Analysis of signed chromatic dispersion monitoring by waveform asymmetry for differentially-coherent phase-modulated systems

We analytically study received waveform asymmetries induced by chromatic dispersion (CD) for signed CD monitoring in differential quadrature phase-shift keying (DQPSK) systems and show that the asymmetries are results of differential detection and the ±π/4 phase shifters used in conventional DQPSK receivers. The theoretical insights developed help explain various published results on signed CD monitoring based on waveform asymmetries and allow us to further propose signed CD monitoring for differential eight phase-shift keying (D8PSK) systems without any modification to the receiver. Simulation results also show that the CD-induced waveform asymmetric features are preserved in presence of self-phase modulation (SPM) and polarization-mode dispersion (PMD). © 2011 Optical Society of America OCIS codes: (060.2330) Fiber optics communications; (060.5060) Phase modulation. 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). 2. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol. 4(10), 15520–1555 (1986). 3. D. C. Kilper, R. Bach, D. J. Blumenthal, D. Einstein, T. Landolsi, L. Ostar, M. Preiss, and A. E. Willner, “Optical Performance Monitoring,” J. Lightwave Technol. 22(1), 294–304 (2004). 4. A. R. Chraplyvy, R. W. Tkach, L. L. Buhl and R. C. Alferness, “Phase modulation to amplitude modulation conversion of CW laser light in optical fibers,” Electron. Lett. 22(8), 409–411 (1986). 5. M. N. Petersen, Z. Pan, S. Lee, S. A. Havstad and A. E. Willner, “Online chromatic dispersion monitoring and compensation using a single inband subcarrier tone,” IEEE Photon. Technol. Lett. 14(4), 570–572 (2002). 6. T. E. Dimmick, G. Rossi and D. J. Blumenthal, “Optical Dispersion Monitoring Technique Using Double Sideband Subcarriers,” IEEE Photon. Technol. Lett. 12(7), 900–902 (2000). 7. G. Rossi, T. E. Dimmick, and D. J. Blumenthal, “Optical performance monitoring in reconfigurable WDM optical networks using subcarrier multiplexing,” J. Lightwave Technol. 18(12), 1639–1648 (2000). 8. K. J. Park, C. J. Youn, J. H. Lee, and Y. C. Chung, “Performance Comparisons of Chromatic DispesionMonitoring Techniques Using Pilot Tones,” IEEE Photon. Technol. Lett. 15(6), 873–875 (2003). 9. Z. Pan, Y. Xie, S. A. Havstad, Q. Yub, A. E. Willner, V. Grubsky, D. S. Starodubovc, and J. Feinberg, “Real-time group-velocity dispersion monitoring and automated compensation without modifications of the transmitter,” Opt. Commun. 230, 145–149 (2003). #140759 $15.00 USD Received 11 Jan 2011; revised 9 Feb 2011; accepted 9 Feb 2011; published 16 Feb 2011 (C) 2011 OSA 28 February 2011 / Vol. 19, No. 5/ OPTICS EXPRESS 4147 10. N. Liu, W. Zhong, Y. Wen, and Z. Li, “New transmitter configuration for subcarrier multiplexed DPSK systems and its applications to chromatic dispersion monitoring,” Opt. Express 15(3), 839–844 (2007). 11. B. Fu and R. Hui, “Fiber chromatic dispersion and polarization-mode dispersion monitoring using coherent detection,” IEEE Photon. Technol. Lett. 17(7), 1561–1563 (2005). 12. Y. K. Lizé, J.-Y. Yang, L. Christen, X. Wu, S. Nuccio, T. Wu, A.E. Willner, R. Kashyap, and F. Seguin, “Simultaneous and Independent Monitoring of OSNR, Chromatic and Polarization Mode Dispersion for NRZ-OOK, DPSK and Duobinary,” OFC/NFOEC 2007, Paper OThN2. 13. B. Kozicki, A. Maruta, and K. Kitayama, “Transparent performance monitoring of RZ-DQPSK systems employing delay-tap sampling,” J. Opt. Netw. 6, 1257–1269 (2007). 14. B. Kozicki, A. Maruta, and K. Kitayama, “Experimental Demonstration of optical performance monitoring for RZ-DPSK signal using delay tap sampling,” Opt. Express 16(6), 3566–3576 (2008). 15. T. Anderson, K. Clarke, D. Beaman, H. Ferra, M. Birk, G. Zhang, and P. Magill, “Experimental Demonstration of Multi-Impairment Monitoring on a commercial 10 Gbit/s NRZ WDM channel,” OFC/NFOEC 2009, paper OTHh7. 16. Z. Li and G. Li, “In-line performance monitoring for RZ-DPSK signals using asynchronous amplitude histogram evaluation,” IEEE Photon. Technol. Lett. 18(3), 472–474 (2006). 17. D. Sandel, V. Mirvoda, F. Wust, R. Noe, and C.-J. Weiske, “Signed online chromatic dispersion detection at 40 Gbit/s based on arrival time detection with 60 attosecond dynamic accuracy,” Electron. Lett. 38(17), 984–985 (2002). 18. R. Noé, D. Sandel, S. Bhandare, F. Wst, B. Milivojevic, and V. Mirvoda, “Signed online chromatic-dispersion monitoring by synchronous detection of FM-induced arrival-time modulations in the clock-recovery phaselocked loop,” J. Opt. Netw. 3(8), 589–600 (2004). 19. Y. Takushima, H. Yoshimi, Y. Ozeki, K. Kikuchi, H. Yamauchi, and H. Taga, “In-Service Dispersion Monitoring in 32× 10.7 Gbps WDM Transmission System Over Transatlantic Distance Using Optical Frequency-Modulation Method,” J. Lightwave Technol. 22(1), 257–265 (2004). 20. N. Liu, W. Zhong, X. Yi, Y. Wang, and C. Lu, “Chromatic dispersion monitoring using the power ratio of two RF tones with a dispersion offset,” OFC 2004, paper MF 81. 21. H. Kawakami, E. Yoshida, H. Kubota, and Y. Miyamoto, “Novel signed chromatic dispersion monitoring technique based on asymmetric waveform distortion in DQPSK receiver,” OECC 2008, paper WeK-3. 22. Z. Li, J. Zhao, L. Cheng, Y. Yang, C. Lu, A. P. T. Lau, C. Yu, H. Y. Tam and P. K. A. Wai, “Signed CD Monitoring of 100-Gbit/s CS-RZ DQPSK Signal by Evaluating the Asymmetry Ratio of Delay-tap Sampling,” Opt. Express 18(3), 3149–3157 (2010). 23. J. Zhao, Z. Li, D. Liu, L. Cheng, C. Lu and H. Y. Tam, “NRZ-DPSK and RZ-DPSK Signals Signed Chromatic Dispersion Monitoring Using Asynchronous Delay-Tap Sampling,” J. Lightwave Technol. 27(23), 5295–5301 (2009). 24. Y. Han, C. Kim, and G. Li, “Simplified receiver implementation for optical differential 8-level phase-shift keying,” Electron. Lett. 40(21), 1372–1373 (2004). 25. K. J. Park, C. J. Youn, J. H. Lee, and Y. C. Chung, “Effect of self-phase modulation on group-velocity dispersion measurement technique using PM-AM conversion,” Electron. Lett. 38(21), 1247–1248 (2002). 26. C. Youn, “Effects of SPM and PMD on chromatic dispersion monitoring techniques using pilot tones,” OFC 2003, paper WP2.


