Fourth-order Dispersion Coefficient Research Articles

Strong and weak interactions of rational vector rogue waves and solitons to anyn-component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate anyn-component fourth-order nonlinear Schrödinger (n-FONLS) system with non-zero backgrounds containing then-Hirota equation and then-Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of then-FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of then-FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of then-FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

Gain-saturated two-pump fiber optical parametric amplifiers in the presence of dispersion fluctuation and fourth-order dispersion coefficient

The saturated gain performance of the two-pump fiber optical parametric amplifier (2-P FOPA) in the presence of random dispersion fluctuation as well as fourth-order dispersion coefficient are investigated. The gain spectra of the 2-P FOPA is simulated by solving the coupled equations numerically, considering the dispersion fluctuations along the fiber as a stochastic process with a given standard deviation and correlation length. The gain performance of 2-P FOPA is analysed with the variation of correlation length, standard deviation and fourth-order dispersion coefficient, respectively. It shows that, the gain saturation curves of 2-P FOPA exhibit obvious differences in the presence of dispersion fluctuations and fourth-order dispersion coefficients. The peak gain reduces with the increase of the standard deviation and the reduction is higher in case of shorter correlation lengths, also the variations of 2-P FOPA saturation powers are affected by the standard deviation and correlation length significantly. In addition, the performance of the 2-P FOPA varies greatly while considering the fourth-order dispersion coefficient, where the peak gain remains the same, but the 3 dB bandwidth is wider with considering the fourth-order dispersion coefficient. The numerical analysis is meaningful for the research of 2-P FOPA saturated gain performance, which having attracted great research interests in the optical fiber communication systems.

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrödinger system in an inhomogeneous optical fiber**Project supported by the BUPT Excellent Ph.D. Students Foundation (Grant No. CX2019201), the National Natural Science Foundation of China (Grant Nos. 11772017 and 11805020), the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schrödinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion (GVD) and fourth-order dispersion (FOD) coefficients are the constants, we exhibit the first- and second-order vector semi-rational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first- and second-order periodic vector semi-rational rogue waves, first- and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.

Breathers, multi-peak solitons, breather-to-soliton transitions and modulation instability of the variable-coefficient fourth-order nonlinear Schrödinger system for an inhomogeneous optical fiber

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a variable-coefficient fourth-order nonlinear Schrödinger system, which describes the simultaneous propagation of the optical pulses in an inhomogeneous optical fiber. The first-order breathers are constructed. Breathers are converted into several types of the nonlinear waves, i.e., multi-peak solitons, antidark solitons, periodic waves and W-shaped solitons, under the transition condition. With the decrease of |λr+a2|, the peak number of a multi-peak soliton increases, where λr represents the real part of the spectral parameter and a is a real constant. Peak number reaches the maximum since the multi-peak soliton is transformed into a set of the periodic waves with λr+a2=0. Widths and velocities of the multi-peak solitons are related to the group velocity dispersion and fourth-order dispersion coefficients. We find that the condition of baseband modulational instability coincides with the transition condition when we use the rogue-wave eigenvalues, i.e., the transition between the rogue waves and multi-peak solitons can occur in the baseband modulational instability.

Saturation behavior of a one-pump fiber optical parametric amplifier in the presence of the fourth-order dispersion coefficient and dispersion fluctuation

The influence of the fourth-order dispersion coefficient on the behavior of parametric gain and saturation power of a one-pump fiber optical parametric amplifier over a signal wavelength span in the presence of fiber random dispersion fluctuations was investigated. The output signal power for the parametric gain calculation was obtained by numerically solving the three-coupled amplitude equations. Based on the calculations of the parametric gain over a variation of the signal wavelength, it is found that the saturation power behavior is dependent on the behavior of parametric gain. The manipulations of signal wavelength and the fourth-order dispersion coefficient changed the phase-matching condition, thereby affecting the resulting parametric gain and saturation power.

Periodic soliton trains and informational code structures in an improved soliton model for biomembranes and nerves.

