pulse broadening in graded-index optical fibers

Pulse broadening in graded-index optical fibers

Robert Oishansky and Donald B. Keck

This paper reports on some theoretical and experimental investigations of the radial refractive index gradi-ent that maximizes the information-carrying capacity of a multimode optical waveguide. The primary dif-ference between this work and previous studies is that the dispersive nature of core and cladding materialsis taken into consideration. This leads to a new expression for the index gradient parameter cr whichcharacterizes the optimal profile. Using the best available refractive index data, it is found that in high-silica waveguides, the dispersive properties of the glasses significantly influence the pulse broadening ofnear-parabolic fibers, and that the parameter ac must be altered by 10-20% to compensate for dispersiondifferences between core and cladding glasses. These predictions are supported by pulse broadening mea-surements of two graded-index fibers. A comparison is made between the widths and shapes of measuredpulses and pulses calculated using the WKB approximation and the near-field measurement of the indexprofiles. The good agreement found between theory and experiment not only supports the predictionsmade for the value of ac, but demonstrates an ability to predict pulse broadening in fibers having generalindex gradients.

1. Introduction

Communication systems of the near future will re-quire information-carrying capacities of several hun-dred Mbits/sec over distances of several kilometers.Over the past few years the optical fiber waveguidehas established itself as a leading contender for thetransmission medium of such systems. Much efforthas been expended on designing optical waveguidesthat will have sufficient information-carrying capaci-ty to meet these requirements. A number of studies'-6 have predicted that multimode fibers having a par-abolic or near-parabolic radial index profile will havebandwidths 2-3 orders of magnitude greater than thebandwidth of a comparable step-index fiber. In-deed, bandwidth increases of an order of magnitudeor greater have been observed7 -1 0 in state-of-the-artgraded-index fibers.

The primary thrust of this paper is to show thatfactors that have heretofore been ignored can have aprofound effect on the radial index profile that maxi-mizes information-carrying capacity. It differs fromthe previous studies primarily in that the dispersivenature of the core and cladding glasses is taken intoconsideration. Using the best refractive index dataavailable, it is found that these dispersive propertieshave a significant effect on the pulse broadening.The calculations indicate that the index gradient pa-rameter a,, which characterizes the optimal profile

The authors are with Corning Glass Works, Corning, New York14830.

Received 15 May 1975.

shape, should be altered by 10-20% to compensatefor dispersion in the core and cladding glasses.These conclusions are supported by a number ofmeasurements of pulse broadening in graded-indexfibers.

To present a clearer picture, a general discussion ofpulse dispersion and its characterization in the multi-mode optical waveguide has been included. This isfound in Sec. II, which presents a general formalismfor characterizing pulse broadening and identifiesdifferent mechanisms producing pulse broadening.

The WKB solution for cylindrical waveguides isdiscussed in Sec. III. This solution can be used tocalculate pulse shapes for an arbitrary index profile.The class of index profiles introduced by Gloge andMarcatili4 is then analyzed, and it is shown that in-cluding the effects of material dispersion leads to asignificant correction to the a value that minimizespulse broadening. Index data are presented andused to evaluate the size of this correction.

In Sec. IV, the results of measurements of twowaveguides at five different wavelengths are present-ed and analyzed. It is shown that the pulse broaden-ing depends strongly on- the source wavelength. Cal-culations of the shape, width, and wavelength depen-dence of the pulses are presented. These show thatif one accounts for differential modal excitation andattenuation and the dispersive properties of theglasses, the WKB solution gives good agreement withthe observations. Hence, an interesting by-productof this work is a demonstrated capability to predictthe pulse broadening in general graded-index wave-guides.

February 1976 / Vol. 15, No. 2 / APPLIED OPTICS 483

II. Pulse Propagation in Graded-index Fibers

A formalism is presented for describing the propa-gation characteristics of graded-index, multimode fi-bers. The index profiles of cylindrically symmetricwaveguides can be conveniently specified by theequation

n2(r) = n 2[1 - 2Af(r/a)], (1)

where n(r) is the refractive index of the waveguide asa function of distance r from the axis, and nj is theindex along the axis. The profile function f(ra) isdefined so that it is zero on axis

f(0) = 0 (2)

and becomes equal to unity at the core-claddingboundary located at r = a,

f(r/a) = 1 forr- a. (3)

The cladding index n2 is thus defined to be

n2 = n(1 - 2A)1 /2 . (4)

