Το παρακάτω κείμενο αποτελεί διπλωματική εργασία που εκπονήθηκε στο Εργαστήριο Ασύρματης Επικοινωνίας του τμήματος Ηλεκτρολόγων Μηχανικών και Τεχνολογίας Υπολογιστών του Πανεπιστημίου Πατρών. Σκοπός μας, ήταν η επίλυση των εξισώσεων του Maxwell σε προβλήματα που αφορούν το σφαιρικό και το κυλινδρικό σύστημα συντεταγμένων. Στην προσπάθεια αυτή κάναμε χρήση της μεθόδου επίλυσης των πεπερασμένων διαφορών στο πεδίο του χρόνου (F.D.T.D) σε δύο ειδών εφαρμογές: η πρώτη αφορά έναν κυλινδρικό κυματοδηγό, τον οποίο μοντελοποιήσαμε με τη βοήθεια του κυλινδρικού συστήματος συντεταγμένων και η δεύτερη αφορά μια κωνική κεραία UWB, την οποία μοντελοποιήσαμε με τη βοήθεια του σφαιρικού συστήματος συντεταγμένων. Η προσομοίωση αυτών των δύο εφαρμογών γίνεται με τη βοήθεια του προγραμματιστικού περιβάλλοντος της Matlab. Επειδή η μέθοδος F.D.T.D επιλύει τις εξισώσεις του Maxwell στο χρόνο μας προσφέρει τη δυνατότητα της οπτικής απεικόνισης των πεδίων σε διάφορες χρονικές στιγμές, κάτι που μας επιτρέπει να παρατηρούμε τη χρονική εξέλιξη των φαινομένων. Η μέθοδος γίνεται ιδιαίτερα ελκυστική λόγω του επιπρόσθετου χαρακτηριστικού της απευθείας επίλυσης των εξισώσεων στροβιλισμού του Maxwell, καθιστώντας παράλληλα περιττή την επίλυση της κυματικής εξίσωσης. Στο πρώτο κεφάλαιο γίνεται μια εισαγωγή στις υπολογιστικές τεχνικές στον ηλεκτρομαγνητισμό. Επίσης γίνεται μια πρώτη αναφορά στη μέθοδο των πεπερασμένων διαφορών στο πεδίο του χρόνου (F.D.T.D), στις δυνατότητες της μεθόδου, στο πεδίο εφαρμογής της καθώς και στα πλεονεκτήματά της. Στο κεφάλαιο δύο παρουσιάζονται οι εξισώσεις του Maxwell. Συγκεκριμένα παρουσιάζονται οι εξισώσεις στροβιλισμού και οι βαθμωτές διαφορικές εξισώσεις που προκύπτουν από αυτές στις τρεις και δύο διαστάσεις. Στις δύο διαστάσεις γίνεται διάκριση σε εγκάρσιο ηλεκτρικό ρυθμό (Transverse Electric) και εγκάρσιο μαγνητικό ρυθμό (Transverse Magnetic). Τέλος παρουσιάζονται οι εξισώσεις του Maxwell που ισχύουν για τα σκεδαζόμενα πεδία. Στο κεφάλαιο τρία παρουσιάζονται τα βασικά στοιχεία της μεθόδου F.D.T.D, τα οποία πρέπει να γίνουν κατανοητά προκειμένου να αναδειχθούν τα πλεονεκτήματα και τα μειονεκτήματά της. Συγκεκριμένα παρουσιάζονται οι εξισώσεις πεπερασμένων διαφορών, που προκύπτουν από τις βαθμωτές διαφορικές εξισώσεις, που προκύπτουν από τις εξισώσεις στροβιλισμού του Maxwell. Στη συνέχεια παρουσιάζονται βασικά χαρακτηριστικά της μεθόδου, όπως η επιλογή του χωρικού και χρονικού βήματος και η δημιουργία πηγών. Μετά παρουσιάζεται η σημαντικότερη απορροφητική οριακή συνθήκη PML του Berenger και τέλος οι υπολογιστικές απαιτήσεις του αλγορίθμου F.D.T.D. Στο κεφάλαιο τέσσερα επιλύονται προβλήματα σε δύο διαστάσεις και συγκεκριμένα το πρόβλημα των ρυθμών διάδοσης TM και ΤΕ εντός κυλινδρικού κυματοδηγού με υλικό εντός του τον αέρα. Στο κεφάλαιο πέντε επιλύονται προβλήματα σε δύο διαστάσεις. Συγκεκριμένα παρουσιάζει την ανάπτυξη ενός σετ εργαλείων λογισμικού που είναι χρήσιμα στην ανάλυση κεραιών και δομών εξαιρετικά ευρείας ζώνης (UWB). Αυτά τα εργαλεία χρησιμοποιούνται στην εκτέλεση προσομοίωσης με τη μέθοδο των πεπερασμένων διαφορών στο πεδίο του χρόνου (FDTD) μίας κωνικής κεραίας με συνεχές κύμα (CW) και παλμικές διεγέρσεις UWB. Η κεραία αναλύεται με τη χρήση εξισώσεων σφαιρικών συντεταγμένων FDTD που προέρχονται από τις βασικές αρχές. Τα αποτελέσματα της προσομοίωσης για τη διέγερση τύπου συνεχούς κύματος (CW) συγκρίνονται με τα αποτελέσματα από προσομοιώσεις και μετρήσεις σε δημοσιευμένες πηγές· τα αποτελέσματα της διέγερσης UWB είναι νέα. Τα παραπάνω προβλήματα κάνουν σαφές το πόσο σημαντικό είναι να γνωρίζουμε σε βάθος τα χαρακτηριστικά της μεθόδου προκειμένου να φτάσουμε στην λύση τους. Σε περιπτώσεις όπου γνωρίζουμε τη λύση εκ των προτέρων (είτε ποιοτικά ή ποσοτικά) έχουμε τη δυνατότητα να επαληθεύσουμε την ορθότητα των αποτελεσμάτων της F.D.T.D. Οι λύσεις των προβλημάτων β ασίζονται στην εύρεση των ολικών πεδίων. Ο εναλλακτικός τρόπος της εύρεσης των σκεδαζόμενων πεδίων δεν χρησιμοποιείται. Βέβαια στα προβλήματα ακτινοβολίας κεραιών υποχρεωτικά εφαρμόζεται η διατύπωση των ολικών πεδίων. Μέσω αυτής της εργασίας έγινε σαφής η ικανότητα της F.D.T.D να εφαρμόζεται σε μεγάλη ποικιλία προβλημάτων. Κάτι που δεν έγινε σαφές είναι η δυνατότητα της μεθόδου να συνδυάζεται με άλλες μεθόδους, κάτι που μπορεί να επιφέρει σημαντική καταστολή ή και εξάλειψη των μειονεκτημάτων της. Με αυτό τον τρόπο δημιουργούνται νέες υβριδικές μέθοδοι. Με τις μεθόδους εύρεσης των ηλεκτρομαγνητικών πεδίων (όπως είναι η F.D.T.D) έχουμε τη δυνατότητα να δούμε τον ηλεκτρομαγνητισμό από νέα σκοπιά, κατανοώντας τον καλύτερα και προσαρμόζοντάς τον στις σύγχρονες ανάγκες της εποχής. / -

