the measurement of radiance and the van cittert–zernike theorem

The measurement of radiance and the vanCittert–Zernike Theorem

Roland Winstona,*, Yupin Suna, Robert G. Littlejohnb

a The Enrico Fermi Institute, The University of Chicago, 5640 South Ellis Avenue, Illinois, Chicago, IL 60637, USAb Department of Physics, University of California, Berkeley, CA 94720, USA

Received 17 December 2001; received in revised form 26 March 2002; accepted 10 April 2002

Abstract

We demonstrate a remarkable analogy between the measurement of radiance and the well-known van Cittert–

Zernike Theorem. � 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

It is obvious that the measurement of radiancecan be understood in terms of the statisticalproperties of the electromagnetic field and theproperties of the instrument. In an accompanyingpaper [1] we attempt to bridge the gap betweenhigh resolution radiometric measurements and anaccessible theory. In this contribution, we exhibit asymmetry between the incoherent source whoseradiance is being measured, and the detector whosesignal is the measurement. There results a re-markable analogy between the result of measuringradiance and the van Cittert–Zernike Theorem.

2. The van Cittert–Zernike Theorem

The well-known van Cittert–Zernike Theoremstates that for an incoherent, quasi-monochro-

matic source of radiation, the equal-time degree ofcoherence (2-point correlation function) Cðr; r0Þ isproportional to the complex amplitude in a certaindiffraction pattern: the amplitude at r formed by aspherical wave converging to r0 and diffracted byan aperture the same size, shape and location asthe source [2]. The source could, for example be athermal black body followed by a filter that selectsa small wavelength range. A familiar geometry is acircular source. Then, apart from a normalizingfactor, Cðr; r0Þ in a transverse plane becomes thewell-known Airy diffraction amplitude

Cðr; r0Þ ¼ ðconst:ÞF ðkshsÞ; ð1Þ

where F ðxÞ ¼ 2J1ðxÞ=x, k ¼ 2p=k, hs is the anglesubtended by the source at r or r0, and where s ¼jr� r0j. Recall that Cðr; r0Þ has its first zero at s1 ¼0:61k=hs. For a numerical example, we considerterrestrial sunlight. Then hs is 4.7 mrad, so that fork ¼ 0:5 lm, s1 is approximately 65 lm. This is thescale of the transverse correlation of sunlight.The function F ðxÞ ¼ 2J1ðxÞ=x will play an im-

portant role in the following. We note here its

15 June 2002

Optics Communications 207 (2002) 41–48

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*Corresponding author. Tel.: +1-773-7027756; fax: +1-773-

7026317.

E-mail address: [email protected] (R. Winston).

0030-4018/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

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principal properties. First, we have F ð0Þ ¼ 1.Next, we note the limit,

lima!1

a2F ðasÞ ¼ 4pd2ðsÞ; ð2Þ

where s ¼ ðx; yÞ is a two-dimensional vector ands ¼ jsj.

3. Connection with measuring radiance

In a previous paper [3], we examined the rela-tionship between the generalized radiance and themeasuring process. We showed how this processcan be quantified by introducing the instrumentfunction, which is a property of the measuringapparatus [1]. We showed that the result of themeasurement is represented by the quantity

Q ¼ TrðMMCCÞ; ð3Þ

where MM is a nonnegative-definite Hermitian op-erator that characterizes the measuring apparatus,and CC is the 2-point correlation function of theincident light, viewed as an operator. The instru-ment function itself is a coordinate representationof the measurement operator MM , for example, itsmatrix element or its Weyl transform. The Weyltransform maps an operator to a Wigner function(for a discussion of the Wigner–Weyl formalism inoptics see [3]). It is appropriate to associate Q withthe signal. We then derived an analytical form forthe instrument function for a simple radiometer inone space dimension.One difficulty in using Eq. (3) is that it may not

