928 Volume 56, Number 7, 2002 APPLIED SPECTROSCOPY0003-7028 / 02 / 5607-0928$2.00 / 0q 2002 Society for Applied Spectroscopy

New Technique for Determining the Optical Constants ofLiquids

C. DALE KEEFE* and JASON K. PEARSONDepartment of Physical and Applied Sciences, University College of Cape Breton, Sydney, Nova Scotia, Canada B1P 6L2

The traditional techniques of transmission and attenuated total re-� ectance (ATR) spectroscopy for determining the optical constantsof liquids are not practical or reliable for very strong absorptionbands. Specular re� ectance can be used in these cases, but for vol-atile liquids it is impossible to separate the re� ectance spectrum ofthe liquid from the absorption spectrum of the vapor above theliquid. Methods using special cells have been described in the lit-erature to prevent the liquid from evaporating. In this paper, asimilar technique that makes use of traditional transmission cells ispresented. It is shown that this new technique generates k ( ) spectran

for strong absorption bands that are accurate to approximately 2%.

Index Headings: Optical constants; Liquids; Infrared; Re� ectionmeasurements; Complex refractive index measurements.

INTRODUCTION

The current experimental methods of obtaining the op-tical constants of liquids include transmission, attenuatedtotal re� ection (ATR), and specular re� ection. All ofthese, however, have their drawbacks, particularly forstrong absorption bands. In this paper, a new method todetermine optical constants of these bands accurately isdiscussed.

The optical constants are the real, n, and imaginary, k,components of the complex refractive index:1–3

n( ) 5 n( ) 1 ik( )n n n (1)

where i 5 Ï(21).†The standard transmission technique of determining

optical constants has been widely used 4–28 for weak andmoderately strong bands (typically k , 0.5). The al-gorithm for this process is well de� ned in the litera-ture.10 –15,29 –32 It is based on a general iterative proce-dure:

(1) An approximate k spectrum is calculated from theexperimental absorbance spectrum15 assuming aconstant real refractive index of the liquid.

(2) An approximate n spectrum is calculated from thek spectrum by a Kramers–Kronig transform.1,2,33–38

(3) These n and k spectra, along with the n spectrumof the windows (nw( )) and the liquid thickness,nare used in Fresnel’s equations1,10,39,42 (vide infra)to calculate12,28 the transmission through the cell.

(4) The k spectrum is adjusted byln 10(EA (n) 2 EA (n))m ck 5 k + (2)i+1 i 4pdn

to improve the � t between the calculated and mea-sured transmissions. In Eq. 2, EAm and EAc are the

Received 6 November 2001; accepted 4 March 2002.* Author to whom correspondence should be sent.† Optical constants have also been de� ned as n 5 n(1 1 ik), n 5 n 2

ik and n 5 n(1 2 ik).

measured and calculated experimental absorbancespectra, respectively.

Steps 2 to 4 are repeated until the average magnitude ofthe change in k( ) is less than a predetermined conver-ngence limit. Jones and Young31 used 2 3 1025.

This method, however, can not determine the opticalconstants in regions of very strong absorption due to thefact that very thin cells are required to allow suf� cientenergy to reach the detector. Standard transmission cellsare commercially available in a wide variety of path-lengths greater than 10 mm. However, pathlengths assmall as 1 mm or less are required to determine the com-plex refractive index spectra in regions of strong absorp-tion.23 Such cells are very dif� cult to construct and use.First of all, variations on the surface of the windows ofcommon transmission cells could result in a complete oralmost complete closure internally, thus making the cellvery dif� cult to � ll. Also, it is necessary to overcome thesurface tension of the liquid in order to � ll a 1-mm cell,which can be very dif� cult. Therefore, it is desirable tohave a method that can utilize larger cells.

Attenuated total re� ection40 is another method current-ly available. It utilizes total internal re� ection by sur-rounding a crystal of known refractive index with theliquid sample, which has a smaller refractive index thanthe crystal. At wavenumbers where the liquid absorbs thelight, the re� ection becomes attenuated; hence the nameattenuated total re� ection. Algorithms for determining theoptical constants of a liquid using ATR41 are very similarto that of transmission. An approximate k spectrum iscalculated using the pATR spectrum (2log10 of the ATR)and the corresponding n spectrum is generated via theKramers–Kronig transform. These n and k spectra arethen used to calculate the pATR spectrum using Fresnel’sequations1,39,42 and the k spectrum is adjusted to improvethe � t between the measured and calculated pATR spectraby reducing it when the calculated pATR spectrum is toohigh and vice versa. This is repeated to conver-gence.41,43,44 In regions of strong absorption the re� ectioncan become normal rather than attenuated due to thechange in refractive index of the liquid making the de-termination of the optical constants unreliable.41 Also, thepATR does not vary proportionate to k for strong ab-sorption bands,45 and so adjusting k in a proportionalmanner fails to produce convergence. However, Dignam46

and Urban45 and their co-workers have developed algo-rithms that work well for strong ATR bands.

