J. H. W. G. den Boer, G. M. W. Kroesen, and F. J. de Hoog

Transcription

1 Measurement of the complex refractive index of liquids in the infrared using spectroscopic attenuated total reflection ellipsometry: correction for depolarization by scattering J. H. W. G. den Boer, G. M. W. Kroesen, and F. J. de Hoog With spectroscopic ellipsometry one can measure the real and imaginary parts of the refractive index of a medium simultaneously. To determine this index in the infrared for a number of technical liquids, use was made of attenuated total internal reflection at the glass liquid interface of a specially designed prism. This attenuated total reflection approach warrants minimal signal loss and is, for strongly absorbing liquids, the only way to measure the complex refractive index. A surprising phenomenon, observed when BK-7 prism glass was used, is scattering in the vicinity of the absorption wavelengths of the glass. A simple model that can be used to describe the relations among absorption, scattering, and depolarization was successfully used to correct the measurements. Refractive indices for demineralized water, Freon 113, heptane, benzene, gas oil, and crude oil in the wave number range from 5000 to 10,000 cm µm2 are presented. Key words: Ellipsometry, attenuated total reflection, infrared, liquid, refractive index, scattering, absorption, depolarization, water, Freon, heptane, benzene, gas oil, crude oil. 1. Introduction Most methods that determine the refractive index of a liquid can be used to measure only either the real part or the imaginary part. One can usually determine the imaginary part using an absorption measurement 1,2 through a sample of known thickness. For liquids with a high absorption coefficient this means that the samples must be very thin 1several micrometers2 to measure any intensity at all. However, the required thickness and parallelism of the front and backside of a sample cause multiple reflections to occur and therefore limit the accuracy. The problem of absorption becomes even more urgent when one determines the real part of the refractive index using a refractometer, 3 an interferometric technique, 4,5 or a beam deviation technique. 6,7 In this case the light travels a relatively long distance in the material and almost the entire intensity may be absorbed. Fur- The authors are with the Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Received 25 July 1994; revised manuscript received 14 March @95@ $06.00@0. r 1995 Optical Society of America. thermore, in the case of absorption the angle of refraction depends on both the real and imaginary parts of the refractive index, so it is impossible to distinguish between them. One method that is capable of measuring the real and imaginary parts of the refractive index of a liquid simultaneously is ellipsometry. Havstad et al. 8 thus measured the refractive index of liquid uranium under ultrahigh vacuum conditions for wavelengths in the range from 400 nm to 10 µm. Brückner et al. 9 did the same for liquid gold and tin at 10.6 µm. The problem with these measurements is that the free liquid vacuum interface is sensitive to vibrations from the environment, and, more important, for materials other than metals most of the incident intensity is lost as it is transmitted into the liquid. To solve this problem we devised an attenuated total reflectancelike method 10 in which the liquid is locked behind a glass prism in a sample holder and ellipsometry is performed on the stable glass liquid interface. Besides the gain in stability and ease of handling the major advantage is the possibility to measure under the condition of total internal reflection. If the real part of the refractive index of the prism is higher than that of the liquid and the angle of incidence is sufficiently high, the full intensity is reflected toward the detector APPLIED Vol. 34, No. 1 September 1995

2 2. Theoretical Background A. Ellipsometry Ellipsometry is a technique that measures the change of polarization of light that reflects at a surface. In our rotating analyzer ellipsometer a linear polarizer is used to bring the light to a linear polarization state just before reflection. A second rotating linear polarizer, the analyzer, is used to determine the polarization state after reflection. The change of polarization is characterized by C and D, the relative amplitude change and the phase shift between the two polarization directions, respectively. Angles C and D can be determined by measurement of the reflected intensity while the analyzer is rotated. This intensity I will then vary harmonically with analyzer angle A and can be written as I 5 g11 1 a cos 2A 1 b