Stray-light corrections in integrating-sphere measurements on low-scattering samples

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 components 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 characterization since early in the century.' There are several different geometrical designs as well as different 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.2 3 Single-beam and double-beam instruments have their advantages and disadvantages. Several papers have presented detailed 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. 8 9 All measurements with integrating spheres require some correction not only because of these geometrical imperfections but also because the detected signal depends on how the incident light is scattered by the sample. This has been treated in earlier research,10 "'1in which the importance of distinguishing between the specular and diffuse components 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. tD1994 Optical Society of America. 6092 APPLIED OPTICS / Vol. 33, No. 25 / 1 September 1994 detected signal level. For most routine measurements 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 measurements. In this paper we extend the models previously presented' 0 "' to take stray light into consideration. 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 corrections leads to errors greater than 100%. It is shown that for low-scattering samples these correction factors play a far more important role than the geometrical 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 approximately proportional to the square of the root-meansquare roughness. It is thus important to eliminate stray light in the measurements or to make the appropriate corrections. An incorrect diffuse reflectance 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- sented are therefore valid for this number of ports. The basic principles are, however, universal, and the method can easily be adapted to any design of a double-beam integrating sphere. If the same entry port is used for both the sample and reference beam, the appropriate term in the equations below simply vanishes. A schematic drawing of the Beckman integrating sphere is shown in Fig. 1. The sphere wall and the reference plates in this sphere are coated with BaSO 4 paint. Mathematical Model To characterize scattering samples accurately, particularly low-scattering ones, some corrections have to be made. These corrections include measurements of stray light from imperfect optical components, which in the ideal case should be zero. The stray light enters all open ports of the sphere. A number of measurements are listed below together with the corresponding equations for the detected signals, which, if correctly interpreted, make it possible to correct for these stray-light components. The method is applicable to transmitting as well as reflecting samples. Before the measurements are taken, it is important to establish the exact zero level of the detector, i.e., the signal output from the detector circuits when no light reaches the detector. This can be done with the instrument set in a single-beam mode so that the reference beam can be blocked. With both the sample and reference beams blocked and making sure no stray light can enter the sphere from the outside, one can obtain the true zero reading. In the normal double-beam mode the recorded instrument reading is always proportional to the ratio between the detected signals caused by the sample-beam intensity and the reference-beam intensity with a constant of proportionality A. Covering port 4 with a BaSO 4 reference, assumed to be Lambertian, we detect the signal S1. Port 3 should in this 7 6 4 5 Fig. 1. Integrating sphere: 1, entrance port-sample beam; 2, entrance port-reference beam; 3, specular exit port; 4, sample port; 5, reference port; 6, sample beam; 7, reference beam. case be covered by a BaSO 4 plate: S, = A (IOFBRB + Il + I2) (IrFBRB + I' + I2') (la) Io and Ir represent the sample- and reference-beam power, respectively. RB is the reflectance of the BaSO 4 reference. FB is the fraction of the reflected radiation from the reference plate that is contained in the sphere, i.e., the fraction that does not immediately escape through the opposite ports. I and I2 represent the stray-light power entering ports 1 and 2, respectively, during the sample-beam reading and I,' 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, which includes the sensitivity of the detector, the power of the incident sample beam, and division with the reference beam signal. It should be pointed out that the term FBRB is dimensionless. D