(E)ESI Determination From Mode-field Diameter and Refractive Index Profile Measurements on Single-mode Fibres

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E)ESI determination from mode-field diameter andrefractive index profile measurementsonsingle-mode fibresF. Martinez, MScC D . Hussey, PhD

Indexing terms: Measurements andmeasuring,Opticalfibres,Waveguides andwaveguide components

Abstract: We show, both experimentally andtheoretically, that for single-mode fibres, the(E)ESI and MFD methods are interrelated in aself-consistent model with the theoretical cutoffwavelength playing a pivotal role. Three indepen-dent measurement approaches are examined:mode-field diameter measurements, preformprofile measurements and fibre profile measure-ments.

refractiveindexdifference

1 IntroductionCurrently, there are two trends in characterising single-mode fibres. They are:(a) from refractive index profile measurements and(b)from mode-field diameter (MFD) measurements.*Between these two there exist equivalent step index (ESI)methods. The two ESI parameters (equivalent core-radiusand equivalent numerical aperture) can be obtained fromeither the refractive index profile or the MFD. Havingthe ESI parameters and the MFD, we can predict mostof the essential characteristics of single-mode fibres, suchas bending loss, splice loss, microbending loss, waveguidedispersion (and therefore total dispersion). However,there is still confusion because there are several ESI defi-nitions and measurement techniques, and several MFDdefinitions and measurement techniques, which havebeen proposed and none of them has been recognised asa standard method for characterising single-mode fibres.In this paper, we propose a theory, which links theESI parameters obtained from measurements of the ref-ractive index profile to the same parameters derived frommeasurements of the MFD. We can therefore, in prin-ciple, unify both trends in characterising single-modefibres. Measurements of refractive index profile andmode-field diameter are performed on MCVD fibres, toprovide experimental support for this theory.1.1 quivalent step index ESI) techniquesIn general, the refractive index profile of a single-modefibre deviates from the ideal step index as shown in Fig.* N o t e : In addition to the term mode fielddiameter' MFD, wewill useth e term mode fieldradius 'co (i.e. MFD = 2oi).coh as also been referredas spot size' in the literature.Paper 5954J(El3), received 19th October 1987Th e authors ar e with th e Optical Fibre Group , Department of Elec-tronics an d Computer Science, Th e Univers i ty , Southampton, Hamp-shire SO9 5NH, United Kingdom

radius,|jm

Fig. 1 Typical refractive index profile of a MCVD single-mode fibreand apossible equivalent step index representationae is theequivalent core radiusand hetheequivalent index differenceactual refractive index profileequivalent step index profile

1. The problem is then to determine the propagationcharacteristics of a single-mode fibre with an arbitraryrefractive index profile. Exact solutions are difficult toimplement, and good approximate methods are preferred.It has been observed that the fields of all single-modefibres look similar. Since the analytical solutions for thestep-index fibre are already available and well known, itis then very convenient to have a method which has as itsreference a step-index fibre. This gives rise to equivalentstep index methods.The ESI fibre should have a second mode cutoff wave-length, fundamental mode propagation constant, MFD,and evanescent field as close as possible to the corre-sponding parameters of the actual fibre. If this is the case,the bending losses, microbending losses, and splice lossescan be predicted from the ESI fibre [1].There exist, however, some problems including [2]:(a)Several ESI techniques are available, each of whichgives different values for the two ESI parameters.(b) The accuracy of the predicted propagation charac-teristics is not always good, particularly with the predic-tion of waveguide dispersion.

Referring to the latter problem, we have shown thatone particular ESI method can be readily enhanced fromthe accurate prediction of waveguide dispersion [3], we202 IEE PROCEEDINGS, Vol. 135, Pt. J, No. 3,JUNE 1988



