JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Detectable Information in a Photon Beam*

ELLEN HISDAL

Institute of Physics, University of Oslo, Blindern, Oslo 3, Norway(Received 1 July 1966)

The information content in a thermal photon beam and in a single-mode maser beam is computed, takingaccount of the limited quantum efficiency of the detector, and extending the results to polychromaticradiation in the thermal case. Formulas for fluctuations of the measured quantity in a polychromaticthermal beam are found, both for a photoelectric and for a thermal type of detector. Numerical examples ofthe information content are given for three common thermal light sources, using the eye as detector. Forall three sources the information is found to be of the order of 10'-105 bits/sec for a sample of 1 cm' at 1 mdistance from the the eye.INDEx HEADINGS: Information theory; Maser; Detection.

1. INTRODUCTION

Ia previous paper1 we computed the information con-tent in a quasi-monochromatic, polarized photon

beam, referred to a detector array of a given resolution.Each component detector was assumed to be of thephotoelectric type with unit quantum efficiency. Theformulas found in Ref. 1 represent an upper limit to theinformation content in a quasi-monochromatic, polar-ized photon beam which cannot be exceeded by the useof any type of detector.

The only information loss considered in Ref. 1 wasdue to radiation fluctuations in the photon beam. Inthis paper we consider the additional information lossor "detector noise" due to a limited quantum efficiencya of the detectors. The problem of the informationcapacity of a radiation detector has previously beentreated by Toraldo di Francia2 and R. Clark Jones.3

Both authors consider the case of an additive detectornoise which is superposed on the signal. Here we con-sider only the effect of the limited quantum efficiencyof the detector, treating it directly as an integral partof the whole problem. Thereby we avoid the restrictionof Jones, that the detector noise must be independent ofthe signal. Also, instead of working with continuousquantities, the quantum nature of the radiation is takeninto account from the beginning. The same problemhas also been treated from the point of view ofelectrical communication theory by Gordon4 and byGoodwin and Bolgiano,5 the former having taken intoaccount the quantum nature of the radiation.

Fluctuations of the measured quantity are computedboth for a photoelectric type of detector and for a ther-mal detector, and the results are extended to the case ofpolychromatic beams. The "black-white" informationcontent in a polychromatic beam is computed only forthe case of a photoelectric detector.

* Work supported in part by the Norwegian Research Councilfor Science and the Humanities.

I E. Hisdal, J. Opt. Soc. Am. 55, 1446 (1965).2 G. Toraldo di Francia, Optica Acta 2, 5 (1955).3 R. Clark Jones, J. Opt. Soc. Am. 50, 1166 (1960).4J. P. Gordon, Quantum Electronics and Coherent Light (Aca-

demic Press, New York, 1964), p. 170.5 B. E. Goodwin and L. P. Bolgiano, Jr., Proc. IEEE 53, 1745

(1965).

We assume that the intensity of the information-carrying beam may vary not only in the lateral directionbut also with time. Each detector reading refers to atime interval I, and the readings are repeated at equaltime intervals t during a total observation time T. Weuse capital letters for quantities relating to the wholebeam, i.e., the beam which covers the whole detectorarray and which extends over the total time of observa-tion T. That part of the beam which impinges on onedetector during one measuring interval / is called amacrocell, to distinguish it from a microcell in phasespace. Quantities referring to one macrocell are denotedby small letters. The total number of macrocells in thebeam is given by

r=FQT/(fwl) (1.1)

for a tightly stacked detector array and no intervals be-between successive scanning periods; F and 5 are, re-spectively, the focal area and angular divergence for thetotal beam, with corresponding small letters for onemacrocell. In practical arrangements we usually haveQ=co.

The formulas derived for the information content arevalid for n>>1, where is is the number of photons permacrocell. For a thermal beam we have the additionalrestriction z>>1, z being the number of microcells permacrocell.

An increase of the measuring interval I results in a de-crease of the relative fluctuation and therefore in an in-crease of the information content per macrocell in thebeam. This result depends, however, on a measuringapparatus which counts the total number of pulses dur-ing the time t. If the measured quantity is a current intowhich the succession of pulses is converted, we must,according to the sampling theorem, replace I by a re-solving time 1/(2B), where B is the bandwidth of themeasuring apparatus.

When we come to practical examples, we also extendthe results to the case where the incident beam fromthe source is no longer focused on the sample, so that wecan apply the theory to a sample illuminated directlyby the sun or by an artificial illuminant. We also allowfor either specular or diffuse samples.

35

VOLUM E 5 7, NUM BER I JANUARY 1967

ELLEN HISDAL

The following is a list of the main symbols used in thispaper:

N-number of photons or pulsesF-focal area of beam

2-its (solid) angular divergenceT-time of observationZ-number of microcellsAll the above quantities refer to the whole beam. Correspond-ing lower case letters refer to one macrocell.At-frequency range of beamc-number of macrocells in the beama(v)-quantum efficiency of detector1-subscript for quantities referring to photons2-subscript for quantities referring to measured pulses. 2

goes over into 1 for a= 1a(q)-root mean square deviation of q from its average( )-ensemble average or time average for stationary beami-subscript for incident beam in case of specular sample;

subscript for beam reflected from a perfect diffuse samplewith r = 1 in case of diffuse sample

r-subscript for reflected beamL,L'-radiance, and luminance of beam, respectivelyP,,-subscript which indicates that quantity refers to unit

frequency, and wavelength interval, respectivelyI g(0) I = L>/fo'L~dv-normalized spectral density of beam

