1. INTRODUCTION

The signal-to-noise ratio in long-haul high-speed optical communication systems can only be increased to a certain level by increasing the power since nonlinear effects will generate pulse distortions that eventually become the limiting factor. This is of great practical importance and has been carefully studied. Of special interest for the present paper is intrachannel cross-phase modulation (XPM) and four-wave mixing (FWM) in return-to-zero on–off keying data, i.e., nonlinear interaction due to the Kerr nonlinearity between pulses belonging to the same channel. It is well-known that this leads to timing jitter, amplitude jitter, and the generation of noise pulses in empty bit slots, referred to as ghost pulses.[[1], [2]] Although the complete dynamics are difficult to calculate analytically from the nonlinear Schrödinger equation (NLSE), different methods of performing approximate analysis have been suggested. Timing jitter has, e.g., been studied using a variational procedure,[[3], [4]] and an analysis based on calculations analogous to traditional FWM is found in Ref. [[5]]. Of greatest interest for the present paper is a perturbation analysis[[6], [7], [8]] based on the assumption that the nonlinear effects are weak, which is true in pseudolinear transmission. The nonlinear interaction is described by a perturbation, which is added to the linearly propagating signal pulses, and this allows many properties of the intrachannel effects to be described. However, in a general pulse train, the perturbation is built up from a large number of coherent interactions between different combinations of signal pulses. This is difficult to describe analytically, and there are few results covering this topic in the literature. The present work partially fills this gap by studying the interaction range and the perturbation growth. The former is defined as the separation, measured in bit slots, between a given bit slot and the most distant bit slot that must be accounted for in order to make the nonlinear signal distortion in the studied bit slot show a converged behavior. The latter is the rate of generation of the nonlinear distortion as a function of propagation distance.

It is emphasized that the aim of the current investigation is not to estimate the nonlinear system penalty. A lot of information about this is found in the literature. Instead, the current analysis should serve as a means of understanding the nonlinear generation process.

The perturbation analysis is presented and commented on in Section 2, and some general conclusions are drawn. The studied system is described in Section 3 and the reasons for the particular choices are made clear. In Section 4, numerical simulations showing the word length dependence are presented and interpreted using approximate analytical results. In Section 5, the generation rate is described using numerical simulations of how the nonlinear perturbation increases its strength along a communication fiber. The work is concluded in Section 6.

2. PERTURBATION ANALYSIS

Consider a transmission system consisting of a number of fibers with given parameters, and a number of lumped, noiseless amplifiers. The pulse propagation is described by the NLSE, and between two amplifiers it has the form

The exact analytical solution can be obtained for a general contribution generated by the term $-\gamma {\psi }_k{\psi }_l{\psi }_m^{*}$, which will be referred to as the $(k,l,m)$ contribution. The power variation due to loss and lumped amplification is expressed in terms of the power ratio $p\left(z\right)$, which is normalized to the initial value, i.e., $p(z=0)=1$. The accumulated dispersion, $B\left(z\right)$, is here defined in dimensionless form as

The exponential function in the integrand of Eq. (5) is step shaped, and, as discussed in Appendix A, using this observation Eq. (5) can be approximated according to

3. SYSTEM SETUP

The total perturbation is the sum of the generated distortion in all fibers of the system. Since the effect from using a periodic system and different fibers is predicted by the perturbation analysis,[[8]] the simulations have been made for a simplified system consisting of a single SMF, and the dispersion is compensated for using a linear lossless dispersion-compensating fiber (DCF). Such a system does not include the effects from using a nonzero RDPS, but in order to obtain a system that can be analytically interpreted in a straightforward manner, this effect has been neglected. The SMF has the following parameters: the bit slot is $25\mathrm{ps}$ corresponding to $40\mathrm{Gbit}∕\mathrm{s}$ and the Gaussian pulse $1∕e$ width is $5\mathrm{ps}$ ($8.3\mathrm{ps}$ FWHM), making $\nu =T_B∕t_0=5$. The pulse amplitude is chosen to make the energy content of a pulse equal to $50\mathrm{fJ}$, which gives $A_0\approx 0.075{\mathrm{W}}^{1∕2}$. The dispersion parameter is $D=16\mathrm{ps}{\mathrm{nm}}^{-1}{\mathrm{km}}^{-7}$ and the nonlinear parameter is $\gamma =2{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$.