call this the enhanced ESI, or (E)ESI model. This modelis based on the moments of the refractive index profile,and it proposes the use of an additional third parametercalled the enhancement parameter. Unfortunately, per-forming refractive index profile measurements on thefibre is difficult and, occasionally, it is difficult to estimatethe core diameter (i.e. core-cladding boundary) from theprofile measurements. Because of these problems, thealternative of characterising single-mode fibres fromMFD measurements has received a lot of attention.1.2 Mode-field diameter measu rement {MFD)techniquesThe MFD is the width of the fundamental mode, guidedby a single-mode fibre above its cutoff wavelength. MFDis in principle a very useful parameter, as it allows theprediction of splicing and microbending losses. In addi-tion to that, the wavelength dependence of the MFDallows us to predict the bending losses (through the ESIparameters), and the waveguide dispersion.However, quoting Reference 4, 'MFD appears to be aparameter in turmoil. Its importance is well understood,but no consensus on its fundamental definition or mea-surement method has yet emerged. Several measurementtechniques and definitions have been proposed during thepast few years. None of these methods is universallyaccepted as a reference test method, nor is any methodmore or less fundamentally correct than the others.'The degree of consistency between the different tech-niques is a direct result of the choice of a definition forthe MFD. The Gaussian approximation, as produced byMarcuse [5, 6] and illustrated in Fig. 2, so widely used

U 6 8 10radius, pmFig. 2 Field amplitude distribution of the fundamental m ode, in a step-index single-mode fibre at V = 2.1, and its Gaussian approximationM F D f is the Gaussian Mode-field diameteractual field amplide distributiongaussian approximation

for the first generation of single-mode fibres, is nowseverely questioned. Small systematic errors result evenfor quasistep index fibres when the Gaussian approx-imation is used, especially at longer wavelengths [7].Also, the second generation, single-mode fibres withnonstep refractive indices for dispersion shifting and dis-persion flattening do not exhibit Gaussian fields at anywavelength.An alternative definition of MFD based, on themoments of the near-field, which is frequently called thePetermann 2 definition, can, in principle, solve theproblem of consistency, and reduce the systematic errors[8]. The primary difference with respect to the Gaussiandefinition is that the Petermann definition uses the mea-sured field directly, making no assumption about the

shape of thefield.Because ofthis,the Petermann 2 defini-tion is becoming more widely accepted.Cutoff wavelength is also essential in characterisingsingle-mode fibres, but some problems arise because thesecond-mode becomes highly attenuated at wavelengthslower than the theoretical cutoff wavelength. This effec-tive cutoff isdependent on the length and the bending ofthe fibre. There is some agreement on the effective cutoffdefinition and measurement techniques [9], but the rela-tion between the effective and theoretical cutoff is pre-sumably very difficult to determine. This suggests thatone has to be careful when using the measured cutoffwavelength, as an input in the determination of the ESIparameters from the spectral variation of MFD, forexample.2 Unifying ESI and MFD methods in single-modefibresMillar's method is perhaps the most standard techniquefor determining the two ESI parameters and the second-mode cutoff wavelength for single-mode fibres. One basicassumption in the Millar method is that the MFD forarbitrary profiles behaves simply as a scaled version ofthe step-index MFD [10]. While this is true for mostfibres used for telecom munications, it is not true ingeneral, and in particular the method breaks down whenapplied to the triangular-type profile fibres and dualshape profile fibres [11] which are receiving a lot ofattention for dispersion shifting.In its simplest form, the Millar method measures thecutoff wavelength, together with the MFD. Unfor-tunately it is at the cutoff wavelength where the MFD ismost sensitive (over the single-mode regime) to vagariesin the refractive index profile. Moreover, the Millarmethod has no built-in alarm to show that it breaksdown for a specific fibre under measurement.Here we show how the Millar method can be adaptedto cater for a more complex MFD structure. Ourapproach uses the Petermann MFD [12, 13] (or far-fieldRMS width) to find the two parameters for the ESI orthe three parameters necessary to specify the enhancedESI (or (E)ESI). Having determined the three param eters,we can then decide whether to use all three parameters inthe (E)ESI, or only two in the simple ESI approximation,but now we have a greater appreciation of the errorsinvolved, ifwechoose the latter.The implementation of our approach is aided by ouranalytic approximation to the Petermann MFD, which isoutlined in Appendix 8.2.2 Petermann MFD and (E)ESI approximationThe (E)ESI approxim ation is outlined in Appendix 8, andis specified by three parameters. For our purpose we willfind it convenient to use the parametersV ae, and |Aft41,where V= (2n/X)aeNAe = V(2fio)K , ae = ayj2Cl2, an dNA e = NAy/lo/fi21 a is t n e c o r e radius, NA is thenumerical aperture, Q o is the guidance factor and fin isthe moment of profile. We can see from Appendix 7 thatA 4 is the enhancement parameter; its magnitude givesan indication of the deviation of the refractive index fromthe step index (for which Aft4 = 0). From Table 6, forexample, we find that for the parabolic profile fibre, theenhancement parameter is 0.125, for the triangular profilefibre it is 0.190, and for cusp-like profiles, its value canapproach 0.3.In terms of our three parameters, the normalised near-field Petermann mode-field radius for fibres of arbitrary