The quantity A provides a useful measure of thecore-cladding index difference. From Eq. (4) onefinds

A = (n12 - n 2

2 )/2n,2. (5)

Each mode of the waveguide can be specified bythe pair of integers g and v, which, respectively, spec-ify the number of radial nodes and azimuthal nodesin the transverse electromagnetic fields of that mode.The propagation constant , of each mode dependsexplicitly on all quantities that specify the waveguidestructure and on the wavelength X of the propagatinglight,

idL = ala, (l, A, a, ). (6)

The propagation constants depend on the wave-length explicitly and implicitly through the wave-length variation of nj and A. Although the indexprofile f(ra) may also vary slightly with wavelength,such effects will not be considered here.

For analyzing pulse transmission, one is concernedwith the group delay time per unit length for themode g,v. This is given by

dw, . (7)If the free space propagation constant, k = 2r/X, isintroduced, Eq. (7) can be rewritten as

T Idak (8)IL.], = c

If the fiber is excited by an impulse excitation andif there is no mode coupling, the impulse response,P(t,z,X), for spectral component X at position z canbe written as

P(tzX) = PL(X)6[t - Z T,, (X)i], (9)

where the summation extends over all guided modes.The distribution functions Pl^(X,z) describe thepower in mode ,,v as a function of wavelength andposition. At z = 0 the distribution function will be

determined by the spatial, angular, and spectral dis-tribution of the source, as well as by the source-fibercoupling configuration. As the impulse propagatesalong the waveguide the power in each mode willchange according to the attenuation occurring in thatmode. It will be assumed throughout that no modecoupling occurs.

In general, only the total power integrated over allsource wavelengths will be detected. Hence, thequantity of practical interest is the full impulse re-sponse

P(t,z) = f dXP(t,z,X). (10)

The propagation characteristics of the fiber can bedescribed by specifying the moments Mn(z) of thefull impulse response. These moments are definedby

v1W (z) = f dttnP(t,z). (11)

In some situations, knowledge of only the first fewmoments is sufficient for system design consider-ations. If this is the case, the required amount ofpulse broadening information is reduced considera-bly.

Equations (9)-(11) can be combined to yield

Mn(Z) = znf dX.P,,V(Xz)Th n (). (12)

The predominant wavelength dependence of thedistribution function P(X,z) is determined by thespectral distribution S(X) of the source. Even for therelatively broad LED sources, S(X) is a sharplypeaked function whose rms width is, at most, a fewpercent of the mean source wavelength. One canthus define a new distribution function p,,( X,z) bythe expression

P11.(X1z) = SXP1(1) (13)

where p, ( Xz) is a slowly varying function of X overthe range where S(X) is nonzero.

The main wavelength dependence of the distribu-tion function Pgv is expected to come from such fac-tors of the X-4 dependence of the Rayleigh scatteringlosses or possible variations with wavelength of thespatial and angular distribution of the source. Theseeffects are generally small and will be ignored in theremainder of the paper.

Proceeding with the analysis, it can be assumedthat S(X) is normalized so that

J dXS(X) = 1. (14)

Consequently, the mean source wavelength X isgiven by

X0 = f dXXS(X),0

(15)

and the root mean square (rms) spectral width of thesource as is given by

= [ dX(X - X)2S(X)] (16)

484 APPLIED OPTICS / Vol. 15, No. 2 / February 1976

The influence of the source spectral distribution onthe fiber's transmission properties can be studied byexpanding the delay per unit length of the ,vthmode, r,,(X), in a Taylor series about X0. Substitut-ing this series into Eq. (12) and using Eq. (13) give

Mn(z) = f dXS(X)p",(z) {T~' ()

+ n(X - X) Tan- (XA) T.' (Xo)

+ n(X - X) 2/2Tr IV()T(X "(X0 )

+ n(n - 1)( - Xo)2/27iV (XO)[Tv (X)] + . .} (17)

Treating puv as independent of X, Eqs. (14)-(16) canbe used to integrate Eq. (17) to find that

Mn(Z) = Zn;p",(Z)(rxn(X0)

+ U52/(2XO2 ) X {nT7 ."n-(X0)X02 TW"(X0 )

+ n(n - )T "n-2(X 0)[X0 7T,'(Xo)]2}) + 0(u53 /X0

3). (18)

The small size of o-/Xo allows the neglect of higherorder terms.

