Loop-gap resonator


A loop-gap resonator 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 making them useful for electron spin resonance experiments, and precision measurements of electromagnetic material properties.

Background

Loop-gap resonators can be modelled as lumped-element circuits. The slit along the length of the resonator has an effective capacitance and the bore of the resonator has effective inductance. At, or near, the resonance frequency, a circumferential current is established along the inner wall of the resonator. The effective resistance that limits this current is, in part, determined by the resistivity and electromagnetic skin depth of the conductor used to make the LGR. It is, therefore, possible to model the LGR as an circuit. Since the LGR current is a maximum at the resonant frequency, the equivalent circuit model is a series 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 and quality factor of the resonance. Measurements of the changes in and 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 and 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
where is the permittivity of free space, is the thickness of the bore wall, is the gap width, and is the length of the resonator. The resonator bore acts as a single-turn solenoid with inductance given by

where is the permeability of free space and is the inner radius of the LGR bore. For a high- resonator, the resonant frequency is, to an approximation, given by
where is the vacuum speed of light. Therefore, the resonant frequency of the LGR is determined from its geometry and is, to first approximation, independent of its length.

Quality factor

For a highly underdamped series circuit, the quality factor, which determines the sharpness of the resonance, is given by
The effective resistance of a LGR can be estimated by considering the length of conductor through which the current travels and the cross-sectional area available to it. The relevant conductor length is the circumference of the conductor's inner surface. The depth that the current penetrates into the inner surface of the LGR bore is determined by the electromagnetic skin depth. Therefore, the cross-sectional area through which charge flows is. Combining these results gives an effective resistance
where is the resistivity of the conductor. The effective capacitance, inductance, and resistance then lead to a simple expression for the expected quality factor of the LGR
where, for a good conductor, the electromagnetic skin depth at the resonance frequency is given by
and. For an aluminum resonator with and the above analysis predicts.

Radiative losses

In practice, the measured quality factor a cylindrical LGR, without additional electromagnetic shielding, will be much less than the predicted value of. The suppression of the quality factor is due to radiative power loss from magnetic field lines that extend out of LGR bore and into free space. An order-of-magnitude estimate of the effective radiation resistance can be made by treating the LGR as a conducting loop. In the limit that the wavelength of the radiation is much larger than the loop radius, the radiation resistance is
and can be much larger than the resistance due to the resistivity of the LGR conductor.
The radiative losses can be suppressed by placing the LGR inside a circular waveguide. Provided that the cutoff frequency of the lowest TE11 waveguide mode is well above the resonant frequency of the LGR, the magnetic field lines will be prevented from propagating into free space. The presence of the electromagnetic shield will alter the resonant frequency and quality factor of the LGR, but typically by only a few percent.

Toroidal LGR

In some applications requiring high quality factors, the electromagnetic shielding provided by a concentric circular waveguide surrounding a cylindrical LGR can be bulky and awkward to work. A toroidal LGR can be used for high- measurements without requiring additional electromagnetic shielding. In the toroidal geometry the two ends of a cylindrical LGR are joined to form a completely closed structure. In this case, the magnetic field is completely confined within the bore of the resonator and there is no radiative power loss. The toroidal LGR consists of two halves that are bolted together along the outer diameter of the structure.
Like the cylindrical LGR, the toroidal LGR can be modelled as a series circuit. In general, the effective capacitance, inductance, and resistance of the toroidal LGR will differ from the expressions given above for the cylindrical LGR. However, in limit that the radius of the torus is large compared to the bore radius, the capacitance, inductance, and resistance of the toroidal LGR are approximated by the expressions above if one takes to be equal to the circumference of the torus.
The toroidal LGR is particularly convenient when characterizing the electromagnetic properties of liquid samples or particles suspended in a liquid. In these cases, the bore of the toroidal LGR can be partially filled with the liquid sample without requiring a special sample holder. This setup allows one to characterize the magnetic properties of, for example, a ferrofluid. Alternatively, if the liquid sample is nonmagnetic, the entire toroidal LGR can be submerged in the liquid. In this case, the dielectric properties of the sample only modify the effective capacitance of the resonator and the changes in and can be used to determine the complex permittivity of the sample.

Coupling to a LGR

loops are typically used to couple magnetic flux into and out of the LGR. The coupling loops are made by first removing a length of outer conductor and dielectric from a semi-rigid coaxial cable. The exposed centre conductor is then bent into a loop and short-circuited to the outer conductor. The opposite end of the coaxial cable is connected to either a signal generator or a receiver. In the case of a signal generator, an oscillating current is established in the coupling loop. By Faraday's law of induction, this current creates and oscillating magnetic flux which can be coupled into the bore of the LGR. This magnetic flux, in turn, induces circumferential currents along the inner wall of the LGR. The induced current, once again by Faraday's law, creates an approximately uniform oscillating magnetic field in the bore of the LGR. A second coupling loop, connected to a receiver, can be used to detect the magnetic flux produced by the LGR. Alternatively, using a vector network analyzer, a single coupling loop can be used to both inject a signal into the LGR and measure its response. The VNA can measure the ratio of the forward and reflected voltages as a function of microwave frequency. Far away from resonance, the magnitude of the reflection coefficient will be close to one since very little power is coupled into the LGR at these frequencies. However, near the resonance frequency, the magnitude of the reflection coefficient will fall below one as power is transferred into the LGR. The coupling between the external circuits and the LGR can be tuned by adjusting the relative positions and orientations of the coupling loop and LGR. At critical coupling, impedance matching is achieved and the reflection coefficient approaches zero.
It is also possible to capacitively couple electric fields into and out of the gap of the LGR using suitably-fashioned electrodes at the end of a coaxial cable.

Multi-Loop, Multi-Gap LGRs

Multi-loop, multi-gap LGRs have also been developed. The simplest of these is the two-loop, one-gap LGR. In this case, magnetic field lines form closed loops by passing through each of the bores of the LGR and the currents on the inner walls propagate in opposite directions - clockwise in one bore and counterclockwise in the other. The equivalent circuit, neglecting losses, is a parallel combination of inductors and in series with capacitance. If, then the resonant frequency of the two-loop, one-gap LGR is times greater than that of the conventional one-loop, one-gap LGR having the same bore and gap dimensions. It is also worth noting that, since magnetic field lines pass from one bore to the other, radiative power losses are strongly suppressed and the resonator maintains a high quality factor without requiring additional electromagnetic shielding.
The multi-loop, multi-gap LGRs with more than two loops have more than one resonant mode. If the central bore is singled out as having inductance, then one of the resonant modes is one in which all of the magnetic flux from each of the external loops of inductance is shared with the central loop. For this mode, the resonant frequency of an -loop, -gap LGR is given by
where it has been assumed that all loops have the same inductance.