Principal Investigator: Prof. Dr. Daniel Erni, UDE

Achieved results and methods

Multiscale modeling of skin tissues as complex composites As an outflow from the 1^st and 2^nd phase, the FEM-based simulation workbench has been perfected to cover all relevant scales of the skin tissue's underlying microstructure, starting at the sub-cellular and cellular features up to the skin's specific layer structure, while using a quasi-static EM solver in the microstructure, which is then «calibrated» to the macroscopic EM exposure model of the skin surface [12]. In addition to the spectroscopic evaluation of skin tissue properties using reflectometry [14], the method also enables virtual microdosimetry at the cellular level – based on the microscopic representation of the specific absorption rate (SAR) [12] – and for the first time also takes into account the influence of subcellular structures (i.e. organelles) [13]. Nevertheless, the planned analysis and design of functional surface systems revealed that the FEM-based workbench is still too slow to meet the efficiency needed to solve the associated inverse problem.

Slim and efficient simulation engine based on lattice sums starting in the 1^st phase we developed a high-speed computational EM simulator for (quasi-) periodic and partly irregular complex scattering structures, applicable in designing compact devices for mm-wave, THz, [15-17] and photonic/plasmonic applications [5-7,9,10,18-20]. The 2D frequency-domain implementation employs cylindrical wave expansions for each scatterer, utilizing lattice sums (LS) technique, generalized reflection/transmission matrices, and a recursive algorithm for multilayered composite structures [17,19,20]. 3D implementations for multilayer composites are available [8]. The method extends to dissipative or highly dispersive structures and complex eigenvalue problems (e.g., leaky waves) using (higher-order) Ewald's formulation [20] for convergent LSs, improving the handling of long-range evanescent field interactions and speeding up the simulation engine. Compared to the FEM-based workbench speed-up factors of 27-40 were already achieved for (GA resp. ANN optimized) planar filter structures [15-17] or 2D [20] respective 3D aggregates of scatterers [8]. As efficient as this simulation engine is, it comes not without its limitations according to the restriction to piecewise (quasi) periodic surface systems, which is inherently demanded by the LS technique. The generalization of such type of efficient simulation engine defines therefore the most important achievement of the 2^nd phase. Nevertheless, the LS-based simulation approach served as a powerful numerical framework in which nonlinear and topological material properties could be implemented aiming at advanced applications in e.g. all-optical digital photonics [18] and electrically tunable soliton propagation [7].

Extended material properties for the simulation engine based on lattice sums Nonlinear material properties: The LS-simulation method has been extended by implementing a Kerr-type nonlinearity in order to support envelope soliton propagation in channel waveguides of a planar photonic crystal (PhC) structure. Stable slow-light gap solitons in triple channels have been achieved in by tuning the thickness of the planar PhC slab accordingly [9]. With the extended LS-simulation, intensity-dependent perturbations of the propagating mode's dispersion characteristics could be analyzed respective designed in order to get a sensitive (super-) mode selection (i.e. by pushing propagating super-modes in to the photonic bandgap respective into an evanescent state and vice versa) within a triple-parallel-waveguide mode coupling scenario in order to achieve full-optical on-off switching of super-modes dependent of the resulting intensity of the interfering, individually excited super-modes. Hence, based on coupled triple channel-waveguide configurations, ultra-compact all-optical NOT, AND, and NAND logic gates have been proposed and their digital signal pulse (soliton) operation was demonstrated within comprehensive finite-difference time-domain (FDTD) simulations of these nonlinear PhC devices. A review of all our unique soliton-based, all-optical logic gates is given in [18] (and in our references therein). Topological material properties: A very recent effort is now focused on the implementation of topological, namely multiferroic [6,7,10] and in particular ferroelectric material properties [5,10] into the LS-based simulator. Polar vortex textures offer potential interaction pathways with chiral, topological, and photonic materials. We investigated the interaction between electromagnetic waves and a helically ordered spin lattice with emerging ferroelectric polarization analyzing the formation of helical spin-photonic band gaps (and their spectral location) using our extended LS-method [10]. We explored circular dichroism in the scattering of electromagnetic waves from this type of multiferroic lattice, which is tunable by a static electric field. The setting can be extended to a tunable cavity when two of such lattice slabs (consisting of helical multiferroic oxides) are used as confining boundaries to form a Fabry-Pérot-like cavity structure [6]. Adding even more complexity to the system we analyzed a configuration comprising a dielectric planar waveguide with Kerr-type nonlinearity sandwiched between two of such helical multiferroic oxide layers, which demonstrated to sustain electrically-controllable chiral solitons [7]. Moving now from optical frequencies to the mm-wave/THz range we numerically explored the sub-THz collective response of a ferroelectric vortex lattice that allows coupling to electric and magnetic fields, giving rise to a novel photonic lattice system with distinctive dispersion. The numerical simulations are carried out by two independent simulation approaches, based on the LS-technique respective the rigorous coupled-wave method. The frequency responses of the surface reflection and the transmission from such ferroelectric vortex lattice slab exhibit various distinct mini-stopbands emerging from crossings in the associated band diagram [5]. This goes along with a corresponding set of very anomalous diffraction orders, the functionalization of which will be part of our research in WP1 of the 3^rd phase.