The following quantities are most useful in de-scribing the energy distribution at z. By definition,the total power arriving at z is given by

MO(z) = zpss(Z); (19)

the mean delay time of the pulse 7r(z) is given by

T(Z) = M1(z)/M 0(z); (20)

and the rms pulse width U(z) by

U(z) = [M2(z)/MO(z) - T2(Z)] /2. (21)

Combinations of higher moments further describethe power distribution, but the first three are themost important.

To simplify the notation required in the followingexpressions, the symbol ( ) will be used to indicatethe average value of a quantity with respect to thedistribution Pave so, for example,

(A) =_ Yp,,(z)A,,,M0. (22)

From Eqs. (19)-(22), the full pulse delay time isfound to be

T(Z) = z[(T(X0 )) + tu52/(2XO

2) (XO2 Tr'(O))]. (23)

For the purpose of specifying the fiber bandwidthfor digital systems, Personick" has shown that one isprimarily concerned with the rms width o(z). This isgiven by

Cr(Z) = (oINTERMAL + UINTRAMODAL) + 0(o53/X0 ), (24)

where the definitions

cINTERMODAL = Z {@(SO)) (T(X0 ))

+ -y [(X02'"( 0 )rG'()) - ( 2T"(X))(T(XA))]} (25)

and

INTRAMODAL = Z X([OT (xO)]2) (26)

have been introduced.

The square of the rms width has been separatedinto an intermodal and an intramodal component.The intermodal term [Eq. (25)] results from delaydifferences among the modes and vanishes only if alldelay differences vanish. This term is found to con-tain a dominant term and a small correction that isproportional to the square of the relative source spec-tral width (/Xo). For the refractive index profilesconsidered in this paper, this term is found to be neg-ligible.

The intramodal term [Eq. (26)] represents an aver-age of the pulse broadening within each mode. It be-comes the only term present in the dispersion of asingle-mode waveguide. The intramodal dispersionarises from two distinct effects, a pure material effectthat corresponds to the pulse broadening in bulk ma-terial and a waveguiding effect. This separation canbe made explicit by writing the modal delay time inthe form

T. = N1/C + TaV

where N, is material group index,

N = - Xdn1/dX,

(27)

(28)

and 6-rV represents the correction to this introducedby the waveguide structure. The derivative rzJ canbe written as

v = -nl + (29)

Since the intramodal contribution to the total rmspulse width is obtained by squaring and averagingover Eq. (29), one can write the intramodal contribu-tion as

2 2 U 2 i)

UINTRAMCDAL = Z v [(X0 n 1")

- 2XA2n1" (X 06T) + ((Xo6T') 2)]. (30)

Hence, the total intramodal contribution has a purematerial component, a waveguide component, and amixed component arising from the cross product.

In the remainder of this paper, the primary inter-est will be in graded-index waveguides in which theintermodal delay differences are small. In this case,the derivative of the intermodal delay differences6-rw' is also small, and, as will be shown explicitly inthe next section, the intramodal contribution is thendominated by the pure material term

0 INTRAMODAL ' ( 2n 1"). (31)

This quantity represents the ultimate lower limit ofthe pulse broadening.

Ill. WKB Approximation

A. General Solution

The WKB approximation can be used to deter-mine propagation constants of the cylindrically sym-metric multimode waveguide having an arbitraryindex profile. As shown by Gloge and Marcatili,4 theWKB solutions are


(jA + 1/2) = fR 2 drK,(r), (32)R1

where

KJ(r) = [k2n2 (r) - V2/y2 - (3"2]1/2 (33)

and R, and R2 are the zeros of Eq. (33). The delaytime r can be determined by taking the derivativeof Eq. (32) with respect to k, while holding ,u and vfixed. Using Eq. (1) for the index profile and takingthe derivative give

R2=k/fpt,, [ (Ni/ni) fR drn2(r)/K,,,(r)RR

-(A'/2 A) fR 2 dr[n2 (r)

- n2 ]/K (rjf 2 dr/K,,"(r). (34)

Equation (34) is equivalent to Eq. (42) of Ref. 4 if onesets N, = nj and XA' = 0. Equation (34) can be usedfor making numerical predictions of pulse shapesonce the index profile is specified. Such calculationshave been carried out for two fibers whose index pro-files were measured by the near field technique.12These calculations as well as the measured pulses willbe discussed in Sec. IV.

