Minimum Pulse Broadening in Multimode Fibers with ... - IEEE Xplore

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IEEE JOURNAL OF QUANTUM

Arthur H. Hartog received the B.S. degree in electronics and the Ph.D. degree from the UniEnversity ofSouthampton,Southampton, gland, in 1976 and 1981,respectively. Hewas a Research Student with the Optical FiberGroup atSouthampton University, and later became a Research Fellow. He has studied propagation in optical fibers, including material and intermodal dispersion, and the OTDR and POTDR techniques for fiber evaluation. Dr. Hartog was awarded the 1979 John Logie Baird Traveling Scholarship by the Royal Television Society.

M. J. Adams, for a photograph and biography, see p. 535 of the April 1982 issue of this JOURNAL.

F.M.E. Sladen, photograph and biography not available at the time of publication.

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D. N. Payne, for a photograph and biography, see p. 487 of the April 1982 issue of this JOURNAL.

Adrian Ankiewicz was born in Sydney, N.S.W., Australia, on August2, 1952. Hereceived the BSc. and B.E. (Hons) degrees in electrical engineering fromthe University of New South Wales, Australia, in 1973 and 1975,respectively, and the Ph.D.degree from the Australian National University, Canberra, for his thesis on thetheory of graded index optical fibers in 1979. He worked in the Optical Fiber Communications Section of the Department of Electronics, Southampton University, Southampton, England, in 1978 and 1979, f i s t o n a C.S.I.R.O. (Australian) postdoctoral fellowship, and then as a Research Fellow. Currently, heis in the Department of Applied Mathematics of the Institute of Advanced Studies of the Australian National University, Canberra, Australia, as a Radio ResearchBoard Fellow. His research interests include optical waveguide theory and vision and mathematical problems in electrical engineering generally.

Minimum Pulse Broadening in Multimode Fibers with Index Imperfections

Abstract-Minimum pulse broadening of multimode graded-index fibers is investigated theoretically. Exact solutions of the r m s pulsewidth u for a fiber with a power-law index profileis obtained by using the integral expression of group delay time based the multilayer approximation. It is shown thatthe minimum value of u andthe optimum cy for the power-law index profile calculated by the multilayer approximation in the steady-state modal power distribution are nearly equal to the well-known WKB solution. Next,urninandthe corresponding cyopt. are evaluated for certain index imperfections such as a central indexdlp and sinusoidal index ripple along a radial direction. The results offer the possibility to avoid the bandwidth degradation due to an inevitably generated index ripple in an MCVD graded-index fiber.

I. INTRODUCTION T iswell known that pulse broadening due to modal dispersion in a multimode fiber canbe greatly reduced with the choice of the optimum index profile [l], [2] . However, inthe conventional MCVD technique [3], inevitable profile deviations from theoptimum shape, such as central index

I

Manuscript received September 9, 1981; revised November 24,1981. K.-I. Kitayama and S. Seikai are with the Ibaraki Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Ibaraki 319-11, Japan. K. Morishita is with the Osaka Electro-Communication University, Neyagawa-shi,Osaka 572, Japan.

depression and periodic index ripples along a radial direction as wellas complicated combinations of power-law index profiles, cause degradation of the bandwidth. Therefore, it is of practical importance to know exactly to what extent the pulse broadening can be minimized for fibers with the abovementioned index imperfections. Marcuse [4] has developed a computational technique based on the WKB method [ 1] for calculating the impulse response of actual multimode fibers with index deviation from a powerlaw profile. Using this method, effects of various kinds of deviation from the perfect index profile have been numerically investigated [4] - [ 7 ] . Olshansky [8] has also made a similar calculation for pulse broadening by using a perturbation theory. This paper describessome results for the minimum pulse broadening of multimode fibers. First, exact solutions are given for the pulse broadening of fibers with a power-law index profile. The calculation is made by usinganewly derived integral expression for group delay time, based on the multilayer approximation of the scalar wave equation [9], and the results are compared with the well-known WKB solution [2]. Next,the minimum pulsewidth is numerically obtained by using the multilayer approximation for fibers having index profiles with a central dip and a sinusoidal ripple oscillation

