JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Use of an Auxiliary Sphere with a Spectroreflectometer to ObtainAbsolute Reflectance

DAVID G. GOEBEL, B. PATRICK CALDWELL, AND HARRY K. HAMMOND, III

Photometry & Colorimetry Section, National Bureau of Standards, Washington, D. C. 20234(Received 27 November 1965)

Reflectance measurements that are made on a scale that is not relative to an arbitrary standard are oftencalled "absolute" measurements. The method presented here uses an auxiliary sphere with a double-beamintegrating-sphere spectrophotometer to make measurements on an absolute basis. The basic requirementsare: (1) The auxiliary sphere must be uniformly coated with a highly-reflecting, highly-diffusing material;(2) a flat plate must be coated in an identical manner to provide a measure of reflectance of the coating;(3) the interior-surface area of the sphere and the area of the entrance port must be measured.

The theory of the method is discussed and an error analysis is made. Reflectance data are reported forspecimens of smoked MgO and pressed powders of MgO and BaSO4 .

The precision of repeatability has been evaluated from measurements of a Vitrolite reference standard.More than a dozen measurements at each of eight wavelengths made over a 3-year period exhibited astandard deviation of 0.003 for the spectral reflectance.INDEX HEADINGS: Reflectance; Spectrophotometry.

1. INTRODUCTION

SPECTRAL-REFLECTANCE measurements madeon an "absolute" scale by use of an auxiliary sphere

were described by Van den Akker and associates in1956.1 Few laboratories have attempted to use the Vanden Akker method, however, because it required ademountable auxiliary sphere equipped with 13 remov-able disks which had to be individually measured toobtain the average reflectance of the sphere coatingand an index of its uniformity. If it were certain thatthe coating is uniform, measurement of only one diskwould be sufficient. If the coating technique is suf-ficiently well controlled so that the reflectance of aseparate flat plate, coated at the same time as thesphere, is essentially identical, then a removable diskis not required.

2. THEORY

Spectrophotometers designed for reflectance measure-ments are generally provided with integrating spheresto collect reflected flux. Two basic optical designs areused for specimen irradiation: double-beam and single-beam.2 Only the double-beam design permits makingabsolute reflectance measurements with a high degreeof accuracy by utilizing an auxiliary sphere with a singlecircular opening (See Fig. 1). When a given quantityof flux enters a sphere (or any enclosure) a certainfraction is reflected out through the opening. For asphere with a uniform and perfectly diffusing coating,the "effective" reflectance of the opening due to multiplereflections of the flux within the sphere can be calculatedfrom the measured reflectance of the coating and theareas of the sphere and the opening.1' 3 In analyzing theexchange of flux between the instrument sphere and the

'J. A. Van den Akker, L. R. Dearth, and W. M. Shillcox, J.Opt. Soc. Am. 46, 378A (1956); 56, 250 (1966).

2 Also referred to as the comparison mode, and the substitutionmode, respectively. A

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

auxiliary sphere, the important concept to keep in mindis that the opening of the auxiliary sphere appears thesame to the measuring sphere as does a flat, diffuselyreflecting specimen.

An integrating-sphere reflectometer can be used tomeasure the ratio of two reflectances. If the auxiliarysphere has as part of its wall a removable flat plate,then the ratio of the reflectance of the opening of theintact auxiliary sphere to the reflectance of the coatingon the flat plate can be measured. From this ratio andthe geometry of the sphere, the reflectance of the coatingcan be obtained on an absolute scale. The reflectanceof the coating can then be used to obtain the reflectanceof a reference standard of permanent material.

The effective reflectance of a port in a sphere p, iscomputed from the reflectance of the wall pp and theratio f of the area of the port to the area of the interiorof the entire sphere, by using the following equation3 :

Ps-= PF/ (1-Pp) (1-A). (1)

The spectrophotometer reading proportional to theratio of the reflectance of the auxiliary sphere and acomparison specimen is

Q8 = K (p/pc) .

