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Enhanced energy localization in electromagnetically thick metallic mesoshells

Nitish Chandra, Wiktor Walasik, and Natalia M. Litchinitser

Open Access Open Access

Abstract

While subwavelength dielectric structures enclosed by a thin metallic nanoshell have found a wide range of applications, their wavelength-scale counterparts have not been addressed. Conventionally, a dielectric enclosed by a thick metallic shell is considered isolated as the fields attenuate to a negligible value. Here, we show that, due to the Mie resonances of the wavelength-scale dielectric core, the energy density in the core can be enhanced by six orders of magnitude as compared to the off-resonance case, despite the presence of a thick metallic shell. In contrast to the widely studied case of plasmonic core-shell subwavelength particles, where the field enhancement occurs at the boundary of the metallic shell, the thick metallic shell surrounding the wavelength-scale dielectric core provides a strong energy confinement at the center of the core at longer wavelengths, where plasmonic effects are negligible. The observed enhancement can find applications for the probing of shielded materials and designing structures with engineered electromagnetic responses.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The effect of geometry on linear properties of subwavelength metal nanoshells has been shown to enable a number of functionalities, such as, tuning the surface-plasmon [1,2] and Fano [3,4] resonances, designing meta-atoms [5], and improving the surface-enhanced Raman sensors [6]. It has been shown that adding a metallic shell to a dielectric nanoantenna improves the nonlinear response [7]. Multilayered planar structures have been suggested to enhance the evanescent waves for nonlinear applications [8,9]. The dependence of the optical properties of subwavelength plasmonic nanoshells on geometry has been studied at optical frequencies [10]. In that case, the electric field intensities is strongly enhanced at the two metal surfaces and the scattering cross-section is enhanced at the resonant frequencies [11,12]. At lower frequencies, these plasmonic effects disappear and the field at the inner surface of the shell is evanescent. The possibility of the field enhancement of the evanescent waves at microwave frequencies in thick metallic wavelength-scale structures using bulk materials has not been investigated. As the plasmonic effects are absent at lower frequencies, a different mechanism is required to obtain a strong field localization at the wavelength scale. We propose to use a mesoshell with the thickness comparable to the skin depth in order to achieve energy confinement in a small volume. The enhancement of electromagnetic (EM) fields in the core is not plasmonic in nature and can be used to design new family of core–shell structures. Moreover, understanding of EM field penetration within thick conducting shells is important because it allows us to avoid unwanted coupling of radiation with antennas [13,14].

The interaction of EM waves with wavelength-scale structures can be analyzed using Mie formalism [15]. The wavelength-scale dielectric structures have been shown, for instance, to support hybrid Mie–Fabry-Perot modes for multimode directional scattering [16]. Traditionally, scattering from highly conducting bulk structures is analyzed under perfectly conducting approximation [17,18]. Therefore, it is assumed that the material contained inside the shell which is much thicker than the skin depth does not interact with incident EM fields. However, perfect conduction is an idealization and in reality, for planar structures, the penetration (or skin) depth is defined as the distance $(\delta )$ at which the amplitude of EM fields within a material attenuates to $1/e$ of its original value [19]. In general, a conducting medium is considered to be electromagnetically thick if the magnitude of the EM fields behind the medium is negligible.

In this article, we study scattering from a wavelength-scale dielectric core surrounded by an electromagnetically thick metallic shell, referred to as a mesoshell. We find that at the frequencies corresponding to the Mie resonances of the core, the magnitude of EM fields in the core increases by six orders of magnitude. The enhancement of the evanescent waves in the core shows that the incoming EM waves are not fully isolated by the electromagnetically thick mesoshell. This result is counterintuitive as at microwave frequencies the material dispersion in metals is ignored, and often treated as ideal very high reflection and negligible absorption. We observe that the field distribution in the metallic shell at resonant frequency differs from exponential attenuation. We analytically analyze the impact of the geometry and materials on the resonant frequency and on the magnitude of fields in the core.

2. Scattering from mesoshell structures

In this section, we present a general mathematical formalism used to analyze scattering from mesoshell structures. Let us consider a plane wave $\{\mathbf {E}_\textrm {inc}(\mathbf {r}, t), \mathbf {H}_\textrm {inc}(\mathbf {r}, t) \} = \{\hat {\mathbf {z}}E_0, -\hat {\mathbf {y}}H_0\}e^{i(kx - \omega t)}$ incident on a metal–dielectric mesoshell structure as illustrated in Fig. 1. Here, $\omega$ defines the angular frequency of the incoming radiation and $k$ denotes the wavenumber. In region 1 $(r_1 < r < \infty )$, the EM fields in the surrounding medium are described as $\{\mathbf {E}_1(\mathbf {r}, t), \mathbf {H}_1(\mathbf {r}, t)\}$, which are the sum of incident and scattered fields. In region 2 ($r_2 < r < r_1$), the fields in the conducting medium are given by $\{\mathbf {E}_2(\mathbf {r}, t), \mathbf {H}_2(\mathbf {r}, t)\}$. In region 3 $(0 < r <r_2)$, the transmitted fields are described by $\{\mathbf {E}_3(\mathbf {r}, t), \mathbf {H}_3(\mathbf {r}, t)\}$.

 figure: Fig. 1.

Fig. 1. (a) Schematic of a plane wave incident on the cross-section of the core-shell structures. (b) Infinite cylindrical mesoshell. (c) Spherical mesoshell.

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The EM fields in the three regions are the solutions of wave equations in space-frequency domain,

