Mode-coupling processes in polymethylmethacrylate-core optical fibers
Jacques Dugas and Gilles Maurel
Aperture, attenuation, and far-field radiation diagrams have been studied on short lengths of commercialpolymethyl methacrylate- (PMMA-) core optical fibers. The spectral attenuation of the PMMAconstituting the core has been carefully measured and compared with the attenuation that may becalculated from optical properties of PMMA. Accurate extrinsic attenuation spectra have beenobtained. Moreover, the dependence of the far-field radiation diagram on the fiber length and on thelaunching incidence of a laser beam has been studied on this side of the mode-equilibrium length. Theanalysis of these diagrams, first performed with the Gloge mode-coupling model, has been improved withthe hypothesis that the mode-coupling processes are a result of light diffraction by structural anomalies inthe core. The average size and form of these structural anomalies has been evaluated. They may belongitudinal microcracks of the PMMA coming from stress relaxation, which occurs during thefiber-drawing process.
Introduction
Plastic optical fibers (POF) and especially polymethylmethacrylate- (PMMA-) core ones are now of generaluse. Because of their attenuations of 100 dB/km(in the better cases), they are much worse than silicafibers for light transmission; of course, they areabsolutely inadequate for long-range links. How-ever, they possess specific abilities that make themcomplementary to the silica fibers for short links.
Because of the lower price of constituting materi-als, POF's may be manufactured with large diame-ters (0.5-1 mm or larger). The use of organic glassesgives them an excellent flexibility. Because of theirlarge diameters, POF's are more easily handled thansilica fibers. They use connectors with a rougherprecision, which are consequently cheaper. At thesame time they have large apertures, so their cou-pling to light sources such as LED's is made easier,and a larger part of the emitted beam is alwayslaunched.
POF's may be used without problems for opticallinks with a length of 100 m. However, for theselengths, their principal optical characteristics cannot
The authors are with the Laboratoire de Physique des Solides,Associ6 au Centre National de la Recherche Scientifique; Univer-sit6 Paul Sabatier, 118, Route de Narbonne, 31062 ToulouseCedex, France.
be accurately defined, because at least for the betterPMMA fibers the so-called mode equilibrium is notyet completely achieved.
For these step-index optical fibers with large diam-eters and a large N.A. (e.g., 0.5), the number ofpropagating modes is large so that their distributionmay be considered as continuous. Under these con-ditions, the formalism of the geometrical optics be-comes convenient to describe the light propagationwith rays and to take into account the anomalies thatperturb their path. However, even if we use thisformalism, we shall continue to use the mode-equilibrium vocable to describe the situation wherethe evolution of the light propagation may be charac-terized by only one parameter, the attenuation coeffi-cient. In addition, we shall not consider the prob-lems of modal dispersion, because for short links,except for special cases, the bandwidth is alwayssufficient.
The aim of this research was to study how, inPOF's, the propagation goes to its equilibrium condi-tions. From there we have tried to understand theprincipal physical processes that limit their perfor-mances. The power losses in step-index optical fi-bers essentially result from absorption and scatteringin the core and scattering on the core-claddinginterface. The intrinsic losses in the core have beenevaluated in the case of PMMA and PS. The otherlosses come from impurities or defects and depend onthe manufacturing conditions. Structural defects,either in the core material or on the core-cladding
interface, modify the ray paths and cause the so-called mode couplings. The analysis of the variousprocesses of mode couplings that may be foreseennecessitates finding a way to separate what comesfrom the core and what comes from the interface.
In a previous study, Dugas et al.1 tentatively ex-plained how the rays launched with a large incidencemay be more strongly attenuated. These rays un-dergo an important number of reflections on thecore-cladding interface, where the reflection coeffi-cient is made slightly lower than 1 by taking intoaccount a nonzero attenuation for the cladding mate-rial. However, this model only gives a good agree-ment with the experimental results when attributingtoo large and nonrealistic values to the claddingabsorption coefficient. Therefore, it was necessaryto look for another process that allows us to deter-mine whether the extrinsic attenuation essentiallycomes from the core material or from the core-cladding interface.
The studies previously devoted to mode-couplingprocesses generally concern silica fibers, in which thenumber of modes is usually much smaller, and whichare often graded-index fibers.
The essential research concerning the power propa-gation in multimode fibers was done by Gloge. Inhis model, the light power is progressively spread onvarious modes by considering that modes only couplebetween next neighbors. The power fraction thatpasses from one mode to another by the unit of fiberlength is characterized by a coupling coefficient D,which may be determined by a comparison betweenthe calculated and the experimental results (see, e.g.,Refs. 3-5). However, this description does not giveany information about the physical mechanisms thatpermit the power to change the propagating mode.In addition, Gloge's calculations have been developedwith the hypothesis that launching angles 0-andconsequently the aperture-always remain smallenough to replace sin 0 by 0 and cos 0 by 1, so thequestion arises about the consequences of such simpli-fications when the angular aperture of PMMA fibersapproaches - 300.