B. Intermodal Broadening for a Class of Index ProfilesFor the class of index profiles given by

f (ra) = (r/a)', (35)

Gloge and Marcatili have derived a simple closedform approximation to the WKB solution. Thepropagation constant f is found to be

= nok{l - 2A[n/N(a)] 'a/ 2}1 1/2, (36)

whereN(oa) = y a2k2n 2A (37)a + 2 f 1 .(7

The designation n used in Eq. (36) is not a true modelabel as is the pair (g,v). Rather, n counts the num-ber of modes with propagation constant f in therange

n1k ' 1 n (38)

The delay time for the modes can be determinedfrom Eqs. (36)-(38) as

Tn= NI + Aa 2 E_) (n/NY"1012

l [ (a + 2 )(/)2 (3a -2 - 2E) (n/N)2 a/+ 23 + 0(A3 ), (39)

where-2n XA' (40)N A

To first order in A the delay differences in Eq. (39)are zero if

a = 2 (41)

Assuming that all modes propagate equal power (v= 1), Eqs. (25) and (39) can be used to calculate therms width due to intermodal broadening. If thesummation required in evaluating Eq. (25) is approx-imated by an integration, one finds

LNIA a l a + 2 \1/2ANTERMODAL 2c a + 1 3a + 2 )

x 2+ 4CC 2A(a + 1)+ 4A2 C 2 (2co + 2)2 1/2

X 2cz + 1 (5a + 2)(3 + 2)] '

wherec = -2- E

+ 2

and3a - 2 - 2E2(a + 2)

(42)

(43)

The minimun of the intermodal rms with width oc-curs for'3

a = 2 + - A (4 + E)(3 + E)(5 2E) (44)

The term on the right of Eq. (44) represents asmall correction to the optimal a given by Eq. (41).This correction results from a partial cancellationthat occurs between the two mode dependent termsin Eq. (39) if a - 2 - E is of order A. The size of thiscorrection changes as the distribution of poweramong the modes varies. Since corrections to theapproximate WKB solution [Eq. (36)] can be of thissame magnitude, and inaccuracies in the value of determined from measured refractive indices are alsothis large, Eq. (44) is adequate for the present discus-sion.

To enable computations to be made, it will be as-sumed in the remainder of this paper that n,(X) andn2(X) correspond to the refractive indices of TiO2doped fused silica and fused silica, respectively.This choice is made simply because the best availabledispersion data exist for these two materials. Forthe index n2(X) of the cladding the Sellmeier fit ofMalitson14 for fused silica has been used. For nj(X)a two-term Sellmeier fit has been made to refractiveindex measurements made on a bulk sample of 3.4 wt% TiO2 doped silica at eight wavelengths in the 546-1083-nm range. The estimated precision of thesemeasurements is a few parts in 105.

Figure 1(A) shows the fitted values of nj(X) andn2(X) in the range from 0.5 m to 1.1 im. Figures1(B) and 1(C), respectively, show the first and secondderivatives of the refractive indices of these twoglasses. Figure 2 shows the values of A and XA' de-termined from the data. It can be observed in Fig.2(A) that, over the range of wavelengths shown, A de-creases by about 15%.

From the index data on Figs. 1 and 2, E(X) can becalculated, and a plot of ac vs A is shown in Fig. 3.The value of a, minimizing the intermodal rms pulsewidth departs significantly from the optimal profile a= 2(1 - 6A/5), which is predicted if the effect of ma-terial dispersion is ignored.

The optimal a varies quite strongly with wave-


REFRACTIVE INDEX DATA

.5 .6 .7 .8 .9 1.0WAVELENGTH (MICRONS)

Fig. 1. Refractive index data for 3.4 wt /6 TiO2 doped silica (ni)and fused silica (n2) are shown for (A) Sellmeier fit to the refrac-

tive index, (B) Xdn/dX, (C) X 2d2 n/dX2 .

length, decreasing from about 2.5 at 500 nm to 2.2 at1000 nm. A waveguide with a given index profile a isthus predicted to show different pulse widths accord-ing to the source wavelength used. This wavelength-dependent pulse broadening provides a tool for ob-serving these effects and is discussed at length in Sec.IV.

