JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Network Representation of the Integrating Sphere

PHILIP F. O'BRIENDepartment of Engineering, University of California, Los Angeles 24, California

(Received December 1, 1955)

The interreflected-flux distribution in an integrating sphere equipped with apertures for the introduction,

reflection and absorption of radiant flux may be represented by a network. Sphere apertures of finite size,

nonblack apertures and sphere ports defined by plane surfaces are represented in the network as resistance

elements. For the case of black entrance and exit ports, an expression for the flux incident on the exit port

is computed by circuit analysis and is identical to the expression derived by Jacques and Kuppenheim using

the integral equation technique. A solution of the network representing a sphere with a nonblack exit port

of various relative sizes is presented as obtained using a network analyzer.

PROBLEMS concerned with the prediction of radiantflux transfer arise in the design of systems where

the flux is of interest either because of its heating effector its information content. This dual classification offlux transfer problems is according to the propertiesof the flux receiver or analyzer but the same geometricaland physical properties and the associated analyticaltools are employed regardless of the way the flux isutilized at the receiving element. For example, the totalradiant power incident on a man exposed to a radiantpanel heater may be computed using the same methodsemployed to predict the illumination within the room.

The distribution of radiant flux within an enclosuremay be predicted by using one or more of the followinganalytical methods: (1) the summation of infinite seriesof rays that are traced according to their order ofreflection,' (2) the integral equation method,2 (3) thephotic field method,3 and (4) the network method.4

Although these analytical tools are closely related, thenetwork method is particularly suited to flux-transfersystems whose geometry is not ideal in the mathematicalsense, whose surface reflectances are distributed in somenonuniform way, and whose reflectances are highenough to produce significant interreflections. Method1, is largely limited to flux transfer between infiniteparallel planes, infinite concentric cylinders, and con-centric spheres. The integral equation, Method 2, hasbeen utilized by Moon and Spencer for a wide varietyof problems but certain integrals that may arise requiremuch inventiveness and/or expense to solve even withmodern computing machinery. Moon and Spencer3

point out that the range of application of the photicfield theory (i.e., Method 3) is definitely limited...

"largely because of present limitations in availablecoordinate systems . . ."

Multi-variable flux transfer systems with irregulargeometries and reflectance distributions seem to be therule rather than the exception in design problems. Anetwork representation of this type of problem is

I Parry Moon, Lighting Design (Addison-Wesley Press, Inc.Cambridge, 1948), pp. 171-173.

2 Parry Moon, J. Opt. Soc. Am. 31, 374 (1941).3 Parry Moon and D. E. Spencer, J. Franklin Inst. 255, 33

(1953).4P. F. O'Brien, J. Opt. Soc. Am. 45, 419 (1955).

particularly helpful to the designer because eachvariable is represented by resistance elements whoserelative importance may be visualized by viewing acircuit diagram. In addition, analog and/or digitalcomputers are well suited to solving network problems.Elementary methods of network analysis may be sup-plemented by matrix operations and modern tools thathave recently been devised to reduce large networks.Because the network is a lumped parameter repre-sentation of the distributed photic field the accuracyof the network is improved by the expensive method ofincreasing its size.

THE INTEGRATING SPHERE PROBLEM

Flux distribution in an integrating sphere is, at leastin the highly idealized cases, subject to analysis by thefour methods listed above. Jacquez and Kuppenheim5

have recently reviewed the literature of integrating-sphere analysis and have extended the formulation ofthe integral equation method as applied to this problem.

An integrating sphere may be utilized in several ways.but the substitution method of reflectance measurementwith a sphere is considered in this paper as an exampleof the manner in which the network method may beemployed. In the substitution technique flux enters thesphere via a port, is incident of a specimen or standard,and is then reflected directly and indirectly via inter-reflections to a measuring port where some or all of theflux is utilized. Some postulates of the system are (1) all

reflections are perfectly diffused, (2) the sphere portsare of a finite size, (3) only the entrance port is perfectlyblack, (4) the specimen, standard and measuring portsurface conform to the spherical geometry, and (5)the ports need not be symmetrically located with respectto each other. The case of sphere ports which do notconform to the spherical geometry but are flat may betreated by the network method but the shape factorsare more complicated and may require empiricalevaluation.

