Phase-coherent lightwave communications with frequency combs

TL;DR

This paper introduces two innovative signal processing schemes leveraging the phase coherence of optical frequency combs to improve phase noise estimation and compensation in fiber-optic communications, potentially simplifying systems and enhancing performance.

Contribution

It presents novel methods that utilize broadband phase coherence of frequency combs for more efficient phase noise management in coherent optical communication systems.

Findings

Enhanced phase noise estimation accuracy

Potential for simplified signal processing

Improved transmission performance

Abstract

Fiber-optical networks are a crucial telecommunication infrastructure in society. Wavelength division multiplexing allows for transmitting parallel data streams over the fiber bandwidth, and coherent detection enables the use of sophisticated modulation formats and electronic compensation of signal impairments. In the future, optical frequency combs may replace multiple lasers used for the different wavelength channels. We demonstrate two novel signal processing schemes that take advantage of the broadband phase coherence of optical frequency combs. This approach allows for a more efficient estimation and compensation of optical phase noise in coherent communication systems, which can significantly simplify the signal processing or increase the transmission performance. With further advances in space division multiplexing and chip-scale frequency comb sources, these findings pave the way for compact energy-efficient optical transceivers.

Figures

e). This form of processing is particularly useful to track and compensate the fast phase noise variations that result from the nonlinear interaction in the transmission fiber (Fig. 3b). As before, two neighboring channels in the center of the comb are detected and processed with either conventional single-channel independent phase estimation or joint phase estimation. The same total number of symbols are used for the phase estimation in both cases. In the independent case, all the symbols are taken from one polarization of one wavelength channel, while in the joint case the symbols are distri

a), all the comb lines are modulated with 8 amplitude levels in each quadrature of both polarizations of the electric field, creating 25 channels with polarization multiplexed 64-ary quadrature amplitude modulation (PM-64QAM) at 20 GBaud spaced 25 GHz apart (giving raw bit-rate of 0.24 Tb/s per channel or 6 Tb/s in the fiber, before coding). The signals are then transmitted through up to two spans of 80 km standard single-mode fiber (SMF) (Fig. 2b). In the receiver (Fig. 2c), two of the channels are jointly received using two synchronized standard coherent receivers. A second, independent, fre

a and 3b the phase curves for the center channel and its neighbor after 80 km transmission are plotted for two launch powers. The curves show high visual similarity, also in the higher launch power case where nonlinear distortions cause rapid phase fluctuations. The cross-correlation (Fig. 3c) of the two phase traces confirms the high correlation. The decrease in correlation length at the higher launch power is due to the shorter correlation of the nonlinear phase noise.

a indicate that joint processing can achieve a similar performance to individual processing, in spite of the fact that the phase estimation is only done once. Slight penalties are observed for propagation lengths beyond 80 km and for the outermost channels. This is due to a complex interplay between dispersive walk-off among the channels and the nonlinearity of the fiber. Adding a time offset electronically can partly counteract the walk-off and minimize the penalty, but not completely eliminate it.

Equations

$$E_n(t)=|E_n(t)|\exp (2\pi {\nu }_{0}t+{\varphi }_0(t)+n(2\pi ft+\psi (t)),$$

$$\begin{array}{r}{\varphi }_n(t)=2\pi ({\nu }_{0,S}-{\nu }_{0,LO})t+({\varphi }_{0,S}(t)-{\varphi }_{0,LO}(t))+\\ +n(2\pi (f_S-f_{LO})t+{\psi }_S(t)-{\psi }_{LO}(t)),\end{array}$$

$${\varphi }_k(t)={\varphi }_n(t)+\frac{k-n}{m-n}[{\varphi }_m(t)-{\varphi }_n(t)].$$

Full text

1 Introduction

Optical frequency combs were originally conceived for establishing comparisons between atomic clocks [1] and as a tool to synthesize optical frequencies [2, 3], but they are also becoming an attractive light source for coherent fiber optical communications, where they can replace the hundreds of lasers used to carry digital data [4]. One of the key advantages of frequency combs in optical communication is that the separation between consecutive lines is extremely stable. This enables high-spectral-efficiency transmission by minimizing the spectral guard bands between wavelength channels [5, 6], and allows for an efficient pre-compensation of fiber nonlinearities [7]. Next in the hierarchy of the comb properties to be exploited is the broadband phase coherence (comb lines are phase locked to each other). This characteristic has been instrumental in expanding the portfolio of comb-based applications [8, 9, 10], but its use in lightwave communication systems has been limited to inflexible analog methods to lock the transmitter and receiver [11, 12, 13]. Here, we present a different way to harness the phase coherence of frequency combs in wavelength division multiplexing (WDM). Our approach allows for a more efficient estimation and compensation of the optical phase noise – a fundamental noise source that results in one of the predominant impairments in coherent optical receivers.