In this study we first performed a series of originalexperiments on PMMA-core optical fibers. Theiranalysis essentially allows us, on one hand, to deter-mine the part of the core spectral attenuation thatcannot be attributed to the intrinsic optical proper-ties of the constituting material, and on the otherhand, to show that the mode couplings are a result ofthe diffraction of the light by microscopic anomaliesrandomly shared in the core. The quantitative com-parison of both series of results has led us to concludethat the extrinsic core attenuation comes from thepresence of these diffracting anomalies, which areprobably longitudinal cracks of the PMMA createdduring or immediately after the fiber-drawing pro-cess.
Experiments
Measurement Devices
Several kinds of experimental results have beenobtained. We studied the variations of the aperturewith the fiber length, the core-material attenuation at543.5 nm, its spectral attenuation, and the far-fieldradiation diagrams.
All of these measurements were performed inconditions particularly suitable for these step-index,large-aperture fibers. Launching or collecting lightbeams that have a 600 aperture is not a trivialoperation. At the least it necessitates the use ofoptical devices whose apertures are sufficient, thathave no aperture aberrations, and whose transmittiv-ity is quite independent of the incidence of the rays.The light source must also be considered as a Lamber-tian one.
To get over these difficulties, we have alwayslaunched a collimated beam with variable incidenceinto the fibers and have measured how it is transmit-ted. The variable incidence was performed by usingthe rotating mirror device previously developed in ourlaboratory by Sotom.6 The principal advantage ofthe device comes from its ability to modify thelaunching incidence without moving either the sourceor the fiber.
For monochromatic measurements, the source wasa He-Ne laser emitting in the green range- =543.5 nm-rather than in the usual red one. Thischoice was justified by the fact that the PMMA, andconsequently the fibers, presented just an attenua-tion peak at 630 nm, whereas it is much moretransparent at 530 nm.
For spectral-attenuation measurements, the sourcewas a W ribbon lamp followed by a monochromator;two microscope objectives were used to form a nearlyparallel beam from the outcoming slit.
All the light coming out from the fiber end wascollected on a Si solar cell for the aperture andmonochromatic attenuation measurements, whereasa photomultiplier tube was used for spectral-attenua-tion ones. As in this last case, the light power to bemeasured was always weak, the beam was modulatedby a mechanical chopper before it was launched intothe fiber, and the photomultiplier was followed by alock-in amplifier.
The far-field radiation diagrams were achieved bylaunching the laser beam with variable incidence.The outcoming beam was scanned by a 0.1-mm-broadslit in front of the photomultiplier, with both set onan arm that rotates about an axis situated along adiameter of the fiber end. The fiber end and the slitwere 13 cm apart, so for a 1-mm-diameter fiber theangular resolution was better than half a degree.In all cases the fibers were first cut and their endsconveniently polished, with care taken to make themperpendicular to the fiber axis. The results pre-sented here essentially concern the Mitsubishi Model
EK40 and Asahi Model TB1000 fibers, both of whichhave a diameter of 1 mm and are uncoated.
Experimental Results
Aperture MeasurementsFigure 1 shows the variations of the relative transmit-ted power versus the incidence for various fiberlengths between 1 and 50 m. From a nearly squareshape obtained for the shorter length, the curves tendto the typical bell shape expected for longer fiberswhen the mode equilibrium is reached. We observethat this evolution is faster in the EK40 fiber than inthe TB1000 one.
From these curves we obtain a value of the experi-mental aperture OM by taking the launching angle, forwhich the transmitted power is 50% of the maximumwhen the launching is normal, to the entrance fiberend. This convention is such that the value of OM,extrapolated when the length tends to zero, accu-rately fits with the theoretical aperture calculated byusing the refractive indices of the constituting materi-
120
3 1003:00_-0 80a 0
-4-
E 60V)C0
-4 40
a)
U 20a)
r I
(a)
120
a 1003:00--0 80CD
-4---4-
E 60
>
C_
au
4.-
Launching incidence(b)
Fig. 1. Variations of the relative transmitted power P(O)IP(O)versus the launching incidence 0 of the laser beam for various fiberlengths for (a) the TB1000 fiber and (b) the EK40 fiber.
als.7 The transmitted light for 0 > M is carried byleaky rays that are slowly attenuated for these shortfiber lengths.8 9
The variations of OM versus the fiber length areshown in Fig. 2. For both fibers it appears that theequilibrium length is not reached, even for 50 m.We observe that the fibers differ very slightly (lessthan 10) by their aperture limit OM(O), which is 28.50(N.A. 0.48). Such a weak difference means thatthe cladding materials are nearly the same or at leasthave similar refractive indices. However, their aper-tures OM(oo) seem quite different. Although we havenot characterized fibers that are long enough tomeasure the apertures accurately, we may say thatthe apertures are 200 (N.A. 0.34) for the TB1000fiber and only 120 (N.A. 0.2) for the EK40 one.This measurement means that if, e.g., the beamemitted by a LED is launched in such short fibers, thepower transmitted by the TB1000 fiber will dependmuch less on its length than in the case of a EK40fiber, where the quality is worse.
