power coupled between partially coherent vector fields in different states of coherence

S. Withington and G. Yassin Vol. 18, No. 12 /December 2001 /J. Opt. Soc. Am. A 3061

Power coupled between partially coherent vectorfields in different states of coherence

Stafford Withington and Ghassan Yassin

Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, UK

Received February 26, 2001; revised manuscript received June 11, 2001; accepted July 6, 2001

A procedure is described for calculating the power coupled between collimated, partially coherent vector fieldsthat are in different states of coherence. This topic is of considerable importance in designing submillimeter-wave optical systems for astronomy. It is shown that if the incoming field S has coherence matrix A, and theoutgoing field D has coherence matrix B, then the power coupled is simply Ps 5 Tr(ATBT†), where the ele-ments of T project the basis functions of B onto those of A. A similar technique can be used to calculate thepower coupled from the background of S to D. The scheme is illustrated by calculating the power coupledbetween two scalar, Gaussian Schell-model beams. The procedure can be incorporated into optical design soft-ware. © 2001 Optical Society of America

OCIS codes: 030.1640; 030.4070; 030.6600; 040.1240; 040.1880; 040.3060.

1. INTRODUCTIONIn designing instruments for submillimeter-wave as-tronomy (300 GHZ–2 THz, 1 mm–150 mm), it is impor-tant to control precisely the optical coupling between thesource, the telescope, and the detector.1 In the case ofsingle-mode systems, such as heterodyne spectral-linereceivers,2 it is simply necessary to match the fully coher-ent field of the detector to the point-spread function of thetelescope. In the case of multimode systems, such as pla-nar bolometers,3,4 the situation is more complicated, be-cause here the detector is sensitive to radiation approach-ing from large angles, and stray light becomes a problem.To alleviate the many problems associated with back-ground loading, single-mode or few-mode bolometers canbe used.5–7

The rapid development of imaging arrays is furthercomplicating the design of long-wavelength opticalsystems.8 In the case of planar bolometers, the indi-vidual pixels in an array are both individually and mutu-ally incoherent; in the case of single-mode waveguide de-tectors, the pixels are individually coherent but mutuallyincoherent; and in the case of heterodyne spectral-line re-ceivers, the local-oscillator beams, which are usually pro-duced from a single source,9 are both individually andmutually coherent. It is clear that in designing imagingarrays for submillimeter-wave astronomy, various statesof coherence may be encountered.

In recent years we have developed a number of tech-niques for modeling the behavior of coherent and partiallycoherent submillimeter-wave optical systems.10–12

Modal optics is particularly effective at these wave-lengths, because diffraction is important, and yet it is notrealistic to evaluate the multiple diffraction integralsneeded to propagate a field through a complex system ofcomponents. In addition, models based on numerical dif-fraction calculations are not suitable for iterative optimi-zation routines. Modal decompositions based on Gauss-ian modes,13 plane waves,14,15 spherical harmonics, and

0740-3232/2001/123061-11$15.00 ©

metallic and dielectric waveguide modes16 can be used togreat effect.

In the case of fully coherent optical systems, a propa-gating beam is characterized by a vector of mode coeffi-cients. The modes diffract in free space through a pro-cess of phase slippage and scaling. At opticalcomponents such as apertures and mirrors, the mode co-efficients scatter linearly. In the case of partially coher-ent fields, the vector of mode coefficients is replaced by amatrix, which describes the correlations between modeamplitudes; and again, these matrices scatter linearly.10

During the course of our work, we have frequentlyfound it desirable to be able to calculate the powercoupled between two collimated, partially coherent vectorfields that are in different states of coherence. In thecase of fully coherent fields, it is simply necessary toevaluate the overlap integral between the two fields con-cerned. In the case of partially coherent fields, the situ-ation is more complicated, and it is not at all clear howthe problem should be solved.

In this paper we describe a procedure for calculatingthe power coupled between two partially coherent beamsthat are in different states of coherence. The scheme isformulated with the language of modal optics. First, inSection 2 we review the modal analysis of partially coher-ent vector fields; then, in Section 3 we discuss the cou-pling problem in detail. We explore the structure of theexpressions found and consider the related question ofhow to calculate the power coupled into a detector from ageneral background field. In Section 4 we illustrate theoverall technique by investigating the power coupled be-tween two scalar Gaussian Schell-model beams.

2. MODAL ANALYSIS OF PARTIALLYCOHERENT VECTOR FIELDSTo characterize the second-order statistical properties of acollimated, partially coherent vector field, we set up anensemble of identical systems. In accordance with the

2001 Optical Society of America

3062 J. Opt. Soc. Am. A/Vol. 18, No. 12 /December 2001 S. Withington and G. Yassin

theory of analytic signals,17 we assume that the band-width of each measurement is sufficiently narrow that therelative phases of field components at different points, inany given member of the ensemble, are well defined.

Following the principles of paraxial optics, we assumethat power flows predominantly in the z direction andconsequently that it is sufficient to work in terms of two-dimensional, rather than three-dimensional, field vectors.The total electric field associated with any given memberof the ensemble can be written as

E~ r t , z ! 5 Ex~ r t , z !x 1 Ey~ r t , z !y, (1)

where x and y are orthogonal unit vectors transverse tothe direction of propagation z, and r t is a general trans-verse position vector: r t 5 xx 1 yy.