The reading for the ratio of the removable plate to the

COMPARISON SPECIMEN

FIG. 1. Schematic diagram of double-beam spectrophotometerwith integrating sphere and auxiliary sphere attached.

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VOLUME 56, NUMBER 6 JUNE 1966

GOEBEL, CALDWELL, AND HAMMOND

TABLE I. Effect of sphere-coating reflectance on instrument reading calculated by use of Eq. (5)for an auxiliary sphere with an f value of 0.01.

Sphere coatingreflectance

PF 0.80 0.85 0.90 0.92 0.94 0.96 0.98 0.99 1.00

Correspondinginstrument reading

Q. 0.02 0.06 0.09 0.11 0.14 0.20 0.34 0.53 1.05n

a Reading above one is obtained because comparison reflectance p, is assumed to be 0.95.

comparison specimen is

QF=K XDPr/P3).

The ratio of Q, to Qp is

Therefore,

QPsQF=Q/Q(1-PF)/(1-f).

PF = (1-f) (Q/1Qs)/ (1-f)

(2)

(3)

Note that Eq. (3) for pp is the same as the equation forRdd derived by Van den Akker.' The ratio of the area onthe sphere occupied by the port Ap to the area of thesphere A, is

f= A p/As= 1/2[1- (1- (r/R)2)-l], (4)where

A p = 27rRk

A S = 4rR2,

in which It is the height of the spherical segment,h=R-(R2-r2)i, r equals the radius of the port Ap,and R equals the radius of the sphere.

There are several important implications in the fore-going equations. Of particular importance is the mannerin which PF affects the value of Q,.

Q.,= K@p/p7) -f /(I-pF) (1-. (5)

To obtain a reading Q, greater than 0.10, it is necessaryfor the auxiliary-sphere coating to have a reflectancepF greater than 0.90. See Table I for the effect of PF

on Q, with f=0.01, K=1.0, and p,=,0.9 5. The effectof variation in the area ratio f is shown in Fig. 2. Todetermine Q, for small values of f different from 0.01simply multiply the values for Q, in Table I by f/0.01,neglecting the change in the pF (1-f) term; for example,if f=0.007, multiply Q, by 0.7.

3. ERROR ANALYSIS

The uncertainty in PF is due to the uncertainties inthe measured values in QP, Q, and f. The general equa-tion for the uncertainty in a computed result Z wherethe independent random variables Xi are uncertain by

AXj is given approximately by:

(AZ)2 = E Xi .i \dXi

The uncertainty in PF is therefore,

( 1 =Q/Q f) </ -AQ )2(APF )2 = (1 - f)2A f+ KQS(1-f)

fQ FAQS \

Qe 2(1-f) /

As an example, for 2.54-cm-diam hole in a 15.25-cm-diam sphere, f has a value of approximately 0.007.Assume that this value can be determined to an accuracyof 1%. Assume further than QF equals 0.99 and Q,equals 0.25, yielding a value of 0.98 for pp from Eq. (3).If Qp and Q, are each uncertain by 0.002, then theuncertainty in pp from Eq. (6) is

(App) 2=4.3X 10- 8+0.32X 10 8+5.0X 10-8

AppO = .0003 1.

The uncertainty in pp can therefore be seen to behighly insensitive to the uncertainties in QF, Q., and ffor the above example. However, the uncertainty inthe determination of pp does depend strongly on itsmagnitude. Take for example, a sphere with values off=0.007, pp=O.9O, Qp2Že0.95, and Qrs0.06 (Q, fromTable I multiplied by 0.7) and with the same uncer-tainties assumed previously,

(App)2 =1.1 X 10-+65.5 X 10-8+ 13.8X 10-6,

so thatApr, = 0.0039.

Note that a decrease in pp from 0.98 to 0.90 causes theuncertainty in its determination to increase by morethan a factor of ten. Therefore, to obtain the most ac-curate value of pF, the auxiliary-sphere coating shouldbe as highly reflecting as possible; this requires theuse of such coating materials as magnesium oxide,magnesium carbonate, or barium sulfate.

