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Modelling of a polarization-division multiplexing soliton transmission system
View the table of contents for this issue, or go to the journal homepage for more 1998 J. Opt. 29 278
(http://iopscience.iop.org/0150-536X/29/4/006)
Modelling of a polarization-division
multiplexing soliton transmission
system
Chi-Feng Chen and Sien Chi†
Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, Republic of China
Received 31 December 1997, accepted 6 April 1998
Abstract. We have simulated numerically a polarization-division multiplexing soliton transmission system by using propagation equations with and without the third-order dispersion and Raman shift terms, respectively. It is found that the Raman shift terms can reduce the soliton interactions and hence the allowed transmission distance increases.
Keywords: Soliton transmission, polarization-division multiplexing
Mod `ele de transmission soliton par
multiplexage `a division de la polarisation
R ´esum ´e. Nous avons simul ´e num ´eriquement un syst `eme de transmission soliton par multiplexage `a division de la polarisation en utilisant des ´equations de
propagation avec et sans terme dispersifs du troisi `eme ordre ou terme de d ´ecalage Raman.
Nous montrons que les termes d’effet Raman peuvent r ´eduire les interactions entre solitons et donc augmenter la distance de transmission.
Mots cl ´es: Transmission soliton, multiplexage `a division de polarisation
1. Introduction
In optical soliton communication systems using optical amplifiers to compensate for fibre loss, the transmission bit-rate–distance product is limited by the soliton interaction [1] and the timing jitter arising from noise introduced by the optical amplifiers [2], from the acoustic interaction among the solitons [3] and from the polarization dispersion [4]. A method to reduce the soliton interaction is provided by polarization-division multiplexing (PDM), for which the adjacent solitons are orthogonally polarized [5–9]. It was demonstrated that the interaction of orthogonally polarized solitons was weaker than that of parallelly polarized solitons [5]. An analytical description of the orthogonally polarized soliton interaction was found using the perturbation method, which neglected the effects of third-order dispersion and Raman shift [6–8]. The modified coupled averaged propagation equations which describe
† Author to whom correspondence should be addressed. E-mail address: [email protected]
the soliton propagation including the effects of the third-order dispersion and Raman shift in a PDM transmission system have been derived [8]. In recent theoretical and experimental works, it was found that the interaction of the orthogonally polarized solitons could also be reduced by a sliding-frequency filter (SFF) as the interaction of the parallelly polarized solitons [9, 10].
In this paper, we will study numerically the interactions of the orthogonally polarized solitons and the allowed transmission distance in a PDM system by using propagation equations with and without the third-order dispersion and Raman shift terms, respectively. We will also show that the soliton interactions are reduced and the allowed transmission distance increases when the Raman shift terms are included, and the third-order dispersion term has negligible effect. Therefore, for ultra-high capacity of the PDM soliton transmission system, the Raman shift terms must be included. On the other hand, we will show that these terms have little effect on the soliton interactions and the allowed transmission distance in the parallelly polarized soliton system.
Modelling of a PDM soliton transmission system
Figure 1. Power evolution of a soliton bitstream along the fibre in a PDM soliton transmission system without the SFF: (a) when the third-order dispersion and Raman shift terms are not included and (b) when the third-order dispersion and Raman shift terms are included.
