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The nonlinear Fourier transform (NFT) decomposes waveforms propagating through optical fiber into nonlinear degrees of freedom, which are preserved during transmission. By encoding information on the nonlinear spectrum, a transmission scheme inherently compatible with the nonlinear fiber is obtained. Despite potential advantages, the periodic NFT (PNFT) has been studied less compared to its counterpart based on vanishing boundary conditions, due to the mathematical complexity of the inverse transform. In this paper we extract the theory of the algebrogeometric integration method underlying the inverse PNFT from the literature, and tailor it to the communication problem. We provide a complete algorithm to compute the inverse PNFT. As an application, we employ the algorithm to design a novel modulation scheme called nonlinear frequency amplitude modulation, where four different nonlinear frequencies are modulated independently. Finally we provide two further modulation schemes that may be considered in future research. The algorithm is further applied in Part II of this paper to the design of a PNFT-based communication experiment., Comment: Significantly extended, final published version. 22 pages, 8 figures

The nonlinear Fourier transform has the potential to overcome limits on performance and achievable data rates which arise in modern optical fiber communication systems when nonlinear interference is treated as noise. The periodic nonlinear Fourier transform (PNFT) has been much less investigated compared to its counterpart based on vanishing boundary conditions. In this paper, we design a first experiment based on the PNFT in which information is encoded in the invariant nonlinear main spectrum. To this end, we propose a method to construct a set of periodic waveforms each having the same fixed period, by employing the exact inverse PNFT algorithm developed in Part I. We demonstrate feasibility of the transmission scheme in experiment in good agreement with simulations and obtain a bit-error ratio of $10^{-3}$ over a distance of 2000 km. It is shown that the transmission reach is significantly longer than expected from a naive estimate based on group velocity dispersion and cyclic prefix length, which is explained through a dominating solitonic component in the transmitted waveform. Our constellation design can be generalized to an arbitrary number of nonlinear degrees of freedom., Comment: Final accepted version. 10 pages, 7 figures

The integrable nonlinear Schr\"odinger equation (NLSE) is a fundamental model of nonlinear science which also has important consequences in engineering. The powerful framework of the periodic inverse scattering transform (IST) provides a description of the nonlinear phenomena modulational instability and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It associates the complex nonlinear dynamics with invariant nonlinear spectral degrees of freedom that may be used to encode information. While optical fiber is an ideal testing ground of its predictions, maintaining integrability over sufficiently long distances to observe recurrence, as well as synthesizing and measuring the field in both amplitude and phase on the picosecond timescales of typical experiments is challenging. Here we report on the experimental realization of FPUT recurrence in terms of an exact space-time-periodic solution of the integrable NLSE in a testbed for optical communication experiments. The complex-valued initial condition is constructed by means of the finite-gap integration method, modulated onto the optical carrier driven by an arbitrary waveform generator and launched into a recirculating fiber loop with periodic amplification. The measurement with an intradyne coherent receiver after a predetermined number of revolutions provides a non-invasive full-field characterization of the space-time dynamics. The recurrent space-time evolution is in close agreement with theoretical predictions over a distance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinear spectrum. The space-time scale exceeds that of previous experiments on FPUT recurrence in fiber by three orders of magnitude., Comment: Version accepted for publication in Scientific Reports

We design a two-dimensional signal constellation based on the exact periodic inverse nonlinear Fourier transform. Feasibility of continuous transmission with periodic signals is experimentally demonstrated over more than 2000 km., Comment: Accepted in the Optical Fiber Communications Conference (OFC) 2019

Polarization-division multiplexed (PDM) transmission based on the nonlinear Fourier transform (NFT) is proposed for optical fiber communication. The NFT algorithms are generalized from the scalar nonlinear Schr\"odinger equation for one polarization to the Manakov system for two polarizations. The transmission performance of the PDM nonlinear frequency-division multiplexing (NFDM) and PDM orthogonal frequency-division multiplexing (OFDM) are determined. It is shown that the transmission performance in terms of Q-factor is approximately the same in PDM-NFDM and single polarization NFDM at twice the data rate and that the polarization-mode dispersion does not seriously degrade system performance. Compared with PDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized to multi-mode fibers in the strong coupling regime, paving the way for the application of the NFT to address the nonlinear effects in space-division multiplexing., Comment: 15 pages, 9 figures

We initiate the systematic computation of the heat-kernel coefficients for Laplacian operators obeying anisotropic dispersion relations in curved spacetime. Our results correctly reproduce the limit where isotropy is restored and special anisotropic cases considered previously in the literature. Subsequently, the heat kernel is used to derive the scalar-induced one-loop effective action and beta functions of Horava-Lifshitz gravity. We identify the Gaussian fixed point which is supposed to provide the UV completion of the theory. In the present setting, this fixed point acts as an infrared attractor for the renormalization group flow of Newton's constant and the high-energy phase of the theory is screened by a Landau pole. We comment on the consequences of these findings for the renormalizability of the theory., Comment: 32 pages, 2 figures