Core Attenuation at 543.5 nmThe attenuation of the core material (i.e., PMMA) isobtained from the comparison of powers P(l) transmit-ted by various fiber lengths when the laser beam islaunched normally to the entrance end. In this casethe beam undergoes only a few reflections on thecore-cladding interface-none, if the fiber is keptperfectly linear-and its attenuation essentially re-sults from the core-material absorption and scatter-ing, whereas the effects of the core-cladding interfaceare virtually removed. This behavior is well verifiedby drawing 10 log[P(1)/Po] versus the fiber length 1,where lo is one of the characterized lengths that istaken as a reference (1 m for instance) and for whichthe transmitted power is Po.
We see in Fig. 3 that the representative points arecorrectly lined up, showing that this attenuationfollows a characteristic Beer-Lambert law. It is
Fig. 2. Variations of the angular aperture OM (measured from thelaunching incidence for which the transmitted power is 50% of thetransmitted power at 0 = 0) versus fiber length.
Fig. 3. Variations of the maximum transmitted power when thelaunching incidence is normal versus the fiber length. The slopedirectly gives the core-material attenuation coefficient act,.
perhaps important to recall that such a behaviorwould not be observed for a beam that would not belaunched normally into the entrance end and afortiori for an uncollimated beam, because for theseshort lengths the mode equilibrium is far from beingreached and the leaks do not follow an exponentiallaw as long as the aperture of the transmitted beamvaries. The slope of these lines gives 110 dB/km forthe TB1000 fiber and 202 dB/km for the EK40 fiberfor the laser wavelength of 543.5 nm. These valueshave to be attributed to the attenuation coefficient ctof the core materials of these fibers, and not to thefibers themselves.
Spectral AttenuationSpectral attenuation was obtained in the same way asmonochromatic attenuation by measuring the trans-mitted P(X) between X = 400 nm and X = 700 nm forvarious fiber lengths and by comparing them. How-ever, because of the lack of accuracy in these measure-ments where the signals were weak, the curves ot.(X)were eventually shifted to fit with the values mea-sured with the laser at X = 543.5 nm. These curvesare given in Fig. 4. We emphasize that these curvesessentially represent the losses in the core materialsand that the effects of the losses at the core-claddinginterface are negligible. As we expected, it appearsthat the spectral core attenuation for the TB1000fiber is lower than for the EK40 fiber. Furthermore,the difference between these attenuations becomesmuch more important on the short-wavelength side,indicating that the scattering phenomena are perhapsmore important in the EK40 fiber.
Far-Field Radiation DiagramsFrom geometrical optics considerations we expectthat, far from the mode equilibrium, when the laserbeam is launched into the fibers with an incidence 0(lower than the fiber aperture OM), the transmittedlight is regularly shared on a cone. The half-vertexangle of the cone is equal to the launching angle.
Fig. 4. Spectral attenuation *t(A) of the core material.
On a screen parallel to the outcoming fiber end, aring is observed. However, even for the shortestfibers, it appears well resolved only when the launch-ing angle is larger than approximately 50 or 6°. Forlower incidences, an inhomogeneously lighted spot isgenerally observed even for long fibers. Such aphenomenon is not inherent in these fibers: It isalso observed with short-silica-silicone fibers thathave the same diameter. For higher incidences, thedependence of the angular width of the ring on thelaunching incidence and the fiber length has beensystematically studied. For every incidence, the ra-diance of the outcoming beam was measured byscanning it with the arm bearing the slit and thephotomultiplier. When a ring is analyzed in such away, the representative curve exhibits two nearlyidentical peaks (Fig. 5). Their peak-to-peak angulardistance fits correctly with twice the launching angleof the laser beam, as long as this angle is lower thanthe fiber aperture.
