Absolute measurements of diffuse reflectance in the aoIdconfiguration

Dan Sheffer, Uri P. Oppenheim, and Adam D. Devir

An improved theory for data reduction of absolute reflectance measurements using the third Taylor methodin the ao/d configuration is presented. A brief description is given of an absolute reflectometer operating inthe 0.8-2.5-,sm region. The reflectometer is operated according to the improved theory. Experimental datafor some widely used samples are given, as well as data showing agreement between the current measurementsand those made by the U.S. National Institute of Standards and Technology.

1. Introduction

The problem of measuring the absolute diffuse reflec-tance of materials has been addressed in the past byseveral investigators.'-6 Of the several methods sug-gested for making such measurements, perhaps themost attractive is the so-called third Taylor meth-od,1'2 5 which allows one to measure many samplesquickly and easily. The most recent theory analyzingsuch measurements was introduced by Budde andDodd 5 (which we refer to as the old theory). Althoughtheir theory was formulated for the d/00 configuration,the Helmholtz reciprocity theorem7 permits the use oftheir results for the 00 /d geometry as well. Otherrelevant studies on integrating spheres have appearedrecently,- 10 showing increased interest in this field.Recently, we described elsewhere'1 the experimentalsystem that was built at the Electro Optics ResearchCenter (EORC) of the Technion for making absolutemeasurements in the 7/d configuration. It wasshown that large discrepancies between our measure-ments and measurements at the U.S. National Insti-tute of Standards and Technology (NIST) on the samestandard sample were observed when the old theorywas used, and that a new theory for data reduction wasneeded. A summary of an improved theory developedby one of the authors (D. Sheffer) was given. The useof the improved theory resulted in good agreementbetween measurements made by EORC and NIST.

The authors are with the Department of Physics, Technion-IsraelInstitute of Technology, Electro Optics Research Center, Haifa32000, Israel.

Received 7 November 1989.0003-6935/91/223181-05$05.00/0.© 1991 Optical Society of America.

In this paper the improved theory for absolute mea-surements of diffuse reflectance in the a0Id configura-tion is derived, and some experimental results are giv-en.

II. Third Taylor Method

A brief description of the third Taylor method follows.According to this method, the sample is placed againstan integrating sphere with an entrance port, a sampleport, and a detector port. Two measurements areneeded to determine the absolute diffuse reflectance ofa given sample: First, the wall of the integratingsphere is illuminated by the entering beam. The areaviewed by the detector is illuminated in this case bothby direct reflection from the sample and by multiplereflections from the sphere wall. Second, the sphere isrotated so that the illuminating beam strikes the sam-ple, only now there is a baffle that prevents the firstreflection from the sample from reaching the areaviewed by the detector (but that area is illuminated bymultiple reflections from the sphere wall). The ratioof the signal from the detector in the second case to thesignal in the first case, which we denote by K, enablesone to calculate the absolute diffuse reflectance of thesample. A schematic drawing of an integrating spheredesigned to operate according to this method appearsin Refs. 1 and 11.

Ill. Theory

In this section a derivation is given of the equations forcalculating the diffuse reflectance of an arbitrary dif-fuse sample, given the signal ratio K. To that end, thefollowing quantities, shown schematically in Fig. 1, aredefined (Fig. 1 is reproduced from Ref. 11 for the sakeof convenience):

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DETECTOR

ILLUMINATINGtLA0

ILLUMINATINGBEAM

#0

SPHEREWALL

SAMPLE

A5

Fig. 1. Schematic drawing of an integrating sphere for makingabsolute reflectance measurements, illustrating the various physicalquantities defined in Section III of the text.

r1 is the (absolute) diffuse reflectance of the sample;r2 is the diffuse reflectance of the wall of the sphere;

AO is the total area of the sphere wall, including allports;

Al is the spherical area of the sample port;A2 is the net area of the wall of the sphere, excluding

all ports;A3 is the spherical area of the entrance port;A4 is the spherical area of the detector port;A5 is the area of the wall of the sphere, which is

shielded by the baffle from direct illumination bythe sample; and

A6 is the area of that part of the sphere wall that isviewed by the detector.