The earliest method used to determine IR optical con-stants37,38 was external re� ection spectroscopy or specularre� ection, which is theoretically the simplest methodavailable. It involves measuring a beam re� ected directlyfrom the surface of the sample. In practice, this method

APPLIED SPECTROSCOPY 929

FIG. 1. Transmittance and re� ectance at a single plate (medium 1)surrounded by two semi-in� nite media (media 0 and 2). n i is the com-plex refractive index of medium i, E0 is the electric � eld vector of theincident radiation, u0 is the incident angle, and t ij and rij are the fractionsof the transmitted and re� ected radiation at interface ij.

is unreliable for measuring the complex refractive indicesof volatile organic liquids because of their high vaporpressures. The gas phase is present above the liquid sam-ple and thus the experimental measurement is an absorp-tion spectrum of the gas superimposed on a re� ectionspectrum of the liquid. These spectra can not be separatedaccurately.

Methods have been described in the literature47–49,51

that use a transparent window to prevent the gas phasefrom interfering and allowing an accurate re� ectancespectrum to be measured. Those methods used specialcells to measure the re� ection of the liquid through atransparent window. The method developed in the currentstudy uses standard transmission cells to achieve thesame results. The method is presented and illustrated inthe infrared region of the electromagnetic spectrum, butit can also be applied to other regions of the electromag-netic spectrum.

DETAILS OF METHOD

Fresnel’s equations for the ratios of the re� ected (r )and transmitted (t) electric � eld vectors to the incidentelectric � eld at an interface are:

E n cos u 2 n cos ur,s j j k kr 5 5 (3)jk,s E n cos u 1 n cos u0,s j j k k

E n cos u 2 n cos ur,p j k k jr 5 5 (4)jk,p E n cos u 1 n cos u0,p j k k j

E 2n cos ut,s j jt 5 5 (5)jk,s E n cos u + n cos u0,s j j k k

E 2n cos ut,p j jt 5 5 (6)jk,p E n cos u + n cos u0,p j k k j

where the subscript jk indicates that the light is travellingfrom medium j to medium k. The subscripts s and p in-dicate the perpendicular and parallel polarizations, re-spectively. The symbol n represents the complex refrac-tive index of the respective medium, and u is the angleof the electric wave vector measured with respect to thenormal in the medium.

Figure 1 illustrates the re� ections that occur when elec-tromagnetic radiation with the electric � eld vector, E0, isincident upon the front half of a standard transmissioncell, where 0 represents the air surrounding the cell, 1represents the window enclosing the liquid sample, and2 represents the liquid sample. Analogous re� ectionswould occur in the other half of the cell. Note that theincident angle is measured with respect to the normal.Also, it can be seen that a portion of the radiation is bothtransmitted and re� ected at each interface. This effectcauses an in� nite number of re� ections inside the windowmaterial of the cell and potentially inside the liquid sampleas well depending on the thickness of the liquid layer andhow strongly the liquid absorbs. While these multiple re-� ections can be accounted for completely,10–12,23,30,32 it com-plicates the iterative procedures discussed later in this pa-per.

In the case shown, the refractive indices of all the me-dia are considered complex to keep the equations general.The electric � eld incident upon the � rst interface is E0.

The electric � eld transmitted from medium 0 to medium1 is E0t01 and the electric � eld re� ected is E0r 01. Thesubscripts indicating s- or p-polarization have been omit-ted for clarity. In practice, all equations must be evaluatedseparately for s- and p-polarizations.

From the coef� cients above, the re� ectance (R) andtransmittance (T ) at each interface can be calculated via:

Ir,w 2R 5 5 zr z (7)jk,w jk,wI0,w

I n cos ut,w k k2T 5 5 zt z (8)jk,w jk,w ) )I n cos u0,w j j

where w 5 s or p and I0, It, and Ir are the intensities ofthe incident, transmitted, and re� ected radiation, respec-tively.