Measured80r depends
1.1
1.0
0.9
0.8
j .7
,0.6a)o 0.5C:
S 0-4
( 0.3
0.2
0.1
0.0
at half-height, the peak angular widthon the launching angle and the fiber
-30 -20 -10 0 10 20scanning angle (deg )
30 40
Fig. 5. Example of the record of the far-field radiation diagramgiven by a 1-m-long TB1000 fiber when the launching angle of thelaser beam is 23°,
length. It is always larger than the expected valuefrom the ratio of the fiber diameter-fiber-end slitdistance. It tends to the corresponding value onlyfor the ring given by a good silica-silicone fiber (insuch a case, we measured 0.70 instead of 0.50). For agiven fiber length that is short enough, (approxi-mately 1 or 2 m), it appears (Fig. 6) that the ringwidth 0
r decreases when the launching incidence 0increases. For any given 0, in contrast, this widthincreases with the fiber length. Figure 7 shows thevariations of 3 0
r versus the fiber length when thelaunching angle is 200; we shall later see the reasonfor this choice. It appears that the ring is alwayswider for the EK40 fiber than for the TB1000 one.Furthermore, the fluctuations of the results concern-ing the former fiber are more important. We alsoobserve that the width extrapolated for a fiber lengthtending to zero takes the same value for both fibers(=- 2.7°). When the fiber is too long the peaks can nolonger be resolved, because their width becomes toolarge and they overlap.
Such a behavior comes from the mode-couplingprocess that gradually modifies the propagation condi-tions and scatters the beam. The question nowconcerns the physical origins of this scattering. Dothey come from the propagation in the core materialor from the reflections on the core-cladding inter-face?
Discussion
Aperture
The major component in our observations is thedifference in the behavior of the experimental aper-tures with the length of both fibers. At first sight itmay be supposed that when the launching incidencebecomes larger, the beam interactions with the core-cladding interface play a more important role.Therefore, the faster decrease of the aperture of theEK40 fiber would be attributed to the poorer qualityof its core-cladding interface, where more important
01
_0 5
.1
C_34
. _01
C
5 10 15 20 25launching angle (deg )
3.0 .5
Fig. 6. Variation of the angular width of the ring versus thelaunching incidence 0 of the laser beam for a 1-m-long TB1000fiber.
15.0
12.5
' 10.0
= 7.5-o
cn 5.0. _
2.5
0.0l 5 10 15 20 25 30 35 40Fiber length (m)
45 50 55 60
Fig. 7. Variations of the angular width of the ring versus the fiberlength. Filled circles and triangles represent the experimentalvalues and curves were calculated by using either Gloge's model formode coupling or a diffraction process.
mode couplings occur and hasten light leaks. Themode equilibrium is reached faster but at the cost ofmore important losses.
Core Attenuation
In our first step we tried to determine the origin ofthe difference between the core attenuations of thetwo fibers that are manufactured with the samematerial, PMMA. First we must consider that thepart of the attenuation that has an intrinsic origin isidentical in both fibers: it is a result of the absorp-tion and the Rayleigh scattering in the PMMA. Todetermine the extrinsic part of the attenuation, wehave calculated the intrinsic spectral-attenuationcurve that gives the better fit with both experimentalones. We used Kaino's model' 0"' for the contribu-tion of the high harmonics of CH absorption andRayleigh scattering. However, the contribution ofthe harmonics of the stretching vibrations and thoseof the combinational band of bending vibration wereestimated again to improve the fitting.
In Table 1 we have reported the experimentalpositions and intensities of the various stretchingvibration peaks and stretching-bending vibrationpeaks, respectively. They have been chosen so thatthe fit is as good as possible. In such a model, the
Table 1. Experimental Positions and Intensities of VariousVibration Peaks
Coefficients (cm-') This Research Kaino This Research
A 2966 3005 3818B 53.1 53.5 172
successive absorption peak wavelengths are given bythe law'2
= 107/(An - Bn2),
where Xn is expressed in nm and the empiricalcoefficients A and B are in cm-'. The best-fittingvalues of the wavelengths are reported in Table 2.The successive peak intensities versus the quantumnumber n are given by the law
I(n) = C exp(- m),
and the best-fitting couples C and r are gathered inTable 3.
To build up the theoretical spectrum we have beenlead to give a Gaussian form to the absorption peaks,every one of which has the same width of 26 nm.The Rayleigh scattering was taken into account byadding the contribution given by Kaino' 0"' to theattenuation:
aR(X) = 13(633/X)'.
We did not find any other expression or coefficientsthat allowed us to obtain a better fit (Fig. 8).
It is interesting to substract this calculated spec-trum shown in Fig. 8 from both of the experimentalones, thus obtaining the curves given in Fig. 9.These curves represent the extrinsic attenuation ofthe oriented PMMA, which constitutes the fibercores. It is quite surprising to observe, especially inthe case of the TB1000 fiber, that the residualspectrum is flat along the whole investigated wave-length range. Its remaining fluctuations essentiallyresult from the experiment.
