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Numerical Methods

We use and extend the Time Domain Discontinuous Galerkin method to perform the challenging multi-scale simulations  in many of our projects.

Next topic: Plasmonics

Related publications by the TET group

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Application of the Discontinuous Galerkin Time Domain Method in Nonlinear Nanoplasmonics

Y. Grynko, J. Förstner, in: 2018 IEEE 17th International Conference on Mathematical Methods in Electromagnetic Theory (MMET), IEEE, 2018

Hybrid coupled-mode modeling in 3D: perturbed and coupled channels, and waveguide crossings

M. Hammer, S. Alhaddad, J. Förstner, Journal of the Optical Society of America B (2017), 34(3), pp. 613-624

The 3D implementation of a hybrid analytical/numerical variant of the coupled-mode theory is discussed. Eigenmodes of the constituting dielectric channels are computed numerically. The frequency-domain coupled-mode models then combine these into fully vectorial approximations for the optical electromagnetic fields of the composite structure. Following a discretization of amplitude functions by 1D finite elements, pro- cedures from the realm of finite-element numerics are applied to establish systems of linear equations for the then- discrete modal amplitudes. Examples substantiate the functioning of the technique and allow for some numerical assessment. The full 3D simulations are highly efficient in memory consumption, moderately demanding in com- putational time, and, in regimes of low radiative losses, sufficiently accurate for practical design. Our results include the perturbation of guided modes by changes of the refractive indices, the interaction of waves in parallel, horizontally or vertically coupled straight waveguides, and a series of crossings of potentially overlapping channels with fairly arbitrary relative positions and orientations.

Guided Wave Interaction in Photonic Integrated Circuits — A Hybrid Analytical/Numerical Approach to Coupled Mode Theory

M. Hammer, in: Recent Trends in Computational Photonics, 204th ed., Springer, 2017, pp. 77-105

Frequently, optical integrated circuits combine elements (waveguide channels, cavities), the simulation of which is well established through mature numerical eigenproblem solvers. It remains to predict the interaction of these modes. We address this task by a general, “Hybrid” variant (HCMT) of Coupled Mode Theory. Using methods from finite-element numerics, the properties of a circuit are approximated by superpositions of eigen-solutions for its constituents, leading to quantitative, computationally cheap, and easily interpretable models.

    Simulation of Second Harmonic Generation from Photonic Nanostructures Using the Discontinuous Galerkin Time Domain Method

    Y. Grynko, J. Förstner, in: Recent Trends in Computational Photonics, Springer International Publishing, 2017, pp. 261-284

    We apply the Discontinuous Galerkin Time Domain (DGTD) method for numerical simulations of the second harmonic generation from various metallic nanostructures. A Maxwell–Vlasov hydrodynamic model is used to describe the nonlinear effects in the motion of the excited free electrons in a metal. The results are compared with the corresponding experimental measurements for split-ring resonators and plasmonic gap antennas.

      Wave interaction in photonic integrated circuits: Hybrid analytical / numerical coupled mode modeling

      M. Hammer, in: Integrated Optics: Devices, Materials, and Technologies XX, SPIE, 2016, pp. 975018-975018-8

      Typical optical integrated circuits combine elements, like straight and curved waveguides, or cavities, the simulation and design of which is well established through numerical eigenproblem-solvers. It remains to predict the interaction of these modes. We address this task by a ”Hybrid” variant (HCMT) of Coupled Mode Theory. Using methods from finite-element numerics, the optical properties of a circuit are approximated by superpositions of eigen-solutions for its constituents, leading to quantitative, low-dimensional, and interpretable models in the frequency domain. Spectral scans are complemented by the direct computation of supermode properties (spectral positions and linewidths, coupling-induced phase shifts). This contribution outlines the theoretical background, and discusses briefly limitations and implementational details, with the help of an example of a 2-D coupled-resonator-optical-waveguide configuration.