I D. C. Baird, Experimzentation: An Introduction to MeasurementTheory and Experiment Design (Prentice-Hall, Inc., EnglewoodCliffs, N. J., 1962).

(6)

784Z Vol. 56

AUXILIARY SPHERE FOR ABSOLUTE REFLECTANCE

A more informative form of Eq. (6) is obtained bysubstituting Eq. (2) for QS/QF.

(1-PF Af2(1-PF(1-f) AQF)2(1-)) QF )

+( 1PF(If A)\Qs )

(1 - D) Q.,

('P f1| ) [( Af) ( QFA)Q (2 Q) ]

for f < 0.04.

The percentage error in pF is therefore proportional,to a good approximation, to the square root of the sum ofthe squares of the percentage errors in the measurements,

APF (l-PF)(i-f)P F,

PF AF (1- f)where

(7)

[= _2 _ __) AQ 2]2The dependence of the percentage error in pF on the

magnitude of pF is well illustrated in a graph of Eq. (7)

0.8

~ 0.6z4

L-J

1=0.4

0)

0.2 - f..

0.05

0.0201 0.0I1.00 .96 .92 .88 .84 .80

COATING REFLECTANCE, p

FIG. 2. Auxiliary-sphere reflectance as a function of coatingreflectance for three values of f, the ratio of the area of the portto the area of the entire sphere.

0s0

z

I I

0

LU(

0.92 0.88

REFLECTANCE, PF

0.80

FIG. 3. Curves showing the percentage error in reflectance100 APF/PF for various percentage errors in f, QF, and Q,, com-puted from Eq. (7) for values of 4 equal to 0.5, 1, 2, 4, and 6%and for f=0.01.

where APF/PF VS PF is plotted for f = 0.01 and for variousvalues for the square root of the sum of the squares ofthe percentage errors in the three measured parameters.See Fig. 3. The graph indicates that the percentage errorin p decreases rapidly with increasing reflectance eventhough the errors in the measured parameters areseveral percent.

The graph can also be used to determine the percent-age error in pF for other values of f which are slightlydifferent from 0.01.

If f= 0.01+ e, then Eq. (7) yields

APF/PF= [(1-PF) (1- 0.01 - E)/pFF(1-0.01 - 6)1@.

Thus,

APFIPF-E (1-PF) (1 - 001)IPF(l - 0.01)+ e]@)

- [(I -PF) (I - 0-01)IPF( (- 0.01)1'b+ e@),where the first term can be obtained from Fig. 3 and thesecond term is a small correction to this term. Forexample, if f=0.005, then e=-0.005. For PF=0. 9 2

and P= 2%,o APF/PFO-.OO 2 -0.005 (0.02)>0.19%.Use of Eq. (4) shows that the percentage error in f

is approximately equal to twice the percentage error inr, the radius of the port, for

Af/ft2[(Ar/r) 2 + (AR/R) 2]I.

Since the percentage error in the determination of theradius of the sphere is likely to be an order of magnitudesmaller than the percentage error in the radius of theport, Af/fi-2 (Ar/r).

The theory of the integrating sphere is based on twoassumptions concerning the coating: (1) perfect dif-fusion, and (2) perfect uniformity of reflectance at allpoints on the surface of the sphere. In actuality, nosurface is perfectly diffusing, and the reflectance of thesphere coating is likely to be somewhat nonuniform. Theuniformity of reflectance of a sphere coating should betested, particularly smoked MgO coatings because of the

June 1966 785

GOEBEL, CALDWELL, AND HAMMOND

difficulty of depositing a uniformly thick layer of MgOin a sphere. To evaluate uniformity, Van den Akker,Dearth and Shillcox3 used a sphere with 13 disks whichthey removed for measurement to provide a representa-tive sampling of the reflectance of the sphere coating.The effects on PF of lack of diffusion and lack of uni-formity of the coating are difficult to evaluate theoreti-cally, but our experience indicates that they may be theprincipal causes of the large discrepancies sometimesencountered in replicate determinations of absolutereflectance. Careful experimental technique in coatingthe sphere minimizes errors from these sources.

