Kramers-Kronig analysis of molecular evanescent-wave absorption spectra obtained by multimode step-index optical fibers

Kramers–Kronig analysis of molecularevanescent-wave absorption spectra obtained bymultimode step-index optical fibers

Radislav A. Potyrailo, Vincent P. Ruddy, and Gary M. Hieftje

Spectral distortions that arise in evanescent-wave absorption spectra obtained with multimodestep-index optical fibers are analyzed both theoretically and experimentally. Theoretical analysis isperformed by the application of Kramers–Kronig relations to the real and the imaginary parts of thecomplex refractive index of an absorbing external medium. It is demonstrated that even when theextinction coefficient of the external medium is small, anomalous dispersion of that medium in thevicinity of an absorption band must be considered. Deviations from Beer’s law, band distortions, andshifts in peak position are quantified theoretically as a function of the refractive index and the extinctioncoefficient of the external medium; the effect of bandwidth for both Lorentzian and Gaussian bands isalso evaluated. Numerical simulations are performed for two types of sensing sections in commonlyused plastic-clad silica optical fibers. These sensors include an unclad fiber in contact with alower-index absorbing liquid and a fiber with the original cladding modified with an absorbingspecies. The numerical results compare favorably with those found experimentally with these types ofsensing sections.Key words: Evanescent wave, absorption spectroscopy, fiber-optic sensors. r1996 Optical Society of

America

1. Introduction

Multimode step-index optical fibers are becomingimportant tools for qualitative and quantitativechemical analysis by means of attenuated totalreflection 1ATR2 spectroscopy. Evanescent-wave ab-sorption spectra obtained with this type of cylindri-cal waveguide have been reported for a wide varietyof gaseous, liquid, and solid species that absorb inspectral regions ranging from the near ultraviolet1UV2 to the middle infrared1–9 1IR2. In some casesthe observed spectra were distorted,1,3,5 whereas inothers the absorption spectra appeared as if they hadbeen obtained by conventional transmission meth-ods.4,7

R. A. Potyrailo and G. M. Hieftje are with the Department ofChemistry, Indiana University, Bloomington, Indiana 47405.V. P. Ruddy is with the School of Physical Sciences, Dublin CityUniversity, Glasnevin, Dublin 9, Ireland.Received 9 November 1995; revised manuscript received 18

March 1996.0003-6935@96@214102-10$10.00@0r 1996 Optical Society of America

4102 APPLIED OPTICS @ Vol. 35, No. 21 @ 20 July 1996

Three general differences exist between evanes-cent-wave absorption spectra and those obtained bytransmissionmethods. These differences have beencharacterized qualitatively in conventionalATR spec-troscopy10,11:

112 The magnitude of evanescent-wave absorp-tion increases at longer wavelengths because thepenetration depth 1and therefore the effective pathlength of absorption2 of the evanescent field is di-rectly proportional to the wavelength of the probingradiation.122 A spectrum will appear distorted if the probe

beam that produces it strikes the reflecting interfaceat an angle close to the critical angle. The criticalangle uc is given by

uc 5 sin211n̂2@n12, 112

where n1 is the refractive index of the optical-waveguide element and n̂2 is the complex refractiveindex of the external absorbing medium. The com-plex refractive index consists of a real part n21refractive index2 and an imaginary part n2* 1extinc-

tion coefficient2:

n̂2 5 n2 1 in2*. 122

To avoid band distortions caused by operating withthe probe beam at angles close to the critical, it iscustomary to launch a probe beam into a conven-tional ATR element at angles far from the criticalangle.132 An increase in absorber concentration can

cause the peak of an evanescent-wave absorptionspectrum to be shifted toward longer wavelengthsand the long-wavelength wing of the absorptionband to be broadened. The origin of these distor-tions is anomalous dispersion in the vicinity of theabsorption band. Samples studied by conventionalATR elements in the middle-IR spectral region haveexhibited such distortions, which complicate chemi-cal identification from spectral libraries obtained inthe conventional transmission mode.