We can define the cross spectral dyadic as the dyadicproduct of the field at one point, E( r t , z), and the complexconjugate of the field at another point, E( r t8 , z), averagedover the ensemble.18 Here we concentrate on the corre-lations between field components at points on planes ofconstant z. The individual terms in the dyadic are thecross spectral power densities of the field components attwo positions:

W~ r t8 , r t , z ! 5 ^E~ r t , z !E* ~ r t8 , z !&. (2)

According to this scheme, the cross correlations be-tween field components are represented by an operator,W( r t8 , r t , z), which can be regarded as an intrinsic prop-erty of the field and independent of the particular trans-verse coordinate system used. The cross spectral dyadicis Hermitian: W( r t8 , r t , z) 5 W( r t , r t8 , z)†, where † de-notes the adjoint. It is also a function of frequency, buthere we shall not refer to frequency explicitly.

To give the scheme practical value, we need to expressthe above forms in terms of some particular representa-tion. Let us expand the field associated with each mem-ber of the ensemble in terms of a set of vector fields thatare appropriate to the system being studied. These arenot necessarily the natural modes of the field but can beany set of functions that are orthonormal and completeover the surface of interest. For example, they may be aset of polarized plane waves or a set of polarizedGaussian–Laguerre modes. Whatever the choice, we canwrite

E~ r t , z ! 5 (n

anCn~ r t , z !, (3)

where the modes are described by

Cn~ r t , z ! 5 cn~ r t , z !u 1 fn~ r t , z !v (4)

and cn( r t , z) and fn( r t , z) are the individual field com-ponents. Here we use a single index to represent what inreality, for a two-dimensional surface, must be a doubleindex. In other words, we label every two-dimensionalmode with a single, unique identifier. Also, we describethe modes in terms of the transverse unit vectors u and v,rather than x and y, to emphasize that any general basisset can be used.

Substituting the expansion of the field, Eq. (3), into thedefinition of the cross-spectral dyadic, Eq. (2), we find

that the statistical properties of the field are representedby a set of coefficients Cnm , where

W~ r t8 , r t , z ! 5 (nm

CnmCn~ r t , z !Cm* ~ r t8 , z !, (5)

Cnm 5 ^anam* &. (6)

For convenience, we can order the coefficients Cnm of thisbimodal expansion into a matrix, which we shall call thecoherence matrix C.

We need an expression for finding the elements of thecoherence matrix when the cross-spectral dyadic isknown. It can be shown that

Cnm 5 ESE

S8Cn* ~ r t , z ! • W~ r t8 , r t , z !

• Cm~ r t8 , z !d2r t8d2r t , (7)

which can be appreciated by substituting the bimodal ex-pansion, Eq. (5) into Eq. (7) and using the orthonormalitycondition for the modes over the surface of integration.Hence we can find the coherence matrix for any vector ba-sis set once the cross spectral dyadic is known.

To use this equation, it is necessary to know the crossspectral dyadic over a primary surface. If there is somesurface z1 over which it is known that the source is fullyincoherent, then under paraxial assumptions, the cross-spectral dyadic takes the form

W~ r t8 , r t , z1! 5 I%I~ r t!d ~ r t 2 r t8!, (8)

where I( r t) is the intensity of each polarization of thefield and I is the two-dimensional idem factor, which canbe represented in any basis set. Hence, if the field is in-coherent over the primary surface and we know the inten-sity as a function of position, then Eq. (7) takes on a spe-cial form.

When the field is fully incoherent, and unpolarized, wehave, substituting Eq. (8) into Eq. (7),

Cnm 5 ES

I~ r t!@ cn* ~ r t , z !cm~ r t , z !

1 fn* ~ r t , z !fm~ r t , z !#d2r t , (9)

where I( r t) is the intensity of each polarization, as de-scribed.

When the field is fully coherent, we can find the modecoefficients an through

an 5 ES

E~ r t , z1! • Cn* ~ r t , z1!d2r t (10)

and then form the coherence matrix by

C 5 aa†, (11)

where a is the column vector of mode coefficients an and †denotes the conjugate transpose.

It is always possible to decompose a field into fully co-herent and fully incoherent parts. It is clear from thedefinition of the cross-spectral dyadic, Eq. (2), that if apartially coherent field is formed from a superposition of


fields that are incoherent with respect to each other,whatever the states of coherence of the individual fields,then the composite cross-spectral dyadic is the sum of theindividual dyadics:

W~ r t8 , r t , z ! 5 (i

Wi~ r t8 , r t , z !. (12)

Equivalently, when a number of fields are incoherentwith respect to each other, then, regardless of the state ofcoherence of each contribution, the overall coherence ma-trix is given by

C 5 (i

Ci , (13)

where Ci are the coherence matrices of the individualparts. In simulating the behavior of optical systems, it isoften convenient to be able to assemble composite coher-ence matrices in this way.

It is important to be able to propagate and scatter co-herence matrices. If we know the scattering matrix as-sociated with the basis set, then the solution is straight-forward, for we have b 5 Sa, where a is the vector ofmode coefficients at the primary surface S1 ; b is the vec-tor of mode coefficients at the secondary surface S2 ; and Sis the scattering matrix that relates them. We can thenwrite

D 5 ^bb†& 5 SCS†. (14)

In certain cases the scattering matrix associated withthe basis set is not known, and then it is desirable to de-compose the field into a set of modes that are individuallyfully coherent but mutually completely uncorrelated.Once this has been done, other methods of propagationcan be used. In accordance with Eq. (12), we thereforelook for an expansion of the form

W~ r t8 , r t , z ! 5 (i

b iC ib~ r t , z !C i

b* ~ r t8 , z !, (15)

where C ib( r t , z) form a complete orthonormal set, and

b iC ib~ r t , z ! 5 E

S8W~ r t8 , r t , z ! • C i

b~ r t8 , z !d2r t8 ,

(16)

which can be appreciated by substituting Eq. (15) into Eq.(16). C i

b( r t , z) are therefore the natural modes of thefield, and they are vector generalizations of the scalarmodes introduced by Wolf.19

It is straightforward to show that the coherence matrixis nonnegative: Z†CZ > 0 for all Z, where Z is any com-plex column vector. Hence, because the coherence matrixis also Hermitian, there is some unitary transform U forwhich U†CU 5 L, where the individual columns of U arethe eigenvectors of the coherence matrix C and the ele-ments of the diagonal matrix L are the eigenvalues of C.We can therefore diagonalize the coherence matrix andassemble the eigenfunctions through

C ib~ r ! 5 (

jUji@ f j~ r !u 1 f j~ r !v#. (17)

These are the natural modes of the field as described byEq. (15).

