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1 Design Aspects of Multi-Soliton Pulses for Optical Fiber Transmission Vahid Aref∗, Zhenhua Dong†, and Henning Buelow∗ ∗ Nokia Bell Labs, Nokia, Stuttgart, Germany, Email: fi[email protected] † Photonics Research Centre, The Hong Kong Polytechnic University, Kowloon, Hong Kong Abstract—Weexplainhowtooptimizethenonlinearspectrum tailored to meet the physical constraints. Consider the on-off ofmulti-solitonpulsesbyconsideringthepracticalconstraintsof keyingofupto10eigenvaluesforatargettransmissionlength 7 transmitter, receiver, and lumped-amplified link. The optimiza- of at most 2100 km. Our objective is to optimize the discrete 1 tion is applied for the experimental transmission of 2ns soliton spectrum such that the following conditions are satisfied for 0 pulseswithindependenton-offkeyingof10eigenvaluesover2000 2 km of NZ-DSF fiber spans. all 210 possible soliton pulses: (i) Pairwise distance between eigenvalues is large enough n a I. INTRODUCTION for reliable detection in the presence of perturbations from J Nonlinear Frequency Division Multiplexing (NFDM) has lumpedamplification[6],ASEnoise,andnumericallimitation 7 recently been proposed to design an optical waveform bet- of current NFT algorithms. 2 ter matched to the nonlinear propagation in a fiber [1]. In (ii) The largest pulse-width (duration) of all pulses must particular, the information bits are modulated in the so-called becomeminimum(highesttransmissionrate).Wetruncatethe ] T nonlinearFourierspectrumwhichisthenmappedtoapulsein tails of each pulse outside a common interval T. To have a I timedomain[1,2].Thespectrumispartitionedintocontinuous negligible √inter-symbol interference (ISI), we choose T such s. part and discrete part. that |s(t)/ Es| < 0.01 for t ∈/ T and for all solitons s(t) c The discrete spectrum consists of a finite set of complex with energy Es. [ values, called eigenvalues, and the corresponding spectral (iii) The bandwidth of mulit-solitons is changing during 1 amplitudes.Thispartrepresentsthesolitoniccomponentofthe the transmission. The largest bandwidth (BW) of all pulses v pulse, in which the effects of Kerr nonlinearity and chromatic must become minimum all over the transmission (highest 1 dispersion are balanced. spectral efficiency). We define bandwidth as the frequency 8 9 Inanideallosslessfiber,modeledbynonlinearSchro¨dinger range holding more than 99% of energy. 7 Equation (NLSE), the propagation of soliton pulses follows (iv) A further ISI may occur when Re{λ } =(cid:54) 0. There i 0 simple principles in the nonlinear Fourier spectrum: the must be a negligible ISI between adjacent pulses during the . 1 eigenvalues remain the same, and each spectral amplitude transmission. 0 transformslinearlyonlybasedonitseigenvalue.Thissuggests It is not yet fully understood which distribution of eigen- 7 to modulate or/and detect data over nonlinear spectrum. values maximizes the spectral efficiency. This is because no 1 The first realization of such a modulation was the on-off analytic expression is yet available for bandwidth and pulse- : v keying of first-order soliton which has been well-studied in widthofmulti-solitonsandforperturbationsofeigenvaluesby Xi thelastthreedecades,see[3]andreferencestherein.However, noiseandthelumpedamplification.Weconsiderherethesim- thespectralefficiencycanbeincreasedbyusingsolitonpulses plest (but not the most efficient) distribution for eigenvalues: r a withseveraleigenvalues:On-offkeyingofupto4eigenvalues, λ=ω+jσ, with ω ={±2,±1,0} and σ ={1,2}. located on imaginary axis, was experimentally shown [4] as well as the QPSK-modulation of spectral amplitudes for 2- solitonpulses[5,6].Morerecently,twooftheauthorsshowed 1550nm 2ns the transmission of soliton pulses with seven eigenvalues and ∆ν=1kHz 1550nm QPISnKt-hmisodpualapteerd, swpeectbrarileflamypelixtupdlaeisn[7h]o.w to optimize the 88DGASCa/s QI I/Qdulator EDFA PC ∆ν=1kHz nonlinear spectrum of a multi-soliton pulse in order to re- mo Coherent Normalization Receiver duce the perturbations caused by the practical constraints of β2,γ OBPF,50GHz transmitter, link and receiver. Applying the optimization, we 3×24kmNZ-DSF offlineprocessing demonstratetheexperimentaltransmissionof2nsmulti-soliton INFT } r(t) 0 pulses carrying 10 information bits by on-off keying of 10 1 λ eigenvalues over 2033 km of NZ-DSF fiber spans. Normali FT ., zation N . EDFA EDFA . II. OPTIMIZATIONOFMULTI-SOLITONPULSES λ,1 β2,γ,P0,T0 { We explain our sub-optimal method for the on-off keying modulation in which the spectral amplitudes can be freely Fig.1. ExperimentalsetupwithofflineNFT-baseddetection 2 z)) 2 15 A B C 2.5 ( λg(i 11..26 D Hz) 13 2 Ima 0.80 .5 1.0 1.5 2.0 dth(G 11 D λag() 1.5 z)) 2 dwi Im λ(i 1.6 A B C Ban 9 1 g( 1.2 a m 