## Getting around the capacity crunch

The growing appetite for an ever-faster Internet and enhanced long-haul communication requires the pumping of more light down optic fibers. However, light-induced nonlinearities limit how much light can be pumped into the fiber without compromising the signal. This limitation has led to the prospect of a “capacity crunch.” Temprana *et al.* eliminated the effects of nonlinearity by using digital back-propagation methods with mutually coherent laser pulses from a single frequency comb.

*Science*, this issue p. 1445

## Abstract

Nonlinear optical response of silica imposes a fundamental limit on the information transfer capacity in optical fibers. Communication beyond this limit requires higher signal power and suppression of nonlinear distortions to prevent irreversible information loss. The nonlinear interaction in silica is a deterministic phenomenon that can, in principle, be completely reversed. However, attempts to remove the effects of nonlinear propagation have led to only modest improvements, and the precise physical mechanism preventing nonlinear cancellation remains unknown. We demonstrate that optical carrier stability plays a critical role in canceling Kerr-induced distortions and that nonlinear wave interaction in silica can be substantially reverted if optical carriers possess a sufficient degree of mutual coherence. These measurements indicate that fiber information capacity can be notably increased over previous estimates.

High-capacity optical communication is made possible by a nearly ideal physical platform: An exceptionally low fiber attenuation (α = 4.6 × 10^{−5} m^{−1}) (*1*) is matched by a small nonlinear refractive index (*n*_{2} = 2.5 × 10^{−20} m^{2} W^{−1}) in silica (*2*). However, over sufficiently long transmission distances, this combination of near-transparency and low nonlinearity still leads to a distributed, Kerr-mediated wave interaction that degrades (*3*) the optical signal. The nonlinear impairment has been identified as the primary physical mechanism (*4*–*6*) that imposes a strict limit on the achievable information capacity in optical fiber transmission. This limit is specific to the optically guided channel and has no direct analog in either wireless (*7*) or free-space photon communications (*8*). At present, the fiber capacity limit is engineered by compromising between the optical power and acceptable nonlinear distortions (*5*). To reach the full potential of this transmission medium, a true linear (*n*_{2} = 0) waveguide should be realized in order to suppress the onset of the nonlinear distortion. Unfortunately, the latter is an unphysical requirement (*2*). Consequently, a suppression or outright cancellation of the nonlinear impairment in optical fibers is seen as the main challenge in multiple disciplines (*4*–*6*, *9*–*12*) that are hampered by the waveguide’s nonlinear response to the propagating electromagnetic waves. Here, we demonstrate nonlinear impairment compensation by inverting the effects of distributed, nonlinear interaction among multiple frequency-stabilized optical signals. We show that the nonlinear noise-imposed uncertainty remains the only physical mechanism on the path to Shannon-limited fiber communication.

The precise knowledge of the optical carrier frequency is critical for a successful inversion of the nonlinear interaction effects. To clarify this important requirement, consider the interaction of *N* copolarized frequency modes in a single-mode waveguide, described by a set of coupled scalar nonlinear Schrödinger (NLS) relations (*2*): (1)where *A*_{l} represents the mode complex amplitude (each having carrier frequency ω* _{l}*), γ is the nonlinear coefficient, and α

*,*

_{I}*v*, and β

_{gl}*are the mode attenuation, group velocities, and dispersion orders, respectively. The latter is defined by the standard expansion: β*

_{kl}*=*

_{k}*d*(ω)

^{k}n*/d*ω

*, where*

^{k}*n*is the effective waveguide refractive index.

The wave interaction defined by Eq. 1 is deterministic and stable (*13*, *14*), allowing, in principle, for computational inversion of distributed Kerr interaction. Indeed, this notion was the basis for recent nonlinear cancellation (NLC) efforts that solved the inverse-propagating NLS relation (*15*, *16*). However, in practice these attempts have led to a limited impairment suppression (*17*–*19*). We note that a separate class of nonlinear mitigation research, not considered here, relies on phase conjugation (*20*, *21*) that can rely on free-running transmitters. These approaches, however, have restricted applicability because they require symmetrical power evolution (*20*), vanishing third-order dispersion, or a trade-off of spectral efficiency for performance (*21*, *22*).

