J. Nathan Kutz, Cheryl V. Hile,* and William L. Kath
Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science,Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3125
Ruo-Ding Li and Prem Kumar
Department of Electrical Engineering and Computer Science, McCormick School of Engineering and Applied Science,Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118
Received January 21, 1994; revised manuscript received May 3, 1994
Recently we proposed using periodically spaced, phase-sensitive optical parametric amplifiers to balance linearloss in a nonlinear fiber-optic communication line [Opt. Lett. 18, 803 (1993)]. We present a detailed analysisof pulse propagation in such a fiber line. Our analysis and numerical simulations show that the length scaleover which the pulse evolution occurs is significantly increased beyond a soliton period. This is because of theattenuation of phase variations across the pulse's profile by the amplifiers. Analytical evidence is presentedthat indicates that stable pulse evolution occurs on length scales much longer than the soliton period. This isconfirmed through extensive numerical simulation, and the region of stable pulse propagation is found. Theaverage evolution of such pulses is governed by a fourth-order nonlinear diffusion equation, which describesthe exponential decay of arbitrary initial pulses into stable, steady-state, solitonlike pulses.
1. INTRODUCTION
The use of lumped erbium-doped fiber amplifiers has beendemonstrated to be an effective method for compensatingfor loss in long-distance optical communication sys-tems."q In addition, for soliton-based systems severalfiltering techniques4 have been developed that decreasethe Gordon-Haus jitter 5 -the random walk of solitonscaused by spontaneous-emission noise present in theerbium-doped amplifiers or by acoustic perturbations 6-thereby increasing the maximum allowable bit rate.
As a possible alternative to erbium-doped fiber ampli-fiers, the use of lumped phase-sensitive amplifiers (PSA's)has been proposed7 as a method for compensating for loss.Because PSA's are free of spontaneous-emission noise7
(they are ideal quantum-limited amplifiers with a 0-dBnoise figure), they add no Gordon-Haus jitter to thepropagating solitons and therefore lead to a significantincrease in the maximum bit rate.8 A PSA can also bethought of as a combination of an amplifier and a filterintegrated into one device. In this sense they are analo-gous to the erbium amplifiers and passive filters usedin the schemes mentioned above. For PSA's, however,the filtering is done in the signal's optical phase, ratherthan only in the frequency domain, since only one phasequadrature is amplified and the other quadrature is at-tenuated (or filtered out) by the PSA's.
Quantum-limited phase-sensitive parametric amplifi-cation in bulk x(2) materials was recently demonstrated,9
and waveguide parametric amplifiers based on the x(2)nonlinearity were built.10 For application in a commu-nication link, however, phase-sensitive parametric ampli-fiers exploiting the nonlinear refractive index n2 of thefiber potentially hold more promise.1 113
Recently we proposed using PSA's to compensate forloss in nonlinear optical communication systems.1 4 Inthe present paper we present a detailed theoretical analy-sis and extensive numerical simulations of pulse propa-gation in a nonlinear optical fiber line in which linearloss in the fiber is balanced by a chain of periodicallyspaced phase-sensitive parametric amplifiers as sketchedin Fig. 1. We consider the case in which the amplifierspacing is much smaller than the dispersion length; i.e.,the loss experienced by the pulse because of the fiber andthe gain associated with the PSA occur on a length scalethat is much shorter than that of the dispersion and non-linearity. In our approach we average over the rapidfluctuations that are due to the loss and the gain2 andanalyze the averaged equation governing the pulse evo-lution over distances much greater than the dispersionlength. The averaged envelope equation supports stablepulse propagation, and initial pulses are shown to decayexponentially to steady-state solutions. This is in con-trast to the case of a fiber line with erbium amplifiers,for which pulse stability is reached by the shedding ofdispersive radiation.2 3
Our analysis also provides a physical explanation forthe above results. On propagation through a segment ofthe fiber, the pulse is attenuated by the loss and developsa quadratic phase sweep across its profile, since group-velocity dispersion and self-phase modulation do not ex-actly balance each other as the pulse decays. The PSA's,however, work to produce an output pulse that is uniformin phase; the phase sweep induced in the pulse is there-fore attenuated by the PSA's, canceling the effects of thedispersion and self-phase modulation. Thus the PSA'sact as phase-sensitive filters (analogously to lock-in am-plifiers) that fight dispersion and other pulse-deforming
Fig. 1. Schematic of a nonlinear optical fiber transmission linein which loss is balanced by a chain of periodically spaced PSA's.
effects. The analysis also implies that this effect doesnot necessarily depend on self-phase modulation's beingpresent. As a result PSA's can be used to compensatefor dispersion in fiber-optic communication systems whenthe nonlinearity of the fiber plays no role.' 5"16
We begin our analysis in Section 2 by setting up thepertinent equations describing the nonlinear fiber andthe PSA's. Since the PSA's amplify one quadrature anddeamplify the other, we decompose the pulse envelope intothese in- and out-of-phase quadratures. In Section 3 weemploy a multiple-scale expansion method to determinethe pulse evolution, and in Section 4 we analyze the sta-bility of the resulting pulses. In Section 5 we comparethe results of our analysis with detailed numerical simu-lations of the nonlinear Schrodinger (NLS) equation, in-cluding linear loss and periodically spaced PSA's. Weconclude in Section 6 with a discussion of our results.
2. FORMULATION
The evolution of an optical soliton propagating in a fiberwith linear loss is governed by the NLS equation' 7 '9
___ i a2 q 1=q - d + iq q- yq.(1
Here T is the physical time divided by the pulse widthr (full width at half-maximum), q is the field envelopedivided by the peak field amplitude Eo, and Z is thephysical distance divided by the dispersion length Zo,where
tween the amplifiers.2 3 To take advantage of this math-ematically, we define a small parameter e,
el-= Z 1/Zo, (3)
where e << 1,1 is an 0(1) parameter, and Z is the spacingof the amplifiers (e.g., 36 km).4 The parameter I is some-what arbitrary and is included merely for convenience sothat the effect of varying the amplifier spacing can be eas-ily evaluated in what follows. (Because e is used as anexpansion parameter, it drops out of the ordered equa-tions that are obtained; including permits dependenceon the amplifier spacing to be retained. Note, however,that only the combination el has physical significance.)
Since e < 1, the loss and the gain experienced by apropagating pulse occur on a much shorter length scale,Z 5 e;, than the effects of dispersion and nonlinear self-phase modulation. This implies that y >> 1; i.e., theloss is large in comparison with the nonlinear self-phasemodulation and dispersion. We write y = F/e to showthis explicitly; between amplifiers, pulse propagation istherefore governed by Eq. (1) with y = F/e.
