Closed-form, localized wave solutions in optical fiber ...ziolkows/research/papers/Localized... · in optical fiber waveguides. As with the free-space case, these optical waveguide

Vol. 9, No. 6/June 1992/J. Opt. Soc. Am. A 937

Closed-form, localized wave solutions in opticalfiber waveguides

Ashish M. Vengsarkar* and loannis M. Besieris

Bradley Department of Electrical Engineering, Virginia Polytechnic Institute and State University,Blacksburg, Virginia 24061-0111

Amr M. Shaarawi

Department of Engineering Physics and Mathematics, Faculty of Engineering, Cairo University, Giza, Egypt

Richard W Ziolkowski

Electromagnetics Laboratory, ECE Building 104, Room 422E, University of Arizona, Tucson, Arizona 85721

Received June 6, 1991; accepted September 11, 1991; final revised manuscript received December 18, 1991

A novel bidirectional decomposition of exact solutions to the scalar wave equation has been shown to form anatural basis for synthesizing localized-wave (LW) solutions that describe localized, slowly decaying transmis-sion of energy in free space. We present a theoretical feasibility study that shows the existence of LW solutionsin optical fiber waveguides. As with the free-space case, these optical waveguide LW solutions propagate overlong distances, undergoing only local variations. Four different source modulation spectra that give rise tosolutions similar to focus wave modes, splash pulses, the scalar equivalent of Hillion's spinor modes, and themodified power spectrum pulses are considered. A detailed study of the modified power spectrum pulse is per-formed, the practical issues regarding the source spectra are addressed, and the distances over which suchLW solutions maintain their nondecaying nature are quantified.

1. INTRODUCTION

Brittingham's pioneering work' in finding focus wavemode (FWM) solutions to Maxwell's equations has inspiredseveral researchers to explore the existence of nondis-persive, packetlike solutions to the wave equation thatcan propagate through free space without any decay.Ziolkowski2 has argued that, although the FWM solutionshave an infinite total energy content, a superpositionof FWM's might have an advantage over the standardplane-wave superposition when it comes to describingthe transfer of directed pulses in free space. Such pulses,characterized by high directionality and slow energy decay,have been termed localized waves (LW's).

Experimental investigations of launching acousticalLW's have shown considerable success. It has been estab-lished that a LW pulse launched from a linear syntheticarray propagates with little variation over extended near-field distances. Recent results3 have shown that a 10-foldimprovement over continuous excitations is possible. An-other area of research that is based on similar underlyingprinciples is Durnin's diffraction-free Bessel beam.4

Experimental verification of the larger depth of such abeam, as compared with a Gaussian beam, has been pro-vided by Durnin et al.5

To assimilate all these attempts toward synthesizinghighly directional pulses and beams into one representa-tion, Besieris et al.6 have proposed a novel approach to thesynthesis of wave signals. Within the framework of thisnew approach, exact solutions are decomposed into bidirec-tional, backward and forward, plane waves traveling along

a preferred direction z, viz., exp[-ia(z - ct)]exp[-i3(z +ct)]. These bilinear expressions can be elementary solu-tions to the Fourier-transformed (with respect to x and y)three-dimensional wave equation, provided that a con-straint relationship involving a, /3, and the Fourier vari-ables dual to x and y is satisfied. Such blocks have beenshown to constitute a natural basis for synthesizing Brit-tinghamlike solutions, such as Ziolkowski's splash pulses,Hillion's spinor modes,7 and the Ziolkowski-Belanger-Sezginer scalar FWM's. An application of this techniqueto infinitely long, cylindrical, metallic waveguides has beendiscussed by Shaarawi et al.' By varying the free parame-ters of specific source modulation spectra, one can synthe-size within the waveguide the localized, nondecayingpulses that move predominantly in the positive direction.Questions about the physical nature and the causality ofthe bidirectional representation have been thoroughly ad-dressed and will be presented elsewhere.9

In this paper we apply the method of bidirectional de-composition to optical fiber waveguides and investigatethe possibility of synthesizing pulsed solutions that canpropagate along such guides with only local variations.The analysis is similar to that of the infinite waveguidedescribed by Shaarawi et al.,' with the exception of theboundary conditions that the fields must satisfy at thecore-cladding interface. For simplicity, the core andthe cladding materials are assumed to be both nondisper-sive and nonabsorptive. The only dispersive characteris-tics are due solely to the core-cladding interface. Thegeneral solution to the scalar wave equation in an opticalfiber is analyzed for four source modulation spectra, and

0740-3232/92/060937-13$05.00 © 1992 Optical Society of America

Vengsarkar et al.

938 J. Opt. Soc. Am. A/Vol. 9, No. 6/June 1992

the approximations that need to be made to generate solu-tions similar to FWM's, splash pulses, the scalar equiva-lent of Hillion's spinor modes, and the modified powerspectrum (MPS) pulses are considered.

Current research efforts in the area of high-speed, long-distance fiber-optic communications are concentrated onthe development of new methods that will counter thepulse dispersion effects introduced during informationtransmission. The transmission of stationary, nonlinearoptical pulses, termed solitons, in dispersive optical fiberswas proposed by Hasegawa and Tappert'0 in 1973 anddemonstrated in practice by Mollenauer et al." in 1980.While soliton-based communications systems are on theverge of becoming a practical reality, limitations imposedon system performance by fiber loss, frequency chirp, andthe interaction of neighboring pulses are still being stud-ied. We propose a novel method of wave synthesis thatdoes not invoke the need for nonlinear effects in opticalfibers and that leads to solutions that are capable of coun-tering the dispersive effects introduced by the waveguide.Although only an elementary theoretical discussion ispresented in this paper, this technique may lead to theemergence of a competing technology in the area of high-capacity, fiber-optic communications.

In Section 2 we analyze the optical waveguide by assum-ing a solution of the bidirectional type and arrive at a ge-neric elementary solution to the scalar equation in termsof the newly defined parameters. In Section 3 we takerecourse to the classical waveguide analysis to simplify thegeneric solution further. A one-to-one correspondence isestablished between the two methods, and the conditionsfor the core and the cladding regions are established.This leads to an elimination of the practically unrealizablesolutions and simplifies the mathematical analysis thatfollows. In Section 4 we briefly describe the linearly po-larized (LP) modes in the optical fiber and use the proper-ties derived in the literature to specify the boundaryconditions. The superposition of the elementary solutionsobtained after the boundary conditions have been invokedis described in Section 5. In Section 6 we consider thesingular source spectrum, the splash pulse spectrum, thezero-order Bessel spectrum, and the MPS spectrum andderive expressions for pulselike solutions in optical fibers.In Section 7 we evaluate the nondecaying nature of theMPS pulse shape and relate the free parameters to physi-cally meaningful quantities. We quantify the distance forwhich no decay will be observed in the MPS pulse solutionderived earlier. The feasibility of practical implementa-tion of such spectra as well as future theoretical direc-tions are considered in the concluding section.