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 3, JUNE1988 203



profile can be expressed as [14]

1 [/(io K3/ '(*OM\(bl(V) 4 (i )

where bst{V ) = (1.1428K- 0.996)2/K 2 is the Rudolph-Neumann approximation,f(V) = 0.0313 -0.013K2 (2)

(from Appendix 8.2), andcb st = 0.65 + 1.619K 3/2 + 2 . 879F 6

(0 . 016+ 1.561K 7) (3)(from Appendix 8.1).Since the first term in eqn. 1 represents the simple ESI,the full form of this equation illustrates dramatically thelimitations of any procedure which arbitrarily attemptsto fit a measured mode-field diameter with that of somestep-index fibre. Additionally, the term in square bracketsin eqn. 1 is a mono tonically increasing function of V, sothat by measuring the mode-field diameter at cutoff, weare choosing the worst sampling point for fitting thesimple step-index mode field diameter for fibres whichhave a large enhancement parameter. These points arealso illustrated in Fig. 3, where we plot co as a function ofV for the triangular-profile fibre.

2.0

1.5

g 1-0o

0.5

1.0 1.5 2 .0 2.5 3.0 3.5effective norma lised frequency, V

Fig. 3 Mode-field radiusc against V for the triangular profile fibreexact ESI enhanced ESI

It is convenient to rew rite eqn. 1 as follows:~2 (4)

where

V3f (V)bst(V)

The quantities p and q are specified once V is known.Similarly, the far-field mode radius can be expressed as5 )

2.3 Adapted Millar procedureThe starting point of the Millar method [10] is the factthat the theoretical normalised cutoff frequency Vco isunlike any other fibre characteristic, because it isextremely insensitive to profile shape. The cutoff wave-length Xco should therefore, in principle, be a fixed refer-ence point for all single-mode fibre measurements.Millar's method determines Xco from observation ofthe near-field MFD behaviour. Further refinements haveproduced more accurate determinations of Xco [15]. Thepossibility of using the far-field M F D behav iour in deter-mining kco has also been suggested [16]. Such a tech-nique could depend critically on the detector sensitivity.By combining such a determination of the cutoff wave-length, with the measurement of the Petermann M FD attwo different wavelengths {Xx and X2) in the single-moderegion, i.e. Xt ^ X2^ Xco , the three necessary parametersfor specifying the (E)ESI are obtained as follows:= 2.405

Pi -L, 2

(6a)

(66)

q2) (6c)where K is the ratio of the two mode field radius mea-surements, (col/col), plt 2a nd qlt 2 are the values of p an dq of eqn. 2 evaluated at V12 , and colt 2 are the measuredvalues of the near-field mode field radius at A1( 2 .If we use the far-field RM S wid th mea surem ents at thetwo different wavelengths, then the parameters VY 2 an d| Afi41 are still given by eqns. 6a and b, while the equiva-lent core radius is now:

o j lCO (6d)7 / 2

whileR in eqn. 66 is now defined as ((Offjcol).The two-parameter ESI can be obtained from mea-surements of the Petermann M F D at one wavelength,together with the cutoff wavelength. Neglecting theenhancement parameter (i.e. |A fi41 = 0 in eqn. 6c, thenthe two parameters are given by: = 2.405 -fA:and

ie bM)

(7)

(8)and costVi) can be given by eqn. 3. The subscript i nowcorresponds to any wavelength about the cutoff wave-length.The equivalent numerical aperture NAe is nowobtained simply from NAe = 2A05Xco/2nae.3 MeasurementsIn this paper we implement and critically assess ourapproach using three independent measurement pro-cedures. The (E)ESI parameter s as obtained from M F Dmeasurements, preform profile measurements, and fibre

