Linear and nonlinear propagation of optical Nyquist pulses in fibers

Toshihiko Hirooka; Masataka Nakazawa

doi:10.1364/OE.20.019836

1. Introduction

Return-to-zero (RZ) optical transmission such as an optical time-division multiplexed (OTDM) system has traditionally adopted Gaussian or sech waveforms as an optical pulse shape [1]. For example, as is well known, an optical soliton propagating in a fiber with uniform anomalous dispersion has a sech pulse [2]. In a dispersion-managed fiber, the shape of a dispersion-managed soliton approaches a Gaussian profile as the dispersion management becomes stronger [3]. Gaussian and sech functions are also known to be steady-state waveforms in an actively or passively mode-locked laser cavity [4,5]. However, these waveforms generally occupy a large bandwidth in the frequency domain due to their rapidly decreasing tails in the time domain. Therefore, in terms of increasing the spectral efficiency, they may not always be optimum waveforms.

Recently, we proposed a new type of optical pulse, which we call an “optical Nyquist pulse”, whose shape is given by the sinc-function-like impulse response of the Nyquist filter [6]. Nyquist filters have been widely deployed in wireless or microwave communication to narrow the signal spectrum to the channel bandwidth owing to their rectangular or raised-cosine spectral profiles [7]. It is also well known that the impulse response has a relatively slowly decaying tail compared with the symbol period, but accompanied by periodic zero-crossing points. This property makes it possible to realize the time-interleaving of individual isolated Nyquist pulses to a higher symbol rate without intersymbol interference (ISI), in spite of the strong overlap between adjacent pulses. Therefore, there is a possibility of achieving an ultrahigh data rate and ultrahigh spectral efficiency simultaneously. Our scheme proposed in [6] is different from the conventional Nyquist filtering technique, which is applied to a baseband signal to shape the data sequence [8–11]. It should be noted that this shaped sequence is sometimes called a Nyquist pulse, but this is not an isolated pulse despite the name. The optical Nyquist pulse can be easily generated with a spectrum manipulation technique for example by the spatial intensity and phase modulation of spectral components [6].

To take advantage of these ideal properties of optical Nyquist pulses for long-haul transmission, it is important to maintain them during the propagation. In this paper, we analyze how the periodic zero-crossing property is maintained or affected by fiber dispersion and nonlinearity. We determined that the nonlinear Nyquist pulse propagation depends strongly on the sign of the dispersion, and the ISI-free property is less affected by the nonlinearity in a normal GVD fiber.

2. Optical Nyquist pulse propagation in the presence of dispersion

In a wireless system, a Nyquist pulse is identified as the impulse response of a bandwidth-limited signal-shaping filter for a data sequence in the baseband [12]. The same principle has also been applied in a digital coherent optical transmission, in which the Nyquist filter is employed as a baseband data signal at the transmitter in the digital domain [8–11]. However, there have been no concepts that utilize Nyquist pulses as a pulse train for TDM. The TDM transmission of optical Nyquist pulses that we proposed in [6] features ISI-free bit interleaving in the optical domain. However, with Nyquist pulse transmission in optical fibers it is very important to take account of the propagation characteristics in the presence of dispersion and nonlinearity, which are irrelevant in a wireless system. In this section, we provide an analytical representation and numerical analysis of Nyquist pulse propagation through a dispersive medium.

2.1 Analytical approach

The amplitude waveform and frequency spectrum of the Nyquist pulse are defined as follows:

r (t) = \frac{\sin (π t / T)}{π t / T} \frac{\cos (α π t / T)}{1 - {(2 α t / T)}^{2}}, R (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 T is the symbol period, and α (0 ≤ α ≤ 1) is a roll-off factor. For example, for 160 Gbaud OTDM, we set T = 6.25 ps. The signal bandwidth, defined as the width of both edges of the spectrum tail (not the FWHM) is given by B = (1 + α)/T. It should be noted that the pulse waveform has an oscillating tail, but it becomes zero at t = nT (n: integer), and therefore there is no ISI at any symbol interval in the OTDM pulse train, despite a strong overlap with neighboring pulses. In analytical literature, the ISI-free condition for the pulse u(t):

u (n T) = {\begin{matrix} 1, & n = 0 \\ 0, & n \neq 0 \end{matrix},

is equivalent to the following relationship in terms of the frequency spectrum U(f) [12]:

\sum_{m = - \infty}^{\infty} U (f + m / T) = T .

