Spatial chirp and angular dispersion of a laser crystal for a four-mirror cavity Kerr-lens mode-locked laser

We describe oscillating loops in a laser cavity and optical paths in a laser crystal of different wavelength rays for a four-mirror cavity Kerr-lens mode-locked laser. The relation between different wavelength ray paths and laser resonator parameters is deduced. The analytical expressions of second- and third-order dispersion including angular dispersion of the crystal are presented. The variations of group-delay dispersion (GDD) and third-order dispersion (TOD) with cavity parameters are calculated exactly. The calculation shows that GDD and TOD increase rapidly when the spacing between two folding mirrors approaches the boundary of a cavity stability zone. The rapid dispersion increase influences the mode-locked pulse width and the mode-locked stability.     

超短孤子脉冲在色散缓变光纤中的时间抖动

通过微扰理论分析了超短孤子在色散缓变光纤(DDF)中传输的时间抖动。结果发现三阶色散和拉曼散射分别影响了孤子的位置和频率,是决定时间抖动大小的重要因素。在光纤的拉曼系数一定的情况下,选择不同的三阶色散参数,系统的时间抖动存在一个最小值。当DDF的三阶色散参数为接近零的一个负值的时候,系统时间抖动最小,这个负值与拉曼系数和传输距离有关,其绝对值随传输距离的增大而增大。在采用色散缓变光纤进行超短孤子传输的系统设计中应该重点考虑三阶色散的影响,当DDF的三阶色散参数较大的时候,必须对系统的三阶色散进行补偿,使时间抖动达到最小,能够极大地增加了孤子的传输距离。     

Stochastic dark solitons for a higher-order nonlinear Schrödinger equation in the optical fiber

In this paper, a stochastic higher-order nonlinear Schrödinger equation, which describes the wave propagation in the optical fiber with stochastic dispersion and nonlinearity, is investigated analytically. Via the symbolic computation and white noise functional approach, the stochastic dark one- and two-soliton solutions are obtained, and effects of the Gaussian white noise on the stochastic dark one and two solitons are discussed. For the stochastic dark one soliton, velocity and phase change randomly because of the Gaussian white noise, but the energy, shape and amplitude keep unchanged during the soliton propagation. For the stochastic dark two solitons, effect of the Gaussian white noise leads to the inversion of the velocity directions, while the velocities have the same varying trend so that the interaction appears that the stochastic dark two solitons keep parallel.     

A direct perturbation method investigation of the interplay between self-frequency shift and cross-phase modulation for solitons

The self-frequency shift of solitons, mainly observed in femtosecond pulses and arising from the Raman effect, can be counteracted by the cross-phase modulation (XPM) arising from collisions between pulses of different frequency. The same could be said for the time displacement that follows the frequency shift. The current work is an analytical approach to the interplay between the Raman self-frequency shift and XPM with third-order dispersion (TOD) effects taken into account. By using coupled NLS equations, the effect of XPM between two channels is considered for the cases of complete and incomplete collision. The analysis is based on the direct perturbation method and provides quantitative and qualitative insight of the spectral and temporal evolution of ultra short soliton pulses.     

Dynamics of self-similar sub-10-fs-pulses in an inhomogeneous highly nonlinear fibre amplifier

ABSTRACT

We study the propagation of ultrashort pulses of width around sub-10 femtosecond in an inhomogeneous highly nonlinear single-mode fibre within the framework of a generalized higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms and spatially inhomogeneous coefficients. Additional effects to the cubic model include the distributed third-order dispersion, self-steepening, self-frequency shift due to stimulated Raman scattering, quintic nonKerr nonlinearity, derivative non-Kerr nonlinear terms, and gain or loss. The exact self-similar brightand dark-solitary-wave solutions of the governing equation are derived via a transformation connected with the constant-coefficient higher-order nonlinear Schrödinger equation with non-Kerr nonlinearity. The constraint relations among the optical fibre parameters for the existence of these self-similar structures are also discussed. Based on these exact solutions, we investigate the dynamical behaviours of self-similar localized pulses in a periodic distributed fibre system for different parameters.     