A slight increase appears on the short-wavelengthside that cannot be reduced by modifying the Ray-leigh scattering contribution. For the EK40 fiber,this increase is much more important. However, inthe range 500-700 nm, the curve is nearly flat andquite parallel to the TB1000 fiber. The spectra aresplit by 80 dB/km. The extrinsic core attenua-tions remain at approximately 50 and 130 dB/km forthe TB1000 and the EK40 fibers, respectively, and weshall remember that in this wavelength range their
Table 3. Best-Fitting Values of C and r Coefficients
ratio is approximately 0.38. Therefore, it appearsthat EK40 fibers are worse than TB1000 fibersbecause of the lack of high quality of the corematerial, the core-cladding interface, or both simulta-neously. However, the second defect essentially af-fects the rays launched with a large 0 as observed bythe aperture variations, whereas the beams launchedfairly normally are only affected by the core.
The problem is to find an interpretation of theorigin of these core extrinsic attenuations. Such flatcurves are not helping us to determine it. We do notobserve any characteristic peak that might be relatedto the presence of a specific impurity. Along theselines, the role of water was investigated by Kaino.13It seems quite negligible in this wavelength range.The eventual incidence of core-diameter fluctuationshas also been foreseen. It is quite negligible:Kaino13 showed that the associated losses reach only10 dB/km when the diameter fluctuations exceed10%, which never happens in these commercial fibers.Consequently, it may be thought that these lossescome from structural anomalies dispersed in the core
E00
C4000
a 300
-4-
>1200
U) 100
q0 450 500 550 600 650 700wavelength (nin)
Fig. 9. Extrinsic core attenuation spectra calculated by substract-ing the curve of Fig. 8 from the measured spectra of Fig. 4.
material that induce mode coupling. The results ofthe analysis of the far-field radiation diagrams agreewith this hypothesis.
Far-Field Radiation Diagram Analysis
In the far-field radiation diagram analysis we areconcerned by any incidence, so a priori, the core aswell as the core-cladding interface are liable to act inthe broadening process. We have seen that theangular width 8Or of the ring varies with the fiberlength and the launching angle 0. In this last case, itis interesting to observe the behavior of the productof the peak intensity IM by its width 8Or and by sin 0,which is proportional to the ring radius. This prod-uct is expected to be proportional to the total transmit-ted power. For short fibers and as long as 0 < OM,this total power is approximately independent of thelaunching angle (Fig. 1). The variations Of IM80r sin0 are represented in Fig. 10 for a 1-m-long TB1000fiber. We observe that this product is really roughlyindependent of 0 as long as 0 does not exceed 250,where the effects of the losses at the limit aperture OMbegin to be felt. Therefore, because the rings arenarrower as the launching incidence becomes larger,it may be considered that the mode coupling is weakerfor the higher modes.
Inadequacy of Gloge's Model
To get quantitative characteristics from these far-field radiation diagrams, we used Gloge's model2 tosimulate the effects of mode couplings on the flowpower in these highly multimode fibers. To set theequation that expresses the evolution of the powerdistribution between the various modes as a functionof the distance covered by the light in the fiber, Glogefirst considered that because of the cylindrical symme-try of fibers, all the modes that correspond to thesame propagation internal angle Oi have the samebehavior. Gloge showed that the power distribution
Fig. 10. Evolution of the product (far-field diagram peak inten-sity x angular width x sin 0) versus the launching angle 0 for a1-m-long TB1000 fiber, showing that the total transmitted poweris independent of the launching angle as long as this angle does notexceed 250.
P(0i, z) per unit of solid angle and at the distance zobeys the differential equation
aP(0i, z)
=z -(0i)P(0i, z)+ 802 a 0id(0i) aP(O, Z)]
where a(Oi) is the attenuation coefficient of the modes and d(0) is the coupling coefficient of the Oimodes with their next neighbors. Therefore, he alsosupposed that the modes only couple between proxi-mal neighbors and that all the power is lost as soon asO > OMi where OMi is, of course, the internal anglecorresponding to OM (we know that this last proposi-tion is not true for leaky modes in short fibers).Here 80i is the angular separation between next-neighbor modes. It is constant and equals X/4an,where a is the radius of the core and n is its refractiveindex. In our case, we have considered that theattenuation coefficient a is the same for all the modes,so it is not necessary to take it into account becausehere we are not concerned with the dependence of theattenuation on the length.
In the same way, to simplify the calculations wehave considered, as Gloge previously did, that thecoupling coefficients d(0i) are given by their first-order parts and are taken as a constant D. Withinthese conditions, the equation to solve is reduced to
aP(0i, z) D a [ OP(0i, z)]
az 0, aoi a0i
Its solution has been achieved by using the numericalmethod previously implemented by Rousseau andJeunhomme.5 It involves a segmentation of theranges of the variables Oi and z so that at each point ofthe buildup grid the derivatives of P(0i, z) are approx-imated by the corresponding finite differences.
