Magnetic-Field Transmission Through a Circular Aperture in a Magneto-Conductive Screen: Identification of Aperture Penetration and Field Diffusion Contributions

The magnetic field transmitted through a circular aperture in an infinite magneto-conductive planar screen with finite thickness is studied, considering as a source a circular electric current loop coaxial with the aperture. The field source consists of a circular electric current loop coaxial with the aperture. First, the finite thickness of the aperture is taken into account through suitable generalized boundary conditions. Next, a pair of dual integral equations are obtained whose solutions are obtained by means of an expansion of the unknowns in Neumann series of Bessel functions combined with a Galerkin method of moments. Such solutions furnish equivalent electric and magnetic spectral current densities from which all the field components can easily be obtained. The proposed formulation allows for clearly distinguishing the contributions to the total transmitted field coming from the diffusion in the magneto-conductive screen and the penetration through the aperture, thus gaining physical insight into the transmission mechanism. Numerical results are presented and compared with those obtained through commercial software to show the accuracy and the efficiency of the proposed formulation.


I. INTRODUCTION
T HE transmission of the electromagnetic field through a circular aperture in an infinite perfectly conducting (PEC) plane with infinitesimal thickness is a canonical problem which admits an exact solution in terms of oblate spheroidal vector wavefunctions, after solving the complementary disk problem [1]. Under the assumption of small aperture, different approximate formulations have also been proposed based on electrostatic and magnetostatic approximations of the fields at Manuscript  the aperture [2], [3] or, starting from a rigorous spectral-domain approach, considering the limiting cases of near and/or far-field regions [4], [5]. In [6], the case of a circular electric current loop has instead been addressed through the solution of a second-kind Fredholm integral equation and the application of the Abel integral-transform technique. As said above, most of the works in the literature consider the case of a circular aperture in a PEC plane, i.e., a conducting plane with an ideal infinite electric conductivity. In fact, in such a case, the Babinet principle can be applied and the scattering problem from an aperture in a PEC plane can thus be reduced to the scattering problem from a PEC disk [7], [8]. Other approaches apply the equivalence principle to derive an integral equation whose unknown is the equivalent magnetic current density on the metalized aperture [6].
On the other hand, the diffraction of plane waves from a circular aperture with finite thickness in a PEC plate has been investigated in [9] using the so-called coupled-mode method and, more recently, in [10] through a boundary element method and in [11] using a theory based on the eigenvector expansion and mode matching. Only in connection with the extraordinary optical transmission phenomenon, the diffraction of light from a single aperture in nonideal thick metals (characterized by a complex dielectric constant) has started to be analyzed [12], [13].
In this work, we present an original formulation for the evaluation of the field generated by a circular current loop and transmitted through a circular aperture (coaxial with the loop) cut in a planar magneto-conductive screen of finite thickness. To the best of our knowledge, this practical problem has never been rigorously addressed in the literature, although the aperture in a PEC plate [6], [14] and the infinite planar conductive and magnetic screen [15], [16], [17], [18], [19], [20] have been extensively studied.
Only very recently, the problem of the electromagnetic field penetration through a circular aperture in a infinitesimally-thin plate characterized by a nonzero surface impedance has been addressed through a dual-integral-equation approach [21]. In this work, with respect to [21], the finite thickness of the aperture, the finite conductivity and the permeability of the plate are taken into account through suitable generalized boundary conditions. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Next, a pair of dual integral equations are derived whose solutions are obtained by a Galerkin method of moments in the Hankel spectral domain in which the unknowns are expanded in Neumann series of Bessel functions. Such solutions furnish equivalent electric and magnetic spectral current densities from which all the field components can easily be obtained through suitable inverse Hankel-transform integrals. It is worth mentioning that in [21], only one pair of dual integral equations was sufficient to describe the problem and the unknown was the spectral electric current density on the impedance plate. On the other hand, in the configuration analyzed in the present article, both electric and magnetic current densities need to be introduced making the problem more complex. However, we show that by decomposing the current loop source into even-and odd-symmetric parts, two subproblems can be derived and the relevant pairs of dual integral equations are decoupled. In this framework, [21] is a particular case of the more general problem considered here, which reduces to the problem considered in [21] for nonmagnetic screens with small thicknesses (smaller than the skin depth) and suitably low frequencies.
On the other hand, the use of the adopted generalized boundary conditions limits the analysis to subwavelength thicknesses: The the proposed formulation has thus an upper limit in frequency f max depending on the screen thickness d (in particular f max c/ (2 d), where c is the speed of light in free space). However, it should be noted that all the practical configurations (metallic and/or magnetic screens up to the microwave range) remain well inside this class of structures. In addition, we want to mention a possible limitation of the present study: In fact, for simplicity, the electric current in the loop is assumed to be constant. In principle, the extension of the proposed analysis to nonconstant currents could be feasible, but introducing an azimuthal dependence of the exciting current would destroy the azimuthal invariance of the problem, thereby making the incident and scattered field hybrid. We leave this analysis for a future work. We point out that the analysis presented here is limited to apertures of circular shape: this assumption allows for simplifying the numerical treatment of the problem, taking advantage of the azimuthal symmetry and of the possibility of expanding the unknowns of the integral equations through entire-domain basis functions. Again, the consideration of more complex shapes could be feasible using, e.g., an expansion in a Fourier series to take into account the azimuthal dependence of the problem and an expansion of the unknowns in RWG subdomain basis functions to model the arbitrary shape of the aperture. However, the assumption of circular aperture, while maintaining a numerical simplicity, allows for deriving general considerations on the involved electromagnetic phenomena. Finally, a last limitation of the proposed analysis consists in considering screens of infinite extent: It is understood that such an assumption does not allow for taking into account possible edge diffraction effects from the edges of the screen: However, the latter should be negligible for sufficiently large screens (with respect to the operating wavelength).
One great advantage of the proposed formulation consists also in the possibility of separating the contributions to the total transmitted field coming from the diffusion in the magneto-conductive screen and the penetration through the aperture, thus gaining physical insight into the transmission mechanism.
The rest of this article is organized as follows. The problem is described in Section II, where the adopted GBCs are also discussed and two auxiliary subproblems are introduced, each of which leads to a dual integral equation whose unknown is an equivalent spectral electric or magnetic current density. The procedure for the solution of this pair of dual integral equations is outlined in Section III, where all the quantities of interest are also calculated. As mentioned above, the proposed formulation allows for clearly distinguishing the contributions of the infinite solid screen and the aperture. Numerical results are presented in Section IV to show the accuracy and the efficiency of the proposed approach and to point out possible critical situations in particular configurations. Finally, Section V concludes the article.

