1. Introduction

Chromatic dispersion (CD) is one of the major sources of transmission degradation in high-speed optical communication systems. Neglecting the high order CD effect in a dispersive fiber [e.g., in the single mode fiber (SMF)], the low-pass transfer function of the fiber has the form (cf. e.g., Refs. [1, 2])

with D(λ) the fiber CD parameter at wavelength λ, L the fiber length, and f the frequency deviation with respect to the carrier frequency. Eq. (1) means that, for a system within its CD-limited transmission, the CD-induced signal distortion is mainly caused by those spectral components that are away from the carrier frequency. Due to the bit correlation in an optical duobinary (ODB) system, the NRZ signal spectrum is compressed by a factor of 2, compared to that in the NRZ-OOK system. In a RZ-ODB system, its signal spectrum is also more centered at its carrier frequency than the RZ-OOK one [3, 4]. As a result, an ODB system is more robust to CD than the OOK system, with its demodulation scheme much simplier than that using differential phase-shift keying (DPSK) modulation.

A signal with narrower bandwidth should suffer less from CD. However, whether it is the key reason for the CD tolerance of an ODB system or not, there were two competing views [5]. The results of this work show that the reduced bandwidth effect is not the key factor to explain the measured CD penalties of ODB modulation. Instead, the phase effect of the correlative modulation plays a dominant role. Here we explicitly show how to control the correlation-induced phase effect in reality and how it reduces the CD impact in an ODB system.

A conventional ODB transmitter has been typically implemented by using a Mach-Zehnder (MZ) modulator driven with 3-level signal generated by using electrical low-pass filter (LPF). Although this LPF is not required for generating an ideal 3-level electrical signal, it is important to get the improved CD tolerance [6]. This LPF also causes problems such as word length dependent system performance degradation [7]. To remove this problem and get longer CD-limited transmission, the phase effect of the ODB modulation has attracted growing attention in recent years. In Ref. [5], based on the linear combination of two elementary signals, a new family of combined amplitude-phase shift (CAPS) codes was proposed. CD tolerance better than that of conventional ODB (i.e., a special case of its order-1 code) was theoretically predicted. As a result of one implementation of order-1 CAPS line coding realized by filtering DPSK bits with filter bandwidth equal to 0.67/T_b (T_b is the bit period.) [8], improved CD tolerance was confirmed, although not so good as those plotted in Ref. [5]. In Ref. [9], the concept to optimize the phase difference between marks and spaces was proposed to get the improved differential binary phase-shift keying (DBPSK) bits (without bit correlation). To convert the DBPSK bits into ODB bits, a narrow bandwidth optical filter (~0.56/T_b) was introduced. In Ref. [10], by introducing a dual-arm MZ modulator, an electrical-binary-signal-based duobinary transmitter was proposed to obtain the correlation between adjacent bits with adjusted phase difference between them.

In this work we further show that, a new cost-effective ODB transmitter can be obtained by adjusting the delay time of the delay-and-add circuit in the conventional ODB transmitter without LPF [cf. Fig. 1(b)]. As shown in the next section, the concept behind this partial bit delay correlative modulation (PBDCM) method is the tunable phase correlation between two adjacent bits [see the phase factor in Eq. (3)], rather than the fixed correlation in conventional 1-bit delay modulation.

Note that this PBDCM is not a specific implementation of the CAPS method of Ref. [5]. Indeed, as shown in Eqs. (3)–(5), the PBDCM signal can be viewed as a kind of tunable phase combination of one basic sample, while the CAPS signal can be equivalently expressed as the real combination of two basic samples having two fixed periods. Ref. [9] differs from the PBDCM method in that, like the LPF in the conventional ODB transmitter, its narrow bandwidth optical filter that was required to generate the ODB signal from the uncorrelated DBPSK signal reduced the CD tolerance compared to the conventional ODB with LPF [9, 10], while the PBDCM method does not need such optical filter, as shown in Fig. 1(b). [The optical filter in Fig. 1(a) is used to properly suppress the edged spectra of signal and ASE noise.] Unlike Ref. [10], where any variation in delay time would also change the signal duty cycle [11], which further influenced the measured CD-induced penalty, the duty cycle of any PBDCM signal (NRZ or RZ) is not affected by the delay time.