A comparison of conventionalATR elements with amultimode step-index optical fiber is presented inTable 1. Differences between the light-waveguid-ing conditions and the range of sample extinctioncoefficient over which these two types of opticalwaveguides operate suggest that spectral distortionsencountered with multimode optical fibers cannot beneglected. Although the general features of spec-tral distortions in ATR spectroscopy have been out-lined,10,11 no quantitative description has been pub-lished for either conventional ATR elements ormultimode step-index optical fibers. This lack of aquantitative model hinders the development of fiber-optic sensors for practical applications.The theoretical model developed in this paper

pertains to multimode propagation of light in astep-index optical fiber and assumes a uniform distri-bution of the incident radiation among all boundmodes in the sensing region. Numerical simula-tions were performed for plastic-clad silica 1PCS2

Table 1. Comparison of Conventional ATR Elements with MultimodeStep-Index Optical Fibers

Parameters toCompare

Conventional ATRElements

Multimode Step-Index Optical

Fibers

Waveguiding con-ditions

Use of probe beamwith a well-de-fined incidenceangle

Operation in multi-mode regimewithin a fiber NA1n12 2 n222 1@2 NA

Typical number ofreflections

Less than 100 Thousands

Sample extinctioncoefficient, n2*

0.01–3 0–0.01

Applications Spectral identifica-tion of stronglyabsorbingsamples

Quantitative mea-surements ofconcentrationsdown to parts in109 levelsa

aRefs. 12 and 13.

fibers. These multimode fibers are the ones mostcommonly used in the development of fiber-opticchemical sensors for the near-UV, visible, and near-IRspectral regions.1,2,5,14,15 Two widely used sensingconfigurations are discussed. The first is an uncladfiber in contact with a lower-index absorbingmedium.The latter medium is ordinarily 1but not necessarily2an aqueous solution of absorbing species.2,3 Thesecond arrangement is a fiber surrounded by acladding that has been impregnated with an absorb-ing species. This absorbing species might be gas-eous or liquid analytes that have diffused into thepolymer cladding1,5 or might be reagents that wereintentionally immobilized onto or within the clad-ding for indirect chemical sensing.14,15 The theoreti-cal analysis is performed by the application of Kram-ers–Kronig relations to the real and the imaginaryparts of the complex refractive index of the externalabsorbing medium. Deviations from Beer’s law,band distortions, and shifts in peak position arecharacterized theoretically as a function of the refrac-tive index and the extinction coefficient of the exter-nal medium and of bandwidth for Lorentzian andGaussian absorption features. The numerical simu-lations were tested against experimental results forthe two types of sensing arrangements describedabove.

2. Theory

As shown in Table 1, the conditions for light propaga-tion in multimode optical fibers are very differentfrom those in conventional ATR elements. The nar-row-diameter fiber structure guides a large numberof modes. If a section of optical fiber is exposed toan external absorber over a length L, the transmit-tance for a particular mode Tm is given by16

Tm 5 exp12gL2, 132

where g is the power-attenuation coefficient for thebound mode. The power-attenuation coefficient de-pends on the fundamental characteristics of theexternal absorbing medium 1refractive index n1 andbulk absorption coefficient a2 and on the characteris-tics of the cylindrical waveguide 1profile height pa-rameter D, core parameterU, cladding parameterW,waveguide parameter V, and mode-propagation con-stant b2. The power-attenuation coefficient for abound mode is given by17

g 5 1an2kU22@1bV 2W 2, 142

where k is the wave number in free space, k 5 2p@l,l is the wavelength of the probing radiation invacuum, and

D 5 1n12 2 n̂222@2n12, 152

U 5 r1k2n12 2 b221@2, 162

W 5 r1b2 2 k2n̂2221@2, 172

20 July 1996 @ Vol. 35, No. 21 @ APPLIED OPTICS 4103

V 2 5 U2 1 W 2, 182

where r is the radius of the fiber core.The mode-propagation constant b can be repre-

sented as17

b 5 n1k11 2 2Dx221@2, 192

where x, the relative mode index, is the ratio of aspecific mode number to the total number of modes.For a highly multimode fiber with a large V-number,the relative mode index can be approximated as acontinuous variable that ranges from x < 0 forlow-order modes to x < 1 for higher-order modes.Substituting Eqs. 142–192 into Eq. 132, one can ex-

press the mode transmittance as

Tm 5 exp52 an2Lx2

n1V 311 2 x2211 2 2Dx2241@26 . 1102

The absorbance of the mode is given by

Am 5 0.434an2Lx2

n1V 311 2 x2211 2 2Dx2241@2. 1112

If the incident radiant power is equally distributedamong all bound modes in the fiber, integration overall those modes 1x 5 0–12 yields the sensor absor-bance:

A 5 2log10 e0

1

exp52 an2Lx2

n1V 311 2 x2211 2 [email protected]

The evanescent-wave absorbance in the aboveexpression, pertaining to a fiber-optic sensor, differsfrom that calculated for a transmission method inthe three general waysmentioned above. Thewave-length dependence 11@l2 of the absorbance lies in thewaveguide parameter V. Higher-order modes 1withrelative mode index x < 12 propagate in the fiber atangles close to the critical angle. The bulk absorp-tion coefficient of the sample a is related to its molarabsorptivity, e, and extinction coefficient n2* as

a 5 ec 5 4pn2*@l, 1132

where c is the analyte concentration.Anomalous dispersion of the refractive index n2 in

the vicinity of an absorption band can be taken intoaccount by the use of the relationship between n2 andn2* known in spectroscopy as the Kramers–Kronigrelation and inmathematics as theHilbert transform.The appropriate mathematical relationship is givenby 18

n21n2 5 n2` 12

p e0

1` sn2*1s2

s2 2 n2ds, 1142


n2*1n2 5 22n

pP e

0

1` n21s2

s2 2 n2ds, 1152

where n2` is the refractive index of the sample farfrom the absorption band, P is the principal value ofthe integral,18 and n and s are wave numbers.The most commonly used spectral profiles for

approximating absorption bands in molecular spec-troscopy employ Lorentzian andGaussian line shapesand their combinations.19 The Lorentzian profile isgiven by

eL 5 emax@31 1 41n 2 nmax22@1Dn224, 1162

and the Gaussian profile is given by

eG 5 emax exp321ln 221n 2 [email protected], 1172

where emax and nmax are the extinction coefficient1absorptivity2 and wave number at the peak of theabsorption band, respectively, and Dn is the fullwidth at half maximum 1FWHM2 of the band. Ifwave numbers are plotted on the horizontal axis ofthe spectrum, the Lorentzian and the Gaussianprofiles are symmetrical about nmax. In contrast,when wavelength is plotted on the horizontal axis,both profiles are asymmetrical about lmax.

3. Results of Simulation Experiments

A. Conditions for Computer Simulations

Parameters employed for the numerical simulationsare summarized in Table 2. The bandwidths1FWHM2 used in the simulations are typical foranalyte species and chemical reagents that absorb inthe UV to near-IR spectral regions. The rangeselected for the sample extinction coefficient n2* iswithin the zero approximation11 1n2* , 0.12 and is

Table 2. Parameters for Numerical Simulations

Optical fiber PCS optical fiberRadius of fiber core, r 50 µmRefractive index of fibercore,a n1

1.4618 1fused silica2

External absorbing medium Aqueous solution of an indi-cator and indicator-dopedfiber cladding

Refractive index of externalmedium,a n2

1.33 1water2

Refractive index of fiber clad-ding,a n2

1.41 1siloxane2

Extinction coefficient ofexternal medium, n2*

1025–1022

Absorption band shapes Gaussian and LorentzianAbsorption peak maximum 500 nm 120,000 cm212Band FWHM 20–200 nm 1800–8000 cm212Number of points for Kram-ers–Kronig transform

20,000

Step size in the numericalsimulations

10 cm21 10.25 nm at 500 nm2

aRefractive index at 500 nm.

valid for typical analytes studied with fiber-opticchemical sensors.20 To demonstrate the generaltheoretical characteristics of absorption-band distor-tions, the specific dispersion relations pertaining tothe fiber-optic core, cladding, and the surroundingsolvent have been intentionally omitted.The Kramers–Kronig transform of the simulated

absorption bands, Eq. 1142, was performed by aprogram written in LabVIEW 1Version 3.1, NationalInstruments2. A numerical computer analysis ofEqs. 1112 and 1122 was performed with Maple V1Release 3, Waterloo Maple Software2 and Kaleida-graph 1Version 3.0.2., Abelbeck Software2.

