Recent advances in microresonators and supporting instrumentation for electron paramagnetic resonance spectroscopy

Inductively detected EPR spectroscopy is the most common and broadly applicable form of signal detection for EPR spectroscopy. At resonance, the sample magnetization changes, affecting the electrical characteristics of the resonator. This change is ultimately recorded as a voltage, which is the signal in inductively detected EPR spectroscopy.^28,56,57 28. C. P. Poole, Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques (Dover Publications, Mineola, NY, 1996).56. G. Feher, "Sensitivity considerations in microwave paramagnetic resonance absorption techniques," Bell Syst. Tech. J. 36(2), 449-484(1957). https://doi.org/10.1002/j.1538-7305.1957.tb02406.x57. D. I. Hoult and R. E. Richards, "The signal-to-noise ratio of the nuclear magnetic resonance experiment," J. Magn. Reson. 24(1), 71-85(1976). https://doi.org/10.1016/0022-2364(76)90233-xThe sample signal is only detectable if the voltage produced is greater than the noise floor. The signal voltage in the linear detection regime of a reflecting, critically coupled resonator is given bywhere V_S is the signal voltage, χ″ is the magnetic susceptibility of the sample, η is the fill factor [Eq. (3)], Q is the quality factor [Eqs. (3) and (4)], P is the incident microwave power, and Z₀ is the impedance of the transmission line to which the resonator is attached.

The spatial confinement of B₁ can be visualized through the mode volume. Cavity perturbation theory defines mode volume of an electromagnetic cavity resonator in terms of the effect produced by a tiny dipole on the resonator characteristics when placed at the position (r⃗max ) of maximum B₁.⁵⁸ 58. K. G. Cognée, W. Yan, F. La China, D. Balestri, F. Intonti, M. Gurioli, A. F. Koenderink, and P. Lalanne, "Mapping complex mode volumes with cavity perturbation theory," Optica 6(3), 269-273(2019). https://doi.org/10.1364/optica.6.000269For the case of the magnetic interaction between a tiny magnetic dipole and the oscillating magnetic field in a resonator, the mode volume V_C is given by the following equation:

VC(r⃗max)=∫resonatorB12dV|B1(r⃗max)|2.

(2)

Mode volume is inversely proportional to |B1(r⃗max)|2 , which means B₁ intensity increases with its spatial confinement. For open resonant structures with inhomogeneous B₁ distributions, mode volume is difficult to estimate, but the above explanation attempts to illustrate the concept and encourage interested readers to peruse the indicated references.

The fill factor η [Eq. (3)] can be represented as the ratio of the sample volume to the resonator active volume, where the active volume refers to the volume over which B₁ is confined within the resonator. η is related to the mode volume because placement of a point-like sample in a device with a smaller mode volume produces a larger fill factor. Therefore, the mode volume may also be defined as the effective volume, which is the volume of a point-like sample divided by fill factor of this sample placed at the location of highest B₁ in the resonator.⁵⁹ 59. A. Blank, C. R. Dunnam, P. P. Borbat, and J. H. Freed, "High resolution electron spin resonance microscopy," J. Magn. Reson. 165(1), 116-127(2003). https://doi.org/10.1016/s1090-7807(03)00254-4When the active volume is comparable to the available sample volume, the fill factor approaches 1, which is one of the optimal conditions for maximum signal-to-noise ratio (SNR), assuming that the microwave electric field E₁ is spatially well-separated from the sample. Equation (2) shows that the fill factor is directly related to the volume integral of B1x2+B1z2 over the sample volume, which is proportional to the signal generated by the sample. Here, we assume that the static field is applied along y and EPR spectroscopy is conducted in the conventional perpendicular mode, i.e., only the x and z components of B₁ contribute to the EPR signal. For a cavity resonator, η is

η=∫sample(B1x2+B1z2)dV∫resonator(B1x2+B1z2)dV.

(3)

The quality factor Q is a measure of how efficiently a resonator stores microwave energy. It is the ratio of the average energy stored per cycle in the resonator volume to the average energy dissipated through radiative, dielectric, and conductive losses [Eq. (4)] per cycle.²⁸ 28. C. P. Poole, Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques (Dover Publications, Mineola, NY, 1996).It also determines the sensitivity with which a generated signal can be detected because the detection sensitivity is related to the ability to distinguish the power reflected from the resonator in the presence vs absence of resonance,

Q=2πenergystoredenergydissipatedpercycle.