To start the calculations, we find it necessary tointroduce initial values P(0i, 0) of the launched beam.Following Rousseau and Jeunhomme,5 we have takena Gaussian distribution,
P(0i, 0) = exp - 4 n(2) 60
where O0j is the internal launching angle of themaximum of the beam with the half-height width 800.We should hope that it agreed with the real distribu-tion of the launched laser beam. Rather than di-rectly using the coefficient D, we find it convenient tointroduce the dimensionless coefficient
ApL= (2/D)(AO2/A__),
where AO and Az are the segmentation steps of Oi andz, respectively.
Figure 10 shows an example of the evolution of thepower distribution in the ring when the launchingincidence 0 is 200 and OM = 280. The initial beam
width was o00 = 1.50, as we may see on the first peak.A new curve is obtained every time a new step isperformed on the variable z, and the peak intensitydecreases on each turn while its width increases.However, to make the comparison easier, we normal-ized every peak before using it for the calculation ofthe next one. These successive calculations wereperformed with p. = 0.5 so that the evolution of thepeaks would be slow enough. For the sake of clarity,only every 40th peak has been drawn in Fig. 11. Thelast one is thus the 400th. Many more runs wouldbe necessary to observe the peak overlap, whichmeans that the ring is no longer observed and themode equilibrium would eventually be evidenced stillfarther.
To show the limits of Gloge's model, we havestudied how the calculated ring width 80r varies with:the fiber length for various values of the coefficient p.(i.e., D, Fig. 12), the initial width 800 (Fig. 13), and theincidence 0 of the launched beam (Fig. 14), respec-tively. We observe that the coupling coefficient D(i.e., p.) has a major effect on the ring-width evolutionwith the fiber length z, whereas the width 800 of thelaunched beam mainly acts on 0
r for short lengthsand does not modify the shape of the curves 0r(z).Finally, the most important result is that as long asany peak overlap does not occur (it occurs, of course,the smaller the 0, the sooner),- the launching inci-dence 00, has no effect. This behavior does not agreeat all with the experimental results, meaning that inGloge's model the choice of a coupling coefficientindependent of 0 is not sufficient to describe thecoupling processes that appear in these fibers.
Nevertheless, setting this problem apart for themoment, we have determined the values of p and B0ofor which the best agreement was obtained with theexperimental variations of 5r(Z). This fit was per-formed for the launching incidence 0 = 200 because,about this value, 80r is weakly dependent on 0. Thisfit is shown in Fig. 7. It has been obtained by takingAO = 0.1° and Az = 0.1 m. It leads us to the
1.1
1.0
0.9
CD 0.8
0.7
-0 0.6N- 0.50E0.4L-
z 0.3
0.2
0.1
10.00 15.00 20.boOutput angle (deg )
1716
15-14- 0.413 0.5
Fig 12. -hoeia evlto ofteaglrrigwdhi6h
U)11- :
9
3
far-field diagram in Gloge's model of mode coupling versus the fiberlength for various values of the p.L coefficient.. The launched beamhas an incidence of 200 and its initial width is 1.50.
following parameters: D = 0.38 sr/in for the TB1000fiber and 0.96 sr/in for the EK4O one. For thelatter fiber the results are much more inaccurate.The initial width b0o has to take the value 1.50 in bothcases, because when extrapoling both curves to azero-fiber length, they appear to come from the samevalue of 8(3r.
Two extra comments have to be made here. Thefirst one is in favor of Gloge's model, whereas thesecond one is unfavorable. First, the ratio of thevalues of D is approximately 0.39; it is comparablewith the ratio of the values of the extrinsic attenua-tion of the core materials as they were found earlier.Second, the value of b0o does not fit with the angulardivergence of the launched laser beam, which is muchweaker ( m 0.05°).
Therefore Gloge's model for mode coupling in theseshort fibers, far from the mode equilibrium, appears
13
12
01
a)
-+,
01
.L-
Fig. 11. Theoretical progressive broadening of transmitted ringin the far-field diagram in Gloge's model of mode coupling. Thelaunched beam has an incidence of 200.
11I
10
9
8
7
6
5
4
3
2
10.00 20.00 30.00Fiber length (a.u.)
40.00
Fig. 13. Effect of launched beam angular width (10, 1.5, and 2)on the theoretical variations of the angular ring width in thefar-field diagram in Gloge's model of mode coupling versus the fiberlength. The launched beam has an incidence of 200 and p. = 0.53.
Fig. 14. Effect of the launching incidence of the laser beam on thetheoretical variations of the angular ring width in the far-fielddiagram in Gloge's model of mode coupling versus the fiberlength. The launched beam has an angular width of 1.5° and ,u =0.53.
only to be able to give a good estimation of theevolution of the power angular distribution of a beamwith the fiber length for a given launching incidence.It may be thought that the observed discrepancy,when the launching angle varies, could be correctedby the introduction of a more general form of D(0) inGloge's equation. However, its solution would thenbecome complicated because it seems to be difficult tosuggest the mathematical form of D(0). Further-more, the fitting of the ratio of the D values obtainedfor both fibers with the ratio of the extrinsic attenua-tion of their core leads us to conclude that the modecouplings occur in the core rather than at the core-cladding interface. Consequently, these extrinsiclosses essentially originate from defects in the core.
