JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Information Content of a Beam of Photons*

ELLEN HISDALInstitute of Physics, University of Oslo, Blindern, Oslo 3, Norway

(Revision received 29 June 1965)

A quasimonochromatic photon beam is constructed by the superposition of quasimonochromatic planewaves with certain mutual coherence properties. The information which the photon beam can carry aboutthe reflectivity of a specular sample is investigated, taking radiation fluctuations into account. In Sec. 4an expression is found for the information content of the beam as a function of the resolution of the detectorsystem. In Sec. 5 it is found that there exists a maximum information which a given thermal photon beamcan carry. This is obtained with a detector system of a specified resolution. The information contained in anidealized maser beam is computed in Sec. 6 and compared with the information in a beam from a low-pressuremercury emission line. It is found that the information in the maser beam can be several orders of magnitudehigher. It is limited mainly by the highest obtainable scanning rate.

1. A BEAM OF PHOTONS

THE concept of a beam of photons was defined byPlanck (Refs. 1, 2; § 14, 21) who used the word

"Strahlenbiindel." This may be translated as a "beamof rays" or a "pencil of rays."

Planck defines such a beam as the radiation emanat-ing from a given area, called the focal area of the beam,and contained within a small solid angle fI (Fig. 1). Thefocal area may be imagined as the surface of a thermallyemitting body or alternatively as an arbitrary areawithin a radiation-filled cavity. We shall here use theprojection F of the focal area on a plane perpendicularto the direction of propagation. We assume that thebeam is linearly polarized and quasimonochromaticwith a frequency v in a small interval Av.

By placing F in the focal plane of a lens, the beam ofFig. 1 can be converted into the type of beam repre-sented schematically in Fig. 2. Each pair of parallelarrows in Fig. 2 represents a collimated beam (which weshall call a ray) due to the radiation from one point ofF. For a thermally emitting surface, two rays withdifferent directions, due to two different points of F, areincoherent. If F'= a2 is the focal area of the beam ofFig. 2 and Q'=02 its divergence, we have

SŽF/X2 = Q2'F'/X'2 , (1.1)

where X and X' are the wavelengths of the beams 1 and2. Equation (1.1) follows from the fact that the quantityq2 2F is an invariant for beams which can be convertedreversibly into each other.3' 4 Here q is the refractiveindex of the medium in which the beam propagates.

FFIG. 1. A beam of photons with focal area F and divergence U.

* This work was supported in part by the Norwegian ResearchCouncil for Science and the Humanities.

I M. Planck, Theorie der Warmnestrahlung (Johann AmbrosiusBarth, Leipzig, 1921), 4th ed.

2 M. Planck, The Theory of II eat Radiation (Dover Publications,Inc., New York).

3 M. Laue, Ann. Phys. 20, 365 (1906).4 D. Gabor, Progress in Optics I (North-Holland Publishing

Company, Amsterdam, 1961), p. 111.

a

Laue5 ' 6 has shown, that the conversion of beams intoeach other by absorption-free reflection or refraction isa reversible process. It is assumed here that the con-verting lens subtends a solid angle at F which is largecompared to S2, in order to avoid diffraction effects.Because of this reversibility, it is immaterial whetherwe give a mathematical description of a beam of thetype of Fig. 1 or of Fig. 2.

Our aim is to construct a beam whose intensity is, asfar as possible, constant over the focal area and theangular opening of the beam, and zero outside theseregions. Because of the wave character of light, such asharp drop in the intensity cannot be completelyachieved.

As plane waves are solutions of Maxwell's equations,we shall try to construct a linearly polarized beam ofthe type of Fig. 2 by a superposition of linearly polarizedplane waves. It will be understood, that whenever weuse the expression "plane wave," we mean a quasi-monochromatic plane wave as described by Wolf, 7 all

y

-- 1--

ao

FIG. 2. Schematic representation of the superposition of three rayswhose directions differ by ai, to form a beam of angular width 3.

G M. Laue, Ann. Phys. 23, 1 (1907).6 M. Laue, Ann. Phys. 30, 225 (1909).7 E. Wolf, Nuovo Cimento 12, 1165 (1959).

1446

-0- - '7

5 - - - - -30., '�l

VOLUME 55, NUMBER 11 NOVEMBER 1965

November1965 INFORMATION CONTENT OF A BEAM OF PHOTONS

> x a W | E

FIG. 3. Schematic representation of a ray propagating in x direc-tion on the left, and its amplitude e as a function of y on the right.

the different plane waves having the same frequency vand frequency interval Arv.

To simplify the constructions, we shall assume for thetime being, that the beam, which propagates in the xdirection, is limited to the width a along the y axis, andthat all quantities connected with it are constant in thez direction.

The first step is the construction of a collimated beamor ray. Such a ray is constructed by the superpositionof mutually coherent plane waves.

