1. INTRODUCTIONIn a long-haul optical fiber transmission system, the am-plified spontaneous emission (ASE) noise introduced bythe in-line optical amplifiers is the dominating noisesource. Its characteristics must be known in order tocompute the bit error rate (BER) of the system. In mostof the works on BER computation, the ASE noise is as-sumed to propagate through the fiber without the effect offiber-nonlinearity-induced interaction with the signal.In this case, the noise after transmission is still whiteGaussian noise, which simplifies the BER analysis signifi-cantly (Ref. 1 and references therein). However, the non-linear interaction between the signal and the noise mayin fact significantly change the noise process. In this pa-per, we present a method based on a perturbative ap-proach to analyze the nonlinear interaction between thenoise and the signal when they co-propagate through thefiber.
The nonlinear amplification of noise by the signal, orparametric gain due to the fiber nonlinearity, to put it an-other way, has been found to change the noise signifi-cantly. First, the noise is no longer stationary after itsinteraction with the time-varying signal; second, thenoise obtains a frequency-dependent power; third, thereis an energy transfer between the real and imaginarycomponents of the noise and they are no longer indepen-dent. The nonlinear amplification of noise has been stud-ied with the assumption of a cw pump signal2,3; the noiseprocess remains stationary at the output under this as-sumption. The result in Ref. 3 is further used to derivethe BER performance of optical fiber communication sys-tems where the transmitted signal pulse is long enough tobe approximated by a cw signal.4 However, the cw-pump-signal approximation limits the previous analysisto systems of low bit rate. On the other hand, as statedin Ref. 4, the extension of cw approximation to arbitrarilymodulated signals is a difficult problem because the noiseis no longer stationary; thus, the noise power spectrum
cannot be defined. We present in this paper the first re-sult of the analysis of nonlinear amplification and color-ing of noise by an arbitrarily modulated signal based onperturbation theory.
In this study the signal is modeled as a pseudorandombit sequence with arbitrary modulation, and the nonsta-tionary output noise process is characterized by the use ofcorrelation functions among its frequency components.We first find the nonlinear terms from the nonlinear am-plification of noise by using a perturbative approach inthe frequency domain. We then compute the noise corre-lation functions, which are kept up to the second order infiber nonlinearity for computational simplicity. We checkour results against an estimated noise correlation func-tion by using the split-step Fourier (SSF) method5 withmultiple realization of the amplifier noise. The validrange of our approximation is also discussed in this paper.Our results constitute a new analytical method of com-puting system performance with a more accurate noisemodel for ASE-dominated optical fiber communicationsystems.
The paper is organized as follows. Section 2 is a briefdiscussion of the system model we study in this paper.We then introduce the first- and second-order nonlineareffects in Section 3. Section 4 is dedicated to the study ofthe noise correlation functions in the frequency domain,and is the main result of this paper. Section 5 extendsour result to multispan systems, and also includes a dis-cussion of the computational complexity of our results.We validate our results by numerical simulation andstudy the validation range of our method in Section 6.We give a simple application of our results to find theprobability distribution function of detector statistics inSection 7. Section 8 concludes the paper.
2. SYSTEM MODELFiber nonlinearity originates from the power-dependenceof the refractive index of glass fiber and is a fundamental
2004 Optical Society of America
500 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
phenomenon in optical fibers. It imposes significant limi-tations on optical fiber communication systems by chang-ing the properties of the optical signal and noise. Thepropagation of the nonlinear optical signal through the fi-ber can be described by the nonlinear Schrodinger (NLS)equation. The NLS equation written in a referenceframe moving at the group velocity of the pulse is givenby5
]A
]z5 2
a
2A 1
j
2b2
a2A
at2 2 jguAu2A, (1)
where A 5 A(t, z) is the slowly varying complex enve-lope of the optical field at time t and position z along thefiber. The first term on the right-hand side of Eq. (1) de-scribes the transmission loss of the signal power, where ais the fiber attenuation factor. The second term comesfrom the dispersion of the optical pulses, where b2 is thesecond-order dispersion coefficient of the fiber. The com-bination of the first and second terms is called the linearterm of the NLS equation. The last term on the right-hand side of Eq. (1) is the dominating nonlinear term, andg is the nonlinear coefficient of the fiber. Its dependenceon the signal power is clearly shown in the equation.Even though the NLS equation can be solved numericallyby the SSF method, this method is time-consuming andnot well suited for studying the system performance ana-lytically. Instead, we use a perturbative approach tosolve the NLS equation with the assumption that the non-linear term in Eq. (1) can be treated as a small, perturba-tive term.
The system that we study in this paper is shown in Fig.1 with a single ASE noise source. In this section, we fo-cus on the nonlinear amplification of noise in a single fi-ber span, and we extend the results into multispan sys-tems with and without multiple ASE noise sources later,in Section 5. Within each span, bi-end dispersion com-pensation (DC)6 with both pre- and post-DC is used, andwe model the pre- and post-dispersion compensators withtransfer functions exp( jDprev
2/2) and exp( jDpostv2/2), re-
spectively, where v is the baseband angular frequency.Dpre and Dpost are the amount of DC from the pre- andpost-dispersion compensators, respectively. Meanwhile,the fiber dispersion effect can be characterized by a trans-fer function of exp(2jb2v
2L/2), where b2 is the second-order dispersion of the fiber and L is the span length. Inthis study, we assume that the fiber dispersion is fullycompensated for by the pre- and post-DC within eachspan, that is, Dpre 1 Dpost 5 b2L. A lumped optical am-plifier provides gain G 5 exp(aL) to compensate for thefiber transmission loss. We have neglected the possiblenonlinear effects in the DC components in this simplifiedsystem model.
Fig. 1. Simplified system model for multispan system with asingle ASE noise source. DC is the dispersion compensator.
The input to the single-span system is the sum of amodulated signal and the ASE noise. The ASE noise ismodeled as complex white Gaussian noise with powerspectral density N0 5 (G 2 1)(NF)(hn) where h isPlanck’s constant, n is the optical carrier frequency, andNF is the optical amplifier noise figure.7 We characterizethe noise in the frequency domain in this paper since theperturbative approach solves the NLS equation in the fre-quency domain. For this purpose, we write the observednoise process in the frequency domain as N(v) 5 U(v)1 jV(v), i.e., U(v) and V(v) are the real and imaginaryparts of the noise process at frequency v. For a systemdesigned with reasonable performance, the noise can beassumed to be much smaller than the signal. The outputnoise process then remains Gaussian and it can thus befully characterized by its correlation functions.