sin 2A2, 112 where C and D can be expressed as Fourier coefficients a and b of this harmonic function. When the polarizers are perfectly linearly polarizing, the detector is equally sensitive to all polarization states, the transmission axis of the polarizer makes an angle of 45 with the plane of incidence, and the starting position of the transmission axis of the analyzer is in the plane of incidence, the relations between C and D on the one hand and Fourier coefficients a and b on the other hand become quite simple: cos 2C 52a, cos D5b@Π12a When polarizer, source, and detector imperfections are taken into account, the expressions in Eqs. 122 become implicit and have to be solved numerically for C and D. 11 B. Relation between the Refractive Index and 1C, D2 For reflection at an interface between two semiinfinite media Fresnel defined two reflection coefficients. The first one, r p, is the ratio of the E-field amplitudes after and before reflection of light with the E field in the plane of incidence. The second, r s,is the same ratio, but then for light with the E field perpendicular to the plane of incidence. The plane of incidence contains both incident and outgoing light beams. The Fresnel coefficients for reflection can be expressed as r p 5 n 1 cos u 0 2 n 0 cos u 1 n 1 cos u 0 1 n 0 cos u 1, r s 5 n 0 cos u 0 2 n 1 cos u 1 n 0 cos u 0 1 n 1 cos u 1, 132 with n 0, n 1 being complex refractive indices of the two media and u 0, u 1 being the complex angles of incidence and refraction, respectively. The ratio of these two coefficients defines a complex quantity r that can be written as r ; tan C exp id ; r s. 142 Thus C and D represent the relative changes in amplitude and phase after reflection, as was established in Subsection 2.A. From Eqs. 132 and 142 and Snell s law, n 0 sin u 0 5 n 1 sin u follows an explicit expression for the refractive index of the second medium 12 : n 1 5 n 0 sin u r r2 tan 2 u 0 4 1@ After C and D are determined for a specific angle of incidence u 0 and the refractive index of the incident medium is known, Eq. 162 allows us to calculate the refractive index of the other medium. To gain insight into the relation between the refractive indices and 1C, D23Eq we consider two extreme cases in which the real part of the refractive index of the incident medium is larger than that of the second medium. First the situation of total internal reflection is investigated. When both media are nonabsorbing and the incident angle is larger than the critical angle, the light is totally internally reflected. If this situation occurs, 0r p 0 5 0r s and obviously C545, so that r5exp id. This simplifies Eq. 162 to a point at which D can be expressed in terms of the incident angle and the refractive indices of the prism and the liquid 13 : tan D 2 52 cos u 0 3 sin2 u n1 n @2 sin 2 u If the second medium is slightly absorbing, it can be shown that no total internal reflection can occur. For large angles of incidence, though, there is attenuated total reflection. It is clear from Eq. 172 that also in this case a change in D relates closely to a change in the real part of the refractive index. Similarly a deviation of C from 45 relates to the imaginary part of the refractive index. The second extreme case arises when the angle of incidence is smaller than the critical angle and again both media are nonabsorbing. If the angle of incidence is still higher than the Brewster angle, D50 ; otherwise D In both cases ratio r of the reflection coefficients will be real and equal to 6 tan C, leading to tan C56 sin 2 u 0 2 cos u 0 31 n 1 n sin 2 u 0 4 1@2 sin 2 u 0 1 cos u 0 31 n 1 n sin 2 u 0 4 1@2, September Vol. 34, No. APPLIED OPTICS 5709

3 with 51 : D50 arctan1n 0 2 #u 0, arcsin1n 0 2, 2 : D5180 u 0, arctan1n 0 2. If the second medium is slightly absorbing, it can be shown that the value of D deviates slightly from either 0 or 180. It is clear from Eq. 182 that, as long as this deviation is small, a change in C will relate to a change in the real part of the refractive index. Similarly the deviation of D from 0 1or will relate to the imaginary part of the refractive index. From these two extreme cases one should note the contrast in the role of the ellipsometric quantities C and D. In the case of attenuated total reflection the real part relates to D and the imaginary part relates to C, whereas in the second extreme case the real part relates to C and the imaginary part to D. C. Prism To avoid disturbing vibrations of the liquid surface from the environment we used a prism with a known refractive index to cover the liquid. This prism 1Fig. 12 at the same time provides several other advantages. First, it permits