«fc4. ' H

I
NTH) STATES IRTMENT OF MMERCE

IUCATION

NBS TECHNICAL NOTE 594-7

I.O»

Optical Radiation Measurements:

Approximate Theory of

the Photometric

Integrating Sphere

U.S.

ARTMENT

OF

)MMERCE

National

Bureau

of

Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts

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Optical Radiation Measurements:

Approximate Theory of the

Photometric Integrating Sphere

W. B. Fussell

Heat Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

4 , °**- *

C

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, D/recfor

Issued March 1974
National Bureau of Standards Technical Note 594-7

Nat. Bur. Stand. (U.S.), Tech. Note 594-7, 39 pages (Mar. 1974) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1974

For sale by (he Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SO Catalog No. C13.46:594-7). Price 75 cents.
Preface

This is the seventh issue of a series of Technical Notes entitled

OPTICAL RADIATION MEASUREMENTS. The series will consist primarily of

reports of progress in, or details of, research conducted in radiometry

and photometry in the Optical Radiation Section of the Heat Division.

The level of presentation in OPTICAL RADIATION MEASUREMENTS will be

directed at a general technical audience. The equivalent of an undergraduate degree in engineering or physics, plus familiarity with the basic

concepts of radiometry and photometry £e.g., G. Bauer, Measurement of

Optical Radiations (Focal Press, London, New York, 1965) 3 » should be

sufficient for understanding the vast majority of material in this series.

Occasionally a more specialized background will be required. Even in

such instances, however, a careful reading of the assumptions, approximations, and final conclusions should permit the non-specialist to understand

the gist of the argument if not the details.

At times, certain commercial materials and equipment will be identified in this series in order to adequately specify the experimental

procedure. In no case does such identification imply recommendation or

endorsement by the National Bureau of Standards, nor does it imply that

the material or equipment identified is necessarily the best available for

the purpose.

Any suggestions readers may have to improve the utility of this series

are welcome.

Henry J. Kostkowski, Chief

Optical Radiation Section

National Bureau of Standards

111
Contents

Page

1

.

Introduction 1

2. Current NBS Techniques for Comparing the Total Luminous

Fluxes of Lamps with the Integrating Sphere 3

3

Elementary Sphere Analysis 6

3.1. Empty Sphere Luminance Distribution 6

3.2. Introduction of the Baffle 10

3.3. Effects of Absorbing Objects 12

4. Analysis: Sphere with Coaxial Disc Baffle 13

4.1. First Order Luminance at the Sphere Window

as a Function of Source Point Coordinates 13

4.2. Lamp Flux Reflected from the Baffle 18

4.3. Extended Source Distributions 18

4.4. Extended Source Distributions : Approximate Model 20

4.5. Optimum Baffle and Lamp Positions to Minimize

the Error in Measuring Total Luminous Flux 23

4.6. Upper Bound on the Fractional Sphere Error 25

4.7. Factors Neglected 27

5

Conclusions 30

6 References 31

Figures

Figure 1. Cross section of typical integrating sphere 32

Figure 2. Cross section of empty sphere; viewfactor geometry 33

Figure 3. Integrating sphere geometry 34

IV
Approximate Theory of the Photometric Integrating Sphere

W. B. Fussell

An approximate mathematical theory of the photometric

integrating sphere is developed. The analysis is accurate to

the first order in the ratio of the baffle area to the sphere

wall area. The sphere is assumed to be occupied by a circular

baffle and a spherical lamp; the centers of the baffle and the

lamp lie on a diameter of the sphere. The surfaces of the

sphere and the baffle are assumed to reflect in a uniformly

diffuse manner. The lamp is assumed to absorb a fraction of

the radiation incident upon it, and to transmit (or specularly

reflect) the remainder. The luminance distribution at the

sphere window is derived for a general source input at any

point of the sphere wall. A model lamp illuminance distribution is assumed, and a formula for the fractional error in

comparing the total luminous fluxes of two lamps in the

integrating sphere, is derived. The physical significance of

the formula is described.

Keywords: Illuminance distribution; integrating sphere; lamp

comparisons; photometric accuracy; photometry; total luminous

flux.

1. Introduction

The photometric integrating sphere was invented by R. Ulbricht, who

made the first sphere in 1900 £lj . The usefulness of the integrating

2

sphere for comparing the total luminous fluxes of lamps was quickly

recognized, and many large spheres were soon constructed £33.

In 1915 a large integrating sphere of 2.2-meter (88-inch) diameter

was constructed at NBS by Rosa and Taylor [Vj. They compared total

luminous flux values obtained with this sphere for several incandescent

Figures in square brackets are literature references at the end of the

text.

2The nomenclature of this paper is that of the International Lighting

Vocabulary (1970) [2].
lamp-reflector combinations, against corresponding values obtained bydistribution photometry, and found differences ranging from to 1.6$ in

magnitude. These differences are within acceptable limits for NBS values

of total luminous flux, even today. (The current MBS goal is an accuracy

of 2-§# for reported values of total luminous flux, for incandescent

lamps .

)

Integrating spheres are now universally used in photometric laboratories to compare the total luminous fluxes of lamps of all types. NBS

for example, uses at present two 2-meter (79-inch) diameter spheres, one

76-cra (30-inch) sphere, and one 30-cm (12-inch) sphere, to compare the

total luminous fluxes of incandescent lamps emitting from 10,000 lumens

to 6 lumens.

