We find exact solutions to the nonlinear Schrödinger equation ͑NLSE͒ in the presence of self-steepening and a self-frequency shift. These include periodic solutions and localized solutions of dark-bright type which can be chiral, the chirality being controlled by the sign of the self-steepening term. A form of self-phasemodulation that can be tuned by higher-order nonlinearities as well as by the initial conditions, distinct from the nonlinear Schrödinger equation, characterizes these solutions. In certain nontrivial parameter domains, solutions are found to satisfy the linear Schrödinger equation, indicating the possibility of linear superposition in this nonlinear system. Dark and bright solitons exist in both the anomalous and normal dispersion regimes, and a duality between the dark-bright type of solution and kinematic higher-order chirping is also seen. Localized kink solutions similar to NLSE solitons, but with very different self-phase-modulation, are identified. The nonlinear Schrödinger equationgoverns the dynamics of picosecond pulse propagation in optical fibers ͓1͔, where a 1 is the group velocity dispersion ͑GVD͒ parameter and a 2 specifies the strength of Kerr nonlinearity. As predicted by Hasegawa and Tappert ͓2͔ and experimentally observed by Mollenauer et al. ͓3͔, this system supports stable soliton solutions owing their existence to complete integrability ͓4͔. With the advent of high-intensity laser beams, it has become possible to generate optical pulses with width of the order of 10 fs. Higher-order effects like third-order dispersion, self-steepening of the pulse due to the dependence of the slowly varying part of the nonlinear polarization on time, and the self-frequency shift arising from the delayed Raman response become important in the study of the propagation of these pulses. In order to account for them, Kodama ͓5͔ and Kodama and Hasegawa ͓6͔ proposed a higher-order nonlinear Schrödinger equation as a generalization of the NLSE:where a third-order dispersion with coefficient a 3 , a selfsteepening term with coefficient a 4 , and a self-frequency shift effect with coefficient a 5 have been added. This model, unlike the NLSE, is not integrable in general. A few integrable cases have been identified: ͑i͒ the Sasa-Satsuma case ͓a 3 : a 4 : ͑a 4 + a 5 ͒ =1:6:3͔ ͓7͔, ͑ii͒ the Hirota case ͓a 3 : a 4 : ͑a 4 + a 5 ͒ =1:6:0͔ ͓8͔, and ͑iii͒ derivative NLSEs of types I and II ͓9͔. Many restrictive special solutions of bright and dark type have been obtained ͓10-12͔.The effect of third-order dispersion is significant for femtosecond pulses when the GVD is close to zero. It is negligible for optical pulses whose width is of the order of 100 fs or more, having power of the order of 1 W and GVD far away from zero. However, in this case self-steepening as well as self-frequency shift terms are still dominant and should be retained. The effects of these higher-order terms on pulse propagation have been extensively studied numerically ͓1,13͔, and some special solutions to this system are also known ͓14͔...

Dust particle transport in low-temperature plasmas has received considerable attention due to the desire to minimize contamination of wafers during plasma processing of microelectronic devices and for their use to study nonideal plasmas. Dust particles in radio frequency discharges form Coulomb crystals and display collective behavior under select conditions. In this article, we discuss results from a self-consistent three-dimensional model for dust particle transport in plasma processing reactors. The consequences of varying the bias voltage of the capacitively coupled discharge, plasma density, particle diameter, and the number of particles on the propensity for Coulomb crystal formation in argon plasmas will be discussed. We found that a single one-layer lattice spontaneously breaks up into separate lattices as the substrate bias is increased due to a redistribution of plasma forces. At high substrate biases, a void occurs in the plasma crystal which tends to close upon addition of electronegative gases such as O 2 and Cl 2 to argon. The interparticle spacing in the lattice depends on the number of particles in the lattice due to gravitational compressive forces; and on the plasma density due to the change in shielding length.

Flux and energy analysis of species in hollow cathode magnetron ionized physical vapor deposition of copper Rev. Sci. Instrum. 81, 123502 (2010); 10.1063/1.3504371 In situ plasma diagnostics study of a commercial high-power hollow cathode magnetron deposition tool Atomistic feature scale modeling of the titanium ionized physical vapor deposition process Ionized metal physical vapor deposition is being increasingly used to deposit diffusion barriers and Cu seed layers into high aspect ratio trenches for microelectronics fabrication. Hollow cathode magnetrons ͑HCMs͒ represent a technology capable of depositing metal over large areas at pressures of a few millitorrs. The fundamental mechanisms of these devices are not well understood and so their optimization is difficult. In this article, results from a two-dimensional computational investigation of HCMs are discussed to illuminate scaling issues. The hybrid model incorporates algorithms whereby transport coefficients for use in fluid equations are derived using a kinetic simulation. The goal is to enable the fluid algorithms in the model to be able to more accurately represent low pressure operation. The consequences of power, pressure, and magnitude and orientation of applied magnetic fields were investigated. The authors found that the magnetic field configuration significantly affects the magnitude and distribution of fluxes incident on the substrate. A study of the Cu seed layer deposition process, carried out using a feature scale model, correlates changes in plasma properties with conformal deposition into trenches.

We analyse the structure of the exact, dark and bright soliton solutions of the driven nonlinear Schrödinger equation. A wide class of solutions, phase locked with the source, is identified for distinct parameter ranges. These contain periodic as well as localized solutions, which can be singular implying extreme increase in intensity. Conditions for obtaining non-propagating solutions are also found. A special case, where the scale of the soliton emerges as a free parameter, is obtained. We also study the highly restrictive structure of the localized solutions, when the phase and amplitude get coupled. Numerical solutions are obtained for this case, which reveals presence of periodic solutions. Stability analysis is also carried out through the Crank-Nicolson method.

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