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

G. Baruch, G. Fibich, S. Tsynkov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


We 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.

Original languageEnglish
Pages (from-to)3789-3815
Number of pages27
JournalJournal of Computational Physics
Issue number10
StatePublished - 1 Jun 2009


FundersFunder number
Israel Science Fund123/08
National Science FoundationDMS-0810963, DMS-0509695
National Science Foundation
Directorate for Mathematical and Physical Sciences0810963
Directorate for Mathematical and Physical Sciences
U.S. Air ForceFA9550-07-1-0170
U.S. Air Force


    • (Narrow) solitons
    • Arrest of collapse
    • Artificial boundary conditions (ABCs)
    • Backscattering
    • Compact scheme
    • Complex valued solutions
    • Discontinuous coefficients
    • Finite-difference approximation
    • Forward scattering
    • Frechét differentiability
    • High-order method
    • Inhomogeneous medium
    • Kerr nonlinearity
    • Layered medium
    • Material discontinuities
    • Newton's method
    • Nonlinear Schrödinger equation
    • Nonlinear optics
    • Nonparaxiality
    • Paraxial approximation
    • Traveling waves
    • Two-way ABCs


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