Transmission characteristic of interaction between Airy pulses and solitons in silicon-on-insulator waveguides

In this paper, we numerically simulate the transmission characteristics of Airy pulses and solitons interacting in silicon-on-insulator waveguides by utilizing Split-step Fourier method implemented in MATLAB. During transmission, the energy of Airy pulse is focused, and a new Airy pulse is subsequently separated from frontal edge. The new Airy pulse and the initial pulse exhibit reversal characteristics in time domain. In process of interaction with soliton, partial energy of the main lobe is separated from Airy pulse and transmitted along soliton transmission trajectory. The soliton produce a series of end edge oscillations. When the soliton is affected by Airy pulse, the propagation distance decreases, in constrast to the trailing edge oscillation frequency. In addition, third-order dispersion, Kerr effect and two-photon absorption on Airy pulse and soliton are also mentioned. At the same time, the effect on transmission of Airy pulse and solition will be showed in different truncation coefficients, powers and pulse widths.     

Measurement of group delay dispersion of high numerical aperture objective lenses using two-photon excited fluorescence

We determined the group-delay dispersion (GDD) of five microscope objectives by measuring the second-order autocorrelation at the focal points of the objectives with two-photon excited fluorescence as the power square sensor. We found that typical microscope lens systems introduce significant GDD (2000-6500 fs(2)). The third-order dispersion determined for these objectives limits the minimum obtainable pulse width at the focal point of an objective to 20-30 fs if not compensated. No significant chromatic aberration or higher-order dispersion effects were found for any of the optical components measured within the wavelength range of 700-780 nm and for pulse widths greater than 50-60 fs.     

Effect of intensity dependent higher-order dispersion on femtosecond pulse propagation in quantum well waveguides

We have studied theoretically the dispersion of ultrafast coherent pulses in GaAs/AlGaAs quantum well waveguide structures as a function of optical intensity. Semiconductor Bloch equations are used to obtain the polarization induced in the medium due to an incident Gaussian electromagnetic beam. The partial differential equation describing the pulse propagation in the presence of group velocity dispersion is used to analyze the role of higher-order dispersion on femtosecond pulse propagation in the waveguide. Due consideration has been given to the intensity dependent optical susceptibility of the medium. The results of the numerical analysis manifest significant influence of higher-order dispersion on femtosecond pulse propagation over short waveguide distance.     

Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures

Abstract

The excitation and behaviour of monochromatic and pulsed optical Bloch waves in pure distributed feed-back structures are discussed and analysed. The Bloch wave approach, based on a detailed knowledge of the natural optical modes of the periodic structure, is complementary to the more commonly used coupled-wave approach. The inter-relationship between dispersion, field micro-structure and group velocity is discussed, and the effects of group-velocity and higher-order dispersion on pulse propagation treated. Questions about the usefulness of DFB structures for dispersion correction and soliton formation are addressed.     

Optical solitons in some SIT type equations

Abstract

In this paper, we consider the SIT type equations with group velocity dispersion, Kerr nonlinearity, higher order dispersion, self-steepening and pumping effects and analyse the dynamics of ultra short pulse propagation in nonlinear resonant fibre. The nature of the pulse propagation is governed by the nonlinear Schrödinger-Maxwell Bloch (NLS-MB) type equations. Considering the isospectral and non-isospectral eigen value parameter, we generate the soliton solutions through the Bäcklund transformation method.     

Periodic and solitary wave solutions for ultrashort pulses in negative-index materials

Abstract

We present a detailed analysis for the existence of dark and bright solitary waves as also fractional-transform solutions in a nonlinear Schrödinger equation model for competing cubic–quintic and higher-order nonlinearities with dispersive permittivity and permeability. Parameter domains are delineated in which these ultrashort optical pulses exist in negative-index materials (NIMs). For example, dark solitons exist for the case of normal second-order dispersion, anomalous third-order dispersion, self-focusing Kerr nonlinearity, and non-Kerr nonlinearities, while the bright solitons exist for the case of anomalous second-order dispersion, normal third-order dispersion, self-focusing Kerr nonlinearity, and non-Kerr nonlinearities. This is contrary to the situation in ordinary materials.     

