evanescent waves

Theory of fiber-optic,evanescent-wave spectroscopy and sensors

A. Messica, A. Greenstein, and A. Katzir

2274 APPLIED OPTICS @

A general theory for fiber-optic, evanescent-wave spectroscopy and sensors is presented for straight,uncladded, step-index, multimode fibers. A three-dimensional model is formulated within the frame-work of geometric optics. The model includes various launching conditions, input and output end-faceFresnel transmission losses, multiple Fresnel reflections, bulk absorption, and evanescent-waveabsorption. An evanescent-wave sensor response is analyzed as a function of externally controlledparameters such as coupling angle, f number, fiber length, and diameter. Conclusions are drawn forseveral experimental apparatuses. r 1996 Optical Society of America

1. Introduction

In the last decade fiber-optic, evanescent-wave spec-troscopy 1FEWS2 has become a popular analyticalmethod for IR absorbance spectroscopy.1 FEWS isbased on the attenuated total-internal-reflection 1TIR2effect2 and uses IR transparent optical fibers assensing elements. This method has the advantagesof fast, real-time, in situ, selective, nondestructive,and safe detection. It has been successfully em-ployed in a variety of configurations for studies ofsolid–liquid interfaces,3 chemical reaction rates,4complexmaterial curing,5 organometallic thin films,6liquid7–10 and gas11–15 detection and monitoring, con-centrationmeasurements,16–19 and biological applica-tions.20,21 Usually the sensing elements are in-stalled in a Fourier-transform IR 1FTIR2 spectrometersystem, but recently tunable lasers were incorpo-rated into FEWS systems10,22 instead of conventionalblackbody sources. This has improved the wave-length resolution, sensitivity, and detection limit ofthe method.Although there have been much progress and an

increase in published research, no complete theoreti-cal formulation of this subject exists in the literature.Although wave-optics theory for single-mode fibers

The authors are with the Raymond and Beverly Sackler Facultyof Exact Sciences, School of Physics and Astronomy, Tel AvivUniversity, Tel Aviv 69978, Israel. A. Messica is currently at theWeizmann Institute of Science, Braun Center for SemiconductorSubmicron Research, Rehovot 76100, Israel.Received 25 July 1995; revised manuscript received 24 October

1995.0003-6935@96@132274-11$10.00@0r 1996 Optical Society of America

Vol. 35, No. 13 @ 1 May 1996

has been developed to give an accurate description,this is not the case for multimode optical fibers.From the early research of Elsasser23–25 one canunderstand that an accurate solution of the waveequation for a multimode fiber is not applicable evenwhen a powerful computer is used; therefore onemust resort to a geometric optics description.Kapany and co-workers26–29 and Potter30,31 set theframe, within the framework of geometric optics, fora three-dimensional 13D2 description of waveguideproperties of optical fibers but did not considerattenuated TIR losses specifically. Because FEWSis based on the attenuated TIR effect, one mustdevelop an appropriate description for light–matterinteraction at the fiber–sample interface. Such adescriptionwas developed for planarwaveguides2,32–34and optical fibers35–38 mostly within the frameworkof wave-optics approximations. Recently there hasbeen a great amount of theoretical research sup-ported by experiments and devoted to FEWS sensors.This research gives a good description of attenuatedTIR, both by wave-optics approximations7,37 andby a two-dimensional 12D2 geometric optics ap-proach,1,8,19,22 but does not provide a full descriptionthat takes into account additional factors that affectFEWS sensors such as launching conditions.We present a comprehensive formulation of FEWS

in general and fiber sensors based on the attenuatedTIR effect specifically.2 An extended formulation isdevoted to different launching conditions that arefound to be especially important for the detectionproperties of FEWS sensors. In Section 2 a generic3D optical geometry model for bound and skew raysis formulated. Apart from various launching condi-tions the model accounts for input and output Fres-

nel transmission, multiple Fresnel reflections atboth fiber end faces, and bulk and evanescent-waveabsorption. In Section 3 we present results fordifferent sensor configurations as a function of con-trolled parameters such as coupling angle, spot size,fiber length, and diameter. In Section 4 we discussthe results and measures that can be taken tooptimize the FEWS performance. In Section 5 wesummarize the main results of this study and con-clude with some remarks regarding the model, thenumerical simulations, and the role of the model inthe design and practice of FEWS.

