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December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1351

Phase noise in photonic communications systems using linearamplifiersJ. P. Gordon and L. F. Mollenauer

AT&T Bell Laboratories,Holmdel, New Jersey 07733

Received September 7, 1990; accepted September 28,1990Spontaneous emission noise limits the capacity and range of photonic communications systems that use linearoptical amplifiers. We consider here the question of phase detection in such systems. Amplitude-to-phase-noiseconversion occurs owing to the nonlinear Kerr effect in the transmission fiber, resulting in optimal phase noiseperformance when the nonlinear phase shift of the system is approximately 1 rad. Error-free state-of-the-artsystems that use phase detection at multigigabit rates are thereby limited to a range of a few thousand kilometers.

The range of photonic communications systems thatuse linear amplification is limited by the inevitablepresence of amplified spontaneous emission (ASE)noise. To examine the limits on such systems, it isessential to take into account the Kerr effect in thefiber, which produces such amplitude-dependent non-linear effects as self-phase modulation (SPM) andfour-wave mixing. While group-velocity dispersionalters the temporal shape of a pulse as it traverses thefiber, SPM alters its frequency spectrum. Taken to-gether, they can have complicated effects and affectdifferent modulation schemes in different ways.Phase- or frequency-modulation schemes have theapparent advantage that the signal amplitude may beinitially constant, thus delaying the major effects ofSPM for a while. In this Letter we examine the effectof ASE noise on the phase of the received signals.Since the fiber's refractive index is dependent on opti-cal power through the Kerr effect, fluctuations in opti-cal power, created by the ASE noise, giverise to phasefluctuations at the receiver. This phase noise, whichincreases with increasing signal power, adds to theusual phase noise that the system would have if it werelinear. For phase modulation, we show that thereresults a minimum error rate when the signal powerproduces a total nonlinear phase shift of approximate-ly 1 rad. For simplicity we presume here the use ofpolarization-preserving fiber, or the equivalent, sothat only one polarization state of the optical fieldneed be considered.The nonlinear phase shift is given by

ad01az (2irn2/XCAeff)P= k2P, (1)where P is the optical power' and k2 is the nonlinearcoefficient. Using typical values for the Kerr coeffi-cient (n2 = 3.2 X 10-16 cm2 /W), the free-space wave-length (X0 = 1.55 Am 2), and the effective fiber core areafor dispersion-shifted fibers (Aeff = 35 ,jm 2), we get thevalue 3.7 rad/W/km for k2. A 1-rad nonlinear phaseshift results, for example, from a path-averaged P = 27jiW and a total path of 10,000 km. Although disper-sion in the line is of obvious importance to the opera-tion of any such system, here we ignore its effects in

order to elucidate the basic limitations to phase-sensi-tive detection schemes imposed by these phase shifts.Consider then a transmission line using phase-shift-keying signaling, wherein an otherwise monochromat-ic optical field of constant power is modulated by im-posing on it a phase shift of either 0 or 7r rad duringeach bit period. In what follows, we refer to the fieldof each bit period as a pulse. Along the line, there arelinear optical amplifiers periodically spaced to com-pensate for the loss in the fiber. The receiver mea-sures the time-average phase of each signal pulse byusing a homodyne method.At the output of each amplifier, an ASEnoise field isadded to each pulse. Classical treatment of this noisefield as additive and Gaussian is appropriate, since theenergy per pulse greatly exceeds one photon. Twodegrees of freedom (DOF's) of the noise field are ofmost concern. They have the same form as the pulse.One is in phase with it, and the other in quadrature.The quadrature noise component produces an imme-diate phase shift, and the in-phase one alters the ener-gy of the pulse. Both these changes persist throughoutthe rest of the transmission; the energy change resultsin an additional nonlinear phase shift by the time thepulse reaches the receiver.By a DOF we mean one of a completeorthogonal setof functions needed to describe the field of a pulse.The sampling theorem tells us that a pulse of durationT and bandwidth B has 2BT DOF's. Given one suchset, others can be created by unitary transformations.If the noise is effectively white, as is true of the ASEnoise, then the mean noise energy per DOF is the totalnoise energy in bandwidth B and time Tndividedby thenumber of DOF's and thus is one half the spectralpower density (power per unit bandwidth) of thenoise, independent of the choice of the set. If wechoose the two functions mentioned above as DOF's,then the remaining noise is orthogonal to the signaland has less-significant effects if the optical band-width is not too large. In the further interests ofsimplicity we ignore these.We are thus lead to represent the field of a pulse in asimplified two-dimensional phasor diagram, as in Fig.

