Stray-light corrections in integrating-sphere measurements on low-scattering samples Daniel Rbnnow and Arne Roos A method for correcting integrating-sphere signals that considers differences in the angular distribution of scattered light is extended to sources of errors that are due to stray light from imperfect optical components. We show that it is possible to measure low levels of scattering, below 1%, by using a standard integrating sphere, provided that the various contributions to stray light are taken into account properly. For low-scattering samples these corrections are more important than those from the angular distribution of the scattering. A procedure for the experimental determination of stray-light compo- nents is suggested. Simple, easy to use, compact equations for the diffuse and specular reflectance and transmittance values of the sample as functions of the recorded signals are presented. Introduction Integrating spheres have been used for optical charac- terization since early in the century.' There are several different geometrical designs as well as differ- ent sphere coatings for use in different spectral ranges. Accurate measurements are possible if the sphere is designed with ports, detectors, and internal light shields correctly positioned. 23 Single-beam and double-beam instruments have their advantages and disadvantages. Several papers have presented de- tailed theories of how the light intensity inside the sphere is set up.4- 7 Various imperfections still lead to more or less serious errors that cannot be handled easily. 89 All measurements with integrating spheres require some correction not only because of these geometrical imperfections but also because the de- tected signal depends on how the incident light is scattered by the sample. This has been treated in earlier research, 10 "'1 in which the importance of distin- guishing between the specular and diffuse compo- nents was pointed out. Most spectrophotometer manufacturers provide an integrating sphere as a standard accessory. These instruments are compact and are designed to be used for both transmittance and reflectance measurements. The compact design makes it difficult to keep stray light from entering the sphere and contributing to the The authors are with the Department of Technology, Uppsala University, Box 534, S-751 21 Uppsala, Sweden. Received 20 July 1993; revised manuscript received 4 February 1994. 0003-6935/94/25609206$06.00/0. tD 1994 Optical Society of America. detected signal level. For most routine measure- ments this can be neglected and to some extent be compensated for by setting the zero base line properly. Thus zero adjustment should in general not be the same for transmittance and reflectance measure- ments. In this paper we extend the models previ- ously presented' 0 "' to take stray light into consider- ation. It is shown that the diffuse reflectance and transmittance of samples with scattering levels of less than 1% can be measured with reasonable accuracy. The necessary corrections are in some cases greater than the measured signal, and neglecting these correc- tions leads to errors greater than 100%. It is shown that for low-scattering samples these correction fac- tors play a far more important role than the geometri- cal corrections discussed in previous research,' 0 "' whereas for samples with higher levels of diffuse reflectance or transmittance the situation is reversed. Light-scattering measurements are frequently used for the characterization of optical surfaces.' 2 In particular integrated light-scattering measurements are used for the determination of effective surface roughness.' 3 The diffuse reflectance is approxi- mately proportional to the square of the root-mean- square roughness. It is thus important to eliminate stray light in the measurements or to make the appropriate corrections. An incorrect diffuse reflec- tance value obviously gives an incorrect value for the surface roughness. We used a Beckman 5240 spectrophotometer equipped with a 198851 double-beam integrating sphere. This instrument has two entry ports for the sample and reference beams as well as an exit port for the specularly reflected beam. The formulas pre- 6092 APPLIED OPTICS / Vol. 33, No. 25 / 1 September 1994
Transcript
Stray-light corrections in integrating-spheremeasurements on low-scattering samples
Daniel Rbnnow and Arne Roos
A method for correcting integrating-sphere signals that considers differences in the angular distributionof scattered light is extended to sources of errors that are due to stray light from imperfect opticalcomponents. We show that it is possible to measure low levels of scattering, below 1%, by using astandard integrating sphere, provided that the various contributions to stray light are taken into accountproperly. For low-scattering samples these corrections are more important than those from the angulardistribution of the scattering. A procedure for the experimental determination of stray-light compo-nents is suggested. Simple, easy to use, compact equations for the diffuse and specular reflectance andtransmittance values of the sample as functions of the recorded signals are presented.