2. FLUCTUATIONS OF THE MEASUREDQUANTITY

The mean square deviation o-2(n,) =((nI- (n ,) 2) of

the number of photons nl in a quasi-monochromaticbeam which is emitted by a thermal source during a timeI is given by':

a2(nj) = (n1)+ ((n,)2/z) = (n,)(1+di), (2.1)

whereZ=ZPZIzt (2.2)

is the number of phase-space cells contained in onemacrocell of the beam; z, is the number of phase-spacecells due to the state of polarization of the beam.According to Ref. 6, Eq. (6.11), z,=2/(1+P2 ) whereP is the degree of polarization of the beam. In this paperwe assume a completely polarized beam, i.e., P= 1,z,= 1; zi is the number of lateral cells per macrocell, andis given by'

Zz=(of/X2. (2.3)

The number zt of temporal cells was found to be'

Z,= tAy, (2.4)

where Av is the frequency range of the beam. In thequasi-monochromatic case treated in this section, Avmust be very small compared with the frequency v ofthe beam. When the spectrum is not rectangular, thevalue of z, is given by Eq. (3.1).

The degeneracy or occupation number in phase-spaceis defined as

di= (n,)/z, (2.5)

and is according to Eq. (2.1) equal to the fractional in-crease of the mean square fluctuation above the "coin-cidental" amount (ail) expected according to a pure par-

6 L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

tide picture. It is equal to the ratio between the meansquare fluctuation expected according to the Rayleigh-jeans radiation formula and the fluctuation expectedaccording to the Wien formula.

Let L be the radiance of the source and L, and Lx theradiance per unit frequency interval and per unit wave-length interval, respectively; L, and Lx satisfy therelations

L, is the energy carried by the beam per unit fcotAv. Wecan now write for the total average energy per macrocell:

fF.to L2. = (ne)hav. (2.7)

From Eqs. (2.2) to (2.4r) we have

Lvdv=LxdX and L,= (X2/c)Lx;

fwtAp = X2z,

and it follows from Eqs. (2.7) and (2.8) that

X2zL,= (n)/hv.

(2.6)

(2.8)

(2.9)

Eq. (2.9) gives us the following relation betweendi= (ni)/z and L,

d,= (X3 /hc)L,= (X'/hc2)Lx. (2.10)

Let us now assume that we have a photoemissive typeof detector with quantum efficiency a. We are interestedin the measured number of pulses n2 counted by thedetector, when we assume that each photoelectron givesrise to one count. The fluctuation in n2 can be found byassuming that, as far as the number of counts is con-cerned, the combination of a photon beam with a de-tector of quantum efficiency a is equivalent to the useof a filter with transmittance a in the beam, togetherwith a detector of unit efficiency. This assumption isconfirmed by the result of Purcell7 and Mandel.8' 9

The filter changes the thermal beam with photonnumber (u,) into a thermal beam with photon number

(n 2 ) = a(ni) (2.11)

and we must substitute n2 for nl in Eq. (2.1). This givesfor the mean square fluctuation in n2

r(n 2 ) = (n 2 )+ (a2(nl)2

/Z) = (n 2 )(1 +d 2 ), (2.12)

whered 2 = (n2)/z=0a(nl)/Z- (2.13)

We may call d 2 the degeneracy in the number of counts.For a beam from a well-stabilized maser which oscil-

lates in a single mode, we saw' that a2(n,) = (ni). Assum-ing again that the fluctuation of the number of pulses isthe same as the fluctuation of the number of photons ina maser beam of average photon number (n2)=a(nl),we obtain

2(n2) = (n2). (2.14)

7 E. M. Purcell, Nature 178, 1449 (1956).8 L. Mandel, in Progress in Optics II, E. Wolf, Ed. (North-

Holland Publishing Co., Amsterdam, 1963), p. 181.9 L. Mandel, E.C.G. Sudarshan, and E. Wolf, Proc. Phys. Soc.

(London) 84, 435 (1964).

Vol. 57

January1967 DETECTABLE INFORMATION IN A PHOTON BEAM

This result is rigorously derived by Mandel, Sudarshanand Wolf.9

The results for quasi-monochromatic beams which wehave found in this section are valid also for thermal de-tectors, like a thermocouple, under the assumption thatthe detector is so sensitive that it can measure each ab-sorbed photon; a is now the emissivity of the detector.

3. FLUCTUATIONS OF POLYCHROMATIC BEAMS

Using the theory of random gaussian processes,Purcell7 and Mandel (Ref. 8, p. 230) have derived Eq.(2.12) for quasi-monochromatic thermal beams and forone lateral cell. The requirement of one lateral cell is notexplicitly stated in their papers, but is implied in the re-quirement of lateral coherence. For t>>1/Av, the numberof temporal cells is, according to their result, given by

z=t/( IP(t) 1 2d)4/(f Ig1 2dv), (3.1)

where g(v) is the normalized power spectrum of thebeam, and p(r) is the normalized temporal coherencefunction of the beam. The last equality in (3.1) followsfrom Parseval's theorem, p(r) being the Fourier trans-form of g(Q).

In this section we derive the mean square fluctuationboth in the number of counts and in the energy absorbedby the detector for the case of a polychromatic thermalbeam and several lateral cells. Instead of using thetheory of random gaussian processes, we use onlyEinstein's fluctuation formula, Eq. (2.1), which he de-rived from thermodynamic considerations. Equation(3.1) follows as a special case of our formula for Av<<vand zl= 1.