the glass liquid interface to be placed at any desired orientation. It also facilitates handling, prevents pollution of the liquid, and keeps the liquid from evaporating, thus not requiring extensive safety measures with the more dangerous liquids such as benzene. The most important advantage, though, is that the real part of the refractive index of the prism is higher than that of the liquid so that for large angles of incidence a situation of attenuated total internal reflection arises. This ensures a high intensity at the detector. The incident light beam enters the prism perpendicularly. Both the parallel and perpendicular components experience the same amplitude change so that their ratio remains unchanged. Neither of the components shifts in phase, so the relative phase is unchanged. After entering the prism the beam reflects at the glass liquid interface at the backside of the prism. This is described by Eq. 162 in Subsection 2.B. To prevent multiple reflections in the prism, it was designed such that the light that leaves the prism strikes the glass air interface at an angle that is slightly nonperpendicular. This has as the side effect that it also introduces a relative amplitude change that is given by Fresnel transmission coefficients: t p 5 t s 5 2n 0 n 1 cos u 0 1 n 0 cos u 1, 2n 0 n 0 cos u 0 1 n 1 cos u Note that n 0 and n 1 now represent the refractive indices of the prism glass and air, respectively. Together with Snell s law 3Eq. 1524, for the relative amplitude change this gives tan C t ; 0 t p t s 0 5 n 0n 1 cos u 0 1 n 1 1n n 0 2 sin 2 u 0 2 1@2 n 1 2 cos u 0 1 n 0 1n n 0 2 sin 2 u 0 2 1@ The phase has remained unchanged. The total effective relative amplitude change that the light has experienced in the prism is now given by tan C eff 5 tan C tan C t The angle C eff is measured and C t can be calculated by Eq Thus C can be calculated from the measured C eff and the calculated C t by Eq This C together with D equal to the measured D eff can be inserted into Eq. 142 to calculate the complex quantity r. This, in combination with the refractive index of the prism, the angle of incidence, and Eq. 162, allows us to calculate the refractive index of the liquid. 3. Experimental Setup A. Rotating Analyzer Ellipsometer The light source used in this experiment 1Fig. 22 is a cascaded arc, which emits from the UV to the IR. 14 By means of a mirror the light that comes from the arc is coupled into a Bruker IFS66 Fourier transform IR spectrometer. The spectrometer contains a CaF 2 beam splitter that determines the lower limit of the wave-number range to 1650 cm 21. The light that emerges from the Fourier transform IR spectrometer then strikes a MgF 2 Rochon polarizer, passes through Fig. 1. BK-7 prism, the light is incident through the left side, reflects at the fluid glass interface at the bottom, and leaves the prism through the right side. A small difference in angle between the left and the right sides prevents multiple reflections in the prism. Fig. 2. Spectroscopic ellipsometer setup with 1, cascaded arc light source; 2, Bruker IFS66 spectrometer; 3, Rochon prism polarizer; 4, diaphragm; 5, sample holder and prism; 6, Rochon prism polarizer; 7, diaphragm; 8, imaging lens; 9, HgCdTe detector APPLIED Vol. 34, No. 1 September 1995

4 a diaphragm that blocks the extraordinary beam that also emerges from the Rochon polarizer, and enters the prism from which it reflects at the glass liquid interface. After the light leaves the prism it falls on the analyzer, which is identical to the polarizer, passes another diaphragm and a lens, and finally hits a HgCdTe detector. This detector determines the upper limit of the wave-number range as approximately 10,000 cm 21. The polarizer and analyzer are mounted on stepper motor-controlled rotation tables. The polarizer is held fixed with the transmission axis at an angle of 45 relative to the plane of incidence. The analyzer is rotated in steps. For each step a spectrum is measured. From a cross section through these spectra at a fixed wave number the Fourier coefficients may be determined and, hence, C and D and the refractive index. The sample holder is placed such that the incident light strikes the prism perpendicularly. In this case the angle of incidence on the glass liquid interface is 70. Prior to measurement the position of both polarizer and analyzer with respect to the plane of incidence needs to be determined. This can be done by a method that was described in a previous publication. 