The network that describes the flux transfer in thethree port integrating sphere is shown in Fig. 1.

5 J. A. Jacquez and H. F. Kuppenheim, J. Opt. Soc. Am. 45,460 (1955).

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MIAY, 1956VOLUME 46, NUMBER 5

PHILIP F. O'BRIEN Vol. 46

SPHERE WALL-A \ SPECIMEN PORT-Hod aO Hoe P r,

C/S(I-rc)

FIG. 1. Network for three port sphere where all flux enters viathe radiant potential Hoc. Subscripts a, b, c, d, refer to the entrance,photocell, sample, and sphere wall, respectively. P is the fluxinitially incident on sample or standard; as, bs, cs, and dsare the factional areas of the sphere elements; r is the spectralreflectance; and F is the shape factor.

NETWORK SOLUTION FOR BLACK ENTRANCEAND PHOTOCELL PORTS

The solution of the network of Fig. 1 for the case ofblack entrance and photocell ports (i.e. r= 0 andrb=0) is more easily visualized by reconstructing thenetwork as in Fig. 2.

From the network of Fig. 2, equations describing thecontinuity of radiant flux flow through nodes H, andHd may be written as follows:

(1- r) (c/s)(Ho - H) = (.,,- Hd) (c/s) (d/s)

r7+H{[(a/s) (c/s)+ (b/s) (c/s)] (1)

(H.-Hd) (cls) (d/s)

= Hd[(a/s) (d/s) + (b/s) (d/s)+ ( rd) (ds) (2)rd

Solving Eqs. (1) and (2) simultaneously for H (i.e.,total emittance of specimen after all interreflections)and d (i.e., total emittance of sphere wall), the fol-lowing expressions are obtained:

H.,=

Hd =

Pr,,[l -rd(d/s)]

(c/s)[ - r (c/s) -rd (d/s)]

Prerd

1- r (c/s) - rd(d/s)

(3)

(4)

The total flux incident on the phototube after allinterreflections is

Db(b/s) = HFb.. (b/s) +HdFbl-d(b/s), (5)

where Db= illumination of photocell; Foe= shape factorof c with respect to b; and, F bd = shape factor of d withrespect to b or the fraction of total flux leaving b whichis initially intercepted by d.

Substituting in Eq. (5) the values of H. and Hd givenin Eqs. (3) and (4) yields an expression for the totalflux (B) incident of the phototube as follows:

Pr, (b/s) (6)

1-rc(c/s)-rd(d/s)

Equation (6) is identical to an expression developed byJacquez and Kuppenheim5 using the integral equationmethod.

In order to investigate the effect of photocell relative-size (b/s) and photocell reflectance (rb), the network ofFig. 1 was programmed for the A-C network analyzeravailable at the Department of Engineering, Universityof California, Los Angeles. For this study the postulateswere (1) the relative sizes of the entrance and sampleports were constant at a value of 0.01, (2) the entranceport was black or ra= 0, and (3) the specimen reflectance(r,) 8=0.1, reference standard reflectance (r,) 8t=0.95,and sphere wall reflectance (rd) = 0.95.

The relationship between the relative flux absorbedat the photocell port and the photocell reflectance (rb)for various relative photocell sizes (b/s) is given inFig. 3.

The ordinate of Fig. 3 may be regarded as the sphereefficacy or the ratio of flux absorbed at the photocell tothe initial flux entering the sphere from the standard.The effect of photocell reflectance has been investigatedbecause flux diffusers whose reflectance is not zero may

Fc...a(C/S)

Fd.,.b(d/5)

FIG. 2. Network for sphere with black entrance and photocellports. Note that (a/s)+ (b/s)+ c/s) (d/s) = 1 and the path a tob vanishes because Ha= Hb.

