Evanescent-wave and nonlinear transformation analysis of graded-index fibersGunnar Jacobsen
Electromagnetics Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark(Received 19 February 1980)
For a general class of graded-index fiber profiles, modal propagation constants and group delayswere previously evaluated asymptotically to 0 (k-9 ) by using the systematic calculation scheme ofevanescent wave theory. Here, the numerical results obtained directly are improved by using anonlinear transformation technique developed by Shanks. This technique extracts information fromthe diverging as well as converging part of an asymptotic series, whereas the direct method utilizesthe converging part alone. We obtain accurate results for a profile far from parabolic shape as well asfor one of near-parabolic shape. For the profile far from parabolic shape the results quantify theerror in group delay for the WKB technique for the lowest-order mode to approximately 30 ps/km.For high-order modes the WKB results agree with the transformed results.
INTRODUCTION
Evanescent wave theory' has been used as an efficient toolfor evaluation of accurate values of modal propagation con-stants and group delays for a general class of graded-indexfiber profiles.2 In order to apply the method, it is necessaryto solve transport equations to high order in the large pa-rameter k (- the free space wave number) for the problem,and this has been possible because the formulation of Ref. 1provides a systematic calculation scheme that has been solvedusing an analytical algebraic computer code (FORMAC3).For low-order modes of a near parabolic profile accurate re-sults were obtained using the high-order expansion directly. 2
High-order modes were specified via larger error intervals. Asimple linear transformation improved the results obtainedfor the higher-order modes of the near parabolic profile.4
When profiles of a shape that is far from parabolic wereconsidered, the asymptotic series, giving modal propagationconstants, were diverging from low order in 1/k and the errorintervals specified were so large for all modes that no infor-mation was contained in the results. The linear transfor-mation did not improve the results in this case.
Here it is shown how nonlinear transformations of the typegiven by Shanks 5 are useful for extracting the basis of theasymptotic series that specify propagation constants andgroup delays. The nonlinear transformations which use allterms of the series considered, both in the convergent and thedivergent part of the series, have to be applied with care. Itturns out that the application of a simple successive first-ordernonlinear transformation scheme is very efficient for the caseof a profile far from parabolic shape, whereas higher-ordertransformations have to be applied in the case of a near par-abolic profile. When using the transformations in the caseof a profile far from parabolic shape, very tight error intervalsfor the propagation constants of low-order modes are speci-fied, whereas the intervals are larger for higher-order modes.Similar results are obtained for modal group delays. Acomparison to results obtained using a previously developedWKB calculation method6 specifies the inaccuracy of thelatter method for low-order modes explicitly. For the profileof near parabolic shape we demonstrate that the applicationof the nonlinear transformation schemes is more difficult thanin the case of a profile far from parabolic shape.
Section I contains a brief review of evanescent wave theory(EWT) combined with the nonlinear transformations usedhere. Section II contains detailed results for the modalpropagation constants and group delays of the two fiber pro-files considered. In Sec. III the results are compared with aWKB calculation method. Finally, conclusions are given inSec. IV.
1. THEORY
Here, it is the intention to present explicitly the applicationof the nonlinear transformation method to the results of theevanescent wave theory.", 2 A detailed review of the trans-formation method is given in Ref. 5.