In previous studies on the nonlinear interaction be-tween noise and signal, the signal was assumed to becw.2,3 With a cw signal, the noise is still stationary, ex-periencing only a change in the power spectrum and a cor-relation between the in-phase and out-of-phase compo-nents. However, with a modulated signal, the noise is nolonger stationary since the noise is signal dependent afterits interaction with the signal. In this case, the powerspectrum can no longer be defined.
In this paper, we use an arbitrarily modulated signalwith a pseudorandom input-bit sequence as used in anySSF simulation. To fully characterize the output noiseprocess, we use the correlation functions among the fre-quency components of the noise at the output, and we de-fine the correlation functions of the noise frequency com-ponents as
SXY~v1 , v2! 5 E$X~v1!Y* ~v2!%, (2)
where * denotes complex conjugation. X(v) and Y(v)could each be N(v), U(v), or V(v). E$ • % is the expec-tation operation over the random noise. When X 5 Y,we call the correlation function an autocorrelation func-tion; otherwise, we call it a cross-correlation function. Inthe above definition, we have used the fact that the noiseprocess has zero mean with the perturbation assumption.Since the perturbative approach gives a natural descrip-tion of the output signal in the frequency domain, we fo-cus here on the noise correlation function in the frequencydomain instead of in the time domain. Fourier transfor-mation can be used if the noise correlation function in thetime domain is needed.
In the above definition of correlation functions for noisefrequency components, real and imaginary parts of thenoise process are used instead of its in-phase and out-of-phase components as in Ref. 3 because of the presence ofa time-varying phase on the signal from nonlinearity-induced self-phase modulation (SPM). At the output, thesignal can be written as AP(t) exp@ ju(t)#, where P(t) is itstime-varying power and u(t) is its time-varying phase.Then the in-phase and out-of-phase noise components areR$N(t)exp@2ju(t)#% and I$N(t)exp@2ju(t)#%, respectively. R
stands for the real part and I stands for the imaginarypart. Unfortunately, u(t) cannot be written out in closedform because of the complicated nature of the SPM pro-cess; therefore, to have closed-form expressions, the cor-
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 501
relation functions based on the real and imaginary partscannot be replaced by the in-phase and out-of-phase com-ponents of the noise.
3. NONLINEAR EFFECTSTo solve the NLS equation following the perturbation ap-proach, the fiber is divided into many infinitesimal seg-ments of length Dz. The small contribution of nonlineareffect from such a segment at location z is found to be
DANL~t, z ! 5 2jguA~t, z !u2A~t, z !Dz. (3)
This small contribution then passes though the rest of thefiber from z to L without nonlinearity, through the post-DC, and through the optical amplifier, as shown in Fig. 1.The total nonlinear term at the output is the sum of thesesmall contributions.
To find DANL(t, z), we approximate A(t, z) 5 As(t, z)1 N(t, z) with
As~t, z ! 5 As,0~t, z ! 1 As,1~t, z ! 1 As,2~t, z ! 1 ... ,(4)
N~t, z ! 5 N0~t, z ! 1 N1~t, z ! 1 N2~t, z ! 1 ... .(5)
As,0(t, z) is simply the linear transmission part of the sig-nal up to distance z; that is, the signal that passesthrough the fiber up to distance z without consideration ofnonlinearity. It is easy to show that
As,0~v, z ! 5 exp~2az/2!exp~2jb2v2z/2!
3 exp~ jDprev2/2!As~v, 0!. (6)
N0(t, z) is similarly defined. As,1(t, z) and N1(t, z) arethe first-order nonlinear terms in g for the signal andnoise respectively. As,2(t, z) and N2(t, z) are similarlydefined as the second-order nonlinear terms in g.
A. First-Order Noise Nonlinear EffectsBy substituting As(t, z) and N(t, z) into Eq. (3), we getthe first-order nonlinear term in g as
DAs,1~t, z ! 1 DN1~t, z ! 5 2jDz@ uAs,0~t, z !u2As,0~t, z !
1 2uAs,0~t, z !u2N0~t, z !
1 As,02 ~t, z !N0* ~t, z !
1 H.O.T.#. (7)
H.O.T. stands for higher-order terms of N0(t, z), and theyare neglected in the following on the assumption that thenoise is much smaller than the signal. The first term onthe right-hand side of Eq. (7) is DAs,1(t, z), the SPM ef-fect on the signal. The second and third terms on theright-hand side of Eq. (7) are the first-order nonlinearterms for noise from the signal due to cross-phase modu-lation (XPM) and four-wave mixing (FWM) effects, re-spectively, between the signal and noise. Accordingly, wedefine DNX(t, z) 5 2jg2uAs,0(t, z)u2N0(t, z)Dz andDNF(t, z) 5 2jgAs,0
2(t, z)N0* (t, z)Dz; thus DN1(t, z)5 DNX(t, z) 1 DNF(t, z). At the output of the systemwe have in the frequency domain
DN1~v, z ! 5 expS a
2L D expS j
2Dpostv
2D expF2a
2~L 2 z !G
3 expF2j
2b2v2~L 2 z !GDN1~v, z !. (8a)
Note that a notation without tilde is for nonlinear contri-bution generated at distance z, while a notation with tildeis for the corresponding nonlinear contribution at the out-put, including the linear part of the rest of the fiber, post-DC, and optical amplifier. By defining LD as Dpre5 b2LD and Dpost 5 b2(L 2 LD), we can simplify Eq. (8)to
DN1~v, z ! 5 exp~az/2!exp~2jb2v2~LD 2 z !/2!
3 DN1~v, z ! (8b)
by using our assumption that Dpre 1 Dpost 5 b2L. Then,we obtain the first-order term for nonlinear amplificationof noise at the output as N1(v) 5 *DN1(v, z), where theintegral is taken effectively over Dz from 0 to L.
Note that N1(v) consists of two terms. One of theterms comes from the XPM effect and is found to be
NX~v! 5 E0
L
expS a
2z D expF2
j
2b2v2~LD 2 z !G
3 ~2jg!2F$As,0~t, z !As,0* ~t, z !N0~t, z !%dz (9)
5 E0
L
expS a
2z D expF2
j
2b2v2~LD 2 z !G
3 S 2jg
4p2D 2EE As,0~v, z !As,0* ~v, z !
3 N0~v 2 v1 1 v2 , z !dv1dv2dz. (10)
F $•% stands for the Fourier transform. The other termcomes from the FWM effect and is found to be
NF~v! 5 E0
L
expS a
2z D expF2
j
2b2v2~LD 2 z !G
3 ~2jg!F$As,0~t, z !N0* ~t, z !As,0~t, z !%dz. (11)
Note that in Eq. (10), if we do the integration over zfirst, we will get exactly the same expression as those ob-tained from the Volterra series transfer function (VSTF)proposed in Ref. 8.