Figure 3 represents schematically a ray propagatingin the x direction, with an electric vector E in the x= 0plane given by

whereE(t,y)=cos27rvt e(y),

IEO for -a/2<y<a/2e (y) =

0 for all other y.

(1.2)

(1.3)

As is known from the theory of Fourier transformsapplied to optical imaging systems,8 it is possible toconstruct a sinusoidal standing wave of frequency v andof any desired wavelength between X and co along they axis by superposing two coherent, plane, travellingwaves with frequency v whose directions of propagationmake angles hca with the x axis. The wavelength of thestanding wave along the y axis will then be (Fig. 4)

X,,= X/since. (1.4)

By Fourier's theorem we can therefore construct anydesired amplitude distribution in the x=0 plane bysuperposing plane waves with different directions cx butwith the same frequency. The direction of polarizationof the plane waves is assumed to be the projection of thedesired direction of polarization of the ray on a planenormal to the direction of propagation of the planewave.

Because e(y) is symmetrical around the x axis, all thecomponent travelling plane waves must be in phase atthe origin.

8 A. Marechal and M. Frangon, Dipffaction. Structure des images,Edition de la Revue d'optique th6orique et instrumentale. 165,rue de Sevres, Paris (15e), 1960.

X

y

\ -\ - x

FIG. 4. Wave fronts of a component plane wave propagating indirection a, and wavelength X, of standing wave along y axisproduced by two travelling waves with directions ±ta.

To represent the function e(y) given by Eq. (1.3), wewrite

where

1 WY

e(y)=- f C(k,)coskly dki,7r (/ n

k, =27r/X, = (27r/X) sinat.

We then find by Fourier inversion

C kd) = eoa sin (kya/2)/ (ka/2).

(1.5)

(1.6)

(1.7)

To obtain the standing wave of Eq. (1.2) we musttherefore superpose infinitely many travelling planewaves making different angles h-ca with the x axis. Theamplitude of the component waves is a function ofca and is given by (1/27r)C(k,)dk,. In other words theamplitudes C of the component plane waves varywith a in the same way as the amplitude due to diffrac-tion of a plane wave at an infinitely long slit of width- a/2 < y < a/2. We can therefore obtain a ray by lettinga plane wave go through a slit of width a. At longdistances from the slit the ray will give rise to a Fraun-hofer pattern of angular width a,, from center to firstzero, given by

sinac = X/a. (1.8)

The radiation emerging from an opening of width a in acavity, owing to a given mode of vibration inside thecavity, will therefore also be a ray. A single ray will beproduced by a maser which oscillates in one mode only,a being the width of the window from which the maserradiation escapes.

The order of magnitude of cx,, Eq. (1.8), can also befound from the uncertainty principle for a photon whichis limited to Ay= a.

The ray construction assumes scalar addition of theelectric and magnetic fields. This will be valid only aslong as coscx_ 1 or a<'3 for 5% accuracy. Owing to the

1447

L £ 0 b

ELLEN HISDAL

resulting lack of the high spatial frequencies in theFourier spectrum (1.5) of e(y), we do not get the sharpcut-off of the amplitude distribution (1.3) aroundy= 4a/2. If we wish that e(y) should not be changedappreciably except in a region of the order of Ay= a/maround y= ia/2, we must use plane waves with direc-tions a from 0 up to am where sinamw= mN/a. Together,the two restrictions on a give us the necessary condition

a>>X (1.9)

for the validity of the ray construction.A ray of a different type has been constructed by

Gabor (Ref. 4, p. 143) who calls it a "restricted beam."As a ray is constructed by the superposition of

mutually coherent plane waves, a photon belonging to aray is in a definite but superposed state in the sense usedby Dirac.9 A ray has therefore no entropy (providedthat the time of measurement does not exceed one tem-poral cell as defined in Sec. 2).

The second step in the construction of the beam ofFig. 2 is the superposition of many rays with slightlydifferent directions of propagation, but such that allare limited to the width a along the y axis. In this waywe obtain a bean with a divergence B and with a focalarea of width a. If the different rays which we superposeare mutually incoherent, as they will be for thermalradiation, then the beam will have an entropy becausewe can distribute the photons in different ways betweenthe component rays. A ray has thus a relation to thebeam similar to that of a mode of vibration to theradiation enclosed in a box.

We now ask how many rays are needed to constructa beam of angular width f. Since each ray has, accordingto (1.8) and (1.9), an angular width a,= X/a it is reason-able to say that we need

3/.I=-13(a/X) (1.10)

rays to construct the beam. An optical system with anaperture a will then just be able to resolve two adjoiningrays according to Rayleigh's criterion.

In the two-dimensional case, the number Z1 of raysneeded to construct a beam which is limited to the focalarea F'= a' and the angular opening Q=/3' in both they and z directions is similarly given by

(1.11)

The value (1.11) of Z, agrees with the value of onefactor in the expression for the number of phase spacecells contained in the beam as computed according tothe ordinary methods of statistical mechanics. As Z,rays pass through each cross section of the beam, weshall say that the beam contains Z1 lateral cells.