2. Theoretical Model

Let us consider an ideal straight, uncladded, step-index, multimode fiber, not restricted to weak guid-ance, for which theV number 3V ; 12pa@l21nco2 2 ncl224is much higher than unity. Here a is the fiberradius, l is the wavelength, and nco and ncl are thefiber core and clad refractive indices, respectively.Ray optics is a good approach to considering thesefibers, providing an intuitive and fairly accurateanalytical solution compared with the wave-opticsapproach. Our purpose is to determine the powertransmittance, guided along the fiber by bound andtunneling skew rays. We assume an arbitrary inten-sity angular distribution, I1r, g, u, w2, at the fiber-input end face. Angles u and w define solid angledV, r and g define an area element, dS, at thefiber-input end face as depicted in Fig. 11a2. Thegeneral expression for P, the total power transmitted

1a2

1b2

1c2

Fig. 1. Schematic view of ray propagation: 1a2 Propagationplane defined by g, the skewness angle, and, u, the axial angle; 1b2top view of the propagation plane; 1c2 multiple Fresnel reflectionsat the fiber end faces.

by an optical fiber30,31 of length L and radius a, is

P1L, a2 5 ee T1r, g, u, w2 p I1r, g, u, w2dSdV, 112

where T is a general transmission function whoseform depends on fiber-optic properties and ray direc-tion. T is composed of several factors, e.g., Fresneltransmissions at the fiber end faces, multiple reflec-tions from both fiber end faces, bulk absorption, andattenuated TIR. In Subsections 2.A–2.E we give adetailed description of these factors and a full repre-sentation of T at the end. Various launching condi-tions manifest themselves through I. Angles u andw have the same meaning as in spherical coordi-nates, and the coordinate system is chosen so that uis the inclination angle with respect to the axialdirection 3see Fig. 11a24. Equation 112 assumes nocoherent propagation and therefore contains no phaseinformation. The integral is over the intensity ofeach guided ray, and no possible interference istaken into account.

A. Conditions for Bound and Tunneling Skew-RayPropagation

Each guided ray is characterized by axial angle u andby g, a skewness angle between the fiber radius andthe normal at the point of incidence. These anglesdefine a plane in which a skew ray propagates.From geometry one may verify that in three dimen-sions one must modify Snell’s law for TIR in thefollowing manner30,31,36:

sin u cos g # 31 2 1ncl2@nco2241@2. 122

For g 5 0 one obtains the well-known and familiarexpression for the TIR critical angle in the 2Ddescription. The maximal value uM possible for u,the axial angle, is given by applying Snell’s law for aray that is incident on the fiber input at grazingincidence u8 5 90°:

sin uM 5 nair@nco. 132

Note that by inequality 122 in the 3D case a TIR ispossible even for angles greater than the 2D criticalangle defined by sin uc 5 31 2 1ncl2@[email protected] the allowed values for g are

0 # u # uc ⇒ 0 # g # 90°,

uc # u # uM ⇒ gm # g # 90°,142

where cos gm ; sin uc@sin u, which means that forany inclination angle in the uc # u # uM rangeskewness angle g has lower bound gm. It is clearthat if uM . uc, the integral over the solid angle isdivided into two integrals and the limits are deter-mined according to inequalities 142.

B. Fresnel Reflections at the Fiber End Faces

To take into account the total power transmission ofa FEWS system, one must consider the passage of

1 May 1996 @ Vol. 35, No. 13 @ APPLIED OPTICS 2275

rays into and out of the fiber. Therefore transmis-sion function T should contain the Fresnel transmis-sion coefficients for the fiber input and output endfaces. Because of its helical path a skew ray doesnot have a definite polarization; hence the transmit-ted power is equally weighted over the two possiblepolarizations. The Fresnel transmission coeffi-cients are determined by u, the incidence angle,irrespective of skewness angle g and by the fiber andair refractive indices, which are given by2,23,24,36

t2 5 1⁄21tp2 1 ts22, 152

where the Fresnel power transmission coefficients23are given, by Snell’s law, as

tp2 54nco cos u 3 b

1nco cos u 1 b22, ts2 5

4nconair cos u 3 b

1nair2 cos u 1 ncob22,

b ; 1nair2 2 nco2 sin2 u21@2. 162

C. Bulk and Evanescent-Wave Absorption

Bulk absorption attenuates the power carried byeach guided ray, traversing a fiber of length L andradius a, according to Beer–Lambert’s law. Eachguided ray undergoesN attenuated TIR’s.30,31 FromFig. 11b2 1top viewof thepropagationplane2 it is clear that

N 5L tan u

2a cos g. 172

The path length each ray traverses between two succes-sive internal reflections is

Li 52a cos g

sin u. 182

The total effective optical path is independent of skew-ness angle g:

Lp 5 N 3 Li 5L

cos u. 192

The expression for bulk absorption is thereforeexp12acoLp2, where aco is the bulk absorption coefficient.Attenuated TIR occurs on each reflection at the

fiber–sample interface. Note that the fiber is unclad-ded; hence the sample actually has the role of a lossy

