Compensation of the soliton self-frequencyshift with phase-sensitive amplifiers
Christopher G. Goedde and William L. Kath
Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineeringand Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3125
Prem Kumar
Department of Electrical Engineering and Computer Science, McCormick School of Engineeringand Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3118
Received July 7, 1994
We analyze the effect of Raman scattering and higher-order dispersion on the propagation of short (i-ps) pulses ina nonlinear optical fiber in which the loss is balanced by a periodic chain of phase-sensitive, degenerate parametricamplifiers. The analysis shows that the Raman scattering does not induce a continuous frequency downshifton the pulse but only a small, finite frequency shift.evolution equation that admits stable solutions thatgroup-velocity rest frame.
When short soliton pulses propagate in an opticalfiber, physical effects lead to higher-order pertur-bations that modulate the pulses' shape, velocity,and frequency. These effects include the Ramanself-frequency shift, third-order dispersion, and non-linear dispersion. For typical communication andnetwork applications, the most important of these isthe Raman self-frequency shift, because it leads to acontinuous downshift of the soliton frequency and acontinuous deceleration of the pulse. 1 2 The effects ofthe Raman downshift accumulate as the pulse prop-agates down the fiber, so this effect becomes progres-sively more important as the propagation distanceincreases. On the other hand, it is well known3 thathigher-order dispersion simply leads to a constantvelocity change of the pulse.
Recently, lumped phase-sensitive amplifiers(PSA's) have been proposed as an alternative toerbium-doped fiber amplifiers in optical transmis-sion networks.4 5 Such amplifiers have been shownto suppress dispersive soliton radiation5 as wellas reduce Gordon-Haus jitter. 6 In this Letter weshow that these PSA's also compensate the Ramanself-frequency shift.
The basic equation for the pulse amplitude in theslowly varying envelope approximation is the non-linear Schrbdinger equation. Including the higher-order physical effects and periodic phase-sensitiveamplification leads to the perturbed equation,
=_ 2 ± 2 + idq q + h(e)q+ ± f(e)q*OZ 2 aT2
E if(A q*
-aTrQa_ - aT | (1)
Here T is the physical time divided by the pulsewidth r (FWIHM), q is the field envelope divided bythe peak field amplitude E0, and Z is the physical
Pulse propagation is governed by a nonlinear fourth-orderpropagate with a small, constant, inverse velocity in the
distance divided by the dispersion length Z0. Thehigher-order coefficients are given by7
1.76T, 1.76k"' 2 x 1.76E /33 -61k"Ir Efl war
(2)where Tr is the Raman time constant, 1" and k"' arethe second- and third-order dispersion coefficients,respectively, and ws0 is the carrier frequency. Weconsider short pulses (r = 1 ps) at a wavelength of1.55 ,um.
In the case when the amplifier spacing Z, is shortcompared with the dispersion length (Zz/Za e <<1), there is a natural short length scale 4 = Z/e overwhich the loss and amplification act. In Eq. (1) boththe loss and the gain are accounted for in the expres-sions for h(4) and f (4), and both uniform line loss andlumped loss at each amplifier node are considered.For very low-dispersion fiber (k" = -0.1 ps2/km) andshort pulses, the dispersion length is only 3.228 km.The amplifier spacing is small, of the order of 1 km,since it must be shorter than the dispersion length;such closely spaced nodes would be suitable for lo-cal, high-speed optical networks.8 For typical fiber,for which the uniform line loss is of the order of0.25 dB/km, the line is essentially lossless over suchshort distances, and we neglect the line loss betweenamplifiers and include only the lumped loss at eachnode. With these assumptions, h and f become
N
h(4)q = (cosh y - 1) E 8(4 - n)Fq,n=1
Nf (4)q* = exp(i4,)sinh 'y E (4 - n)Fq*,
n=1
(3a)
(3b)
where F is the fraction of the signal amplitude passedthrough at each node [so that (1 - F2)"2 is the frac-tion tapped off], y is the amplifier gain, and 0 is theamplifier phase.
Following Kutz et al.,5 we perform a multiple-scaleexpansion of Eq. (1) in the short length scale A, thesoliton length scale Z, and a long length scale f eZ:
q = qo(S, Z. a, T) + eq, (S, Z. a, T) + .... (4)
This expansion effectively separates the effects thatare important on short length scales (the loss and theamplification) from those that act on longer lengthscales (the Raman downshift and the higher-orderdispersion). This allows us to average out the pe-riodic loss and gain and examine the effects of thehigher-order terms over long length scales. Becauseit is the dominant physical effect, we allow the Ra-man term to be 0(l) and take the dispersive pertur-bations to be O(E).
Since the amplifier gain is phase sensitive, thepulses will quickly become phase locked to the am-plifier phase qS.5 This allows us to decompose theamplitude q into quadratures, q = (A + iB)exp(i0/2)and to write jump condition for the pulse amplitudeat each amplifier as
Here q± (A, + iB,) represents the pulse ampli-tude immediately after/before an amplifier. Thus,one quadrature is periodically amplified by a factorof eY and the other is periodically attenuated by afactor of eVr. We pick the gain to nearly balance theloss, y = - In F + e2Aa, which results in a periodicsolution for the amplified quadrature and an ex-ponentially decaying solution for the attenuatedquadrature. (A small amount of overamplification,represented by Aa, is necessary for stability of theperiodic solution.) By averaging this periodic solu-tion, we derive a fourth-order evolution equation onthe long length scale 6:
aA C A ]2d+ - [L_(A) - 2orA A
Le aTj
+ 83 A3 + 3,AnlA2aA _ AaA = 0, (6)
where
L_(A) = ( 2 _ - 2 + A 2 (7)
C = (1 + F2 )/(1 - F2 ), and K = dq6/dZ is assumed tobe constant. The pulse evolution occurs only on thislong length scale; there is no pulse evolution on theintermediate soliton length scale, i.e., , aA/aZ = 0.
