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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13860–13868
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A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system

Zhenning Tao, Weizhen Yan, Shoichiro Oda, Takeshi Hoshida, and Jens C. Rasmussen  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13860-13868 (2009)
http://dx.doi.org/10.1364/OE.17.013860


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Abstract

Cross-phase modulation (XPM) has been considered as one of the ultimate obstacles for optical coherent dense wavelength division multiplexing (DWDM) systems. In order to facilitate the XPM analysis, a simplified model was proposed. The model reduced the distributed XPM phenomena to a lumped phase modulation. The XPM phase noise was generated by a linear system which was determined by the DWDM system parameters and whose inputs were undistorted pump channel intensity waveforms. The model limitations induced by the lumped phase modulation and undistorted pumps approximations were intensively discussed and verified. The simplified model showed a good agreement with simulations and experiments for a typical hybrid optical coherent system. Various XPM phenomena were explained by the proposed model.

© 2009 OSA

1. Introduction

The frequency response of XPM was analyzed by the pump and continuous-wave (CW) probe model in [6

6. T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996). [CrossRef]

]. Based on this pioneering work, a brief introduction of a simplified XPM model where the distributed XPM was reduced to a lumped phase modulation was reported in [14

14. W. Yan, Z. Tao, L. Li, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “A Linear Model for Nonlinear Phase Noise Induced by Cross-phase Modulation” in Tech. Digest of the Conference on Optical Fiber Communication, 2009, paper OTuD5.

]. In this paper, in addition to that of [14

14. W. Yan, Z. Tao, L. Li, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “A Linear Model for Nonlinear Phase Noise Induced by Cross-phase Modulation” in Tech. Digest of the Conference on Optical Fiber Communication, 2009, paper OTuD5.

], the fundamental model approximations and the model limitations are intensively discussed and verified. Additionally, simulation demonstrates that the normalized XPM phase noise auto-correlation, i.e. the spectrum shape equivalently, is not dependent on polarizations. Simulation also shows that the model agrees with it not only in terms of 2nd order statistical characteristics, but also the real time phase noise waveform, which implies higher order statistical agreements. With this model, impact of XPM in a phase shift keying (PSK) and on-off keying (OOK) hybrid optical coherent system and statistical characteristics of XPM phase noise are able to be estimated from system configurations. This function facilitates the system design significantly.

2. The simplified model

Fig. 1. Simplified model of XPM phase noise. (a) N-span M-channel DWDM system, (b) System with lumped XPM phase noise modulation, (c) Linear model of XPM phase noise

The principle, fundamental approximations and limitations of the model are explained in this section. Figure 1(a) shows a schematic diagram of an N-span, M-channel, amplified and inline dispersion compensated DWDM system. Figure 1(b) shows the proposed model corresponding to the system. Without losing the universality, the probe channel is assumed to be channel 1, and all the other channels are assumed as pump channels. In the model, the distributed inter-channel XPM of channel 1 is reduced to a lumped phase modulation at the end of fiber link. Figure 1(c) illustrates how to build the modulating signal ΦXPM applied to the phase modulator. In this paper, probe channel is assumed to be QPSK modulated because QPSK is one of the major modulation formats for coherent system. The pump channel is assumed to be OOK modulated, since such hybrid system induces the largest XPM impact. For simplicity, the polarization effect is ignored, which means that the PMD is zero and polarizations of probe and pump channels are aligned with each other. Waveform distortion on the pump channel is also ignored.

By assuming the CW probe channel 1 and the sinusoidally modulated pump channel 2, the XPM phase noise induced in one transmission span has the spectrum of:

H12(f)=2γ1exp(αL+j2πfd12L)αj2πfd122γα×11j2πd12fα
(1)

τ12=[d12L+DDCFLDCF(λ1λ2)]
(2)

where DDCF and LDCF are dispersion coefficient and length of dispersion compensation fiber (DCF) in each span. The phase noise associated with one of the pump channels (exp. Φxpm_2) is simply assumed to be the sum of all spans. Eventually, the total XPM induced phase noise ΦXPM is the sum of ΦXPM_i for all the pump channels.

