Q-factor analysis of nonlinear impairments in ultrahigh-speed Nyquist pulse transmission

Toshihiko Hirooka; Masataka Nakazawa

doi:10.1364/OE.23.033484

1. Introduction

The increase in the single-carrier symbol rate realized with optical time-division multiplexing (OTDM) has made it possible to achieve a serial Tbit/s channel capacity [1–3]. OTDM has traditionally adopted RZ pulses with Gaussian or sech waveforms. However, these waveforms generally occupy a large bandwidth in the frequency domain due to their rapidly decreasing tails in the time domain, and thus may not always be optimum waveforms in terms of increasing the spectral efficiency. An increase in spectral efficiency allows not only capacity expansion within a finite transmission bandwidth, but also an enhanced tolerance to chromatic dispersion (CD) and polarization-mode dispersion (PMD) in such an ultrahigh-speed transmission.

We recently proposed a new TDM transmission scheme using an optical Nyquist pulse [4]. Its waveform is given by a sinc (roll-off factor α = 0) or quasi-sinc (α≠0) function, and its spectrum has a rectangular (α = 0) or raised-cosine (α≠0) profile. The waveform and spectrum, respectively, correspond to an impulse response and transfer function of a Nyquist filter. The tail of the Nyquist pulse decays slowly in the time domain, but because the ringing crosses zero periodically, it is possible to bit-interleave the pulses at a symbol rate equal to the period of the zero crossing, in which the neighboring pulses do not suffer from intersymbol interference (ISI) with each other despite their strong overlap. In the frequency domain, the signal is confined within a finite bandwidth, and the spectral width can be greatly reduced compared with that of RZ pulses under the same symbol rate. This is an attractive feature not only in terms of increasing the spectral efficiency but also in improving CD and PMD tolerance. The advantages of using Nyquist pulses for ultrahigh-speed TDM, such as at 160 Gbaud ~1.28 Tbaud, have been demonstrated for both non-coherent (OOK, DPSK, DQPSK) [5, 6] and coherent (QPSK, QAM) transmissions [7, 8].

We previously reported an analysis and numerical simulations of a single optical Nyquist pulse propagation in optical fibers in the presence of dispersion and nonlinearity [9]. Specifically, we analyzed the way in which the periodic zero-crossing property of a Nyquist pulse is affected by the fiber dispersion and nonlinearity during propagation. In this paper, we analyze the transmission of a Nyquist TDM data sequence and compare the dispersion and nonlinear tolerances with an RZ pulse TDM transmission. A numerical simulation of Q-factor contour mapping (Q map) [10] shows a large system margin for Nyquist pulses with respect to both dispersion and optical power. We also provide an analytical description of nonlinear impairments in a Nyquist TDM transmission in a high power regime.

2. Performance comparison of RZ and Nyquist TDM transmissions using Q-map analysis

Pulse propagation in fibers can be described by a nonlinear Schrödinger equation in the form

i \frac{\partial u}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u}{\partial t^{2}} + γ a^{2} (z) | u |^{2} u = 0,

where u(z,t) is the envelope of a complex field representing an optical pulse, β₂ and γ are the dispersion and nonlinear coefficients, respectively, and a²(z) = exp(−αz) takes account of the loss and gain profiles. We first carried out a direct numerical simulation of Eq. (1) to analyze the transmission impairments in a 640 Gbaud DPSK transmission using Nyquist and Gaussian pulses with a split-step Fourier method. Figure 1 shows the waveform and spectrum of these pulses. Here, the spectrum of a Nyquist pulse is given by