From the point of view of the beam of Fig. 1, thenumber Z, of lateral cells is equal to the number of

9 P. A. M. Dirac, The Principles of Quantumn Mechanics(Clarendon Press, Oxford, England, 1958), p. 13.

resolvable areas of size

LFin= X2/Q (1.12)

discernible in an area F which is viewed with an opticalsystem of opening S. Laue'0 calls the Z, defined in thisway "the number of degrees of freedom" in the beam.

Zz is also of the order of the number of coherenceareas"l contained in a cross section of the beam. Aquantity of the same order of magnitude as (1.11) hasalso been derived by several more recent authors.

We conclude that it is possible, according to the wavetheory of electromagnetic radiation, to construct abeam of quasimonochromatic photons as depictedschematically in Fig. 2, provided that the lineardimensions a of the focal area of the beam are largecompared to the wavelength of the radiation. Theintensity of the beam will be constant over the wholeangular range of the beam except for small local vari-ations over angular regions of the order of A/a. Thesesmall variations are a necessary consequence of theuncertainty principle when the photons are limited toa focal area of width a.

2. ENTROPY AND INFORMATION

The number Zi of lateral cells is an important quan-tity for the determination of both the thermodynamicentropy and the information content of a beam. It is,however, limited to a two-dimensional cross sectionwhile we know that a light beam can carry with itinformation about three-dimensional objects. Althoughan optical system can focus sharply only a two-dimen-sional cross section of an object, a photograph e.g., cangive us much information about a three-dimensionalscene (cf., Gabor's method of wave front reconstructionfrom a hologram). Alternatively we may think of thescanning of a three-dimensional nuclear photographicemulsion in a microscope, where different cross sectionsof the emulsion can be focused successively. In thisarrangement, we can imagine that we observe a givencross section during a time Al, and that we then movethe image plane and observe a new cross section. If thetotal time of observations is t, we can observe in all Ztsuch cross sections where

Zt=t/lAt. (2.1)

We can find the smallest time A/mm which we need inorder to make an observation of a given cross sectionwith quasimonochromatic light of a bandwidth Azv fromthe uncertainty principle. Each photon has a "length"

"OM. Laue, Ann. Phys. 44, 1197 (1914).11 M. Born and E. Wolf, Principles of Optics (Pergamon Press,

Inc., Newv York 1964).12 A. Blanc-Lapierre, Ann. l'Inst. Henri Poincar6 13, 245 (1953).13P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc.

London, Ser. A 247, 369 (1955).14 G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955a).16 E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955a).

1448 Vol. 55

Z, = 021C,� = Q'F'IV- = OF/X2.

November1965 INFORMATION CONTENT

Al given by(2.2)

where Ap==hAv/c is the uncertainty in momentum ofthe photon. The smallest time which we need to observesuch a photon is the time taken by the wavepacket oflength zAl to pass or to be absorbed in the detector

Almin = 4A/c = 1/AV7. (2.3)

Atmjn is now often called the coherence time of theradiation." With this value of At we obtain from (2.1)for the number of longitudinal or temporal cells con-tained in the beam.

Z 6= t1Az. (2.4)

The sampling theorem of information theory16 givesus a value of 2t AP for the number of measurements inthe time t which determine a signal of bandwidth AzVcompletely. This is correct if it is possible to determinethe phase of the signal. We shall assume here, that wecan measure the exact number of photons in the signal.It then follows from the uncertainty principle, that thephase is completely undetermined.1 7 We must thereforeomit the factor 2, and obtain again the result (2.4).

Now we saw that if we "exploit" the photon beam tothe uttermost, we can observe Zt longitudinal crosssections or temporal cells and ZA lateral cells. Therefore,if we make our observations as detailed as possible, wecan observe the number of photons in ZZt cells andusing (1.11) and (2.4) we obtain for the total numberZ of observable cells

Z=ZlZt= (FQ/X2 )1Av. (2.5)

Z is the total number of cells which can be resolved inthe beam. The same number is obtained by the methodof statistical mechanics for photons emerging from afocal area F with momenta p= hv/c, in the rangeAp= hAv/c, limited to the solid angle U. Figure 5 shows,that the volume occupied by the beam in coordinatespace (horizontal shading) is

V= ctF, (2.6)

where t is the time during which the beam is observed,because all photons contained in a length ct pass throughF during the time t. The volume occupied in momentumspace (vertical shading) is

V,=p2gip== (h3v2 xAV/c3)Q (2.7)

The number of phase-space cells occupied by thebeam is therefore

Z= VVl/h 3= (v2 /c2)FQAvt= (FQ/X2 )APt. (2.8)