Introduction
Integrating spheres have been used for optical charac-terization since early in the century.' There areseveral different geometrical designs as well as differ-ent sphere coatings for use in different spectralranges. Accurate measurements are possible if thesphere is designed with ports, detectors, and internallight shields correctly positioned.2 3 Single-beam anddouble-beam instruments have their advantages anddisadvantages. Several papers have presented de-tailed theories of how the light intensity inside thesphere is set up.4-7 Various imperfections still leadto more or less serious errors that cannot be handledeasily.8 9 All measurements with integrating spheresrequire some correction not only because of thesegeometrical imperfections but also because the de-tected signal depends on how the incident light isscattered by the sample. This has been treated inearlier research,10"'1 in which the importance of distin-guishing between the specular and diffuse compo-nents was pointed out.
Most spectrophotometer manufacturers provide anintegrating sphere as a standard accessory. Theseinstruments are compact and are designed to be usedfor both transmittance and reflectance measurements.The compact design makes it difficult to keep straylight from entering the sphere and contributing to the
The authors are with the Department of Technology, UppsalaUniversity, Box 534, S-751 21 Uppsala, Sweden.
Received 20 July 1993; revised manuscript received 4 February1994.
0003-6935/94/25609206$06.00/0.tD 1994 Optical Society of America.
detected signal level. For most routine measure-ments this can be neglected and to some extent becompensated for by setting the zero base line properly.Thus zero adjustment should in general not be thesame for transmittance and reflectance measure-ments. In this paper we extend the models previ-ously presented' 0"' to take stray light into consider-ation. It is shown that the diffuse reflectance andtransmittance of samples with scattering levels of lessthan 1% can be measured with reasonable accuracy.The necessary corrections are in some cases greaterthan the measured signal, and neglecting these correc-tions leads to errors greater than 100%. It is shownthat for low-scattering samples these correction fac-tors play a far more important role than the geometri-cal corrections discussed in previous research,' 0"'whereas for samples with higher levels of diffusereflectance or transmittance the situation is reversed.
Light-scattering measurements are frequently usedfor the characterization of optical surfaces.' 2 Inparticular integrated light-scattering measurementsare used for the determination of effective surfaceroughness.' 3 The diffuse reflectance is approxi-mately proportional to the square of the root-mean-square roughness. It is thus important to eliminatestray light in the measurements or to make theappropriate corrections. An incorrect diffuse reflec-tance value obviously gives an incorrect value for thesurface roughness.
We used a Beckman 5240 spectrophotometerequipped with a 198851 double-beam integratingsphere. This instrument has two entry ports for thesample and reference beams as well as an exit port forthe specularly reflected beam. The formulas pre-
sented are therefore valid for this number of ports.The basic principles are, however, universal, and themethod can easily be adapted to any design of adouble-beam integrating sphere. If the same entryport is used for both the sample and reference beam,the appropriate term in the equations below simplyvanishes. A schematic drawing of the Beckmanintegrating sphere is shown in Fig. 1. The spherewall and the reference plates in this sphere are coatedwith BaSO4 paint.
Mathematical ModelTo characterize scattering samples accurately, particu-larly low-scattering ones, some corrections have to bemade. These corrections include measurements ofstray light from imperfect optical components, whichin the ideal case should be zero. The stray lightenters all open ports of the sphere. A number ofmeasurements are listed below together with thecorresponding equations for the detected signals,which, if correctly interpreted, make it possible tocorrect for these stray-light components. Themethod is applicable to transmitting as well as reflect-ing samples.
Before the measurements are taken, it is importantto establish the exact zero level of the detector, i.e.,the signal output from the detector circuits when nolight reaches the detector. This can be done with theinstrument set in a single-beam mode so that thereference beam can be blocked. With both the sampleand reference beams blocked and making sure nostray light can enter the sphere from the outside, onecan obtain the true zero reading.