B. Simulated Evanescent-Wave Absorbance forBound Modes

The evanescent-wave absorbance, computed fromEq. 1112 for bound modes in an optical fiber, ispresented in Fig. 1. The calculated absorbance isplotted as a function of wave number and therelative mode index x for both Lorentzian 3Fig. 11a24and Gaussian 3Fig. 11b24 band profiles with Dn 5 4000cm21 and n2* 5 1025. The contribution of differentmodes to the evanescent absorbance increases withthe relative mode index; the strongest contributionsare from higher-order modes that propagate in thefiber at angles close to the critical, as described byEq. 112.The general behavior of the evanescent-wave absor-

bance shown in Fig. 1 is similar for unclad andabsorbing-cladding fibers. However, there is an im-portant difference in the magnitude of the absor-bance for corresponding modes in the two types ofsensing elements. This difference can be attributedto three factors in Eq. 1112: n2, V, and D. First, anydrop in the refractive index n2 leads to a proportionalreduction in the mode absorbance Am. Second, ac-cording to Eq. 152, the profile height parameter Dincreases as the sample refractive index is lowered,causing an elevation in Am. Last, according to Eqs.162–182, the waveguide parameter V goes up with adecrease in n2, giving a decline in Am. The netresult of a reduction in the refractive index of theexternal medium from 1.41 1siloxane cladding2 to1.33 1water2 gives a theoretical drop in the modeabsorbance of ,40%.

C. Band Profile Distortions

For visualization and quantification of band distor-tion, a common method21,22 is to linearize the equa-tions that describe the Lorentzian and the Gaussianprofiles in the following way. The Lorentzian pro-file becomes

0.531emax@eL2 2 141@2 5 1n 2 nmax2@Dn, 1182

and the Gaussian profile becomes

0.51ln 2221@23ln1emax@eG241@2 5 1n 2 nmax2Dn. 1192

In the transformed coordinates the regions of the

band profiles that correspond to the respective1Lorentzian or Gaussian2 shape appear as straightlines. The slope of the line is dependent on theabsorption band FWHM, Dn, given by

Dn 5 1@tan w, 1202

where w is the angle between the line and theabscissa axis.

Fig. 1. Calculated evanescent-wave absorbance for bound modesin an optical fiber described by Eq. 1112 as a function of wavenumber and of the relative mode index x for 1a2 Lorentzian,1b2 Gaussian profiles. Simulation parameters are Dn 5 4000cm21 and n2* 5 1025 1see Table 22. The general behavior of theplots is similar for both unclad and absorbing-cladding fibers.


Spectral distortions were simulated by means ofEq. 1122. The resulting plots were transformed withthe linearization procedure given by Eqs. 1182 and1192. The results are plotted in Fig. 2 as trans-formed ordinate versus wave number for differentband FWHM’s and extinction coefficients for uncladand absorbing-cladding fibers. The data for thecorresponding absorption profiles obtained in thetransmission mode are also shown in Fig. 2 forcomparison. Over the range of the extinction coeffi-cient n2* from 1025 and 1022, distortions of the bandprofiles are relatively constant. However, the distor-tions are dependent on the shape of the absorptionband and aremuchmore pronounced for the Lorentz-

Fig. 2. Distortions of absorption bands in evanescent-wavespectra calculated from Eq. 1122. The ordinates of the plots aretransformed according to Eqs. 1182 and 1192 for 1a2 Lorentzian,1b2 Gaussian profiles, respectively. Evanescent-wave profiles:Dn 5 800 cm21, n2* 5 1025 1n2; Dn 5 800 cm21, n2* 5 1022 1m2; Dn 5

4000 cm21, n2* 5 1025 1X2; Dn 5 4000 cm21, n2* 5 1022 1W2. Thecorresponding absorption profiles obtained in the transmissionmode are presented for comparison as solid lines. The generalbehavior of the plots is similar for both unclad and absorbing-cladding fibers.

ian profile 3Fig. 21a24 than for the Gaussian profile3Fig. 21b24.The quantitative effect of the band distortion

shown in Fig. 2 is summarized in Table 3. Not onlyare the red-side wings of the Lorentzian and theGaussian absorption bands broadened, as was de-scribed previously11 qualitatively for conventionalATR spectroscopy, the simulated data here also showa narrowing of the Lorentzian profiles and a broaden-ing of the Gaussian profiles in the blue-side wing ofthe respective bands. However, for the 800-cm21

