1. Introduction

Digital coherent detection has recently attracted extensive attention due to its capability to receive high spectral-efficiency signals with high receiver sensitivity and to digitally compensate for transmission impairments [1,2]. However, the achievable transmitter bandwidth and sampling speed of the analog-to-digital converter (ADC) used in digital coherent detection limit the channel data rate that can be achieved for a given modulation format. To address this problem, a novel technique was recently proposed [3], where two synchronously modulated carriers were spaced exactly at the baud rate, or under the orthogonal frequency-division-multiplexing (OFDM) condition, to double the overall channel data rate and at the same time achieve high spectral efficiency. An aggregate channel rate of 88.8-Gb/s was demonstrated by using two carriers spaced 11.1-GHz apart, with each carrier supporting 44.4-Gb/s through 11.1-Gbaud polarization-division-multiplexing quadrature phase shift keying (PDM-QPSK). This work was followed by several subsequent publications [4–6] from the same group, and a channel rate of 111 Gb/s was demonstrated with a 2-carrier 13.9-Gbaud PDM-QPSK having a carrier spacing at the baud rate and with an ADC sampling speed of 50 Gsamples/s or an oversampling factor of about 4. Small coherent crosstalk between the adjacent modulated carriers was observed. These results demonstrate a useful scheme to address the growing demand to support high data rates at high spectral efficiencies. A similar 2-carrier PDM-QPSK design, with the carriers spaced at about twice the baud rate was reported and demonstrated in Refs [1,2]. with an oversampling factor of about 2, however, the coherent crosstalk penalty incurred when the carrier spacing was reduced to the baud rate was found to be unacceptably high. Wavelength-division multiplexing (WDM) with more than two independent channels spaced at the modulation symbol rate was experimentally studied [7]. Compared to single-channel transmission, experimental results show a Q-factor penalty of 2.8 dB when detecting a 6-Gbaud BPSK channel in the presence of two other channels. Also, large oversampling was found to be desired in order to capture the full spectrum of the interfering channels along with the desired channels. In this paper, we systematically investigate the performance of multi-carrier PDM-QPSK with digital coherent detection, and show the performance dependence on key design parameters such as the time alignment between the modulated carriers, the frequency separation between the carriers, the oversampling factor, and the bandwidth of the digital pre-filter used for carrier separation. Moreover, we quantify the performance of 3-carrier and 5-carrier PDM-QPSK with frequency-locked carriers spaced at the modulation symbol rate.

2. Experimental setup

A hybrid integrated device, consisting of LiNbO₃ phase modulators and planar lightwave circuits (PLC) in a compact package [8] was at the heart of the demonstrations in Refs [3–6]. In the hybrid integrated device, the frequency spacing between the two carriers was locked at the baud rate, and the transmitter characteristics that influence the system performance were difficult to surmise or quantify. In order to be able to quantify several metrics such as the time alignment of information symbols between the two carriers and the frequency separation of the two carriers relative to the baud rate, we used two methods to generate the 2-carrier PDM-OFDM signal. Figure 1(a) shows the methods used to generate the two carriers at wavelengths λ₁ and λ₂ (equivalently the frequencies f₁ and f₂). The first method used independent lasers for each of the carriers. In the second method, a single-drive chirp-free Mach-Zehnder modulator (MZM) was biased at null and driven with a sinusoidal waveform having a clock frequency f_c, nominally equal to half the baud rate, to generate two frequency-locked optical carriers from a single laser source. The frequencies f₁ and f₂ of the two optical carriers were exactly at f₀ + f_c GHz and f₀-f_c GHz, where f₀(λ₀) is the nominal frequency (wavelength) of the laser source. The clock source used for carrier generation was independent of that used for data generation. This approach allowed us to vary the clock frequency to generate the two carriers with a spacing that was more than the baud rate. Following amplification of the two carriers, a carrier separation filter (CSF) was used to separate them before launching into their respective data modulators. The CSF was a 25/50 GHz de-interleaver for a baud rate of 25-Gbaud as well as 28-Gbaud, while it was a dual optical delay interferometer (ODI), one ODI per carrier, tuned to reject the other carrier. (The first approach did not need the CSF). All lasers used in this investigation were tunable external cavity lasers (ECL) with linewidths about 100 KHz.