It can be easily confirmed that r(t) and R(f) in Eq. (1) satisfy this condition.

In the presence of group-velocity dispersion (GVD) and nonlinearity, the condition of Eq. (3) cannot be maintained. For example, chromatic dispersion β(ω) is responsible for the phase change in the frequency domain, U’(ω) = U(ω)exp[iβ(ω)L], whereas fiber nonlinearity modifies both the spectral amplitude and phase. Due to these spectral changes, Eq. (3) can no longer be satisfied. Nevertheless, it is very important to determine in detail how the zero-crossing property and its periodicity are affected by the dispersion and nonlinearity.

As a first step, we analyzed the effect of GVD on optical Nyquist pulse propagation by solving the following evolution equation of the waveform u(z, t):

i \frac{\partial u}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u}{\partial t^{2}} = 0,

where β₂ is the second-order dispersion coefficient (β₂ = d²β/dω²). Equation (4) can be solved analytically, and the waveform at z = L is given by

\begin{array}{l} u (L, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} U (0, ω) \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω \\ = \frac{T}{4 π} \int_{- \frac{π}{T} (1 + α)}^{- \frac{π}{T} (1 - α)} {1 + \sin (\frac{T ω + π}{2 α})} \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω \\ + \frac{T}{2 π} \int_{- \frac{π}{T} (1 - α)}^{\frac{π}{T} (1 - α)} \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω + \frac{T}{4 π} \int_{\frac{π}{T} (1 - α)}^{\frac{π}{T} (1 + α)} {1 - \sin (\frac{T ω - π}{2 α})} \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω, \end{array}

where U(0, ω) is the Fourier transform of u(0, t), and we used Eq. (1). Equation (5) can be simplified into the following form.

\begin{array}{l} u (L, t) = \frac{T}{4 π} \int_{- \frac{π}{T} (1 + α)}^{\frac{π}{T} (1 + α)} \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω + \frac{T}{4 π} \int_{- \frac{π}{T} (1 - α)}^{\frac{π}{T} (1 - α)} \exp (i \frac{β_{2} L}{2} ω^{2} - i ω t) d ω \\ - \frac{T}{2 π} \int_{\frac{π}{T} (1 - α)}^{\frac{π}{T} (1 + α)} \sin (\frac{T ω - π}{2 α}) \cos ω t \exp (i \frac{β_{2} L}{2} ω^{2}) d ω . \end{array}

After some algebra, we finally obtain the following result:

\begin{array}{l} u (L, t) = \frac{T}{8} \sqrt{\frac{1}{i π β_{2} L / 2}} {F (t; \frac{π}{T} (1 + α)) - F (t; - \frac{π}{T} (1 + α)) + F (t; \frac{π}{T} (1 - α)) - F (t; - \frac{π}{T} (1 - α))} \\ + \frac{T}{16} \sqrt{\frac{i}{π β_{2} L / 2}} [\exp (\frac{i π}{2 α}) {F (t + \frac{α}{2}; \frac{π}{T} (1 + α)) - F (t + \frac{α}{2}; \frac{π}{T} (1 - α))} \\ + \exp (- \frac{i π}{2 α}) {F (- (t + \frac{α}{2}); \frac{π}{T} (1 + α)) - F (- (t + \frac{α}{2}); \frac{π}{T} (1 - α))} \\ + \exp (- \frac{i π}{2 α}) {F (t - \frac{α}{2}; \frac{π}{T} (1 + α)) - F (t - \frac{α}{2}; \frac{π}{T} (1 - α))} \\ + \exp (\frac{i π}{2 α}) {F (- (t - \frac{α}{2}); \frac{π}{T} (1 + α)) - F (- (t - \frac{α}{2}); \frac{π}{T} (1 - α))}] . \end{array}

Here, F(τ; β) is defined as

F (τ; β) = \exp (- i \frac{τ^{2}}{2 β_{2} L}) erfi [\sqrt{\frac{i β_{2} L}{2}} (- \frac{τ}{β_{2} L} + β)],

where

erf i (z) = \frac{erf (i z)}{i} = \frac{2}{i \sqrt{π}} \int_{0}^{i z} \exp (- t^{2}) d t

is an imaginary error function. When α = 0, Eq. (7) is simply written as

u (L, t) = \frac{T}{4} \sqrt{\frac{1}{i π β_{2} L / 2}} {F (t; \frac{π}{T}) - F (t; - \frac{π}{T})} .