B. Second-Order Noise Nonlinear EffectsSimilar to the first-order case, we get the second-orderterm in g for noise nonlinear amplification as
DN2~t, z ! 5 2jgDz@As,02 ~t, z !N1* ~t, z !
1 2As,0~t, z !As,1* ~t, z !N0~t, z !
1 2As,0~t, z !N0* ~t, z !As,1~t, z !
1 2As,0~t, z !As,0* ~t, z !N1~t, z !
1 2As,1~t, z !As,0* ~t, z !N0~t, z ! 1 H.O.T.#.
(12)
502 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
Note that As,1(t, z) is the SPM term of the signal gener-ated in the fiber from 0 to z as defined above. Similarly,N1(t, z) is the first-order nonlinear term for the noisegenerated in the fiber from 0 to z. By substitutingN1(t, z) 5 NX(t, z) 1 NF(t, z) into Eq. (12), we haveseven terms in total and we define them as
DNH~t, z ! 5 ~2jg!As,0~t, z !NX* ~t, z !As,0~t, z !Dz, (13)
DNI~t, z ! 5 ~2jg!As,0~t, z !NF* ~t, z !As,0~t, z !Dz,(14)
DNJ~t, z ! 5 ~2jg!2As,0~t, z !As,0* ~t, z !NX~t, z !Dz,(15)
DNK~t, z ! 5 ~2jg!2As,0~t, z !As,0* ~t, z !NF~t, z !Dz,(16)
DNL~t, z ! 5 ~2jg!2N0~t, z !As,0* ~t, z !As,1~t, z !Dz,(17)
DNM~t, z ! 5 ~2jg!2As,0~t, z !N0* ~t, z !As,1~t, z !Dz,(18)
DNN~t, z ! 5 ~2jg!2As,0~t, z !As,1* ~t, z !N0~t, z !Dz.(19)
The corresponding terms at the output of the fiber,DNH(v, z) to DNN(v, z), can be defined similar to Eq.(8).
Finally, we obtain the second-order term for noise non-linear amplification at the output through integrationover Dz as
N2~v! 5 NH~v! 1 NI~v! 1 NJ~v! 1 NK~v! 1 NL~v!
1 NM~v! 1 NN~v! (20)
N~v! ' N0~v! 1 N1~v! 1 N2~v!, (21)
with N0(v) 5 N(v, 0) as a result of the 100% DC used.Note that every term in relation (21) is composed of
both real and imaginary parts U(v) and V(v). In Sec-tion 4, we focus on the computation of the autocorrelationfunctions SUU(v1 , v2) and SVV(v1 , v2), and the cross-correlation function SUV(v1 , v2).
4. NOISE FREQUENCY CORRELATIONFUNCTIONSTo compute the noise frequency correlation functions, wesubstitute U(v) ' U0(v) 1 U1(v) 1 U2(v) and V(v)' V0(v) 1 V1(v) 1 V2(v) into Eq. (2) and we have, forexample,
E$U~v1!U~v2!%
' E$@U0~v1! 1 U1~v1! 1 U2~v1!#
3 @U0~v2! 1 U1~v2! 1 U1~v2!#%
5 E$U0~v1!U0~v2!% (22)
1 E$U0~v1!U1~v2!% 1 E$U1~v1!U0~v2!%
(23)1 E$U0~v1!U2~v2!% 1 E$U1~v1!U1~v2!%
1 E$U2~v1!U0~v2!%
1 H.O.T. (24)
Relation (22) is the zeroth-order nonlinear term in g inthe noise frequency correlation functions, and we call itSUU,0(v1 , v2). Similarly, we define the first- andsecond-order nonlinear terms in the noise frequency cor-relation functions SUU,1(v1 , v2) and SUU,2(v1 , v2) in ex-pressions (23) and (24), respectively.
With our assumptions that the ASE noise is Gaussianwhite noise with independent real and imaginary parts atthe input, we have
where sv2 5 2pN0 because frequency components of the
noise are independent, and
SUV,0~v1 , v2! 5 E$U0~v1!V0~v2!% 5 0 (26)
because the real and imaginary parts are independent.In the remainder of this section, we concentrate on
studying the first- and second-order nonlinear terms inthe noise frequency correlation functions.
A. First-Order Nonlinear Term in the Noise FrequencyCorrelation FunctionsTo compute the first-order nonlinear term in the noise fre-quency correlation functions, we have, for example
After substituting Ui(v) 5 @Ni(v) 1 Ni* (v)#/2 andVi(v) 5 @Ni(v) 2 Ni* (v)#/2j for i 5 0, 1, we find thatSUU,1(v1 , v2), SVV,1(v1 , v2), and SUV,1(v1 , v2) are allcomposed of the four terms E$N0(v1)N1(v2)%,E$N0(v1)N1* (v2)%, E$N0* (v1)N1(v2)%, E$N0* (v1)3 N1* (v2)%. For example,
with E$N0(v1)NX(v2)% 5 0 because E$N0(v1)N0(v2)%5 0. After substituting in the expression for NF(v) andsome mathematical manipulation, we find that
E$N0~v1!N1~v2!% 5 E0
L
expF2j
2b2v2
2~LD 2 z !G3 S 2
jg
4p2Dsv2E As,0~v, z !
3 As,0~v1 1 v2 2 v, z !dvdz,
, T1~v1 , v2!. (29)
Similarly, we obtain that
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 503
E$N0~v1!N1* ~v2!% 5 E0
L
expF2j
2b2~v1
2 2 v22!~LD
2 z !G S jg
4p2Dsv2E 2As,0~v,z !
3 As,0* ~v 1 v1 2 v2 , z !dvdz,
, T2~v1 , v2!, (30)
E$N0* ~v1!N1~v2!% 5 T2* ~v1 , v2!, (31)
E$N0* ~v1!N1* ~v2!% 5 T1* ~v1 , v2!. (32)
With these expressions, we are now ready to obtain thefirst-order nonlinear term in the noise frequency correla-tion function. The first-order noise frequency autocorre-lation function for the real part of the noiseE$U(v1)U(v2)% is
One interesting observation about the first-order non-linear effect in the noise frequency correlation is that itscomplex noise variance is
SNN,1~v, v! 5 SUU,1~v, v! 1 SVV,1~v, v! 5 0. (38)
That is, the first-order nonlinear effect has no amplifica-tion over the complex noise, but it introduces an energytransfer between the real and imaginary parts plus corre-lations between the real and imaginary parts.
