February 1980 / Vol. 5, No. 2 / OPTICS LETTERS 73

Coherent detection of partially coherent sources

D. McGuire

U.S. Army ERADCOM, Harry Diamond Laboratories, Adelphi, Maryland 20783

Received August 15, 1979

The mean square of the beating component of the photodetector output current of a coherent optical or infraredreceiver, illuminated by partially coherent quasi-monochromatic radiation, is expressed in terms of the mutual in-tensity functions of the signal and local reference fields on the detector's active surface. The result is of interestfor heterodyne detection problems in which partially coherent sources occur. The use of the result is illustratedand is compared with a known special case by treating the heterodyne detection of a spatially incoherent source inthe far-field zone of the receiver aperture, a problem previously discussed by H. T. Yura [Appl. Opt. 13, 150 (1974)]from a different viewpoint.

We have been investigating several theoretical prob-lems connected with the coherent detection of quasi-monochromatic radiation sources that exhibit varyingdegrees of spatial coherence. In general, we conceiveof the following problem. Given a spatially extendedquasi-monochromatic source, a, whose coherenceproperties are known [e.g., through a specification nearthe source of the mutual coherence function (MCF) ofits field], and given a coherent receiver system at somedistance, consisting of a local radiation source (here weunderstand a quasi-monochromatic source whose meanfrequency differs slightly from that of a), a mixer, anda photodetector, relate the amplitude of the beatingcomponent of the photodetector output to the coher-ence properties of a.

Several special cases of the foregoing problem havebeen treated by Yura.1 He considers the limiting casesof complete spatial coherence for a and complete spatialincoherence.

Most of the elements that are needed for a generalformulation of the problem are available from standardpartial-coherence theory. If the MCF of the sourcefield is known over a surface surrounding the source, thewell-known Zernike propagation formula 2 can be usedto determine the MCF over the receiver aperture. (Weare not considering atmospheric turbulence effects; forthis, see Fried. 3 ) If the receiver uses an axially sym-metric optical system, the effect of the optics can bedetermined by using well-known conjugacy results. 4

Given the mixing geometry and the MCF of the localradiation source, it is possible in principle to determinethe MCF of both a's field and the local-source field overthe photodetector's active surface.

To determine the mean square of the beating com-ponent of the photodetector output current requires theevaluation of a fourth-order correlation function in thetwo fields for points on the detector surface. If the twofields are effectively statistically independent, then itshould be possible to reduce the needed fourth-ordercorrelation function to a sum of products of second-order correlation functions that do not mix the twofields. This would make it possible to determine the

heterodyne signal power from a knowledge of only theMCF's of the two fields.

We consider the typical heterodyne detection situa-tion and show that the foregoing type of reduction canbe accomplished. A formula is obtained for the heter-odyne signal in terms of the mutual-intensity functionsof a's field (henceforth called the signal field) and thelocal source field on the active photosurface. This'formula permits analysis of the general problem we haveposed and is applicable to heterodyne detection prob-lems in which one'or more sources are most appro-priately characterized by the mutual coherence func-tion, e.g., background sources that could be present forcoherent infrared or millimeter-wave radars. As anillustration, we analyze the heterodyne detection of aspatially incoherent source in the far-field zone of thereceiver aperture, a problem treated by Yural from adifferent point of view. Cohen's 5 results for wave-front-misalignment effects in the mixer also follow fromour formula.

We use the partial coherence-theory notations es-tablished by Born and Wolf 2 Specifically, we denoteby Ps (rr,r2;i) and PL (rl,r2;r), respectively, the MCF'sof the signal and local-source fields for the arbitraryspace-time points (rjn) and (r2,r). [The mutual-coherence function, r(rj,r 2;i), of a field is defined byP(ri,r2 ;'r) = (V(r1,t + -r)V*(r2,t), where V(r,t) de-notes the analytic signal associated with the field atposition r and time t and ( ) denotes a time averageover many oscillation periods of the field.] The corre-sponding mutual-intensity functions, Js and JL, aredefined by

Js(ri,r 2 )- rs(r=ar2;0),JL(rl,r 2 ) = PL(rl,r 2 ;O), (1)

and, in the quasi-monochromatic approximation, wehave

rs(r 1,r2 ;') = JS(r 1 ,r2)exp(-iw-r),

rL(rl,r 2;r) = JL(rl,r2)exp(-iwLT), (2)

0146-9592/80/020073-03$0.50/0 ©) 1980, Optical Society of America

74 OPTICS LETTERS / Vol. 5, No. 2 / February 1980

where w and cL are the mean angular frequencies of therespective fields.

