Long-distance pulse propagation in nonlinear optical fibersby using periodically spaced parametric amplifiers
J. Nathan Kutz, William L. Kath, Ruo-Ding Li, and Prem Kumar
McCormick School of Engineering and Applied Science, Northwestern University,2145 Sheridan Road, Evanston, Illinois 60208-3125
Received December 8, 1992
We analyze pulse propagation in a nonlinear optical fiber in which linear loss in the fiber is balanced by achain of periodically spaced, phase-sensitive, degenerate parametric amplifiers. Our analysis shows that nopulse evolution occurs over a soliton period owing to attenuation in the quadrature orthogonal to the amplifiedquadrature. Evidence is presented that indicates that stable pulse solutions exist on length scales much longerthan the soliton period. These pulses are governed by a nonlinear fourth-order evolution equation, whichdescribes the exponential decay of arbitrary initial pulses (within the stability regime) onto stable, steady-state,solitonlike pulses.
The use of lumped erbium-doped fiber amplifiers hasbeen demonstrated as a method for canceling lossin long-distance optical communication systems. ' 2
More recently, various filtering schemes3 have beenproposed as a way of reducing the Gordon-Hausjitter 4 (i.e., the random walk of solitons that is causedeither by spontaneous emission noise of the erbiumamplifiers or by initial fluctuations in the solitonparameters) present in such systems, thereby in-creasing the maximum allowable bit rate.
As a possible alternative to erbium-doped ampli-fiers, the use of lumped parametric amplifiers hasbeen proposed.5 A chain of such amplifiers shouldlead to a higher-bit-rate limit because no sponta-neous emission noise is present.5 Here we presenta theoretical analysis of pulse evolution in a non-linear optical fiber in which loss is balanced by achain of periodically spaced, phase-matched, degener-ate parametric amplifiers (as shown in Fig. 1). Sucha system, including periodic parametric amplifica-tion, nonlinear propagation, and linear loss, can beshown to be governed by the equation
aq -i a2 q 12 1 2 ± +dq = d q+ ilql q + ht Z q + - f ( q*az 2 a2T qq i()q fj *
(1)
In the above equation, both the uniform loss inthe fiber and the optical phase-sensitive gain of thelumped parametric amplifiers are accounted for inthe expressions for h(g;) and f (i), where = ZIe,which are given by
Here F is the linear loss coefficient in the fiber,Za is the length of each parametric amplifier, I isthe spacing between the amplifiers in terms of thescaled variable ', exp(iq$) = iP/IP!, where P is theamplitude of the pulses that pump the amplifiers,6B is a real constant that depends on the X(2) nonlin-earity of the amplifying medium and the frequencyof the signal, and N is the number of amplifiers.Note that the amplifiers are modeled as periodicallyspaced delta functions in which the pump pulsesare assumed to be undepleted and much wider induration than the signal pulses. When r = IPI = o,then h = f = 0, and Eq. (1) reduces to the stan-dard nonlinear Schrodinger equation describing theevolution of short pulses q(Z, T) in a lossless non-linear [X,(3)] fiber.7 As in the case of erbium-dopedamplifiers,' the length Z in Eq. (1) has been scaledon a typical soliton period (e.g., 500 km for erbiumamplifiers2); the amplifier spacing Z1 = el is assumedto be much shorter than this length (e.g., 33 km).Mathematically, this assumption about the amplifierspacing is made by taking e << 1 and assuming I tobe an 0(1) quantity.
In order to understand the dynamics of the pulsesas dictated by Eq. (1), we perform a multiple-scaleexpansion8 in the short length scale ' = Z/e,the soliton period Z, and the long length scale6 = eZ, i.e.,
q = qo(;, Z, e XT) + eq, (S, Z, e sT) + ±... (3)
This multiple-scale expansion method is similar inspirit to the Lie transform method used by Kodamaand Hasegawa' for the guiding-center soliton. It isclear from Eq. (1) that attenuation and amplifica-tion are the dominant effects on the short ; scale,while dispersion and nonlinearity become significantover the longer length scale Z. The multiple-scaleexpansion method allows the separation of the av-eraged effects of attenuation and amplification thatare important over the longer length scale Z from thefluctuations that they cause, which are merely localon the ; scale.
Fig. 1. Schematic of a nonlinear optical fiber trans-mission line in which loss is balanced by a chain ofperiodically spaced, phase-matched, degenerate opticalparametric amplifiers (DOPA's).