is then the power of the stray light that enters the sphere through port 1 divided by the sample-beam power (=I,/Io) and is thus also dimensionless. D2 is the same as D, but for port 2. Ideally D, and D2 should be zero, but they are not, as we shall see, negligible, because all optical components scatter light more or less. Thus in the mirror chamber there is a level of diffuse light that enters the integrating sphere through the ports. It is assumed below that straylight contributions I,' and I2' can be neglected, which means that the constant K is identical for all measurements. This simplification leads to an error that is of the order of (D, 2)2, which is less than 1-5 for all results in this paper. Covering port 1 with a sample gives the signal S2 = K[FBRBTSb + (1 + TtotD + D2], - B)FBRBTdb + BTdb (2) where Tsb is the collimated (specular) transmittance, defined as the part of the transmitted light that hits the BaSO4 plate covering port 4 and Tdb is the diffuse transmittance. B is the fraction of the diffuse transmittance that is ideally diffuse, and thus (1 - B) is the fraction of the diffusely transmitted light that is not. The 1 - B fraction of Tdb is assumed to be reflected off the sphere wall in the immediate vicinity of port 4, and thus this light enters the sphere in the same way as the specular component. From the detector point of view Tsb and (1 - B)Tdb are equivalent. The diffuse light D, is reduced with the total transmittance of the sample: Ttot = (Tsb + Tdb). The FB factor is missing in the term BTdb since it is assumed that a negligible part of the scattered light from the sample immediately escapes through the reference-beam entry port."I To measure the diffuse transmittance Tdb, one replaces the BaSO 4 reference plate covering port 4 with a beam dump, in this case a cone painted black, 1 September 1994 / Vol. 33, No. 25 / APPLIED OPTICS 6093 With the sample removed, the black cone covering port 4, and still with the specular exit port open, signal S8 is recorded: with reflectance Rdump. The reflectance of the beam dump is assumed to be perfectly diffuse: S3 = K[FBRdumpTsb + (1 + Tt 0tD, + D2]- - B)FBRBTdb + BTdb Expressions for the specular and diffuse transmittance and reflectance can now be obtained easily; Eqs. (lb)-(4) give The sample is then taken away, and the signal becomes S4 = K(FBRdump + D, + D2)- (4) T Tb Port 1 is then covered with a black, completely opaque, plate, and only the diffuse light entering the sphere through port 2 is measured: Tdb Measuring signals Si to S5 provides enough information to calculate the collimated and diffuse transmittance of a sample. If the reflectance of the sample is also desired, further measurements must be carried out. For measurements in the reflectance mode, the sample is placed to cover port 4 and held by the beam dump. Thus the signal S 6 is measured: S5 - Tsb(S4 - S5) S3 =[ Bl-FR)1 RB) ] (10) For the Beckman sphere FB has been experimentally determined to be 0.98.8 RB, the reflectance of the diffuse BaSO 4 reference, has been experimentally determined relative to a white ceramic diffuse reflectance standard.' 0 It is also necessary to determine Rdump, and we show below how this term was experimentally determined. B is a factor that depends on the sample. For some samples it is obvious that it should be 1 (a perfectly diffuse sample) or 0 (perfectly specular). In most cases B must be estimated. By using Eq. (10) with B = 0 and B = 1, we can find the highest and lowest possible values for Tdb. Equations (lb), (4), and (6)-(8) give the following expression for the specular reflectance: (6) where R5 is the specular reflectance of the sample and Rd is the diffuse reflectance. B is the fraction of the diffusely reflected light that is perfectly diffuse. Note that the transmitted light is reflected by the beam dump and then transmitted through the sample again, making a small contribution to the signal. The specular exit port 3 is then opened to allow the specular beam to escape, and the signal S7 can be measured: RS = FB(RB S 7 = K[CRs + (1 -B)RRd + FsBRd + FBRdupTtot2 - RdUmp)[(S6 - S 4 ) - (S7 - S 8 )] (RB - C)(S1- S 4) (11) We show below how C can be experimentally determined. Using Eqs. (lb), (4), (7), and (8), we find the following expression for the diffuse reflectance, where Ttot = Tsb + Tdb is taken from Eqs. (9) and (10), and R5 from Eq. (11): (7) + DI1+ D2 + D3]- (9) S 4 - S5 + (S1 - S4) 1 + B(R - S6 = K[RBRs + (1 - B)RBRd + FBBRd + FBRdumpTtot 2 + DI + D 2] _ (S 2 - S3 ) (S1 - S 4 ) To obtain an expression for the diffuse transmittance, Eqs. (lb) and (3)-(5) are used: (5) S5 = KD2 - (8) S 8 = K(FBRdump + D, + D2 + D3 ). (3) When the specular beam enters the mirror chamber it hits a plate painted black and is partly absorbed and partly reflected. Thus the level of diffuse light in the chamber increases because the plate cannot be per- - FB(RB Rd = - Rdump)(S7 - S 8 ) (S1 - S4 ) (1 - CRs - FBRdump(Ttt 2 B)RB + FBB fectly black. C is then the fraction of the specularly reflected beam that enters the sphere again through the ports. D3 represents the diffuse light in the chamber, which does not have its origin in the specularly reflected beam but enters the sphere through port 3. 