This value of Z refers to the number of phase-space

16 L. Brillouin, Science and Information Theory (Academic PressInc., New York, London, 1962), 2nd ed.

17 W. Heitler, Quantumt Theory of Radiation (Clarendon Press,Oxford, England, 1954), 3rd ed. p. 65.

Al=h/Ap=c/Av,

d =N/Z= (ehvIkT 1)-1' (2.9)

where N is the number of photons traversing a crosssection of the beam during the time I. The thermo-dynamic entropy S of the beam is given by k times thelogarithm of the number of ways of distributing the Nindistinguishable photons among the Z cells. UsingStirling's approximation for 9N and Z! this turns outto be'8

S=kZ[(1+d)ln(l+d)-d lnd]. (2.10)

Suppose now that we use the thermal photon beamto obtain information about the reflectance of a sample.In principle, we should be able to measure the numberof photons in each phase-space cell of the reflected beam.We could then divide the information problem into twoparts, the first being the information content in thedetermination of the total number N, of photonsreflected from the sample as a whole, and the second thedetermination of how these NV, photons are distributedamong the Z phase-space cells of the reflected beam. Itis tempting to identify the information in part 2 withthe logarithm of the number of all the different ways ofdistributing N, photons among Z cells. Except for aconstant factor, identical expressions would then beobtained for the thermodynamic entropy and for theinformation content of the reflected beam of N,. photons.There are, however, several errors in this reasoning.

In the first place it can be shown that the information,as given by the entropy formula (2.10), is a mixedinformation about the reflectivity of the object and themicrostate of the incident beam.

A second error in identifying the entropy formulawith the information content is the assumption that thetotal number Ni of photons in the source beam is aconstant. In reality Ni is a fluctuating quantity and thesame is true of the total number N, of reflected photonsfrom a given object.

The third drawback to the entropy formula (2.10)used as an information measure is, that it refers to adetector whose resolution is the highest obtainableaccording to the uncertainty principle. In practice it isof course very difficult to obtain such a high resolution.

I' M. Planck, Ann. Phys. 4, 553 (1901).

OF A BEAM OF PHOTONS 1449

I - FN-l2 = O 2Q

ct p Ap

FIG. 5. Horizontal shading: Volume V which beam of focal areaF and divergence t2 occupies in coordinate space. Vertical shading:Volume Vp which same beam occupies in momentum space.

cells in a polarized beam. For an unpolarized beam thevalue of Z is twice that given by Eq. (2.8).

Let d be the degeneracy of the beam or the averagenumber of photons per phase-space cell. For thermalradiation from a blackbody of temperature T, the valueof d is given by

ELLEN HISDAL

S

F

D

FIG. 6. Experimental arrangement to transmit informationabout specularly reflecting sample F to a detector D with the aidof a beam from a source S.

'It is therefore advantageous to refer the informationcontent to a detector of a given resolution.

Because the thermodynamic entropy of the beam andits information content are different quantities, we use,in this paper, the word entropy only in its thermo-dynamic sense and never in the sense in which it is usedin information theory.

3. FLUCTUATIONS IN THE NUMBEROF PHOTONS

The number N of photons in a beam emitted by athermal source which is kept at a constant temperature,is a fluctuating quantity. Using entropy considerations,Einstein'9' 2 0 found, that N has a Gaussian probabilitydistribution for small deviations from N. (It can beshown, that the Gaussian approximation holds towithin 5% as long as A./NN<7.5%.) The probabilityP that N deviates from N by the integer number ofphotons AN can be written in the form:

P=El/ (27r)12je- (N)'/ (2) (3.1)

For the mean square fluctuation 22 Einstein finds

Z2= (A\N)2 =RE1+ (NV/Z)]- (3.2)

In the Wien approximation, d<<l, only the first termof the fluctuation formula is significant

;2 = IV. (3.2 W.)

This term represents the quantu nfluctuations, the onlyfluctuation term obtained according to a classicalparticle picture.

In the Rayleigh-Jeans (R.J.) approximation, d>A>,only the second fluctuation term is left

2- = (N)2/Z (3.2 R.J.)

These are the interference fluctuations obtained accord-ing to a classical wave picture.

The fluctuation formula (3.2) has also been derivedby Purcell2' who showed that it holds for any bosonbeam whose spectral components are independent andhave Gaussian probability distributions. Therefore

19 A. Einstein, Physik. Z. 10, 185 (1909).20 A. Einstein, Sitzber. Preuss. Akad. Wiss. 3(I 1925).21 R. Hanbury-Brown and R. Q. Twiss, Nature 178, 1447 (1956).

(3.1) and (3.2) also hold for thermal radiation from anonblackbody. In this case the right-hand side of (2.9)must of course be multiplied by the emissivity of thesource. For maser radiation the fluctuation formula(3.2) does not hold.22 -2 4

We are interested in the fluctuation of the number ofphotons N,. which are reflected from a sample withreflectance R. We assume an incident thermal beam ofNi photons so that

Nr=RNi. (3.3)

The reflected beam can in no way be distinguishedfrom a thermal beam which contains on the averageN,=-RNi photons. Its entropy and its fluctuationproperties are therefore given by Eqs. (2.9), (2.10), and(3.1), (3.2) when we substitute Nr for N. We thusobtain for the mean square fluctuation in the reflectedbeam

2;,2=N,[i+ (Nr1Z)]=RIVi~l+ (Rzl~Z)]. (3.4)

In the R.J. limit of high intensity beams, it followsfrom (3.2 R.J.) and (3.4)

(3.5)

the rms fluctuation in the reflected beam being propor-tional to the rms fluctuation Xi in the incident beam.