In the context of this work, the total power in the beamPt , which must be one of the invariants of lossless propa-gation, is of particular interest. We can find the power ina simple way. By integrating the total intensity over thesurface S at z, we find

Pt 5 ESE~ r t , z ! • E* ~ r t , z !d2r t 5 (

nmCnmdnm 5 Tr C,

(18)

where Tr C represents the trace of C. The total power inthe beam is simply the trace of the coherence matrix. Infact, each term on the leading diagonal of C can be inter-preted as the power in the associated mode. Given thatthe trace is invariant to unitary transforms, the eigenval-ues simply give the power in each of the natural modes.

3. COUPLING POWER BETWEENPARTIALLY COHERENT FIELDSA. Source–Detector Coupling through Natural ModesWe wish to calculate the power coupled between two par-tially coherent fields when the second-order statisticalproperties of each field are known. To this end, considerthe optical system shown in Fig. 1, where a thermalsource S is connected to a multimode detector D throughan optical system comprising two parts. The first partseparates the source from the coupling surface H, and thesecond part separates H from the detector. The field at Hgenerated by the source is described by coherence matrixA, and the field at H representing the detector is de-scribed by coherence matrix B. The question before usis, ‘‘How can we calculate the power coupled from thesource to the detector from knowledge of A and B alone?’’

To give the problem physical form, we shall assumethat the source at z1 is thermal and unpolarized and canbe described by a cross-spectral dyadic having the form

W~ r t8 , r t , z1! 5 I%H hn

expX hn

kTs~ r t!C 2 1

es~ r t!

1hn

expS hn

kTbD 2 1

@1 2 es~ r t!#J3 d ~ r t 2 r t8!, (19)

where Ts( r t) is the temperature of the source and es( r t)the emissivity. In the normal manner, h is Planck’s con-stant, k is Boltzmann’s constant, and n is the frequency.

Fig. 1. Generic optical system comprising an incoherent source,S, a sequence of optical components, S and S8, and a multimodedetector, D. H is a general surface in the system at which wewish to calculate the coupled power.


Expression (19), with the emissivity es( r t) set to unity, fol-lows from evaluating Eq. (6) of Kano and Wolf 20 in theparaxial limit. In other words, for a perfect blackbodysource at uniform temperature, all of the modes are ex-cited equally and incoherently, and in the paraxial limitthis assumption leads to a particularly simple form forthe cross-spectral dyadic. In Eq. (19) both the tempera-ture and emissivity are allowed to be functions of trans-verse position r t . The second term inside the curlybraces of Eq. (19) follows from the need to provide a back-ground source of thermal energy in the case when theemissivity of the primary source is less than unity.Therefore in Eq. (19) Tb is the temperature of the back-ground, which is assumed to be uniform. For the firstpart of the analysis, we will assume that the temperatureof the background, Tb , is sufficiently low that it makes nocontribution to the coupled power. Later we will use thegeneralized source defined by Eq. (19) to evaluate theamount of power coupled from the background to the de-tector.

Equation (19) implies that the intensity of each polar-ization, I( r t), is infinite for all r t , but this unphysical be-havior is simply an artifact of the transverse coherencelength being zero. In reality, the optical system will havelimited throughput, owing to mode filtering, and the co-herence length and intensity at H will be finite. Equa-tion (19) is therefore reasonable as long as we rememberthat the field at S contains more modes than we can col-lect. In Eq. (19), we have assumed that both the sourceand the background are fully incoherent. This assump-tion simplifies the analysis, without reducing the general-ity of what follows.

Using Eqs. (19) and (7), and after ignoring the back-ground, the elements of the source coherence matrix C be-come

Cnm 5 ESF hnes~ r t!

expX hn

kTs~ r t!C 2 1G Cn* ~ r t , z1!

• Cm~ r t , z1!d2r t , (20)

where Cn( r t , z1), $m, n P 1,..., `% are the vector basisfunctions used. We can write a similar expression for thecoherence matrix of the detector, which is, at Z2 ,

Dnm 5 ED

hd~ r t!Cn* ~ r t! • Cm~ r t!d2r t , (21)

were hd( r t) allows for position-dependent sensitivity. Wecan now reference these two coherence matrices to H:A 5 SCS† and B 5 S8†DS8, where S and S8 are the scat-tering matrices of the first and second parts of the opticalsystem, respectively.

