mode-dependent attenuation in optical fibers

1282 J. Opt. Soc. Am.iVol. 73, No. 10/October 1983

Mode-dependent attenuation in optical fibers

Alan R. Mickelson and Morten Eriksrud

Electronics Research Laboratory, Norwegian Institute of Technology, Trondheim N-7034, Norway

Received November 8, 1982; revised manuscript received May 9, 1983

A theory with which one can calculate modal distributions within a multimode fiber excited by a source of arbitraryspatial coherence, but within the mode-continuum limit, is presented. The theory is specialized to the case of inco-herent source excitation, which is arbitrarily limited in both spatial and angular extent. The theory is then appliedto the problem of determining the significance of various experimental techniques designed to measure differentialmode attenuation. It is found that standard measurement techniques measure a quantity that differs from the ac-tual differential mode attenuation. It is also illustrated that it can be difficult to calculate accurately the differen-tial mode-attenuation coefficient from measurement data.

INTRODUCTION

Mode-dependent effects cause the results of optical trans-mission measurements in multimode fibers to become lengthdependent' and therefore hard to interpret. Such mode-dependent effects can be caused by a radially dependentvariation of absorption or Rayleigh scattering. Also, modecoupling in the fiber can lead to transient behavior. The ef-fects of mode coupling in high-quality fibers have been shownto have a negligible effect on most propagation characteris-tics.2-4 The mode-coupling coefficients are not zero,5-6however, and for modes that are close to cutoff even a smalldegree of coupling to radiation modes can lead to a noticeabletransient effect. Coupling to radiation modes can also beaffected by adiabatic parameter variations. In the sense thatthese important mode-coupling effects are loss mechanisms,they can be lumped together with the effects of spatiallyvarying absorption and Rayleigh scattering and placed underthe general heading of differential mode attenuation.

Beginning in 1974, the first studies of differential modeattenuation (DMA) in step-index fibers were carried out.4' 7

The measurement technique used in step-index fibers involvesvarying the input beam angle so as to excite single modes toenable one to measure the propagation loss of each modeseparately. The extension of this step-index measurementtechnique to graded-index fibers is not obvious. It has onlybeen recently that single modes have been excited ingraded-index fibers,8'9 and, at that, only a special class ofmodes can be excited by the technique of Refs. 8 and 9. Toexcite, for example, only one of the radially symmetric (azi-muthal mode number v = 0) modes would indeed be a difficulttask, as electromagnetic studies of the coupling problem haveshown.10-' 3 Indeed, the first measurements14 and the theo-ry' 5 of DMA in graded-index fibers involved the study of onlythe total excess loss rather than of the loss of the individualmodes. Subsequently, various restricted mode-volume re-ception and excitation techniques were devised and carriedout.1"'r' 9 However, a study20 of the sensitivity of suchmethods indicates that they measure loss quantities averagedover a large number of modes and not DMA as it is defined ingeneral or as it is measured in the step-index case. Thiscauses DMA curves derived by selective excitation of

graded-index fibers to be misleading as it is not actually DMAthat they display at all.

Accurate determination of DMA can be important to bothfiber technologists and users of fiber-optic systems. DMAcurves say much about the distribution of dopants and pos-sible contaminants within the core region and therefore canbe used as a diagnostic on the fabrication process. As DMAaffects both average loss and dispersion, it becomes a deter-mining factor for fiber-optic system parameters. A solutionto the problem of standardization of fiber-optic measure-ments2 l certainly will require a clear understanding of thevarious consequences of DMA. Accordingly, it is the purposeof the present work to consider the problem of accurate de-termination of actual DMA curves.

THEORY

Mode-Continuum ApproximationThis section contains the basic relations of the mode-contin-uum approximation2 2 for easy reference. It should be pointedout here that, despite the confusion in the literature,2 32 8 themode-continuum approximation is valid for fibers of arbitraryprofile provided only that the linewidth of the exciting sourcesatisifes the following inequality 29 :

- > -AX akNI

(1)

where a is the core radius, A is the relative refractive-indexdifference, k is the free-space wave number, X is the free-spacewavelength, and N, = nj - X (dnl/dX), the material groupindex expressed in terms of the axial refractive index nj.