 

Fig. 1 Schematic of the experimental setup. (a) Two 2-carrier generation methods; (b) generation of QPSK modulation and polarization multiplexing of the two data streams; and (c) coherent reception of the 2-carrier signal. Clock frequency f_c = |f₂- f₁|/2; PC: polarization controller; MZM: Mach-Zehnder modulator; CSF: carrier-separation filter; PBS(C): polarization beam splitter (combiner); PM; power monitor; VOA: variable optical amplifier; BPF: band-pass filter; OLO: optical local oscillator.

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Two independent nested Mach-Zehnder modulators were used to generate the QPSK modulation, as shown in Fig. 1(b). Each nested MZM was driven by two data streams I and Q, which were pseudo-random bit sequences (PRBS) of length 2¹⁵-1 at the desired baud rate, with appropriate relative delay, to generate a QPSK signal. The outputs of the two MZMs were then controllably delayed with respect to each other with an optical delay placed on one path and then combined in a 3-dB optical coupler. Polarization controllers (PC4 and PC5) were used to ensure that the two modulated signals were co-polarized. This was monitored on one of the two outputs of the 3-dB coupler with a polarization beam splitter (PBS) and power monitors (PM). Following amplification, the 2-carrier QPSK signal was then equally split into two paths with controllable relative delay before being recombined by a polarization beam combiner (PBC) to form a 2-carrier PDM-QPSK signal. Three baud rates were investigated, namely 12.5-, 25.0-, and 28.0-Gbaud.

At the receiver, as shown in Fig. 1(c), the 2-carrier signal was attenuated by a variable optical attenuator (VOA) followed by an Erbium-doped fiber amplifier (EDFA) to obtain different optical signal to noise ratio (OSNR) (referred to in 0.1nm optical bandwidth), filtered with a 0.5-nm band-pass filter (BPF) centered to capture both carriers, and then launched into the signal-port of a free-space-optics-based 2 × 8 polarization-diversity optical hybrid. A separate tunable ECL laser, centered at one of the two carrier wavelengths [3–6], was launched into the optical local oscillator (OLO) port of the hybrid. The OLO power-to-signal power ratio was set to about 18 dB. After coherent mixing, the four pairs of outputs of the hybrid were connected to four balanced photodetectors, whose outputs were connected to the four channels of a 50-GS/s real-time sampling oscilloscope with an analog bandwidth of 20 GHz. Up to five sampled waveforms of length 1x10⁶ each were stored and processed offline with typical digital coherent detection processes [9–11], which included resampling, constant modulus algorithm (CMA) based blind equalization for crosstalk suppression, polarization demultiplexing, frequency and phase estimation, and data recovery. In the case of 2 × oversampling, the CMA-based blind equalizer is based on 15 T/2-spaced taps. In addition, a half-symbol (T/2) delay-and-add filter (DAF), as described in [3], was used as a pre-filter for subcarrier separation in the digital signal processor (DSP) prior to the CMA-based equalizer. The DAF is a simplest type of finite impulse response filter (FIR). In the case of 4 × oversampling, the CMA-based equalizer is based on 15 T/4-spaced taps. Also, the carrier-separation pre-filter can be a T/4 DAF that will be discussed in detail in section 3.3. In effect, the coherent crosstalk among modulated carriers is suppressed by the collective use of the DAF and the CMA-based multi-tap equalizer.

As described later, sampling at 25-GS/s and 50-GS/s were done for the 12.5-Gbaud rate and at 50-GS/s for the 25-Gbaud and 28-Gbaud rates. Performances were quantified in terms of the required OSNR (R-OSNR) for a bit error ratio (BER) of 1x10⁻³, as well as in terms of Q (expressed in dB) at the full signal OSNR, typically equal to about 35 dB. This Q value at full OSNR was estimated using the measured standard deviation of the recovered symbols with respect to their original locations in the constellation. After confirming the performance of each of the two carriers for one typical case were within ± 0.5-dB of each other, all subsequent measurements were performed on only one of the two carriers.