The clash between the value of the initial angularwidth 8Oi, which has to be given to the beam, and thedivergence of the laser beam actually launched ismore troublesome. It may not be considered thatthe laser beam is broadened when it passes throughthe entrance fiber end. This fiber is always planedand polished as carefully as possible so that noanomaly can perturb the beam too much.
Diffraction Model
Taking into account the previous remarks and search-ing to determine the physical origin of the modecouplings in these fibers, we put forward the follow-ing hypothesis. In the PMMA that constitutes thefiber core and that has been drawn during themanufacturing process, structural anomalies existthat diffract light. An initially parallel launchedbeam is gradually spread out by successive diffractions.For fibers that are too short, the number of interac-tions between the beam and these anomalies is notsufficient to permit the beam to become uniformlybroadened: The ring does not have a regular width.To observe a regular ring, we find it necessary for allthe constituting infinitesimal beams to meet at least
one anomaly. This occurence depends on the anom-aly density. In the TB1000 fibers, it is such that thelength between 1 and 2 m is necessary. This mini-mal length is smaller than 1 m for EK40 fibers.
Because for a given fiber length the beam broaden-ing is higher when its launching incidence is lower,i.e., when the number of reflections on the core-cladding interface is lower, we think that theseanomalies are not on this interface. They are, on thecontrary, randomly distributed in the core material.Furthermore, to explain the dependence of this broad-ening on the launching incidence 0, we consider thattheir apparent dimension is lower perpendicularly tothe fiber axis. Therefore, they may be considered tohave an oblong form with their axis lying along thefiber axis, i.e., along the drawing direction. Theanomalies may then be oriented impurities, dusts, ormore probably, longitudinal internal microbreaks ormicrobubbles in the material.
To estimate in a simple way the average dimensionsof these defects, we suppose that they have the formof small identical rods, parallel to the fiber axis.Then we consider the case of a fiber with a length thatis short enough so that only one interaction hasoccured. We have calculated the diffraction effect asa function of the launching angle 0 for an initiallycollimated beam. Then the internal angle Oi corre-sponding to 0 is also the angle between the beam andthe rod axis. Because of the axial symmetry theazimuthal effects cannot be observed in the ring, andwe only need to perform calculations in a meridianplane. Within this condition the diffracted intensityin the direction, which makes an angle P13 with theincident direction 0, was calculated as resulting fromthe Fraunhofer diffraction by all the points belongingto the cylinder that constitutes the rod. This inten-sity is proportional to
(sin2 u/u2 )[2J(v)Iv] 2 ,
where
u = (QrLn/X)[cos(0i + 3i) - cos il,
v = (2'wpn/X)[sin(0i + 1i) - sin Oi].
Here J, is the first-order Bessel function and L and 2pare the length and the diameter of the rod, respec-tively.
For various values of 0, the variations of theintensity have been determined when varying 1i andhave been expressed versus the corresponding exter-nal angle 1:
n sin(0i + 3i) = sin(0 + 13).
The angular width of the principal diffraction maxi-mum was calculated at half-height and comparedwith the experimental ring widths corresponding tothe same 0 and given by a 1-m-long TB1000 fiber.A correct agreement was obtained (Fig. 15) when the
Fig. 15. Comparison of the calculated (curve) and experimental(triangles) values of the angular ring width in the far-field diagramversus the launching incidence of the laser beam in the model ofthe diffraction by 37-pm-long and 7-pim-diam. rods.
rods were given the following average dimensions:
L = 37 1 pm, p 2 ,5 0.3 pim.
For the smaller values of 0 the diffraction maximumis too broad to permit the observation of a ring,whereas for the higher values the role of the system-atic measurement errors caused by the finite dimen-sion of the beam at the fiber output end explains theincrease of the discrepancy between the experimentalpoints and the calculated curve.
The evolution of the angular width of the ring withthe fiber length is calculated from the same hypothe-sis in the following way. The beam, broadened by afirst series of interactions, is used to calculate theeffects of a new series of diffractions on identicalobjects. We take into account their random distribu-tion and suppose that no phase coherence existsbetween both series of interactions.
As the calculations are performed the former diffrac-tion peak is angularly resolved in a juxtaposition ofrectangular peaks that are separately diffracted, andthe intensities given by each one are added up foreach 13 direction. A new, broader, diffraction peak isthus built up. The process is repeated as long asnecessary. We determine the number of runs perunit of fiber length necessary to obtain widths that fitwith the experimental values for every length of fiber.