In contrast, the NLC that relies on the Eq. 1 inversion faces a fundamental challenge that was not addressed in prior experimental studies. As predicted by a recent theoretical study (*23*), the NLS inversion requires precise knowledge of modal carrier frequencies. During propagation, any frequency uncertainty is mapped to mode velocity ambiguity via waveguide chromatic dispersion (*23*), as illustrated in Fig. 1. Although Kerr-mediated process is deterministic, the resulting nonlinear interaction appears random and leads to an underestimation of the transmission information capacity. Indeed, the experimental demonstrations to date (*17*, *18*) have relied on uncorrelated (free-running) emitters, inherently reducing the modal frequency (and phase) stability. In this case, the carrier frequency uncertainty is transformed to a stochastically varying walk-off rate between the modes, leading to a diverging inversion of Eq. 1. Consequently, the reversibility of multifrequency nonlinear interaction mandates a high degree of mutual coherence and the knowledge of the carrier offset from the absolute reference.

Recognizing this basic requirement, we devised NLC experiments to quantify the role of mutually correlated emitters. Specifically, the carrier frequencies were referenced to a parametric frequency comb derived from a single, continuous-wave master oscillator (*24*). In contrast to postcompensation techniques (*15*–*19*), predistorted channel launch results in signal reception that is free from nonlinear modal cross-talk (*23*). The respective input waveforms were synthesized by inverting the NLS propagation model given by Eq. 1 (*25*).

As with any Kerr-mediated interaction, the presence of noise limits the ideal distortion reversal. Consequently, the NLC compensation is demonstrated in two distinct experiments. The first experiment illustrates Kerr-inversion physics and nonlinear reversal in a pump-probe configuration in the absence of noise. Both the intense (pump) and weak (probe) waves had a high signal-to-noise ratio (SNR) and propagated over a short, nearly lossless, highly nonlinear fiber (HNLF) segment to guarantee that Kerr-induced impairment would dominate over stochastic, noise-induced distortion. Pump and probe waves, separated by 30 nm, were derived from the parametric comb source and had SNR of more than 40 dB. The pump and probe were launched into a HNLF 1100 m in length, with nonlinear parameter of 7 W^{−1} km^{−1}, dispersive parameters β_{2} = 37.9 ps^{2}/km and β_{3} = –0.06 ps^{3}/km, and transmission loss α = 0.6 dB/km. This segment was specifically selected to guarantee a sufficient walk-off between the pump and probe and to provide a clear distinction among the nonlinear interaction mechanisms. The pump beam, centered at 1588 nm, was amplified to a power level of 250 mW and was amplitude-modulated to achieve strong cross-phase modulation (CPM) (*2*, *3*). The signal wave, centered at 1558 nm and copolarized with the pump, had two orders of magnitude less power (1 mW) and was amplitude-modulated, as shown in Fig. 2. The weak (probe) wave experienced considerable distortion (red curve in Fig. 2A) that could be completely reversed by NLC in the high-SNR regime, erasing any distinction between the launched (black) and compensated (green) waveforms (*25*). This contrast is even more apparent in the spectral domain (Fig. 2B).

In the second experiment, we demonstrated the reversal of nonlinear distortion in a three-channel coherent wavelength division multiplex (WDM) transmission. In this case, the NLC is performed in a loop (*26*) emulating a modern communication link: Signal is sent over a total distance of 1020 km and re-amplified periodically after each span of 85 km of the standard single-mode fiber, as shown in Fig. 3. To gauge the role of frequency stabilization in the inversion of Eq. 1, we performed the first measurement with lasers with uncorrelated carrier frequencies. The emitters were modulated at four amplitude levels in each electric field quadrature, generating a (two-dimensional) 16-level quadrature amplitude modulation (*27*) (16-QAM) at 16 GBaud rate. In this case, the optimal channel launch power was 200 μW (Fig. 4A). The second measurement was performed with mutually coherent channels by deriving carriers from frequency comb tones centered at 1549.3 nm and separated by 25 GHz. Each carrier was independently modulated by synchronized patterns, imparting real and imaginary parts of the electric field defined by inversion of Eq. 1. The degree of nonlinear compensation is measured by the performance (*28*) of the central channel at the loop output in terms of the *Q* factor
(2)(*29*) while varying the signal launch powers from –9 to 2 dBm, where BER denotes the bit error ratio and erfc^{−1} is the inverse error function complement. The measurements in Fig. 4A clearly demonstrate the effective suppression of the nonlinear interaction: Even after distributed nonlinear interaction at a distance of 1020 km and an order of magnitude increase in the signal launch power, transmission quality (*Q*) is maintained. The saturation in performance (*30*, *31*) and its eventual decline are attributed to the Kerr-induced signal-noise interaction (*32*) (which is an inherent fundamental limit on the achievable performance), as well as the limitations of the experimental setup. No apparent jitter attributed to the Gordon-Haus effect (*33*) has been noticed in the experiments. It is nonetheless important to note a critical role of fast arbitrary waveform shaping, necessary to launch the inverted NLS solution. Electronic generators capable of operating over tens of GHz necessary to match a high-capacity coherent channel have become available only recently (*34*).