0018-9197/82/0500-0838$00.75 0 1982 IEEE

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KITAYAMA e t al.: MINIMUM PULSE BROADENING IN MULTIMODE FIBERS

along a radial direction.The results offer the possibility of avoiding bandwidth degradation due to aninevitably generated index ripple in an MCVD graded-index fiber. 11. COMPARISONWITH WKB SOLUTION In scalar approximation, the groupdelay of a guided mode is expressed in terms of the propagation constant /3 along with the fiber axis and integral forms offield energy $2 [ 101 . When a multilayer approximationmethod is applied fordetermining p and @ of each guided mode by using an explicit expression of group delay, the impulse response of a multimode fiber can becalculated with sufficient accuracy [9] . First, the rmswidthu is calculated by the multilayer approximation in thecase of the power-law index profile

n(r) =

I

n l [ l - A(r/a)"]

r Ia

nz(=nl[l - A])

r2a

(1)

and the results are compared to those of the WKB method [ 2 ] . Fig. 1 numerically shows u as afunction of the power-law parameter a. Here, the fiber parameters are 2a = 50 pm and A = 0.01 where Ge02-doped fused silica core [l I] and pure silica cladding are considered. The solid and dashed curves are the results obtained by the multilayer approximation and the WKB method, respectively. Lossless, zero mode coupling, and uniform excitation of all modes are assumed. A factor of 4 is multiplied to each azimuthally dependent mode by taking two possible polarizations and sine or cosine dependence, while a weight factor of2 is applied to azimuthally independent modes. Thus, 156 modes are counted to be guided for the present operation wavelength of 1.3 /m( Y = 25). In the multilayer approximation,theindex profile is approximated by dividing it into 25 layers. It is noted that u is almost invariant within 0.25 percent for a division layer number larger than 25 up to 80. Then the result in Fig. 1 is regarded as the exact solutionof scalar wave equation for the power-law profile. It isseen from Fig. 1 that the minimum rms pulsewidth of 0.87 ns/km at a = 2.22 calculated by the multilayer approximation is far larger than 0.02 nslkm at a = 1.93 by the WKB method. This seems to be due to the fact that in the multilayer approximation, a few higher order modes near cutoff, having considerably smaller group delay times than those of well-confined lower order modes, are calculated rigorously, while they are ignored in the WKB method. The impulse response shape will give a satisfactory explanation for the effect of higher modes near cutoff. In Fig. 2 the impulse responses calculated by both methods are illustrated. Delay time of the lowest order mode is taken as the origin of the horizontal time coordinate. The power-law parameter is chosen to be 1.93 which is theoptimum value for urnin in the WKB solution. In the impulse response obtained by the multilayer approximation shown bythehatched lines, the leading precursor edge of the modes near cutoff is apparent, while the WKB solution shows a sharp impulse shape. The decrease in group delay time of the higher modes near cutoff is predicted in the corrected WKB solution taking the cladding effect into account [12]. Also, this has been noted by the analysis based on a variational method [ 131.

0.0 1 1.6

I

I

I

1.8

I

2.0

-

I

2.2

a Fig. 1. Numerical rms pulsewidth u as a function of the power-law parameter a. The solid and dashed curves denote the results by the multilayer approximation andthe WKB method, respectively. Lossless and uniform excitation of all modes are assumed.

7

I

DELAY TIME ( N S )

Fig. 2. Impulseresponsesobtained by themultilayerapproximation and the WKB method. All the conditions are the same as in Fig. 1. Delay time of thelowest order mode is taken as f = 0 in the horizontal coordinate.