We now calculate the signal from the detector for thetwo cases of sample illumination and wall illumination.The following assumptions are made:

(1) The sample and the sphere wall are completelydiffuse, that is, are completely Lambertian.

(2) The baffle is completely diffuse and its reflec-tance is identical to that of the sphere wall.

(3) The detector response depends linearly on theincident flux.

A. Calculation of the Detector Signal During WallIllumination

In what follows the total radiant flux falling on A6, Vtot,is calculated.

Let 4b be the flux (in watts) entering the sphere. -P0is reflected from the sphere wall, having a reflectanceof r2. Thus, after the first reflection from the spherewall, flux -1 is given by cb = r2 10. (P is uniformlydistributed over the sphere wall and undergoes multi-ple reflections from the wall of the sphere, each withaverage reflectance of r. Thus, after the nth reflec-tion from the wall, the flux in the sphere, caused bythat reflection, is given by

BAFFLE

Fig. 2. Schematic drawing of an absolute integrating sphere show-ing the various physical quantities used to calculate Utt.

n+l 4)lrn.,

or, expressed in terms of (D, by

n+l = r2 4'0r,.

The area viewed by detector A6 is illuminated bothby -1 and by 4'2, t3, etc. It follows that the total fluxfalling on A6 is given by the expression

A6Vtot = A ((> + (I? + 1)3 + "n +--)

After summation, this becomes

Vtt A6 r2( irm) (1)

Here the average reflectance r, assuming zero reflec-tance of the sphere ports, is given by

Ajrj + A2r2rm=A,

B. Calculation of the Detector Signal During SampleIllumination

(2)

The total flux, falling on A6 in this case, is denoted byUtot, and the various physical quantities, which will beused in the following discussion, are shown schemati-cally in Fig. 2. The calculation is similar to that of Vt(t,with a few important differences:

(a) The entering beam is reflected from the sample,having reflectance rl, and not from the sphere wall.

(b) A6 is not illuminated by any part of 4bl, becauseof the baffle.

(c) A6 is not illuminated by that part of 4b reflectedfrom the baffle, since A6 lies in the shadow of the baffle.

It is assumed that the baffle does not affect thedistribution of the flux in any way other than theeffects mentioned in items (b) and (c) above. Further-more, the sample is assumed to be concave, with thesame radius of curvature as the sphere. Any devi-ations of the real system from any of the assumptionsmade so far will cause errors in the final equations.These errors will be dealt with elsewhere. Their esti-

3182 APPLIED OPTICS / Vol. 30, No. 22 / 1 August 1991

R

mated combined effect can be kept below +0.005 in theabsolute value of ri. Under these assumptions, Uttcan be calculated as follows:

The flux entering the sphere is o0 and is reflected bythe sample. Therefore 1' = r1b0 . Had the baffle beenabsent, (b would have caused a uniform irradiance ofthe wall of the sphere. This irradiance (in watts/cm2 )would have been

El* =P -_ = °riAO AO

In the presence of the baffle, a flux equal to P isdistributed over the portion of the wall of the spherenot including A5 and is given byPi = Ei*(A2 -A A). Ofthis flux an amount P1 * is reflected after a first reflec-tion from the wall ofthe sphere: P1 * =E*(A 2 -A 5)r2.The baffle receives all the flux that the sample hasreflected and that would have reached A5 in the ab-sence of the baffle, that is, a flux equal to P2 = E*A5,and this flux is reflected back into the sphere, withreflectance r2. Therefore, a flux equal to P2 * is reflect-ed by the baffle into the sphere: P2* = Ei*Asr2. Thesample itself, being concave, is also irradiated by P1 *and reflects back into the sphere an amount of fluxequal to P3*: P3* = Ei*Ajrj. Summing up Pi*, P 2*,and P3 *, it follows that, after the first reflection of 4bfrom the various parts of the sphere, the total fluxcaused by that reflection is

2 = PI* + P2* + P3* = E,*(A2r2 + Air,),

or

'12 =- (A2r2 + Air,).