We first derive the fluctuation of the number of counts1i2 during the time I, assuming that the detector is ofthe photoemissive or photoconductive type, giving riseto one pulse for each photoelectron. No distinction ismade between pulses resulting from photons of differentenergies. The probability a(r) of the ejection of a photo-electron when a photon of frequency v is incident uponthe detector is now, however, a function of i.

Let us divide the frequency range of the beam intosmall, equal intervals 3v, and let ni, be the number ofphotons in the jth interval. If n,, is the average numberof photons per unit frequency interval, we have

Kuj) = 111,5V- (3.2)

We assume that a(v) and ip may be considered constantwith respect to v inside each interval 3 v. At the sametime it must be possible to choose 3v in such a way thatni'ap>>1 so that we still have a large number of photonsper 3v. We also require 16vz,>>1, i.e., a large number ofphase-space cells per 6v. The fluctuation formula Eq.(2.12) for a rectangular spectrum then holds for eachinterval separately. Substituting from Eq. (3.2) intothe term a'(ni)'/z in Eq. (2.12) ,we obtain for the fluc-

tuation of the number of pulses in the jth frequencyinterval

5-(t j= (n 2j)+(1/cft)A2 &2(V)u2 bV, (3.3)

where we have used Eqs. (2.2) to (2.4) for z.Now, for thermal radiation, the different spectral

components are statistically independent. This fact wasused by Planck'0 when he put the entropy of blackbodyradiation equal to the sum of the entropies of its spectralcomponents. A thermal beam may also be defined asa beam whose electric vector, as a function of time,is a random gaussian process.8 For such a process, theamplitudes in different spectral regions are independ-ent." The mean square fluctuation in the total numberof pulses n2 counted by the detector when the wholespectrum of the beam is allowed to fall upon it, istherefore equal to the sum of the mean square fluctua-tions of the different spectral components

a'(72n) == F2 (3.4)

and, substituting from Eq. (3.3),

172(n12) = (K2)+ (1cof1)Jf X2a2(v)n 2lvdv. (3.5)

where the first term on the right is the sum of Kn21) overall frequency intervals. In the second term we have re-placed the sum by an integral.

If we wish to use the spectral radiance L, instead ofni, we have, from Eqs. (2.7) and (3.2),

nl,= (cwftlhc)XL,.

From Eqs. (2.11) and (3.2) we have

n12j) = aunlj) = anl,6v,

and summing over all frequency intervals

(3.6)

(3.7)

(3.8)Ku2) = f aQYn:4v.

Substituting from Eq. (3.6) into (3.8), we obtain for (n 2)

(n2) = (cof//hc) Xa(v)Ldv. (3.9)

Eq. (3.5) can therefore be written in the form

r2(n,2)=(n 2){1+(1/hc)j X4a2G()L2vd/f XaQ()Lshij

(3.10)

We have here taken (u2) outside the bracket, and havesubstituted for nuw and (n2) from Eqs. (3.6) and (3.9).

10 M. Planck, Tkeorie der Warmeslrahlzung (Johann AmbrosiusBarth, Leipzig, 1921), 4th ed.

" J. L. Lawson and G. E. Uhlenbeck, Threshold Signals(McGraw-Hill Book Co., New York, 1950).

37

ELLEN HISDAL

If we again wish to write 0 2(1n2) in a form equivalent toEq. (2.12)

F2(112) = (n12)±+(2 2 /Z2= (n2)(1+d 2 ), (3.11)

we see from Eq. (3.10) that the degeneracy of the num-ber of counts in the polychromatic case is given by

= zX4a%)L2dv f X6a2(A)Lx 2dX

d2=- = ,(3.12)kzc ktfc2 rfXa(v)Ldv f Xa(X)LxdX

where we now define the degeneracy d2 as the fractionalincrease of the number of counts above the "coinci-dental" amount. The last form of Eq. (3.12) was ob-tained with the aid of Eq. (2.6).

We can also define a polychromatic z; but if a(v) is notconstant with respect to u, we must now distinguish be-tween zi and Z2, these being defined by

di= (n1)/z d 2 = (n2 )/Z2 . (3.13)

From Eq. (3.13) we obtain for Z 2 , using Eqs. (3.12) and(3.9)

[f Xa(v)L4dv] [f Xa(X)LxdX]

z2=wft cofifc . (3.14)

f Xka2 (v)LA2d v a a2(\)Lx2dX

If a((P) is independent of v, Z2 is independent of the quan-tum efficiency of the detector, and equal to z1, the num-ber of phase-space cells per macrocell in the photonbeam.