2. The modified propagation equations
Soliton propagation in single-mode fibre with random birefringence can be described by the modified coupled averaged propagation equations [8]
i∂u ∂z+ 1 2 ∂2u ∂T2− i C3 6 ∂3u ∂T3+ |u| 2+ |v|2u − τRu ∂ ∂T|u| 2−1 2τRu ∂ ∂T|v| 2 −1 2τRv ∂ ∂T(v ∗u)= iγ u (1a) i∂v ∂z+ 1 2 ∂2v ∂T2− i C3 6 ∂3v ∂T3+ |u| 2+ |v|2v − τRv ∂ ∂T|v| 2−1 2τRv ∂ ∂T|u| 2 −1 2τRu ∂ ∂T(u ∗v)= iγ v (1b)
where u and v are two polarization components of the electric field envelope normalized by the electric field scale
Q; Z and T are normalized by the dispersion length LD
and time scale T0, respectively. Q, LD and T0 are related
by Q= 9 8 λ|β2|Aeff 2π n2T02 1/2 LD= T2 0 |β2|
where λ is the wavelength, β2 is the second-order
dispersion, Aeffis the effective fibre cross section area and n2is the Kerr coefficient. T0= TW/1.763, where TWis the
initial full pulse width at half magnitude. The coefficients in equations (1) are C3= β3LD T03 τR= TR T0 γ = αLD
where β3 is the third-order dispersion, α is the fibre loss, TR is the slope of the Raman gain profile at the carrier
frequency. The last three terms on the left-hand side of equations (1) describe the averaged Raman effect, in which the first term is the self-frequency shift term, and the other two are the cross-frequency shift terms. The parallelly polarized soliton propagation in a single-mode fibre can be described by equation (1a) without a v component or equation (1b) without a u component. The transfer function of the optical filter placed after every amplifier is taken as
H (− f)= 1
1+ 2i( − f)/B (2)
where = ω − ω0 and ω0 is the input soliton carrier
frequency, f is the centre frequency of the filter and B is
the filter bandwidth.
3. Numerical results
The typical fibre parameters for solving equations (1) are: soliton wavelength λ = 1.55 µm, β2 = −0.32 ps2 km−1
(D = 0.25 ps km−1 nm−1), β3 = 0.1 ps3 km−1, α =
0.21 dB km−1, n2= 2.3 × 10−20m2 W−1, TR= 3 fs and Aeff= 50 µm2. The fibre loss is periodically compensated
by the optical amplifiers with 30 km spacing. We consider the soliton pulsewidth TW = 12 ps. To show the multiple
soliton interactions, we consider the soliton trains with patternh0011011101111011111011111100i and take a 3.5 pulse width separation between the neighbouring solitons. The corresponding bit rate is 23.8 Gbits s−1. To reduce the soliton interaction and timing jitter, we insert the up-sliding filter after every optical amplifier. For the maximum transmission distance, the bandwidths of the filters and the sliding rate are taken to be 190 GHz and 4 GHz Mm−1 in a PDM soliton transmission system, and 225 GHz and 8 GHz Mm−1in a parallelly polarized soliton transmission system, respectively.
Figure 2. Power evolution of a soliton bitstream along the fibre in a PDM soliton transmission system with the SFF: (a) when the third-order dispersion and Raman shift terms are not included and (b) when the third-order dispersion and Raman shift terms are included.
Figures 1(a) and (b) show the power evolution|u|2+
|v|2 of a soliton bitstream in a PDM soliton transmission
system when the third-order dispersion and Raman shift terms are not and are included, respectively. It is seen in figure 1(a) that the two central solitons in bit patterns (011110) coalesce at a distance of 11 Mm. In figure 1(b), the same solitons get close at 11 Mm but do not coalesce. Therefore, the interactions of PDM solitons are weaker when the third-order dispersion and Raman shift terms are considered. Figure 2 shows the same evolution as in figure 1 except that a SFF is placed after each amplifier. We can see from figure 2(a) that the solitons in the bit patterns (011110) and (0111110) coalesce around 21 Mm. In figure 2(b), it is seen that the separations of the solitons are well maintained even up to a distance of 30 Mm. As the numerical results obtained with β3 = 0 are almost
identical to those shown in figures 1(b) and 2(b), the reduction of the soliton interaction is mainly due to the Raman shift terms. Therefore, when the SFF is used, the interactions of PDM solitons are greatly reduced, and the interactions are much weaker when the Raman shift terms are considered.
