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

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Recent Trends in Computational Photonics

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 204))

Abstract

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.

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Notes

  1. 1.

    The term ab initio is here used to indicate simulations that predict the optical electromagnetic field for given structural data (geometry, material properties, and excitation, if applicable), without any further fit parameters.

  2. 2.

    There are as many supermodes as there are unknowns in (3.24), in principle. The relevant ones need to be filtered out, typically by specifying a range of resonance frequencies, or a maximum level of attenuation.

  3. 3.

    One regards the entire structure as one composite waveguide with three interior layers that supports, per polarization, two “supermodes” of different parity with slightly different propagation constants \(\beta _0\) and \(\beta _1\). These determine the coupling length as \(L_{\text{ c }} = \pi /|\beta _0-\beta _1|\). The supermodes are computed by a solver (cf. e.g. [42]) for the modes of dielectric multilayer slabs.

  4. 4.

    Certainly we do not intend to recommend the HCMT approach as the “method of choice” for this particular waveguide crossing.

  5. 5.

    In special cases radiated fields can also be incorporated [27].

  6. 6.

    Although the forward and backward propagating modes of the same channel share, up to the signs of certain field components identical profile shapes, the combination of electric and magnetic parts of the mode profile, as applied here, ensures orthogonality of the directional modes with respect to a suitable inner product [1, 43] (“power orthogonality”).

  7. 7.

    The overall phase of the solution has been adjusted to exhibit the maximum amplitude of the standing waves in the field plots Fig. 3.8a, b, therefore \(f(z_{\text{ l }})\) differs from 1.

  8. 8.

    The computational setting, with, e.g. the rectangular computational window, and quadrature rules applied subsequently along the x- and z-coordinates, does not respect the triangular symmetry, i.e. must be expected to numerically lift the degeneracy.

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Acknowledgements

Financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft DFG, projects HA 7314/1-1 and TRR 142) is gratefully acknowledged.

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Authors and Affiliations

  1. Theoretical Electrical Engineering, University of Paderborn, Warburger Straße 100, 33098, Paderborn, Germany

    M. Hammer

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  1. M. Hammer
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Editors and Affiliations

  1. Department of Electrical and Electronic Engineering, City, University of London, London, United Kingdom

    Arti Agrawal

  2. School of Electrical and Electronic Engineering, University of Nottingham, Nottingham, United Kingdom

    Trevor Benson

  3. School of Engineering, University of Glasgow, Glasgow, United Kingdom

    Richard M. De La Rue

  4. Department of Physics, King’s College, London, United Kingdom

    Gregory A. Wurtz

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Hammer, M. (2017). Guided Wave Interaction in Photonic Integrated Circuits — A Hybrid Analytical/Numerical Approach to Coupled Mode Theory. In: Agrawal, A., Benson, T., De La Rue, R., Wurtz, G. (eds) Recent Trends in Computational Photonics. Springer Series in Optical Sciences, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-55438-9_3

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