11 B. Liquid Sample Holder and Prism The liquids are contained in an aluminum sample holder with a volume of approximately 20 cm 3. The sample holder is closed by the prism and sealed by an O-ring. The prism material is BK-7 glass manufactured and certified by Schott Glaswerke. The refractive index of BK-7, close to 1.5 and expressed in a Sellmeier dispersion formula, 15 is accurate to the fifth decimal. The BK-7 prism further limits the wavenumber range from approximately 5000 cm 21 upward. Liquids that have been investigated are water, Freon, heptane, benzene, gas oil, and crude oil. 4. Results A. Scattering For wave numbers that approach the cutoff wavelength of the prism BK-7 glass cm 21 2, spectra of the refractive index show a sudden drop in value. This decrease is specifically observed for the real part of the refractive index and can be traced back to a similar decrease in Fourier coefficient b 1Fig. 32. As this phenomenon is independent of the liquid under observation and coincides with the cutoff wavelength of BK-7 glass, it must be attributed to absorption in the BK-7 glass. Further proof of this assumption is obtained from an IR transmission spectrum of BK-7 glass, which reveals a structure similar to that observed in the refractive-index spectra. This observation also rules out a possible surface phenomenon. Absorption on its own is not sufficient to explain the decrease in refractive-index values, but absorption is usually accompanied by scattering. Scattering lowers the polarization degree of the light that Fig. 3. Fourier coefficient b versus wave number calculated from a measurement with air behind the prism. Note the sudden drop in value in the vicinity of s55000 cm 21. reflects at the sample, and as such disturbs the ellipsometric measurement. 16 A decreased polarization degree P leads to Fourier coefficients a* and b* related to the undisturbed coefficients a and b in Eq. 112 through a* 5 Pa 52Pcos 2C, b* 5 P 2 b 5 P 2 cosd sin 2C These relations are obtained from complete evaluation of an ideal rotating analyzer ellipsometer into which depolarizing components in front of and behind the reflecting sample are introduced. It is obvious that the b coefficient is affected stronger than the a coefficient, especially when a < 0 and b < 1, as is the case with attenuated total reflection. To describe the scattering we used a simple model that was derived from a description of scattering by small spherical particles by van de Hulst. 17 For small particles with a small absorption coefficient we can assume that the cross section for scattering Q has a linear relation with the wave number s and the imaginary part of refractive index k: Q 5 2prjsk Here r is the radius of the scattering particles and j is a constant factor. As the absorption and subsequent scattering increase, the polarization degree decreases. When the polarization degree reaches a value of 0.9 the absorption becomes so high that no more light reaches the detector. This means that we are interested only in the values of P close to 1, that is, in small values of the scattering cross section. For small values of the scattering cross section the relation with the polarization degree can be approximated by any power series that satisifes the following two boundary conditions. The polarization degree must decrease monotonically with increased scattering cross section, and for zero scattering cross section the polarization degree must be unaffected. As a first estimate an exponential function is chosen to reflect the relation P 5 exp12q For zero cross section the polarization degree is 1 September Vol. 34, No. APPLIED OPTICS 5711

5 indeed unaffected whereas for large scatter cross section the light depolarizes completely 1P To describe the cutoff of BK-7 prism glass at 5000 cm 21, it is necessary to realize that the absorption is caused by vibration of several chemical bonds. Absorption of this kind can be described by a harmonic oscillator model resulting in a Lorentz profile. The Lorentz profile is characterized by three parameters. These parameters are the absorbing wave number, the width, and the strength of the absorption. In our case, though, more than one absorption plays a role, so the relation between the imaginary part of the refractive index of BK-7, k, and wave number s,isthe sum of several Lorentz profiles: 6 k1s2 5 oi51 k i a i 2 1 1s 2s i A choice was made for summation of six Lorentz profiles, each centered at another wave number, because this best approaches the observed structure in the real part of the refractive index. The parameter k i relates to the strength, a i to the width, and s i to the central wave number of a specific absorption. Combining Eqs yields P 5 exp3 22prj o i51 6 k i s a i 2 1 1s 2s i This relation between the polarization degree and the wave number allows us to fit a