3. Experimental results

3.1 Dependence on the relative symbol alignment between the modulated carriers

We first investigated the dependence of the performance of 2-carrier PDM-QPSK on the symbol alignment between the two modulated carriers. The measurements were done for 12.5-Gbaud data rate and 12.5-GHz carrier spacing, using independent lasers. As described earlier, the symbol alignment of one modulated carrier was varied with respect to the other carrier [see Fig. 1(b)] using the variable optical delay following modulation. The results of these measurements are plotted in Fig. 2 , where the Q at OSNR = 35 dB is plotted as a function of the relative symbol alignment between the two carriers, measured over one full symbol period (80 ps for this baud rate). The recovered constellations for one polarization for the symbol-aligned and symbol-interleaved cases are also shown as insets in the figure. In addition, eye diagrams of the 2-carrier signal detected with a 30-GHz photodetector (before polarization multiplexing) are also shown for these two cases. The additional transitions seen at the middle of the symbol period are quite evident. When the symbols in the two carriers are well aligned in time (at 0 and 1 in the figure), the Q was 16.5 ± 0.2 dB and the Q starts to degrade as the two symbols start to interleave, with the worst performance seen at half-symbol spacing. This feature was previously observed and reported in Ref [7]. This observation is of key importance to show the benefit of OFDM, where the symbols of the modulated carriers are time-aligned, in suppressing the coherent crosstalk between the modulated carriers. When the modulated symbols are interleaved, the OFDM condition is violated, resulting in a coherent crosstalk induced Q penalty of over 4 dB, as shown in Fig. 2. Thus, in all subsequent measurements, the symbols of the modulated carriers were set to be fully aligned in time.

 

Fig. 2 Measured Q at OSNR = 35 dB as a function of the relative symbol alignment of one carrier with respect to the second carrier for the two carriers spaced at 12.5 GHz and at 12.5 Gbaud. Insets show recovered constellations for one polarization and directly detected eye diagrams of the 2-carrier system before polarization multiplexing for the two extreme alignment cases.

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3.2. Dependence of 2-carrier PDM-QPSK performance on carrier separation and oversampling factor

We then investigated the performance of the 2-carrier signal for different carrier frequency separation. The measurements were done with two independent lasers at 12.5 Gbaud. The commonly used T/2 DAF was used as the pre-filter for carrier separation. The frequency separation between the two carriers was varied from 50 GHz to 10 GHz. The data was sampled at 25 GS/s (corresponding to 2 × oversampling) and at 50-GS/s (corresponding to 4 × oversampling) for each of the frequency separation. The measured Q factor as a function of frequency separation is shown in Fig. 3(a) , the optical spectra of the 2-carrier PDM-QPSK signal at three characteristic frequency separations, (1) 18 GHz, (2) 12.5 GHz, and (3) 10 GHz, are shown in Fig. 3(c), while the recovered constellations for one polarization at these characteristic frequency separations are shown in Fig. 3(b). There are several interesting features seen in Fig. 3(a). At 50-GHz spacing, the Q factors were measured to be about 22 dB, which is close to that of a single carrier (see Table 2 in section 3.6), indicating coherent crosstalk penalties for both oversampling cases are minimal. As the frequency separation decreases to 18 GHz, indicated in Fig. 3 (a) as characteristic separation (1), the Q performance reaches a local minimum for both oversampling cases. The Q penalties are about 5.5 dB and 2.5 dB for the 2 × and 4 × oversampling cases, respectively. This can be explained by the maximized coherent crosstalk between a carrier and the first side lobe of its neighboring carrier, as shown in Fig. 3(c).

 

Fig. 3 (a) Measured Q at OSNR = 35 dB as a function of the frequency separation between the two carriers under 2 × oversampling and 4 × oversampling. (b) Recovered constellations for one polarization at three characteristic frequency separations, (1) 18 GHz, (2) 12.5 GHz, and (3) 10 GHz (3). (c) Modulated spectra of the two-carrier PDM-QPSK signal corresponding to the three characteristic frequency separations.

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Table 2. Required OSNR, excess OSNR penalty, and Q at OSNR = 35 dB for three different baud rates.