C. Intramodal Broadening for a Class of Profiles

Equation (39) for the delay time can be used toevaluate X-n', which is required for predicting the in-tramodal broadening. Keeping only the largestterms gives

XT,,' = -X2n111

D. Predicted Pulse Widths

The total rms pulse widths for a profiles can bepredicted from Eqs. (42) and (46). In Fig. 4, this rmspulse width is shown as a function of a for threetypes of GaAs sources, all operating at X = 0.9 gim,but having different spectral bandwidths. The threecurves correspond to an LED, an injection laser, anda distributed-feedback laser having typical rms spec-tral widths of 150 A, 10 A, and 2 A, respectively. Thewaveguide is assumed to have n,(X) and n2(X) of Fig.1(A) and to propagate equal power in all modes. Forcomparison, a dashed curve represents the rms widthpredicted if material dispersion and intramodalbroadening are ignored.

INDEX DIFFERENCE AND ITS DERIVATIVE

.5 6 .7 .8 .9( RWAVELENGTH ( MICRONS)

Fig. 2. The index difference A determined from the data of Fig.I (A) is shown in (A), and the derivative XdA/dX is shown in (B).

.0066

.0064

.0062

.006C

.0058

.0056

-00081

-00101

-. 00121

0zU.

0a

x

0z

+NlA( ~ (x 2 ) ( 2a ) ( )/2 (45)

Since X2n," and A are the same order of magnitude,both terms contribute to Xn' for large a (but withopposite signs). For a near a, the pure materialterm in Eq. (45) dominates. This reflects the factthat when the intermodal delay differences are small,the derivatives of these differences are also small.

Assuming equal excitation of the modes, one canperform the summation over all modes required byEq. (26) and find that

UINTRAMODAL = A [(X2 nii")2 - 2X2n"K(N1A) ( )

x (2a 2 + 2) + (NA)2(a 2 )2 2a 2]/2

a 2.5

zw0 2.4

x

W 2.3a

-J

E 2.2I-0~

2.1500 600 700 800 900

WAVELENGTH nm)1000 1100

Fig. 3. The a value that minimizes the pulse broadening is shownas a function of wavelength.


I I I ~I I I I

\ =2n,2

A

/ \dA/dX

I I I I ,1- -1,

I I I I I I I

IxII

-t014

-.0016

1.0 1.1

2.6r

z

IU)

C

-J

U)

RMS PULSE WIDTH VS. INDEX GRADIENT

INDEX GRADIENT a

Fig. 4. Assuming equal power in all modes, the rms pulse width isshown as a function of a for three different sources, all operatingat 0.9 ,um. The sources are taken to be an LED, a gallium arsenideinjection laser, and a distributed feedback laser having rms spec-tral widths of 150 A, 10 A, and 2 A, respectively. The dashedcurve shows the pulse width that would be predicted if all material

dispersion effects were neglected.

For the LED source, pulse broadening of less than1.5 nsec/km can be achieved if a is within 25% of theoptimal value. For the injection laser, an a within5% of the optimal will give pulse widths less than 0.2nsec/km, and for the distributed-feedback laser,widths of 0.05 nsec/km are predicted if 1% control ona can be achieved.

IV. Wavelength-Dependent Pulse Broadening

In the previous sections it was found that the avalue that minimizes the intermodal pulse dispersiondeparts significantly from the parabolic profile andthat the magnitude of this departure depends on thewavelength of the source. In Fig. 4, one sees that thecalculated rms pulse width depends very strongly onthe difference between the a of the waveguide andthe optimal a. As a result of these features, the rmspulse width will depend on the source wavelength forwaveguides having an a value not too different fromac. This wavelength-dependent pulse broadening ismore fully analyzed in this section, and some experi-mental results are reported that support the analysis.

In Fig. 5, the calculated rms width is shown as afunction of source wavelength for waveguides withindex profiles in the range 1.5 < a < 2.9. For thepurpose of illustration, it is assumed in these calcula-tions that the rms spectral width of the source is 2 A.For this range of a values both the intermodal andintramodal contributions must be considered.

The wavelength dependence of the intermodal

term depends on the specific a value. For a < 2.2,the intermodal pulse width decreases as X increases,because for these a's the difference a a - a decreaseswith X. The opposite occurs for a> 2 .5, and the in-termodal pulse width increases with X. For wave-guides with 2.2 < a < 2.5, the minimal intermodalbroadening occurs at one wavelength in the range 500nm < X < 1100 nm. As X is varied through thisrange, the intermodal broadening decreases until theoptimal wavelength is reached, and thereafter it in-creases.