0146-9592/90/231351-03$2.00/0 1990 Optical Society of America

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1352 OPTICS LETTERS / Vol. 15, No. 23 / December 1, 1990y

110 t

noise +nonlinear\ phase shiftsignal "1" x

Fig. 1. Phasor diagram for pulse propagation. The axeshave been rotated to eliminate the nonlinear phase shift atthe signal energy.

1. The radius vector is the square root of the pulseenergy at an amplifier output, and the angle is thephase. At each amplifier a two-dimensional Gaussiannoise field is added, a vector of random direction onthe figure whose mean square length is the amplifier'soutput noise spectral density. As a pulse progresses,phase shifts accrue both by the simple linear additionof the noise and by the nonlinear effect, which is pro-portional to the pulse energy and thus to the square ofthe radius to the point representing the pulse.Even this simplified picture is analytically compli-cated; however, we can see qualitatively that the errorsin such a system will be significantly increased by thenonlinear phase shifts when the total nonlinear phaseshift at the original signal energy exceeds approxi-mately 1 rad. In the absence of nonlinearity, the noisecontinues to add linearly. The result at the receiver isa Gaussian distribution of points around the pointthat represents the original signal pulse. Errors arerepresented by the part of the distribution that ex-tends to the wrong side of the y axis, and this will occurmainly for points near the origin, representing pulsesof small energy. However, by symmetry there are asmany points on the other side of the distribution, withenergies four times the original signal energy. In thepresence of nonlinearity these will be suffering nonlin-ear phase shifts at a rate four times faster than a pulseat the signal energy. If a pulse at the signal energysuffers a phase shift of approximately 1 rad, then thesehigher-energy pulses will have a good chance of havingadvanced enough in total phase to cause an error.Note in this regard that while the nonlinear phaseshifts distort the angular distribution of the pulses,they do not affect the radial, or energy, distribution.While we cannot solve this problem analytically tofind precise error rates, we can solve for the variance ofthe phase distribution at the receiver. This gives ameasure of the relative importance of the nonlinearphase shifts, in agreement with the above qualitativediscussion. In doing this we assume a large signal-to-noise ratio, since this will be so whenthe error rates aresmall.Let L be the length of a fiber span between adjacentamplifiers. If the fiber's loss coefficient is oz,hen the

amplifier gain that offsets the loss in the fiber span isthen given by G = exp(cL). If Ltot is the overallsystem length, then the number of amplifiers Na isgiven by Na = Lt,0tL = aLtt/ln(G). Let EBa e thesignal energy at the output of an amplifier. The pathaverage of the signal energy over a fiber span, B, deter-mines the nonlinear phase shift. This is given byL= (EaIL) J dz exp(-az) = Ea(G - 1)I[G ln(G)].

(2)Now each amplifier produces an ASE noise outputwhosespectral density per polarization is given by ,3(G- 1)hv, where (G - 1)hv is the noise that would beproduced by an ideal amplifier of gain G and j3 s theexcess noise factor, 2 sometimes called n5p. Thus, atthe receiver input, the optical signal-to-noise ratio(signal energy divided by noise power per unit band-width per polarization state) is given by

Q Baa[Ni(G - 1)hv] = E/[#F(G)aLt0thvj, (3)where F(G) = [(G - 1)/ln G]21G. The second form ofthe signal-to-noise ratio is obtained by eliminating Eaand Na in favor of E and G. For a given path-averagedsignal energy, the quantity F(G) represents a noisepenalty paid simply for using amplifiers of high gain.Its value is unity for small gain but grows rather rapid-ly forgains above approximately 10 dB. The problemhere is that if the amplifiers have high gain, it followsthat their input power level is small relative to thepath-averaged power, and hence the signal-to-noiseratio suffers.

