## Probability-density function for energy perturbations of isolated optical pulses

Optics Express, Vol. 11, Issue 26, pp. 3628-3648 (2003)

http://dx.doi.org/10.1364/OE.11.003628

Acrobat PDF (259 KB)

### Abstract

The mathematical methods required to model simple stochastic processes are reviewed briefly. These methods are used to determine the probability-density function (PDF) for noise-induced energy perturbations of isolated (solitary) optical pulses in fiber communication systems. The analytical formula is consistent with the numerical solution of the energy-moment equation. System failures are caused by large energy perturbations. For such perturbations the actual PDF differs significantly from the (ideal-ized) Gauss PDF that is often used to predict system performance.

© 2003 Optical Society of America

## 1. Introduction

5. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. **11**, 665–667 (1986). [CrossRef] [PubMed]

6. C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. **200**, 165–177 (2001) and references therein. [CrossRef]

7. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. **23**, 1351–1353 (1990). [CrossRef]

8. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. **27**, 1887–1889 (2002)and references therein. [CrossRef]

9. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. **14**, 1062–1070 (1971). [CrossRef]

10. W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. **22**, 226–230 (1977). [CrossRef]

11. D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A **27**, 3135–3145 (1983). [CrossRef]

12. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A **42**, 5689–5694 (1990). [CrossRef] [PubMed]

13. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B **7**, 386–392 (1990). [CrossRef]

14. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. **8**, 1816–1823 (1990). [CrossRef]

15. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. **9**, 170–173 (1991). [CrossRef]

16. P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. **9**, 1576–1582 (1991). [CrossRef]

17. J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. **12**, 1224–1229 (1994). [CrossRef]

18. T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A **51**, 1662–1668 (1995). [CrossRef] [PubMed]

19. B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E **48**, R5–R8 (1993). [CrossRef]

20. G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E **63**, 25601R (2001). [CrossRef]

21. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. **20**, 389–400 (2002). [CrossRef]

22. R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. **28**, 105–107 (2003). [CrossRef] [PubMed]

23. C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.

24. H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. **15**, 320–322 (2003). [CrossRef]

## 2. Simple stochastic process

*X*. Suppose that this process is modeled by the SDE

*Ẋ*=

*dX/dz, a*and

*b*are deterministic functions of

*X*and

*z*, and

*r*is a random function of

*z*.

*r*is white noise, which is characterized by the equations

*δz*be a short, but finite, distance interval and let

*δW*=

*W*(

*δz*). Then it follows from Eqs. (5) and (6) that a typical noise increment

*δW*~

*δz*

^{1/2}.

*r*(

*z*) represents a stationary stochastic process, the ensemble-autocorrelation is a function of

*ζ*=

*z-z*′ (rather than

*z*and

*z*′ separately) and the space average (7) equals the ensemble average (8):

*G*

_{e}=

*G*

_{z}=

*G*[26]. Define the Fourier transform

*r*|

^{2}is (noise) power,

*S*is the power per unit inverse-wavelength.] Then the Wiener-Khinchin (statistical-autocorrelation) theorem [26] states that

*G*(

*ζ*)=

*δ*(

*ζ*), so the associated spectral density

*S*(

*k*)=1: The spectrum has infinite bandwidth and contains infinite power. Such a spectrum cannot exist. [In transmission systems the frequency spectra are limited by the amplifier (or filter) bandwidths, which are finite. They are converted to wavenumber spectra by propagation.]

*r*is weakly-colored noise, the simplest example of which is the rectangular spectral density

*K*(which we assume is much broader than any other relevant bandwidth) and contains finite power. The associated autocorrelation function

*K*(which is much shorter than any other relevant distance scale). One can still use Eqs. (2) and (3) to characterize weakly-colored noise, provided that one interprets

*δ*as the sinc function in Eq. (14). For weakly-colored noise Eq. (4) does not define a Wiener process. However, Eq. (5) is still satisfied exactly and Eq. (6) is satisfied approximately. In particular, the increment

*r*(

*z*′)

*dz*′~

*δz*

^{1/2}, provided that

*δz*is much longer than the correlation distance.

*X*(

*z*) is one member of an ensemble of solutions. This description of the stochastic process is based on an SDE for the dependent variable

*X*. An alternative description is based on the PDF for the independent variable

*x*. Let

*P*(

*x, z*)

*dx*be the probability that

*X*(

*z*) is in the range (

*x,x*+

*dx*). Then

*P*(

*x, z*) satisfies a probability-diffusion equation, which is called the Fokker-Planck equation (FPE). (In the mathematics literature the probability-diffusion equation is called the Kolmogorov equation.)