Before proceeding further, let us consider how toproject a coherence matrix from one basis set to another.Throughout this section we will use the notation T:C→ F to mean that the operator T maps the coherencematrix from basis set $Cm( r t , z)% to basis set $Fm( r t , z)%.Assume that we have an ensemble of identical optical sys-

tems. For every member of the ensemble, we can writethe field at H, E( r t , z), as a linear combination of basisfunctions,

E~ r t , z ! 5 (n50

`

anCn~ r t , z !, (22)

but we require that

E~ r t , z ! 5 (m50

`

am8 Fm~ r t , z !, (23)

where C( r t , z) is the original basis set and F( r t , z) thenew basis set. The coefficients of the new expansion aregiven by

am8 5 ESFm* ~ r t , z ! • E~ r t , z !d2r t . (24)

Substituting Eq. (22) into Eq. (24) gives

a8 5 Ta, (25)

where a8 and a are column vectors of mode coefficients,and

Tmn 5 ESFm* ~ r t , z ! • Cn~ r t , z !d2r t (26)

are the elements of the projection matrix T. Finally, thecoherence matrix of the field in the new mode set is

A8 5 ^a8a8†& 5 T^aa†&T† 5 TAT†. (27)

The matrix elements Tmn transform the coherence matrixfrom one basis set to another, and T:C → F.

Now assume that the basis set of the source field at Hconsists of those modes Cn( r t , z) for which A is diagonal;in other words, they are the natural modes of the scat-tered source field. We will use the notation L for the co-herence matrix to emphasize that A is diagonal in this ba-sis set; the elements of L are the eigenvalues of A.Likewise, assume that Fm( r t ,z) are those modes forwhich B is diagonal: they are the natural modes of thedetector at H. We will use the notation Y for the coher-ence matrix of the detector to emphasise that B is diago-nal in this basis set; the elements of Y are the eigenvaluesof B. We can now expand the correlation matrix of thesource in terms of the natural modes of the detector. Thenew correlation matrix A8 still contains complete infor-mation about the statistical properties of the source. Us-ing the transform described by Eq. (27), T:L → Y, whereL and Y represent the basis functions of L and Y, we get

A8 5 TLT†, (28)

which of course is no longer diagonal, unless the eigen-modes of the source are the same as those of the detector.

The elements on the leading diagonal of A8 give theamount of power coupled to the eigenmodes of the detec-tor. Expanding Eq. (28), we find that

Amm8 5 (n50

`

uTmnu2an , (29)

where an are the eigenvalues of the source coherence ma-trix at H. Note, however, that Amm8 do not correspond to


the powers that are actually coupled to the detector, asthe detector is not equally sensitive to each of its naturalmodes.

To find the power actually coupled, we use the followingargument. Place a perfect blackbody source at H, andimagine that the absorbing element of the detector is ra-diating. If the detector and the blackbody source are atthe same, uniform temperature, no net power can betransferred between them. If the natural modes of thedetector are used as the basis set, the coherence matrix ofthe detector is diagonal. The elements of Y are the ei-genvalues of B, which we will call bm . The power radi-ated by the detector into mode Fm( r) is therefore propor-tional to bm , and this power is completely absorbed bythe blackbody source at H. Not only is the coherence ma-trix of the blackbody source at H diagonal in Fm( r t , z),but all of the diagonal elements are equal—the coherencematrix of a perfect blackbody source is a scaled identitymatrix whatever the basis set used. To maintain thermo-dynamic equilibrium, therefore, between the blackbodysource at H and the detector, the sensitivity of the detec-tor to power in its natural mode Fm( r) must be propor-tional to bm .

Returning to the original problem, we can now find ameasure of the total coupled power Ps by adding togetherthe power in each of the natural modes of the detector, asgiven by Eq. (29), and applying the weightings bm :

Ps 5 (m50

`

(n50

`

uTmnu2anbm . (30)

A signal-flow graph representing this behavior is shownin Fig. 2. This signal-flow graph shows only the powertraveling in the forward direction. For every forward-traveling mode, however, we assume that there is abackward-traveling mode having the same spatial form.These backward-traveling modes are not referred to ex-plicitly in this paper, as the extra terms that would beneeded in the various equations would only complicatethe expressions without contributing anything more tothe underlying physics.

Equation (30) appears to be physically reasonable.First, the expression is symmetrical in such a way that itdoes not matter which of the two beams is the source and

Fig. 2. Flow diagram showing the way in which power is scat-tered from the natural modes of the source field to the naturalmodes of the detector. an are the eigenvalues of the source co-herence matrix, and bn the eigenvalues of the detector coherencematrix. Tmn are the elements of the projection matrix betweenthe two basis sets.

which is the detector. This symmetry is needed to ensurethat thermodynamic equilibrium is maintained when thedetector and source are at the same uniform temperature.Second, from a quantum statistical point of view, theprobability that an incoming photon is in source mode n isproportional to an , and the probability that an outgoingphoton in detector mode m is absorbed is proportional tobm . The matrix element Tmn is simply the probabilityamplitude that a photon in source mode n is found in de-tector mode m. Equation (30) can therefore be inter-preted in terms of the probability of transferring a photonfrom the source to the detector through the paths avail-able. Third, when the two beams are identical and fullycoherent, the projection matrix is diagonal, Tmn 5 dmn ,and only one natural mode is present in each of the sourceand detector fields. If the temperature of the source isuniform, and the emissivity equal to unity, the maximumpower transferred per unit bandwidth is given by Eq. (30)as

Pt 5hn

exp@~hn!/~kTs!# 2 1, (31)

as expected. This situation would occur, for example, ifthe scattering components S and S8 each consisted of alength of single-mode metallic waveguide with horns ateach end. If we have the situation where all the of eigen-values below some critical mode number N are unity andall the eigenvalues above are zero, we will have N timesmore power than given by Eq. (31); this situation wouldoccur if the metallic waveguide allowed N modes to propa-gate. For partially coherent fields, Eq. (30) can thereforehave a value greater than unity, showing that a few-modedetector is capable of absorbing more power than a single-mode detector.

