Luminous Transfer in Discrete Spaces

Philip F. O'Brien

The specification of the transfer function for luminous flux in an enclosure is a necessary component of the

lighting design process. Daylight and lamplight excitations are coupled to the luminance and illuminancedistributions in space by the luminous refiectances of the surfaces and the geometrical parameters. Anidealized representation of this luminous transfer utilizes a postulate of uniform excitation and responseover discrete spatial regions. This leads to a matrix equation and an equivalent circuit or network rep-resentation of the light distributions. The discrete formulation is particularly adapted to numerical spec-ification of the transfer function for luminous flux and thus to the lighting design process.

I. IntroductionLuminous flux transfer in an enclosure or room is

associated with the luminous source, the room geometry,the surface reflectances, multiple reflections among thesurfaces, and the total resulting spatial distribution ofluminance and illuminance. The interrelationships ofthese various elements can be illustrated by a flowdiagram (Fig. 1). The excitation is constituted by theluminous source whose total luminous output, direc-tional characteristics, spectral distribution, temperalvariation, and location in the enclosure of room must bespecified. For example, the excitation of a room fordaylighting may be a window whose size, location in theroom, and apparent luminance distribution in space andtime and the spectral energy distribution must be knownif the response or luminance distribution within theroom is to be computed.

The system is the space or room that couples theexcitation to the response which for lighting systems isthe luminance of every surface element within theroom when multiple reflections are included. The sys-tem is characterized by both the reflectances p of eachsurface element and the relative geometry of the sur-faces which define the space. Although the surfacereflectance may be treated as a continuous functionof position in space, engineering representations of fluxtransfer often require that the reflectance be regardedas constant over some finite interval of space. That is,the field or continuous system is replaced by an ap-proximate lumped or discretized system whose reflec-tances and luminances are averaged and postulated tobe constant over finite areas. This lumping process

The author is with the College of Engineering, University ofCalifornia, Los Angeles, California 90024.

Received 14 June 1967.

also requires that the excitation or initial luminousemittance L(O) be averaged over some finite areas.

Luminous flux flows in a room by the process of actionat a distance and, for this reason, surface elements, notin actual contact, are in virtual or eff ective contact.That is, one finite area of a room may, for the purposesof analysis, be regarded as being in contact with manyother finite surfaces which are viewed directly (i.e.,surfaces view each other when a straight line can bedrawn within the space so as to connect the surfaces).Briefly, the action at a distance process causes certainlumped luminous systems to be multidimensional.

The response of a room to a luminous excitation isdefined simply as the luminous emittance distributionover all surface elements. This luminous emittancedistribution includes the multiple reflections which oc-curred within the room. Although a finite time intervalexists between the initiation of the luminous excitationfunction L(O) and the steady state response L of thesystem, this time constant is very short because of theshort distances in a room and the high speed of lightand may be ignored for room lighting. The illumina-tion distribution and the average illumination on theworking plane are not specified as the primary responsesbecause these quantities may be directly computedfrom the luminous emittance distribution.

The transfer functions which link the luminous re-sponse function L to the luminous excitation functionL(O) have been obtained both by experiment and byanalysis. Experimental works by Harrison and Ander-son' and later by Potter and Russell' produced extensivetables of luminous transfer functions for lamplightingexcitation. Similar experimental work with daylight-ing excitation has been reported by Griffith' and others.Analytical methods were employed by Moon andSpencer 4 to compute luminous transfer functions forsymmetrical rooms.

In the luminous field or the lumped representation ofthe field, the through variable or quantity which flows is

September 1967 / Vol. 6, No. 9 / APPLIED OPTICS 1469

simply the luminous flux or lumens. The across vari-able or potential function is the luminous emittancedistribution L which is also the response function of thetransfer system. The parameters of the field are energydissipators whose values are determined by the geom-etry and the reflectance distribution. These quanti-ties which define the field or lumped problem are mademore specific by the analytical development whichfollows. The boundary conditions which must bespecified for each finite or lumped surface are the surfacereflectance p and the initial luminous emittance L(O).Because the initial luminous emittance L(O) is also thesystem excitation, each surface element may experiencea net loss or gain of flux. This condition of distributedinputs or sinks at each element of the system leads to amathematical formulation similar to the Poisson equa-tion as pointed out by O'Brien and Howard.'