(4)

Because it is difficult to directly measure the energy stored and dissipated in a circuit, Q also is defined [Eq. (4)] in terms of measurable quantities, namely, resonator frequency (ν_res) and the resonator bandwidth (Δν),The resonator bandwidth is the full-width at half-maximum (FWHM, or 3 dB linewidth) of the frequency profile of the reflected microwave power (S₁₁). The S₁₁ return-loss measurement is carried out using a vector network analyzer (VNA), and the minimum in the trace (a dip) is the resonance frequency, ν_res. Optimal coupling of the resonator to the incident microwaves in most reflection-mode EPR spectrometers is signaled by generation of the narrowest, deepest minimum in the S₁₁ trace during the tuning (coupling) procedure. Because the signal intensity is proportional to Q [Eq. (1)], samples that degrade Q will result in lower sensitivity. Table I provides typical values of η and Q for the conventional resonators described in Sec. I D.

Q
and η (or mode volume) further determine the following additional properties that are important to consider:

•	Conversion factor (CP): The conversion factor or conversion efficiency is the field produced per square root unit of power incident on the sample. For a cavity critically coupled to a waveguide,28 28. C. P. Poole, Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques (Dover Publications, Mineola, NY, 1996).CP may be represented in terms of fill factor, as given by the following equation:where P is the incident power, QL is the loaded quality factor, and A is the ratio of the volume of one length of waveguide to sample volume. CP can also be stated in terms of mode volume as in the following equation:where μ0 is the magnetic permeability and ωmv is the microwave frequency. Experimentally, conversion factor (or power-to-field conversion efficiency) c is defined as c=B1/QP , where P is the power output of the microwave source and B1 is the field generated in the active volume of the resonator, and the factor 1Q is introduced for normalization to the unloaded Q-factor. (Often, the conversion factor is not normalized to Q. Instead, it is reported as c′=B1/P , which is an experimental quantity specific for the reported resonator design and measurement setup.) A high conversion factor means that even a small output power from the bridge can generate a large B1 in the active volume. As shown by Eq. (6), the conversion factor is determined principally by the fill factor (mode volume) and quality factor (resonator losses). In general, smaller mode volumes and lower losses result in higher conversion factors with the dominant determinant of the conversion factor being mode volume. For example, smaller wavelengths in dielectric media result in correspondingly smaller mode volumes. This scaling is used to increase resonator conversion factors by dielectric loading.61-63 61. A. G. Webb, "Dielectric materials in magnetic resonance," Concepts Magn. Reson. 38A(4), 148-184(2011). https://doi.org/10.1002/cmr.a.2021962. R. R. Mett, J. W. Sidabras, I. S. Golovina, and J. S. Hyde, "Dielectric microwave resonators in TE011 cavities for electron paramagnetic resonance spectroscopy," Rev. Sci. Instrum. 79(9), 094702 (2008). https://doi.org/10.1063/1.297603363. J. S. Hyde and R. R. Mett, "EPR uniform field signal enhancement by dielectric tubes in cavities," Appl. Magn. Reson. 48(11-12), 1185-1204(2017). https://doi.org/10.1007/s00723-017-0935-4Section II A 1 explains how the intrinsic loss of Q in microresonators places an asymptotic limit on the scaling of conversion efficiency with mode-volume confinement.
•	Bandwidth: For pulse EPR measurements, the resonator bandwidth determines the fraction of an EPR spectrum that can be simultaneously excited by a microwave pulse. Bandwidth is directly related to Q in that low-Q resonators provide higher bandwidths. For pulse EPR using resonators with Q > 1000, the resonator must be over-coupled to the microwave feed line to decrease Q and increase bandwidth. Microresonators display intrinsically low Q-factors combined with high conversion factors (see below), which permits pulse EPR experiments at critical coupling.
•	B1 homogeneity: A homogeneous B1 distribution over the sample volume is an important factor for quantitative EPR spectroscopy.
•	Spatial separation of B1 and E1: Aqueous solutions and other dielectric lossy samples or even sample holders may interact with the electric-field component E1 of the incident microwaves. These lossy dielectric interactions deteriorate Q and shift the resonant frequency. The reduction in Q results in a loss of sensitivity. To avoid dielectric losses, resonators are typically designed to separate B1 and E1 maxima spatially, and the extent to which such spatial separation can be achieved directly affects resonator performance.