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<div>Multiscale and multiphysics have been two major challenges in analyzing and designing new emerging engineering devices, materials, circuits, and systems. When simulating a multiscale problem, numerical methods have to overcome the challenges in both space and time to account for the scales spanning many orders of magnitude difference. In the finite-difference time-domain (FDTD) method, subgridding techniques have been developed to address the multiscale challenge. However, the accuracy and stability in existing subgridding algorithms have always been two competing factors. In terms of the analysis of a multiphysics problem, it involves the solution of multiple partial differential equations. Existing partial differential equation solvers require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. When the problem size, and hence the matrix size, is large, existing methods become highly inefficient.</div><div><br></div><div>In this work, a symmetric positive semi-definite FDTD subgridding algorithm in both space and time is developed for fast transient simulations of multiscale problems. This algorithm is stable and accurate by construction. Moreover, the method is further made unconditionally stable, by analytically finding unstable modes, and subsequently deducting them from the system matrix. To address the multiphysics simulation challenge, we develop a matrix-free time domain method for solving thermal diffusion equation, and the combined Maxwell-thermal equations, in arbitrary unstructured meshes. The counterpart of the method in frequency domain is also developed for fast frequency-domain analysis. In addition, a generic time marching scheme is proposed for simulating unsymmetrical systems to guarantee their stability in time domain. </div>

The Finite Difference Time Domain (FDTD) Method has been a powerful tool in numerical simulation of electromagnetic (EM) problems for decades. In recent years, it has also been applied to biomedical research to investigate the interaction between EM waves and biological tissues. In Wireless Body Area Networks (WBANs) studies, to better understand the localization problem within the body, an accurate source/receiver model must be investigated. However, the traditional source models in FDTD involve effective volume and may cause error in near field arbitrary direction. This thesis reviews the basic mathematical and numerical foundation of the Finite Difference Time Domain method and the material properties needed when modeling a human body in FDTD. Then Coincident Phase Centers (CPCs) point sources models have been introduced which provide nearly the same accuracy at the distances as small as 3 unit cells from the phase center. Simultaneously, this model outperforms the usual sources in the near field when an arbitrary direction of the electric or magnetic dipole moment is required.

