Thickness and refractive index measurements using multiple beam interference fringes (FECO)

Transcription

1 Journal of Colloid and Interface Science Note Thickness and refractive index measurements using multile beam interference fringes FECO Rafael Tadmor, 1 Nianhuan Chen, and Jacob N. Israelachvili Materials Research Laboratory and Deartment of Chemical Engineering, University of California, Santa Barbara, CA 93106, USA Received 28 January 2002; acceted 11 Aril 2003 Abstract We reort on the use of otical interferometry emloying fringes of equal chromatic order FECO in a surface force aaratus SFA to determine film thicknesses and refractive indices of confined media for a wide range of searations. In articular, we show how to calculate the surface searation film thickness based on two fringes whose contact osition was not measured. We discuss the measurement accuracy, and though the theoretical accuracy is 1 Å for all searations, we show that in ractice, for large searations, it is very hard to get to this accuracy Elsevier Inc. All rights reserved. Keywords: Large searations; Thick films; Three-layer interferometer; Disersion; Disersive hase change In a SFA surface force aaratus [1] exeriment, there is usually a three- or five-layer interferometer [2]. For simlicity we assume here a three-layer interferometer, although the analysis alies to any number of layers. When the transarent substrate layers tyically mica sheets are brought into contact, the ositions λ 0 of the th-order fringes is usually measured in the visible range of wavelengths. After the substrate surfaces are searated by a distance D, a three-layer interferometer is formed. The fringes shift to longer wavelengths and their new ositions λ D are given see Ref. [3] by tan 2πµ med D/λ D 2 µ sin4πµy/λ D = 1 µ µ 2 cos4πµy/λ D, which, assuming no disersion we discuss this oint later in the text, reduces to the more useful form tan 2πµ med D/λ D 2 µ sin π 1 λ0 /λd = 1 + µ 2 cos 1 λ 0 /λ0 1 π 1 λ0 /λd 1 λ 0 /λ0 1 ± µ 2 1, 1a * Corresonding author. address: jacob@engineering.ucsb.edu J.N. Israelachvili. 1 Present address: Det. of Chemical Engineering, Lamar University, Beaumont, TX where Y is the otical thickness of each substrate we discuss later the difference between the otical and hysical thicknesses and µ = µ/µ med,whereµµ γ or µ β and µ med are, resectively, the refractive indices of the mica and the intervening medium at λ D.InEq.1a,+ and refer to odd and even order fringes, resectively. Other ways of calculating thicknesses are available see, for examle, Ref. [4]; however, the simlicity of Eq. 1a made it the most common method. Indeed, due to the increased use of Eq. 1a in SFA measurements, eole have addressed some secial cases which were not addressed by the original ublication [3], such as reflecting metallic media [5], nonsymmetric three-layer interferometry [6], and most recently the slitting of the fringes as a result of a liquid crystalline medium [7]. Currently Eq. 1a is limited due to the fact that the fringes at contact need to be of the same order as the fringes of measurements. Aart from imosing inconvenience, this also limits the range of measurable searations, since at far searations, the contact fringes are no longer in the visible range. Different aroaches to this roblem were addressed by Horn and Smith [6], by Heuberger et al. [4], and by Farrell et al. [8]. In this study, however, we resent a very simle account for this roblem. An even simler account than the one resented here is given in the thesis of H. Christenson [9]. This solution, however, is extremely sensitive to the osition of the fringes, sub-ångstrom errors in the wavelength measurement can roduce a micrometer-scale /$ see front matter 2003 Elsevier Inc. All rights reserved. doi: /s

