Loop-gap resonator

A loop-gap resonator (LGR) is an electromagnetic resonator that operates in the radio and microwave frequency ranges. The simplest LGRs are made from a conducting tube with a narrow slit cut along its length. The LGR dimensions are typically much smaller than the free-space wavelength of the electromagnetic fields at the resonant frequency. Therefore, relatively compact LGRs can be designed to operate at frequencies that are too low to be accessed using, for example, cavity resonators. These structures can have very sharp resonances (high quality factors) making them useful for electron spin resonance (ESR) experiments, and precision measurements of electromagnetic material properties (permittivity and permeability).

Background Loop-gap resonators (LGRs) can be modelled as lumped-element circuits. The slit along the length of the resonator has an effective capacitance C {\displaystyle C} and the bore of the resonator has effective inductance L {\displaystyle L} . At, or near, the resonance frequency, a circumferential current is established along the inner wall of the resonator. The effective resistance R {\displaystyle R} that limits this current is, in part, determined by the resistivity ρ {\displaystyle \rho } and electromagnetic skin depth δ {\displaystyle \delta } of the conductor used to make the LGR. It is, therefore, possible to model the LGR as an L R C {\displaystyle LRC} circuit. Since the LGR current is a maximum at the resonant frequency, the equivalent circuit model is a series L R C {\displaystyle LRC} circuit. This circuit model works well provided the dimensions of the resonator remain small compared to the free-space wavelength of the electromagnetic fields. One advantage of the LGR is that it produces regions of uniform electric and magnetic fields that are isolated from one another. A uniform electric field exists within the slit of the LGR and a uniform magnetic field exists within the bore of the resonator. The uniform magnetic field makes the LGR a good source of microwave magnetic fields in ESR experiments. Furthermore, because the electric and magnetic fields are isolated from one another, one can use the LGR to independently probe the electric and magnetic properties of materials. For example, if the gap of the LGR is filled with a dielectric material, the effective capacitance of the LGR will be modified which will change the frequency f 0 {\displaystyle f_{0}} and quality factor Q {\displaystyle Q} of the resonance. Measurements of the changes in f 0 {\displaystyle f_{0}} and Q {\displaystyle Q} can be used to fully determine the complex permittivity of the dielectric material. Likewise, if the bore of the LGR is filled with a magnetic material, the effective inductance of the LGR will be modified and the resulting changes in f 0 {\displaystyle f_{0}} and Q {\displaystyle Q} can be used to extract the complex permeability of the magnetic material.

Resonant Frequency and Quality Factor

Resonance frequency The capacitance of the gap of the LGR is given by

C = ε 0 w ℓ t , {\displaystyle C=\varepsilon _{0}{\frac {w\,\ell }{t}}\,,}

where ε 0 {\displaystyle \varepsilon _{0}} is the permittivity of free space, w {\displaystyle w} is the thickness of the bore wall, t {\displaystyle t} is the gap width, and ℓ {\displaystyle \ell } is the length of the resonator. The resonator bore acts as a single-turn solenoid with inductance given by

L = μ 0 π r 0 2 ℓ , {\displaystyle L=\mu _{0}{\frac {\pi \,r_{0}^{2}}{\ell }}\,,}

where μ 0 {\displaystyle \mu _{0}} is the permeability of free space and r 0 {\displaystyle r_{0}} is the inner radius of the LGR bore. For a high- Q {\displaystyle Q} resonator, the resonant frequency is, to an approximation, given by

f 0 ≈ 1 2 π 1 L C = c 2 π r 0 t π w , {\displaystyle f_{0}\approx {\frac {1}{2\pi }}{\frac {1}{\sqrt {LC}}}={\frac {c}{2\pi r_{0}}}{\sqrt {\frac {t}{\pi w}}}\,,}

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