6094 - APPLIED OPTICS / Vol. 33, No. 25 / 1 September 1994 - 1) (12) Parameter Calibration In Eqs. (9)-(12) the constants Rdump and C need to be known. To determine Rdump, we replaced the black cone with a 56-cm baffled black cylinder with the black cone at the end. The cylinder with the black cone was assumed to have negligible reflectance, and thus signal S4 ' was measured: S 4 ' = K(D1 + D2)- (13) Equations (lb), (4), and (13) give Rdump = RB (S - Experimental Results 5') (14) The reflectance of the beam dump was found from Eq. (14): Rdump 0.003. This value is constant throughout the whole spectral range, 0.3-2.5 mm. It should be pointed out that the reason the inherently much better beam dump used for signal S 4 ' is not used for all measurements is that it cannot be positioned on the sphere without considerable trouble. With this beam dump in position the compartment lid cannot be closed and stray light from the room may enter the sphere. The only way to avoid this is to perform this measurement in complete darkness, which, for routine measurements, is quite inconvenient. It now remains to determine the constant C, which was done with an almost perfectly specular sample, in this case a silicon wafer with a sputtered aluminum film. With port 4 covered with the aluminized silicon wafer and port 3 closed, S 9 was measured: S 9 = K(FBRBRspe + D + D2 ). (15) Port 3 was opened and the following signal was recorded: S1 = K(CRspec + D + D2 + D3 )- (16) The specular reflectance of the aluminized silicon wafer can now be calculated with Eqs. (lb), (4), (14), and (15): (S9 - 4)(RB - Rdump) (S - S4 ) Rspek = reflectance over the whole spectral range. After this modification C was low and constant from 0.3 to 2.5 rim. Rdm + Rdump RB Since Rdump and C are now known, we have no unknown parameters in Eqs. (9)-(12) except parameter B. To evaluate the procedure above and to investigate the influence from the stray-light corrections and the B factor, we prepared some scattering samples. Signals Si to S 8 were recorded, and the specular and diffuse reflectance and transmittance values were calculated according to Eqs. (9)-(12) with and without the stray-light corrections. The uncorrected spectra were simply calculated with the signals S4, S 5 , and S 8 as well as Rdump and C all set to zero. In this case Eqs. (9)-(12) are identical to those reported earlier.' 0 ,1 The samples used in the measurements were tin-oxide-coated glass substrates, one with a relatively high level of scattering and one with relatively low scattering. The tin-oxide films were pyrolytically deposited in a spray oven and were a few hundred nanometers thick. Such tin-oxide films tend to be slightly hazy owing to the surface roughness of the crystalline tin oxide.' 4"15 In Fig. 2 the specular reflectance and collimated transmittance spectra of one of the samples are shown. As can be seen, the corrections are negligible in this case. In Figs. 3 and 4 the diffuse transmittance and reflectance spectra of the same sample are shown. The calculations have been made with the B = 1 factor as well as with B = 0; i.e., the scattered light as recorded by the sphere is assumed to be either totally diffuse or totally specular. Thus the curves corresponding to these two extremes set the limit within which the true diffuse reflectance or transmittance value is to be found. In Figs. 3 and 4 we can see that for the higher scattering levels the stray-light corrections are not as important as the influence from the B factor, whereas for lower scattering levels the stray-light corrections are the most important. In Figs. 5 and 6 the diffuse transmittance and reflectance of a low scattering sample are shown. For this sample the Equation (17) together with Eqs. (8) and (16) now gives c) FB (Sio-S8)(RB - Rdump) B (S - S4) 0.8 + Rdumpl . (18) 0.6 Rspec The constant C was thus found to be C 0.001 throughout the whole spectral range. We should mention that the spectrophotometer's mirror chamber had to be slightly modified to achieve a C value that did not vary with wavelength. The black plate, which the specular beam hits when it leaves the sphere, was originally found to have high reflectance in the IR; the reflectance was greater than 60%, which noticeably influenced C. The plate was therefore painted with black paint with the same low c) ) 0 0.4 0.2 0 300 1000 3000 Wavelength/nm Fig. 2. Specular reflectance and collimated transmittance of a tin-oxide-coated glass. The difference between corrected and uncorrected spectra is negligible. 