## Abstract

A two-channel long-haul soliton propagation system is demonstrated. Extensive variations of system parameters were numerically studied to provide the optimized system reported. In this system each soliton carries 80 Gb/s pseudo-random bit signal over 14,000 km. Key system parameters, such as the input peak power, the non linear coefficient, average dispersion and the number of sections of fiber in one amplifier length, are optimized, while the Q factor is monitored. As the Q factor falls below the threshold of 6.0 (corresponding to a BER of one error in a billion bits) the propagation distance is noted. For this optimized system a Q-factor threshold distance of 12700 km was obtained. This yields a bit-rate-distance (BL) product in excess of 2 ∙ 10^{15} b/s km.

©2007 Optical Society of America

## 1. Introduction

Dispersion managed solitons (DMS) are finding increased applicability in ultra-large-capacity (BL > 1000 Tb/s km) fiber optic transmission systems [1-4]. Simulated and experimental studies of single channel long-haul optical systems have been extensively reported [5-8]. At these higher bit rates, dense dispersion management (DDM), in which the dispersion map consists of multiple periods, is often needed. However, high data-rate DDM soliton systems are adversely affected by nonlinear interpulse interaction and Gordon-Haus timing jitter [9] thus limiting the transmission distance. Polarization multiplexing is used to reduce the nonlinear interaction between soliton pulses. Optical band-pass filtering reduces the effect of timing jitter [10].

This study combines the above techniques for two-channel dense dispersion-managed soliton (DDMS) systems. Extensive variations of system parameters were studied to produce the reported BL product. Here we present the results of numerical studies for a fiber transmission system consisting of two DDM solitons waves. Each soliton carries 80-Gb/s pseudo random digital signal. The two signals are wavelength-division multiplexed (WDM). However to provide additional isolation the two soliton waves are orthogonally polarized as well. This form of polarization-mode dispersion (PMD) resulted in a four-to-one improvement in BL product than without PMD. The numerical studies are based on a comprehensive simulation package [11] that takes in to account Kerr nonlinearity, amplified spontaneous emission (ASE), and second- and third-order dispersion (GVD and TOD).

## 2. Theory of operation

For 80-Gb/s operation and a soliton separation parameter q = 8, the full width at half maxima (FWHM) of a pulse is *T _{FWHM}* = 1.375 ps. For such a short pulse, higher-order-nonlinearity and dispersive effects must be taken into account. The generalized nonlinear Schrödinger equation (NLS) [12] governs this system:

for each soliton, where

and where δ_{3}, s and *τ _{R}* account for TOD, soliton self steeping and Raman Effect, respectively [12]. Raman Effect also leads to the soliton self frequency shift; however, the effect of soliton self steeping can be safely neglected, since it becomes important only for the pulse < 50 fs and is a small perturbation for the pulses around 1 ps [13]. (See the Glossary of acronyms).

A necessary condition for solitons to survive is that the dispersion length *L _{D}* exceed the amplifier length

*L*. At 80-Gb/s it is difficult to satisfy this condition because TOD normally comes into play for such short pulses. However, this issue was resolved with judicious dispersion management design. Dispersion management lowers the average GVD of entire link while keeping the GVD of each section high enough so that the effect of TOD is rendered negligible. Agrawal address these issues with the following statement: “At 80-Gb/s, a soliton operates in the quasi adiabatic regime in which considerable dynamic evolution of the soliton occurs over one amplifier spacing. The fate of the soliton in such a regime strongly depends on the loss per dispersion length

_{A}*L*. In this regime the soliton can adapt to losses adiabatically by increasing its width and decreasing its peak power while preserving its soliton nature.” [13].

_{D}Numerous numerical experiments were made for dual channel soliton systems. In these experiments the frequency separation is 0.08%. Consequently, it is assumed that amplifier gain and noise-figure values for each of the channels are the same. Details of the experimental setups and their respective system performances are in Fig. 1.

## 3. Dual-soliton experiments

The system diagram in Fig. 1 for the two-soliton experiments is comprised of two optical channels with the center frequency of 193.026 THz and 193.174 THz. Even though the signal frequencies are separated by 148 GHz, corresponding to a wavelength of Δ*λ* = 1.19 nm for WDM, the system uses polarization-division multiplexing (PDM) to further reduce crosstalk. For this purpose a polarizer is used to orthogonalize the electric-field polarization of one channel with respect to the other as shown. The effects of ASE noise on each component of the optical signal spectrum are taken into account in the simulation. An optical in-line filter with a Gaussian shaped band-pass characteristic is used to reduce the effect of timing jitter. Although a sliding frequency filter at 80 Gb/s would be a more suitable choice, it is difficult to implement in the practice because of the need to maintain precise frequency control [13]. The pseudo-random 16 bit pattern ‘1010110101110110’ is assumed in the simulation for both the channels. This assumption was actually more conservative than when differing bit patterns were used, because there is a greater chance of channel coupling with identical bit patterns.