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The second characteristic frequency separation (2) corresponds to the OFDM condition where the carrier spacing equals the baud rate at 12.5 GHz. At this frequency separation, the first null of the interferer occurs exactly at the peak of the other carrier, resulting in the lowest level of crosstalk at the center of the second carrier. For the 2 × oversampling case, the Q is ~16.5 dB, corresponding to a crosstalk penalty of ~5.5 dB. One contribution to the penalty may come from improper anti-aliasing filtering in the 2 × oversampling case, where the RF bandwidth before ADC (20 GHz) is larger than half of the sampling rate. The alias components are present in two spectral ranges: [-20 GHz, −5 GHz] and [5 GHz, 20 GHz]. Since we applied a CMA-based blind equalizer, the alias components can be suppressed through tight filtering, but this in turn causes filtering penalty. Due to the limitation of the experimental setup, we are unable to quantify the penalty contribution from the improper anti-aliasing filtering.

On the other hand, when 4 × oversampling is applied, dramatic performance improvement is observed at the OFDM condition: the measured Q factor is increased by 4.2 dB to 20.7 dB. The above observations agree with the previous 2-carrier PDM-QPSK reports where large crosstalk penalty was found when ~2 × oversampling was applied [1,2], and small crosstalk penalty was found when ~4 × oversampling was used [3–6]. In fact, the performance at the OFDM condition for the 4 × oversampling case reaches a local maximum, and the penalty due to coherent crosstalk from other modulated carriers is negligibly small at BER around 10⁻³, as to be shown later. This distinct peaking in the performance clearly shows the benefit of OFDM in suppressing the coherent crosstalk. Note that this observation is different from that reported in Ref [7], where moderate penalty (2.8 dB) was found when the channels were spaced at the modulation symbol rate (or baud rate), and the penalty was minimized to 0.15 dB when the channel spacing was 25% larger than the baud rate. This performance difference may be due to different digital signal processes used in these two works. For example, this work uses a CMA-based multi-tap equalizer that was not used in Ref [7]. One key takeaway from the above observation is that for OFDM to be more effective in suppressing the crosstalk penalty, large oversampling is desired. An intuitive picture for this is as follows. The oversampling can be thought of as a narrow time gate, which allows one to discard the interference artifacts seen at the boundaries of each symbol period and pick a central portion, where due to orthogonality conditions, the interference is minimized. This effect has also been reported in the pioneering work on coherent wavelength-division multiplexing (Co-WDM) by Ellis and Gunning [12], where a narrow optical time gate was used to minimize the coherent crosstalk between adjacent on-off-keyed (OOK) channels. In this paper, we limit our study to the experimental configuration where the OLO is centered at one of the modulated carriers, as done in Refs [3–6]. With the OLO centered between the two carriers, we found that 3 × oversampling is sufficient to achieve virtually the same performance as 4 × oversampling. Detailed results on this are beyond the scope of this paper, and will be reported elsewhere.

The third characteristic frequency separation (3) is 10 GHz, which is 20% narrower than that at the OFDM condition. The interferer power spectrum starts to fill the central portion of the target carrier, as shown in Fig. 3(c). The crosstalk penalties at the both oversampling cases quickly increase. We will show more results when the pre-filter is changed from T/2-DAF to T/4-DAF in the next subsection.

3.3 Dependence of 2-carrier PDM-QPSK performance on the filtering function of the digital pre-filter used for carrier separation

As stated earlier, a T/2-DAF was used as a pre-filter, before CMA-based blind equalizer, for carrier separation in the DSP for all the measurements described up to now. However, with 4 × oversampling, since there are four samples per symbol, one has the option to use a quarter-symbol (T/4) DAF. The 4 × oversampled data shown in Fig. 3 were re-processed with a T/4-DAF and the results are shown in Fig. 4 , along with the results using the T/2-DAF shown earlier. For this comparison, we limited the carrier frequency separation range to between 10 GHz and 18 GHz, which is of interest for high spectral efficiency transmission.

 

Fig. 4 (a) Measured Q at OSNR = 35 dB as a function of the frequency separation of the two carriers, with 4 × oversampling and two different pre-filter for carrier separation: T/4-DAF and T/2-DAF. The inset shows recovered constellation for one polarization at carrier frequency separation of 12.5 GHz using T/4-DAF. (b) The measured BER as a function of OSNR for the case of two carriers spaced 10-GHz apart. Inset shows recovered constellation at OSNR = 18 dB for one polarization at carrier frequency separation of 10 GHz.