2276 APPLIED OPTICS @ Vol. 35, No. 13 @ 1 May 1996

cladding 1as long as its thickness is greater than thewavelength2.Once again, equal weight is assumed for two

possible polarizations36:

rATR2 5 1⁄21rs2 1 rp22. 1102

Then, employing Fresnel reflection coefficients for aray that is incident on the boundary between thefiber and a lossy cladding 1i.e., sample2, we can writedown, in the local plane-wave approximation36 andin the limit of weak attenuated TIR, the evanescent-wave absorption expressions for each polarization:

rs2 5 1 2

4nconcl2 cos c1kcl

ncl2

kco

nco21nco2 2 ncl221nco2 sin2 c 2 ncl221@2

,

rp2 5 1 2

4nco2ncl2 cos c1kcl

ncl2

kco

nco2 3 12nco2 sin2 c 2 ncl22

1nco2 2 ncl221nco2 sin2 c 2 ncl221@2 3 1nco2 sin2 c 2 ncl2 cos2 c2, 1112

where cos c 5 sin u cos g and ki 5 ail@4p so thatindex i stands for core or clad and a the usualnomenclature for the absorption coefficient. An ex-plicit derivation of these expressions is in AppendixA. At this stage we are in a position to write theevanescent-wave absorption contribution toT. Thisis just 1rATR22N. This is the most important part ofthis study because all the spectroscopic informationis contained within Eqs. 1112 whereas all others maybe normalized experimentally. Because Eqs. 1112express the reflected power on each reflection at thecore–sample interface, one can intuitively regard thesecond term of the right-hand side as the powerfraction evanescently absorbed by the sample.Note that, unlike transmission spectroscopy, whenthe Beer–Lambert law is obeyed, FEWS does nothave a simple exponential form. Rather the compli-cated dependence on wavelength, absorption coeffi-cients, and refractive indices gives, in general, a lineshape different from that recorded by transmissionspectroscopy.

D. Multiple Successive Reflections at Fiber End Faces

Each ray incident on the fiber output end face ispartially backreflected. Figure 11c2 illustrates theprocess of multiple successive reflections at bothfiber end faces. Each ray propagating to and froundergoes bulk absorption and attenuated TIR losses.Considering these successive reflections, it is simpleto make the summation and write the outcome:

M1u, g2 5 1 2 3r4 3 1rATR22Nexp12acoLp242. 1122

Here r2 is the power reflected from both fiber end

faces at the multiple-reflections path and has thesame form as Eq. 1102.These reflection coefficients are taken as23,24,36

rs2 5 3nco sin c 2 1nair2 2 nco2 cos2 c21@2

nco sin c 1 1nair2 2 nco2 cos2 c21@242,

rp2 5 3nair2 sin c 2 nco1nair2 2 nco2 cos2 c21@2

nair2 sin c 1 nco1nair2 2 nco2 cos2 c21@242. 1132

Here we emphasize that, although considered, thisfactor hardly affects the total power transmission asone would a priori suspect.

E. Launching Conditions

Coupling radiation into the fiber is usually per-formed by positioning the fiber input end face in focalplane f of a mirror or a lens. The simplest launch-ing condition that we can think of we term a non-tilted fiber and a centered spot, i.e., one in which thefiber and the coupling lens are aligned on the sameoptical axis and the spot center is located at thefiber-input end-face center 1us 5 02. This launchingcondition is primarily employed in FTIR FEWS.If the focused spot has a relatively large diameter,compared with the fiber diameter, such a launchingcondition excites both meridional and skew rays.Amore general launching condition, which we call atilted fiber and a centered spot, is one in which thefiber axis makes an angle, us fi 0, with respect to theoptical axis as depicted in Fig. 2. Such a launchingcondition is primarily employed in laser FEWS.Because the focused laser spot diameter is relativelysmall, the former launching condition excites primar-ily meridional or nearly meridional rays and thesystem response might be low. Hence tilted cou-pling is employed so as to enlarge the axial angle toenhance evanescent-wave absorption. Figure 2 il-lustrates the setup that is considered in the follow-ing formulation of a general tilted coupling. Thelast launching condition we consider is a variation ofthe preceding two, namely, a nontilted fiber and anoff-centered spot. We are not familiar with reportsregarding this launching condition; nevertheless weconsider it because it enables the excitment of skewrays alone and therefore could be of great impor-tance in FEWS sensors. Here we stress that thelaunching condition has an important effect on FEWS

Fig. 2. Tilted fiber and centered-spot launching conditions: us,angle between the optical axis and the fiber axis; u8, angle ofincidence.

performance and therefore is the most importantfactor of all.Let us consider a uniformly illuminated focusing

lens and fiber positioned at its focal plane with a ustilting angle. To calculate the intensity I1u82 distribu-tion 1the prime stands for the intensity distributionin the medium outside the fiber input, usually air2 ofeach area element of the fiber input end face, weconvert the radial integration at the lens plane in thefollowing way:

I1u82du8 5 e0

2p

I0r81u8, f82dr8df8, 1142

where I0 is the intensity incident on the lens, r8dr8df8

is an area element in the lens plane, and r81u8, f82 isgiven by

r81u8, f82 5 r cos f8 6 3F1u82 2 r2 sin2 f841@2, 1152

F1u82 ;f 2 sin21u8 2 us2

cos2 u8

. 1162

The positive root is taken for 2p@2 # f8 # p@2 andthe negative one for p@2 # f8 # [email protected] that f8 is an azimuthal angle in the plane of

the lens, but it also defines the angle between theprojection of r8 in the lens plane and r, the radialposition of the incident ray at the fiber input. I1u82 isthe intensity distribution incident on the fiber-inputend face and is related to intensity distribution I1u2inside the fiber when a Jacobian transformationthrough Snell’s law is employed. Hence, given theangular intensity distribution outside the fiber-input end face, one may transform to the angularintensity distribution inside the fiber by

I1u2 5 I1u82nco2 cos u

b. 1172

Note that I1u2 does not depend on fiber length22 aslong as L # Lc > 1@aco. For fibers longer thancoupling length Lc, scattering mechanisms inducemode mixing that redistributes the direction of theray 1and power2. At that point the ideal fiber as-sumption is no longer valid, and one must take thisinto account through modification of I1u2 5 I1u, L2.Experimentally, the tilting angle may be changed

to optimize the system response in terms of evanes-cent absorption and signal-to-noise 1S@N2 consider-ations. Another important fact is that for a narrow-angular-width coupling, as is the case for laserFEWS, onemaywell approximate the incident inten-sity distribution by a constant. Last, to enhanceevanescent absorption, onemaywant to launch skewrays alone by coupling to gm # g # 90° angles.Experimentally this can be done with a spatial filterpositioned at the entrance cone. The experimental-ist who wants to avoid introducing this filter cancircumscribe spatial filtering by considering a spotthat is focused in an offset position on the fiber-input


end face; i.e., the spot center is located a distance r0from the input end-face center. Analytically thiskind of launching condition manifests as a change inthe integration limits of the r, f, and g variables.Expressions for the integration limits and areaelements of the different launching conditions aregiven inAppendix B.Note that from the launching condition point of

view, employing lasers in FEWS is far superior toconventional blackbody sources, such as those em-ployed in FTIR or other spectrometers. This is dueto the narrow beamwidth and angular distributionthat enable better control of experimental conditionssuch as sensor response or signal-to-noise con-straints.To conclude and summarize, we have presented a

general 3D model for a FEWS system. The differ-ent launching conditions, input and output Fresneltransmission, multiple Fresnel reflections, and bulkand evanescent-wave absorption, have been formu-lated explicitly, giving the main result for transmis-sion function T:

T1r, g, u, w2 5t41rATR22Nexp12acoLp2

M1g, u2. 1182

The numerical calculation of the integral defined inEq. 112 is therefore straightforward.

3. Results

Three different apparatuses, A–C in our nomencla-ture, were considered, and parameters were takenfrom experimental data.10,15,18 The integration ofEq. 112 was carried out along the following lines withan eight-point Gauss–Legendre integration scheme.The evanescent-wave absorption, A ; 2ln1Tsample@-Tbackground2, was calculated in a manner that matchesthe experimental procedure. First, the fiber trans-mission was calculated without the presence of thesample 1which we term a background measurement,i.e., kcl 5 02, and then the fiber transmission wascalculated with the sample present 1termed a samplemeasurement, e.g., for a gas or a liquid, kcl 5 finiteand small2. Evanescent-wave absorptionA is there-fore calculated as a function of externally controlledparameters and presented for the three apparatusesdescribed below. Experimental parameters15 usedfor the calculation of SF6 gas absorption were nair 51, nco 5 2.15, l 5 10.562 µm, a 5 0.05 cm21,kcl 5 0.0307, us 5 p@9.

A. Laser FEWS Tilted Fiber and Centered SpotConfiguration

For evanescent-wave sensors based on laser sources10the common practice is to launch laser radiation atangle us with respect to the fiber axis. Becauselaser sources excite a relatively low number of modeswith a narrow axial angle spread, then for the us 5 0case, the evanescent absorption response is low.Therefore it is useful to employ a tilted coupling toimprove evanescent-wave absorption. This proce-


dure, easily performed experimentally, usually im-proves evanescent-wave absorption without signal-to-noise degradation.

B. FTIR FEWS Nontilted Fiber and Centered-SpotConfiguration

For an FTIR FEWS apparatus a common practice isto focus the blackbody source ‘s radiation onto thefiber-input end face by a coupling lens with a large fnumber 1F#2, e.g., 1, to maintain a reasonable S@Nratio 1,1002. The source, the coupling lens, and theoptical fiber are all aligned to the same optical axis.Therefore this kind of coupling involves, relativelyspeaking, a wide entrance cone as does the spread inthe axial angle.