Indeed, for a given CD penalty, e.g. ~3dB, the PBDCM method can increase the CD-limited transmission to 2~3 times of that using 1-bit delay modulation, depending on the format of the input signal s⃗_in(t) in Fig. 1(a). For comparison, we also consider the optical and receiver filter bandwidth effects on the CD tolerance, which are related with the reduced signal bandwidth caused by correlation. As indicated below, the optimized filter bandwidth (OFB) method can increase the CD-limited transmission by less than 50%.

Polarization-mode dispersion (PMD) and polarization dependent loss (PDL) further impair the system performance at high data rates. Here, their first order effects on the CD tolerance are studied by extending the Karhunen-Loève series expansion method [1, 12] to case of ODB. In fact above properties of the PBDCM and OFB methods are still valid for the systems with PMD and PDL.

Our calculations yield basic CD penalty features of other simulation and experimental results [6, 10, 13, 14, 15, 16].

2. Theoretical models

Figure 1(a) shows the low-pass equivalent model used for our study. At the output of the pulse carver, which is used to transform a NRZ signal into a RZ one [16], the optical signal s⃗_in(t) is launched into the system with lumped CD, (first-order) PMD and PDL, in the linear regime. The ASE noise from the flat gain amplifier G is partially polarized due to the PDL [12]. The optical and electrical (i.e., receiver) filters are used to limit the spectral bandwidths of the signal and ASE noise. In this work, B_o (B_r) represents the 3dB bandwidth of the Fabry-Pèrot optical filter (fifth-order Bessel receiver filter), respectively [2].

 

Fig. 1. (a) The low-pass equivalent optical model used to study the performance of an optical duobinary (ODB) system with partially polarized ASE noise. The correlative modulation can be realized either by a duobinary encoder (i.e., a delay-and-add circuit) in (b) or by a Mach-Zehnder (1-bit) delay-interferometer (DI) in (c). Low-pass filter (LPF) in conventional ODB transmitter is used to further improve the CD tolerance. Without the LPF, Tx (b) and (c) are equivalent. Details of d^ODB(t) and d^DPSK(t) in (d) are given in Appendix.

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To our knowledge, the most cost-effective ODB transmitters (Tx) is the one shown in Fig. 1(b), where the ideal binary NRZ electrical signal is correlatively modulated by a delay-and-add circuit, with T_b the bit period. Without the LPF, an ideal (no band limiting) ODB signal can be obtained by applying the correlated electrical signal to the RF input of a Mach-Zehnder modulator (MZM). Note that in Fig. 1(b) [also (c)], the DPSK encoder is used to denote a NRZ data generator followed by an invertor and a differential encoder introduced in Fig. 7 of Ref. [17]. This is because, according to Ref. [17], the output of this differential encoder, i.e., the input of the delay-and-add circuit in Fig. 1(b), will produce the same codes as the DPSK encoder used in a balanced DPSK system (see the relevant discussion in Appendix or Ref. [12]).

Obviously the same ODB input signal can also be generated by the Tx shown in Fig. 1(c), where the constructively correlated optical DPSK bits can be obtained at the C-port [2, 12]. Assuming linear modulation, both Tx implementations in Figs. 1(b) and 1(c) yield the same BER at the receiver. In this work, model in Fig. 1(a) with Tx in Fig. 1(c) is used for the following discussion. Conclusions thus obtained can be applied to the case using Tx (b) (without LPF).

Tx implementation in Fig. 1(c) directly indicates that its output s^ODB(t) relates with MZM output s^DPSK(t) by

which means that in frequency domain we have

with $A_l^{\mathrm{ODB}}=A_l^{\mathrm{DPSK}}{e}^{\frac{-\mathrm{j\pi lT}}{N{T}_b}}\cos \left(\frac{\mathrm{\pi lT}}{N{T}_b}\right)\mathrm{and}T=T_b.$ and T=T_b. In (3), P(l/NT_b) is the Fourier coefficient of the elementary input pulse p(t) at f_l=1/NT_b and N is the bit number of the de Bruijn sequence considered in Appendix. Eq. (3) shows the phase effect of the correlative modulation in an explicit way.