The intramodal contribution to the rms width isdominated by material dispersion. It is largest at theshorter wavelengths and decreases quite rapidly withwavelength. For a source spectral width of 2 A, therms width decreases from 0.1 nsec/km to 0.006 nsec/km between 500 nm and 1100 nm. The pulse widthbehavior shown in Fig. 5 reflects the combined effectof intermodal and intramodal pulse broadening.

A. Experimental ObservationsPulse broadening measurements were made on two

1-km long, graded-index, germania-doped silica fi-bers to observe this wavelength dependence and toattempt to verify the prediction made here for theoptimal a value.

The experimental setup is shown schematically inFig. 6. Pulse broadening was measured at 531 nm,568 nm, 676 nm, and 799 nm using a mode-lockedkrypton laser and at 900 nm using a self-pulsing LOCgallium arsenide injection laser. The signal was de-tected with a germanium avalanche photodiode.

Light was injected into the test fiber through a 5-msegment of fiber in which strong mode coupling wasgenerated by an irregular epoxy coating. The in-duced mode coupling loss was 7 dB. This method ofinput coupling insured that the same excitation con-

RMS PULSE WIDTH VS WAVELENGTHI I I I I

1.6-

1.4 5

~ .2-

~~~~ .0~ ~ ~ ~ ~~.0-

0.8 oU)-J

a, 0.6 U)

Er0.4-

0.2-2.3

.6 .7 .8 .9 1.0 I.IWAVELENGTH (MICRONS)

Fig. 5. For a source with 2-A spectral width, the rms pulse widthis shown as a function of wavelength for several different values of

a.


EPOXYLASE COVEREDSOURCE SECTION

MICROSCOPEOBJECTIVES

WVAVEGI

BUTTJOINT

UIDE

schematic representation of themeasurement.

NEAR-FIELD INDEX PROFILE

0 10 eU 3uRADIUS (MICRONS)

/ Ge-APDMICROSCOPE DETECTOROBJECTIVES

SO SAMPLINGOSCILLOSCOPE

pulse broadening

FIBER A- RADIUS = 42.8

NA =.16- BEST FIT ALPHA=.74 \

I i I I I I i I I

FIBER BRADIUS = 31.5NA =.16BEST FIT ALPHA=2.51

.I i I nIrt |u

Fig. 7. The index profile n(r) - n2 (a), determined by a near-field measurement at 900 nm, is shown for two fibers. Fiber A isbest fit by a 1.7 and fiber B by a 2.5. The dashed curves showthe optimal profile that passes through the end points determined

from the best a fit.

ditions prevailed at all wavelengths regardless ofvariations in the spatial and angular distribution ofthe sources. Observations indicated that, withoutthis method of input, large changes in the transmit-ted pulse could be obtained by off-axis illuminationof the test waveguide. No such changes were ob-served with the mode coupled input.

Mode coupling in the test waveguide due to surfaceirregularities was minimized by wrapping the fiberson a special drum.15

The index profiles of the fibers were measured by anear-field technique,'2 and the measured profiles areshown in Figs. 7(A) and 7(B). The profile of fiber Ais best fitted by a 1.7 and fiber B by a 2.5.

The output pulses observed for the two fibers atthe five wavelengths are shown in Figs. 8(A) and8(B). Fiber A exhibits about a 30% decrease in pulsewidth over the range of wavelengths measured, ingood agreement with the prediction shown in Fig. 5

for a = 1.7. An increase of about 20% over the rangeof measured wavelengths was expected for the fiberwith a = 2.5. Only a slight change in pulse widthwas actually observed, however, and it was in opposi-tion to this expected increase.

B. Analysis

An understanding of these measurements requiresa more detailed theoretical treatment of pulse broad-ening. The predictions of Fig. 5 are for waveguideshaving an exact a-type profile. The measured pro-files show considerable variation from the fitted aprofiles. The effect of these variations on the modaldelays can be calculated by using the WKB solution[Eq. (34)]. Such a calculation has been carried outby making a fifteen-term polynomial fit to the near-field data and then numerically calculating the delaytime for each mode.

Theoretical output pulses were generated by as-suming that the response of each mode was a Gauss-ian pulse and summing over all modes using an ap-propriate power distribution function. For the rmswidth of the Gaussian pulse, 0.4 nsec was chosen as areasonable estimate of the combined effects of thetemporal width of the source, the response time ofthe detector, and the intramodal pulse broadening.The distribution function Pwv was chosen to reflectthe steady-state modal distribution of the epoxiedsection used for input coupling and to reflect thestrong attenuation of the higher-order modes typical-ly observed in optical fibers.16 From the calculatedsteady-state distributions in graded-index fibers withrandom bends,17 it is found that the simplified form

