Soliton Propagation in Birefringent Fibers

In this article, the propagation of solitons in a single mode fiber with polarization mode dispersion (PMD) is analyzed. In optical fibers, the randomly varying birefringence degrades soliton transmission system in two aspects. First, the dispersive waves cause pulse broadening. Second, the dispersive waves interact with other soliton pulses. Here we studied the effects of PMD on a single pulse and the variation of pulse broadening, energy decay, and degree of polarization on a single soliton pulse propagating over a very long distance.     

Soliton interaction in birefringent optical fibers: Effects of nonlinear gain devices

This paper presents the influences of polarization mode dispersion (PMD) on the performance of soliton transmission system in birefringent fibers. Dispersive waves generated in single mode fibers due to PMD degrade the soliton transmission system in two aspects. First, solitons continuously lose their energy, thus cause enhancement in pulse width. Second, the dispersive waves interact with neighboring pulses and cause distortion in a sequence of pulses. Both these effects reduce the effective bit-rate and degrade the performance of high-speed optical transmission systems. Optical fibers with large group velocity dispersion (GVD) have less dispersive waves and are relatively robust to pulse broadening, but it enhances the interaction between the adjacent pulses. In this paper, we analyzed these effects of PMD on soliton propagation in birefringent fibers and introduced nonlinear gain devices with perturbation terms proportional to second and fourth power of amplitudes to reduce these effects. We proposed Symmetric Split-Step Fourier Method to solve the coupled nonlinear Schrödinger equations (CNLSE); which yields better results over the existing Split-Step Fourier Method.     

Analytical theory for pulse broadening induced by all-order polarization mode dispersion combined with frequency chirp and group-velocity dispersion

We derive an analytical expression for the expected root-mean-square (rms) pulse broadening with considering the combined effects of all-order polarization mode dispersion (PMD), group velocity dispersion (GVD) and frequency chirp. It is shown that two commonly used first-order PMD compensators lose their efficiency quickly with chirp parameter increasing if GVD is ignored. When GVD is added, prechirp technique is helpful for the enhancement of PMD tolerance for both uncompensated case and first-order compensators when GVD and PMD coefficient satisfy some special relationship.     

Solitons in a linearly coupled system with separated dispersion and nonlinearity

We introduce a model of dual-core waveguide with the cubic nonlinearity and group-velocity dispersion (GVD) confined to different cores, with the linear coupling between them. The model can be realized in terms of photonic-crystal fibers. It opens a way to understand how solitons are sustained by the interplay between the nonlinearity and GVD which are not "mixed" in a single nonlinear Schrodinger (NLS) equation, but are instead separated and mix indirectly, through the linear coupling between the two cores. The spectrum of the system contains two gaps, semi-infinite and finite ones. In the case of anomalous GVD in the dispersive core, the solitons fill the semi-infinite gap, leaving the finite one empty. This soliton family is entirely stable, and is qualitatively similar to the ordinary NLS solitons, although shapes of the soliton's components in the nonlinear and dispersive cores are very different, the latter one being much weaker and broader. In the case of the normal GVD, the situation is completely different: the semi-infinite gap is empty, but the finite one is filled with a family of stable gap solitons featuring a two-tier shape, with a sharp peak on top of a broad "pedestal." This case has no counterpart in the usual NLS model. An extended system, including weak GVD in the nonlinear core, is analyzed too. In either case, when the solitons reside in the semi-infinite or finite gap, they persist if the extra GVD is anomalous, and completely disappear if it is normal.     

Single- and multi-peak solitons in two-component models of metamaterials and photonic crystals

We report results of the study of solitons in a system of two nonlinear-Schrödinger (NLS) equations coupled by the XPM interaction, which models the co-propagation of two waves in metamaterials (MMs). The same model applies to photonic crystals (PCs), as well as to ordinary optical fibers, close to the zero-dispersion point. A peculiarity of the system is a small positive or negative value of the relative group-velocity dispersion (GVD) coefficient in one equation, assuming that the dispersion is anomalous in the other. In contrast to earlier studied systems of nonlinearly coupled NLS equations with equal GVD coefficients, which generate only simple single-peak solitons, the present model gives rise to families of solitons with complex shapes, which feature extended oscillatory tails and/or a double-peak structure at the center. Regions of existence are identified for single- and double-peak bimodal solitons, demonstrating a broad bistability in the system. Behind the existence border, they degenerate into single-component solutions. Direct simulations demonstrate stability of the solitons in the entire existence regions. Effects of the group-velocity mismatch (GVM) and optical loss are considered too. It is demonstrated that the solitons can be stabilized against the GVM by means of the respective “management” scheme. Under the action of the loss, complex shapes of the solitons degenerate into simple ones, but periodic compensation of the loss supports the complexity.     