204 IEE PROCEED INGS, Vol. 135, Pt. J, No. 3, JUNE 1988



profile measurements,arecomparedfor anominally stepindex fibre fabricated by theMCV D process.A statistical approach is adopted in analysing themeasurement data, thereby reducing the influence ofmeasurement error and other inherent uncertainties.Results show that whilethepreform and thefibre profilemeasurements produce essentially the same equivalentcore radiusand thesame equivalent numerical aperture(with allowancefordiffusion effects), theESIparametersproducedby the M FD are substantially different.The discrepancy arises from the use of the effectivecutoff wavelength, as derived from the MFDmeasure-ments, instead of the theoretical cutoff wavelength(correspondingto aVvalueof2.405)inth eevaluationoftheESIparameters.However,itisfound that ifthe theoretical cutoffwave-length,asderived from theprofile measurem ents,isusedin the interpretation of the MFDdata, then the threemeasurement approaches predict very similar ESIpara-meters. While only one fibre-preform is reported here,one other nominally step-index fibre-preform exhibitedsimilar trends.

Fig.4.When usingthepreform data,the same measuredcore radiiasusedforpoints 1,2,3,4, 5 and 6 inFig. Abwere assigned for the corresponding points in Fig. 4a.Assuming circular symmetry only,the right hand halfofeach profileha sbeen used.The (E)ESI parameters resulting from the profilemea-surements also differ slightly from onechoiceofprofiletoanother. Table 1shows the average (E)ESI parametersTable 1: Average (E)ESI parameters as obtained from ref-ractive index profi le measurements and moments theoryParameter Profi le me asurementsFibre PreformAcon m) 1226 25a8//m) 2.86 0.12NA e 0.164 0.005|A Q 4 | 0.08 0.02

12171 12.71 0 10.1722 0.00070.030.01(Average theoretical cutoff is l c o =1222 nm). The uncertaintyquotedisthe standard deviationresulting from the sixprofiles, bothfor thefibreand thepreform.

3.1 ProfilemeasurementsFigs. AaandAbshow the refractive index profilesof thepreform and the fibre, respectively, as measured usingcommercial equipment (spatial filtering technique andrefracted near-field technique, respectively). The(E)ESIparameters were derived from themoment analysisofth erefractive index profile for the preform and thefibre.Problems in implementing this procedure arise becauseboth thecore radiusand the refractive index levelsofth ecoreand thecladdingare not well defined. Becauseofthis,sixlimits were proposed,assketchedin the insetsof

a; 0.01

0.005

0.012

-6 -4 -2 0 2r ad ius ,mm

3.2 Mode-field diameter measurementsThe near-field Petermann's MFD wasobtained fromtheinverseofthe RM S far-field width measuredby the vari-able aperture method using commercial equipment.Thecutoff wavelengthwasmeasured usingtwotechniques:(a)th eMillar procedure from thevariationofM F Dasa function ofwavelength(b)thetransm itted power technique(asrecommendedby CCITT)[17].Table 2 shows the values measured for the Peter-mann's MFD at four wavelengths. In theory, for thedetermination ofthe (E)ESI parameters,thecutoff wave-length together with the MFD at anyother twowave-lengths are sufficient. In practice, if we try different

Table 2: Near-field Petermann s MF D at four wavelengthsfor the fibre used in the comparative analysisA nm) Petermann's MFD jjm)1300135014001450

6.47 %6.64 %6.85 %7.07 %

wavelength pairs we obta in different results. Ourapproach is to take the average value of the (E)ESIparameters predicted from different pairs of the fourwavelengthsinTable2.Table3showstheaverage value

0.01

0.005

-6 0 -4 0 -20 0 20 ACradius,urn

bFig . 4 Refractive index profile of (a) preform and(b )fibreThe insets showthe six different core sizesandcladding levels usedIEE PROCEEDINGS, Vol.135,Pt. J, No. 3,JUNE 1988

60

Table 3: Average (E)ESI parameters as obtained from MFDmeasurementsParameter Aco,=1275 25 nm) Aco2 =1295 25 nm)ae jjm) 3.16 0. 12NA a 0.155 0.00 60.13 0.08

3.22 0.140.154 0.0070.14 0.0 9Aco, wasmeasured u singthetransmitted po wer techn ique.Aco2 wasmeasured usingtheva riationofM FD as a funct ionofwavelengthof these (E)ESI parametersforthe two different measuredcutoff wavelengths.If wechoose toignoretheenhancement parameterindetermining the twoparameters of the simple ESI, werequire thecutoff wavelength and theM F Datonlyonewavelength.We again determinetheaverage values from