4. EXPERIMENTAL METHODS

All of the reflectance measurements with an auxiliarysphere were made on a Hardy-type spectrophotometeroperated by the Spectrophotometry Group at NBS.Most of the measurements have been made from 0.4 to0.75 um; a few measurements have been made to 1.08um. An auxiliary sphere with thirteen removable disksfollowing the design of Van den Akker was measured inMay 1962. The sphere was coated with BaSO4 , approxi-mately 1-mm thick, applied by spraying a suspension ofthe powder in alcohol without any binder. Consider-able difficulty was encountered, however, in coatingthe sphere, in positioning it for measurement, and inremoving the disks for measurement. Most of our meas-urements were made by using a 15-cm-diam spherewith one removable plate since measurement of thethirteen disks indicated that the coating was uniformin reflectance to within approximately -t0.2%.

One 15-cm-diam sphere was precision-made bymachining hemispheres from blocks of aluminum to atolerance of approximately 0.002 cm on the internalsphere diameter and the entrance-port diameter. Sixother spheres were made from spun-aluminum hemi-spheres; three had nominal 10-cm diameters and threehad nominal 15-cm diameters. All of these sphereswere supplied with 2.54-cm-diam entrance ports andwith separate flat plates to be coated simultaneouslywith the spheres.

Over the past three years twenty-two separate meas-urements were made utilizing various auxiliary spheres.Eight sets of data were discarded as being self-contra-dictory, the primary cause probably being nonuniform-ity in coating reflectance. Three of the discarded sets ofdata were obtained with a sphere coating of com-mercial, diffuse-white paint which had a reflectance ofless than 0.90. The five other discarded sets of data weredetermined with spheres coated with BaSO4 or MgO,applied by spraying powder suspended in alcohol. Thecoatings were rather thin (less than 1 mm) and conse-quently had low reflectances, of the order of 0.90. Ofthe fourteen usable sets of data, seven were obtainedwith coatings of BaSO4 , applied by spraying, and sevenwere obtained with coatings of MgO powder pressedinto the precision 15-cm-diam sphere.

FIG. 4. Photograph of the disassembled precision-auxiliarysphere with jig used to provide accurately dimensioned interior.coating for pressed-power specimens.

The pressed coatings were made by filling each hemi-sphere with powder and then packing it into the spherewall. The excess powder was scraped out by using aLucite disk mounted on a jig designed for the purpose(See Fig. 4). The coating layer was about 2.5 mm thick.Pressed coatings have three advantages over paintedcoatings: (1) higher reflectance, (2) better uniformity,and (3) more accurate sphere dimensions. Some of thepowder removed while forming the sphere coating waspressed to form two flat samples, eliminating the needfor a removable-flat sample in the sphere wall. Thespectrophotometer curves for a sphere coated in thismanner are shown in Fig. 5 for the visible spectrum andin Fig. 6 for the near-infrared spectrum. Note that the100-percent curve is obtained from the two flat samplespressed from the material removed from the sphere andthat the curve was rerun with the sample positions inter-changed showing that the two were quite identical.

5. RESULTS

From each set of data the absolute reflectance of thesphere coating on the flat plate was obtained, and thiswas then used to calculate the absolute reflectance of awhite-Vitrolite-reference standard. From the measure-ments of its reflectance a determination was made of theprecision of the values. If Qv is the reading proportionalto the ratio of the reflectance of the Vitrolite standardrelative to the comparison specimen, and QF is the read-ing for the flat plate relative to the same comparisonspecimen, then the absolute reflectance of the Vitroliteis given by

P. = (P F/Q F)Qv-

A particular Vitrolite, V1-G3, has been used as thereference standard for reflectance measurements for alarge number of specimens over the past twenty years.Thus, by now determining the absolute value of thereflectance of the Vitrolite we can compute the absolutereflectance of these specimens. In particular, a largenumber of freshly smoked MgO standards 1.0 mm thickhas been prepared and measured by the Spectrophotom-etry Group. We have also measured the reflectance from

786 Vol. 56

AUXILIARY SPHERE FOR ABSOLUTE REFLECTANCE

0.4 to 0.75 usm of a large number of pressed reagent-grade MgO and BaSO4 samples. The pressed sampleswere prepared in accordance with the ASTM recom-mended practice. Two pressed-MgO samples and twopressed-BaSO 4 samples have been measured from 0.73to 1.08 ,m.