The purpose of the analysis presented in this paper is: (1), to

pinpoint the major sources of error in comparing the total luminous

fluxes of lamps with the integrating sphere; (2), to estimate the magnitude of the errors in the present NBS techniques for comparing the total

luminous fluxes of incandescent lamps with the integrating sphere; (3),

to show how the errors described in (1) and (2) may be minimized.
2. Current NBS Techniques for Comparing the Total

Luminous Fluxes of Lamps with the Integrating Sphere

Figure 1 shows the cross section of a typical integrating sphere

with the baffle and a lamp inside the sphere. The lamp (whose total

luminous flux is to be compared with that of another lamp) is located at

the center of the sphere; the baffle is placed between the sphere window

and the lamp so that it completely screens the lamp from the window.

The baffle is usually placed roughly l/3 the sphere radius from the

lamp, since calculations have shown that this distance will approximately

minimize the error in comparing the total luminous fluxes of two lamps

£V], Usually, the substitution technique is employed; that is, if the

total flux of lamp A is to be compared with that of lamp B, the detector

output is measured with only lamp A in the sphere, and then with only

lamp B in the sphere.

The correction for the difference between the absorptances of lamps

A and B is found approximately by Helwig's method ^53* which uses an

auxiliary lamp placed in the sphere near the wall. Lamps A and B are

successively placed in the sphere with the auxiliary lamp (which is

shielded so that none of its flux strikes either the sphere window, or A

or B, directly). The lamps A and B are unlit , and the correction for

the difference between the absorptances of A and B is approximately the

ratio of the detector outputs produced by the auxiliary lamp when first

A, and then B, is placed in the sphere.

The diameter of the integrating sphere used to measure the total

luminous flux of a lamp, depends upon the magnitude of the lamp's flux

output. As will be shown later, if the sphere is too large, the
detector signal will be too small, if the sphere is too small, the sphere

error will be too large. At NBS, lamps emitting between 270 and 10,000

lumens are measured in a 2-meter diameter sphere; lamps of smaller flux

output are measured in either the 76-cm or the 30-cm sphere.

At present, only tungsten-filament incandescent lamps are calibrated

for total luminous flux on a regular basis at NBS [[63.

The internal surfaces of the NBS integrating spheres (and the surfaces of the baffles) are coated with paints of high reflectance which

reflect in an approximately uniformly diffuse manner. (A surface which

reflects - or emits - in a uniformly diffuse manner is defined as a surface which has the same luminance in all directions.) One 2-*ieter diameter sphere is painted with a thick coating of barium sulfate; the other

2-meter sphere - and the 76-cm and 30-cm spheres - is coated with Burch

sphere paint (Burch Paint Mfg. Co., 10609 Briggs Road, Cleveland, Ohio

Vflll) . The spectral reflectance of Burch sphere paint varies with

wavelength, falling off towards the blue end of the visible spectrum £7j.

(The spectral reflectance of barium sulfate is more uniform over the visible spectrum than that of Burch sphere paint C8j.) The net effect of

this varying spectral reflectance is to reduce the color temperature of

the illumination at the detector from that of the lamp in the sphere, by

a large amount. (For example, the color temperature of the illumination

at the detector, produced by a 2856 K lamp in the sphere, is reduced by

700 K - including the effect of the sphere window £9].)

This reduction in the color temperature of the illumination at the

detector from that of the lamp, due to the decreasing spectral reflectance

of Burch sphere paint towards the blue end of the visible spectrum,
produces an error in comparing the total luminous fluxes of two lamps with

different color temperatures. To compensate for this effect, a blue

filter (Corning 5900 glass; Corning Glass Works, Corning, N.Y. 14830) is

used to make the net spectral transmittance of the sphere (from the lamp

to the detector) approximately independent of wavelength. Calculations

show that the spectral reflectance of an integrating sphere coated with

Burch sphere paint can be neutralized sufficiently by this technique, to

reduce the error due to this cause , in comparing lamps having color

temperatures of 2360 K and 3000 K, to less than 0.2$.
3. Elementary Sphere Analysis

Elementary integrating sphere analysis has been presented in the

literature ClOJj. The first step is to compute the luminance distribution in an empty sphere.

3.1. Empty Sphere Luminance Distribution

Refer to figure 2 and let a be the colatitude angle of a source point

P(a) on the sphere wall; let 6 be the colatitude angle of a detection

point on the sphere wall. (The azimuth angles of the source point and

the detection point are ignored, since the sphere wall reflectance is

assumed to be spatially uniform, and hence the sphere is symmetric about

the polar axis.)

It is assumed throughout this paper that the sphere wall and the

baffle surfaces reflect in a uniformly diffuse manner. This means that

the luminance of an element of surface area is independent of the direction

from which it is viewed.

(The luminance, in a given direction, of an element of surface area,

is defined to be the quotient of the luminous flux leaving the surface

through an element of solid angle in the given direction, divided by the

product of the element of solid angle and the projection of the element

of surface area in the given direction. Symbolically, the luminance L

is defined as

L = lim [(6

20)/(6a*6Acos0)],

6a ->0

6A->0

where 6 is the element of luminous flux leaving the element of surface

area 6A, through the element of solid angle 6a, in a direction inclined

6
at angle 6 to the normal to the element of surface area.)

Let the reflectance of the sphere wall be r, and let the luminance

of the sphere wall at D(0), produced by a 1 lumen source input at P(<x)

,

be denoted L(cc,0). The 1 lumen source input at P(a) produces a secondary source of luminous intensity r/n candelas at P(a); this secondary

source illuminates the wall of the, sphere with a uniform illuminance of

r/(4nR ) lux (neglecting multiple reflections). To demonstrate that

this illuminance is uniform, refer to figure 2 and consider the chord PD.

The length of ED, denoted Vd, is 2Rsin(p/2) ; the angle between the normal

to the sphere at E(a) and the chord PD is £ = tt/2 - fci/2 ; this is also

the angle between the normal to the sphere at D(0) and the chord PD.

