Читать книгу Magnetic Resonance Microscopy - Группа авторов - Страница 34

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.

(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.

(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.

(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:

(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. It is the same principle for transverse magnetic (TM) and hybrid (HEM) mode types.

When the resonant structure has dielectric/dielectric or dielectric/air interfaces with the surrounding medium, this quantification with integer numbers is no longer valid since the electromagnetic (EM) field does not vanish at the boundaries. In such situations, the EM field outside the resonator must be explicitly expressed. An approximation consists of writing the field outside with similar variations as inside the resonator with evanescent terms, neglecting the contribution of other modes to the decaying part. This can be made part by part, as developed in [4,7] and as shown in Figure 2.2 for the disk resonator.

Figure 2.2 Resonant mode field estimation: part-by-part decomposition of the computation volume, case of the disk resonator.

For example, the field expression remains unchanged within the dielectric disk, while outside it is written as in Equation 2.6, with f a decaying function depending on the computation volume boundaries, Kn the modified Bessel function of the first kind and order n, and ν the radial wavenumber in the region where the EM field is radially decaying:

(2.6)

Combining the boundary conditions at the interface air/dielectric (continuity of the tangential field component) as well as the wavenumber decomposition in each region provides a set of equations describing the modes arising from the combination of a planar dielectric waveguide with a circular dielectric waveguide.

Examples of electromagnetic field maps obtained with the same method extended to the case of a ring resonator are shown in Figure 2.3. As expected from the mode description, the electric field minimum coincides with the magnetic field maximum.

Figure 2.3 Electromagnetic field distribution of the TE_01δ mode of a ring resonator (relative permittivity 500, outer radius 10 mm, height 10 mm) filled with a sample (relative permittivity 50) for varying inner to outer radii ratio: magnetic field (first line) and electric field (second line) field maps and lines (gray arrows), and both fields profiles (third line). The 2D maps are plotted in grayscale with a linear value distribution. From [21].