In the numerical simulations, the attenuation of the fiber has been set to zero, which eliminates the impact from a specific choice of amplifier positions. If the system is not lossless, the nonlinear interaction is suppressed in the parts carrying little power, and this will limit the interaction range if no amplifiers are present in the regions of highest accumulated dispersion. This important topic is thoroughly discussed in Subsection 4D. In the same way, the generation rate is suppressed when the power decreases. The aim of the current investigation is to make a comparison of the nonlinear interaction in different parts of the fiber, and the assumption of a lossless system is then crucial.

It should be noticed that the signal distortion examined in the current paper is deterministic and evaluated within a single bit slot. The term jitter is therefore avoided, and the changes in pulse location and amplitude are instead referred to as timing and amplitude shifts.

The numerical results have been calculated from Eq. (1) using the split-step Fourier method, and the sampling procedure used depends on whether the bit slot under consideration contains a zero or a one. In the former case, ψ is simply sampled in the middle of the bit slot. In the latter case, the mean pulse position is calculated according to

The nonlinear interaction affecting bit slot zero is investigated in all numerical simulations. This bit slot carries either a zero or a one depending on what effect is currently being examined. To describe the surrounding bit train, the word “symmetric” is used if the bit train has a mirror symmetry in bit slot zero. If the bit train contains no signal pulses in bit slots with negative indices, the bit train is described as “asymmetric.” Furthermore, when the interaction range is investigated the bit train length is indicated by N, which means that bit slots with indices $1\le n\le N$ contain signal pulses. In all simulations, sufficiently many empty bit slots are added to avoid effects from the periodicity.

4. WORD LENGTH DEPENDENCE

The nonlinear interaction is a long-range effect that acts over a significant number of bit slots. Using common system parameters, it was found experimentally in Ref. [[9]] that a pseudorandom bit sequence (PRBS) must contain at least $2^{15}$ pulses before the bit error rate (BER) shows a convergent behavior, which has far-reaching implications for the interpretation of numerical simulation data. The aim of the current investigation was to examine the word length dependence in detail. This is done by calculating the interaction range and using the fact that the transmitted word must be able to account statistically correctly also for the most long-range effects.

4A. Ghost Pulse Generation

Figure 2 shows the ghost pulse amplitude generated in four SMFs of length $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$ as a function of the bit train length N using asymmetric bit trains. No precompensation is used, but it is known that it is not possible to counteract the generation of ghost pulses efficiently with this technique. (This is illustrated in Fig. 9.) The amplitude is low for $N=1$ since a single signal pulse cannot give rise to a ghost pulse. The curves then rise steeply, reaching a maximum and starting to oscillate before they converge to the final value. This clearly shows that many contributions must be taken into account also for the shortest fiber. In the case of the $100\mathrm{km}$ long fiber, range is roughly 25 bit slots. Thus the interaction is even more long range here than indicated by the result in Ref. [[9]], since a PRBS must include the sequence 25 zeroes, 1 zero, 25 ones, and to do this, the length of the PRBS must be $2^{51}$. The reason is that attenuation is included in Ref. [[9]], and a discussion about how to account for this effect is found in Subsection 4D.

The value of N at the first maximum of the different curves is predicted by the perturbation analysis using the fact that when N is increased, the newly added contributions will eventually interfere destructively with the total perturbation. Studying the argument of the first $E_1$ function of Eq. (8) it is realized that a contribution is strong if $\mid kl\mid$ is small, and consequently the strongest contributions have $\mid kl\mid =1$. (For the present purposes, the second $E_1$ function can be approximated with zero since the corresponding value typically falls in the spiral-shaped part of the $E_1$ curve.) The x values (see Fig. 1) to use as arguments for the $E_1$ function are found in the third column of Table 1 , and the $x^{\prime }$ arguments at which a contribution will be $\pi ∕2$ out of phase with the strongest contributions are found in the fourth column. When increasing the sequence of ones from $N-1$ to N, the value of $\mid kl\mid$ for the newly added contributions is minimized by $(1,N-1,N)$. Thus using $x^{\prime }∕x=N-1$, the values in column six are found, and they describe the location of the peaks in Fig. 2 well.