*z*-dependence of

*a*and

*b*was suppressed and the initial position was denoted by 0 (rather than

*z*). The first term on the right side of Eq. (15) is of order

*δz*, whereas the second term is of order

*δz*

^{1/2}. By omitting terms of order

*δz*

^{3/2}and higher, one can make the simplification

*a*

_{0}=

*a*(

*X*

_{0})=

*a*[

*X*(0)]: The first contribution only depends on the value of

*X*at the beginning of the interval. In contrast, because the second term is larger than the first, one must account for the cumulative change in

*X*. By doing so, one finds that

*δz*. The second is random (with zero mean) and of order

*δz*

^{1/2}. The third is also random, but of order

*δz*. Consequently, its contribution to the random part of

*δX*is insignificant. However, its mean value, if nonzero, is of order

*δz*. It follows from Eq. (18) that

*l*

_{c}=0. As we explained above, no real physical process can have

*l*

_{c}=0. The requirement that

*l*

_{c}be much shorter than the next-shortest length

*δz*introduces an ambiguity into the right side of Eq. (15), as viewed from the perspective of deterministic calculus. Indeed, if

*r*were a deterministic function of

*z*Eq. (15) could be rewritten as

*l*≪

*δz*. The ambiguity, which exists for stochastic processes, is whether one treats

*l*≫

*l*

_{c}or

*l*≪

*l*

_{c}. The choice one makes determines whether one obtains Ito or Stratonovich calculus, respectively.

*r*and

*b*are meant to be evaluated at the same position

*z*, so the parameter

*l*, defined in Eq. (20), should satisfy

*l*≪

*l*

_{c}. In the Stratonovich formalism

*l*=0. Let

*I*and

*J*denote the double integrals that appear in Eqs. (19) and (22), respectively. Then, by using the sinc function in Eq. (14) to evaluate these integrals, one finds that

*J*=

*δz*+

*O*(1/

*K*) and

*I*=

*J*/2. Because the correlation distance is much shorter than the interval under consideration (

*Kδz*≫

*1*)

*δX*(which is called the drift), but does not affect its variance. These conclusions do not depend sensitively on the choice of autocorrelation function. For the special case in which the noise is additive (

*b*′=0), the drift formulas (21) and (24) are identical.

*f*

_{X}=

*df/dX*. It follows from the discussion of Eq. (18) that the larger deterministic contribution to

*δX*

^{2}must be retained [Eq. (23)], whereas the other (smaller deterministic and random) contributions need not. Consequently,

*X*-derivatives on the right side of Eq. (30) by

*Y*-derivatives. Let

*g*be the inverse of

*f*, so that

*f*

_{X}=1/

*g*

_{Y}and

*f*

_{XX}=-

*g*

_{YY}/

*T*is the transition-probability function and the initial position was denoted by 0 (rather than

*z*). Suppose that

*X*(0) has the value

*x*. Then

*T*(

*δx,δz*|

*x*,0)

*d*(

*δx*) is the probability that

*X*(

*δz*) has a value between

*x*+

*δx*and

*x*+

*δx*+

*d*(

*δx*). Because

*δx*is a small increment (the transition probability is only significant for small values of

*δx*), one can expand the integrand

*TP*in a Taylor series about

*x*. The result is

*P*and

*T*on

*x*and

*z*=0 was suppressed for simplicity of notation. For white noise the random contribution to

*δX*is

*bδW*[Eqs. (4) and (18)]. The increment of a Wiener process (

*δW*) has Gaussian statistics [25]. It follows from this fact and Eq. (18) that the Ito transition probability

*aδz*and the third is

*b*

^{2}

*δz*. By cancelling like terms one obtains the Ito FPE

*T*to be known exactly: The cumulative transition probability is always 1, and the mean and variance of

*δX*follow directly from Eqs. (19) and (22), respectively.

*Y*be an arbitrary function of

*X*. Then the FPEs for

*P*(

*y*) are also identical. The proof of this statement is similar to the derivations of the Ito and Stratonovich change-of-variables rules.

