1. Introduction

Electromagnetic (EM) waves scatter when interacting with differing material structures; a phenomenon which can be exploited and forms the basis of numerous applications. Scattering of EM waves closely depends on the properties of the source, such as radiation type (plane wave, line-source, etc.), polarization (e.g., transverse electric (TE), transverse magnetic (TM), etc.), as well as the geometrical and material properties of a scatterer. Standard scattering theory shows that, for a subwavelength homogeneous structure, the absolute scattering cross section (σ_sca) describing the scattering capacity of a scatterer is primarily constrained by the so-called single-channel limit [1]. In particular, the maximum cross section of an atom in two- (three-) dimensional space cannot exceed 2λ/π (3λ²/π) where λ is the wavelength in vacuum. Recently, Ruan and Fan have theoretically shown that σ_sca of a subwavelength structure can be significantly higher than the single-channel limit, a phenomenon referred to as superscattering [2]. The concept of superscattering stems from the fact that although cross sections in each individual scattering channel are bounded, one can maximize the total σ_sca by leveraging contributions from several channels (resonances). To overcome the limitations of homogeneous scatters, subwavelength structures with higher structural and material complexity are required to manipulate distinct scattering channels. In particular, exploiting the dispersion engineering enabled by surface plasmons, scatterers based on alternative plasmonic/dielectric layered structures have been used to investigate superscattering phenomenon in the optical regime [3–7]. Recently, superscattering effects based on the Fano-like resonance in individual plasmonic nanodisks [8] and the resonances in isolated resonators associated with bound states in the continuum (BICs) [9] have been reported. Interestingly, on top of the total scattering cross section engineering, a precise mode design can also lead to highly directive optical elements [10,11].

Practical scatterers that can be represented by cylinders (e.g., the fuselage of airplanes) are of particular interest. Recently, Qian et al. have experimentally demonstrated superscattering from a metasurface-based multilayer structure [12], while Shcherbinin et al. have reported superscattering from subwavelength corrugated cylinders [13]. Both studies were based on the excitation of spoof surface plasmons which are polarization sensitive. Developing in parallel with plasmonic architectures, dielectric subwavelength structures have been demonstrated as an attractive platform for manipulating light with extremely low loss [14,15]. In contrast to the limited material candidates in the optical regime, at lower frequencies (such as microwave and THz) an abundance of materials, especially low-loss, high-permittivity (high-κ) ferroelectric ceramics (such as Ba_0.5Sr_0.5TiO₃ (BST), MgTiO₃-CaTiO₃, etc.), offer opportunities for comprehensive engineering of both the magnetic and electric response from pure dielectric subwavelength structures [16]. For instance, Zhao et al. have demonstrated isotropic negative permeability in an array of BST cubes [17], while Peng et al. have observed left-handed behavior in an array of BST resonators [18]. Furthermore, given the fact that the dielectric properties of ferroelectric ceramics can be easily tuned by varying the temperature or chemical doping, subwavelength structures of high-κ ceramics have been used to achieve reconfigurable meta-devices [19–22]. It should also be noted that, compared with their optical counterparts, dielectric structures operating at lower frequencies can be free standing, in which asymmetries introduced by a substrate can be avoided. Therefore, dielectric cylindrical systems can be a good candidate for exotic scattering responses. Nevertheless, a delicate balance between distinct resonance modes generally requires precise engineering of a complex cylindrical structure, which is an enormous challenge for conventional optimization methods [23–25]. A new strategy for obtaining superscattering from dielectric structures is highly desirable but has so far remained elusive.

In this work, starting from a mode analysis of cylindrical systems and leveraging an optimization approach based on gradient descent and the Gram-Schmidt process, we numerically validate superscattering from individual multilayered dielectric cylinders illuminated by a TM^z incident plane wave. By directly performing mode optimizations to allow individual channels to approach the single-channel limit, we show that a maximum normalized σ_sca of ∼5 can be achieved based on resonances associated with the lowest five angular momentum channels (0, ± 1, and ±2). Hereafter being referred to as the “pure” mode configuration, this scenario simultaneously results in high directivity along the propagation direction of the incident wave. Furthermore, by switching the optimization objective to maximize the value of σ_sca with more channels involved (referred to as the “complex” mode configuration), we obtained optimized σ_sca as large as ∼ 6.46 and ∼ 6.71 for two- and three- layer cylinders, respectively. It should be noted that the superscattering behavior from the proposed cylindrical systems are largely dependent on the allowed permittivity ranges. Additionally, we devise a superscattering device based on experimentally verified permittivity values which provides insight into the practical implementation of structures that exhibit this phenomenon.