Another approximation is to ignore the pump channel intensity waveform distortion. The approximation of undistorted pumps is usually valid when the transmission length is shorter than the dispersion length or the nonlinear length. However, the dispersion length of a 10 Gbaud signal in a single mode fiber is about 100 km, which is much shorter than the typical several hundred kilometer transmission length. The typical nonlinear length is also usually shorter than the transmission length. As a result, the mechanism of this approximation in this model would be different. It is attributed to the low-pass characteristic of XPM and the white characteristic of the low frequency spectrum of OOK waveforms. It is clear that the spectrum of undistorted OOK waveform is white in the low frequency part. In other words, the low frequency components may be regarded as a random noise. Pump channel transmission distortions, such as SPM, CD and PMD, introduce the inter-symbol-interference (ISI). The ISI makes the determined bit pattern, or intensity waveform, more random. However, it does not affect the white low frequency components of the pump channel. The essential fact is that a distorted random noise would be still a random noise. Owning to the low-pass filtering of H 12(f), the XPM phase noise spectrum is determined by the shape of H 12(f) in either case with or without transmission distortion. Eventually, the pump channel intensity waveform distortion may be ignored. Nevertheless, this approximation is no longer valid when the pump channel is in PSK, such as RZ-QPSK. Without transmission distortion, the intensity waveform of RZ-QPSK is periodic, and the low frequency spectrum components should be near zero. With distortion, the ISI makes the spectrum whiter and the low frequency spectrum components turns higher. The essential requirement of this approximation is that the low frequency spectrum of undistorted pump channel intensity waveform is white. More complicated model would be required with PSK pump channel, which is out of the scope of this paper.

In short, the lumped phase modulation and undistorted pumps approximations lead to the model limitations as following: 1) The probe channel has to be PSK modulated. 2) The XPM has to be the dominant source of distortions to the probe channel. 3) The pump channel have OOK or other non-constant-envelop modulation format, such as quadrature amplitude modulation and amplitude-phase shift keying modulation. 4) The dispersion map seems not to be a major limitation. A typical terrestrial QPSK and OOK hybrid system usually satisfies above requests.

The proposed model would be verified in following section 3 and 4. Generally, the XPM phase noise ΦXPM is determined by the bit patterns of the pump channels. It looks like a random noise, whose probability density function is assumed to Gaussian distribution [15

15. K. P. Ho, “Error Probability of DPSK Signals With Cross-Phase Modulation Induced Nonlinear Phase Noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).

]. For simplicity, the discussion in following mainly focuses on the standard deviation and auto-correlation of XPM phase noise.

3. Simulation

Fig. 2. System setup of hybrid transmission NZ-DSF: non-zero dispersion shifted fiber, DCF: dispersion compensation fiber, PC: polarization controller, MUX: optical multiplexer

In order to verify the proposed model, a 9-channel, 50 GHz spacing DWDM transmission system as shown in Fig. 2 was investigated by numerical simulation. The channel number was limited to 9 for the reason of nonlinear simulation speed, whereas 80-channel experiment will be demonstrated in section 4. The CW probe channel lay in the middle, and the pump channels were 11 Gb/s NRZ modulated. The 900 km fiber link consisted of 12 spans. Each span had 75 km non-zero dispersion shifted fiber with a local CD coefficient of 4ps/nm/km. Eighty percent of the span CD was post-compensated by a DCF for each span. The span launch power was -4 dBm/ch. For simplicity, ASE noise was ignored in simulation. The polarization of each channel was aligned by polarization controllers and the polarization mode dispersion (PMD) of fiber link was assumed to be zero, so that the scalar nonlinear Schrödinger equation could be used. The coherent receiver, including the carrier phase recovery, was assumed to be ideal. The residual dispersion of the whole transmission link was also compensated completely at the receiver.

With such ideal coherent receiver, the output of digital storage oscilloscope was the XPM phase noise directly, and the auto-correlation could be calculated accordingly. The results of the simplified model were calculated according to Fig. 1(c). The undistorted pump channel intensity waveforms were obtained by simulation. H 12(f), τ12 and other parameters could be calculated from system configurations according to Eqs. (1) and (2). Figure 3(a) shows the auto-correlation of the XPM phase noise. It is clear that the proposed model agrees with the numerical simulation in terms of 2nd order statistical characteristics. Figure 3(b) shows the waveform comparison. It demonstrates that the agreement of the model and the simulation is not only in 2nd order statistical characteristics, but also in the real time waveform, which would lead to higher order statistical agreements.