\hat{s} (f) = {\begin{matrix} T, 0 \leq | f | \leq \frac{1 - α}{2 T} \\ \frac{T}{2} {1 - \sin [\frac{π}{2 α} (2 T | f | - 1)]}, \frac{1 - α}{2 T} \leq | f | \leq \frac{1 + α}{2 T} \\ 0, | f | \geq \frac{1 + α}{2 T} \end{matrix},

where α (0 ≤ α ≤ 1) is a roll-off factor. In the simulation, a data stream consisting of a 2¹³−1 PRBS was transmitted over 7 spans of a 75 km dispersion-managed link consisting of 50 km anomalous GVD (D = 20 + δD ps/nm/km, A_eff = 100 μm²) and 25 km normal GVD (D = −40 + δD ps/nm/km, A_eff = 30 μm²) fibers, as used in our previously reported experiment [2], in which δD is the residual path-average dispersion. Here we do not assume that the residual dispersion is compensated for with DSP. Figure 2 shows a schematic configuration of a dispersion-managed fiber span. The fiber loss (0.2 dB/km) was compensated for by installing a lumped EDFA with a noise figure of 5.0 dB every 75 km, followed by an in-line optical filter with 8.4 nm (edge to edge) and 20 nm (− 3 dB) bandwidths for Nyquist and Gaussian pulses, respectively. It should be noted that, because the spectrum of a Nyquist pulse can be confined within a finite bandwidth, it allows us to install a rectangular filter with a much narrower bandwidth than a Gaussian pulse.

Fig. 1 Waveform (a) and spectrum (b) of Nyquist (α = 0 and 0.5) and Gaussian (0.8 and 0.6 ps) pulses for 640 Gbaud transmission.

Download Full Size | PDF

Fig. 2 Schematic configuration of a dispersion-managed fiber span.

Download Full Size | PDF

Figure 3(a) and 3(b) show a Q map for a 640 Gbaud transmission over 525 km using a Nyquist pulse with α = 0 and 0.5. The Q map is a contour profile of Q factors as a function of the average dispersion and the transmitted power, and it allows us to evaluate easily the system tolerance to CD and nonlinearity [10]. It can be seen that the largest Q factor is obtained at δD = 0 and a transmission power of ~7 dBm. A Q factor of > 6 (Q² > 15.6 dB), which corresponds to a bit error rate (BER) of < 10⁻⁹, is obtained over a region of approximately |δD| < 1x10⁻³ ps/nm/km and with a power of 2~10 dBm as shown by the green curve. Without giving δD such a low value, a large transmission impairment can occur at a symbol rate of 640 Gbaud. Although we do not assume that δD is compensated for with DSP, it can be easily brought close to zero in the optical domain by using, for example, a liquid crystal on silicon (LCoS) spectral phase manipulator [11]. It should also be noted that, when comparing Figs. 3(a) and 3(b), the Q map for α = 0 exhibits a larger margin in terms of dispersion but the power margin is somewhat decreased compared with α = 0.5. The increased dispersion margin with α = 0 is a consequence of the spectral width being narrower than α = 0.5, while the lower power margin may be attributed to the larger overlap between neighboring pulses, resulting in larger nonlinear impairments. A detailed analysis of the nonlinear impairments between bit-interleaved Nyquist pulses will be presented in Sec. 3.

Fig. 3 Q maps for 640 Gbaud-525 km transmissions using a Nyquist pulse with α = 0 (a) and 0.5 (b).

Download Full Size | PDF

For comparison, we evaluated the performance of a Gaussian pulse with two different pulse widths, Δτ. First we set Δτ at 0.8 ps, for which the spectrum has the same 3 dB bandwidth as that of the Nyquist pulse. The corresponding Q map is shown in Fig. 4(a), where it can be clearly seen that the margin is substantially decreased. This is because the pulse width is too broad against the pulse interval (1.56 ps) and the strong overlap between adjacent pulses induces large ISI, whereas the Nyquist pulse does not suffer from ISI despite having the same 3 dB bandwidth. To separate adjacent pulses, we next considered a Gaussian pulse with Δτ = 0.6 ps, and the result is shown in Fig. 4(b). The largest Q factor was obtained at δD = 0 and a power of ~9 dBm. Here, the power margin was slightly improved as indicated by the contour distribution expanded in the vertical direction compared with Fig. 4(a). However, the dispersion tolerance is still much lower than that of Nyquist pulses. These results show that a Nyquist pulse has the largest Q margin with respect to both dispersion and power.

Fig. 4 Q maps for 640 Gbaud-525 km transmissions using a Gaussian pulse with Δτ = 0.8 ps (a) and 0.6 ps (b).