The total power P(s) crossing an annular core region be-tween s = rna, where r is the radial coordinate, and unity is

dP(S) = ff 1/2( ) p(R)m(R, s)dR. (2a)

when expressed in terms of the mode-continuum param-eter

R = (1 -. 2 /n, 2 k 2 )'/2,2A

(2b)

0030-3941/83/101282-09$01.00 © 1983 Optical Society of America

A. R. Mickelson and M. Eriksrud

Vol. 73, No. 10/October 1983/J. Opt. Soc. Am. 1283

where p(R) is the modal distribution, f(s) is the profile func-tion, which is defined by the expression for the radial variationof the refractive index

n(s) = n[1 - 2Af(s)], (3)

m(R, s) is the modal density, which is defined by the rela-tion2 9

m(R, s) = V2RIVf -'(R2 )]2 - S2}

where V is the fiber's V number, which is defined by

V = s/2A nika,

(4)

(5)

and f3 is the modal propagation constant in the fiber.By using the above relations and the fact that in the

mode-continuum limit the near-field intensity is radiallysymmetric,2 9 it can be shown that the relation between themodal distribution and near-field intensity is

V2 II(s) =T Xt/2 p(R)RdS. (6)

Modal Distribution of Spot ExcitationNow it is necessary to relate the modal distribution in the fiberto the excitation within the limitations of the continuum ap-proximation. Figure 1 illustrates the situation and thequantities to be dealt with. The quantity E(s, t) is the (nor-malized) generalized radiance30 (or specific intensity3 l) of thesource and optical system, where t, the normalized solid anglewithin the fiber, and s, the normalized radial coordinate, canbe defined by

sin 0t =, 2' (7a)

S = -, (7b)a

where 0 is the usual angular coordinate.If one assumes that the coupling is representable as a linear

system, one can write

p(R) = f K(R, s', t')E(s', t')ds'2 dt'2 , (8)

where K(R, s', t') is a radiance (to modal distribution) transferfunction. It should be noted here that within the fiber themodal distribution and the radiance are related by the simpleexpression

E(s, t) = 2AV 2 p hff(s) + t2j11/2 (9)

where [ffs) + t2 ]"/2 is an expression for R in terms of ray-tracing variables. Note that R is one of the constants of themotion along a paraxial ray path that, together with Eq. (9),

I zZz = o

Fig. 1. Diagram indicating the quantities associated with problemof fiber excitation. Note that the incident radiance is denoted E(s,t) and the resultant modal distribution within the fiber is denoted byp(R).

I I I I X\I L I I I I

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 2. Diagram illustrating the forms of the excited modal powerdistribution p(R) and corresponding near-field intensity i(s) inparabolic-index multimode fibers. The near fields correspondingto the modal distributions in (a), (c), and (e) are contained in (b), (d),and (f), respectively.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3. Modal power distributions p(R) and near fields I(s) corre-sponding to varying beam-width excitations.

indicates that the specific intensity at a given value of R is aconstant with a longitudinal coordinate within the fiber.

In an earlier work, 32 intensity transfer functions were em-


1284 J. Opt. Soc. Am.Irol. 73, No. 10/October 1983

pirically derived and experimentally verified for completelyangularly unbounded, incoherent sources (sources that areindependent of t). The form of the radiance transfer functionthat reproduces these results is

K(R, s', t') = 2R 62 [R - f (s') - t'2], (10)m (R)

where 62 is the two-dimensional delta function. This transferfunction has a clear physical meaning. It consists of a modalweighting factor times a function that takes a ray of infini-tesimal spatial extent and zero numerical aperture and placesit in the corresponding mode (ray path) within the fiber.

Figures 2 and 3 illustrate some predictions of Eqs. (8) and(10) for parabolic profile fibers. It should be noted here thatall further considerations will be limited to the case of fiberswith near-parabolic index profiles. The quantities u, p, andr are normalized launch coordinates and denote the offset ofthe beam center from the fiber center, the beam radius(waste), and the beam numerical aperture, respectively. Theycan be defined in terms of the fiber parameters by the rela-tions

