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[ 101 W. R. King, Maximum entropy spectral analysis in the spati alarray data, Geophys ic s ,vo l . 39 , pp. 843-851, Dec. 1974.[ 111 L. J.Griffiths,Rapidmeasurement of digital nstantaneousdomain, Naval Res. Lab. Rep. 8298, Mar. 1979.frequency, IEEE Trans Aco ust., peech , Signal Processing,[12] D. R. Morgan and S. E. Craig, Real-timeadaptive inearpre-vol. ASSP-23, pp. 207-222, Apr. 1975.diction using the leastmeansquaregradientalgorithm, IEEETrans. Acoust . Speech, Signal Processing, vol. ASSP-24, pp. 494-507, Dec. 1976.[ 13 1 M. A. Alam,Adaptive pectralestimation, n Prm. 1977J o i n t Au t o m a t i c Co n t r o l Co n $ ,une 1971.[ 141 A. Van den Bos , Alternative interpretation of maximum entropyspectralanalysis, IEEE Trans. In form. Theory , vol. IT-17, pp[ 151 P. W. Howells, Exp lorat ions in fixed and adaptiv e resolution atGE and S U RC, I EEE Trans. AntennasPropaga t . , vol. AP-24, pp.575-584, Sept. 1976.[ l a ] S. P. Applebaum, Adaptive rrays, IEEE Trans. AntennasPropaga t . ,vo l . AP-24, pp. 585-598, Sept. 1976.[17] I. S. Reed, J . 0. Mallett, and L.E. Brennan, Rapid convergencerate in adaptive arrays, EE E Trans.Aerosp . Elec t ron . Sys t . ,vol.[ 181 W. F. Gabriel, Adaptive arrays-An intr oduc tion , Proc. IEEE,

493-494, July 1971.

AES-10,PP. 853-863, NOV.1974.

[ 191 0. . Frost, An algorithm for linearly constrained adaptive arrayvol. 64, pp. 239-212, Feb. 1976.[ 20 ] S.P. Applebaum and D. J.Chapman, Adaptive arrays with main-processing,Proc. IEEE, vol. 60, pp. 926-935, Aug. 1912.beam constraints, IEEE Trans. Antenn as Propag at., vol. AP-24,pp. 650-662, Sept. 1976.[ 2 l ] G .V. Borgiotti and L. J. Kaplan,Superresolution of uncorrelatedinterference sources by using adaptiv e array echniques, IEEE1221 M. A. A l a m , Orthonormal attice iiter-Amultistage,multi-Trans.Antennas Propagat . ,vol . AP-27, pp. 842-845, Nov. 1979.channel estimation echnique, Geophys ic s , vol. 43, pp. 1368-1383, Dec. 1978.[ 23 ] B. D. Steinberg, Pnnciples of Aper ture & A m y S y s te m D esignNew York : Wiley, 1976, reference ch. 10.I241 h o c . R AD CS p e c t n r m Es ti m cl ti on W o r ks ho p (Rome Air Develop-ment Cente r) Griff= AFB, NY, Oct . 1979.I251 J . E.Evans, Aperture sampling techniques for recision directionfinding, IEEE Trans. Aerosp . Elec tron. Syst . , vol. AES-15, pp.I261 W. D.White, Angular spectra in radar applications, EEE Trans.891-895, Nov. 1919. See also pp. 899-903.[27] L. E. Brennan, J . D. Mallett, and I. S. Reed, Adaptive arrays inAerosp. Electron. Syst.,vol.AES-15, pp. 895-899, Nov. 1979.airborne MTI radar, IEEE Trans. Antenn as Propag at., vol. AP-24 , pp. 601-615, Sept. 1916.