scatter structure observed in a measured Fourier coefficient. In order to eliminate possible effects of the liquid behind the prism during this procedure, we chose to measure with air behind the BK-7 prism. Subsequently the logarithm of Eq was fitted to the logarithm of the Fourier coefficient b. Figure 4 shows the results of the measurement and the fit. With the polarization degree as a function of wave number established, measurements of liquids can be corrected for depolarization. This was done by dividing the measured Fourier coefficients a* and b* bythe Fig. 5. Fourier coefficient b versus wave number before 1b*2 and after 1b2 correction for depolarization. Since the ellipsometer was not well calibrated during this experiment, the absolute values are too low. Translating these results to the refractive index of air would give n values slightly below 1. This does not, however, affect the polarization degree. polarization degree according to Eq Figure 5 shows the results of the correction procedure. Because of bad calibration of the ellipsometer during this measurement the absolute value of the b coefficients is too low. Therefore translation of these results to refractive-index values also gives values that are slightly below 1. Since only the structure seen in this measurement is used to fit Eq to, the absolute value plays no role, and this does not affect the correction procedure. B. Refractive Indices Figures 6 11 show the refractive indices versus wave number for the real and imaginary parts of water, Freon, and heptane and only the real part of benzene, gas oil, and crude oil. The reason for this is that the real part of the refractive index of the latter three is so close to that of the prism that there no longer exists a situation of attenuated total reflection. This has two consequences. As most of the intensity is Fig. 4. Logarithm of Fourier coefficient b and matching fit of Lorentz profiles plotted versus wave number. Note the structure that relates to Lorentz profiles centered at 4024, 4158, 4296, 4454, 4719, and 4897 cm 21. The Fourier coefficient was obtained by a measurement with air behind the prism. The value of is the average value of the Fourier coefficient in an undisturbed range of the spectrum from 5000 to 6000 cm 21. Fig. 6. Real and imaginary parts of the refractive index of demineralized water plotted versus wave number. The structure in the imaginary part is in agreement with absorption measurements of water APPLIED Vol. 34, No. 1 September 1995

6 Fig. 10. Real part of the refractive index of benzene 1C 6 H 6 2 plotted versus wave number. Fig. 7. Real and imaginary parts of the refractive index of heptane 3CH 3 1CH CH 3 4 plotted versus wave number. Fig. 11. Real part of the refractive index of crude oil 1Eider 892 plotted versus wave number. Fig. 8. Real and imaginary parts of the refractive index of Freon 113 plotted versus wave number. ellipsometry, it is obvious that in the vicinity of D50 and D5180 the accuracy is not good. In this case only C is accurately measured, and thus only the real part, which relates to C, is determined. Values below 5000 cm 21 are shown to demonstrate the correct working of the correction procedure. The intensity on the detector in this region has dropped dramatically as can be seen from the increased noise level. The average value in this region, though, gives an idea of the refractive-index inclination. The accuracy of the refractive-index values is directly connected to the accuracy in C and D, which is determined by the angle of incidence and the exact positions of the polarizer and analyzer. The angle of incidence is known to within an accuracy of 1@60. The positions of polarizer and analyzer are known to within an accuracy of This translates to an accuracy of in the refractive-index values. The reproducibility or precision is largely determined by the stability of the light source on a time scale of minutes 1time of measurement2. As can be seen from the graphs the precision is approximately for n and 0.01 for k. Note that this implies that the relative precision is much better for n than for k. Fig. 9. Real part of the refractive index of gas oil plotted versus wave number. transmitted into the liquid, the intensity at the detector decreases dramatically. Furthermore D is close to either 0 or 180. Since it is not D but the cosine of D that is determined by rotating analyzer 5. Conclusion Spectroscopic ellipsometry is a powerful tool for determining the complex refractive index of a liquid. Especially in combination with a prism, which effectively makes it into an attenuated total reflection method, many advantages are gained over more traditional methods of refractive-index determination. 1 September Vol. 34, No. APPLIED OPTICS 5713