(11a)a

abeam (11b)a

NAbeam (1NA (lic)

where rS is the radial launch position, abeam is the half-spotsize, and NAbeam is the effective numerical aperture of thelaunching system. For ease of computation, it is assumed thatthe beam is an image of an incoherent source, i.e., the energydistribution within both the spot and the solid angle is con-stant and is zero outside. Although this model is clearly notcorrect, it is probably not too bad for images of arc lampsources. In the continuum limit, one can find the effectiveradiance on the fiber end face by averaging the spot aroundthe end face of the fiber, that is, by calculating the intersectionof the spot with each circle of constant radius on the fiber endface and using this function (after normalization) as thesymmetrized (and therefore effective) radiance. This pro-cedure, however, is mathematically messy and does not leadto analytical results. Although this calculation is necessaryfor large spots or spots near the origin, it is useful to considera limit of small spot size far from the origin in which one canassume that energy within the excited ring falls off as the in-verse of the radius. Also, if the spot is centered on zero, it iseasy to see that the effective radiance is constant within thespot. In these special limits, the modal distributions can beexpressed, respectively, in the forms

small off-axis spot:


where MIN (MAX) represents the usual minimum (maxi-mum) function and where the subscripts u, r, and p are at-tached to the modal distribution to indicate its explicit de-pendence on all three. The near-field intensity for such anexcitation can be found by substituting Eqs. (12) into Eq.(6).

Figure 2 illustrates the effect of limiting the numerical ap-erture [under the approximations inherent in Eqs. (12)]. Aswould be expected, smaller numerical apertures lead to nar-rower mode distributions and faster roll-off of near-field in-tensities.

Figure 3 illustrates the effect of varying the beam width forfixed spot center and numerical aperture. Although narrowerbeam widths always lead to strictly narrower mode distribu-tions, the near field, although of smaller spatial extent, mayactually appear to be fatter for the more-restricted launch.

As the results of Eqs. (8) and (10) have been verified onlyfor the strictly incoherent case,3 2 it is of interest to comparethose results with experiments for some restricted launches.This is done in Fig. 4. In spite of crude source representationdiscussed above, however, the theoretical-experimentalagreement in Fig. 4 is seen to be quite good. The disagree-ment near the center of the fiber can be attributed to thepresence of a dip in the test fiber, an effect not accounted forin the model. It should be noted here that the calculationspresented in Fig. 4 employed the same approximations asembodied in Eqs. (12).

Differential Mode AttenuationWith the radiance transfer function in hand, it is now possibleto consider the actual DMA problem. One can define the(length-dependent) fiber-power transmission TL by

TL = PinPout(L)

(13)

where Pin is the power coupled into the fiber (at z = 0) andPout(L) is the power that emerges from the fiber after apropagation distance of L (at z = L). One can therefore ex-press the measured (reference-length-dependent) attenuationby

e-aL, Lo(u, T, p)(L - Lo) = TLITLo, (14)

where a has u, p, Xr as an argument to indicate its dependenceon launch conditions, L is the test length, and Lo is the ref-erence length. One now considers limited mode-volume ex-citations centered on coordinate u with launch N.A. i- and spotsize p. The general idea of restricted launch DMA measure-ments is to make many such excitations at different u's andthereby be able to construct the DMA curve as an (almost)continuous function of u. By combining Eqs. (13) and (14)with Eqs. (8) and (10), one can relate the actual DMA coeffi-cient called y(R) with CLLo (u, p, r) in the followingmanner:

{2AMIN[R, u + p) - MAX[(R2 - 72

)1/2

, u -P

p.,7 ,v(R) = V2 R2

centered spot:otherwise

A MIN[R, p] - MAX[R2 - r 2, 0] forR2]p

2 +r 2

pa,,,p(R) = V2 R2

o, otherwise

for (u - p)2 <R 2 < (u + p)2 + r2 and u <p(12a)

(12b)


1.0 0.5 0.0 0.5 1.0Normalized radial position, s

Fig. 4. Comparisons of the theoretical predictions (dashed lines)with actual measured near fields (solid lines) for three different setsof excitation parameters.