Another interesting observation is that the XPM effectdoes not contribute any effect to the noise frequency cor-relation functions up to the first-order since the termT2(v1 , v2) that comes from the XPM effect cancels out.
B. Second-Order Nonlinear Terms in the NoiseFrequency Correlation FunctionsFrom expression (24), we have the second-order nonlinearterm
and similar expressions for SVV,2(v1 , v2) andSUV,2(v1 , v2). After substituting Ui(v) 5 @Ni(v)1 Ni* (v)#/2 and Vi(v) 5 @Ni(v) 2 Ni* (v)#/2j for i5 0,1,2 into these equations, we find that SUU,2(v1 , v2),SVV,2(v1 , v2), and SUV,2(v1 , v2) are all composed of twogroups of terms of form E$U0(v1)U2(v2)% or of formE$U1(v1)U1(v2)%. We first study these two groups ofterms separately in Subsections 4.B.1 and 4.B.2, and wecombine them in Subsection 4.B.3 to obtain the second-order terms in the correlation functions.
1. Terms of the Form E$U0(v1)U2(v2)%From expression (24), the first term of SUU,2(v1 , v2) is
E$U0~v1!U2~v2!% 5 ~1/4!E$@N0~v1! 1 N0* ~v1!#
3 @N2~v2! 1 N2* ~v2!#%. (39b)
Note that N2(v) is composed of seven terms from NH(v)to NN(v).
First, we derive E$N0(v1)NH(v2)%; the result is
E$N0~v1!NH~v2!%
5 2g2sv2 E
0
LEy
L
expF2j
2b2v1
2~LD 2 y !G3 expF2
j
2b2v2
2~LD 2 z !G3 H 1
4p2 E C~v2 1 v, z !B~v1 2 v, y !
3 expF j
2b2v2~z 2 y !GdvJ dzdy,
, TH~v1 , v2!, (39c)
where we have used the definition of C(v, z) from Eq.(35) and a new definition of B(v, z) 5 1/2p3 *As,0(v, z)As,0* (v 2 v, z)dv or B(t, z) 5 uAs,0(t, z)u2.
Equation (39c) has a clear physical meaning. Wewould expect a factor of C(t, z) from the FWM effect inEq. (13). We also expect a factor of B(t, y) from the XPMeffect for NX(t, z). C(t, z) and B(t, y) are the nonlinearterms generated at positions z and y, respectively, and thefactor exp@ jb2v
2(z 2 y)/2# is the dispersion-induced phaseinteraction between them. The other two dispersion fac-tors are simply the linear propagation after the nonlinearterms are generated.
The other term for NH(v) is E$N0(v1)NH* (v2)% 5 0since E$N0(v1)N0(v2)% 5 0.
504 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
Similarly, we obtain other nonzero terms of formE$U0(v1)U2(v2)%, and we list them in Table 1. A newexpression D(t, z) 5 As,0(t, z)As,0* (t, z)As,0(t, z) is used;it originates from the SPM of the signal. It is easy toprove that TN* (v1 , v2) 5 2TL(v1 , v2) simply by substi-tution.
2. Terms of the Form E$U1(v1)U1(v2)%The second term of SUU,2(v1 , v2) is of the formE$U1(v1)U1(v2)% or 1/4E$@N1(v1) 1 N1* (v1)#@N1(v2)1 N1* (v2)#%. Note that N1(v) is composed of two termsNX(v) and NF(v). We list the nonzero terms in Table 2.
Table 1. Second-Order Nonlinear Terms of the Noise Frequency Correlation Functionsfrom the Second-Order Noise Nonlinear Terms NH(v) to NN(v)
Second-Order NoiseNonlinear Term
Second-Order Nonlinear Term of Noise FrequencyCorrelation Function
E$N0(v1)NH(v2)%, TH(v1 , v2) 2g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 y!GexpF2 j
2b2v2
2~LD 2 z!G3 H 1
4p2 EC~v2 1 v, z!B~v1 2 v, y!expF j
2b2v
2~z 2 y!GdvJ dzdy
E$N0(v1)NI* (v2)%, TI(v1 , v2) g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 y!GexpF j
2b2v2
2~LD 2 z!G3 H 1
4p2 EC~v 1 v1 , y!C* ~v 1 v2 , z!expF2 j
2b2v
2~z 2 y!GdvJ dzdy
E$N0(v1)NJ* (v2)%, TJ(v1 , v2) 2 4g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 y!GexpF j
2b2v2
2~LD 2 z!G3 H 1
4p2 EB~v1 2 v, y!B~v 2 v2 , z!expF j
2b2v
2~z 2 y!GdvJ dzdy
E$N0(v1)NK(v2)%, TK(v1 , v2) 2 2g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 y!GexpF2 j
2b2v2
2~LD 2 z!G3 H 1
4p2 EC~v1 1 v, y!B~v2 2 v, z!expF2 j
2b2v
2~z 2 y!GdvJ dzdy
E$N0(v1)NL* (v2)%, TL(v1 , v2) 2 2g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 z!GexpF j
2b2v2
2~LD 2 z!G3 H 1
4p2 ED* ~v, y!As,0~v1 2 v2 1 v, z !expF j
2b2v
2~z 2 y!GdvJ3 expF2a
2~z 2 y!Gdzdy
E$N0(v1)NM(v2)%, TM(v1 , v2) 2 2g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 z!GexpF2 j
2b2v2
2~LD 2 z!G3 H 1
4p2 ED~v, y!As,0~v1 1 v2 2 v, z !expF2 j
2b2v
2~z 2 y!GdvJ3 expF2a
2~z 2 y!Gdzdy
E$N0(v1)NN* (v2)%, TN(v1 , v2) 2g2sv
2E0
LEy
L
expF2 j
2b2v1
2~LD 2 z!GexpF j
2b2v2
2~LD 2 z!G3 H 1
4p2 ED~v, y!As,0* ~v2 2 v1 1 v, z !expF2 j
2b2v
2~z 2 y!GdvJ3 expF2a
2~z 2 y!Gdzdy
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 505
Note that in Table 2, the integration for z and y areboth from 0 to L. Now, we try to use TH(v1 , v2) toTN(v1 , v2) to express these three terms by dividing *0
L*0L
into two parts *0L*y
L and *0L*0
y . After careful mathemati-cal manipulation, we obtain
3. Combined Second-Order Nonlinear Terms in theCorrelationsNow we combine the results in Subsections 4.B.1 and4.B.2 to obtain the final results on the second-order non-linear terms of the noise frequency correlation functions:
SUU,2~v1 , v2! 5 R$TI~v1 , v2! 1 TI~v2 , v1!%
1 R$TK~v1 , v2! 1 TK~v2 , v1!%
1 ~1/2!R$TM~v1 , v2! 1 TM~v2 , v1!%,
(43)
SVV,2~v1 , v2! 5 R$TI~v1 , v2! 1 TI~v2 , v1!%
2 R$TK~v1 , v2! 1 TK~v2 , v1!%
2 ~1/2!R$TM~v1 , v2! 1 TM~v2 , v1!%,
(44)
SUV,2~v1 , v2! 5 2I$TI~v1 , v2! 2 TI~v2 , v1!%
1 I$TK~v1 , v2! 1 TK~v2 , v1!%
1 ~1/2!I$TM~v1 , v2! 1 TM~v2 , v1!%.