In the special case of a quasi-monochromatic beam forwhich a(v) and A may be considered constant over thefrequency range of the beam, Eq. (3.14) reduces to

Z2=Zl=Z=coft/(2 j g(g) 2dv), (3.15)

in agreement with Eq. (3.1). The degeneracy d2 in thequasi-monochromatic case is therefore, according toEqs. (3.13) and (3.15)

d2= (1/zI)((ni2)/)f jg(P) [ 'dP, (3.16)

where we have replaced wf/X 2 by z,. For zi= 1 this agreeswith the result of Purcell7 and Mandel.8

Let us now suppose that we have a detector of thethermal kind, e.g., a thermocouple, which integrates theenergies of the absorbed photons. The measured quan-tity is now e2, the energy per macrocell absorbed in thedetector. If e2; is the energy in the jth frequency in-

terval, we have for the mean square fluctuation of e2

C02(e2) = E 0.2(e2j) = X 1r02v(1 2j). (3.17)

Using Eq. (2.12) for each interval j and substituting fora2(U2j) in Eq. (3.17) we obtain

T(e2) = E Izv6e2 j)+Y [a2(v)(eIj)2/z;], (3.18)i i

where c(v) is now the emissivity of the detector. Sub-stituting for zj and ej in terms of L, and replacing thesum by integral, we obtain the final result for the meansquare fluctuation of the absorbed energy

02(e2) = WftI {fla(v)Lvdv+ X2 a 2(v)Ltpdu . (3.19)

Eq. (3.19) is valid as long as the temperature of the ther-mal detector is low compared with the radiation tem-perature of the beam, so that the fluctuations due to thethermal radiation emitted by the detector itself can beneglected.

It is easily shown that in the quasi-monochromaticcase the ratio between the two terms which contributeto the mean square energy fluctuation in Eq. (3.19) isthe same as d2, Eq. (3.16), for a pulse-counting detector.

4. INFORMATION CONTENT OF A QUASI-MONOCHROMATIC THERMAL BEAM

WITH DETECTOR NOISE

We now wish to generalize the results of the previouspaper for the information content in a photon beam,taking into account that the detector has a limited quan-tum efficiency a&().

We start with the quasi-monochromatic case, and as-sume linearly polarized incident light with nil incidentphotons per macrocell. The photoelectric detector has aquantum efficiency a, and is assumed to record withcertainty one pulse for each emitted photoelectron. Fora thermal detector, a must be interpreted as the enis-sivity of the detector, and we assume that each absorbedphoton is measured by the detector. The measuring ar-rangement is the same as in the previous paper (Ref.1, Fig. 6). Let n, 2 be the number of pulses recorded bythe detector for one macrocell, the r subscript indicatingradiation reflected from the sample. If there had beenno fluctuations, the a priori expected possible values ofZr2, and of the corresponding reflectance r of the sam-

ple, would be

(4.1)

r= 0, 1/(ni2), 2/(ti2 ), * * -(n 2)/(ni2 ), (4.2)

where(n1i22) = t(Oi,) (4.3)

and (nai) is the average number of photons incident onthe sample per macrocell.

38 Vol. 57

nrv= 0) 1; 2) ' ' ') �11i2)

January1967 DETECTABLE INFORMATION IN A PHOTON BEAM 39

TABLE I. 12 =information content in a polychromatic thermal beam combined with a detector of quantum efficiencyaCZ(). Jr2= infor-mation loss due to limited efficiency of detector. Quantities with subscript I are obtained from corresponding quantities with subscript2 by setting a(p) = 1. When the detector resolution is properly chosen, the maximum information I2, max for the given beam and a(s), isobtained with the corresponding values Zr2,m and (ni2s); (ni2), Zr2, and dr2 are given in Eqs. (5.1), (5.5), (5.7), and (5.9). In the quasi-monochromatic case, (ni2) is given in Eq. (4.3) and Zr2 and d,2 may be replaced by z, Eq. (3.15), and ada, respectively.

Wien, di2<<1or Rayleigh-Jeans,

General case single-mode maser di2>>1

Is ii{ln(ni,)-ln(2r/e)-[1 + (l/dri)]ln(l+drl) } ln((ni,)/27r) b lnEzr,/(2xr/e)]12 2k{ln(n,2Y-ln(27r/e)-[1 +(1/dr 2)]ln(1+dr2 )} 3 In ((ni2)/27r) Ir ln[z-2/(27r/e)]J12= 17-12 4V{ln((nil)/(ni2))-)(1+1/dri)ln(1+dri)-(1+1/dr)ln(1+dr]2)l } V ln((nin)/(Qi2)) 4D ln(zrl/zr,2)J12 quasi- 411n(1/a) -[(I +1/dj)1n(1 +da) - (1-+ 1/adi)ln(1 +adil)]} 41- ln(1/a) 0

monochromatic12,-a (Ni2)/{4--E1+d,2]1+(l1 drl)) (Ni2)14,re Zr2/4-

Zr2, (2

1r/dr2) (1 +dr 2 )1+(l/dr2) 2

re/dr2 27r

(1i2)m 27r(l+d,2) l+(14dr2)

27re 27rd,2

The a priori expected information content in a meas-urement is now given by

i02= ln((%ni2)+ 1) = ln(ni2), (4.4)

corresponding to the (ni2)+1 possibilities for a reflec-tance measurement according to (4.2).

A measurement of nr2 pulses gives a most-probablevalue of nr2/(ni2) for r, but there are also certain proba-bilities Pz(n72) that the reflectance has any one of thevalues

r= l/(nX2>? l= 0, 1, 2, ** (in2) (4.5)

when we measure iir2 pulses. These probabilities aregiven by

PI(n7 2 ) = [I/(2ir)2 (1)] eXp[-(l-nr2)2 /2o, 2 (l)] (4.6)

as long as ll-nr2K <<a(l); l2(l) is the mean squarefluctuation of the number of counts for a beam whichgives I pulses on the average, and is according to (2.12)and (2.13) given by

12(j) = 1(1+1/z). (4.7)

We now proceed as in the previous paper1 to find theinformation loss due to the fluctuations (4.6) and (4.7).The results are similar to the previous ones, and arevalid within 5% for

1%2> 2a(nr2) and (ni 2>nr2 Žn2c2(nr2). (4.8)

The information loss due to the fluctuations is found tobe

jo2(trs2)=-2Pl lnP,= 4+ln[(2-r)r(nr2)]. (4.9)

Averaging this expression over all values (4.1) of nr2which are assumed to be equally probable a priori, weobtain

102= ln(2ir)'+42Eln(ni 2)+ (1/di2) (1 +di2 )ln(l+di2)-1]

(4.10)where

di2=adi (ni2)/Z. (4.11)

For simplicity we have left out the average sign overjo2 in Eq. (4.10), and shall continue to do so for all sub-sequent information and information-loss quantities.