2. SCALAR WAVE EQUATION

The scalar wave equation for an idealized optical fibermade of a nondispersive and nonabsorptive material canbe written as

( ni2) i = 1 > core,

i = 2 > cladding, (1)where V2 is the three-dimensional Laplacian, n is the re-fractive index (n > n2; n, n2 assumed to be constant),and c is the velocity of light in vacuo.

According to the bidirectional decomposition,6 we as-sume a solution of the type

T(p, 4, z, t) = I)(p, )exp[if3(z + ct)]exp[-ia(z - ct)]

= 1D(p,0)exp(i83q)exp(-iaY), (2)

where -q= z + ct and; = z - ct. Breaking up theLaplacian into its transverse component V,2 and a longitu-dinal component Oz2, we can write the following:

az2 = -( - a)2-

d 2, = -C2 (3 + a) 2P.

Equation (1) then becomes

(V,2 + K 2) ID = 0,

(3)

(4)

(5)

where K 2= ni 2 (a + 63)2 - (a - p)2. In cylindrical coor-

dinates, V2 is given by

V,2 = 2/ap2 + p-la/ap + p-2 2 /a42. (6)

Substituting Eq. (6) into Eq. (5) and assuming separabilitywith respect to the variables p and , viz., cD(p,4) =R(p)F()), we get a set of ordinary differential equations;specifically,

d2R/dp2 + p-'dR/dp + (K,2- v2/p2)R = 0,

d 2F/d4)2 + v2F = 0.

(7)

(8)

Equation (7) is the Bessel differential equation, andEq. (8) is the second-order, ordinary, harmonic differentialequation. Solving Eqs. (7) and (8) and recombining vari-ables, we can write a general solution to the Helmholtzequation [Eq. (5)] as

c'(po) = {AJp(Kjp) + BYXKp) COS V,LCI,cKp) + DKV(Kp)J

Ki 2 > O

Ki2 < 0(9)

where A, B, C, and D are constants and J,, Y I,, and K,are the ordinary and the modified Bessel functions. Inthe classical analysis of a waveguide solution, one woulddetermine which of these constants need to be zero so asto acquire practically realizable solutions. To do that, wefirst determine the respective regions of the waveguide inwhich Ki > 0 and K,2 < 0.

3. CLASSICAL WAVEGUIDE ANALYSIS

We seek to establish a one-to-one correspondence betweenthe parameters in the bidirectional approach and the clas-sical analysis. Since the waveguiding conditions that themode propagation constant ,Bcl should satisfy are known ina classical analysis, the one-to-one correspondence will de-termine the constraints on the parameters a and ,B as wellas help in establishing the regions denoted by KI2 > 0 andKi

2 < 0. This is a valid means of obtaining some of thedesired conditions because the bidirectional decomposi-tion is an alternative method for the synthesis of pulselikesolutions and not a replacement for the classical Fouriersynthesis. There exists a parallel between the two meth-ods, and it is most advantageous to extract maximum in-formation from each of these approaches.

In the classical analysis we consider solutions ofthe form

P(p, 4, z, t) = (D(p, 0)exp(tif,3jz)exp(+iwsjt), (10)

Vengsarkar et al.


where 3 is the propagation constant, &) is the angular fre-quency given by 2'irc/A in terms of the wavelength, and thesubscript cl is used to denote the classical approach. Com-paring Eqs. (10) and (2), we can show the correspondence

j01 (/3 - a), wli - -(a + /3)c (11a)

or

(lib)

The classical waveguiding condition in this case is given by

(12)wcc o < aec < n cto e

which can also be expressed as the set of expressions

(n 1 ).I)' - (3~,c)2 > 0,

(/361c) 2 - (n2 &o 1)2 > 0.

(13a)

(13b)

These properties of LP modes imply that the solutionCD(p,4) in Eq. (15) can be considered a representation ofthe LP01 mode of a fiber, the first subscript denoting thatv = 0. To find a relation between the constants A and Din Eq. (15), we use the property of LP modes that, to meetall boundary conditions, it is sufficient that d1 and d/dpbe continuous at p = a, where a is the radius of the fiber. 2

This leads us to our final expression for the elementarysolution for the Helmholtz equation [Eq. (5)], viz.,

D(p) = AJo(Kip), p < a,

AKo(K2P) Jo(Kla), p > a

as well as to the characteristic equation of the fiber

KiJi(Kla) K2K1(K2a)

Jo(Kla) Ko(K2a)

(17)

(18)

The corresponding expressions for the bidirectional ap-proach give us

n,2(a + 3)2 _ (a _ )2 > 0,

n22(a + 3)2 _ (a - /3)2 < o,

(14a)

(14b)

thereby implying that K 2 > 0 and K22 < 0. Elementary

solutions for the Helmholtz equation [Eq. (5)] in the tworegions of interest, the core and the cladding, are given byEq. (9). Since Y,(K1p) -- - for p = 0 and I,(K2P) -- - asp -a> c, the constants B and C are taken to be zero so thatphysically realizable solutions are obtained. Further, forsimplicity of analysis we assume that there is no azimuthaldependence of the field (v = 0). Hence the solution sim-plifies to

qD(p) = {o(KIP), p < aDKo(K2 p), p > a

The constants A and D are determined in Section 4.

(15)

4. LINEARLY POLARIZED MODES INWEAKLY GUIDING FIBERS

In most practical optical fibers, the refractive indices ofthe core and of the cladding are nearly equal. If the pa-rameter , is defined as

n2 -n,2 2_ - n2, (16)

then the condition A << 1 is called the weak-guidancecondition, and an optical fiber with this property is calleda weakly guiding fiber. With this weak-guidance condi-tion, we can simplify the exact solutions of Maxwell'sequations to give an insight into the properties of certaincombinations of modes, termed LP modes. Apart fromthe simplification in the analysis and the ease in under-standing the mode phenomena, the concept of LP modeshas other useful properties. Gloge12 has shown that it ispossible to calculate the fields in the fiber and the charac-teristic equations of LP modes directly from Maxwell'sequations. The solutions so obtained are based on thescalar wave equation, and for this reason the LP modesare sometimes referred to as scalar modes.

where we have used the property of the Bessel functions,Jo'(Kla) = -J(Kla) and Ko'(K 2a) = -K,(K 2a), where theprimes indicate derivatives with respect to the argumentsof the Bessel functions.