(45)
By comparing with the first-order nonlinear effects, wefind that the second-order effect for the complex noise fre-quency domain variance is
That is, unlike the first-order effect, the second-order non-linear effect has nonzero amplification over the complexnoise. Moreover, it introduces energy transfer and corre-lation between the real and imaginary parts.
5. EXTENSION TO MULTIPLE SPANSIn this section, we extend our results on noise nonlinearamplification to multispan systems. We also discuss thecomputational complexity associated with our methodand compare it with the SSF method.
A. Noise Correlations for Multiple SpansFor usual multispan systems, we have both multiple fiberspans and multiple optical amplifiers. Each optical am-plifier introduces ASE noise independent of the others,and the ASE noise from each optical amplifier experiences
nonlinear amplification by the signal during the trans-mission through the fiber spans after its generation.Since the ASE noise from each optical amplifier is inde-pendent of that of the others, the output noise correlationfunctions are the sums of the correlation functions foreach input ASE noise. As a first step to study the non-linear amplification of noise in multispan systems, westudy the noise frequency correlation functions in a sys-tem with a single ASE noise source with multiple spansas in Fig. 1.
Since 100% DC and perfect loss compensation are usedwithin each span, we have the following relationships:
As,0~t, z 1 mL ! 5 As,0~t, z !, (47)
N0~t, z 1 mL ! 5 N0~t, z !, (48)
for different span number m 5 1,2,... for both the modu-lated signal and the noise. It is then easy to show that
NX,m~v! 5 mNX~v!, NF,m~v! 5 mNF~v!, (49)
where NX,m(v) and NF,m(v) are the first-order terms fornoise nonlinear amplification from the XPM and FWM ef-fects, respectively, for m spans, and NX(v) and NF(v) arethe corresponding terms from the single-span case. As aresult, we have the first-order nonlinear terms for thenoise frequency correlation functions as
The first-order noise nonlinear terms NX(v) and NF(v)contribute not only to the first-order nonlinear terms inthe noise frequency correlation functions, i.e., terms of theform E$N0(v1)N1(v2)%, but also to the second-order non-linear terms for the noise frequency correlation functionsthrough terms of the form E$N1(v1)N1(v2)%. It is easyto see that their contributions to the second-order nonlin-ear terms for the noise frequency correlation functionsare the same as those from the single span except for afactor of m2 from relations (49).
Now we study the second-order nonlinear terms in thenoise frequency correlation functions [that is, terms of theform E$N0(v1)N2(v2)%] from the second-order noise non-linear terms NH,m(v) to NN,m(v). For multispan sys-tems, we have NH,m(v) as an integration over (0, mL),that is,
506 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
Table 2. Second-Order Nonlinear Terms of the Noise Frequency Correlation Functions from the Beatingof the First-Order Noise Nonlinear Terms NX(v) and NF(v)
First-Order NoiseNonlinear Term
Second-Order Nonlinear Term of Beat NoiseFrequency Correlation Function
E$NX(v1)NF(v2)%22g2sv
2E0
LE0
L
expF2 j
2b2v1
2~LD 2 z!GexpF2 j
2b2v2
2~LD 2 y!G3 H 1
4p2 EB~v1 2 v, z!C~v2 1 v, y!expF2 j
2b2v
2~z 2 y!GdvJ dzdy
E$NX(v1)NX* (v2)%2 4g2sv
2E0
LE0
L
expF2 j
2b2v1
2~LD 2 z!GexpF j
2b2v2
2~LD 2 y!G3 H 1
4p2 EB~v1 2 v, z!C~v 2 v2 , z!expF2 j
2b2v
2~z 2 y!GdvJ dzdy
E$NF(v1)NF* (v2)%g2sv
2E0
LE0
L
expF2 j
2b2v1
2~LD 2 z!GexpF j
2b2v2
2~LD 2 y!G3 H 1
4p2 EC~v1 1 v, z!C* ~v2 1 v, y!expF j
2b2v
2~z 2 y!GdvJ dzdy
NH,m~v! 5 E0
mL
expS a
2z D expF2
j
2b2v2~LD 2 z !G
3 S 2jg
4p2D EE As,0~v3 , z !
3 As,0~v 2 v3 1 v4 , z !
3 E0
z
expF j
2b2v4
2~ z 2 y !G3 expF2
a
2~ z 2 y !G jg
4p2
3 2EE As,0~v1 , y !As,0~v2 , y !
3 N0* ~v4 2 v1 1 v2 , y !d4vdydz, (53)
with z 5 z-floor(z/L)L, y 5 y-floor( y/L)L as the trans-mission distance after the last optical amplifier. By us-ing relations (47) and (48) and integrating first over thefour frequency variables, Eq. (53) can be separated intotwo terms with integration limits of *0
L*0z and *0
L*zL ; the
result is
NH,m~v! 5m2 1 m
2E
0
LE0
z
DNH~v, y, z !dydz
1m2 2 m
2E
0
LEz
L
DNH~v, y, z !dydz,
(54)
with DNH(v, y, z) as the total integrand in Eq. (53), in-cluding integration over the frequency variables. Forother terms from NI,m(v) to NN,m(v), similar things canbe done.
The first double integration in Eq. (54) is simplyNH(v); the second term is a new term, which we name asNH8 (v). We substitute Eq. (54) into E$N0(v1)NH,m(v2)%
for the second-order nonlinear effects for the noise fre-quency correlation and obtain
E$N0~v1!NH,m~v2!% 5m2 1 m
2TH~v1 , v2!