The PSA's restore the signal pulse after the linear losshas caused it to decay significantly, which occurs over adistance that is a small fraction of the dispersion length.While the internal details of specific PSA's may be differ-ent, the qualitative features of each are the same.16 Eachexploits a pump pulse to amplify the signal. In contrastwith phase-insensitive amplifiers such as erbium-dopedfiber amplifiers, PSA's amplify only the parts of the sig-nal that are in phase with the pump; the parts of thesignal pulse that are 900 out of phase with the pump areattenuated.
If the signal pulse just before an amplifier is decom-posed into quadratures (orthogonal complex phases),
q- = (A_ + iB_)exp(i4/2),
ao w (1.76 )2(k IEo = a (-k"),
Z= ( )
y = ZO.
(2a) where qS is a reference phase associated with the pumppulse driving the PSA, then after amplification by the PSAthe output can be written as
(2b)
(2c)
Here is the linear field-amplitude loss rate, " =d2k/dw2 I"o is the group-velocity-dispersion coefficient, n2
is the nonlinear coefficient of the fiber (in square metersper square volt), a is a geometric factor depending on theindex profile in the fiber, and wo and c are the optical car-rier's angular frequency and the free-space speed of light,respectively. For a typical dispersion-shifted fiber at awavelength of A = 1.55 ,um, for example, Zo = 500 km forTr 50 ps and k" = -1.6 ps2 /km, ao is roughly 1/2,19 and3 0.02763 km-' gives a power loss rate of 0.24 dB/km.Note that k" < 0, so the optical fiber is operated in theanomalous-dispersion regime as is necessary for solitonpropagation. Alternatively, Z = 500 km for T = 25 psand k" = -0.4 ps2/km, or Zo = 250 km for = 20 ps andk" = -0.5 ps2/km.
In typical long-distance communication situations, thedispersion length is much longer than the spacing be-
q+ = [A_ exp(a) + iB- exp(-a)]exp(i0/2), (5)
where a is the field gain of the PSA. This explicitlyshows the gain experienced by the in-phase quadrature(A) and the attentuation experienced by the orthogonalquadrature (B). Since in practice the length of each PSAis likely to be much shorter than either the linear decaylength or the dispersion length (e.g., 100 m; Ref. 13), theabove jump conditions can be used at fixed locations tomodel the action of the PSAs. In essence this assump-tion treats each amplifier as a delta function that givesrise to an 0(1) amplitude change over a length scale thatis negligible in comparison with the other length scalesin the problem.2
As a specific case, a degenerate optical parametricamplifier2 0 whose pump pulses are assumed to be un-depleted and much wider in duration than the signalpulses 2 ' is a PSAwith a = CJPIza. Here Za is the lengthof each parametric amplifier segment, PJ is the pump-pulse peak power, and C is a real constant that depends
(4)
Kutz et al.
2114 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
on the x(2) nonlinearity of the amplifying medium andthe frequency of the signal.
In the analysis that follows it will be convenient touse the quadrature variables A and B, where q = (A +iB)exp(i0/2). Since the reference phase 0 at each PSAwill change from one amplifier to the next, we let it varywith distance, 0 = 4 (Z). With this assumption Eq. (1)becomes
aA = A - 1 a2 B- (A + B2)B + 2 B. (6a)e 2aT 2 2
aB F l a2 A 2_ B + - + (A2 + B2)A - K A, (6b)
OZ e 2 aT 2 2
where K = do/dZ. In addition, from Eqs. (4) and (5), ateach amplifier we have the jump conditions
A+ = exp(a)A-, (7a)B = exp(-a)B. (7b)
3. AVERAGING
To understand the pulse dynamics determined by Eqs. (6)and (7), we perform a multiple-scale expansion 2 2 (i.e., av-eraging) in the short length scale ; = Zie, the dispersionlength scale Z, and the long length scale 6 = eZ by ex-panding
A= Ao(;, Z, eT) + eA(;, Z, 6,T) +.*-, (8a)
B = Bo(;, Z. , T) + eBi(;, Z. , T) + ... (8b)
This multiple-scale expansion method is similar in spiritto the Lie transform method used by Hasegawa andKodama2 for the guiding-center soliton. It is clear fromEqs. (6) and (7) that attenuation and amplification are thedominant effects on the short ; scale, whereas dispersionand nonlinear self-phase modulation become significantover the longer length scale Z. The multiple-scale expan-sion method permits the separation of the averaged effectsof attenuation and amplification that are important overthe longer length scales Z and e from the merely localfluctuations that are caused on the T scale.
Another way to look at this multiple-scale expansion isas follows: the differently sized deviations from an exactbalance between loss and amplification build up to pro-duce an 0(1) effect over different length scales; an 0(e)deviation in the balance will build up to produce an 0(1)effect over the ;/e = Z length scale, whereas an 0(e 2)deviation in the balance will produce an 0(1) effect af-ter distances that are of the order of {/e2 = Z/e = .These deviations are produced by dispersion and nonlin-ear self-phase modulation and by the interaction of theseeffects with the amplifiers. The multiple-scale expansionmerely separates these different deviations and examinesthem individually over their appropriate length scales.
To leading order, 0(1/e), and between the amplifiers,the multiple-scale expansion gives
aBo + rAo = 0.
d ;o + rBO = 0.
As the pulse travels between a pair of amplifiers it isattenuated by a factor exp(-Vl). Across the amplifiersthe pulse must satisfy the jump conditions given by theleading-order approximation of Eqs. (7). Therefore thesolution of Eqs. (9) that is commensurate with the jumpconditions is
AO = R(Z, , T)exp(na - FS),
Bo = Q(Z, , T)exp(-na - F;),
(lOa)
(lOb)
where n is the number of amplifiers that have been passedby the pulse, n = 1V/l (the greatest integer less thanor equal to ;/l). Note that at ; = nl+, just after thenth amplifier, Eqs. (10) become AO = R(Z, e, T)exp[n(a -
VI)] and Bo = Q(Z, 6, T)exp[-n(a + Vl)].Equation (lOb) implies that the quadrature associated
with B is attenuated by both the amplifiers and the linearloss in the fiber. Therefore after only a few amplifiersthis quadrature quickly decays to zero. We ignore thisinitial transient by taking Q = 0 and focus on the Aquadrature, which experiences both loss and gain. If weset the gain a equal to the linear loss over one amplifierspacing, i, then we obtain a periodic solution of Eqs. (9),where loss and gain are balanced in the A quadrature onthe ; length scale. Note that, because the B quadraturequickly decays to zero, the pulses lock onto and followthe phase of the amplifiers; i.e., just after an amplifierqO = (Ao + iBo)exp(i0/2) = R(Z, 6, T)exp[iO(Z)/2].