0.8 7 0.5 I 0 .5 1.0 1.5 2.0 0 .5 1.0 1.5 2.0 -2 -1 0 1 2 fiberlengthz,(×1000km) fiberlengthz,(×1000km) Re(λ) Fig.2. Simulatedtransmissionoftwosolitonpulseswiththesame8eigenvaluesbutdifferentinitialspectralamplitudesoveranEDFAamplifiedNZ-DSF link.Left:Intheabsenceofnoise,perturbationsofimaginarypartofeigenvaluesforbothpulsesMiddle:Thebandwidthvariationofpulsesanditsrelation toperturbationsofeigenvalues.Right:Inthepresenceofnoise,eigenvaluesofthereceivedpulsesforbothsolitonsafter2000kmwithOSNR≈34dB. The magnitude of spectral amplitudes, q (λ), mainly con- d trols the pulse-width and the phase of qd(λ) controls the 2 bandwidth. To avoid ISI (condition (iv)), we set |q (λ )| = Ad(λi)|exp(+2jλ2izL)|,wherezListhemaximumlindkleingth λg() a (2000 km) and A (λ ) is |q (λ )| in z /2. In this case, a m d i d i L I 1 pulse will first be contracted and then eventually broadened to the same pulse-width it had at transmitter. To minimize thepulse-width,weoptimizeA (λ ).Wenumericallyobserve d i 0 that the pulse-width becomes minimum if |B | = |A | = 1 −3 −2 −1 0 1 2 3 i i Re(λ) in Darboux transformation for computing inverse nonlinear Fig.3. Theeigenvaluesofreceivedpulsesafter2033km. Fouriertransform[1,8].Finally,wecomputedthepulse-width T = 12 (for standard nonlinear Schro¨dinger equation) which scales down to 2 ns in our transmission setup (5 Gbit/s). with 80GSa/s and 33 GHz bandwidth, followed by an offline The bandwidth (BW) of each soliton is important not only data-aided phase and carrier offset correction. becauseofspectralefficiency,butalsobecauseofdeterministic Asubsetof28solitonswererandomlychosensuchthateach perturbations of eigenvalues in a lumped amplified link. It is eigenvalue is “on” in one-half of pulses, and a “fair” number shownin[6]thattheeigenvaluesofthepath-averagedsolitons of k−soliton pulses are chosen, 1 ≤ k ≤ 10. The pulse train may fluctuate when their instantaneous BW are large. The is repetitively transmitted. We used Fourier Collocation (FC) same effect is shown here in Fig. 2 for two soliton pulse with method [1] to detect the eigenvalues of each received pulse. thesame8eigenvaluesbutdifferentspectralamplitudes.Split- The eigenvalues of all received pulses after 28 loops (≈2033 step Fourier method is used to simulate the pulse propagation km) are plotted in Fig. 3. By mapping each pulse to 10 bits, in our experimental setup (Fig. 1) with NZ-DSF fiber with we found the BER≈8×10−3. β ≈−5.75ps2/km and γ ≈1.6W−1/km and span length of Inthispaper,webrieflyexplainedhowtheeigenvaluesofa 2 24.2km.Wefirstexcludenoiseinoursimulation.Fig.2shows multi-soliton are sensitive to the BW in a periodically lumped that the eigenvalues fluctuate when the BW gets large. The amplified link and how to optimize a soliton pulse using the larger BW is, the larger fluctuations of eigenvalues are. In the available degrees of freedom in spectral amplitudes. presence of noise (assumed 50 GHz filtered noise with NF=5 dB), the soliton with larger eigenvalue fluctuations is more REFERENCES distorted. For each soliton, we should then find the spectral [1] M.Yousefi,“Informationtransmissionusingthenonlinearfourier amplitudes which minimize the maximum BW over the link. transform,” Ph.D. dissertation, University of Toronto, 2013. We quatized the phase of q to levels of π/4 and used the [2] J.E.Prilepsky,etal.“Nonlinearinversesynthesisandeigenvalue d division multiplexing in optical fiber channels,” Phys. review exhaustive search to find the soliton with minimum BW for letters, vol. 113, no. 1, p. 013901, 2014. all spans. [3] L. F. Mollenauer and J. P. Gordon, Solitons in optical fibers: fundamentals and applications. Academic Press, 2006. [4] Z.Dong,etal.“Nonlinearfrequencydivisionmultiplexedtrans- III. EXPERIMENTALRESULTSANDCONCLUSION missions based on nft,” IEEE PTL, vol. 27, no. 15, 2015. The experimental setup is shown in Fig. 1. Following a [5] V. Aref, et al. “Experimental demonstration of nonlinear fre- 88GSa/s digital-to-analog converter (DAC), a drive signal is quencydivisionmultiplexedtransmission,”inProc.ECOC,2015. [6] V. Aref, H. Buelow, “Design of 2-soliton spectral phase modu- providedforaMach-ZehnderIQmodulatorwhichtransmitsa lated pulses over lumped amplified link,” in Proc. ECOC, 2016. singlepolarization0.5GBdstreamintoalinkofupto28loops [7] H. Buelow, et al. “Transmission of waveforms determined by (3×Lspan= 72.6 km) of NZ-DSF fiber with a mean launch 7 eigenvalues with psk-modulated spectral amplitudes,” in Proc. power of about −2.7dBm. We used homodyne detection with ECOC, 2016. a low phase noise fiber laser (1 kHz linewidth). The received [8] V.Aref,“Controlanddetectionofdiscretespectralamplitudesin nonlinear fourier spectrum,” arXiv preprint:1605.06328, 2016. signal is coherently detected and sampled by an oscilloscope