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Fachgebiet Theoretische Elektrotechnik (TET)
Prof. Dr. Jens Förstner

Dielectric Waveguides

Based on the coupled mode theory and on other analytical and semi-analytical methods we model the propagation of electromagnetic waves in dielectric integrated optical circuits.

Next topic: Optical Antennas

Related publications by the TET group

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Light diffraction in slab waveguide lenses simulated with the stepwise angular spectrum method

L. Ebers, M. Hammer, J. Förstner, Optics Express (2020), 28(24), pp. 36361

A stepwise angular spectrum method (SASM) for curved interfaces is presented to calculate the wave propagation in planar lens-like integrated optical structures based on photonic slab waveguides. The method is derived and illustrated for an effective 2D setup first and then for 3D slab waveguide lenses. We employ slab waveguides of different thicknesses connected by curved surfaces to realize a lens-like structure. To simulate the wave propagation in 3D including reflection and scattering losses, the stepwise angular spectrum method is combined with full vectorial finite element computations for subproblems with lower complexity. Our SASM results show excellent agreement with rigorous numerical simulations of the full structures with a substantially lower computational effort and can be utilized for the simulation-based design and optimization of complex and large scale setups.

    Hybrid coupled mode modelling of the evanescent excitation of a dielectric tube by semi-guided waves at oblique angles

    M. Hammer, L. Ebers, J. Förstner, Optical and Quantum Electronics (2020), 52

    A dielectric step-index optical fiber with tube-like profile is considered, being positioned with a small gap on top of a dielectric slab waveguide. We propose a 2.5-D hybrid analytical/numerical coupled mode model for the evanescent excitation of the tube through semi-guided waves propagating in the slab at oblique angles. The model combines the directional polarized modes supported by the slab with analytic solutions for the TE-, TM-, and orbital-angular-momentum (OAM) modes of the tube-shaped fiber. Implementational details of the scheme are discussed, complemented by finite-element simulations for verification purposes. Our results include configurations with resonant in-fiber excitation of OAM modes with large orbital angular momentum and strong field enhancement.

      Oblique quasi-lossless excitation of a thin silicon slab waveguide: a guided-wave variant of an anti-reflection coating

      M. Hammer, L. Ebers, J. Förstner, Journal of the Optical Society of America B (2019), 36, pp. 2395

      Coupled microstrip-cavities under oblique incidence of semi-guided waves: a lossless integrated optical add-drop filter

      L. Ebers, M. Hammer, M.B. Berkemeier, A. Menzel, J. Förstner, OSA Continuum (2019), 2, pp. 3288

      We investigate optical microresonators consisting of either one or two coupled rectangular strips between upper and lower slab waveguides. The cavities are evanescently excited under oblique angles by thin-film guided, in-plane unguided waves supported by one of the slab waveguides. Beyond a specific incidence angle, losses are fully suppressed. The interaction between the guided mode of the cavity-strip and the incoming slab modes leads to resonant behavior for specific incidence angles and gaps. For a single cavity, at resonance, the input power is equally split among each of the four output ports, while for two cavities an add-drop filter can be realized that, at resonance, routes the incoming power completely to the forward drop waveguide via the cavity. For both applications, the strength of the interaction is controlled by the gaps between cavities and waveguides.

      Oblique evanescent excitation of a dielectric strip: A model resonator with an open optical cavity of unlimited Q

      M. Hammer, L. Ebers, J. Förstner, Optics Express (2019), 27(7), pp. 8

      A rectangular dielectric strip at some distance above an optical slab waveguide is being considered, for evanescent excitation of the strip through the semi-guided waves supported by the slab, at specific oblique angles. The 2.5-D configuration shows resonant transmission properties with respect to variations of the angle of incidence, or of the excitation frequency, respectively. The strength of the interaction can be controlled by the gap between strip and slab. For increasing distance, our simulations predict resonant states with unit extremal reflectance of an angular or spectral width that tends to zero, i.e. resonances with a Q-factor that tends to infinity, while the resonance position approaches the level of the guided mode of the strip. This exceptionally simple system realizes what might be termed a “bound state coupled to the continuum”.

        Optical transition between two optical waveguides layer and method for transmitting light

        M. Hammer, J. Förstner, L. Ebers. Optical transition between two optical waveguides layer and method for transmitting light, Patent DE102018108110B3. 2019.