I. Discrete Formulation

The well known integral equation form of the lumi-nous transfer in an enclosure is

LdA = LOdA +space p0(dA.,dA,)LdAd,. (1)

Following the suggestion of O'Brien and Bobco6 andO'Brien,7 Eq. (1) is written in finite difference form suchthat the diffuse luminous emittance of a discrete sur-face area A, forming some part of the boundary of aroom or other enclosure defined by n discrete surfaces,is

L = Lo + PD11[(1,1)L + (1,2)L2 + ... + (l,n)Ln], (2)

where L, L, Ln = total diffuse luminous emittance ofA, A2, and A, L,, = the diffuse emittance originatingat Al, PD1 = the diffuse reflectance component of Al, and0(1,1) = the sum of the direct form factor from Alto Al and the product of the form factors to the images

of Al and the specular reflectances raised to the powerof the order of reflection (the properties of this functionare defined in Ref. 6 and in the following text).

The quantity in brackets that appears as the productof PD, in Eq. (2) is the total illumination of Al bothdirectly from the discrete surfaces and from all imagesof the discrete surfaces as seen via the specular or mirrorcomponent of each surface. As shown in Refs. 6 and 7,the function (nn) allows the computation of the totalillumination of A. directly from Am and from all theimages of A, seen when viewing half space from An.The 0 functions in enclosures bounded by plane specularsurfaces are described i Refs. 6 and 7, while the 0 func-tions in symmetrical cylindrical enclosures are given inRef. S. The functions to specular surfaces of arbi-trary shape are best obtained using digital computerprograms that employ stochastic methods.

The total illumination E(2,1) of Al from A42 and allimages of A2 as viewed from Al is

E(2,1) = (1,2)L2 , (3)where

0(1,2) = (1,2) + EF[1,2(1 + 2 + 3

+ )]PSIPs2Ps.. . (4)

The summation of Eq. (4) contains the product of theform factor F to each image of A2 as seen from Al andthe specular reflectance components Ps necessary to theformation of each image.

The functions are related by the reciprocity ex-pression

And (n,m) = A,,,(?n,n). (5)

In addition, a useful conservation relationship for the4 functions in an enclosure of n discrete surfaces is

(1 - PSI)q(l,1) + (1 - PS2)9(1,2) +

+ (1 - psn)9(1,n) = 1. (6)

An expression in the form of Eq. (2) may be written forthe total radiosity of each discrete surface that definesthe enclosure. This set of linear simultaneous equationsis written in matrix equation form as

+ [(11PD1) - (,l)] - (1,2) - -p(1,3)0(2,1) + [(1/PD2) - (2,2)] - (2,3)0(3,1) - (3,2) + [(1/PD3) - (3,3)] + ...

Transfer Matrix

Li LO1 /PDI

L2 Lo02/PD2X L3 = LO3/PD3

Response ExcitationVector Vector

The inverse of the transfer matrix of Eq. (7) contains the diffuse transfer functions (Ln,m/LOm) as follows:

+ PD1 Ll

+ PD1 L2)

(L,

+ PD2(2)L02

+ PD2 L2)+P 2D

+ PD2 I~.2L02/

+ PD3( L,.) +L 03 1

+ PD3L2,3) + Go3)

+ PD3 (L,9 +L 03I

PD1

0Lo2

I PD2

PD3i

1470 APPLIED OPTICS / Vol. 6, No. 9 / September 1967

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EXCITATION

[ L ] A

ENCLOSURESYSTEM

[ F] [ P ]RESPONSE

[ L ]

Fig. 1. Flow diagram.