Index Profile Measurements of Fibers andTheir EvaluationDIETRICH MARCUSE, FELLOW, IEEE, A N D HERMAN M. RESBY

Invited Paper

Ahmct-lhe refractive index distriiution in the core of amulthnodeoptical-fiber waveguide playsan mportantrd e in determining he t r a mmissionproperties of the guide. be doser the index profile is to therequired i d 4 d i d i t i o n , the great- the resulting nformation ary-ing capacity of the fik. Ihis review paper discosaes methods formeasuring the refractive index distnbution n optical bbers and for pr edicting their impulse response and signnt bandwidth from the measuredprofiles. Some attention is also given to preform and dngle-mode fiberprofiling.

Manuscript received January 3, 1980; evised March 4,1980.The authors are with Bell Laboratories, Crawford Hill Laboratory,Holmdel, NJ 07733.

I. INTRODUCTIONPTICAL FIBERSare an attractivemedium or rans-mitting light signals that are used as carriers of widebandcommunications [ 1 . Fibers werebeginning toattract the attention of communications engineers when theirlosses were reducedfromapproximately1000dB/km to 20dB/km [21 . One decade after his achievement their losseshave been reduced by two more orders f magnitude; losses aslow as 0.2 dB/km have been reported in the literature [ 3 1 .As their losses decrease, fiber cable spans between repeaters0018-9219/80/0600-0666$00.75 1980 IEEE

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MARCUSE AND PRESBY: INDEX PROFILE MEASUREMENTS FOR FIBERS 66 1

can bemade even longerand-insome iber ypes-becomelimited less by fiber loss than by the bandwidth of the signalthat can be transmitted through the fibers.Optical ibersare basically of two ypes 141: Single-modefibers which can support only the HEll-mode, albeit in twomutuallyorthogonalpolarizations, nd have thepotentialfor ransmitt ing signals with bandwidths n the hundreds oreven thousands of GHz . km range. However, because of theirsmall core diameters single-mode fibers are somewhat difficultto workwith.Theymustbe excitedwith laser light andsplicing becomesa challenging problem. For his reason theother fiber type, multimode fibers, are a prime considerationfor applications where signal bandwidth is limited t o below athousand MHz * km. In this article we are primarily concernedwith multimode fibers.In multimode fibers the signal bandwidth is critically depen-dent on the refractive index distribution of the fiber core [5 1 ,[61 . With ideal refractive index distributions multimode fiberscan, in principle, have signal bandwidths on the order of tenGHz .km. However, extremely careful control of the indexprofile is required to achieve this theoretical goal. It is clearthat precise methods for measuring index profiles are requiredif the desired ideal index profiles are to be produced. This isespecially so sincemode-couplingeffects [7 ] dueeither toindex fluctuation or microbending can substantially ncreasethebandwidth n a generally unpredictablemanner.Thusinferring tha t a fiber possesses a good index profile becauseit exhibits a large bandwidth can, in general, e misleading.It is the purpose of this paper to providean overview ofmethods or measuring efractive index profiles of opticalfiberswithsome attention also given to he preforms romwhich the fibers re rawn.Thehape of the refractiveindexprofile, the diameter of the guide, and he loss anddispersion of the material from which the fiber is made com-pletelydetermine heproperties of lightpropagation n thefiber.Thus it is possible to predict theperformance of thefiber in the absence of mode coupling once its index profile isaccuratelyknown [81, [91 . We shall alsodiscuss how thesignal bandwidth can be calculated from the measured indexprofile.The various methods for determining index profiles arebasedon different physical principles. Some methods utilize modeguiding properties by measuring the light intensity at the out-put end of a short length of fiber. If leaky modes and differ-ential mode losses can be neglected, the output light intensityis proportional to the difference of the refractive index of thefibercoreand the cladding [51, [ 101. A complementarymethod utilizes the light that is not guided but escapes fromthe iber core to determine the ndex profile [ 111 . Theprofile shape is established by moving a tiny light spot, illumi-nating the fiber input , across its face and measuring the lightintensity escaping sideways hrough he core boundary as afunction of the radial position of the sharply focused input

The refractive index can, in principle, be obtained by utilizingthedependence of the reflectivity ofglass on ts refractiveindex. For this purpose the reflectivity of a fiber endface ismeasured as a function of radial position [ 121.Interferometry can be used in different ways. An accurate,but destructive, method consists in cutting a slab out of thefiber, polishing its nds nd observing interference fringesacross the faceof the slab using an nterference microscope

light.