*z*=0

*a*,

*b*′ and

*f*″ were omitted from Eqs. (41) and (42) because they are of higher order in

*δz*. By assumption, the correlation distance of the noise is shorter than

*δz*, so 〈

*f*[

*X*(-

*δz*)]

*r*(0)〉=0. It follows from Eq. (14) that

*r*(

*z*′)

*r*(0)〉

*dz*′=1/2. Consequently, in the Stratonovich formulation

*δr*

_{x}denote the deviatoric rate of change

*r*

_{x}-〈

*r*

_{x}〉. Then

## 3. Energy equation

*∂*

_{z}=

*∂/∂z*,

*A*is the (complex) electric-field amplitude,

*g*is the amplifier gain rate,

*α, β*and

*γ*are the fiber loss, dispersion and nonlinearity coefficients, respectively, and

*R*is a random driving (source) term that models the effects of amplifier noise. Because

*R*is independent of

*A*, the noise is said to be additive. At each position the rate at which noise changes the amplitude is a random function of time. Furthermore the rates of change at different positions are independent. These properties are quantified by the equations

*δ*denotes a delta function. (Boldface was used to distinguish the symbol for a delta function from the symbol for a small change in value.) The source strength

*S*=

*n*

_{sp}

*h̄ωg*, where

*n*

_{sp}is the spontaneous noise factor (which has a typical value in the range 1.1–1.3) and

*h̄ω*is the photon energy.

*z*=0 and the neighboring point

*z*=

*δz*, in the absence of a signal pulse. It follows from Eq. (47) that

*T*be the bit period. Then the noise energy in a bit period is

*δE*〉=

*SδzTδ*(0). If one tries to interpret

*δ*(

*t*) as the Dirac delta function

*δ*

_{D}(

*t*), one obtains the unphysical result that the frequency bandwidth and energy of the noise are infinite. In real systems the noise bandwidth and energy are limited by the (common) bandwidth of the amplifiers (or filters). Consequently, one should interpret

*δ*(

*t*) as the delta-like function

*F*sinc(

*πFt*), where

*F*is the amplifier (or filter) bandwidth. By doing so, one obtains the physical result

*Sδz*is the energy per noise mode and

*FT*is the number of modes [14

14. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. **8**, 1816–1823 (1990). [CrossRef]

15. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. **9**, 170–173 (1991). [CrossRef]

16. P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. **9**, 1576–1582 (1991). [CrossRef]

*δ*(

*z*) as the Dirac delta function

*δ*

_{D}(

*z*), and use Ito calculus, or one can interpret

*δ*(

*z*) as the delta-like function

*K*sinc(

*πKz*), where

*K*is the noise bandwidth, and use Stratonovich calculus. The former assumption implies that the correlation distance of the noise is zero, whereas the latter assumption implies that the correlation distance is short, but finite. We will illustrate both approaches in this section. The required elements of stochastic calulus were reviewed in Section 2.

*A*. By applying the Ito change-of-variable rule [Eq. (28)] to the real and imaginary parts of Eq. (47), combining the results and integrating the transformed equation over a bit period, one can show that

*SFT*is present in Eq. (56) because the Ito rule differs from the deterministic rule. No energy-flux terms are present. On average, the noise-energy flux at the time boundaries (-

*T*/2 and

*T*/2) is zero. The signal-energy flux can be neglected if the signal pulse is isolated (does not interact with other pulses in the same channel or other channels).

*a*and

*b*are deterministic functions and

*r*is a random driving term with unit strength [〈

*r*(

*z*)〉=0 and 〈

*r*(

*z*)

*r*(

*z*′)〉=

*δ*(

*z*-

*z*′)]. Henceforth, the explicit dependence of

*a*and

*b*on

*z*will be suppressed. Let

*r*

_{e}denote the total rate of change of the bit energy [right side of Eq. (56) or (57)] and let

*δr*

_{e}denote the deviatoric rate of change

*r*

_{e}-〈

*r*

_{e}〉. Then, for an Ito SDE,

*r*

_{e}has the canonical properties [Eqs. (44) and (46)]

*a*and

*b*by calculating the requisite moments of the right side of Eq. (56). In the Ito formulation the correlation distance of the noise is zero and

*A*(

*z, t*) depends only on the noise emitted at previous positions (

*z*′<

*z*). It follows from these facts that

*g*and

*S*depend on

*z*. Notice that the rate at which noise changes the bit energy depends on the current energy. Equation (63) is valid for any combination of distributed and lumped amplification.