At the next order, 0(1), from Eqs. (6)-(8) we obtain
a~i + VAl = '
aB+ 1 a2A 0 3 -
g ~2 aT 2 2
(lla)
(llb)
with the jump conditions across the amplifiers being
Al+ = exp(Vl)Al-,
Bi+ = exp(-l)B,-.(12a)
(12b)
We solve Eqs. (11) so that loss and gain are balancedon the ; length scale on passage through the fiber-segment-amplifier combinations. The effects of any0(e) deviations in the balance between loss and amplifi-cation show up as part of the evolution on the Z lengthscale. This is a result of using the multiple-scale ex-pansion. We obtain the proper evolution by noting thatonly a specific choice for the Z derivative in Eq. (lla) willallow the solution for Al to be periodic on the ; lengthscale. This specific choice can be found either by solvingEq. (lla) and the jump condition (12a) with the Z deriva-tive arbitrary and then requiring the resulting solutionto be periodic or by applying a solvability condition2 3 toEq. (lla), which is necessary for periodic solutions in (see Appendix A for the details of this approach).
Whichever method is used, the result is
dZ ;
(9a)(13)
i.e., R and hence Ao are independent of Z. This im-(9b) plies that, unlike in the case of a fiber line with er-
bium amplifiers,2 no pulse evolution occurs on the length
Kutz et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2115
scale of the dispersion or nonlinear self-phase modulation.The evolution of R = R(e, T) occurs on the longer lengthscale (e.g., 6900 km for a 500-km dispersion length anda 36-km amplifier spacing, 2500 km for a 500-km disper-sion length and a 100-km amplifier spacing, or 1725 kmfor a 250-km dispersion length and a 36-km amplifierspacing). Physically this is because any phase variationsin the pulses that are caused by the dispersion and thenonlinearity in the fiber between the amplifiers are at-tenuated by the PSA's (this is further explained below).In addition, since R andAO are independent of Z, Eq. (la)for Al therefore becomes homogeneous, and we can takeA l 0.
Solving Eqs. (lib) and (12b) over one fiber-segment-amplifier combination and looking for the periodic solu-tion gives the next-order correction in the B quadrature,
B = exp(-rl) rl( 2R2r sinh r 2 \aT 2 KR) + sinh(2Fl)R3]
x exp(- ) + - exp(-Ff) a2 - KR)R 3
- exp(-317). (14)
This solution for B provides direct evidence for thesmoothing and attenuation of phase variations in a propa-gating pulse. In particular, just after an amplifier onefinds
B, = exp(-rl)4V sinh t
x{[1 - exp(-2V1)]R3 + rl(a2 KR)}- (15)
Since Al = 0, the above equation shows that pulse defor-mations that are due to nonlinear self-phase modulationand dispersion (the two terms in the braces) are in thequadrature that is attenuated [by a factor of exp(-rl)]by the PSA's. In addition, since the deformations causedby each effect are separately attenuated, we note thatthe presence of the self-phase modulation is not neces-sary. Thus the proposed amplifier system may also beused to compensate for the effects of dispersion in fiber-optic communication systems when the nonlinearity ofthe fiber plays no role. An analysis of this situation hasshown that PSA's can extend the effective propagationdistance of pulses to many times the dispersion length ofthe fiber. 15,16
At the next order, 0(e), from Eqs. (6)-(8) we find
aA2 + A2 = a 1 a -2B, K + 2+A 2= --- --- --A B, + 1a6 2 aT2 2
(16a)
(16b)aB2 + r2 = aB,+ VB2
with the jump conditions across the amplifiers being
A2+ = exp(rl)A2 + Ay exp(Vl)Ao-,
B2+ = exp(-rl)B 2- -
(17a)
(17b)
Here we have allowed for the possibility that the gain maynot exactly balance the loss, i.e., a = V + e2Ay. Note
that now the fluctuations in the attenuated quadrature,B1, feed back into the amplified quadrature at this order.This means that the effects of dispersion and nonlinearself-phase modulation should be seen at this order.
In what follows we assume that K = dq5/dZ is constant.In a physically realizable phase-sensitive amplificationscheme (such as the one sketched in Fig. 2) a portion ofthe input signal pulses will be tapped to track the signal'soptical phase. This is done to set the phase of the pumppulses properly, which one can accomplish, for example,by injection locking the pump laser to the tapped signal.For most phase fluctuations the phase-recovery electron-ics does not need to be fast, since it does not need torespond to the phase variations of individual pulses butrather only to the slow drifting of the phase from one pulseto the next caused by acoustic and other sources that pro-duce relatively slow phase changes. The phase-recoveryelectronics, therefore, will respond to the ensemble av-erage of a number of pulses that have already passedthrough the amplifiers. Residual fluctuations caused byrelatively fast processes such as guided acoustic-waveBrillouin scattering,2 4 which the phase-recovery elec-tronics may not be able to track, will eventually deter-mine the overall performance of fiber lines that employPSA's. Such issues will be the subject of our futureinvestigations.
Therefore, if a number of solitons with average phaserotation rate d/dZ = K have passed through thefiber-PSA line and we wish to study the stability andevolution of a signal pulse that follows them, then as faras the signal pulse is concerned K can be regarded asbeing constant. The phases of the amplifiers will havealready been set at the time that the signal pulse reachesthem, and this signal pulse cannot significantly alter theamplifier phases as it passes, because the feedback elec-tronics will not respond quickly enough. In addition,we note for future reference that, even if K were not as-sumed to be constant, no additional terms would appearin Eq. (16a). Such terms show up only in the equationfor the out-of-phase quadrature, Eq. (16b).
At the previous order, 0(1), we solved Eqs. (1a) and(12a) over one fiber-segment-amplifier combination andrequired the solution to be periodic on the ; length scale.Similarly, at 0(e) the effects of any 0(e 2 ) deviations in thebalance between loss and amplification show up as partof the evolution on the length scale. As before, only aspecific choice for the derivative in Eq. (16a) will permitthe solution for A2 to be periodic on the length scale.
The result for this evolution on the length scale is sub-stantially simplified if we rescale the envelope by using
= - exp(-21) 1R
Phaer - Pu r --- FieG--- a
PSASoliton Fiber Fiber Fiber Fiber Soliton
In Tap Coupler Coupler Tap Out
PumpDump
Fig. 2. Physically realizable PSA, employing a degenerate op-tical parametric amplifier for use in long-distance fiber-optictransmission.
Kutz et al.
2116 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
this is the same resealing that was used to analyze solitonpropagation in fiber lines with erbium amplifiers.2 Inaddition, we rescale the independent variable 6 by letting
= fI/2 tanh Vi.
With these substitutions we obtain
aU 1 2 U K a2U (K2
7+ -___ -T-+I AaUae 4 AT 2 +t4- cU
-KU' + U5 + lU a- 0 + 2U2 a2 = , (18)
where
,Aa = 2 tanh(rl)Ay/12
is a parameter that measures the amount of overamplifi-cation, ,1 = 6 - 3 tanh(Vl)/Fl, and /2 = 3 - tanh(rl)/rl.