Simulating wave propagation in microstrip lines with Gyrotropic magnetic substrate is considered in this thesis. Since the static internal field distribution has an important effect on the device behavior, accurate determination of the internal fields are considered as well. To avoid the losses at microwave frequencies it is assumed that the magnetic substrate is saturated in the direction of local internal field. An iterative method to obtain the magnetization distribution has been developed. It is applied to a variety of nonlinear nonuniform magnetic material configurations that one may encounter in the design stage, subject to a nonuniform applied field. One of the main characteristics of the proposed iterative method to obtain the static internal field is that the results are supported by a uniqueness theorem in magnetostatics. The series of solutions Mn,Hn, where n is the iteration number, minimize the free Gibbs energy G(M) in sequence. They also satisfy the constitutive equation M = χH at the end of each iteration better than the previous one. Therefore based on the given uniqueness theorem, the unique stable equilibrium state M is determined. To simulate wave propagation in the Gyrotropic magnetic media a new FDTD formulation is proposed. The proposed formulation considers the static part of the electromagnetic field, obtained by using the iterative approach, as parameters and updates the dynamic parts in time. It solves the Landau-Lifshitz-Gilbert equation in consistency with Maxwell’s equations in time domain. The stability of the initial static field distribution ensures that the superposition of the time varying parts due to the propagating wave will not destabilize the code. Resonances in a cavity filled with YIG are obtained. Wave propagation through a microstrip line with YIG substrate is simulated. Magnetization oscillations around local internal field are visualized. It is proved that the excitation of magnetization precession which is accompanied by the excitation of magnetostatic waves is responsible for the gap in the scattering parameter S12. Key characteristics of the wide microstrip lines are verified in a full wave FDTD simulation. These characteristics are utilized in a variety of nonreciprocal devices like edgemode isolators and phase shifters.

Simulating wave propagation in microstrip lines with Gyrotropic magnetic substrate is considered in this thesis. Since the static internal field distribution has an important effect on the device behavior, accurate determination of the internal fields are considered as well. To avoid the losses at microwave frequencies it is assumed that the magnetic substrate is saturated in the direction of local internal field. An iterative method to obtain the magnetization distribution has been developed. It is applied to a variety of nonlinear nonuniform magnetic material configurations that one may encounter in the design stage, subject to a nonuniform applied field. One of the main characteristics of the proposed iterative method to obtain the static internal field is that the results are supported by a uniqueness theorem in magnetostatics. The series of solutions Mn,Hn, where n is the iteration number, minimize the free Gibbs energy G(M) in sequence. They also satisfy the constitutive equation M = χH at the end of each iteration better than the previous one. Therefore based on the given uniqueness theorem, the unique stable equilibrium state M is determined. To simulate wave propagation in the Gyrotropic magnetic media a new FDTD formulation is proposed. The proposed formulation considers the static part of the electromagnetic field, obtained by using the iterative approach, as parameters and updates the dynamic parts in time. It solves the Landau-Lifshitz-Gilbert equation in consistency with Maxwell’s equations in time domain. The stability of the initial static field distribution ensures that the superposition of the time varying parts due to the propagating wave will not destabilize the code. Resonances in a cavity filled with YIG are obtained. Wave propagation through a microstrip line with YIG substrate is simulated. Magnetization oscillations around local internal field are visualized. It is proved that the excitation of magnetization precession which is accompanied by the excitation of magnetostatic waves is responsible for the gap in the scattering parameter S12. Key characteristics of the wide microstrip lines are verified in a full wave FDTD simulation. These characteristics are utilized in a variety of nonreciprocal devices like edgemode isolators and phase shifters.