B. Source–Detector Coupling in a General Basis SetIn Subsection 3.A we derived an expression, Eq. (30), forthe power coupled between two fields when the eigen-modes and eigenvalues of the coherence matrices A and Bare known. Obviously, when performing simulations, itis desirable that we not have to diagonalize the coherencematrices numerically. In this subsection we considerhow to calculate the coupled power directly in terms of Aand B.

First, because L and Y are diagonal, it is possible towrite Eq. (30) in the form

Ps 5 Tr~TLT†Y!. (32)

The Hermitian coherence matrix A, described in termsof a general basis set C( r t ,z), can be diagonalized by aunitary transform U8,

U8AU8† 5 L, (33)

where U8:C → L. Substituting Eq. (33) into Eq. (32),we find that

Ps 5 Tr~T8AT8†Y!, (34)

where T8 5 TU8. T8 is a product of unitary operatorsand is therefore itself unitary. Previously, we hadT : L → Y, and therefore T8 : C → Y. If only one ofthe coherence matrices is diagonal, we can use Eq. (34) to


find the coupled power. Clearly, we are simply projectingthe incoming field, described in terms of coherence matrixA, onto the eigenmodes of the detector.

Now, for the detector field, described in terms of gen-eral basis set F( r t , z), we can write

V8BV8† 5 Y, (35)

where V8 5 V† is the operator that diagonalizes B; or,equivalently, V8 : F → Y.

Substituting Eq. (35) into Eq. (34), we find that

Ps 5 Tr~AT9†BT9!, (36)

where T9 5 VT8. Here we have premultiplied by T8†

and postmultiplied by T8 on the understanding that T8 isunitary and the trace of a matrix is invariant to unitarytransforms. It is clear that because V : Y → F, thenT9 : C → F. Also, because T8 and V are unitary, T9 isunitary, and we can premultiply Eq. (36) by T9 and post-multiply by T9† to get

Ps 5 Tr~T9AT9†B!. (37)

Hence we can calculate the coupled power in terms of thecoherence matrices alone.

Equations (36) and (37) have physical significance inthat they show that the coherence matrix of the sourcecan be projected onto the detector, or the coherence ma-trix of the detector can be projected onto the source, with-out changing the result.

A particular case arises when A and B use the same ba-sis set: Cn( r t , z) 5 Fn( r t , z) for all n. Because the ba-sis sets are equal, T9 5 I, and

Ps 5 Tr~AB!. (38)

This result shows that we can calculate the coupled poweronce A and B are known.

It is beneficial to analyze various limiting cases of Eq.(38). First, when A and B are simultaneously diagonal,Eq. (38) reduces to the result given previously: Eq. (30).In other words, power simply passes through the couplingsurface H without being scattered between modes. It isstraightforward to show that the same result is givenwhen the beams have the same natural modes, eventhough the coherence matrices may be expressed in termsof some other general basis set.

Second, we would expect Eq. (38) to reduce to the clas-sical expression for the coupled power when both fieldsare coherent. This behavior can be verified as follows.When the fields are fully coherent, the coherence matricesare given by A 5 aa† and B 5 bb†, where a and b arecolumn vectors of mode coefficients an and bm . Substi-tuting these forms into Eq. (38), we find that

Ps 5 U(n

anbn* U2

. (39)

For a coherent system, we would evaluate the coupledpower through the classical expression

Ps 5 U EH

EA • EB* dHU2

. (40)

EA and EB are the fields on each side of the interface H.Substituting modal expansions of the form (22), we findthat

Ps 5 U(n

(m

anbm* ESCn~ r t , z ! • Fm~ r t , z !* dsU2

, (41)

which is equal to Eq. (37) or to Eq. (38) if the same modeset is being used for both fields, supporting the integrityof the result.

C. Stray-Light CouplingIn designing low-noise submillimeter-wave instruments,it is important to be able to predict how much stray lightwill enter the detector. Stray light can come from eitherthe region surrounding the source or from losses withinthe optical system itself. To understand this problem, wewill use the full expression for the cross-spectral dyadicgiven by Eq. (19). The emissivity term in Eq. (19),es( r t , z1), allows for those situations where the intensityacross the surface of the source is not uniform eventhough the temperature of the source is uniform. Thissituation can occur if the emissivity of the surface is lessthan unity in parts or if the source is partially transpar-ent. In both cases, background noise will be fed into thesystem. In the first case, noise will be reflected off thesurface of the source, and in the second case, noise will bepartially transmitted through the source.

Let us assume that we can calculate the source coher-ence matrix C8 when the temperature of the source is uni-form and equal to that of the background: Ts( r t) 5 Tb .Let us also assume that we can diagonalize this matrix tofind its eigenvalues. In general, the source coherencematrix for uniform temperature, C8, will not be the sameas the source coherence matrix for the temperature distri-bution of primary interest, C. For example, if the sourceis a disk of radiating material, then we may be interestedin the signal coupled from a Gaussian temperature distri-bution rather than the uniform temperature needed tofind C8. Because, in general, some of the eigenvalues ofC8 will be less than others, the incoming field must com-prise two parts: one that is perfectly coupled to thesource and one that is not coupled at all. The field that isnot coupled to the source must be coupled to some otherabsorption mechanism, and this mechanism may injectnoise of its own. Again, if the source comprises a disk ofabsorbing material, then the region surrounding the diskmay feed noise into those modes not perfectly coupled tothe disk.