        Oblique incidence of semi-guided waves on rectangular slab waveguide discontinuities: A vectorial QUEP solver

        M. Hammer, Optics Communications (2014), 338, pp. 447-456

        The incidenceofthin-film-guided, in-planeunguidedwavesatobliqueanglesonstraightdiscontinuities of dielectricslabwaveguides,anearlyproblemofintegratedoptics,isbeingre-considered.The3-D frequencydomainMaxwellequationsreducetoaparametrizedinhomogeneousvectorialproblemona 2-D computationaldomain,withtransparent-influx boundaryconditions.Weproposearigorousvec- torial solverbasedonsimultaneousexpansionsintopolarizedlocalslabeigenmodesalongthetwo orthogonal crosssectioncoordinates(quadridirectionaleigenmodepropagationQUEP).Thequasi-ana- lytical schemeisapplicabletoconfigurations with — in principle — arbitrary crosssectiongeometries. Examples forahigh-contrastfacetofanasymmetricslabwaveguide,forthelateralexcitationofa channel waveguide,andforastepdiscontinuitybetweenslabwaveguidesofdifferentthicknessesare discussed.

          Application of the discontinous Galerkin time domain method to the optics of metallic nanostructures

          Y. Grynko, J. Förstner, T. Meier, AAPP | Atti della Accademia Peloritana dei Pericolanti (2011), 89(1)

          A simulation environment for metallic nanostructures based on the Discontinuous Galerkin Time Domain method is presented. The model is used to compute the linear and nonlinear optical response of split ring resonators and to study physical mechanisms that contribute to second harmonic generation.

          Enhanced FDTD edge correction for nonlinear effects calculation

          C. Classen, J. Förstner, T. Meier, R. Schuhmann, in: 2010 IEEE Antennas and Propagation Society International Symposium, IEEE, 2010

          The electromagnetic field in the vicinity of sharp edges needs a special treatment in numeric calculation whenever accurate, fast converging results are necessary. One of the fundamental works concerning field singularities has been proposed in 1972 [1] and states that the electromagnetic energy density must be integrable over any finite domain, even if this domain contains singularities. It is shown, that the magnetic field H(, ϕ) and electric field E(, ϕ) are proportional to ∝ (t−1) for  → 0. The variable  is the distance to the edge and t has to fulfill the integrability condition and thus is restricted to 0 < t < 1. This result is often used to reduce the error corresponding to the singularity without increasing the numerical effort [2 - 5]. For this purpose, a correction factor K is estimated by inserting the proportionality into the wave equation. It is shown, that this method improves the accuracy of the result significantly, however the order of convergence is often not studied. In [4] a method to modify the material parameters in order to use analytic results to improve the numeric calculation is presented. In this contribution we will - inspired by the scheme given in [4] - develop a new method to estimate a correction factor for perfect conducting materials (PEC) and demonstrate the improvement of the results compared to the standard edge correction. Therefore analytic results (comparable to [1]) are consequently merged with the scheme in [4]. The main goal of this work is the calculation of the second harmonic generation (SHG) in the wave response of so-called metamaterials [6]. Frequently these structures contain sharp metallic edges with field singularities at the interfaces which have a strong impact on the SHG signals. Thus, an accurate simulation of singularities is highly important. However, the following approach can also be applied to many other setups, and one of them is shown in the example below.

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          Funding, co-operations


          Dr. Yevgen Grynko

          Theoretical Electrical Engineering

          Yevgen Grynko
          +49 5251 60-3815
          +49 5251 60-3524

          Dr. Manfred Hammer

          Theoretical Electrical Engineering

          Manfred Hammer
          +49 5251 60-3560
          +49 5251 60-3524

          Office hours:

          on appointment

          Head of the group

          Prof. Dr. Jens Förstner

          Theoretical Electrical Engineering

          Jens Förstner
          +49 5251 60-3013
          +49 5251 60-3524

          Office hours:

          on request (during lecture break)

          The University for the Information Society