$$(\nabla^2 + k_j^2) \Big \{ \mathbf{E}_j(\mathbf{r}, t), \mathbf{H}_j(\mathbf{r}, t) \Big \}^\textrm{T} = 0,$$
where $(k_j^2)$ is the square of the wavenumber and subscript $(j)$ denotes medium 1, 2 or 3. The wavenumber for surrounding and core dielectric material is given by, $k_1^2 = \omega ^2\epsilon _1 \epsilon _0\mu _0$ and $k_3^2 = \omega ^2\epsilon _3 \epsilon _0\mu _0$, respectively. Here, $\mu _0$ and $\epsilon _0$ are the vacuum permeability and permittivity, respectively, and $\epsilon _j$ is the relative permittivity of the medium. At microwave frequencies, the relative permittivity of metals with conductivity $\sigma$ is given by $\epsilon _2 = (1 - \frac {i\sigma }{\omega \epsilon _0})$ [20]. The square of wavenumber within the metal is given by $k_2 = \omega ^2 \epsilon_0\mu _0 - i\omega \mu _0 \sigma$. For highly conducting materials the relative permittivity is imaginary as $\frac {\sigma }{\omega \epsilon _0} >> 1$, and the wavenumber becomes $k_2 = e^{i \frac {\pi }{4}}\sqrt {\omega \mu _0 \sigma }$. The solution of the system of equations given by Eq. (1) is obtained using the boundary conditions relating the fields at the interface between two media:
$$\hat{\mathbf{n}} \times [\mathbf{E}_{j+1}(\mathbf{r}, t) - \mathbf{E}_j(\mathbf{r}, t)] = 0, \\$$
$$\hat{\mathbf{n}} \times [\mathbf{H}_{j+1}(\mathbf{r}, t) - \mathbf{H}_j(\mathbf{r}, t)] = 0,$$
where $\hat {\mathbf {n}}$ is the normal vector to the surface [21]. It is important to note that the boundary conditions for fields at the surface with a finite conductivity differ from those for the perfect electric conductor leading to a distinct solution [21]. In the following sections, we show through the example of an infinite cylindrical shell and a spherical shell that the penetration of EM field through highly lossy material can be significantly enhanced. The analytical solution for the scattering from wavelength-scale structures derived here using first principles analysis remains valid for the entire EM spectrum. However, we have restricted the discussions to a frequency range corresponding to the scattering parameter, $1 \lesssim \chi _s = \frac {2 \pi \sqrt {\epsilon_3}}{\lambda _0}r \lesssim 10$ [15], where $\lambda _0$ is the free-space wavelength of the incoming radiation.

3. Infinite cylindrical shell

In this section, we limit the discussion to transverse-magnetic (TM) waves. For infinite cylindrical mesoshell structure, Jacobi-Anger expansion in terms of Bessel functions $(J_n(kr))$ is used to determine the representation of the plane wave in cylindrical coordinates [22]

$$\mathbf{E}_\textrm{inc}(\mathbf{r}, t) =\hat{\mathbf{z}}E_0 e^{i(k_1x - \omega t)} = \hat{\mathbf{z}}E_0e^{{-}i \omega t} \sum_{n ={-} \infty}^{\infty}i^nJ_n(k_1 r)e^{in \phi}.$$
Components of the electric fields in each region where the time dependence $e^{-i \omega t}$ is omitted are given by,
$$E_{1,z}(r, \phi) = E_0 \sum_{n ={-} \infty}^{\infty}i^n\Big \{J_n(k_1 r) + a_n H_n^{(1)}(k_1 r)\Big \}e^{in \phi},$$
$$E_{2,z}(r, \phi) = E_0 \sum_{n ={-} \infty}^{\infty}i^n\Big \{b_n J_n(k_2 r) + c_n H_n^{(1)}(k_2 r)\Big \}e^{in \phi},$$
$$E_{3,z}(r, \phi) = E_0 \sum_{n ={-} \infty}^{\infty}i^n\Big \{d_n J_n(k_3 r)\Big \}e^{in \phi},$$
where Hankel function of first kind, $H^{(1)}_n(kr) = J_n(kr) + iY_n(kr)$ represent waves traveling radially outward for time dependence of form $e^{-i\omega t}$. Maxwell-Faraday equation $(\nabla \times \mathbf {E} = - \mu _0\frac {\partial \mathbf {H}}{\partial t})$ determines the corresponding magnetic field in each region. The system of equations of form $\mathbb {M}\mathbf {x} = \mathbf {y}$ for the coefficients governing the complex amplitudes of the field is obtained using boundary conditions given by Eq. (2) and Eq. (3) in the electric and magnetic field equations
$$\left[ \begin{array}{cccc} -H_n(\rho_{11}) & J_n(\rho_{21}) & H_n(\rho_{21}) & 0 \\ 0 & J_n(\rho_{22}) & H_n(\rho_{22}) & - J_n(\rho_{32}) \\ -k_1 \dot{H_n}(\rho_{11}) & k_2 \dot{J_n}(\rho_{21}) & k_2 \dot{H_n}(\rho_{21}) & 0 \\ 0 & k_2 \dot{J_n}(\rho_{22}) & k_2 \dot{H_n}(\rho_{22}) & - k_3 \dot{J_n}(\rho_{32}) \end{array} \right] \left[ \begin{array}{c} a_n \\ b_n \\ c_n \\ d_n \end{array} \right] = \left[ \begin{array}{c} J_n(\rho_{11}) \\ 0 \\ k_1 \dot{J_n}(\rho_{11}) \\ 0 \end{array} \right].$$
To simplify the notations, we have substituted $\rho _{i,j} = k_ir_j$, where ${i,j}\in {1,2,3}, (.) = \frac{d()}{d\rho_{ij}}$ and omitted the order of Hankel function as only the first order is used. The exact expressions of the coefficients as solutions of linear equations are obtained using Cramer’s rule.

3.1 Fields in the core region

The amplitude of the fields in the core is governed by the transmission coefficients which are calculated as $d_n = \frac {Det(\mathbb {D})}{Det(\mathbb {M})}$, where $\mathbb {D}$ is given by