        Die Erfindung betrifft einen optischen Übergang zwischen zwei optischen Schichtwellenleitern. Dazu ist eine Anordnung vorgesehen aus einem ersten optischen Schichtwellenleiter (2) und einem zweiten optischen Schichtwellenleiter (3), wobei der erste optische Schichtwellenleiter (2) und der zweite optische Schichtwellenleiter (3) voneinander verschiedene über ihre jeweilige Länge konstante Dicken (d, r) aufweisen, der erste optische Schichtwellenleiter (2) mit dem zweiten optischen Schichtwellenleiter (3) mittels einer optischen Schichtwellenleiterstruktur (4) verbunden ist, die über ihre gesamte Länge (w) eine Dicke (h) aufweist, die zwischen der Dicke (d) des ersten optischen Schichtwellenleiters (2) und der Dicke (r) des zweiten optischen Schichtwellenleiters (3) liegt. Erfindungsgemäß ist die Dicke (h) der optischen Schichtwellenleiterstruktur (4) über die gesamte Länge (w) der optischen Schichtwellenleiterstruktur (4) konstant. Damit wird eine Möglichkeit für einen effizienten und mit geringen Verlusten behafteten Übergang zwischen zwei optischen Schichtwellenleitern mit unterschiedlicher Dicke bereitgestellt.

          Oblique Semi-Guided Waves: 2-D Integrated Photonics with Negative Effective Permittivity

          M. Hammer, L. Ebers, A. Hildebrandt, S. Alhaddad, J. Förstner, in: 2018 IEEE 17th International Conference on Mathematical Methods in Electromagnetic Theory (MMET), IEEE, 2018

          Semi-guided waves confined in dielectric slab waveguides are being considered for oblique angles of propagation. If the waves encounter a linear discontinuity of (mostly) arbitrary shape and extension, a variant of Snell's law applies, separately for each pair of incoming and outgoing modes. Depending on the effective indices involved, and on the angle of incidence, power transfer to specific outgoing waves can be allowed or forbidden. In particular, critical angles of incidence can be identified, beyond which any power transfer to non-guided waves is forbidden, i.e. all radiative losses are suppressed. In that case the input power is carried away from the discontinuity exclusively by reflected semi-guided waves in the input slab, or by semi-guided waves that are transmitted into other outgoing slab waveguides. Vectorial equations on a 2-D cross sectional domain apply. These are formally identical to the equations that govern the eigenmodes of 3-D channel waveguides. Here, however, these need to be solved not as an eigenvalue problem, but as an inhomogeneous problem with a right-hand-side that is given by the incoming semi-guided wave, and subject to transparent boundary conditions. The equations resemble a standard 2-D Helmholtz problem, with an effective permittivity in place of the actual relative permittivity. Depending on the properties of the incoming wave, including the angle of incidence, this effective permittivity can become locally negative, causing the suppression of propagating outgoing waves. A series of high-contrast example configurations are discussed, where these effects lead to - in some respects - quite surprising transmission characteristics.

            Oblique incidence of semi-guided planar waves on slab waveguide steps: effects of rounded edges

            L. Ebers, M. Hammer, J. Förstner, Optics Express (2018), 26(14), pp. 18621-18632

            Oblique propagation of semi-guided waves across slab waveguide structures with bent corners is investigated. A critical angle can be defined beyond which all radiation losses are suppressed. Additionally an increase of the curvature radius of the bends also leads to low-loss configurations for incidence angles below that critical angle. A combination of two bent corner systems represents a step-like structure, behaving like a Fabry-Perot interferometer, with two partial reflectors separated by the vertical height between the horizontal slabs. We numerically analyse typical high-index-contrast Si/SiO2 structures for their reflectance and transmittance properties. When increasing the curvature radius the resonant effect becomes less relevant such that full transmittance is reached with less critical conditions on the vertical distance or the incidence angle. For practical interest 3-D problems are considered, where the structures are excited by the fundamental mode of a wide, shallow rib waveguide. High transmittance levels can be observed also for these 3-D configurations depending on the width of the rib.

            Spiral modes supported by circular dielectric tubes and tube segments

            L. Ebers, M. Hammer, J. Förstner, Optical and Quantum Electronics (2017), 49(4), pp. 49:176

            The modal properties of curved dielectric slab waveguides are investigated. We consider quasi-confined, attenuated modes that propagate at oblique angles with respect to the axis through the center of curvature. Our analytical model describes the transition from scalar 2-D TE/TM bend modes to lossless spiral waves at near-axis propagation angles, with a continuum of vectorial attenuated spiral modes in between. Modal solutions are characterized in terms of directional wavenumbers and attenuation constants. Examples for vectorial mode profiles illustrate the effects of oblique wave propagation along the curved slab segments. For the regime of lossless spiral waves, the relation with the guided modes of corresponding dielectric tubes is demonstrated.