Soliton stability against polarization-mode-dispersion in the split-step system

We consider polarization dynamics of solitons in the split-step system (SSM), built as a periodic concatenation of dispersive and nonlinear segments. The model is based on coupled equations for two polarizations, which include birefringence and PMD (polarization-mode dispersion) in the form of random misalignments of the principal polarization axes at junctions between fiber segments. By means of direct simulations, we identify a full stability region for solitons (RZ signals) in the system, and compare it with that in the regular SSM. Beyond the stability border, pulses suffer splitting (which is a characteristic feature of the SSM). Considering co-transmission of soliton pairs, we conclude that the minimum separation between the RZ signals necessary to prevent their interaction increases by ?25% in comparison with the regular (single-polarization) SSM.     

高速光纤通信系统中抑制偏振模色散的新机制

重点研究了偏振模色散、群速度色散、自相位调制三者之间在高速光纤通信系统中的相互作用，从时域角度分析脉冲的演变，从频域角度分析频谱的变化，提出一定条件下，啁啾、色散、自相位调制可以部分补偿偏振模色散的思想。通过对40Gbit／s系统进行偏振模色散、群速度色散和自相位调制共同作用的仿真，从统计意义上验证了它们之间的相互影响，并找到最佳传输方案，对系统设计提供了参考。     

A scheme for pre-shaping of dispersion-managed solitons

We demonstrate efficacy of shaping Gaussian pulses into dispersion-managed (DM) solitons by dint of a device which includes two fibers with the same group-velocity-dispersion (GVD) coefficients as in the DM system, a frequency-domain filter, and an amplifier. The shaping of a given input pulse is optimized by adjusting values of four free parameters of the cell, viz., lengths of the two fibers, filtering coefficient, and amplification gain, with the objective to achieve the best fit of the transformed pulse to the (numerically found) DM soliton, including systems with the third-order GVD, and with three-step DM maps. Launching reshaped pulses into the DM system, we demonstrate, in direct simulations, their complete stability over indefinitely long propagation distances. We examine sensitivity of the reshaping to deviations of the control parameters from their optimum values, a noteworthy results being very weak dependence on variations of the filtering strength. The dependence of the four control parameters on the width and chirp of the Gaussian input signal is also studied in detail.     

GVD compensation schemes with considering PMD

Three group velocity dispersion (GVD) compensation schemes, i.e., the post-compensation, pre-compensation and hybrid-compensation schemes, are discussed with considering polarization mode dispersion (PMD). In the 10- and 40-Gbit/s non-return-zero (NRZ) on-off-key (OOK) systems, three physical factors, Kerr effect, GVD and PMD are considered. The numerical results show that, when the impact of PMD is taken into account, the GVD pre-compensation scheme performs best with more than 1 dB better of average eye-opening penalty (EOP) when input power is up to 10 dBm in the 10-Gbit/s system. However the GVD post-compensation scheme performs best for the case of 40 Gbit/s with input power less than 13 dBm, and GVD pre-compensation will be better if the input power increased beyond this range. The results are different from those already reported under the assumption that the impact of PMD is neglected. Therefore, the research in this paper provide a different insight into the system optimization when PMD, Kerr e     

Stability of temporal solitons in uniform and “managed” quadratic nonlinear media with opposite group-velocity dispersions at fundamental and second harmonics

The problem of the stability of solitons in second-harmonic-generating media with normal group-velocity dispersion (GVD) in the second-harmonic (SH) field, which is generic to available χ⁽²⁾ materials, is revisited. Using an iterative numerical scheme to construct stationary soliton solutions, and direct simulations to test their stability, we identify a full soliton-stability range in the space of the system’s parameters, including the coefficient of the group-velocity-mismatch (GVM). The soliton stability is limited by an abrupt onset of growth of tails in the SH component, the relevant stability region being defined as that in which the energy loss to the tail generation is negligible under experimentally relevant conditions. We demonstrate that the stability domain can be readily expanded with the help of two “management” techniques (spatially periodic compensation of destabilizing effects) - the dispersion management (DM) and GVM management. In comparison with their counterparts in optical fibers, DM solitons in the χ⁽²⁾ medium feature very weak intrinsic oscillations.     

