1492 J. Opt. Soc. Am./Vol. 71, No. 12/December 1981
Multimode graded-index optical fibers: comparison of twoWentzel-Kramers-Brillouin formulations
Gunnar Jacobsen
Electromagnetics Institute, Technical University of Denmark, DK-2800 Lyngby, Denmark
Received February 23, 1981; revised manuscript received July 10, 1981
Light propagation in graded-index optical fibers is described by two WKB methods, a classical formulation andthe Langer-McKelvey approach. Explicit errors in these methods are evaluated for a fiber profile of near-parabol-ic shape and for one far from parabolic shape by comparing the methods with the results of evanescent wave theorythat involve asymptotic error estimates. The most accurate formulation in general is the classical method, but inspecial cases the Langer-McKelvey approach is slightly more accurate. One example shows an inaccuracy of-672.5 to -672.1 psec/km in modal group delay for the latter method when the maximum intermodal group delaydifference is 2.84 nsec/km.
BACKGROUND AND INTRODUCTION
Two different WKB formulations were used recently forevaluation of modal propagation-characterized by meansof modal propagation constants and modal group delays-ingraded-index optical-fiber profiles. One method is the clas-sical WKB formulation,1' 2 which uses an oscillating radialeigenfield between two caustics for determining these quan-tities. This method has been able to describe measurements. 3
The other WKB method is the so-called Langer-McKelveyapproach4' 5 formulated for graded-index optical fibers in Ref.6. In the latter method one radial eigenfunction is used in auniform asymptotic formulation that reaches from the fiberaxis past the internal modal caustic (for azimuthal modenumber v > 0) into the oscillating region.6 One recent ap-plication of the Langer-McKelvey approximation was givenby Olshansky.7 No explicit comparison of the two formula-tions has been possible so far because a direct comparisonrequires a more accurate reference solution that has not beenavailable.
In 1977 a new high-frequency formulation of the propaga-tion problem was stated in evanescent wave theory.8 Thisformulation is solvable in analytical form for special graded-index fiber profiles to high order in a large parameter of theproblem, the large parameter being k (- the free-space wavenumber).9,IO From these solutions asymptotic error estimatesfor a modal propagation constant or group delay can be givenas a maximum value and a minimum value that are known toinclude the exact result. By using nonlinear transformationssystematically"l 2 it was verified that the series for the modalpropagation constants or group delays in terms of powers of1/k are given well within an exponential behavior, and bysuitable use of transforms the asymptotic error estimates areshrunk consistently to give rather accurate results for the 40lowest-order modes of a near-parabolic untruncated fiberprofile and for the 42 lowest-order modes of a profile far fromparabolic shape."" 3 From a comparison of the transformedresults with results of the classical WKB formulation it hasbeen possible for the first time to our knowledge to specifyexplicitly the errors involved in the latter method.'4 As ex-pected, it turns out that the WKB model fails most drasticallyfor the fundamental mode,,where the two modal caustics are
not properly separated, whereas the method works better formodes of higher order."" 4 Also, the method is less accuratefor a profile far from parabolic shape than for a profile closeto a parabola, which is to be expected because in the (non-truncated) parabolic case the WKB results are exact.
It is our aim to compare the Langer-McKelvey formulationwith the evanescent wave theory results of Refs. 11, 13, and14 and to use our results for a detailed comparison betweenthe two WKB formulations. In Ref. 15 the classical WKBformulation was compared with the corrected method of 01-shansky' 6 in order to determine the influence of a fibercladding. Only modes with azimuthal mode number v > 0were considered, and only a truncated parabolic index profilewas considered. Also, the Langer-McKelvey formulation wasused for truncated nonparabolic ae profiles' and comparedwith a numerical power-series expansion method. The twoWKB methods were not compared directly.
Section 1 of this paper lists the formulas to be compared,and Section 2 provides detailed numerical comparisons of thetwo WKB formulations. Conclusions are given in Section3.