Fig. 3. (a) auto-correlation and (b) waveform of the XPM phase noise. simulation: results obtained by solving the nonlinear Schrödinger Equation, model: results by the proposed model.

The discussion in section 2 shows the limitations of the model. The PSK probe and XPM dominant distortion limitations seem to be straightforward. In order to verify the other two limitations, systems of zero in-line dispersion compensation with 11 Gbit/s OOK and 11 Gbaud RZ-QPSK pumps were simulated, whereas the other system parameters kept same. Figure 4(a) shows the XPM phase noise auto-correlation of OOK pump system. The result of the model agrees with simulation well, which demonstrates the model functions under various dispersion maps. Figure 4(b) shows the auto-correlation when the 11 Gb/s OOK pump channel is replaced by the 11 Gbaud RZ-QPSK pump. There is significant difference between the simulation result and the model result. It provides a clear evidence of the limitation, which is only valid with OOK pump channels.

Fig. 4. XPM phase noise auto-correlation of zero in-line dispersion compensation system with (a) 11 Gb/s NRZ pump and (b) 11 Gbaud RZ-QPSK pump, simulation: results obtained by solving the nonlinear Schrödinger Equation, model: results by the proposed model

Fig. 5. Impact of polarization: (a) reduction of XPM phase noise, (b) independence of normalized auto-correlation

The result of Fig. 5(a) is consistent with the common understanding that XPM depends on polarizations. However, the answer of which kind of XPM characteristic does not depend on polarizations has been still unclear. Figure 5(b) shows that the normalized auto-correlation of XPM phase noise, or phase noise spectrum shape equivalently, is not dependent on polarizations. The diamonds and circles corresponds to the cases θ=0 and 60 degree, when PMD was 0. There is no difference between them. In order to reproduce more complicated and realistic scenarios, the ideal zero PMD was replaced by a realistic value of 0.1 ps/sqrt(km) and two cases were investigated: the case with aligned polarization at the transmitter (shown by triangles in Fig. 5(b)) and the case with randomized polarization at the transmitter (shown in squares). There is no difference between those four polarization settings, and all of them agree with the proposed model well.

In short, simulation results demonstrate the capabilities and limitations of the proposed model. Simulation results also show that polarization does not affect the shape of auto-correlation, or the spectrum, of XPM phase noise. It only affects the amplitude, or power. As a result, the proposed model may be applied to non-ideal polarization scenarios, even though the model ignores the polarization effects.

4. Experiment

In order to verify the proposed model, 80 channels, PSK-OOK hybrid DWDM transmission experiment was done. The experimental setup was the same as simulation setup in Fig. 2 except that the probe channel was 43 Gb/s QPSK modulated and the number of DWDM channels was 80. The processing algorithm in the coherent detection digital receiver was based on the Viterbi-Viterbi carrier phase recovery [8

8. T. Tanimura, S. Oda, M. Yuki, H. Zhang, L. Li, Z. Tao, H. Nakashima, T. Hoshida, K. Nakamura, and J. C. Rasmussen, “Non-linearity tolerance of direct detection and coherent receivers for 43 Gb/s RZ-DQPSK signals with co-propagating 11.1 Gb/s NRZ signals over NZ-DSF” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OTuM4.

]. Figure 6 shows the Q values measured as a function of phase recovery average length after 450km (Fig. 6(a)) and 900km (Fig. 6(b)) transmission at the launch power of -4dBm/ch. The OSNR was 12.3dB for the both cases. The simplified model results were obtained by single channel experiment with lumped phase noise modulation calculated by the proposed model. In order to reduce the experimental difficulties, the phase modulation was emulated in electrical digital domain after coherent detection and analog to digital convertors with a speed of 2 samplers per symbol. Since the coherent detection linearly converted an optical domain signal into the electrical domain, the emulated phase modulation was equivalent to a real phase modulation as shown in Fig. 1(b). The results of proposed model were confirmed to agree with real experiment in its tendency and the maximum deviation between the two as 0.5 dB.