1

fr Pulp (Ram (Rate -r(R)L

- exp[-aL,L0 (u, p, T)(L - Lo)]X exp[-,y(R)Lo]JdR = 0


the limit of zero spot size and zero numerical aperture onerapidly finds the result that

,yR = aL,LO(U, 0, 0)1.=R- (17)

Equation (17) states that the requirement for the measured,limited mode-volume DMA curves to match the actualmode-dependent attenuation is that the launch violate dif-fraction theory (i.e., zero spot size with zero N.A.). One canalso consider a limit in which the spot recedes to zero as thelaunch N.A. goes to unity. In this limit one finds that

y(R) = GL,Lo(U, 1, 0)

Of,+ u(L -Lo)}1Lln {1 + [1 -f(u)] UL,LO(U, 1, 0) (L-(18)L -Lo aflau au =R8

It can be instructive at this point to examine the shape of theIlL,LO(U, T, p) for some simple forms of the actual DMA coef-ficient -y(R). This comparison is carried out in Fig. 5. As ismanifest from Fig. 5, we have employed two different modelsfor y, one of them a so-called hard-cutoff approximation andthe other a linear increase in attenuation for near-cutoff valuesof the mode coordinate R. As can be seen from Fig. 5, ratherlarge changes in -y can lead to only modest changes in themeasured attenuation a. This is not especially surprising ifone judges from the differential nature of Eq. (18) the solution

0.0 0.2 0.4 0.6 0.8

(15)

in terms of the modal distribution or

SS EUp(s', tl) -e-(R')L - exp[- aL,Lo(u, p, T)

X (L - Lo)Iexp[-y(R')LoII ds' 2dt' 2 = 0 (16)

in terms of the source optical system radiance. Note that inEq. (16) the notation R' = [f(s') + t' 2]1/2 has been used. Bothp(R) and E(s, t) have been given the subscripts u, r, and p toindicate that they are dependent on all three.

Equations (15) and (16) are quite difficult to solve in gen-eral, but one can consider certain limiting cases to gain insightinto the physical content of these relations. For example, in

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 5. Predictions of what the measured DMA curves [a(u)] shouldlook like for two arbitrarily assumed forms for the actual DMA curve[,y(R)].

1286 J. Opt. Soc. Am./Vol. 73, No. 10/October 1983

for y in the case of overfilled N.A. It is also evident from Fig.5 that the detailed form of ai can be vastly different from thatof 'y.

One can cast Eq. (15) into a form that is amenable to nu-merical solution for the actual attenuation fumction -'(R). For

example, consider a limit in which Lo can be considered neg-ligible. In this limit, one could rewrite Eq. (15) in the form

1fo k(x, R)g(R)dR = h(x), (19)

where

k(x, R) = purp(R)m(R), (20a)

g(R) = e-y(R)L, (20b)

h(x) = exp[-cLLo(U, p, 7-)L] f purp(R)m(R)dR,

(20c)

and x is used to indicate which one (or combination) of theparameters u, p, and r is allowed to vary during the mea-surement. Unfortunately, Eq. (19) is in the form of a Fred-holm integral equation of the first kind, and therefore its so-lution embodies an ill-posed problem 33 [i.e., infinitesimalchanges in h(x) can cause finite changes in the desired g(R)].Often, attempts to invert the results of filtering processes leadto such ill-posed problems, and indeed, in the present case thefiber itself is acting as a filter. The desired attenuation at agiven value of R is masked by the fact that each measurementinvolves an average of the actual attenuation at many valuesof R. Problems of the mathematical form of Eq. (19) havereceived much attention in recent years. For example, theproblem of determination of the current density on an antennafrom far-field measurements is of the form of Eq. (19) and, asis well known, can be solved uniquely with the momentmethod.34 Unfortunately, the moment method cannot beapplied in the present case. In the antenna problem, thekernel (Green's function) k(x, R) is an oscillatory function.This oscillatory nature allows one to select a sampling length,that is, to determine a minimum spacing between samplingpoints on the antenna, such that the far-field contributionsof these sampling points will be independent. In the presentproblem the kernel k (x) is quite smooth. There is thereforeno rule of thumb to apply to the numerical sampling problem.This smoothness also raises questions about the uniquenessof the solution, which are addressed below. But as Eq. (19)does represent the results of a physical process, it would in-deed be surprising if some solution did not exist. Indeed,there are various ways to attempt to effect a so-called mini-mum norm solution [a solution that excludes the rapidly os-cillating unphysical solutions to Eq. (19)33]. The one em-ployed here is variously ascribed to Landweber or Fridmanand is aptly discussed in Ref. 33 (pp. 184-185). This iterativesolution takes the form

g(N)(R) = g(N-l)(R) + 6 [h(N)(X)

'1- oJ k(x, R)g(N-1)(R)dR]k(x, R)dx, (21)

where 3 is a parameter that must be chosen small enough soas to suppress oscillations and the superscripts N and N - 1indicate the number of the iteration. Note that Eq. (21) can


be used to solve the problem of Eq. (15) if one modifies thesubstitutions of Eqs. (20) to read

k(x, R) = p.Jp(R)m(R),

g(N)(R) = exp[-,y(N)(R)(L -Lo)],

h(N)(x) = exp[-CL,L,(U, r, p)(L - Lo)

x S exp[-y(N- 1)(R)Lo]h(x, R)dR.