1. ANALYTICAL RESULTS
The untruncated optical-fiber profiles considered throughoutthis paper are of the form
At the wavelength A evanescent wave theory gives asymptoticexpansions for propagation constants-taken as solutions ofthe scalar-wave equation-as'0
or
fP 2 =E Bi(no, ao, a,, g, v)
A/V = E pi(no, a0, a,, gA, P)j=0 k
(2)
(3)
where /,,. is the effective modal refraction index for a modehaving radial mode number p and azimuthal mode numberv. Thus (g, P) are nonnegative integers, and (g, P) = (0, 0) is
the fundamental mode. The quantities #,,2 and OA. areasymptotically expanded with respect to 1/k, where k = 27r/Xis the free-space wave number and the Eqs. (2) and (3) areconnected by
Bi = L Pipj-i.j=o
(4)
We have evaluated Bo - B8 of Eq. (2) analytically, solvingeikonal and transport equations successively on the com-puter.9"0
From Eqs. (2)-(4) we obtain by formal differentiation twoalternative formulas for the modal group delays, namely,
1 (2 - i)Bi / 1 E ) 1/2TAP, IkZj .1
c Li=O 2ki / 1 oJ
and1 at(1-i)pC 6
T,.POpi,(6)c i=o hi
with c specifying the free-space velocity of light.From the classical WKB approximation we obtain the fol-
lowing eigenvalue equation for the modal propagation con-stant 13P1-3:
where the turning points r1 and r2 are defined as zeros of theintegrand. At the turning points (caustics) the eigenfieldchanges from exponentially decaying behavior to oscillatingbehavior. In the special case of azimuthal mode number v =0, the inner caustic r1 vanishes, and Eq. (7) reads (followingRefs. 2 and 3)
From formal differentiation we obtain, for determination ofthe modal group delay r, after determination of 0,,, using Eq.(7),
TAP
1 f n2(r) / 1k 2n
2 (r) -k 20AY
2 -1_ dr
1 JriI[ - r2]
I/f3Al,,/1k2n2(r) -k 2flA2_ 2 1-2 dr
J r i r r21
- (9)
with Eq. (9) modified according to Eq. (8) for v = 0. TheLanger-McKelvey formulation gives, corresponding to Eq.(7), for implicit determination of fl,, (Ref. 6)
where rt denotes the zero of the integrand. For v = 0, Eq. (10)is identical with the classical WKB formulation [Eq. (8)]. Forv > 0, Eqs. (7) and (10) differ mainly in that the inner causticr, of Eq. (7) is not present in Eq. (10) because in this case theradial modal eigenfunction describes both the exponentiallydecaying behavior 0 < r < r, and the oscillating behavior forr1 < r < r2 (crt). By formal differentiation one obtains forthe modal group delay
JO ! s n2(r)/[k2n2(r) - k2 3l, 2 ]1/ 2 drft =-- . (11)
For v = 0, Eqs. (9) and (11) are identical.At this point it is worth noting that for a parabolic fiber
profile [a, = 0 of Eq. (1)], the two WKB formulations andevanescent wave theory both yield the well-known exact re-sults using Eqs. (7)-(11)
and
0,, = [no 2 - 2(2A + v + l)ao/k]l/ 2
kno2 - ao(2A + v + 1)T ck[no2 - 2(2A + v + 1)ao/k] 1/2
(12)
(13)
It is noteworthy that all modes within a mode group with
2, + v + 1 = H = constant (14)
are degenerate; they have identical values of f3, and TAP.For a nonparabolic profile the Langer-McKelvey approx-
imation still retains the parabolic degeneracy within modegroups H = constant [see Eqs. (10)-(14)]. For H odd, all-modes in a group have values of O,, and TAP that correspondto (,u v) = [(H - 1)/2, 0] of the classical WKB approximation.In the following section this will be commented on using nu-merical examples.
In Ref. 7 it is concluded that the two WKB formulationsgive the same condition for minimal pulse dispersion for re-fractive-index profiles of the a type.' This can most probablybe traced back to the approximative treatment of the classicalWKB formulation used to obtain analytical expressions ofmodal propagation constants and group delays.1"' 7 The ap-proximations in this step transform the radial integrationbetween r1 and r2 of Eq. (7) to radial integration between r =0 and rt of Eq. (10).1 In the final results the degeneracy be-tween modes of the mode groups of Eq. (14) is also kept fornonparabolic profiles.1"' 7 Thus this procedure is closely re-lated to the approximation involved in the Langer-McKelveyformulation and explains the conclusion reached in Ref. 7.