$$\mathbb{D} = \left[ \begin{array}{cccc} -H_n(\rho_{11}) & J_n(\rho_{21}) & H_n(\rho_{21}) & J_n(\rho_{11}) \\ 0 & J_n(\rho_{22}) & H_n(\rho_{22}) & 0 \\ -k_1 \dot{H_n}(\rho_{11}) & k_2 \dot{J_n}(\rho_{21}) & k_2 \dot{H_n}(\rho_{21}) & k_1 \dot{J_n}(\rho_{11}) \\ 0 & k_2 \dot{J_n}(\rho_{22}) & k_2 \dot{H_n}(\rho_{22}) & 0 \end{array} \right].$$
Using the identity for Wronskain of Bessel and first order Hankel functions [22], $J_n(x)\dot {H}_n(x) - H_n(x)\dot {J}_n(x) = \frac {2i}{\pi x}$ we obtain the numerator for the transmission coefficient,
$$Det{(\mathbb{D})} ={-}\frac{4}{\pi^2 r_1r_2}.$$
Similarly, the denominator $Det{(\mathbb {M})}$ is given by
$$\begin{aligned}Det{(\mathbb{M})} & = k_2^2H_n(\rho_{11})J_n(\rho_{32})[\dot{J_n}(\rho_{21})\dot{H_n}(\rho_{22}) - \dot{J_n}(\rho_{22})\dot{H_n}(\rho_{21})]\\ & - k_3k_2H_n(\rho_{11})\dot{J_n}(\rho_{32})[\dot{J_n}(\rho_{21})H_n(\rho_{22}) - J_n(\rho_{22})\dot{H_n}(\rho_{21})]\\ & - k_1k_2\dot{H_n}(\rho_{11})J_n(\rho_{32})[J_n(\rho_{21})\dot{H_n}(\rho_{22}) - \dot{J_n}(\rho_{22})H_n(\rho_{21})]\\ & + k_1k_2\dot{H_n}(\rho_{11})\dot{J_n}(\rho_{32})[J_n(\rho_{21})H_n(\rho_{22}) - J_n(\rho_{22})H_n(\rho_{21})]. \end{aligned}$$
We have obtained an analytical expression for the $d_n$ transmission coefficients for a mesoshell structure. The electric field in the core can be computed using Eq. (7). As an example, let us consider an aluminum mesoshell enclosing a dielectric core subject to an incident plane wave with a frequency $\omega = 20$ GHz. At this frequency, the conductivity of aluminum is $\sigma = 3.65 \times 10^7 \Omega ^{-1}\rm {m}^{-1}$ and the corresponding skin depth, given by $\delta = \sqrt {2/\mu \sigma \omega }$, is $1.48 \mu m$. The intensity of electric field inside a planar conducting medium attenuates exponentially and is given by
$$I = |\mathbf{E}|^2 = |E_0|^2e^{-\frac{2x}{\delta}}.$$
Thus, the intensity becomes negligible in a conducting medium much thicker than the skin depth. Hence, the material inside a mesoshell of such a thickness is assumed to be shielded and is not considered to be affected by incoming radiation. Equation (12) provides a good first approximation for field intensities in cylindrical or spherical shells with a radius larger than the wavelength, such that locally the curvature is small, and can be approximated as a planar surface. However, obtaining exact distribution of fields for various geometry requires solution of Maxwell’s equations with appropriate boundary conditions as shown in following sections.

In order to compute the fields in the core, we need to numerically calculate the values of the transmission coefficients $d_n$. For a shell of the outer radius comparable to the wavelength $r_1 = 3.3$ cm, and thickness $\Delta \approx 10\delta$, the argument of the Bessel functions is complex and has a large magnitude:

$$\rho_{21} \approx \rho_{22} = (1 + i) \times 2.2 \times 10^4.$$
For this argument, the values of Bessel $(J_n(\rho ))$ and Hankel $(H_n(\rho ))$ functions are too large to be handled numerically. A direct use of the expression Eq. (11) in numerical calculations leads to $Det{(\mathbb {M})} \rightarrow \infty$ and transmission coefficients $d_n \rightarrow 0$. This points to and erroneous conclusion that the core of an electromagnetically thick mesoshell is isolated.

However, for large arguments the asymptotic form of the Bessel functions is given by [22],

$$J_n(\rho) \cong \sqrt{\frac{2}{\pi \rho}}\cos \left[\rho - (n + \frac{1}{2})\frac{\pi}{2} \right] = \frac{1}{2}\left[\sqrt{\frac{2}{\pi \rho}}e^{i[{\rho - (n + \frac{1}{2})\frac{\pi}{2}}]} + \sqrt{\frac{2}{\pi \rho}}e^{{-}i[{\rho - (n + \frac{1}{2})\frac{\pi}{2}}]}\right].$$
For large complex argument $(\rho = \alpha + i\alpha )$, $\alpha \in \mathbb {R}^+$, the values of the first term on the right hand side of Eq. (14) is small compared to the second term. Thus, the Bessel function for the large complex arguments can be approximated by $J_n(\rho ) \cong \sqrt {\frac {1}{2 \pi \rho }}e^{-i({\rho - (n + \frac {1}{2})\frac {\pi }{2}})}$. Similarly, asymptotic form of Hankel function of first kind is given by [22], $H_n(\rho ) \cong \sqrt {\frac {2}{\pi \rho }}e^{i({\rho - (n + \frac {1}{2})\frac {\pi }{2}})}$. Using the asymptotic representation and derivatives of Bessel and Hankel functions, we obtain the expression for denominator of transmission coefficient
$$\begin{aligned} Det{(\mathbb{M})} & = k_2^2H_n(\rho_{11})J_n(\rho_{32})[\frac{8i}{\pi}\frac{1}{\sqrt{\rho_{21}\rho_{22}}}\sin(\rho_{22} - \rho_{21})]\\ & - k_3k_2H_n(\rho_{11})\dot{J_n}(\rho_{32})[\frac{4i}{\pi}\frac{1}{\sqrt{\rho_{21}\rho_{22}}}\cos(\rho_{22} - \rho_{21})]\\ & - k_1k_2\dot{H_n}(\rho_{11})J_n(\rho_{32})[\frac{4i}{\pi}\frac{1}{\sqrt{\rho_{21}\rho_{22}}}\cos(\rho_{22} - \rho_{21})]\\ & + k_1k_2\dot{H_n}(\rho_{11})\dot{J_n}(\rho_{32})[\frac{2i}{\pi}\frac{1}{\sqrt{\rho_{21}\rho_{22}}}\sin(\rho_{22} - \rho_{21})]. \end{aligned}$$
Thus, the coefficients governing the transmission are given by
$$\begin{aligned} d_n &= \frac{4ik_2}{\pi \sqrt{r_1r_2}} \Big \{[4k_2^2H_n(\rho_{11})J_n(\rho_{32}) + k_1k_2\dot{H_n}(\rho_{11})\dot{J_n}(\rho_{32})]\sin(k_2 \Delta)\\ &\quad+ 2[k_3k_2H_n(\rho_{11})\dot{J_n}(\rho_{32}) - k_1k_2\dot{H_n}(\rho_{11})J_n(\rho_{32})]\cos(k_2 \Delta) \Big \}^{{-}1}. \end{aligned}$$
The transmission coefficients determine the energy density in the core dielectric.