Manakov solitons and polarization mode dispersion

The effect of polarization mode dispersion (PMD) on Manakov solitons and dispersion managed solitons is treated analytically and by numerical simulation. In the analytic approach the internal motion of the Manakov soliton is represented as a damped harmonic oscillator. The PMD functions as a white noise source driving the oscillations. It is shown that the solitons can withstand PMD up to a certain instability threshold for which an analytic expression is obtained. This threshold is also evaluated for dispersion managed solitons. (c) 2000 American Institute of Physics.     

Impacts of polarization mode dispersion on the pulse-width with considering initial pulse chirp and fiber GVD

The impacts of polarization mode dispersion (PMD) on the pulse-width in linear systems have been investigated. The analytical solutions, including the effects of initial frequency chirp and group velocity dispersion (GVD), are derived. Analyses show that the pulse broadening effects induced by the second-order PMD depend on GVD and chirp parameter, which are different from those induced by the first-order PMD. An initially chirped Gaussian pulse is taken as an example, upon which analytical solutions of rms pulse-width are derived before and after the first-order PMD compensation. The first-order PMD compensator is also evaluated based on these solutions. The results show that the pulse broadening effects will be resisted efficiently by choosing appropriate GVD and chirp parameter; in general, the post-transmission compensation method will be less efficient than the PSP-transmission method.     

Interactions between two-dimensional solitons in the diffractive-diffusive Ginzburg-Landau equation with the cubic-quintic nonlinearity

We report the results of systematic numerical analysis of collisions between two and three stable dissipative solitons in the two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) with the cubic-quintic (CQ) combination of gain and loss terms. The equation may be realized as a model of a laser cavity which includes the spatial diffraction, together with the anomalous group-velocity dispersion (GVD) and spectral filtering acting in the temporal direction. Collisions between solitons are possible due to the Galilean invariance along the spatial axis. Outcomes of the collisions are identified by varying the GVD coefficient, β, and the collision “velocity” (actually, it is the spatial slope of the soliton’s trajectory). At small velocities, two or three in-phase solitons merge into a single standing one. At larger velocities, both in-phase soliton pairs and pairs of solitons with opposite signs suffer a transition into a delocalized chaotic state. At still larger velocities, all collisions become quasi-elastic. A new outcome is revealed by collisions between slow solitons with opposite signs: they self-trap into persistent wobbling dipoles, which are found in two modifications — horizontal at smaller β, and vertical if β is larger (the horizontal ones resemble “zigzag” bound states of two solitons known in the 1D CGL equation of the CQ type). Collisions between solitons with a finite mismatch between their trajectories are studied too.     

The PMD of sinusoidally spun fibers in the presence of random birefringence perturbations

Although fiber spinning is known to reduce polarization mode dispersion (PMD) effects in optical fibers, relatively few studies have been performed of the dependence of the reduction factor on the strength of random birefringence fluctuations. In this paper, we apply a general mathematical model of random fiber birefringence to sinusoidally spun fibers. We find that while even in the presence of random birefringence perturbations the maximum reduction of PMD is still obtained when the phase matching condition is satisfied, the degree of PMD reduction and the probability distribution function of the DGD both vary with the random birefringence profiles.     

Observation of dark pulse in a dispersion-managed fiber ring laser

The observation of dark pulse in a dispersion-managed fiber ring laser with net negative cavity group velocity dispersion (GVD) is reported. Both bright and dark pulses can be obtained in our fiber laser. When we appropriately adjust the cavity birefringence to achieve triple-wavelength mode-locked operation in the laser by rotating the polarization controller, the bright pulse could be switched to dark pulse. It is believed that the dark dispersion-managed (DM) pulse generation is caused by the linear and nonlinear intermodulation effects among the three wavelength pulses.     

Effects of residual stress on polarization mode dispersion of fibers made with different types of spinning

We analyze the effects of residual stress on the polarization mode dispersion (PMD) of fibers made with different types of spinning. A theoretical scheme is developed from a previous model by the incorporation of a circular birefringence term contributed by residual torsional stress. It is found that the residual stress can significantly affect the PMD of unidirectionally spun fibers when the fiber birefringence is low, but it has little effect on the PMD of bidirectionally spun fibers.     