AO

This may be written as

'I2 = A (A2r2 + Air,).AO

It should be noted that it follows from Eq. (3) that 4'2 =

P0r1rm, and this result is what we would have expectedin the absence of the baffle. Thus, the introduction ofthe baffle into the sphere does not affect the value of42, but it does affect the distribution of 4'2 over the wallof the sphere.

Assuming that 2 is reflected by the wall of thesphere with average reflectance r and observing thatafterward it is distributed over the whole sphere wall,including A6 it follows that (b3 = (b2rm, 4 = b2rm, etc.

Here we assume that the baffle causes only a smalland negligible disturbance in the uniformity of thesphere illumination. This is because the radiation 4'3is no longer concentrated on the sample and the areaclose to the baffle, but originates in the hemispherereflecting '2. Now we are in a position to calculateUtot, the power received by A6 during sample illumina-tion. The contribution of (P to Utot is given by U1 = 0,because A5 does not receive direct radiation from thesample.

The contribution to UtOt of the first reflection fromthe sphere as a whole is

A6U2 Ob 2 -P2 * - P3*)

The negative terms P2 * and P3* appear because P2* iscaused by the baffle, which does not reflect 41 in thedirection of A6, and P3 * is the reflection from thesample that is not seen by A6. The contribution of thesecond reflection from the sphere is

A6

where we have assumed uniform illumination of thesphere. Similarly,

A6

etc. Summing up, we obtain

A6 ~~~~A6Utot =Ui A (t2 + P3+...+n + ..* (P2* + P3*)

-K 2(1 _:r) - AEl*(r 2A5 + r1A1).

The last expression may be reduced to

U A6 [(A2 r2 +Airi) (r2A + Al)]. (4)

If the detection system is linear, the signals from thedetector are proportional to the total flux falling on it,that is, to the flux reflected by A6 toward the detector.That flux is proportional to the total flux falling uponA6. Hence, ratio K of the signals in the two measure-ment positions, sample illumination and wall illumina-tion, is

A, (A22+Ar,) +r](3) K - utot

Vt01

(5)4A6 1\A6 r2 1-rmJ

C. Finding r2

The value of r2 can be found by covering the sampleport of the sphere with a sample having a reflectance ofr2, that is, a sample identical in reflectance to that ofthe wall of the sphere. In this case, Eq. (5) simplifiesto

(A + Al)(A + A2) 2 (A2 - A)A2

2 AK-~~ A0 r,+which may be written as

ar2 + br2 + c =0, (6)

where

(A1 + A 5)(A1 + A2 )

AO

(A2 - A,)AO

c = -K.

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Equation (6) for r2 is of the second degree and has twosolutions, one always positive and the other alwaysnegative. Hence, there is always only one physicalsolution to r2. This procedure of measuring r2 has tobe carried out only once (for each wavelength), since r2is a constant of the sphere.

D. Calculation of r1

Introducing the value of r2cubic equation for rj:

Erl + Fr2 + Gr, + H = 0,

whereA 12E =-;AO

F = Ar (A2 +A,);

G = r 2(A 2 -A 5 + 5A2 );

H = -AOr2 K.

into Eq. (5) results in a

Equation (7) generally has three (complex) solutions.By using the Cardan method to solve this equation, onecan easily show that, in the case of the actual integrat-ing spheres built in our laboratory, there is always oneand only one real solution for rl.E. Summary of the Current Method for the AbsoluteMeasurement of Diffuse Reflectance

The current method for making absolute measure-ments can be summarized as follows:

(1) The principal measuring device is an integratingsphere, operating in the a/d configuration, that canassume two positions: sample illumination and wallillumination.

