Optical vortices in dispersive nonlinear Kerr-type media

The applied method of slowly varying amplitudes gives us the possibility to reduce the nonlinear vector integrodifferential wave equation of the electrical and magnetic vector fields to the amplitude vector nonlinear differential equations. Using this approximation, different orders of dispersion of the linear and nonlinear susceptibility can be estimated. Critical values of parameters to observe different linear and nonlinear effects are determined. The obtained amplitude equations are a vector version of3D+1nonlinear Schrödinger equation (VNSE) describing the evolution of slowly varying amplitudes of electrical and magnetic fields in dispersive nonlinear Kerr-type media. We show that VNSE admits exact vortex solutions with classical orbital momentumℓ=1and finite energy. Dispersion region and medium parameters necessary for experimental observation of these vortices are determined.

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Continuous Analog of Accelerated OS-EM Algorithm for Computed Tomography

Mathematical Problems in Engineering ◽

10.1155/2017/1564123 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

Kiyoko Tateishi ◽

Yusaku Yamaguchi ◽

Omar M. Abou Al-Ola ◽

Tetsuya Yoshinaga

Keyword(s):

Computed Tomography ◽

Image Reconstruction ◽

Em Algorithm ◽

Expectation Maximization ◽

Vector Fields ◽

Nonlinear Differential Equations ◽

Continuous System ◽

Continuous Analog ◽

Nonlinear Vector ◽

Constrained Solutions

The maximum-likelihood expectation-maximization (ML-EM) algorithm is used for an iterative image reconstruction (IIR) method and performs well with respect to the inverse problem as cross-entropy minimization in computed tomography. For accelerating the convergence rate of the ML-EM, the ordered-subsets expectation-maximization (OS-EM) with a power factor is effective. In this paper, we propose a continuous analog to the power-based accelerated OS-EM algorithm. The continuous-time image reconstruction (CIR) system is described by nonlinear differential equations with piecewise smooth vector fields by a cyclic switching process. A numerical discretization of the differential equation by using the geometric multiplicative first-order expansion of the nonlinear vector field leads to an exact equivalent iterative formula of the power-based OS-EM. The convergence of nonnegatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem for consistent inverse problems. We illustrate through numerical experiments that the convergence characteristics of the continuous system have the highest quality compared with that of discretization methods. We clarify how important the discretization method approximates the solution of the CIR to design a better IIR method.

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Linear and Nonlinear Normal Interface Stiffness in Dry Rough Surface Contact Measured Using Longitudinal Ultrasonic Waves

Applied Sciences ◽

10.3390/app11125720 ◽

2021 ◽

Vol 11 (12) ◽

pp. 5720

Author(s):

Saeid Taghizadeh ◽

Robert Sean Dwyer-Joyce

Keyword(s):

Rough Surface ◽

Contact Pressure ◽

Ultrasonic Wave ◽

Bulk Material ◽

Nonlinear Differential Equations ◽

Small Deformation ◽

Ultrasonic Waves ◽

Solid Contact ◽

Interfacial Stiffness ◽

Linear And Nonlinear

When two rough surfaces are loaded together contact occurs at asperity peaks. An interface of solid contact regions and air gaps is formed that is less stiff than the bulk material. The stiffness of a structure thus depends on the interface conditions; this is particularly critical when high stiffness is required, for example in precision systems such as machine tool spindles. The rough surface interface can be modelled as a distributed spring. For small deformation, the spring can be assumed to be linear; whilst for large deformations the spring gets stiffer as the amount of solid contact increases. One method to measure the spring stiffness, both the linear and nonlinear aspect, is by the reflection of ultrasound. An ultrasonic wave causes a perturbation of the contact and the reflection depends on the stiffness of the interface. In most conventional applications, the ultrasonic wave is low power, deformation is small and entirely elastic, and the linear stiffness is measured. However, if a high-powered ultrasonic wave is used, this changes the geometry of the contact and induces nonlinear response. In previous studies through transmission methods were used to measure the nonlinear interfacial stiffness. This approach is inconvenient for the study of machine elements where only one side of the interface is accessible. In this study a reflection method is undertaken, and the results are compared to existing experimental work with through transmission. The variation of both linear and nonlinear interfacial stiffnesses was measured as the nominal contact pressure was increased. In both cases interfacial stiffness was expressed as nonlinear differential equations and solved to deduce the contact pressure-relative surface approach relationships. The relationships derived from linear and nonlinear measurements were similar, indicating the validity of the presented methods.

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