We show that the minimum EOP (eyeopening penalty) obtained by tunable dispersion compensation is a function of a figure of merit for a nonlinear process, I0Leff, where I0 is the optical intensity and L
_{eff}
is the effective length of the interaction region. Using this rule, we do not need to conduct nonlinear simulations in all the cases of signal power and transmission length to obtain the signal distortion in dispersionmanaged optical transmission. Instead, we need to conduct a simulation in only one case of a signal power and find the functional relation, and then we can obtain the values of the signal distortion in other cases using the discovered functional relation. This technique can reduce the number of nonlinear simulations to less than 10%.
I. INTRODUCTION
Nonlinear effects in optical fibers such as selfphase modulation (SPM) have been considered as a major impairment in the dispersionmanaged longhaul optical fiber transmission systems. There has been much work in attempting to understand the nonlinear effects as a function of system parameters such as optical power per channel, transmission distance, and chromatic dispersion
[1

5]
. In particular, there have been a lot of efforts to find the optimal dispersioncompensation ratio in nonlinear optical transmission
[6

9]
.
To investigate the nonlinear effects as a function of the system parameters, we usually conduct numerical simulations because it is not easy to change all the system parameters freely in a longhaul transmission experiment. However, nonlinear simulation also takes a lot of time if we want to conduct for all the possible cases. Therefore, if we find a way to reduce the nonlinear simulation cases, it will be very helpful for doing research on nonlinear effects.
In this paper, we report that the signal distortion in dispersionmanaged optical transmission systems can be expressed by using a figure of merit for the efficiency of a nonlinear process
[10]
, I
_{0}
L
_{eff}
, where I
_{0}
is the optical intensity and L
_{eff}
is the effective length of the interaction region. In other words, the signal distortion at the optimal dispersion compensation is only a function of I
_{0}
L
_{eff}
although the optical power per channel or transmission length is changed. Even though we have found this phenomenon from nonlinear simulation results, we will also show an analytical derivation that the signal distortion at the optimal dispersion compensation is a function of I
_{0}
L
_{eff}
in the case of a Gaussian pulse.
II. SIGNAL DISTORTION IN DISPERSIONMANAGED TRANSMISSION
It is well known that signal distortion is minimized when dispersion of the transmission fiber is exactly compensated in the linear optical transmission. However, in the nonlinear transmission case, the optimal value of dispersion compensation is changed due to the effect of SPM
[11]
.
Fig. 1
shows an example of eyeopening penalty (EOP), a measure
Eyeopening penalty (EOP) due to dispersion and SPM in 40Gb/s transmission over 720 km of nonzero dispersionshifted fiber (NZDSF, D = 4 ps/nm/km) with a launch power of  4dBm. Solid line is the simulation result without considering the nonlinear effect and dotted line is with considering the nonlinear effect. The definition of the minimum EOP is also illustrated.
The minimum EOP with the optimal dispersion compensation as a function of signal power and transmission length in the dispersionmanaged 40Gb/s nonlinear transmission. In this simulation, the use of NZDSF (D = 4 ps/nm/km, loss = 0.23 dB/km) and Raman amplifier with 80km span is assumed.
of signal distortion, as a function of residual dispersion in the linear and nonlinear transmission cases. In
Fig. 1
, the solid line is the simulation result without considering the nonlinear effect and the dotted line is with consideration of the nonlinear effect. In this paper, we define
the minimum EOP
as the EOP at the optimal dispersion compensation. In other words, the minimum EOP is the minimal EOP which can be obtained by using tunable dispersion compensation. And, note that the optimal dispersioncompensation ratio is different for each nonlinear transmission case.
Fig. 2
shows a contour graph of
the minimum EOP
at the optimal dispersion compensation as a function of signal power and transmission length in a dispersionmanaged nonlinear transmission. It can be recognized that the minimum
A figure of merit for the efficiency of a nonlinear process, I_{0}L_{eff}, as a function of signal power and transmission length in the same condition as Fig. 2. I_{0} is the optical intensity and L_{eff} is the effective length of the interaction region.
EOP increases when the signal power or transmission length increases. In the simulation, the use of NZDSF (nonzero dispersionshifted fiber, D = 4 ps/ nm/km, loss = 0.23 dB/km) and Raman amplifier with 80km span is assumed. Because we need to conduct nonlinear simulations in many different cases of dispersion compensation to find the optimal dispersion compensation at each point, it takes lots of time to obtain the contour graph of the minimum EOP, such as
Fig. 2
. For example, the simulation results of
Fig. 2
were obtained through more than 6000 nonlinear simulations.
In
Fig. 3
, we illustrate a contour graph of a figure of merit for the efficiency of a nonlinear process, I
_{0}
L
_{eff}
, as a function of signal power and transmission length in the same condition as
Fig. 2
. Surprisingly, the contour graphs of
Fig. 2
and
Fig. 3
look similar, which makes us realize that there can be a strong relationship between
the minimum EOP
and a figure of merit for the efficiency of a nonlinear process, I
_{0}
L
_{eff}
. From here in this paper, we will shorten the term, a figure of merit for the efficiency of a nonlinear process, to
a figure of merit for a nonlinear process
III. FUNCTIONAL RELATION BETWEEN THE MINIMUM EOP AND A FIGURE OF MERIT FOR A NONLINEAR PROCESS
We investigated the relation between the minimum EOP and a figure of merit for a nonlinear process, I
_{0}
L
_{eff}
.