(2) To perform a reflectance measurement, thesample port of the integrating sphere is covered by thesample, and two measurements of the detector signalare made, one in each of the illumination positions.The signal ratio K (= sample illumination/wall illumi-nation) is calculated.

(3) A sample that is identical to the sphere wall isintroduced, and signal ratio K2 is measured. The val-ue of K 2 is used in Eq. (6) and the value of r2 isobtained.

(4) The sample with unknown reflectance is placedat the sample port of the sphere, and signal ratio K ismeasured. The values of K1 and of r2 are then used inEq. (7), and the value of ri is found by solving it.

(5) Steps (3) and (4) are repeated for each wave-length at which r is desired.

Several points about this procedure should be noted.First, the procedure for finding r2 need be performednot each time a sample is measured but only periodi-cally, depending on the degree of confidence one hasthat the reflectance of the wall of the sphere staysconstant.

Second, area A5 may be measured in the followingway. The lower half of the sphere, containing thesample port, is detached from the reflectometer and adiffuse sample is placed at the sample port. The cen-ter of the sample is now illuminated by an intense pointsource (in the visible). The shadow of the baffle isclearly visible and A5 is easily measured. The preci-sion with which A5 must be obtained need not be veryhigh, as an error of 5% in A5 produces an error of+0.001 in the value of rl. This procedure has to beperformed only once, at a single wavelength and forone calibration sample. In subsequent reflectancemeasurements, one may use an extended illuminationspot on the sample. In this case, an effective A5 has tobe calculated. This is easily accomplished once r 2andr1 for the calibration sample are known by using Eqs.(6) and (7).

Third, the method described above for measuring r,is indeed absolute, because the experimenter does nothave to make assumptions, either about the sphere orabout the sample, to determine rl. The only assump-tion required is that the sphere wall and the sample arediffuse. Also, a sample with reflectance identical tothat of the sphere wall must be available. All the otherfactors appearing in Eqs. (6) and (7) are purely geo-metric, depending only on the design of the sphere, andmay be determined by the experimenter. No refer-ence sample is needed.

Fourth, it is clear from Eqs. (6) and (7) that r isdependent both on A5 and on r2. It can be shown thatthis dependence was the cause for the large deviationsbetween the EORC and NIST measurements. Intro-ducing it into the theory enables one to extend themeasurements into wavelength regions where r 2 is notconstant.

IV. Experimental Results

The experimental setup is that of the absolute re-flectometer at EORC and has been described else-where.' Briefly, itis composed of asystem of sphericaland plane mirrors, several radiation sources, detectors,and circular variable filters that serve as monochroma-tors. By choosing the right combination of these ele-ments, the operator can set up the system to performmeasurements in various wavelength regions and de-liver monochromatic radiation into one of two absoluteintegrating spheres, each of which is built according tothe general design described in Section II.

The measurements now described were made usinga sphere of 10-cm diameter, which was coated byBaSO4 paint. The relevant data for this sphere are:

A = 314.2 cm2; Al = 3.2 cm 2; A2 = 300.3 cm 2;

A 3 = 8.0 cm 2; A4 = 1.1 cm2 ; A 5 = 186.9 cm 2.

To verify the new theory, a standard sample wasobtained from NIST. This sample is a white ceramictile carrying the designation Standard Reference Ma-terial 2019b. Large deviations between NIST andEORC measurements on this sample were observedwhen the old theory was used for the reduction of the

3184 APPLIED OPTICS / Vol. 30, No. 22 / 1 August 1991

Table 1. Absolute Reflectance Values of Some Samples; Explanations InText

r (0.01)Wavelength

(Mm) BaSO 4 Powder BaSO 4 Paint Black Sample

0.8 1.005 0.985 0.0401.0 1.005 0.985 0.0251.2 1.000 0.980 0.0201.4 0.985 0.970 0.0201.6 0.980 0.980 0.0151.8 0.965 0.950 0.0152.0 0.920 0.900 0.0152.2 0.925 0.910 0.0152.4 0.890 0.870 0.0152.5 0.865 0.845 0.015

data. When the improved theory was used with thesame K values, good agreement was obtained."'