*E*. There is an alternative description, which is based on the PDF for the independent variable

*e*. Let

*P*(

*e, z*)

*de*be the probability that

*E*(

*z*) is in the range (

*e,e*+

*de*). Then [according to Eq. (37)] the energy PDF satisfies the Ito FPE

*SFT*is absent because the Stratonovich rule is the same as the deterministic rule. Our goal is to rewrite Eq. (65) in the canonical form (57). For a Stratonovich SDE

*r*

_{e}has the canonical properties [Eqs. (45) and (46)]

*b*′=

*∂b/∂E*and non-delta-like terms of order 1 were omitted. In the Stratonovich formulation the correlation distance of the noise is short, but finite, and

*A*(

*z, t*) depends on the noise emitted at all previous positions and the current position (

*z*′≤

*z*). It follows from these facts [and a short calculation similar to that which produced Eq. (43)] that

*g*and

*S*depend on

*z*. It follows from Eq. (71) [and Eq. (38)] that the energy PDF satisfies the Stratonovich FPE

## 4. Analysis of energy jitter

*g*=

*α*). The effects of NDA (

*g*≠

*a*) will be discussed in Section 6. Let

*E*

_{0}denote the initial pulse energy, and let

*X*=

*E*/

*E*

_{0},

*µ*=

*FT*and

*ζ*=

*Sz*/

*E*

_{0}denote the normalized bit energy, mode number and normalized distance, respectively. Then, for systems with UDA, the normalized energy is governed by the Ito SDE

*r*is a random driving term with unit strength. (In the Stratonovich SDE the parameter

*µ*is replaced by

*µ*-1/2.) By definition, the initial condition is

*X*(0)=1. Equation (73) is difficult to solve exactly.

*X*differs significantly from 1 is small. By making the (standard) approximation

*X*

^{1/2}≈1 in Eq. (73), one obtains the linearized equation

*X*(approximately) as a Gauss (normal) random variable with mean

*m*

_{n}=1+

*µζ*and variance

*ν*

_{n}=2

*ζ*. The associated PDF is

*X*cannot be exactly Gaussian, because, if it were, the probability of

*X*<0 would be finite for all

*ζ*>0. From a practical standpoint this inconsistency is tolerable if the probability of

*X*<0 is exponentially small for system lengths of interest.)

*α*=0.21 dB/Km,

*β*=-0.30 ps

^{2}/Km (

*D*=0.38 ps/Km-nm) and

*γ*=1.7/Km-W. Then a soliton (sech pulse) with a full-width at half-maximum of 30 ps has an input energy of 21 fJ (time-averaged power of 0.21 mW). If the system length

*l*=10 Mm, the output noise power in both polarizations, in a frequency bandwidth of 12 GHz (wavelength bandwidth of 0.1 nm), is 1.7

*µ*W: The (optical) signal-to-noise ratio (SNR) is 21 dB. (Systems with NDA produce the same noise power in shorter distances.) For this system the normalized output-energy variance 2

*Sl*/

*E*

_{0}is 6.6×10

^{-3}and the output-energy deviation is 8.1×10

^{-2}.

*x*(relative to the mean value). Equation (73) defines

*X*as the sum of independent random increments. Because the size of each increment depends on the current value of

*X*, the actual PDF cannot be a symmetric function of

*x*. Thus, the standard approximation is inadequate. One can remove the coefficient

*X*

^{1/2}from the random term in Eq. (73) by making the change of variables

*Y*=

*X*

^{1/2}, in which case

*=(2*µ ¯

*µ*-1)/4. By making the approximation

*Y*≈1 in the drift term, one obtains the linearized equation

*Y*(approximately) as a normal random variable with mean

*m*

_{n}=1+

*and variance*µ ¯
ζ

*v*

_{n}=

*ζ*/2. It follows from this result that

*X*is a non-central chi-squared random variable with mean

*m*

_{x}=

*v*

_{n}and variance

*v*

_{x}=2

*v*

_{n}[27]. For the process under consideration

*m*

_{x}≈1+

*µζ*and

*v*

_{x}≈2

*ζ*. The associated PDF can be written approximately as

*X*(0)=1 are equivalent to the FPE

*P*(

*x*,0)=

*δ*(

*x*-1). The moments of

*x*are defined by the formula

*µζ*and the variance is 2

*ζ*+

*µζ*

^{2}≈2

*ζ*. For typical systems

*ζ*

^{2}~10

^{-5}, so the approximate PDF (78) predicts the energy mean and variance accurately.