C. Laser FEWS Nontilted Fiber and Off-Centered SpotConfiguration

The third and last apparatus considered has neverbeen reported on, as far as we know. We analyze itto check the effect of skew-ray-based coupling inFEWS systems. The excitation of skew rays alonecan be achieved by focusing the beam to an off-centerposition on the fiber-input end face. We computedthe evanescent-wave absorption of an off-centeredfocused spot of a laser FEWS sensor for the nontiltedfiber case, i.e., us 5 0 and took r0 5 a@2 in allcalculations.Figure 3 presents evanescent-wave absorption as

a function of fiber length for the three differentlaunching configurations. A remarkable linear de-pendence of absorbance versus fiber length is evident.This indicates that fiber transmission is an exponen-tial decaying function of the fiber length usuallycharacteristic of transmission spectroscopy.Figures 4 and 5 are closely related because both

depict the tilting angle dependence of the two majorFEWS systems employed currently, i.e., A and Bconfigurations. Figure 4 presents the laser FEWS,

Fig. 3. Evanescent-wave absorbance versus fiber length forthree different experimental apparatuses. A, laser FEWS of atilted fiber and centered-spot coupling; B, FTIR FEWS of anontilted fiber and a centered spot; C, laser FEWS for a nontiltedfiber and an off-centered spot.

configuration A, evanescent-wave absorbance as afunction of tilting angle. Note that in this case thefiber and the coupling lens are positioned to makeangle us between their optical axes. The laser FEWSsensor exhibits a sigmoid behavior in general and aparabolic one for low values of us. An approximatelinear region exists in the 0.3 # us # 0.8 range and isimportant for practical applications when measure-ment repeatability is crucial and one wants to main-tain high precision. Therefore coupling the laserradiation at angles in this regime overcomes nonlin-earity in the sensor response to this external param-eter. In fact the coupling angle can be changedeasily and tuned as a preliminary step for S@Noptimization before measurement. Therefore itserves as a convenient external controlling param-eter and may improve system performance if onetakes into account its response curve.Figure 5 depicts a similar calculation for a FTIR

FEWS, configuration B, but now the evanescentabsorbance is calculated versus the lens F#. Herewe change the entrance cone of the radiation focused

Fig. 4. Evanescent-wave absorbance versus coupling angle for alaser FEWS sensor with centered-spot coupling.

Fig. 5. Evanescent-wave absorbance versus coupling lens F# fora FTIR FEWS apparatus with centered-spot coupling.

onto the fiber-input end face. Therefore the highvalues of F# in Fig. 5 correspond to large entrancecones and vice versa. Aparabolic dependence of thesystem response curve can be observed, which iscompatible with recent reports.19,39Figure 6 depicts the evanescent-wave absorbance

versus fiber radius for the three different configura-tions discussed. Here also we find that the laserFEWS tilted fiber and centered spot configurationfeature the best performance. In terms of sensorsensitivity both A and B configurations exhibit asimilar performance, but the detection limit of theformer is better. The laser FEWS nontilted fiberand off-centered spot, the C configuration, is inferiorto both.

4. Discussion

The pseudo-Beer–Lambert behavior observed is notintuitive because, as argued above, FEWS transmis-sion does not in general possess a simple exponentialform. The linear absorbance versus length depen-dence may be expected for laser FEWS sensors,because with these sensors all laser energy islaunched within a very narrow 1a few degrees2 en-trance cone and a relatively small spot size. There-fore rays excited inside the fiber suffer similarabsorption, effectively making the sensor obey apseudo-Beer–Lambert law. Such behavior for laserFEWS sensors has been predicted both by 2D 1Ref.222 and by 3D 1Ref. 102models. Note that configura-tion A 1the laser FEWS tilted fiber and centered spot2is superior to both other configurations in terms ofsensitivity and detection limit. The linear depen-dence of the FTIR FEWS apparatus is in contrast to2Dmodel predictions10,22 and has been verified experi-mentally.19 Although one could think that by cou-pling to skew rays alone, the C configuration, thenumber of reflections 3Eq. 1724 each ray suffers at thefiber–sample interface is increased, therefore increas-

Fig. 6. Evanescent-wave absorbance versus fiber radius forthree different FEWS apparatuses: A, laser FEWS of a tiltedfiber and centered-spot coupling; B, FTIR FEWS of a nontiltedfiber and a centered spot; C, laser FEWS for a nontilted fiber andan off-centered spot.