In a soliton communication system using optical amplifiers to compensate for the fibre loss, noise will be introduced. The noise randomly modulates the soliton carrier frequency and causes the timing jitter of the system. The noise-induced timing jitter will influence the soliton interaction and complicate the problem. In addition, the acoustic interaction among the solitons and the polarization dispersion also induce timing jitters. Thus, the total standard deviation of timing jitter can be written as [9]
σ = σc2+ σa2+ σp21/2 (3) where σc, σa, σp represent the standard deviation in
picoseconds of the combined soliton interaction and noise-induced jitter, the acoustic-interaction-noise-induced jitter and the
polarization-dispersion-induced jitter, respectively. σa can
be estimated by [3] σa= 140A−3/4eff D2 TW γ−1z R 1− r− 9.33 A1/2eff 1/2 (4) where D is the group-delay dispersion in ps km−1 nm−1,
TWis the FWHM pulsewidth in ps, Aeffis the effective fibre
cross section in µm2, r is the intensity reflection coefficient
of sound from the cladding–coating boundary, R is the bit rate in Gbits s−1, z is total path in Mm and γ−1 is the damping length in Mm. In addition, σp can be estimated
by [4] σp= π 16 (G− 1)nsphν Wsol D2 pz2 Lamp 1/2 (5) where G is the amplifier power gain, nspis the spontaneous
emission factor of an amplifier, Wsol is the soliton pulse
energy at an amplifier output, Dp is the fibre polarization
dispersion parameter and Lamp is amplifier spacing. In
our system the other parameters are taken as r = 0.25,
R = 23.8 Gbits s−1, γ−1 = 0.37 Mm, G = 4, nsp= 1.2, hν/Wsol = 2 × 10−6 and Dp = 0.1 ps km−1/2, we obtain σa= 0.8 ps and σp= 0.23 ps at z = 10 Mm.
We calculate the standard deviation by simulating the transmission of 512 pseudorandom bits. Figure 3 shows the evolutions of the standard deviation of the timing jitters of solitons in a PDM soliton system and a parallelly polarized soliton system when the third-order dispersion and Raman shift terms are and are not considered, respectively. It is seen that the difference caused by these terms is very little for the parallelly polarized soliton system and is substantial for the PDM soliton system. In our case, the allowed 10−9 bit error rate corresponds to a 2.3 ps standard deviation of the timing jitter. The allowed transmission distance for the PDM soliton system is 12.8 Mm and 11 Mm when the
Modelling of a PDM soliton transmission system
Figure 3. Evolutions of the standard deviation of the timing jitter of solitons. Curves (a) and (b) are the evolutions when the third-order dispersion and Raman shift terms are not and are included in a parallelly polarized soliton transmission system, respectively. Curves (c) and (d) are the evolutions when the third-order dispersion and Raman shift terms are not and are included in a PDM soliton transmission system, respectively.
third-order dispersion and Raman shift terms are and are not considered, respectively. Furthermore, it is found that the allowed transmission distance is almost identical with and without the third-order dispersion. Therefore, for the increment of the allowed transmission distance the effect of the Raman shift terms is mainly due to the Raman-shift terms. On the other hand, the allowed transmission distance for the parallelly polarized solitons system is only around 2.4 Mm.
4. Conclusion
In conclusion, we have simulated numerically the interactions of the orthogonally polarized solitons and the allowed transmission distance in a PDM transmission system by using propagation equations with and without the third-order dispersion and Raman shift terms, respectively. It is found that the soliton interactions are reduced and the allowed transmission distance increases when the Raman shift terms are included, and the third-order dispersion term has a negligible effect. Therefore, for ultra-high capacity PDM soliton transmission systems, the Raman shift terms must be included. We have also simulated the parallelly polarized soliton system with the same system parameters, and it is found that the allowed transmission distance is much shorter than the PDM system.
Acknowledgment
This work is partially supported by National Science Council, Republic of China, under contract no NSC 86-2811-E009-002R.
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