5. GENERALIZED SOLUTION

We now develop the generalized solution to the scalar waveequation by using the bidirectional representation. Inthe analysis that follows, the constraint on a and /3, givenby the equation K,

2= n,2(a + /3)2 _ (a - 3)2, is difficult

to handle. For this reason, we model the waveguide asone with a core of refractive index nl = 1 and a claddingof effective refractive index n2 = ne(ne < 1). It is fairlysimple to calculate this effective index ne, as shown inAppendix A. This change in notation for the refractiveindices alters the constraints to

K12= 4a/3,

K22 = ne2(a + 3)2 - (a _ 3)2

(19a)

(19b)

A general solution to the scalar wave equation can be writ-ten in terms of the superposition

p(PX tm) ( )2 f dKif da fd A(a,, ,K)KiJo(KP)

x exp(-iafexp(i/37)8(4aj6 - K 2) (20)

for p s a, where the constraint of Eq. (19a) has been incor-porated within the integral with the help of the delta func-tion. Similarly, for the region p 2 a we have

IP(p4 Cm) = (X2 IdKi dK 2 f daf d3A(a, ,K)KKo(K 2P)

x K () exp(-ia;)exp(i,6,)8[K 2 - f(K,)]

x 8(4a3 - K12)8K2

2 -ne 2(a + 3)2 + (a - /3)2], (21)

which includes all the boundary conditions that 4 shouldsatisfy at p = a as well as the constraints of Eqs. (18) and(19). In Eq. (21) f(Kj) denotes the constraint imposed onthe choice of K2 once a value is chosen for K, and can beevaluated from the characteristic equation (18). A nu-merical evaluation of Eq. (18) gives the functional relation-ship between K2 and K, and is shown in Fig. 1.

Vengsarkar et al.

,Sri - (a - 0), 0).1 - (a PC.

940 J. Opt. Soc. Am. A/Vol. 9, No. 6/June 1992 Vengsarkar et al.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

which, for ni = 1 and n2 = ne, gives us the two conditions

a// > s, a > 0. (27)

As a result, the limits of the integrals in Eqs. (20) and (21)can be written, for the case when both a and / are greaterthan zero, as

fdKif d,3f daI0 o Us

(28)

or

f dKi J da f d,/I,0 0 0o(29)

Kappa I x a

Fig. 1. Dependence of the cladding parameter K2 on the choice ofvalues for the core parameter Ki. The graph defines the func-tion f(KI) in the text.

The limits on the integrals in Eqs. (20) and (21) will bedependent on the constraints given by expression (12)along with the two choices for the one-to-one correspon-dence between the classical analysis and the bidirectionalapproach, expressions (a) and (lib):

Choice 1: Expressions (a) reduce constraint expres-sion (12) to

-n2(a + /) < ( - a) < -n,(a + /3), (22)

which, for n = 1 and n2 = , gives us the two conditions

a/ < s, a < 0, (23)

where s = [(1 + ne)/(l - ne)]. As a consequence, the lim-its of the integrals in Eq. (20) and (21) can be written as

r 0 A'reJdKiJ d/PJ daI (24)

or

f dzi f daf d/3I, (25)

where I represents the integrand in each of the integrals.The order in which the integrations are carried out (firstover the variable a and then over / or vice versa) deter-mines which of the above two expressions is applicable. Inexpression (24) we note, however, that, since /3 is alwaysnegative, the integration over a is from zero to a negativenumber, /3s (s > 0). But a should always be positive tosatisfy the condition a < 0. Hence expression (24) givesus a contradiction in terms and leads to a trivial solutionthat is always equal to zero. This is a mathematical veri-fication of the fact that the boundary conditions do notmatch. Similarly, we can show that changing the order ofintegration [cf. expression (25)] will also lead to a situationin which both conditions (23) cannot be satisfied.

Choice 2: Expressions (lib) reduce constraint expres-sion (12) to

n2(a + /3) < (a - /3) < n(a + P),

where I represents the integrand in each of the integrals.With this choice of the one-to-one correspondence, it ispossible to satisfy both conditions (27). Also, in the limitn- 1, we would expect that the solution should vanishsince the waveguiding constraint is no longer met. Thisis evident from the limits, since s -a oo, and integrals (28)and (29) go to zero for any integrand.

The generalized solutions [Eqs. (20) and (21)] were ob-tained under the assumption that the constraints [expres-sion (26)] are satisfied in all the regions. Hence the limitsin integrals (28) and (29) are applicable to the evaluationof the generalized solutions in the core as well as the clad-ding region.

6. PULSELIKE SOLUTIONS IN OPTICALFIBERSDifferent choices of the modulation spectrum A(a, ,/, K) en-tering into the bidirectional representations [Eqs. (20)and (21)] can now lead to the desired packetlike solutions.In this section we consider four specific spectra and derivesolutions that resemble the ideal FWM, the scalar analogto Hillion's spinor modes, the ideal splash pulse, and theideal MPS pulse. It is interesting to note in the analysisthat follows that the simplicity of the final expressions forthe localized solutions depends on the order in which theintegrations in Eq. (21) are carried out. For instance, theintegration, first over a and then over /3, described byexpression (28) would likely lead to a compact solution inthe form of an integral that may need numerical evalua-tion. On the other hnd, if, the order of integration isreversed, as described by expression (29), we may be ableto extract information about the ideal free-space solutionthat the pulse would resemble. In such cases the solutionT is broken into two parts, viz., = , + ,, where thesubscript u indicates the unperturbed packetlike solutionand the subscript w stands for the additional wall termintroduced by the waveguiding constraint.

In the following subsections we summarize the resultsobtained from the synthesis of four different spectra; thedetails of the derivations are provided in Appendix B.A discussion of the physical implications is given inSection 7.

A. Focus Wave ModesWe first consider the singular spectrum

A(a,,/,K,) = (872 2 al)8(/3 - l3')exp(-aaj), (30)

7.1

6/

4./

3 . .- ..-. . .

2 . . ..... :I ........ .

Q .: .

14

P.=1

�2

(26)


which, in free space, has given rise to FWM's, as intro-duced by Belanger," Sezginer,'4 and Ziolkowski.'5 Theconstants a, and 3' are free parameters in the sourcemodulation spectrum that will be useful in adjusting thesynthesized solutions to give directed energy transfer inthe waveguide.

Substituting the singular spectrum [Eq. (30)] into thebidirectional representation [Eq. (20)] and carrying outthe details shown in Appendix B, we get the expression forthe ideal FWM solution [see Eq. (B6)]:

al exp(i/3f7) I- fp2 1,'.=(al + i) ex (al + i) ) (31)

which is identical to that given by Ziolkowski.'5 The wallterm is found to be, from Eq. (B10),

where tan 0, = s, p = (al + i)cos 0 - i sin 0, q =2p(sin 0 cos 0)1/2, d = b sin 0, and Qn (z) is an associatedLegendre function described in Appendix B. To obtain aphysical understanding of the pulse behavior describedby Eq. (38), we reverse the order of integration and ob-tain the expressions for %, and t,, given by Eqs. (B24)and (B26):

alb P{ 2 12 2-1/2

(al + i) (a, + i) J

t = (a +i) d,3 exp{-,3[(al + i)s - i]}

x {U,[-2i/3(a, + is,2p,/3V]+ iU2[-2i,/(al + is,2p/3Vs]}.