To prove that the core is not fully isolated from the incident radiation, we study the dependence of the energy density in the core as a function of the frequency of incident radiation. The energy density of the EM wave is given by $U = \frac {1}{2}\epsilon \epsilon _0E_0^2 + \frac {1}{2}\mu _0H_0^2$. From Eq. (7) we see that for the TM-polarized radiation, the electric-field at the center of core depends on $J_0$ as $J_n(r = 0) = 0$ for $n = \pm 1, \pm 2 \dots$. Using the symmetry considerations the magnetic field at the center of the core does not contribute to the energy density. Thus, the energy density at the center is given by,

$$U(r = 0) = \frac{1}{2}\epsilon_0\epsilon_3E_0^2 |d_0|^2.$$
For TM waves, the energy density at the center relative to the incident energy density is given by $|d_0|^2$. Estimating the order of magnitude of the terms in Eq. (16), we see that the leading term in the denominator is $\propto H_n(\rho _{11})J_n(\rho _{32})$ as $k_2 >> k_1, k_3$. The remaining terms have a relatively small magnitude and, due to the properties of the Bessel functions, the zeros of these terms do not overlap with the zeros of the leading term. Additionally, the Hankel function $H_n(\rho _{11})$ is complex-valued and its absolute value remains non-zero. Therefore, the behavior of the transmission coefficients $d_n$ is determined by $J_0(\rho _{32})$ and the peaks of transmission appear at the zeros of $J_0(\rho _{32})$. In Table 1, we summarize the values of $k_3r_2$ required to satisfy the resonance conditions.

Tables Icon

Table 1. Resonance conditions for TM waves

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In Fig. 2, we plot the variation of zeroth-order transmission coefficient $d_0$ as a function of frequency and the scattering parameter $\chi _s = k_3r_2$. Figure 2(a) shows the energy in the center of the air core surrounded by an aluminum shell with thickness $\Delta$ and air. We observe an increase of the energy density in the core at the frequencies corresponding to the Mie resonances of the dielectric core. The spectral location of the resonances, in terms of the scattering parameter, agrees well with the analytical predictions. The absolute amplitude of the resonant peaks decreases with the increase of the shell thickness, but the energy density enhancement with respect to the off-resonant frequencies remains the same. The quality factor of the resonances in the Mie resonators surrounded by a metallic shell is much higher than for the purely dielectric resonator. Similarly, Fig. 2(b) shows the energy in the center of the FR4 core for the same two shell thicknesses. The increase in the core permittivity leads to the shift of the resonance towards lower frequency. The Mie resonances of the dielectric core result in an enhancement of the energy density by six orders of magnitude comparing to the off-resonant frequencies. In both cases presented in Figs. 2(a) and 2(b), the change in the surrounding medium does not influence the resonant frequencies and leads to a small change in the energy density inside the mesoshell. Figures 2(c)–(h) show the energy density distribution in the core at different frequencies indicated in Figs. 2(a) and 2(b). We find that the amplitude of field at the center increases by two orders of magnitude with an increase in permittivity of core from $\epsilon _3=1$ to $\epsilon _3=4.28$. For the shell thickness below the skin depth, the quality factor of the resonances remains comparable to the case shown in Fig. 2, while the amplitude of the fields in the resonator can be amplified compared to the incident wave. This provides a mechanism for increased energy confinement alternative to plasmonic core–shell nanoparticles. In this case, the energy localization takes place at the center of the dielectric core [see Figs. 2(c)–(h)] and is not limited to the immediate vicinity of the metal/dielectric interfaces. Such mesoshells may offer a new platform for enhanced nonlinear light–matter interactions.

 figure: Fig. 2.

Fig. 2. Relative energy density at the center of the cylindrical mesoshell with two different thicknesses $\Delta = 2 \mu$m (blue) and $\Delta = 10 \mu$m (orange) for different core materials: (a) air core, (b) FR4 dielectric core with $\epsilon _3 = 4.28$. (c)–(h) Electric field intensity distributions in the core of radius $3.299$ cm and thickness $\Delta =10$ $\mu$m for air core (c)–(e) and FR4 core (f)–(h). The field intensity is normalized to the intensity of the incident plane wave. The values on the radial axis are given in centimeters. In all the calculations the dispersion of the conductivity is neglected.

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Figure 3 shows the variation of the position and amplitude of the first resonant peak, as a function of geometry of the mesoshell. As the inner radius of the mesoshell is increased, the resonant frequency red-shifts, as seen in Fig. 3(a). The changes in the outer radius do not affect the spectral position of the resonance, as the size of the resonating core is not modified [see Fig. 3(b)]. In both cases, the increase in the shell thickness leads to the decrease in the amplitude of the fields penetrating to the core region.

 figure: Fig. 3.

Fig. 3. Spectral dependence of the transmission coefficient on the geometry of the mesoshell with an air core. (a) $\Delta$ is increased by decreasing the inner radius $r_2$ while keeping the outer radius $r_1$ constant. (b) $\Delta$ is increased by increasing the outer radius while $r_2=\rm {const}$. The permittivity of the surrounding is taken as $\epsilon _1=1$.

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3.2 Fields in the conducting shell region

The electric field in the conducting mesoshell given by Eq. (7) is proportional to the coefficients $b_n$ and $c_n$. These coefficients are determined by solving Eq. (15) and computed numerically using the same approximation for large complex arguments for Bessel and Hankel functions as in Section 3.1. The electric field in region 2 is given by

$$\begin{aligned}E_{2,z}(r, \phi) &= E_0\sum_{n ={-}\infty}^{\infty}i^ne^{in \phi}\Big\{\frac{2i}{\pi^2r_1\sqrt{r_2r}} \Big[(k_3 \dot{J}_n(\rho_{32}) - 2ik_2J_n(\rho_{32}))e^{ik_2(r_2-r)}\\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + (k_3 \dot{J}_n(\rho_{32}) + 2ik_2J_n(\rho_{32}))e^{{-}ik_2(r_2-r)}\Big]\Big\}\\ & \quad \times \Big \{[4k_2^2H_n(\rho_{11})J_n(\rho_{32}) + k_1k_2\dot{H_n}(\rho_{11})\dot{J_n}(\rho_{32})]\sin(k_2 \Delta)\\ & \quad + 2[k_3k_2H_n(\rho_{11})\dot{J_n}(\rho_{32}) - k_1k_2\dot{H_n}(\rho_{11})J_n(\rho_{32})]\cos(k_2 \Delta) \Big \}^{{-}1}. \end{aligned}$$
From Eq. (12), we know that amplitude of the electric field intensity decays exponentially with the distance in the conducting medium. In Fig. 4, we show the evolution of electric intensity (in the logarithmic scale) with the radius as the field propagates from the outer surface to the inner surface. The solid lines show the field distributions at resonant frequencies for two different core materials. The rate of the exponential decay is different as the skin depth is inversely proportional to square root of the frequency of the incident wave. Due to the non-zero value of the field in the core, the behavior of the field in the metal region close to the inner surface deviates from the exponential decay. We also find that the field intensity in the core region increases towards the center at the resonant frequency and appears to be constant (seen on the same scale) in the off-resonant case as confirmed by the plots shown in Fig. 2(c)–(h). For the FR4 core with high permittivity, the field at the core/shell boundary is two orders of magnitude larger than for the air core.

 figure: Fig. 4.