PLASTIC JACKET

Fig. 1 . Cross section of a multimodegraded-indexoptical fiber. Theplastic jacket which preserves the pristine strength of the fiber, is notdrawn to scale. It is typically over 50 pm thick.

with light illumination parallel to the axis of the slab 1131 ,[ 141. A nondestructive interferometric methoduses transverseillumination of the unbroken fiber, gain under an interferencemicroscope [ 151, [ 161. The strong influencef the outer fiberboundary must be eliminated by immersion in index matchingfluid.The refractive-index distribution can also be obtained fromobservation of the pattern of scattered light when a collimated,coherent light beam is incident at right angles to the fiber axis[ 171, 181. A different method, also using transverse illumina-tion, relies on the focusing power of the fiber core which is re-garded as a cylindrical lens [ 191, [201. The refractive indexdistribution can be computed from the intensi ty distributionof the focused light. This latter method is also applicable tothe much larger preforms from which the fibers are produced[211. In preformsone may alternatelycompute he ndexdistribution from measurements of the deflection angle of alightay passing transverselyhrough the core [2 2] . Al lthese methods require immersion of the fiber or preform inindex matching fluid, to achieve high accuracy.Finally,nonopticalmethods canbe employed to measurethe concentration of dopant materials which are responsiblefor thechange of the refractive index of the Si02 host material .Such methods can employ scanning electron microscopy [23 ]X-ray microprobe analysis [241 or chemical etching analysis[2 5]. However, we shall concentrate on optical methods andpay particular attention t o those methods with which we aremost intimately acquainted, such as the interferometric meth-ods and the focusing method.A comparison of accuracies expected from each method isexceedingly hard to give. It is hard enough to get a feeling forthe accuracy of the method which one is familiar with, but itis far more difficult to assess some other method which oneonly knows heoretically. For this reason no comparison ofaccuracies, other than those reported in the literature, hall beattempted.A , Multimode Fibers

Fig. 1 is a schematic drawing of the cross section of a multi-mode, graded-index fiber. The central region, the core, typi-cally consists of doped silica glass. The low refractive index



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I STEP NDEX

II ba r

Flg.2. Refractive-indexdistributions of step-index and graded-indexoptical fibers.The cladding ndex is that of SiO, and is equal to about1.450 at a wavelength of 1 pm .of pure Si02, no = 1.450 at a wavelength of A = 1pm , is modi-fied by doping with materials such as oxides of germanium,phosphorous ndboron [261. Germanium [271 and phos-phorous [ 281 increase the refractive index of SiOz, boron [ 291decreases it. The fiber can guide light if the core has a higherrefractive indexhanhe urrounding region [ 3 0 ] . Thusmost fibers consist of a cladding region of pure Si0 2 and acore whose index is increased by addition of germanium orphosphorous oxides. In some fibers the situation is reversed,the cladding is doped with boronoxide to lower ts ndexrelative to the undoped fibercore.