Stabilization of spatiotemporal solitons in second-harmonic-generating media

We report the results of a systematic analysis of the existence and stability of spatiotemporal (two-dimensional) solitons (STSs) in the model of a planar waveguide with the intrinsic χ⁽²⁾ nonlinearity. Fundamental obstacles to the creation of STSs under physically realistic conditions are the normal sign of the group-velocity dispersion (GVD) at the second harmonic (SH), and the significant group-velocity mismatch (GVM) between the SH and fundamental-frequency (FF) components. To construct STS solutions in a numerical form, we adjust the iterative method, which was recently used for finding temporal (one-dimensional) χ⁽²⁾ solitons in a similar setting. We identify effective existence borders for the STSs, within which the energy loss to the generation of extended “tails” in the SH component (due to the normal sign of the GVD) is negligible. It is demonstrated that the existence region can be made much broader by means of the GVD-management and GVM-management techniques. We also explore interactions between the STSs, and find robust two-soliton bound states, with a moderate separation in the longitudinal (temporal) direction. Head-on collisions between the STSs are always destructive.     

Multidimensional semi-gap solitons in a periodic potential

The existence, stability and other dynamical properties of a new type of multi-dimensional (2D or 3D) solitons supported by a transverse low-dimensional (1D or 2D, respectively) periodic potential in the nonlinear Schr?dinger equation with the self-defocusing cubic nonlinearity are studied. The equation describes propagation of light in a medium with normal group-velocity dispersion (GVD). Strictly speaking, solitons cannot exist in the model, as its spectrum does not support a true bandgap. Nevertheless, the variational approximation (VA) and numerical computations reveal stable solutions that seem as completely localized ones, an explanation to which is given. The solutions are of the gap-soliton type in the transverse direction(s), in which the periodic potential acts in combination with the diffraction and self-defocusing nonlinearity. Simultaneously, in the longitudinal (temporal) direction these are ordinary solitons, supported by the balance of the normal GVD and defocusing nonlinearity. Stability of the solitons is predicted by the VA, and corroborated by direct simulations.     

Filter control polarization mode dispersion in dispersion managed soliton systems

In this paper, the dispersion managed soliton (DMS) transmission equation is built on considering the effects of polarization mode dispersion (PMD) and filter control. The DMS transmission of filtering control in constant birefringence fibers is firstly analyzed by varitional method, from which the evolving rules of characteristical DMS parameters are obtained. Secondly, the stability of DMS transmission and its timing jitter are investigated in the random varying birefringence fibers with the conventional model of PMD. The results reveal that filter control DMS system has powerful robustness to PMD effects and DMS's timing jitter can be decreased considerably with the help of filters.     

Modelling of polarization mode dispersion in optical communications systems

With the rapid increase in the data rates transmitted over optical systems, as well as with the recent extension of terrestrial systems to ultra-long haul reach, polarization mode dispersion (PMD) has become one of the most important and interesting limitations to system performance. This phenomenon originates from mechanical and geometrical distortions that break the cylindrical symmetry of optical fibers and create birefringence. It is the random variations of the local birefringence along the propagation axis of the optical fiber that create the rich and complicated bulk of phenomena that is attributed to PMD. The detailed statistical properties of the local birefringence and its dependence on position are only important as long as the overall system length is comparable with the correlation length of the birefringence in the fiber. In typical systems, however, the latter is smaller by more than three orders of magnitude so that the specific properties of the local birefringence become irrelevant. Instead, the fiber can be viewed as a concatenation of a large number of statistically independent birefringent sections characterized only by the mean square value of their birefringence. This model has been used extensively in the study of PMD and its predictions have been demonstrated to be in excellent agreement with experimental results. This approach opens the door to the world of stochastic calculus, which offers many convenient tools for studying the PMD problem. In this article we review the modelling of PMD and discuss the properties of this phenomenon as a stochastic process. We explain the use of stochastic calculus for the analysis of PMD and describe the derivation of the frequency autocorrelation functions of the PMD vector, its modulus and the principal states. Those quantities are then related to commonly used parameters such as the bandwidth of the first order PMD approximation, the bandwidth of the principal states and to the accuracy of PMD measurements.