Representation of grossone-based arithmetic in simulink for scientific computing

AbstractNumerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent

Rogue waves driven by polarization instabilities in a long ring fiber oscillator

We present an experimental and theoretical results of a study of a complex nonlinear polarization dynamics in a passively self-mode-locked erbium-doped fiber oscillator implemented in a ring configuration and operating near lasing threshold. The theoretical model consists of seven coupled non-linear equations and takes into account both orthogonal states of polarizations in the fiber. The experiment confirmed the existence of seven eigenfrequencies, predicted by the model due to polarization instability near lasing threshold. By adjusting the state of polarization of the pump and in-cavity birefringence we changed some eigenfrequencies from being different (non-degenerate state) to matching (degenerate state). The non-degenerate states of oscillator lead to the L-shaped probability distribution function and true rogue wave regime with a positive dominant Lyapunov exponent value between 1.4 and 2.6. Small detuning from partially degenerate case also leads to L-shaped probability distribution function with the tail trespassing eight standard deviations threshold, giving periodic patterns of pulses along with positive dominant Lyapunov exponent of a filtered signal between 0.6 and 3.2. The partial degeneration, in turn, guides to quasi-symmetric distribution and the value of dominant Lyapunov exponent of 42 which is a typical value for systems with a source of the strongly nonhomogeneous external noise

Excitation back transfer in a statistical model for upconversion in Er-doped fibres

We report a new analytical approach to evaluate the accuracy of a statistical model of the migration assisted upconversion in Er-doped fibres. Unlike the mean-field approach to the excitation back transfer which was used in previous statistical model, we use a new approximation accounting for the variance of population of the first excited level. Implementing these results, we find that the accuracy of upconversion rate calculations is within 13 % if the concentration of erbium ions is smaller than the critical one

Changes in normalized permutation entropy during non-coherent pulsepulse interaction in the laser cavity

Summary form only given. Statistically extreme events, that is to say, the events which have L-shaped probability distribution function (PDF, or in other words L-statistics) and, in particular, the rogue wave (RW) events, constitute a special family of natural phenomena. Curiously, this kind of events is more common than it is believed. It was observed that the behavior of the systems which obey L-shaped statistics is clustered. Periods of high predictability and low volatility are followed by periods of drastic changes and high volatility, for example, RWs [1], earthquakes, epileptic seizures etc. Permutation entropy (PE) is an indicator of dynamical predictability of the system [2, 3] which allows quantization of the predictability or volatility. We report an investigation of using it to characterize the uncertainty period which was observed during pulse-pulse interaction in a long ring laser resonator without a saturable absorber. These resonators, in particular, the resonator built using Er3+ doped fiber (Fig.1), have polarization dependent instability near the lasing threshold [4] which leads to the generation of pulses propagating with different speed, either randomly or regularly. Using this property, an investigator can observe the mechanism of the pulse-pulse interaction in detai

Trapping polarization of light in nonlinear optical fibers: An ideal Raman polarizer

The main subject of this contribution is the all-optical control over the state of polarization (SOP) of light, understood as the control over the SOP of a signal beam by the SOP of a pump beam. We will show how the possibility of such control arises naturally from a vectorial study of pump-probe Raman interactions in optical fibers. Most studies on the Raman effect in optical fibers assume a scalar model, which is only valid for high-PMD fibers (here, PMD stands for the polarization-mode dispersion). Modern technology enables manufacturing of low-PMD fibers, the description of which requires a full vectorial model. Within this model we gain full control over the SOP of the signal beam. In particular we show how the signal SOP is pulled towards and trapped by the pump SOP. The isotropic symmetry of the fiber is broken by the presence of the polarized pump. This trapping effect is used in experiments for the design of new nonlinear optical devices named Raman polarizers. Along with the property of improved signal amplification, these devices transform an arbitrary input SOP of the signal beam into one and the same SOP towards the output end. This output SOP is fully controlled by the SOP of the pump beam. We overview the sate-of-the-art of the subject and introduce the notion of an "ideal Raman polarizer"

Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409-436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming

Rigorous modeling and physical interpretation of terahertz near-field imaging

Apertureless scanning near-field optical microscopy (SNOM) operating with terahertz (THz) laser pulses is a subject of great research interest. The Mie scattering theory is commonly used to explain the features of the optical waves produced by field interactions with SNOM tips and microstructures. However, since Mie scattering fails with SNOMs at submillimeter wavelengths, a rigorous model and analysis are desirable to assess the feasibility of the THz tip-enhanced scanning near-field techniques. In this paper, we present a numerical simulation of an apertureless SNOM imaging system in the THz band. A 2-dimensional model based on the finite element method (FEM) is investigated and discussed. The modeling results are in good agreement with the experimental data obtained for this system at 2 THz radiation [H.-T. Chen at al., Phys. Rev. Lett. 93, 267401 (2004)]. Additionally, a physical interpretation using the antenna theory is successfully confirmed by the simulation results