Let Vs and VL denote the analytic signals associatedwith the signal and local-source fields, respectively. Onthe active photosurface, we assume that they have theform

Vs(r,t) = Es(r,t)exp&-ict),

VL(r,t) = EL(r,t)exp-icoLt), (3)

where wo = C-L - w is very small compared with both CLand c and where the complex amplitudes ES and ELvary slowly with time compared with cos w0t. Theforegoing assumptions typically hold for heterodynedetection and make it possible to take time averagesover a period that is simultaneously long compared witha beat period and short compared with a period of sen-sible variation of ES and EL. We will denote such anaverage by either an angle bracket or an overbar,whichever is more convenient.

An immediate consequence of our assumptions is thatVs and VL are effectively statistically independentrelative to the time average in question, since one readilyverifies that, for example, ( VSVL* ) and (/Vs) (VL*)are both sensibly zero.

The details of an analysis of the beating component,ib, of the output photocurrent depend on the type ofphotodetector that one has in mind. For photocon-ductive detectors and photomultipliers (in general,photoemissive detectors), one has

ib cc J dA 7 Re(VsVL*), (4)

where qj is the quantum efficiency, * denotes complexconjugation, and the integration is carried out over theactive photosurface area, AD .36 Absorbing the pro-portionality constant into 7 (and denoting the new etaby '), we have

ib 2= |S dAq' Re(VsVL,) (5)

Averaging Eq. (5), we obtain

(ib2 ) JAD dAq' Re(VsVL*)| (6)

which can be written in the expanded form

(ib2 ) = f d2r, TA d2 r2 ij'(r1)77'(r2 )e.AD fAD

X Re[Vs(r 1,t)VL*(ri,t)]ReIVs(r 2 ,t)VL*(r2,t)]. (7)

We will now show that the time-averaged quantity inthe foregoing integrand (denote it by L) equals 1/2Re[JS (r2, rl)JL (r1, r2)].

Using the fact that

Re(VsVL*) = 2 (VSVL* + VS*VL), (8)2

we may write L as

L = I Re[< (Vs(ri,t) Vs(r2,0 VL* (rl,t)VI,* (r2,t))

+ ( Vs(r2,t)Vs*(rlt)VL(rl,t)VL*(r 2,t))]. (9)

The first term in the bracket (denote it by f) is

42j f dtEs(r1,t)ES(r2,t)

X EL*(rl,t)EL*(r 2 ,t)exp(2ico0t), (10)

where (- T, T) is the time interval for the averaging andEqs. (3) have been used. Thus f is proportional tosin(2w0T)/2w0T. The second term in the bracket (de-note it by s) is just [again using Eqs. (3)]

s = ES(r2,t)Es*(rlt)EL(rlt)EL*(r 2 ,t), (11)

so f is clearly negligible compared with s. Therefore,L = 1/2 Re s, and all that remains to be done is to rein-terpret the factors composing s.

Observe that

(Vs(r 2,t)Vs*(ri,t)) = s(r2,t)Es*(r1,t) (12)and

(VL (r1,t) VL* (r2 ,t) = = EL (rl,t)EL* (r2 ,t) (13)

Time averages do not appear on the right-hand sides ofEqs. (11)-(13) because we have assumed that the E's aresensibly constant over the averaging period. Since thetime averages involved are clearly over an extremelylarge number of optical periods, the basic definition ofthe scalar MCF (Ref. 7) shows that the left-hand sidesof Eqs. '(12) and (13) are rs(r 2,ri;O) and rL(rl,r2;0),respectively. Equations (1) now show that s =

Js(r 2rl)JL(rl,r 2).To summarize, we have shown that

(ib 2 ) = - S d2 r1 S2' AD A,

X d2r 2 iq'(ri)7,'(r 2 )Re[Js (r2 , rl)JL (r1 ,r2 )], (14)

which expresses the fourth-order correlation functionsought in terms of the second-order correlation func-tions Js and JL. In general, JL can be determined onAD from the configuration of the receiver and thespecifications of the local radiation source. On theother hand, Js can be determined in principle from theMCF of the signal source over its boundary by using(generally difficult) coherence-propagation calcula-tions.