4. INFORMATION CONTENT IN ATHERMAL BEAM

The problem of the information content in a photonbeam has previously been attacked by Gabor4 ,

25 and byJones.26 In order to have a well-defined experimentalsituation, let us consider the arrangement shown inFig. 6.

We illuminate a specularly reflecting surface F of asample with a photon beam from a source S. The sourceis imaged on the sample surface. The beam has a smallsolid-angle opening Q, is linearly polarized and quasi-monochromatic with a frequency v within a smallbandwidth Av. The specularly reflected beam is focusedon a detector D, e.g., a photocell, where it is measuredduring a time interval t. The incident number of photonsduring the time t is Ni. Only Ni, the average number ofphotons in the incident beam, is assumed to be knownexactly.

The detector D is assumed to consist of an array ofsmall detectors, e.g., photocells, which are placed besideeach other. The different photocells measure the re-flectances of different areas on the illuminated part ofthe sample surface. An alternative arrangement con-

" A. B. Smith and G. W. Williams, J. Opt. Soc. Am. 52, 337(1962).

2" R. Glauber, Phys. Rev. Letters 10, 84 (1963).21 L. Malidel, 3e COqfeyewCe interalioiiale d'dealromiqic qilanhi-

que, Paris, 1963 (Dunod Cie., Paris, and Columbia UniversityPress, New York, 1963), p. 1 1.

25 D. Gabor, Phil. Mag. 41, 1161 (1950).26 R. C. Jones, J. Opt. Soc. Am. 52, 493 (1962).

1450 Vol. 55

2;,=R2;j,

November1965 INFORMATION CONTENT OF A BEAM OF PHOTONS

sists of one photocell and a scanning incident beamwhich illuminates only a small area of the sample sur-face at a time.

The information obtained at the detector about thereflectances of different parts of the sample will dependon the degree of detail which the photocell array canresolve. If there were no fluctuations, the maximuminformation would be obtained according to Eq. (1.12)when we use photocells with surface areas Fmin.=X2/&each, which are closely stacked. We would then measurethe reflectances of Zi object areas or lateral cells whereZi is given by (1.11). Similarly the incident photonbeam would be exploited to the utmost according to(2.3) if each detector could measure the number ofphotons received during the time Ztmin= 1/Ar. Wewould then have the maximum resolution allowableaccording to the uncertainty principle, so that we couldmeasure the number of photons in every phase-spacecell. In a practical case it is of course very difficult toachieve such a high resolution.

We therefore refer the information content of thebeam to a detector of a given resolution. Let z be thesmallest number of phase-space cells which the detectorcan resolve. Such a group of z phase-space cells ormicrocells we shall call a macrocell. N is the totalnumber of photons in the beam, and n the number ofphotons per macrocell with the subscripts i and rreferring, respectively, to the incident and reflectedbeams. If r is the number of macrocells contained in thebeam, we have

(4.1)

The expected information content in a reflectionmeasurement is therefore2 7

jo= lniii. (4.5)

Here we have assumed nii>»l, so that n7+l can bereplaced by fi in the logarithm.

Equation (4.4) gives the expected measurable valuesof the reflectance r if there had been no fluctuations.However, because of the fluctuations in the number ofphotons, we can only say that when we measure n,reflected photons, the most probable value for thereflectance is n,/ni but there are also certain prob-abilities that the sample has other ones of the reflec-tance values of (4.4).

Let Pl (n,) be the probability that r has the value

r1= I/ni,

where I can take the values

l==0, 1, 2, *-, f i,

(4.6)

(4.7)

when we measure nr reflected photons.Assuming that all values (4.4) of r have the same

a priori probabilities, we derive from the last section thatPI is given by (3.1) when we replace zAN by l-n, and 2by the fluctuation a (1) in the number of photons in amacrocell with an average number of photons 1. Thisgives for PI

Pi (nr) [= E1/ (27w)ia (l)]e,[lnr)2/12d 2 (1)] (4.8)

wherea2 (l) = l[+ J (l/z)] (4.9)

We assume that the detector can measure the exactnumber n, of reflected photons per macrocell, and thatthe temperature of the sample is so low compared withthe temperature of the beam, that the thermal radiationemitted by the sample can be neglected.

Let us start with the case == 1, Z= z, so that thewhole beam is considered as one macrocell; i.e., wemake only one measurement of the reflectance of thewhole sample.