These analytical representations provide useful insights as regards the influence of GVD on Nyquist pulse propagation. In particular, when α = 0, we can derive an explicit formula as regards the periodicity in the tail and the increase in the zero level in the presence of dispersion. We first take a derivative of |u(L, t)|² from Eq. (10) by using the formula:

\frac{d}{d z} erfi (z) = \frac{2 \exp (z^{2})}{\sqrt{π}} .

After some algebra, we obtain the following result:

\begin{array}{l} \frac{\partial}{\partial t} {| u (L, t) |}^{2} = \frac{i T^{2}}{2 β_{2} L π \sqrt{π}} {erfi [\sqrt{- \frac{i β_{2} L}{2}} (- \frac{t}{β_{2} L} + \frac{π}{T})] - erfi [\sqrt{- \frac{i β_{2} L}{2}} (- \frac{t}{β_{2} L} - \frac{π}{T})]} \\ \times \exp [- i \frac{t^{2}}{2 β_{2} L} - i \frac{β_{2} L}{2} {(\frac{π}{T})}^{2}] \sin \frac{π t}{T} . \end{array}

Equation (12) indicates that (∂/∂t)|u(L, t)|² = 0 at t = nT because of the factor sin(πt/T), and thus |u(L, t)|² takes a minimum value at t = nT, i.e., at every symbol interval. Therefore, the periodicity in the tail is preserved even in the presence of dispersion.

As regards the increase in the zero level, we approximate Eq. (10) for β₂L << 1 by using the following series expansion:

erfi (z) = - i + \frac{\exp (z^{2})}{\sqrt{π}} (\frac{1}{z} + \frac{1}{2 z^{3}} + \dots) for z > > 1.

By applying Eq. (13) to Eq. (10) up to the order 1/z, we obtain

u (t) ~ \frac{T \exp (i \frac{β_{2} L}{2} {(\frac{π}{T})}^{2})}{4 π \sqrt{i β_{2} L / 2}} {\frac{\exp (- i \frac{π}{T} t)}{- \sqrt{\frac{i}{2 β_{2} L}} t + \sqrt{\frac{i β_{2} L}{2}} \frac{π}{T}} + \frac{\exp (i \frac{π}{T} t)}{\sqrt{\frac{i}{2 β_{2} L}} t + \sqrt{\frac{i β_{2} L}{2}} \frac{π}{T}}} .

At t = nT, the intensity of Eq. (14) is given by

| u (n T) |^{2} = \frac{{(β_{2} L)}^{2}}{{{(n T)}^{2} - {(β_{2} L \frac{π}{T})}^{2}}^{2}} .

This gives us a simple and useful description of how gradually the intensity grows from zero as the dispersion increases as we describe below.

2.2 Numerical analysis

Figure 1 shows numerical results for the propagation of an optical Nyquist pulse with α = 0, 0.5, 1, and T = 6.25 ps, which corresponds to a symbol rate of 160 Gbaud, as a function of the cumulative dispersion |β₂L|, where L is the fiber length. |β₂L| = 10 ps² is equivalent to a 0.46 km-long standard single-mode fiber (β₂ = − 21.67 ps²/km) without any dispersion compensation, and corresponds to the dispersion length for a 5.3 ps Gaussian pulse. In practice, such a short optical pulse is transmitted over a much longer distance with dispersion compensation. It can be seen that, as the pulse propagates, the bottom of the oscillating tail increases gradually and does not cross zero. With α = 1, the periodic tail oscillation is too small to be visible even before transmission, and the pulse distortion resembles that with conventional pulses. Although the ISI-free property is not maintained in the presence of GVD, the period of the tail oscillation remains the same as the symbol period, as we derived analytically above. The pulse broadening is slower with a lower α, which can be understood from Eq. (1) because the bandwidth B is narrower. The analytical result for α = 0 shown in Eq. (10) is also plotted by the dots on the right in Fig. 1(a), which coincides exactly with the numerical result.