When the fixed delay time T_b in Fig. 1(c) is replaced with a tunable parameter T, which can be realized in reality, Eq. (3) still works. In this case, the standard ODB signal Eq. (2) changes to

As usual, rect(t) equals one when 0≤t<T_b. Outside this time interval, it is zero. Logic values {c^DPSK_k} are simply the repetition of {a^DPSK_m} introduced in the Appendix, while E_b is the optical energy per transmitted bit [1, 2]. By comparing Eq. (4) with the signal modulated with order-1 CAPS approach, which can be equivalently expressed as (cf. Eq. (10a) of Ref. [5] with α=0.5)

we can view the PBDCM method as a kind of tunable time shift combination of one basic sample, which results in phase shift in frequency domain, as shown in Eq. (3), whereas the CAPS approach is equivalent to a real combination of two basic samples with two fixed periods.

3. CD tolerance improved by optimizing filter bandwidths

One way to study the CD impact on the system performance is to calculate the CD-induced power penalty, which concerns, for a given CD effect, how much signal power (or OSNR) needs to be increased so that the BER can keep its target value. In this work, the CD-induced penalty is calculated as a function of CD index ξ defined as ξ=10^-4 D(λ)LR ² in the unit of 10⁴(Gb/s)²ps/nm [2], where R=1/T_b is the bit rates in Gb/s. A L=100km SMF with R=10Gb/s and λ=1550nm yields D(λ)≈17.5 ps/nm/k and ξ≈17.5[10⁴(Gb/s)²ps/nm]. Note that such CD penalty-ξ curve can be applied to various systems with different fiber types, transmission distances, and bit rates. Alternatively, the CD impact can be studied by calculating the required OSNR at the target BER as a function of ξ. Here the required OSNR is defined to be E_p/N ₀, with N ₀ the two-sided power spectral density of ASE noise [1, 2, 12].

Table 1. Bandwidths of five ODB systems

View Table

To get the lowest required OSNR at BER=10^-9, the bandwidths of the optical and receiver filters are adjusted to their optimized values. Since the optimized filter bandwidths will be affected by parameters like CD index ξ, PMD value τ (τ≡DGD), and PDL value α, we consider some NRZ and RZ ODB systems with their filter bandwidths detailed in Table 1. Note that NRZ-DuoB and NRZ-DuoB^* denote their bandwidths are optimized at ξ=0 and ξ=12.5 (with τ=0 and α=0), whereas the bandwidths of the RZ-DuoB, RZ-DuoB^*, and RZ-DuoB⁺ systems are optimized at ξ=0 (τ=0), ξ=12.5 (ξ=0), and ξ=12.5 (τ=0.2T_b~0.4T_b), respectively, with α=0. The penalties of NRZ- and RZ-OOK systems, with their filter bandwidths optimized at ξ=0, τ=0, and α=0 and detailed in Ref. [2], are calculated for comparison.

 

Fig. 2. CD-induced power penalty at BER=10^-9 with negligible PMD (DGD≡τ=0) and PDL (α=0) [(a), (b)], and non-negligible PMD and PDL [(c), (d)]. The CD penalties of the NRZ- and RZ-OOK systems, with their filter bandwidths {B_o, B_r} being {1.6/T_b,0.60/T_b} (NRZ) and {1.8/T_b,0.65/T_b} (RZ) [2], are plotted for comparison. The filter bandwidths of the NRZ and RZ ODB systems, optimized at different values of ξ and τ, are given in Table 1. Insets: the required OSNRs vs ξ at BER=10^-9 with negligible PMD and PDL (a)–(b), τ/T_b=0.2 (c), and τ/T_b=0.4 (d). CD index ξ=17.5 [10⁴(Gb/s)²ps/nm] corresponds to a SMF of ~100km long, when R=10Gb/s and λ=1550nm.

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As mentioned in section 1, the CD penalty of an ODB system using NRZ or RZ format is smaller than that of NRZ- or RZ-OOK, which is confirmed in Figs. 2(a) and 2(b). In addition, the thick solid (NRZ-DuoB) and thin solid (NRZ-DuoB^*) curves in Fig. 2 (a) show that, for an ODB system using NRZ format, filter bandwidths optimized at different ξ result in almost the same CD penalty. However, as depicted in Fig. 2(b) for the RZ format, the CD penalty curve with bandwidths optimized at nonzero ξ (e.g., ξ= 12.5, thin solid) can be significantly lower than the case optimized at ξ=0 (thick solid). To understand the different effects of the optimized bandwidth method on NRZ and RZ formats, we consider the input field spectra in these two cases. As shown in Fig. 3, near f_l≈1.0, the RZ spectrum amplitude is obviously larger than that of the NRZ one. When the CD-induced phase impact on this part of the RZ (NRZ) spectrum is more than (almost the same as) the filter-induced amplitude impact on it, the increased CD-limited transmission with RZ format will be more than that with NRZ format. This is confirmed by the thin and thick solid curves in Figs. 2(a) and 2(b).