(38)

(39)

(32)

where s = [(1 + ne)/(1 - ne)] and U is the Lommel func-tion of two variables defined in Appendix B.

A similar analysis can now be extended to the scalarfield solution in the cladding. The constraints imposed bythe delta functions in Eq. (21), however, necessitate theuse of a numerical integration, and we do not address thatissue in this paper.

B. Splash PulsesWe consider next the spectrum

Ao(a,,8,i<) = 8i 2ala 28(K, - y)exp[-(aal + 3a2)], (33)

which, without the delta function, yields Ziolkowski'ssplash pulse in free space. Equations (B13) and (B14)from Appendix B give us

'P = aa 2YJo(yp)Ko{y[(a + i)(a 2 - i1qW'21,

= - a1a2v fy da exp[-a(al + i)]2 0 a

x exp Y2 (a2 - in)

D. Modified Power Spectrum PulseWe now consider the MPS pulse spectrum

A(a,/,, K)

Pexp[-aal - (p/B - b)a2], /3 > b/p

.0, b/p > 3 > O0(40)

which leads to localized, slowly decaying solutions in freespace.6 Substituting the spectrum from Eq. (40) intoexpression (28), we arrive at the expressions for P, and T Wof the MPS pulse given by Eq. (B29) and expression (B32)in Appendix B:

W. (P, ad-0 = exp[- (bx/p)]=47r(a, + i)(a2 + X/P)

'P _ip exp(ba2) (- b f)21-(al + i) P /

X {Ui[2ik(al + it)s,2pkvN1

+ iU2 2ib(al + i)s, 2p- V]} (34)

(41)

(42)

where

(35)

in the fiber core. The expressions for the cladding regioncan be obtained by numerical methods, similar to the caseof the FWM solutions in Section 5.

P2

al + i)fl= (pa2 - i) + s(al + i.

(43)

(44)

C. Scalar Analog to Hillion's Spinor ModesWe consider next the Bessel spectrum of order zero,

Ao(a, /3, Kl) = 87r2albJo(/3b)exp(-aa1), (36)

which, in free space, yields the scalar analog to Hillion'sspinor modes.7 Integrating first over /3 and then over a,we find the total solution to be, from Eq. (B19) of Appen-dix B,

IF= alb I dOJo (qd)"/2[(p2 + q

2 + d2) - 4q2d2]1/2

X Q-,/2' (37)

7. DETAILED STUDY OF THE MODIFIEDPOWER SPECTRUM PULSE

Equations (31), (34), (38), and (41) for P, represent thenondecaying, ideal, LW solutions in the core for theFWM's, the splash pulses, the scalar equivalent of Hillion'sspinor modes, and the MPS pulses, respectively, whileEqs. (32), (35), and (39) and relation (42) for t, describethe deviations in those free-space solutions that are due tothe waveguiding constraint. The behavior of these solu-tions is sensitive to the values of the free parameters al,a2, b, p, 'y, and /3' To understand the properties of thesesolutions, we analyze the pulses as they propagate throughthe waveguide and quantify the distances over which theymaintain their nondecaying nature. Substituting values

'w = ( + exp[/3'(al + i)s](a + i2x {UJ[-2i/3'(ai + is,2p/3'V-s]+ iU2[-2i,3'(al + i)s,2p/3'V1]},

Vengsarkar et al.


of the free parameters in the source modulation spectrainto the equations, we will be able to determine theamount of localization of the initial pulse and the behaviorof the pulses traveling in the positive and the negative di-rections. By doing so, we will relate the free parametersin the spectra to physically meaningful quantities, such asthe width of the initial pulse and its amplitude.

We note that Eqs. (32), (35), (39), and (40) for tI can beevaluated numerically only for specific values of the freeparameters. In this section we restrict ourselves to adetailed analysis and quantification of the MPS pulse forwhich a tractable, closed-form solution is available. How-ever, to gain a better understanding of the properties ofthe ideal pulses resulting from the remaining spectra, wehave determined the nature of the free parameters in thespectra and their effect on the unperturbed, ideal pulseshapes and amplitudes. A description is given inAppendix C.

constants. We intentionally fix the constant a2 = 1.0 m;the other constants control the following properties.

Normalizing the magnitude of the solution T at p = 0,z = ct, to its initial value 4rala 2 , we obtain

IP(p = 0,z = ct)I = [1 + (2z/pa2) 2 ] 112

This means that the amplitude of the solution is main-tained to the distance L pa2 . Since we desire to main-tain the initial amplitude over distances L 40,000 km,we take p to be very large, i.e., p = 108.

Note that the choice of values for p is not completelydetermined by the localization distance L. The waist ofthe beam, as well as the minimum angular frequency°m in, controls the value of p as shown below. The maxi-mum angular frequency of interest, to,,a, is fixed by alto be

A. Similarity to Free-Space Modified Power SpectrumPulseIt is known that the unperturbed MPS pulse is localized inspace and can propagate without decay for thousands ofkilometers (for an appropriate choice of the free parame-ters). We therefore compare the relative amplitudes ofthe unperturbed and the wall terms for a specific case,namely, at p = 0 and z = 0. We use the Bessel-functionexpansion of Lommel functions [See Eq. (B9) of Ap-pendix B] to evaluate the wall term at p = 0. Usingl'H6pital's rule and the recursive relations for Bessel func-tions, we can show that

lim (az) 'm (45)Z- ZIT =m!2m

) max = c/a , (50)

and the minimum angular frequency Wmin is fixed by theratio b/p to be

Wmin = cbIp. (51)

The transverse localization of the unperturbed MPS pulseis given by its initial waist,

wo2 = pal Amin Amaxb 2- r i (52)

where Amax = 2TC/&)min and min = 2 7Tc/Wmax. Hence the

waist is fixed if the minimum and the maximum wave-lengths are specified. On the other hand,

The wall term tk can be expressed at p = 0 and z = 0,from expression (42) and Eq. (45), as

I'IW( = O Z = 0)I = [1 - exp _ )] (46)27rala2 P

where we have assumed thatpa 2 » sa,. The ratio of theunperturbed term to the wall term then becomes

I1'PUI _ 1 , (47)iI'T 2[1 - exp(-x)]

where x = sba/p. For small values of x, this ratiobecomes

b a, 2'ir= = .p W02 Am.x (53)