2. NUMERICAL EXAMPLES
We consider two profiles of the class defined by Eq. (1)with'0,1""13,14
Profile 1: no = 1.5, ao = 6.207 X 10-3, m-1,a, = 5.215 X 10-4, m-2 ,
Profile 2: no = 1.5, ao = 2.5 X 10-3 nm-1,a, = 5.0 X 10-3, m-2
at the wavelength
X= 0.86 Am,
and for the free-space velocity of light we use
c = 2.9979 m/sec.
(15)
(16)
(17)
(18)
First let us comment briefly on the numerical accuracy of ourcomputer programs. By comparing direct numerical evalu-ation of Eqs. (7)-(9) with Eqs. (12) and (13) for a parabolicindex profile we have found relative numerical inaccuraciesfor determination of modal propagation constants to be betterthan 10-7 for the 40 lowest-order modes with modes of highestorder specifying the worst cases.14 Similar calculations formodal group delays specify accuracies of 0.5-1 psec/km.Using a new numerical implementation of the Langer-McKelvey approach, we have for the same parabolic case
1494 J. Opt. Soc. Am./Vol. 71, No. 12/December 1981
Table 1. Propagation Constants of Evanescent WaveTheorya
jC(EWT)
p min max
0 0 1.49942089562577 1.49942089562619 4.2 X 10-135 0 1.492768480 1.492768587 1.07 X 10-7
a (p, iv) are mode numbers.
Table 2. Group Delays of Evanescent Wave TheoryaTr,,(EWT) A'rv
found relative inaccuracies in determining %,, to be smallerthan 3 X 10-12; on s,> the inaccuracy is smaller than 0.003psec/km. The differences in accuracies of the two imple-mentations can be removed by a careful reimplementation ofthe classical approach. However, for our purpose the accuracyof this method is sufficient as it stands.
For evanescent wave theory [Eqs. (2)-(6)] no numericalintegrations are required, and the direct evaluation of thepropagation constants and group delays is therefore numer-ically accurate. Improvements of the results for profile 1 [Eq.(15)3 by means of suitable nonlinear transforms still give rel-ative inaccuracies of O,,, smaller than 10-14 and accuracies forgroup delays better than 10-20 sec/km.14 For profile 2 thecorresponding accuracies are better than 10-1o and 0.001psec/km.
From evanescent wave theory and a third-order nonlineartransformation we have computed propagation constants andgroup delays of the lowest-order modes of profile 1. Tables1 and 2 show results for propagation constants and groupdelays of the fundamental mode and for the mode of thehighest order considered, namely, mode (g, v) = (5, 0). Thetables show asymptotic error estimates AO,,V and ATTI. Thelargest intermodal group delay difference is seen to be r5,0 -To,o = 2.84 nsec/km.