According to Ref. [2], the NRZ format is more CD tolerant but less PMD tolerant than the RZ format. For the NRZ and RZ formats with non-negligible PMD and PDL, their CD penalties and required OSNRs are compared in Figs. 2(c) (normalized DGD τ/T_b=0.2) and 2(d) (τ/T_b=0.4), from where one can see, with lumped τ≤0.4T_b and α≤1.5dB, the NRZ format (dashed) is still more CD tolerant than the RZ format (dash-dotted), because in this range the PMD and PDL effects are less important than the CD effect. The solid curves (RZ-DuoB) in Figs. 2(c) and 2(d) are plotted for comparison.

In Figs. 2(c) and 2(d), the CD penalties correspond to the worst-case of PMD-PDL-induced signal distortion, i.e., in the 3D Poincaré sphere, the three Stokes vectors, used to represent signal polarization, PMD, and PDL, are perpendicular to each other. Thus, for a given ξ, the curves in the insets of Figs. 2(c) and 2(d) are the worst (highest) required OSNR at BER=10^-9.

 

Fig. 3. Numerically calculated input signal spectrum amplitude as a function of frequency f_l=1/(NT_b) in NRZ-DuoB (dashed) and RZ-DuoB (dash-dotted). The NRZ-OOK curve (thick solid) is plotted for comparison. To show clearly the NRZ-DuoB spectrum is compressed by two, compared with the NRZ-OOK one, the x-axis is scaled differently for DuoB and OOK. Notice that, near f_l=1.0, the RZ-DuoB spectrum amplitude is obviously larger than the NRZ-DuoB one.

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Fig. 4. Required OSNR vs ξ curves (at BER=10^-9) with different bit delay time T. The CD-limited transmission distance with T=0.6T_b is 2~3 times of that with T=T_b.

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4. CD tolerance improved by partial bit delay modulation

The CD-limited transmissions shown in Fig. 2 are far less than the usual experimental results. This means the correlation-induced reduced bandwidth effect is not the key reason for the real CD tolerance of an ODB system. In this section, we further show that the CD tolerance can be dramatically improved by the PBDCM-induced phase effect, i.e., by treating the T_b in Fig. 1(c) as a tunable parameter T. For the following discussion, assuming 0≤T/T_b≤1 is enough.

The evaluated OSNRs (at BER=10^-9) for the NRZ and RZ systems with different bit delay time T are plotted in Fig. 4, from where one can see that, for an NRZ-ODB system [(a) and (c)] using SMF at R=10Gb/s and λ=1550nm, the CD-limited transmission distance with T=T_b is ~86km (ξ~15), whereas the transmission distance with T=0.6T_b is ~211km (ξ~37), assuming the CD penalty is less than ~3dB. Also, as shown in Figs. 4(b) and 4(d), the transmission distance using RZ format can be increased from 63km (ξ~11) with T=T_b to~217km (ξ~38) with T=0.6T_b. Again, the required OSNRs in Figs. 4(c) (NRZ) and 4(d) (RZ) with α=1.5dB and τ/T_b=0.2 correspond to the worst-case of PMD-PDL-induced signal distortion mentioned in section 3.

 

Fig. 5. Received signal-signal beating as a function of time in the NRZ-DuoB^* system with (a) 1-bit delay modulation and (b) PBDCM of T=0.6T_b. To show the phase effects of these two modulations, the CD-induced phase deviations of the received s-s beating spectrum Y^ss_l (ξ , T) in NRZ-DuoB^*, obtained using Eq. (6), are plotted as a function of frequency f_l=l/(NT_b) with (c) ξ=15 and (d) ξ=30 [10⁴(Gb/s)²ps/nm]. As shown in (c) and (d), within f_l<0.5 the phase deviation with T=0.6T_b is obviously smaller than that with T=T_b.