Many experiments have shown that the action potential propagating in a nerve fiber is an electromechanical density pulse. A mathematical model proposed by Heimburg and Jackson is an important step in explaining the propagation of electromechanical pulses in nerves. In this work, we consider the dynamics of modulated waves in an improved soliton model for nerve pulses. Application of the reductive perturbation method on the resulting generalized Boussinesq equation in the low-amplitude and weak damping limit yields a damped nonlinear Schrödinger equation that is shown to admit soliton trains. This solution contains an undershoot beneath the baseline ("hyperpolarization") and a "refractory period," i.e., a minimum distance between pulses, and therefore it represents typical nerve profiles. Likewise, the linear stability of wave trains is analyzed. It is shown that the amplitude of the fourth-order mixed dispersive term introduced here can be used to control the amount of information transmitted along the nerve fiber. The results from the linear stability analysis show that, in addition to the main periodic wave trains observed in most nerve experiments, five other localized background modes can copropagate along the nerve. These modes could eventually be responsible for various fundamental processes in the nerve, such as phase transitions and electrical and mechanical changes. Furthermore, analytical and numerical analyses show that increasing the fourth-order mixed dispersion coefficient improves the stability of the nerve signal.

Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrödinger equations in an inhomogeneous optical fiber

In this paper, investigation is made on the coupled variable-coefficient fourth-order nonlinear Schrödinger equations, which describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Via the generalized Darboux transformation, the first- and second-order rogue wave solutions are constructed. Based on such solutions, effects of the group velocity dispersion coefficient and the fourth-order dispersion coefficient on the rogue waves are graphically analyzed. The first-order rogue waves with the eye-shaped distribution, the interactions between the first-order rogue waves with solitons, and the second-order rogue waves with one highest peak and with the triangular structure are displayed. When the value of the group velocity dispersion coefficient or the fourth-order dispersion increases, range of the first-order rogue wave increases. Composite rogue waves are obtained, where the temporal separation between two adjacent rogue waves can be changed if we adjust the group velocity dispersion coefficient and fourth-order dispersion coefficient. Periodic rogue waves are presented. Periods of such rogue waves decrease with the periods of the group-velocity dispersion and fourth-order dispersion coefficient decreasing.

One-pump fiber optical parametric amplifier in presence of dispersion fluctuations: impact of fourth-order dispersion coefficient.

The impact of random dispersion fluctuations on the gain and saturation behavior of a one-pump fiber optical parametric amplifier (1-P FOPA) in the presence of fourth-order dispersion coefficient (β4) is investigated. Three coupled amplitude equations with fiber losses are solved numerically for the calculation of the pump, signal, and idler. The performances of 1-P FOPA are also analyzed with variation of dispersion fluctuation amplitude (σ) and correlation length (Lc). Based on the numerical results, it is found that the gain spectra and saturation curves exhibit some differences when β4 is considered. In comparison with the case where β4 is ignored, the peak gain remains the same, but the 3dB bandwidth increases when β4 exists. Another notable difference is that the saturation power is shifted to lower or higher values, depending on the σ and Lc parameters. In general, the peak gain reduces as σ increases, and the peak gain reduction is greater for the case of shorter Lc. The numerical analysis is probably useful, especially for the case where the signal wavelength is detuned far from the pump wavelength.

Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects

In this paper, we study a variable-coefficient nonlinear Schrodinger (vc-NLS) equation with fourth-order effects describing an inhomogeneous one-dimensional continuum anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. The first-order nonautonomous breather solution of the fourth-order vc-NLS equation is derived. The state transition between nonautonomous breather and nonautonomous multi-peak soliton can be realized when group velocity dispersion (GVD) coefficient is proportional to the fourth-order dispersion (FOD) coefficient. We also display how the higher-order effects influence the nonautonomous multi-peak solitons. Our results show that the velocity and localization of the nonautonomous multi-peak soliton are affected by the FOD coefficient, and the peak number is controlled by the GVD coefficient. Further, we also show the compression effect and motion with variable velocity of nonautonomous multi-peak soliton in two kinds of dispersion management systems. Finally, we reveal the relation between the state transition and the modulation instability (MI) analysis.