Fig. 4. Logarithm of electric field intensity in the conducting medium as the wave propagates to the core region. As the permittivity of the core materials increases the field on the inner surface increases, deviating from exponential attenutaion thoughout the conducting region. The solid line shows the distribution of electric fields at the resonant frequency and dashed represents fields away from resonance.

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3.3 Surrounding region

In this section, we compare the scattering coefficients of the core-shell structure and of the solid metallic cylinder to understand the impact of enhanced penetration. The difference between the boundary conditions for perfectly conducting surface and finitely conducting surface can be seen in the the scattering coefficients for solid conducting cylinder, respectively

$$a_{n, \rm{solid}}^{\sigma \rightarrow \infty} ={-}J_n(k_1r_1)/H_n(k_1r_1),$$
$$a_{n, \rm{solid}} ={-}\frac{ik_2 J_n(k_1r_1)\dot{J}_n(k_2r_1) + k_1\dot{J}_n(k_1 r_1)J_n(k_2r_1)}{ik_2 H_n(k_1r_1)\dot{J}_n(k_2r_1) - k_1\dot{H}_n(k_1 r_1)J_n(k_1r_1)}.$$
The expression for scattering coefficient for the core-shell structure is calculated similar to the transmission coefficients. The scattering coefficient, $a_n = \frac {Det(\mathbb {A})}{Det(\mathbb {M})}$, where $\mathbb {A}$
$$\mathbb{A} = \left[ \begin{array}{cccc} J_n(\rho_{11}) & J_n(\rho_{21}) & H_n(\rho_{21}) & 0 \\ 0 & J_n(\rho_{22}) & H_n(\rho_{22}) & - J_n(\rho_{32}) \\ k_1 \dot{J_n}(\rho_{11}) & k_2 \dot{J_n}(\rho_{21}) & k_2 \dot{H_n}(\rho_{21}) & 0 \\ 0 & k_2 \dot{J_n}(\rho_{22}) & k_2 \dot{H_n}(\rho_{22}) & - k_3 \dot{J_n}(\rho_{32}) \end{array} \right].$$
Using the asymptotic approximation Eq. (11) the scattering coefficient can be computed as
$$\begin{aligned} a_n &= \quad\Big \{4[{-}k_2^2J_n(\rho_{11})J_n(\rho_{32}) + k_1k_2\dot{J_n}(\rho_{11})\dot{J_n}(\rho_{32})]\sin(k_2 \Delta)\\ & \quad\quad+ 2[k_3k_2J_n(\rho_{11})\dot{J_n}(\rho_{32}) - k_1k_2\dot{J_n}(\rho_{11})J_n(\rho_{32})]\cos(k_2 \Delta)\Big \}\\ & \quad\quad\times \Big \{[4k_2^2H_n(\rho_{11})J_n(\rho_{32}) + k_1k_2\dot{H_n}(\rho_{11})\dot{J_n}(\rho_{32})]\sin(k_2 \Delta)\\ & \quad\quad+ 2[k_3k_2H_n(\rho_{11})\dot{J_n}(\rho_{32}) - k_1k_2\dot{H_n}(\rho_{11})J_n(\rho_{32})]\cos(k_2 \Delta) \Big \}^{{-}1}. \end{aligned}$$
Using the same order of magnitude approximation as in the previous sections, it can be seen that the behavior of $a_n$ is dictated by the terms multiplying $k_2^2$, namely $J_n(\rho _{11})J_n(\rho _{32})/(H_n(\rho _{11})J_n(\rho _{32}))$.

As it can be seen from Fig. 5(a) the zeroth order coefficient of the scattered field for the mesoshell and the bulk cylinder show a similar behavior. A close inspection of the vicinity of the first minimum of $|a_0|^2$ reveals a small spectral shift and a change in the magnitude. The difference between the total scattering cross sections $\propto \sum _{n = - \infty }^{n=\infty } |a_n|^2$ [see Fig. 5(b)] in these two cases is minimal. This suggests that the scattering is dictated by the outer surface of the mesoshell and is only slightly affected by the changes in the core. Similar calculation can be repeated for transverse electric case.

 figure: Fig. 5.

Fig. 5. The comparison between zeroth order scattering coefficients. The peak position of peak shifts by $0.5$ MHz.

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4. Spherical shell

The analysis for the EM wave scattering from the infinite cylinder is essentially two-dimensional. In this section, we extend the formalism to the three-dimensional case using an example of a spherical mesoshell structure. Let us consider a plane wave $(\mathbf {E}_\textrm {inc} = \hat {\mathbf {x}}E_0e^{ikz})$ incident on a wavelength-scale spherical mesoshell structure. For spherical geometry, we can rewrite the incident electric field as [21]