In the approximation di2<<1, which we now call theWien approximation, Eq. (4.10) becomes

02= ln[2rr(nj2s)I= ln[(2ir)k(nj 2)], (4.10W.)

and for di2>>1, now called the Rayleigh-Jeans approxi-mation, we obtain

jo2= ln[(2ir/e)i(ni2)/z] = In[(2ir/e)Wio(ni2)]. (4.10R.-J.)

The average information in a reflectance measure-ment of r macrocells is

(4.12)

and substituting from Eqs. (4.4) and (4.10) we obtainfor I2 the expressions given in the second row of TableI. The symbols z,2 and d,2 which occur in the tableshould be replaced by z and di2 in the quasi-monochro-matic case.' 2 The information loss due to the limitedquantum efficiency of the detector is J12 = 1- 12, whereIA is the value of 12 for a= 1. J12 is listed in the fourthrow of Table I. We see that in the Rayleigh-Jeans case,the limited efficiency of the detector does not introduceany information loss, because in this case we can saywith certainty that a measurement of nr2 pulses wascaused by nrl=n1 2/a photons. All uncertainty in theinformation is due to the fluctuations of the incidentbeam.

The maximum information 12,maw that can be ob-

tained with a given beam and a detector array of a givena by varying the resolution or the number of macrocellsr, can be computed as before. Again we obtain

12, max= 2 m , (4.13)

where Vm is the number of macrocells which must beused to obtain maximum 12. We see that also in the pres-ent case we obtain a maximum information of 2 unit,

12 There is an error in the first expression for ii in Eq. (4.21 W.)of Ref. 1, which should have been ln(ii/27r)i.

I2= �i2=�(iO2_ jO2),

ELLEN HISDAL

or 0.72 bits per macrocell. Except in the Rayleigh-Jeanslimit, a macrocell is now, however, larger than in thecase without detector noise.

The results for I2,m,,,: and the corresponding values ofZn, the number of microcells per macrocell, and of(ni2 )n, the average number of pulses per macrocellwhich maximize 12, are summarized in Table I. By set-ting a= 1 in the formulas for jo 2, 12, 12.max, and z,n, weobtain the quantities jol, IA, 'i1mv,, and zn for the casewithout detector noise. A comparison of the correspond-ing expressions shows that the case with detector noisecan be obtained from that without noise by replacingnii, Nil, and din, respectively, by ansi, al~il, and adil.This means that, so far as the information content of athermal beam is concerned, the combination of an in-cident beam of spectral radiance L, with a detector ofefficiency a is equivalent to a beam with spectral radi-ance 4/oa, combined with a detector of unit efficiency.We might have guessed this result, but we have givena short derivation here which is used again in the poly-chromatic case, for which the results are not so obvious.

5. INFORMATION CONTENT OF A POLY-CHROMATIC THERMAL BEAM

We assume that we have a completely polarized,polychromatic thermal beam of spectral radiance L,,combined with a photodetector of quantum efficiencya(®). We wish to find the "black-white" informationcontent of such a beam, i.e., we assume, that the detec-tion of a pulse gives us no information about the fre-quency of the photon which gave rise to the pulse. Wewill obtain information about the value of the "effectivereflectance" of the sample. This effective reflectance isdefined below, being a suitable average over v of thespectral reflectance r(v).

If the sample had a reflectance r(Q)= 1 for all v forwhich ni>ca(v)#-O, the most probable value for the meas-ured number of pulses per macrocell would be

(14 2 )= a(v)n1 ldv, (5.1)

where na, is given by Eq. (3.6). In general, for a sampleof reflectance r(v), the most probable number of meas-ured pulses is

(11r2) = frQP~aQ~ni~dv. (5.2)

beam of spectral radiance r(v)L,, because a thermalbeam L, reflected from a sample with reflectance r(v)can in no way be distinguished from a thermal beamwhich comes directly from a source of spectral radiancer(v)L,. According to Eqs. (3.11) and (3.14) we havetherefore for the mean square fluctuation of n,-2:

(5.4)where

ZrWf Xa(v)r(v)Lvdv]

X al~x(y),r2(v)L;>2dv

o2

[f Xa(X)r(X)LxdX]

=wXftc . (5.5)

J XV6a2%(\r2(X.)LxdX

We see that Z,2 now depends on the shape of the func-tion r(v). We therefore must assume that we have someprior information about the relative shape of the func-tion r(v), before we can compute the information whichthe beam can supply about the magnitude of reff.This difficulty does not arise in the Wien case, which isthe usual one in the optical region of the spectrum. Inthis case the last term of Eq. (5.4) can be neglected, andZ42 does not enter into the information formulas.