Ultra-fast full-wave electromagnetic scattering simulator RACTMA The main research activity within the 2^nd phase was the development of a generalized, ultra-fast, inherently multiscale and in particular accurate, full-wave EM scattering simulation platform for very large aggregations of randomly distributed, individually shaped (cylindrical) scatterers. Our method is based on the Recursive Aggregated Centered T-Matrix Algorithm (RACTMA) [1,2], which has a numerical complexity of O(N ²) (having N scatterers). It can be viewed as a significant extension of our prior LS-technique [8,15-17,20] while conceptually aiming at both, (i) the potentially fastest, but less accurate Fast Multipole Method (FMM) [21] (with its generalization, the Multi-Level Fast Multipole Algorithm (MLFMA) [22,23] and a numerical complexity of O(N∙logN), and (ii) the butterfly-based block-iterative multi-stage scheme reminiscent of the FFT algorithm [24] O(N∙1.5 logN) or, as an alternative in conjunction with a direct matrix solver (LU factorization) [24]. RACTMA relies on a scatterer-centered transition-matrix formalism using generalized T-matrices to relate the complex expansion coefficients of the incidence field excitation to the expansion coefficients of the various scatterers' scattered field (composed of corresponding scalar or vector wave basis functions) and gets thus independent of the illumination [1]. With multiple scatterers, incoming and scattered field have to be related to the origin of each scatterer, which is realized using the translational addition theorem (AT). The computational complexity to calculate the resulting T-matrix of the multiple scatterer system can be reduced using recursive T-matrix generation schemes, where the fast methods violates the AT generally and in particular when looking at multi-stage scatterer aggregation. In [1] we therefore developed unique and reliable aggregation schemes with distinct selection rules for scatterers particularly designed to avoid violation of the AT. In this respect RACTMA [1] has already outperformed comparable (canonical) algorithms [25,27] in terms of speed and accuracy [cf. Fig. 1 (c)]. Further acceleration is achieved when looking at multi-scale/multi-stage implementations using e.g. a 2-layer T-matrix algorithm [2], where RACTMA may achieve acceleration factors up to 1000 (cf. Fig. 1). Hence, an n-layer RACTMA implementation dealing with a clever bottom-up, local-to-global aggregation scheme of scatterer clusters bears the promise of a reduced numer-ical complexity below the already achieved O(N ^5/3) together with a massive speedup due to parallelization, which will be explored in WP1. Given the T-matrices, any further analysis of different illumination scenarios reduces to simple matrix-vector multiplications, which renders RACTMA both fast and accurate.

In order to get a reliable trade-off between accuracy and numerical complexity we developed a robust criterion that relates the truncation number of the field expansions (i.e. size of the T-matrix) to the approximation error in the near-field given a single-particle Mie scattering scenario [2]. This is absolutely new, as the commonly used Wiscombe criterion [26] relies on remote far-field quantities only, whereas our criterion provides explicit minimal radial distances (ranges), where the demanded accuracy is met. With this, we got an additional grip on the reduction of numerical complexity, which may enable a further speed-up. A convincing numerical example [2] is displayed in Fig. 1 for total 2025 scatterers (confined into the overall cluster layer 2) that are associated with 45 sub-aggregates (clusters layer 1) each of which contains 45 different cylindrical scatterers as described in Fig. 1. The analysis shows a great mutual agreement in the scattering far-fields for an oblique plane wave excitation but demonstrates the superiority of the 2-layer RACTMA in performance and accuracy (in particular for large aggregations) when checking the fulfillment of both the reciprocity relation and the optical theorem [2]. All simulations are run on a PC with an AMD Ryzen Threadripper PRO 3995WX 64-Core processor (2.7 GHz) with 490 GB RAM using MATLAB numerical codes at standard double precision arithmetic.

Microfluidic test structures made e.g. of a heterogeneously filled 3D-printed PLA cylinder hosting 13 different microfluidic channels (cf. Fig. 2) are (i) used as experimental validation of the RACTMA scattering analysis and (ii) to reconstruct the inner structure from measured radar cross section (RCS) data within an inverse scattering analysis. The results are shown in Fig. 2 for RCS measurements at (a) 120 GHz and (b) 160 GHz. The structure may resemble e.g. a plant stem dummy for the development of a THz plant monitoring system where water and nutrient transport have to be monitored within the Stem's inner vascular bundles (xylem and phloem) [4]. In this regard highly-accurate material characterization is key. We use here the waveguide-based reflection/transmission material characterization kit (MCK) from SWISSto12, where its deficiencies, namely the inherent power leakage from the sample volume, had to be carefully analyzed in advance [32]. A flexible setup with controllable water filled microtubes is shown in Fig. 2 (left) and an additional microfluidic verification test set for scattering measurements at 1-40 GHz using a rectangular group of 250 randomly distributed water-filled straws is discussed in [11]. Recently we discovered an extremely sensitive, reflection-based method to characterize liquids using a rotary cylindrical setup that carries one (or more) probe vial(s) with optimized size(s) position(s) as shown in Fig. 3. Using RACTMA, the liquid's permittivity may be retrieved in real-time. This will be explored within WP2.