The amount of dopant that is added to the core is subject toa variety of tradeoffs. On the one hand, the more dopant thegreater the acceptance angleof the iberand ts resultingnumerical aperture. This increases the number of modes thatcan propagate, as will be shortly described, enables the fiber togather more light from an LED sok ce and reduces the fiberssensitivity to bending loss. On he other hand, increasing thedopant increases compositional fluctuations which leado fiberlossesvia scattering, ncreases group delay spread and ntro-duces fabrication difficulties due o mismatches in the physicalproperties of the core and cladding glasses. Typical gemanium-doped fibers have a maximum index difference f 0.02. Fibers,with germanium dopant, however, have been fabricated withmaximum ndexdifferences of 0.05 [ 3 11. For comparison,single-mode fibers typically have index differences an order ofmagnitude lower.Depending on the hape of the refractive-index profile of thefiber core, we distinguish stepindex and graded-index fiberswhose profiles are schematically shown n Fig. 2 . Both types offibers can be fabricated by various deposition processes. Cur-rently the lowest-loss highest-bandwidth fibers are fabricatedby the modified chemical-vapor deposition (MCVD) process[ 2 7 ] . Losses as low as 0 .2 dB/km at a wavelength of 1.55 pmhave been achieved [ 3 ] andbandwidths of 1 GHz km havebeen realized in production [33 I .In the MCVD process, shown schematically n Fig. 3 , a fused-quartz tube is mounted on a glass-working lathe and slowly

Rg. 3. Schematic diagram of modifiedhemical vapor depositionprocess.rotated while reactants, usuallySiCl4, and dopant reactantssuch as GeC14 and BCIB, flow through it in an oxygen stream.An oxy-hydrogen burner is slowly traversed along the outsideof the tube toprovide simultaneous deposition and fusion f alayer of the reacting materials. On the order of 50 layers aredepositedbymultiple passes of the burner. A borosilicatebarrier ayer is generally ncorporated at he core-claddinginterface to preventdiffusion of impurities rom theoutersupport tube to the core [ 3 4 ] . To fabricate step-index fibersthe dopant concentration is held fixed as a function of thelayer deposited whereas for graded-index fibers it is graduallyincreased with increasing layer number.At the conclusion of deposition the temperaturef the burneris raised to collapse the tube into a solid preform. The high-temperaturesexperiencedby the preformduring this stagelead to the vaporization of the dopants from he nnermostlayers causing an index depression in the center. The magni-tude of this dip is such that the index value a t the center isthe sameas that of the cladding [351.A thin slice of an MCVD fabricatedpreform is shown nFig. 4 as observed by optical microscopy. In order to bringout he details of the depositionprocess, he ample wasetched ndilutedhydrofluricacid. This particularpreformwas fabricated with a ten-step linear increase f GeC14, chosenas the simplest variation which would aid in analyzing the re-sulting structure and ndex distribution. The cladding, boro-silicate layer and he ten tages of increasing germanium dopantare abeled. Each step is composed of a number of layersformedduringeach ranversal of the oxy-hydrogenburner.For example, step 3 is composed of 15 layers deposited whilethe GeC14 flow was maintainedconstant.The ntegrity ofthese layers in the preform indicates relatively litt le diffusionof the dopant.A photomicrograph of a polished transverse cross sectionof the fiber pulled from the preform is shown in Fig. 5. Thestep structure is maintained through the drawing process. Toobserve the layers within each step, scanning electron micro-scopy was utilized on an etched fiber end. Results seen in Fig.6 also indicate he preservation of the layerstructure.Theindex depression on axis appears as the tapered elevation inthe center.The fiberorpreform, then, whose indexdistribution wewould like to measure can be rich in structure. Generally itsscale in the fiber is on the orderof less than a wavelength, andthe observed structure is smoothed out by the measurement.Notable exceptions can occur near the center here the deposi-tion layers are thickened and n any region where several layersmay have the same index due to fabrication faults.



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LAYERH .5mmFig. 4. Slice of MCVD fabricatedpreform having a linear refractive-index profrle asobserved by optical microscopy.