As shown by Jones and co-workers,10–12,32 the re� ec-tance of a plate surrounded by two semi-in� nite media isgiven by:

2T R01 12R 5 R 5 (9)01 1 2 R R10 12

Figure 2a shows the typical placement of a transmis-sion cell in the beam in a spectrometer. Note that theresulting transmitted intensity, It, is a combination of themultiple re� ections within each layer. For clarity, the an-gles of re� ection and refraction have been exaggeratedand only a single re� ection is shown for each interface;in reality, a portion of each beam will be re� ected and

930 Volume 56, Number 7, 2002

FIG. 3. Calculated re� ectance spectra of benzene for pathlengths of10, 50, 100, 500, and 1000 mm in a KBr cell at an incident angle of10 degrees. The bottom spectrum is the 1000-mm pathlength; the nextfrom the bottom is 500 mm, and so on.

FIG. 2. (a) Shown is the transmittance through a transmission cell inthe usual arrangement. Only a single re� ection is shown for each in-terface and the angle of re� ection is exaggerated. In reality, in� nitere� ections will occur within each layer and the re� ected, transmitted,and incident beams will all be normal to the surface. (b) The novelarrangement proposed in the current work. The measured re� ection isthe sum of the re� ection from the window and that re� ected from theliquid for each re� ection within the window. It is assumed that any

¬

radiation transmitted into the liquid is completely absorbed before it canbe re� ected back into the window.

transmitted at each interface. Figure 2b shows the ar-rangement presented in this paper. Instead of measuringthe transmitted intensity, the re� ected intensity is mea-sured. Only the re� ections off the surface of the liquidare shown because it is assumed that any radiation thatis transmitted into the liquid is completely absorbed be-fore it is re� ected back to the � rst window.

The optical constants of benzene,4 particularly those ofthe strong out-of-plane vibration at 670 cm21, were usedto calculate re� ectance spectra for a variety of path-lengths, incident angles, and window materials for thearrangement in Fig. 2b. By examining all variables, theoptimal conditions for the experiment were determinedand are discussed below. Once the re� ectance spectrawere calculated, the task turned to determining if the op-tical constants could be obtained accurately from the re-� ectance spectra.

Selection of Pathlength. Figure 3 shows re� ectancespectra for pathlengths of 10, 50, 100, 500, and 1000 mmin a KBr cell at an incident angle of 10 degrees, startingwith the 10 mm spectrum at the top. Large interferencefringes are noticeable. These fringes are due to the fact

APPLIED SPECTROSCOPY 931

FIG. 4. Re� ectance spectra of benzene in an in� nite KBr cell at variousincident angles. Starting from the top they are 85, 75, 65, 55, 45, 35,25, 15, and 58, respectively. The curves for 25, 15, and 58 are basicallysuperimposed.

that the different beams resulting from the multiple re-� ections inside the cell (see Fig. 2) have all travelleddifferent distances, so they are out of phase. In the cal-culation, it was assumed that the windows are suf� cientlythick that interference fringes resulting from multiple re-� ections within the windows are too close to be resolved.All interference fringes are due to re� ection within theliquid layer. Just as easily noticed in Fig. 3 is the smooth-ing of the fringes as the pathlength of the cell is increaseddue to the fact that increasing the thickness of the liquidsample decreases the amplitude of the re� ected rays with-in the liquid. Calculation of k( ) with these fringes pre-nsent produces similar fringes in the k( ) spectrum, whichnwe have not succeeded in removing.

In regions of strong absorption, a liquid need not bevery thick to absorb all incident radiation and thus elim-inate multiple re� ections within the liquid layer. This ef-fect is desirable when trying to simplify calculations asmuch as possible. The thinner the liquid layer, the morelikely that the incident radiation will travel to the windowbehind the liquid sample and be re� ected back. There-fore, to eliminate these fringes and to allow for simplercalculation methods, an in� nite cell was assumed andthus only a single re� ection off the surface of the liquidwas considered for the remainder of the calculations.From Fig. 3, it is evident that for regions of strong ab-sorption, cells with pathlengths greater than 500 mm canbe considered to be in� nite cells.