FWHM band, the Gaussian profile also demon-strates a narrowing of the red side of the band.This difference in the behavior and magnitude of thespectral distortions for the Lorentzian and theGauss-ian line shapes is due to the slower roll-off of theLorentzian profile compared with that of the Gauss-ian 1see Fig. 12.These band distortions are independent of the

refractive index n2 of the external medium withinthe range of the extinction coefficient n2* that wasinvestigated. However, additional simulations showthat with a further increase of n2* from 1022 to 1021,which is still in the zero-absorption approximationrange,11 the band distortions become more pro-nounced, for both the Lorentzian and the Gaussianprofiles in the case of an absorbing-cladding fiber,when the difference between the refractive indices ofthe fiber core and the absorbing medium becomesless.

D. Peak Shifts

The magnitude of the peak shift between the trans-mission and evanescent-wave absorption spectrawas calculated with Eq. 1122 in combination with Eqs.1182 and 1192 for Lorentzian and Gaussian bandshapes, respectively. The resulting peak shift, dn,as a function of the band FWHM Dn and extinctioncoefficient n2* of the external medium for Lorentzianand Gaussian absorption bands and for unclad andabsorbing-cladding fibers is presented in Fig. 3.In all cases, the peak shifts are strongly dependenton the FWHM of the absorption band. For n2*values ranging from 1025 to 1024, the shift is less

Table 3. Distortions of the Lorentzian and the Gaussian Absorption-Band Profiles for the Unclad and the Absorbing-Cladding Fibers

Band RegionExtinction

Coefficient, n2*

Absorption-Band FWHM inEvanescent-Wave Mode 1cm212

Lorentzian Profile Gaussian Profile

800a 8000a 800a 8000a

Unclad fiber, blue-side wing 1025

1022740740

74297398

800802

81078180

Unclad fiber, red-side wing 1025

1022932931

94119361

800799

82628324

Absorbing-cladding fiber, blue-side wing 1025

1022739737

74137341

800804

81078290

Absorbing-cladding fiber, red-side wing 1025

1022917914

92539330

800798

82578499

aAbsorption-band FWHM in transmission mode.


tWaiGt

b

FbL1s

han the step size for the calculations 110 cm212.ith an increase in extinction coefficient, the shiftsre more pronounced for both Gaussian and Lorentz-an absorption bands. In all cases, the peak of theaussian profile is shifted slightly more than that ofhe Lorentzian profile.The data for the peak shifts, dn, as a function ofand FWHM Dn, were fitted with a second-order

ig. 3. Calculated absorption peak shifts as a function of theand FWHM and sample extinction coefficient n2* for 1a2, 1c2orentzian, 1b2, 1d2 Gaussian absorption bands for 1a2, 1b2 unclad, 1c2,d2 absorbing-cladding fibers. Extinction coefficient n2*: #1024,olid curve; 1023, dashed–dotted curve; 1022, dotted curve.

polynomial:

dn 5 a0 1 a1Dn 1 a21Dn22, 1212

where a0, a1, and a2 are the polynomial coefficients.The results of the polynomial fits and the correlationcoefficients R are presented in Tables 4 and 5 forunclad and absorbing-cladding fibers, respectively.

E. Deviations from Beer’s Law

Absorption-band distortions caused by the evanes-cent-wave measurement mode result in deviationsfrom Beer’s law when one operates with monochro-matic light in either wing of the absorption band.To estimate the magnitude of the deviations, simula-tions were performed for different band FWHM’sand assuming monochromatic radiation located atthree specific positions in the absorption bands,namely, nmax 1 0.5Dn 1blue wing of the absorptionband2, nmax 1absorption peak2, and nmax 2 0.5Dn 1redwing of the absorption band2. The resulting absor-bance A at each frequency was represented as apower function of the extinction coefficient n2* overthe range from 1025 to 1022:

A 5 M1n2*2N, 1222

where M and N are coefficients of the power fit. Itwas found that when the probe frequency is locatedat the absorption peak 1see Table 22, Beer’s law isfollowed, no matter what the band FWHM. Forprobe-radiation frequencies other than at the absorp-tion peak, deviations from Beer’s law are observedthat result in a change in the coefficient N of thepower fit as a function of band FWHM for both theblue- and the red-side wings of the Lorentzian andthe Gaussian profiles 1Fig. 42. The deviation is