In the absence of nonlinear phase shifts, the simpleGaussian distribution would yield a theoretical errorrate for this system given by erfc(4Q)/2,where erfc isthe complementary error function. If the phase isapproximated by y/s in Fig. 1, then the variance of thephase fluctuations is given by(6,2) = 112Q, (4)

which would give an error rate of erfc(7rVQ/2) on thenaive assumption of a Gaussian distribution of phase,with errors when the phase exceeds +4r/2. Note thatuse of the variance does not give an accurate value of Qfor a prescribed error rate; hence it can only be used togive a rough estimate. For the purposes of this Letter,this may be good enough.The variance of the phase fluctuations at the receiv-er produced by the amplitude fluctuations along theline is easily found. Ifwe treat the energy fluctuationsas small, then each amplifier introduces energy fluctu-ations that can be described by

(5)and which then persist until the pulse reaches thereceiver. The resulting phase fluctuation at the re-ceiver produced by the nth amplifier, counting back-ward from the receiver, is given by

6qNL,amp = (6E/T)k 2 nL. (6)Squaring and averaging this gives the variance of thenonlinear phase shift from the nth amplifier. Since

= 21N,,Q

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December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1353variances of independent processes add, and here thesum of n2 is approximately Na3/3, the variance of thetotal nonlinear phase shift at the receiver is

(&ONL%)= (2/3)XNL2 /Q, (7)where we have used Eq. (5) and 'kNL is the total nonlin-ear phase shift at the signal energy, given by k2 (E/7)Ltot.Comparing Eqs. (4) and (7), we note that the linearand nonlinear phase variances are proportional to 1/Eand E, respectively. Thus the total variance is mini-mized when the two terms are equal, and this occurswhen the nonlinear phase shift 4NL is approximately 1rad. While this result does not tell us preciselyhowtominimize the error rate, as can be surmised from thediscussion of errors in the linear system, neverthelessit should be roughly correct.A recent experiments using frequency-shift-keyingmodulation with a modulation index of 0.7 (maximumphase excursion 0.7 rad) succeeded in signaling at a2.5-GHz bit rate over r2200 km with an error rate ofapproximately 10-12. For comparison, this scheme isroughly equivalent to phase-shift-keying modulationwith phases of 0.35rad. The given parameters im-ply a nonlinear phase shift ofapproximately 1.5rad, in

reasonable agreement with the optimum predictedhere. Increasing the system length would require alowering of the signalpower to maintain this optimum,which would result in a rapidly increasing error rate.With careful attention to minimizing the ASE noise,however, somewhat increased distances could be ob-tained.In summary, we have shown that for phase-shiftkeying or small-index frequency-shift-keying modula-tion in a photonic transmission system using linearamplifiers, errors are minimized when the total non-linear phase shift is in the neighborhood of 1 rad. TheASE noise from the amplifiers limits the range of suchsystems, since this noise increases with length whilethe optimum path-averaged signal power decreases.References1. L F. Mollenauer, J. P. Gordon, and M. N. Islam, IEEE J.Quantum Electron. QE-22, 157 (1986).2. E. Desurvire, Appl. Opt. 29, 3118 (1990).3. S. Saito, T. Imai, T. Sugie, N. Ohakawa,Y. Ichihashi, andT. Ito, in Optical Fiber Communication, Vol. 1 of OSATechnical Digest Series (Optical Society of America,Washington, D.C., 1990), paper PD2.