II. FORMULATION OF THE PROBLEM
The configuration under analysis is reported in Fig. 1 and it consists of an infinite conductive plate characterized by a conductivity σ, a magnetic permeability μ = μ 0 μ r , a finite thickness d, and symmetrically placed around the plane z = 0 of a cylindrical coordinate system (ρ, φ, z). A circular aperture of radius a is cut in it with center at the origin and a current loop of radius R and constant current I 0 (coaxial with the aperture) is placed at a distance h from it with a time-harmonic e jωt behavior. Such a source produces a purely incident TE z field with nonzero components E inc φ , H inc ρ , and H inc z .

A. Generalized Boundary Conditions
The finite thickness of the screen makes the analysis more complex with respect to the infinitesimally-thin structure case. However, if the thickness d is sufficiently small (in particular, where k 0 is the free-space wavenumber), it is possible to shrink the thickness to zero and adopt generalized boundary conditions (GBCs), which relate the tangential average electric (magnetic) fields to the tangential magnetic (electric) field jump across an infinitely thin surface, i.e., Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. and where ζ 0 is the free-space impedance. Such GBCs have recently be used successfully in different scattering problems [22], [23], [24], [25], [26]. The coefficientsẐ S andŶ S are related to the thickness and the electromagnetic parameters of the screen and different expressions have been proposed in the literature. In his seminal paper [27], Mitzner derived where ε cr = 1 − jσ/(ωε 0 ) indicates the relative complex permittivity.
More recently, Karlsson [28] proposed more refined expressions which take into account the compensation for the reduction of the screen volume to a surface of infinitesimal thickness and which lead to more accurate results, i.e., where It is worth noting that the condition k 0 d 1 is not a limiting factor in all the practical configurations, where the thickness d of the screen is much smaller than the operating wavelength up to the microwave range.

B. Decomposition Into Even-and Odd-Symmetric Subproblems
The GBCs in (1) and (2) define a combined electrically resistive/magnetically conductive sheet with infinitesimal thickness, which supports both electric and magnetic surface currents, related to the discontinuities across the sheet of the tangential magnetic and electric fields, respectively. Since any plane containing the vertical z-axis is an odd symmetry plane, the electric surface current density J s is azimuthally directed, whereas the magnetic current density M s is radially directed (see Fig. 2).
Electric and magnetic current densities defined over curved surfaces generally interact (i.e., the field produced by one of the currents couple to the other current). However, for a planar sheet, the reflection symmetry of the configuration with respect to the normal z-axis makes the two currents uncoupled [29]. The problem can thus be conveniently decoupled into two distinct subproblems; for instance, by decomposing the original electric current loop source into even-and odd-symmetric parts, the former is readily seen to excite only electric currents on the sheet, whereas the latter excites only magnetic currents, as shown in Fig. 2(b) and (c), respectively.
In the following subsections, the two subproblems will be separately discussed.

