)==e2 1 Ii(.) 1 2E(,7j8)2be(W) (17)

DELAY

We now show that if the photons are subjected to asystematic delay before detection, the effect is observedto alter only the portion of the power-spectrum densityrepresenting the statistical dependence of the photons.The most important sources of delay come from at-mospheric turbulence, photon to photoelectron conver-sion, and the extended size of the source. These can allbe accounted for at the output of the photodetectorby rewriting I(1) as

(18)I (t) = F, ef (I- t. 5 Xi..),nM t

where

picOw)= pi(xi)c-i~xid,~

and pji~) is the probability density of the variablexi- As an example, consider a single delay caused bythe photon to photoelectron conversion. It is usuallyassumed that

p(x) = -le7xIT 0<X°

from which

4(co) - e lAl(c,) I 2{n'1

+ (?i1n)246Q')[1± ((wT)2]-1}. (20)

From this we can see that the cutoff frequency (-3 dB)would be wco= T'. As a second example, consider alight source being equally split into two different paths.If the time difference in the paths is To we can writep () as

p (X1=2 ()+ 12 (XT-From this

(.) =e e2lAl(@) 12 En Ft+ (,n )

214 (c) cos2 wro]. (21)

TWO LIGHT SOURCES

Consider now the mixing of light from two sources

1 (i)=Z ef(t-tm),I n

Ft =a ef (I -t)

where the Xi are random variables, each representing adifferent source of delay. If the xi are independent, thepower-spectrum density can be written in a compactform as (see Appendix),

tD (X) = e2 1| (W) | 2[nS+ ( W1I1)24( 171 I Pi(X,) 12], (19)

i

I S(M) lim I()+Sr,(-)][S1l*(W)+S7,*W]lim = im

(T+@2() Tim 2 -TT a22T

T (v) K(n) 112 As

X [ee-iCase I: The variations of the two sources are independent.

p(Iinxtnz) = pi (hn)p2 (In) = [1 +el (im)]1/2 TE1 + e2 (In)]/2T,

where

DI(w) =41I(w)+Ib 1 2(w+)+ I rn e2 Il( ) 2 E E E E 2Re[(b12(c-)],T-K (in) K (n) (2T) 2 -1 It

tI12( )= HM l[ + El (Im)]e jwtrndImjj E+2(t11)]e iJtfltn]j | 2T

= 2irS (cX)+ lim [F,, (w)F2 2* (cw)]/2 T,

and

Now

S Il (co) = eel (X) 57 e- jl t1"'

S12 (w)=ehl(co) Aejt1S)

Sr2(CO) = elt (w) 57 e@-"'

tmt) T e (t'tm)

-T J-T _2 T

:(tm-t.)+e-j@(t-1-t)]dtmndt1,, (22)

(23)

688 Vol. 55

June1965 SPECTRUM OF DETECTED LIGHT: A PARTICLE MODEL

andT

- Twith

(FE i(co))avei= 0

Case II: The statistics of the light sources are also independent.

P[K(m),K(n)]= P[K(m)]P[K(n)],c1(w) = Ill(w)+d2 (w)Je 2 h(.) 2 (Xi~ m) (n71X)2 Re[Ebl2 (w)].

N LIGHT SOURCES

In general, the sum of N independent light sourcesIi(t) yields

4 (co)= Jim -~ F_ (Sri (w)SIj* (W))avT= 1 2T i=i i

N N N

=j a|inhjReE41j(()1.

i=l i= j =1i 5;j .

(25)CONCLUSIONS

A spectral analysis of detected light has been demon-strated by use of a particle model. The equivocationintroduced by the photodetector has also been discussedwith regard to both the analog process and its power-spectrum representation. The conditions have beenexhibited under which we can associate the statisticsand spectral properties of the photoelectrons with thoseof the photons. For the analog process, an intrinsicsignal-to-noise ratio for photon to photoelectron conver-sion has been defined which determines the "quality"of reproduction. The power spectrum is shown to bedegraded only in its definition and not in its generalcharacteristics. This degradation can be succinctlyexpressed as the noise figure of the photodetector or theratio of the signal-power to noise-power ratios underperfect (X1= 1) and imperfect conditions. In all spectralcalculations a clear separation is shown between thecorrelated and uncorrelated parts. This is particularlynoted in the calculation of the effects of statistical delay.As expected, the delay imposes constraints on thecorrelated portion of the spectrum. For a rule of thumb,the bandwidth constraint is proportional to the inverseof the average delay. Finally, the power spectrum forthe sum of light beams has been obtained.

APPENDIX

We are interested in obtaining the power-spectrumdensity of a random function of time I (t). If

rS

then it can be shown3 that the power-spectrum densityc)(W) is

4I(w)= Jim (IS(w)1 2 2T)aV2T-wc

Now,

I (t) = 5, m (t-E im)m i

S(c) = f I(t)eijwtdt

= ek(co) E- e-jit1- exp (- jx E_ xsm))mi

(X) = f(t)e-i' t dt.

where

Now

ISK(-)I 2=e2 Ij(.)I2 E e-Z -mn n

xexpl-j-M (i7-Xn3;

= e4I (w) { 2K+K(K- 1)e-jw(11-2)

Xexpf-jwE xix')}i

If P(K) is the probability of K arrivals (with mean K),Kp (t)dt is the probability of an event (associated with t)occurring in the interval (t,t+dt) (see Rice3 ), and pi(xi) is the probability density of the event xi, then

lim (IS(c) 2 /2T)= e I 4(c) I2[Xjn+ (rns)2

X {2?r6()+4(C)} II I Pi ()2],

wherewn, = K/2T, K=E[K],

1 rT 2

If (c)-= lim - E(t)e-jltdt2T-- 2T JT

and

Pi(X)= pj(xj)e-jllXidxi.