*x*, one can solve Eq. (79) by Laplace transforming in

*x*, making the

*a priori*assumption that

*P*(0,

*ζ*)=0, and using the method of characteristics to solve the transformed equation. The result is

*µζ*and 2

*ζ*+

*µζ*

^{2}, respectively, in agreement with the moment results. One can invert the transformed solution by rewriting the numerator as exp(-1/

*ζ*)exp[1/(

*ζ*

^{2}

*s*+

*ζ*)]. By using the shift theorem and the relation [28]

*I*

_{n}(

*z*) is the modified Bessel function of order

*n*, one finds that

*P*(0,

*ζ*)=0, as assumed. For short distances, one can use the relation

*I*

_{n}(

*z*)~exp(

*z*)/(2

*πz*)

^{1/2}as

*z*→∞ to rewrite solution (86) as

*x*

^{µ}/

^{2}in formula (87) is to shift the mean from 1 to 1+

*µζ*.]

*P*(

*x,ζ*|

*x*

_{0},0) of observing the value

*x*at the position

*ζ*, provided that

*x*

_{0}was observed at

*ζ*=0. (In this context one should consider the normalization energy

*E*

_{0}as a reference energy rather than the initial energy.) The conditional PDF satisfies the forward FPE (79) with respect to the variable

*x*. Moreover, it can be shown [25], similarly to Section 2, that the backward FPE (or backward Kolmogorov equation), which

*P*(

*x,ζ*|

*x*

_{0},0) satisfies with respect to

*x*

_{0}, has the form of the equation adjoint to (79). Specifically,

30. R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A **25**, 3234–3258 (1982). [CrossRef]

*q*(

*x*) is the particular (but not necessarily normalizable) solution of the FPE (79) satisfying

*p*(

*x,λ*) and

*p̃*(

*x,λ*) must be complete. The completeness relation has the form

*f*(

*x*) is any function for which the following integrals exist, then

31. R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A **25**, 1731–1754 (1982). [CrossRef]

30. R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A **25**, 3234–3258 (1982). [CrossRef]

*J*

_{n}(

*z*) is the Bessel function of order

*n*. The completeness relation for the eigen-functions (101) and (102) follows from the orthogonality relation (94), because the arguments of the Bessel functions depend symmetrically on

*λ*and

*x*. Notice that the functions

*x*

^{(µ-1)/2}

*N*

_{|µ-1|}[2(

*λx*)

^{1/2}], where

*N*

_{n}(

*z*) is the Neumann function of order

*n*, also solve (92) and are bounded at

*x*=0. However, they are not included in the set of eigenfunctions

*p*(

*x,λ*) because they are not required for completeness. Notice also that for

*µ*>3/2 the individual eigenfunctions

*p*(

*x,λ*) are not normalizable due to their divergence at

*x*=∞. This behavior does not invalidate expansion (91), but does make its convergence non-uniform in

*ζ*: One cannot interchange the limits

*x*→∞ and

*ζ*→∞ in the resulting solution [31

31. R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A **25**, 1731–1754 (1982). [CrossRef]

*x*=0 and

*x*=∞. Not only does the Sturm-Liouville analysis validate the Laplace-transform analysis (

*x*

_{0}=1), it also obviates the need for

*a priori*boundary conditions.

14. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. **8**, 1816–1823 (1990). [CrossRef]

15. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. **9**, 170–173 (1991). [CrossRef]

16. P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. **9**, 1576–1582 (1991). [CrossRef]

*µ*=5 and

*ζ*=3.3×10

^{-3}, which correspond to the physical parameters described in Section 3. The solid, dot-dashed and dashed curves represent the exact PDF (86), the approximate PDF (78) and the Gauss PDF (75), respectively. All three PDFs have (approximately) the same mean and variance. When plotted on a linear scale (Fig. 1

*a*), the differences between the PDFs are barely perceptible. When plotted on a logarithmic scale (Fig. 1

*b*), it is clear that the exact and approximate PDFs have enhanced high-energy tails and diminished low-energy tails relative to the Gauss PDF. In particular, at high energies these (logarithmic) PDFs decrease linearly with energy, rather than quadratically [20

20. G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E **63**, 25601R (2001). [CrossRef]

*a*). For the stated parameters, the differences between the exact and approximate PDFs are insignificant. The probability of observing the low (normalized) energy

*X*=0.5 is about three orders of magnitude smaller than that predicted by the Gauss PDF and the probability of observing the high energy

*X*=1.5 is about two orders of magnitude larger. As the energy deviation |

*E*-

*E*

_{0}| increases, so also do the differences between the PDFs. These results show the danger of using Gaussian statistics to predict system performance.