Fig. 5 . Slice of fiber pulled from preform shown n Fig. 4. The GeO,dopant was increasedin 10 steps.In addition, as will be seen, the refractive index is no t uni-form within each layer and this variation, with the exceptionof the above cases, is also smoothed over by the measurement.Thewidthandheight of the ndex profiledetermine henumber of modes tha t can be guided by the fiber. To quantifythis relationship it is convenient to int roduce the V-parameterdefined as follows:

V = n1 k n a .In this formula, n l ndicates the maximum value of the refrac-tive index profile near the axis of the fiber core with radius u.The parameterk is the propagation constant of a plane wave invacuum,

k = - 71h(h is the vacuum wavelengthof the light). Finally, A is de-

fined as the relative difference between the maximum value ofthe index in theiber core and its value n 2 n the cladding,(3 1



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610

2

I1 9fig. 7. Relationship of propagation vector and propagation constant.

s" 1.8 -1,6 - PRESBY - KAMINOW- -FLEMING - -- --SLAMN. PAYNE et rl---1.4 I I I I I I0.3 0.5 0.7 0.9 1.1 1.3 1.5

1 pm)Ftg. 8. Optimum profile constant g versus wavelength for Ge 0, -S O,fibem as determined by several investigators.

In graded-index fibers the number of guided modes also de-pends on V z but, n addition, depends on the shape of theindex profile. For parabolic index profiles thenumber ofguided modes is half that of the step index fiber [SI . Typi-cally, thisnumber is on the order f one thousand.In addition to guided modes the fiber supports leaky modeswhich continuously radiate power from he core into hecladding [371 . But the radiation losses of some of these leakymodes can be quite low. Modes are characterized by two num-bers, the radial mode number p, which is a measure of thenumber of zero crossings of the electric field as a function ofradius, and the azimuthal mode number v , which describes theazimuthal variation of the mode field. The radiation losses ofleaky modes decrease with increasing azimuthal mode number[ 3 8 1 .An important parameter,characterizing guided modes, isthe propagation constant &,,,. Each mode can be represented

PROCEEDINGS O F THE IEEE, VOL. 68 , NO. , JUNE 1980as a superposition of locally plane waves whose propagationvector has the magnitude n ( r ) k . Its direction may be regardedas the direction of light rays propagating in he fiber. Theray and mode picture of light guidance in fibers is often usedinterchangeably. The magnitude and direction of the propaga-tion vector is dependent on the position of the quasiplanewave or ray in the fiber core. The projection of the propaga-tion vector on the fiber axis is the propagation constant pv,,of the mode [391 . Its value is constant for each mode becauseof the change of direction of the light rays. This relationshipbetween the propagation vector of the quasiplane wave andits projection on the z-axis is shown in Fig. 7. The importanceof the propagation constant for our discussion is the fact thatit is confined to the range [301, [ 391

n z k < & , , , < n l k . ( 5 )The upper limit n k of Pu r is approached by the modes withthe lowest possible values of Y and p. The lower value n z k isthe cutoff value which is approached by the modes of highestorder. Waves with flu,, < n z k are the aforementioned eakymodes.Before discussing the various measurementprocedures itwould be helpful to consider the ideal profile, that one wouldlike to be measuring in graded-index fibers, and how accuratelythis profile need be measured. Modal delay distortion occursin multimode optical fibers because the many different modes,as discussed above, travel at different group velocities, spread-ing an impulse over a time interval that is equal to the differ-ence of the arrival times of the slowest and astestmodes.This pulse spreading is accompanied by a reduction of thesignal bandwidth.It is well known that the fiber bandwidth can be maximizedby optimizing the shape of the refractive index distributionof the fiber ore [S I, [61. Near optimumbandwidth isachieved with a power-law index profile of the form

n ( r ) = nl [ 1 - (r /aIgAIin which g is the exponent of the power law.' In the absenceof chromatic dispersion the bandwidth is maximized for

125g = 2 - - A .