If we may assume that r(v) is independent of v, thenZ,2 is independent of r and equal to z2 of Eq. (3.14). Ifca(v)r(v) may be assumed to be independent of v, thenZr2 is independent of r and a, and equal to zi, the numberof phase-space cells per macrocell in the incident photonbeam.

In general we must make an assumption of the kind

r(v)= = V)reff, (5.6)

where f(v) is known, and rff is independent of v and isthe quantity about which we wish to obtain informationby our measurement. Substituting Eq. (5.6) into Eq.(5.5), we obtain for Z42 in the general case

L =(cf(f)Ltdv

,Zr = c.ft- (5.7)

We define the effective reflectance of the sample by theequation

rdff= 0t,2)/(1i2)- (5.3)

The a priori expected values of n1,2, of rff and of theinformation i02 are again given by Eqs. (4.1), (4.2), and(4.4) with (ni 2) given by Eq. (5.1).

An average measurement of (u,-2) pulses has associatedwith it the same fluctuation as that due to an incident

X4

a2(v)f

2(v)L,2dv

A normalization condition for f(v) is obtained by sub-stituting Eq. (5.6) into Eq. (5.2) and using the resultingexpression for (1,2) in Eq. (5.3). This gives the condition

j f(v)a(v)nlvdv= f a(v)nldv. (5.8)

40 Vol. 57

2(_0, k,.,2) == (nrl)+ (�41)2/Zrl,

Jo

January 1967 DETECTABLE INFORMATION IN A PHOTON BEAM 4

The average information content I2 is now computedin the same way as for the quasi-monochromatic case.The result is given in the second row of Table I. Thefirst row gives the value of I,, the information contentof a polychromatic beam combined with an ideal de-tector having a(v)- 1. The difference J12 between thetwo rows gives the information loss due to the limitedefficiency of the detector. (ni2) is given by Eq. (5.1) and

dr 2 = (ni2)/Zr2, (5.9)

where Z,2 is given by Eq. (5.5) or (5.7) We note thatZr2 and dr2 depend only on the relative shape of the func-tion r(v) but not on its magnitude. In the Wien casethese two quantities do not enter into the formulas forthe information content, and the latter is therefore in-dependent also of the shape of r(v).

Again we have computed the maximum informationI2,max obtainable by varying the resolution ¢, whileQFT, LVX, C(V), f(V), and therefore also dr2 and (Ni2 ) arekept constant. As in the quasi-monochromatic case wefind I2,max= 21m-

The values of I2,max and the corresponding values ofZ2r and of (ni2) are listed in the last three rows of TableI. We see that 27re pulses per macrocell give maximuminformation in the Wien case.

6. EXAMPLES

In this section we wish to compute numerical valuesfor the information content of the image of a scene il-luminated by sunlight or by artificial light sources.There are two main differences between this case, andthe experimental arrangement assumed up to now forcomputing the information content of the beam.

The first difference concerns the mode of illumination.We have assumed that the light source was imaged onthe sample (Ref. 1, Fig. 6), so that a lateral cell couldbe identified with the smallest resolvable area in thesample. Sunlight reflected from a specular surface oflimited dimensions does, however, give rise to a beamof the type of Fig. 2, Ref. 1; i.e., a superposition of col-limated rays or lateral cells with slightly different direc-tions. These two types of illumination correspond tocritical illumination and Koehler illumination, respec-tively, of the object in a microscope. It is shown in Bornand Wolf,'" paragraph 10.5.2, that the light incident onthe object plane has the same degree of coherence forboth types of illumination. We wish here to give a physi-cal argument why a thermal beam of the type of Fig. 1,Ref. 1, where each point of the focal area F radiatesindependently, is completely equivalent to a collimatedthermal beam of the type of Fig. 2, Ref. 1.

Blackbody radiation can be obtained in two ways,both giving radiation of the same properties. It can beconsidered as the radiation sent out by a black surfaceof area F which is held at a given temperature. Such a

13 M. Born and E. Wolf, Principles of Optics (Pergamon Press,Inc., New York, 1964), 2nd ed.

radiating surface gives a beam of the type of Fig. 1 ofRef. 1. But exactly the same radiation is obtained froma large cavity whose walls are held at the same tempera-ture as the black surface of the previous case, the beamnow emanating from a hole of area F in the walls ofthe cavity. Each mode of vibration inside the cavity nowgives a contribution to the radiation from the hole inthe form of a collimated ray, as represented in Fig. 3 ofRef. 1, and the radiation due to the different modes isa superposition of such collimated rays of the type ofFig. 2, Ref. 1. This proves that there is no physical dif-ference between a focused thermal ray and a collimatedone of the same Q and F. We can therefore compute theinformation content of a beam of sunlight just as if thebeam had been of the focused type, and of radianceequal to that of a sunlight beam which is reflected froma sample with reflectance r= 1.

The second difference between the case which wehave considered up to now, and a practical one, con-cerns the surface properties of the sample. We have as-sumed that the sample reflects specularly. Most naturalobjects are, however, more nearly diffuse reflectors. Intheory, there is no difficulty in extending the presenttreatment to the case of samples with arbitrary surfaceproperties. We can imagine the sample to be introduced,very slightly off-center, into a large hemisphere whoseinner wall is a complete, specular reflector. The incidentbeam enters through a small hole in the wall of thesphere. In such an arrangement, the radiation reflectedfrom a given point in the sample is focused at a pointwhich is symmetrical with respect to the center of thesphere. The aberrations decrease with increasing sphereradius for a given sample size and distance from the cen-ter. A detector placed in a position symmetrical to thesample records pointwise the radiation reflected by thesample points over a half sphere. The information con-tent of the beam now refers to the diffuse reflectance ofthe sample.14 Our information-content measures 11, 12

are therefore valid without assuming a specularly re-flecting sample.