Table 4. Parameters of Polynomial Fit CEq. A21BD for Shifts in the Peak of an Absorption Band as a Function of Band FWHM and Sample ExtinctionCoefficient n2* for Lorentzian and Gaussian Band Shapes a

Parameter

Lorentzian ProfileExtinction Coefficient n2*

Gaussian ProfileExtinction Coefficient n2*

0.0001 0.001 0.01 0.0001 0.001 0.01

a0 2.4298 2.4378 6.1947 6.4797 8.8776 9.2818a1 20.0018385 4.7763 3 1025 0.0043225 20.0051722 20.0050215 0.0069743a2 6.7248 3 1026 6.8203 3 1026 7.3741 3 1026 1.0212 3 1025 1.0315 3 1025 1.0478 3 1025

R 0.99992 0.99997 1 0.99997 0.99995 0.99996

aUnclad fiber in aqueous solution 1n2 5 1.332.

Table 5. Parameters of Polynomial Fit CEq. A21BD for Shifts in the Peak of an Absorption Band as a Function of Band FWHM and Sample ExtinctionCoefficient n2* for Lorentzian and Gaussian Band Shapes a

Parameter

Lorentzian ProfileExtinction Coefficient n2*

Gaussian ProfileExtinction Coefficient n2*

0.0001 0.001 0.01 0.0001 0.001 0.01

a0 2.867 4.6018 7.6212 3.7953 4.4444 4.568a1 20.0011265 0.00025839 0.020636 20.0037843 20.0010333 0.028871a2 6.6256 3 1026 6.7586 3 1026 6.9918 3 1026 1.0046 3 1025 1.006 3 1025 1.0523 3 1025

R 0.99975 0.99987 0.99997 0.99998 1 0.99993

aAbsorbing-cladding fiber 1n2 5 1.412.


usually positive 1N . 12 if the probe frequency is atthe red-sidewing of the absorption band 1nmax 2 0.5Dn2and negative 1N , 12 if the probe frequency is at theblue-side wing of the absorption band 1nmax 1 0.5Dn2.The magnitude of the deviation from Beer’s lawdepends slightly on the band FWHM but strongly onthe shape of the band, i.e., Lorentzian or Gaussian.

4. Experimental Results and Discussion

A PCS fiber 1Superguide SPC100@200N, FiberguideIndustries, Inc., core diameter 100 µm, claddingdiameter 200 µm, NA of 0.42 that has the samecharacteristics as the fiber used in the numericalsimulations 1Table 22 was employed in the measure-ment system depicted in Fig. 5. Light from a 65-Wquartz-halogen lamp was collimated, chopped at 250Hz, and focused into the fiber by means of a micro-scope objective 1Newport, 0.4 NA2. The sensingportion of the fiber was coiled and placed in a glasssample chamber. The light transmitted throughthe fiber was directed onto the 100-µm entrance slit

Fig. 4. Deviations from Beer’s law caused by distortions inevanescent-wave absorption spectra. Magnitude of deviation isexpressed as a change in the coefficient N of the power fit 3see Eq.12224 versus band FWHM for the blue- and red-side wings of theLorentzian and Gaussian profiles in the 1a2 unclad, 1b2 absorbing-cladding fibers. The assumed monochromatic probe frequenciesare set in specific regions of the absorption bands: 1nmax 1 0.5Dn2is a probe frequency in the blue-side wing of the Lorentzian 1I2 andthe Gaussian 1II2 profiles, and 1nmax 2 0.5Dn2 is a probe frequencyin the red-side wing of the Lorentzian 1III2 and Gaussian 1IV2profiles.


of a 0.35-m grating monochromator 1Model EU-700,GCA-McPherson2 and detected by a photomultipliertube 1R928, Hamamatsu2. Data were collected overthe 350–750-nm spectral range in 0.25-nm incre-ments and sampled at regular intervals by a 16-bitanalog-to-digital converter board, which was in-stalled in a Macintosh computer. Monochromatorcontrol and data acquisition were achieved withLabVIEW software. Conventional absorption mea-surements of an indicator solution and of absorbing-cladding fibers were performed in the transmissionmode with the same experimental arrangement butwith the microscope objective and optical fiber re-moved.