The uncertainties of the reflectance values for thesamples are indicated in two ways: (1) The reproduci-bility of the samples is indicated in terms of the standarddeviation of the individual measurements,6 (2) anestimate of the "accuracy" of the reflectance values isbased on the combined uncertainty of the samplesevaluated relative to the Vitrolite standard and theuncertainty of the Vitrolite absolute-reflectance cali-bration. Table II lists the absolute-reflectance values inthe visible spectrum for an NBS-Vitrolite standard andthe values for the average reflectance of a number of

0z

I.-0-JU-Wi

0.40 0.50 0.60WAVELENGTH, I.L m

0.70

FIG. 5. Tracing of the curve sheet from a General Electric re-cording spectrophotometer used to compute absolute-reflectancedata for the visible spectrum, 0.40 to 0.75 pm. All of the specimenswere measured relative to the same comparison specimen, a speci-men pressed from powder removed from the sphere: Curve (1) isthe 100% curve obtained with the two pressed-powder specimens;Curve (2) was obtained with the Vitrolite-reference standardV1-G3; Curve (3) was obtained for a 15-cm auxiliary sphere witha pressed-MgO coating and a 2.54-cm port.

I "Tentative Recommended Practice for Preparation of Refer-ence White Reflectance Standards," ASTM Designation E 259-65T, Book of ASTM Standards, Part 30 (1966).

6 W. J. Dixon and F. J. Massey, Jr., Introduction to StatisticalAnalysis (McGraw-Hill Book Co., Inc., New York, 1957), 2nd ed.

.00

Q-e

0zF-0-JU-w

80[

60-

40[

C0.80 0.90

WAVELENGTH, A.m1.00

FIG. 6. Tracing of the near-infrared spectrum, 0.73 to 1.08 um,for the same specimens as Fig. 5.

freshly smoked MgO specimens. This table also con-tains the values for the average reflectance of a numberof pressed MgO and BaSO4 specimens together withvalues for the specimen of highest reflectance that wehave measured, a smoked-MgO specimen prepared inan argon-oxygen atmosphere. Table III contains datafor the spectral range 0.73 to 1.08 ,m.

6. CONCLUSIONS

From the data presented, we draw the followingconclusions:

(1) The auxiliary-sphere technique provides a simplemethod for determining the absolute reflectance of areference standard if the coating is uniform andperfectly diffusing.

(2) The precision of the auxiliary-sphere techniqueis such that the reflectance of a reference standard canbe determined from a relatively small number of meas-urements with a standard deviation of 0.003.

(3) Absolute-reflectance values for pressed MgO andBaSO4 determined by the auxiliary-sphere techniqueare higher than 0.98 for the entire portion of the visiblespectrum and are above 0.99 from 0.55 to 1.08 Am.

(4) The absolute-reflectance values for the averageof 20 smoked-MgO specimens in Table II beyond 0.55

I I I1/ I1

-i

=/

201-10 I 1 I'/ 1

80 --

20 --

On I I l

787June 1966

l

I

I

II

GOEBEL, CALDWELL, AND HAMMOND

TABLE II. Absolute reflectance values 0.4 to 0.75jum.