Yura1 treats the general problem outlined at the be-ginning of this Letter for the case in which the source,r, exhibits complete spatial incoherence. To compareour result, Eq. (14), with Yura's analysis, it is convenientto consider a further specialization of the problem, forwhich Yura develops explicit analytical formulas for themean-square power in the beating component of thephotodetector output.

We consider a flat, circular, quasi-monochromaticsource of radius a, radiating into a half-space and ex-hibiting complete spatial incoherence. We assume aplane, circular receiver aperture of radius b parallel tothe source plane, as illustrated in Fig. 1. If R, the dis-tance from the source to the receiver, is sufficiently largecompared with both a and b, the Van Cittert-Zerniketheorem can be used to determine the mutual intensityof the source field over the receiver aperture as

February 1980 / Vol. 5, No. 2 / OPTICS LETTERS 75

x I*R HI

SOURCE RECEIVER APERTURE

Fig. 1. Source and detector geometry.

where a denotes the source area (i.e., x varies over thesource). Substituting Eq. (18) into Eq. (17), there re-sults

(ib2 ) = 2f ISI X d2x

X A d2rexpvik x-rj2fA D R

, (19)

Js(r 2,ri) =7ra'2Is

R2

2J1

(15)haR

where ri and r2 are any two points in the receiver ap-erture, Is is the source intensity, which is assumeduniform, J4 is the first-order Bessel function of the firstkind, and k is the wave number of the radiation.Equation (15) assumes that R >> kb2, i.e., that thesource is in the far-field zone of the receiver aperture.

We next assume that the receiver aperture is beingilluminated by a plane-wave local radiation sourcewhose equiphase surfaces are parallel to the apertureplane. Letting IL denote the uniform value of the in-tensity of this field over the aperture, it follows thatJL (r1,r2) = IL. Thus

Re[Js(r 2,rO)JL (ri,r2 )]

7ra 2ISIL

R2 haR 2-r

. (16)

This result and Eq. (14) can now be used to calculate(ib 2), assuming that the receiver aperture is the activephotosurface (See Ref. 1, Sec. II, paragraph 2).

Take eit in Eq. (14) to be independent of position.Substituting Eq. (16) into Eq. (14) then yields

ib2

) 1 (71 2 XSA 2 SAD

2J1 (ka r2 - rix d 2r 2 R (17)

ha.R r2 - ri

By a standard integral representation for Bessel func-tions, it follows that

2Jj kar2-ri) 1

h~~~~ia2 Jar - X dxr2- r]

x d2x exp |i R-x - (r2 - rl)l,(8

where the integration variable r ranges over the receiveraperture. The integrations in Eq. (19) are readily done.The result is

(i5 2) = 1 (2rn)2 1 1

x -J0 2 ( ) J2(kab)j (20)

where Jo is the zeroth-order Bessel function of the firstkind.

To compare Eq. (20) with Yura's corresponding result[Ref. 1, Eq. (25)], it is necessary to divide Eq. (20) by itsasymptotic value as a - 0 and Is - - in such a waythat wra2ls remains fixed; more specifically, we mustdivide Eq. (20) by the limit

lim 1 )w' ra2

X 12 [i- Jo2 (ktb) J12 tb

which equals

I (2wr'b)2 2 k( 022 l ~| ra 1ISILl R

By performing the division, Yura's reduction factor /B[see Ref. 1, Eq. (25)] is obtained.

In closing, it should be pointed out that Eq. (14) isalso valid in the simple case in which all fields areplane-wave monochromatic fields. It is straightfor-ward, for example, to obtain one of Cohen's 5 results[specifically, Ref. 5, Eq. (5)] for wave-front-misalign-ment effects in the mixer from Eq. (14).

References

1. H. T. Yura, Appl. Opt. 13,150 (1974).2. M. Born and E. Wolf, Principles of Optics, 4th ed. (Per-

gamon, Oxford, 1970), Chap. 10, pp. 516-518.3. D. L. Fried, Proc. IEEE 55, 57 (1967).4. Ref. 2, pp. 519-520.5. S. C. Cohen, Appl. Opt. 14,1953 (1975).6. A. Yariv, Introduction to Optical Electronics (Holt,

Rinehart and Winston, New York, 1971), pp. 276-278,285-287.

7. Ref. 2, pp. 494-500.

-

2J, r2 - ri