Download Full Size | PDF

3. Analysis of nonlinear impairments in Nyquist TDM transmission

The Q maps shown in Fig. 3(a) and 3(b) show that the Q factor distribution for a Nyquist pulse is located in a power regime approximately 2 dB lower than that of Gaussian pulses. This indicates that a Nyquist pulse is affected by fiber nonlinearities in a different way from a typical Gaussian pulse. Figure 5 shows the Q factor from Figs. 3 and 4 plotted as a function of the transmission power when δD = 0. It can be seen that a Nyquist pulse performs better than a Gaussian pulse in the lower power regime, while the Q factor degrades faster in the higher power regime. To clarify the influence of nonlinearity, we applied the analytical model for nonlinear interactions developed in [12] to a Nyquist TDM transmission with a random data sequence, and evaluated the impairments due to intra-channel cross-phase modulation (IXPM) and intra-channel four-wave mixing (IFWM) [13–15].

Fig. 5 Q factor in Figs. 3 and 4 plotted as a function of transmission power when δD = 0.

Download Full Size | PDF

Following [12], we treat the nonlinearity as a perturbation (γ << 1) and write the optical field as u(z, t) = u_L(z, t) + γΔu(z, t), in which u_L is the solution of the linear propagation (γ = 0 in Eq. (1)). The spectrum of the nonlinear perturbation, Δû(z, f), caused by a nonlinear interaction term between three pulses, γa²(z)u_lu_mu_n* (u_k: a pulse centered at t = kT, T: symbol interval), is then obtained as [12]

Δ \hat{u} (z, f) = i P_{p}^{3 / 2} \sum_{l = - N / 2}^{N / 2} \sum_{m = - N / 2}^{N / 2} \sum_{n = - N / 2}^{N / 2} a_{l} a_{m} a_{n}^{*} \int_{0}^{z} a^{2} (z^{'}) X_{l, m, n} (z^{'}, f) \exp [- i \frac{β_{2} z^{'}}{2} {(2 π f)}^{2}] d z^{'}

X_{l, m, n} (z, f) = \frac{1}{2 π | β_{2} z |} \hat{s} (f - \frac{l T}{2 π β_{2} z}) \hat{s} (f - \frac{m T}{2 π β_{2} z}) \hat{s} (- f + \frac{n T}{2 π β_{2} z}) \exp [i \frac{β_{2} z}{2} {(2 π f)}^{2} + i \frac{l m T^{2}}{β_{2} z}]

where P_p is the peak power of each pulse, a_l, a_m, and a_n are the random data patterns at t = lT, mT, and nT, and N is the number of pulses under consideration. ŝ(f) is the spectrum of the signal pulse. The nonlinear interaction term, γa²(z)u_lu_mu_n*, induces a perturbation at a temporal location of t = (l + m−n)T. The nonlinear impairments can be described as a noise with a power spectral density of

\begin{array}{l} ρ_{N L} (f) = \lim_{N \to \infty} \frac{1}{(N + 1) T} 〈 | γ Δ \hat{u} (z, f) |^{2} 〉 \\ = γ^{2} P_{p}^{3} \lim_{N \to \infty} \frac{1}{(N + 1) T} \sum_{l, m, n = - N / 2}^{N / 2} \sum_{l^{'}, m^{'}, n^{'} = - N / 2}^{N / 2} 〈 a_{l} a_{m} a_{n}^{*} a_{l^{'}}^{*} a_{m^{'}}^{*} a_{n^{'}}^{} 〉 Y_{l, m, n} (f) Y_{l^{'}, m^{'}, n^{'}}^{*} (f) \end{array}

Y_{l, m, n} (f) = \int_{0}^{z} a^{2} (z^{'}) X_{l, m, n} (z^{'}, f) \exp [- i \frac{β_{2} z^{'}}{2} {(2 π f)}^{2}] d z^{'}