Wavelength Vitrolite V1-G31 Smoked MgOb Best MgOa Pressed MgO'l Pressed BaSO4 '(#m) P SI' P S P P S P S

0.40 0.9027 0.0046 0.9886 0.0021 0.9931 0.9841 0.0014 0.9828 0.00250.43 0.8893 0.0035 0.9916 0.0014 0.9934 0.9877 0.0016 0.9857 0.00190.49 0.9101 0.0024 0.9909 0.0014 0.9970 0.9902 0.0012 0.9894 0.00160.55 0.9190 0.0021 0.9892 0.0021 0.9954 0.9908 0.0012 0.9906 0.00110.61 0.9063 0.0022 0.9889 0.0018 0.9950 0.9914 0.0015 0.9901 0.00140.67 0.8952 0.0025 0.9874 0.0018 0.9948 0.9920 0.0018 0.9905 0.00180.72 0.8877 0.0028 0.9861 0.0021 0.9943 0.9912 0.0014 0.9906 0.00210.75 0.8804 0.0024 0.9851 0.0021 0.9949 0.9913 0.0015 0.9908 0.0017

Pooled std dcv S, 0.0029 0.0019 0.0015 0.0018Uncertainty 6t 0.0035 0.0033 0.0034

a Average values of reflectance p and the standard deviation of a single measurement computed from 12 measurements.b Average values of reflectance p computed from 20 measurements and the estimated standard deviation of a single measurement as determined by

multiplying the range of values by 0.307 as shown in Ref. 6, Table 8b(S), p. 404.e Reflectance p of a smoked-MgO specimen (1.0 mm thick) prepared in an argon-oxygen atmosphere to eliminate the possibility of the formation of

magnesium nitride.d Average values of reflectance p and the standard deviation of a single measurement computed from 30 specimens of reagent-grade MgO (5 mm thick);

powder from four different suppliers.o Average values of reflectance p and the standard deviation of a single measurement computed from 21 specimens of reagent-grade BaSO4 (5 mm thick);

powder from four different suppliers.f The accuracy of specimens measured relative to Vitrolite involves the uncertainty of both determinations; 6 is computed as [(Sp)2+Sp2]i.

TABLE III. Absolute reflectance values 0.73 to 1.08 pum.

Wave-length

(pm)

0.730.750.800.850.900.951.001.051.08

Vitrolite V1G3ap Range

0.8869 0.00000.8811 0.00220.8673 0.00020.8543 0.00230.8437 0.00230.8366 0.00060.8289 0.00050.8213 0.00160.8183 0.0000

Smoked MgObP S

0.9871 0.00240.9860 0.00270.9832 0.00280.9813 0.00320.9802 0.00320.9812 0.00280.9798 0.00330.9767 0.00430.9758 0.0043

PressedMgOc

p

0.99360.99370.99410.99340.99250.98820.99130.99130.9909

PressedBaSOad

p

0.99560.99420.99460.99440.99460.99450.99240.99360.9938

a Average values of reflectance p and range for two determinations.b Average values of reflectance p and estimated standard deviation S for

18 specimens.e Reflectance values p for one specimen. Note that this specimen has

values higher than the average values reported for pressed specimens inTable II.

d Average values of reflectance p for two specimens; powder from twosuppliers.

pm are low compared to values for the average of 30pressed-MgO specimens. This is probably due to thefact that the smoked-MgO specimens were only 1.0 mmthick whereas a thickness of 2.0 mm would be more likelyto produce the maximum reflectance of the materialwith thermal deposition. Zeiss literature on theirElrepho reflectometer indicates that the critical-layerthickness is 1.5 mm in the red end of the spectrum andfor this reason they use 1.8 mm.7

(5) Pressed-MgO and pressed-BaSO 4 specimens havenearly the same reflectance on the average and differin the visible spectrum from the average values for the1 mm thick layers of freshly smoked MgO shown inTable II by less than 0.005. In the near infrared theabsolute-reflectance values of pressed MgO shown inTable III are 1% higher than those for smoked MgO at0.8 pm and 1.5% higher at 1.08 pm.

7 Calibration of thse Reflectance Standard for the Elrepizo (CarlZeiss, Oberkochen, Germany and New York, 1963).

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788 Vol. 56