2 R. Tadmor et al. / Journal of Colloid and Interface Science error in the searation. This aer describes an easy and accurate way of calculating the searation from any order of fringes at any surface searation. From Eq. 1a it is clear that if the refractive indices of the mica substrate and the medium are known then the searation between the surfaces D can be obtained from λ D, λ0, and λ 0 1. In the case where the medium refractive index µ is not known [10,11], there is a need to solve two equations with two unknowns, D and µ med. The two equations are Eq. 1a for an odd fringe and Eq. 1a for an even fringe. In order to solve these simultaneous equations, we need to know the location of two adjacent fringes, λ D and λd 1,at finite D and the contact ositions of three adjacent fringes, λ 0, λ0 1,andλ0 2,atD = 0. The limitation of the above conventional method is that λ D, λd 1, λ0, λ0 1,andλ0 2 need to be in the visible for them to be measured. This limits the range of measurable searations D obtained using this aroach to D<1µm for tyical substrate thicknesses thicker substrates would increase the range but decrease the accuracy of the measurements. In this aer we describe a rocedure that allows accurate and unambiguous measurements of distance and refractive index to be made using the contact osition of only two adjacent fringes, λ 0 and λ0 1, and more imortantly using two adjacent fringes of any order, λ D m and λd m 1, and not necessarily λ D and λd General rocedure for measuring D and µ med of films of arbitrary thickness The contact ositions, λ 0, corresond to a single-layer interferometer. A one-layer interferometer made of a substrate with a refractive index µ and hysical thickness Y, obeys the relation 2µY = λ, = 1, 2, 3,...,, 2 where λ is the wavelength that corresonds to constructive interference and is a natural number the fringe order. In ractice there is never a one-layer interferometer, since the reflecting layer results in a hase change of the light at the substrate reflector interface say a mica silver interface. Such a hase change for the mica silver interface reflection has been studied [12], and it has been shown that it can be viewed as an aarent small change in the otical thickness of the substrate to Y instead of Y [13]: 2µY = λ, = 1, 2, 3,...,. 2a It is intuitive that the solution to the roosed roblem may be rovided using this equation. Secifically, using Eq. 2a to substitute the thickness of the layer Y for the contact osition of the fringe in Eq. 1a, we should be able to relate one contact fringe osition to any other contact fringe osition. Hence, the roblem becomes one of how to exress a singlelayer interferometer in terms of a relation between different fringe orders rather than the resentation of Eq. 2a. Using Eq. 2a and assuming constant Y [13], we write for three arbitrary fringes of order, 1, and m 2µ Y = λ 0, 3a 3b 2µ 1 Y = 1λ 0 1, 2µ m Y = mλ 0 m, 3c where we write µ to emhasize the disersive nature of the substrate, i.e., that each refractive index µ corresonds to a wavelength λ 0. Equations 3a and 3b have just two unknowns, Y and, and solving for them we get 1 = 1 µ 1λ 0 µ λ 0. 1 From Eqs. 3a and 3c and using Eq. 4 we obtain λ 0 m = µ m λ 0. 5 µ 1 m 1 µ 1λ 0 µ λ 0 1 Equation 5 relates the osition of the mth fringe to that of the th fringe as a function of the number of fringes m between them, as well as five other arameters that need to be known, including µ m. Note that m is an integer, which may be ositive, negative, or zero. Now, we may use any fringe in order to use Eq. 1a for the calculation of searations and refractive indices: we simly need to count the number of fringes from m to, ut this number as m into Eq. 5, and calculate a new set of contact ositions λ 0 m, λ 0 m 1,andλ0 m 2. As we show later in the worked Examle 1, the value of µ m can be obtained by substitution of an aroximated λ m given by Eq. 6 into Eq. 12, which we introduce below. Exerimentally, if a measurement is done at a large searation on a fringe λ D m whose contact osition λ 0 m is not visible, one way of finding m after comleting a measurement using the mth fringe is to bring the surfaces to substrate substrate contact while counting the assing fringes until one finds the -fringe. This number is the integer m. Equation 5 requires knowledge of µ m. To obtain this, note that Eq. 5 has a very weak deendence on the refractive index of the substrate. Neglecting the disersion, for examle, results in λ 0 m = λ m1 λ 0 /λ0 1, which is an excellent aroximation for many cases. Equation 6 can also be used for a first guess for the wavelength at which one needs to look for µ m in Eq. 5. In worked Examle 1, we show how this can be done. Similarly, Eq. 5 or its aroximated version, Eq. 6, which is written for any λ 0 m, can also give λ0 2, thus reducing the minimal number of required contact fringes to 2. In fact, there are cases where only two fringes are in the visible range for very thin mica, for which Eq. 5 may be very useful. In worked Examle 2 we show that for a three-layer interferometer of known 4