1m2 2 m
2TH8 ~v1 , v2!,
(55)
with TH8 (v1 , v2) defined similarly as TH(v1 , v2) in Eq.(39c) using NH8 (v). With some mathematical manipula-tion, we can show that
TH8 ~v1 , v2! 5 2TK~v2 , v1!. (56)
For other terms from NI,m(v) to NN,m(v), we obtain simi-lar results, summarized as
TI8~v1 , v2! 5 TI* ~v2 , v1!, (57)
TJ8 ~v1 , v2! 5 TJ* ~v2 , v1!, (58)
TK8 ~v1 , v2! 5 TH8 ~v2 , v1!, (59)
TL8 ~v1 , v2! 5 2TN8* ~v2 , v1!. (60)
Following the same procedures as for the single-spancase, we obtain
SUU,2,m~v1 , v2! 51
4 F2m2TI~v1 , v2!
2 ~m2 2 m !TH~v1 , v2!
1 ~m2 2 m !TK~v1 , v2!
1m2 1 m
2TM~v1 , v2!
1m2 2 m
2TM8 ~v1 , v2!G , (61)
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 507
For the second-order nonlinear effects on the complexnoise variance E$N(v)N* (v)%, we still have
SUU,2,m~v, v! 1 SVV,2,m~v, v! 5 m2TI~v, v!. (67)
That is, the noise nonlinear amplification is simply a fac-tor of m2 larger, but the noise energy transfer and corre-lations are more complicated.
Now we are ready to compute the noise frequency cor-relation functions with multiple ASE noise sources. Thesystem model is shown in Fig. 2. With the ASE noisefrom each optical amplifier being independent of the oth-ers’, it is easy to show that
SSXY,i,M~v1 , v2! 5 (k51
M
SXY,i,M2k~v1 , v2!, i 5 0, 1, 2,
(68)
for a system with M spans in total. X and Y can be eitherN, U, or V. Note that we have used notationSXY,i,m(v1 , v2) as the ith order nonlinear term for thenoise frequency correlation functions with single ASEnoise source passing through m spans of fibers with the
signal. The double-S notation SSXY,i,m(v1 , v2) is forthe noise frequency correlation functions with multipleASE noise sources.
This paper has focused on one type of system with100% DC per span. For systems with residual disper-sion, we still have, for example, the noise term NH,m(v)as an integration over (0, mL)
NH,m~v! 5 E0
mL
G~z, mL !expF2j
2D~z, mL !v2G
3 S 2jg
4p2D EE As,0~v3 , z !
3 As,0~v 2 v3 1 v4 , z !
3 E0
z
expF j
2D~ y, z !v4
2G3 G~ y, z !
jg
4p2 2EE As,0~v1 , y !As,0~v2 , y !
3 N0* ~v4 2 v1 1 v2 , y !d4vdydz, (69)
where D(x1 , x2) is the accumulated dispersion intro-duced between location x1 and x2 including fiber disper-sion and pre- and post-DC, and G(x1 , x2) is the accumu-lated amplification or attenuation introduced betweenlocation x1 and x2 including fiber attenuation and lumpedor distributed amplification. G(x1 , x2) , 1 if the opticalsignal has a power loss when propagating from x1 to x2 .Other noise nonlinear terms can be similarly extended tothis case. With residual dispersion, relations (47), (48),and (49) no longer hold, and Eq. (69) cannot be simplifiedas in Eq. (54) with 100% DC per span. But our analysiscan easily be modified to accommodate this general casewith modification in the integrand of those nonlinearterms for the noise frequency correlation functions.
B. Computational ComplexityTo build the correlation matrix for the noise frequencycomponents, we need to compute the following terms:T1(v1, v2), TI(v1, v2), TH(v1, v2), TK(v1, v2),TM(v1, v2), and TM8(v1 , v2). We discuss here the com-putational complexity associated with computing theseterms.
We first rewrite T1(v1, v2) as follows:
T1~v1 , v2! 5 E0
L
expF2j
2b2v2
2~LD 2 z !G3 S 2
jg
2pDsv
2C~v1 1 v2 , z !dz,(70)
Fig. 2. Simplified system model for multispan system with mul-tiple ASE noise sources.
508 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
using the definition of C(v) in Eq. (35). To find C(v),note that C(t, z) 5 As,0
2(t, z) and As,0(v, z) is the linearpropagation of the signal up to distance z, which can beeasily computed in the frequency domain. In the follow-ing discussion, we assume that both As,0(t, z) andAs,0(v, z) are available for every z we use in the numeri-cal integration, and so is the term C(t, z) and C(v, z).As a result, T1(v1 , v2) can be computed easily by nu-merical integration.
For the second-order nonlinear terms in Table 1, thekey is how to compute the integral over v with the help ofthe fast Fourier transform (FFT) and the inverse FFT(IFFT). Taking TI(v1 v2) as an example, we can rewritethe integral over v as a convolution over v for any fixed v1and any ( y, z), that is,
F HF 21H expF2j
2b2v2~z 2 y !GC~v 1 v1 , y !J
3 F 21$C* ~v 1 v2 , z !%. (71)
In total, we need 3Nz2Nv FFTs or IFFTs with Nz and Nv
being the number of steps in the transmission distanceand frequency, respectively. However, we can do muchbetter. For each v1 and z, we integrate over y first, thatis, we rewrite TI(v1 v2) as
TI~v1 , v2! 5 g2sv2E
0
L
expF j
2b2v2
2~LD 2 z !G3
1
4p2 E C* ~v 1 v2 , z !expS 2j
2b2v2z D
3 E0
z
expF2j
2b2v1
2~LD 2 y !G3 expS j
2b2v2y DC~v 1 v1 , y !dydvdz.
(72a)
Note that
E0
z
expF2j
2b2v1
2~LD 2 y !G3 expS j
2b2v2y DC~v 1 v1 , y !dy (72b)
is simply a function of v if v1 is fixed; then we can com-bine it with exp@(2j/2)b2v2z# and convolve the productwith C* (v 1 v2 , z) as in relation (71). Supposing thatthe FFT and IFFT are the most time-consuming manipu-lation, we need only 3NzNv FFTs and IFFTs with someextra computation for addition.
For all other terms, we have confirmed that everysecond-order nonlinear term can be computed in a similarway with the number of FFTs or IFFTs ;3NzNv .
Now, we see how many FFTs or IFFTs are needed forthe SSF simulation as a comparison. At each step, weneed at least one FFT to transform the signal to thefrequency-domain and compute the linear effect. Then,one IFFT is needed to transform the signal back to the
time-domain and add the nonlinear effect. As a result,each SSF simulation needs 2Nz FFTs or IFFTs.