Equation (18) is a fourth-order nonlinear diffusionequation describing the evolution of the amplified quadra-ture on the long length scale T. This equation describes,in an averaged sense, the interaction among the PSA's,the linear dispersion, and the nonlinear self-phase modu-lation in the fiber over long distances.' 4
Once the e derivative in Eq. (16a) is chosen properly, aperiodic solution for A2 can be found. Since it representsan 0(e 2) correction to the leading-order part of the ampli-fied quadrature, however, we omit this solution here. Inaddition, it is possible to solve for the deamplified quadra-ture B2 at this order. When dK/dZ = 0, the solutionB 2 0 is obtained.
4. STABILITY ANALYSIS
As a preliminary investigation of the stability of a propa-gating pulse, we examine Eq. (18) when the amplifiersare closely spaced, i.e., when Vi is small. Although thatis somewhat unrealistic physically, it is mathematicallyconvenient because a simple solution is easily found inclosed form in this limit. Since 8,1 and 2 in Eq. (18)are even functions of V7, we let a = (Vl)2 and expandU = U0 + aU, + 2U2 + ... and Aa = Bal + 82a2 + ---In addition, we let o- = 3T and make use of anothermultiple-scale expansion to capture any slow growth inU caused by the perturbation measured by 6.
At the leading order in 8, 0(1), Eq. (18) becomes
a U0 + 1 a2 + U02 _ K(1 a2U0 + U 3 _-K U)= O (19)
which has a hyperbolic-secant steady-state solutionU0 = -q sech y7 T, where 72 = K. Note that the shape ofthis pulse agrees with what is expected physically as 6 ap-proaches zero-a limit where the fluctuations caused bythe attenuation and parametric amplification are negli-gible-namely, the hyperbolic-secant shape associatedwith a soliton solution of the NLS equation. Thestructure of Eq. (19) is clearly inherited from the NLSequation.
At the next order, 0(8), we find
au, ( _,i oauo 2 2 0 2O io=0al aT ) 3 aT2
(20)
whereI a2
U2-KL --+ 3Uo 2 aT2 2
1 82 2 K
2 aT2 2
are the real and the imaginary parts of the linearizedoperators associated with the NLS equation. Many prop-erties of these operators have been determined2 5 ; in par-ticular, it can be shown that the spectrum of the operator-ULL+ is bounded to the left of the origin, with the ex-ception of two zero eigenvalues. Since the homogeneouspart of the linear equation for U1,
au,d~ + L-L+ U= ,
has the symbolic solution
U1(6) = exp(-eL-L+)U,(0),
it is necessary only to determine the effect of the perturba-tion on these two zero eigenvalues to ascertain the stabil-ity of Eq. (18). One of these two eigenvalues arises fromthe translation invariance of Eq. (19), and therefore thiseigenvalue remains zero under the perturbation, sinceEq. (18) is also translation invariant. The other eigen-value is affected by the perturbation, however, and thestability of a pulse is determined by this single eigenvalue.
Since Eq. (20) has homogeneous solutions, solvabilityconditions are necessary for a solution of the perturbedproblem to exist.23 A condition is associated with eachof the zero eigenvalues. The one corresponding to thetranslation invariance is automatically satisfied, but theother condition is satisfied only if al = 0. With al = 0,we obtain
U, = -(7 7/6)sech y7T
+ /3(c)(sech 77T - -qT tanh AqT sech -qT),
where ,8 is arbitrary at this order.At the second order, 0(82), the situation is similar in
that a forced (inhomogeneous) equation is obtained thatcan be solved only if a certain solvability condition issatisfied. At the second order, however, this conditionnow determines f3(o-) by leading to the equation
d/ =77 (22 - 2X22 + 7 35 77 ). (21)
It is straightforward to show that a steady-state solutionof Eq. (21) exists and is stable as long as a 2 > (4/405)74,or, equivalently, Aa > (4/405)K2 (Fl)4 Aa, (when Fl issmall). When this condition is satisfied, therefore, theabove analysis implies that a stable steady-state pulsesolution of Eq. (18) should exist.
Kutz et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2117
The critical value Aa determines the minimumamount of overamplification that is necessary for stablepulse solutions to occur. The need for a small amount ofoveramplification is consistent with the use of PSA's, be-cause there is a small additional amount.of decay that isdue to losses in the out-of-phase quadrature. For valuesof A a below A a, it is expected that a pulse will decay tozero. Of course, these results are valid only when V issmall, but they are nonetheless indicative of the resultsobtained by numerical simulation for 0(1) values of Vi.
Furthermore, if one linearizes Eq. (18) about the trivialsolution U = 0, one finds that it is unstable for Aa > K 2/4.This implies that if too much overamplification is used,then a steady pulse solution is no longer possible. Thepulse continually grows with distance as it propagates.One expects that the stability range should be substan-tially widened, however, if optical gain control on the am-plifiers is employed. Analysis of the system with suchgain control is in progress.
Note that within the stability regime an initial non-steady-state pulse decays exponentially to the stable so-lution in an entirely local manner because of the diffusivenature of the envelope equation (18), which is in contrastto the stability of erbium amplifier systems for which asteady-state soliton pulse is reached by the shedding ofdispersive radiation.2 3
5. NUMERICAL RESULTSWhen Vi is an 0(1) quantity, it is easiest to determinethe evolution and stability of a propagating pulse numeri-cally. Such values of Vi correspond to physically realiz-able values of the amplifier gain, fiber loss, dispersionlength, and amplifier spacing. In what follows we as-sume the fiber power loss rate to be 0.24 dB/km and thedispersion length to be 500 km. Stable pulse solutionsare shown to exist by numerical solution of Eq. (18).
The numerical procedure employed to solve Eq. (18)uses a fourth-order Runge-Kutta method in time and afiltered pseudospectral method in space.26 This proce-dure combines the advantages of split-step1 7 8 27 andexplicit Runge-Kutta 2 8 methods, giving a relativelysimple fourth-order scheme with improved numericalstability properties. In all the numerical runs the com-putational region was taken to be larger than the regionof interest, and an absorbing boundary layer was added toeliminate any reflections from the edges of the computa-tional region. We carefully tested the results by varyingthe number of Fourier modes and the time step as wellas the size of the computational region.
Figure 3 shows two representative numerical solutionsof Eq. (18). Figure 3(a) is for an initial pulse U(T, 0) =sech T, and Fig. 3(b) is for U(T, 0) = 1.8 sech T. Inboth cases the solution exponentially approaches a stablesteady state as it evolves. The parameters used in thiscomputation are Vi = 1 (which corresponds to an ampli-fier spacing of roughly 36 km), K = 1, and A a = 0.1. Asis expected from the stability analysis of Section 4, posi-tive values of Aa (i.e., overamplification) are necessaryfor the stable pulse solutions to be obtained. The pulsesin these simulations propagate 10 units in the long lengthscale e, which corresponds physically to a pulse travelingthrough 2,936 amplifiers for a total distance of roughly
212 dispersion lengths. Such a long distance was cho-sen to show the stability of the pulses explicitly.
