The physics and mathematical theory of nano-scaled ring resonators and loop antennas
This thesis is based on the realisation that no analytical theory of loop antennas and rings exists that is at once applicable to the Radio Frequency (RF), Micro-wave (MW), TeraHertz (THz), Infra-red (IR), and Optical (OR) regions. Nor is there any Electrical Engineering circuit model, rigorously developed from the results of that theory, that generates results which match numerical simulations and experimental work in the literature across all of these regimes. This thesis fills that gap. Maxwell’s equations for perfectly conducting, closed circular loops are presented, and then solved, using standard RF andMWantenna theory. The governing equation is then extended to include real, lossy metals with focus on the noble metals, gold, silver and copper. The solution to the extended equation yields results for rings in the THZ, IR and OR. Next, the governing equation is extended to include a single impedance on the periphery. The solution is studied using a capacitive reactance, in particular. These results are compared to simulations of illuminated rings with a single gap, and a relationship is developed between the width of the gap and its capacitive reactance. Primary results are these: • An analytical set of mathematical functions derived from Maxwell’s equations now exist that give the current distribution on closed and single gapped loops at all frequency regimes from the RF through OR, constructed of any metal for which the index of refraction is known. • A detailed RLC circuit model has been derived from these functions, accurate at all frequency bands, from which the total R, L and C of the loop at any frequency or wavelength, and the R, L and C of any modal resonance, can be calculated. The model yields the functions R(w), L(w), C(w) from which radiation resistance, power loss, radiation efficiency, radar cross-section, and the quality factor (Q) of any resonance can be calculated. • The input impedance of the circuit model representing the loop can be calculated as a function of wavelength for closed loops and single gap loops. • The introduction of a single gap in the periphery of a loop will cause a very high-Q resonance in the sub-wavelength region. This is due to the zero-order mode inductance of the loop resonating with a combination of the gap capacitance and the closed loop capacitance. The Q is on the order of several thousand. • Gap width and capacitance value of the gap are closely related. However, none of the simple models suggested in the literature, such as the flat-plate capacitance model, generates the correct relationship, at least for gaps in rings.