In Table 3 we list a detailed comparison among the resultsof evanescent wave theory, the classical WKB formulation(superscript WKB), and the Langer-McKelvey formulation(superscript LM) for 36 modes of profile 1. The modes arearranged with decreasing propagation constants and areidentified by their radial and azimuthal mode numbers (g, a')and compound mode numbers H [Eq. (14)]. For the WKBformulations we list differences between WKB and evanescentwave theory (EWT) results for propagation constants and groupdelays. We also give asymptotic error estimates for modalpropagation constants whenever they are larger than 10-7 andfor group delays when they are larger than 0.1 psec/km. Thismeans, for example, for mode (A, v) = (3, 4) that the tablespecifies the modal group delay calculated by the classicalWKB method to deviate between 4.5 and 5.2 psec/km fromthe exact value. The asymptotic error estimates are largest
Gunnar Jacobsen
for high-order modes and most significant when modal groupdelays are considered. For propagation constants calculatedby the classical WKB method (column 5) we see how the re-sult, as expected, deviates most from the exact value for thefundamental mode (3.1 X 10-6), whereas the deviation forhigher-order modes decreases to the order of 2.1 X 10-6. Thedegeneration within mode groups in the Langer-McKelveyapproximation leads to much larger deviations (column 6),especially for modes with large v values. See, for instance,mode (p, ') = (0,10), which deviates 249 X 10-6 from the exactvalue. For modes with v = 0, the two WKB methods agreenicely, as expected. Only the modes (ji, a) = (0, 1), (A' a) =(1, 1), (.a, v) = (2, 1), (A, v) (3, 1), and (p ') = (4, 1) have O.",(LM)closer to the exact value than fl",(WKB). It is seen that for afixed value of P > 0 the Langer-McKelvey result approachesthe exact value when p is increased. See, for instance, for v= 2 how the deviation between mode (0, 2) and (4, 2) decreasesfrom 9.2 X 10-6 to between 7.9 X 10-6 and 8.0 X 10-6 .
The problem of the uniform Langer-McKelvey approxi-mation for large azimuthal mode numbers has been pointedout by Arnold.' 8
Similar comments apply to columns 7 and 8 of Table 3,which deal with group delays. As expected, the classicalWKB formulation is least accurate for the fundamental mode(deviation 9.4 psec/km), and for higher-order modes the de-viation decreases to 2.6-3.3 psec/km for (y, V) = (5, 0). TheLanger-McKelvey formulation agrees with the othermethod-within the numerical errors stated earlier in thisparagraph-for all modes having v = 0. For large v values thedegeneracy within mode groups causes large errors. [Seeagain (ji, a) = (0, 10) with deviation between -672.5 and-672.1 psec/lkm.] The deviations are comparable with themaximum intermodal group delay difference (-2.84 nsec/km).This makes the application of the Langer-McKelvey formu-lation for pulse-spreading calculations questionable, whereasthe errors of the classical formulation-as was concludedearlier' 4 -are 2 orders of magnitude smaller and do not con-tribute significantly to errors in pulse-spreading evalua-tion.
Again for modes having (gi, v) = (0, 1), (A, ') = (1, 1), (p, ')= (2, 1), (A, a) = (3, 1), and (A, v) = (4, 1) the Langer-McKel-vey results are closer to the exact value than the classical WKBresults. Improvements are of the order of 2 psec/km. For v> 0 the deviation decreases for increasing Ai values [see zt = 2,where the deviation for mode (0, 2) is -28.6 psec/km and formode (4, 2) it is between -22.8 and -22.0 psec/km].
From the calculations performed here it appears that theuse of the classical WKB formulation for a truncated parabolicprofile in Ref. 15 cannot be expected to be more accurate thanthe Langer-McKelvey formulation. For truncated nonpar-abolic profiles it is, in general, recommended to use the clas-sical formulation instead of the Langer-McKelvey procedure,although the matching procedures at the two caustics and thecore-cladding boundary are more tedious to perform.
For profile 2 [Eq. (16)] far from parabolic shape we nowsummarize a discussion similar to the above one for profile 1.Results published earlier 1l show that the classical WKB ap-proximation agrees with the asymptotic error estimates forEWT for propagation constants of all the 42 lowest-ordermodes except the fundamental one and for group delays of allmodes except the fundamental one and the one in which (m,
Table 3. Comparison among Results of Evanescent Wave Theory (EWT), the Langer-McKelvey Approximations(LM), and Classical WKB Theory (WKB) for Propagation Constants and Group Delays0
M Mode ordered after decreasing propagation-constant values. (u, v) are mode numbers and H = 2v + v + 1.
v) = (0, 1). If we assume that the fundamental mode specifiesthe largest deviation between classical WKB theory and theexact result, this indicates a relative inaccuracy of propagationconstants smaller than 1.1 X 10-5 and for modal group delaysan inaccuracy smaller than 30 psec/km.