To further corroborate the critical importance of frequency referencing and its impact on the nonlinearity compensation ability, we repeated the experiment with three frequency-uncorrelated (free-running) carriers with linewidths of ~100 kHz and with a combination of correlated and uncorrelated carriers. The latter was realized by two comb-referenced channels while substituting the middle carrier by a single free-running laser. The performance of the system at the launch power of 1.6 mW (2 dBm) is shown in Fig. 4B, and the resulting constellation diagrams are given in Fig. 4C. The results in Fig. 4B clearly show that as the level of the mutual coherence between the interacting waves is increased (i.e., from the complete lack of coherence for three free-running oscillators, all the way to the fully frequency-referenced system), a qualitative improvement of the signal restoration is obtained, fully attesting to the gradual increase in the ability of the ensuing nonlinear interaction reversal. In particular, relative to the fully referenced system, the frequency uncertainty of only the middle carrier suffices to prevent a stable nonlinear compensation (Fig. 4B). Indeed, we observed a considerable variation of the output signal condition that is reflected by a widely varying figure of merit accumulated over 2200 measurements, as shown by the histogram in Fig. 4B (inset). [See (*25*) for additional measurements with free-running carriers.]

We note that the phases of the transmitters’ paths, although frequency-referenced, were not stabilized in the experiment. As a consequence, the experimental results serve to corroborate the claim from (*23*) that phase of the carriers indeed plays only a minor role in an effective NLC. We emphasize that, contrary to the widespread opinion that nonlinear interaction reversal is only a matter of computation, our results clearly demonstrate that no amount of computational complexity can make up for the mutual coherence of the modes (in the interaction reversal). Although the second experiment encompasses three interacting modes, it qualitatively captures all of the relevant physical effects with the exception of polarization mode dispersion, whose random time variation will cause a variation in the nonlinear interaction in time and will affect the integrity of the interaction reversal, the quantification of which is beyond the scope of this report. The extension to higher mode counts, as well as to polarization multiplexed systems, is straightforward (*25*). We note that information capacity in the strict sense (*7*) cannot be measured experimentally. However, the experiments attest to the capacity increase beyond the currently accepted limits by demonstrating reversal of signal-signal interactions, assumed to be nonviable in previous information capacity treatments (*4*)*.*

Our findings demonstrate the inversion of Kerr-induced interaction among multiple optical (frequency) modes in an optical fiber. The experiments have identified mutual carrier coherence as the critical requirement for substantial cancellation of nonlinear transmission effects. The compensation method relies on frequency comb–referenced carriers and enables an immediate increase of information capacity (*35*) and transmission reach in fiber communications beyond previously established limit estimates. By eliminating the stochastic contribution to Kerr-mediated wave interaction, this approach can be used to eliminate highly dissipative regeneration electronics from fiber networks and completely redefine the economy on which the present data traffic rests.

## Supplementary Materials

www.sciencemag.org/content/348/6242/1445/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

## References and Notes

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**Acknowledgments:**We thank Sumitomo Electric Industries for fibers used in the experiments, and Google Inc. for support of this work. The University of California has filed a patent on the method and applications of frequency-referenced carriers for compensation of nonlinear impairments in transmission.