In the normal double-beam mode the recordedinstrument reading is always proportional to theratio between the detected signals caused by thesample-beam intensity and the reference-beam inten-sity with a constant of proportionality A. Coveringport 4 with a BaSO4 reference, assumed to be Lamber-tian, we detect the signal S1. Port 3 should in this
Io and Ir represent the sample- and reference-beampower, respectively. RB is the reflectance of theBaSO4 reference. FB is the fraction of the reflectedradiation from the reference plate that is contained inthe sphere, i.e., the fraction that does not immedi-ately escape through the opposite ports. I and I2represent the stray-light power entering ports 1 and2, respectively, during the sample-beam reading andI,' and I2 during the reference-beam reading.Equation (la) can be simplified to
S1 = K(FBRB + D + D2), (lb)
where K is a constant of proportionality, whichincludes the sensitivity of the detector, the power ofthe incident sample beam, and division with thereference beam signal. It should be pointed out thatthe term FBRB is dimensionless. D is then thepower of the stray light that enters the spherethrough port 1 divided by the sample-beam power(=I,/Io) and is thus also dimensionless. D2 is thesame as D, but for port 2. Ideally D, and D2 shouldbe zero, but they are not, as we shall see, negligible,because all optical components scatter light more orless. Thus in the mirror chamber there is a level ofdiffuse light that enters the integrating spherethrough the ports. It is assumed below that stray-light contributions I,' and I2' can be neglected, whichmeans that the constant K is identical for all measure-ments. This simplification leads to an error that isof the order of (D, 2)2, which is less than 1-5 for allresults in this paper.
Covering port 1 with a sample gives the signal
S2 = K[FBRBTSb + (1 - B)FBRBTdb + BTdb
+ TtotD + D2], (2)
where Tsb is the collimated (specular) transmittance,defined as the part of the transmitted light that hitsthe BaSO4 plate covering port 4 and Tdb is the diffusetransmittance. B is the fraction of the diffuse trans-mittance that is ideally diffuse, and thus (1 - B) is thefraction of the diffusely transmitted light that is not.The 1 - B fraction of Tdb is assumed to be reflected offthe sphere wall in the immediate vicinity of port 4,and thus this light enters the sphere in the same wayas the specular component. From the detector pointof view Tsb and (1 - B)Tdb are equivalent. The dif-fuse light D, is reduced with the total transmittanceof the sample: Ttot = (Tsb + Tdb). The FB factor ismissing in the term BTdb since it is assumed that anegligible part of the scattered light from the sampleimmediately escapes through the reference-beam en-try port."I
To measure the diffuse transmittance Tdb, onereplaces the BaSO4 reference plate covering port 4with a beam dump, in this case a cone painted black,
with reflectance Rdump. The reflectance of the beamdump is assumed to be perfectly diffuse:
S3 = K[FBRdumpTsb + (1 - B)FBRBTdb + BTdb
+ Tt0tD, + D2]- (3)
The sample is then taken away, and the signalbecomes
With the sample removed, the black cone coveringport 4, and still with the specular exit port open,signal S8 is recorded:
S8 = K(FBRdump + D, + D2 + D3 ). (8)
Expressions for the specular and diffuse transmit-tance and reflectance can now be obtained easily; Eqs.(lb)-(4) give
S4 = K(FBRdump + D, + D2)- (4)
Port 1 is then covered with a black, completelyopaque, plate, and only the diffuse light entering thesphere through port 2 is measured:
S5 = KD2 - (5)
Measuring signals Si to S5 provides enough infor-mation to calculate the collimated and diffuse trans-mittance of a sample. If the reflectance of thesample is also desired, further measurements mustbe carried out. For measurements in the reflectancemode, the sample is placed to cover port 4 and held bythe beam dump. Thus the signal S6 is measured:
S6 = K[RBRs + (1 - B)RBRd + FBBRd
+ FBRdumpTtot2 + DI + D2] (6)
where R5 is the specular reflectance of the sample andRd is the diffuse reflectance. B is the fraction of thediffusely reflected light that is perfectly diffuse. Notethat the transmitted light is reflected by the beamdump and then transmitted through the sampleagain, making a small contribution to the signal.
The specular exit port 3 is then opened to allow thespecular beam to escape, and the signal S7 can bemeasured:
S 7 = K[CRs + (1 -B)RRd + FsBRd + FBRdupTtot 2
+ DI1+ D2 + D3]- (7)
When the specular beam enters the mirror chamber ithits a plate painted black and is partly absorbed andpartly reflected. Thus the level of diffuse light in thechamber increases because the plate cannot be per-
T _ (S2 - S3)Tb (S1 - S4) (9)
To obtain an expression for the diffuse transmit-tance, Eqs. (lb) and (3)-(5) are used:
S 3 S5 - Tsb(S4 - S5)Tdb =[ Bl-FR)1
S4 - S5 + (S1 - S4) 1 + B(R - RB) ]
(10)
For the Beckman sphere FB has been experimentallydetermined to be 0.98.8 RB, the reflectance of thediffuse BaSO4 reference, has been experimentallydetermined relative to a white ceramic diffuse reflec-tance standard.' 0 It is also necessary to determineRdump, and we show below how this term was experi-mentally determined. B is a factor that depends onthe sample. For some samples it is obvious that itshould be 1 (a perfectly diffuse sample) or 0 (perfectlyspecular). In most cases B must be estimated. Byusing Eq. (10) with B = 0 and B = 1, we can find thehighest and lowest possible values for Tdb.