Numerical Simulation of Highly-Nonlinear Dispersion-Shifted Fiber Optical Parametric Gain Spectrum with Fiber Loss and Higher-Order Dispersion Coefficients

The previous study investigated the fiber parameters on the analytical one-pump fiber optical parametric amplifier (FOPA) gain spectrum of a loss-free highly-nonlinear dispersion-shifted fiber (HNL-DSF) and got the optimum results for each parameter. However, the FOPA gain of the combination of all the optimum values of the considered parameters was not reported. Hence, this paper intends to investigate the analytical FOPA gain of the combination of all the optimum values of the considered parameters from the previous study. Later, the analytical gain was compared with the numerical gain from the fourth-order Runge-Kutta (RK4) method and Matlab built-in function, ode45. Next, the effect of fiber loss and higher order dispersion coefficients such as fourth and sixth-order dispersion coefficients were studied. It is found that RK4 gives a smaller error and the gain reduces while the bandwidth remains same in presence of fiber loss. The fourth-order dispersion coefficient broadens the bandwidth a bit while maintaining the gain and there are two narrow-band gains generated to the left and right-side of the broad-band gain. The sixth-order dispersion coefficient just shifts the two narrow-band gains toward or away from the broadband gain depending on the positive or negative signs of the sixth-order dispersion coefficient.

Optimizing bandwidth of fiber optical parametric amplifier with different combinations of higher-order dispersion coefficient signs

This paper investigates the influence of different combinations of second-order (β2 ), fourth-order (β4 ) and sixth-order (β6 ) dispersion coefficient signs (negativity/positivity) to the fiber optical parametric amplifier gain performance. The numerical simulation has exploited the fourth-order Runge-Kutta method to solve the coupled amplitude equations which basically represent the parametric process of four-wave mixing. In the normal regime at which β2 is positive, the fiber optical parametric amplifier exhibits a poor gain spectrum and it is well performed once the pump is presumed to be in anomalous regime i.e. β2 is negative. The effect of β4 is significant when the signal wavelength is further from the pump wavelength and a wide amplification bandwidth is produced when the fiber have positive β4 in the anomalous regime. As for β6 , unfortunately in this simulation its influences are hardly can be seen probably because of its small value regardless its sign, but, from phase-mismatch equation it is shown that its impact is prominent when the signal is positioned way far from the pump. All in all, this shown that the signs combination of the higher-order dispersion coefficients are significant in order to optimize the FOPA amplification bandwidth.

Designing a two-octave spanning flat-top supercontinuum source by control of nonlinear dynamics through multi-order dispersion engineering in binary multi-clad microstructured fiber

A flat-top, coherent supercontinuum generation (SCG) spanning ∼1540  nm from the near-infrared to short-wave infrared (NIR–SWIR) band in a host lead–silicate-based binary multi-clad microstructure fiber (BMMF) is analyzed and reported. This ultra wide band (903–2443 nm) SCG with flatness <5  dB is theoretically achieved with a combination of a low input pump power source (peak power=25  kW, pulse width=75  fs) and a short fiber length (∼8  cm) by controlling the nonlinear dynamics of propagating ultrashort pulses accurately through multi-order dispersion engineering. Simulations reveal that by appropriately controlling the fourth-order dispersion coefficient, a great enhancement in the spectral flatness can be achieved when the device is operated close to the maximum dispersion wavelength in the all-normal dispersion regime.

Optimization of gain bandwidth and gain ripple of a hybrid Raman/parametric amplifier for access network applications

We describe the mathematical model and present simulation results for the optimization of a hybrid Raman/optical parametric amplifier (HROPA), exhibiting a bandwidth of 170 nm and low ripple that covers the top half of the wavelength plan (e.g., 1441 to 1611 nm) of next generation coarse wavelength division multiplexed passive optical network systems. We show that a critical parameter in the proper amplifier parameter optimization is the inclusion of the fourth-order dispersion coefficient (β(4)). Omission of β(4) can lead to over-estimation or underestimation of the gain bandwidth, and hence its inclusion in the analysis of the HROPA is necessary.