A. Unclad Fiber

The cladding was removed from the central 75-cmregion of the fiber by the use of a standard procedure.2The exposed sensing region of the fiber was coiledwith a radius of R 5 2.5 cm, yielding the ratio R@r 5500, which is great enough to ensure that no enhance-ment of the evanescent-field amplitude should occurbecause of the fiber bending.16 Bromothymol blueindicator 1Aldrich2 of the sulfonphthalein class wasselected for examination because the red side of theabsorption band of its basic form is not significantlydistorted by neighboring peaks and can be approxi-mated by a Gaussian.22 The bulk absorption coeffi-cient of the indicator solution was a 5 34 cm21.Absorption spectra obtained in the transmission

and evanescent-wave modes are presented in Fig. 6;the linearized functions are compared with the corre-sponding Gaussian bands in Fig. 7. The ordinateon the plots of Fig. 7 was transformed according toEq. 1192 in order to perform the linearization. Theabsorption-band maxima observed experimentally

Fig. 5. Experimental setup for evanescent-wave sensing withstep-index multimode optical fibers. PMT, photomultiplier tube.

Fig. 6. Absorption spectrum of the basic form of bromothymolblue indicator solution in water: spectrum obtained in transmis-sionmode 1dashed curve2 and evanescent-wave spectrum of indica-tor obtained with an unclad optical fiber 1solid curve2.

were 617.5 and 621.0 nm for the transmission-modeand the evanescent-wave measurements, respec-tively. In order to compare the theoretical peakshift with this experimental value 13.5 nm2, theFWHM of each band was found from the slope of theGaussian fit of the red side of the corresponding bandby the use of Eq. 1202. The FWHM’s were calculatedto be 1660 and 1630 cm21 for the transmission-modeand the evanescent-wave absorption profiles, respec-tively. From Eq. 1212 and Table 4, the theoreticalpeak shift was determined to be 25 cm21 or 0.97 nm.The difference between the experimentally observed13.5-nm2 and predicted 10.97-nm2 shifts might be dueto any of several factors related to the propagation oflight in an optical fiber that were not considered inthe applied model. For example, scattering of lightfrom the core-liquid interface by contaminants onthe polar silica surface2 can contribute to the localgeneration of high-order bound or radiation modesthat exaggerate the peak shift. Further, the narrow-ing of the band in the evanescent-wave absorptionspectrum might be caused by adsorption of indicatormolecules on the polar fiber surface, a factor that isnot considered in the simple model developed here.

B. Absorbing-Cladding Fiber

The sensing region 175 cm2 of the fiber was coiledwith a radius R of 3.8 cm, yielding the ratio R@r 5760. The cladding in the sensing region of the fiberwas doped with an indicator by a technique de-scribed elsewhere.15 For a sensing reagent, phenolred indicator 1Aldrich2 of the sulfonphthalein classwas selected because the absorption band of itsacidic form is not significantly distorted by neighbor-ing peaks when the chemically modified fiber is inair.15 The bulk absorption coefficient of the indica-tor solution used for chemically modifying the fiberwas a 5 17 cm21. The absorption spectrum of the

Fig. 7. Comparison of linearized absorption spectra of an aque-ous solution of bromothymol blue indicator obtained in thetransmission and the evanescent-wave modes. The ordinate onthe plots is transformed according to Eq. 1192. Experimentaldata: transmission-mode spectrum of indicator 1W2 and evanes-cent-wave spectrum of indicator with an unclad optical fiber1X2. Gaussian fits: transmission-mode spectrum fit accordingto Eq. 1172 1dashed line2 and evanescent-wave spectrum fit accord-ing to Eq. 1122 1solid line2.