205



Table 4: Average E)ESI parameters derived from MF D mea-surements assuming|A0d | =0Parameter Acoy =1275 + 25 (nm) Aco2 = 1295 2 5 (nm)aajjmN

2 940 0050 16620 0003

2 970 0080 16680 0004

y\ co 1 a nd Aco2 as in Table 3

the measurements in T able 1. These are shown in Table 4for the two different measured cutoff wavelengths.4 Discussion of resultsTaking the results for the profile measurements in Table1, it is found that the two parameters aeand NAe are ingood agreement between the fibre and the preform.However, there is a slight increase in ae and a slightdecrease in NAe in the transitio n from preform to fibre,this is consistent w ith the occurrence of diffusion duringfibre drawing. The fact that there is only a slight varia-tion in these parameters from preform to fibre is inter-esting, since these two parameters are derived from thefirst two even moments of the profile shape function, andas such they rely on the average properties of the refrac-tive index profile which are not expected to change verymuch in the fibre draw. It should be pointed out that thepoor resolution in the fibre profile measurement will alsoappear as diffusion, however, even within the error boundthere is an overall true diffusion effect. In addition, thetheoretical cutoff wavelengths, as derived from:

kco = 2naeNAJ2A05 (9)are in good agreement between fibre and preform.The enhancement parameter |AQ 4 | is small in bothcases of preform and fibre profile measurements. Theaverage value of the enhancement parameter of thepreform is|AQ41= 0.03, and according to Table 6, it cor-Table 5: Average E)ESI parameters as obtained from MF Dmeasurements, but using now the average theoreticalcutoff wavelength Xco 1222 nm) derived from the refrac-tive index profile and moments theoryParameter (E)ESI ESINA 3.0 0.070.156 0.0030.09 0.05

2.840.010.1649 0.0004

Table 6: Data for the first tw o moments ft0 andCt2 and forthe enhancement parameter Aft4 for several values of theexponent a of the po wer-la w profiles

oo1684214

i

0.5000.4440.4000.3330.2500.1670.1360.1000.056

0.5000.4500.4170.3750.3330.3000.2890.2780.265

0.0000.0100.0290.0670.1250.1900.2150.2460.285responds approximately to a power-law profile witha = 8, which is very step-like. The enhancement para-meter in the fibre (|Af241= 0.08) again, according toTable 6, corresponds approximately to a power-lawprofile with a 4 which is also very step-like. Theenhancement parameter is therefore small enough to beneglected, because its effect on the dispersion parametersb, bi and b2 is small (see Appendix 8), and therefore its

influence on the behaviour of the MFD is also very small.It is also interesting to note that |Aft 4 | in the fibre isslightly larger than in the preform. This gives a measure-ment of the slight diffusion that occurs in the profile of asingle-mode fibre when it is drawn from the preform. Thecomparatively larger standard deviation in the determi-nation of the (E)ESI parameters for the fibre is an indica-tion of the intrinsic difficulty when working with thesmall dimensions involved.On examining the results from the MFD measure-ments in Table 3, it is found that the results are com-pletely at odds with those from the profiles. Thediscrepancy is due to the use of the measured cutoffwavelength (Xmco)in determining the fibre param etersfrom eqn. 9. On substituting the average theoreticalcutoff value Xco deduced from the profiles (i.e. Xco=1222 nm) into the M FD analysis, the parameter values asshown in Table 5 are obtained. These parameters arenow in good agreement with those derived from the fibreprofile.From Table 5 we can conclude, as we have done forthe profiles, that the simple two parameter ESI should beadequate for this fibre. Indeed the simple two parameterESI results in this Table are found to be in remarkablygood agreement with those in Table 1, once we use thetheoretical cutoff wavelength in the analysis.5 ConclusionsWe show that the theoretical cutoff wavelength, asdefined by eqn. 9, is the key to obtaining a self-consistentmodel for characterising single-mode fibres which unitesboth the ESI and MFD models. The measured cutoffwavelength, which for a given fibre remains consistentbetween different reference measurement techniques, doesnot have a definite relationship to the theoretical cutoffwavelength when comparing different fibres. As such, themeasured cutoff wavelength can lead only to confusionwhen used as a reference wavelength in determining theESI parameters. This conclusion is drawn not only fromthe results from a single fibre as presented here, but alsofrom other experimental results in our laboratory, andfrom results quoted by other workers (such as in Refer-ences 18 and 19).One of the limitations of the MFD measurements pre-sented here is the restricted set of data. Ideally, a largernumber of measurements would improve the results forthe average and the standard deviation of the (E)ESIparameters. This, however, would increase the time spentperforming the measurements.6 AcknowledgmentsThis work was supported by the SERC. The authorswould like to thank Dr. N. McFarlane from York Tech-nology for performing the mode-field diameter measure-ments. Mr. Martinez would also like to thank theMexican Institutions CONACYT and HE for theirsupport.7 References