Equations (52) and (53) imply that the maximum andthe minimum wavelengths Amin and Amax are fixed if aland wo are specified. Since in practice the available fre-quencies in the fiber are limited, we have chosen, for thesake of demonstration, Amin = 0.3 Am and Amrax = 30 ,m.The fiber dimensions limit the allowed choices for thewaists. Since our wavelength selection gives wo =0.50 ,m, the initial waist is smaller than the chosen fiberradius (r 2-5 Am),. Then the ratio b/p = 2.1 X 105 fi-nally fixes the remaining constant b to be very large:b = 2.1 X 1013. Therefore the assumption that al << p/a 2or, since

IP.I 1

71W 1l: 2 xI

sbal (al 2X S _ << ,P W

(54)(48)

which implies that, for a more pronounced localization ofthe total solution, one should choose the factor x tobe small.

that x << b/a2 , which was used to obtain Eq. (49), is valid.In this small-x limit, expression (48) reduces to the form

IT" I 1 _ 1 W2_ I Amax_ 1 &JmaxITP1 2x 2s al2 2s Amin 2s min (55)

B. Practical Implications of Free ParametersUsing our knowledge from the analysis of the free-spaceMPS pulse,6 we can now connect the behavior of theguided LW solution directly to the values of its defining

This result indicates that an increase in the bandwidth or,equivalently, an increase in the number of minimumwavelengths in the waist will increase the difference be-tween the unperturbed MPS core term and the wall term,

(49)


thus making the guided LW solution more like its free-space counterpart. The requirement that x << 1 sets apractical limitation on the fiber structure: Either thefiber material should be highly transparent to a broad win-dow of wavelengths or the fiber should be strongly guiding.One would therefore have to explore materials with largetransparent windows of transmission or reevaluate theentire analysis without using the LP mode formulation.The latter approach would necessitate the use of smaller-core fibers to remain close to single-mode operation andwould in turn imply added complexity of source design.Equation (55) also underscores the contribution of thewall term as a function of the weak-guidance condition ofthe fiber. For example, if the index of refraction of thefiber cladding were made closer to the core refractive index,the fiber would guide the modes in a weaker fashion, andwe would expect the contribution from the wall term toincrease. This physical insight is confirmed by Eq. (55)since, as ne -> 1, s increases and the ratio ITP,/ITP, de-creases, implying a growing influence of the wall term.

0.60.402

(a)

2.5 10'

Returning to the explicit representation of the wall term[Eq. (43)], one finds that its transverse variations are con-trolled by the Lommel-function terms. Since

= 4rx-p Amin

(56)

except when p < Amin, an explicit numerical evaluation ofthose terms is necessary. These terms yield an oscilla-tory, rapidly damped function of p. The decay rate can beobtained since, as z --- c, the function

Jn(az) -*(az2n+l)-1/2(57)

so that only the n = 1 term survives, and it goes to zerolike (/Amin)F' 2

. In contrast, the perturbed core termdecays transversely as exp[-(p2 /wo2)] except in the (for-ward and backward) tail regions of the MPS pulse, whereIZ - ctl >> a,, so that the term's decay is more gradual,

0.6

0 .2 7. ~0~~~~~~~~~~~~~~~~~~~~~~~~~~

0 Rais(in)

(C)

adius () i Radius ()Zeta (m )l U ~

(b) (d)

Fig. 2. Evolution of the MPS pulse as it propagates through the fiber. (a) z = 0. Normalized unperturbed free-space equivalent, Ilk,,12.(b) z = 0 m. Normalized wall term, [T I . (c) z = 0 m. Normalized total solution, I1yu + %rW 2. (d) z = 40,000 km. The plots show thatthe MPS pulse remains localized over distances of the order of several thousand kilometers, starts decaying slowly when z = 10,000 km,and spreads out significantly (with a drop in peak amplitude) after propagating over a distance of 40,000 km. The parametric valuesused are a, = 5 x 10-8 m, a2 = 1 m, b = 2.1 X 103 m 1',p = 108, and s = 10. These values correspond to Amax/Amin = 100.

Vengsarkar et al.


similar to its axial decay:

exp(-{p 2 /w 02 [1 + (z -t)2/a2])

1 + (z - Ct)2/a,2

exp{[p2a,2/wO2(z - t)2]} + (z - ct)2t11 + (Z - t)2/a, 2 l a, J

Let us consider another example, in particular, thesplash pulse spectrum. As we show in Appendix C, Sub-section 4, for longitudinal amplitude maintenance along

(58)

Thus, if the waist is many wavelengths in size, the wallterm will be much smaller than the core term everywhereexcept near the wall, where its decay rate becomes slightlysmaller than the core's. This comparison illustrates thatthe wall term in the forward tail region of the core MPScomponent will lead the central peak. This means thatone can visualize the wall component of the guided LWsolution as a low-level background field propagating in thefiber with little significant contribution except near thewall, where it provides a guiding and renovating mecha-nism for the central, core component. Heuristically, itappears that the core component surfs along in the fiberon the background wall field.

C. Graphic Description of Modified Power SpectrumThe variation of the square of the amplitude of the unper-turbed solution, i1,l2, for Amax/Amin = 100 is shown inFig. 2(a). The corresponding wall term is shown inFig. 2(b). Figure 2(c) shows the total launched pulse atthe input end of the fiber (z = 0); for a fiber core diameterof the order of a few micrometers, the pulse is largely con-fined to the core. The total pulse looks exactly like theunperturbed solution, indicating that the effect of the wallterm is negligible for this choice of free parameters.Local variations in the pulse shape are seen as a functionof the parameter . At distances of propagation of theorder of several thousand kilometers, there is no variationin the amplitude of the pulse. At z = 40,000 km [shownin Fig. 2(d)], the localized nature of the solution startscollapsing, the pulse shape changes, and the peak ampli-tude drops to less than half the original amplitude. Theplots in Fig. 2 were obtained for an x value of 0.2, andhence the contribution that is due to the wall term is notnoticeable for a ratio IP./11 P. = 5.0. To emphasize theheuristic surfing nature of the unperturbed core termalong the wall term, we plot the different components ofthe MPS pulse for Aa/Amin = 25 or, equivalently, forIIPI/I'IPI = 1.25. These plots are shown in Fig. 3. Onceagain, the normalized unperturbed and wall terms behavelike their counterparts in Fig. 2; however, the total pulsenow appears to possess a noticeable contribution from thewall term. Note that in Figs. 2 and 3 the squares of theamplitudes are normalized; as a result, the total pulse so-lution should not be considered to be a mere sum of theindividual contributions from the P, and the t, terms.