From previous workl we have, for untruncated fiber pro-files
n2(r) = no- ar2(1 air2)2 0 < r < (1)
where r denotes radial distance, no is the on-axis refractive-index value, ao a positive real constant, and a I a real constant,determined modal eigenfield solutions to the scalar waveequation
[V2 n 2(r)k 2] u(r) = 0, (2)
where in a cylindrical coordinate system, with z giving thepropagation direction,
r = (r,O,z). (3)
Here, k is the free-space wave number and u (r) is the modaleigenfunction. Each mode is distinguished via a radial modenumber yu and an azimuthal mode number v, ,p,v being non-negative integers. The modal eigenfunction specifies prop-agation along the z axis (- the fiber axis) via the propagationconstant 3,, that is given on normalized form (correspondingto the effective modal refractive index) as'
2 "= E Bi(noaoal,,A) I(4)A i=0 k 4
where in the asymptotic series of Eq. (4), Bo - B8 has beenevaluated analytically. 2 From Eq. (4) we obtain the sequenceof partial sums:
Note that we consider O3, instead of f directly in Eqs. (4)-(5).Instead of transforming the results of Eq. (4) for 32 we havealso considered transformations of
pi(noao,al ,,v)i0 (6)
where the connection between Eq. (4) and Eq. (6) is givenby
Bi= pjpi-j. (7)H=o
From Eq. (6) we obtain a sequence of partial sums similar toEq. (5). The partial sum of the series for 32 and 3 will now berecalculated using the nonlinear transforms developed byShanks.5 The mathematical details for the transformationsare given in an appendix. As explained below, the nonlineartransformations are able to filter out "spatial transients," i.e.,if the successive sequences are well described within theframework of exponential behavior, we may improve theconvergence of a series significantly using the transforms. InSec. II we see that this assumption concerning the sequentialbehavior is justified.
A transform of order one [denoted e1 (Ben) for the sequenceof Eq. (5), see Appendix] is able to filter out one transient,whereas a transform of order k > 1 [denoted ek (Ben)] can filterout k or less transients.
Considering Bej (j = 0, 1, . . . , 8) of Eq. (5) directly, 2 we letthe two partial sums that have the least absolute differencespecify the asymptotic error interval of the calculation. Wenow consider ek (Bej) instead of Bej and look at the error in-terval specified in this way. To my knowledge there is nomathematical proof that justifies the assumption that ek (Bej)specifies an error interval asymptotically when Bej does this,but we see below that the results obtained with this assump-tion are meaningful. We furthermore, whenever possible, alsoconsider successive transforms like ek (ek (Bej)), etc. and theerror intervals obtained from doing this.
In Sec. II we consider profiles of Eq. (1) with ao, a, > 0.This case specifies an oscillating behavior of Bej whenever i> 1 and since the general behavior of the summation is verydifferent for j = 0,1 we apply the transformation for j > 1 anduse Be1 (the parabolic part of the result) as the first memberof the sequence to be transformed. We thus have Be1,...,Be 8 for transformations. This allows us to make three suc-cessive first-order transformations or one second-order orthird-order transformation that contains an error interval.
For the modal group delay
1 6 (k 2f 2) 2_ 1' ___1_
/2k =- n2 + _ -c (3kk) / c(=1 2 ij o= hk,(8)
or
1 pi(1-i)Typ = E - .o
c i=O kit
obtained in this way for the denominator is given by the pos-itive numbers D 1, D 2 (D1 < D 2) and the interval given for thenumerator is given by N1 , N 2 (N1 < N2 ) we let the intervalgiven by N 2/D1 , N 1 /D 2 be a worst-case error estimate. Theapplication of Eqs. (8)-(9) will be commented upon in Sec.II.
11. NUMERICAL RESULTS
Earlier 2' 4 we have considered two profiles of parabolic shapenear the axis [given in Eq. (1)]. Profile 1 was nearly parabolicwith
no = 1.5, ao = 6.207 X 10-3 gm-l,
a, = 5.215 X 10-4 y-2 (10)
and profile 2 was far from parabolic shape with
no = 1.5, a0 = 2.5 X 0-3O ,m- 1, a, = 5.0 X 10-3 pm-2 .
(11)
For profile 1 we obtained rather accurate results with mean-ingful error estimates for modal propagation constants usingEqs. (4)-(5) directly for all the 40 lowest-order LP modesconsidered.2 The results were improved and smaller errorintervals were specified for the modes of high order using asimple linear transformation, whereas this transformation didnot improve the results for the lowest-order modes.4 Forprofile 2 neither the direct summation of Eqs. (4)-(5) nor thelinear transformation of Eq. (4) gave results with a reasonablysmall error interval. Similar conclusions were drawn withrespect to the modal group delays of the modes of profiles 1and 2.