1 JE UN HO M M E, L.B.: 'Single-mode f ibre opt ics : Pr inciples andapplications' (Marcel Dekkr. Ing, New York, 1983)2 SA MS ON , P.J.: 'Usage-based com parison of ESI techniques', J.Lightwave Tech., 1985,LT-3,pp. 165-1753 MA RTIN EZ, F., and HUSS EY, C D .: 'Enhanced ESI for the pre-diction of waveguide dispersion in single-mode optical fibres', Elec-tron. Lett., 1984,24, pp. 1019-1021

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4 DIC K, J .M., and SHAAR , C : 'M ode f ie ld diameter : toward a s tan-dard definition', Laser and Appi, 1986, 5, pp. 91-945 MARCUSE, D.: 'Loss analysis of single-mode fibre splices' , BellSyst. Tech. J., 1977, 56, pp. 703-7186 MA RC US E, D.: 'Gaussian appro ximation for single-mode fibres' , J.Opt. Soc. Am., 1978, 68, pp. 103-1097 SARAVAN OS, C, and LO WE , R.S.: 'The measurement of non-gaussian mode fields by the far-field axial scanning technique', J.Lightwave Tech., 1986,LT-4,pp. 1563-15668 ANDERSON, W.T.: 'Status of single-mode fibre measurements' .Technical Digest Symposium on Optical Fibre Measurements,Natio nal Bureau of Standard s, U.S. Depa rtme nt of Comm erce,1986, pp. 77-909 FRANZEN, O.C.: 'Determining the effective cutoff wavelength ofsingle-mode fibres: An interlaboratory comparison', J. LightwaveTech., 1985,LT-3,pp. 128-13410 MIL LA R, C : 'Direct method for determining equivalent step indexprofiles for monomode fibres', Electron. Lett., 1981,17, pp. 458-46011 OG AI , M, KINO SHITA , E. , TAM URA, J , NA KAM URA , S. , andHIGASHIMOTO, M.: 'Low-loss dispersion shifted fibre with dualshape refractive index profile', Tech. Dig. ECOC '87,1, pp. 171-17412 PETERMANN, K.: 'Constraints for fundamental modal spot sizefor broadband dispersion compensated single-mode fibres' , Electron.Lett., 1984,19 , pp. 712-71413 HU SSE Y, C D .: 'Field to dispersion relationships in single-modefibres', ibid., 1984, 20, pp. 1051-105214 HUSSEY , CD ., and MA RTIN EZ, F .: 'New interpreta tion of spot-size measurements on singly-clad single-mode fibres', ibid.,1986,22,pp . 28-3015 C AM PO S, A.C., SRI VAST AV A, R., and ROV ERSI, J .A.: 'Charac-terization of single-mode fibres from wavelength dependence ofmodal field and far field', J. Lightwave Tech., 1984, LT-2, pp .334-34016 PAS K, C , and RU HL , F.: 'New method for equivalent-step indexfibre determ ination', Electron. Lett., 1983,19 , pp. 658-65917 CCITT: Recommendation G.652: 'Characterization of single-modeoptical fibre cable', Section III: 'Test methods for the cutoff wave-length'18 PASK , C , and RU HL , F.: 'Effects of loss on equivalent-step indexfibre determ ination', Electron. Lett., 1983,19 , pp. 643-64419 FO X, M .: 'Calculation of equivalent-step-index param eters forsingle-mode fibres',O pt. and Quant. Electron., 1983,15, pp. 451-45520 HUS SEY, C D ., and MA RT INEZ , F.: 'Approximate analyt icalforms for the propagation characteristics of single-mode opticalfibres', Electron. Lett., 1985,21 , pp. 1103-110421 RU DO LP H, H.D., and NEU MA NN , E.G.: 'Approximation for theeigenvalues of the fundamental mode of a step index glass fibrewaveguide', Nachrichtentech. Z., 1976,29 ,pp. 328-32922 HUSS EY, C D ., and PASK , C : 'Theory of prof ile moments descrip-tion of single-mode fibres', IEE Proc. H., 1982,129, pp. 123-13423 GAMBLING, W.A. , MATSUMURA, H. , and RAGDALE, CM.:'Mode dispersion, material dispersion and profile dispersion ingraded index single-mode fibres', Microwave, Optics and Acoustics,1979, 3, pp. 239-245