**8**, 1816–1823 (1990). [CrossRef]

**9**, 170–173 (1991). [CrossRef]

**9**, 1576–1582 (1991). [CrossRef]

33. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. **15**, 840–842 (2003) and references therein. [CrossRef]

*b*), but underestimates the probability of a 0 by a similar amount, and the two errors cancel. For DPSK systems with balanced detectors the Gauss approximation overestimates the BER by several orders of magnitude [33

33. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. **15**, 840–842 (2003) and references therein. [CrossRef]

## 5. Simulation of energy jitter

*n*denotes the distance

*nδz*. The noise increments have zero mean and satisfy the correlation equation

*µ*=5 and

*ζ*=3.3×10

^{-3}, which correspond to the physical parameters described in Section 3. The solid curve denotes the analytical solution (86), whereas the dashed curve denotes the numerical solution of the FPE (79) and the dots represent the results of numerical simulations based on the Stratonovich FDE (113), with

*a*=

*µ*-1/2 and

*b*=(2

*X*)

^{1/2}, and the correlation equation (107). The simulations were made with an ensemble of 3×10

^{6}pulses. Importance sampling [22

22. R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. **28**, 105–107 (2003). [CrossRef] [PubMed]

23. C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002.

22. R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. **28**, 105–107 (2003). [CrossRef] [PubMed]

## 6. Effects of nonuniformly-distributed amplification

*X*=

*E/E*(0) is governed by the Ito SDE

*r*is a random driving term with unit strength. The excess gain rate ν(

*z*)=

*g*(

*z*)-

*α*and the source strength

*z*.

*l*is the distance between the Raman pumps (or the Erbium amplifiers). In the presence of noise the energy undergoes an oscillatory random walk. Energy decision thresholds (for the detection of 1s) are defined as fractions of the mean energy. In the context of such thresholds, what is important is not the absolute energy variance, but the relative energy variance (which is measured relative to the mean energy). Thus, one can simplify the mathematical analysis of energy jitter and clarify the physical significance of the results by writing the energy variable as a product of oscillatory and non-oscillatory factors. Let

*X*=

*Y*exp[

*z*′)

*dz*′]. Then the modified energy

*Y*is governed by the equation

*ζ*and rescales the energy variable.

*E*

_{a}equals the input energy

*E*(0) and the path-averaged source strength

*σ*

_{a}=(

*n*

_{sp}

*h̅ωα*/

*E*

_{a}). For systems with NDA, the path-averaged energy

*ζ*, so one can make the approximation

*ρ*is the product of the integrals in Eq. (121) divided by

*l*

^{2}. It follows from this definition that

*ρ*≥1, which reflects the fact that NDA produces more noise than UDA.

*α*=0.21 dB/Km,

*β*=-0.30 ps

^{2}/Km (

*D*=0.38 ps/Km-nm),

*γ*=1.7/Km-W and backward Raman amplification. The gain rate

*α*

_{p}=0.25 dB/Km is the fiber loss rate at the pump wavelength. A soliton with a full-width at half-maximum of 30 ps has a path-averaged energy of 21 fJ and an input energy of 75 fJ (time-averaged input power of 0.75 mW). If the system length

*l*=5 Mm the output noise power in both polarizations, in a frequency bandwidth of 12 GHz (wavelength bandwidth of 0.1 nm), is 5.4

*µ*W: The (optical) SNR is 21 dB. (The SNR of this 5-Mm system with NDA is comparable to the SNR of the 10-Mm system with UDA that was discussed in Section 4.)

*X*are plotted as functions of distance in Fig. 3 for the aforementioned physical parameters and

*µ*=5. One can determine the mean and variance by using the formulas of Section 4, and evaluating the scale factor exp[

*z*′)

*dz*′] and the distance variable

*ζ*numerically. One can also determine them by solving numerically the moment equations

*Y*is plotted as a function of distance in Fig. 4. The solid curve was obtained by solving Eqs. (124) and (125) numerically, whereas the dashed line was obtained from the formula

*v*

_{y}=2

*ζ*+

*µζ*

^{2}. The distance variable

*ζ*was evaluated by using Eq. (122), with

*ρ*=1.8. As stated above, the oscillations in

*ζ*and (hence)

*v*

_{y}are small: From a practical standpoint

*v*

_{y}grows monotonically.