Mixed-type solitons for the coupled higher-order nonlinear Schrödinger equations in multi-mode and birefringent fibers

Abstract

In this paper, coupled higher-order nonlinear Schrödinger equations are studied for the ultrashort pulse propagation in such optical media as the multi-mode fibers and birefringent fibers. Through the Hirota method and symbolic computation, analytic mixed-type one- and two-soliton solutions are derived, and three sets of conditions for the non-singular solutions are given as well. Via the one-soliton solutions obtained, critical condition for the black and gray solitons is obtained analytically. Asymptotic analysis is carried out on the two-soliton solutions to derive the condition for the inelastic interaction. Evolution of the bound vector solitons, elastic and inelastic interactions between the two vector solitons are also investigated graphically. Moreover, after the inelastic interaction, the two bright solitons are observed to disappear, while the two dark ones form the new bound soliton.     

Influence of nonlinear coupling of pulses on spatio-temporal compression

Abstract

Properties of a novel configuration of an optical (spatio-temporal) pulse compressor, that is based on a Kerr-type planar waveguide into which two pulses are simultaneously launched, are studied. It is assumed that the pulse which is the subject of the compression propagates in the anomalous dispersion regime, while the auxiliary pulse is in normal dispersion. The best parameters of the proposed compressor are obtained when duration of the auxiliary pulse is so large that this dispersion can be neglected, while energy of the second pulse is above the threshold of first-order soliton generation. It is observed that in such a configuration the compression occurs simultaneously with the generation of a soliton-like solution. It is argued that the proposed configuration with two simultaneously propagating pulses has advantages over the configuration with a single pulse, namely the maximal compression factor and the optimal length of the compressor is, respectively, more than 3 times larger and, at least, 10 times greater than the corresponding values of the compressor with a single pulse. It is also demonstrated that such a compressor can be considered as a universal device, since its operation depends only slightly on the initial parameters of the pulse subject to the compression.     

Analytical expressions for group-delay dispersion and third-order dispersion of a reflection grism-pair compressor

We provide a detailed analytical expression of group-delay dispersion (GDD) and third-order dispersion (TOD) for a reflection grism-pair compressor without the first-order approximation of grating diffraction. The analytical expressions can be used to design a grism-pair compressor for compensating the dispersive material without ray tracing. Furthermore, the dispersion performance of the grism pair compressor, depending on compressor parameters, is comprehensively analyzed. Results are shown that we can adjust several parameters to obtain a certain GDD and TOD, such as the incidence angle of the beam, refractive index of the prism, grating constant, and the separation of the grism pair.     

The multiple-scale averaging and dynamics of dispersion-managed optical solitons

Multiple-scale averaging is applied to the nonlinear Schrödinger equation with rapidly varying coefficients, and use the results to analyze pulse propagation in an optical fiber when a periodic dispersion map is employed. The effects of fiber loss and repeated amplification are taken into account by use of a coordinate transformation to relate the pulse dynamics in lossy fibers to that in equivalent lossless fibers. Second-order averaging leads to a general evolution equation that is applicable to both return-to-zero (soliton) and non-return-to-zero encoding schemes. The resulting equation is then applied to the specific case of solitons, and an asymptotic theory for the pulse dynamics is developed. Based upon the theory, a simple and effective design of two-step dispersion maps that are advantageous for wavelength-division-multiplexed soliton transmission is proposed. Theuse of these specifically designed dispersion maps allows simultaneous minimization of dispersive radiation in several different channels.     