be easy to compute the instrument function. Al-though the one-dimensional calculation in [3] wasnot too hard, we do not expect it to be easy tocompute the instrument function for many realisticradiometers, which are two-dimensional in cross-section and which may have complicated geome-try. Therefore we have considered other means fordetermining the instrument function. In a previouspublication [4] we considered the possibility thatthe instrument function could be measured. In thispaper we present an alternative approach. That is,we point out a physical interpretation of the in-strument function which is similar to the vanCittert–Zernike Theorem. We do this initially by

working through the example of a simple ‘‘pin-hole’’ radiometer, and then we comment aboutgeneralizations.Radiance is the power per unit volume in phase

space. Therefore an instrument for measuring ra-diance (called a radiometer) has to select a windowfunction in phase space. For measurements closeto the diffraction limit, the exact shape of thewindow function is not critical. For this reason weexamine a simple radiometer, illustrated in Fig. 1.The dotted line in the figure is the axis of the ra-diometer. Light enters from the left and passesthrough the circular pinhole of radius a. It thenpasses through a drift space of length L, beforepassing through another circular aperture of ra-dius b. We assume L a; b, so the rays are par-axial. The detector is assumed to measure the totalpower passing through the aperture b and can bethought of as composed of tiny, densely packed,independent absorbing particles (which is a fairlygood approximation to what commonly used de-tectors like photon detectors, thermal detectors orphotographic film do).As explained in [3] the effect of the radiometer

on the radiation field is described by the operator

PP ¼ AAðbÞDDðLÞ AAðaÞ ð4Þ

which maps the wave field at the entrance aperturea into the wave field at the exit aperture b. Here

Fig. 1. A pinhole radiometer. The dotted line is the axis. Light

enters from the left, passing through circular pinhole a, drift

space of length L, and finally circular aperture b.

42 R. Winston et al. / Optics Communications 207 (2002) 41–48

AAðaÞ is the aperture or ‘‘cookie cutter’’ operatorrepresenting the pinhole, DDðLÞ is the Huygens–Fresnel operator representing the drift space, andAAðbÞ is the aperture operator for the aperture b. Inthe approximation L k the drift operator hasthe kernel (or matrix element),

hr?jDDðLÞjr0?i ¼ � ikL2p

eikR

R2; ð5Þ

where r? ¼ ðx; yÞ, r0? ¼ ðx0; y0Þ, r ¼ ðx; y; zÞ, r0 ¼ðx0; y 0; z0Þ, L ¼ z� z0 and R ¼ jr� r0j. Here z is thecoordinate along the optical axis and it is assumedthat z > z0. If in addition we assume rays are par-axial ðr?; r0? LÞ, then the kernel can be writtenas

hr?jDDðLÞjr0?i ¼ � ik2pL

eikL expik2L

jr?�

� r0?j2

�:

ð6ÞThis is a two-dimensional version of Eq. (15) of [3].(Note that the equation was incorrect, because itomitted a phase factor of eik0l, in the notation ofthat paper. The error does not affect the otherconclusions of that paper, however.)Following the methods of [3], the instrument is

represented by the operator MM ¼ PP yPP whose ma-trix elements are

hr?jMM jr0?i ¼kL2p

� �2 Zjr00?j6 b

d2r00?exp½ikðR1 � R2Þ�

R21R22

;

ð7Þwhere R1 ¼ jr0 � r00j, R2 ¼ jr� r00j, and the inte-gration is over the detector area. In Eq. (7) thetransverse variables r? and r

0? are understood to lie

in the entrance plane (the pinhole), so thatr?; r0? 6 a. If this condition is not met, the matrixelement is understood to be zero.Expression (7) is identical (up to constants) to

the mutual intensity evaluated at aperture a of auniform, delta-correlated source at aperture b(the location of the detector). Thus, to form thevan Cittert–Zernike interpretation of the instru-ment function, we replace the detector (at aper-ture b, in this example) by a delta-correlatedsource, and measure the radiation field emanatingfrom the entrace aperture of the instrument (thepinhole in this example). The instrument function

