High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

Baruch, Guy; Fibich, Gadi; Tsynkov, Semyon

doi:10.1016/j.cam.2006.01.048

“…The new computational methodology for the NLH that we present builds up on our previous work [19,2,20,21] and extends it substantially. We introduce a new semi-compact discretization and a new Newton's solver, and the ensuing capabilities include an explicit demonstration of the removal of singularity that "plagues" the NLS, and the computation of narrow nonparaxial solitons.…”

Section: Methodsmentioning

confidence: 94%

“…In [19], we have demonstrated that the iterations' convergence in [2,20,21] breaks down far below the power threshold for non-uniqueness of the one-dimensional problem. This suggested that the convergence difficulties in [2,20,21] were not related to the loss of uniqueness by the solution [18,8], but rather to the deficiencies of the iteration scheme itself. The latter may be (partially) accounted for by the known convergence limitations of the Born approximations, because they can be interpreted as a Neumann series [27] for the corresponding integral operator [26,Section 13.1.4].…”

Section: Methodsmentioning

confidence: 94%

“…In doing so, the discontinuities that are confined to transverse planes will always be aligned with the grid. [2,20,21]; B) Compact 3 × 3 fourth-order stencil for linear operators, as, e.g., in [23,24]; C) Semi-compact stencil used in this work.…”

Section: Methodsmentioning

confidence: 99%

“…The NLH (20) subject to local transverse boundary conditions (21) and nonlocal two-way longitudinal boundary conditions (25) is discretized on the grid (26) or (34). In the Cartesian case, we obtain semi-compact schemes (30) and (31) in the interior and exterior of the Kerr material, respectively, and discretization (33) for the continuity conditions at the interface.…”

Section: Summary Of the Numerical Methodsmentioning

confidence: 99%

“…In this section, we briefly describe a discrete approximation of the transverse boundary conditions (21). In doing so, we follow the approach of [21], where additional details can be found. Let us first consider the radiation boundary conditions (21a) and (21b) at the "upper" boundary m = M − 1/2.…”

Section: Local Transverse Boundary Conditionsmentioning

confidence: 99%

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A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

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a b s t r a c tWe present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method.The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the ''augmented" paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.

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High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

Baruch

¹

,

Fibich

²

,

Tsynkov

³

2007

Journal of Computational Physics

Self Cite

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a b s t r a c tWe present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method.The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the ''augmented" paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.

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Skew-Symmetric Difference Analogs of the Fourth-Order Approximation to the First Derivative

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High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

Cited by 9 publications

References 16 publications

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

Skew-Symmetric Difference Analogs of the Fourth-Order Approximation to the First Derivative

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