Sameh Y. Elnaggar

Coupled modes, frequencies and fields of a dielectric resonator and a cavity using coupled mode theory

Probes consisting of a dielectric resonator (DR) inserted in a cavity are important integral components of electron paramagnetic resonance (EPR) spectrometers because of their high signal-to-noise ratio. This article studies the behavior of this system, based on the coupling between its dielectric and cavity modes. Coupled-mode theory (CMT) is used to determine the frequencies and electromagnetic fields of this coupled system. General expressions for the frequencies and field distributions are derived for both the resulting symmetric and anti-symmetric modes. These expressions are applicable to a wide range of frequencies (from MHz to THz). The coupling of cavities and DRs of various sizes and their resonant frequencies are studied in detail. Since the DR is situated within the cavity then the coupling between them is strong. In some cases the coupling coefficient, κ, is found to be as high as 0.4 even though the frequency difference between the uncoupled modes is large. This is directly attributed to the strong overlap between the fields of the uncoupled DR and cavity modes. In most cases, this improves the signal to noise ratio of the spectrometer. When the DR and the cavity have the same frequency, the coupled electromagnetic fields are found to contain equal contributions from the fields of the two uncoupled modes. This situation is ideal for the excitation of the probe through an iris on the cavity wall. To verify and validate the results, finite element simulations are carried out. This is achieved by simulating the coupling between a cylindrical cavitys TE011 and the dielectric inserts TE01δ modes. Coupling between the modes of higher order is also investigated and discussed. Based on CMT, closed form expressions for the fields of the coupled system are proposed. These expressions are crucial in the analysis of the probes performance.

Analysis of two stacked cylindrical dielectric resonators in a TE102 microwave cavity for magnetic resonance spectroscopy

The frequency, field distributions and filling factors of a DR/TE₁₀₂ probe, consisting of two cylindrical dielectric resonators (DR1 and DR2) in a rectangular TE₁₀₂ cavity, are simulated and analyzed by finite element methods. The TE(+++) mode formed by the in-phase coupling of the TE₀₁(δ)(DR1), TE₀₁(δ)(DR2) and TE₁₀₂ basic modes, is the most appropriate mode for X-band EPR experiments. The corresponding simulated B(+++) fields of the TE(+++) mode have significant amplitudes at DR1, DR2 and the cavitys iris resulting in efficient coupling between the DR/TE₁₀₂ probe and the microwave bridge. At the experimental configuration, B(+++) in the vicinity of DR2 is much larger than that around DR1 indicating that DR1 mainly acts as a frequency tuner. In contrast to a simple microwave shield, the resonant cavity is an essential component of the probe that affects its frequency. The two dielectric resonators are always coupled and this is enhanced by the cavity. When DR1 and DR2 are close to the cavity walls, the TE(+++) frequency and B(+++) distribution are very similar to that of the empty TE₁₀₂ cavity. When all the experimental details are taken into account, the agreement between the experimental and simulated TE(+++) frequencies is excellent. This confirms that the resonating mode of the spectrometers DR/TE₁₀₂ probe is the TE(+++) mode. Additional proof is obtained from B₁(x), which is the calculated maximum x component of B(+++). It is predominantly due to DR2 and is approximately 4.4 G. The B₁(x) maximum value of the DR/TE₁₀₂ probe is found to be slightly larger than that for a single resonator in a cavity because DR1 further concentrates the cavitys magnetic field along its x axis. Even though DR1 slightly enhances the performance of the DR/TE₁₀₂ probe its main benefit is to act as a frequency tuner. A waveguide iris can be used to over-couple the DR/TE₁₀₂ probe and lower its Q to ≈150. Under these conditions, the probe has a short dead time and a large bandwidth. The DR/TE₁₀₂ probes calculated conversion factor is approximately three times that of a regular cavity making it a good candidate for pulsed EPR experiments.