The result for a symmetric pulse train is plotted in Fig. 3 . In this case, with a single zero between long arrays of ones, a larger ghost pulse is obtained. The interaction range is similar to the asymmetric case, but the global maxima occurs at larger values for N.

The analytical interpretation is slightly different in this case, because when a contribution $(k,l,k+l)$ is generated, $(k,-l,k-l)$ is also present. [Contributions of the type $(k,k,2k)$ do not have such a counterpart, but these constitute a small minority among the large number of contributions.] When studying these contributions using Eq. (8), it is noticed that (i) the lower integration limits are different (leading to very small changes) and (ii) the sign of the x argument is changed. This implies that the two contributions have similar strengths and are the mirror images of each other with respect to the imaginary axis. Therefore only the projection on the imaginary axis contributes to the ghost pulse.

Studying the curve for the $200\mathrm{km}$ long fiber, it is seen that it is close to zero for $N=5$, and the reason is that the strongest contributions have a positive imaginary part. As contributions with a negative imaginary part are generated, the ghost pulse decreases its strength due to destructive interference, but eventually it starts to grow again. This effect is not seen for the $50\mathrm{km}$ long fiber because in this case the strongest contributions also have a negative imaginary part.

The global maximum is reached when newly generated contributions again have a positive imaginary part. This occurs for $\mid x\mid \ge 3.38$ (see Fig. 1), and using Eq. (8), it is obtained that

Equation (12) constitutes an analytical way of predicting the word length needed before the obtained amplitude of the ghost pulse starts to show an oscillating, and eventually convergent, behavior. The exact value of the interaction range depends on what one considers to be a converged result. As is seen in, e.g., Figs. 3 and 6, it is necessary to double the value for N at the peak before the amplitude is within $\pm 10\%$ of the converged value. It should also be noticed that a small error in the interaction range estimate leads to a large error in the predicted word length due to their exponential relation. These uncertainties imply that these analytical results are guidelines, and it could prove useful to reproduce, e.g., Fig. 3 when carrying out system simulations with an intended high degree of accuracy. Such a simulation is not computationally demanding since the length of the bit train is moderate and could show that the nonlinear effects are properly accounted for.

The fact that the argument of the first (but not the second) $E_1$ function of Eq. (8) has no dependence on $t_0$ (since $B_{\min }$ and ${\nu }^2$ depend on $t_0$ in identical ways) implies that the perturbation analysis predicts the interaction range to be very weakly affected by a pulse width change. This has been examined numerically by reproducing Fig. 3 with different values of $t_0$. The amplitude of the perturbation is then changed, but the interaction range is almost constant. The estimate based on pulse overlap in Ref. [[9]] predicts a much stronger dependence. Although this argument is intuitive, the interaction range is not simply determined by the number of neighboring pulses that overlap during propagation. Instead, one must study how the strength of the contributions fall off with increasing signal pulse separation, and how different contributions interfere.

4B. Amplitude Shifts

Next we study amplitude shifts using the definition given in Section 3. Figure 4 shows the resulting total amplitude from a simulation using asymmetric pulse trains.

In the case of amplitude and timing shifts, the following facts must be taken into account. (i) The signal pulse is affected by self-phase modulation (SPM), (ii) there are contributions of the type $(0,k,k)$ and $(k,0,k)$, i.e., XPM, (iii) there are contributions of the type $(k,-k,0)$, and (iv) the contributions interfere with the signal pulse.

The impact from SPM is identical in all cases and the different effects (for $N>0$) seen in Fig. 4 are therefore not caused by SPM. XPM does not cause any significant amplitude shift, and this has been confirmed by two observations. The case $N=1$ in Fig. 4 corresponds to one single contribution caused by XPM, and it is seen that the amplitude shift is small. Second, by phase shifting every second bit slot [using the alternate-phase return-to-zero (RZ) modulation format[[10]]] it is seen that the amplitude shift in Fig. 4 oscillates as N increases. Since the XPM contributions are not affected by phase shifting, it is again concluded that the generating effect is not XPM. Using an asymmetric pulse train, no contributions of the type $(k,-k,0)$ are generated.

The first maximum of the different curves in Fig. 4 occur at similar values for N as in Fig. 2, which should be expected since the perturbation is added to the signal pulse, and at the peaks of Fig. 2, the perturbation is starting to decrease its amplitude.