The dimensionless parameters used in these runs (andin the ones to follow) can be more easily compared withthe physical parameters when we note that
ri = 3Z1 ,
where 8 is the linear field-amplitude loss rate (e.g., 8 =0.02763 km-' for a power loss rate of 0.24 dB/km) and Zis the amplifier spacing in kilometers. In addition,
1 ZI-Z2 tanh(8Z) ZO
gives T in terms of Z, and
tZ1 2 Aa= ZI + ZoJ 2 tanh(8Z)
gives the total amplifier gain.
(a)
1.6 ~~~~~~~~~~~~~(b)1.4e.e
.9:
Fig. 3. Evolution of initial pulses (a) U(T, 0) = sech T and (b)U(T, 0) = 1.8 sech T, showing exponential decay to the stablepulse solution. The parameters are rl = 1 (corresponding toan amplifier spacing of 36 km), K = 1, and Aa = 0.1. Thecomputations were run to T = 10 (i.e., a distance of 212 dispersionlengths) to show the stability of the pulses explicitly.
Kutz et al.
2118 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
1.8 demonstrating that the averaged equation is an accurateapproximation.
1.4 Most of the difference between the two solutions canbe attributed to second-order terms in the perturbation
12 expansion (i.e., A2), which have been ignored in this com-parison. In addition, a small amount of linear dispersive
a. 1.0
0.6 .6
0.6 ~~~~~~~~~~~~~~~~~~~~~1.400.6 .0
z 1.20 -= 0.050.4 -
UNSTABLE 1.000.2 LAz
D 0.80-
0.0 1.0 2.0 3.0 4.0 5.0 0.6
INITIAL WIDTH 0.40
Fig. 4. Initial pulse amplitudes A and widths To that give stablepulse solutions for Ft = 1, K = 1, and Aa = 0.1. The initial 0.20 /conditions U(T, 0) = A sech(T/To), with different values of Aand To, were used. For all initial conditions within the hatched 0.00-region the same final steady state was reached. -0.20- .....................
-12.0 -8.0 -4.0 0.0 4.0 8.0 12.0
Figure 4 shows that a wide range of initial pulse ampli- DIMENSIONLESS TIME
tudes and widths can be used to produce stable pulse solu-tions. We obtained these data simply by solving Eq. (18) 1.60
for many different initial pulses of the form U(T, 0) = 1.40
A sech(T/To) with different values ofA and To and record-ing the cases in which the stable steady-state pulse solu- 1.20 = 0.1tion was reached. Note that all initial pulses within the
1.00 hatched region are asymptotic to the same stable steady aUstate. The numerical simulations were performed with D0.80 lthe same parameter values as in Fig. 3, i.e., Vl = 1, K = 080
1, and Aa = 0.1. Similar numerical runs indicate thatstable pulse solutions are also obtained for a wide range 0.40
of Ul values, such as Vi = 2, which corresponds to an am-plifier spacing of 72 km. 0.20
Figure 5 shows the steady-state pulse shapes that areobtained by solution of Eq. (18) for different values of 0.00
overamplification, Aa. Note that for larger values of -0.20
overamplification small wings develop in the pulse's pro- 12.0 -8.0 -4.0 0.0 4.0 8.0 12.0
file. This is similar to what is observed when phase- DIMENSIONLESS TIME
sensitive amplifiers are used in linear systems (fiber-PSA 1.60
lines in which the nonlinearity plays no role).'5 6
A measure of the accuracy of the averaged envelope 1.40-
equation, Eq. (18), is obtained by comparison of its so- 120 LVa = 0.2lutions with numerical solutions of the full NLS equa- l
tion with loss and periodic phase-sensitive amplification, 1.00
Eqs. (6) and (7). Figure 6(a) shows such a comparison for C 0.8
a total propagation distance of 10,000 km for the sameinitial pulse and physical parameters as those used in X0.60
the simulation in Fig. 3(a). Note that in this figure only l
the in-phase quadrature of the full simulation of the NLS 0.40
equation, i.e., A in Eqs. (6) and (7), is plotted, and it has 0.20
for comparison with Eq. (18). Because the two solutions Fig. 5. Final steady-state pulse profiles for different values ofare indistinguishable when plotted together, the differ- the overamplification parameter, Aa = 0.05, 0.10, 0.20. Note
ence between the two pulses is shown in Fig. 6(b). Note the small wings in the pulse's profile that develop for the larger
that tho difforonco is quite small, of the order of 10', values.
Kutz et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2119
1.2
0
E
0.8
0.6
0.4
0.2
0
3
2
f-x..
2
10
-1
-2
-3
-10 -5 0 5 10Dimensionless time
-10 -5 0 5 10Dimensionless time
Fig. 6. Comparison of the solutions of the averaged envelopeequation and the full NLS equation with loss and periodicphase-sensitive amplification, showing (a) the in-phase quadra-ture and (b) the difference between the two solutions. Theparameters are rl = 1.0, corresponding to an amplifier spacingof 36 km, a dispersion length of 500 km, K = 1, and Aa = 0.1.The solutions are plotted after a total propagation distance of10,000 km, or 275 amplifiers.
radiation can be seen in Fig. 6(b), which is largest in thevicinity of the main pulse and decreases away from it.This linear dispersive radiation does not show up in themultiple-scale expansion because it is exponentially smallin the perturbation parameter2 9 ; such exponentially smallterms typically do not show up in perturbation expansionsthat use powers of the small parameter2 2 unless specialtechniques are employed.3 0
Not all frequencies are present in the linear dispersiveradiation. A detailed analysis of the linear response ofan optical fiber line employing PSA's16 shows that onlycertain frequencies are able to maintain phase matchingwith the amplifiers as they propagate, and thus only thesefrequencies experience an overall gain close to unity asthey pass through an optical fiber-PSA segment. Thesefrequencies are strongly dependent on the spacing be-tween the amplifiers. Therefore, in a more realistic situ-ation, when the amplifier spacing varies somewhat withdistance along the fiber line or the PSA bandwidth is lim-ited, this linear dispersive radiation is expected to be au-tomatically eliminated. For the calculations presentedhere the amplifier spacing was taken to be exactly pe-riodic and the PSA bandwidth was assumed to be infinite[cf. Eq. (5)], and thus the dispersive radiation is able tosurvive. The effect of variable amplifier spacing on the
solitons is expected to be minimal, however, because thesolitons and the PSA's will be phase locked.
