Δυαδικές συναρτήσεις Green πολυκεντρικών σφαιρικών σωμάτων: μικροκυματικές, βιο-ιατρικές & οπτικές εφαρμογές

Abstract

Dyadic Green’s Functions (DGF) are very important analytical tools for radiation problems involving a source of electromagnetic (EM) radiation and a scattering body. A DGF incorporates the electrical and geometrical properties of the body which obstructs the EM radiation emitted by a point source (i.e., a Hertz dipole). The source can be placed anywhere can be placed anywhere in space, even within the scattering body, and it can have any orientation. Knowledge of the DGF allows for the evaluation of the electric- and magnetic-field intensities associated with any physical source of EM radiation, localized or distributed in space, and the specific body that the DGF applies to. Determination of those intensities requires integration of the internal product of the DGF with the current density throughout the space occupied by the source. This thesis presents analytical steps leading to the DGF of non-spherical scatterers, i.e. bodies defined by several, non-concentric, spherical boundaries. The main part of the analysis yields the DGF of three, relative simple, non-spherical bodies: (a) the cluster of spheres, (b) the sphere with eccentric spherical inclusions and (c) the eccentrically stratified sphere; all three DGFs appear for the first time. There are no limitations with regard to the geometrical features of any one of those bodies apart from the condition that the spherical interfaces thereof must not intersect. The electrical properties in every part of those bodies are represented by an electric permittivity, which may be complex. These three analytical solutions are used to build an algorithm which allows for the formulation and determination of the DGF of non-spherical bodies of higher order, e.g., non-spherical bodies built as aggregates of (b) or (c). That algorithm is new, as well as user-friendly, because it can be applied without actual knowledge of the details of the analysis required to formulate the DGF. Superposition is applied to formulate the DGF outside and within the bodies of cases (a)-(c). Those formulations come as infinite series of dyads, each composed of a vector spherical wave-function and an unknown vector wave amplitude. The latter are determined by enforcement of the appropriate boundary conditions on all of the spherical interfaces that define the scattering body. The boundary conditions are applied indirectly, as integrals over the spherical interfaces. The aforesaid integrals require use of the translational addition theorem for vector spherical harmonics because the various spherical surfaces are not concentric. The end-result of this analytical procedure is a set of linear equations for the vector wave amplitudes and that set is solved by truncation and matrix-inversion. The solutions given for cases (a)-(c) have been checked analytically and numerically. The analytical checks aim at the correct reproduction of existing results for marginal geometries. On the other hand, the numerical checks are aimed at proving that the analytical solutions developed in this thesis conform to the principles of energy conservation and reciprocity; both are necessary conditions for the validity of any solution. All checks have been successful. The energy conservation checks have given, as a side-effect, useful information about the truncation number required for the convergence of the solution. The most important piece of information is that the truncation number is increased, in some cases dramatically, when the source of radiation approaches any spherical interface. Moreover, it has been found independently that a point source cannot be placed within a conductive medium, because in this case the energy conservation law is violated. The analysis occupies the major part of the thesis; the rest is occupied by four numerical applications aimed at manifesting the potentials of the analytical solutions. The first application is a study of the effect of a line cluster of rexolite spheres on the EM radiation emitted by a point source. The number of spheres, their size, the spacing between them, and the position of the source have been tuned in order to create a directive radiating system out of an isotropic point source. By use of a specific design methodology, the radiating system can achieve high gain, almost 18dBi, good front-to-back ratio and low side lobes. The second application is aimed at achieving increased gain and reduced weight of a radiating system composed of a focusing sphere and a nearby point source. An eccentric spherical cavity is considered inside a rexolite sphere and the properties (position and size) of that cavity are determined so that the radiator is optimized. It has been proven that the mass of a rexolite sphere of normalized size 3 6. can be reduced up to 33% without compromising the gain, which is almost 8dBi. Furthermore, a rexolite sphere of normalized size 4.2 with an implanted point source can be reduced in mass by 50% and still provide substantial suppression of the back lobe of radiation of that source. The third application concerns an implanted point source within a human head model. The model allows for the consideration of a spherical bone shell (i.e., the scull) which accommodates a major spherical core for the brain and two minor spherical inclusions for the eyes. The size and the electrical properties of each part of the model have been adapted to anatomical reality as much as possible. The numerical calculations have shown (a) how much energy is absorbed by the scull, the brain and the eyes for two positions of the implant, and (b) that absorption can be controlled by proper orientation of the implant. Radiation patterns have been drawn for the implant from within the head, for telemetry applications. The last numerical application involves layered nano-spheres made of gold and glass. The incident, plane EM wave spans the optical band of the spectrum. The numerical calculations are aimed at the conditions favoring the appearance of surface-enhanced Raman scattering (SERS). A specific design methodology involving several steps is proposed, which results in a hot spot on the surface of the gold layer. The electric-field intensity at the heart of that hot spot can be as high as 35dB above the corresponding value of the incident wave. The thesis includes a description of the current state-of-the-art in the introductory chapter, a final chapter devoted to conclusions and future work, four appendices with support mathematical material, and a list of references.

Στην παρούσα διατριβή παρουσιάζεται ο αναλυτικός υπολογισμός της Δυαδικής Συνάρτησης Green (DGF) πολυκεντρικών σφαιρικών σκεδαστών, δηλ. σωμάτων που ορίζονται από μη τεμνόμενες σφαιρικές επιφάνειες. Στο κύριο μέρος της ανάλυσης υπολογίζονται για πρώτη φορά οι DGF τριών πολυκεντρικών σφαιρικών σωμάτων α)του συσσωματώματος διηλεκτρικών σφαιρών, β) της σφαίρας με έκκεντρες σφαιρικές ανομοιογένειες και γ) της έκκεντρα στρωματοποιημένης σφαίρας. Οι παραπάνω αναλυτικές λύσεις χρησιμοποιούνται ώστε να διαμορφωθεί ένας αλγόριθμος που επιτρέπει τον υπολογισμό της DGF οποιουδήποτε πολυκεντρικού σφαιρικού σώματος. Οι λύσεις (α)-(γ) ελέγχθηκαν επιτυχώς τόσο αναλυτικά όσο και αριθμητικά. Η διατριβή ολοκληρώνεται με τέσσερεις αριθμητικές εφαρμογές της ανάλυσης. Στην πρώτη μελετάται η επίδραση μιας γραμμικής διπόλου. Η δεύτερη αποσκοπεί στον περιορισμό του βάρους της σφαίρας που εστιάζει την προσπίπτουσα ακτινοβολία, θεωρώντας μια σφαιρική κοιλότητα στο εσωτερικό της. Στην τρίτη εφαρμογή ελέγχεται η επίδραση ενός ενεργού εμφυτεύματος σε ένα μοντέλο του ανθρώπινου κεφαλιού. Τέλος, στην τέταρτη αριθμητική εφαρμογή μελετάται το φαινόμενο της επιφανειακά ενισχυμένης σκέδασης Raman (surface enhanced Raman scattering) σε έκκεντρα στρωματοποιημένα νανο-σφαιρίδια που αποτελούνται από χρυσό και γυαλί.