Let us now look at profile 2 since the transformation methodis most easily demonstrated here. This will create the basisfor application of the transformation of the results of profile1.
For all the modes of profile 2, except for the fundamentalone LPoo, the tightest error estimate of the direct summationresult of Eq. (4) is specified via Be1 ,Be2 [Eq. (5)]. (For LPooit is specified via Be2,Be3, see Table I.) In Fig. 1 is shown howin principle the sequence Be1,...,Be8 behaves. It appears
1 2 3 4 5 6 7 8
j
FIG. 1. Sketch of Be, - Be8 [Eq. (5)] for a mode of profile 2.
Gunnar Jacobsen 1339
U)
-d
Ed_
(9)
where c is the free-space velocity of light, the application ofthe transformations is more difficult. We may transform Eq.(9) directly or for Eq. (8) we may transform the square of thedenominator and the numerator directly. If the error interval
1339 J. Opt. Soc. Am., Vol. 70, No. 11, November 1980
TABLE I. Three successive first-order transforms for 32 of mode LPOO (profile 2).
Basisa One transforma Two transforms 5 Three transforms
from the figure that a simple first-order nonlinear transfor-mation, which is able to filter out one transient, 5 should revealthe true basis of the sequence. Using three successive first-order transformations we obtain the results for mode LPooshown in Table I. Here column 1 shows Bel,...,B 8, column2 shows results of el(Bej), column 3 shows results ofe1(e1 (Bej)), and column 4 shows the final result after threesuccessive transformations. Asterisks in columns 1, 2, and3 show terms that specify tightest error estimates. Note thatthe error interval of the direct result (column 1) contains theone obtained by one transformation, which contains the oneobtained by two successive transforms, which again containsthe one obtained by means of three successive transforms.Thus tighter and tighter error intervals are specified via suc-cessive transformations. Since the sequence of Table I col-umn 1 is first converging and thereafter diverging, one mayargue that at least two transients are present and have to befiltered out by means of a second-order or even higher-ordertransformation. However, as will be clear from the followingcalculations, the presence of more than one transient appearsto be of no serious harm in this case.
Evaluating 3 instead of 32 we obtain Table II, where it ap-pears that the first error interval is almost two orders ofmagnitude smaller than the one given by the original se-quence.
We obtain similar results using three successive first-ordertransformations for all other modes of profile 2. In TablesIII and IV are shown conclusive results for mode LP50 wherewe improve the accuracy of the results by one order of mag-nitude (again asterisks in table denote tightest error esti-mates). LP5 0 is a mode of highest order considered and here,we obtain a broad error interval. For this mode the power ofthe method of transformation appears very clearly from TableIII where the three successive transformations are able toextract information even from the rapidly varying terms ofthe sequence of column 1.
A second-order nonlinear transformation is able5 to filterout two transients and should give results that agree with theones obtained from a first-order transformation when only onetransient is present. So should a third-order transformation.In Table V is shown the result of a second- and third-ordertransformation applied on Be,... ., Be8 of mode LPoo. Againasterisks denote tightest error estimates. It is seen fromBe, . . . , Be8 that we can obtain only one transformation ofsecond order since each step requires five terms in the se-quence, and one transformation of third order since each step
1340 J. Opt. Soc. Am., Vol. 70, No. 11, November 1980
requires seven terms in this case. Because the third-ordertransformation uses two terms more in each step, it is expectedto be more accurate than one second-order transformation andthis is indeed the case. We see from Table V that the tightesterror from the second-order transformation contains the oneof the third-order transformation, which (see Table I) containsthe one obtained by three successive first-order transforma-tions. In Fig. 2 is shown the placement of error intervals for/2 of mode LPoo using evanescent wave theory directly andthe nonlinear transformations of Tables I and V. We see thatthe results obtained agree with one another, which shows thatone transient is dominating in the behavior of Be,, . . ., Be8.The figure clearly demonstrates the advantage of using threesuccessive first-order transformations over the other possi-bilities. Similar conclusions appear for all other modes ofprofile 2 and this establishes confidence in the application ofthree successive first-order nonlinear transforms to improvethe direct result of Eq. (4) in this case.