Similar results are also found when one examines theout-of-phase quadrature B (which, of course, is muchsmaller than the in-phase quadrature). Figure 7 shows acomparison of the out-of-phase quadrature obtained fromthe averaged equation, a suitably rescaled Eq. (15), withthe result for the out-of-phase quadrature obtained fromthe numerical solution of the NLS equation with loss andPSA's, i.e., B in Eqs. (6) and (7). The two (indistinguish-able) curves are plotted just after an amplifier in Fig. 7(a),where the parameters and the total propagation distanceare the same as those for Fig. 6. Since the two curvesare indistinguishable when they are plotted together, thedifference between the two is plotted in Fig. 7(b). As forthe in-phase quadrature, here a small amount of lineardispersive radiation is also seen.
It is also illustrative to examine the stabilizing effect ofthe amplifiers directly by plotting the magnitude of theout-of-phase quadrature between the amplifiers. Thisplot is shown in Fig. 8, which provides clear evidence thatafter an amplifier the out-of-phase quadrature grows be-cause of forcing from the dispersion and the nonlinearself-phase modulation but that on reaching the next am-plifier the out-of-phase quadrature is sharply attenuated.(Note that in this figure the exponential decay that is dueto the loss between the amplifiers has been factored out.)
3
To 2
=o 10.E
0
0.2
0
-0.2
u;- -0.4
'x, -0.6- -0.8LUi
-1
-1.2
-1.4
-1.6
-10 -5 0 5Dimensionless time
10
-10 -5 0 5 10Dimensionless time
Fig. 7. Comparison of the solutions of the averaged envelopeequation and the full NLS equation with loss and periodicphase-sensitive amplification, showing (a) the out-of-phasequadrature and (b) the difference between the two solutions.The parameters are the same as in Fig. 6.
I i I
i' \
2 " i | |~~~~~~~~~b
Kutz et al.
2120 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
0.1
0.08
0.06
E 0.04
0.02
0 . . . . ....... .... ... ... . ...... .
0 0.1 0.2 0.3
Distance (Z)
Fig. 8. Midpoint value of the out-of-phase quadas a function of distance, showing the evolutionamplifiers (calculated from the full NLS equationperiodic phase-sensitive amplification). The expthat is due to loss between the amplifiers has beeThe magnitude grows after an amplifier, but on reEamplifier it is sharply attenuated. Here the amp]50 km, and the distance is in terms of dispersioncorresponds to 500 km).
1.3
a)'O 1.2
E
1.1
0 20 40 60 80 100 120 140 160 180 200
Distance (Z)
Fig. 9. Midpoint value of the in-phase quadrature just after anamplifier, plotted as a function of distance (in terms of dispersionlengths). Results from both the averaged equation (dashedcurve), Eq. (18), and the full NLS equation with loss and periodicphase-sensitive amplification (solid curve) are plotted; the twoare virtually indistinguishable. The parameters are rl = 1.0,corresponding to an amplifier spacing of 36 km, a dispersionlength of 500 km, K = 1, and Aa = 0.1. A total propagationdistance of 2750 amplifiers or 200 dispersion lengths is shown.Note that an approximate steady state is not reached untilafter the pulse has propagated roughly 100 dispersion lengths,showing the degree to which PSA's are able to suppress theeffects of dispersion and self-phase modulation.
In the numerical simulations shown in Figs. 6-8 thesolution is not close to a steady state. Because of the sup-pression of the dispersion and the self-phase modulationby the amplifiers much longer distances are necessaryfor a true steady state to be reached. As an example,in Fig. 9 the value at the center of the pulse just afteran amplifier is plotted as a function of distance (in dis-persion lengths). Results from both the averaged equa-tion, Eq. (18), and the full NLS simulations, Eqs. (6) and(7), are plotted. The curves are again almost indistin-guishable. Note that the solution is not even close tothe steady state until the pulse has propagated approxi-
mately 100 dispersion lengths, showing the degree towhich the phase-sensitive amplifiers are able to eliminatethe effects of dispersion and self-phase modulation.
We also investigated a 100-km amplifier spacing, andthe results for the in-phase and the out-of-phase quadra-tures after 10,000 km are shown in Figs. 10(a) and 10(b),respectively. For both figures l = 2.76, K = 1, andAa = 0.05. Here the dispersive radiation generated asa result of the periodic forcing by the loss and PSA's isrelatively more pronounced, although it is still limited toa narrow range of frequencies by the action of the PSA's.
..-. _As is mentioned above, this radiation is expected to belargely eliminated when the amplifier spacing is allowed
0.4 0.5 to vary along the length of the fiber and a finite PSAbandwidth is used.
rature plotted It is also of interest to compare these numerical resultsi between the with similar results obtained from the equations that arel with loss and used to describe a communication system employing soli-
nfactored out. tons and lumped erbium-doped fiber amplifiers.2 TheLching the next physical parameters in the numerical simulation of theifier spacing is erbium amplifier system include a dispersion distance oflengths (Z = 1 411 km and an amplifier spacing of 50 km. If the am-
plitudes of the initial pulses in both cases are taken to
1.2
1
0.8(D
:- 0.6
E< 0.4
0.2
0
-10 -5 0 5 10
Dimensionless time
3
2
x
O
E -1
-2
-3
I I I (b)
o 1
-10 -5 0 5 10
Dimensionless time
Fig. 10. Comparison of the solutions of the averaged envelopeequation (dashed curves) and the full NLS equation with loss andperiodic phase-sensitive amplification (solid curves) for both (a)the in-phase (b) the out-of-phase quadatues. The parametersare rl = 2.76, corresponding to an amplifier spacing of 100 km,a dispersion length of 500 km, K = 1, and Aa = 0.05. The solu-tions are plotted after a total propagation distance of 10,000 kmor 100 amplifiers.
Kutz et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2121
1.40
1.20
1.00
0.80
0.0
0-.0.40
0.20
0.00
-0.20
1.40
1.20
1.00I
UL0Z)-J
IL
0.80
0.60
0.40
0.20
0.00
-0.20 ;1-r-12.0
(a)
-8.0 -4.0 0.0 4.0 8.0
DIMENSIONLESS TIME
(b)
-8.0 -4.0 0.0 4.0 8.0
DIMENSIONLESS TIME
12.0
Fig. 11. Qualitative comparison of pulse solutions, showing theamount of dispersive radiation shed by the soliton-based com-munication systems employing (a) erbium amplifiers and (b)PSA's. In both cases the initial pulse amplitude was takento be 10% higher than the optimum (for a fixed width). Thesystem employing PSA's, (b), generates considerably less lineardispersive radiation with such an initial condition. For thesesimulations the dispersion length was taken to be 411 km, theamplifier spacing was 50 km, and the gain of the amplifiers wasset exactly to cancel the fiber loss between the amplifiers.2
be precisely those required for the corresponding steady-state solutions (a one-soliton solution in the erbium am-plifier case), then the amount of dispersive radiation gen-erated in each case is roughly of the same magnitude. Ifthe amplitudes of the initial pulses are taken to be 10%higher than those required for the corresponding steady-state solutions, however, then the system employing er-bium amplifiers generates more dispersive radiation, asis illustrated in Fig. 11. In the system employing PSA's,Fig. 11(b), the amplifiers attenuate most of the linear dis-persive radiation that is shed by the pulse as it adjustsits amplitude.