Phase noise arises mainly from random phase variations of the carrier and local-oscillator (LO) light sources, which are usually semiconductor lasers, as well as from the nonlinear interaction among wavelength channels [14]. Modern WDM systems compensate for phase noise digitally but treat each channel as a separate entity. While the lines of frequency combs also suffer from random phase variations, the broadband phase coherence correlates the variations between WDM channels (Fig. 1a). As a result, the traditional techniques that realize phase tracking on a channel-by-channel basis (Fig. 1c) are redundant when the light source is an optical frequency comb. Our proposal consists of viewing the WDM transmission system as a single coherent entity and perform channel processing jointly. Having access to multiple channels impaired by the same phase noise means that the phase estimation can be made more efficient in terms of phase tracking capabilities [15, 16, 17], or power consumption of the digital electronics [18]. It has been suggested that joint phase processing can be implemented with optical frequency comb sources [19, 20, 21, 22] but a proof in a transmission experiment is still outstanding.

2 Results

2.1 Joint phase processing

We study two different schemes for joint phase processing. The first scheme is best described as master-slave carrier recovery and is based on reusing the phase estimated from one master channel to compensate the phase variations of several slave channels, as illustrated in Fig. 1d. This has the benefit of sharing the relatively demanding phase estimation block between several channels, thereby reducing the complexity and power consumption of the DSP electronics. The second scheme we propose, illustrated in Fig. 1e, instead uses the multiple versions of the phase noise to improve the phase estimate, which means that the same system can tolerate faster phase variations without incurring penalties. The algorithms are described in more detail in the Methods section.

2.2 Experimental setup

Our proof-of-principle experiments are based on the joint reception of two WDM channels, originating from an electro-optic frequency comb [23]. In the transmitter (Fig. 2a), all the comb lines are modulated with 8 amplitude levels in each quadrature of both polarizations of the electric field, creating 25 channels with polarization multiplexed 64-ary quadrature amplitude modulation (PM-64QAM) at 20 GBaud spaced 25 GHz apart (giving raw bit-rate of 0.24 Tb/s per channel or 6 Tb/s in the fiber, before coding). The signals are then transmitted through up to two spans of 80 km standard single-mode fiber (SMF) (Fig. 2b). In the receiver (Fig. 2c), two of the channels are jointly received using two synchronized standard coherent receivers. A second, independent, frequency comb acts as a source for the LO. The two frequency combs are seeded from independent continuous-wave (CW) lasers and are not synchronized to each other. Since two channels are simultaneously received and recorded, this scheme allows for establishing a quantitative comparison between individual and joint phase tracking. The setup is described in more detail in the Methods section.

2.3 Phase-noise correlation

In the first set of experiments, we verify that the phase noise remains correlated also after transmission. This can be qualitatively assessed by comparing the phase recovered from the two channels with conventional, independent phase estimation. In Fig. 3a and 3b the phase curves for the center channel and its neighbor after 80 km transmission are plotted for two launch powers. The curves show high visual similarity, also in the higher launch power case where nonlinear distortions cause rapid phase fluctuations. The cross-correlation (Fig. 3c) of the two phase traces confirms the high correlation. The decrease in correlation length at the higher launch power is due to the shorter correlation of the nonlinear phase noise.