To leading order 0(1/e), we get
aqo - h(;)qo + f (f)qo*. (4)
Solving the above equation with appropriate jumpconditions across the amplifiers leads to a recursionrelation for q0 between the successive amplifiers.We can show that fixing the value of the quantityBIP Iz, (parametric gain) in terms of F l (linearloss) allows for the existence of one periodicand one exponentially decaying solution of Eq. (4).Physically, one quadrature of the pulses is amplifiedand the other attenuated. Therefore, if the loss inthe fiber between successive amplifiers is balancedexactly with the gain experienced by the amplifiedquadrature, a periodic solution will result. Likewise,the deamplified quadrature decays exponentially byboth the loss in the fiber and attenuation in theamplifiers. Consequently, initial pulses lock ontoand follow the phase of the amplifiers, i.e., just afteran amplifier q0 = R(Z, , T)exp(i0(Z)/2). {Note:this means that just before the next amplifier,qo = R (Z, 6, T)exp[i b (Z)/2]exp(- Fl) because of thelinear loss.}
Solvability of Eqs. (1), (2a), and (2b) for the aver-aged effects at the next order 0(1) shows that
aR dR= 0 , (5)
i.e., R is independent of Z. Therefore, unlike witherbium-doped amplifiers,' in this case no pulse evo-lution occurs on the scale of the soliton period. Theevolution R(6, T) occurs on the longer length scale4 (e.g., 7500 km if the soliton period and amplifierspacing are as indicated above). Physically, this isbecause any phase variations in the pulses causedby dispersion and nonlinearity in the fiber betweenthe amplifiers is smoothed out by the fixed phaseassociated with the gain quadrature of the parametricamplifiers.
Further evidence for this smoothing and attenu-ation of variations in a propagating pulse is givenby solving for the fluctuations q, in the expansion,Eq. (3), relative to the main part of the pulse, q0.Just before an amplifier one finds
q= i exp[ir(Z)/2] 1 [1 - exp(-2F1)]R3
4F7 sinh FN
( IF2I R - _'R)} (6)+ a(T2 dZR) 6
Because each term is precisely 90° out of phase withthe main part of the pulse, this explicitly shows that
the fluctuations in the pulse caused by dispersion andnonlinearity are in the quadrature, which is atten-uated [by a factor of exp(- F 1)] by the parametricamplifiers. Since the nonlinearity is not essential forthis, we note here that the proposed amplifier systemmay also be used to overcome the effects of dispersionin fiber systems where the nonlinearity of the fiberplays no role. Our analysis therefore suggests thepossibility of utilizing linear optical fibers for undis-persed pulse propagation over distances many timeslonger than the dispersion length of the fiber.
Continuing the multiple-scale expansion of Eqs. (1),(2a), and (2b) to the next higher order 0(e) gives theaveraged evolution equation
2 aR 1 a4 R K a2R (K' 2-tanh(rF) - + - + -+ (4 2iR1 a 4 aT 4 2aT 2 4 /
- /31KR3 + /32R5 + 63 R( ) + 64R 2 =
(7)
where K = d4k/dZ is assumed constant, the ,Gi'sdepend on the parameters r and 1, and F2 is an 0(e2)deviation from the exact balance between amplifica-tion and decay.
As a preliminary investigation of the stability of thepropagating pulses, we examine the above equationfor small amplifier spacing 1. Although this may besomewhat unrealistic physically, it is mathematicallyconvenient since a simple solution of Eq. (7) is knownin closed form in this limit. We therefore expandr2 = K2[(rl) 2y2 + (rl)4y4 + ... ] and R = a[Ro +(r l)2R2 + . . .], where a = 1 + r 1/2 - (r 1)2/8 -5(r 1)3/48 + ... , collecting terms of equal powers in r Fto determine the stability of the pulses. At leadingorder 0(1), the steady-state solution of Eq. (7) isa hyperbolic secant of the form Ro = A sech AT,where A2
= K is determined by the phase rotation ofthe parametric amplifiers. Here the phase rotationof the parametric amplifiers acts as a surrogatefor the rotation that would normally be caused bythe nonlinear self-phase-modulation.7 The shape ofthe pulses agrees with what is expected physicallyas the amplifier spacing l approaches zero (a limitwhere the fluctuations caused by the attenuationand parametric amplification are negligible), namely,the hyperbolic-secant shape associated with a solitonsolution of the nonlinear Schrodinger equation. Atthe next order 0(r 2 12), solvability of the inhomogeoussteady-state equation requires that Y2 = 0, and weobtain R2 = aA(sech AT - AT tanh AT sech AT),where a is an arbitrary parameter. Continuing theexpansion to the next order 0(r414) then gives Y4 =
-(8/135) + (4/9)a - a2.We now present an analysis of the linearized sta-
bility of the steady-state propagating pulses foundabove. We set R = R8 + R, where R, = a[Ro +(r l)2R2] and R << R8, and obtain the equation
Fig. 2. Example of stable pulse propagation for Fl = 1(corresponding to a parametric gain of 2.7), 1 = 1.5, K = 1,F2 = -0.1, and an initial pulse R(T, 0) = 1.6 sech(T).