2. Technical discussion

2.1 Overview

The problem addressed in this work concerns the realization of superscattering from an individual multilayer dielectric cylinder in free space which is excited by a plane wave. We chose an operating frequency of 2.2 GHz and a cylinder with an external radius of 18 mm (∼ 0.13λ). When studying the two-layer cylinder, the inner radius was set to 9 mm, while the three-layer cylinder was given inner and middle radii of 6 and 12 mm, respectively. Figure 1(a) shows an overview of the problem, with the scattered radiation from a dielectric cylinder and a cross section showing the material parameters and radii definitions. The polarization of the incident wave is oriented parallel to the cylinder (TM^z). Given this specific polarization, the system exhibits circular symmetry, i.e., the angle of the incident wave does not affect scattering. We note that Mie resonances are sensitive to the geometries of dielectric structures; for the considered cylinder system, changes in polarization can dramatically change the scattering properties. Cylindrical scatterers intended for operation at other polarizations (e.g., TE^z) require additional optimizations and are not considered in this work.

 

Fig. 1. Overview of the problem and core concepts. (a) Diagram of our design. A plane wave is incident upon a multilayer dielectric cylinder with an electric field polarized parallel to the cylinder axis (TM^z). The cylinder is defined by multiple radii and material parameters. (b) Intrinsic response of the cylinder’s exterior modes. (c) Parametric study of scattering cross section versus relative permittivity of a homogeneous cylinder. The distributions of the total electric field are shown on top for three permittivities where peak σ_sca is observed. (d) Scattered electric far field resulting from different numbers of consecutive modes being excited.

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An important metric to consider is the angle-dependent scattering cross section (σ(θ)). Shown in Eq. (1), σ(θ) corresponds to the ratio between the power of the scattered field and that of the incident field and is normalized to 2λ/π. Next, in Eq. (2), we calculate the absolute scattering cross section (σ_sca), and it will be our primary figure of merit when assessing different cylinder designs. For context, a perfectly electrically conducting (PEC) cylinder with a radius of 0.13λ has an σ_sca ∼1.3.

The value of σ_sca is dependent on the scattered electric field $({{E^{scat}}} )$, which can be calculated using Mie theory. This is a common approach used in similar problems for cylindrical and spherical geometries [26]. Given that the incident field $({{E^{inc}}} )$ is a plane wave in our study, the scattered electric field is represented as a summation of weighted Bessel or Hankel functions as shown in Eq. (3). We refer to these weightings b_n as the “modal coefficients” where n represents the mode number, and a mode is considered fully excited when |b_n| = 1. Each solution region (the cylinder’s core, annular layers, and the exterior) has its own summation term for the scattered electric field. In our study, we are solely concerned with the space outside of the cylinder since the modal coefficients in this location can be used to fully characterize the σ_sca of a particular cylinder design. Figure 1(b) shows the normalized scattered field of the lowest five individual modes calculated using Eq. (2), with n = 1 representing a dipole, n = 2 representing a quadrupole, and so on with higher order multipoles.

If the modes are allowed to interfere constructively, then adding more terms will augment the scattered electric field. Figure 1(d) depicts several electric far fields calculated from Eq. (3) using different numbers of consecutive modes and demonstrates how utilizing additional terms enhances scattering phenomena and improves directivity. Nonetheless, the modes must have the correct phase or else they may interfere destructively which produces sidelobes and/or backlobes. When they add together favorably, the values for σ_sca and forward directivity are maximized simultaneously.

We note that it is possible to model this problem using a simulation package such as COMSOL, but an open-source analytical method is preferable since it would allow increased control over various parameters, faster execution time, and better integration with optimization methods. The MieSolver by Hawkins is a suitable option [27]. It provides a MATLAB interface for fast computation of scattering from multilayer dielectric cylinders in two dimensions, and we can extract the modal coefficients corresponding to the region exterior to the cylinder from n = -5 to n = 5 for use in our custom optimization approach.

2.2 Optimization procedure

As an initial study, consider a one-layer (homogenous) cylinder whose relative permittivity (ε_r) is swept from 1 to 200. The results are displayed in Fig. 1(c). The maximum σ_sca is approximately 3, which is already over double the σ_sca of an equivalently sized PEC cylinder. Peaks occur at ε_r = 8.2, 44.1 and 108.7, which represent cases where multiple exterior modes are excited, leading to larger scattering. There are other narrower peaks that correspond to the interaction of different modes. We see in the corresponding near field distributions in Fig. 1(c) that these three cases of high scattering have identical E-field distributions outside of the cylinder, though inside of the cylinder, higher order resonances are excited as the permittivity increases. Another interesting outcome is that even though the permittivity becomes larger, σ_sca does not continue to rise in turn, i.e., a larger σ_sca cannot be realized with a homogeneous cylinder by increasing its permittivity value alone. Lastly, the regions around ε_r = 22.8, 72.8, and 151.8 are of note because they display low scattering. While not discussed in this work, these solutions show potential for scattering reduction which is of interest in areas such as stealth and cloaking technology [28,29].