Thus the illuminance at D(0) produced by the 1 lumen source input at

P(a) , denoted E(a,6), is given by

E(a,9) «= (r/n)(PD)"2cos

2

E;

since

ED

2 - 4R2cos

2

2,

the equation for E(a,0) becomes

E(a,9) = rC^TTR

2 )*"1

,

and so it is seen that E(a,0) is indpendent of the positions of F(a) and

D(0) , as desired.

Thus the illuminance at any point on the sphere wall, produced by a

source input at any other point on the wall, is independent of the positions of the source point and the detection point, when the reflectance
of the sphere wall is spatially uniform and uniformly diffuse. Stated

in another way, the viewfactor of unit area on the sphere, as viewed from

any point on the sphere, is constant and equal to 1/(4ttR ). (The viewfactor of a surface A, , as viewed from surface Ap, is defined as the

fraction of the total flux emitted by Ap which is incident upon A-,, assuming that Ap radiates in a uniformly diffuse manner. The concept of viewfactor is also useful in radiometry and heat exchange.)

The constant viewfactor of any. area of the sphere, as viewed from

any point of the sphere, is the basic property which makes the empty

sphere, with walls of spatially uniform and uniformly diffuse reflectance,

a perfect flux integrator.

Now it is necessary to consider the interreflections in the sphere.

It is seen that the uniform illuminance over the wall of the sphere, E

(dropping the functional dependence notation), produced by the 1 lumen

source input at P(a), in turn produces a luminance distribution L' which

is also uniform. It is clear that

L" = rE/ir « r

2/(^t#R2

).

The uniform luminance distribution L' in turn produces a uniform

illuminance over the sphere wall, E', which is given by

E« « 2TT(i4ft

2 )-1

/ dG'sine'L'R2

;

this reduces to

E.i = r

2 MnR2

),

which is equivalent to

8
E« = rE.

Thus it is seen that the uniform illuminance in the sphere produced

by the n-th reflection from the wall, E , is related to the initial illuminance from the source point, E, by the equation,

Therefore the total illuminance in the sphere, E

TQT , is also uniform

and is given by

oo ETPfT = Er,£nr » TOT n^0J

which reduces to

E

TOr =E/(l-r).

Consider now instead of a point source input, the input illuminance

distribution E ( a) . The total input flux, 0* , is

0*{j «= 2TTR^/ da'sina'E

IN

(a»).

The total illuminance over the sphere wall produced by E

IN (a) is therefore

( excluding the input illuminance)

-i tt -. E

Ta]? = r(l-r)"x

/ da"sina'E

IN

(a«)2"-L

.

In terms of 0|N

» it is seen from the preceding two equations that

E

TOT

= ^r/Md-r)].

This is the basic integrating sphere equation ; it states that the

illuminance inside an empty sphere, with walls whose reflectance is

9
spatially uniform and uniformly diffuse, is proportional to the total

source luminous flux in any region of the sphere which is not directly

illuminated by the source.

Note that in this simple model, E^^p becomes very large as the wall

reflectance r approaches 1; note also that EfOT decreases rapidly with

increasing sphere radius R.

A necessary condition on Emgj, is that it satisfy flux conservation;

that is, the total flux absorbed at the wall of the sphere, 0^,* must

equal the source input flux, 0* . To show that E satisfies flux

conservation, note that

d* = E 4nR^(l-r)/r «= i*

*OUT TOT v ;/ ^BJ

3.2. Introduction of the Baffle

Consider a source point P(a) which is screened from a detection point

D(9«) by an infinitesimal baffle. The effect of such a baffle is to

eliminate the initial illuminance due to the source point P(a) at D(9").

Thus the total illuminance distribution in the sphere is no longer uniform,

and it is now a function of 6 as well as a. It is seen that, at a detection point D(8) which is not screened from the source point by the baffle,

the total illuminance (now denoted E(9) since it is a function of 6)

for a 1 lumen source input, is

EO)TOT

= r/[>TR*(l-r)],

identical to the illuminance equation for the empty sphere given above.

However, for the screened detection point D(9'), it is clear that the

total illuminance, E(9') , for a 1 lumen source input, is

10
E(e«)

T0T »E(9)

TOr

- r/(4TTR

2

),

since the illuminance from the source point which is screened by the

baffle from the detection point is r/(4nR ). Thus one arrives at the

result,

E(6«) T0T

= r

2 /[W(l-r)].

The basic effect, therefore, of even an infinitesimal baffle is to screen

every point on the sphere from another point on the sphere; if a detection

point is screened from a source point, the total illuminance at the detection point due to the screened source point, is equal to the total

unscreened illuminance due to the source point multiplied by the spherewall reflectance r.

Consider now the effect of a baffle of finite size. The baffle

clearly screens an area of the sphere wall from each point on the wall;

the screened area associated with each point is defined by the projection

of the baffle boundary from the point to the sphere wall. For a source

point P(cc) and a detection point D(6") screened by the baffle, the major

effect of the baffle is the screening of the initial illuminance from

P(a) at D(6'). This effect is termed a "zero-order" effect, since it is

independent of the baffle area. For an accurate "first order" analysis

("first order" as used in this paper means the ratio of the baffle area

to the total sphere-wall area, a quantity assumed to be much less than 1)

,

several other effects must be considered (such as the flux reflected from

baffle surfaces, and the screening of sphere-wall areas which are not

source areas). These effects, however, vary as the baffle area and thus

are excluded from the elementary analysis of this section.

11
3.3. Effects of Absorbing Objects

The effects of introducing absorbing objects into the sphere are of

two types: first, the screening effect which was treated in an elementary

fashion above, and second, the absorption effect. This effect depends

upon the area of the absorbing object, and the absorptance of the object.

An elementary expression for the absorption effect can be derived from the

flux conservation requirement discussed in section 3.1.