|Phase-ReceveryL - Pump Laser | - Feebac Gain IElecton -c (775 om) Con--trolba k --
> ~~DOPA _Soliton Fiber WD Fiber Soliton
In Tap Coupler coupler Tap outPumpDump
Fig. 3. Physically realizable degenerate optical paramet-ric amplification (DOPA) scheme for use in long-distancefiber-optic transmission.
where L is a linear operator that depends on thesteady-state solution R, through its coefficients. Set-ting R = exp(-Ae)u yields the eigenvalue problemLu = Au, where A = 2 tanh(F l)A/l. By expandingL = K2[Lo + (rl)2L2 + ...], u = UO + (rl) 2 U2 + ... , andA = K2[Ao + (r l)2A2 + .. .], we can evaluate the eigen-values and their correction terms. It can be shownthat the spectrum of eigenvalues for the leading-orderproblem, LouO = AOuO, is positive apart from two zeroeigenvalues.9 Therefore stability is guaranteed forall but two eigenmodes, and for these two modes itbecomes necessary to evaluate their correction termsA2. We find in this case that one eigenvalue re-mains zero as a result of the translation invari-ance of the evolution Eq. (7) and that the other hasthe correction A2 = 4(a - 2/9). Stable solutions existfor A2 positive and, as is evident from the above equa-tion and the result for Y4, there exists a critical valuefor the perturbation parameter 74 (yc = -4/405)below which a branch of the steady-state pulse so-lutions are stable. A negative value of 74 indicatesoveramplification, and we see that the critical valueyc determines the minimum amount of overamplifi-cation necessary for stable pulse solutions to occur.This critical value translates into the approximatestability condition F2 < -(4/405)K 2 (r l)4 (again, thisis valid for small r l); for values of F2 above this value,it is expected that all pulses decay to zero.
Numerical analysis is in progress to determine thefull stability range for the parameter F2 and the sta-bility regimes for the case r 1 - 0(l). For example,preliminary results indicate stable pulse solutions forr F = 1 (corresponding to a parametric gain of 2.7) andF2 = -0.1, as shown in Fig. 2. Note that within the
exponentially onto the stable solution in an entirelylocal manner owing to the diffusive nature of theenvelope Eq. (7); this is in contrast to the stability oferbium-doped amplifier systems where steady-statesoliton pulses are reached by means of the sheddingof dispersive radiation.
A physically realizable degenerate optical para-metric amplification scheme is shown in Fig. 3. Aportion of the input signal pulses is tapped off to trackthe optical phase in order to set properly the phase ofthe pump pulses. One way to accomplish this wouldbe to injection lock the pump laser to the secondharmonic of\ the tapped-off signal. A wavelength-dependent (WD) coupler is used to input the pumplight, at twice the frequency of the signal pulses,into the main fiber. An optical gain control on theparametric amplifier can be implemented by tappingoff some of the signal power at the output of theamplifier.
In conclusion, we have investigated the possibilityof implementing phase-sensitive parametric ampli-fiers in a long-distance fiber-optic communicationsystem. Owing to the nature of the amplifiers, i.e.,amplification in one quadrature and attenuation inthe other, no pulse evolution occurs on the lengthscale of the soliton period. On the longer lengthscale A, the diffusive nature of the derived non-linear evolution Eq. (7) shows that an initial pulsedecays exponentially onto its stable state (transientsdie off on the 6 scale), in contrast to the shed-ding of transients as dispersive radiation that occursfor solutions of the nonlinear Schrodinger equation.Also, as shown by Yuen,5 the use of parametric am-plifiers suppresses the Gordon-Haus timing jitter.Furthermore, the proposed method indicates the pos-sibility of overcoming the effects of dispersion inoptical-fiber systems where the nonlinearity of thefiber plays no role.
The authors acknowledge the help of Cheryl V. Hilein generating the data in Fig. 2. This study wassupported in part by the U.S. Air Force Office ofScientific Research, the Defense Sciences Office of theDefense Advanced Research Projects Agency, and theNational Science Foundation.
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