Fig. 6. System performance of simplified model and real experiment (a) 450km (b) 900km

In addition to the Q value, the XPM phase noise auto-correlations obtained by the experiment and the proposed model were compared. By using the method in [14

14. W. Yan, Z. Tao, L. Li, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “A Linear Model for Nonlinear Phase Noise Induced by Cross-phase Modulation” in Tech. Digest of the Conference on Optical Fiber Communication, 2009, paper OTuD5.

], the normalized auto-correlation of XPM phase noise was measured in the experiment. The results of simplified model were calculated according to Fig. 1(c), Eq. (1) and Eq. (2). Figure 7 shows the normalized auto-correlation after 450km (Fig. 7(a)) and 900km (Fig. 7(b)) transmission at a launch power of -4dBm/ch. The experiment and proposed model results agreed well.

Fig. 7. Normalized auto-correlations of XPM phase noise after transmission over (a) 450km and (b) 900km.

In short, the Q value under XPM distortion and auto-correlation of XPM phase noise were measured in the experiment. The results of experiment and proposed model agree well with each other in both cases of 450 km and 900 km transmission, which demonstrates that the proposed model functions under various system conditions.

5. Statistical characteristics of XPM phase noise

Fig. 8. Dependences of XPM phase noise on (a) fiber launch power, (b) dispersion compensation (DC) ratio, (c) DWDM channel spacing, (d) guard spacing, (e) total channel number, (f) pump OOK channel bit rate. HWHM: the half-width half-magnitude of auto-correlation. std. the standard deviation of phase noise

Figure 8 presents the standard deviation and auto-correlation length as a function of various system parameters, when a system configuration similar to Fig. 2 is assumed. Figure 8(a) shows the impact of fiber launch power. The phase noise increases in proportion to the launch power in linear unit (mW), and the auto-correlation length does not depend on it. The fact that the transfer function in Fig. 1(c) is independent on the launch power explains this nature. Figure 8(b) shows the impact of inline dispersion compensation ratio. It has been reported that low dispersion compensation ratio improves transmission performance with regards to XPM [9

9. X. Liu and S. Chandrasekhar, “Suppression of XPM penalty on 40-Gb/s DQPSK resulting from 10-Gb/s OOK channels by dispersion management” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OMQ6.

,10

10. S. Chandrasekhar and X. Liu, “Impact of channel plan and dispersion map on hybrid DWDM transmission of 42.7-Gb/s DQPSK and 10.7-Gb/s OOK on 50-GHz grid,” IEEE Photon. Technol. Lett. 19(22), 1801–1803 (2007). [CrossRef]

]. The proposed model supports such a conclusion that may be explained by the impact of accumulated group delay τ 12. With the reduction of dispersion compensation ratio, the accumulated group delay becomes larger, so that XPM phase noise is reduced. Figures 8(c) and 8(d) show the impacts of channel spacing and guard spacing, which is defined as the frequency spacing between the interested channel and the nearest pump channel. XPM phase noise is reduced with large channel spacing and guard spacing. This is consistent with the results reported previously [9

9. X. Liu and S. Chandrasekhar, “Suppression of XPM penalty on 40-Gb/s DQPSK resulting from 10-Gb/s OOK channels by dispersion management” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OMQ6.

,11

11. A. S. Lenihan, G. E. Tudury, W. Astar, and G. M. Carter, “XPM-induced impairments in RZ-DPSK transmission in a multi-modulation format WDM system” in Tech. Digest of Conference on the Lasers and Electro-Optics, (CLEO) 2005. paper CWO5.

]. By enlarging channel spacing or guard spacing, the walk-off parameter is increased. As a result, smaller phase noise and longer auto-correlation are achieved. Figure 8(e) shows that the standard deviation and auto-correlation length saturate with increasing number of channels, just as reported in [12

12. X. Huang, L. Zhang, M. Zhang, and P. Ye, “Impact of nonlinear phase noise on direct-detection DQPSK WDM systems,” IEEE Photon. Technol. Lett. 17(7), 1423–1425(2005). [CrossRef]

]. In DWDM system, the channels far-away from the wavelength of interest tend to have larger walk-off, which leads to much smaller XPM phase noise contribution. Consequently, the near channels dominate the XPM phase noise, and far-away channels may be ignored. Figure 8(f) shows the impact of pump channel bit rate. It shows that the phase noise decreases with pump channel bit rate. Similar conclusion was also reported in [13

13. O. Vassilieva, T. Hoshida, J. C. Rasmussen, and T. Naito, “Symbol rate dependency of XPM phase noise penalty on QPSK-based modulation formats” in Tech. Digest of the European Conference on Optical Communication, 2008, paper We.1.E.4. [CrossRef]

]. It may also be explained by the proposed model. Given the launch power, the pump channel spectrum density reduces with the bit rate. Owning to low-pass filtering characteristic of XPM, only the low frequency component contributes to the phase noise. As a result, the standard deviation reduces with the pump channel bit rate.