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Evidently, the use of a T/4-DAF leads to better overall performance across the frequency separation range than a T/2-DAF, particularly at 10-GHz and 18-GHz separations where the Q performances are about 2.5 dB better. The Q penalty at 18-GHz separation is essentially eliminated, and the penalty at 10-GHz separation is only about 2 dB. One plausible explanation for this is that the most significant crosstalk at 18-GHz separation is at the center of the modulated carrier under test [as shown in the subplot of Fig. 3(c)], and it can be effectively suppressed by the CMA-based equalizer without much penalty when the outer region of the carrier spectrum is not severely attenuated, as in the case of the T/4-DAF (rather than the T/2-DAF). The above explanation is verified by inspecting the frequency-domain transfer function of the CMA-based equalizer, which exhibits a deemphasizing regime at the center of the carrier to be recovered and an emphasizing regime on the side of carrier that is away from the other carrier. Detailed discussion on the filter response is beyond the scope of this paper. The 4 × oversampling coupled with the T/4 DAF as the pre-filter is thus found to be a very effective way to achieve high performance for such a two-carrier system.

We further explored the noise loaded performance of the 2-carrier signal with the carriers spaced at 10 GHz. The results are shown in Fig. 4(b), where we also plot the performance with carriers spaced at 12.5-GHz for comparison. It is remarkable to note that there is only 0.7-dB additional OSNR penalty at a BER of 1x10⁻³ for the two carriers spaced 10-GHz apart when compared with 12.5-GHz spacing. We again attribute this good performance to the high effectiveness of the CMA-based equalizer in suppressing the coherent crosstalk when the outer region of the carrier spectrum is not attenuated. We do not, however, have rigorous theoretical explanation for this, and hope that the experimental results may stimulate further theoretical and numerical studies on this subject.

3.4 Impact of frequency locking between the carriers on 2-carrier PDM-QPSK performance

The measurements presented above were performed with independent lasers as the carriers. One interesting question is if similar performances are obtained when the carriers are frequency locked. To answer this question, we used the second method of carrier generation described earlier in section 2, namely, frequency-locked 2-carrier generation by a MZM driven by a clock signal, whose frequency was varied to achieve different carrier spacings. The results of this set of measurement are shown in Fig. 5 , plotted in comparison with the previous set of measurements made with independent lasers. 4 × oversampling and T/4-DAF were used in the DSP. Each of the two plots has been normalized to the Q of its single carrier case. The measurements again were limited to the frequency range of interest from 10 GHz to 18 GHz.

 

Fig. 5 Measured Q (normalized to the Q of its single carrier) as a function of the frequency separation between the two carriers for the case with frequency-locked carriers and for the case with independent carriers. 4 × oversampling and T/4-DAF are used.

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It is clear from the figure that there is no performance advantage seen by using carriers obtained from the same laser source. The signature of the two curves is similar, within experimental errors. Our results here indicate that frequency locking between the carriers is not a necessity for 2-carrier PDM-QPSK as long as the frequency spacing is accurately obtained. Nevertheless, the comb-generation method produces frequency-locked carriers with stable frequency separation, determined by the clock frequency, and it can be readily used to generate more than two carriers, as we will show in the next section.