The assumption of lossless, zero mode coupling, and uniform excitation of all modes seems to overestimate the pulse broadening for actual fibers. Then thetransient- and steady-state modal power distributions [I41 are introduced to develop the discussion closely related to the actual fibers. Fig. 3 shows plots of u after 1 km propagation versus a in the multilayer approximationfortransient and steady statesby solid and dashed curves, respectively. Thetransientstate is assumed to be with the case of uniform excitation of all modes with the mode-dependent loss ~ ( mshown ) in the inset of Fig. 3 . Here, behavior of ~ ( mversus ) m is based on the experimental

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0

-1 0

0.01

1.6

I

I

I

I

2.0

1.8

0.5 m/M

2.2

1

J

1

'10

Fig. 4. Model of the index profile with a dip. 2d and A' denote the dip diameter and depth at the core center, respectively.

a Fig. 3. u as a function of 01 for transient- and steady-state modal power distributions by solid and dashedcurves,respectively. Mode-dependent loss shown in the inset is used for both distributions.

result [15]. The steadystatecorresponds to the casewith the experimental steady-state modal power distribution [ 161 with the same ~ ( r n ) . Under the transient state, urnin = 0.11 n s / h with aopt = 1.96, whileundersteady state, u is considerably reduced to as small as 0.03 ns/km with aOpt = 1.94, which are nearly the same as the WKB solution shown in Fig. 1 under the assumption of lossless and uniform excitation. This feature originates from the fact that the higher order modes near cutoff are most susceptible to attenuation,as seen from the ~ ( m curve, ) and do not affect the pulsebroadening for the steady-state condition. The rms widths of0.1 1 and 0.03 n s / h correspond to 3 dB bandwidths of 4.4 and 14 GHz . km, re- Fig. 5 . spectively, in thefrequency response obtained by Fouriertransforming the impulse response. 111. INFLUENCE

OF

2d

2

*IW

I

Li* A/2

0.01

1.6

2 .o

1.6

2.2

0 CJ

as a function of 01 for 2d = 2, 5, and 10 um with A' = A/2. The transient state is assumed.

DIP

In the MCVD technique, deviations from the optimum index profile suchas central index dipandperiodic index ripples arise inevitably. Therefore, it is of practical importance to investigate theoptimum shape for minimizing u for such index imperfections. First, the influence of the central dip is studied for thispurpose. For an example, the index profile shown in Fig. 4 is considered. The profile is basically a power-law shape, but with a central dip. It is assumed that the dip diameter varies with keeping the dip depthconstant and the top of refractive index at Y = d is equal to A(= 0.01). Fig. 5 shows plots of u versus CY for 2 d = 2 , 5 , and 10 pm with A' = A/2. Here, the transientstate is assumed, that is, uniform excitation ofall modes with the mode -dependentloss ~ ( r nshown ) in the inset of Fig. 3 . It is seen from the figure that as d increases, urnin obviously increases and the optimum a! value slightly becomes small. For typical valuesof dip parameters, A' = A/2 and 2d = 5 pm for a current MCVD fiber; u,in = 0.40 ns/km at a! = 1.94. Impulseresponses are plotted in Fig. 6 for fibers with and without the dip of A' = Aj2 and 2 d = 5 pm for a! = 1.96. The pulseshape deformation due to the leadinglower modes is clearly observed for fiber with the dip. The tendency

I

TRANSIENT-STATE

WlTKWT DIP

a=1.96

-2

-1

1

0

DELAY TIME

2

(NSI

Fig. 6 . Impulse responses for fibers with and without the dip of A ' = Ai2 and 2d = 5 prn for a = 1.96. The transient state is assumed.

of u against d in Fig. 5 isin qualitative agreement with the result by the WKB approximation [4] . This seems to be due to the fact that since the field energy of the lower modes is still confined well in the core in the existence of the central index dip, the cladding effect which is not taken into account in the WKB approximation remains to be small. In Fig. 7, the basebandlossesare plotted as a function of

KITAYAMA et al.: MINIMUM PULSE BROADENING IN MULTIMODE FIBERS

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FREQUENCY f (GHz)

0 0

1

2

3

2

12‘

Fig. 7. Baseband loss as a function of frequency f for fibers with and without the dip of A’ = A / 2 and 2d = 5 pm for cr = 2.0. The solid anddashedcurvesrepresent the results for the transient and steady states, respectively.

