Magnetic Resonance Microscopy. Группа авторов

Magnetic Resonance Microscopy

applied to any higher-order or different mode type.

Figure 2.1 TE_01δ mode of a cylindrical resonator: field-line schematic and notations.

2.3.1 Dielectric Cylindrical Resonator Modes

Describing the eigenmodes of dielectric structures relies on a similar methodology as for metallic cavities. Here we focus on resonators with circular cross sections, therefore all the equations will be expressed in a cylindrical coordinates system (ρ, θ, y) with ρ the radius, θ the azimuth, and y the altitude. ρ and θ relate to the Cartesian coordinates system (x, y, z) according to and θ = arctan z / x. The unit vectors associated with this coordinates system are denoted eρ, eθ, and ey.

We start with the Helmholtz equation, in a source-free, linear, homogeneous, and isotropic medium, verified by the electric E and magnetic H fields denoted indifferently U [20, p. 16]. ω is the angular frequency, μ the medium permeability, the medium complex permittivity (with j the unit imaginary number), and k is the complex wavenumber such that with ky and kρ the wavenumber components along the y- and the ρ-directions, respectively. The method of longitudinal components [17] uses the decomposition of the vector field U into its longitudinal Uyey and transverse UT = U − Uyey components. These components are solutions of the scalar and vector Helmholtz equations, respectively.

$table row cell nabla squared U plus k squared U equals 0 text end text end cell row cell table row rightwards double arrow row cell text along end text y end cell end table text end text nabla squared U subscript y plus k squared U subscript y equals 0 text end text table row rightwards double arrow row cell left parenthesis rho comma text end text theta comma text end text y right parenthesis end cell end table text end text nabla subscript T superscript 2 U subscript y plus fraction numerator partial differential squared U subscript y over denominator partial differential y squared end fraction plus k squared U subscript y equals 0 end cell end table.$ (2.1)

Assuming that the longitudinal component Uy can be written as the product of three functions, one for each space coordinate (in a cylindrical coordinates system in this case), the method of separation of variables allows the partial differential equation to be transformed into three independent ordinary differential equations [18]. The boundary conditions of the cavity, imposing the continuity of the tangential field components at interfaces, are then used to solve these new equations. The contributions of the axial components Uy are then separated from the transverse ones UT. The last equation for which Uy is the solution can then be solved independently in Ey, Hy, ky, and kρ. The expressions for the other field components are deduced by applying the Maxwell equations, with the resulting expressions in Equation 2.2.

$table row cell nabla cross times bold italic E equals negative j omega mu bold italic H end cell row cell nabla cross times bold italic H equals plus j omega bold italic element of bold italic E end cell end table rightwards double arrow table row cell table attributes columnalign left end attributes row cell E subscript rho equals negative fraction numerator 1 over denominator k subscript rho superscript 2 end fraction fraction numerator partial differential squared E subscript y over denominator partial differential rho partial differential y end fraction plus fraction numerator j omega mu over denominator rho k subscript rho superscript 2 end fraction fraction numerator partial differential H subscript y over denominator partial differential theta end fraction end cell row cell E subscript theta equals negative fraction numerator 1 over denominator rho k subscript rho superscript 2 end fraction fraction numerator partial differential squared E subscript y over denominator partial differential theta partial differential y end fraction minus fraction numerator j omega mu over denominator k subscript rho superscript 2 end fraction fraction numerator partial differential H subscript y over denominator partial differential rho end fraction end cell row cell H subscript rho equals negative fraction numerator 1 over denominator k subscript rho superscript 2 end fraction fraction numerator partial differential squared H subscript y over denominator partial differential rho partial differential y end fraction minus fraction numerator j omega over denominator rho k subscript rho superscript 2 end fraction fraction numerator partial differential E subscript y over denominator partial differential theta end fraction end cell row cell H subscript theta equals negative fraction numerator 1 over denominator rho k subscript rho superscript 2 end fraction fraction numerator partial differential squared H subscript y over denominator partial differential theta partial differential y end fraction plus fraction numerator j omega over denominator k subscript rho superscript 2 end fraction fraction numerator partial differential E subscript y over denominator partial differential rho end fraction end cell end table end cell end table$ (2.2)

Here we add another simplification and consider only TE modes, meaning that the axial electric field component, Ey, is equal to zero everywhere. In this case, the mode field distribution is deduced from the solution Hy of Equation 2.1.

As an example, let us consider the TE modes of a disk resonator with radius r, height L, and its symmetry axis corresponding to the y-axis. Solving the above-mentioned problem leads to a solution inside the resonator of the form described in Equation 2.3 (magnetic field axial component inside the disk resonator for the TE modes) with A the amplitude coefficient, n an integer describing the azimuthal mode order, ϕ and ψ constant phase shifts deduced from the boundary conditions, and Jn the Bessel function of the first kind and order n.

H subscript y equals A cos left parenthesis n theta plus ? right parenthesis J subscript n left parenthesis k subscript rho rho right parenthesis cos left parenthesis k subscript y y plus psi right parenthesis. (2.3)

If the resonant structure has metallic borders, boundary conditions impose the field cancellation at the interfaces:

In ρ = r, the tangential components of the magnetic field vanish, for example Hy, which quantifies the radial wavenumber: (2.4)with xnm the m-th zeros of the n-th order Bessel function. This boundary condition gives the radial variation order m, which is an integer.

In the magnetic field tangential components vanish, for example Hρ, which quantifies the axial wavenumber:

$table row cell H subscript rho left parenthesis rho comma theta comma text end text y equals plus-or-minus L over 2 right parenthesis equals 0 end cell row cell rightwards double arrow sin left parenthesis plus-or-minus k subscript y L over 2 plus psi right parenthesis equals 0 end cell row cell rightwards double arrow k subscript y comma p end subscript equals minus-or-plus left parenthesis fraction numerator 2 p pi over denominator L end fraction minus fraction numerator 2 psi over denominator L end fraction right parenthesis end cell end table comma$ (2.5)

with p an integer defining the axial variation order.

For a metallic cavity, the mode variations are quantified by three integers, n, m, and p, and is therefore named TE_nmp.

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