3 550 R. Tadmor et al. / Journal of Colloid and Interface Science medium refractive index and substrate disersion [14], one can theoretically calculate the ositions of the contacting fringes even without counting the integer m. However, in ractice this can be used only for small m; therefore, for large searations, m usually needs to be counted from the assing fringes as described above. The simlified result of Eq. 6 can also be obtained by utting D = 0 in Eq. 1a. This gives 1 λ 0 /λd 7 1 λ 0 = z, /λ0 1 where z is an integer. In Eq. 7, λ D is not the contact osition λ 0,butany contact osition λ0 m. This is because any such fringe would corresond to tan2πµ med D/λ D = 0in Eq. 1a. Thus, solving for Eq. 7, we again get Eq. 6. It is clear from the above treatment that Eq. 1a does not take into account the disersion of the substrate as was noted already in the original aer [3]. The disersion of mica has been studied [15,16], and below we derive the equation corresonding to Eq. 1 for disersive substrates and media. Using the more fundamental form of Eq. 1, rather than 1a, we note that a one-layer interferometer imlies that 4πµY 8 λ D = π λ 0 /λd, where µ is the refractive index of the mica at λ D as noted in Eq. 1. Substituting Eqs. 4 and 8 into Eq. 1, we obtain the disersive version of 1a as tan 2πµ med D/λ D = 2 µ sin π 1 λ 0 /λd 1 µ 1λ 0 1 µ λ 0 1 [ 1 + µ 2 cos π 1 λ 0 /λd 1 µ 1λ 0 1 µ λ 0 1 and then use Eq. 5 to relace the unknowns λ 0 m and λ0 m 1 with the measured λ 0 and λ0 1 and the calculated λ0 2 in the case where the refractive index of the medium is not known. In certain situations, the integer number m may not be available or measurable. Since we have two different equations 10 for odd and even fringes, we can use the measured λ D m and λd m 1, and simly guess a value for m. Then, using Eq. 5 with λ 0 and λ0 1, we calculate the contact ositions which corresond to fringes m, m 1, and m 2. Aarently, only if the guess is correct will one obtain the same searation D using both forms of Eq. 10 for even and for odd fringes. In ractice, however, this method which anyway works only for a three-layer interferometer can be done only for very small m values, while for large m other factors such as the disersion of the medium µ med and the disersive hase change at the reflector substrate interface should also be known very accurately. This is discussed further after Examle 2. The examles below are based on measured values. 2. Worked examles 2.1. Examle 1 Estimating the value of λ 0 2 from λ0 and λ0 1. In an exeriment using brownish mica as substrate, the contact wavelengths of three adjacent fringes of unknown order, 1, and 2 are measured Fig. 1 and found to be at λ 0 = ± 0.11 Å, λ0 1 = ± 0.16 Å, and λ 0 2 = ± 0.21 Å. If we want to estimate the osition of λ 0 2 from λ0 and λ0 1, then utting m = 2 into Eq. 6 we obtain λ 0 2 = / = Å, neglecting disersion. 11 ± µ 2 1] 1. 9 Equation 9 is the analogue of Eq. 1a with a correction for the disersion in the substrate refractive index, and as in Eq. 1a, + and refer to odd and even order fringes, resectively. Finally, we may write the analogue to Eq. 9 for any order of fringe, tan 2πµ med D/λ D m = 2 µ sin π 1 λ 0 m /λd m 1 µ m 1λ 0 1 m µ m λ 0 [ m µ 2 cos π 1 λ 0 m /λd m 1 µ m 1λ 0 1 m µ m λ 0 m 1 ± µ 2 1] 1, 10 Fig. 1. Schematic reresentation of the locations and shaes of the FECO in Examles 1 and 2. The flat arts in λ 0, λ0 1,andλ0 2, result from the elastic distortion of the glue which suorts the mica [3]. For such a flat contact of mica in air, odd...,, 2,... fringes always have a similar shae, which is different from that of the even..., 1, 3,... fringes. Secifically, the lace at which the fringes sto being flat and start bending has a bigger discontinuity in its derivative for odd fringes than for even fringes. Also note that since m>, λ D m, λd m 1,andλD m 2 are closer together than λ 0, λ0 1,andλ0 2.