Introduction
Long-haul fiber-optic communication systems are currently moving towards differentiallycoherent as well as coherent detection where the information is encoded in the phase of the optical carrier [1].In particular, modulation formats such as polarization-multiplexed differential quadrature phase-shift keying (PM-DQPSK) are believed to be the standard for nextgeneration systems operating at 100 Gb/s and are currently under intensive research.Signal transmission in optical fibers is subjected to impairments such as chromatic dispersion (CD) and polarization-mode dispersion (PMD) which vary with temperature [2], component aging, fiber plant maintenance and dynamic routing in optical networks.Therefore, monitoring of the channel parameters such as CD is required for their appropriate compensation using optical or electronic techniques.Consequently, CD monitoring is a vital part of optical networks [3], especially for next-generation systems in which the tolerance to residual CD of the link decreases dramatically with increasing symbol rate.
Traditionally, CD can be monitored by phase-modulating the optical carrier [4] or inserting tones at the transmitter [5][6][7][8] and measuring the power of a CD-induced RF tone at the receiver.Alternatively, such RF tone can be obtained from clock tones for pulse shapes such as Non Return-to-Zero (NRZ)-, Return-to-Zero (RZ)-and Carrier-Suppressed (CS)-RZ On-Off Keying (OOK) [9].CD monitoring techniques for coherent or differentially-coherent systems have also been studied [10][11][12][13][14][15][16].However, the majority of the proposed techniques are not able to distinguish between the sign of CD which is vital for residual CD compensation.Sandel [17], Noé [18] and Takushima [19] studied signed CD monitoring by amplitude modulating a CW laser and measuring the signal arrival time or frequency modulating a CW laser and measuring received power fluctuations.However, complicated phase-lock loops are required.Liu et al. [20] added an additional dispersion element at the receiver to induce asymmetric dependence of the power ratio between two received RF tones on the sign of CD.On the other hand, Kawakami [21] proposed the use of asymmetric waveform distortions as a simple signed CD monitoring technique for RZ-DQPSK systems and our previous works [22,23] also exploited such asymmetric features for signed CD monitoring for DQPSK and DPSK systems.However, no in-depth study on the nature of this asymmetry has been undertaken and the applicability and limitations of signed CD monitoring using waveform asymmetry are mostly unknown.
In this paper, we analytically study received waveform asymmetries induced by CD for DQPSK signals and show that such asymmetries are results of differential detection and the presence of ±π/4 phase shifters in conventional DQPSK demodulators.The theoretical insights developed thus explain the origins of waveform asymmetry obtained from DPSK receivers with an imperfectly tuned phase shifter [23].In addition, the underlying principles can be similarly exploited for D8PSK systems without any modifications to the receiver.We study the use of delay-tap plots to exploit this asymmetry for signed CD monitoring through simulations and note that many other techniques can be used for exploiting the asymmetry.Furthermore, simulation results show that CD-induced waveform asymmetries are preserved in long-haul systems with SPM and PMD.