C. Even-Symmetric Subproblem
The boundary condition on the plate for the even-symmetry subproblem is The incident field E (1)inc φ is that produced by the original current loop of current I 0 /2 at z = h and by its mirror image at z = −h in free space, i.e., where ν is the normalized radial wavenumber, J 1 (·) is the Bessel function of order 1, and is the z-component of the normalized spectral wavenumber. The scattered field E (1)scat φ is that radiated by the electric current density J Sφ induced on the screen, i.e., whereJ Sφ (ν/a) is the Hankel transform of order 1 of the surface current density J Sφ (ρ).
Using (9), (10), and expressing the electric current density J Sφ (ρ) as an inverse Hankel transform and rearranging some terms, (7) and (8) can be rewritten as a first pair of dual integral equations as where r = ρ/a and the spectral currentJ Sφ is the unknown function. Once the spectral electric current is known, any component of the scattered field E

D. Odd-Symmetric Subproblem
Dually, the boundary condition on the plate for the oddsymmetry subproblem is The incident field H (2)inc ρ is that produced by the original current loop with current I 0 /2 at z = h and by its antimirror image with current −I 0 /2 at z = −h in free space, i.e., while the scattered field H (2)scat ρ is that radiated by the magnetic current density M Sρ induced on the screen, i.e., whereM Sρ (ν/a) is the Hankel transform of order 1 of the surface current density M Sρ (ρ).
Using (16), (17), and by expressing the magnetic current density M Sρ (ρ) as an inverse Hankel transform and rearranging some terms, (14) and (15) can be rewritten as a second pair of dual integral equations as where the spectral currentM Sρ is the unknown function. Once the spectral magnetic current is known, any component of the scattered field E can be calculated through a suitable inverse Hankel-transform integral [30].

III. SOLUTION OF THE DUAL INTEGRAL EQUATIONS
The dual integral (12), (13) and (19), (20) are both in the same form as those studied in [21]. In particular, they can both be reduced to the canonical form where i = 1 and i = 2 denote the even-symmetrical and oddsymmetrical problem, respectively, and and In particular, for the even-symmetry subproblem i = 1 we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
and (29) whereas for the odd-symmetry problem i = 2 it results and As described in [21], in order to solve the dual integral (21), (22), the ψ(ν) function (which is related either to theJ Sφ or to theM Sρ functions) is expanded in a Neumann series of Bessel functions, i.e, In fact, thanks to the well-known discontinuous Sonine-Schafheitlin integral [31, 6.574 Equation (34) automatically satisfies (22). The parameter μ i in (34) is arbitrary (provided that μ i > 0) and can be used to guarantee the convergence of the integrals arising in the analysis or to enforce a specific order of singularity of the spatial currents at the edge of the aperture [21]. Using the expansion (34) and following the same steps as in [21], the dual integral equations are reduced to the matrix system +∞ n=0 where the matrix elements are given by and From (37), and the expressions of γ i (ν) it is easy to derive that it must be μ 1 > 1/2 and μ 2 > 0, so that we can set μ 1 = μ 2 = 1 and therefore and Equations (39) and (40) can explicitly be written as (42) and Once the system is solved, the coefficients ψ (i) n furnish the unknown spectral currents through and Such spectral currents can then be used to calculate any component of the field. In particular, the z-components of the scattered magnetic fields H (1)scat z and H (2)scat z for z < 0 are given by and so that The z-component of the magnetic field radiated by the original current loop source is instead (50)

A. Field Calculation
Now, we consider the magnetic-field calculation along the z-axis, which is the critical case for shielding-effectiveness evaluations. Therefore, by assuming and suppressing ρ = 0 throughout, we have Moreover i.e., so that with and Analogously i.e.,