Consider a source point P(a) in an empty sphere which generates the

uniform illuminance E

TOT over the wall of the sphere. If an absorbing

object of area A and absorptance k is introduced into the sphere, then the

illuminance in the sphere is no longer uniform, due to the screening effect

of the object. If the object is assumed to be opaque . with reflecting

surfaces of absorptance X (at any angle of incidence) , then the illuminance

at the detection point D(6), denoted E (a,8,X), is given approximately by

[l-(l-r)S(a,0)]

E

TOT

(a » 6 ^- E

TOT

^

where S(a,0) is the screening function of the object. This function is

defined to be 1 if the object intercepts the line connecting the source

point F(a) and the detection point D(6), and otherwise.

If the object is assumed to be transparent, with non-reflecting

surfaces, and to absorb the fraction X of the incident flux which strikes

it from any direction, then the illuminance at the detection point is

given approximately by

[l-X(l-r)S(a,e)]

E (a,9,X) = 5——-. T0T [l+U/(4TTR2(l-r))]

12
k. Analysis: Sphere with Coaxial Disc Baffle

k.l. First Order Luminance at the Sphere Window

as a Function of Source Point Coordinates

In figure 3, a is the radius of the disc baffle B; the baffle axis

goes through the centers of the sphere and the lamp L; the plane of the

baffle is a distance h from the center of the sphere; the center of the

lamp is a distance 1 from the center of the sphere (1 is measured in the

opposite sense from h) ; the radius of the lamp is b, and R is the radius

of the sphere. The position of the source point P(a) is defined by the

colatitude angle a; a" is the colatitude angle of the projection on the

sphere wall of the center of the lamp, from P(a) ; a is the colatitude

angle of the projection on the sphere wall of the center of the baffle,

from P(a). The detection point is now taken to be the sphere window W

at the south pole of the sphere (that is, at 6=tt) . The luminance of the

sphere wall at the window, produced by a 1 lumen source input at P(cx) , is

denoted L, (a,n,\) ; k is the absorptance of the lamp. The surface of the

baffle visible from the lamp is labeled the "north" surface, and has

reflectance !•„. The surface of the baffle visible from the sphere window is labeled the "south" surface, and has reflectance r

s

. The lamp is

assumed to be transparent and non-reflecting, and to absorb the fraction

\ of radiation incident upon it from any direction.

The viewfactor of the baffle from the point P(a) is defined as the

fraction of the luminous flux reflected from P(a) which strikes the

baffle; this viewfactor is denoted FBp

(a). Similarly, the viewfactor of

the lamp from P(a) is denoted F^Ca). The viewfactors of the baffle and

the lamp from the detection point at the sphere window, are denoted

13
F_. r,(ir) and FT (tt) , respectively,

BD LdJ At this point it is assumed (as suggested by Safwat £ll3) that the

baffle radius a is much less than the sphere radius R; it is also assumed

that the radius of the lamp b is sufficiently small to ensure that no

luminous flux from the lamp reaches the sphere window directly (this is

the "screening condition"); thus the lamp radius b is also much less than

the sphere radius R. On the basis of these assumptions, therefore, it

is reasonable to compute the luminance produced at the sphere window by

a 1 lumen source input at P(a) , L-(a,T7,X,), to terms of the first order in

the ratio of the area of the baffle to the sphere-wall area, and to neglect

terms of higher order in this ratio. Thus the first order luminance at

the sphere window produced by a 1 lumen source input at P(a) is found to

be ( note that this is the luminance within the sphere at the window)

< q < 26 .

L^a.TT.X) - ^{kih^G)'1

x[l-(l-r

s

)FBD(n)-(l-r

N

)FBp(a)-XFLp(a)]; (1)

26 < a < v .

L

L (a,TT,X) « rC^Ti

2!^})"1

x[l-r(l-r

s )FBD

(TT)-r(l-r

N

)FBp

(a)

+r(2-r

s-r

N )a»V1

+rX(b' 2

-F

Lp

(a))]; (2)

L4-
y < a < n .

L^a.n.X) - r^TT^G) -1

x[l-r(l-r

s

)FBD(n)-r(l-r

s

)FBp(a)

+r(2-r

s-r

N

)a' V^rXCb^-F^Ca)

)

-MKl-r)r

sFBD

(n)FBp

(a)a'"2

]. (3)

In eqs (l)-(3)» the quantity G is

G = [(l-r)+r(2-r

s-rN )a»V1

+r*b»

2

],

and

a' - a/R, b' - b/R;

in addition, referring again to figure 3. it is seen that 6 is the half

angle subtended by the baffle at the sphere window W, while Y is the

colatitude angle of the baffle plane.

The physical significance of the various terms in eqs (l)-(3) is as

follows

a. The term "+rrNF

B

p(a)" represents luminous flux from the source

point which strikes the north surface of the baffle, is reflected to the

sphere wall, and then is reflected to the sphere window.

b. The term "-rF _(tt)" represents luminous flux from the sphere

wall (excluding the direct flux from the source point) which is screened

from the sphere window by the baffle.

c. The term "-rFD_(a)" represents luminous flux from the source

15
point which is screened from the sphere wall by the baffle, and is thus

screened indirectly from the sphere window. (Note that the terms "a."

and "c." represent positive and negative effects, at the window, of the

same flux; for example, if r =1, then term "a." cancels term "c.".)

2 -1

d. The term M -r(r

s+rN

)a' *f ", together with the same term in the

quantity G in the denominator, represents the luminous flux from the

sphere wall (excluding the direct flux from the source point) which

strikes the north and the south surfaces of the baffle, is reflected to

the sphere wall, and then is reflected to the sphere window.

