Pulse Transmission Through a Dielectric Optical Waveguide

F. P. Kapron and D. B. Keck

Waveguide propagation of a pulse-modulated carrier wave is formulated to include distortion due to dis-persion in both attentuation and phase velocity. An optimum input gaussian pulse width exists formaximum information carrying capacity. Results are applied to a numerical study of several single-mode glass optical waveguides in which mode and dielectric dispersion may total zero at some wavelength.For our low-loss (20 dB/km) guides in kilometer lengths, information rates of at least 3 X 1010 bits/secshould be attainable.

IntroductionThe necessity of medium to long distance information

transmission at ever increasing rates has led to a grow-ing interest in optical communication systems." 2 Withthe eventual system as yet undefined, work is progress-ing on several aspects of optical circuitry,3 such assources, modulators, amplifiers, detectors, and thepropagation medium. An important candidate for thislast component is the cladded glass fiber, especiallysince it has been demonstrated that such fibers can befabricated in at least hundred-meter lengths with at-tenuations on the order of 20 dB/km. 4

With the feasibility of transmission over sizablelengths assured, it becomes desirable to known the in-formation carrying capacity of fiber waveguides. Thisultimate capacity is the subject of this paper. We shallshow that long fiber lengths will be required for readilymeasureable effects; the unavailability of such low-losslengths has heretofore prevented this determination.Some investigation has been made of pulse broadeningover multimode glass fibers,', but short lengths andlimited detector resolution have allowed only upperbound estimates of about 1.5 X 10-' sec/km pulsebroadening. However, in the case of single-modeguides, our calculations below will show broadeningsabout three orders of magnitude less. These guidesshould therefore be all the more advantageous in large-capacity systems.

Pulse Broadening in Waveguides

Proceeding down the z axis of a waveguide, a single-mode harmonic wave of arbitrary transverse variationB(x,y) and initial phase is specified by the amplitude

The authors are with the Research & Development Labora-tories, Corning Glass Works, Corning, New York 18830.

Received 18 December 1970.

A(xy,z) = B(xy) exp(-acez) exp[i(hz - wt)] . (1)

Here a(w) and h(wo) are, respectively, the intensity ab-sorption coefficient and wavenumber of the guide, andco is the circular frequency of the light wave. After aguide length L has been traversed, the output wave isrelated to the input via

A(x,y,L) = S(w)A(x,y,O), (2)

where

S(co) = exp[(-2c + ih)L] (3)

is the amplitude transfer function (ATF) at this fre-quency. The ATF is the spectral response to a unitstrength, infinitely sharp impulse and contains the dis-persive properties of the guide both with respect toattentuation and phase delay. It is important to notethat this dispersion is due to the nature of waveguidemodes7 and bulk dielectric properties combined.' Thiswill be elaborated upon later.

We assume that if a mathematically well-behavedcomplex amplitude f(t) is incident upon the guide, theneach component of its fourier spectrum,

F(w) = f(t) exp(iwt)dt, (4)

is independently acted upon by the ATF as in Eq. (2).Superposition in frequency space and an inverse trans-form yield the output

q(t) = (1/27r) f F(w)S(w) exp(-icot)dco. (5)

Waveguide transmission is said to be distortionless ifthe temporal output is a replica of the temporal input,apart from a possible alteration in magnitude and delayin time, viz.,

q(t) = kf(t -to)

This implies that in Eq. (2)

k,to const. (6)

July 1971 / Vol. 10, No. 7 / APPLIED OPTICS 1519

a = -(2/L) logek, h _ vp = Lto, (7)

so both the absorption coefficient a and phase velocityv, must be constant across the spectral range of thelight input. The extent to which these conditions areviolated determines the amount by which the trans-mitted signal is deformed.

For information transmission, one is interested inpreserving the characteristics of an envelope wave g(t)modulating a comparatively rapidly oscillating carrierwave of frequency co. We Taylor-expand the ATFabout this frequency:

m=O

where

8m(wtwo) = exp{-! [-dm(-a + ih) L)1AA wave

f(t) = g(t) exp(-iwot)

by Eq. (5) will emerge as

q(t) = exp(-t aoL) exp [-icoo(t - L/vo)] r(t - L/v0o),

(8) Fig. 1. Schematic representation of input and output pulse

train intensities vs time. Note the time lag and intensity de-crease of the output. The full width at half maximum is2 vlog,2 times the l/e half-width. If is the maximum allowed

(9) intensity fraction at pulse intersection, then 1/N is the corres-ponding minimum temporal separation.