From Eq. (8), the total luminous emittance Li is ob-tained in the form of linear superposition,

L = (Li,,) X L + L ) X Lo + L1)(Lo) \L02/Q03

X Lo 3 +.... (9)

The reciprocity and conservation relationships for thetransfer functions (Lnm/Lon m) may be derived by writingthe equations for net radiant transfer in a uniform tem-perature enclosure or blackbody and, also, by notingthat the sum of the net radiant transfer from all surfacesof an enclosure must be equal to zero by conservation.The details of these derivations are omitted here forbrevity. The reciprocity relationship for the transferfunctions is

(10)

The conservation relationship may be expressed inseveral forms:

(1 - PSI - pD)(Lj,i/Lo,) + (1 - PS2 - D2)(L,,2/LO2)

+ ... = -psi, (11)

( - ps, - PDI) L1 ,)\ 1 (1- PS2_ PD2) 02/

PD1 L [Lo, PD1i(L 02)

faces may be developed. This network is similar to theform of the network for flux transfer in enclosures con-taining only diffuse surfaces. The form of this networkfor surfaces that exhibit both specular and diffusecomponents is developed here in the context of luminousflux transfer in an infinite room (RCR = 0) with ceilingA, and floor A2. The luminous emittance of the ceilingis, in this case of a two-surface enclosure, written in theform of Eq. (2),

L = Loi + PD[o(l,l)Li + p(L,2)1L2]. (15)

The function (1,2) is, from Eq. (4),

p(1,2) = F(1,2) + F[1,2(1,2)]ps1ps2

+ F[1,2(1,2,1,2)Jps1,ps,2 + ... (16)The form factor F(1,2) and all form factors to imagesof A2 seen apparently behind A2 are equal to unity.The infinite series of Eq. (16) is then collected as

o(1,2) = 1/(1 - psips2). (17)

By the reciprocity relationship in Eq. (), and the factthat Al = A2 ,

0(1,2) = (2,1) = 1/(1 - PSIPS2).

By the conservation Eq. (6)

r(l1,) =1 - (1 - s2)0(l,2)

(1 - psi)P82

- PSIPS2

Combining Eqs. (18) and (19),

0(1,1) = PS20(1,2 ) = pS2q(2

, I),

0(1,2) - ps,0(l,l) = (1,2)[ - psIps2 = .

(18)

(19)

(20)

(21)

+ (1 - PS3 - D3) (L3) + =1PDi LL 03

(12)

[ Lio L2,1 (1- ps -PDI) p(ll) - + (1,2) -+ ..

_ Lo, Lo,

+ (1 PS2 PD2) (1,l) 2 + (1,2)2.2+ ]L0o2 L02

+ . = 1. (13)

Using the reciprocity relationship of Eq. (10), alltransfer functions below the diagonal of the inversetransfer matrix (8) may be computed from a knowledgeof the transfer functions above the diagonal. Thetransfer functions on the diagonal can then be com-puted using the conservation Eqs. (11), (12), or (13). Aconservation equation for the transfer functions on thediagonal of the inverse transfer matrix is

(1 -PS - PDI)2Ai LiI1 \ (1 -PS2 - D2)2A2 /L2,2\- + _IIPD1 \L o PD2 \L02J

A,(1 - PS,)(1 - PSI - PDI) A2(1 PS2)(1 - PS2 - PD2)

PD1

111. The Network RepresentationAs shown in Ref. 6, a network or equiv

for flux flow in enclosures with specular and

Equation (15) is rearranged and combined with Eq.(20)

Li(1 - psi) _ Lo,(1 - PSI PDI) + PD1pS2,P(l,2 )Lt

(i-psi) (1 - Psi - PD1)

+ pD10(1,2)L2. (22>

Adding the quantity -L(1 - p - pD,)/(l - ps,) toboth sides of Eq. (22), introducing the identities ofEq. (21), and multiplying both sides of the equation by(1 - ps,)Aj/PD1, the network form of Eq. (15) is ob-tained,

Lo , L 1(1 - psi - PD) (I- ps)

_PDi/Ai(1 - psi)(1 - p.i - pDI)_

F_ L, _ £2 1(1 - PSI) (I - PS2)

_L1/1A(1 - ps,)(1 - Ps2)0(1,2)_

(2:3)

The terms in te numerators of Eq. (23) represent po-D2 tentials or luminous flux now, while the denominators+ ... . (14) are analogous to resistance elements. When the spec-

ular reflectances are zero, the network reduces to theform for the all diffuse case. By symmetry, the net-work expression derived from the total emittance L.

alent circuit at A2 is obtained by interchanging the subscripts i Eq.1 diffuse sur- (23). The total network thus derived for the infinite

September 1967 / Vol. 6, No. 9 / APPLIED OPTICS 1471

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A. Q...) = A. (L.,.PD. Lo. PD,,, Loj

