Lossy Optical Fibers Research Articles

On dynamics of elliptic solitons in lossy optical fibers

By exploiting the theory of electromagnetic waves from Maxwell’s equations, the damped nonlinear Schrödinger (DNLS) equation is shown to govern the evolution of nonlinear periodic optical signals in a lossy optical fiber. These optical periodic pulses are mainly generated by the classical process of modulational instability (MI) in which nonlinearity is balanced by chromatic dispersion in the anomalous regime, with the linear loss generally suppressing the existence of soliton trains during propagation down the lossy fiber. When the periodic optical wave trains are subjected to weak external perturbations, this leads to the exposure of some internal modes of the system which are bound states solutions of the first order Lamé equation. These modes generally characterize various fundamental background excitations that co-propagate with the optical periodic signals in the fiber. Direct numerical simulations of the DNLS amplitude equation depict the exponential decrease in the amplitude and corresponding increase in the width of the wave trains during propagation. Power lasers are used in order to compensate for fiber losses; this is realized via time-division multiplexing of optical pulses which are periodically pumped into the lossy fiber at regular distances within the framework of a distributed amplification scheme. This leads to the regular energy restoration in the lossy fiber as a result of the interactions between the energized multiplexed light signals (generated by the power lasers) and the propagating damped optical pulses, hence ensuring effective transmission over long distances.

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

In this paper, we present a novel extension of the well-known split-step Fourier transform (SSFT) approach for solving the one-dimensional nonlinear Schrödinger equation (NLSE), which incorporates the fiber loss term. While this essential equation governs the pulse propagation in a lossy optical fiber, it is not supported by an exact analytical solution. In this regard, extended versions of the Fourier pseudospectral method (FPSM) and Hopscotch method (HSM) are effectively established as well to cope with the fiber losses effects associated with the pulses’ propagation through the fiber optics, and thus, numerous comparisons are exhaustively conducted among these three compelling numerical approaches to validate their reliability, stability, and accuracy. Based on this, the MATLAB numerical findings bolster that the extended version of the SSFT approach demonstrates superior performance over the other suggested schemes in simulating the solitons propagation in a lossy optical fiber.

Entanglement robustness of continuous variable Einstein-Podolsky-Rosen-entangled state distributed over optical fiber channel

Einstein-Podolsky-Rosen (EPR)-entangled state light field at a telecommunication wavelength of 1.5 μm is an important quantum source for realizing the continuous variable quantum information processing and some quantum protocols over optical fiber channel. When the EPR-entangled state light field is distributed over the optical fiber channel, the disentanglement is always present because the the EPR entangled state interacts with the fiber channel. It affects the performance of quantum information processing. In this paper, we theoretically calculate the positive partial transposition (PPT) of the entangled state distributed over the optical fiber channel in the single-channel and dual-channel distribution scheme, respectively. Three types of initial entangled light field are considered and analyzed, they being an initial EPR entangled state, an EPR entangled state with asymmetric quadratures, and an EPR entangled state with asymmetric modes. Furthermore, the influence of the extra noise in the optical fiber on the transmission distance of EPR entangled state over the optical fiber channel is investigated. In the single-channel scheme or dual-channel scheme, the extra noise in the optical fiber channel leads the entangled state light field to be disentangled, and the transmission distance of EPR entangled state over the optical fiber channel to decrease rapidly with the increase of the extra noise. For maintaining the robustness of EPR entangled states in lossy optical fiber channels, the dual-channel scheme has more stringent requirements for the correlation quadrature symmetry and purity of the initial entangled state than the single-channel scheme. In the single fiber noise channel scheme, the maximum transmission distance and the robustness of the EPR entangled states with asymmetric modes are not sensitive to the asymmetry between modes. The change of asymmetry between modes does not lead to being disentangled. The maximum transmission distance does not change either. However, the decrease of asymmetry between modes results in the disentanglement in the double fiber noise channels’ scheme. The maximum transmission distance is reduced and the sudden death occurs to the entanglement. The present results will lay a foundation for continuous variables quantum information processing based on optical fiber, such as realizing continuous variables quantum communication over optical fiber and constructing metropolitan quantum network over optical fiber.

Periodic and nonperiodic amplifications of attosecond solitons in an inhomogeneous lossy optical fiber

We investigate the variable-coefficient generalized higher order nonlinear Schrödinger equation which governing the attosecond optical soliton propagation in an inhomogeneous fiber system. Three-soliton solutions are attained through Darboux transformation based on the Lax pair. Using three soliton solutions, we investigate the periodic and nonperiodic amplification of three solitons by adopting certain functions to variable coefficients. Various amplification schemes are demonstrated graphically via symbolic computation. Especially, periodic and nonperiodic amplifications are attained under dispersion and nonlinearity management. In order to realize the soliton amplifications, two different set of variable coefficients profiles are employed. The results obtained in this work will be helpful for designing the optical amplification devices by soliton control technology.