frequency f for the power-law index profiles a! = 2.0 with and without dip. The dip parameters are A’(= A / 2 ) = 0.005 and 2d = 5 p.The solid and dashed curves represent the results for the transient- and steady-state modal power distributions, respectively. It is foundfromthe figure thatthe central index dip causes complex variation of baseband loss against f, especially in the frequency range of interest where baseband -1 0 1 loss amounts to about 3 dB, while for fiber without the dip, ‘/a the loss is in proportion to fz in the frequency range lower cb) than approximately 2 GHz. Therefore,it is notedthat for Fig. 8. Power-law index profdes with sinusoidal index ripple. (a) 6n(r) fibers with theindexdip, simple bandwidth evaluation is in (3). (b) 6n(r) in (4). insufficient forthe full description of fiber transmission quality.Furthermore,the frequency response fortheindex profile with the dip changes markedly with the variation of [ 181, and the fiber preform after collapsing exhibits the index ripple given by (4). modal power distribution, while it is rather stable forthe First, the N dependence of u is investigated for two cases of case without the dip. 6n(r). In previous works treating theindex ripple given by ( 3 ) , [4], [8], there remains uncertainty as to wh.ether or not Iv. INFLUENCE OF RIPPLE Next, periodic ripple dong a radial direction [ 4 ] , [8] is u decreases for larger N . The solution by themultilayer approxitreated as the second kind of index profile deviation from the mation answers the question because numerical precision can optimum power-law profile. Influence of the ripple cm pulse be improved by increasing the division layer number for the broadening is numerically evaluated for the following two index profre. Fig. 9 shows the plots of u versus N for the two kinds of ripples. Here, the transient-state modal power types of the index profile n(r) given by distribution is considered, that is, uniform excitation of all ) by the n(r) = n , [ 1 - A(r/a)OL]+ 6n(r) r I a (2) modes with the mode-dependent loss ~ ( mrepresented inset of Fig. 3 . In the calculations, the division layer number where the index deviation 6n(r) is modeled by sinusoidal ripple is carefully examined to obtain asufficient numerical precision, oscillations as and the number of 80 is adopted. This value is found to be large enough for the number of ripples N < 16. It is found 6n(r) = A sin [27rN(r/a)] ( 3 ) from Fig. 9 that u is maximized around N = 2. As is expected, the excesspulse broadening gradually decreases to zero for and larger N . Generally, behavior of u versus N for the sinusoidal 6n(r) = A * sin [27rN(r/a)’ ] . (4) index oscillation of ( 3 ) is in good quantitative agreement with that calculated by Olshansky’s perturbation theory [8]. Fig. 8(a) and (b) illustrates examples of an index profile with Comparing two cases of given by ( 3 ) and (4), u drops 6n(r) represented by (3) and (4), respectively. Here, N = 10, more rapidly in the large N region for ( 3 ) . The excess pulse CY = 1.96, and A is chosen as 1 percent of the maximum index broadening is reduced to zero at N = 8. Thus, it is useful to difference n1 - n 2 . Actual graded-index fibers fabricated by adopt the deposition technique to form each layer so that the the MCVD technique may have various kinds of index ripple deposition layer thicknesses becomes equal to each other after generated in the deposition process. The present model of collapsing. (4) seems rather realistic for theconventional MCVD technique In the existence of sinusoidal index ripple, the optimum a! by comparing it with that expressed by (3) because each layer for minimizing u is investigated. Fig. 10 shows plots of u is formed by the same amount of pure silica deposition [ 171, versus CY for two types of sinusoidal oscillations given by (3)

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3 2 1 (w

-E

. n P

-b

mNT-SATE n v

0

-

2

4

6

0.1

Qas

0

1

0

1

2

(

4

MEW-STATE

N

Fig. 9. u as a function of N for two types of sinusoidal ripple given by (3) and (4). The transient state is assumed.