By taking the sum of ρ_NL(f) and the power spectral density of the ASE noise, ρ_ASE = n_sphν(G − 1)N_A (n_sp = NF/2, G: amplifier gain, N_A: number of amplifiers), it is possible to evaluate the SNR and Q factor analytically. Specifically, the Q factor can be obtained from Q² = SNR = P_av/σ², where P_av and σ² are the signal (average) power and noise power, respectively. The noise power is calculated from ρ_ASE and ρ_NL(f) as

σ^{2} = σ_{ASE}^{2} + σ_{NL}^{2} = \int_{- \infty}^{\infty} [ρ_{ASE} + ρ_{NL} (f)] H (f) d f

where H(f) is the transfer function of the optical filters. From Eqs. (5)-(7), the noise power resulting from fiber nonlinearities, σ_NL², is proportional to the transmission power cubed P_av³,

σ_{N L}^{2} = (η_{S P M} + η_{X P M} + η_{F W M}) P_{a v}^{3}

where the coefficient is given by

\begin{array}{l} η = η_{SPM} + η_{XPM} + η_{FWM} \\ = \lim_{N \to \infty} \frac{γ^{2} {(P_{p} / P_{a v})}^{3}}{(N + 1) T} \int_{- \infty}^{\infty} \sum_{l, m, n = - N / 2}^{N / 2} \sum_{l^{'}, m^{'}, n^{'} = - N / 2}^{N / 2} 〈 a_{l} a_{m} a_{n}^{*} a_{l^{'}}^{*} a_{m^{'}}^{*} a_{n^{'}}^{} 〉 Y_{l, m, n} (f) Y_{l^{'}, m^{'}, n^{'}}^{*} (f) H (f) d f \end{array}

Here we consider the particular case of n = l + m to observe the nonlinear perturbation at t = 0. With a proper choice of l and m in Eqs. (3) and (4), it is possible to identify the individual contribution of nonlinearities, such as self-phase modulation (SPM): l = m = 0, i.e. |u₀|²u₀, IXPM: l = 0, m ≠ 0 or l ≠ 0, m = 0, i.e. |u_l|²u₀ or |u_m|²u₀, degenerate IFWM: l = m ≠ 0, i.e. u_l²u₂_l*, and non-degenerate IFWM: l ≠ m ≠ 0, i.e. u_lu_mu_{l + m}*. Since other N pulses located at t = (l + m−n)T suffer from the same amount of nonlinear perturbation as that of t = 0, ρ_NL(f) can be obtained by calculating the power spectral density for n = l + m and multiplying it by (N + 1). As a result, Eqs. (5) and (9) are simplified as follows:

ρ_{N L} (f) = \frac{γ^{2} P_{p}^{3}}{T} \sum_{l, m = - N / 2}^{N / 2} \sum_{l^{'}, m^{'} = - N / 2}^{N / 2} 〈 a_{l} a_{m} a_{l + m}^{*} a_{l^{'}}^{*} a_{m^{'}}^{*} a_{l^{'} + m^{'}}^{} 〉 Y_{l, m, l + m} (f) Y_{l^{'}, m^{'}, l^{'} + m^{'}}^{*} (f)

η = \frac{γ^{2} {(P_{p} / P_{a v})}^{3}}{T} \int_{- \infty}^{\infty} \sum_{l, m = - N / 2}^{N / 2} \sum_{l^{'}, m^{'} = - N / 2}^{N / 2} 〈 a_{l} a_{m} a_{l + m}^{*} a_{l^{'}}^{*} a_{m^{'}}^{*} a_{l^{'} + m^{'}}^{} 〉 Y_{l, m, l + m} (f) Y_{l^{'}, m^{'}, l^{'} + m^{'}}^{*} (f) H (f) d f

Figure 6 is a schematic showing an example of the way in which the overlap between individual Nyquist pulses induces IXPM and IFWM in a pulse at t = 0. The overlap between u₀ and u₁ introduces IXPM at t = 0 through the term |u₁|²u₀ as indicated in yellow, while the overlap between u₁ and u₂ results in IFWM at t = 0 through u₁²u₂* as shown in green. Figure 5 shows an example of four identical Nyquist pulses for simplicity, but actually they are encoded with a random data sequence, which translates these nonlinearities into stochastic noise.

Fig. 6 Example of IXPM and IFWM contributions from Nyquist pulse overlap.