2 -1

(Note that the net effect of the terms 1, -r(r

s +rN )a l 4 " in the

numerator and the denominator is positive . For example, if the quantity

(1-r) is much larger than the magnitude of the remaining terms in G, then

G~ may be written to first order accuracy as

G"1 « (l-r)":LCl-r(2-r

s-r

N )a'V1

(l-r)"1

-r2ib'

2(l-r)":L].

When this expansion of G~" is multiplied the numerator, the term

2 -1

"-r(r

s +rM

)a* k " in the numerator combines with the term

2 -1 -1 -1

"+r(r

s +rN

)a' k (1-r) " in the expansion of G to yield the net term

2 2 —1 —1

"+r (ro+Oa 1 4~ (1-r) " in the first order expansion of the product.)

2 -1

e. The term "+ra* 2 ", together with the same term in the quantity

G in the denominator, represents the luminous flux from the sphere wall

(excluding the direct flux from the source point) which is screened from

the sphere wall by the baffle, and is thus screened indirectly from the

sphere window.

2

f. The term "+rXb' ", together with the same term in the quantity G

in the denominator, represents the luminous flux from the sphere wall

16
(excluding the direct flux from the source point) which is partly

screened (since the lamp absorptance X is less than 1) from the sphere

wall by the lamp, and is thus screened indirectly from the sphere window.

g. The term "-rA.F (a)" represents luminous flux from the source "~ LP

point which is partly screened from the sphere wall by the lamp, and is

thus screened indirectly from the sphere window.

h. The term n +rrgFBD

(n)" represents luminous flux from the sphere

wall (excluding the direct flux from the source point) which strikes the

south surface of the baffle, and is reflected directly to the sphere

window.

i. The term "+rr

sFB

p(a) u has the same physical significance as term

"a.", "+rr

NFB

p(a)" , except that the source point now illuminates the south

surface of the baffle.

j. The term "+4(l-r)rQFBD(^)FBp(a)a , " represents luminous flux

from the source point which strikes the south surface of the baffle, and

then is reflected directly to the sphere window.

Referring again to figure 3» it is convenient to denote the intersections of the baffle axis with the sphere as the "north" and "south"

poles; the north pole is marked N in figure 3» the south pole is marked

W, since the sphere window is located at this point. The photometric

detector D (in the typical photometric integrating sphere) is located at

the sphere window, but is placed outside the sphere; the detector views

the luminous flux transmitted through the translucent (diffusely transmitting and reflecting) window. (Filters required to match the spectral

responsivity of the sphere-detector system to the CLE. photopic luminosity function £l2], are placed between the detector and the sphere window.)

17
4.2. Lamp Flux Reflected from the Baffle

In figure 3» it is seen that if 1 lumen is incident upon the north

surface of the baffle, the resulting luminance at the sphere window,

denoted L,,.., is

XD

In this equation, F,^ is the viewfactor of the lamp as viewed from the

baffle, L^CG'tHjX) is the luminance produced at the sphere window by a 1

lumen source input to the sphere wall at a point whose colatitude angle

is 8', and L-.(0,tt,X) is the average luminance produced at the sphere

window, by a 1 lumen source input which illuminates uniformly the area of

the sphere wall defined by the projection of the lamp from the center of

the baffle, onto the sphere wall. (The preceding equation for L^g is

approximate, and valid to first order only. Note that F^ is of first

2

order in a* , and that for an extended source distribution, the fraction

of the total source flux emitted by the baffle can reasonably be assumed

to be also of first order; thus for an extended source distribution, the

term involving F, R becomes of second order and hence may be discarded in

a first order analysis.)

4.3. Extended Source Distributions

If the source illuminance on the sphere wall is denoted E(a) , the

total lamp flux 0£ (lumens) is given by the equation,

$1 «= 2TiR'

i/ de«sine'E(e«)+^£

B (l-rN

), (4)

where 0* is the lamp flux which strikes the baffle. The luminance L^

18
produced at the sphere window by this lamp flux is therefore

TT l

t

- 2ra'7 de«sine»E(e')L1

(e>,TT,A).

It is useful to define the ratio of the luminance at the sphere

window to the lamp flux which produces that luminance; denote this ratio

k, so that

It is seen from the preceding equations that k is given in terms of E(a)

by the equation,

2TTR

2

/ndG »sin6 «E( • )L. ( 6 • ,tt,X)

k 5-S

i . (5)

2nR^/ d9 'sine •£( 6 ' J-H^Cl-r,,)

It is convenient now to normalize the lamp flux to 1 lumen, for

comparing lamps of different illuminance distributions. Then eq (5)

becomes

,

9 tt k - 2m'7 de«sine«E»(e , );L1

(e«,iT,x) f

where E'(8') is the normalized source illuminance distribution on the

wall of the sphere.

Consider now two different lamps, A and B, whose total luminous flux

ratio is desired. The the fractional error 6k ' in the ratio of the

sphere-window luminances, ^gA^A* compared with the true total luminous

flux ratio, 0Jq/0£a » is defined,

6k' = (kg-V/V

19
If the normalized illuminance distribution of lamp A is EJ(0')t and

that of lamp B is E'(6'), then it is seen that

/Sde«sin6«[E^(e')L

IB

(e«,n,A

s

)-E/(e')L1A(G',TT,XA )]

6k' - . (6)

/Sde«sine«E^(e»)L

:iA (e',TT,xA

)

kA. Extended Source Distributions: Approximate Model

To compute the fractional error in the ratio of the sphere-window

luminances of two different lamps, with respect to the ratio of the total

luminous fluxes of the lamps, one may of course use experimental lamp-illuminance distribution data, if these are available. Another approach is

to use a model distribution to approximate the true lamp illuminance distribution. A useful one-parameter model distribution is the cardioid,

E»(6) = (l4g(e)cos6)/(lfTiR2

);

the parameter g(6) is assumed to be a random variable such that

/ode»sine«g(e«)cose' = o. (7)

If the above cardioid lamp illuminance distribution is substituted

into eq (^) for the total lamp flux 0£, then it is seen that the total

lamp flux is independent of g(6), provided it satisfies eq (7) and that

the lamp flux which strikes the baffle, 0j*=g, is also independent of g(6).