Application of similar transforms on sequences definedfrom Eq. (6) for : gives results for error estimates that agreevery closely with the intervals already presented using Eq. (4).All intervals specified lie to the order of 1% of the intervallength within the ones given so far. Therefore, detailed nu-merical results are not presented here.
Turning to the modal group delays, a plot of any of the se-quences of the numerator of Eq. (8) shows a similar behaviorto the one showed in Fig. 1. Application of transforms oforder 1, 2, and 3 give results that fit nicely into one anotherwith respect to error intervals and again three successivefirst-order transformations specify the tightest error esti-mates. However, application of Eq. (9) specifies tighter in-tervals in agreement with the ones found using Eq. (8) withthe best result obtained via three successive first-ordertransforms. In Sec. III these results are compared with WKBcalculations.
For all the modes of profile 2 we see how three successive
TABLE II. / of mode LPoo (profile 2).
LPoo 01 /2 1 all/:
Direct 1.499750 1.499647 6.87 X 10-6
Nonlinear 1.499 688 381 1.499 688 055 2.17 X 10-7
transform
Gunnar Jacobsen 1340
TABLE ll. Three successive first-order transforms for 32 of mode LPso (profile 2).
Basisa One transforma Two transformsa Three transforms
first-order transformations specify much improved results forpropagation constants and group delays compared with thedirect result of the asymptotic series. It should be noted thatthe transformations use all the terms in the divergent part ofthe original series. Therefore the systematic calculationscheme of the evanescent wave theory is of importance sincethe calculation of more terms of Eq. (4) beyond B8/k8, possiblyby means of more successive transformations, may improvethe results even more. In principle any desired calculationaccuracy can be obtained in this way.
Now, let us look at the propagation constants of the modesof index profile 1. The principle behavior of Be,...,Be 8 inthis case is shown in Fig. 3. It is seen that at least two tran-sients-one that specifies the convergent part of the sequencea and one that specifies the divergent part-are present, andthis makes the application of successive first-order transfor-mations questionable. Indeed for higher-order modes, wherethe presence of the transient of the diverging sequence be-comes more and more significant, the error interval specifiedby three successive first-order transformations is not con-tained within the one obtained by one third-order transfor-mation.
Let us now concentrate on the application of second- andthird-order transformations. For all modes the third-ordertransformation specifies tighter error estimates than thesecond-order one and the intervals are completely containedwithin each other. In Table VI this is shown for 32 of modeLPoo. Again asterisks denote tightest error estimates. Thepropagation constant of this mode and of mode LP50 is shownin Table VII compared with the direct result, which is im-proved by two orders of magnitude in both cases. From theresults of Table VII compared with the result of Table I in Ref.4, it appears that the error interval of Ref. 4 is specified cor-rectly by the two terms at N = 6 and 8. This gives an intervalthat is tighter than the direct one but less tight than the oneobtained in Table VII. The application of one third-ordernonlinear transformation is in all cases more efficient than theapplication of the linear transformation of Ref. 4.
TABLE IV. 3 of mode LP50 (profiie 2).
LP5 0 1 02 |Aflh/0
Direct 1.497 1.486 7.3 X 10-3Nonlinear transform 1.493926 1.493 225 4.70 X 10-4
1341 J. Opt. Soc. Am., Vol. 70, No. 11, November 1980
As was the case for profile 2 we obtain for profile 1 niceagreement between results for /3 obtained by transforming asequence of Eq. (6) and the sequence of Eq. (4). Numericalresults using Eq. (6) are not given.
For modal group delays of profile 1 a plot of the sequenceof the numerator of Eq. (8) looks like Fig. 3 and a second- orthird-order transformation is expected to work well in thiscase. Transforming this quantity and using the transformedresults for : found earlier, we find error intervals in agreementwith the ones obtained by transforming Eq. (9). The resultusing a third-order transform is most tight and the tightesterror intervals for lowest-order modes are obtained using Eq.(8), whereas the tightest intervals for high-order modes aregiven via Eq. (9). We have chosen here to concentrate onnumerical results of profile 2 and do not list results of modalgroup delays of profile 1 in detail.