In our numerical simulation, the two coupled Schrödinger equations describing the propagation of two orthogonally polarized optical signals in the fiber with randomly varying birefringence are solved based on the coarse-step method. This method treats the random variations of birefringence over many short polarization sections with constant value of birefringence. At the end of each scattering section, the polarization state of the optical field is scattered to a new point on the Poincaré sphere. The mathematical model for the pulse propagation is based on the set of modified non-linear Schrödinger equations written for the orthogonal polarization component of the electric field [14]. This takes into account polarization-dependent loss (PDL), the nonlinear effects of PMD, self- and cross-phase modulation (SPM and XPM), GVD, TOD and fiber attenuation.

Figure 2 depicts the dispersion map used in the experiment. Dispersion of the first and second fiber are +0.25 ps/nm km and -0.23 ps/nm km, respectively. We set the value of dispersion map strength S to be equal to 1.60 as stated in Refs. [15] and [16] and the total average dispersion of the link to be 0.039 ps/nm km. From dispersion map strength *S* given by

and average dispersion

the lengths of the positive and negative dispersion shifted fibers is 4.80 km and 5.05 km respectively.

The effective area of the fibers is held fixed at 70 μm^{2} and the average fiber loss is set at 0.2-dB/km. An amplifier with a noise figure of 4.5 dB is used to compensate the total fiber loss within each fiber span and an optical band pass Gaussian filter with filter order 5 is used before each amplifier to reduce the effect of timing jitter. For the given dispersion map parameters shown in Fig. 2, the effective area, and the nonlinear coefficient, all other parameters are calculated using the analytical procedure given by Agarwal [11, 12]. Here all the parameters are referred to the center reference frequency of 193.1 THz.

The peak power that maximizes the transmission distance is *P*
_{0} = *N*
^{2} ∣β_{2}∣ /(*γT*
_{0}
^{2}) for a single channel. But for dual channel at 80-Gb/s with dense dispersion map, the peak power is higher than this. Once the optimum power is found, the next step is to optimize the number of fiber sections in an amplifier length. That optimum number was determined to be five. This yields a total amplifier length of 49.25 km. The nonlinear coefficient, fiber core area, amplifier gain, initial phase, and the width of the solitons are optimized to achieve the maximum Q-threshold distance. The dispersion length for the DDMS system is 74.36 km, which is greater than the amplifier length as required for the solitons to survive.

In this quasi adiabatic regime the soliton waves are able to complete 257 loops. The system was found to give a better performance when the peak powers of both the channels vary about 10% and when an initial phase of 180 degrees is applied, as part of the optimization process.

## 4. Experimental results and commentary

Using the optimal values cited above, the resulting BER/Q-factor vs. transmission distance characteristics are obtained for the 2 × 80-Gb/s DDMS system and plotted in Fig. 3. At the Q=6.0 threshold 12,700 km transmission distance is obtained as shown by the highlighted values noted in the figure.

Channel 1 and 2 eye diagrams at the intermediate distance of 6,451 km are presented in Fig. 4, and again in Fig. 5 at 12,854 km, a distance slightly beyond the Q-factor threshold. Further, frequency spectrum traces of the dual-soliton signal are in a three-dimensional graph at 15 outer-loop intervals along the 12 thousand km system (Fig. 6).

We assume the use of polarization-maintaining fibers (PMF) neglecting the effect of polarization-mode dispersion (PMD); however, using a typical PMD value of 0.05 ps/√km for non-PMF fibers resulted in a Q-threshold distance of 5150 km.

In the absence of the in-line filter shown in the outer loop of Fig. 1, the Q-factor threshold transmission distance was limited to 2500 km. This can be clearly seen from Fig. 7. Without the filter, timing jitter is around 12 ps at 3000 km. However with filter, timing jitter is less than 4 ps.

## 5. Conclusion

A dual soliton dense dispersion-managed transmission system experiment was simulated with wavelength- and polarization-division-multiplexing schemes. Each channel carrying 80-Gb/s signals was optimized against key fiber-system parameters for maximum transmission distance. Optimization parameters were channel spacing; amplifier distance; peak power, initial phase, effective core area, non linear index; and optical amplifier gain. The transmission distance attained corresponding to a Q-factor threshold of 6.0 was 12700 km yielding a BL product of 2032 Tb/s km.

The next study to be reported is a quad-channel WDM soliton DDMS system with alternate polarizations using the scheme described here.

## Glossary of acronyms:

ASE—amplified spontaneous emission

BER—bit error rate

BL—bit-rate distance product

DDM—dense dispersion management

DDMS—dense dispersion-managed soliton

DMS—dispersion managed soliton

FWHM—full width at half maxima

GVD—group-velocity dispersion

NLS—nonlinear Schrödinger equation

PDL—polarization-dependent loss

PMD—polarization-mode dispersion

PMF—polarization-maintaining fibers

Q-factor—quality factor

SPM—self-phase modulation

TOD—third-order dispersion

WDM—wavelength-division multiplexing

XPM—cross-phase modulation

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