6. SUMMARY AND DISCUSSION
In conclusion, we have investigated the possibility ofimplementing phase-sensitive amplifiers (PSA's) in asoliton-based long-distance fiber-optic communicationsystem. We have shown that linear dispersion and non-linear self-phase modulation are largely suppressed be-cause of the nature of these amplifiers, i.e., amplificationin one quadrature and attenuation in the other. In par-ticular, when the amplifier spacing is much less than thedispersion length of the fiber, no pulse evolution occurs onthe length scale of the dispersion or nonlinear self-phasemodulation.
On the longer length scale we derived an averagednonlinear evolution equation, Eq. (18). This equationshows that an initial pulse smoothly and cleanly decaysto its stable state (i.e., transients decay exponentiallyon the t length scale), in contrast to the transient shed-ding of dispersive radiation that occurs for solutions of thenonlinear Schrodinger (NLS) equation. Comparisons be-tween this averaged evolution equation and simulations ofthe full NLS equation with loss and PSA's are also given,with excellent agreement obtained between the two.
Yuen7 has suggested that PSA's be used as a means forcompensating for loss in communication systems becausePSA's are ideal quantum-limited amplifiers with a 0-dBnoise figure. As a result they are free from spontaneous-emission noise and add no Gordon-Haus timing jitter tothe propagating solitons, leading to a potentially sig-nificant increase in the maximum bit rate. Here wehave investigated the classical behavior of a soliton fiberline employing PSA's. Detailed investigations into thequantum theory of such a line that goes beyond thesingle-mode analysis of Yuen7 and the potential bit-rate limitations imposed by other fluctuations (such asdissipation-induced amplitude and phase fluctuations)are under way.
A PSA can be thought of as a combination of an ampli-fier and a filter integrated into one device. In this sensethe use of PSA's is analogous to schemes employing er-bium amplifiers followed by passive optical filters. In thecase of PSA's, however, the filtering is done in the signal'soptical phase rather than only in the frequency domain.
Finally, the results provide a physical explanation forthe benefits of the PSA's. On propagation through asegment of the fiber, the pulse is attenuated by lossand develops a quadratic phase sweep across its profile,since group-velocity dispersion and self-phase modulationdo not exactly balance each other as the pulse decays.PSA's, however, operate to produce an output pulse thatis uniform in phase; the phase sweep induced in the pulseis therefore attenuated by the PSA's, canceling the effectsof dispersion and self-phase modulation. This argumentalso shows that the cancellation effect does not necessar-ily depend on self-phase modulation's being present. Asa result PSA's can also be used to compensate for disper-sion in optical-fiber systems when the nonlinearity of thefiber plays no role. 5
16
APPENDIX A: SOLVABILITY CONDITIONS
In this appendix we show that there is a solvability condi-tion for the equation describing the amplified quadrature,
Kutz et al.
2122 J. Opt. Soc. Am. B/Vol. 11, No. 10/October 1994
Eq. (11a) with jump condition (12a), but that the equa-tion describing the deamplified quadrature has no suchrequirement. In addition we explain how this solvabil-ity condition is used to simplify the derivation of the ap-propriate evolution equations on the longer length scales,Eqs. (13) and (18).
First we consider the amplified quadrature with a gen-eral periodic forcing,
aA (Al)
for 0 < < I, with the jump condition at the amplifiers
A = exp(Vi)A-. (A2)
Here f (4) is periodic in ; with period 1, as is the casefor the forcings for both the 0(1) and 0(e) problems,Eqs. (la) and (16a). Since we want the loss and the gainto balance each other, we require an i-periodic solution ofEq. (Al) with jump condition (A2). If the amplifiers arelocated at ; = 0, = 1, ; = 21, etc., then the periodicityrequirement means that A+ = A(O+) = A(i+), etc. Sim-ilarly, A_ = A(O-) = A(i-), etc. Thus jump condition(A2) can be replaced by the equivalent condition
A(O+) = exp(Vi)A(i-), (A3)
and we can restrict the entire analysis to the range 0 <- < 1.
The solution of Eq. (Al) is
A(Ol = C exp(-I`;) + exp(-Fr) f exp(rz)f (z)dz.
(A4)
Substituting this solution into jump condition (A3), weimmediately find that C is arbitrary and that f (f) mustsatisfy the condition
f 1 exp(Vz)f(z)dz = 0.
This is the solvability condition associated with the am-plified quadrature. We can also derive this condition bynoting that exp(V`) is the homogeneous solution of theadjoint equation associated with Eqs. (Al) and (A3).23
This solvability condition is a natural tool for use withEqs. (lla) and (16a). The integral averages (with the ap-propriate weighting) the I-dependent fluctuations arisingfrom the loss and gain and the interaction of the lossand gain with the dispersion and the nonlinear self-phasemodulation to give the appropriate evolution on the longerlength scales. In this sense the solvability conditionseparates the effects of these fluctuations into two parts,one that remains merely local on the 2 length scale andanother that can build up to produce a significant resultafter a long distance.
Repeating the above analysis for the out-of-phasequadrature, we have
aBaB + rB =Wg(;) (A6)
with the jump condition
B(O+) = exp(-Vi)B(I-). (A7)
Note that the exponent in the jump condition has theopposite sign because this quadrature is attenuated bythe PSA's.
As before, the solution of Eq. (A6) is
B(;) = D exp(- F2) + exp(-V) f exp(Vz)g(z)dz,
(A8)
but now when jump condition (A7) is applied one merelyobtains a result for the constant D, that is,
D = exp(-i) f1 exp(Vz)g(z)dz. (A9)
In this case no condition on the forcing function g(z) isnecessary for there to be a solution.
ACKNOWLEDGMENTSThis work was supported in part by the U.S. Air ForceOffice of Scientific Research, the Defense Sciences Officeof the Advanced Research Projects Agency, and the Na-tional Science Foundation.
*Permanent address, Department of Mathematics andCenter for Applied Mathematics and Statistics, New Jer-sey Institute of Technology, University Heights, Newark,New Jersey 07802-1982.
