Improved Fast Inverse Nonlinear Fourier Transform for Multi-solitons

Abstract

The relation between the input of an ideal optical single-mode fiber and the corresponding fiber output constitutes a nonlinear system that can be described using the nonlinear Schrödinger equation. This nonlinear system has the interesting property that it can be solved analytically using nonlinear Fourier transforms. To utilize this property, new methods of optical communication are being developed by embedding information in the nonlinear Fourier domain and employing fast nonlinear Fourier transforms. Many of the recent works use a specialized form of inverse nonlinear Fourier transform to generate information-bearing fiber inputs in the form of so-called multi-soliton pulses. Recently, multiple fast inverse nonlinear Fourier transform algorithms that can generate multi-solitons have been proposed. The goal of this thesis is to study and improve these algorithms, in particular, with respect to their computational complexity.
Based on the literature survey, discrete Darboux transform combined with other discrete techniques is studied and a new algorithm is proposed. The algorithm employs a single-start approach in which discrete Darboux matrix is computed at only one sample point and rest of the samples are computed by evolution of the Darboux matrix. The algorithm is hence named as discrete Darboux evolution algorithm (DDE). The errors in the generated signal and run-time are studied by comparison with the classical Darboux transform (CDT). The DDE algorithm is shown to have floating point operations complexity of O(KN) for K eigenvalues and N samples. However, in a limited precision environment the number of samples that can be generated is found to be limited. To better understand the effects of machine precision, both the CDT and DDE algorithms are studied in a multi-precision environment. Certain insights from the study are used to develop two modifications to overcome the limitations. The first modification computes the signal using multiple single-start runs while the second one uses a multi-start approach. The second modification is shown to have errors comparable with other fast algorithms in literature. Additionally, in a qualitative comparison it is shown to be potentially faster than existing algorithms in a certain regime.