Selection of Angle. Figure 4 shows re� ectance spectrafor various incident angles on an in� nite KBr cell. Whilethere is an increased amount of re� ection at larger anglesof incidence, this is due to re� ection from the nonab-sorbing window. The contribution to re� ection from thecomplex refractive index of the liquid is seen much moreevidently at small incident angles. Angles smaller than10 degrees are not practical due to space limitations ofthe arrangement shown in Fig. 2b, and thus, angles be-tween 108 and 258 are recommended. For the remainderof this paper, a 108 angle is used.

DETERMINING OPTICAL CONSTANTS FROMREFLECTANCE SPECTRA

Algorithm. As previously mentioned, the task here isto calculate the optical constants from a re� ectance spec-trum. While the opposite calculation is almost trivial, thelatter has proven to be quite challenging. In the pursuitof an accurate method, one generates a nonlinear equa-tion and consequently expects to use an iterative processin the algorithm.

If we consider a function, DR, such that

DR(n, k, ) 5 Rcalc(n, k, ) 2 Rexp( )n n n (10)

where Rcalc and Rexp are the calculated and experimentalre� ectance spectra, respectively, then DR(n, k, ) 5 0nwhen the real and imaginary refractive indices used tocalculate Rcalc(n, k, ) match those of the liquid. This isnthe basis of the algorithm.

The � rst step is to calculate an approximate imaginaryrefractive index spectrum, kapp( ), of the liquid. Sincenboth n( ) and k( ) are unknown, the approximate imag-n ninary refractive index spectrum is calculated using the

Newton–Raphson method while holding the real refrac-tive indices constant. Once kapp( ) is obtained, n( ) is cal-n nculated via the KK transform, a new k( ) calculated bynthe Newton–Raphson method with this n( ), and the pro-ncess is repeated until k( ) is converged.n

The Newton–Raphson method uses the slope at onepoint to predict where the function will cross the x-axisor where f (x) 5 0. When used in an iterative way, it willgenerate the roots of a nonlinear equation. The next pointin a series of iterations is determined by taking the pre-vious point and subtracting the ratio of the function andits � rst derivative at that point.

f (x )ix 5 x 2 (11)i+1 i f 9(x )i

The problem, however, is that it requires the � rst de-rivative of the function which is not practical for ourpurpose due to its complexity. The secant approximationcan be used to approximate the � rst derivative of a func-tion:

f (x ) 2 f (x )i21 if 9(x ) ø (12)i x 2 xi21 i

Substitution of Eq. 12 into Eq. 11 yields:

x f (x ) 2 x f (x )i21 i i i21x 5 (13)i+1 f (x ) 2 f (x )i i21

932 Volume 56, Number 7, 2002

FIG. 5. Flow chart describing the calculation procedure in programR2K.

FIG. 6. Shown are the synthetic and re� ned imaginary refractive indexspectra for 3 test cases (case 1, top box; case 2, middle box; case 3,bottom box). In each box, the two are indistinguishable over most ofthe range. At the high wavenumber wings and at the peaks, the synthetick( ) are the lower curves while at the lower wavenumber wings theynare the higher curves.

When the proper optical constants and re� ection valuesare substituted into Eq. 13 one gets:

k (n)DR (n,k , n) 2 k (n)DR(n, k , n)i21 i i i21k (n) 5 (14)i+1 DR (n, k , n) 2 DR (n, k , n)i i21

Once converged, this generates an approximate imagi-nary refractive index spectrum, kapp( ). Since n( ) is as-n nsumed to be constant at this stage, the kapp( ) that is cal-nculated is an approximate spectrum, which will be im-proved with further iterations.

From kapp( ), n( ) is generated by a Kramers–Kronign ntransformation and the two are used to calculate the re-� ectance spectrum. This re� ectance spectrum is thencompared to the experimental re� ectance spectrum andk( ) is adjusted by:n

R (n) 2 R (n)exp calck (n) 5 k (n) 1 1 (15)j j21 1 2R (n)calc

where k j21( ) and k j( ) are the imaginary refractive indi-n nces before and after the adjustment, respectively. Thisnew k( ) is then used to generate a new n( ) by the Kra-n nmers–Kronig transformation and the process is repeated.This is continued until the deviation in k( ) is less thanna predetermined limit or a maximum number of iterationsare performed.