Pi

CEILING FLAl

POTENTIALS

I L-p L02I[ \ P P \ _ P 7 Ps 2) I Zs pT 2

Ro, R12 0o2

of (At Is ( |PD -S)

02 A, -PS,)( I -oI -X2 ))

02 0A S2)(I 2 -2

I RESISTANCES

Fig. 2. The network representation of lmitions emittanceand flux flow i an infinite room whose surfaces display a specular

and a diffuse component.

room is shown in Fig. 2. With reference to Fig. 2,the potentials are written above the iietwork whilethe resistances are written below and an abbreviatednotation for the resistances is defined. A continuityequation can be written from the structure of the net-work for the infinite room as

[Lol/(1 - PS - PD)] - [ 02/(1 - P2 - PD2)]Roi + R1, + R 0 2

[iL,/(1 - p.S)] - [7,2/(1 - pS2)] (24)R?12

If L02 is made zero, Eq. (24) may be solved for the fol-lowing transfer functions:

(Lj,,/Lo,) = [1 -pD2A(2,2)/Zl, (25)

(L2,1/Lol) = [PD2b(2,1)1Z], (26)

Z = 1 - PDIO(l,l) - p2i0(2,2) + PDIPD2

X [(1,1)(2,2) - (2,1)0(1,2) . (27)

Similarly when Lo, is set at zero, the remaining twotransfer functions are obtained from Eq. (24):

(JI,2,/LO2) = [1 - pDI0(ll)/Zl, (28)

(L,, 2 /Lo,) = 1pDilj(l,2)/Z1. (29)

The transfer functions of Eqs. (25)-(29) also applyto the cases of concentric spheres or infinitely longconcentric cylinders. The luminous flux is exchangedbetweeni the intier cylinder Al and the inner surface of

co the outer enclosing cylinder A2. An incandescent lampLOOR in a spherical glass or plastic sphere might be repre-

A 2 sented by this idealized geometry. The functionsthat apply in this case are slightly different than forthe infinite room because the ratio of the surface areas(A,/A 2) is not unity. The function 0(1,2) is obtainedby noting that any ray incident on the enclosing surfaceA2 that originates at the inner surface Al is directly re-flected back to Al by a specular reflection at A2. Thisis identical to the case of the infinite room and k(1,2)is obtained as in Eqs. (16) and (17). The remainingtransfer functions for this case of concentric spheres orcylinders are obtained from (1,2) combined with thereciprocity Eq. (5) and the conservation Eq. (6) asfollows:

YO(,M) = PS2/(1 - PSIPS2),

0(1,2) = 1/(1 - pSIPS2),

0(2,1) = [(Al/A 2)/(1 - SlpS2)l,

(30)

(31)

(32)

k0(2 ,2) = [ - PSIPS2 - (1 - psi)(Ai/A2 )l/

[(1 - S2)( - SlPS2)]. (33)

IV. Conclusion

A general approach to the prediction of luminoustransfer in spaces containing both specular and diffusereflecting surfaces is presented in the context of thelighting design problem. Both matrix and networkformulations of the problem are available. In thecomplete design process the transfer methods presentedhere are combined with physical statements of visualcriteria which lighting must satisfy in order to formulateeffective and economical visual displays and lightingsystems in general.

References1. W. Harrison and E. A. Anderson, Trans. Ilum. Eng. Soc. 11,

67 (1916).2. W. M. Potter and A. H. Russell, Illum. Engr. 46, 619 (1951).3. J. W. Griffith, W. J. Arner, and W. E. Conover, Illum. Engr.

50, 103 (1955).4. P. Moon and D. E. Spencer, Lighting Design (Addison-Wesley

Publ. Co., Cambridge 1948).5. P. F. O'Brien and J. A. Howard, Illum. Engr. 54, 209

(1959).6. P. F. O'Brien and R. P. Bobco, Illum. Engr. 59, 337 (1964).7. P. F. O'Brien, Illum. Engr. 61, 198 (1966).8. P. F. O'Brien and E. F. Sowell, J. Opt. Soc. Am. 57, 28

(1967).

Frederick Seitz, President of theNational Academy of Sciencesand Research Professor of Phys-ics at the University of Illinois.

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co