Waveform asymmetry for DQPSK systems in presence of chromatic dispersion
Consider a fiber-optic transmission system with transmitted signal where x k ∈ {1, −1, j, − j} denotes QPSK symbols that are independent and identically distributed (i.i.d.) random variables, T is the symbol period and b(t) is the pulse shape centered at t = 0 which can be NRZ, RZ or CS-RZ.Polarization, multi-channel effects and self-phase modulation (SPM) are neglected for simplicity.In addition, noise from optical amplifiers and other components are also excluded in our analysis to highlight the origins of waveform asymmetry caused by CD.We note that the analytical derivations in this paper equally applies to general DMPSK systems.
A channel with CD can be modeled as a linear and time-invariant (LTI) system with transfer function where β 2 is the group velocity dispersion (GVD) parameter and L is the length of the fiber.In this case, the corresponding impulse response h(t) is given by Denoting the collective effect of optical band-pass filters along the transmission link and electrical low-pass filters at the receiver as a LTI system with impulse response g(t), the received signal is given by where and p(t) = b(t) * g(t) denotes the signal pulse shape with optical band-pass and electrical lowpass filtering.For DQPSK systems, E r (t) will be mixed with a delayed version of itself plus a ±π/4 phase shift using a demodulator shown in Fig. 1.The delayed signal is given by In this configuration, the output of the balanced receiver is given by We will focus on y I (t) for the rest of the paper and note that subsequent results will be similar to those from y Q (t).Now, The terms in Eq. ( 8) can be rewritten as In Eq. ( 9), the terms in the first line correspond to differentially encoded information to be detected and products of each individual pulse with the conjugate of its delayed self.As the expected value of two information symbols the expected value of the balanced detector output is given by where Re{•} and Im{•} correspond to real and imaginary part respectively.If we let Since E[y I (t)] is periodic with period T , we will just focus on the symbol centered at t = 0 and consider the term as p(t) is real for practical pulse shapes and filters.In this case, let (14) Now, we note that If we re-define the dummy variables of integration τ 3 = −τ 1 and τ 4 = −τ 2 , where the equality in (a) stems from the fact that any practical pulse shape is symmetric about its center.Similarly, one can show that w 2 (−t) = −w 1 (t) and therefore is an odd function of t and hence asymmetric about t = 0. On the other hand, it can be shown that Re{z(t)} is an even function of t and will not result in any waveform asymmetry.Since the sum of an symmetric and asymmetric waveform is still asymmetric, we have just shown that the expected value of the received waveform E[y I (t)] of a DQPSK demodulator exhibits an asymmetry about the center of each bit.Furthermore, Im{z(t)} and Re{z(t)} is also an odd and even function of β 2 respectively.Therefore, any waveform asymmetric feature for a given β 2 will be reversed about the center of the bit when the sign of β 2 is changed.Such asymmetry reversal is exactly the feature that allows one to distinguish the sign of CD from the received waveform.To illustrate the analytical insights with an example, consider the transmission of a pulse train with Gaussian pulse shape Neglecting the effect of optical and electrical filters for simplicity, it can be shown that Note that Im{z(t)} is an odd function of t and β 2 and is shown in Fig. 2 for a 20G Sym/s system with T o = 15 ps and ± 136 ps/nm of accumulated CD.It is clear from Eq. ( 18) that when the sign of β 2 changes, the direction of asymmetry also changes, enabling one to distinguish the sign of CD from the received waveform.As y I (t) is an ergodic process, the ensemble average E[y I (t)] will be equal to the time-averaged waveform of one particular random realization of y I (t) and therefore the CD-induced asymmetric features can be extracted from an eye diagram.
Note that E[y I (t)] is non-zero only because of the second line of Eq. ( 9) i.e. the product of each individual pulse with the conjugate of its delayed self.Consequently, CD-induced waveform asymmetries should only be present for differentially-coherent phase-modulated systems and not present for coherent modulation formats such as BPSK, QPSK or 16-Quadrature Amplitude Modulation(QAM).In addition, the asymmetry is due to the presence of the term ∑ ∞ k=−∞ Im{z(t − kT )} which is brought about by the ±π/4 phase shifters used in a typical DQPSK demodulator.The two above insights together explain the waveform asymmetries reported in [21,22] for DQPSK systems.Furthermore, they also explain how an intentional phase Fig. 2. The term Im{z(t)} for a 40Gb/s (20GSym/s) RZ-DQPSK system with Gaussian pulse shape and ±136 ps/nm of accumulated CD.For a given CD value, the waveform is an odd function of t and hence asymmetric about t = 0.In addition, since Im{z(t)} is also an odd function of β 2 , the direction of the time asymmetry will be reversed when the sign of β 2 is changed, thus explaining the observed waveform asymmetry that changes with the sign of accumulated CD. mismatch of π/4 in one arm of a DPSK receiver enables signed CD monitoring for DPSK systems [23].In addition, Fig. 3 shows the received eye diagrams for 40 Gb/s (20 GSym/s) NRZ-, RZ-and CS-RZ-DQPSK systems with an accumulated dispersion of ±136 ps/nm obtained from simulations.Noises from lasers, optical amplifiers and receiver circuitries are neglected to illustrate the CD-induced waveform asymmetries.From the figure, it can be seen that asymmetries are present for all pulse shapes.This is in agreement with expectations as the analytical derivations above only assume the pulse shape p(t) to be real and symmetric about its center.Therefore, the waveform asymmetric features should be present for all types of pulse shapes commonly used in communication systems.