$$\mathbf{E}_\textrm{inc} = \sum_{n = 1}^{\infty}\zeta_n [\textbf{M}_{1n}^j + i\textbf{N}^j_{1n}]$$
where $\zeta _n = E_0i^n\frac {2n+1}{n(n+1)}$ and,
$$\mathbf{M}_{1n}^{j , h} = z_n^{j , h}(kr)[i\pi_{1n}(\theta) \hat{\boldsymbol{\theta}} - \tau_{1n}(\theta) \hat{\boldsymbol{\phi}}]e^{i \phi},$$
$$\mathbf{N}_{1n}^{j , h} = \frac{z_n^{j , h}(kr)}{r}n(n+1)P_n^1(\cos \theta)\hat{\mathbf{r}} + \frac{1}{r}\frac{\partial {\bigg[}rz_n(kr) {\bigg]}}{\partial r}{\bigg[} \tau_{1n}(\theta) \hat{\boldsymbol{\theta}} + i\pi_{1n}(\theta) \hat{\boldsymbol{\phi}} {\bigg]} e^{i \phi}.$$
$z_n^j(kr) = j_n(kr)$ and $z_n^h(kr) = h_n(kr)$. The expressions are written in terms of $\pi _{1n}(\theta )$ and $\tau _{1n}(\theta )$ which are called scalar tesseral function related to the Legendre function $(P^1_n(\cos \theta ))$ [18,23].
$$\pi_{1n}(\theta) = \frac{P^1_n(\cos \theta)}{\sin \theta}, ~~~~~~~~\tau_{1n}(\theta) = \frac{\partial}{\partial \theta} P^1_n(\cos \theta).$$
Similar to cylindrical case, outgoing scattered vector wavefunctions $\{\textbf {M}_{1n}^h, \textbf {N}^h_{1n}\}$ depend on the spherical Hankel functions $h_n(kr)$. Electric field is each region has the following form:
$$\mathbf{E}_1 = \sum_{n = 1}^{\infty}\zeta_n[\textbf{M}_{1n}^j + a_n \textbf{N}_{1n}^h - i\textbf{N}_{1n}^j - ib_n \textbf{N}_{1n}^h],$$
$$\mathbf{E}_2 = \sum_{n = 1}^{\infty}\zeta_n[c_n\textbf{M}_{1n}^j + d_n \textbf{M}_{1n}^h - if_n\textbf{N}_{1n}^j - ig_n \textbf{N}_{1n}^h],$$
$$\mathbf{E}_3 = \sum_{n = 1}^{\infty}\zeta_n[p_n\textbf{M}_{1n}^j - iq_n\textbf{N}_{1n}^h],$$
The scattered electric and magnetic fields are dependent on $(a_n, b_n)$ and transmitted fields are governed by $(p_n, q_n)$. The magnetic fields can be determined by Maxwell-Faraday equation and the coefficients of $\{\textbf {M}_{1n}, \textbf {N}_{1n}\}$ are interchanged due to curl properties. The coefficients can be determined using boundary conditions which lead to a system of 8 linear equations. Using substitution, we eliminate 4 variables and obtain the system of the following four equations
$$\left[ \begin{array}{cc} \mathbb{M}_1 & \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right] \\ \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right] & \mathbb{M}_2 \end{array} \right] \left[ \begin{array}{c} c_n \\ d_n \\ f_n \\ g_n \end{array} \right] = \left[ \begin{array}{c} -\frac{ik_1}{\rho_{11}} \\ 0 \\ -\frac{ik_2}{\rho_{11}} \\ 0 \end{array} \right],$$
where,
$$\mathbb{M}_1 = \left[ \begin{array}{cc} \{\Gamma_h(\rho_{21})h(\rho_{11}) - \Gamma_h(\rho_{11})h(\rho_{21}) \}& \{\Gamma_j(\rho_{21})h(\rho_{11}) - \Gamma_h(\rho_{11})j(\rho_{21})\} \\ \{\Gamma_h(\rho_{22})j(\rho_{32}) - \Gamma_j(\rho_{32})h(\rho_{22})\}& \{\Gamma_j(\rho_{22})j(\rho_{32}) - \Gamma_j(\rho_{32})j(\rho_{22})\} \end{array} \right],$$
$$\mathbb{M}_2 = \left[ \begin{array}{cc} \{k_1^2\Gamma_h(\rho_{21})h(\rho_{11}) - k_2^2\Gamma_h(\rho_{11})h(\rho_{21}) \}& \{k_1^2\Gamma_j(\rho_{21})h(\rho_{11}) - k_2^2\Gamma_h(\rho_{11})j(\rho_{21})\} \\ \{k_3^2\Gamma_h(\rho_{22})j(\rho_{32}) - k_2^2\Gamma_j(\rho_{32})h(\rho_{22})\}& \{k_3^2\Gamma_j(\rho_{22})j(\rho_{32}) - k_2^2\Gamma_j(\rho_{32})j(\rho_{22})\} \end{array} \right].$$
For numerical calculations, high conductivity of the shell requires the use of the asymptotic form of spherical Bessel and Hankel functions. We use an order of magnitude approximations in the asymptotic form of the spherical Bessel and Hankel functions to simplify the coefficients in Eqs. (33) and (34), where $Det[\mathbb {M}_1] = h(\rho _{11})j(\rho _{32})[i\sin [k_2 \Delta ]]$ and $Det[\mathbb {M}_2] = \frac {k_2^4}{\rho _{21}\rho _{22}}\Gamma _h(\rho _{11}) \Gamma _j(\rho _{32})[i\sin [k_2 \Delta ]]$. Thus, coefficients governing the transmitted field are given by
$$p_n = \left \{-\rho_{11}\rho_{22}Det\mathbb{M}_1\right \}^{{-}1} = i\Big \{ \rho_{11}\rho_{22}h(\rho_{11})j_n(\rho_{32})\sin(k_2 \Delta) \Big \}^{{-}1},$$
$$q_n = \left \{-\rho_{11}\rho_{22}Det\mathbb{M}_2\right \}^{{-}1} = i\frac{\rho_{21}}{\rho_{11}}\Big \{ k_2^4\Gamma_h(\rho_{11})\Gamma_j(\rho_{32})\sin(k_2 \Delta)\Big \}^{{-}1}.$$
The resonance condition for $p_1$ is given by zeros of $j_1(k_3r_2)$ and for $q_1$ it is given by zeros of $\Gamma _j(\rho _{32}) = \frac {\partial }{\partial \rho _{32}}\rho _{32}j_1(\rho _{32})$. In order to determine the energy density at the center of the shell, we have to evaluate $\mathbf {M}$ and $\mathbf {N}$ at $r = 0$. We know that $j_n(r) = 0$ for $n = 1, 2 \dots$, therefore all $\mathbf {M}$ type wavefunctions are zero at the center. Also $\lim _{\rho \rightarrow 0}\frac {j_n(\rho )}{\rho } = \frac {1}{3}$ for $n = 1$ and zero otherwise. Along with the identity $\Gamma _j(\rho ) = \frac {\rho }{(2n + 1)}[(n + 1)j_{n-1}(\rho ) - nj_{n+1}(\rho )]$, we get, $\mathbf {N}^j_{11}(r = 0) = \frac {2}{3}P_1^1(\cos \theta )\hat {\mathbf {r}} + \frac {2}{3}[\tau _{11}(\theta ) \hat {\boldsymbol {\theta }} + i\pi _{11}(\theta ) \hat {\boldsymbol {\phi }}]e^{i \phi }$. Thus, electric and magnetic energy density at the center are given by
$$|\mathbf{E}|^2 = |q_1 E_0 \mathbf{N}^j_{11}|^2,$$
$$|\mathbf{H}|^2 = |p_1 H_0 \mathbf{N}^j_{11}|^2.$$
Hence, the electromagnetic energy density at the center relative to incident density is given by $|p_1|^2 + |q_1|^2$.

In Fig. 6, we plot the variation of energy density at the center of the spherical mesoshell as a function of frequency and the scattering parameter $\chi_s = k_3r_2$. Figure 6 shows the energy in the center of the air core and the FR4 core, respectively for the surrounding medium taken as air. Similar to the cylindrical case, the resonant frequencies red shift as the permittivity of the core dielectric is increased. For the spherical geometry, the resonant increase in the energy density in the center of the core is even larger than for the cylindrical case. The resonant enhancement reaches ten orders of magnitude, leading to a relatively large amount of energy being stored in the core.

 figure: Fig. 6.