to= 1 - (ni/Me) m ' •(47)

PIV= 0 m > Mc,

gives a good approximation to the steady-state distri-bution for graded-index fibers with 1.5 < a < 3.0. InEq. (47), m is the principal mode number defined by

MEASURED OUTPUT PULSES

FIBER A FIBER B

DELAY TIME (I NSEC/DIV)

Fig. 8. Output pulses measured at five wavelengths are shown forfibers A and B.


Fig. 6. A

I

z:a

NZ

-+U UV

CALCULATED AND MEASURED PULSE SHAPES

A

-9 -7.0 -5.0 -3.0 -1.0 1.0

B

-2.0 0 2.0 4.0

C

-7.0-5.0 -3.0 -1.0 1.0 -2.0 0 2.0 4.0

RELATIVE DELAY (NSEC)

Fig. 9. Two calculated pulses for each fiber are shown in (A) and(B), and the corresponding measured pulses are shown below in(C) and (D). The larger of the two calculated pulses correspondsto equal power in all modes and the smaller pulse to the power dis-

tribution given by Eqs. (47)-(49).

m = 2 + v, (48)

and MC represents the maximum value of m thatpropagates over 1 km. From studies of a fiber's ef-fective numerical aperture as a function oflength,16"8 it is found that after 1 km,

Ml ; 0 9MMAX,1 (49)where MMAX is the theoretical maximum value of mthat can occur for the guided modes.

Figures 9(A) and 9(B) each shows two calculatedpulses for fibers A and B at 0.9 gm, and Figs. 9 (C)and 9(D) show the corresponding measured pulses.For each fiber, the larger of the two calculated pulsescorresponds to equal excitation of all modes, and thesmaller pulse is calculated using Eqs. (47)-(49) forthe power distribution. It is seen that the form usedfor the output power distribution greatly improvesthe agreement between the predicted and measuredpulse shapes. If all modes are equally excited, thetheoretical pulse shape for either fiber has two peaks.In fiber A, the earlier arriving peak is composed ofhigher-order modes and the later arriving peak oflower-order modes. The opposite holds true for fiberB. This can qualitatively be predicted from the bestfit a and Eq. (39). As a result of the reduced powerin the higher-order modes implicit in Eq. (47), thepeak composed of the higher-order modes is greatlyreduced in the pulses calculated using Eqs. (47)-(49).Although other forms for the output power distribu-tion can improve the agreement between the predict-ed and measured pulse shapes, Eq. (47) has beenused since it is a more justifiable choice for the powerdistribution based on present understanding of thefactors involved.

Figures 10(A) and 10(B) show the predicted pulsesat X = 0.5 ,im, 0.7 gm, and 0.9 gim. The horizontalscale is in nanoseconds, and the delay time is mea-sured relative to the arrival time of the lowest-ordermode. A strong wavelength dependence of the pulsewidth is calculated for fiber A, and only very slightdependence is calculated for fiber B. This is inagreement with the observations.

Examination of the calculated modal delay timesreveals the explanation for the absence of appreciable

FIBER A

I--' 1 1 I I

. .

FIBER B o

I. .

I I

I'REL DELAY (NS)

Fig. 10. Calculated pulses for fibers A and B are shown at threewavelengths. The pulses are shown for 500 n (solid lines), 700nm (dashed lines), and 900 nm (dotted lines). Zero relative delay

corresponds to the arrival of the lowest order mode.

2.0

U)

z

U)-J

a.U)

1.5

.0

0.5

CALCULATED AND MEASURED PULSE WIDTHS

l I ~ I I I I I -

-~~~~~~~~0 -

- FIBER A o

- FIBER B 0

71I1I I I1 I I1 I r.5 .6 .7

WAVELENGTH (MICRONS).8 .9

Fig. 11. Calculated rms pulse widths (solid lines) and the rmswidth determined from experiment (dashed lines) are shown.

a:

SQ

E-0


. . . . . . . . .