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The OSNR vs ξ curves in Figs. 4(a) and 4(b) agree reasonably well with the simulation and experimental results for systems with various R,D(λ) and L (cf. e.g., Refs. [6, 10, 13, 14, 15]).

To see how the PBDCM can improve the CD tolerance, we plot the signal part of the photoelectric current y(t), i.e., the electrically filtered signal-signal (s-s) beating y^ss(t) of y(t) [1, 12], in the NRZ-DuoB^* system with T=T_b and T=0.6T_b in Figs. 5(a) and 5(b). Notice that, at the time near 18, 24, or 32 (T_b), the eye opening with T=T_b is obviously smaller than that with T=0.6T_b.

To further explore the difference between the 1-bit and partial-bit delay modulations, we consider the spectrum of the received s-s beating, denoted as Y^ss_l (ξ,T). Due to the direct detection, the s-s beating satisfies Y ^ss*_l (ξ,T)=Y ^ss _-l(ξ,T). So, we only need to concern Y^ss_l (ξ,T) with positive l. We find that, in the region of f_l≤2 [f_l=l/(NT_b)], although the amplitudes of the two received s-s spectra, |Y^ss_l(ξ,T_b)| and |Y^ss_l(ξ,0.6T_b)|, are distributed in the similar way, their phase parts are quite different. To gain the physics insight, we consider the difference between the phases with and without CD, i.e.,

where θ(ξ,T)=arg[Y^ss_l(ξ,T)] is the phase part of the received s-s spectrum. Thus Δθ(ξ,T) in Eq. (6) is used to show the CD-induced phase deviation. It relates with the distorted s-s beating by $y^{\mathrm{ss}}\left(t_k\right)=Y_0^{\mathrm{ss}}(\xi ,T)+{\sum }_{l=1}^{2L_0}2\mathrm{Re}\left[Y_l^{\mathrm{ss}}(\xi ,T)e^{\frac{j2\mathrm{\pi l}{t}_k}{N{T}_b}}\right]\left(L_0=\mathrm{\eta N}{B}_o{T}_b\left[12\right]\right).$

As shown in Fig. 5(c) for the NRZ-DuoB^* system with ξ=15[10⁴(Gb/s)²ps/nm], the dotted curve (the CD-induced phase deviation with T=T_b) varies more than the solid curve (the phase deviation with T=0.6T_b), when f_l<0.85. This means that, before |Y^ss_l(15,0.6T_b)| decreases to small enough value, because of the square-law photodetection, the CD-induced phase effect is partly compensated by the PBDCM-related factor e ^-j0.6πl/Ncos(0.6πl/N) in Eq. (3) (T=0.6T_b). However, for the case of T=T_b, the CD-induced phase effect is not so efficiently “absorbed” by the 1-bit delay modulation, which corresponds to the correlation-related factor of e ^-jπl/Ncos(πl/N) in Eq. (3). In Fig. 5(d) for ξ=30 [10⁴(Gb/s)²ps/nm], the CD-induced phase deviation with T=T_b (dotted) also varies significantly more than the one with T=0.6T_b (solid) when f_l<0.5. Notice that, in the case of NRZ format and no signal distortion, the first node (zero point) of the ODB spectrum intensity locates at f_l=0.25.

5. Conclusions

To show that it is the phase effect of the ODB modulation, rather than the reduced-bandwidth effect, plays an important role in improving the CD tolerance of an ODB system, the CD-induced penalties obtained using OFB and PBDCM methods have been studied separately. The OFB approach can increase the CD-limited transmission of a RZ-ODB system by up to ~50%. However it is not very effective to a NRZ-ODB system. This is because the input spectra of the two formats distribute differently. By adjusting the delay time between the correlated bits, the PBDCM approach can increase the CD-limited transmission of a NRZ-ODB (RZ-ODB) system to ~2 (3) times of that using standard 1-bit delay modulation, respectively, because the CD-induced phase effect can be partly compensated by the PBDCM-induced phase effect. The calculated CD penalties agree reasonably well with the experimental results [6, 10, 13, 14, 15, 16]. For ODB systems with non-negligible PMD and PDL, above conclusions are also valid.

We suggest that a more cost-effective and CD-tolerant ODB transmitter than the conventional one can be obtained by adjusting the delay time of the delay-and-add circuit in Fig. 1(b) (without the LPF).