ing sensor sensitivity and detection limit. This ismore than compensated when the low axial angle isexcited, thus degrading sensor performance. Thismight drastically change if one couples radiation at amodified C configuration of an off-centered positionwith a tilting angle, but it may turn out to be difficultto utilize and optimize experimentally.In terms of absolute absorbance, FTIR FEWS is

inferior to laser FEWS. This is due to the largeentrance cone, because different rays suffer differentevanescent absorption. Rays excited at low axialangles suffer little absorption, rays excited at higherangles contribute more to the sensor response, butthe overall absorption is averaged over all axialangles excited, resulting in reduced absorption, hencereduced response. For laser FEWS the equivalentcone angle carries all the energy and is thereforemore efficient in terms of evanescent absorption.Moreover S@N consideration has not been taken intoaccount. If one wants to couple radiation at aspecific entrance angle with an iris or a spatial filter,the trade-off in the reduced signal intensity launchedinto the fiber will come into play and manifest itselfas a reduced S@N leading to a degraded measure-ment precision. In this sense laser FEWS sensorshave the advantage over FTIR sensors as reportedby Schnitzer et al.22 who applied a 2D model.22Needless to add is the fact that tuning the couplingangle in the laser FEWS sensor is by far easier thanreplacing the coupling lens in the FTIR FEWS.From Fig. 6 one can deduce that a significant

improvement in the sensor detection limit is ob-tained for all systems at a fiber radius smaller than350 µm. This should be taken into account in viewof coupling efficiency and convenience when oneseeks to improve the detection limit by utilizingoptical fibers of smaller diameter. Although the Cconfiguration has an approximate linear response asa function of fiber radius, both of the other configura-tions have a 1@a dependence as recently reported byKatz et al.19 This makes it beneficial to use fibers ofsmaller diameter because the overall amount ofreflection each ray suffers at the fiber–sample bound-ary is increased, hence improving sensitivity anddetection limit.To gain more insight into the subject, we find it

worthwhile to crunch mathematically integral 112 bymaking some crude approximations, obtaining aqualitative analytical expression for the absorbance.We focus on the two major FEWS apparatuses 1AandB2. Consider a laser FEWS apparatus with a cen-tered spot and tilted fiber launching condition.Even for relatively high values of us 1but still far fromuM2 the axial angle is small enough that sin u < u andcos c < sin u < u. For the centered spot launchingcondition cos g < 1. To a reasonable approximationit can be shown that the multiple reflections termM1u, g2, the Fresnel transmission coefficients, andthe bulk absorptionmay be set as constants through-out the integration and can be factored out of inte-gral 112.


If that is the case, the FEWS termmay be approxi-mated as follows:

1RATR2N , 11 2 Ku2N , exp12K

Lu2

2a 2 , 1192

where K is a constant dependent on fiber and samplerefractive indices and absorption constants.The intensity distribution inside the fiber may be

approximated as

I1u2 < C1u 2 us2, 1202

where C is a constant dependent on the beamintensity, lens focal length, fiber and air refractiveindices, and tilting angle.Therefore it is possible to write an approximate

expression for the relative transmittance of a FEWSsensor as

T1L, a2 ~ e du1u 2 us2exp12KLu2

a 2 . 1212

The integration limits are us 2 d # u # us 1 d for thelaser FEWS and 0 # u # uT for FTIR FEWS. uTdefines the entrance cone of the FTIR system and d

the angular spread for the laser system, i.e., the wideand narrow entrance cones, respectively. Thereforefor the laser FEWS system this integral may beapproximated as

T1L, a, us2 ~ d2 exp32KL1us 1 d22

a 4 . 1222

For the FTIR FEWS system one obtains

T1L, a, uT2 ~ e0

uT

du p u p exp12KLu2

a 2 , 1232

which may be approximated to a similar result.Note that this crude approximation is reasonablycompatible with the numerical results. One caneasily calculate the absorbance and obtain lineardependence on fiber length, inverse dependence onfiber radius, and parabolic dependence for the tiltingangle. This by no means is a trivial result. Usu-ally one cannot factor out the contributing terms ofintegral 112. Hence one cannot expect such a re-sponse unless small values of the tilting angle areconsidered. For laser FEWS this condition is notalways met, and indeed one can see that for hightilting-angle values this approximation fails to fol-low the sigmoid behavior that stems from numericalcomputations. For FTIR FEWS this approximationis fairly adequate even for a coupling lens with [email protected] is because the entrance cone angle, uT 5arctan1RL@F2, although relatively wide, still meetsour constraint. We have also performed a numeri-cal computation of Eq. 1102 in this spirit, i.e., discard-ing the mentioned terms in the integration andobtaining similar results with a discrepancy of a fewpercent. Therefore one can use the approximationabove for practical applications by employing a re-

verse engineering procedure to the experimentaldata with K as a fitting parameter. We emphasizethat this approximation is valid only for the experi-mental parameters characterizing the experimentalapparatus discussed and fail for an apparatus inwhich high tilting angles are involved. To conclude,let us point out that the absorbance is computed as afunction of the relative transmittance, i.e.,

A ; 2ln3 T1L, a, us, kcl2

T1L, a, us, kcl 5 024 .Therefore numerically it is possible to obtain highabsorbance values whereas experimentally the valueof the numerator must exceed the system’s noise.Therefore interpretation of the simulation resultsshould always bemade in view of S@N considerationsthat are closely connected with bulk absorption andalways present. In fact S@N and bulk absorptionlimit our capability to enhance evanescent absorp-tion with longer optical fibers. Therefore it is notpossible to reach any desired detection limit bysimply using longer sensors. Moreover the model isformulated in the geometric optics approach; thus itis not valid for fibers with radii smaller than 0.2 mm.For these diameters a local plane-wave approxima-tion breaks down.