To ensure propagation of electromagnetic energy in thepositive direction, it is instructional to analyze the be-havior of the pulse traveling in the negative direction. Weevaluated the behavior of the MPS pulse in the negativedirection and found that the solution in the negative direc-tion decays rapidly over distances of the order of 10'8 mand is of the order of 10-'9 times the original pulse ampli-tude after propagating a distance of 1 m. This result fur-ther confirms that the choice of the free parameter valuescan lead to positive pulse propagation without any nega-tive component's being supported by the waveguide.

0 .0 .

(a)

(b)

"2.5 lo 7

Radius (m)

(C)Fig. 3. MPS pulse composition for Amax/Amin = 25. (a) Normal-ized unperturbed free-space equivalent, JrI2. (b) Normalizedwall term, .1t2. (e) Normalled total solution, ITU + t 12. Notethat the effect of the wall term is noticeable in the final solution,in contrast to the case depicted in Fig. 2. All parameters are thesame as in Fig. 2, except that b = 8.4 X 1013 m 1 .

2.5 o

Radius (m)


p = 0 for z 1000 km, the free parameter a 2 in thesplash pulse spectrum should be of the order of 10'. Atthe same time, we also find that, for the energy in thepulse to be maintained within the core (transverse con-finement), a2 should be in the 10-4_10-3 range. An imme-diate conclusion drawn from this result is that the freeparameter a2 dictates not only the decay of the pulse butalso the field confinement in the core. Hence the require-ments imposed on this parameter for a well-confined, lo-calized pulse to propagate over long distances cannot bemet, as seen from the wide variance in the desired valuesof a2 . This example proves that the synthesis processmust be carried out with caution by balancing the require-ments on the different free parameters.

8. CONCLUDING REMARKS

In summary, we have shown the existence of slowly decay-ing, nondispersive solutions in an optical fiber waveguide.A novel bidirectional decomposition of solutions to thescalar wave equation into multiplicatively backward- andforward-traveling plane waves was applied to an opticalfiber operated in the linear region. Four choices of thesource spectra were considered, and closed-form solutionswere obtained. The MPS pulse proved that, even thoughthe solution deviated from the ideal MPS pulse owing tothe waveguiding constraint, a proper choice of the free pa-rameters could lead to large distances over which the pulsewould maintain its original amplitude. We also showedthat the synthesis process leading to exact solutions needsto be performed with caution and demonstrated this inthe case of the splash pulse spectrum without the deltafunction when it was not possible to tweak the free pa-rameters to obtain nondecaying localized pulses confinedto the fiber core. This example underscores the fact thatall the spectra that have given rise to LW solutions in freespace need not be suitable for generating nondispersive so-lutions in waveguides.

In the practical realization of localized pulse-launchingschemes, the source spectra that are most easily imple-mentable should be considered. It is not clear whetherthe four spectra considered in this paper will prove to bethe right candidates. However, a rich class of other re-lated spectra A(a,/3,K) of the form 6(/3 - /')exp(-aa,)Io(a 2 K), 8(,3 - /')exp(-aa,)Jo(a2K), 6(, - /')exp(-aaj)Jo(a 2 K2 /4), or a exp(-aa)exp(-,/a 2 )exp(-a3/a), whichare theoretically more cumbersome, may lend themselvesto easier implementation because of the extra free pa-rameters available for tweaking. Such spectra may alsoreduce the restriction on the highly broadband nature ofthe MPS pulse spectrum described in this paper. Utiliz-ing the full potential of the bidirectional superpositionmay thus pave the way for future experimental systems.

In practical fiber-optic systems, pulse dispersion is sig-nificantly affected by material properties, especially whena broadband spectrum is used as the source. To aid ouranalysis when material dispersion is present, we havelooked at a much simpler system, namely, a plasma-filledcylindrical metallic waveguide. Our results indicate that,despite the presence of material dispersion, the localizedpropagation of pulses is feasible; these results will be pre-sented elsewhere. Future theoretical efforts will includea more detailed analysis that incorporates material disper-

sion and losses in the fiber. Several numerical methodsor asymptotic techniques available in the literature can beused to perform this analysis.'6 Despite the centrally lo-calized nature of these solutions, the possibility of generat-ing higher-order modes in the fiber cannot be ignored.Higher-order modes will act as noise in the system, andtheir effect on the localization of the fundamental solutionneeds to be analyzed. In the practical arena the avail-ability of the optical equivalents of the arrays used forgenerating the acoustic LW's will finally determine the re-alizability of such schemes.

APPENDIX A: EFFECTIVE INDEXCALCULATION

A method to calculate the effective refractive index, ne, ofthe cladding for a unity core refractive index is now given.The analysis is carried out in the classical notation. Theunderlying principle is that the expressions for the fields inthe core and the cladding, as well as the eigenvalue equa-tion, should remain unchanged by this transformation.

We first define the parameters U W and V as follows:

U = koa(nl - 32)1/2

W. = koa(f32 _ ,2)1/2

V = (U2 + W2)1/2.

(Al)

(A2)

(A3)

Here ,B = ,/ko is the normalized propagation constant,ko = 2wr/A is the free-space propagation constant, and a isthe fiber core radius. The fields in the core and the clad-ding can be described completely in terms of U and W.Eigenvalue equation (18) can be expressed in the formf(U,W) = 0. This implies that, if by a simple transforma-tion we can express U and W in terms of ni = 1 and n2 =

ne we will have an equivalent waveguide.Consider the transformation, ko' = konj. We can write

the transformed normalized propagation constant /3' as,' = 0//ko' = /3/konj. U and W can then be expressed as

U = ko'a(l _ p2)1/2

W = k'a[ 2 -

(A4)

(A5)

For n = 1 and ne = n2/n,, we now have an equivalentwaveguide in a frequency-scaled system that maintainsthe eigenvalue equation of the fiber as well as the expres-sions for the fields in the core and the cladding regions.

APPENDIX B: DETAILS OFLOCALIZED-WAVE PULSE DERIVATIONS

1. Focus Wave Modelike SolutionAfter integrating over K, and using the sifting property ofthe delta function, we find %U to be

"u(p, ,77) = a,f daJo[(4aj8')11 2p]exp(-iat - aal)

X exp(+iffq7), (B1)

Vengsarkar et al.


and Tw, is expressed as

Pw(p, ;,,q) = -a, f daJo[(4a/')"p]exp(-ia; - aal)

X exp(+il3"'r),

equation [Eq. (20)] and integrating over ,B and Kj, we get

T(p, = 2aja 2Jo(YP) - exp[-a(al + i)]fJ / 2 4 a

(B2)

where a' = ,'s.We first obtain a closed-form solution for PU. Using

identity (6.614.1) from Gradshteyn and Ryzhik,7 viz.,) 7 1/2 1' b2

fd/Jo(Vb)exp(-/3a) = 3 (i3 ex - )

8a) (8a)(B3)

and the Bessel-function relation

[I-/2(z) - I 2(z)] = ( ))exp(-z),

we obtain the relation

I e(b2fd/3Jo(b N/i)exp(-/3a) = expi - - .fo, a \4a

(B4)

(B5)

From Eqs. (Bi) and (B5) we then obtain the FWMsolution

,P ) = exp(i/3",j) [ p32 ~,) =a, l+ i,) exp (a i) .