However, if we wish to obtain a measure of the infor-mation content of a sample, as seen, e.g., by the eye, wemust make some assumption about the angular distri-bution of the reflected radiation. Such an assumption isnecessary because we observe the light reflected in onedirection only, and the measurement cannot furnish uswith information about the angular distribution of thereflected light. We therefore compute the informationcontent of a photon beam for the following two cases:(1) The detector is placed at the angle of specular re-flectance. The sample is assumed to be flat and specu-larly reflecting. (2) The second case corresponds morenearly to natural scenes. Here the sample is assumedto be flat and diffusely reflecting, i.e., each small areaof the sample is assumed to have a Lambert distribu-tion of the reflected radiation. The detector may be

14 B. Hisdal, J. Opt. Soc. Am. 55, 1255 (1965).

41

ELLEN HISDAL

TABLE It. Numerical values for different light sources. 12= information received by eye about a sample of 1 cm2 at a distance of1 m during 1 sec; i2 =information content per macrocell which the eye can resolve. An asterisk indicates that the luminances given in thecorresponding columns are too high to be tolerated by the eye.

Light source Sun Mercury arc Tungsten filament,frosted bulb

Sample Specular* Diffuse Specular* Diffuse Specular DiffuseUnit

U2SA-SO sr 6.76 X105 6.76X10- 5 3.14X10'G 3.25X10-6 2.83X10- 3 2.83X10-3

L' stilb 5.47 X104 1.18 1.25X104 1.29X10-2 1.70 1.53X10-3dil 1/sr 0.96 X10-2 2.07X10-7 2.47X10-3 2.56X10-9 6.85X10-7 6.16X10-1°5 mm 2.00 2.00 2.00 3.14 2.00 4.33Wft sr cm' sec 5.32 X10-' 0 5.32X10-1o 5.32X10-10 13.1X10-1" 5.32X10-1" 2.49X10-9(nil) 3.33 XI011 7.18X106 5.OOX1010 1.27XI0 1.30XI07

5.49X104

Zil sr 3.46 X1013 3.46X103 2.02X10=L3 4.98X1013 1.90X1013 8.91X1013(Nei) 1.97 XI0'5 4.24X1010 2.95XlO14 7.53X10 7.69X10' 0 3.24X108I1.max 1.44 bits 5.77 X10 3 1.24X109 8.64X1012 2.20X107 2.25X109 0.95X10 7

I, 1.44 bits 7.30 X104 4.12X104 6.74X10 4 2.93X10 4 4.30X104 2.68X10 4

assumed a 0.5 X10-2 0.5 X10-2 0.5 X102 1 X10-2 0.5 X10-2 1 X10-212,max 1.44 bits 2.88 X1011 0.62X107 4.32X10" 2.20X105 1.13X10 7 0.95X105

12 1.44 bits 5.73 X104 2.56X104 5.17X104 1.57X104 2.73X104 1.32X104

i2 1.44 bits 9.70 4.33 8.75 2.66 4.62 2.24

placed anywhere within the half space above the sample.The numbers Nil, nil of photons in the incident beamwhich have entered our formulas are now replaced bythe numbers of photons which would be reflectedtowards the detector by a completely white and diffusesurface. For both the specular and the diffuse case theradiation reflected from the sample is assumed to befocused on the detector. The reflectance may vary withposition on the sample surface.

The numerical results for three different light sourceswith the eye as detector, are summarized in Table II.The sample is assumed to have an area of 1 cm2 and tobe placed 1 m from the detector (eye) lens DL, Fig. 1.For the tungsten and mercury lamps, the distance fromthe light source SO to the sample SA is assumed to be1 m. In all cases we assume practically normal incidenceand normal direction of observation. For clarity we havedrawn the case of a transmitting instead of a reflectingsample. All our formulas hold equally well for this caseif r is interpreted as the transmittance of the sample. Drepresents the detector or retina.

In front of the source we assume a polarizer whichtransmits only one half of the total flux. The radianceand luminance values assumed in the table are therefore2 of the actual ones. Using the results of Rose,"5 thequantum efficiency a of the eye was assumed to be 0.5%in all cases except for the two lowest luminances, for

QSA-SO QSA-DL

SO SA DL D

FIG. 1. Source (SO), sample (SA), detectorlens (DL), and detector (D).

1A. Rose, J. Opt. Soc. Am. 38, 196 (1948).

which we assumed a= 1%. These values of a refer tothe eye as a whole, and include such effects as absorp-tion, scattering, and reflection in the various media ofthe eye. We also assume that a is independent of v. Thisis of course not correct, but Rose's experiment refersto the information obtained with white light, and tothe average a which can be deduced from this informa-tion. It therefore seems reasonable that this averagevalue of a will also give approximately correct resultshere for sunlight and tungsten light. For the unevenspectrum of the mercury arc, the assumed constancy ofa may introduce some error into the values of i2 and I2.