Fig. 6. Relative energy density at the center of the spherical mesoshell proportional to transmission coefficients for two different core materials: Air (a), FR4 dielectric (b). The shell thickness $\Delta = 10 \mu$m and the outer radius $r_1=3.3$ cm.

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5. Discussion

We have analyzed the interaction of EM waves with metal-dielectric mesoshell structures with dimensions comparable to the incident wavelength. We show that despite the presence of a highly conducting electromagnetically thick mesoshell, there exist frequencies where the amplitude of the EM field in the core is increased by six orders of magnitude in case of an infinite cylinder and by up to ten orders of magnitude in the case of a spherical mesoshell comparing to off-resonant frequencies. This study discovers a new mechanism for increased energy confinement, alternative to plasmonic core–shell nanoparticles. The energy localization is not limited to the immediate vicinity of the metal/dielectric interfaces, but rather occurs at the center of the dielectric core. These results were obtained using an extended Mie formalism for finitely conducting metallic shells employing the asymptotic form of Bessel and Hankel functions for large imaginary arguments. The asymptotic formalism enables us to numerically observe the enhancement in the transmission of EM fields through thick metallic shells at resonant frequencies. The validity of the approximation requires the dimension of the structure to be comparable to wavelength. Therefore, the behavior of the mesoshell differs from the nanoshells, where the structure is much smaller that the wavelength. We can expect to find a similar resonant energy penetration enhancement for other geometries of wavelength-scale scatterers. The formalism provides a fundamental insight into the effect of changes in material and geometrical shapes, which can also be used for building a new class of meta-atoms or meta-molecules for novel metamaterials’ designs. The results obtained can be experimentally accessed at microwave frequencies by using a vector network analyzer with a measuring probe inside the cylindrical cavity. The change in resonance frequencies due to change in permittivity of core dielectric and accurate knowledge of fields in metallic shells can be leveraged to build sensors to detect material distribution at higher accuracy. Similar effects should be observable at optical frequencies, opening a way for engineering enhanced energy localization in dielectric structures in the presence of the metallic shells.

Funding

Army Research Office (W911NF1810348).

Acknowledgments

This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

References

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Figures (6)
Fig. 1.
Fig. 1. (a) Schematic of a plane wave incident on the cross-section of the core-shell structures. (b) Infinite cylindrical mesoshell. (c) Spherical mesoshell.

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Fig. 2.
Fig. 2. Relative energy density at the center of the cylindrical mesoshell with two different thicknesses $\Delta = 2 \mu$m (blue) and $\Delta = 10 \mu$m (orange) for different core materials: (a) air core, (b) FR4 dielectric core with $\epsilon _3 = 4.28$. (c)–(h) Electric field intensity distributions in the core of radius $3.299$ cm and thickness $\Delta =10$ $\mu$m for air core (c)–(e) and FR4 core (f)–(h). The field intensity is normalized to the intensity of the incident plane wave. The values on the radial axis are given in centimeters. In all the calculations the dispersion of the conductivity is neglected.

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Fig. 3.
Fig. 3. Spectral dependence of the transmission coefficient on the geometry of the mesoshell with an air core. (a) $\Delta$ is increased by decreasing the inner radius $r_2$ while keeping the outer radius $r_1$ constant. (b) $\Delta$ is increased by increasing the outer radius while $r_2=\rm {const}$. The permittivity of the surrounding is taken as $\epsilon _1=1$.

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Fig. 4.
Fig. 4. Logarithm of electric field intensity in the conducting medium as the wave propagates to the core region. As the permittivity of the core materials increases the field on the inner surface increases, deviating from exponential attenutaion thoughout the conducting region. The solid line shows the distribution of electric fields at the resonant frequency and dashed represents fields away from resonance.

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Fig. 5.
Fig. 5. The comparison between zeroth order scattering coefficients. The peak position of peak shifts by $0.5$ MHz.

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Fig. 6.
Fig. 6. Relative energy density at the center of the spherical mesoshell proportional to transmission coefficients for two different core materials: Air (a), FR4 dielectric (b). The shell thickness $\Delta = 10 \mu$m and the outer radius $r_1=3.3$ cm.

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Tables (1)

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Table 1. Resonance conditions for TM waves

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Equations (36)