The final result for the information content i1 in sucha measurement can then be represented as the differencebetween two terms:

il = io- iol(4.2)where io is the a priori expected information contentwhen we neglect fluctuations, and jo, is the informationloss due to "noise" or fluctuations in the incident beam.

The incident number of photons is Ni= ni, and theexpected number of equally probable possibilities for areflection measurement, assuming no fluctuations, isfzi+ 1 corresponding to

rf=c0, 1, 2, t v of the(4r3)

reflected photons and to values of the reflectance

r = 0, l1/ii, 2/nhi, -* *, X ilfi. 44

We thus know the probabilities Pi (nr) of differentvalues rj of r when we measure n, reflected photons. Ifwe could have determined the exact value of r, theinformation contained in such a determination wouldbe2 7

(4.10)

jol is therefore a measure of our ignorance of r after ameasurement of n, photons, and the information contentin such a measurement is given by (4.2) when we sub-stitute for io and jo, from (4.5) and (4.10).

We shall now make certain approximations to com-pute jol, Eq. (4.10): Approximation 1: We assume thatthe width of all the Gaussian curves, PI(,r), contribut-ing to (4.10) is the same and equal to the width of aGaussian curve with an average photon number nr.Equation (4.9) is therefore replaced by

(4.11)

This approximation is valid for o- (lr)<<«r. As explainedin Sec. 6, the approximation is unnecessary in the R.J.limit. Approximation 2: We assume that the sum in

27 C. E. Shannon and W. Weaver, The Mathematical Theory ofCommunication (University of Illinois Press, Urbana, 1949). Weuse throughout this paper natural logarithms in the informationformulae. To obtain the information in bits, multiplybylog2e= 1.44.

1451

�=z1z=9iIii'l,

i0i (n,) = - 2;Pi lnP1.

a 2 (1) = a'(n,) = nEl + (n,/z)].

ELLEN HISDAL

(4.10) can be replaced by an integral, where wiintegrate over the continuous variable 1. Becauinterval between successive values of I in the sumis unity, this sum goes directly over into the in

jo(nr)= -JP, lnP~dl.

This approximation is valid within 1 part in 1a (n,) = 1 and becomes better for larger a. A pproxi.3: We assume that the limits of the integral can beas

- co <l-On,< 0°

instead of the limits -it, to fti-it, given by (4.7last approximation can be shown to hold withinbetter if t,> 2a (or) and fii- ,Ž> 2a (n,).

With these approximations we obtain for jol (r,(4.12), (4.8), (4.11), and (4.13)

jOl(n.)= ( ln2 2)r1o] e-x2I2a2dx

1 X.+- I x2ee-z2/dX2

20.2 J2 _

where we have replaced I-n,. by x and -(aIr) byThe first integral equals (27r)o- and the s

(27r)*o3. This gives for jol

a nowse the(4.10).tegral

error introduced into (4.17) and (4.18) by this proceduredoes not exceed a(ni)/iii.

The integral in (4.18) is evaluated by using 1+ (x/z)as a new variable, obtaining

(41) ln[l+ (n,1z)]== (z/fti)[1+ (jti/z)j ln[1+ Oill)- 1.(4.12)(4.19)

04 f or The final expression for Joi is now obtained fromfation (4.16), (4.17), and (4.19)

taken 3oi==In(27r)l+-[lnin+ (l/d)(l+d) In(1+d)-1],

(4.13) (4.20)where we have replaced gi/z by d. In the Wien approx-

)-The imation, (4.20) becomes

Joi= ln[(27r)(nii)] = ln[(27r)' oj. (4.20 W.)

fro In the R.J. approximation it becomes:

joi= k[(2ir/e) (nQ7i (z)i)] = In[(2wr/e)'oij, (4.20 R.J.)

where o-i is the rms fluctuation of ni.

The average information in a reflection measurement(4-14) is given according to (4.2), (4.5), and (4.20) by

T.

econd

-T = Elnfii-In (27r/e)- (1/d) (1+d)ln (1+d)]

i = in (fi/27r) = In (iiil (27r)'oi)

is = InEz/ (27rle)]'l= InE,9j/ (27r/e) o-j].

(4.21)

(4.21 W.)

(4.21R.J.)

joj(,nr)= =+ln[(27)1o(n,.)], (4.15)

where o(n,.) is given by (4.11).We have now a measure for the information il

[Eqs. (4.2), (4.5), and (4.15)] obtained in a measure-ment of nt, reflected photons with a beam having anaverage number Ti of incident photons.