Ill. DETAILED COMPARISON TO WKB RESULTS
This paragraph contains a detailed comparison to a previ-ously developed WKB method.6 Earlier, for profile 1 we haveperformed such a comparison. 2' 4 Here, we concentrate onresults for profile 2 where no detailed comparison could beperformed so far. In Table VIII we have listed the evanescentwave theory result of the 42 lowest-order modes for propa-gation constants and group delays obtained by three succes-sive first-order nonlinear transformations. The results arecompared with WKB results. For the modes of lowest order,where the WKB calculation is known to be least accurate, theintervals specified by EWT calculations are very tight andthey do not contain the WKB result. For LPoo the deviation
TABLE V. 32 of mode LP00 specified directly and using second- andthird-order transforms (profile 2). 4
FIG. 2. Error intervals for 32 of mode LP00 (profile 2) specified directly(1), via one (2), two (3) or three (4) successive first-order transforms, viaone second-order transform (5), or via one third-order transform (6).
in group delay corresponds to 30 ps/km. For profile 1 thedeviation is2 10 ps/km and this makes sense because profile1 is chosen closer to a parabolic profile (where both WKB andEWT give exact results) than is profile 2. For higher-ordermodes the error intervals specified by EWT grow and theycontain the WKB result. As mentioned earlier, the EWTresults in this case may be improved by performing the EWTcalculations to higher order in 1/k.
IV. CONCLUSIONS
We have demonstrated the applicability of nonlineartransformation schemes for extracting the basis of the as-
, bass
co
1 2 3 4 5 6 7 8
j
FIG. 3. Sketch of Be1 - Be8 [Eq. (5)] for a mode of profile 1.
1342 J. Opt. Soc. Am., Vol. 70, No. 11, November 1980
�
Gunnar Jacobsen 1342
TABLE VI. 32 of mode LP00 specified directly and using second- andthird-order transforms (profile 1).
ymptotic series that specify modal propagation constants andgroup delays of modes of a general class of graded-index fiberprofiles. Simple, successive first-order transforms are veryefficient for fibers that are far from parabolic in shape wherethe series are diverging from the very beginning and thereforecontain one dominating transient. For a profile closer toparabolic shape the series are first converging and thereafterdiverging. Thus more transients are inherent and transformsof higher order than one have to be applied. For the modalgroup delays special care has to be taken in order to transformcorrectly. It turns out that transformation of denominatorand numerator of the series obtained for -r [Eq. (8)] by formaldifferentiation of the asymptotic series for 32 gives consistentresults. Those results agree with the ones obtained fromtransformation of the series for T [Eq. (9)] given by formaldifferentiation of the asymptotic series for /3. The tightesterror intervals for a profile far from parabolic shape is ob-tained using Eq. (9). For low-order modes of a near parabolicprofile the use of Eq. (8) gives the tightest intervals, whereasthe use of Eq. (9) gives the tightest intervals for higher-ordermodes.
For the profile far from parabolic shape a comparison to aWKB calculation method quantifies the error inherent in thelatter method for modes of low order. For the lowest-ordermode the WKB result for the group delay deviates by 30ps/km from the one obtained by the transformations. Formodes of higher order the WKB results are within the errorintervals specified by the transformations.
The accuracy of the result obtained by the transformationsmay be improved by inclusion of more terms in the asymptoticseries for propagation constants and group delays. Thisemphasizes the importance of the systematic calculationscheme that is inherent in the EWT formulation and the im-portance of the application of the computer for performingsuch simple but tedious systematic calculations.1-3
For a more general profile than the one of Eq. (1) thepropagation constants have been evaluated to 0 (k -4) usingMaslov Theory.7 In order to apply the nonlinear transfor-mation technique efficiently, the results presented heredemonstrate that evaluation to higher order in 1/k is of im-portance.