(at a given plane) is then proportional to theamplitude at r0 formed by a spherical wave con-verging to r and diffracted by an aperture thesame size, shape, and location as the detector.The detector emulates a delta-correlated source.In a sense, this model involves running the radi-ometer backwards (exchanging the detector for asource).This interpretation applies also to other radi-

ometers, for example, those with lenses. The es-sential property is that the operator PP y shouldserve as a propagator for wave fields travelling tothe left (in the negative z direction), just as PP servesas a propagator for waves travelling to the right.The situation is rather much like time reversal inquantum mechanics. Not all time evolutions inquantum mechanics are time reversal invariant(only those for which the Hamiltonian commuteswith time reversal). In the case of optical fields, it isa kind of ‘‘z-reversal’’ that we need. Lenses, driftspaces and apertures are ‘‘z-reversal invariant,’’ aslong as evanescent waves can be ignored. We re-mark that the same conditions apply to the usualvan Cittert–Zernike Theorem.

4. Near-field and far-field limits

It is useful to examine the pinhole radiometerand the formalism we have presented in two lim-iting cases. First, however, we explain some nota-tion regarding the correlation operator andfunction (see Ref. [3] for more details). The mutualcoherence is defined by Cðr; r0Þ ¼ wðrÞw�ðr0Þ, wherethe overbar means a statistical or ensemble aver-age, and where we use a scalar model for the wavefield w, which in electromagnetic applications canbe loosely identified with one of the components ofthe electric field. When z ¼ z0 we can associate themutual coherence with an operator CCðzÞ byCðr; r0Þ ¼ Cðr?; z; r0?; zÞ ¼ hr?jCCðzÞjr0?i, so that

CCðzÞ ¼ jwðzÞihwðzÞj. Thus

TrCCðzÞ ¼Zd2r?jwðr?; zÞj2: ð8Þ

Thus, if we identify jwj2 with 4p times the energydensity, then for paraxial rays ðc=4pÞTrCCðzÞ is thepower crossing the plane z.

R. Winston et al. / Optics Communications 207 (2002) 41–48 43

Now let us consider the case that a uniform,thermal source is very close to the radiometer,which is useful for normalizing the signal. Thenat the entrance aperture the mutual intensity isproportional to a delta function, Cðr; r0Þ ¼I0k

2dðr? � r0?Þ, where I0 is a constant with dimen-sions of energy/vol. A thermal source is not reallydelta correlated, of course; the spatial correlationis really a sinc function with a width of the order ofa wavelength. It is for this reason that we insert thefactor of k2 into the formula for the mutual co-herence, so that if we need to set r? ¼ r0? for thepurposes of taking the trace, we can interpretk2dð0Þ as being of order unity. In any case, whenwe compute the signal according to Eq. (3), weobtain

TrðMMCCÞ ¼ I0k2TrMM ¼ I0k

2Ni; ð9Þwhere we set

TrMM ¼ Ni ð10Þfor the number of phase space cells in the accep-tance region of the instrument (this point is dis-cussed more fully in [3]). The trace of MM is easy tocompute. We first make the paraxial approxima-tion in Eq. (7), which gives

hr?jMM jr0?i

¼ k2pL

� �2 Zd2r00? exp

ikLr00? � ðr?

�� r0?Þ

�; ð11Þ

where the r00? integration is taken over a circle ofradius b. Now setting r? ¼ r0? and integrating r?over a circle of radius a, we obtain

Ni ¼ TrMM ¼ kab2L

� �2¼ pah0

k

� �2; ð12Þ

where h0 ¼ b=L.The number Ni has a simple interpretation. A

phase space cell in the four-dimensional k? � r?phase space has volume ð2pÞ2. As viewed from thestandpoint of the exit aperture b of the radiometer,the rays passing through each point of the aper-ture b occupy a solid angle of pða=LÞ2, or a regionof k?-space of area pðka=LÞ2. The region of r?-space is just the exit aperture, of area pb2. Multi-plying these areas and dividing by ð2pÞ2 givesprecisely Ni.