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May1956 NETWORK REPRESENTATION OF INTEGRATING SPHERE

be used over the photocell to improve the utilizationof flux at near grazing incidence. Note that at a photo-cell reflectance of 50% a factor of twenty increase inphotocell relative-size (i.e. from (b/s)=0.01 to (b/s)=0.2) causes a factor of seven increase in the sphereefficacy.

In Fig. 4 the error associated with measuring thespecimen of 10% reflectance compared to a standard of95% when the substitution method is used is plottedas a function of photocell relative-size (bis). For smallphototube ports (e.g., less than 0.02) the error is quitesensitive to phototube size. The error in reflectancemeasurement has been shown by Jacquez6 to be equalto [si/7t) -1] where 7,s is the sphere efficacy withspecimen in place and 7at is the efficacy with standard

in place.

EVALUATION OF SHAPE FACTORS

The shape factors used in the problem of a spherewith ports whose surfaces conform to the spherical

1.0

0.

o;_ PHOTOCELL RELATIVE-SIZE

0 0.2 0.4 0.6 0.8 1X0PHOTOCELL-PORT REFLECTANCE, rb

FIG. 3. Efficacy of flux utilization in an integrating sphere usedto measure reflectance by the substitution method. 78 is the ratioof flux incident on the photocell to flux reflected from the re-flectance standard.

geometry are simply given by the ratios of the area ofsphere elements to the total area of the sphere. Whenthe ports are plane surfaces or when any "nonideal"geometry is involved the evaluation of shape factorsbecomes more difficult.

When the techniques of analytical surface integrationbecome infeasible, several empirical methods for thedetermination of shape factors are available and havebeen employed by those interested in radiant powertransfer problems. These methods depend largely onmeasurements obtained with models that performaccording to the solid angle projection or unit-sphere

D, -010o

0

-0.05o 0.10

0

0 0.05 0.1 0.15 0.2PHOTOCELL RELATIVE SIZE, b/s

FIG. 4. Errors in reflectance measurement when substitutionmethod in integrating sphere is used; specimen reflectance= 0.10,standard reflectance = 0.95; negative reflectance error (-a)

= 1-(,7./7); the parameter 1b is the photocell reflectance.

concept.6 The optical projection methods of Eckert7

and Benford,5 the mechanical projection methods ofHottel,9 Boelter,10 and Townend" and the photo-graphic technique of Jakob" are outstanding examplesof model measurements that allow easy determinationof shape factors from a differential area to a finite area.Graphical methods for shape factors to irregular shapeshave been suggested by Waldram" and Greenberg.4

An averaging of the differential-area shape factorsallows the calculation of finite-area to finite-area shapefactors. An excellent review of the mathematics of shapefactors and of the experimental methods for theirdetermination is presented by Hamilton.'5

CONCLUSION

A network which is an analog to the spatial andsurface reflectance characteristics of an integratingsphere may be easily constructed and solved with ananalog computer or numerical methods. The effects offinite-size sphere openings, flat specimens, nonuniformreflectances, nongrey surfaces, and interreflections maybe determined by this network method.

Some numerical examples of the application of thismethod to a simple sphere problem are included.

6 Parry Moon, The Scientific Basis of Illuminating Engineering(McGraw-Hill Book Company, Inc., New York, 1936), pp. 294-298.

7 E. R. G. Eckert, Introduction to the Transfer of Heat and Mass(McGraw-Hill Book Company, Inc., New York, 1950), p. 214.

8 F. Benford, J. Opt. Soc. Am. 33, 440 (1943).9 H. C. Hottel, Mech. Eng. 52, 699-704 (1930).1L. M. K. Boelter, Trans. Illum. Eng. Soc. 34, 1085-1092

(1939)."H. Townend, J. Sci. Instr. 8, 177 (1931).12 M. Jakob and G. A. Hawkins, J. Appl. Phys. 13, 246-254

(1942).13 P. J. Waldram, Illum. Engineer (London) 16, 90 (1923).14 B. F. Greenberg, Illum. Eng. 35, 629 (1940).15 D. C. Hamilton and W. R. Morgan, National Advisory

Committee for Aeronautics, Technical Note 2836 (December,1952).

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