(22a)

(22b)

(22c)

It should be reemphasized that the series of Eq. (21) takentogether with the substitutions of Eqs. (22) represents onlyan attempt to obtain a physical solution of the problem posedin Eq. (15). The achievement of a true result depends cru-cially on the selection of 6, which can be done only by trial anderror.

Before the algorithm of Eqs. (21) and (22) was applied tothe determination of y(R) from ciLLO(U, p, r), numerous cal-ibration runs were made in which various y's were used togenerate a's (as in Fig. 5), which were in turn fed into the al-gorithm of Eqs. (21) and (22) to see how well the originalfunction y(R) could be retrieved. As would be expected, thesolution (and/or its rate of convergence) depends on the 3value. If 6 exceeded some value [which was a strong functionof the initial form of y(R)], two or three iterations would besufficient to obtain a result whose values oscillated wildlybetween positive and negative values. Further iterationsdiverged rapidly. Small enough values of 3 would lead to asolution that, after 50 or so iterations, would be within 1% ofthe actual y at smooth points of y and differ from y by 4 or5% near points that were located near large changes in slope.Further iteration was found to be counterproductive, as theconvergence was extremely slow and nonuniform. Rippleson the curve appeared soon after 50 iterations. These ripplescould grow to major oscillations at more than 500 iterations.Despite these drawbacks and somewhat poor accuracy, thetruly surprising thing was the efficacy of this numericaltechnique compared to any others available. Moment tech-niques, with their matrix-inversion requirement, showedhundreds-of-per-cent inaccuracy because of the near-zeroeigenvalues of the generated matrices.

EXPERIMENTS

Traversing-Spot MeasurementsIn Fig. 6 the results of a set of typical restricted-launch, tra-versing-spot DMA measurements are depicted. Thelaunching beam was obtained by exciting a single-mode fiberwith a He-Ne laser. The beam radius p was 0.15 and thenormalized spot N.A. was 0.4. As is readily evidenced fromthe figure, DMA curves measured on the same fiber arestrongly dependent on both test and reference lengths.

Figures 7(a), 7(b), and 7(c) depict the actual loss curves (,y's)calculated from the measured loss curves of Figs. 6(a), 6(b),and 6(c), respectively. The measured points are also includedin each of the figures to help give perspective to the varyingscales on these figures. Each of the curves was computed byusing 3 values of 5 X 10-4 and 50 iterations [refer to Eq. (21)and discussion of the preceding section]. The 6 value waschosen to ensure the fastest possible convergence. This choicewas effected by using ever-increasing values of 3 until the first


30E

m

0

co

c20

t)CF20

a)c: 0)

'a

10

0.0 0.2 0.4 0.6 0.8

Normalized beam position, u


tiplied by exponential decays. The modal eigenvectors,however, correspond to linear combinations of modes (the onewith the smallest exponent corresponds to the so-calledsteady-state condition). The decay in any one given modemust correspond to a sum of exponental decays. Unfortu-nately, a sum of exponentals is not expressible as a single ex-ponential with an averaged exponent. Therefore, by tryingto express the decay of single mode by an exponential, onemust find the exponent itself to vary with length. The cir-cumvention of this problem of length dependence of DMA isa formidable task that requires a careful rethinking of exactlywhat information is desired. If one were interested in thefabrication process, i.e., in measuring contaminant levels atdifferent radii, then one would probably want near-steady-state propagation conditions to be produced, and thereforeone would want to use a long reference length. However, ifone were interested in realistic loss estimates of practical links,the situation would be quite different. It is unlikely that asteady state could ever be achieved in a practical link as splicesmust placed at distances less than the steady-state length, and

1 1

1.0

Fig. 6. Plots of traversing spot DMA measurements using the samefiber but various fiber lengths (L) and reference lengths (Lo).

iteration became unstable. Backing off from this unstablevalue by a factor of roughly 2 ensured that the iterations re-mained stable for a considerable period. However, the iter-ations could not be continued indefinitely because of a secondtype of instability inherent in Fredholm equations of the typein Eq. (19). That is that the operator in Eq. (19) can admitof highly oscillatory eigenfunctions g with near-zero eigen-values if the kernel k is smooth and positive definite. Theoperator k in the present problem is indeed smooth and pos-itive definite. Dips and bumps on these curves that generallyprecede or follow slope changes are most likely of the kind thatwere observed in the previously discussed calibration test(4-5% errors near points of rapid slope change). Realistically,therefore, these dips and bumps should be smoothed over.