## 7. Summary

*X*, in which

*X*is the dependent variable. The second involves a probability-diffusion, or Fokker-Planck, equation (FPE) for the probability-density function (PDF)

*P*(

*x*), in which

*x*is an independent variable. This paper contained brief descriptions of stochastic calculus, which facilitates the solution of SDEs, and the relation between SDEs and their associated FPEs, for which many solution methods exist. The Ito formulation is based on the assumption that the correlation-time of the noise is zero, whereas the Stratonovich formulation is based on the complementary assumption that the correlation-time is nonzero, but short. Both formulations were described.

*E*

_{0}be the (unperturbed) output energy in a noiseless system and let

*E*be the (perturbed) output energy in a noisy system. Then the probability of observing the low energy

*E*=0.5

*E*

_{0}is about three orders of magnitude smaller than that predicted by the Gauss PDF and the about two orders of magnitude larger: One cannot predict system performance accurately by well for ASK systems (because the receiver eyes are closed by low energies, which are less likely), whereas increased high-energy probabilities bode ill for DPSK systems [because energy (power) jitter drives phase jitter and phase shifts of either sign close the receiver eyes.]

## References and links

1. | L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in |

2. | E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, |

3. | K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, and T. Misuochi, “A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,” OFC 2003, paper ThE2. |

4. | J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov, and N. S. Bergano, “A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,” OFC 2003, paper PD22. |

5. | J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. |

6. | C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. |

7. | J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. |

8. | C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. |

9. | S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged dersciption of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. |

10. | W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. |

11. | D. Anderson, “Variational approach to pulse propagation in optical fibers,” Phys. Rev. A |

12. | D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A |

13. | H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B |

14. | D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. |

15. | J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,” J. Lightwave Technol. |

16. | P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. |

17. | J. S. Lee and C. S. Shim, “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,” J. Lightwave Technol. |

18. | T. Yoshino and G. P. Agrawal, “Photoelectron statistics of solitons corrupted by amplified spontaneous emission,” Phys. Rev. A |

19. | B. A. Malomed and N. Flytzanis, “Fluctuational distribution function of solitons in the nonlinear Schrödinger system,” Phys. Rev. E |

20. | G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, “Statistics of soliton-bearing systems with additive noise,” Phys. Rev. E |

21. | R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. |

22. | R. O. Moore, G. Biondini, and W. L. Kath, “Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,” Opt. Lett. |

23. | C. J. McKinstrie and P. J. Winzer, “How to apply importance-sampling techniques to simulations of optical systems,” http://arxiv.org/physics/0309002. |

24. | H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. |

25. | C. W. Gardiner, |

26. | J. W. Goodman, |

27. | J. G. Proakis, |

28. | M. Abramowitz and I. A. Stegun, |

29. | W. Horsthemke and R. Lefever, |

30. | R. Graham, “Hopf bifurcation with fluctuating control parameter,” Phys. Rev. A |

31. | R. Graham and A. Schenzle, “Carleman imbedding of multiplicative stochastic processes,” Phys. Rev. A |

32. | I. S. Gradsteyn and I. M. Rhyzhik, |

33. | P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. |

34. | J. D. Hoffman, |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 9, 2003

Revised Manuscript: December 22, 2003

Published: December 29, 2003

**Citation**

C. McKinstrie and T. Lakoba, "Probability-density function for energy perturbations of isolated optical pulses," Opt. Express **11**, 3628-3648 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3628