The single layer cylinder is clearly limited in terms of scattering performance. Adding an additional layer gives more control over the modes and grants the potential for obtaining stronger scattering. To better understand the solution space, a parametric sweep of both the inner and outer permittivities ([ε₁, ε₂]) in a two-layer dielectric cylinder was performed, taking note of the corresponding values of σ_sca and the modal coefficients for each test. The results are shown in Fig. 2(a), where the peaks of each modal coefficient are superimposed over a surface plot of σ_sca. Regions where modal coefficient peaks intersect correspond to areas of high scattering.

 

Fig. 2. Optimization procedure. (a) Two-dimensional parametric sweep showing the peaks of modes n = 1 through n = 3. (b) Example of optimization procedure for a two-layer problem. (c) Gradient descent optimization making use of the Gram-Schmidt process. (d) Example of optimization procedure for a three-layer problem.

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There are different approaches to finding solutions with high σ_sca, with the most straightforward being a brute-force parametric sweep. Nevertheless, it has major limitations due to slow execution time and cumbersomeness of the resulting datasets. Additionally, evaluating a cylinder with more layers vastly increases the number of iterations required, leading to poor scalability. Furthermore, parametric sweeps may not be able to capture the fine features in the parameter space which are critical for leveraging high order modes as we will discuss in a later section. Another possibility is to implement optimization to directly find the maximum σ_sca, which can, however, introduce a new set of issues. The direct optimization of σ_sca as a function of two-layer or three-layer permittivity is problematic. In a global optimization scenario, such as with a genetic algorithm [4] or simulated annealing [30], the entire solution space is searched, and the optimizer can often become stuck in one of the numerous local minima. Likewise, with local optimization, where only solutions in a certain region are considered, the optimizer would find the closest minimum and halt. This would again provide a suboptimal solution. These approaches can be improved by utilizing a type of guess and check method where one layer is optimized at a time and other layers are consistently retuned [11]. Even so, this is still an inefficient process because it does not make explicit use of the physics of the system.

The main disadvantage of these techniques is that they solely consider the σ_sca or directivity, which are functions that are difficult to work with, and the problem becomes akin to finding a “needle in a haystack.” In local and global optimizations of σ_sca, the lowest order modes dominate the structure of the cost surface. Because optimal solutions use sensitive higher order modes, it is unlikely that these optimization methods will converge to one of the said solutions. To this end, we propose an alternative concept where optimization is based on the modal coefficients, which do not suffer from the local minima issue. This new approach guarantees that higher order modes will be used because it works from most sensitive to least sensitive modes, and successive local optimizations on lower order modes are bound to the optimal subspace of the previous higher order modes. Furthermore, the proposed physics-informed optimization strategy is far more reliable and robust than global or local optimization approaches because it leverages an understanding of the physics of Mie scattering rather than assuming a black-box system. At optical frequencies, physics-informed optimization has proven effective for designing asymmetric superscattering structures based on silicon (Si) and titanium dioxide (TiO₂) [29,31,32]. Our approach further leverages this concept to create scatterers that utilize higher index materials only available at RF frequencies.

As mentioned before, the individual modes are well behaved, with fewer local minima in addition to being continuous and differentiable. Likewise, we found that regions where modal amplitude maxima intersect correspond to areas of high σ_sca (see Fig. 2(a)). Considering this information, we formulate a custom optimization algorithm that searches for intersections between modal maxima rather than directly optimizing σ_sca or directivity. Since we have a fast evaluation method in the MieSolver, where a single iteration is measured in milliseconds, local optimization based on gradient descent is an appealing solution [33]. There are multiple regions where modal peaks intersect, so restarting with different initial conditions is necessary. The function ${f_n}$ is provided in Eq. (4) and evaluates the magnitude of mode n for a given set of permittivities [ε₁, ε₂, ε₃]. Equation (5) calculates the direction of maximum change (i.e., gradient) given the permittivity of each of up to three layers. The derivative of ${f_n}$ relative to ${\varepsilon _m}$ is determined in Eq. (6), where m is the layer number, and h is the finite difference. Finally, Eq. (7) provides the next set of permittivity values ${\vec{\varepsilon }^{({i + 1} )}}$ to step towards based on the previous set ${\vec{\varepsilon }^i}$, the gradient $\nabla {f_n}$, and the step size s.

Still, implementing a single gradient descent will only give a solitary point on one mode’s peak. To find intersections, one can instead perform a series of local optimizations while sequentially reducing the dimensionality of the problem. Our process applied to a two-layer cylinder can be visualized as searching a plane for a curve, then searching a curve for a point. By adding an additional layer, the procedure transforms into searching a volume for a surface, searching a surface for a curve, and then searching a curve for a point.