However, to obtain the noise frequency correlationfunction, simulation with different realization of the noisemust be used. To get an idea of how many noise realiza-tions are needed, we study the estimation confidence ofestimating the variance of a zero-mean, Gaussian randomvariable. Suppose the variance of the Gaussian randomvariable is s 2, and the number of observations is M, thenthe unbiased estimation for the variance is 1/(M2 1)(yi
2, where yi is one independent observation. Forlarge M, (1/M)(yi
2 can be used instead. Now, we needto solve for M from the bound
P~ u~1/M!Syi2 2 s 2u < bs 2! . a (73)
with fixed (a, b). The parameter b gives the tolerablefluctuation range of our estimation. The smaller b is, theless fluctuation is tolerable and the more observations areneeded. The parameter a gives the estimation confi-dence. The larger a is, the more confidence we have inour estimation and the more observations are needed.The result is included in Table 3. From the table, weknow that over 3000 observations are needed to get thevariance estimation below 5% fluctuation with a confi-dence of only 95%. Practically, we need many more simu-lations than that to get a good enough estimation for theentire correlation function.
There are ways to reduce the computational cost for ourmethod. First, an adaptive step size can be used as inthe adaptive SSF method.5 Second, we do not need tocompute the whole frequency range since the noise ampli-fication is band limited as in the cw pump case. As a re-sult, we need only to compute the noise frequency corre-lation function in the frequency range where thenonlinear effect is significant. For those frequency com-ponents outside the range, we can simply assume thatthey are not affected by the signal.
Based on the above discussion of the computationalcomplexity of our method, we conclude that our methodprovides a more practical way to estimate the correlationfunctions of the output noise process than by using simu-lation based on the SSF method.
6. SIMULATION VALIDATION ANDAPPLICATION RANGEA. Validation through SimulationIn this section, we compare our analytical results for thenoise frequency correlation function with that estimated
Table 3. Number of Trials Associated withDifferent Estimation Confidence
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 509
by using the SSF simulation. The simulation param-eters we use are b2 5 22.6 ps2/km, g 5 2.2 W21 km21,a 5 0.25 dB/km, and a two-span system with span lengthL 5 50 km. The signal we use is a stream of Gaussianpulses at a rate of 40 Gb/s with pulse width T0 5 9 ps.The peak power is set to 5 mW at the input. A pseudo-random sequence of 64 bits is used with 8 samples per bit;our results show that 8 samples per bit are enough tocover the noise nonlinear amplification range at this rate.
The noise frequency correlation functions are esti-mated from simulation by using
SXY~v1 , v2! 5 1/M( Xi~v1!Yi* ~v2!
2 F1/M( Xi~v1!GF1/M( Yi* ~v2!G ,(74)
where X and Y are N, U, or V. The estimations are thennormalized with respect to the input noise variance in thefrequency domain sv
2. Twelve thousand simulation tri-als were used for this paper.
We first show SNN(v, v) in Fig. 3, which is the normal-ized variance for the complex noise at each frequency andis dependent only on the noise FWM effect as SN(v, v)5 4T(v, v) for a two-span system, a second-order termin the nonlinearity. It can be seen from the plot that ouranalytical result matches the estimations quite well.The fluctuation of the estimation about the simulation isalso clearly seen even for such a large number of differentnoise realizations.
We then show SUU(v, v) and SVV(v, v) in Fig. 4,where we clearly see the energy transfer between the real
Fig. 3. Complex noise variance SNN(v, v) at each frequency.
and imaginary parts of the noise and the nonlinear am-plification on the noise. The fluctuations near the carrierfrequency in the analytical results are due to the shortsignal length. The fluctuations in the simulation esti-mate are due to the high variance of the estimate from allfrequencies.
Then, in Fig. 5 we focus on the offset autocorrelationfunctions
Fig. 4. Noise variance at each frequency for real and imaginaryparts: (a) normalized SUU(v, v), (b) normalized SVV(v, v).
510 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
and the cross-correlation function SUV(v1 v2) with fixedv1 5 0 and different v2 . Note that we have subtractedthe linear contribution from SUU(v1 , v2) andSVV(v1 , v2) to emphasize the nonlinearity-induced cor-relation. Our analytical results fit the estimated resultsvery well in the whole frequency range.
We have also confirmed the special case of cw pumpand our results are found to agree with previous resultsfrom Ref. 3 very well. The details are included in Ref. 9.
B. Parameter Range ValidationIt is known that the autocorrelation matrix is alwayspositive semidefinite for any noise; that is, the eigenval-ues of this matrix should all be nonnegative. However,as the span number or signal power increases, our ap-proximated noise frequency correlation matrix mighthave negative eigenvalues. In Figs. 6 and 7, we show theeigenvalues of the approximated noise frequency correla-tion matrix for the previous two-span case and its exten-sion to four-span. In the two-span case, all the eigenval-ues of the correlation matrix are nonnegative, and thisagrees with our previous finding reported in Subsection6.A that our analytical results in this case match the es-timates well. However, some of the eigenvalues are
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 511
negative in the four-span case; consequently, our analyti-cal results cannot be used to approximate the real noisefrequency correlation functions in this case.
The reason for this failure is that as signal power or thenumber of spans increases, the nonlinear term can nolonger be treated as a small, perturbative term. We showhere a simple analysis of the valid range of perturbationtheory for a simple case with b2 5 0 where closed-formexpressions for the signal at the fiber output exist. As-sume that the input is A0(t) 5 AP0s(t) where P0 is thepeak power at the input and s(t) is the normalized pulseshape; then the output signal after m spans is
Fig. 6. Eigenvalues of the approximate correlation matrix fortwo spans.
Fig. 7. Eigenvalues of the approximate correlation matrix forfour spans.
A~t, mL ! 5 AP0s~t !
3 expH 2jgmP0us~t !u2
a@1 2 exp~2aL !#J .
(77)
With perturbation theory we try to approximate Am(t) bya linear expansion as
Am~t ! 5 A0~t !F1 2 jgmP0
aUs~t !U2 2
1
2 S gmP0
aUs~t !U2D 2
1j
6 S gmP0
aUs~t !U2D 3
1 ¯G , (78)
where we have used the simplification that exp(2aL)! 1. When mP0 is small the approximation works verywell, but it fails as mP0 increases. Even though exp(u)can be expanded into (n50
` un/n! for any u, the dominatingterms certainly are not the first two terms if u is largeenough. For our simulation parameters, we havemgP0 /a 5 0.6 for the four-span case, which is quite alarge value.