3.5 Multi-carrier PDM-QPSK with more than two carriers

An interesting question is whether the 2-carrier PDM-QPSK with closely-spaced carriers can be extended beyond two carriers while retaining the benefit of OFDM in suppressing the coherent crosstalk. To answer this question, we investigated the performance of 3-carrier and 5-carrier PDM-QPSK at 12.5 Gbaud under the OFDM condition. For this set of investigations, we used the frequency-locked carriers generated by a MZM, as shown in Fig. 6 . (The figure depicts 5-carrier generation and modulation, however, the same setup was used to generate 3 carriers by suitably changing the driving conditions of the MZM). The MZM was driven at 12.5-GHz to generate three or five carriers spaced at 12.5 GHz, which were de-interleaved into an odd group (containing odd numbered carriers) and an even group of carriers by using a 1x2 splitter followed by two optical delay interferometers (ODIs) with 25-GHz free spectral range (FSR). The ODIs were tuned so as to reject the unwanted carrier set. The odd and even numbered carriers were modulated by two respective QPSK modulators. At the receiver, the 0.5-nm BPF was centered at the middle carrier and the OLO laser was tuned to this middle carrier. Measurements were made on only this middle carrier as it experiences most of the coherent crosstalk. All data were sampled at 50-GS/s (4 × oversampling). For the 2-carrier case, T/4-DAF was found to give better performance than T/2-DAF, however for the 3-carrier and 5-carrier cases, the T/2-DAF was found to give more stable CMA-based equalization than T/4-DAF, so T/2-DFA was used here. This is understandable as T/2-DFA effectively nulls the center high-power portions of the adjacent carriers and makes the convergence of the blind equalization more stable. Optical noise was loaded at the receiver and the OSNR was measured with all carriers present before the BPF [as shown in Fig. 1(c)]. The performance results are shown in Fig. 7(a) and 7(b).

 

Fig. 6 Experimental setup to generate 5-carriers PDM-QPSK at 12.5-Gbaud per carrier. Insets show optical spectra of the carriers before and after modulation. 3-carrier PDM-QPSK can be similarly generated with the comb-generator producing 3 (instead of 5) carriers.

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Fig. 7 (a) Measured BER as a function of OSNR for 1-carrier, 2-carrier, 3-carrier and 5-carrier systems, with measurement made on the central carrier; (b) BER as a function of the normalized OSNR per 50-Gb/s carrier.

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In Fig. 7(a), the measured BER as a function of the OSNR is plotted for each one of the four cases investigated, while in Fig. 7(b), the data is re-plotted after normalizing the OSNR on a per carrier basis, i.e., the 2-carrier data is shifted left by 3 dB, the 3-carrier data by 4.8 dB, and the 5-carrier data by 7 dB. The plots in Fig. 7(b) have displacement at a BER of 1x10⁻³, indicating increasing relative OSNR penalty, a gradual departure from steepness due to crosstalk, and an increasing BER floor. In Table 1 , these parameters have been quantified. R-OSNR is defined as the required OSNR for a BER of 1x10⁻³ [as read from Fig. 7(a)], and the relative OSNR penalty is the penalty with respect to the 1-carrier case normalized by the number of carriers. The OSNR penalties at BER of 1x10⁻³ for the 2-, 3-, and 5-carrier PDM-QPSK are ~0 dB, 0.4 dB, and ~0.7 dB respectively, So, the effect of OFDM in suppressing the coherent crosstalk penalty remains for the 3-carrier and 5-carrier cases, when sufficient oversampling (e.g., 4 × ) is applied.

Table 1. Required OSNR, relative OSNR penalty and Q at OSNR = 35 dB.

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3.6 Dependence on the signal baud rate

Finally, we investigated the scaling behavior of the 2-carrier system with the signal baud rate. We measured the performance at 25 Gbaud with carriers spaced 25-GHz apart and at 28 Gbaud with carriers spaced at 28-GHz. Frequency-locked carriers were used for this investigation. Since the real-time oscilloscope had sampling speed limit of 50 GS/s, the results for 25-Gbaud correspond to 2 × oversampling and those for 28-Gbaud correspond to 1.8 × oversampling. T/2-DAF carrier separation filters were used in the DSP. The optical BPF was 0.6-nm for this set of measurements. The noise loaded BER as a function of OSNR is shown in Fig. 8(a) and the normalized measured modulation spectra (normalized to the baud rate) of a single carrier for each of the three baud rates is shown in Fig. 8(b).

 

Fig. 8 (a) Measured BER as a function of OSNR for the 2-carrier system at 12.5-Gbaud, 25-Gbaud and 28-Gbaud, respectively. (c) Normalized modulation spectra of a single carrier for each of the three baud rates.