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( ∇ 2 + k j 2 ) { E j ( r , t ) , H j ( r , t ) } T = 0 ,
n ^ × [ E j + 1 ( r , t ) − E j ( r , t ) ] = 0 ,
n ^ × [ H j + 1 ( r , t ) − H j ( r , t ) ] = 0 ,
E inc ( r , t ) = z ^ E 0 e i ( k 1 x − ω t ) = z ^ E 0 e − i ω t ∑ n = − ∞ ∞ i n J n ( k 1 r ) e i n ϕ .
E 1 , z ( r , ϕ ) = E 0 ∑ n = − ∞ ∞ i n { J n ( k 1 r ) + a n H n ( 1 ) ( k 1 r ) } e i n ϕ ,
E 2 , z ( r , ϕ ) = E 0 ∑ n = − ∞ ∞ i n { b n J n ( k 2 r ) + c n H n ( 1 ) ( k 2 r ) } e i n ϕ ,
E 3 , z ( r , ϕ ) = E 0 ∑ n = − ∞ ∞ i n { d n J n ( k 3 r ) } e i n ϕ ,
[ − H n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) 0 0 J n ( ρ 22 ) H n ( ρ 22 ) − J n ( ρ 32 ) − k 1 H n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) 0 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) − k 3 J n ˙ ( ρ 32 ) ] [ a n b n c n d n ] = [ J n ( ρ 11 ) 0 k 1 J n ˙ ( ρ 11 ) 0 ] .
D = [ − H n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) J n ( ρ 11 ) 0 J n ( ρ 22 ) H n ( ρ 22 ) 0 − k 1 H n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) k 1 J n ˙ ( ρ 11 ) 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) 0 ] .
D e t ( D ) = − 4 π 2 r 1 r 2 .
D e t ( M ) = k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) [ J n ˙ ( ρ 21 ) H n ˙ ( ρ 22 ) − J n ˙ ( ρ 22 ) H n ˙ ( ρ 21 ) ] − k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) [ J n ˙ ( ρ 21 ) H n ( ρ 22 ) − J n ( ρ 22 ) H n ˙ ( ρ 21 ) ] − k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) [ J n ( ρ 21 ) H n ˙ ( ρ 22 ) − J n ˙ ( ρ 22 ) H n ( ρ 21 ) ] + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) [ J n ( ρ 21 ) H n ( ρ 22 ) − J n ( ρ 22 ) H n ( ρ 21 ) ] .
I = | E | 2 = | E 0 | 2 e − 2 x δ .
ρ 21 ≈ ρ 22 = ( 1 + i ) × 2.2 × 10 4 .
J n ( ρ ) ≅ 2 π ρ cos ⁡ [ ρ − ( n + 1 2 ) π 2 ] = 1 2 [ 2 π ρ e i [ ρ − ( n + 1 2 ) π 2 ] + 2 π ρ e − i [ ρ − ( n + 1 2 ) π 2 ] ] .
D e t ( M ) = k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) [ 8 i π 1 ρ 21 ρ 22 sin ⁡ ( ρ 22 − ρ 21 ) ] − k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) [ 4 i π 1 ρ 21 ρ 22 cos ⁡ ( ρ 22 − ρ 21 ) ] − k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) [ 4 i π 1 ρ 21 ρ 22 cos ⁡ ( ρ 22 − ρ 21 ) ] + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) [ 2 i π 1 ρ 21 ρ 22 sin ⁡ ( ρ 22 − ρ 21 ) ] .
d n = 4 i k 2 π r 1 r 2 { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ⁡ ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) − k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ⁡ ( k 2 Δ ) } − 1 .
U ( r = 0 ) = 1 2 ϵ 0 ϵ 3 E 0 2 | d 0 | 2 .
E 2 , z ( r , ϕ ) = E 0 ∑ n = − ∞ ∞ i n e i n ϕ { 2 i π 2 r 1 r 2 r [ ( k 3 J ˙ n ( ρ 32 ) − 2 i k 2 J n ( ρ 32 ) ) e i k 2 ( r 2 − r )                                                         + ( k 3 J ˙ n ( ρ 32 ) + 2 i k 2 J n ( ρ 32 ) ) e − i k 2 ( r 2 − r ) ] } × { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ⁡ ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) − k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ⁡ ( k 2 Δ ) } − 1 .
a n , s o l i d σ → ∞ = − J n ( k 1 r 1 ) / H n ( k 1 r 1 ) ,
a n , s o l i d = − i k 2 J n ( k 1 r 1 ) J ˙ n ( k 2 r 1 ) + k 1 J ˙ n ( k 1 r 1 ) J n ( k 2 r 1 ) i k 2 H n ( k 1 r 1 ) J ˙ n ( k 2 r 1 ) − k 1 H ˙ n ( k 1 r 1 ) J n ( k 1 r 1 ) .
A = [ J n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) 0 0 J n ( ρ 22 ) H n ( ρ 22 ) − J n ( ρ 32 ) k 1 J n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) 0 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) − k 3 J n ˙ ( ρ 32 ) ] .
a n = { 4 [ − k 2 2 J n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 J n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ⁡ ( k 2 Δ ) + 2 [ k 3 k 2 J n ( ρ 11 ) J n ˙ ( ρ 32 ) − k 1 k 2 J n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ⁡ ( k 2 Δ ) } × { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ⁡ ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) − k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ⁡ ( k 2 Δ ) } − 1 .
E inc = ∑ n = 1 ∞ ζ n [ M 1 n j + i N 1 n j ]
M 1 n j , h = z n j , h ( k r ) [ i π 1 n ( θ ) θ ^ − τ 1 n ( θ ) ϕ ^ ] e i ϕ ,
N 1 n j , h = z n j , h ( k r ) r n ( n + 1 ) P n 1 ( cos ⁡ θ ) r ^ + 1 r ∂ [ r z n ( k r ) ] ∂ r [ τ 1 n ( θ ) θ ^ + i π 1 n ( θ ) ϕ ^ ] e i ϕ .
π 1 n ( θ ) = P n 1 ( cos ⁡ θ ) sin ⁡ θ ,                 τ 1 n ( θ ) = ∂ ∂ θ P n 1 ( cos ⁡ θ ) .
E 1 = ∑ n = 1 ∞ ζ n [ M 1 n j + a n N 1 n h − i N 1 n j − i b n N 1 n h ] ,
E 2 = ∑ n = 1 ∞ ζ n [ c n M 1 n j + d n M 1 n h − i f n N 1 n j − i g n N 1 n h ] ,
E 3 = ∑ n = 1 ∞ ζ n [ p n M 1 n j − i q n N 1 n h ] ,
[ M 1 [ 0 0 0 0 ] [ 0 0 0 0 ] M 2 ] [ c n d n f n g n ] = [ − i k 1 ρ 11 0 − i k 2 ρ 11 0 ] ,
M 1 = [ { Γ h ( ρ 21 ) h ( ρ 11 ) − Γ h ( ρ 11 ) h ( ρ 21 ) } { Γ j ( ρ 21 ) h ( ρ 11 ) − Γ h ( ρ 11 ) j ( ρ 21 ) } { Γ h ( ρ 22 ) j ( ρ 32 ) − Γ j ( ρ 32 ) h ( ρ 22 ) } { Γ j ( ρ 22 ) j ( ρ 32 ) − Γ j ( ρ 32 ) j ( ρ 22 ) } ] ,
M 2 = [ { k 1 2 Γ h ( ρ 21 ) h ( ρ 11 ) − k 2 2 Γ h ( ρ 11 ) h ( ρ 21 ) } { k 1 2 Γ j ( ρ 21 ) h ( ρ 11 ) − k 2 2 Γ h ( ρ 11 ) j ( ρ 21 ) } { k 3 2 Γ h ( ρ 22 ) j ( ρ 32 ) − k 2 2 Γ j ( ρ 32 ) h ( ρ 22 ) } { k 3 2 Γ j ( ρ 22 ) j ( ρ 32 ) − k 2 2 Γ j ( ρ 32 ) j ( ρ 22 ) } ] .
p n = { − ρ 11 ρ 22 D e t M 1 } − 1 = i { ρ 11 ρ 22 h ( ρ 11 ) j n ( ρ 32 ) sin ⁡ ( k 2 Δ ) } − 1 ,
q n = { − ρ 11 ρ 22 D e t M 2 } − 1 = i ρ 21 ρ 11 { k 2 4 Γ h ( ρ 11 ) Γ j ( ρ 32 ) sin ⁡ ( k 2 Δ ) } − 1 .
| E | 2 = | q 1 E 0 N 11 j | 2 ,
| H | 2 = | p 1 H 0 N 11 j | 2 .
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