The optimum value of g also depends on chromatic dispersionwhich arises from the wavelength dependence of the refractiveindex of the dopant material [ 6 ] , 401 ;i t is shown for germa-nium in Fig. 8, based on measurements by various authors inboth fibers (Presby and Kaminow [401, Sladen 141I , and bulk(Fleming [421) samples.The heoreticalbandwidth hat can be realized with anoptimum profile is about 8500 MHz * km ora iberwithn l - n z = 0.02. Fig. 9, shows that a departure of only 0.05from the optimum g-value of 2-2.4A is sufficient to reducethe fiber performance by more than one order of magnitude.Clearly, techniques to determine g to better han0.05 arerequired if meaningful correlation between fiber performanceand index profiles is to be obtained.An appreciation of the sensitivity required in index profilingto achieve this accuracy in g can be gained from Fig. 10, whichshows various gprofilesnormalized to the same maximumbeen replaced by g to avoid confusion with the attenuation constant.'At the suggestion of S. E. Miller, he commonly used symbol u has



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2 g 3Fig. 9. rms pulse width of step over graded index fibers versus g, assum-ing an optimum value near 2 and a profde dispersion parameterP = @/A) (dA/dA) = 0.

index value and radius. Notice that the difference between theg = 2 curve and the g = 2.05 curve is barely distinguishable onthis scale. In order t o determine g accurate to 0.05, the preci-sion required in the measurement of An(r) must be about 1part in io4.It is also important to note that even very slight local dis-tortions of the refractive indexprofile rom tsoptimumshape decreases the fiber bandwidthmarkedly [8], 43], [44].Consider, forexample, a inusoidal indexdistortionon anoptimum profile

n(r )=nl [ l - ( r / a )g A l + A sin [2nNr/a].Let there be 10 sinusoidal periods, N = 10, in the radius 0 2 all modes arrive later. The width of the pulses canbe gleaned by lookingat thedifferent time scales of the figure,but a much better mpression is conveyed by Fig. 41 whichshows the rms pulsewidth multipliedby c / L to render itdimensionless) as a unction of the power-law exponent g.The dotted curve was computed numerically using (53) withD X 0 nd D2 0. The solid curve represents the result of ananalytical evaluation tha t is applicable for power law profiles[61. The dash dot ted curve in Fig. 41 represents a titanium-doped fiber and akes material dispersion into account, it isshown only to demonstrate that material dispersion cannot beneglected [6 .The fiber bandwidth is obtained as the frequency at whichtheFourier ransform of the impulse esponse decreases toone half of its maximum amplitude. Fig. 4 2 shows the band-width as a funct ion of the power law exponent g. If the fibermaterials were not dispersive the maximum bandwidth wouldbe ndependent of wavelength. For actual, dispersive fibermaterials thebandwidth becomesa unction of wavelengthand we may speak of a bandwidth spectrum 1861. The curvelabeled A = 0 in Fig. 43 shows such a bandwidth spectrum for


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686 PROCEEDINGS O F THE IEEE, VOL. 68, NO. 6 , JUNE 1980

c= 3 -2 -

-THEORY--- PROGRAM2I O-sh.4 368 20 2.2 2.4 2 6

eFig. 41 . r m s pulsewidth imesvelocity of light as a functionof thepower law exponent g. The dash-dot curve takes profile dispersioninto account for a TiO, dopant.

A 0.0435 Iid c I8 :6 -E -; -

m 2 -I

:6 -4 -3 -2 - xo2 1.4 1.6 1.8 2 2.2.4 2.6aFig. 42 . Bandwidth of fiberasa function ofg, with no profile dispersion.a germanium-doped fiber with a pure power-law profile whoseexponent is g = 1.9071. Phosphorus-doped fibers have band-width spectra hat aremore han wice as wide, but highlydoped phosphorus fibers are much harder to produce and aretherefore not popular at the moment.We have seen that slight departures from the optimum index

profile ead toadramatic decrease of thefiberbandwidth.This is true not only for departures of the optimum exponentvalue of a pure power law profile, butt happens whenever theidealprofile is distorted in any way [ 4 3 ] . Sinusoidal per-turbations of the ideal power law profile as mentioned earlier,forexample,causeasubstantialdecrease of thebandwidth[81, [ 4 3 ] . This effect is shown in Fig. 44 where bandwidth isplotted as a function of the number of periods N of a sinu-soidal perturbation of the ideal index profile, up to N = 10.The amplitude of the sinusoidal distortion is 1 percent of thedifference between the maximum core index and the claddingvalue. The effect on the ideal bandwidth spectrumof a N = 10perturbation for two different amplitudes is shown in Fi g 43.The main value of a computer program capable f computingfiber bandwidth from refractiveindex profiles is its ability to