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It is clear from Fig. 8(b) that there is more transmitter bandwidth limitation with the increase of the baud rate, as seen by the decrease of the peak power of the first side lobes and the 3-dB width of the spectrum with the increase of the band rate. This limitation originates both in the modulator as well as the drive electronics. With the larger bandwidth limitation at the higher baud rate, there is larger inter-symbol interference (ISI) which in effect reduces the “orthogonality” between the carriers under the OFDM condition, and thus leads to larger coherent crosstalk penalty. Indeed, as summarized in Table 2, the excess OSNR penalty in the 25-Gbaud case is 2.8 dB, which is 1.7 dB larger than that in the 12.5-Gaud case under the same 2 × oversampling factor. Here, the excess OSNR penalty is defined as the additional OSNR (in dB) required for BER = 10⁻³ relative to 2-carrier case at 12.5 Gbaud with 4 × oversampling and normalized to the same data rate. This extra 1.7-dB penalty is a direct result of the increased transmitter bandwidth limitation at 25 Gbaud than at 12.5 Gbaud.

Further increasing the baud rate from 25 Gbaud to 28 Gbaud (for the same 50-GS/s sampling speed), the excess OSNR penalty increases by an additional amount of 2 dB, which is due to (1) further reduced oversampling factor and (2) slightly larger transmitter bandwidth limitation for the 28-Gbaud case. This implies that to obtain the full benefit of OFDM in minimizing the coherent crosstalk penalty in multi-carrier PDM-QPSK at high baud rates, proportionally high sampling speed and transmitter bandwidth are needed.

It is also interesting to note from the table the R-OSNR for a 2-carrier signal at 12.5-Gbaud with 4 × oversampling is as good as that for a single carrier signal at 25-Gbaud, with both formats having a net data rate of 100 Gb/s. The Q factor also reflects similar performance behavior.

Table 2 also shows the Q values at OSNR = 35 dB for the three different baud rates investigated. The bandwidth limitation induced Q penalty at the higher baud rates is also clearly seen, even for the single carrier case.

4. Discussion and conclusion

We have systematically investigated the multi-carrier PDM-QPSK system consisting of two or more optical carriers spaced around the OFDM condition as a function of several design parameters. We find that performance is optimal when the carriers are symbol aligned (Fig. 2), that large oversampling (e.g., 4 × ) with proper anti-aliasing filtering helps minimize coherent-crosstalk-induced impairments (Fig. 3), and that the use of appropriate electronic pre-filter for carrier separation is beneficial (Fig. 4). In effect, the OFDM condition is satisfied only when (1) the carrier spacing equals to the baud rate of each modulated carrier; (2) the symbols in the modulated carriers are time-aligned; (3) sufficient transmitter bandwidth is used, and (4) sufficient sampling speed and proper anti-aliasing filtering are applied. The performance of 3-carrier and 5-carrier PDM-QPSK when the OFDM condition is not satisfied is under further investigation. When the 2-carrier system concept is extended to higher baud rates, we find the limitations from inadequate oversampling and limited transmitter bandwidth together result in excessive crosstalk penalties (Fig. 8), indicating the need to proportionally increase the sampling speed and transmitter bandwidth in order to support these high-baud-rate channels. For the same aggregate channel data rate, the use of more carriers helps reduce the baud rate per carrier and thus relax the requirement on the transmitter bandwidth and the sampling speed if the carriers are separated into a few sub-bands, each of which is independently sampled and received. However, this relaxed sampling speed requirement is at the expense of increased hardware for receiving all the subbands in parallel. This approach has been recently used in the detection of a coherent optical OFDM (CO-OFDM) channel with aggregate data rate at or beyond 1 Tb/s/ [13,14]. More recently, the subband detection scheme has been experimentally demonstrated with a 1.2-Tb/s 24-carrier no-guard-interval (NGI) CO-OFDM superchannel [14] by recovering two modulated carriers per subband detection. The 1.2-Tb/s NGI-CO-OFDM channel occupied a spectral bandwidth of 300 GHz and was successfully transmitted over 7,200 km of ultra-large area fiber, achieving a channel spectral-efficiency (SE) of 3.7 b/s/Hz and a record SE-distance product of 27,000 km⋅b/s/Hz [15]. The choice between single-carrier and multi-carrier (including 2-carrier) modulation formats may depend on the particular application.