wavelength dependence in waveguide B. Becausethe index profile varies about the optimal profile[Fig. 7(B)], the modes divide into two groups accord-ing to whether they are faster or slower than the low-est-order mode. Because of the wavelength depen-dence of the delay times, contained in Eq. (34), it isfound that as the wavelength is increased, the fastermodes arrive more nearly at the arrival time, r = 0, ofthe lowest-order mode. For the slower modes therelative arrival time increases. Therefore, as thewavelength is increased, the pulse tends to shift to alater arrival time, rather than to narrow. In fiber Aall modes belong to the group of faster modes as isqualitatively expected for a = 1.7. Consequently, asX increases, all delay differences decrease, and thetotal pulse narrows.

Finally, in Fig. 11, the rms widths determined fromthe measured pulses are compared to the calculatedrms widths based on the WKB theory and power dis-tribution of Eq. (47). The slopes of the two curvesare in excellent agreement. The calculated pulses lieabout 20% higher than the measured pulses. Thecause of this discrepancy is not known, although an-effect of this size could easily be caused by a smallamount of mode coupling.

V. Conclusions

Pulse broadening in two germania-doped, graded-index, multimode waveguides has been measured atfive wavelengths. The measured pulse widths werefound to vary strongly with wavelength in one fiber,but not in the other. A theoretical analysis based onthe WKB method and near-field index profile wasable to account for the observed wavelength depen-dence in both fibers.

On the basis of these results, the following conclu-sions can be drawn. First, the observed wavelengthdependence is a result of the differing dispersiveproperties of core and cladding glasses. In order toachieve maximum information carrying capacity, theradial index profile must be modified to compensatefor this. The good agreement found between the ob-served and calculated pulse broadening indicates

that titania and germania-doped silica waveguideshave nearly the same wavelength dependent disper-sion. Therefore, an a about 15% greater than 2 is re-quired for optimal compensation for sources operat-ing near 900 nm. Second, a WKB calculation thatincludes the effect of dispersive properties of coreand cladding glasses and the modal power distribu-tion gives good predictions of the output pulse shapeas well as the wavelength dependence of the rmswidth. Since the small systematic discrepancy be-tween the calculated and measured rms pulse widthsmay be caused by the presence of a small amount ofmode coupling, there is no reason at present to at-tribute this discrepancy to inaccuracies in the theo-retical analysis.

References1. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).2. S. Kawakami and J. Nishizawa, IEEE Trans. Microwave

Theory Tech. MIT-16, 814 (1968).3. T. Ochida, M. Furukawa, J. Kitano, K. Koizumi, and H. Mat-

sumura, IEEE J. Quantum Electron. QE-6, 606 (1970).4. D. Gloge and E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563

(1973).5. M. Ikeda, IEEE J. Quantum Electron. QE-10, 362 (1974).6. C. C. Timmermann, AEU 28, 344 (1974).7. R. Bouille and J. R. Andrews, Electron. Lett. 8, 309 (1972).8. D. Gloge, E. L. Chinnock, and K. Koizumi, Electron. Lett. 8,

562 (1972).9. D. B. Keck and R. D. Maurer, in (to be published) Proc. Mi-

crowave Research Institute Int. Symp. (1975), Vol. 13.10. L. G. Cohen, P. Kaiser, J. B. MacChesney, P. B. O'Connor, and

H. M. Presby, in Technical Digest of OSA Topical Meeting onOptical Fiber Transmission (Optical Society of America,Wash. D.C., 1975).

11. S. D. Personick, Bell Syst. Tech. J. 52, 843 (1973).12. M. C. Hudson, D. B. Keck, and R. Olshansky (investigation of

the near-field technique is currently in progress and will be re-ported).

13. R. Olshansky and D. B. Keck, in Technical Digest of OSATopical Meeting an Optical Fiber Transmission (Optical So-ciety of America, Wash. D.C., 1975).

14. I. M. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).15. D. B. Keck, Proc. IEEE 62, 649 (1974).16. D. B. Keck, Appl. Opt. 13, 1882 (1974).17. R. Olshansky, Appl. Opt. 14, 935 (1975).18. M. C. Hudson, Corning Glass Works; private communication.

A function from feeling inferiorFelt life monotonically drearierWith a hell of a yellIt jumped into LAnd converged to the limit superior.

Leo Moser. This is one of the verses found in the papersof the late Leo H. Moser; it is printed here by permissionof William Moser of McGill University.


pulse broadening in graded-index optical fibers

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