## Kyle Miller and Faculty Mentor: John Colton, Physics and Astronomy

### Introduction:

Optically Detected Magnetic Resonance is one method of performing Electron Spin Resonance (ESR) on a material. ESR is used to determine the electron spin lifetime of a material, an important parameter for use in quantum computing. Resonant cavities are conducting containers that are frequently used in ESR to create a strong magnetic field near the sample. As such it is valuable to design a resonant cavity and predict its resonant frequency. Cylindrical cavities modified with dielectric resonators (DRs) are viable for such experiments.

Many methods exist to predict the resonant frequency of a modified cylindrical cavity. Some involve simple numerical approaches, such as solving transcendental equations.^{1} Others use computational packages that perform complex calculations.^{2} These techniques are often expensive, time intensive, or require restrictions on the geometry of the cavity. The method presented in this paper seeks to bridge the two types of techniques: we seek to accurately calculate resonant frequencies and associated field patterns for arbitrary cylindrically symmetric geometries in a reasonable amount of time and without the need to purchase commercial products.

### Methodology:

The resonant frequency and corresponding electric and magnetic fields are calculated by eigenfunction expansion of the field patterns using cylindrical empty cavity modes as basis functions. These modes are well known and easily calculated.^{3} Any number of DRs or discontinuities in the dielectric constant can be included in the geometry as long as cylindrical symmetry is preserved. To improve the accuracy of the resonant frequency prediction, the calculation is carried out several times with a geometrically increasing number of eigenfunctions. The convergence of the series is accelerated through Aitken extrapolation to predict the resonant frequency obtained from the eigenvalues. The eigenvectors specify coefficients of the empty cavity modes that sum to compose the electric and magnetic field patterns.

Because the DRs are not magnetically active, the magnetic field is continuous at all points in the cylinder, with a possibly discontinuous first derivative. The electric field is discontinuous at various points, which causes inaccuracies in the solved field pattern. The accuracy can be improved by first smoothing the continuous magnetic field, then differentiating to obtain a smoother electric field by Ampere’s law.

### Results

First, the method was tested on a modified cylindrical cavity for which the resonant frequencies are known analytically, namely an infinite cylindrical dielectric inside an infinite cylindrical conductor. For nearly all the modes tested, the error in the resonant frequency was less than 0.4%. Next, the method was used to predict the resonant frequencies of a physical cavity described in detail by Colton and Wienkes.^{4} The lowest resonant frequency was calculated for 11 different configurations of DRs and compared with the experimental value. The error for all 11 cases was less than 1.0%, which is comparable with other methods.^{1,2} Typical total CPU time for the calculations was about 30 hours.

### Discussion:

Solving the infinite cylinder case revealed a limitation of the method. Two modes found analytically were nearly degenerate. Their frequencies were close in value and the field patterns were very similar. Our computational method predicted only one frequency (located between the other two frequencies), resulting in an error of about 11%. In order to resolve the two modes, a very large number of eigenfuctions must be used, which would dramatically increase the computation time. Nearly degenerate modes may not be easily predicted with this method.

Another limitation was found when calculating the resonant frequencies of configurations with very small DRs. The field patterns for these configurations change rapidly in space, requiring a large number of eigenfunctions to be resolved accurately. With more eigenfunctions, an error of less than 1.0% was still obtained for these frequencies, but the computation time rose to about 90 hours. In all cases, error likely arises from uncertainties in the physical dimensions and dielectric constants, as well as departures of the actual configuration from the cylindrically symmetric model.

### Conclusion:

We have presented a new method for calculating the resonant frequencies and field patterns of cylindrical cavities. The modes of modified resonant cavities are expressed as sums of empty cavity modes through eigenfunction expansion. The method is robust in that the amount and placement of cylindrically symmetric dielectric material may be specified, and computation time is relatively short.

^{1} M. Jaworski, A. Sienkiewicz and C. P. Scholes, “Double-stacked dielectric resonator for sensitive EPR measurements,” *J. Magn. Reson.*, vol. 124, no. 1, pp. 87-96, 1997.

^{2} D. Baillargear, S. Verdeyme, M. Aubourg and P. Guillon, “CAD applying the finite-element method for dielectricresonator filters,” *IEEE Trans. Microw. Theory Techn.*, vol. 46, no. 1, pp. 10-17, 1998.

^{3} J. D. Jackson, Chapter 8, “Waveguides, Resonant Cavities, and Optical Fibers,” in *Classical electrodynamics,* 3 ed., (Wiley, New York, 1962), pp. 352-406.

^{4} J. S. Colton and L. R. Wienkes, “Resonant microwave cavity for 8.5–12 GHz optically detected electron spin resonance with simultaneous nuclear magnetic resonance,” *Rev. Sci. Instrum.*, vol. 80, no. 3, p. 035106, 2009.