As is immediately evident from Fig. 7, the actual loss is alsodependent on the measurement lengths, an observation thatseems to raise a contradiction, as the actual loss should rep-resent the actual loss independently of the measurementtechnique. The answer to this paradox lies within the defi-nition of the loss coefficient y(R). It is tacitly assumed that,for each value of R, the loss has an exponential dependenceon length. This would be true if the only loss mechanism wereabsorption. However, the mechanism for excess loss over thefirst tens or hundreds of meters of propagation is most likelycoupling of near-cutoff modes to radiation modes. Lossesthat are due to coupling out of higher-order modes, however,do not effect each value of R as a pure exponential decay.Mode coupling is a diffusion process, and therefore the changein power level of a given mode depends on the power in theadjacent modes. The draining of power from the highest-order mode effects the power occupancy of all other modes.A means of analyzing this process is Gloge's coupled powerequations.3 5 Marcuse36 shows that the solution to theseequations corresponds to a sum of modal eigenvectors mul-

E

co0

a

V

C

0

CIDC

co

Rr

0

Cu

a)

;Z

l (a)

10k

9

1

1

3

2

2

2

0.5 1.0

15 (b)

4-

13 -y(R)o/

2-

1 /

0

9

8 I I I I0.5 1.0

2 (c)

8 y(R)

4-0

0 °

6 -,

2

A I I I I I I

0.5 1.0R or u

Fig. 7. Calculations of the actual DMA coefficient y(R) from themeasurement data presented in Fig. 6. (a), (b), and (c) correspondto (a), (b), and (c) curves of Fig. 6, respectively. The measured dataof Fig. 6 are also plotted in these figures to help to realize the alteredscales of these figures.

o Measured values c(u)

y(R)

I I I I I I


these splices induce sufficient mode coupling so as to reinitiatethe excess loss mechanism.

Another interesting point about the curves of Fig. 7 is thatthey agree well for u < 0.5. This is probably because theprimary excess loss mechanism in this region of the fiber isabsorption, and therefore the theoretically assumed expo-nential form of the loss function should agree well with theactual form. Small parameter variations, which give rise tocoupling to radiation modes and therefore cause the loss inhigher-order modes to exhibit transient behavior, should havelittle effect on the strongly bound modes that lie closer to thecore axis.

A final interesting point that is evident from Fig. 7 is the factthat the actual DMA curves are always steeper near thecore-cladding interface than those measured by the traversingspot technique, that is to say, that the actual DMA alwaysapproaches the hard-cutoff limit more closely than the mea-sured curves. This tends to justify previous studies37 that hadassumed this hard-cutoff limit to be a valid approximation.

Varying Numerical-Aperture MeasurementsFigure 8 illustrates a set of DMA curves made by varying theN.A. of the launch for centered excitation (u = 0.0) and usingthe normalized beam width as a parameter. The excitingsource was a tungsten lamp, which was imaged onto the fiberthrough a two-lens, one-variable-aperture imaging system.The incident light was filtered with a monochromator so asto have a 10-nm bandwidth centered on 837 nm. The totalfiber length was 880 m, and the employed reference length was2 m.

Figure 9 shows the actual DMA curves derived from the twoexperimental curves of Fig. 8. Unfortunately, these curvesdo not agree with each other as they should. The problemhere is a rather basic one. The diffraction limit precludes onefrom using an arbitrarily small N.A. with a fixed beam radius.Indeed, this is the reason why there are not data points on Fig.8 for r < 0.4. However, the algorithm of Eqs. (21) and (22)requires an integration of r for the full range of r, that is, fromzero to one. This requires an extrapolation of T values intothe unphysical regime. The error induced into the mathe-matically unstable system of Eqs. (21) and (22) by this ex-trapolation is essentially incalculable but could easily be 100%or more. Evidently, direct solution for -y(R) from u(r) is notan efficient means to find the fiber's true loss function.