4 R. Tadmor et al. / Journal of Colloid and Interface Science If we include disersion in the refractive index, then we need to know the relation for the refractive index of the fringe β or γ that is being measured as a function of λ. For brownish mica, this is [17] µ β = /λ 2 for β fringes, 12 where λ is the wavelength in Å. We first calculate the refractive indices at the three contact wavelengths, and obtain µ β λ 0 = , µ βλ 0 1 = , µ βλ 0 2 = Now, using Eq. 5, we obtain λ 0 2 = [ ] = Å, 13 which is in excellent agreement with the measured value of ± 0.21 Å, indeed, within the exerimental error. Actually, one could use the measured value of λ 0 2 to determine one of the constants in Eq. 12 or, if we also measured λ 0 +1 or λ0 3, to get both constants. Note that it is not necessary to know the value of, nor does it matter if is odd or even the equations are identical. Only for the searation measurements of the three-layer interferometer is it imortant to know whether is odd or even it haened to be odd for this secific examle. We also note that this method can be used to calculate the contact wavelength of any fringe order, not just λ Examle 2 Calculating distances D from λ 0 and λ0 1, and the measured ositions λ D m and λd m 1 of any two adjacent fringes of unknown order m and m 1. A measured contact osition of a fringe of unknown order is at λ 0 = Å and that of order 1isat λ 0 1 = Å. The substrate is brownish mica, whose refractive index is given by Eq. 12. The surfaces are well searated and we want to calculate the distance D between the two surfaces. We know that the medium between the two mica substrates has a refractive index of [14], and we erform simultaneous measurements of λ D m and λd m 1 and obtain λ D m = Å and λd m 1 = Å see Fig. 1 for the qualitative relative ositions of λ 0, λ0 1, λ D m, λd m 1 in this examle. For a quick estimate of D, one can use Eq. 6 to calculate the contact ositions of various fringes + 1,+ 2,... see Examle 1 and Eq. 1a to calculate the distances of our three-layer interferometer until + m is found. A more accurate aroach is to use Eqs. 5 and 6 for the calculation of the ositions + 1, + 2,..., as described in Examle 1, and Eq. 9 for the distances. In this examle we show a case where Eq. 5 is used with Eq. 1a [14]. Obviously m ; hence our first guess is m =. Using this guess, we ut the values for λ 0, λ0 1,andλD m which we guess to be λd in the odd version of Eq. 1a and obtain D = 1363 Å. We then erform the same calculation using λ D m 1 which we guess to be λ D 1 and the values for λ0 1 and λ0 2 using the even version of Eq. 1a, and obtain D = 1310 Å. Since , our guess was wrong. Our second guess would be m = + 1wherem is now even, and we use the even version of Eq. 1a with the calculated value of λ 0 +1, measured λ0 and λd m which we guess to be λd +1. We obtain D = 3288 Å. The odd version of Eq. 1a with λ 0, λ0 1,andλD m 1 which we guess to be λd,nowresults in D = 3280 Å. Since 3288 = 3280, we conclude that m = + 1. The difference of 8 Å is in art because the refractive index of the medium was assumed to be nondisersive, and in art because the mica silver hase change was also assumed to be nondisersive. We should also note that for m = + 2, we obtain D = 5217 Å using the odd version of Eq. 1a with λ 0 +2, λ0 +1,andλD m guessedtobe λ D +2,andD = 5248 Å using the even version of Eq. 1a with λ 0 +1, λ0,andλd m 1 guessed to be λd +1,andwe get Dλ D m,λ0 +2,λ0 +1 <DλD m 1,λ0 +1,λ0. Indeed, for all guesses above + 1 the second estimate is higher than the first, while for all guesses below + 1 the reverse is true, and this observation is general. In other words, the searation calculated using a higher order contact fringe coule is bigger than that using a lower order contact fringe coule for m guesses which are smaller than the real m, whereas the searation calculated using a higher order contact fringe coule is smaller than that using a lower order contact fringe coule for m guesses which are bigger than the real m. Formulating this would look like this: D λ D m,λ0 +i +i 1,λ0 >D λ D m 1,λ 0 +i 1,λ0 +i 2 for i<m, D λ D m,λ0 +i +i 1,λ0 = D λ D m 1,λ 0 +i 1,λ0 +i 2 for i = m, D λ D m,λ0 +i +i 1,λ0 >D λ D m 1,λ 0 +i 1,λ0 +i 2 for i>m. The above examle uses only m = + 1 i = 1 and would work for most cases only for very small values of m. Thus, in general it is advised to count the number of assing fringes in order to determine m. Theoretically, it should be ossible to include the disersive refractive index of the medium and the disersive hase change at the reflector substrate interface. However, as we discuss below, incororating a disersive hase change, while theoretically desirable, is difficult in ractice. Since the above did not include the disersive hase change, the similarity obtained for the two calculations of D does not suggest that this is the error in the searation, which could be larger. As we shall see, incororation of the hase change gives a formally accurate solution; however, there are good reasons to use only the substrate disersive refractive index and neglect the disersive hase change.