The corresponding result for a symmetric pulse train has been plotted in Fig. 5 . It is seen that the worst case occurs for a short sequence of ones and compared with the asymmetric bit train, the amount of amplitude shift is small for long bit trains. It has been mentioned above that for a symmetric pulse train, pairs of contributions add up in such a way that only the projection on the imaginary axis is important. This implies that there will be a $\pi ∕2$ phase difference between the signal pulse and the perturbation, leading to small amplitude changes. This explains the low amplitude shift for large N, but this argument cannot be used for the short sequences of ones where the worst case occurs. As an example, the two contributions $(k,k,2k)$ and $(k,-k,0)$ are not both present if bit slot $2k$ is empty. Thus studying contributions of the types $(k,-k,0)$, it is obtained that $kl=-k^2<0$, and the curve in Fig. 1 should be mirrored in the imaginary axis. It is seen that the real part of the $(k,-k,0)$ contributions change signs when

4C. Timing Shifts

Timing shifts are generated when an asymmetric pulse train is used, and the result of a numerical simulation is seen in Fig. 6 . The interaction range is similar to what was found above, but the timing shift is monotonically increasing. The reason for this is that XPM, which is known to cause timing shifts,[[4]] gives rise to contributions of the type $(0,k,k)$ [and $(k,0,k)$], which have $kl=0$. Thus, the argument is fixed for the first $E_1$ function of Eq. (8), and the second argument first falls in the spiral-shaped part and then follows the curve as k increases. This shows that all contributions have approximately the same phase, which leads to monotonic growth.

When comparing the amplitude and the timing shifts, it is useful to notice that the maximum amplitude shift is roughly 10% of the initial amplitude. The amount of timing shift needed to give rise to a similar amplitude difference is approximately $t_0∕2$ (or $T_B∕10$), and one concludes that the two effects can be of similar importance.

4D. Effect of Attenuation

The interaction range found above is very long and the reason is that by neglecting the attenuation, the interaction is considerable also in the regions of high accumulated dispersion. However, it is emphasized that these long-range effects should be possible to see experimentally by placing an amplifier at a position where $\mid B\left(z\right)\mid$ is large, but if this is not the case, the loss will suppress the interaction range.

As seen from Eq. (6), the power ratio $p\left(z\right)$ acts as a weighting function for the nonlinear interaction. Thus to predict the needed word length for a real system, the value for B in Eq. (12) must be chosen within the regions of the system that carries significant power.

The system used to produce Fig. 5 of Ref. [[9]] serves as a good example system. The dispersion of the SMF is $17\mathrm{ps}{\mathrm{nm}}^{-1}{\mathrm{km}}^{-1}$, and when using the RZ modulation format, a 33% duty cycle is used, which implies that the FWHM width $(25\times 0.33\approx 8.3\mathrm{ps})$ is the same as used here. Thus the values for ν and B are similar. The attenuation is $0.2\mathrm{dB}∕\mathrm{km}$. Precompensation with $-950\mathrm{ps}∕\mathrm{nm}$, corresponding to $56\mathrm{km}$ of fiber propagation, has been used and since this point of the system contains the maximum pulse power, the value for B is taken there. This value coincides rather well with one of the chosen fiber lengths of Fig. 3, and for this fiber, the value $N=7$ was found as a prediction of the peak position. Thus the PRBS length should be approximately $2^{15}$ $(7+1+7)$ long before convergence can be expected. This is in good agreement with the results found in Ref. [[9]].

As a further example of interaction range prediction, the same system without precompensation could be treated. It is then necessary to decide the level of approximation that is acceptable. By defining “significant power” to mean more than 10% of the initial power, the power becomes too low after $50\mathrm{km}$ using the current parameters. This effective fiber length is again close to the case discussed above, leading to similar predictions for the interaction range. This result has been confirmed by including $0.2\mathrm{dB}∕\mathrm{km}$ attenuation and repeating the calculation leading to Fig. 3. All four curves then converge to their final values at a similar value for N that was obtained earlier using the $50\mathrm{km}$ long fiber, i.e., the peak has been passed when $N=10$, and the following oscillations are suppressed due to the loss.