Signed CD monitoring for D8PSK systems using delay-tap sampling
The theoretical insights developed above indicate that CD-induced waveform asymmetry will be present as long as the modulation format is differentially-coherent and ±π/4 phase shifters are used in the receiver structures.In fact, Eqs. ( 10) and ( 11) actually suggest that the term ∑ ∞ k=−∞ Im{z(t −kT )}, which causes the asymmetry, will be present as long as the phase shift of the 3-dB coupler is not 0. Consequently, for a D8PSK system with proposed receiver structures [24] such as the one shown in Fig. 4, one should be able to exploit the waveform asymmetric features in similar manners for signed CD monitoring.In particular, one of the balanced-detector output can be 'tapped' for signed CD monitoring without modifying the receiver structure.Fig. 5 shows the eye diagrams corresponding to the output signal y 3π/8 (t) of a 60 Gb/s (20 GSym/s) RZ-D8PSK system with -136 ps/nm, 0 ps/nm and 136 ps/nm of CD, indicating the presence of CD-induced waveform asymmetry.Similar to our previous studies, one can generate delay-tap plots from y 3π/8 (t) or y −π/8 (t), evaluate the distances of sample pairs d 1 , d 2 furthest away on both sides of the diagonal D and calculate the distance ratio d 1 /d 2 for signed CD monitoring [22,23].Some delay-tap plots generated from y 3π/8 (t) are shown in Fig. 6 and the distance ratio d 1 /d 2 vs. CD is shown in Fig. 7.It can be seen that the distance ratio corresponding to y −π/8 (t) is not as sensitive to CD compared with that from y 3π/8 (t).This is in agreement with theoretical predictions as the coefficient of ∑ ∞ k=−∞ Im{z(t − kT )} in Eqs. ( 10) and (11) will become large when the phase shift of the 3-dB coupler deviates away from 0. Moreover, in addition to delay-tap plots, we emphasize that many other techniques such as amplitude histograms and artificial neural networks can also be used to extract the CD-induced waveform asymmetric features for signed CD monitoring.