Download Full Size | PDF

We calculated the Q factor of a 640 Gbaud Nyquist pulse with α = 0 and 0.5 after a 525 km transmission with δD = 0 ps/nm/km from Eqs. (3)-(11). The result is shown in Fig. 7, in which the red curves are the analytical results obtained from Eqs. (3)-(11) and the dots are numerical results extracted from Fig. 3 when δD = 0 ps/nm/km. We set N = 1000, i.e., a total of 1000 pulses are included in the nonlinear interaction. The analytical and numerical results agree well. It is important to note that the analysis enables us to decompose the nonlinearity into SPM, IXPM, and IFWM. Figure 8 shows the decomposition of σ_NL² into IXPM and IFWM. SPM is not shown as it is much lower than IXPM and IFWM. These results indicate that IXPM is the most dominant factor with respect to the nonlinear impairments, and it becomes comparable to the ASE noise level at P_av ~8 dBm. We also evaluated the Q factor analytically by including only IXPM terms in σ_NL² and found that it is almost indistinguishable from the red curves in Fig. 7, and therefore the IXPM contribution is dominant. Such a large IXPM contribution can be attributed to the strong overlap between neighboring bits especially when the transmission power increases. In fact, a Nyquist pulse with α = 0 is more susceptible to nonlinear impairments than one where α = 0.5 as seen in Fig. 8. Figure 8 also shows IXPM and IFWM for a Gaussian pulse. It can be seen that both IXPM and IFWM are reduced with a Gaussian pulse, which confirms that a Gaussian pulse has a higher optimum transmission power than a Nyquist pulse as seen in Fig. 5. It can also be seen in Fig. 8 that the reduction of IFWM with a Gaussian pulse is less significant than that of IXPM, which indicates that a Nyquist pulse is more susceptible to IXPM than IFWM.

Fig. 7 Relationship between Q factor and transmission power for Nyquist pulses in a 640 Gbaud-525 km transmission when δD = 0 ps/nm/km. The red curves show the analytical results obtained from Eqs. (3)-(11).

Download Full Size | PDF

Fig. 8 Individual contributions of IXPM and IFWM to the nonlinear impairments of Nyquist and Gaussian pulses.

Download Full Size | PDF

It is also important to note that the origin of σ_NL² in Eq. (8) is attributed mainly to the factor X_l_,_m_,_n(z,f) in Eq. (4), which is given by the product of the three interleaved spectral profiles: ŝ(f − lT/2πβ₂z), ŝ(f − mT/2πβ₂z), and ŝ(f − nT/2πβ₂z). The product is greatly affected by the spectral shape itself ŝ(f) and the combination of (l, m, n). For example, a Gaussian pulse or a Nyquist pulse with larger α has a spectral shape whose tail decays faster, and therefore the product of interleaved spectra can be less significant. On the other hand, a Nyquist pulse with α = 0 has a rectangular profile and thus can yield a large overlap among the three interleaved spectra especially when l, m, and n are close. We also note that the IXPM term, which is given by the combination of (l, m, n) = (l, 0, l) and (0, m, m), is generally much larger than the IFWM term, since two of the three spectra are identical and thus their product becomes larger.

4. Conclusion

We presented a Q map analysis of dispersion and nonlinear tolerance for Nyquist and Gaussian pulses. By virtue of the ISI-free overlapped TDM, the Nyquist pulse showed a larger Q margin with respect to both dispersion and optical power. This clearly shows the advantage of Nyquist pulses in terms of both dispersion and nonlinear tolerance. We also showed that the optimum transmission power for Nyquist pulses is 2 dB lower than for RZ pulses. The perturbation analysis of nonlinear pulse interactions revealed that the strong overlap between neighboring symbols results in large impairments due to IXPM and IFWM in a high power regime in a Nyquist TDM transmission.

Acknowledgment

This work was supported by the JSPS Grant-in-Aid for Specially Promoted Research (26000009).