If the source and its surroundings are at the same uni-form temperature Tb , the coherence matrix of the totalincoming field Ib must be diagonal:

Ib 5hn

exp@~hn!/~kTb!#I, (42)

where I is the identity matrix. When two mutually inco-herent fields are added, the composite coherence matrix issimply the sum of the individual coherence matrices, and


therefore we know that the coherence matrix of the fieldgenerated by the background alone, N, must be

N 5 Ib 2 C8. (43)

If we knew the eigenmodes and eigenvalues of the back-ground coherence matrix N, we would be able to calculate,using Eq. (30), the power coupled to the detector.

Suppose that U† is the unitary transform that diago-nalizes the source coherence matrix C8:

U†C8U 5 L8. (44)

If we apply this transform to the background coherencematrix N, we find that

U†NU 5 U†~Ib 2 C8!U 5 Ib 2 L8. (45)

Because the column vectors of U† are simply the eigenvec-tors of the source field, we see that the eigenmodes of thebackground are the same as those of the source when thesource is at uniform temperature. The eigenvalues of thebackground, hn are given by

hn 5 1 2 an8 , (46)

where an8 are the diagonal elements of L8. This result isphysically reasonable, as all of the coherence present inthe background is determined by the physical form of thesource, which also determines C8.

If the first part of the optical system, described by S, islossless, that is, S is unitary, then the scattering by theoptical system simply represents a change in the basisset, and we could have carried out the procedure de-scribed above at H without changing the result. In otherwords, the power coupled into the detector from the back-ground can be written as

Pn 5 Tr $~Ib 2 A8!B%, (47)

where A8 is the coherence matrix of the source at H whenthe temperature of the source is uniform. Clearly, inthose cases where the signal itself originates from asource having uniform temperature A 5 A8, the expres-sion for the coupled power is even more straightforward:

Pn 5 Tr $~Ib 2 A!B%. (48)

Finally, we can apply the same reasoning to the case inwhich S is lossy. We then find that the total injectednoise is given by

Pn 5 Tr $S~Ib 2 C8!S†B 1 ~Io 2 SIoS†!B%, (49)

where the first term describes the power from the back-ground and the second term describes stray light fromlosses within the optical system itself. Io has the form ofEq. (42), but the temperature of the optical system is usedrather than that of the background. In many instru-ments, losses in the optical system are the only source ofstray light, and then the coupled power has the form

Pn 5 Tr $~Io 2 SIoS†!B%. (50)

This analysis has not considered the noise that may beassociated with the detector—as suggested by the eigen-values bm being less than unity—or indeed the noise fromthe second part of the optical system, S8. Clearly, how-ever, a complete optical system can be built up with the

formalism described above, and it is simply necessary touse the reference plane that is most appropriate for thephysical problem being studied.

4. COUPLING POWER BETWEENGAUSSIAN SCHELL-MODEL BEAMSThe procedure for calculating the power coupled betweentwo beams has been described in terms of partially coher-ent vector fields, where the degree of coherence and thenature of the polarization can vary as a function of posi-tion. To illustrate the method, we will now calculate thepower coupled between two scalar, one-dimensionalGaussian Schell-model beams.21 Gaussian Schell-modelbeams provide a rough, first-order description of partiallycoherent scalar fields, in the same way that Gaussianbeams provide a rough, first-order description of fully co-herent fields.

The cross-spectral density of a Gaussian Schell-modelbeam has the form

W~x1 , x2! 5 @S~x1!#1/2@S~x2!#1/2g~x2 2 x1!, (51)

where the average intensity of the field at x is given by

S~x ! 5 expS 2x2

2ss2 D , (52)

and the cross correlation between the fields at x1 and x2by

g~x2 2 x1! 5 expF2~x2 2 x1!2

2sg2 G . (53)

In other words, the intensity and the transverse coher-ence are characterized by Gaussian functions havingscale sizes of ss and sg , respectively.

As discussed earlier in the paper, we wish to find thenatural modes fn(x) of this field:

W~x1 , x2! 5 (n50

`

bnfn* ~x1!fn~x2!, (54)

where bn are the eigenvalues defined by the scalar formof Eq. (16).

For Gaussian Schell-model sources, these modes havethe conveniently simple forms21

fn~x ! 5 F2c

pG1/4 1

~2nn! !1/2 Hn@x~2c !1/2#exp~2cx2!,

(55)

where

c 5 ~a2 1 2ab !1/2, (56)

a 5 1/4ss2, (57)

b 5 1/2sg2. (58)

If we normalize the spectrum to the lowest-order eigen-value, we get

bm

b05 S b

a 1 b 1 c Dm

. (59)

Now suppose that we wish to couple two fields, each ofwhich has a Gaussian Schell-model form. The incoming


field has eigenfunctions cn(x) and eigenvalues an , andthe outgoing field has eigenfunctions fm(x) and eigenval-ues bm . In general, the incoming and outgoing fieldswill be in different states of coherence, and thereforefn(x) Þ cn(x) and bn Þ an for all n.

According to our scheme, the power coupled from thefirst field to the second is

Ps 5 (m50

`

(n50

`

uTmnu2anbm , (60)

where the projection elements are given by

Tmn 5 E2`

1`

fm~x !cn~x !dx. (61)

Using the substitution u 5 xA2c for the input field andu8 5 xA2c8 for the output field—an unprimed quantityalways refers to the input field and a primed quantity tothe output field—we get

Tmn 5 AjE2`

1`

hm~ku !hn~u !du, (62)

where the parameter j is given by

j 5 ~c8/c !1/2 (63)

and the natural modes by

hm~u ! 5Hm~u !exp~2u2/2!