Next we consider the case of a very distantthermal source, which effectively produces a co-herent plane wave at the entrance aperture, say,ffiffiffiffiI0

p ffiffiffiffiffiffiffiph2s

qeikz, so that hr?jCCjr0?i ¼ I0ðph2s Þ. The di-

mensionless factor ph2s will be explained below.Using this and Eq. (11), we obtain

Q ¼ TrðMMCCÞ

¼ I0ph2sk2pL

� �2 Zd2r? d

2r0? d2r00?

� expikLr? � r00?

� �exp

�� ik

Lr0? � r00?

�: ð13Þ

It is easiest to do the r? and r0? integrals first (both

of which go out to radius a). These are identical,and are given byZd2s exp

�� ik

Ls � r00?

�¼ pa2F ðkar00?=LÞ; ð14Þ

where s ¼ r? or r0?. This leaves only the r00? inte-gration (taken out to radius b)

Q ¼ TrðMMCCÞ

¼ I0ph2spa2

kL

� �2 Zd2r00?

2J1ðkar00?=LÞðkar00?=LÞ

� �2ð15Þ

which agrees with the expected fraction of the Airydiffraction pattern contained by the detector ofradius b.

5. Calculations using axially symmetric models for

source and instrument

We now apply this formalism to the radiometricmeasurement of a circular, thermal source by ourpinhole radiometer. The correlation function atthe entrance aperture is given by Eq. (1). We nowwrite out the constant in that equation in a certainform,

hr?jCCjr0?i ¼ I0ph2sF ðkshsÞ; ð16Þwhere as above I0 has dimensions of energy/voland a factor of h2s is split off so that I0 is a propertyof the source, independent of the distance from thesource to the radiometer. The trace of CC (timesc=4p) is the power entering the radiometer; weeasily compute this, finding


TrCC ¼ I0ðpahsÞ2 ¼ I0k2Nr; ð17Þ

where

Nr ¼pahs

k

� �2: ð18Þ

The number Nr is similar to Ni, defined in Eq. (10),and it has a similar interpretation: Nr is the num-ber of phase space cells occupied by the radiationentering the radiometer. That is, if the generalizedradiance (Wigner function) is computed, it will besubstantially nonzero over a region of phase spaceof volume ð2pÞ2Nr. When either Nr 1 or Ni 1it is no longer appropriate to interpret these asnumber of phase space cells. One finds that onlyone spatial mode accounts for most of the power.In this limit Nr is more properly interpreted as ameasure of power intercepted by the instrumentand Ni as a measure of the acceptance of the in-strument. A detailed discussion can be found in[3].As for the instrument function, it was given in

the paraxial approximation by Eq. (11). Now us-ing Eq. (14), we obtain

hr?jMM jr0?i ¼ ph20k2

F ðksh0Þ: ð19Þ

Thus we have

Q ¼ TrðCCMMÞ

¼ I0k2 NrNi

ðpa2Þ2Zd2r? d

2r0?F ðkshsÞF ðksh0Þ: ð20Þ

This integral cannot be done analytically, butthe result can be studied in various limits. First wenote that the arguments of the F functions in theintegrand can be written in terms of the dimen-sionless numbers Ni, Nr

ksh0 ¼ 2ðs=aÞffiffiffiffiffiNi

p; kshr ¼ 2ðs=aÞ

ffiffiffiffiffiNr

p: ð21Þ

Thus, if Nr 1, we have F ðkshrÞ � F ð0Þ ¼ 1. Thephysical meaning of this condition is that thetransverse correlation length at the entrance ap-erture to the radiometer is large compared to theaperture itself, hence the radiation at the entranceaperture is coherent. The integral (20) can be donein this limit, although it is easier to go back to