The spectral radiance of the sun was computed fromthe data for the irradiance Hx as given in Koller,"5 p.125, for sunlight. All data refer to the spectral region4250-7250 A. Lx for a specular sample was computedfrom the equation

(6.1)

where QSA-SO (Fig. 1) is the solid angle subtended bythe source at the sample. For the specular case, theradiance of the source and the radiance L of a samplewith r= 1 are the same. For a diffuse sample the corre-sponding specular values of L,L', and d, must be multi-plied by 2sA-so/1r. The tungsten source was assumed tobe a frosted bulb, 60-W lamp operated at 2800'K andgiving 509 lumen. The spectral distribution was takenfrom Kinglsake,17 p. 52. The radiance was computedunder the assumption that the flux from the filament dis-tributes itself isotropically over a spherical envelope ofradius 3 cm, which itself radiates diffusely according toLambert's law.

The high-pressure mercury lamp was assumed to beof the HBO 200 Osram type, the spectral distribution

1" L. R. Koller, Ultraviolet Radiation (John Wiley & Sons, NewYork, 1952).

"1 R. Kingslake Editor, Applied Optics and Optical Engineering,V~ol. I (Academic Press, New York and London, 1965).

Vol. 57

LX='(Hx/QSA-SO),

January 1967 DETECTABLE INFORMATION IN A PHOTON BEAM 4

being taken from an Osram catalogue. The lamp has apower of 200 W, a luminous flux of 9500 lumen, and anarc luminance of 25 000 stilb. For simplicity it was as-sumed that USA-SO= 2SA-DL in the specular case, so thatthe light beam reflected from the sample just fills thepupil of the eye completely, although the shortest di-mension of the rectangular arc is slightly too small tofullfil this requirement. To compute QSA-sO for the dif-fuse sample, the actual size of the arc of 2.5 mmX 1.3mm was used.

The pupil diameters a for the given luminances weretaken from Fig. I in Reeves's paper'8 for both eyes open.

The value of FQT for the whole sample is 0.7854X 10-6cm 2 sr- sec (82 mm-2), using a sample area FSA= 1 cm2,a divergence of the beam from the sample to the de-tector (eye) lens of Qs&A-DL=O.7 854XlO6 sr (32mm-2),corresponding to a 1-m distance from sample to eyelens, and a total observation time T= 1 sec. The lateralsize of a macrocell was assumed to be determined by aminimum angle of resolution of 1' for the eye. This cor-responds on the sample to a square area fSA=0.8462

X 10- cm2 ; WSA-DL is equal to USA-DL and the scan-ning period or storage time of the eye was taken to be1=0.2 sec." This gives fi4=1.3292X10-2 cm2 sr * sec(82mm 2). The number r of macrocells contained in thebeam detected by the eye is then, according to Eq. (1.1),equal to 5.9088X10'.

The values of diL and (nil) in Table II were computedfrom Eqs. (3.12) and (3.9) with a= 1. zi, was computedfrom Eq. (3.13), and the information-content valueswere computed from the Wien formulas of Table I. Thespecular cases for the sun and the mercury lamp are in-cluded in the table even though the luminances are toohigh to be tolerated by the eye.

We see from the I2 row of Table II, that the informa-tion received by the eye about a sample of 1-cm 2 sur-face area at a distance of 1 m, when it is observed during1 sec, is of the order of magnitude of 104 to 105 bits forall three light sources. If the sample surface does notchange with time, the listed values of I2 must be dividedby 5, corresponding to a resolving time of 0.2 sec for theeye.

The maximum information-content values, 12,max,

which a detector of suitable resolution, and of the samequantum efficiency as that assumed for the eye, could

8 P. Reeves, J. Opt. Soc. Am. 4, 35 (1920).

extract from the beam are from 4 to 400 times greaterthan I2, for those luminances which the eye can tolerate.The large differences occur for the high luminanceswhere especially a larger time resolution would extractmore information from a moving picture.

The method used in this section for the computationof the information content of a photon beam combinedwith the eye as detector can, of course, be used for anyphotoelectric detector array whose quantum efficiency isindependent of the intensity of the incident radiation.For thermal infrared radiation, di, is computed as above.It may, however, no longer be small compared with 1so that the general formulas for the information-contentmust be used instead of the Wien approximations.

7. SINGLE-MODE MASER BEAM

The treatment of a beam from a well-stablilzed maserwhich oscillates continuously in a single mode is differentfrom that given in the previous paper.' We saw thatthe photon numbers nil in such a beam follow a Poissondistribution or, for large photon numbers, a gaussiandistribution with a root mean square fluctuation (nil)1.If the beam is reflected from a surface with reflectancer, it has the same physical properties as a beam froma maser which oscillates in one mode, but whose averagephoton number is reduced to (nrl)=r(ni,) The rootmean square fluctuation of the reflected beam is there-fore oa(n1 l) = (nr,)A= rko(nil) and not, as assumed in Ref.1, ra(nil), as it would be for a thermal beam in theRayleigh-Jeans region. Proceeding in the same way asin the derivation of Eq. (2.14), we find that the rmsfluctuation of the number of counts is (n72) 1 = [(nri)]

4 .Thus the fluctuation of the number of counts for a

single-mode maser beam follows the same laws as fora thermal beam in the Wien limit, and the results foris and i2 are the same as for a thermal beam in thislimit.

A maser beam oscillating in a single mode is made upof only one microcell. When it is focused, the spot diam-eter is determined by diffraction effects only. Thus z andZ have no meaning for such a beam. v is now equal tothe number of scanning intervals during the measuringtime, and nil is the number of photons in the beam perscanning interval. The Wien-column values of Table Iare now directly applicable to an ideal maser beam, ex-cept for Zr2,m which has no meaning in this case.

43