Computer simulation of electromagnetic behaviour of a device is a common practice in modern engineering. Maxwell's equations are solved on a computer with help of numerical methods. Contemporary devices constantly grow in size and complexity. Therefore, new numerical methods should be highly efficient. Many industrial and research applications of numerical methods need to account for the frequency dependent materials. The Finite-Difference Time-Domain (FDTD) method is one of the most widely adopted algorithms for the numerical solution of Maxwell's equations. A major drawback of the FDTD method is the interdependence of the spatial and temporal discretisation steps, known as the Courant-Friedrichs-Lewy (CFL) stability criterion. Due to the CFL condition the simulation of a large object with delicate geometry will require a high spatio-temporal resolution everywhere in the FDTD grid. Application of subgridding increases the efficiency of the FDTD method. Subgridding decomposes the simulation domain into several subdomains with different spatio-temporal resolutions. The research project described in this dissertation uses the Huygens Subgridding (HSG) method. The frequency dependence is included with the Auxiliary Differential Equation (ADE) approach based on the one-pole Debye relaxation model. The main contributions of this work are (i) extension of the one-dimensional (1D) frequency-dependent HSG method to three dimensions (3D), (ii) implementation of the frequency-dependent HSG method, termed the dispersive HSG, in Fortran 90, (iii) implementation of the radio environment setting from the PGM-files, (iv) simulation of the electromagnetic wave propagating from the defibrillator through the human torso and (v) analysis of the computational requirements of the dispersive HSG program.

In the present thesis, computational methods for theanalysis and design of photonic bandgap structure areconsidered. Many numerical methods have been used to study suchstructures. Among them, the plane wave expansion method is veryoften used. Using this method, we show that inclusions ofelliptic air holes can be used effectively to obtain a largercomplete band gap for two-dimensional (2D) photonic crystals.An optimal design of a 2D photonic crystal is also consideredin the thesis using a combination of the plane wave expansionmethod and the conjugate gradient method. We find that amaximum complete 2D band gap can be obtained by connectingdielectric rods with veins for a photonic crystal with a squarelattice of air holes in GaAs. For some problems, such as defect modes, the plane waveexpansion method is extremely time-consuming. It seems that thefinite-difference time-domain (FDTD) method is promising, sincethe computational time is proportional to the number of thediscretization points in the computation domain (i.e., it is oforderN). A FDTD scheme in a nonorthogonal coordinate systemis presented in the thesis to calculate the band structure of a2D photonic crystal consisting of askew lattice. The algorithmcan easily be used for any complicated inclusion configuration,which can have both the dielectric and metallic constituents.The FDTD method is also applied to calculate the off-plane bandstructures of 2D photonic crystals in the present thesis. Wealso propose a numerical method for computing defect modes in2D crystals (with dielectric or metallic inclusions). Comparedto the FDTD transmission spectra method, our method reduces thecomputation time and memory significantly, and finds as manydefect modes as possible, including those that are not excitedby an incident plane wave in the FDTD transmission spectramethod. The FDTD method has also been applied to calculateguided modes and surface modes in 2D photonic crystals using acombination of the periodic boundary condition and theperfectly matched layer for the boundary treatment. Anefficient FDTD method, in which only real variables are used,is also proposed for the full-wave analysis of guided modes inphotonic crystal fibers. / QC 20100629

Size reduction is a key issue in the development of contemporary integrated photonics. This thesis is mainly devoted to study some integrated photonic components in sub-wavelength or nanometric scales, both theoretically and experimentally. The possible approaches to reduce the sizes or to increase the functionalities of photonic components are discussed, including waveguides and devices based on silicon nanowires, photonic crystals, surface plasmons, and some near-field plasmonic components. First, some numerical methods, including the finite-difference time-domain method and the full-vectorial finite-difference mode solver, are introduced. The finite-difference time-domain method can be used to investigate the interaction of light fields with virtually arbitrary structures. The full-vectorial finite-difference mode solver is mainly used for calculating the eigenmodes of a waveguide structure. The fabrication and characterization technologies for nano-photonic components are reviewed. The fabrications are mainly based on semiconductor cleanroom facilities, which include thin film deposition, electron beam lithography, and etching. The characterization setups with the end-fire coupling and the vertical grating coupling are also described. Silicon nanowire waveguides and related devices are studied. Arrayed waveguide gratings with 11nm and 1.6nm channel spacing are fabricated and characterized. The dimension of these arrayed waveguide gratings is around 100 μm, which is 1--2 order of magnitude smaller than conventional silica based arrayed waveguide gratings. A compact polarization beam splitter employing positive/negative refraction based on a photonic crystal of silicon pillars is designed and demonstrated. Extinction ratio of ~15dB is achieved experimentally in a wide wavelength range. Surface plasmon waveguides and devices are analyzed theoretically. With surface plasmons the light field can be confined in a sub-wavelength dimension. Some related photonic devices, e.g., directional couplers and ring resonators, are studied. We also show that some ideas and principles of microwave devices, e.g., a branch-line coupler, can be borrowed for building corresponding surface plasmon based devices. Near-field plasmonic components, including near-field scanning optical microscope probes and left handed material slab lenses, are also analyzed. Some novel designs are introduced to enhance the corresponding systems. / QC 20100908