Assume now that the two different lamps, A and B, have the following

parameters: lamp A is characterized by X., b. , and sA®)'> lamp B is characterized by k„ 9 bp, and gR

(6). Now substitute these values into eq (6)

for 6k 1 , refer to eqs (l)-(3) for L-,(6,tt,\), and use the normalization

condition (that is, the total flux from both lamp A and lamp B is 1 lumen)

20
to obtain the approximation,

6k «G"1[r(XAb/-^2

)+(l-r)

2

/ode«sin0«cose'(gA

(e»)-gB(e'))2-1

3, (8)

where

G

B

= [(l-rJ+rCE-rs-r^a'V^rXgb^2

],

and it has been assumed further that lamps A and B have similar illuminance distributions in the sense that the average value of the difference

between their normalized fluxes, over any area of the sphere, is never

larger in magnitude than a first order quantity.

It is helpful to consider the physical significance of the terms in

eq (8) at this point. The first term,

represents the difference between the fraction of its total luminous flux

absorbed by lamp A, and the fraction of its total luminous flux absorbed

by lamp B. This is seen from the fact that the flux absorbed by the

sphere wall is proportional to tyrtR (1-r) , the flux absorbed by lamp A is

2 proportional to 4nrX.b

A

, the flux absorbed by lamp B is proportional to

2

AurrXgbg, and the flux absorbed by the baffle is proportional to

nra

2 (2-r

s-r

N

).

The second term,

(l-r)

<iG--

L/ de'sine»cose'(gA

(6')-gB (e»))2 \

represents the difference between the fraction of its total luminous flux

absorbed by the screened region of the sphere wall for lamp A, and the

21
fraction of its total luminous flux absorbed by the screened region of

the sphere wall for lamp B.

It should be noted that this analysis of the physical significance of

6k' (for sources of similar illuminance distributions) is independent of

the positions of the lamps and the baffles within the sphere, provided

that the first order approximation is valid; that is, the lamps and the

baffles may have any position within the sphere for which the lamps are

screened from the sphere window and from each other, and for which all

the interaction viewfactors (the viewfactors of the baffles and the lamps

as viewed from the sphere wall and from each other) are of the first order.

If these conditions are fulfilled, then the sphere error 6k 1 in comparing

the total luminous fluxes of lamps A and B is given to first order accuracy by the formula,

6k 1 « [(Fractional Flux Absorbed by A)

-(Fractional Flux Absorbed by B)

+( Fractional Screened Flux of A Absorbed)

-(Fractional Screened Flux of B Absorbed)]. (9)

Eq (9) is not valid, of course, if the illuminance distributions of lamps

A and B differ widely.

In most practical cases, the sphere-wall absorptance is much larger

than the lamp and the baffle absorptances ; this means that

(1-r) » [r(2-r

s-r

N

)a« V^+rXb'2

].

If this inequality holds, then 6k* becomes approximately,

6k' « [r(^2-^2 )(l-r)"1+(l-r)/ode«sine»cose'(gA

(e')-gB(e'))2"1

]. (10)

22
New define an average value of (g. (9)-gB

(6)) over the screened

region of the sphere wall by the formula,

(gA

(0)-gB

(e)) sc= (2sin

z6)~^ de'sine»coseHg

A

(e')-gB

(e')).

Then eq (10) for 6k' can be written,

6k« * [r(X

Ay

2

-X

B

b^2 )(l-r)~

1

+(l-r)(gA

(e)-gB

(e))scsin

2

6];

this formula assumes that the magnitude and the sign of the average value

of (g. (0)-gR (9)) over the screened region of the sphere wall are known;

usually this information is not available, and it is reasonable to consider (gA(9)-gB

(6)) sc

as a random variable. In this case, the two components of 6k' should be added in quadrature, so that the formula for 6k 1

becomes,

6k« * [(r(X

Ay

2-^2 )(l-r)-1

) 2+((l-r)(gA

(e)-g

B

(e)) sc

sin

2

6)

2

]. (11)

4.5. Optimum Baffle and Lamp Positions to Minimize

the Error in Measuring Total Luminous Flux

Referring to figure 3 and eq (9), it is seen that the sphere error

6k 1 can be reduced by decreasing the magnitude of the fractional screened

fluxes of lamps A and B. It is also clear from figure 3 that the fractional screened flux is a function of the positions of the baffle and the

lamp within the sphere. Note particularly that in the general case,

where a fraction of the lamp flux is directly incident upon the north

surface of the baffle, the fractional screened flux includes that incident upon the baffle. If the lamp illuminance distribution is spherically uniform, then the fractional screened flux is equal to the sum of:

23
(1), the fractional solid angle (that is, the fraction of a complete sphere)

subtended by the projection of the baffle from the sphere window onto the

sphere wall, at the lamp center; plus (2), the fractional solid angle

subtended by the baffle at the lamp center. (A more realistic situation

is a spherically uniform lamp illuminance distribution up to a limiting

colatitude angle which may - or may not - include part of the baffle.)

Thus it is desired to find the baffle and lamp positions which will minimize the sura of (1) and (2) above; this is equivalent to minimizing the

total screened solid angle, as viewed from the lamp center. It can be

shown that this total screened solid angle is approximately equal to

TTa ,2[(h , +l , )"2^(l-h , )"2(l-l')"2

]. where h'=h/R and l'=l/R.