7 One advantage is that both the real and imaginary parts are measured simultaneously, but a more important advantage is that it permits the refractive-index determination even for strongly absorbing liquids such as crude oil. In this situation it is the only available method that is capable of measuring the complex refractive index. However, for liquids that are not or only slightly absorbing, the traditional specialized methods 1 7 may give more accurate results. The results for the real part of the refractive index for water are close to what we expect in this region of the infrared, although few accurate data are available in this spectral region. The spectroscopic features in the imaginary part are in good agreement with the values obtained for absorption measurements. For all other liquids, to the best of our knowledge, there is nothing available in the literature, which was the main reason Shell Research Rijswijk requested us to start this investigation. Ellipsometry appears to be sensitive to the depolarization and scattering that accompany absorption in the prism. Although the effects of scattering were not anticipated before the experiment, a closer study of the phenomenon yielded a surprising and new insight in the relations among absorption, scattering, and depolarization. This was laid down in a simple model that was used to correct the refractive-index measurements. The model may serve as a starting point for further investigation into the rather unknown relation among absorption, scattering, and depolarization. Although the BK-7 prism we used allowed us to detect, quantify, and correct for scattering, for a case in which scattering is not desired, another material may be better suited for use as a prism. KRS-5, for example, is transparent to wavelengths far into the middle infrared and thus will not scatter until far into the middle infrared. At the same time the higher refractive index 1approximately 2.4 in the infrared2 of KRS-5 ensures the working of the prism under attenuated total reflectance conditions even for liquids with a high refractive index. This research in the program of the Foundation for Fundamental Research of Matter has been funded in part by The Netherlands Technology Foundation. Specific research into the measurement of the complex refractive index of liquids was started at the request of and supported by Shell Research Rijswijk. References 1. E. E. Jelley, in Physical Methods of Organic Chemistry, A. Weissberger, ed. 1Interscience, New York, 19602, Part 2, Chap. 21, p W. L. Wolfe, Properties of optical materials, in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. 1McGraw Hill, New York, 19782, Chap. 7, p W. Nebe, Routine- und Präzisionsmessungen an Flüssigkeiten und Gläsern, Mess. Steuern Regeln 9, ; 11, Ph. Marteau, G. Montixi, J. Obriot, T. K. Bose, and J. M. St Arnaud, Simple method for the accurate determination of the refractive index of liquids in the infrared, in Infrared Technology XVI, I. J. Spiro, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1341, T. Li and X. Tan, Stepwise interferometric method of measuring the refractive index of liquid samples, Appl. Opt. 32, E. Moreels, C. de Greef, and R. Finsy, Laser light refractometer, Appl. Opt. 23, M. V. R. K. Murty and R. P. Shukla, Simple method for measuring the refractive index of a liquid or glass wedge, Opt. Eng. 22, M. A. Havstad, W. McLean II, and S. A. Self, Measurement of the thermal radiative properties of liquid uranium, in Developments in Radiative Heat Transfer, S. T. Thynell, ed., HTD Vol American Society of Mechanical Engineers, New York, 19922, pp M. Brückner, J. H. Schäfer, C. Schiffer, and J. Uhlenbusch, Measurement of the optical constants of solid and molten gold and tin at l510.6 µm, J. Appl. Phys. 70, N. J. Harrick, Internal Reflection Spectroscopy 1Interscience, New York, 19672, p J. H. W. G. den Boer, G. M. W. Kroesen, M. Haverlag, and F. J. de Hoog, Spectroscopic IR ellipsometry with imperfect components, Thin Solid Films 234, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light 1North-Holland, Amsterdam, 19792, p M. Born and E. Wolf, Principles of Optics 1Pergamon, New York, 19592, p A. T. M. Wilbers, G. M. W. Kroesen, C. J. Timmermans, and D. C. Schram, Characteristic quantities of a cascade arc used as a light source for spectroscopic techniques, Meas. Sci. Technol. 1, Ref. 13, p A. Röseler, Infrared Spectroscopic Ellipsometry 1Akademie- Verlag, Berlin, 19902, p H. C. van de Hulst, Light Scattering by Small Particles 1Wiley, New York, 19572, p APPLIED Vol. 34, No. 1 September 1995