The average information in a reflection measurementwhen all values (4.4) of r are assumed to be equallyprobable a priori, is obtained by averaging (4.15) overall values (4.3) of ii,. From (4.15) and (4.11) follows

Jo'= 2±ln (2wr)i+-{lnzr+ln (1+flr/Z)}. (4.16)

If ii>>1 we can approximate the average values inthe square brackets by integrals, using ir as a continu-ous variable. We then obtain

and

1 r /iInn = -| Inx d += Ini-) 1,

In + J =-J In 1+- dx.Z) ni oZ

Actually we should have averaged only over thosevalues of n, for which approximation 3 is valid, namely2a (n,)<rt<nii-2or(n,). It can, however, be shown thatfor a (n)<<«fi we may average instead over values of i,.from 0 to fit. The order of magnitude of the relative

The last expression of (4.21 W.) shows, that in theWien approximation the fluctuations are equivalent toassuming that we can measure the number of reflectedphotons in steps of (27r)ai-, while the first expression(4.21 W.) for is shows that the information is inde-pendent of the number of microcells z; it depends onlyon the incident number of photons.

In the R.J. approximation it follows from (4.21 R.J.)that the fluctuations are equivalent to assuming thatwe can measure the number of photons in steps of(27r/e)1ai. The information is independent of d, itdepends only on the number of phase space cells z. Thisis a result of the fact that the fluctuations increasedirectly as the number of photons for a given z.

We now go back to the case that the detector canresolve t=Z/z=NRl/ii macrocells [Eq. (4.1)]. Thenumbers of photons it in the different macrocells areindependent for thermal radiation although this maynot be the case for a maser source. For thermal beamsthe total a priori information Io, the total ignoranceafter a measurement Jo0 and the information contentI when fluctuations are taken into account are thensimply obtained by multiplying to, jol, and is respec-tively by the number of macrocells P. Thus

II= l1. (4.22)

The complete formulae for I1 are given in the first rowof Table:l. Since iij=z d, we see, that the information

1452 Vol. 55

November1965 INFORMATION CONTENT OF A BEAM OF PHOTONS 1453

TABLE I. Al represents information in a thermal beam with degeneracy d, referred to a detector giving ni incident photons per macro-cell and r macrocells (see Eq. (4.1). I1 max is maximum information obtainable with same beam; Si is average number of photons inncident beam during measuring time. Zm is number of microcells per macrocell; i*im is average number of incident photons per macrocell,both for a detector giving 1,=.I max. elmIl/I, maX is information efficiency of detector system with respect to given beam.

Wien R. J.

General case d<<1 d>>1

iA '2¢lnjIi-ln(2ir/e) - (1/d) (I-+d)ln(I-+d)] ln[A/27r] r lnCz/ (2r/e)]

1 Max Ni/ (47r[ (1 +d)l+(l/d)]) NS/4ree= NSi/34.15K<Z/47re Z/47r=aZ/12.57<<«i/47r

Zim 2r(1/d)1(+d)l+((ld) 2lre/d>>17 227r =6.28

flin 27(l+d)l+(I/d) 2re= 17.08 2rd>>6.28

ln(Ai/E (27r/e) (1+d)l+(Ild)] } In (nAi/27r) ln~z/ (27r/e)]

Ai/E27r(1 +d) I+(l/d)j Ai/27re z/27r

obtainable with a beam of a given d, or a given frequencyand radiance per frequency interval, depends on thenumber z of microcells which each individual detectorcan resolve. The information depends only on z, not onthe individual factors zi or Zt according to (2.5). It istherefore immaterial for the information contentwhether the lateral or the time resolution of the detec-tors is biggest as long as z is kept constant.

5. MAXIMUM INFORMATION CONTENTIN A THERMAL BEAM

Suppose now that we have a beam of a given de-generacy d and a given number of microcells Z [Eq.(2.5)], i.e., we assume that Z, the total average numberof incident photons Ni and

d=N9/Z=ii/z (5.1)

are fixed quantities. We may now ask what resolutionthe detector system must have or how many macrocellsP it must resolve if I1 is to have its maximum valueIimax. This value of 11 then gives us the maximuminformation which may be obtained with a beam of thegiven d and Z, where Z, according to (2.5), is againdetermined by the parameters v, AP', F, 2, and I of thebeam.

According to Table I we have for Ii

I1= "vEln (SV/l)- ln (2-/e) - (1/d) (1+d) ln (1+d)].(5.2)

We now find what value Rm of r gives the maximumvalue I1 max of Ii when r is varied and all other quanti-ties are kept constant. From (5.2) we obtain

d~ljdD = I1/D- , (5-3)giving

Ii max='4P. (5.4)

We see, that the maximum information obtainable witha given beam corresponds to 2 information unit or0.72 bits per macrocell. I1 max is obtained by substitut-ing Am from (5.4) into (5.2). The result for I1 max as well

as for ini,, the average number of photons per macrocelland Zm, the number of microcells per macrocell whichmaximize I1 are summarized in Table I.

This table shows that in the Wien case of smallintensities we obtain maximum information with adetector system which gives an average number27rem17 incident photons per detector and scanningperiod, while in the R.J. case the best detector systemis such that each detector resolves 27r microcells.

The physical interpretation of the variation of I1 withv for a given beam is the following: Starting with asmall number of macrocells I, we can increase v byincreasing the lateral resolution or the scanning rate,thereby increasing the information. However, it is notworth while to increase the resolution or scanning ratebeyond a certain limit because for P>r, Z<Zm, theinaccuracy due to the fluctuation noise Jo, increasesfaster than the increase of the information Io owing tothe higher resolution.