ACKNOWLEDGMENTS
This work was supported in part by the Danish GovernmentFund for Scientific and Industrial Research. The author
1
TABLE VII. 3 of mode LPOO and LP50 (profile 1).
132
LP0oDirect 1.499420895646 1,499420895622 1.60 X 10-11Third-order transform 1.499420895626 160 1.499420895625856 2.03 X 10-13
LP5 0
Direct 1.492 818 1.492 718 6.67 X i0-5Third-order transform 1.492 768576 1.492 768477 6.60 X 10-8
thanks Professor W. Streifer, Xerox Corporation, Palo Alto, APPENDIXCalifornia and Professor N. C. Albertsen, Technical University Following Shanks 5 and defining [see Eq. (5)]of Denmark, for drawing his attention to Ref. 5. The WKB ABe = Be,+, - Bej, (Al)calculations used in Table VIII were kindly performed by Dr.J. J. Ramskov Hansen, University of Southampton. we have a kth-order nonlinear transform of the sequence
TABLE VIII. Comparison between WKB results and EWT results of profile 2 using three successive first-order nonlinear transformations.
ii V OWKB OEWT1 OEWT2 TWKB X 109 TEWT1 X 109 TEWT2 X 109
1343 J. Opt. Soc. Am., Vol. 70, No. 11, November 1980 Gunnar Jacobsen 1343
around partial sum n given as
Ben-a B
ABen-k ... A1en- 1 ... Ben
Ben- *--ABen
&Be,.,- 1.*-- Be.+h-2 ... Ben+k-1,
1 *.. 1 *.. 1
ABen-k... ABen-1 ... ABen
ABen- *- I..Aen+k-2 ... ABen+k-I
(A2)
where we apply 2k + 1 partial sums simultaneously.
For k = 1 the simplest transform is given as
el(Ben) Ben + AXBenABen-1 (A3)ABe,,-1 - ABen
Let us demonstrate the effect of a first-order nonlineartransform on a sequence having one "spatial transient" inorder to demonstrate the usefulness of the transformationmethod. Thus assume
Ben = A + aqn, (A4)
where A, a, and q are real constants. The sequence of Eq.(A4) may specify a converging or diverging series for differentq values. For the special sequence of Eq. (A4) we obtain
el(Ben) = A + aqn +a(qn+l - qn)a(qn - qn-1)
,(qn - qn-l) - a(qn+l - qn)A n - n (q - 1)(qn - qn-l) A
qnl qn-1 - 2qn
We thus see how a first-order nonlinear transform is able tofilter out one "spatial transient" to give the basis A exactly.Applying a transform of higher order than one to the sequenceof Eq. (A4) we obtain again the result of Eq. (A5). In fact itcan be shown that if the sequence is given in terms of i tran-sients, i.e.,
Ben = A E ajqq (A, caj, qj real quantities), (A6)j=1
a transform of order k > i can filter out all the transients,giving
ek(Ben) = A. (A7)
IS. Choudhary and L. B. Felsen, "Guided modes in graded index op-tical fibers," J. Opt. Soc. Am. 67, 1192-1196 (1977).
2 G. Jacobsen and J. J. Ramskov Hansen, "Propagation constants andgroup delays of guided modes in graded index fibers: a comparisonof three theories," Appl. Opt. 18, 2837-2842 (1979).
4G. Jacobsen and J. J. Ramskov Hansen, "Modified evanescent wavetheory for evaluation of propagation constants and group delaysof graded index fibers," Appl. Opt. 18, 3719-3720 (1979).
5D. Shanks, "Non-linear transforms of divergent and slowly conver-gent sequences," J. Math. Phys. 1-42 (1955).
6J. J. Ramskov Hansen and E. Nicolaisen, "Propagation in gradedindex fibers: comparison between experiment and three theories,"Appl. Opt. 17, 2831-2835 (1978).
7 H. Ikuno, "Propagation constants of guided modes in graded indexfiber with polynomial-profile core," Electron. Lett. 15(23), 762-763(1979).
