Mett and Hyde [1
] have studied the influence of round and slotted irises on microwave leakage from magnetic field modulation slots cut perpendicular to the axis of TE011
cavities at Q-band. They found that a thin slotted iris has significantly less leakage than a round iris because the slotted iris makes a smaller perturbation on the TE011
mode than the round iris. A similar observation was made using the finite element computer program Ansoft High Frequency Structure Simulator (HFSS) (Version 10.1, Ansoft Corporation, Pittsburgh, PA) in loop-gap resonators (LGRs) of significant length. The authors were surprised at the level of mode perturbation, even at the sample, caused by round and slotted irises in a 10-mm-long 3-loop-2-gap resonator at Q-band (), despite the expected shielding effects of the gaps. The ratio of this LGR length to free space wavelength is 114% and other dimensions are given in . Radio frequency magnetic field energy density uniformity was diminished by 8-15% compared to eigenmode by these couplers, . Uniform magnetic field at the sample is required to achieve uniform spin saturation. In an attempt to lower the level of mode perturbation caused by the iris, narrower and longer irises were tried. It was found possible to extend the iris length across the full WR-28 waveguide width as shown in with iris dimensions given in . This long iris was observed to have many properties opposite or dual to those of a conventional slotted iris of length less than half the long waveguide dimension and those of a round iris. These properties include the phases of the rf fields and currents in the resonator relative to those near the iris, the stored energy type in the iris, the frequency at match relative to the resonator natural resonance frequency (frequency shift), and the sign of the iris reactance (inductive vs. capacitive). In addition, the rf magnetic energy density uniformity at the sample was improved by the long iris compared to the eigenmode solution, . This paper presents an analysis and circuit model of iris coupling of a waveguide to an LGR. Although the analysis is done specifically for an LGR, treatment of a cavity resonator follows with little modification. The model was developed in close conjunction with Ansoft HFSS.
Mechanical drawing of bisected 10-mm-long Q-band 3-loop-2-gap LGR. Resonator body is shown in gray and the sample tube in blue. Gaps face bisecting plane. Coupling iris slot appears on nearest edge.
LGR properties and sample dimensions (mm).
Fig. 2 Radio frequency magnetic field energy density profile at sample center and LGR axis as predicted by Ansoft HFSS. Solid line indicates eigenmode (no iris) solution. Short dashes indicate conventional iris and long dashes long iris. a Iris and LGR dimensions (more ...)
Coupled LGR-iris properties.
The LGR was introduced for use in EPR spectroscopy in the simplest possible cross-sectional geometry, , [2
] and later extended to numerous other cross-sections including those shown in . The literature has been reviewed by Hyde and Froncisz [3
] and by Rinard and Eaton [4
]. Iris coupling between a waveguide and a 3-loop-2-gap LGR of 1-mm length has been done at Q-band, but there is no rationale given for the design [5
LGR cross-sections. a 1-loop-1-gap, no return flux loop. b 1-loop-1-gap. Sample is placed in smaller loop and larger loop is flux return path. c 3-loop-2-gap. d 5-loop-4-gap.
Iris coupling between waveguide and cavity is typically modeled by a mutual inductance M
between the iris and the cavity inductance [1
]. After extensive investigation using Ansoft HFSS, models with mutual inductance were found to be inadequate to explain the iris coupling behavior between waveguide and 3-loop-2-gap LGR for irises of different sizes at Q-band. A simple mutual inductance model following ref. 1
was found to mimic some of the observations, including phases of rf currents and the input impedance, but not others such as frequency shift. The addition of the distributed nature of the iris into the model, Sect. 2.2, including mutual inductance between iris and LGR outer loop, was found to increase disparities between the model and HFSS observations, and a rationale for choosing the sign and value of M
In the present work, the metallic wall high frequency rf boundary condition, which relates surface current to magnetic field [8
], was used to define surface currents, distinct regions, and circuit topology of the coupled iris-LGR with lumped-circuit values of capacitance, resistance, and self-inductance with no mutual inductance. That it is possible to obtain a complete circuit model of the coupled iris-LGR without mutual inductance is perhaps surprising. However, we show in Appendix A
that the equivalent self-inductance of a coil in the presence of a metallic rf shield is equivalently expressible in terms of mutual inductance or self-inductance. The effect of the mutual inductance of the shield on the coil in the high frequency rf limit is to induce an rf current on the outside surface of the coil. The separation of the total current on the coil into an inner current and an outer current leads to the ability to treat the coil as two self-inductances in parallel with no mutual inductance between them. The mutual and self inductance models differ in the definition of the coil currents. The results are consistent with a statement by Grover [10
]: “Self-inductance is merely a special case of mutual inductance.”
In the present work, the circuit model geometry reflects flux conservation between iris and LGR, includes capacitive and inductive circuit coupling, and accounts for the influence of the iris on the LGR outer loop currents. In this model, the geometrical dimensions of the LGR, iris, and waveguide are used to calculate circuit values of self-inductance, capacitance, and resistance, including the effects of sample. These circuit values then determine the solution to the circuit equations and predict the input impedance, rf currents, frequency shift, and magnetic and electric stored energies. The circuit is a pi network with the bridge element an inductance formed by the part of the LGR outer loop cut by the iris. The circuit can be cast into an equivalent form having an effective
mutual coupling between two resonant circuits described by Terman [9
]. The effective mutual coupling represents combined capacitive and self-inductive (complex) coupling between primary (waveguide/iris) and secondary (LGR), although with no mutual inductance.
Two conditions must be met to achieve critical coupling (match). First, the equivalent resistance of the LGR as seen by the iris must be transformed into the waveguide characteristic impedance. Due to the behavior of the LGR resistance with frequency, this transformation typically occurs at a particular frequency relative to the natural LGR resonance frequency. This frequency shift magnitude is strongly dependent on the LGR and sample dimensions, weakly dependent on the iris length and placement, and independent of other iris dimensions. The second condition for match is that the iris reactance at this frequency shift must cancel the residual reactance of the LGR. This second condition is sensitive to the iris dimensions. If both conditions are not simultaneously satisfied, overcoupling or undercoupling results. The first match condition completely determines the frequency shift magnitude, and so LGR and iris design can reduce or eliminate this frequency shift, or tailor it for example to the needs of the EPR spectroscopist, who prefers no difference between the frequencies at match with and without sample.