REFERENCES AND NOTES
1. N. S. Bergano, J. Aspell, C. R. Davidson, P. R. Trischitta,B. M. Nyman, and F. W. Kerfoot, "Bit error-rate mea-surements of 14,000 km 5 Gb/sec fiber-amplifier trans-mission system using circulating loop," Electron. Lett. 27,1889-1890 (1991).
2. A. Hasegawa and Y. Kodama, "Guiding-center soliton inoptical fibers," Opt. Lett. 15, 1443-1445 (1990); "Guiding-center soliton," Phys. Rev. Lett. 66, 161-164 (1991);Y. Kodama and A. Hasegawa, "Theoretical foundations ofoptical-soliton concept in fibers," in Progress in Optics XXX,E. Wolf, ed. (Elsevier, Amsterdam, 1992).
3. L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, "Long-distance soliton propagation using lumped amplifiers anddispersion shifted fiber," J. Lightwave Technol. 9, 194-197(1991).
4. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, "Soli-ton transmission control," Opt. Lett. 16, 1841-1843 (1991);Y. Kodama and A. Hasegawa, "Generation of asymptoticallystable optical solitons and suppression of the Gordon-Hauseffect," Opt. Lett. 17, 31-33 (1992); L. F. Mollenauer,J. P. Gordon, and S. G. Evangelides, "The sliding-frequencyguiding filter: an improved form of soliton jitter control,"Opt. Lett. 17, 1575-1577 (1992).
5. J. P. Gordon and H. A. Haus, "Random walk of coherentlyamplified solitons in optical fiber transmission," Opt. Lett.11,665-667 (1986).
6. E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A.N. Starodumov, "Electrostriction mechanism of soliton in-teraction in optical fibers," Opt. Lett. 15, 314-316 (1990);"Long-range interaction of solitons in ultra-long communica-tion systems," Sov. Lightwave Commun. 1,235-246 (1991).
7. H. Yuen, "Reduction of quantum fluctuation and suppres-sion of the Gordon-Haus effect with phase-sensitive linearamplifiers," Opt. Lett. 17, 73-75 (1992).
Kutz et al.
Vol. 11, No. 10/October 1994/J. Opt. Soc. Am. B 2123
8. S.-H. Lee, "Ultra-long distance optical communication usingphase-sensitive linear amplifier chains," Ph.D. dissertation(Northwestern University, Evanston, Ill., 1992).
9. 0. Aytir and P. Kumar, "Pulsed twin beams of light," Phys.Rev. Lett. 13, 1551-1554 (1990); C. Kim, R. D. Li, andP. Kumar, "Spatial effects in traveling-wave squeezed-stategeneration: detection using a self-generated matched localoscillator," in Quantum Electronics and Laser Science,Vol. 12 of 1993 OSA Technical Digest Series (Optical Societyof America, Washington, D.C., 1993), p. 214.
10. W. Sohler and H. Suche, "Optical parametric amplificationin Ti-diffused LiNbO3 waveguides," Appl. Phys. Lett. 37,255-257 (1980); S. Helmfrid, F. Laurell, and G. Arvidsson,"Optical parametric amplification of a 1.54 1am single-modeDFB laser in a Ti:LiNbO3 waveguide," J. Lightwave Technol.11,1459-1469 (1993).
11. M. Shirasaki and H. A. Haus, "Squeezing of pulses in a non-linear interferometer," J. Opt. Soc. Am. B 7, 30-34 (1990);M. Rosenbluh and R. M. Shelby, "Squeezed optical solitons,"Phys. Rev. Lett. 66, 153 (1991); K. Bergman and H. A. Haus,"Squeezing in fibers with optical pulses," Opt. Lett. 16, 663(1991).
12. M. E. Marhic and C.-H. Hsia, "Optical amplification in anonlinear interferometer," Elect. Lett. 27, 210 (1991).
13. G. Bartolini, R.-D. Li, P. Kumar, W. Riha, and K V. Reddy,"1.5 uctm phase-sensitive amplifier for ultra-high speed com-munications," in Optical Fiber Communication Conference,Vol. 4 of 1994 OSA Technical Digest Series (Optical Societyof America, Washington, D.C., 1994), pp. 202-203.
14. J. N. Kutz, W. L. Kath, R.-D. Li, and P. Kumar, "Long-distance pulse propagation in nonlinear optical fibers us-ing periodically-spaced parametric amplifiers," Opt. Lett. 18,802-804 (1993).
15. R.-D. Li, P. Kumar, W. L. Kath, J. N. Kutz, "Combating dis-persion with parametric amplifiers," IEEE Photon. Technol.Lett. 5, 669-672 (1993).
16. R.-D. Li, P. Kumar, and W. L. Kath, "Dispersion compensa-tion with phase-sensitive amplifiers," J. Lightwave Technol.12, 541-549 (1994).
17. A. Hasegawa and F. D. Tappert, "Transmission of station-ary nonlinear optical pulses in dispersive dielectric fibers.I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144(1973).
18. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York,1989).
19. A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).
20. R. W. Boyd, Nonlinear Optics (Academic, Orlando, Fla.,1992), Chap. 2.
21. If the carrier frequency of the evolving signal pulse is o,then the pump pulses are at 2wo in the case of parametricamplifiers. In fiber amplifiers,13 however, the pump pulsesare also at oo.
22. C. M. Bender and S. A. Orszag, Advanced MathematicalMethods for Scientists and Engineers (McGraw-Hill, NewYork, 1978), Chap. 11; J. Kevorkian and J. D. Cole, Pertur-bation Methods in Applied Mathematics (Springer-Verlag,Berlin, 1981), Chap. 3.
23. B. Friedman, Principles and Techniques of Applied Mathe-matics (Wiley, New York, 1956).
24. R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guidedacoustic-wave forward Brillouin scattering," Phys. Rev. B31, 5244-5252 (1985).
25. M. Weinstein, Modulational stability of ground states ofnonlinear Schrodinger equations," SIAM (Soc. Ind. Appl.Math.) J. Math. Anal. 16, 472-491 (1985).
26. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, "Removingthe stiffness from interfacial flows with surface tension,"submitted to J. Comput. Phys.
27. M. D. Feit and J. A. Fleck, "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978);P. E. Lagasse and R. Baets, "Application of propagatingbeam methods to electromagnetic and acoustic wave propa-gation problems: a review," Radio Sci. 22, 1225-1233(1987).
28. T. R. Taha and M. J. Ablowitz, "Analytical and numericalaspects of certain nonlinear evolution equations. II. Nu-merical, nonlinear Schr6dinger equation," J. Comput. Phys.55, 203-230 (1984).
29. J. P. Gordon, "Dispersive perturbations of solitons of the non-linear Schrodinger equation," J. Opt. Soc. Am. B 9, 91-97(1992).
30. H. Segur, S. Tanveer, and H. Levine, eds., Asymptotics Be-yond All Orders (Plenum, New York, 1992).