5. PERTURBATION GENERATION RATE

As seen in Eq. (10), the asymptotic behavior of the function $E_1\left(ix\right)$ is logarithmic when $x\to 0$, which has led to the suggestion that the growth of the nonlinear perturbation can be made small by designing the dispersion map in such a way that the pulses are strongly overlapping everywhere except close to the transmitter and receiver.[[11]] This is, however, not true in a long pulse train, since many contributions are generated, which tend to align their phases (as they move from the spiral-shaped part of $E_1$ toward the logarithmic part during propagation). This implies a faster growth than expected from Ref. [[11]].

Figure 7 shows a numerical simulation for the growth of the amplitude of a ghost pulse along an SMF using symmetric bit trains and the parameters given in Section 3. The initial field has been propagated according to Eq. (1), at chosen positions the dispersion has been compensated for using a linear lossless DCF, and the resulting field has been sampled. In this way, the perturbation amplitude as a function of propagation distance is obtained.

The curve ending at 0.33 corresponds to a bit train consisting of a single zero between ones. It is a somewhat unexpected result that the growth rate of the ghost pulse is the same along the fiber except for the initial part. The threshold value, $z_{\mathrm{th}}$, where the linear growth starts, corresponds to the fiber length where the strongest contribution has left the spiral-shaped part of $E_1$. Using that the strongest contribution has $\mid kl\mid =1$ [occurring, e.g., for (1,1,2)], the threshold occurs when the argument in Eq. (8) is

It is well known that the strongest ghost pulses tend to appear between long sequences of ones,[[2], [9]] but as seen from Fig. 3, this is actually not the worst case. By using a symmetric pulse train with $N=24$ (i.e., the pulse train consists of 24 ones, 1 zero, 24 ones, and many zeroes), the curve ending close to 0.40 in Fig. 7 is obtained. This corresponds to selecting the strongest contributions that interfere constructively. An even stronger ghost pulse can be produced by including a sequence of ones located further away, which again create contributions that are in phase with the total perturbation. This results in the curve in Fig. 7 ending close to 0.45, i.e., the amplitude is more than 30% larger than in the case of an infinite train of ones.

The difference between the ghost pulse amplitudes at $L_{\mathrm{SMF}}=200\mathrm{km}$ in Fig. 7 is caused by the different generation rates between 100 and $200\mathrm{km}$ of propagation. If the power is low in this region due to attenuation, the generation rates will be close to zero, and the three curves will follow each other. In general, if the amplifiers are placed at points where $\mid B\left(z\right)\mid$ is small, the impact seen in Fig. 7 from pulses located many bit slots away will be suppressed due to the attenuation. In such a case, a single zero between a long sequence of ones is a good indication of the worst case.

The effect from changing the system parameters has been investigated with similar numerical simulations. The effect from a pulse width change is directly predicted by Eq. (8) since the argument is not changed due to the fact that ${\nu }^2$ and B have the same dependence on $t_0$. With the condition that the pulse energy $W=\sqrt{\pi }{A}_0^2{t}_0$ should be constant, it is found that $\mid {\psi }_p\mid \propto \sqrt{t_0}$, which has been confirmed by numerical simulations.

Changing the dispersion parameter does not have a significant impact on the ghost pulse generation. This is illustrated in Fig. 8 where the ghost pulse for a bit train with a single zero between ones has been plotted for a dispersion parameter that is $\frac{1}{4}$, $\frac{1}{2}$, 1, 2, and 4 times the value in Fig. 7. It is seen that the growth rate is similar in all cases but in the case of a small dispersion parameter the start of the strong interaction is delayed to a later position in the fiber. The threshold values are predicted by Eq. (15) to be $z_{\mathrm{th}}=3.8$, 7.7, 15.3, 30.6, and $61.3\mathrm{km}$.

The fact that the perturbation grows linearly has implications for the possibilities of counteracting the ghost pulse generation, since, with exception for the initial region, it should be of little consequence whether the pulses are strongly or weakly dispersed during propagation (without leaving the pseudolinear regime) or if the locations of the lumped amplifiers are changed. Precompensation of the pulse train is also of little use for counteracting the ghost pulses, which is seen in Fig. 9 , where the SMF is preceded by a linear lossless DCF that compensates for $0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$, and 1 times the length of the SMF. (The remaining accumulated dispersion is compensated for at the end of the system.) It is seen that the difference in the final perturbation amplitude is because the precompensation increases the propagation in the region of weak interaction by a factor of 2.