1 September 1994 / Vol. 33, No. 25 / APPLIED OPTICS 6095 0.02 0.15 8) Rd(B=O) corrected * . -.-.Rd(B=1) corrected | Rd(B=0) uncorrected ..-.-a. Rd(B= 1) uncorrected 0.015 0.1 e) C 88 8) 8) 0.01 8) 0.05 0.005 0 1000 300 3000 Wavelength/nm Fig. 3. Diffuse transmittance of a high-scattering tin-oxide-coated glass. Corrected and uncorrected spectra are shown. At shorter wavelengths where the scattering is high the B factor is most significant, whereas at longer wavelengths where the scattering is low the stray-light correction is most significant. B-factor corrections are negligible compared with those caused by stray light. The periodic variations in the diffuse signals are due to interference effects.16 It is noticeable that the signal is readily resolved at 0.08 1 I | 1 1000 3000 0.06 8) 8) 0.04 0.02 300 Wavelength/nm Fig. 4. Diffuse reflectance of a high-scattering tin-oxide-coated glass. Corrected and uncorrected spectra are shown. At shorter wavelengths where the scattering is high the factor B is most significant, whereas at longer wavelengths where the scattering is low the stray-light correction is most significant. 0.02 0.015 .8 a 88 0.01 0.005 300 1000 3000 Wavelength/nm Fig. 5. Diffuse transmittance of a low-scattering tin-oxide-coated glass. Corrected and uncorrected spectra are shown. The importance of stray-light correction is obvious, whereas the B factor is less important. 6096 APPLIED OPTICS / Vol. 33, No. 25 / 1 September 1994 300 1000 3000 Wavelength/nm Fig. 6. Diffuse reflectance of a low-scattering tin-oxide-coated glass. Corrected and uncorrected spectra are shown. The importance of stray-light correction is obvious, whereas the B factor is less important. this low level and that the level of the uncorrected spectrum is more than twice the corrected spectrum. This is a clear indication of the importance of interpreting the instrument signals correctly. The instrument noise is of the order of < 0.001, and the results of the measurements of the tin-oxide samples show that details in the diffuse spectra well below 1% can be detected and evaluated correctly, provided that stray-light contributions are taken into consideration correctly. For levels of scattering below 10-3 the sensitivity of the integrating sphere is insufficient and another technique must be applied. A new spectroscopic total-integrated-scattering instrument based on a focusing Coblentz sphere' 7 has been constructed in our laboratory, and the initial results indicate that it is possible to measure scattering levels below 10-4 in the wavelength range of 400-1000 nm.18 Conclusions Accurate measurements of low-level scattering from reflecting and transmitting samples can be performed with a standard double-beam integrating sphere provided all possible stray-light contributions are taken into account correctly. An experimental procedure for the determination of such stray-light contributions has been suggested, and the experimental results presented indicate that levels of scattering below 1% can be determined. Neglecting the straylight corrections can lead to errors of the order of a factor of 2. The procedure suggested appears tedious at first sight and is also more time-consuming than the standard procedure. We must remember, however, that the stray-light contributions as defined in this paper are constant and do not change with time. As the amount of dust and dirt on the mirrors slowly changes over months or even years, the stray-light levels change only slowly. This means that, once the signal levels without the sample in position have been measured, they can be stored on a computer file and 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 absolute zero level. This research was sponsored by the Swedish Council for Building Research. References 1. R. Ulbricht, "Die Bestimmung der mittleren rumlichen Lichtintensitat durch nur eine Messung," Elektrotech. Z. 21, 595-597 (1900). 2. K. A. Snail and L. M. Hanssen, "Integrating sphere designs with isotropic throughput," Appl. Opt. 28, 1793-1799 (1989). 3. L. M. 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