Eq. (15), which is equal to it (the two differ only bythe order of the integrations). Thus we obtain,

Q ¼ I0k2Nr½1� J 20 ðkah0Þ � J 21 ðkah0Þ�; ð22Þ

valid when Nr 1. Note that kah0 ¼ 2ffiffiffiffiffiNi

p. This is

a text-book result that can be obtained if one as-sumes that all the radiation that hits the aperturecan be subsequently analyzed using plane wavediffraction. The naive expectation that this resultmight approximately account for diffraction effectsin radiometry is not correct. In fact, outside itsrange of validity ðNr 1Þ, the signal is signifi-cantly overestimated as shown in Fig. 2, whichgives the exact and approximate values of thenormalized response Qn ¼ Q=ðI0k2Þ as a functionof Nr for different values of Ni.Similarly, if Ni 1 we have F ðksh0Þ � F ð0Þ ¼

1. Thus one easy limit is Nr;Ni 1, in which casewe have Q ¼ I0k

2NrNi. This also comes out ofEq. (22) in the limit Ni 1.On the other hand, if Nr 1, we can use Eq. (2)

to replace F ðkshrÞ by ð1=k2h2r Þdðr? � r0?Þ, at leastwhenever the rest of the integrand, namely,F ðksh0Þ, is slowly varying on the scale of F ðkshrÞ.This will be the case if Nr Ni. Thus, if bothNr 1 and Nr Ni, we have Q ¼ I0k

2Ni. Con-versely, if Ni 1 and Ni Nr, we have Q ¼I0k

2Nr. This latter result is valid for arbitrarilyvalues of Nr, as long as Ni 1;Nr. For small val-ues of Nr, it also follows from Eq. (22), since largeNi means the entire Airy diffraction pattern iscaptured and the Bessel functions in Eq. (22) go tozero.Combining these results, we have the following,

crude approximation to the integral (20):

Q ¼ I0k2 NrNi if Nr;Ni 1;

minðNr;NiÞ otherwise:

ð23Þ

The expression minðNr;NiÞ is easy to understand.If one phase space region (Nr or Ni) is much biggerthan the other, then the acceptance is determinedby the smaller region, which is contained in thelarger. There is no reason for this approximationto be good near Nr ¼ 1 or Ni ¼ 1, but in fact it isnot bad, as shown in Fig. 2.Note that both approximations, (22) and (23),

give larger Qn values for specified Nr and Ni valuescompared with the exact calculation (20). Varying


Nr for fixed Ni corresponds physically to varyingthe distance of the thermal source for a fixed ra-diometer. If these curves are utilized as means ofmeasuring distance using a source of known sizethrough measuring the signal strength, then theapproximations, in particular the text-book result,Eq. (22), always yield a longer distance at a givensignal strength. Equivalently, for a known distancebetween source and instrument, the approxima-tions yield smaller source sizes compared with theexact calculation at a given signal strength. See [1]

for experimental measurements of Qn versus Nr forfixed Ni. A succinct characterization of these ap-proximations can be stated colloquially; the radi-ation source is closer than you think.

6. Practical radiometers

Our model of the radiometer is that of a simplecollimator taken to the far-field limit a b.However, practical radiometers use a lens at ap-

Fig. 2. Plots of the normalized measurement Qn ¼ Q=ðI0k2Þ as a function of Nr for different values of Ni. The broadly dashed line is the

approximation (23), the tightly dashed line is the approximation (22), and the solid line is the exact value.


erture a with the distance L between lens and de-tector equal to the focal length f. In this case theinstrument operator, MM ¼ PP yPP , follows from