Figure 5 illustrates the algorithm as implemented inprogram R2K.‡ Initially, n( ) is assigned a constant valuen

‡ Program R2K is available from C.D.K.’s website (http://faculty.uccb.ns.ca/dkeefe).

for all wavenumbers and kapp( ) is calculated with thenNewton–Raphson method. Should the Newton–Raphsonalgorithm fail to converge at any given wavenumber after150 tries, the approximate imaginary refractive index atthat wavenumber is determined by a bisection searchmethod. Then, kapp( ) is used as the � rst k value in theniterative process, k0( ). This process begins with a cal-nculation of n( ) by a Kramers–Kronig transform. Thennext step is to calculate the re� ectance, Rcalc, via Eqs. 3–9 with k0( ) and n( ), then use Eq. 15 to adjust k0( ).n n nDuring the course of writing and optimizing the algo-rithm (vide infra), oscillations were observed to occur inthe resulting k( ) spectra. That is, the calculated k( ) val-n nues, and consequently the n( ) values, oscillate back andnforth around the true value. To correct this problem, teniterations were averaged to insure that at least two oscil-lations were included, as an oscillation occurred roughlyevery four iterations. This generated a new k( ) spectrum,nwhich was signi� cantly better than any of the ten spectraused to generate it. If the root mean square adjustmentto k( ) is less than a predetermined limit, the iterationnprocess stops. However, if this does not happen, the it-eration process continues until a maximum number ofiterations are performed, as mentioned above.

APPLIED SPECTROSCOPY 933

FIG. 7. Shown are the synthetic re� ectance spectra in an in� nite cellwith KBr windows at a 108 incident angle for the 3 test cases (case 1,upper box; case 2, middle box; case 3, lower box). See text for a de-scription of the calculation.

TABLE I. Parameters used to calculate the synthetic imaginaryrefractive index and re� ectance spectra.

Parameter Test case 1 Test case 2 Test case 3(benzene)

mj / (D AÊ 21 amu21/2) 1.4 1.2Gj / cm21 10 8

j / cm 21n 670 1500Vm / (cm3 mol21) 60 60«` 2.25 2.25nref

a 1.434 1.5 1.434refn 750 1580 750

a Real refractive index at ref, required by programs KKTRANS andnR2K.

TEST OF ALGORITHM

To determine the ef� ciency and accuracy of the abovealgorithm, three complex refractive index spectra wereused to calculate the expected re� ectance spectra usingEq. 9 for an in� nite cell with KBr windows and a 108incident angle. For the � rst two cases, CDHO band pa-rameters were selected, the complex molar polarizabilityspectra were calculated using Eqs. 13a and 13b of Ref.55, and the complex dielectric constant spectra were cal-culated via Eqs. 21a and 21b of Ref. 52. Note that theright-hand side of Eq. 21b in Ref. 52 should be multipliedby 3. From the complex dielectric constant spectra, theimaginary refractive index spectra can be calculated byEq. 13 of Ref. 41. In the third case, the published 4 imag-inary refractive index spectrum of benzene was truncatedto the range 750 to 600 cm21. The real refractive indexspectra were calculated from the imaginary refractive in-dex spectra using program KKTRANS. 26 The parametersused to calculate the spectra are listed in Table I. Theresulting imaginary refractive index spectra are shown inFig. 6 and the corresponding re� ection spectra are shownin Fig. 7.

These re� ectance spectra were used as input to pro-gram R2K to test how well the algorithm could determinethe optical constants. Also shown in Fig. 6 are the � nalconverged imaginary refractive index spectra from R2K.In case one, the largest error in the re� ned k( ) spectrumn

is 0.014 or 1.2% of the peak height; in case two thelargest error is 0.0069 or 1.2% of the peak height; andin the third test case the largest error is 0.022 or 1.8% ofthe peak height. Thus, it can be expected that this methodwill give k( ) spectra accurate to within 2% when usednfor very strong absorption bands.

CONCLUSION

In this paper, the theoretical basis for a new experi-mental technique to measure the optical constants of liq-uids, particularly in the regions of strong absorption, waspresented. The method enables the optical constants tobe determined with an accuracy of 2%. The next stageof this project is to build the re� ection attachment andtest the technique experimentally. This will be carried outover the next little while in this laboratory, and the resultswill be published in a future paper.

ACKNOWLEDGMENTS

The authors thank the Natural Sciences & Engineering ResearchCouncil of Canada, the of� ce of Research and Academic Institutes atthe University College of Cape Breton, and Human Resources Devel-opment Canada for their support of this research.

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