We can rewrite the expression for P,, Eq. (B2), as

Tw(p, ,77) = 2a, exp(+i,3'77) f wdwJo[(4')/2pw]

X exp[-w2 (a + i)],

(B6)

X [exp E 2 (a2 ]i)) (B11)

Breaking the solution into %, and T, and using identity(2.3.16.1) from Prudnikov et al., 8 namely,

Yx exp(-xa)exp (- -) = 2Ko[2(ab)/2]

we can arrive at the expressions

'.u = aja2yJo(yp)Ko{y[(aj + i) (a 2 - i7)] 2}s

'P = - a J -d exp[-a(al + i)]

[ 2 a4a

(B12)

(B13)

(B14)

3. Hillion's Splash Modes

a. Integrate over ,3 FirstSubstituting the Bessel spectrum [Eq. (37)] into expres-sion (28) and integrating over K, we obtain

T(p, ,77) = alb f da exp[-a(a + i]

" d/3Jo[(4a/3)"2p]Jo(/3b)exp(ipq,7),

(B7)

where we have made the substitution w2 = a. Usingidentity (1.8.2.4) from Prudnikov et al., 8 namely,

J xV+ exp(ax2)J,(bx)dx = (2iax+l)

X [U,+(2iax 2, bx) + iUv+2 (2iax 2, bx)], (B8)

(B15)

where a' = a/s. When we substitute = r cos 0, a =r sin 0, and dad3 = rdrd0, Eq. (B15) becomes

T(pV,71) = alb f do frdrJo[(4 sin 0 cos 0)1"2pr]

X J(br sin 0)

X exp{-r[cos 0(a + i) - i sin ]}, (B16)

where tan ' = 1/s. Using identity (2.12.38.2) fromPrudnikov et al., 8 namely,

where f dxxJo(bx)Jo(cx)exp(-px) = Ao o2 ,

j -Ik Z 2 k + P U. (Z, 0 = k- 2+~

is the Lommel function of two variables, we obtain

ia,Pw= + exp[-/3'(a + i)s](a, + i)

X {Uj[-2i,'(a, + i)s,2p3'Vs]+ iU2[-2i/3'(a + i)s,2p,/'V-s]},

(B9)

(B17)

where

A02 =-pk 2 ( qd) -3 2 , (2 k2

Ao o2 = 2 - q))/ Q-1/2( k k)we- ko2) 1 2ak 2n

we obtain

(B10)

which gives us the final closed-form solution for thewall term.

2. Splash PulseSubstituting the splash pulse spectrum [Eq. (34)] into theexpression for the generalized solution of the scalar wave

(B18)

o (qd) 2[(p2 + q2 + d2) 42d2]1/2Q-/2

( p2 + q2 + d22qd (B19)

Here k2 = 4qc/[p2 + (b + d)2], p = (al + i)cos 0 -iq sin 0, q = 2p(sin 0 cos 0)1/2, and d = b sin 0. Q(z),an associated Legendre function, is a solution of the dif-


ferential equation

(1 _ Z2)d2Qm _ 2zdQnm + [n(n + 1) M2 Q nm = 0.

(B20)

b. Integrate over a FirstSubstituting the Bessel spectrum [Eq. (37)] into expres-sion (29), integrating over K1, and breaking the solutioninto two parts, %, and T,,,, we obtain

"'u(P, ,71) = baif d/fB da exp[-a(al + i)]

X J[(4a/3)"l 2p]Jo(/3b)exp(i/3'q). (B21)

Using identities (2.12.39.10) and (2.12.8.3) from Prudnikovet al.," namely,

1| exp(-ipx)Jo(c )Jo(bx)dx = (b2 _P 2)1/2

[ 4(b2 _ P2)}K4(b2 _p2)] (B22)

| exp(-ipx)Jo(cx)dx = (c2 - 2)1/2 (B23)

respectively, we obtain the following expression for theunperturbed pulse:

bal [ P 2 2 2 b2(a, + i~) (a, 2 j' (B24)(al+ t) al+ i) - +b

The contribution from the wall term, TPL, can be written as

P-(Ap, ';,7) = -alb db/Jo(b)exp(i37 )

X J da exp[-a(al + i)]Jo[(4a/3) 12p].

(B25)

The integral over a is of the same form as Eq. (B2). Usingan approach similar to that adopted for solving Eq. (B2),we obtain

t = (a + i ) | d exp{-,/[(al + is - i77]}

X {UJ[-2i/3(ai + is,2p,8]

+ iU2[-2i/3(al + it)s,2p/3\s]}. (B26)

4. Modified Power SpectrumWe substitute the MPS pulse spectrum [Eq. (41)] into ex-pression (29) and separate the integral into two parts, viz.,

T = T. + P1 = fdKJ daf dol

- f dKJ fdaf d,I, (B27)

where W. and %, correspond to the first and the secondterms, respectively. We first evaluate T,. Integrating

over a first, we can write I, as

'U = 8i7 b/p - exp[(-/3p + b)a2 + i71

lofwdKKJoK~pexp[~K, (al/+ i[ )] (B28)

Using the identity equation (B5) and integrating over Ki

and ,3, we obtain

exp[-(bx/p)]Pu(P4X1) 47r(al + i) (a2 + /P) (B29)

which is identical to the MPS pulse derived by Besieriset al.6

To evaluate 4,1 we switch the order of integration of aand 3 and use the identity equation (B8) to obtain

=ip exp(ba2) fd3x(/fP = 2Pr(a, + i | dp exp(-Q)

X {U,[2i/3(a1 + i)s,2p,6/]

+ iU2[2i,/(al + i)s,2p,3]}, (B30)

where Q = (pa2 - i) + s(a + i). By expressing theLommel functions U and U2 in terms of a series of Besselfunctions [see Eq. (B9)] and switching the summation andthe integral evaluation, one can show that the integrand isbounded and that an end-point evaluation of the form

£ exp[-/3f(x)]g(x)dx -- exp[-/3f(a)]g(a) + o(,-2)a /[(df/dx) (a)]

(B31)

is justified. Using the result from such an end-pointevaluation, we arrive at the final expression for Pt,, viz.,

'P ip exp(ba 2) (_ b Q=2,7(a, +it) Px~-

x {U, [2ik(aI + i)s, 2 pV ]

+ iU2[2ib(aI + i)s,2pb N/]}- (B32)

APPENDIX C: EVALUATION OFNONDECAYING PULSE DISTANCES

1. Focus Wave Mode SolutionThe initial pulse amplitude of the ideal FWM solution isgiven by

( ° (P2 1%,01 = exp - -al (Cl)

The variation of the field amplitude in the radial directionis dictated by the magnitude of the ratio 6/a1. A pulsepropagating in the positive z direction has a pulse peakgiven by

18 + = exp ( p2 ) (C2)

Vengsarkar et al.