Fig. 1 Propagation of an optical Nyquist pulse in the presence of GVD with T = 6.25 ps and α = 0 (a), 0.5 (b), and 1 (c). The left figures show the pulse evolution as a function of the cumulative dispersion |β₂L|, and the right figures are the waveforms for |β₂L| = 0 (black) and 10 ps² (red). The dots in the right figure of (a) show the analytical result given by Eq. (10).

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Figure 2 shows the propagation of an optical Nyquist pulse (α = 0.5) in a periodically dispersion-managed fiber where each span (75 km) is composed of a 50 km anomalous dispersion (D = 21 ps/nm/km) segment and a 25 km normal dispersion (D = −40 ps/nm/km) segment so that the cumulative dispersion per span is 1 ps/nm. In this case, |β₂L| = 10 ps² corresponds to propagation over 600 km. The waveform changes quickly within a period, but the long-scale waveform evolution is the same as in Fig. 1(b). It should be noted that the practical residual dispersion can be greatly reduced (e.g., < 0.1 ps/nm) by controlling the dispersion or the length of each segment more precisely, so that the transmission reach can be further extended.

Fig. 2 Propagation of an optical Nyquist pulse (T = 6.25 ps and α = 0.5) in a periodically dispersion-managed fiber. The left figure shows the pulse evolution within a 75 km span, and the right figure shows the pulse waveform at each period.

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As a measure of the influence of ISI, we define an “extinction ratio” of the signal intensity at the adjacent symbol point t = T with respect to the peak intensity:

η = \frac{{| u (T) |}^{2}}{{| u (0) |}^{2}},

and evaluated its dependence on the pulse waveform and the amount of GVD. Figure 3 shows the numerical result of η for optical Nyquist pulses with α = 0, 0.5, and 1, and Gaussian and sech pulses with 2.5, 3.5, and 4.5 ps FWHM (corresponding to a 160-Gbaud duty cycle of 40, 56, and 72%, respectively). As shown in Fig. 3(b), the zero level starts to rise as the dispersion increases, and thus the zero-ISI property is not rigorously maintained. But nevertheless, the growth in the zero level is slow as shown in Fig. 3(a), and its effect on ISI is still smaller than that of a Gaussian or sech pulse. These results clearly indicate the benefit of optical Nyquist pulses in terms of dispersion tolerance.

Fig. 3 Extinction ratio of the pulse intensity between the center and the adjacent symbol point, η, as a function of the cumulative dispersion |β₂L|. Red, blue, and green curves are the results obtained with optical Nyquist, Gaussian, and sech pulses, respectively. (b) An expanded view of the dotted area in (a).

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Figure 4 compares η with the analytical result, where η can be written analytically as

η = \frac{{(β_{2} L)}^{2}}{{T^{2} - {(β_{2} L \frac{π}{T})}^{2}}^{2}} .

To derive Eq. (17), we referred to Eq. (15) for |u(T)|², and used |u(0)|² ~1 for β₂L << 1 in Eq. (10). It can be seen that the initial growth can be well described by the analytical representation given by Eq. (17) in the |β₂L| < 4 ps² regime.

Fig. 4 Comparison of η in the numerical and analytical results (Eq. (17)) for an optical Nyquist pulse with α = 0.

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To evaluate the transmission performance of a Nyquist TDM signal, it is also very important to carry out a numerical simulation with an optical Nyquist pulse data train. Figure 5 shows the eye diagram of a 160 Gbaud Nyquist pulse data sequence under the same condition as Fig. 1, in which the pulses are OOK modulated with a 2⁷−1 PRBS at 40 Gbit/s and OTDM multiplexed with a non-uniform phase relationship between adjacent tributaries. The figures on the right show the eye diagram of the demultiplexed 40 Gbaud signal. Here the gate width was 1.0 ps, which was used in the experiments [6,13] as an ultrafast optical sampler to extract only the ISI-free point from the overlapped TDM signal. The simulation only includes the dispersion, and other effects such as ASE noise or filtering are disregarded for the sake of simplicity. The eye is clearly open in Fig. 5(a), and gradually degrades as the dispersion increases as shown in Fig. 5(b) and 5(c).