A. Purely Conductive Screens
The first configuration consists of an electric current loop with current I 0 = 1 A of radius R = 5 cm coaxial with an aperture of radius a in a purely conductive screen with conductivity σ = 2.9 · 10 7 S/m, relative permeability μ r = 1 (aluminum alloy), and thickness d = 1 mm. The electric current loop is placed at h = R. In Fig. 3, the absolute value of the z-component of the total magnetic field H z is reported as a function of frequency from dc to 20 MHz at the observation point z = −h for three different values of the aperture radius, i.e., a = 1.5 cm, a = 2.5 cm, and a = 5 cm. As it can be seen, there is a very good agreement between the results of the proposed exact formulation (solid lines) and those of FEKO (gray dots) . As shown in the inset of Fig. 3, the simulation through FEKO requires the accurate modeling of the source loop and aperture through suitable control of the mesh. The modeling of the source using several current ports placed along the loop does not allow to use symmetry planes. On a modern Intel desktop computer (I9-9900 K CPU, 64 GB RAM), the computational times range around 5 h over 301 frequency points using a mesh of 14.442 triangles with first order basis functions and double precision, while the proposed formulation takes only few seconds for each point.
The convergence properties of the exact formulation can be studied through the relative error ε err between the the N th and (N + 1)th partial sums of the Neumann series in (34), as a function of N [21]. In general, very fast decrease of the relative error ε err can be observed, with a number of basis functions N which obviously depends on the involved geometrical and physical parameters. In the cases of Fig. 3, N = 15 basis functions have been used to obtain a relative error ε err less than 10 −3 .
As it can be seen, at low frequencies the level of the transmitted field is the same in all the aperture cases, while, by increasing frequency, the transmitted field in the case of the largest aperture (a = 5 cm) is much larger than in the other cases. To better understand the underlying physical mechanism, in Fig. 4 the contributions of the infinite-plate field H solid z and the aperture field H ap z to the total magnetic field are reported in the case of a = 5 cm (a) and a = 2.5 cm (b). Since the solid screen, the excitation configuration, and the observation point are the same in both the cases, the solid-screen field contribution is also the same and what is different is the aperture-field contribution, which is larger in the largest aperture case, as expected. In Fig. 4, for a fair comparison, the field transmitted through an aperture in a PEC screen is also reported [6]. As it can be seen this is the limiting case for large frequencies and it coincides with the aperture-field contribution. In practice, at low frequencies, the magnetic field is dominated by the solid-screen contribution(i.e., the same screen without any aperture) while at high frequencies is dominated by the aperture contribution (and, for very high frequencies it coincides with the magnetic field penetrating the same aperture in a PEC screen). However, in the intermediate frequency range, there is a transition region where both the contributions are important and shape the total transmitted magnetic field.
Finally, Fig. 5 shows the magnetic shielding effectiveness SE H , SE plate , and SE ap for the same configurations. It can thus be seen that in the low-frequency range, since the contribution of the aperture field is negligible, the magnetic SE coincides with that of a solid screen with the same parameters. However, by increasing frequency, the aperture-field contribution becomes more important and this strongly limits the values of the SE. Obviously, in the case of a smaller aperture the maximum SE (which is obtained at high frequencies) is larger.
Dual results are obtained by changing the thickness of the plate. In fact this mainly influences the solid-screen contribution negligibly affecting the aperture contribution, so that for smaller thickness the transition region is reached at lower frequencies.
The relevant results are not reported here for conciseness.

B. Magneto-Conductive Screens
The second configuration consists of the same setup considered in the previous section, but with a magneto-conductive screen (iron) having conductivity σ = 1.8 · 10 6 S/m, relative permeability μ r = 500 (and again thickness d = 1 mm). In Fig. 6, the absolute value of the z-component of the total magnetic field H z is reported as a function of frequency from dc to 20 MHz at the observation point z = −h for a = 1.5 cm, a = 2.5 cm, and a = 5 cm. As in the case of purely conductive screens, there is a very good agreement between the results of the proposed exact formulation (solid lines) and those of FEKO (gray dots).    For these configurations N = 9 basis functions have been used in order to obtain a relative error ε err less than 10 −3 .
In Fig. 7, the contributions of the solid-screen field H solid z and the aperture field H ap z to the total magnetic field are reported for the cases a = 5 cm and a = 2.5 cm. In the case of magneto-conductive screen, the transmitted magnetic-field at low frequencies is much smaller (because of the presence of a large permeability) and this makes the relevant solid-screen contribution much smaller and comparable with the aperture contribution thus determining, e.g., a destructive interference of the two contributions around f = 2 kHz in the case of a = 5 cm.
Finally, in Fig. 8, the magnetic shielding effectiveness SE H , SE solid , and SE ap for the two configurations are reported. It can be seen that the destructive interference in the case a = 5 cm gives rise to an unexpected increasing of the SE in the kHz region, also with respect to the solid screen.

V. CONCLUSION
The diffusion and penetration of the magnetic field generated by a circular electric current loop through a coaxial circular aperture in a magneto-conductive screen of finite thickness is studied through an original formulation based on generalized boundary conditions and dual integral equations. The magnetic shielding effectiveness is studied in different configurations pointing out possible critical situations. Numerical results are also compared with those obtained through commercial software to show the accuracy and the efficiency of the proposed formulation. One remarkable advantage of the proposed formulation is that it allows for clearly distinguishing the contribution of the magnetic-field diffusion through the solid screen and of the penetration through the aperture thus gaining physical insight into the shielding performance. The relative weight of such contributions strongly depends on all the parameters of the considered configuration and it will be the subject of a future investigation.