~Ap2mm! !1/2, (64)

where Hm(u) is the Hermite polynomial order m in u.These equations can be used to find the coupling once

the scale factors ss and sg are known. At this stage, it isconvenient to express Eqs. (56) and (59) in terms of theparameters q 5 sg /ss for the input field, q8 5 sg8/ss8 forthe output field, and r 5 ss8/ss for the relative sizes. Itcan then be shown that

c8

c5

1

r2

q

q8F1 1 ~q8/2!2

1 1 ~q/2!2 G1/2

, (65)

and

an

a05 F 1

q2/2 1 1 1 q@~q/2!2 1 1#1/2Gn

, (66)

bn

b05 F 1

q82/2 1 1 1 q8@~q8/2!2 1 1#1/2Gn

.

(67)

It is interesting to notice, looking at Eq. (65), that if weset the normalized transverse coherence lengths equal,q 5 q8, j does not depend on q or q8. Thus the couplingintegrals, Eq. (62), do not depend on the absolute state ofdisorder of each field but only on the relative scale sizes.In other words, for a given scale-size difference, there arean infinite number of states of coherence that all give thesame projection-matrix elements.

The above equations can now be used to calculate thepower coupled between two Gaussian Schell-modelbeams. For convenience, we shall refer to the beams asthe source and the detector. In Fig. 3, we have assumedthat the input and output beams are identical in allrespects—r 5 1, q 5 q8—and we show the power coupled

to the load through each of the output eigenmodes. Eachcurve corresponds to a different state of coherence q. Itis clear that when both fields are fully coherent, q 5 q8@ 1, only one mode is available for transferring power.Also, because the fields have the same size, they couplewith unit efficiency. As the two fields are made increas-ingly incoherent, q 5 q8 ! 1, more and more modescouple power from the source to the detector. In addi-tion, for small values of q and q8, all of the low-ordermodes carry the same amount of power. The limitq 5 q8 → 0 corresponds to coupling a perfect blackbodysource to a perfect blackbody detector. For intermediatevalues of coherence q 5 q8 . 0.5, a few modes propagate,with the high-order modes carrying less power than thelow-order modes. We see that for q 5 q8 5 0.1 andabove, only 30 modes couple the two fields. In the calcu-lations that follow, we keep below this limit to ensure thatconvergence is achieved with only 30 Hermite modes.

In Fig. 4 we use the same parameters as Fig. 3, but nowwe show the power coupled to the load from the back-ground: The eigenvalues of the incoming field weretaken to be 1 2 an rather than an . When the fields arehighly coherent, the two beams couple with high effi-ciency, and little background noise is injected; the plot forwhich q 5 q8 5 0.5 is approaching this limit. Figure 4also shows that as the fields become increasingly incoher-ent, significant amounts of noise are injected through thehigh-order modes. It is interesting that for this model,the only case that eliminates background noise com-pletely is when the fields are fully coherent. This effectoccurs because the eigenvalues of the background fieldwere calculated directly in terms of the eigenvalues of thesource, which implies that the intensity of the source isGaussian, even though the temperature is uniform. Thesource must therefore be partially reflective or partiallytransparent, and background noise will enter accordingly.At first sight it seems surprising that when the source isfully coherent, no background power enters the detector.We must remember, however, that the spectrum has beennormalized to the lowest-order eigenvalue, which implies

Fig. 3. Power coupled through each of the natural modes of theoutput field. The source field has a total of 30 natural modes.Curves a, b, c, d, e, and f correspond to different degrees of co-herence: q 5 q8 5 0.01, 0.025, 0.05, 0.1, 0.25, and 0.5, respec-tively. The input and output fields have the same scale size,r 5 1.0.


that the coupling through the lowest-order mode is loss-less. This example emphasizes the need for great carewhen one is normalizing partially coherent coupling cal-culations.

Figure 5 shows the total signal and noise powercoupled between two fields having equal size r 5 1 andequal states of coherence q 5 q8. This plot has been lim-ited to q 5 q8 . 0.1 to ensure that 30 eigenmodes aresufficient to describe the behavior. At one extreme,q 5 q8 5 2, where both fields are essentially fully coher-ent, the source is coupled with unit efficiency, and there islittle background noise. As we move toward the other ex-treme, q 5 q8 5 0.1, where the fields are essentially in-coherent, the coupled power increases. Clearly, becausethe intensity of the source is not uniform, backgroundpower must enter the system also. If we had chosen notto normalize the spectrum to the lowest-order eigenvalue,the plots would be offset, showing that background noise

Fig. 4. Background power coupled through each of the naturalmodes of the output field. The source field has a total of 30natural modes. Curves a, b, c, d, e, and f correspond to differentdegrees of coherence: q 5 q8 5 0.01, 0.025, 0.05, 0.1, 0.25, and0.5, respectively. The input and output fields have the samescale size, r 5 1.0.

Fig. 5. Power coupled between two identical Gaussian Schell-model beams, r 5 1 and q 5 q8, as a function of the degree ofcoherence. The input and output beams are each limited to hav-ing no more than 30 natural modes at most. The solid curveshows the power coupled between the two beams, and the dashedcurve shows the power coupled from the background.

enters the detector even when the fields are fully coher-ent. Also, if we had calculated the background power onthe basis of an opaque disk, we would have arrived at adifferent result for the background power.

In Fig. 6 the sizes of the intensity distributions areequal to r 5 1, but the coherence of the input beam is var-ied while the output beam is kept fully coherent,q8 5 20. At the right of the plot, q 5 2.0, where bothbeams are essentially fully coherent, the signal is coupledwith unit efficiency, and there is no background noise. Atthe left of the plot, q 5 0.1, where the input field is essen-tially incoherent, the source is coupled with less than unitefficiency, and noise power is injected into the system.Clearly, if we had used a source with a uniform intensitydistribution, more signal power would have been coupledinto the detector.