(10)

(11)

where subscript 0 means evaluation at wo. The groupvelocity is

v = 1/h', (12)

where primes indicate differentiation with respect to c.The spectrum of r(t) is

R(w) = G(w) exp)(-,wao'L) II 'm(c,cdo). (13)m=2

Equation (11) shows that after passage through thewaveguide, the carrier wave is reduced in amplitude byao and retarded in phase by vo, while the envelope isdelayed in time by v0o and distorted [Eq. (13)] by dis-persion in both absorption and group velocity (ve' =

-v, 2h"l).

Information Rates

Applying the above exact results to the practical situ-ation of an input envelope with spectral extent narrowcompared to the carrier frequency, we choose a normal-ized gaussian input pulse amplitude

g(t) = (aV7r)-l exp(-t2/2a0), a>>wo'3 X 10-'1 see,

(14)

which has the advantage of a minimal time-frequencyproduct width. Using only the m = 2 term of Eq. (13)and inverting, we have

r(t) = (aV/7r) (1 + ) exp[( a0'L + it)'/2a(1 + A)],(15)

where A = a-2( a0 " - iho')L; so the detected pulse is

| r(t) 2 = (b/,r)-'( + o"L/2a0)- exp[1(eo'L) 2/(2a2 + ao'L)'Iexp -b-2[t + ao'hoL2/(2a0 + ao'L)2}. (16)

Apart from uniform attentuation and time delay [asalready occurred in Eq. (11) and which are of no directconcern in pulse close-packing], the feature distinguish-ing Igal from 1rl2 is broadening of the pulse width froma to the value

b = [a2 + ao"L + (ho"L)2/(a2 + ao'L)] 2 (17)

as shown in Fig. 1.We have found that for our low-loss guides, the term

involving dispersion in the absorption coefficient,(yao'L)I, ranges from 10-1" sec/km to 10-1O sec/kmoutside any anomalous dispersion region. This isnegligible for guide lengths and pulse durations ofinterest here9 ; comparison with dispersion in wavenum-ber will be made later. (The approximate result of Ref.8 overestimates pulse width there.)

Suppose that the resolvability criterion of an infor-mation-carrying pulse train is formulated in terms ofthe maximum allowed intensity fraction overlap 6 as inFig. 1. The resulting minimum pulse separation 1/Nand Eq. (17) yield the information rate

N = !2a[a4 + (ho0'L)'] - IlogI - bits/sec (18)

in the absence of absorption dispersion. DifferentiatingEq. (17) with respect to a yields the minimum outputwidth

bmin = aV2, where am- (ho"L)3 (19)

is the optimum input pulse width taking maximum ad-vantage of guide bandwidth. Any shorter pulse willexceed the guide's high-frequency capability, while anylonger one will not utilize it fully. The correspondingmaximum pulse rate is

Nm = 8ho"L loge -3 bits/sec. (20)

Numerical ResultsWe now examine specifically a dielectric waveguide

consisting of a cylindrical core of radius a and refrac-tive index n,(X) surrounded by an essentially infinite

1520 APPLIED OPTICS / Vol. 10, No. 7 / July 1971

Ig(t I'

0.5

-_ 6-1.665aI I-

/ 1I

t -

t = t-L/v -

Ir(t')I'

0.5

0-

n E

I- I/N-1

cladding of index n2 (X). Here X is the vacuum wave-length, and for the no-cutoff HE,, mode the guide wave-number h(X) is contained in the characteristic equation'"

t(u) - 1(K2 + 1)27(w) + [2

+ (K2 + )W-2]

+ { (2 + 1)2(4X72 + nw-2

)

+ [2

+ (K2 + )w2

] } = 0, (21)

where

(u/a) 2= (27rnl/)2 - h2 , (w/a)2 = h 2

- (2 7rn2/X)2,

Jo(u) Ko(w)uJi(u) wKI(w)

n2K = -

n,

Here J and K are Bessel and modified Hankel functionsof the first kind, respectively. Single-mode propagationis ensured if

V = (u2 + W2) = (2ira/X)(n1 2 - n2 2)2 < 2.405. (22)