5 552 R. Tadmor et al. / Journal of Colloid and Interface Science Comaring disersion and hase change If we consider a normal SFA exeriment with mica as a substrate and a silver layer of 550 Å as a reflector, then out of the two corrections to the ideal osition of the fringes substrate refractive index disersion and substrate reflector disersive hase change, the substrate disersion is usually the bigger contributor, as demonstrated in Fig. 2. In addition to having a smaller effect, the hase change adds comlexity and requires the use of henomenological relations. The above discussion does not neglect hase change comletely, but rather uses a constant value for the hase change at D = 0 as exlained by Bailey and co-workers [8], since the wavelength ositions measured in contact corresond to the reflector hase changing system that roduced them. Thus, for many systems, this constant value correction suffices. However, there are situations in which one would like to incororate the disersive hase change. 4. Incororating the hase change at the substrate reflector mica silver interface For better accuracy there is a need to incororate the hase change at the substrate reflector interface. We use the idea suggested by Bailey and co-workers [8], and we follow their methodology to incororate the hase change contribution exlicitly. The hase change has the effect of adding or subtracting an aarent thickness y to the real thickness, and hence mica of real thickness Y will, due to the hase change, aear to be of thickness Y = Y + y. Note that y is a function of the wavelength. We start again with the single layer interferometer, and following Bailey and co-workers [8], we can rewrite Eq. 3 as 2µ Y + y = λ 0, 14a 14b 2µ 1 Y + y 1 = 1λ 0 1, 2µ m Y + y m = mλ 0 m. 14c Solving Eq. 14 for, weget 1 = λ0 1 λ0 µ 1 µ λ 0 1 µ 1y y 1, 15 where y i is the value of y for the wavelength λ 0 i that is, yλ 0 i and it can easily be seen that for a constant nondisersive y, Eq. 15 reduces to Eq. 4. Generally yλ i and articularly yλ 0 i is a function of the disersive hase change φλ i, λ i yλ i = φλ i 16 4πµλ i, where φλ i was shown by Bailey and co-workers [8] to be given by the exression kλ i φλ i = arctan nλ i µλ i kλ i arctan, 17 nλ i + µλ i where kλ i and nλ i are the imaginary and real arts of the comlex refractive index of the reflector usually silver in our case. Tabulated values for kλ i and nλ i can be found in ordinary handbooks see, for examle, Ref. [18]. In Fig. 2, we used the heuristic relation from Ref. [19], which was obtained from measurements with a silver layer Å thick deosited by evaoration: Fig. 2. The difference [Å] between the calculated and measured exerimental contact wavelengths for three calculations: square use Eq. 6 neglecting both disersion of mica and of mica silver interface; triangle use Eq. 5 neglecting only disersion of mica silver interface; circle use Eq. 20 considering disersion both of mica and of mica silver interface. The filled squares on the vertical line across the 0 of the ordinate corresond to the measured exerimental value. All the calculations were done using the two rightmost measured exerimental data oints. kλ i = λ i λ 2 i, nλ i = In Ref. [19], silver films of thickness Å were measured, and no deendence on film thickness was observed for films thicker than 250 Å. Note that Bailey and co-workers also do not consider the hase change at the mica silver interface to be a strong function of the silver thickness [8]; rather, the silver roughness, which deends on the way of silver deosition, the temerature and ressure, and the duration of the deosition, is the dominant factor in the hase change. Now, the value of y y 1 can easily be calculated from y y 1 = y λ 0 y λ 0 1 = φ λ 0 λ πµ 0 φ λ 0 λ πµ 0, 1 where φλ i is given by Eq. 17. This in turn can be substituted into Eq. 15 to calculate 1/.