The positions of the baffle and the lamp which minimize the preceding

expression for the total screened solid angle, viewed from the lamp center,

are found to be

h» « 1» - 0.212.

Thus the optimum positions of the baffle and the lamp when the lamp

illuminance distribution is spherically uniform, are: (1), the baffle

placed 0.212 of the sphere radius from the center of the sphere, and 0.788

of the sphere radius from the sphere window; and (2), the lamp center

placed 0.212 of the sphere radius from the center of the sphere, and 1.212

of the sphere radius from the sphere window; both the baffle and the lamp

are assumed to be coaxial with the sphere axis which passes through the

center of the sphere window.

If the lamp position is fixed, and only the baffle position is varied,

then to every lamp position corresponds a baffle position which minimizes

24
the total screened solid angle; for a spherically uniform lamp illuminance

distribution, the optimum values of h* and l

1 are related by the equation,

h- - E(i-i')?/3^

1/.?]/Ei-i, ) 2/W/3].

If, for example, l'=0 (the lamp at the center of the sphere), then the

optimum value of h* is 0.386.

Now assume that the lamp has a spherically uniform illuminance distribution up to a limiting colatitude angle which excludes the baffle; in

other words, no lamp flux strikes the baffle directly. Then it is found

that the optimum baffle and lamp positions are

h» - i

1 .- 0;

that is, the baffle and the lamp should both be located at the center of

the sphere. (Note that the optimum values of h* and l

1 derived above

for a coaxial baffle and lamp configuration, also hold for baffle and

lamp positions along any chord of the sphere emanating from the sphere

window, except that h' and 1' must be replaced by h'cos£ and l'cosl,

respectively, and these displacements along the chord are referenced to

the chord center, which is a distance RcosE from the sphere window; here

2 is the angle between the chord and the sphere axis coaxial with the

sphere window.)

4.6. Upper Bound on the Fractional Sphere Error

Eq (11) shows that: (1), if two lamps have the same fractional

absorptance, the dominant term in the error in comparing their total

luminous fluxes in an integrating sphere (if they have similar illuminance

distributions) is the difference in their fractional screened fluxes; (2),

25
if two lamps have the same fractional screened flux, the dominant term in

the error in comparing their total luminous fluxes is the difference in

their fractional absorptances

.

It is useful to estimate an upper bound for the fractional sphere

error in present NBS techniques for comparing the total luminous fluxes

of 1000-watt incandescent lamps in a 2-meter integrating sphere £°0.

The geometry employed is as follows: the baffle is placed 0.55 meter from

the sphere window and is shaped roughly like a projection of the 1000-watt

lamps; the baffle dimensions are 0.15 meter by 0.23 meter (the equivalent

radius is 0.093 meter, to yield the same area) ; the lamps are separately

placed at the center of the sphere for total flux measurement (that is,

only one lamp is in the sphere at the time of measurement) ; the radius of

the lamps is roughly 0.08 meter. The reflectance of the sphere wall and

of the baffle surfaces is assumed to be 0.90 £?J. To proceed further,

it is necessary to introduce the following reasonable assumptions: (1),

the difference in the absorptance k of the lamps is less than 0.05; (2),

the magnitude of the quantity (g.(9)-g^6)) is less than 0.5 (that is,

the fractional screened fluxes of the lamps differ by less than half the

fractional screened flux of a lamp of spherically uniform illuminance

distribution)

.

With the preceding assumptions, eq (11) for the fractional sphere

error 6k 1 yields 0.29% for the component of the sphere error due to the

difference in fractional lamp absorptance, and 0.13$ for the component of

the sphere error due to the difference in fractional screened flux.

Adding these two components in quadrature, as indicated in eq (11), yields

an estimated upper bound of 0.32% for the fractional sphere error in

26
present NBS techniques for comparing the total luminous fluxes of

1000-watt incandescent lamps in a 2-meter integrating sphere.

4.7. Factors Neglected

The preceding analysis has neglected several factors and their

effects on the sphere error in comparing the total luminous fluxes of

lamps in the integrating sphere. These factors are: (1), varying spectral reflectance of the sphere and the baffle coatings, (2), varying

spectral absorptance of the lamp, (3)» spatial variations in the spherewall reflectance, (4), specular component of the sphere-wall reflectance,

(5)» reflectance of the lamp, (6), deviations of the sphere-window transmittance from the cosine law, (7), detector signal-to-noise considerations,

(8), deviations of the sphere wall from perfect sphericity.

Referring to eq (11), it is seen that the effect of increasing the

spectral reflectance of the sphere wall is to increase the fractional

fluxes absorbed by the lamps (whose total luminous fluxes are to be compared) , and to decrease the absorption of the fractional screened fluxes

of the lamps. It is also seen from eq (11) that the effect of increasing

the spectral absorptances of the lamps is to increase the fractional

fluxes absorbed by the lamps.

Theoretically, if the spectral distributions in the visible are

given of, (1), the spectral reflectance r(X) of the sphere wall (X is the

wavelength), (2), the spectral absorptances A^A.) and Xg(X.) of the lamps

A and B, (3), the spectral transmittance t(X) of the sphere window, and

(4), the spectral luminous fluxes of the lamps, JJ.M and J'(X), normalized to unit luminous flux, then the average sphere error over the visible

27
spectrum, 6k 1 , can be computed

.