The variation of effective guide index with wave-length,

ne(X) 9 h(X)/27,

I.I:0

I 1

I

(23)

is schematically indicated in Fig. 2(a) for nondispersivedielectric materials. The curve is asymptotic to thecore (cladding) refractive index toward shorter (longer)wavelengths with a decreasing (increasing) fraction ofthe mode power contained in the cladding. Increasingthe core radius (from a, to a2 > a) has the effect of ex-panding the curve toward longer wavelengths; increas-ing the index difference (e.g., by increasing n1 to ni >

X (m)

Fig. 3. Calculated difference of guide and cladding indices,ne - 2 vs wavelength X for several optical waveguides. Thefirst number is the bulk core-cladding index difference at 632.8nm followed by the core radius in um: guide A-0.0034, 2;B-0.0034, 3; C-0.0094, 1; D-0.0094, 2; E-0.0287, 1.

nj) has a similar effect. This effective index variationdue to mode dispersion is superimposed upon the naturaldielectric dispersion of the two materials from which theguide is constructed. 8 The true guide index is deformedfrom that of Fig. 2(a) by the necessity of being asymp-totic to n,(X) for short X and to n2(X) for long X. Forglass constituents, this is shown schematically in Fig.2(b).

Dispersion properties have been calculated for fivewaveguides utilizing parameters believed to approxi-mate closely those of actual waveguides made in thislaboratory. For each guide, the numerical solutions toEq. (21) as a function of wavelength were calculated ona GE Mark II time-shared computer. It is to be notedthat the measured bulk indices of the core and claddingmaterials at each wavelength were inputted, and thisprocedure generated the guide wavenumber h = h(X)curve from X = 0.4 um to 1.0 Am in 0.5-,um increments.Core radii were 1 m, 2 um, and 3 cm, and core-to-cladding index differences at 632.8 nm were 0.0034,0.0094, and 0.0287. The effective refractive indices ofEq. (23) were then fit to a Sellmeier equation:

Fig. 2. Schematic representation of the effective dielectricguide index n, vs wavelength . (a) For two core radii a, a2,dispersionless indices n, i7, and dispersionless cladding index

n2. (b) For glasslike core and cladding indices ni, n2.

(24)fl(X) = A + E A j2 j = 1 2, * -3 X2- j

on a Univac 1108 computer, to the accuracy limit of themeasured index values and the solution of Eq. (21).This effective index was constrained to approach the

July 1971 / Vol. 10, No. 7 / APPLIED OPTICS 1521

(a)

I

(b)

n2X\Xn

\ _/ 'n

\N1

measured bulk cladding index at sufficiently long wave-lengths where essentially all power is evanescent. Thedifference of guide and cladding indices is shown in Fig.3 for the wavelength range 0.3-1.6 /um. For a givencore material this difference is less for the smaller coreradius, since V of Eq. (22) decreases, In the case ofguide C the latter decrease with larger wavelength issuch that the difference curve crosses below curves forboth lower core index guides A and B.

From Eqs, (23) and (24) we computer calculated thewavenumber dispersion

h"(X) X3

d2n,

27rC2 dX2(25)

for which results are shown in Fig. 4. The ordering ofthe dispersion curves differs from the ordering of theindices in the previous figure. In general, one cannotexplain this by simple arguments. On the basis of theexamples presented here, however, it appears that guidedispersion will most closely approach cladding disper-sion for small core-cladding index differences and forlarge core radii. Toward longer wavelengths, totalguide dispersion decreases substantially, and mode dis-persion predominates over dielectric dispersion. Thefact that the former is positive when the latter becomesnegative accounts for the shift of zero guide dispersiontoward longer wavelengths. In this region the informa-tion carrying capacity is essentially carrier-limited, thecase for propagation through free space." However,

20

18

16

14

12

10

E8 E

6C

4 C

2

0

-2

OA 0.6 0.8 LO 1.2 [.4 1.6(m)

Fig. 4. Calculated second derivative h of guide wavenumberwith respect to circular frequency plotted vs wavelength X forthe same quides as in Fig. 3 and for the bulk cladding (CL).

(1,6)

10 0.4 0.6 0.8 1.0 1.2 14 1.6X(m)

Fig 5. Calculated maximum information rate of waveguide Afor two guide lengths L in kilometers and two gaussian pulseseparation parameters A = -10 logloa in dB vs wavelength X.

this may require operation in a range of high dielectricabsorption.

For waveguide A, the theoretical maximum informa-tion carrying capacity given by Eq. (20) is plotted vswavelength for two guide lengths and two overlaptolerances in Fig. 5.