In Fig. 10 , the amplitude shift for the center pulse using a symmetric bit train with $N=0,1,2,\dots ,6$ is plotted. The effect from the SPM is seen (using the $N=0$ curve) to have saturated after approximately $10\mathrm{km}$. As discussed in connection with Fig. 5, the worst case after $200\mathrm{km}$ corresponds to the case $N=3$, i.e., a signal pulse surrounded by three signal pulses on each side. All curves have qualitatively similar shapes, and the worst case, which depends on the fiber length, can be predicted using Eq. (14).

The case generating the most timing shift is an infinite asymmetric pulse train. In this case, a plot of $⟨t⟩$ as a function of z is almost linear and is included in Fig. 12. The significant interaction starts when the pulses are partially overlapping,[[4]] which they are after approximately $3\mathrm{km}$ with the present parameters.

In Figs. 11, 12 the effect from precompensation on the amplitude and timing shift generation is illustrated using a numerical simulation analogous to that in Fig. 9. The worst cases are considered for both effects, i.e., the amplitude shift is calculated using a symmetric bit train with $N=3$, and the timing shift is obtained from a long asymmetric bit train. It is known that both effects can be counteracted by proper precompensation,[[12]] but it is seen that the optimal amount of precompensation is different in the two cases. In a lossless system, the timing shift is almost completely canceled in a symmetric dispersion map, i.e., when the dispersion compensation is equally divided between the start and the end of the system. However, also when attenuation is included, it is possible to compensate for the timing shift by choosing the precompensation properly.[[4], [13], [14]]

6. CONCLUSION

Intrachannel XPM and FWM have been studied analytically and numerically. A lossless system has been used to find the interaction range without suppressive effects from the attenuation, and analytical estimates for the needed word length have been obtained. A method to account for the losses when predicting the interaction range has been given. The results, which show that the nonlinear interaction is long range, are in agreement with the experimental data from Ref. [[9]]. Using typical system parameters, it has been found that using a PRBS of length $2^{10}$ is not sufficient, and it can be necessary to use a PRBS of length up to $2^{20}$ before the resulting BER is fully converged. If an amplifier is placed in a region of high accumulated dispersion, the interaction range can be increased considerably, causing a need for a significantly longer word.

The generation rate of the nonlinear distortion has been compared at different values of the accumulated dispersion, and it has been found that the strongest ghost pulse grows almost linearly. This implies that it is hard to counteract the nonlinear distortion by designing the dispersion map and the locations of the lumped amplifiers.

Appendix A

To prepare for the approximation that leads from Eq. (5) to Eq. (6), it is convenient to split the exponential function in the integrand of Eq. (5) according to $e^z=e^{\mathrm{Re}z}{e}^{i\mathrm{Im}z}$, with the result

(A1)$$\exp \{-\frac{{\nu }^2}{2}\frac{1}{1+B^2}[{(k+l)}^2\frac{1-3B^2}{1+9B^2}+k^2+l^2]\}\times \exp \left\{i\frac{{\nu }^2}{2}\frac{B}{1+B^2}[{(k+l)}^2\frac{5+9B^2}{1+9B^2}-(k^2+l^2)]\right\}.$$

The integrand of Eq. (5) then consists of a product of three factors, and the observation that the second factor is step shaped allows significant simplifications. This is seen by examining the two limits $\mid B\mid \ll 1$ and $\mid B\mid \gg 1$. In the first case, it is approximately equal to $\exp [-{\nu }^2(k^2+l^2+kl\left)\right]$, which is a very small number. In the second case, the approximate result is $\exp [-{\nu }^2(k^2+l^2-kl)∕(3B^2\left)\right]$, which rapidly approaches unity as $B^2$ increases. A reasonable threshold value for B is found by setting

(A2)$$\exp \{-\frac{{\nu }^2}{2}\frac{1}{1+B^2}[{(k+l)}^2\frac{1-3B^2}{1+9B^2}+k^2+l^2]\}=e^{-1}.$$

This quadratic equation for $B^2$ can be solved exactly, but the solution is somewhat complicated. A good approximation is instead obtained by assuming that $B^2\gg 1$, which implies that

(A3)$$\frac{1-3B^2}{1+9B^2}\approx -\frac{1}{3}.$$

This yields the result stated in Eq. (7), and the step-shaped function is approximated by unity in the regions where $B^2>B_{\mathrm{th}}^2$ and by zero elsewhere.