The curves shown in Fig. 7 are comparable withthose that have been obtained from Gloge's mode-coupling model. They have been obtained by using0.53 run/m and 1.36 run/m for the TB1000 and theEK40 fibers, respectively. These characteristic pa-rameters may be considered as proportional to thenumber of defects by unit of fiber length. Theirratio, equal to 0.39, is still identical to the ratio of theprevious mode-coupling coefficients D, and conse-quently to the ratio of the core extrinsic attenuations.This fact confirms that these diffracting defects wouldbe essentially responsible for the core extrinsic atten-uation.
The effects of wavelength might be taken intoaccount in the diffraction model. On the wavelengthrange where the spectral attenuation has been deter-mined, it is roughly estimated that the ring broaden-ing would be 25-30% higher at 700 nm would besmaller in about the same proportion at 400 nm.The sensitivity of our device was not sufficient toallow us to carry out any direct measurement of theangular width of the ring at any wavelength otherthan the laser one. However, if we- consider that theextrinsic attenuation is likely to come from diffrac-tion, we can observe in Fig. 9 that this attenuationseems, for the EK40 fiber, to undergo a slight in-crease on the long-wavelength side in a proportioncompatible with the estimated 25-30%. On theshort-wavelength side, the expected decrease, if any,is screened by the effects of the fast increase of theattenuation at the shorter wavelengths and for whichwe have no explanation (the increase could be a resultof scattering, foot of the electronic absorption, orimpurities).
What Is the Origin of the Diffraction?The important problem that arises is to know thenature of these defects that cannot be directly ob-served. Some preliminary experiments have shownthat fiber density increases when the fibers are agedby long heating at 85 0C, although their apparentsize, determined from the ring angular width given byshort fibers, does not change. Therefore the defectscannot be impurities such as dust or bubbles. Thedensity or the size of the impurities cannot change; itis probably the size rather than the number of theimpurities that is modified during such an agingprocess. We think that the defects are structuraldefects of the PMMA, such as longitudinal micro-breaks or cracks, that are induced during the fiber-drawing process and are promoted by the relaxationof stresses.
During the manufacturing process, the fibers areindeed quenched when their temperatures steeplycross the PMMA glass transition (Tg 105 0C). Themolecular chains remain preferentially oriented fol-lowing the drawing direction, i.e., the fiber axis, andthere are less entanglements in the radial directions.The PMMA in the fibers is not in its usual amorphousand isotropic equilibrium state. The fibers present aslight uniaxial birefringence and have an anomalousrheological behavior. When the fibers are forciblybent, they are less easily snapped than amorphousPMMA-made threads or rods that have the samediameter would be. Their mechanical resistance isworse in a radial direction than in an axial one. As aresult of the relatively weak thickness of the cladding,its role can be considered as quite negligible, and thefiber behavior is essentially the behavior of the PMMAcore. This PMMA tends to evolve to a more stablestate even at room temperature by stress relaxations,whereas small cracks appear in the directions wherethe mechanical resistance is weaker. This evolutionis supported by an increasing temperature without
i. . . . . . . . . . . . . . ------------l l l l l l l g 4 | x { s l l l .......... .
stepping over Tg, where the material begins its soften-ing that is accompanied by an important shortening(until 50%) of the fiber. We suggest that thesecracks would be responsible, on one hand, for theexcellent flexibility of the fibers. However, on theother hand, they cause an important increase in fiberattenuation.
Conclusion
The specificities of the plastic-core optical fibers maketheir use particularly interesting for short links.Their important attenuation limits their uses tolengths for which the mode equilibrium is notachieved. Therefore, it is not possible to character-ize their aperture or their attenuation accurately in away that would be independent of their length and ofthe form of the beam to be launched.
We have systematically studied how an initiallyparallel beam is transmitted and broadened throughPMMA-core fibers on this side of their mode-equilibrium length as a function of its launchingincidence and the fiber length. When comparing theexperimental results given by two series of commer-cial fibers, we find that the good agreement observedbetween the ratio of the extrinsic attenuations of thecore PMMA's and the ratio of the mode-couplingcoefficients obtained from Gloge's model as well asthe ratio of the densities of the diffracting defects,which we have introduced to explain the evolution ofthe far-field diagrams, allows us to propose that theextrinsic losses are essentially caused by mode cou-plings that occur inside the core material rather thanat the core-cladding interface.
We believe that if the density of such diffractingdefects (we think that they are microcracks prolatealong the fiber-drawing direction) is minimized, theattenuation might go down to 50 dB/km. How-
ever, one would then risk the fibers becoming morebrittle. Perhaps an arrangement could be made inthe manufacturing process to determine the optimaldensity of cracks that allows us to improve the opticalattenuation while keeping acceptable mechanical prop-erties for the fibers.
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