References and links

1. M. Nakazawa, T. Yamamoto, and K. R. Tamura, “1.28 Tbit/s-70 km OTDM transmission using third- and fourth-order simultaneous dispersion compensation with a phase modulator,” Electron. Lett. 36(24), 2027–2029 (2000). [CrossRef]

2. H. G. Weber, S. Ferber, M. Kroh, C. Schmidt-Langhorst, R. Ludwig, V. Marembert, C. Boerner, F. Futami, S. Watanabe, and C. Schubert, “Single channel 1.28 Tbit/s and 2.56 Tbit/s DQPSK transmission,” Electron. Lett. 42(3), 178–179 (2006). [CrossRef]

3. H. C. H. Mulvad, M. Galili, L. K. Oxenløwe, H. Hu, A. T. Clausen, J. B. Jensen, C. Peucheret, and P. Jeppesen, “Demonstration of 5.1 Tbit/s data capacity on a single-wavelength channel,” Opt. Express 18(2), 1438–1443 (2010). [CrossRef] [PubMed]

4. M. Nakazawa, T. Hirooka, P. Ruan, and P. Guan, “Ultrahigh-speed “orthogonal” TDM transmission with an optical Nyquist pulse train,” Opt. Express 20(2), 1129–1140 (2012). [CrossRef] [PubMed]

5. K. Harako, D. Seya, T. Hirooka, and M. Nakazawa, “640 Gbaud (1.28 Tbit/s/ch) optical Nyquist pulse transmission over 525 km with substantial PMD tolerance,” Opt. Express 21(18), 21062–21075 (2013). [CrossRef] [PubMed]

6. H. Hu, D. Kong, E. Palushani, J. D. Andersen, A. Rasmussen, B. M. Sorensen, M. Galili, H. C. Hansen Mulvad, K. J. Larsen, S. Forchhammer, P. Jeppesen, and L. K. Oxenløwe, “1.28 Tbaud Nyquist signal transmission using time-domain optical Fourier transformation based receiver,” in CLEO: 2013 Postdeadline, OSA Postdeadline Paper Digest (online) (Optical Society of America, 2013), paper CTh5D.5.

7. H. N. Tan, T. Inoue, T. Kurosu, and S. Namiki, “Transmission and pass-drop operations of mixed baudrate Nyquist OTDM-WDM signals for all-optical elastic network,” Opt. Express 21(17), 20313–20321 (2013). [CrossRef] [PubMed]

8. D. O. Otuya, K. Harako, K. Kasai, T. Hirooka, and M. Nakazawa, “Single-channel 1.92 Tbit/s, 64 QAM coherent orthogonal TDM transmission of 160 Gbaud optical Nyquist pulses with 10.6 bit/s/Hz spectral efficiency,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper M3G.2. [CrossRef]

9. T. Hirooka and M. Nakazawa, “Linear and nonlinear propagation of optical Nyquist pulses in fibers,” Opt. Express 20(18), 19836–19849 (2012). [CrossRef] [PubMed]

10. A. Sahara, H. Kubota, and M. Nakazawa, “Q-factor contour mapping for evaluation of optical transmission systems: soliton against NRZ against RZ pulse at zero group velocity dispersion,” Electron. Lett. 32(10), 915–916 (1996). [CrossRef]

11. G. Baxter, S. Frisken, D. Abakoumov, H. Zhou, I. Clarke, A. Bartos, and S. Poole, “Highly programmable wavelength selective switch based on liquid crystal on silicon switching elements,” in Optical Fiber Communication Conference (OSA, 2006), paper OTuF2. [CrossRef]

12. S. Kumar, S. N. Shahi, and D. Yang, “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20(25), 27740–27755 (2012). [CrossRef] [PubMed]

13. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photonics Technol. Lett. 12(4), 392–394 (2000). [CrossRef]

14. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightwave Technol. 27(14), 2916–2923 (2009). [CrossRef]

15. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

Q-factor analysis of nonlinear impairments in ultrahigh-speed Nyquist pulse transmission

Abstract

1. Introduction

2. Performance comparison of RZ and Nyquist TDM transmissions using Q-map analysis

3. Analysis of nonlinear impairments in Nyquist TDM transmission

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (8)

Equations (11)

Optics Express