Fig. 4
shows the xy relation between the minimum EOP and a figure of merit for a nonlinear process by using all the simulation results of
Fig. 2
and
3
. From
Fig. 4
, we can say that the minimum EOP is a function of a figure of merit for a nonlinear process. In other words, although signal power or transmission length is changed, the minimum EOPs will be the same if a figure of merit for a nonlinear
The functional relation between the minimum EOP and a figure of merit for a nonlinear process, I_{0}L_{eff}, using all the simulation results of Fig. 2 and 3.
The simulation result of only one case of a signal power, 1 dBm, in the same condition as Fig. 2. The solid line is the 2^{nd} order polynomial fitting graph.
process, I
_{0}
L
_{eff}
, is the same. For example, the minimum EOP at 480 km dispersionmanaged transmission with a signal power of 0 dBm (= 1.05 dB) is almost the same as the minimum EOP at 1520 km dispersionmanaged transmission with a signal power of  5 dBm (= 1.01 dB) because the values of I
_{0}
L
_{eff}
for these two cases are almost the same to a value of 0.175. Some variation points in the functional relation in
Fig. 4
seem to be caused by the discrete value of simulation parameters for dispersion compensation.
Therefore, we do not need to conduct nonlinear simulations in all the cases of signal power and transmission length to obtain the signal distortion in the dispersionmanaged optical transmission. We need to conduct a simulation in only one case of a signal power, and then we can obtain the values of signal distortion in other cases using the functional relation shown in
Fig. 4
.
IV. USE OF THE FUNCTIONAL RELATION
In this section, we will show an example of how to use the functional relation.
Fig. 5
shows the simulation result in only one case of a signal power,  1dBm, in the same
Calculated EOPs using the functional relation between EOP and a figure of merit for a nonlinear process, I_{0}L_{eff}, in the same condition as Fig. 2.
condition of
Fig. 2
. Using the simulation results in
Fig. 5
, we can make a functional for I
_{0}
L
_{eff}
(x) and EOP (y). If we use the 2
^{nd}
order polynomial fit, we can obtain a fitting function, y = Ax + Bx + C, with the coefficients of A = 9.87, B = 4.07, and C = 0.01. The fitting function is illustrated as a solid line in
Fig. 5
.
Using the fitting function and the data of a figure of merit for a nonlinear process in
Fig. 3
, we can calculate EOPs in other cases of signal power and transmission length, and plot a contour graph of them, as shown in
Fig. 6
. The calculated EOPs in
Fig. 6
are very similar to the EOPs obtained by a lot of nonlinear simulations in
Fig. 2
.
Therefore, we can obtain the signal distortion in the dispersionmanaged optical transmission system by conducting nonlinear simulations in only one case of a signal power. We do not need to conduct nonlinear simulations in all the cases of signal power and transmission length. Following the procedure of this section, anyone can use the proposed technique easily. In this example, the proposed technique reduces the number of nonlinear simulations to 5% of
Fig. 2
.
V. FUNCTIONAL RELATIONS IN OTHER SYSTEM CONDITIONS
In this section, we investigate whether the functional relation between the minimum EOP and a figure of merit for a nonlinear process can be applied to other simulation conditions.
Fig. 7
shows the relations between the minimum EOP and a figure of merit for a nonlinear process, I
_{0}
L
_{eff}
, in other simulation conditions. In the simulation of
Fig. 7
, we conducted several nonlinear simulations by changing one of system parameters: (1) the same condition as
Fig. 4
(Raman amplifier with 80km span, NZDSF (D = 4 ps/nm/km, 0.23 dB loss)), (2) 60km amplifier span, (3) 100km amplifier span, (4) 0.2 dB fiber loss, (5) 0.3 dB
The functional relations between the minimum EOP and a figure of merit for a nonlinear process, I0Leff, in other simulation conditions. (1) The same condition as Fig. 4 (Raman amplifier with 80km span, NZDSF (D = 4 ps/nm/km, 0.23 dB loss)), (2) 60km amplifier span, (3) 100km amplifier span, (4) 0.2 dB fiber loss, (5) 0.3 dB fiber loss, (6) SMF (D = 17 ps/nm/km), and (7) EDFA.
fiber loss, (6) SMF (single mode fiber, D = 17 ps/nm/km), and (7) using EDFA (erbiumdoped fiber amplifier) instead of Raman amplifier.
From
Fig. 7
, we recognize that although the function between the minimum EOP and a figure of merit for a nonlinear process is changed depending on the system parameters, the functional relation itself still remains between the minimum EOP and a figure of merit for a nonlinear process. Therefore, in one nonlinear simulation condition, we can obtain the function for the minimum EOP and a figure of merit for a nonlinear process, and use it to reduce the nonlinear simulations. Even though we do not show all the simulation results we have conducted due to the lack of space, we confirmed the functional relation between the minimum EOP and a figure of merit for a nonlinear process in lots of different simulation conditions besides
Fig. 7
.