Soliton propagation in lossy optical fibers

Neste trabalho estudamos a propagação de solitons em fibras ópticas com perdas. O principal objetivo deste trabalho é estudar a perda de energia da onda soliton durante a propagação e avaliar o impacto dessa perda na transmissão do sinal soliton. Neste contexto, um esquema numérico foi desenvolvido para resolver um sistema de equações diferenciais parciais complexas (EDPC), que descreve a propagação de solitons em fibras óticas com mecanismos de perdas e de amplificações não-lineares. O procedimento numérico é baseado na teoria matemática das séries de Taylor para funções complexas. Adaptamos o método de diferenças finitas (MDF) para aproximar derivadas de funções complexas. Em seguida, resolvemos o sistema algébrico resultante da discretização, implicitamente, por meio do método de Gauss-Seidel com relaxamento (MGSR). O estudo numérico do sistema de EDPC com atenuação linear e cúbica mostrou que ondas soliton sofrem efeitos de atenuação, dispersão e oscilação. Por outro lado, verificamos que ao considerar o termo não linear (termo cúbico) como uma amplificação ótica é possível compensar parcialmente a atenuação do sinal ótico. Finalmente, mostramos que um ganho de 9 % triplica a distância de propagação da onda soliton fundamental, quando a taxa de dissipação é de 1 %.

Analytical model and experimental verification of the critical power for modulation instability in optical fibers.

Modulation instability is thoroughly investigated and a simple analytical model for its power critically modifying the wave properties in terms of system parameters is derived and experimentally validated. The differences on the modulation instability gain spectrum in lossless and lossy optical fibers are analyzed based on theoretical models and numerical simulations. In particular the impact of background noise on the behavior of modulation instability is studied analytically and verified by measurements and simulations. The proposed analytical model is experimentally validated by monitoring the wave propagation along an optical fiber using a Brillouin optical time-domain analyzer. This way, the evolution of the optical signal traveling through optical fibers, especially, the pump depletion and the recurrence phenomenon are investigated.

Strong Converse for the Classical Capacity of Optical Quantum Communication Channels

We establish the classical capacity of optical quantum channels as a sharp transition between two regimes-one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical capacity. This result is obtained by proving a strong converse theorem for the classical capacity of all phase-insensitive bosonic Gaussian channels, a well-established model of optical quantum communication channels, such as lossy optical fibers, amplifier, and free-space communication. The theorem holds under a particular photon-number occupation constraint, which we describe in detail in this paper. Our result bolsters the understanding of the classical capacity of these channels and opens the path to applications, such as proving the security of noisy quantum storage models of cryptography with optical links.

Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation

We consider optical pulse propagation in an Erbium doped inhomogeneous lossy optical fiber with time dependent phase modulation, which is governed by a system of Generalized Inhomogeneous Nonlinear Schrödinger Maxwell–Bloch (GINLS–MB) equation. Multi-soliton propagation is studied analytically by means of deriving associated Lax pair and the soliton solutions are obtained using Darboux transformation. By suitably adjusting the group velocity dispersion and nonlinearity parameter, we discuss various soliton dynamics such as periodic distributed amplification, pulse compression etc. In each case, we demonstrate the influence of inhomogeneous parameter. Finally we investigate the pulse compression through nonlinear tunneling.

Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans

The formation of single-soliton or bound-multisoliton states from a single linearly chirped Gaussian pulse in quasi-lossless and lossy fiber spans is examined. The conversion of an input-chirped pulse into soliton states is carried out by virtue of the so-called direct Zakharov-Shabat spectral problem, the solution of which allows one to single out the radiative (dispersive) and soliton constituents of the beam and determine the parameters of the emerging bound state(s). We describe here how the emerging pulse characteristics (the number of bound solitons, the relative soliton power) depend on the input pulse chirp and amplitude.

Variational approach to dynamics of bright solitons in lossy optical fibers

A variational analysis of dynamics of soliton solution of coupled nonlinear Schrödinger equations with oscillating terms is made, considering a birefringent fiber with a third‐order nonlinearity in the anomalous dispersion frequency region. This theoretical model predicts optical soliton oscillations in lossy fibers.