Table I lists pulse characteristics of four mode-locked lasers. Assuming bandwidth-limited inputpulses and two lengths of guide A, the output pulsespread expected from Eq. (17) is listed, a quantity thatmight be measured experimentally to determine wave-number dispersion. The attendant pulse rate is shownfollowed by the maximum attainable rate (from Fig. 5)assuming optimum input pulse width. It is apparentthat significant broadening will occur only if rather nar-row input pulses can be generated and detected and/orrather long lengths of waveguide are used. The 20-dB/km fibers being made in this laboratory would makedispersion measurements more feasible.

Conclusions

An analytic formulation of pulse broadening and in-formation capacity for waveguides has been derived andapplied to dispersion calculations for glass fiber guides.For low-loss guides in kilometer lengths operating withina low-dispersion wavelength region and with optimuminput ulse widths, information rates of at least 3 X 1010bits/sec should be attainable.

1522 APPLIED OPTICS / Vol. 10, No. 7 / July 1971

Table 1. Calculated Pulse Transmission Characteristics for Waveguide A with Several Mode-Locked Laser Sourcesa

Laser type He-Ne GaAs Nd:glass Frequency-doubled Nd

Central wavelength (nm) 632.8 900 1060 530Input pulse duration (psec) 50012 180's 814 514Output pulse spread (psec) 10im 1.1 X 10-6 4.0 X 10-6 3.8 X 104 0.011

1km 1.1 X 10-4 4.0 X 10-4 1.85 37.9Pulse rate (sec') 3.6 X 108 109 1.6 X 1010 4.2 X 108Maximum pulse rate (sec') 2.1 X 1010 3.2 X 1010 3.5 X 1010 1.9 X 1010

I The input pulses are taken to be bandwidth-limited and pulse durations and spreads are full width at half-maximum. Pulse ratesare given for a 1-km guide with 20-dB output pulse separation. The maximum rate assumes attainability of the optimum input pulseduration at the indicated wavelength.

The authors wish to thank J. A. Gawthrop and M. P.Teter for valuable assistance in the numerical computa-tions. We also acknowledge useful discussions withD. Gloge of Bell Telephone Laboratories.

References1. Conference on Trunk Telecommunications by Guided Waves,

Electronics Division of the Institution of Electrical Engineers,29 Sept.-2 Oct. 1970, London.

2. 72nd Annual Meeting of the Amer. Ceram. Soc., May 1970,Philadelphia.

3. Proc. IEEE, Special Issue on Optical Communication 58,No. 10 (1970).

4. F. P. Kapron, D. B. Keck, and R. D. Maurer, Appl. Phys.Lett. 17, 423 (1970).

5. D. Williams and K. C. Kao, Proc. IEEE 56, 197 (1968).6. W. A. Gambling, P. J. R. Laybourn, and M. D. Lee, Elec-

tronics Lett. 6, 364 (1970).7. P. J. R. Laybourn, Electronics Lett. 4, 508 (1968).

8. R. B. Dyott and J. R. Stern, presented at the Conference onTrunk Telecommunications by Guided Waves, ElectronicsDivision of the Institution of Electrical Engineers, 29 Sept.-2 Oct. 1970, London.

9. D. Gloge, Bell Telephone Laboratories, private communica-tion, has pointed out that a more general discussion of thistopic is given by C. G. B. Garrett and D. E. McCumber,Phys. Rev. 1, 305 (1970).

10. N. S. Kapany, Fiber Optics-Principles and Applications(Academic, New York, 1967).

11. W. A. Gambling and P. J. R. Laybourn, Electronics Lett. 6,661 (1970) have used an approximate pulse rate equation anderroneously consider dispersion in phase velocity rather thangroup velocity and have apparently missed some of the abovepoints.

12. M .A. Duguay and J. W. Hansen, IEEE Conference on LaserEngineering and Applicatiorns, paper 5.5, Washington, D.C.,May 1969.

13. T. L. Paoli and J. E. Ripper, Appl. Phys. Lett. 15, 105 (1969).14. M. A. Duguay and J. W. Hansen, Appl. Phys. Lett. 15, 192

(1969).

D. A. Worth Perkin-Elmer has been elected chairman of theLaser Subdivision of the Electronics Industries Association.He is also a member of the board of directors of the Laser In-

dustry Association.

July 1971 / Vol. 10, No. 7 / APPLIED OPTICS 1523