Nonlinear Fourier transform (NFT) can convert a signal into a nonlinear spectrum, including continuous spectrum and discrete spectrum, where the eigenvalues are located in the upper half of the complex plane. With the approach of NFT, information is encoded into the nonlinear spectrum of a signal, which can effectively address the nonlinear transmission impairments arising in standard single-mode fiber. At the same time, as a new effective signal processing tool, NFT can be used to analyze pulses in fiber lasers. For a pure soliton, its nonlinear spectrum only contains the discrete spectrum. The eigenvalues in the discrete spectrum can then be used to characterize soliton pulses, wherein the real and imaginary parts refer to the frequency and amplitude of soliton pulses, respectively. This methodology provides a new perspective for the study of laser dynamics. The soliton and continuous wave background can be separated based on their different eigenvalue distributions after the obtainment of full-field information of pulses. We summarize the principle of NFT and its applications in the field of optical communications and fiber lasers, specifically the "soliton distillation" technology based on NFT.
In optical fiber communication systems, there are transmission impairments such as loss, dispersion, and nonlinearity. Loss and dispersion can be compensated by optical amplification and dispersion compensation technology. Pulse broadening and distortion caused by nonlinear effects related to optical pulse intensity have, however, become the main factors limiting the system communication capacity improvement. As a powerful mathematical tool, NFT can effectively solve the problem of lightwave propagation in a nonlinear medium such as an optical fiber. Recently, a new framework of optical fiber communication systems based on NFT has begun receiving extensive attention. NFT can decompose a signal into a continuous spectrum (nonsoliton component) and discrete spectrum (soliton component), which are considered nonlinear spectra. With this method, information can be encoded into the nonlinear spectrum of the signal, and the technique of doing so is known as nonlinear frequency division multiplexing (NFDM). Compatible with the nonlinear response characteristics of optical fibers, NFDM can effectively address the dispersion and nonlinear impairments arising in standard single-mode fiber (SMF) transmission.
Optical soliton is a special light field that does not change during transmission under the dispersion and nonlinear effects, and it can be generated and spread in optical fiber systems. Periodically repeating stable pulses generated in fiber lasers can also be considered as solitons, more specifically known as dissipative solitons. The output signal can be transformed into a nonlinear spectrum (including continuous spectrum, discrete spectrum, and corresponding eigenvalues) through NFT. In the nonlinear Fourier domain, nonlinear phenomena in optical fiber systems are investigated, such as rouge wave, optical frequency combs, and cavity solitons. Lately, NFT has been applied to laser pulse analysis. For pure soliton, its nonlinear spectrum only contains a discrete spectrum and the eigenvalues in the discrete spectrum correspond to the characteristics of the soliton. For example, the real and imaginary parts of an eigenvalue correspond to the frequency and amplitude of the soliton, respectively. When the dynamic characteristics of the pulse are dominated by solitons, that is, when the discrete spectrum cumulates most of the pulse energy, the eigenvalue distribution can reflect the pulse feature, as shown in Fig. 13. At the same time, pure soliton and the resonant continuous-wave background can be separated according to different eigenvalue distributions, to realize soliton distillation. These findings show how NFT provides new insights into ultrafast transient dynamics in fiber systems.
In an NFDM system (Fig. 3), after the nonlinear spectrum of the signal is recovered at the receiving end, the effects of dispersion and nonlinearity can be eliminated by simple phase compensation, and the system performance can be improved. According to the different modulation spectrum, the nonlinear spectrum can be divided into discrete spectrum communication and continuous spectrum communication. Research on discrete spectrum communication has focused on the number of multiplexed eigenvalues and advanced modulation formats for discrete spectrum. As for continuous spectrum communication, compared to the traditional OFDM systems, NFDM systems have higher quality factors under the optimal condition of fiber launching power. However, the NFT-based optical communication system still suffers from random noise. Meanwhile, problems, such as channel integrability and algorithm complexity, still exist, which greatly restrict the performance of the system.
To perform NFT analysis experimentally in the field of fiber lasers, full-field information, including the amplitude and phase of the pulse, must first be obtained. The current acquisition methods mainly include density functional theory and temporal lensing technology, combined with Gerchberg-Saxton phase recovery (Fig. 9) and coherent homodyne detection technologies (Fig. 11). After the full-field information of pulse is obtained, the pulse features can be projected onto the eigenvalue distribution by NFT. NFT can be used as an analysis tool to reduce the complexity of describing pulse evolution, whether for nonstationary (Fig. 5) or stationary pulses (Fig. 6). At the same time, according to the distribution of eigenvalues and their corresponding discrete spectra, the temporal evolution process of a pulse can be reconstructed using inverse nonlinear Fourier transform (INFT), which also shows that the NFT method can effectively characterize the laser pulse. INFT is not only effective for a single pulse (Fig. 7) but also achieves a good reconstruction effect for multiple pulses (Fig. 8). Further, the sideband eigenvalues can be removed and only the soliton eigenvalues are retained in the nonlinear spectrum. Through INFT, pure soliton can be recovered in the time domain (Fig 14). NFT can perform pure soliton distillation and reconstruction on various pulses generated in fiber lasers, including single pulse, single pulse in period-doubling, different double pulses (Fig. 15), and multiple pulses. The transient nonlinear dynamic analysis based on NFT can deepen the knowledge on soliton formation and its interaction process, and also clarify the transient working mechanism of fiber laser.
As an emerging signal processing tool, NFT provides new system design solutions in the field of optical communication, which is fully compatible with optical fiber. Transient nonlinear dynamics analysis based on NFT can describe laser pulses theoretically and completely, providing a basic overview on ultrafast nonlinear dynamics, and its application in ultra high-power fiber lasers.