2.4 Master-slave performance

We then study the performance of the master-slave carrier recovery. We quantify the performance by using the generalized mutual information (GMI), which is the maximum data throughput attainable for a bit-wise receiver [24], accounting for the redundancy of an ideal forward-error correction (FEC) code. Today’s soft-decision codes can come quite close to this, making GMI a good characteristic of the physical channel, in contrast to the bit-error-rate that must assume a specific FEC code. The maximum GMI is modulation format dependent, and in this case with PM-64QAM it is close to 12 bits per four-dimensional symbol (6 bits in each polarization) at the transmitter and is reduced by all signal impairment during propagation through the channel. We compare the performance of traditional independent carrier recovery with joint carrier recovery by comparing the GMI of the same measurement either processed separately, or with the phase information extracted from the other received channel. The impact of frequency separation between the master and the slave was studied by measuring different channel pairs. Since the performance of the channels varied slightly due to power variations of the comb lines, the center channel was always used as the slave channel, while different channels were used as master. The results in Fig. 4a indicate that joint processing can achieve a similar performance to individual processing, in spite of the fact that the phase estimation is only done once. Slight penalties are observed for propagation lengths beyond 80 km and for the outermost channels. This is due to a complex interplay between dispersive walk-off among the channels and the nonlinearity of the fiber. Adding a time offset electronically can partly counteract the walk-off and minimize the penalty, but not completely eliminate it.

2.5 Joint phase-estimation performance

We next study the possibility of realizing joint phase estimation (see Fig. 1e). This form of processing is particularly useful to track and compensate the fast phase noise variations that result from the nonlinear interaction in the transmission fiber (Fig. 3b). As before, two neighboring channels in the center of the comb are detected and processed with either conventional single-channel independent phase estimation or joint phase estimation. The same total number of symbols are used for the phase estimation in both cases. In the independent case, all the symbols are taken from one polarization of one wavelength channel, while in the joint case the symbols are distributed over both polarizations of two wavelength channels. This means that the joint scheme uses a four times shorter time averaging, while maintaining the same number of total symbols and the same tolerance to additive noise. At the highest launch power, the performance is improved, and the optimal launch power is increased by 1 dB (Fig. 4b). Constellation diagrams (Fig. 4c) show a noticeable reduction of phase noise in the joint phase estimation case. The reduced impact of nonlinear phase noise and increased GMI can be translated into an increased data throughput or an increased transmission distance.

3 Discussion

Fundamentally, joint phase processing presented here will be limited by the phase coherence of the comb sources, which have a line dependent phase noise term [25] that will cause a phase noise difference that increases with the separation between the channels [20]. This term is dependent on the comb generation technique. The electro-optic comb used in this work had a correlation coefficient that was above 99.99% for any line combinations [26]. Furthermore, chromatic dispersion in the fiber causes decorrelation of the phase noise between the different wavelength channels. This will cause the carrier phase noise and the LO phase noise to mix with temporally different parts, incurring phase noise differences in the detected signals [27] that are not possible to compensate electronically. The penalty caused by this effect will depend on the coherence length of the carrier and LO sources, the wavelength separation between the jointly processed channels, the transmitted distance and the dispersion properties of the fiber. Unless optical dispersion compensation is used, this will fundamentally limit the bandwidth and transmission distance over which joint carrier recovery is reasonable. However, with today’s technology, practical limitations in the digital electronics are likely to limit the optimal number of jointly channels more than the coherence properties of the comb lines.

In summary, we have demonstrated two methods for joint phase processing that utilize the broadband phase coherence of frequency combs for multi-wavelength lightwave communications. This is a fundamental change from the traditional method of treating different wavelength channels independently. Optical frequency combs establish a fully coherent relationship between WDM channels, that can be exploited to significantly reduce receiver complexity or overcome the nonlinear impairments introduced by the fiber link. Although here demonstrated with single-mode fibers, the scheme could be scaled up thanks to the development of new optical fibers [28, 29, 30, 31], allowing to increase the number of jointly processed channels from $1\times N$ to $L\times N$ by unleashing the space dimension. The optical frequency combs here used are based on benchtop electro-optic comb sources, but the findings are in principle independent of the platform. For example, the joint processing scheme could benefit from further advances in soliton microcombs and integrated photonics [32]. Together, chip-scale optical frequency combs and joint signal processing have the potential to be a key technology in high-performance, energy-efficient optical transceivers.

4 Methods

4.1 Phase relations of detected channels in frequency comb-based systems

The field of the $n$th line of a frequency comb can be written

[TABLE]

where $\nu_{0}$ and $\phi_{0}(t)$ is the center frequency and phase noise of the center line of the comb, $n$ is the line index, $f$ is the frequency spacing and $\psi(t)$ is a phase noise term related to the timing jitter of the comb. This is a general expression valid for different implementations of optical frequency combs [12, 33, 26, 34, 35].