Ge02 -doped4 -

Z r

ArF0.02N=10g=1.9071

A k m )Fig. 43. Bandwidth spectrum of GeO, doped fiberwith g = 1.9071and effects of sinusoidal perturbation.

50 0114968

0i 5 I O 15

UFig. 44. Bandwidth as a function of the number of periods of a sinu-soidal perturbation added o an ideal index profde..

f 800EEz 60024005s 20 0

0 1 2 3 4 5 6GERMANIUM-DOPED FIBERS

Fig. 45 . Comparison of predicted (shaded bars) and measured (solidbars) impulse responsesof GeO, -doped iber.

predict the performance of a fiber before it is put into acableor, if the prediction is based on the preform, even before thefiber is drawn. F i g . 45 demonstrates hatmeaningfulband-width predictions of real fibers are possible. The black barsrepresent the results of direct bandwidth measurements [ 9 ] .[87], [88] . The shaded bars are calculated using index pro-files obtainedfromeitherend of thefiberby he focus ingmethod.The ibersused or this comparisondidnot haveplastic jackets. This is important since jackets tend o increase


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MARCUSE AND PRESBY: INDEX PROFILE MEASUREMENTS FOR FIBERS 687

fiber bandwidths by introducing microscopically small bends(microbends) that couple-guided modes and thus improve theimpulse esponse of the fibers. This improved mpulseper-formancedue to microbending of the iber is one of thereasons why bandwidthpredictions based on indexprofilemeasurements tend to come out low. Bandwidth calculationsfrom index profiles may thus be regarded as estimates of thelower imit of expected fiber bandwidth, actual bandwidthsare almost always higher. In addition to bandwidth improve-ments due to microbending of the fiber, delay compensationsoccur if the index profile changes gradually along the lengthof the fiber. Such gradual changes do not cause mode couplingbut they can be responsible for partial compensation of differ-ential time delays of t he different modes [891.REFERENCES

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pp. 106-108, eb. 15,1979.[4] H.G. Unger, Plnnar Optical Waveguides and Fibres. Oxford,England: Clarendon Press,1977.[5] D. Gloge and E. A. J. Marcatili, Multimode heory of graded-core fibers, Bell Syst. Tech. J., vol. 52, pp. 1563-1578,Nov.1973.[61 R. Olshansky and D.B. Keck, Pulse broadening in graded-index[7] S. D.Personick, Time dispersion in dielectric waveguides, Benoptical fibers, AppL Opt., vol. 15, pp. 483-491, eb. 1976.Syst. Tech. J.,vol. 50 pp. 843-859,1971.L.G. Cohen and S. D. Personick, Length dependence of pulsedisperion in long multimode optical fiber, AppL Opt., vol.14,(81 D. Marcuse, Calculationofbandwidth rom ndexprofiles ofPp. 1357-1360,1975.optical fibers. Part I : Theory, Appl. Opt., June 15, 1979.[91 H.M. Presby , D. Marcuse, and L. G. Cohen, Calculation of band-width from index profiles of o ptical fibers. Part11: Experiment,

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[20] D. Marcuse and H.M. Presby, Focusing method for nondestruc-method,AppL Opt.,vol. 18, p. 9-13, an. 1, 1979.tive measurem ent of optical f iber index profiles, ppL Opt., vol.18,pp. 14-22, an. 1, 1979.(211 H. M. Presby and D. Marcuse, Preform ndex p r o f ~ gPI),

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