absorbing-cladding fiber in the transmission modewas measured by cutting the fiber into 1.2-cm seg-ments, arranging them side by side on a flat transpar-ent surface, and illuminating that surface withnormally incident light.The absorption spectra obtained in the transmis-

sion and evanescent-wave modes are presented inFig. 8. A comparison of these experimental datawith the corresponding Gaussian fits is found in thelinearized plots of Fig. 9. Again, the ordinate onthese plots was transformed by means of Eq. 1192.The peak positions observed experimentally were437.5 and 440.5 nm for the transmission-mode andthe evanescent-wave absorption profiles, respec-tively, corresponding to a peak shift of 3.0 nm. Thecalculated FWHM’s of these bands were found fromthe slopes of the Gaussian fits of their red wings bythe use of Eq. 1202. These half-widths were 4700and 4900 cm21 for the transmission-mode and theevanescent-wave absorption profiles, respectively.From Eq. 1212 and Table 5, the estimated peak shiftwas 208 cm21 or 3.97 nm, in good agreement with theexperimental value of 3.0 nm. The greater FWHMin the evanescent-wave absorption spectrum, com-

Fig. 8. Absorption spectrum of the acidic form of phenol redindicator immobilized on the cladding of a PCS fiber: transmis-sion-mode spectrum 1dashed curve2 and evanescent-wave spec-trum 1solid curve2.

Fig. 9. Comparison of linearized absorption spectra of phenol redobtained in the transmission and the evanescent-wave modes.The ordinate on the plots has been transformed according to Eq.1192. Experimental data: transmission-mode spectrum 1W2 andevanescent-wave spectrum obtained with indicator immobilizedon the cladding of the PCS fiber 1X2. Gaussian fits: transmis-sion spectrumfit according to Eq. 1172 1dashed line2 and evanescent-wave spectrum fit according to Eq. 1122 1solid line2.


pared with that of the transmission-mode spectrum,is also consistent with the calculated results.

5. Conclusions

Peak shifts and band distortions that occur in evanes-cent-wave absorption spectra have been quantita-tively analyzed by use of data collected with step-index multimode PCS fibers. The experimentalresults are in good agreement with a derived modelfor an absorbing-cladding fiber. However, presum-ably because of surface imperfections on the uncladfiber, the experimentally observed 3.5-nm shift waslarger than that predicted from theory 10.97 nm2.The practical importance of this study might be in

distinguishing between spectroscopic and chemicalcauses of absorption-band distortions and peak shifts.For example, it is known that the degree of chemicalbinding of an immobilized indicator causes both ashift of an absorption peak to longer wavelengthsand a broadening of the absorption band.22,23Distinguishing between these chemically inducedeffects and those caused by the evanescent-wavemeasurement process is difficult experimentally.This problem is all the more troublesome because ofthe promising new method of indicator immobiliza-tion directly onto an existing fiber cladding.13,15,24Similarly, in an aqueous solution, the position of anabsorption peak may be a function of concentrationof the analyte or of another species because ofhydrogen-bonding effects; this behavior has beenobserved in mixtures of water with ethanol8 andacetone.9 The results of this study might also beuseful for the understanding or characterization ofmulticomponent spectrophotometric measurements;spectral displacements of only tenths of a nanom-eter, much less than those introduced by an evanes-cent-wave measurement, can produce significanterrors in applications in which a multiwavelengthcalibration procedure is employed.25It was found that when the probe frequency is

located at the absorption peak, Beer’s law is fol-lowed, no matter what the FWHM of the band. Forprobe-radiation frequencies other than at the absorp-tion peak, deviations from Beer’s law are observedbecause of distortions of band profiles with varia-tions of the sample extinction coefficient. Thesedeviations add to the complexity of calibrating asensor based on a step-index multimode fiber.Other factors that affect Beer’s law behavior includesurface contamination,2 mode-dependent attenua-tion,26 and matching the NA of a fiber with light-coupling optics.17Although the developed model demonstrates good

agreement with experimental data for the absorbing-cladding fiber, it still does not provide a completetheoretical description of the evanescent-wave absor-bance spectrum. It considers only the totally re-flected part of the light in an unbent fiber andassumes equal radiation-field intensities in all opti-cal modes. Improvements of the basic model toaccommodate these realities will be discussed in afuture paper.


The authors are grateful to T. Starn andN. Sesi forhelpful discussions in different phases of the study,which was supported in part by Boehringer Mann-heim Corporation 1USA2 and by the National Insti-tutes of Health.

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Kramers-Kronig analysis of molecular evanescent-wave absorption spectra obtained by multimode step-index optical fibers