6 R. Tadmor et al. / Journal of Colloid and Interface Science We can now use Eq. 14 to calculate the osition of any contact fringe λ 0 m based on two measured fringes λ0 and λ 0 1, λ 0 m = µ m λ 0 µ 1 m λ 0 1 µ 1 y y 1 λ 0 1 λ0 µ 1/µ 2µ m y y m λ 0, 1 µ 1 y y 1 λ 0 1 λ0 µ m 1/µ 20 and one can see that for a nondisersive y, Eq. 20 reduces to Eq. 5. Figure 2 shows the calculated ositions of λ 0 m using Eqs. 20, 5, and 6. Equation 6 is an aroximate solution which neglects both mica disersion and mica silver hase disersion; Eq. 5 accounts for the substrate mica disersion but neglects the hase change at the substrate reflector mica silver interface; and Eq. 20 accounts for both substrate and substrate reflector hase disersions. We can see that as the equation used is more exact the calculated wavelength osition becomes indeed increasingly close to the measured one. Aarently, however, Eq. 20 fails to calculate the exact osition of wavelength and does not rovide an absolutely accurate solution. This is not surrising for two reasons. One, as noted earlier, is that the relations in Eq. 18 are deendent on the exact exerimental conditions that formed the silver layer, which can be different from exeriment to exeriment. Another reason is that Eq. 20 has two oosite large corrective terms, which eventually result in a small correction. Hence a small inaccuracy in the measurement of the wavelength or in Eq. 18 could result in a rather big error if Eq. 20 were used. Unlike Eq. 5, where the corrective term merely multilies an existing term by a small number. Thus, relying on literature values for the reflector hase change is roblematic since they may use different conditions than the exerimental ones, which cannot necessarily be accounted for e.g., silver roughness at the interface. We therefore suggest measuring the values of kλ i and nλ i using the known m values in Eq. 20 for every exeriment in which it may be necessary and recalculating the constants using Eq. 18. The only roblem with this aroach is that unlike the disersive refractive index, which uses the rather well-established Cauchy relation, the equations for the disersive hase change are urely heuristic emirical relations, and hence the extraolation and erhas even the interolation of the exerimental values is not well-founded and could be erroneous. To comlete the icture, we write the equation for the three-layer interferometer that accounts for the hase change. Here again we follow Bailey and co-workers [8], who noted that in Eq. 1 the term 4πµY/λ D can be relaced by [ 4πµ D Y/λ D = πλ0 21 λ D µ D 4µ D y µ 0 λ 0 y D ], where µ D is µλ D, µ 0 is µλ 0, yd = yλd, and y = yλ 0. Substituting from Eq. 15 into Eq. 21 and substituting that into Eq. 1 would give the correct exression for the three-layer interferometer, which accounts for both the disersion and the hase change. Acknowledgments We thank Qi Lin, Peter Ercius, and Linh Tran for useful discussions and comutational assistance. NASA Grant NAG is gratefully acknowledged. References [1] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, [2] S. Tolansky, Interferometry, Longman, London, [3] J.N. Israelachvili, J. Colloid Interface Sci [4] M. Heuberger, G. Luengo, J. Israelachvili, Langmuir [5] M.T. Clarkson, J. Phys. D Al. Phys [6] R.G. Horn, D.T. Smith, Al. Ot [7] V. Kitaev, E. Kumacheva, J. Phys. Chem. B [8] B. Farrell, A.I. Bailey, D. Chaman, Al. Ot [9] H.K. Christenson, Ph.D. thesis, Australian National University, [10] J. Janik, R. Tadmor, J. Klein, Langmuir [11] J. Janik, R. Tadmor, J. Klein, Langmuir [12] E. Eisner, Research [13] The otical thickness Y which takes into account the hase change of light at the reflector substrate interface, as oosed to the hysical thickness Y, is a function of the wavelength and the thickness of the reflector silver layer. Throughout the first art of the aer the aroximation of a constant Y is made. Although later in the aer the disersive Y is accounted for, note that in ractice, the use of a constant Y is more useful for most cases. [14] In rincile, one should use the disersive form of the refractive index for the medium as well; however, from Eq. 10 one sees that the refractive index of the medium is only relevant for the measured fringes which, by definition, are all in the visible range. Thus, the error in the disersion of the medium is limited to the narrow visible range. On the other hand, the refractive index of the substrate should be known at the measured visible fringe as well as at the contact ositions which could be well outside the visible range. Thus, the disersion of the substrate does need to be known. [15] A.I. Bailey, S.M. Kay, Br. J. Al. Phys [16] S.Y. El-Zaiat, Ot. Laser Technol [17] J.N. Israelachvili, G.E. Adams, J. Chem. Soc. Faraday Trans [18] D.R. Lide Ed., CRC Handbook of Chemistry and Physics, CRC Press, London, [19] P.B. Johnson, R.W. Christy, Phys. Rev. B