Optimal dielectric and cavity configurations for improving the efficiency of electron paramagnetic resonance probes.

An electron paramagnetic resonance (EPR) spectrometers lambda efficiency parameter (Λ) is one of the most important parameters that govern its sensitivity. It is studied for an EPR probe consisting of a dielectric resonator (DR) in a cavity (CV). Expressions for Λ are derived in terms of the probes individual DR and CV components, Λ1 and Λ2 respectively. Two important cases are considered. In the first, a probe consisting of a CV is improved by incorporating a DR. The sensitivity enhancement depends on the relative rather than the absolute values of the individual components. This renders the analysis general. The optimal configuration occurs when the CV and DR modes are nearly degenerate. This configuration guarantees that the probe can be easily coupled to the microwave bridge while maintaining a large Λ. It is shown that for a lossy CV with a small quality factor Q2, one chooses a DR that has the highest filling factor, η1, regardless of its Λ1 and Q1. On the other hand, if the CV has a large Q2, the optimum DR is the one which has the highest Λ1. This is regardless of its η1 and relative dielectric constant, ɛr. When the quality factors of both the CV and DR are comparable, the lambda efficiency is reduced by a factor of 2. Thus the signal intensity for an unsaturated sample is cut in half. The second case is the design of an optimum shield to house a DR. Besides preventing radiation leakage, it is shown that for a high loss DR, the shield can actually boost Λ above the DR value. This can also be very helpful for relatively low efficiency dielectrics as well as lossy samples, such as polar liquids.

General expressions for the coupling coefficient, quality and filling factors for a cavity with an insert using energy coupled mode theory

A cavity (CV) with a dielectric resonator (DR) insert forms an excellent probe for the use in electron paramagnetic resonance (EPR) spectrometers. The probes coupling coefficient, κ, the quality factor, Q, and the filling factor, η are vital in assessing the EPR spectrometers performance. Coupled mode theory (CMT) is used to derive general expressions for these parameters. For large permittivity the dominating factor in κ is the ratio of the DR and CV cross sectional areas rather than the dielectric constant. Thus in some cases, resonators with low dielectric constant can couple much stronger with the cavity than do resonators with a high dielectric constant. When the DR and CV frequencies are degenerate, the coupled η is the average of the two uncoupled ones. In practical EPR probes the coupled η is approximately half of that of the DR. The Q of the coupled system generally depends on the eigenvectors, uncoupled frequencies (ω1,ω2) and the individual quality factors (Q1,Q2). It is calculated for different probe configurations and found to agree with the corresponding HFSS® simulations. Provided there is a large difference between the Q1, Q2 pair and the frequencies of DR and CV are degenerate, Q is approximately equal to double the minimum of Q1 and Q2. In general, the signal enhancement ratio, Iwithinsert/Iempty, is obtained from Q and η. For low loss DRs it only depends on η1/η2. However, when the DR has a low Q, the uncoupled Qs are also needed. In EPR spectroscopy it is desirable to excite only a single mode. The separation between the modes, Φ, is calculated as a function of κ and Q. It is found to be significantly greater than five times the average bandwidth. Thus for practical probes, it is possible to excite one of the coupled modes without exciting the other. The CMT expressions derived in this article are quite general and are in excellent agreement with the lumped circuit approach and finite numerical simulations. Hence they can also be applied to a loop-gap resonator in a cavity. For the design effective EPR probes, one needs to consider the κ, Q and η parameters.

An electromagnetic induced transparency-like scheme for wireless power transfer using dielectric resonators

Similar to the hybridization of three atoms, three coupled resonators interact to form bonding, anti-bonding, and non-bonding modes. The non-bonding mode enables an electromagnetic induced transparency like transfer of energy. Here, the non-bonding mode, resulting from the strong electric coupling of two dielectric resonators and an enclosure, is exploited to show that it is feasible to transfer power over a distance comparable to the operating wavelength. In this scheme, the enclosure acts as a mediator. The strong coupling permits the excitation of the non-bonding mode with high purity. This approach is different from resonant inductive coupling, which works in the sub-wavelength regime. Optimal loads and the corresponding maximum efficiency are determined using two independent methods: Coupled Mode Theory and Circuit modelling. It is shown that, unlike resonant inductive coupling, the figure of merit depends on the enclosure quality and not on the load, which emphasizes the role of the enclosure as a m...