Equations (lb), (4), and (6)-(8) give the followingexpression for the specular reflectance:
We show below how C can be experimentally deter-mined.
Using Eqs. (lb), (4), (7), and (8), we find the fol-lowing expression for the diffuse reflectance, whereTtot = Tsb + Tdb is taken from Eqs. (9) and (10), and R5from Eq. (11):
FB(RB - Rdump)(S7 - S8)
(S1 - S4)- CRs - FBRdump(Ttt 2
- 1)
(1 - B)RB + FBB
fectly black. C is then the fraction of the specularlyreflected beam that enters the sphere again throughthe ports. D3 represents the diffuse light in thechamber, which does not have its origin in thespecularly reflected beam but enters the spherethrough port 3.
Parameter CalibrationIn Eqs. (9)-(12) the constants Rdump and C need to beknown. To determine Rdump, we replaced the blackcone with a 56-cm baffled black cylinder with theblack cone at the end. The cylinder with the blackcone was assumed to have negligible reflectance, and
The reflectance of the beam dump was found fromEq. (14): Rdump 0.003. This value is constantthroughout the whole spectral range, 0.3-2.5 mm.It should be pointed out that the reason the inher-ently much better beam dump used for signal S4' isnot used for all measurements is that it cannot bepositioned on the sphere without considerable trouble.With this beam dump in position the compartment lidcannot be closed and stray light from the room mayenter the sphere. The only way to avoid this is toperform this measurement in complete darkness,which, for routine measurements, is quite inconve-nient.
It now remains to determine the constant C, whichwas done with an almost perfectly specular sample, inthis case a silicon wafer with a sputtered aluminumfilm. With port 4 covered with the aluminized sili-con wafer and port 3 closed, S9 was measured:
S9 = K(FBRBRspe + D + D2 ). (15)
Port 3 was opened and the following signal wasrecorded:
S1 = K(CRspec + D + D2 + D3 )- (16)
The specular reflectance of the aluminized siliconwafer can now be calculated with Eqs. (lb), (4), (14),and (15):
Rspek =
(S9 - 4)(RB - Rdump) Rdm(S - S4 ) + Rdump
RB
Equation (17) together with Eqs. (8) and (16) nowgives
FB (Sio-S8)(RB - Rdump)B (S - S4) + Rdumpl
Rspec. (18)
The constant C was thus found to be C 0.001throughout the whole spectral range. We shouldmention that the spectrophotometer's mirror cham-ber had to be slightly modified to achieve a C valuethat did not vary with wavelength. The black plate,which the specular beam hits when it leaves thesphere, was originally found to have high reflectancein the IR; the reflectance was greater than 60%,which noticeably influenced C. The plate was there-fore painted with black paint with the same low
reflectance over the whole spectral range. After thismodification C was low and constant from 0.3 to2.5 rim.