8 Appendix8.1 Approxima te analytical forms for the propagationcharacteristics of the single-mode fibresWe present an approximate analytic form for the Peter-mann mode field radius, as required in the paper. Thenew equation adds two extra terms to the old Marcusemode field radius expression [7] .8.1.1 Mode field radius and eigenvalue: The exactanalytical formula for the normalised Petermann 2 modefield radius, d>p, Q2/W)(J\(U)/J0(U)) can be expressedapproximately as [20]:

w=cbM- (0.016 + 1.561K 7) (10)where cbMis the Marcuse formula for the normalisedmode field radius of an optimally exciting Gaussianbeam, and which we repeat here for convenience [5 ]:

(bM= 0.650 + 1.619K 3/2+ 2.879K 6 (11)For our purposes we have optimised eqn. 10 to be accu-rate to within 1% in the region 1.5 ^ V^ 2.5, since thisis the range of most practical interest in single-mode fibreIEE PROCEED INGS, Vol. 135, Pt. J, No. 3, JUNE 1988

transmission. Like the Marcuse formula, eqn. 10 wasdetermined empirically. Increased accuracy over a moreextended range would require additional terms, andwould be of questionable advantage.Fig. 5 showsd)M, and both the exact and approximatecurves for d>p. The very good accuracy of eqn. 10 forlarger V values (V >2.5), and the good qualitative3.0

2 2.0ooES 1.0

01.0 1.5 2.0 2.5 3.0normalised frequency V

Fig . 5 Petermann mode-field radiusc bp and Marcuse mode-field radius



(1.5 ^ V ^ 2.5). The percen tage accuracy at large Vvalues becomes meaningless, since the function isapproaching zero; however, a good indication of its

1.0 3.0.5 2.0 2.5norma l ised f requency V

Fig . 6 Approximate and exact dispersion and core power curvesplotted as functions of normalised frequency V

exact approximate

validity is that the approximation predicts the zero dis-persion point (Vzd 3) to within 2% .

and a profile height of:(186)

The average waveguide parameter, V, is related to theusual waveguide parameter V by^ (19)

Using the average waveguide parameter and the ESI,defined by eqn. 18, the dispersio n expres sions tak e thefollowing forms:

(20a)

(20b)

(20c)

bx{V ) =

b2(V ) =

d{Vb)dV

Vd{b,)_ fCldV

where bst(V ) is evaluated for the step index fibre at V =V.Any error introduced in using this ESI, is caused byneglecting the difference between the higher moments ofthe profile Q M and the higher moments of ESI,SlMst. Wedefine the normalised difference as follows:

8.2 ESI and E) ESI based on the moments of therefractive index profileThe model we use is based on the moments of the refrac-tive index shape function, and is much easier to use thana similar model propo sed in Reference 22.5.2.7 Mom ents definition: The profile mo men ts aredefined as

(21)

Jo ,M+ i dRwheres(R) is the profile shape function

= s(R)R>

which can be rewritten as

(14)

(15)

(16)Since the fibres are weakly guiding,R = r/p (r is the usualradial variable), p is the core radius, n{r) is the refractiveindex distribution, n0 and ncl are the maximum core anduniform cladding refractive indices, respectively.The shape function s(R) has a ma xim um of 1, i.e.0 ^ s{R)^ 1, in the region 0 ) a r e required to specify any refractiveindex distribution [22 ].The ESI requires that the first two even moments (fl 0>ft2) f both the actual profile and the equivalent stepprofile are equal. This condition renders an equivalentstep with a core radius of:

Afsrwhere M ^ 4.M is even and

\M/2M/2 + 1 (22)

In enhancing the simple ESI model, we introduce theeffects of adding only one extra parameter, this is theenhancement parameter AQ 4 .Using eqns. 21 and 22, Aft4 can be expressed morespecifically as:

( 2 3 )

We have, therefore adopted the following functional formfor the dispersion parameter b(V):

Pe = (18a)

(24)This is an enhancement of the simpler expression of eqn.20a where f(V) is our enhanc ement function. We willrefer to eqn. 24 as the Enhanced ESI, or (E)ESI approx-imation forb(V).Table 6 shows AJ4 for several a values of the powerlaw profiles. AQ 4decreases as we approach the step-indexfibre; the correction_ term in eqn. 24 decreases accord-ingly. The size of Afi 4 can be used to decide whether ornot the functionf(V) is required.8.2.2 The (E)ESI and its prediction of disper-sion: Fro m the exact calculation of b{V) for a range ofprofiles, we found the lowest-order polynomial approx-imated for/(K) to be the quadratic

/ ( F ) = 0 . 3 1 3 K - 0 . 0 1 3 K 2 (25)The power law profiles with exponent a have been usedfor testing the (E)ESI model, and the main results aredescribed as follows:

208 IE E PROCEEDINGS, Vol.135,Pt. J, No.3,JUNE1988



norm alised frequency VFig. 7 Comparison between the dispersion curves b, bx, and b2obtained by using:a An exact numerical techniqueb The Enhanced ESIc The simple ESI approximationThe r esults are for the pow er-law profile with a = 8. Vco indicates the limit ofsingle-mode operationexact EESI ESI

1.6r

) 1 2 3 ' Unormalised frequency VFig . 8 As Fig. 7, witha = 2

exact EESI ES I

6 0 r

50

| 30inaS 20a.* 10

10 12 14 \ \ 16wavelength, um-1 0

-20LFig. 9 Total dispersion for the parabo lic index profile fibre (a = 2)

exact EESI ESI

(a) Profiles with a ^ 2. For profiles which range fromthe clad parabola (a = 2) to the step index (a = oo), theerror in predicting the dispersion parameters is negligiblein the single-mode region. Figs. 7 and 8 show the exactESI and (E)ESI dispersion curves for a = 8 and a = 2,respectively.Fig. 9 shows the ESI, (E)ESI and exact total dispersioncurves for the clad parabolic core profile (a = 2). Thecurves were obtained using the equation of the com-ponents of total dispersion, and data of Reference 23,andthe ESI and (E)ESI dispersion parameters b, blt and b2.It is seen that the total dispersion curve predicted by theEESI and the exact curve are, for all practical purposes,the same. The wavelength of zero total dispersion(X0TD )is predicted exactly by the (E)ESI, while the X0TD is inerror by 5.4% for the simple ESI.(b )Profiles with a < 2. For these extreme profiles ithas been found that the (E)ESI provides the bulk of thecorrection term for waveguide dispersion in the range ofK-values for which the simple ESI breaks down. Fig. 10

normalised frequency VFig . 1 0 As Fig. 7 with a = 1

exact EESI ES I

-20LFig . 11 Total dispersion for the triangular index profile fibre (a = 1)

exact EESI ESIIEE PROC EEDINGS, Vol. 135, Pt. J, No. 3, JUNE 1988 209



illustrates the triangular profile (a = 1) case. For the tri-angular profile, the (E)ESI provides an estimate of thewaveguide dispersion paramater, b2(V), of 92% of itsexact value at the cutoff of the second mode, andimproves rapidly for smaller K-values. In this case, theestimate ofbandbxare accurate.

Fig. 11 shows the ESI, (E)ESI, and exact total disper-sion curves for the triangular core profile fibre. The(E)ESI total dispersion curve is in good agreement withthe exact calculation, and X0TD is the same for both theexact and (E)ESI curves. The ESI approx imation predictsthe zero of total dispersion with an error of 6.7%.

ErratumTWYNA M, J.K., and WO ODS , R.C.: 'Current crowdingeffects in GaAs/AlGaAs heteroj unction phototra nsistors',IEE Proc. J, Optoelectron., 1988,135,(1),pp. 52-55Section 1, paragraph 2, line 5 should read '

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