4E - op=1.1

m * p=0.7 u=0.0

b 0

.0 0 03 - 0

0.0 0.2 0.4 6 0.8 1.0 1.2Normalized beam numerical aperture, T

Fig. 8. Plots of varying N.A. DMA measurements using two differentbeam radii.

5.0

m

co2

a0

a

4.0-

p=1.

=0.73.0 k

2.0

0.5 R 1.0

Fig. 9. Plots of the actual DMA coefficients [y(R)J computed fromthe measured values in Fig. 8.

M

0

7;aa

3

0.0 0.2 0.4 0.6 0.8 1.0Normalized beam numerical aperture,-c

1.2

Fig. 10. Data points of Fig. 8 plotted together with a theoreticalprediction (dashed lines) computed with the assumption that theactual DMA takes the form of hard cutoff [y(R) = 2.6 dB/km for R< 0.9 and y(R) = 10.4 dB/km for R 2 0.9].

Figure 10 illustrates the results of another attempt to finda reliable y. The method used here was to assume a form fory with free parameters and to optimize these parametersthrough optimizing the fit of the af calculated from ,Eq. (14)to the measured values. The form assumed for y here wasthat illustrated in Fig. 5(a), the hard-cutoff model. Thevalues of the loss levels were optimized and found to be y(O)= 2.6 dB/km in the middle of fiber and 4y(0) = 10.4 dB/kmnear the core-cladding interface (R 2 0.9). As can be seenfrom Fig. 10, even this crude loss model gives a satisfactoryfit to the experimental curves. When compared with thecomputed curves of Fig. 9, however, this raises even moreuncertainty about the accuracy to which Sy can be calculatedfrom variable N.A. measurements.

CONCLUSION

The above considerations have shown that the actual DMAcoefficient is an elusive quantity. Although the coefficientis easy to define theoretically, it is not easy to determine fromthe standard measurement data. In the case of traversingspot measurements, there is a serious problem with decidingwhen to stop iterating the solution as high-frequency oscil-lations set in with large numbers of iterations. The truncationerror in this case can easily be several per cent. In the caseof varying N.A. measurements the error is much greater,

. p= } Measured values

-<7.---

- 0 ° -0 * 0\ Calculated act

curves ,/

. -

z . .. . .


I I I I I - I I I I I


perhaps as much as 100%. However, it is clearly better to tryto process the data than not, as is well illustrated by the tra-versing-spot results. The actual DMA coefficient for nor-malized radii greater than 0.5 is clearly different from themeasurement data. For normalized radii less than 0.5, inwhich the loss is dominated by absorption and Rayleighscattering, measurement data can give a clear indication ofwhat the DMA coefficient is. For larger radii, in which modecoupling and adiabatic cutoff also contribute the loss, calcu-lations indicate that the error in the measurement data canbe large. Therefore it is clearly better to try to calculate theactual DMA coefficient from the data, as the uncertainty inthe calculation of the DMA coefficient is clearly less than theerror accrued in using the measurement data themselves asthe actual DMA coefficient.

Many of the problems met in the calculation of the DMAcoefficient have their basis in the very definition of the coef-ficient. There is no reason to assume that the loss in an op-tical fiber should be a single-variable exponential function oflength. Certainly the loss that is due to absorption andRayleigh scattering should be exponential, but it is not easyto separate the effects of these mechanisms from the effectsof mode-coupling-induced loss or radiation loss induced byparameter fluctuation. A possible solution to these problemswould be to define the DMA coefficient as a two-variabletransmission function -y(R, R'), where R' denotes the input-mode parameter and R the output-mode parameter. In thisway one could lump together mode coupling and absorption.The problem of the coefficient's uniqueness would also beovercome, although more information would be required forits determination. An alternative technique could be basedon defining separate DMA coefficients for each loss mecha-nism. However, such an approach would require a techniqueto separate the exponential, Rayleigh scattering, absorp-tion-type loss from the diffusive, mode-coupling-generatedexcess loss.

ACKNOWLEDGMENT

We would like to acknowledge the support of the NorwegianTelecommunications Research Establishment during thecourse of this study.

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mode-dependent attenuation in optical fibers

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