Experimental ResultsSince Rdump and C are now known, we have nounknown parameters in Eqs. (9)-(12) except param-eter B. To evaluate the procedure above and toinvestigate the influence from the stray-light correc-tions and the B factor, we prepared some scatteringsamples. Signals Si to S8 were recorded, and thespecular and diffuse reflectance and transmittancevalues were calculated according to Eqs. (9)-(12) withand without the stray-light corrections. The uncor-rected spectra were simply calculated with the signalsS4, S5, and S8 as well as Rdump and C all set to zero.In this case Eqs. (9)-(12) are identical to thosereported earlier.' 0,1 The samples used in the mea-surements were tin-oxide-coated glass substrates,one with a relatively high level of scattering and onewith relatively low scattering. The tin-oxide filmswere pyrolytically deposited in a spray oven and werea few hundred nanometers thick. Such tin-oxidefilms tend to be slightly hazy owing to the surfaceroughness of the crystalline tin oxide.'4"15 In Fig. 2the specular reflectance and collimated transmittancespectra of one of the samples are shown. As can beseen, the corrections are negligible in this case. InFigs. 3 and 4 the diffuse transmittance and reflec-tance spectra of the same sample are shown. Thecalculations have been made with the B = 1 factor aswell as with B = 0; i.e., the scattered light as recordedby the sphere is assumed to be either totally diffuse ortotally specular. Thus the curves corresponding tothese two extremes set the limit within which thetrue diffuse reflectance or transmittance value is tobe found. In Figs. 3 and 4 we can see that for thehigher scattering levels the stray-light corrections arenot as important as the influence from the B factor,whereas for lower scattering levels the stray-lightcorrections are the most important. In Figs. 5 and 6the diffuse transmittance and reflectance of a lowscattering sample are shown. For this sample the
c)
c))0
0.8
0.6
0.4
0.2
0
300 1000 3000
Wavelength/nm
Fig. 2. Specular reflectance and collimated transmittance of atin-oxide-coated glass. The difference between corrected and un-corrected spectra is negligible.
Fig. 3. Diffuse transmittance of a high-scattering tin-oxide-coatedglass. Corrected and uncorrected spectra are shown. At shorterwavelengths where the scattering is high the B factor is mostsignificant, whereas at longer wavelengths where the scattering islow the stray-light correction is most significant.
B-factor corrections are negligible compared withthose caused by stray light. The periodic variationsin the diffuse signals are due to interference effects.16
It is noticeable that the signal is readily resolved at
0.08 1 1 | I
8)
8)
0.06
0.04
0.02
Ce)
8)
8)
0.015
0.01
0.005
0
300 1000
Wavelength/nm3000
Fig. 6. Diffuse reflectance of a low-scattering tin-oxide-coatedglass. Corrected and uncorrected spectra are shown. The impor-tance of stray-light correction is obvious, whereas the B factor isless important.
this low level and that the level of the uncorrectedspectrum is more than twice the corrected spectrum.This is a clear indication of the importance of inter-preting the instrument signals correctly. The instru-ment noise is of the order of < 0.001, and the resultsof the measurements of the tin-oxide samples showthat details in the diffuse spectra well below 1% canbe detected and evaluated correctly, provided thatstray-light contributions are taken into considerationcorrectly. For levels of scattering below 10-3 thesensitivity of the integrating sphere is insufficientand another technique must be applied. A newspectroscopic total-integrated-scattering instrumentbased on a focusing Coblentz sphere' 7 has beenconstructed in our laboratory, and the initial resultsindicate that it is possible to measure scattering levelsbelow 10-4 in the wavelength range of 400-1000nm.18
300 1000
Wavelength/nm3000
Fig. 4. Diffuse reflectance of a high-scattering tin-oxide-coatedglass. Corrected and uncorrected spectra are shown. At shorterwavelengths where the scattering is high the factor B is mostsignificant, whereas at longer wavelengths where the scattering islow the stray-light correction is most significant.
0.02
.8
a88
0.015
0.01
0.005
300 1000
Wavelength/nm3000
Fig. 5. Diffuse transmittance of a low-scattering tin-oxide-coatedglass. Corrected and uncorrected spectra are shown. The impor-tance of stray-light correction is obvious, whereas the B factor isless important.
ConclusionsAccurate measurements of low-level scattering fromreflecting and transmitting samples can be performedwith a standard double-beam integrating sphere pro-vided all possible stray-light contributions are takeninto account correctly. An experimental procedurefor the determination of such stray-light contribu-tions has been suggested, and the experimental re-sults presented indicate that levels of scatteringbelow 1% can be determined. Neglecting the stray-light corrections can lead to errors of the order of afactor of 2.
The procedure suggested appears tedious at firstsight and is also more time-consuming than thestandard procedure. We must remember, however,that the stray-light contributions as defined in thispaper are constant and do not change with time.As the amount of dust and dirt on the mirrors slowlychanges over months or even years, the stray-lightlevels change only slowly. This means that, oncethe signal levels without the sample in position havebeen measured, they can be stored on a computer fileand used on a routine basis to calculate the reflec-
tance and transmittance values from Eqs. (9)-(12).The only calibration needed is to establish the abso-lute zero level.
This research was sponsored by the Swedish Coun-cil for Building Research.
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