PP ¼ AAðbÞDDðf ÞLLðf Þ AAðaÞ; ð24Þ

where the matrix elements for the lens operator,LLðf Þ, in the paraxial approximation are given by

hr?jLLðf Þjr0?i ¼ exp�ikjr?j2

2f

!dðr? � r0?Þ: ð25Þ

Following [3], a straightforward calculation showsthat the matrix elements of the instrument opera-tor are still given by Eq. (19) but with h0 ¼ b=f . Inother words, a detector at the focal plane isequivalent to a collimator in the far-field limit.This is a reasonable result because the far-fieldlimit means that rays from any point at the de-tector to the lens are approximately parallel. Allresults and analysis done for the collimator in-strument are applicable to this case without mod-ification.When the radiation source is not at infinity, the

experimental approach is to focus the radiometeron the source. This has the effect of placing thesource at infinity as can be shown mathematicallyby explicitly recalculating MM or physically, by in-serting a large weak lens in front of the radiometerat a distance from the source equal to its focallength. The only modification is a negligible cor-rection to the value of Ni (for details see [1]).

7. Discussion

The instrument operator MM is defined throughPP , Eq. (4) or Eq. (24), by MM ¼ PP yPP with PP being thepropagator from the entrance of the apparatus tothe detector. In Eq. (3), which is associated withthe signal, the trace operation is performed at theentrance plane of the instrument. However, it doesnot matter at what stage from source to detectorone identifies subsequent propagation with MM . Thetrace operation would be carried out at the planeat which one begins the association of propagationwith MM . If MM is just the bare detector, the calcu-lation for the signal is equivalent to performing thetrace of CCðzÞ with,

hr?jMM jr0?i ¼ dðr? � r0?Þ; jr?j; jr0?j6 b; ð26Þwhere b is the radius of the detector. One couldalso associate all propagation with MM . For a delta-correlated source, for which the van Cittert–Zer-nicke Theorem applies

hr?jCCð0Þjr0?i ¼ I0k2dðr? � r0?Þ; jr?j; jr0?j6 rs;

ð27Þwhere rs is the source radius. It is this equivalencebetween a delta-correlated source and a delta-correlated detector, apart from a multiplyingconstant, that gives rise to a van Cittert–ZernikeTheorem type result for the instrument if propa-gation is associated with MM from the entranceaperture plane to the detector.

8. Conclusion

We have exploited the symmetry between anincoherent source whose radiance is being mea-sured, and the detector whose signal represents themeasurement. We find a remarkable analogy be-tween the result of measuring radiance and thevan Cittert–Zernike Theorem. In fact, the mea-sured radiance is represented (up to an overallconstant) by the double integral over the instru-ment aperture of the mutual intensity of the fieldand the mutual intensity of a delta-correlatedsource the same size, shape and location as thedetector. While we have expressed our results inthe context of radiometry, one would go througha similar analysis in analyzing the detection of anypartially coherent wave. The signal is representedby the double integral of two mutual coherencefunctions. One of these is for the incident wave,the other arising from the detector considered as asource. It is likely that entirely similar consider-ations may apply to other signal detectionprocesses.

Acknowledgements

We thank Emil Wolf for reading the manuscriptand motivating us to probe beyond the formalanalogy. This work was supported in part by the


US Department of Energy, Division of MaterialsSciences and Engineering, Office of Basic EnergyScience.

References

[1] Y. Sun, R. Winston, J.J. O’Gallagher, K.A. Snail, Opt.

Commun. 206 (2002) 241.

[2] For an excellent, accessible discussion of this theorem, see

L. Mandel, E. Wolf, Optical Coherence and Quantum

Optics, Cambridge University Press, Cambridge, 1995;

also M. Born, E. Wolf, Principles of Optics, seventh

(Expanded ed.), Cambridge University Press, Cambridge,

1999.

[3] R.G. Littlejohn, R. Winston, J. Opt. Soc. Am. A 12 (1995)

2736.

[4] R. Winston, R.G. Littlejohn, J. Opt. Soc. Am. A 14 (1997)

3099.


the measurement of radiance and the van cittert–zernike theorem

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