In the absence of the higher-order physical effects,Eq. (6) has an exact steady-state solution that we useas the basis for a perturbation expansion. Define
A(6, T) = Ao(6 T) + yA1(6, T) + ... ,
which is satisfied by the steady-state solutionAO(6,T) = - sech?7T if 772 = K + (8Aa/C)112.
The higher-order physical effects act at the next or-der of the expansion and induce a constant velocityshift on the pulse. In anticipation of this, we trans-form to a frame that is slowly moving with inversevelocity AA: t = T - AAt. Finding a steady-statesolution in this frame corresponds to finding a pulsethat is moving slowly with constant inverse velocity,A relative to the group-velocity rest frame. To firstorder in ,u, we obtain
aA1 ± {C L_(Ao)[L_(Ao) + 2A2 - A
2aAo a3A0CrL_(Ao)Ao 2 at /33 at 3
2aAo aA*- 3JnA t+ A at(10)
This is a linear, nonhomogeneous equation for Al,and the driving terms must satisfy a solvabilityconditions in order for a solution to exist. This con-dition determines the inverse velocity of the pulse tofirst order in ,u and must be evaluated numericallybecause of the inclusion of the overamplification, Aca,in the homogeneous part of Eq. (10). We obtain
A = KG3/33 + 3KGnlfnl + K2 CGrcr, (11)
where G3, Gn1, and Gr are numerically computedfunctions of Aa. The stability of this pulse can inprinciple be determined at next order; here we sim-ply note that our numerical results indicate thatthe pulse is stable for a range of small overamplifi-cation values.
Amplification with PSA's produces a stable, steady-state pulse that moves with a small constant inversevelocity in the group-velocity rest frame. This in-verse velocity depends linearly on the amplitude ofthe perturbations and is thus inversely proportionalto the pulse width. We remark that this is the ex-pected result if only the effects of the third-order andnonlinear dispersion, are included3 ; thus the PSA'sare effectively transparent to these terms. On theother hand, the Raman downshift has been neutral-ized. Instead of a continuous frequency downshift,and a corresponding deceleration of the pulse, thedownshift stabilizes and the pulse propagates witha constant velocity. This is because the PSA's act as
(8)
where ,A is an expansionorder of the perturbation.be O(/tt); the lowest-order
parameter that tracks theWe takeP3, Pn, and o-, to
equation becomes
aA + [ 2 L-(Ao)2 - Aa]Ao = 0,Fig. 1. Propagation of a 1-ps pulse, including the effects
(9) of higher-order dispersion, in a line with periodic (a)erbium-doped amplifiers and (b) PSA's.
Fig. 2. Effect of the Raman self-frequency shift on a 1-pspulse in a line with periodic (a) erbium-doped amplifiersand (b) PSA's.
200
_ 150N
2 100
so
0.5
,0.375N
2 0.25
0.125
0 v .0 30 60 90 120
zfZO
n0 30 60 90 120
Fig. 3. Frequency shift induced on a i-ps pulse withperiodically spaced (a) erbium-doped amplifiers and(b) PSA's.
a restoring force in frequency, constraining the pulseto remain near its initial carrier frequency.
We compare these results for a PSA line with thosefor a line amplified with erbium-doped amplifiers.We choose a i-ps pulse traveling down a -0.1-ps 2/kmfiber with an amplifier spacing of Z1 = 0.3Z0 . Wetake k"' = 0.01 ps'/km and Tr = 5 fs. We also tap offhalf of the pulse amplitude at each amplifier node, sothat F = 0.5 (25% loss). For the PSA's we choose anoveramplification of Aca = 0.1, and for the erbium-doped line we choose the gain to exactly balancethe loss.
We first examine the effects of the higher-orderdispersive terms; typical pulse propagation is shownin Fig. 1. For these parameters, Eq. (11) predictsan inverse velocity of 7.50 fs/km, which agrees wellwith the result of a numerical simulation of Eq. (1)of 7.10 fs/km. By comparison, the erbium-doped lineshows an inverse velocity of 5.74 fs/km.
We next examine the effect of the Raman down-shift, shown in Fig. 2. The constant inverse ve-locity for the PSA line is clearly evident in thisfigure, and the numerical value of 0.176 fs/km agrees
well with the inverse velocity of 0.174 fs/km pre-dicted by Eq. (11). This leads to a time shift of lessthan 70 fs over 120 soliton periods, as opposed to atime shift of 19 ps over the same distance in a linewith erbium amplifiers.
The frequency downshift can also be calculated,and it is shown in Fig. 3. After a short transient,the frequency downshift in the PSA line stabilizes at8v = 0.350 GHz. By comparison, a i-ps pulse trav-
eling down the line with erbium amplifiers under-goes a downshift of 0.407 GHz/km, for a total shiftof 157 GHz after 120 dispersion lengths. Thus thetotal frequency downshift with PSA's is less than thedownshift per kilometer in the erbium line.
We expect this suppression of the Raman down-shift to carry over to noise-induced frequency shifts.PSA's have been shown to compensate for Gor-don-Haus jitter.' 0 The same mechanism shouldalso compensate for the Raman-induced coupling ofamplitude fluctuations to timing jitter; research onthis is currently in progress.
This research was supported in part by the U.S. AirForce Office of Scientific Research, the National Sci-ence Foundation, and Rome Laboratory, Air ForceMateriel Command, USAF, under grant F30602-94-1-0003. The U.S. Government is authorized toreproduce and distribute reprints for governmen-tal purposes notwithstanding any copyright notationthereon.
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