Using the improved theory, many samples have beenmeasured routinely. Of general interest are, perhaps,samples of BaSO4 powder and paint, and a sample oflow reflectance (black). The BaSO 4 powder sampleconsisted of Eastman White Reflectance Standardcatalog No. 6091, lot No. 307-8. It was compressed in aspecial container provided by Kodak. The BaSO4paint sample was fabricated by painting an aluminumsheet with Kodak White Reflectance Coating catalogNo. 6080. The low reflectance sample was a light trappainted with 2010 Velvet Coating manufactured by3M.

The results for these samples are given in Table I.The accuracy of the results is estimated at 10.01. Acomparison of the current BaSO4 powder results andthose made at EORC using the old theoryl shows thatthe reflectance values obtained now are even higherthan those of Ref. 1 especially in the wavelength regionabove 2.0 jum. This is the result of low values of thereflectance of the sphere coating in the latter region.This effect is not accounted for by the old theory.Further discussion of the differences between the oldand improved theories, as well as a detailed analysis ofthe accuracy of our results, will be given in a futurepaper.

V. Summary and Conclusions

An improved theory for the absolute measurement ofdiffuse reflectance in the a0 /d configuration, using thethird Taylor method, has been described. Experi-mental verification of the theory was carried out andgood agreement with the results of NIST was obtained.The new theory takes into account the effects of thereflectance of the wall of the sphere and permits theextension of the wavelength region to longer wave-lengths. Recently results at EORC have been ob-tained up to 5.5 ,um (see Ref. 11).

References1. D. Sheffer, U. P. Oppenheim, D. Clement, and A. D. Devir,

"Absolute reflectometer for the 0.8- 2 .5-Mm region," Appl. Opt.26, 583-586 (1987).

2. W. Budde, "Calibration of reflectance standards," J. Res. Natl.Bur. Stand. Sect. A 80, 585-595 (1976).

3. F. J. J. Clarke, "Measurement of the radiometric properties ofmaterials for building and aerospace applications," Proc. Soc.Photo-Opt. Instrum. Eng. 234, 40-47 (1980).

4. W. Erb, "Requirements for reflection standards and the mea-surement of their reflection values," Appl. Opt. 14, 493-499(1975).

5. W. Budde and C. X. Dodd, "Absolute reflectance measurementsin the D/0 0 geometry," Die Farbe 19, 94-102 (1970).

6. W. H. Venable, Jr., J. J. Hsia, and V. R. Weidner, "Developmentof an NBS reference spectrophotometer for diffuse transmit-tance and reflectance," Natl. Bur. Stand. (U.S.) Tech. Note 594-11 (1976).

7. F. J. J. Clarke and D. J. Parry, "Helmholtz reciprocity: itsvalidity and application to reflectometry," Light. Res. Technol.17, 1-11 (1985).

8. L. M. Hanssen, "Effects of restricting the detector field of viewwhen using integrating spheres," Appl. Opt. 28, 2097-2103(1989).

9. K. A. Snail and L. M. Hanssen, "Integrating sphere designs withisotropic throughput," Appl. Opt. 28, 1793-1799 (1989).

10. H. L. Tardy, "Flat sample and limited-field effects in integrat-ing sphere measurements," J. Opt. Soc. Am. A 5,241-245 (1988).

11. D. Sheffer, U. P. Oppenheim, and A. D. Devir, "Absolute reflec-tometer for the mid infrared region," Appl. Opt. 29, 129-132(1990).

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