5. Summary

We have presented a comprehensive 3D model for aFEWS sensor based on straight, uncladded, step-indexmultimode optical fiber immersed in an absorb-ing media, i.e., a sample. The model takes intoaccount contributions of Fresnel transmission coeffi-cients at the fiber end faces, multiple reflections,bulk absorption, and most important evanescent-wave absorption and the various launching condi-tions usually employed. Our model indicates that,as indicated by experiment, one may disregard sev-eral terms and focus on evanescent-wave absorptionand launching condition alone. In certain condi-tions it is possible to use an approximate expression,A 5 K1L@a2usm, for the absorbance wherem , 2 andKis a fitting parameter. Checking for conventionalexperimental setups, we find that a laser FEWSsensor with a tilted fiber launching condition has abetter response than its FTIR counterpart. Bothsetups are superior, in terms of evanescent absorp-tion response, to the off-centered spot and nontiltedfiber laser configuration. Surprisingly, both FTIRand laser FEWS sensors feature a pseudo-Beer–Lambert dependence on fiber length. 1Note that thefiber itself is not the absorbing medium.2 We findthat the evanescent-wave absorbance is inversely

dependent on fiber radius and possesses almost aparabolic dependence on coupling angle. Thesemaybe optimized to enhance evanescent-wave absorp-tion to improve the sensor performance in terms ofsensitivity and detection limit. The use of thismodel is important for understanding, design, andconstruction of an optimal FEWS sensor for use in allfields of science, medicine, and industry.

Appendix A.

The Fresnel reflection coefficient of s 1perpendicular,normal2 polarization and p 1parallel2 polarization fora ray incident on the boundary between two mediacharacterized by n1 and n2 refractive indices are

rs 5n1 sin ui 2 n2 sin ut

n1 sin ui 1 n2 sin ut

,

rp 5n2 sin ui 2 n1 sin ut

n2 sin ui 1 n1 sin ut

, 1A12

where ui is the incidence angle of a ray in a mediumcharacterized by an n1 refractive index and ut is therefracted angle in an n2 medium. Because thereflected and refracted rays are coplanar, this descrip-tion does not involve skewness angle g; thus thegeneralization of Fresnel coefficients to the 3D fiberoptic is straightforward. The coefficients above aregiven for the amplitude of the reflected wave. Thepower fraction reflected, which is of our interest, isthe absolute square of these expressions. For TIR utis imaginary but still may be expressed as

cos ut 5 i3sin2 ui 2 1n2@n12241@2

1n2@n12. 1A22

Substituting Eq. 1A22 into Eq. 1A12, we obtain theFresnel reflection coefficients for the TIR case:

rs 5n1 cos u1 2 i1n12 sin2 ui 2 n2221@2

n1 cos u1 1 i1n12 sin2 ui 2 n2221@2,

rp 5n22 cos ui 2 i1n12 sin2 ui 2 n2221@2

n22 cos ui 1 i1n12 sin2 ui 2 n2221@2. 1A32

From these equations we can understand that theabsolute value of the reflection coefficients is alwaysunity. Hence no attenuation occurs during reflec-tion, i.e., TIR. Generalizing this formalism to ac-count for losses in both media is done by assigning acomplex value for refractive index n1l2 = n1l2 2 ik1l2.The imaginary part of the refractive index is propor-tional to absorption coefficient a1l2. Making thissubstitution into Eqs. 1A32 gives the power fractionreflected for the attenuated TIR case:

0rs 02 5 01n1 2 ik12cos ui 2 i31n1 2 ik122sin2 ui 2 1n2 2 ik22

241@2

1n1 2 ik12cos ui 1 i31n1 2 ik122sin2 ui 2 1n2 2 ik22

241@202,

0rp 02 5 01n2 2 ik222cos ui 2 i1n1 2 ik1231n1 2 ik12

2sin2 ui 2 1n2 2 ik22241@2

1n2 2 ik222cos ui 1 i1n1 2 ik1231n12 ik12

2sin2 ui 2 1n2 2 ik22241@2 0

2. 1A42


Although it is not too evident, studying these equa-tions, we can figure that for incidence angles close tothe critical angle the power reflection coefficientschange abruptly. As far as low-loss fibers and weakattenuation are concerned, for incidence angles farfrom the critical angle it is possible to employfirst-order approximations to these equations andgain some insight into the attenuated TIR effect.In the following, we expand Eqs. 1A42 to first-order

terms in k. We assume a low-loss fiber and a weakattenuated TIR so that the following conditions arevalid:

ki 9 ni for i 5 1, 2,

0n2k2 2 n1k1 sin2 u 0

0n12 sin2 u 2 n22 09 1 1A52

that usually hold for FEWS measurements. The iindex that stands for incidence was omitted forclarity but should be considered as present.On expanding the square root in Eqs. 1A42, we

make use of a complex functions theory that statesthat to express the form 1a 1 ib21@2 the proper solu-tion is