Stability analysis of nonlinear left handed transmission lines using Floquet multipliers and bifurcation theory

Nonlinear dynamics and control theory are applied to characterise the stability of Nonlinear Composite Right Left Handed Transmission Lines. The parametric generation process is linked to the instability of a limit cycle attractor. Based on the values of the Floquet multipliers, different bifurcation mechanisms can be identified. The theoretical analysis is confirmed using numerical simulations. The analysis and results demonstrate that nonlinear dynamics and phase space portraits provide qualitative and quantitative answers to questions regarding the performance of NL CRLH TLs.

Experimental and numerical analysis of nonlinear left handed transmission lines using three wave mixing

In this paper, we demonstrate the use of three wave mixing and quasi phase matching to describe the parametric behaviour of a Nonlinear Composite Right-Left Handed Transmission Line. The varactors configuration plays an essential role in determining the parametric frequencies. The findings are verified using simulation and measurement.

Coupled Mode Theory Applied to Resonators in the Presence of Conductors

Using the method of images, Energy Coupled Mode Theory (ECMT), a coupled mode equation in the frequency domain, is extended to deal with important cases where resonators are in close proximity to conducting surfaces. Depending on the type of conductors and the orientation of the resonators, the method of images determines the relative phases of the images. Using the formed images, the coupled frequencies and fields can be determined by applying ECMT. Two cases are studied. In the first case, it is shown that a dielectric resonator inserted in a cavity couples with both the mirror image and the cavity. The frequency behavior is described by the interaction with the image which counteracts that with the cavity. The second case is that of resonators sandwiched between conducting plates. It is shown that an infinite array of stacked images is formed. The coupling of the resonator with its images determines the coupled frequencies and fields. In this context, the main advantage of ECMT is its ability to separate the effects of the walls from the uncoupled system. This means that the system parameters are independent of the separation distances and/or the type of conductors, which renders the post processing analyses easier and predictable. Provided that the behaviors of the uncoupled resonators are known, ECMT is general and can be applied to more complex systems.

Description and stability analysis of nonlinear transmission line type metamaterials using nonlinear dynamics theory

Nonlinear metamaterials offer a potential technology to realize applications at microwave, terahertz, and optical frequencies. However, due to the strong and controlled nonlinearity, the wave interactions can be quite complex. In the current article, a framework based on nonlinear dynamics theory is developed to describe such complex interactions. This is demonstrated for the case of a harmonically pumped nonlinear left handed transmission line through the use of bifurcation theory, stability analysis, and linearization about the limit cycle to calculate the autonomously generated frequencies and their spatial distributions. Higher order parametric interactions, which can be mediated by the strong nonlinearity, are automatically included in the model. It is demonstrated that autonomous components can be visualized in both the phase and the set of solution spaces. The framework is general in terms of the transmission line configuration, the nature and strength of the nonlinearity, and the number of stages....

Forced response of an arbitrary number of electromagnetic coupled resonators

Coupled resonators appear as the building blocks of many systems and devices. Recently, Energy Coupled Mode Theory (ECMT), a general coupled mode formalism in an eigen-value problem form, was introduced to estimate the frequencies and fields of an arbitrary number of coupled resonators. In the current article, impressed sources are introduced to generalize ECMT to cover cases where arbitrary input excitations are applied. The system is represented by a matrix transfer function in the complex domain. To demonstrate how the transfer function can be applied, the efficiency of a classical system of two inductively coupled resonators is calculated under different load values and input frequencies.