Modeling optical solitons in a practical lossy fiber

An analytical investigation is presented that is based on the Lagrangian variational formulation of the interaction of two orthogonally polarized optical pulses. The pulses are governed by a pair of coupled nonlinear Schrödinger equations in a practical lossy optical fiber. PACS Nos.: 03.40Kf, 42.65Tg, 42.81Dp

Soliton amplification and reshaping in optical fibers with variable dispersion

We look for a setup providing optimum amplification and reshaping of short solitons in a lossy optical fiber. We consider a reshaper model combining a pointlike amplifier and a segment of a variable-dispersion fiber whose length is comparable with the soliton’s dispersion length. The objective is to find reshaping configurations with a minimum length providing for release of a chirpless duly amplified soliton into the bulk fiber. In most cases the input pulse is taken as a soliton with no chirp, but chirped input pulses are tested as well. Two particular types of variable dispersion (dispersion management) are considered: piecewise constant and linear. The main part of the analysis is done semianalytically by means of the variational approximation. Direct numerical simulations are also performed at some values of the parameters to permit us to examine the accuracy of the approximation (which proves to be good). It is found that the amplifier placed at the input edge of the reshaper always gives better results than the one at the output. The minimum necessary length of the variable-dispersion segment proves to be a decreasing function of the amplification factor. It is found that the performance characteristics are only weakly sensitive to a particular choice of the configuration within a given type of variable dispersion, so the actual choice can be determined by convenience of fabrication. The obtained results can be applied as well to optimize compression of solitons (without amplification) by variable-dispersion fibers.

Modulational instability in lossy optical fibers

The modulational instability in an optical fiber, with loss taken into account, is analytically investigated. It is demonstrated how the linearized equation for perturbation can be solved exactly in terms of Bessel functions, and qualitative differences from previous analytic approaches are pointed out. Finally, the limits imposed by this instability on optical communication systems are discussed. The use of this theory in system design is examined.

Strong periodic amplification of solitons in a lossy optical fiber: analytical results

Transmission of a femtosecond soliton is analyzed in a model of a lossy fiber with periodically installed pointlike amplifiers. The model incorporates dispersion of the losses, the Raman frequency downshift, and a finite amplification bandwidth. The analysis is focused on the case in which spacing between amplifying pulses is large, so that the pulse must be strong enough to reshape the attenuated soliton. It is demonstrated that evolution of the soliton in this model is governed by a two-dimensional map that defines the soliton’s peak power and central frequency after a given amplification pulse as functions of their values after a previous pulse. The map is considered analytically for the limiting case in which the frequency shift between the spectral maximum of the amplifying gain and the minimum of the dissipative absorption is large enough to permit reduction to a one-dimensional map in terms of the soliton’s peak power only. It is demonstrated that the one-dimensional map has a nontrivial stable fixed point, providing stable transmission of the soliton. An important generic feature of the reshaping regime with large spacing between amplifiers is that the strong amplification pulses inevitably produce additional parasitic (secondary) solitons. In terms of the same one-dimensional map, a regime of operation is found in which only transmission of the primary soliton is supported while all the parasites are completely destroyed by the losses. This regime is based on a sufficiently narrow gain bandwidth.

Nonlinear self-phase modulation in optical soliton systems with lumped amplifiers

The nonlinear Schrödinger equation, governing pulse propagation in lossy optical fibers with periodic amplification, is analyzed under the average soliton concept. Bright and dark solitons solutions, corresponding to separatrices of one-dimensional hamiltonian dynamical systems, have been derived. The shift of the soliton velocity, relative to the group velocity, as well as the effect of nonlinear self-phase modulation on the angular frequency and the wavenumber are investigated. The obtained results are connected with the initial pulse amplitude and its spatial and temporal derivatives.

The propagation of bright and dark solitons in lossy optical fibers

An analytic perturbation solution to the nonlinear Schrodinger equation with loss Gamma for both normal and anomalous dispersion is developed. Explicit results are obtained through second order in the perturbation Gamma . The results show that the dark pulse spreads less rapidly than the bright one and that total spreading as well as the difference in spreading rate for the two types of pulses decreases with loss. Comparisons are made with a zeroth-order perturbation theory and with numerical simulations, which are found to bracket the second-order results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Effect of fifth-order nonlinearity in refractive index on Gaussian pulse propagation in lossy optical fibers

The effect of fifth-order nonlinearity in refractive index on Gaussian pulse propagation in lossy optical fibers is considered. It is shown that fifth-order nonlinearity considerably modifies the pulse propagation, as a result of which the frequency at which the reshaping of the pulse should be done to have a distortionless propagation is reduced significantly (by a factor of about 3 compared with the case of cubic nonlinearity). The peak power for which fifth-order nonlinearity becomes significant is of the same order of magnitude that is required for cubic nonlinearity.