Fig. 5 Eye diagram of a 160 Gbaud OOK optical Nyquist pulse train with α = 0.5 (left) and the demultiplexed 40 Gbaud signal (right) in the presence of GVD. The power is normalized to the peak intensity of a single Nyquist pulse without dispersion as indicated by the blue lines.

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For comparison, eye diagrams with a 2.5 ps Gaussian pulse are also provided in Fig. 6 . Here we chose a gate width of 4.5 ps, which is a typical value for demultiplexing from 160 to 40 Gbaud for example with an EA modulator. With a dispersion of |β₂L| = 5 ps², Gaussian pulses are already greatly distorted as can be seen in Fig. 6(b), while with Nyquist pulses “1” and “0” are still sufficiently separated as shown in Fig. 5(b). When the dispersion is increased to |β₂L| = 10 ps², the eye opening becomes much worse as shown in Fig. 6(c). However, the eye is still open with Nyquist pulses as seen in Fig. 5(c).

Fig. 6 Eye diagram of a 160 Gbaud OOK Gaussian pulse train (left) and the demultiplexed 40 Gbaud signal (right) in the presence of GVD.

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3. Optical Nyquist pulse propagation in the presence of nonlinearity

Next we considered the effect of self-phase modulation (SPM) on optical Nyquist pulse propagation by solving the following evolution:

i \frac{\partial u}{\partial z} + γ {| u |}^{2} u = - i \frac{Γ}{2} u .

Here, γ is a nonlinear coefficient defined as γ = ωn₂/cA_eff, where n₂ is the Kerr coefficient and A_eff is the effective core area. We also took account of the fiber loss Γ, which was amplified periodically at an interval of L_amp where the amplified spontaneous emission noise was neglected for simplicity. Since the pulse intensity is invariant with SPM, we focus on the evolution of the optical spectrum.

Figure 7 shows the evolution of the optical Nyquist pulse spectrum (α = 0.5, T = 6.25 ps) as a function of distance, where we set n₂ = 2.3 × 10⁻²⁰ m²/W, A_eff = 60 μm², Γ = 0.2 dB/km, and L_amp = 75 km. The power was set at 6, 10, and 13 dBm assuming a 160 Gbit/s DPSK transmission. The corresponding nonlinear lengths, z_NL = 1/γP_{p eff} (P_{p eff} = P_p[{1−exp(−ΓL_amp)} / ΓL_amp] is the effective power taking account of the loss against the input peak power P_p), are 504, 201, and 101 km, respectively, which is also shown explicitly in Fig. 7. These figures indicate that the spectrum is distorted gradually during the propagation until z ~z_NL. This degradation is slower than SPM for a Gaussian pulse [14]. However, for longer distances z > z_NL, the distortion becomes much faster.

Fig. 7 Evolution of an optical Nyquist pulse spectrum with T = 6.25 ps and α = 0.5 in the presence of SPM. The left figures show the spectrum evolution as a function of distance, and the right figures are the input spectrum (black) and the spectrum after 525 km propagation (red).

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SPM alone does not modify the pulse intensity in the time domain, and thus the ISI-free property is maintained. However, when combined with GVD, the evolution becomes more complicated. As a next step, we analyzed the effect of the interplay between GVD and SPM in an optical Nyquist pulse transmission by solving the following nonlinear Schrödinger equation:

i \frac{\partial u}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u}{\partial t^{2}} + γ {| u |}^{2} u = - i \frac{Γ}{2} u .

Figure 8 shows the evolution of an optical Nyquist pulse and spectrum in the presence of SPM and an anomalous GVD of 0.1 ps/nm/km. Here, the transmission power was set at 6, 10, 13 dBm in Fig. 8(a)–8(c), respectively, as in Fig. 7. The dispersion length in the present case is z_dis = t²_FWHM / |β₂| = 214 km as explicitly shown in Fig. 8. Namely, Fig. 8(a)–8(c) correspond to z_dis < z_NL, z_dis ~z_NL, and z_dis > z_NL, respectively. The individual waveforms in Fig. 8 are plotted after compensating for the cumulative dispersion at each distance. It can be seen that the spectrum becomes narrower as the nonlinearity increases in contrast to Fig. 7, and the waveform is broadened accordingly. In addition, the oscillation tail gradually disappears, and it becomes more difficult to maintain the periodic zero-crossing property with larger SPM.