For a given beam and detector system, we can definethe information efficiency El of the detector system withrespect to the beam by the equation

el= Ia/1h max- (5.5)

El is listed in the last row of Table I.

6. HIGH INTENSITY, NONTHERMAL SOURCES

The above considerations are valid for beams fromthermal sources. Under certain conditions it is, how-ever, possible to apply some of the results to the case ofmaser sources.

Let us assume that the fluctuations in the number ofphotons in a maser source also have a Gaussian prob-ability distribution of the type (3.1) although the meansquare fluctuation will no longer be given by (3.2). Thedegeneracy d is for a maser source of the order of107-1012 (Ref. 28). We saw in Sec. 3, Eq. (3.5), that forthermal sources in the R. J. limit, d>>», the fluctuationsin the reflected number of photons follow the fluctu-

28 L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).

1453

ELLEN HISDAL

ations in the incident beam, so that we always have,r= rni. This is also the case for maser radiation

because of its high intensity, and we can always cor-relate a fluctuation At, in the reflected beam with afluctuation

a~n1= (1/r)A~nr (6.1)

in the incident beam. The information content in ameasurement of i,. reflected photons from a macrocellcan be computed as before and leads to (4.15) forjoil(nr) to be substituted into (4.2). a (nr) in (4.15) canbe replaced by

o(fir)= ( 0i r/ ('i)n(i) (6.2)

according to (6.1), giving

jol (or) = 2+ln[(27r)i (nt/fi)o (ne)]. (6.3)

Equation (6.3) can also be derived directly byascribing the uncertainty in our knowledge of thereflectance to an uncertainty in the incident number ofphotons. The approximation (4.11) is then unnecessary.(6.3) can now be averaged over all possible values of n,.Using (4.17), we then obtain from (6.3)

J.01 = In[ (2,r/e)4l (,it)], (6.4)

which gives for the average information content in themeasurement of a macrocell according to (4.2) and (4.5)

il = In (nii/ (27r/e) lai). (6.5)

It has been shown by Mandel,24 that for an idealmaser which oscillates in one mode only, the probabilitythat the beam contains fi+An photons follows a Poissondistribution.

For fi>>1, and A~n/i<<1 the Poisson distributiontends towards a Gaussian with rms deviation ft. We cantherefore put

vi= (m)~, (6.6)

and using (6.5) we obtain

!,= lnEi/ (2wr/e)]J (6.7)

in this ideal case.A beam from a maser oscillating in one mode only,

contains only one ray in the sense of Sec. 1, only onelateral cell can be resolved when the ray is focused onthe sample. However, we can scan the sample, so that

we use the beam of fij photons v times, obtaining anaverage information

(6.8)

The maximizing of 71 is here limited by practicalconsiderations, namely the highest obtainable scanningrate. According to Eq. (1.8), the ray emerging from themaser has an angular width of the order of X/a, wherea is the diameter of the resonating cavity. For X; 6000 Aand at 1 cm we get an angular dispersion of 6X 10-5 radfor the maser ray. If we assume that the scanningmotion is produced by a mirror drum rotating with500 rps, it follows that we can scan P=27rX2X500/6X 10-5= 108 macrocells/sec. 10-8 sec is also of the orderof the limiting time resolution obtainable with a vacuumphotocell.

According to Lipsett and Mandel,29 the coherencetime for one mode of a ruby laser is 0.5X 10-6 sec. Wehave now the case that a macrocell, determined by thehighest practically realizable scanning rate, is a fraction10-8/0.5 X 10-6=2>X 10-2 of a microcell. Assuming5X107 photons/microcell in a ruby laser,2 8 we havenfii=5X107X2X 10-2= 106 photons/macrocell or scan-ning period. Substituting for r and fiu into (6.8) weobtain

I,= 2 X 108XIn(106 /27re) = 5.5X 108 (6.9)

for the information per second assuming continuousoperation of the laser mode.

If we had used the green line from low pressure 198Hgemission, we would have had a degeneracy of the orderof 10-3 and Av 109 cps.28 This gives 10-3 photons perlateral cell per 10-9 sec, or NV -106 photons/sec. Themaximum obtainable information per lateral cell persecond is, according to Table I:

(6.10)

We see that with the ideal maser, the information isincreased by a factor of the order 104 per lateral cell ascompared with the mercury source. This increase is dueto the fact that a much higher scanning rate can be usedwith the maser because of the larger photon density andsmaller relative fluctuations. The order of magnitude ofthe information is, indeed, determined by the scanningrate in the maser case.

29 M. S. Lipsett and L. Mandel, Nature 199, 553 (1963).

1454 Vol. 55

1i = M = 2' � InEfli/ (27r/e) ].

11 .. x= Nj/4re-: 101/34;zz: 3 X 104.