Estimation of the fourth-order dispersion coefficient \beta4

The fourth-order dispersion coefficient of fibers are estimated by the iterations around the third-order dispersion and the high-order nonlinear items in the nonlinear Schordinger equation solved by Green's function approach. Our theoretical evaluation demonstrates that the fourth-order dispersion coefficient slightly varies with distance. The fibers also record β4 values of about 0.002, 0.003, and 0.00032 ps4/km for SMF, NZDSF and DCF, respectively. In the zero-dispersion regime, the high-order nonlinear effect (higher than self-steepening) has a strong impact on the transmitted short pulse. This red-shifts accelerates the symmetrical split of the pulse, although this effect is degraded rapidly with the increase of β2. Thus, the contributions to β4 of SMF, NZDSF, and DCF can be neglected.

Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides

We demonstrate the generation of a supercontinuum in a 2 cm long silicon wire by pumping the wire with mid-infrared picosecond pulses in the anomalous dispersion regime. The supercontinuum extends from 1535 nm up to 2525 nm for a coupled peak power of 12.7 W. It is shown that the supercontinuum originates primarily from the amplification of background noise. A detailed analysis of the spectral components which are generated through phase-matched processes is applied to extract the group velocity dispersion and fourth-order dispersion coefficient of the silicon wire waveguide.

Design of fiber parametric wavelength converters based on photonic crystal fibers with two zero-dispersion wavelengths

We propose the design method of fiber parametric wavelength converters based on dispersion-flattened photonic crystal fibers (PCFs) with two zero-dispersion wavelengths (ZDWs). Analytical expressions of the optimum signal frequency and maximal pump tuning range are deduced. By our method, the tuning ranges can be considerably broadened with a relatively low pump power and short fiber. This is because the fourth-order dispersion coefficient of two ZDW PCFs can be 1-2 orders of magnitude larger than those of one ZDW fiber and are effectively utilized to compensate the linear phase mismatch due to the second-order dispersion, resulting in a low phase mismatch for a widely tunable pump. To exemplify the effectiveness of our method, a PCF based on lead-silicate glasses with two ZDWs spaced 127-nm apart is presented. Numerical simulations show that based on this PCF a transparent wavelength conversion with a 117-nm pump tuning range can be achieved with only a 3.7-m-long fiber and 0.532 W pump power.

Generation of pulse trains in nonlinear optical fibers through the generalized complex Ginzburg-Landau equation

We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement.

Fourth-order dispersion mediated solitonic radiations in HC-PCF cladding

We observe experimentally, for the first time to our knowledge, the simultaneous emission of two strong conjugate resonant dispersive waves by optical solitons. The effect is observed in a small waveguiding glass feature within the cladding of a Kagome hollow-core photonic crystal fiber. We demonstrate theoretically that the phenomenon is attributed to the unusually high fourth-order dispersion coefficient of the waveguiding feature.

730-nm optical parametric conversion from near- to short-wave infrared band

A record 730 nm parametric conversion in silica fiber from the near-infrared to the short-wave infrared band is reported and analyzed. A parametric gain in excess of 30 dB was measured for a signal at 1300 nm (with corresponding idler at 2030 nm). This conversion was performed in a travelling single-pass one-pump parametric architecture and high efficiency is achieved by a combination of high peak power and a nonlinear fiber with a reduced fourth-order dispersion coefficient.

Simple four-wave-mixing-based method for measuring the ratio between the third- and fourth-order dispersion in optical fibers

A simple four-wave-mixing (FWM)-based method for measuring the ratio between the third- and the fourth-order dispersion coefficients (β3/β4) in optical fibers is reported. The FWM interaction involves a low power laser and a low power amplified spontaneous emission noise source. The method is applied in several dispersion-shifted and non-zero-dispersion-shifted fibers with lengths varying from 0.03 to 25km, and we have obtained an error of less than 3% in measuring β3/β4. In highly nonlinear fibers the error has presented a strong dependency with longitudinal variations of the zero-dispersion wavelength (λ0); an error less than 20% could be obtained in most of the tested fibers. Our method also allows for a fast estimation of the distribution of fluctuations of λ0 along the fiber.