and (4) with iV = 8. The transient state is assumed again. By comparing the two results for u, an improvement is found in the sinusoidal oscillation function of (3) for minimizing the pulse broadening. u is reduced to as smallas 0.10 ns/km at a = 1.955. These u and 01 values are nearly equal to the minimum u and the optimum a(= 1.96) without an index ripple. For the oscillation function of (4), u is minimized at most to 0.18 ns/km at a = 1.98. These features are clearly seen from the impulse response shapes shown in Fig. 11. Here, N and 01 are chosen as 8 and 1.96, respectively. As shown in Fig. 1l(b), pulse deformation due to higher modes is serious forthe oscillation function of (4). As for the influence of the sinusoidal index ripple on fiber frequency response, a similar tendency to the case ofthe central index dip was observed. Theindex ripple results in the complex variation of baseband loss against frequency and in the considerable changeof frequency response curve for different modal power distributions, especiallyin the region of small N .

a0 1

2 .o

1.8

1.6

22

a Fig. 10. u as a function of D for two types of sinusoidal index ripple given by (3) and (4) in the case of the transient-state modal power distribution.

-I

0

Asin{ExN(r/al}

k

N=8

3

A=1.4~10-' o=1.96

81

1 T

-2

-1

1

1

0

DELAY TIME (NS)

(a)

V. CONCLUSION An exact solution for pulse broadening has been obtained for multimode fibers with a power-law index profile. The calculation has been made by using an integral expression for group delay time, based on the multilayer approximation. The resuit has been compared with the well-known WKB solution. It has been clarified that a few higher order modes exist near cutoff, having a considerably smaller group delay time than those of the modes confined well in the core, while they are neglected in the WKB solution. It has shown that the minimum value of u and the optimum valueof a for the power-2 -1 0 1 law index profile calculated by the multilayer approximation DELAY TIME (NSI inthe experimentally determined steady-state modal power cb) distribution are nearly equal to the WKB solutions under the Fig. 11. Impulse responsesforfiber with sinusoidal index ripple. (a) 6n(r) in (3). (b) 6n(r) in (4). The transient stateis assumed. conditions of lossless and uniform excitation of all modes. Next,the minimum u and the corresponding optimum 01 have been evaluated for certain index imperfections. It has A * sin [27rN(r/0)~ have been evaluated. It has beenshown excesspulse broadening is been clarified that for typical valuesofdip parameters, dip that for A = (nl - n,)/100,the depth of 0.005 (= A/2) and dip diameter of 5 pm, u is de- maximized around N = 2 and is reduced to zero for larger N . The results are in good quantitative agreement with those of graded to 0.40 ns/km which is four times larger thanthat Olshansky's perturbation theory. For &n(r)given by A .sin without an index dip, with a slight decrease in the optimum power-law index parameter. Influences of thetwotypes of [27rN(r/a)],u rapidly becomes equal to that with N = 0 in sinusoidal index ripple 6n(r) givenby A . sin [27~N(r/a)]and comparison to the case of &n(r)given by A . sin [ 2 ~ r N ( u / a ). ~ ]

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KITAYAMA e t a[.: MINIMUM PULSE BROADENING IN MULTIMODE FIBERS

Thus, in order to reduce the number of deposition. layers, it is useful to adopt a deposition technique so that thedeposition layer thicknesses becomes equal to each other after collapsing.

ACKNOWLEDGMENT Theauthors wish to thank N. Uchida for stimulating discussions and his constant interest in this work. Sincere appreciation is also due to Prof. N. Kumagai of Osaka University for useful suggestions, and to H. Fukutomi and Y. Kat0 for their encouragement.

