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on top rho J subscript 1 superscript 2 left parenthesis x subscript 01 rho over r subscript text d end text end subscript right parenthesis d rho end cell row cell P subscript text loss end text end subscript superscript text sample end text end superscript equals left parenthesis H subscript 0 superscript text disk end text end superscript right parenthesis squared tau squared P subscript text loss end text comma text norm end text end subscript superscript text sample end text end superscript space end cell row cell text with end text space space P subscript text loss end text comma text norm end text end subscript superscript text sample end text end superscript equals space 2 pi left parenthesis fraction numerator omega mu subscript 0 over denominator 2 k subscript rho end fraction right parenthesis squared sigma subscript text sample end text end subscript left parenthesis 1 plus fraction numerator sin k subscript y L over denominator k subscript y L end fraction right parenthesis L blank with 0 below and r subscript text h end text end subscript on top rho J subscript 1 superscript 2 left parenthesis x subscript 01 rho over r subscript text d end text end subscript right parenthesis d rho space end cell end table"/> (2.8)

The so-called Lommel’s integrals, involving Bessel functions, can be found in [22,23].

2.3.3 SNR Estimation

The SNR is proportional to the transmit efficiency as follows [24]:

$text SNR end text tilde text end text fraction numerator H subscript 0 over denominator square root of P subscript text loss end text end subscript end root end fraction comma$ (2.9)

with H₀ the magnetic field amplitude induced in the sample by the probe and P_loss the total power losses. For ceramic probes working with the TE_01δ mode, the latter includes the ceramic resonator and the sample contributions. Assuming the magnetic field amplitude equal to that of a disk resonator in its center, weighted by the penalty coefficient τ, Equation 2.9 becomes:

$text SNR end text subscript text ceramic end text end subscript superscript text TE end text subscript 01 text δd end text end subscript end superscript ∝? fraction numerator τ tau H subscript 0 superscript text disk end text end superscript over denominator square root of P subscript text loss end text end subscript superscript text ring end text end superscript plus P subscript text loss end text end subscript superscript text sample end text end superscript end root end fraction.$

The above equation can be simplified to:

$text SNR end text subscript text ceramic end text end subscript superscript text TE end text subscript 01 text δd end text end subscript end superscript ∝? fraction numerator τ tau over denominator square root of P subscript text loss end text comma text norm end text end subscript superscript text ring end text end superscript plus τ tau squared P subscript text loss end text comma text norm end text end subscript superscript text sample end text end superscript end root end fraction.$ (2.10)

2.3.4 Mode Frequency

To be used as a coil for MR imaging, the exploited mode of the resonator must be adjusted to the working frequency. In this framework estimating the mode frequency with a simple method is very useful for predesigning the ceramic probe. Methods with low computational costs provide an approximated value of the TE_01δ mode frequency for disk resonators. For example, a commonly used expression, derived from simulation data, is the following:

$f subscript text MHz end text end subscript superscript text TE end text subscript 01 text δd end text end subscript end superscript equals fraction numerator 3400 over denominator square root of ϵ ? subscript straight r end root r subscript straight d comma text cm end text end subscript end fraction left parenthesis r subscript straight d over L plus 3.45 right parenthesis.$ (2.11)

The expression in Equation 2.11 is valid with an accuracy around 2% or below in the following domain: [1]. A diversity of alternative simplified methods can be found in the literature, some extrapolated from experimental data [25,26], some using rough approximations about the boundary conditions at the dielectric interfaces [27] or implying the numerical solving of a transcendental equation [17], and others combining the above-mentioned approaches [21].

These estimation methods are valid for disk resonators, while the dielectric probe is described as a high-permittivity (ϵ_d = ϵ_rϵ₀) ring filled with a lower permittivity (ϵ_d = ϵ_r,sampϵ₀) sample. In the specific case of the TE_01δ mode, equating the field distribution of the probe and the disk resonators is a reasonable approximation. Indeed, between the disk and the probe, the TE_01δ mode features are affected in proportion to the electric and magnetic energies stored in the sample volume V_samp [28]. The mode frequency varies from the value f_disk for the disk to f_probe for the probe according to:

$fraction numerator f subscript text probe end text end subscript minus f subscript text disk end text end subscript over denominator f subscript text disk end text end subscript end fraction almost equal to fraction numerator negative begin display style triple integral subscript V subscript text samp end text end subscript end subscript left parenthesis ? subscript straight s minus ? subscript straight d right parenthesis vertical line E subscript text disk end text end subscript vertical line squared d v end style text end text over denominator begin display style integral integral integral subscript V subscript text samp end text end subscript union to the power of text ? end text end exponent text end text V subscript text ring end text end subscript end subscript left parenthesis ? subscript straight d vertical line E subscript text disk end text end subscript vertical line squared plus mu subscript 0 vertical line H subscript text disk end text end subscript vertical line squared right parenthesis d v text end text end style end fraction.$ (2.12)

As the region where the sample is located corresponds to the lowest values of the E-field distribution, the frequency shift due to the sample is limited. For the TE_01δ mode, the ceramic probe is expected to be robust with respect to the sample loading while the permittivity contrast and the sample volume (or sample over disk radii ratio) have reasonable values. This is demonstrated in Figure 2.4.

Figure 2.4 Quantification of the TE_01δ mode frequency shift between the disk resonator and the dielectric probe (for both, the numerical simulations results were obtained with the CST Eigenmode Solver). The probe ring has its relative permittivity equal to 200, 500, and 800. The frequency variation is plotted as a function of the sample permittivity and for several discrete values of the radii ratio. Curves in dashed lines correspond to systematic frequency shift inferior to 5%. From [21].

2.3.4 Application: Design Guidelines

One way of designing a ceramic probe as depicted in Figure 2.5 consists of studying how the achievable SNR varies with the probe dimensions and/or material properties for a given sample. In the following, the required field of view constrains the ring’s inner diameter and height, the ceramic material properties optimize the SNR value, and the outer diameter and the permittivity are used for tuning the resonance at the Larmor frequency. Therefore, designing the ceramic probe

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