Fig. 6. Power coupled between two Gaussian Schell-modelbeams of equal size, r 5 1. The output beam is fully coherent,q8 5 20, and the coherence of the input beam q is varied. Theinput and output beams are each limited to having no more than30 natural modes at most. The solid curve shows the powercoupled between the two beams, and the dashed curve shows thepower coupled from the background.

Fig. 7. Power coupled between two Gaussian Schell-modelbeams. The input beam is four times larger than the outputbeam, r 5 0.25, and the output beam is fully coherent,q8 5 20; the coherence of the input beam q is varied. The inputand output beams are each limited to having no more than 30natural modes at most. The solid curve shows the powercoupled between the two beams, and the dashed curve shows thepower coupled from the background.


The calculation used for Fig. 7 was similar to that ofFig. 6, but the source was made four times larger than thedetector, r 5 0.25. When the source is essentially fullycoherent, q 5 2, we see that the source is not coupled wellto the detector. For the particular geometry used here,there are equal amounts of signal and noise as q → `.As the source becomes increasingly incoherent, q → 0,the single mode of the detector is excited with unit effi-ciency, essentially because the detector ‘‘sees’’ to the innerregion of the source, which behaves locally as a blackbody.

In Fig. 8 the detector was made incoherent, q8 5 0.1,while the source was varied from being fully coherent tofully incoherent. The size of the intensity distributionsare equal, r 5 1. This model contrasts with Fig. 6,where the detector was fully coherent. When the sourcefield is essentially fully coherent, q 5 2, the detector col-lects almost all of the power available from the singlemode of the source. The coupling is not perfect, however,as parts of the incoming beam illuminate the outer re-gions of the detector, where the absorption efficiency islow. Many modes are available in the output field, andtherefore a large amount of background noise couples intothe detector also. This situation is similar, for example,to the case where a multimoded bolometer is used to ob-serve a point source in the far field of a submillimeter-wave telescope. When the source is incoherent,q 5 0.1, many detector modes can couple power from thesource. This situation is similar to the case where a mul-timoded bolometer is used to observe an extended source.When the source is significantly more incoherent than thedetector—the left extreme of the plot—all of the modes ofthe source that couple to the detector are excited withunit efficiency and the background power falls to zero.

Finally, Fig. 9 shows the same calculation as Fig. 8, butnow the detector field is made much larger than thesource: r 5 4. Again, when the source is coherent,q 5 2, all of the signal power that is available is collected.In fact, the incoming beam simply illuminates the centralregion of the detector field, which has a uniformly high

Fig. 8. Power coupled between two Gaussian Schell-modelbeams of equal size, r 5 1. The output beam is essentially fullyincoherent, q8 5 0.1, and the coherence of the input beam q isvaried. The input and output beams are each limited to havingno more than 30 natural modes at most. The solid curve showsthe power coupled between the two beams, and the dashed curveshows the power coupled from the background.

sensitivity. Background power is injected through theunused modes of the input field; and because of the sizedifference, more background power is collected than inthe case of Fig. 8. As the source becomes increasingly in-coherent, q → 0, increasing amounts of signal power be-come available, to the point where all of the incomingmodes carry signal power and the background noise fallsto zero. The size difference causes the coupling effi-ciency, in terms of what could be achieved given that thedetector is multimoded, to be relatively low. Obviously,for two fields to couple well, both the scale sizes and thetransverse coherence lengths must be equal. In otherwords, the incoming field can be highly incoherent, but itcannot transfer the power available if it does not extendwell beyond the transverse coherence length of the outputbeam. Indeed, this argument shows that the transversecoherence length of a few-moded bolometer must bematched to the scale size of the point-spread function ofthe telescope.

Although this is not explained in detail in this paper,we have also verified the expression Ps 5 Tr AB for thecoupled power by carrying out a selection of calculations,similar to those that produced results shown in Figs. 1 to8, after representing the coherence matrices of the twoSchell beams in terms of arbitrary sets of Hermite modes.Precisely the same results were found.

5. CONCLUSIONIn the context of submillimeter-wave optical systems, wehave considered how to calculate the power coupled be-tween two vector fields that are in different states of co-herence. If the second-order statistical properties of thefields are described by coherence matrices A and B, wherethe matrix elements are ensemble averages of products ofmode coefficients, then the total coupled power isPs 5 Tr AB. This expression corresponds to the case inwhich the two fields are represented in terms of the same

Fig. 9. Power coupled between two Gaussian Schell-modelbeams. The output beam is four times larger than the inputbeam, r 5 4, and the output beam is essentially fully incoherent,q8 5 0.1; the coherence of the input beam q is varied. The inputand output beams are each limited to having no more than 30natural modes at most. The solid curve shows the powercoupled between the two beams, and the dashed curve shows thepower coupled from the background.


mode set. When different mode sets are used, a similarexpression applies, but now one of the coherence matricesmust be projected onto the other through the use of theappropriate overlap integrals. Although the procedure isapplicable to vector fields, we illustrated the method byconsidering the coupling between two scalar, GaussianSchell-model beams. This simple model characterizesthe first-order behavior of many partially coherent long-wavelength optical systems.

ACKNOWLEDGMENTThe authors thank Anthony Murphy, of the National Uni-versity of Ireland, Maynooth, for a number of importantdiscussions during the course of this work.

Address correspondence to Stafford Withington, Astro-physics Group, Cavendish Laboratory, Madingley Road,Cambridge CB3 0HE, UK. Phone, 44-1223-337393; fax,44-1223-354599; e-mail: [email protected].

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power coupled between partially coherent vector fields in different states of coherence

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