REFERENCES [ 11 D. Gloge and E.A.J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J.,vol. 52, pp. 1563-1578, 1973. [2] R. Olshansky and D. B. Keck, “Pulse broadening in graded-index optical fibers,” Appl. Opt., vol. 15, pp. 483-491, 1977. [ 3 ] J. B. MacChesney, “Materials and process for preform fabricationmodified chemical vapor deposition,” Proc. ZEEE, vol. 68, pp. 1181-1184,1980. [ 4 ] D. Marcuse, “Calculation of bandwidthfromindexprofilesof optical fibers, 1: Theory,” Appl. Opt., vol. 18, pp. 2073-2080, 1979. [5] H. M. Presby, D. Marcuse, and L.G. Cohen,“Calculation of bandwidth from index profiles of optical fibers, 2: Experiment,” Appl. Opt., V O ~ . 18, pp. 3249-3255,1979. [6] D. Marcuse and H. M. Presby, “Fiber bandwidth-spectrum studies,”Appl. Opt., vol. 18, pp. 3242-3248, 1979. [ 71 -, “Effects of profile deformations of fiber bandwidth,” Appl. Opt.,vol. 1 8 , ~3758-3763,1979. ~ . [8] R. Olshansky, “Pulse broadening caused by deviations from the optimal index profile,”Appl. Opt., vol. 15, pp. 782-788, 1976. [9] K. Morishita, “Numerical analysis of pulse broadening in gradedindexopticalfibers,” IEEE Trans. Microwave Theory Tech., V O ~ .MTT-29, pp. 348-352, 1981. [ 101 K. M. Case, “On wave propagation in inhomogeneousmedia,” J. Math. Phys., vol. 13, pp. 360-367, 1972. of [ l l ] N. Shibata and T. Edahiro,“Refractive-indexdispersion doped silicaglasses foropticalfibers” (in Japanese), Paper, Tech. Group on Opto-Quantum Electron., IECE, Japan, OQE80114, pp. 85-90,1980. [12] R. Olshansky,“Effect of the cladding on pulsebroadening in graded-index optical waveguides,” Appl. Opt., vol. 16, pp. 21712174, 1977. [13] K. Okamoto and T. Okoshi, “Analysis ofwave propagation in optical fibers having core with a-power refractiveindex distributionanduniformcladding,” IEEEPans. Microwave Theory Tech., vol. MTT-24, pp. 416-421, 1976.

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[ 141 A. W. Snyder and C. Pask, “Optical fibre: Spatial transient and steady state,” Opt. Cornmun., vol. 15, pp. 314-316, 1975. [ 15J K. Kitayama, M. Tateda, S. Seikai, and N. Uchida, “Determination of mode power distribution in a parabolic-index optical fibers: Theory and application,” IEEE J. Quantum Electron., vol. QE15, pp. 1161-1165,1979. [16] K. Kitayama, S. Seikai, and N. Uchida,“Impulseresponse prein a diction based on experimental mode coupling coefficients 10 km-long graded-index fiber,” ZEEE J. QuantumElectron., V O ~ .QE-16, pp. 356-362, 1980. [17] B. Bendow and S . S . Mitra,FiberOptics. New York:Plenum, 1979, pp. 269-275. [18] M. J. Saunders, “Optical fiber profiles using Iefracted near-field technique: A comparison with othermethods,”Appl. Opt., V O ~ .20, pp. 1645-1651, 1981.

Ken-ichi Kitayama (S’75-M’76)was born in Kobe, Japan, on October 28,1950. Hereceived the B.E., M.E., and Ph.D. degrees in communication engineering from Osaka University, Osaka, Japan, in 1974, 1976, and 1981,respectively. In 1976 he joined the Ibaraki Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Ibaraki, Japan, where he has been engaged in research work on transmission characteristicsfor designing optical fiber cables. His current interests include graded-index fiber and single- and two-mode fibers. Dr. Kitayama received the 1980 Young Engineer Award from the Institute of ElectronicsandCommunication Engineers of Japan, of which he is a member.

ShigeyukiSeikai, for a photograph and biography, see p. 58 of the January 1982 issue of this JOURNAL.

KatsumiMorishita (S’74-MY77) was born in Fukui,Japan, on February24, 1949. He received the B.E.,M.E., and Ph.D. degrees in electrical communication engineering from Osaka University, Osaka, Japan, in 1972,1974, and 1977, respectively. From1977 to1978 he was aPostdoctoral Fellow of the Japan Society for the Promotion of Science. Since 1981 he has been with Osaka Electro-CommunicationUniversity, Neyagawashi, Osaka, Japan, where he is now a Lecturer in the Department of Precision Engineering. His research interests are in the areas of electromagnetic field analyses and optical waveguides. Dr. Morishita is a member of the Institute of Electronics and Communication Engineers of Japan and theSociety of Instrumentand Control Engineers of Japan.