In short, the proposed single model explains various XPM phenomena in Fig. 8, although each of them has been investigated and reported [9

9. X. Liu and S. Chandrasekhar, “Suppression of XPM penalty on 40-Gb/s DQPSK resulting from 10-Gb/s OOK channels by dispersion management” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OMQ6.

13

13. O. Vassilieva, T. Hoshida, J. C. Rasmussen, and T. Naito, “Symbol rate dependency of XPM phase noise penalty on QPSK-based modulation formats” in Tech. Digest of the European Conference on Optical Communication, 2008, paper We.1.E.4. [CrossRef]

] separately. With the proposed model, the complex XPM may be estimated from system configurations easily.

6. Conclusion

In optical coherent transmission systems, especially hybrid PSK-OOK systems, the XPM has strong impact on system performance. The XPM phenomenon is complex, and strongly dependent on the system parameters. In order to facilitate its analysis, a simplified model was proposed. It converts the distributed nonlinear XPM phenomena into a lumped phase modulation at the end of the transmission link. The phase modulating signal could be calculated through a linear system whose inputs are undistorted pump channel intensity waveforms. The fundamental lumped phase modulation and undistorted pumps approximations limit the proposed model in the typical PSK-OOK hybrid systems where XPM is the dominant distortion, whereas the dispersion map is not a major limitation. Both simulation and experiment show a good agreement with the results of the proposed model. It is demonstrated that various XPM characteristics may be explained by this simplified model, which should be useful for designing of optical coherent transmission systems.

References and links

1.

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-753. [CrossRef]

2.

S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver” in Tech. Digest of the Conference on Optical Fiber Communication, 2009, paper OThR6.

3.

E Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Compensation of chromatic dispersion and nonlinearity using simplified digital backpropagation” in Coherent Optical Technologies and Applications (Optical Society of America, 2008), paper CWB1.

4.

K. Kikuchi, “Electronic post-compensation for nonlinear phase fluctuations in a 1000-km 20-Gbit/s optical quadrature phase-shift keying transmission system using the digital coherent receiver,” Opt. Express 16(2), 889–896 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-2-889. [CrossRef]

5.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-2-880. [CrossRef]

6.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996). [CrossRef]

7.

K. P. Ho, Phase-Modulated Optical Communication Systems, (Springer, 2005), Chap. 8.3.

8.

T. Tanimura, S. Oda, M. Yuki, H. Zhang, L. Li, Z. Tao, H. Nakashima, T. Hoshida, K. Nakamura, and J. C. Rasmussen, “Non-linearity tolerance of direct detection and coherent receivers for 43 Gb/s RZ-DQPSK signals with co-propagating 11.1 Gb/s NRZ signals over NZ-DSF” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OTuM4.

9.

X. Liu and S. Chandrasekhar, “Suppression of XPM penalty on 40-Gb/s DQPSK resulting from 10-Gb/s OOK channels by dispersion management” in Tech. Digest of the Conference on Optical Fiber Communication, 2008, paper OMQ6.

10.

S. Chandrasekhar and X. Liu, “Impact of channel plan and dispersion map on hybrid DWDM transmission of 42.7-Gb/s DQPSK and 10.7-Gb/s OOK on 50-GHz grid,” IEEE Photon. Technol. Lett. 19(22), 1801–1803 (2007). [CrossRef]

11.

A. S. Lenihan, G. E. Tudury, W. Astar, and G. M. Carter, “XPM-induced impairments in RZ-DPSK transmission in a multi-modulation format WDM system” in Tech. Digest of Conference on the Lasers and Electro-Optics, (CLEO) 2005. paper CWO5.

12.