Innovations and novel methods Regarding the outcomes in the 2^nd phase we have several innovations to report: (i) The microdosimetric assessment of power absorption within human tissue's sub-cellular micro­structure [13], and the published validity limits for the effective material properties in these settings [14]. In an additional effort we characterized and modeled the material properties of honey bees [4,32].
(ii) The inclusion of topological (i.e. multiferroic) material properties into our numerical scattering analysis while proposing new functionalities in resulting surface systems in particular at (sub-) THz frequencies [5]. (iii) The inclusion of nonlinear (i.e. Kerr-type) material properties into our LS-based analysis that yields to the proposal of new types of PhC-based all-optical logic gates [18]. (iv) The currently unsurpassed ultra-fast and accurate, multi-stage RACTMA simulator [1], where we circumvented the violation of the addition theorem [1,2], and provided for the first time a near-field criterion to tradeoff accuracy against low truncation numbers of the field expansions [3]. (v) The proposal of an extremely sensitive microfluidic setup for the characterization of complex permittivities of liquids in the (sub-) THz frequency range.

Selected project-related publications

[1]     M. Degen, V. Jandieri, B. Sievert, J. T. Svejda, A. Rennings, and D. Erni, "An efficient analysis of scattering from randomly distributed obstacles using an accurate recursive aggregated centered T-matrix algorithm," IEEE Trans. Antennas Propagat., 2023, doi: 10.1109/TAP.2023.3316792.

[2]     M. Degen, V. Jandieri, A. Rennings, and D. Erni, "Scattering analysis of ensembles of cylinders using a 2-layer T-matrix algorithm," IEEE CAMA 2023, Genoa, Italy, Nov. 15-17, 2023, doi: 10.1109/CAMA57522.2023.10352852

[3]     M. Degen, A. Rennings, and D. Erni, "Near-field truncation error of a Mie-series for a perfect electric con-ducting cylinder," IWMTS 2023, Bonn, Germany, July 3-5, 2023, doi 10.1109/IWMTS58186.2023.10207855.

[4]     F. Sheikh, A. Prokscha, A. Batra, D. Lessy, B. Salah, B. Sievert, M. Degen, A. Rennings, M. Jalali, J. T. Svejda, P. Alibeigloo, E. Mutlu, C. Preuss, R. Kress, S. Clochiatti, K. Kolpatzeck, T. Kubiczek, I. Ullmann, K. Root, F. Brix, U. Kraemer, M. Vossiek, J. C. Balzer, N. G. Weimann, T. Kaiser, and D. Erni, "Towards continuous real-time plant and insect monitoring by miniaturized THz systems," IEEE J. Microwaves, 2023, doi: 10.1109/JMW.2023.3278237.

[5]     R. Khomeriki, V. Jandieri, K. Watanabe, D. Erni, D. H. Werner, M. Alexe, and J. Berakdar, "Photonic ferroelectric vortex lattice," Phys. Rev. B., 2024, doi 10.1103/PhysRevB.109.045428.

[6]     V. Jandieri, R. Khomeriki, K. Watanabe, D. Erni, D. H. Werner, and J. Berakdar, "Tunable chiral photonic cavity based on multiferroic layers," Opt. Express, 2023, doi: 10.1364/OE.489612.

[7]     V. Jandieri, R. Khomeriki, K. Watanabe, D. Erni, D. H. Werner, and J. Berakdar, "Chiral optical solitons in electrically active multiferroic guiding structure," Opt. Express, 2024, doi: 10.1364/OE.507216.

[8]     V. Jandieri, S. Ceccuzzi, P. Baccarelli, G. Schettini, D. Erni, W. Hong, D. H. Werner, and K. Yasumoto, "Efficient analysis of radiation from a dipole source in woodpile EBG structures," IEEE Trans. Antennas Propagat., 2022, doi: 10.1109/TAP.2021.3098578.

[9]     C. Bohley, V. Jandieri, B. Schwager, R. Khomeriki, D. Schulz, D. Erni, D. H. Werner, and J. Berakdar, "Thickness-dependent slow-light gap solitons in three-dimensional coupled photonic crystal waveguides," Opt. Lett., 2022, doi: 10.1364/OL.457044.

[10]   V. Jandieri, R. Khomeriki, L. Chotorlishvili, K. Watanabe, D. Erni, D. H. Werner, and J. Berakdar, "Photonic signatures of spin-driven ferroelectricity in multiferroic dielectric oxides," Phys. Rev. Lett., 2021, doi: 10.1103/PhysRevLett.127.127601.