              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.

                Oblique incidence of semi-guided waves on step-like folds in planar dielectric slabs: Lossless vertical interconnects in 3D integrated photonic circuits

                A. Hildebrandt, S. Alhaddad, M. Hammer, J. Förstner, in: Integrated Optics: Devices, Materials, and Technologies XX, SPIE, 2016

                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.

                  How planar optical waves can be made to climb dielectric steps

                  M. Hammer, A. Hildebrandt, J. Förstner, Optics Letters (2015), 40(16), pp. 3711-3714

                  We show how to optically connect guiding layers at different elevations in a 3-D integrated photonic circuit. Transfer of optical power carried by planar, semi-guided waves is possible without reflections or radiation losses, and over large vertical distances. This functionality is realized through simple step-like folds of high-contrast dielectric slab waveguides, in combination with oblique wave incidence, and fulfilling a resonance condition. Radiation losses vanish, and polarization conversion is suppressed for TE wave incidence beyond certain critical angles. This can be understood by fundamental arguments resting on a version of Snell’s law. The two 90° corners of a step act as identical partial reflectors in a Fabry–Perot-like resonator setup. By selecting the step height, i.e., the distance between the reflectors, one realizes resonant states with full transmission. Rigorous quasi-analytical simulations for typical silicon/silica parameters demonstrate the functioning. Combinations of several step junctions can lead to other types of optical on-chip connects, e.g., U-turn- or bridge-like configurations.

                  Full Resonant Transmission of Semiguided Planar Waves Through Slab Waveguide Steps at Oblique Incidence

                  M. Hammer, A. Hildebrandt, J. Förstner, Journal of Lightwave Technology (2015), 34(3), pp. 997-1005

                  Sheets of slab waveguides with sharp corners are investigated. By means of rigorous numerical experiments, we look at oblique incidence of semi-guided plane waves. Radiation losses vanish beyond a certain critical angle of incidence. One can thus realize lossless propagation through 90-degree corner configurations, where the remaining guided waves are still subject to pronounced reflection and polarization conversion. A system of two corners can be viewed as a structure akin to a Fabry-Perot-interferometer. By adjusting the distance between the two partial reflectors, here the 90-degree corners, one identifies step-like configurations that transmit the semi-guided plane waves without radiation losses, and virtually without reflections. Simulations of semi-guided beams with in-plane wide Gaussian profiles show that the effect survives in a true 3-D framework.

                    Planar prism spectrometer based on adiabatically connected waveguiding slabs

                    F. Civitci, M. Hammer, H. Hoekstra, Optics Communications (2015), 365, pp. 29-37

                    The device principle of a prism-based on-chip spectrometer for TE polarization is introduced. The spectrometer exploits the modal dispersion in planar waveguides in a layout with slab regions having two different thicknesses of the guiding layer. The set-up uses parabolic mirrors, for the collimation of light of the input waveguide and focusing of the light to the receiver waveguides, which relies on total internal reflection at the interface between two such regions. These regions are connected adiabatically to prevent unwanted mode conversion and loss at the edges of the prism. The structure can be fabricated with two wet etching steps. The paper presents basic theory and a general approach for device optimization. The latter is illustrated with a numerical example assuming SiON technology.

                      General relation for group delay and the relevance of group delay for refractometric sensing

                      H.J.W.M. Hoekstra, M. Hammer, Journal of the Optical Society of America B (2014), 31(7)

                      The relevance of our definition for sensitivity in refractometric sensing, being the relative change in the transmittance of a certain output channel of an optical device over the change in the refractive index of the probed material, is discussed. It is compared to one based on spectral shift per refractive index unit change. Further, there is discussion on how group delay and sensitivity are interrelated and can be converted into each other and which physical quantities are relevant for high sensitivity. As a by-product of the theory presented, a general expression relating group delay and the ratio of the time-averaged optical energy and the input power is presented.

                        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.

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                          Funding, co-operations


                          Dr. Manfred Hammer

                          Theoretische Elektrotechnik (TET)

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


                          nach Vereinbarung

                          Lena Ebers

                          Theoretische Elektrotechnik (TET)

                          Lena Ebers
                          +49 5251 60-3147

                          Head of the group

                          Prof. Dr. Jens Förstner

                          Theoretische Elektrotechnik (TET)

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


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