CD-induced waveform asymmetry in presence of SPM and PMD
In long-haul systems, signal transmission will also be affected by SPM as well as PMD and the performance of any CD monitoring technique maybe affected [25,26].To investigate the applicability of the signed CD monitoring technique for long-haul systems, Fig. 8 shows the distance ratio as a function of residual CD obtained from simulations for a 40 Gb/s RZ-DQPSK system with length 500 km and 1000 km.The transmission link consists of multiple spans of fiber with span length around 60 km and dispersion compensating fibers at each span such that the accumulated residual CD is equally distributed across the spans.The coarse-step model with 1-km step size is used for the simulation and the birefringence is random across different sections.The birefringence beat length is 15 m.The distance ratio corresponding to a link without SPM and PMD is also shown as reference.The input power, fiber nonlinear coefficient γ and the PMD coefficient is 0 dBm, 1.2/W • km and 0.1 ps/ √ km respectively.From the figure, it can be seen that the CD-induced asymmetric features are still present for systems with SPM and PMD.Although the mapping between the distance ratio and residual CD will generally be different than the case without SPM and PMD and may need to be calibrated for each system setup, such mappings will still be able to distinguish between positive and negative CD. Consequently, the proposed signed CD monitoring technique will be applicable in realistic long-haul systems with high data rates in which residual CD compensation plays an important role in the overall performance of the transmission link.

Conclusions
In this paper, we analytically studied signed CD monitoring using waveform asymmetries for DQPSK systems and showed that such asymmetries are results of differential detection and ±π/4 phase shifters present in conventional DQPSK receivers.The theoretical insights developed help elucidate the operating principles behind other signed CD monitoring techniques based on waveform asymmetries reported in the literature and naturally suggest that similar methods can be used for D8PSK systems without modifications to the receiver.Signed CD monitoring using delay-tap plots is investigated for D8PSK systems through simulations and results are in agreement with theoretical predictions.In addition, the asymmetric features are shown to be preserved in presence of SPM and PMD and therefore the proposed technique can be used for signed CD monitoring in practical long-haul systems.

Fig. 1 .
Fig. 1.Typical configuration for a DQPSK demodulator.±π/4 phase shifters in the two arms of the 3-dB couplers are used to detect the in-phase and quadrature component of the received signal.The T here represents one-bit delay.

Fig. 3 .
Fig. 3. Received eye diagrams for 40 Gb/s (20 GSym/s) DQPSK systems with various pulse shapes and residual dispersion of ±136 ps/nm.The eyes are asymmetric about their centers and such asymmetries change with the sign of β 2 .

Fig. 6 .Fig. 7 .
Fig. 6.Delay-tap plots obtained from the received signal y 3π/8 (t) for a 60 Gb/s (20 GSym/s) RZ-D8PSK system with (a) -136 ps/nm (b) 0 ps/nm and (c) 136 ps/nm of residual CD with delay value τ = 24 ps.The distances d 1 , d 2 of sample pairs furthest away from both sides of the diagonal D (shown in (b))can be used to calculate a distance ratio d 1 /d 2 for signed CD monitoring.

Fig. 8 .
Fig. 8. Distance ratio d 1 /d 2 as a function of residual CD for a 40 Gb/s RZ-DQPSK system in presence of SPM and PMD.