1a 1 ib21@2 5 x 1 iy, 1A62

where x and y are

x 5 230a 6 1a2 1 b221@2

2 041@2

,

y 5 2302a 6 1a2 1 b221@2

2 041@2

. 1A72

In the conditions specified in 1A52 and using a first-order expansion

31 1 1ba22

41@2

< 1 1b2

2a2,

we can write the approximations for x and y as

x < 2Œa 5 21n12 sin2 u 2 n2221@2,

y < 2b

2Œa5 2

n2k2 2 n1k1 sin2 u

1n12 sin2 u 2 n2221@2; 1A82

then the final expression for 0rs 02 is

0rs 02 5 01n1 2 ik12cos u 1 i1n12 sin2 u 2 n2221@2 2

n2k2 2 n1k1 sin2 u

1n12 sin2 u 2 n2221@2

1n1 2 ik12cos u 2 i1n12 sin2 u 2 n2221@2 1n2k2 2 n1k1 sin2 u

1n12 sin2 u 2 n2221@202

. 1A92


Taking the absolute value of Eq. 1A92 and retainingonly first-order terms of k, we obtain

0rs 02 5

1n12 2 n222 2 2 cos un1n2k2 2 n22k1

1n12 sin2 u 2 n2221@2

1n12 2 n222 1 2 cos un1n2k2 2 n22k1

1n12 sin2 u 2 n2221@2

. 1A102

Rearranging the nominator by subtraction and addi-tion of the second term while neglecting the secondterm in the denominator with comparison to thefirst, we obtain the final result:

0rs 02 5 1 2

4n1n221k2

n22

k1

n12cos u

1n12 2 n2221n12 sin2 u 2 n2221@2. 1A112

In a similar manner one may approximate theexpression for 0rp 02 given by Eqs. 1112.

Appendix B.

Considering the general case of a tilted fiber and acentered spot, it is of great use to employ the polarrepresentation for the ellipse, because the projectionof the spot on the input endface is an ellipse withsemiaxes A 5 rs@1cos us2 and B 5 rs. Using a sin g 5r sin w between w and g, we write the area integra-tion needed for Eq. 112 computation in terms ofskewness angle g:

e0

arcsin1B@a2

4a cos gdg ea sin g

3A2B221A22B22a2 sin2 g41@2

3rdr

1r2 2 a2 sin2 g21@2. 1B12

The nontilted fiber and a centered spot launchingcondition is achieved by taking us 5 0. The limitsfor the axial incidence angle are determined byp

2, f8 #

3p

2:

r

f cos us# tan u8 2 tan us #

1RL2 1 r2 2 2RLr cosf821@2

f cos us

,

2p

2, f8 #

p

2:

r

f cos us$ tan u8 2 tan us .

r sin f8

f cos us

r sinf8

f cos us, tan u8 2 tan us #

1RL2 1 r2 2 2RLr cosf821@2

f cos us

.

1B22

Conversion of u8 1the incidence angle2 limits to u 1theray propagation axial angle2 is through Snell’s law.RL is the lens radius or the beam radius 1for acollimated beam incident on the lens with a beamdiameter smaller than the lens diameter2. The lim-its for f8 are 30, 2p4.Considering the case in which one desires to excite

skew rays alone by an off-centered spot 1no tiltingangle2, the integration limit for the skewness angle isdetermined by

sin gmin 5r0 sin w 2 rsr0 cos w

,

sin gmax 5r0 sin w 1 rsr0 cos w

. 1B32

Here 1r0, w2 is the vector position of the center of theoff-centered spot and rs is the spot radius.The limits of the radial integration are given by

R̃min2 5 5r0 cos w 2 3rs2 2 1r0 sin w 2 a sin g2241@262

1 a2 sin2 g,

R̃max2 5 5r0 cos w 1 3rs2 2 1r0 sin w 2 a sin g2241@262

1 a2 sin2 g. 1B42

The integration limits for the incidence angle are thesame as above, given by 1B22 with the difference thatnow the integration is performed with an areaelement:

dS 5 a cos gdgR̃dR̃

1R̃2 2 a2 sin2 g21@2. 1B52

Therefore one must transform r in Eq. 1B22 by

r2 5 r02 1 R̃2 2 2r0R̃ cos1w 2 f82, 1B62

where R̃ denotes the radius vector to the incidentposition taken from the center of the input end faceand r is the same taken from the center of the beamspot.

A. Messica thanks David and Suzan Messica forsupport. This research is dedicated to the memoryof B. Bar.

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