UNCONDITIONALLY STABLE HIGHER-ORDER ACCURATE HERMITIAN TIME FINITE ELEMENTS

In this paper, single step time finite elements using the cubic Hermitian shape functions to interpolate the solution over a time interval are considered. The second-order differential equations are manipulated directly. Both the effects of modal damping and external excitation are considered. The accuracy of the solutions at the end of the time interval and the interpolated solutions within the time interval is investigated. The weighted residual approach is adopted to derive the time-integration algorithms. Instead of specifying the weighting functions, the weighting parameters are used to control the characteristics of the time finite elements. The weighting parameters are chosen to eliminate the higher-order truncation error terms or to enforce the asymptotic annihilation condition. A one-parameter family of third-order accurate asymptotically annihilating algorithms and another one-parameter family of fourth-order accurate non-dissipative algorithms are presented. The ranges of the weighting parameters for unconditionally stable algorithms are given. It is found that one of the members in each family corresponds to the Padé approximants of the exponential function in solving the first-order differential equations. Some of the existing unconditionally stable higher-order accurate algorithms are re-derived by the present unified approach.     

Analysis of Modulational Instability in Monomode Optical Fibres with High-order Nonlinearity

Abstract

The modulational instability (MI) in monomode optical fibres with fifth-order nonlinearity, fibre loss, higher-order dispersion, and the temporal variation of third-order nonlinearity is studied theoretically. The conditions for the existence of the MI and the maximal modulational growth are given and discussed in detail. The results obtained show that the key factor dominating the producing condition of the MI is the power P of the continuous wave initially launched into the optical fibres. If P falls into 3/10<P/P ₀ <1/2 where P ₀ is defined as characteristic power, the MI can be produced in the range of not only anomalous group velocity dispersion but also the normal in which the final evolution state of the modulated wave is dark soliton.     

Multiple-Pole soliton interactions in optical fibres with higher-order effects

Abstract

We consider the Hirota equation which governs the propagation of nonlinear waves in optical fibres with higher-order effects like third-order dispersion, self-steepening and delayed nonlinear response. By using the Hirota bilinear method a special two-soliton solution is derived. We also undertake a special limit corresponding to the multiple-pole case. Physically, this special pair of solitons separates very slowly.     

Tucker compression and local optima

This paper deals with a particular Tucker compression problem. Formally, given a higher-order tensor, we are interested in its best low multilinear rank approximation. We study the local minima of the cost function that measures the quality of putative low multilinear rank approximations. This discussion is specific to higher-order tensors since, in the matrix case, the low rank approximation problem has only one local, hence global, minimum. Furthermore, in the higher-order tensor case, convergence to the solution with lowest value of the cost function cannot be guaranteed with existing algorithms even if they are initialized with the truncated higher-order singular value decomposition.We observed that the values of the cost function at different local minima can be very close to each other. Thus, for memory savings based on Tucker compression, the choice between these local minima does not really matter. On the other hand, we found that the subspaces on which the original tensor is projected can be very different. If the subspaces are of importance, different local minima may yield very different results. We provide numerical examples and indicate a relation between the number of local minima that are found and the distribution of the higher-order singular values of the tensor.     

Impact of two-photon absorption on soliton trapping in silicon-on-insulator waveguides

In this paper, we present a numerical simulation study of the effects of two-photon absorption on soliton trapping occurring between two components decomposed from a soliton pulse. The two pulse components polarize along the polarization directions of the quasi-TE mode and quasi-TM mode in a silicon-on-insulator waveguide. The effects of free carriers generated by two-photon absorption are also investigated, leading to free-carrier absorption and free-carrier dispersion. The simulation image of the propagation of solitons is provided by solving the coupled nonlinear Schrödinger equation satisfied by the quasi-TE mode component and the quasi-TM mode component utilizing the split-step Fourier method with MATLAB. The simulation shows that two-photon absorption and free carriers will hinder the formation of soliton trapping. As the incident pulse power increases, the soliton trapping will occur, inspite of the presence of two-photon absorption and free carriers. The simulation also indicates that if the value of the free-carrier’s lifetime is comparable to the value of the pulse width, the formation of soliton trapping is susceptible to the lifetime of free carriers. With the incident pulse width increasing, soliton trapping forms between the mean peak of two polarization components, whereas it originally formed between the mean peak of TE mode and secondary peak of TM mode at small pulse width.