Jose Nathan Kutz

Stability of attractive Bose-Einstein condensates in a periodic potential

Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.

Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose-Einstein condensate.

The cubic nonlinear Schrödinger equation is the quasi-one-dimensional limit of the mean-field theory which models dilute gas Bose-Einstein condensates. Stationary solutions of this equation can be characterized as soliton trains. It is demonstrated that for repulsive nonlinearity a soliton train is stable to initial stochastic perturbation, while for attractive nonlinearity its behavior depends on the spacing between individual solitons in the train. Toroidal and harmonic confinement, both of experimental interest for Bose-Einstein condensates, are considered.

Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries.

We demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters. First, we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low-rank coherent structures. Here, libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points. The selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics, stability, and bifurcations of nonlinear dynamical systems. The use of empirical interpolation points and sparse representation facilitates a family of local reduced-order models for each physical regime, rather than a higher-order global model, which has the benefit of physical interpretability of energy transfer between coherent structures. The method advocated also allows for orders-of-magnitude improvement in computational speed and memory requirements. To illustrate the method, the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex Ginzburg-Landau equation. It is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present.

Scaling Fiber Lasers to Large Mode Area: An Investigation of Passive Mode-Locking Using a Multi-Mode Fiber

The mode-locking of dissipative soliton fiber lasers using large mode area fiber supporting multiple transverse modes is studied experimentally and theoretically. The averaged mode-locking dynamics in a multi-mode fiber are studied using a distributed model. The co-propagation of multiple transverse modes is governed by a system of coupled Ginzburg-Landau equations. Simulations show that stable and robust mode-locked pulses can be produced. However, the mode-locking can be destabilized by excessive higher-order mode content. Experiments using large core step-index fiber, photonic crystal fiber, and chirally-coupled core fiber show that mode-locking can be significantly disturbed in the presence of higher-order modes, resulting in lower maximum single-pulse energies. In practice, spatial mode content must be carefully controlled to achieve full pulse energy scaling. This paper demonstrates that mode-locking performance is very sensitive to the presence of multiple waveguide modes when compared to systems such as amplifiers and continuous-wave lasers.

Extremum-Seeking Control of a Mode-Locked Laser

An adaptive controller is demonstrated that is capable of both obtaining and maintaining high-energy, single-pulse states in a mode-locked fiber laser. In particular, a multi-parameter extremum-seeking control (ESC) algorithm is used on a nonlinear polarization rotation (NPR) based laser using waveplate and polarizer angles to achieve optimal passive mode-locking despite large disturbances to the system. The physically realizable objective function introduced divides the energy output by the kurtosis of the pulse spectrum, thus balancing the total energy with the coherence of the mode-locked solution. In addition, its peaks are high-energy mode-locked states that have a safety margin near parameter regimes where mode-locking breaks down or the multipulsing instability occurs. The ESC is demonstrated by numerical simulations of a single-NPR mode-locked laser and is able to track locally maximal mode-locked states despite significant disturbances to parameters such as the fiber birefringence.

Theory and simulation of passive modelocking dynamics using a long-period fiber grating

A novel modelocking technique is presented in which the intensity-dependent mode-coupling dynamics of a long-period fiber grating is used to achieve modelocking in a passive optical fiber laser. By an appropriate choice of the grating period, a resonant coupling occurs between co-propagating core and cladding modes, causing the low-intensity wings of the pulse to be transferred to the cladding mode and be attenuated. In contrast, the higher intensity peaks of a pulse are detuned from resonance by the nonlinearity and remain largely unaffected. Numerical studies of this pulse-shaping mechanism show that the laser produces stable mode-locked soliton-like pulses which are limited in bandwidth by the smaller of either the grating transmission bandwidth or amplifier bandwidth.

Dual transmission filters for enhanced energy in mode-locked fiber lasers.

We theoretically demonstrate that in a laser cavity mode-locked by a set of waveplates and passive polarizer, the energy performance can be increased by incorporating a second set of waveplates and polarizer in the cavity. The two nonlinear transmission functions acting in combination can be engineered so as to suppress the multi-pulsing instability responsible for limiting the single pulse per round trip energy in a myriad of mode-locked cavities. In a single parameter sweep, the energy is demonstrated to double. It is anticipated that further engineering and optimization of the transmission functions by tuning the eight waveplates, fiber birefringence, two polarizers and two lengths of transmission fiber can lead to further significant increases. Moreover, the analysis suggests a general design and engineering principle that can potentially realize the goal of making fiber based lasers directly competitive with solid state devices. The technique is feasible and easy to implement without requiring a new cavity design paradigm.

Gordon-Haus timing jitter reduction in dispersion-managed soliton communications

An analytic theory is presented which demonstrates that the noise induced Gordon-Haus timing jitter in arbitrary dispersion-managed transmission systems is reduced by the power-enhancement factor required to support a dispersion-managed solitons provided the path-average soliton period is much greater than the dispersion-map period. The analysis further predicts the behaviour of the amplitude, width, and quadratic chirp fluctuations due to the amplified spontaneous emission (ASE) noise.

Model selection for dynamical systems via sparse regression and information criteria

We develop an algorithm for model selection which allows for the consideration of a combinatorially large number of candidate models governing a dynamical system. The innovation circumvents a disadvantage of standard model selection which typically limits the number of candidate models considered due to the intractability of computing information criteria. Using a recently developed sparse identification of nonlinear dynamics algorithm, the sub-selection of candidate models near the Pareto frontier allows feasible computation of Akaike information criteria (AIC) or Bayes information criteria scores for the remaining candidate models. The information criteria hierarchically ranks the most informative models, enabling the automatic and principled selection of the model with the strongest support in relation to the time-series data. Specifically, we show that AIC scores place each candidate model in the strong support, weak support or no support category. The method correctly recovers several canonical dynamical systems, including a susceptible-exposed-infectious-recovered disease model, Burgers’ equation and the Lorenz equations, identifying the correct dynamical system as the only candidate model with strong support.

Sparse Sensor Placement Optimization for Classification

Choosing a limited set of sensor locations to characterize or classify a high-dimensional system is an important challenge in engineering design. Traditionally, optimizing the sensor locations involves a brute-force, combinatorial search, which is NP-hard and is computationally intractable for even moderately large problems. Using recent advances in sparsity-promoting techniques, we present a novel algorithm to solve this sparse sensor placement optimization for classification (SSPOC) that exploits low-dimensional structure exhibited by many high-dimensional systems. Our approach is inspired by compressed sensing, a framework that reconstructs data from few measurements. If only classification is required, reconstruction can be circumvented and the measurements needed are orders-of-magnitude fewer still. Our algorithm solves an