948 J. Opt. Soc. Am. A/Vol. 9, No. 6/June 1992Vngakreal

which is identical to the initial pulse amplitude. The dis-tance over which this pulse can be utilized for communica-tions purposes depends on the losses and the materialdispersion of the waveguide. Transmission of energy inthe positive direction seems possible with little constrainton the free parameters of the source spectra. The wallterm can be expected to impose further constraints on thepulse parameters.

2. Splash Pulse SolutionThe amplitude of the unperturbed splash pulse solution,PT, evaluated at z = t = 0 can be expressed as

'.= aa 2yJ0(yp)Ko[y(aja 2 )12 ]. (C3)

The radial dependence of the field is governed by the mag-nitude of y. We need to ensure during synthesis that wedo not impose any further constraints on the free parame-ter when considering the distances over which this pulsecan remain nondecaying. At z = t = 0, P,,01 is given atthe center of the fiber (p = 0) by the equation

1 =,1 3 2 al a23 -- exp[-4y(ala 2)1 2 ], (C4)

where we have used the large-argument expansion of theBessel function and the fourth power of the amplitude istaken only for convenience in deriving the results that fol-low. At z = ct, the amplitude of the pulse traveling in thepositive direction, P,j'I, can be written as

~UI = '4(a 22 + 4"

X exp[-4,y\/__(a 22 + 4 2 )1"4 COS 4)], (CS)

where 4) is given by the relation tan 24) = 2/a 2 . Ex-pressing cos 4) in terms of z and a 2 , using the approxima-tion a 2 »> z of interest, and defining a decay factor F,4 asI~p Pu 4

/hIi 'p 01, we obtain an estimate for the nondecayingpulse distances. The decay factor F,4 can be written as

F4=

4Z2

a22 ,1/expI -2y(a,a 2)1/2 2 1

(C6)

which shows that, for the decay factor to remain 1, thevalue of the free parameter a 2 should be a few orders ofmagnitude greater than the distance, z, of interest. Notealso that the nondecaying nature of the solution does notdepend on the value of y, which defined the confinement ofthe field in the fiber core.

3. Scalar Analog to Hillion's Spinor ModesThe amplitude of the unperturbed pulse expression for theBessel spectrum can be written, at z = t = 0, as

'jU= [I + 2 2 12

The corresponding expressions for Pu'J and PU,+j are

IPY"0= 1I

The decay factor F" has the same expression as P(,givenby Eq. (C9), and we note that the pulse in the positive di-rection is singular at p = 0 and z = b/2. This behavior ofthe unperturbed, ideal pulse is physically unrealizable. Ifthe wall term P,, cancels this singularity, the sensitivityto variations in initial distribution may make this pulseextremely difficult to realize in a practical sense.

4. Splash Pulse Spectrum without the Delta FunctionIn all the previous examples in this appendix we consid-ered the effect of the free parameters on the ideal pulseshapes. The only total pulse ( 0 + P%) that we consideredwas the MPS pulse in the main text of this paper. We nowinvestigate the case in which the free parameters cannotbe tweaked to generate a pulse that will propagate for longdistances and at the same time be confined to the fibercore. Consider the splash pulse spectrum without thedelta function 6(K - y). After going through lengthy alge-bra we can show that the final expression for this pulse isgiven by

41 aja2 '.{ps+ [(a, ~s a - 12(a + .)PS +&Ts k2

(ClO)

The initial pulse amplitude is given by

1P0 I = a2[4p 2s + (ais + a 2 )2] -1

2. (Cli)

At the center of the fiber core, the amplitude of the pulsetraveling in the positive direction can be written as

1qP+1 = a2[4z2s + (ais + a 2 )

2]"-1

2, (C12)

As a result, the decay factor, , can be expressed in the form

r,= i~iP = I1 + 4zA 11/2IPI L (ais + a2)

(C13)

From Eq. (Cli), the variation of the initial field amplitudein the core can be plotted versus the radius p as shown inFig. 4. The variation of the decay factor, F, is given inFig. 5, where we have made the assumption that as «<a2. Figure 4 shows that, to confine the field to the core,the free parameter values should be of the order of 1V-i0-'. At the same time, Fig. 5 shows that the decay factor

'0

.2

's

0(C7)

1.0

0.8

0.6-

0.4-

0.2-

-2

a 1 32

-4a2 10

a2 1

0(C) Fiber Core Radius (i ptn)

Fig. 4. Dependence of the field confinement in the fiber on the(C9) free parameter a2 for the splash pulse. For guided confinement

of the field power, a 2 values of the order of 1085 will be preferred.

Vengsarkar et al.


0

QI

1.0 -

0.8 -

0.6

0.4

0.2-

*,, rev , _ _ ~~~~~~~~~~a2 10

a2 10 , -a2= 10

a a2= 0

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

1,000 2,000

Propagation Distance z (in cm)

Fig. 5. Variation of the decay factor r as a function of the freeparameter a2 for the splash pulse. Note that for nondecayingtransmission of energy the values of a2 should be of the order of10

7 , which contradicts the requirement imposed on the field con-finement as seen from Fig. 4.

will remain close to 1 for a2 values of the order of 107 ifnondecaying solutions over 1000-km distances are de-sired. This shows that the free parameter a2 dictatesboth the variation of the field within the core and the dis-tance over which the field remains nondecaying and thatit cannot meet the requirements.

ACKNOWLEDGMENTS

This research was supported in part by the Lawrence Liv-ermore Laboratory and the Virginia Center for InnovativeTechnology.

*Present address, AT&T Bell Laboratories, Murray Hill,New Jersey 07974-0636.

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16. For example, see K. A. Connor and L. B. Felsen, "Complexspace-time rays and their application to pulse propagation inlossy dispersive media," Proc. IEEE 62, 1586-1598 (1974);K. E. Oughstun and G. E. Sherman, "Propagation of electro-magnetic pulses in linear dispersive medium with absorption(the Lorentz medium)," J. Opt. Soc. Am. B 5, 817-849 (1988).

17. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Seriesand Products (Academic, New York, 1965).

18. A. P. Prudnikov, Y A. Bruchkov, and 0. I. Marichev, Integralsand Series. Volume2: Special Functions (Gordon & Breach,New York, 1986).

Vangsarkar et al.

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