Fig. 8 Evolution of an optical Nyquist pulse waveform (left) and spectrum (right) with T = 6.25 ps and α = 0.5 in the presence of SPM and anomalous GVD (0.1 ps/nm/km).

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Next we calculated the optical Nyquist pulse propagation in the presence of SPM and a normal GVD of −0.1 ps/nm/km. The results are shown in Fig. 9 . The deformation in the spectrum is less significant than with anomalous GVD, where the distortion exists mainly around the peak, and the spectral bandwidth remains nearly identical even with strong SPM such as that shown in Fig. 9(c). Furthermore, the oscillation in the tail still exists in the waveform, with small deviations in the periodicity associated with pulse narrowing.

Fig. 9 Evolution of an optical Nyquist pulse waveform (left) and spectrum (right) with T = 6.25 ps and α = 0.5 in the presence of SPM and a normal GVD (−0.1 ps/nm/km).

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To evaluate the way in which the ISI-free property is affected by GVD and SPM, we calculated the change in the zero-crossing point around t = T (6.25 ps) and its deviation for various transmission power values under normal and anomalous GVD. Figure 10(a) shows the numerical result. Here, the zero-crossing deviation is defined as |T′−T|/T, where T′ is the zero crossing location after propagation. We find that an anomalous GVD makes the zero-crossing point shift outwards, while a normal GVD results in an inward shift. In addition, there is less zero-crossing deviation with a normal GVD. The extinction ratio η defined in Eq. (16) is also calculated for these data, which are plotted in Fig. 10(b). It is seen that a lower η can be obtained for a normal GVD especially for a higher power. These results indicate that optical Nyquist pulse propagation tends to be more stationary in a normal GVD fiber, as opposed to conventional RZ pulses such as optical solitons, which are more compatible with an anomalous GVD.

Fig. 10 (a) The change in the zero-crossing point around t = T (6.25 ps) and its deviation for various transmission power values. The open and closed circles correspond to normal and anomalous dispersion (−0.1 and + 0.1 ps/nm/km), respectively. (b) Extinction ratio η corresponding to (a).

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4. Conclusion

We presented analytical and numerical results regarding the role of dispersion and nonlinearity in optical Nyquist pulse propagation, with a special focus on the way in which the periodic zero-crossing property is maintained or affected, which is crucial for ISI-free characteristics in Nyquist TDM transmission. GVD results in an increase in the zero level in the tail oscillation, while the oscillation period remains the same as the symbol period. In the presence of SPM, GVD causes different forms of waveform distortion depending on the sign. Specifically, normal GVD yields more stationary behavior than conventional RZ pulses, which are more compatible with anomalous GVD. These results confirm the large tolerance of optical Nyquist pulses with respect to GVD, as we recently verified experimentally [13], as well as to SPM especially when combined with normal GVD.

References and links

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8. K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express 5(1), 6–10 (2008). [CrossRef]

9. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010). [CrossRef]

10. X. Zhou, L. E. Nelson, P. Magill, B. Zhu, and D. W. Peckham, “8x450-Gb/s, 50-GHz spaced, PDM-32QAM transmission over 400 km and one 50 GHz-grid ROADM,” in Optical Fiber Communication Conference (OFC 2011), paper PDPB3.

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13. T. Hirooka, P. Ruan, P. Guan, and M. Nakazawa, “Highly dispersion-tolerant 160 Gbaud optical Nyquist pulse TDM transmission over 525 km,” Opt. Express 20(14), 15001–15007 (2012). [CrossRef] [PubMed]

14. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

Linear and nonlinear propagation of optical Nyquist pulses in fibers

Abstract

1. Introduction

2. Optical Nyquist pulse propagation in the presence of dispersion

2.1 Analytical approach

2.2 Numerical analysis

3. Optical Nyquist pulse propagation in the presence of nonlinearity

4. Conclusion

References and links

Cited By

Figures (10)

Equations (19)

Optics Express