Note that the above analysis is for the SPM effect. Weknow that the XPM and FWM effects are more efficientthan the SPM effect; thus, the failure of the perturbationtheory up to the second order for the noise nonlinear am-plification analysis is entirely predictable for the inputpower that we are using.
As a simple rule, we will limit the valid range to wheremgP0 /a , 0.3 holds for the systems studied in this pa-per. Note that systems with 100% DC per span repre-sent the worst case in nonlinear effects since the nonlin-ear terms accumulate most efficiently. For other types ofsystems, perturbation theory can be applied to morespans or higher powers. For example, Ref. 10 reports ourstudy of the accuracy of the VSTF method with approxi-mate total transfer function, an approach equivalent tothe perturbative approach to solve the NLS equation formultispan systems with and without DC. There, a nor-malized energy deviation (NED), i.e., the difference in en-ergy between the optical field observed through the SSFsimulation and that observed through the perturbativetheory solution, was used to monitor the accuracy of theperturbative method by comparing the pulse distortionobtained by the two methods. For a 5% NED constraint,the equivalent VSTF method can be used to only 10 spansfor a 100% DC system while it can be used up to 25 spansif no DC is used. The NED analysis in Ref. 10 is based onsingle-channel SPM effect. For the noise nonlinear ef-fects studied here, the perturbative approach fails earlierthan predicted by the above NED analysis because of theefficiency of XPM and FWM. We expect that the pertur-bation theory approach is applicable to more spans orhigher powers for systems with residual dispersion thanfor those with 100% DC per span.
7. APPLICATIONWe show a simple application of our analysis of noise non-linear amplification in this section. With the noise fre-quency correlation matrix at hand, it is possible for us to
512 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 B. Xu and M. Brandt-Pearce
use Monte Carlo (MC) simulation to find the probabilitydensity function (PDF) of the detector statistics. (We callthe MC simulation using the analytical expressions forthe noise statistics the analysis-based MC simulation.)This is not possible to obtain by SSF-based MC simula-tion because a very large number of simulation trialswould be necessary to estimate the tail behavior of thePDFs.
As an example, we show the PDFs for the detectorsamples when either bit 0 or 1 is sent for a two-span op-tical fiber system transmitting at 40 Gb/s. The param-eters used are the same as before except for a higher noiselevel with the signal-to-noise ratio as 10 log PAVE /N05 5 dB. We picked a relatively high noise level to showthe crossover of the two PDFs for bit 0 and 1 so that wecan read the optimal threshold from the plot.
To do the analysis-based MC simulation, we first usethe SSF method to find the signal output without noise.Then, we generate Gaussian noise in the frequency do-main according to the noise frequency correlation matrixfor the exact signal and convert it into the time domain.The sum of the output signal and noise is then passedthrough an optical filter, a photodetector, and an electricalfilter. Both the optical and electrical filters are modeledas Butterworth filters of fifth order with bandwidth of 50GHz and 32 GHz, respectively. About 1.5 million trialsare used to generate the PDFs by histogram and a pseu-dorandom bit sequence of 64 bits is used for each trial.
The result is shown in Fig. 8 together with PDFs basedon a Gaussian approximation in which a bit sequence oflength N 5 1 and 3 was used.11 The Gaussian-approximated PDF’s for the detector statistics—takinginto account the effect of intersymbol interference from adifferent number of adjacent bits—can be written as
Fig. 8. Comparison between PDFs based on MC simulation andGaussian approximation.
fY~ yub0 5 0 or 1 !
51
2N21 (b0P$0,1%N21
N @m~b0 , b0!, s 2~b0 , b0!#
(79)
for bit 0 or 1. b 5 ...,b21 ,b0 ,b1 ,..., P $0,1%N is the bitsequence of length N, b0 is the central bit we are study-ing, and b0 is the bit sequence excluding b0 . The nota-tion N@m(b), s 2(b)# represents a Gaussian PDF withmean m(b) and variance s 2(b) when b is sent. Themeans and variances for the Gaussian approximation areestimated from simulations.
The discrepancies between the true PDFs based on MCsimulation and the Gaussian approximation are clearlyseen, especially near the tails of the PDFs. The thresh-olds for the maximum-likelihood detector associated withthe different PDF approximations are also different. TheMC-based PDFs give an optimal threshold of 1.1 mW,while the Gaussian approximations with 1-bit and 3-bitsequences give thresholds of 0.8 and 0.6 mW, respectively.This indicates that one must be careful in applying theGaussian approximation in such types of fiber optic com-munication systems.
8. CONCLUSIONAn accurate description of the noise is essential for thestudy of system performance of fiber optic communicationsystems. For this purpose, we have introduced aperturbation-theory-based method to analyze the noisenonlinear amplification and coloring in an optical-amplified fiber system used to transmit an arbitrarilymodulated signal. As a result of the nonlinear interac-tion between the noise and the signal, i.e., the parametricgain of the fiber, the noise becomes signal dependent, thusnot stationary. To describe such a noise process, we canno longer use the concept of a power spectrum. Instead,we use the noise correlation functions among its fre-quency components; the noise correlation functionsamong any time samples can be found by Fourier trans-form accordingly.
The analytical results presented in this paper are vali-dated by comparing with numerical simulation, and therange of validity of the method is discussed. Our methodis accurate within the range where perturbation theoryholds but fails for large signal power or large number ofspans. However, our method based on perturbationtheory still gives an idea of how the noise interacts withan arbitrarily modulated signal. A simple application isdemonstrated to show the PDF of the detector statistics,which is impossible to obtain by numerical simulationwith the SSF method.
Even though this paper focuses on a specific type of sys-tem with 100% DC per span and lumped optical amplifi-ers, the analysis can be extended without much difficultyto other systems with different configurations of DC anddistributed amplification from Raman amplifiers. Boththe dispersion and distributed amplification can be ac-counted for by modifying the integrands of the nonlinear
B. Xu and M. Brandt-Pearce Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 513
terms. The analysis can also be applied to WDM systemsand polarization-multiplexed systems similar to the workin Refs. 12 and 13.
This study on noise correlation is a first step toward ac-curately evaluating fiber optic communication systemperformance. The next step is to understand the effect ofthe photodetector on the nonstationary and colored noise.The square-law operation of the photodetector introducesan interaction between the signal and the signal-dependent noise. However, the noise process after thephotodetector is no longer Gaussian distributed and thuscannot be described uniquely by its correlation function.Another method should be used, such as the Karhunen–Loeve expansion as used Ref. 4.
Corresponding author M. Brandt-Pearce’s e-mail ad-dress is [email protected].
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