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### References

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- E. Iannone, F. Matera, A. Mecozzi and M. Settembre, Nonlinear Optical Communications Networks (Wiley, New York, 1998).
- K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda and T. Misuochi, �??A comparative study of 10 Gb/s RZ-DPSK and RZ-ASK WDM transmission over transoceanic distances,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper ThE2.
- J. X. Cai, D. G. Foursa, C. R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W. W. Patterson, A. N. Pilipetskii, M. Nissov and N. S. Bergano, �??A DWDM demonstration of 3.73 Tb/s over 11,000 km using 373 RZ-DPSK channels at 10 Gb/s,�?? OFC (Optical Society of America, Washington, D.C., 2003) paper PD22.
- J. P. Gordon and H. A. Haus, �??Random walk of coherently amplified solitons in optical fiber transmission,�?? Opt. Lett. 11, 665�??667 (1986). [CrossRef] [PubMed]
- C. J. McKinstrie, �??Gordon�??Haus timing jitter in dispersion-managed systems with distributed amplification,�?? Opt. Commun. 200, 165�??177 (2001) and references therein. [CrossRef]
- J. P. Gordon and L. F. Mollenauer, �??Phase noise in photonic communication systems using linear amplifiers,�?? Opt. Lett. 23, 1351�??1353 (1990). [CrossRef]
- C. J. McKinstrie, C. Xie and T. I. Lakoba, �??Efficient modeling of phase jitter in dispersion-managed soliton systems,�?? Opt. Lett. 27, 1887�??1889 (2002) and references therein. [CrossRef]
- S. N. Vlasov, V. A. Petrishchev and V. I. Talanov, �??Averaged dersciption of wave beams in linear and nonlinear media,�?? Radiophys. Quantum Electron. 14, 1062�??1070 (1971). [CrossRef]
- W. J. Firth, �??Propagation of laser beams through inhomogeneous media,�?? Opt. Commun. 22, 226�??230 (1977). [CrossRef]
- D. Anderson, �??Variational approach to pulse propagation in optical fibers,�?? Phys. Rev. A 27, 3135�??3145 (1983). [CrossRef]
- D. J. Kaup, �??Perturbation theory for solitons in optical fibers,�?? Phys. Rev. A 42, 5689�??5694 (1990). [CrossRef] [PubMed]
- H. A. Haus and Y. Lai, �??Quantum theory of soliton squeezing: a linearized approach,�?? J. Opt. Soc. Am. B 7, 386�??392 (1990). [CrossRef]
- D. Marcuse, �??Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 8, 1816�??1823 (1990). [CrossRef]
- J. P. Gordon and L. F. Mollenauer, �??Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission,�?? J. Lightwave Technol. 9, 170�??173 (1991). [CrossRef]
- P. A. Humblet and M. Azizoglu, �??On the bit error rate of lightwave systems with optical amplifiers,�?? J. Lightwave Technol. 9, 1576�??1582 (1991). [CrossRef]
- J. S. Lee and C. S. Shim, �??Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain,�?? J. Lightwave Technol. 12, 1224�??1229 (1994). [CrossRef]
- T. Yoshino and G. P. Agrawal, �??Photoelectron statistics of solitons corrupted by amplified spontaneous emission,�?? Phys. Rev. A 51, 1662�??1668 (1995). [CrossRef] [PubMed]
- B. A. Malomed and N. Flytzanis, �??Fluctuational distribution function of solitons in the nonlinear Schrodinger system,�?? Phys. Rev. E 48, R5�??R8 (1993). [CrossRef]
- G. E. Falkovich, I. Kolokolov, V. Lebedev and S. K. Turitsyn, �??Statistics of soliton-bearing systems with additive noise,�?? Phys. Rev. E 63, 25601R (2001). [CrossRef]
- R. Holzlohner, V. S. Grigoryan, C. R. Menyuk and W. L. Kath, �??Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,�?? J. Lightwave Technol. 20, 389�??400 (2002). [CrossRef]
- R. O. Moore, G. Biondini and W. L. Kath, �??Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems,�?? Opt. Lett. 28, 105�??107 (2003). [CrossRef] [PubMed]
- C. J. McKinstrie and P. J. Winzer, �??How to apply importance-sampling techniques to simulations of optical systems,�?? <a href="http://arxiv.org/physics/0309002">http://arxiv.org/physics/0309002</a>.
- H. Kim and A. H. Gnauck, �??Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,�?? IEEE Photon. Technol. Lett. 15, 320�??322 (2003). [CrossRef]
- C. W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 2002).
- J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
- J. G. Proakis, Digital Communications, 3rd Ed. (McGraw-Hill, New York, 1995).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), result 29.3.81.
- W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1983), Sections 5.5 and 6.7.
- R. Graham, �??Hopf bifurcation with fluctuating control parameter,�?? Phys. Rev. A 25, 3234�??3258 (1982). [CrossRef]
- R. Graham and A. Schenzle, �??Carleman imbedding of multiplicative stochastic processes,�?? Phys. Rev. A 25, 1731�??1754 (1982). [CrossRef]
- I. S. Gradsteyn and I. M. Rhyzhik, Table of Integrals, Series and Products, 5th Ed. (Academic, San Diego, 1994), result 6.633.2.
- P. J.Winzer, S. Chandrasekhar and H. Kim, �??Impact of filtering on RZ-DPSK reception,�?? IEEE Photon. Technol. Lett. 15, 840�??842 (2003) and references therein. [CrossRef]
- J. D. Hoffman, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1992).

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