In a system using frequency combs as signal carrier and local oscillator (LO) light sources, the detected signals will have a phase evolution that will be the difference between the phase evolution corresponding lines of the carrier and LO frequency combs

[TABLE]

where the subscripts $S$ and $LO$ correspond to the signal and LO combs respectively. This equation shows that the channels can be regarded to have the same phase evolution if the combs have sufficiently similar spacing ($f_{s}\approx f_{LO}$) and the timing jitter noise term is negligible ($\psi_{S}(t)-\psi_{LO}(t)\approx 0$). This is the basis for joint carrier recovery. What can be considered "sufficiently similar" and "negligible" depends on the exact carrier recovery scheme and the desired number of jointly processed channels. It should be noted that in addition to the intrinsic comb properties, phase variations will also arise from mechanical and thermal disturbances in the transmission fibers. This means that any joint carrier scheme will need to consider some phase differences between the channels, independent of the comb coherence properties. In the description below of our joint carrier-recovery schemes, the practical implementation is described.

Even in the case where the timing-jitter noise or spacing difference is significant, the phase evolution of any channel can be calculated from any two other channels as

[TABLE]

From a practical perspective, this relation means that a master-slave processing with two master channels could cope with large amounts of timing jitter noise and spacing difference. The allowable spacing difference is however limited by the maximum allowable difference between LO and signal for coherent detection.

4.2 Master-slave carrier recovery

Master-slave carrier recovery relies on using the phase correlations to eliminate redundant carrier recovery blocks, to reduce the complexity and power consumption of the DSP electronics. The basic principle is that the frequency and phase offset is estimated from one channel (the master channel), and that information is used to compensate the other (slave) channels. The master frequency and phase can be estimated with any algorithm. Specifically, in our method we estimate the frequency offset by finding the peak in the 4th power spectrum [36] and use the blind phase search (BPS) algorithm [16] for phase estimation.

As discussed above, some additional functions are needed to compensate for small frequency and phase differences between the channels. This was implemented as a slow phase tracker to compensate remaining phase variations on the slave channels. As it is desirable to keep any additional processing of the slave channel to a minimum, this slow phase tracking is performed by the adaptive equalizer that is also performing polarization demultiplexing and compensation of polarization mode dispersion. This is achieved by using a decision directed update algorithm for the equalizer taps, which is sensitive to phase variations, and performing the phase recovery inside the update loop of the equalizer, as is standard for decision directed algorithms [37]. To realistically evaluate the tracking speed, the equalizer taps were updated every 64th symbol, which emulates hardware parallellization [38]. This solution could track the timing jitter noise of our combs without penalty, and tolerated up to several tens of kHz of remaining frequency offset. However, the spacing of the transmitter and receiver frequency combs had a difference of around 20 kHz, varying a few kHz over several hours. This spacing difference would hinder joint carrier recovery for any channels but the nearest neighbours due to the scaling with line index. This limitation would not be present in a system with more than two coherent receivers as also the spacing difference could be estimated from the received channels, based on equation (3), so in our experiments we measured the frequency difference of the RF clocks and used that information in the signal processing according to Supplementary Figure 2. In Supplementary Note 1 we verify that this approach is valid.

4.3 Joint phase estimation

Joint phase estimation was performed using the BPS algorithm [16], extended to several channels as described below. The BPS algorithm is based on rotating the received signal with a number of test phase angles, after which the distance to the closest constellation point is calculated for each test phase angle and averaged over several consecutive symbols. The test phase angle with the lowest average distance is chosen as the estimated phase. The averaging is needed to minimize the effect of additive noise on the signal, but the phase tracking speed will be limited by the length of the averaging filter. The optimal length of the averaging filter is a trade-off between tolerance to additive noise and phase tracking speed. A multichannel version of the BPS algorithm extends the averaging to include several channels. Then, a shorter filter length can be used while retaining the same tolerance to additive noise. This is illustrated in Supplementary Figure 3. A more detailed description can be found in [20].

Small phase differences between the channels were compensated in a similar way to the master-slave algorithm, but separated from the main polarization demultiplexing equalizer. Instead, a one tap decision-directed equalizer was used separately on all channels, with the joint phase estimation taking place inside of the update loop of the equalizers, as illustrated in Supplementary Figure 4.