The approximate formula for 6k 1 is (accurate to first order, and

assuming that lamps A and B have similar spectral luminous flux distributions) :

6k » « ([A 1 dX'(r(X')t(X')(l-r(X'))"1(J«(X»)-J'(X'))]

2

\ p

^/^dX'r^XOtCX^Cl-rCX*))-2^^')^^')^2-^^')^2

)]

2

•ff;udX'r(X')t(X')J|(X«)sin2

6(gA

(e)-g

B

(e))

sc]

2

)

- 5

>2 1 -1

X(/udX«rU»)t(X«)(l-r<X")) -1

) \ (12)

Eq (12) is analogous to eq (11) for 6k 1 in that the error components are

treated as random variables and hence are added in quadrature to obtain

the total sphere error 6k*; X, and X~ are the limits of the visible spectrum (approximately 0.38-0.78 micrometer £l2]) ; J/!(^) and ^r(^) &re normalized so that

X2 X2

/udx«j|(x«) - ;udX'j^(x») -l.

Spatial variations in the reflectance of the sphere wall may be a

significant source of error. Experimental work is needed to establish

the magnitude of this effect.

The specular component of the sphere-wall reflectance is estimated

to produce a second order sphere error in comparing the total luminous

fluxes of two lamps of similar illuminance distributions; hence this

error can be neglected in a first order analysis. (To ensure the validity of this assumption, the baffle must be of adequate size to make the

28
maximum angle of reflection, from the portions of the sphere wall directly illuminated by the source, to the sphere window, less than the critical angle at which specularity effects become important.)

The effect of luminous flux reflected from the lamp bulb is estimated

to be negligible for extended sources.

The deviations of the sphere window from cosine-law transmittance

(that is, from uniformly diffuse transmittance) are probably a significant

source of error, except where the lamps compared have very similar illuminance distributions. Probably the best procedure (if the illuminance

at the sphere window is high enough to provide an adequate signal-to-noise

ratio at the detector) is to remove the window and replace it with a

small auxiliary integrating sphere; the inner surface of this auxiliary

sphere is then viewed by the detector.

Detector signal-to-noise considerations are important in limiting

the maximum sphere diameter which can be used to compare lamps of a given

flux output. At NBS, for example, the 2-meter diameter integrating

sphere is not used to measure total luminous fluxes of less than 200

lumens C?J. Thus to optimize the accuracy of measurements of total

luminous flux, the sphere diameter should be roughly that which equalizes

the magnitudes of the computed sphere error 6k 1 , and the measurement error

due to detector noise.

The effect of deviations of the sphere wall from perfect sphericity,

on the sphere error in comparing two lamps of similar illuminance distributions, can be shown to be of second order (and hence negligible) for

deviations which consist of flattened portions of first order area (that

is , the area of each flattened portion is comparable with the baffle area)

29
5. Conclusions

The approximate mathematical theory of the photometric integrating

sphere, developed in section 4, leads to the following conclusions:

For typical sphere parameters, the fractional sphere error in comparing the total luminous fluxes of two incandescent lamps is probably

less than 0.2$ if: (1), the viewfactor of the baffle from the sphere window is less than 0.025; (2), the lamp flux directly incident upon the

baffle is negligible; (3)» the lamps compared have similar illuminance

distributions; (4), the lamp dimensions are similar; (5). the lamp absorptances differ by less than 0.02; (6), the total flux of each lamp is measured with only that lamp in the sphere (the screening effect and the

absorptance of the other lamp introduce errors if it is also present in

the sphere).

30
6. References

[l] Ulbricht, R., Die bestimmung der mittleren rattmlichen lichtintensitat durch nur eine messung, Elektrotechnische Zeit. 21, 595

(1900).

f2J International Lighting Vocabulary (Bureau Central de la CIE, Paris,

1970).

[33 Ulbricht, R. , Das Kugelphotometer, pp. 1-12 (R. Oldenbourg, Munich

and Berlin)

.

[^3 Rosa, E. B., and Taylor, A. H., Theory, construction, and use of

the photometric integrating sphere, Sci. Papers Natl. Bur. Stand.

(U.S.) 18, 281 (Aug. 1922).

£53 Keitz, H. A., Light Calculations and Measurements, p. 3^5 (MacMillan

and Co., New York, 1971).

[63 Natl. Bur. Stand. (U.S.), Tech. News Bull. 36, 153 (June 1972).

[73 Private communication frcm V. I. Burns of the NBS Optical Radiation

Section.

[83 McSparron, D., Mohan, K., Raybold, R. , Saunders, R. , and Zalewski,

E., Spectroradiometry and Conventional Photometry, an Interlaboratory Comparison, Natl. Bur. Stand. (U.S.), Tech. Note 559 » p. 186

(Nov. 1970).

[93 Burns, V. I., and McSparron, D. A., Photometric Calibration Procedures, Natl. Bur. Stand. (U.S.), Tech. Note 59^3, p. 21 (Nov. 1972).

£l03 For a typical presentation see: Walsh, J. W. T., Photometry, chap.

(Dover Publications, Inc., New York, 1958).

[ll3 Safwat, H. H., Effect of centrally located samples in the integrating sphere, J. Opt. Soc. Am. 60, 53^ (April 1970).

£l23 American Institute of Physics Handbook, 3rd ed. , chap. 6, p. 184

(McGraw-Hill Book Co., New York, 1972).

31
Figure 1. Cross section of typical integrating sphere:

B-baffle, D-detector, F-filter, L-lamp, W-window.

32
""•-v.

//

/'/

./

„/*' ^

Figure 2. Cross section of empty sphere; viewfactor geometry.

33
D

Figure 3. Integrating sphere geometry: B-baffle, D-detector, L-lamp,

N-north pole, P( a) -source point, R-sphere radius, W-sphere

window; a-baffle radius, b-lamp radius, h-displacement of baffle south of sphere equator, 1-displacement of lamp north of

sphere equator; a-colatitude angle of projection of baffle

center from P(a) to opposite sphere wall, a"-colatitude angle

of projection of lamp center from P(a) to opposite sphere

wall, 6-half angle subtended by baffle at window, Y-colatitude angle of baffle plane intersection with sphere.

3^
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