It is possible to increase the accuracy of the approximate integration by using a better approximation than a simple step. However, to keep the expressions manageable it seems reasonable to accept this choice as sufficient. As will be seen, the other factors in the integrand can be very well approximated in the regions where $B^2>B_{\mathrm{th}}^2$, which means that the main error in the approximate analytical result is the amplitude error close to the step. (Typically the amplitude is too high in the approximation.)

It should be noticed that it is not possible to set $k=l=m=0$ in Eq. (7), i.e., this type of approximation cannot be used when studying SPM contributions. (There is no threshold value for B for SPM contributions, since there is temporal overlap from the start.) This is a minor problem since an alternative approximation is easy to obtain because the exponential function in this case is much simpler; an approximation of the type in Eq. (A3) yields a solvable integral.

The factor in the integrand of Eq. (5) involving the square root is approximated according to

(A4)$$\frac{1}{\sqrt{1+2iB+3B^2}}\approx \frac{1}{\sqrt{3B^2}}(1-\frac{i}{3B})\approx \frac{1}{\sqrt{3B^2}}{e}^{[-i∕\left(3B\right)]},$$

and the exponent in the last factor is approximated according to

(A5)$$e^{\dots }\approx \exp \left\{i\frac{{\nu }^2}{2B}[{(k+l)}^2-(k^2+l^2)]\right\}=e^{i{\nu }^{2}kl∕B}.$$

Inserting the resulting expressions into Eq. (5), one immediately obtains Eq. (6). No further approximations are needed to obtain the final result, Eq. (8), in a lossless situation.

ACKNOWLEDGMENTS

The author thanks Magnus Karlsson of the Department of Microtechnology and Nanoscience, Photonics Laboratory, Chalmers University of Technology and Ulf Österberg of the Thayer School of Engineering, Dartmouth College, for helpful discussions.

P. Johannisson’s e-mail address is elfpj@chalmers.se.

Table 1. Analytically Calculated Values for N at the First Maximum Using Different Fiber Lengths

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Fig. 1 Function $i{E}_1\left(ix\right)$ in the complex plane. The value of the positive real argument x has been indicated at chosen points on the curve.

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Fig. 2 Ghost pulse amplitude using different asymmetric pulse trains with SMF of length (from below) $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$.

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Fig. 3 Ghost pulse amplitude using different symmetric pulse trains with SMF of length (from below) $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$.

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Fig. 4 Amplitude shift using different asymmetric pulse trains with SMF of length (from below) $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$.

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Fig. 5 Amplitude shift using different symmetric pulse trains with SMF of length (from above at the global minima) $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$.

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Fig. 6 Timing shift using different asymmetric pulse trains with SMF of length (from above) $L_{\mathrm{SMF}}=50$, 100, 150, and $200\mathrm{km}$.

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Fig. 7 Ghost pulse amplitude as a function of propagation distance. The bit trains used to generate the different curves are described in the text.

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Fig. 8 Ghost pulse amplitude as a function of propagation distance using (from below) a group-velocity dispersion parameter ${\beta }_2∕4$, ${\beta }_2∕2$, ${\beta }_2$, $2{\beta }_2$, and $4{\beta }_2$.

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Fig. 9 Ghost pulse amplitude as a function of propagation distance using different amounts of precompensation. Counting the horizontal parts of the curves from below the initial DCF compensates for $0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$, and 1 times the length of the SMF.

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Fig. 10 Amplitude shift as a function of propagation distance using symmetric bit trains with (from above at the global maxima) $N=6$, 5, 4,3, 2, 1, 0.

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Fig. 11 Amplitude shift as a function of propagation distance using (from below at $L=200\mathrm{km}$) an initial DCF that compensates for $0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$, and 1 times the length of the SMF.

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Fig. 12 Timing shift as a function of propagation distance using (from below at $L=200\mathrm{km}$) an initial DCF that compensates for $0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$, and 1 times the length of the SMF.

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