The Langer-McKelvey formulation agrees with the otherWKB formulation, within the numerical accuracy, for modeshaving v = 0. For mode (g, v) = (0, 1) both the propagationconstant value and the group delay value are within theasymptotical error limits of EWT, in contrast to the groupdelay result of the classical formulation. For all other modeswith v = 1, results agree with EWT. This indicates for v = 1that the Langer-McKelvey approach might be more accuratethan the other WKB formulation-as was the case for profile1-but no explicit improvement numbers can be given here.For large v values the Langer-McKelvey results deviate fromthe error estimates of EWT. The worst case is (A, v) = (0, 10),where the deviation 1 3olo(EWT) - 1 3o0 lo(LM) is between 0.611 X10-3 and 1.046 X 10-3 and in group delay it is ro,1 o(EWT) -
7 0 ,10(LM), between -0.86 and -3.42 nsec/km. This error iscomparable with the maximum intermodal group delay dif-
ference that, from the EWT results of Ref. 11, is estimated tobe between 10.3 and 8.4 nsec/km and, from the classical WKBmethod, is found to be 9.1 ns/km". As was the case for profile1, the deviation grows for increasing v values within a modegroup 2g + v + 1 = constant [Eq. (14)].
3. CONCLUSIONS
By using evanescent wave theory for the modal propagationconstants and group delays of untruncated graded-indexoptical fibers as a reference solution, it has been possible forthe first time to our knowledge to determine errors in theLanger-McKelvey formulation of the problem. Here, twofiber profiles, one of near-parabolic shape and another farfrom parabolic shape, have been considered. In general, theLanger-McKelvey approach results in degeneracies withinmode groups 2g + v + 1 = constant. This degeneracy leadsto large errors for modes having large v values in nonparabolicfiber profiles. For our example of a near-parabolic profile theerrors are up to 249 X 10-6 in propagation constant values and-672 psec/km for group delays at X = 0.86 Am when the 36
Gunnar Jacobsen
1496 J. Opt. Soc. Am./Vol. 71, No. 12/December 1981
lowest-order modes are considered. For the profile far fromparabolic shape similar errors are between 0.611 X 10-3 and1.046 X 10-3 and between -0.86 and -3.42 nsec/km when the42 lowest-order modes are considered.
The classical WKB formulation has been evaluated ex-plicitly with respect to errors made earlier'4 for the same twoprofiles. For the near-parabolic profile we find propagationconstant errors below 3.1 X 10-6, and for group delays errorsbelow 9.4 psec/km. For the profile far from parabolic shapethe errors are below 1.1 X 10-5 and 30 psec/km, respectively,assuming that the fundamental mode is the worst case."
Comparing the two WKB methods we find, as expected, forv = 0 that they agree within the numerical accuracy of thecomputer implementations. For v = 1 the Langer-McKelveyformulation lies closer to the exact result than the othermethod for modes of the near-parabolic profile with im-provements of the order of 1.5 X 10-6 for propagation con-stants and 2 psec/km for group delays. However, for v > 1 theLanger-McKelvey results for both propagation constants andgroup delays deviate from the exact values by up to 2 ordersof magnitude more than the classical WKB results deviate.For a profile far from the parabolic shape and v = 1 bothmethods yield propagation-constant results within the as-ymptotic error estimates of EWT."" 13 The group delay for(p, a) = (0, 1) is within the EWT estimates when the Lan-ger-McKelvey approach is used and outside it when theclassical formulation is used. For larger u values and a = 1both methods fall within the EWT estimates. For larger avalues the Langer-McKelvey approach gives rather inaccurateresults that are outside the EWT estimates in which theclassical method agrees with EWT.
For practical use in calculations of multimode pulsespreading and pulse response for fibers in which the inter-modal group delay differences are significant 3 our investiga-tions here indicate that the classical WKB approximation isapplicable, whereas the Langer-McKelvey formulation seemsto be questionable.
ACKNOWLEDGMENTS
The Danish Government Fund for Scientific and IndustrialResearch supported this work. The author thanks W.Streifer, Xerox Corporation, Palo Alto, for contributing dis-
Gunnar Jacobsen
cussions. One referee is thanked for helpful suggestions forimproving the form of this manuscript.
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