VI. ANALYTICAL DERIVATION FOR THE FUNCTIONAL RELATION
Until this section, we have shown that the minimum EOP is a function of a figure of merit for a nonlinear process through our simulation results. In this section, we will derive it analytically for the case of a Gaussian pulse. From the earlier work, the analytic expression for the pulse broadening induced by dispersion is given by
[12]
where U(z,
T
) is the normalized amplitude of the pulse, Ũ(z,ω) is the Fourier transform of U(z,
T
), Ũ(0,ω) is the Fourier transform of the incident field at z = 0, and β
_{2}
is the GVD (groupvelocity dispersion) parameter. And, from the earlier work, the analytic expression for the pulse broadening induced by SPM is given by
[13]
where,
where P
_{0}
is the optical peak power, L
_{eff}
is the effective length, ω
_{0}
is the carrier frequency, and A
_{eff}
is the effective mode area. Therefore, the analytic expression for the pulse broadening when dispersion and SPM coexist can be written as
[14]
Consider the case of a Gaussian pulse for which the incident field is of the form
where
σ
_{0}
is the halfwidth of the pulse. When this Gaussian pulse, U(0,
T
), is incident at the input end of a fiber of length L, the broadening factor can be solved by Eq. (5) and is given by
[13]
where
D
represents the normalized distance for dispersion, or it can be considered as a measure of accumulated dispersion. L
_{D}
is the dispersion length.
Now, consider the case of dispersion compensation. By using Eq. (7), the broadening factor of a Gaussian pulse with dispersion compensation can be written as
where D
_{res}
represents the residual dispersion after dispersion compensation. Because the broadening factor is also a measure of signal distortion and is proportional to EOP, we express that “EOP ≡ broadening factor” in Eq. (9). To find the optimum dispersion compensation, we can take the partial derivative of the pulse broadening of Eq. (9) with respect to D
_{res}
and obtain the following equation:
From Eq. (10), we can obtain the value of optimum dispersion compensation:
From Eq. (11), we can recognize that the optimum value of dispersion compensation, D
_{opt}
, is only a function of φ
_{max.}
Now we substitute D
_{opt}
back into Eq. (9) to obtain the signal distortion at the optimum dispersion compensation:
where
Therefore, from Eq. (12), we can say that the pulse broadening, i.e. signal distortion, at the optimum dispersion compensation is a function of a figure of merit for a nonlinear process, I
_{0}
L
_{eff}
.
VII. THE RELATION IN THE CASE OF NONOPTIMUM DISPERSION COMPENSATION
Until this section, we have shown that there exists a functional relation between the minimum EOP and a figure of merit for a nonlinear process when the dispersion is optimally compensated. However, in this section, we investigate the relation between the minimum EOP and a figure of merit for a nonlinear process when the dispersion is not optimally compensated.
Fig. 8
shows the relations between the minimum EOP and a figure of merit for a nonlinear process, I
_{0}
L
_{eff}
, in the case of nonoptimum dispersion compensation. In
Fig. 8
, we show the results in the cases of 99% and
The relations between the minimum EOP and a figure of merit for a nonlinear process, I_{0}L_{eff}, when the dispersion is not optimally compensated. The solid circles are the cases of 100% dispersion compensation. The hollow diamonds are the cases of 99% dispersion compensation.
100% dispersion compensation. The system condition of
Fig. 8
is identical to
Fig. 2
and
Fig. 4
.
From the result of the 99% dispersion compensation case, we can recognize that the functional relation between the minimum EOP and a figure of merit for a nonlinear process does not remain when the dispersion is not optimally compensated. However, when the dispersion is compensated by 100%, the functional relation still remains. We think that these results are somewhat obvious because only a nonlinear effect exists after 100% dispersion compensation. Thus the minimum EOP is also a function of a figure of merit for a nonlinear process in the case of 100% dispersion compensation, but not in other cases. Therefore, we can conclude that the functional relation only remains in the cases of the optimum and 100% dispersion compensation. In other words, we can say that the functional relation only remains in the cases when the nonlinear effect is dominant.
VIII. CONCLUSION
From our nonlinear simulation results for several cases, we showed that
the minimum EOP
obtained by tunable dispersion compensation is a function of a figure of merit for a nonlinear process, I
_{0}
L
_{eff}
, where I0 is the optical intensity and L
_{eff}
is the effective length of the interaction region. We also showed an analytic derivation of the functional relation for the case of a Gaussian pulse.
Using this rule, we do not need to conduct nonlinear simulations in all the cases of signal power and transmission length to obtain the signal distortion in dispersionmanaged optical transmission. Instead, we only need to conduct a simulation for one case of signal power and find the functional relation between the minimum EOP and a figure of merit for a nonlinear process, and then we can obtain the signal distortions in other cases using the discovered functional relation. This technique can reduce the number of nonlinear simulations to less than 10%. This technique will reduce the time of nonlinear simulations in investigating, optimizing, or designing the nonlinear optical transmission systems.
Acknowledgements
This research was supported by Kyungsung University Research Grants in 2012.
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