4.4 Detailed experimental setup

A schematic of the experimental can be seen in Supplementary Figure 5. A frequency comb was created by modulating a continuous wave laser at 1545.32 nm (linewidth $<$100 kHz) with one phase modulator and one intensity modulator, similar to the comb described in [23]. The modulators were driven by a 25 GHz radio frequency (RF) signal. The comb was then amplified in an erbium-doped fiber amplifier (EDFA) and fed through an optical processor for flattening. After the optical processor, a 25 GHz interleaver was used to split the comb lines into even and odd. The even and odd lines were then separately modulated with 20 GBaud 64QAM root-raised cosine pulses with a roll-off factor of 0.05 generated with an arbitrary waveform generator operating at 60 GS/s. After modulation, polarization multiplexing was emulated by splitting the signals and delaying one part around 200 symbols before recombining on orthogonal polarizations. The signals were amplified again and recombined using an interleaver. The even and odd paths were length-matched to within 5 symbols on the x-polarization, but the two delays in the polarization multiplexing emulator differed 10 symbols. The performance was evaluated both in a back-to-back configuration and with up to two 80-km spans of standard single mode fiber (SMF).

In the receiver setup, two channels could be received simultaneously. The two channels were separated using a multi-port optical processor. The LO lines were taken from a second frequency comb similar to the one in the transmitter. The LO lines were separated using a 25 GHz interleaver. This limited the possible channel combinations to a combination of one even and one odd channel. The LOs were additionally filtered to ensure sufficient extinction ratio. The signals and LOs were mixed in two standard coherent receivers and sampled at 50 GS/s using two synchronized digital sampling oscilloscopes with a bandwidth of 23 GHz.

4.5 Digital signal processing

The sampled signals were first normalized and orthogonalized using the Gram-Schmidt method to compensate for imperfections in the optical hybrid. Then matched filtering and downsampling from 50 GS/s to two samples per symbol (40 GS/s) was performed, followed by dispersion compensation. This was followed by compensation of time skew caused by differences in electrical pathlength of the two receivers. After this, adaptive equalization and carrier recovery was performed. The equalizer had 35 $T_{s}/2$-spaced taps, where $T_{s}$ is the symbol time. The taps were pre-converged using the constant modulus algorithm on 400000 symbols. The output from the pre-convergence was used for coarse frequency offset estimation by raising the signal to 4th power and finding the spectral peak. After pre-convergence, the equalizer was switched to decision directed mode. Phase estimation and compensation was performed in the update loop of the equalizer, using the BPS algorithm, either independently or jointly as described above. The equalizer taps were updated every 64th symbol to emulate hardware parallelization. The step size was $10^{-4}$ for the power-normalized signal. After equalization and carrier recovery, orthogonalization to compensate for modulator bias imperfections was performed. Finally, the performance was evaluated by calculating the generalized mutual information (GMI) using the method in [39], using over 1 million bits.

Acknowledgments

This work was supported by the Knut and Alice Wallenberg foundation, the Swedish Research Council and the European Research Council (GA 771410)

Author contributions

Lars Lundberg and Mikael Mazur jointly designed the experimental setup and built it with assistance from Ali Mirani. Lars Lundberg designed the digital processing methods and analyzed the data. Benjamin Foo contributed to the interpretation of the nonlinear phase noise results. Lars Lundberg, Victor Torres-Company, Jochen Schröder and Magnus Karlsson wrote the paper. Magnus Karlsson, Jochen Schröder and Peter A. Andrekson supervised the work and provided technical leadership. All co-authors contributed to the paper with critical comments and suggestions.

Competing interests

The authors declare no competing interests.

5 Supplementary material

Contents:

• •

Supplementary note 1

• •

Supplementary figures 1–5.

Supplementary note 1

In this note we investigate how well the difference in frequency spacing between the carrier and LO combs can be estimated from the two channels. This is done by monitoring the spacing difference by measuring the beatnote between the radio-frequency clocks driving the combs. This measured value was then compared to a value estimated from independent carrier recovery on the two received channels. As seen in Supplementary Figure 1, the frequency difference could be successfully estimated to within a few kHz, which is sufficient to to be within the tracking capabilities of the adaptive equalizer. Therefore, it can be safely assumed that a joint processing scheme with three or more received channels would have access to information about the comb spacing difference.