X. Huang, L. Zhang, M. Zhang, and P. Ye, “Impact of nonlinear phase noise on direct-detection DQPSK WDM systems,” IEEE Photon. Technol. Lett. 17(7), 1423–1425(2005). [CrossRef]

13.

O. Vassilieva, T. Hoshida, J. C. Rasmussen, and T. Naito, “Symbol rate dependency of XPM phase noise penalty on QPSK-based modulation formats” in Tech. Digest of the European Conference on Optical Communication, 2008, paper We.1.E.4. [CrossRef]

14.

W. Yan, Z. Tao, L. Li, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “A Linear Model for Nonlinear Phase Noise Induced by Cross-phase Modulation” in Tech. Digest of the Conference on Optical Fiber Communication, 2009, paper OTuD5.

15.

K. P. Ho, “Error Probability of DPSK Signals With Cross-Phase Modulation Induced Nonlinear Phase Noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 29, 2009
Revised Manuscript: June 12, 2009
Manuscript Accepted: July 22, 2009
Published: July 27, 2009

Citation
Zhenning Tao, Weizhen Yan, Shoichiro Oda, Takeshi Hoshida, and Jens C. Rasmussen, "A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system," Opt. Express 17, 13860-13868 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13860


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References

  1. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-753 . [CrossRef]
  2. S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver” in Tech. Digest of the Conference on Optical Fiber Communication,2009, paper OThR6.
  3. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Compensation of chromatic dispersion and nonlinearity using simplified digital backpropagation” in Coherent Optical Technologies and Applications (Optical Society of America, 2008), paper CWB1.
  4. K. Kikuchi, “Electronic post-compensation for nonlinear phase fluctuations in a 1000-km 20-Gbit/s optical quadrature phase-shift keying transmission system using the digital coherent receiver,” Opt. Express 16(2), 889–896 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-2-889 . [CrossRef]
  5. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-2-880 . [CrossRef]
  6. T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996). [CrossRef]
  7. K. P. Ho, Phase-Modulated Optical Communication Systems, (Springer, 2005), Chap. 8.3.
  8. T. Tanimura, S. Oda, M. Yuki, H. Zhang, L. Li, Z. Tao, H. Nakashima, T. Hoshida, K. Nakamura, and J. C. Rasmussen, “Non-linearity tolerance of direct detection and coherent receivers for 43 Gb/s RZ-DQPSK signals with co-propagating 11.1 Gb/s NRZ signals over NZ-DSF” in Tech. Digest of the Conference on Optical Fiber Communication,2008, paper OTuM4.
  9. X. Liu, and S. Chandrasekhar, “Suppression of XPM penalty on 40-Gb/s DQPSK resulting from 10-Gb/s OOK channels by dispersion management” in Tech. Digest of the Conference on Optical Fiber Communication,2008, paper OMQ6.
  10. S. Chandrasekhar and X. Liu, “Impact of channel plan and dispersion map on hybrid DWDM transmission of 42.7-Gb/s DQPSK and 10.7-Gb/s OOK on 50-GHz grid,” IEEE Photon. Technol. Lett. 19(22), 1801–1803 (2007). [CrossRef]
  11. A. S. Lenihan, G. E. Tudury, W. Astar, and G. M. Carter, “XPM-induced impairments in RZ-DPSK transmission in a multi-modulation format WDM system” in Tech. Digest of Conference on the Lasers and Electro-Optics, (CLEO)2005. paper CWO5.
  12. X. Huang, L. Zhang, M. Zhang, and P. Ye, “Impact of nonlinear phase noise on direct-detection DQPSK WDM systems,” IEEE Photon. Technol. Lett. 17(7), 1423–1425 (2005). [CrossRef]
  13. O. Vassilieva, T. Hoshida, J. C. Rasmussen, and T. Naito, “Symbol rate dependency of XPM phase noise penalty on QPSK-based modulation formats” in Tech. Digest of the European Conference on Optical Communication,2008, paper We.1.E.4. [CrossRef]
  14. W. Yan, Z. Tao, L. Li, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “A Linear Model for Nonlinear Phase Noise Induced by Cross-phase Modulation” in Tech. Digest of the Conference on Optical Fiber Communication,2009, paper OTuD5.
  15. K. P. Ho, “Error Probability of DPSK Signals With Cross-Phase Modulation Induced Nonlinear Phase Noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).

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