This method is promising, but it introduces a new issue that can best be demonstrated with an example. Suppose we aim to fully excite modes n = 3 and n = 2. Starting at an initial point, a gradient descent is performed which lands somewhere on a curve where the n = 3 mode peaks are located. Next, the intersection between the peaks of modes n = 3 and n = 2 is sought. While a gradient descent could be performed to move towards the mode 2 peaks, this would mean that we would no longer reside in the mode 3 optimal subspace. To ensure that we remain in this subspace, we leverage a form of the Gram-Schmidt process [34].

In our modified Gram-Schmidt process, we aim to confine the optimization for each subsequent mode to the optimal subspace for the previous mode. Each instance of the gradient descent is given a policy number p which represents what iteration is currently executing, and we intend to maximize a set of modes $n = [{{n_1},{n_2},{n_3}, \ldots ,{n_N}} ]$. The set of modes can be in any order and do not need to be one away from each other. For example, in a three-layer problem it is possible to optimize for modes 4, 2, and 1. The final gradient for a particular policy is given by ${\vec{G}_p}$. Equations (8), (9), and (10) calculate the gradient for policies $p = [{1,2,3} ]$ while Eq. (11) calculates the gradient for policy p = N, where N is any positive integer.

Figure 2(b) demonstrates the Gram-Schmidt process in two dimensions. First, a series of initial points for each iteration are defined using Latin hypercube sampling to assure they are evenly distributed [35]. Next, we run many instances of gradient descent with different initial conditions to find a point in the optimal subspace of mode n = n₁. We then use this point as the initial condition for another gradient descent. Returning to the earlier example of optimizing modes n = 3 and n = 2, the proposed solution to the issue of “falling off” a previous mode’s optimal subspace is shown in Fig. 2(c). We can subtract the normal component ($\hat{n}$) to the mode 3 optimal subspace to achieve a projection of the mode 2 gradient along the mode 3 subspace. This normal component is found by simply calculating the gradient for mode 3 while located in the mode 3 subspace. By following the projection, we arrive at a design where both modes 2 and 3 are large. In a similar fashion, an example three-layer cylinder optimization is shown in Fig. 2(d). As noted before, the whole process needs to be rerun many times with different initial conditions, and not all of them are guaranteed to find a good solution. Fortunately, this is not a concern because the MieSolver and optimization procedure each have short execution times.

The magnitude of mode 0 is not sensitive to changes in material parameters and is fully excited in most cases. Hence, it is not necessary to try to maximize mode 0 since most optimized designs will lie in an area where it is large. On the other hand, high order modes such as n = 4 are much more sensitive, and in certain cases, the optimizer may still “fall off” the mode 4 subspace despite attempts to confine it. This issue arises due to the finite steps taken by the optimizer. To address it, we can add a final stage to the method where we directly optimize for σ_sca while using a miniscule step size. This is not required for optimizations involving only lower order modes, and it can even be detrimental since it may trend towards a solution which utilizes modes that are not being targeted.

For most problem configurations, there are multiple solutions for attaining a given σ_sca, but many of them have different modal distributions. A criterion is necessary for determining what the best solutions are. Suppose that one aims for a goal of σ_sca = 5. It can be achieved by partially exciting modes n = 0, ± 1, ± 2, ± 3, however this is a non-ideal, “complex” result. A better solution would be if n = 0, ± 1, ± 2 were each fully excited. We refer to this as a “pure” solution, and it is preferable not only because higher order modes possess increased sensitivity to changes in permittivity when compared to lower order modes, but also because a “pure” solution has other desirable properties such as high directivity. In the next section we present several example designs. “Pure” solutions with two- and three-layer cylinders are provided as well as “complex” solutions with the maximum possible σ_sca.

2.3 Results

Figure 3 displays a summary of the initial optimization of a two-layer cylinder. The procedure was run several thousand times over a range of 1 < ε_r < 1000, and we aimed to obtain a “pure” solution with σ_sca = 5. Figure 3(a) shows the two-layer cylinder configuration, and Fig. 3(c) displays the best outcome we achieved for different permittivity ranges. Likewise, Fig. 3(b) depicts each result’s σ(θ) pattern, and Fig. 3(d) illustrates each design’s modal distribution. When allowing each layer to vary between 1 < ε_r < 200, the maximum σ_sca attained was 3.44. This design had an inner permittivity (ε₁) of 1.05 and an outer permittivity (ε₂) of 98. For 1 < ε_r < 400, we attained a maximum σ_sca of 4.77, with [ε₁, ε₂] = [237.6, 21.26], and for 1 < ε_r < 500, the result was σ_sca = 4.96, with [ε₁, ε₂] = [483.83, 25.96]. Therefore, materials with larger ε_r are needed to realize higher σ_sca, but ε_r > 500 is unnecessary in this case.

 

Fig. 3. Initial optimization results. (a) Cylinder configuration. (b) Polar plot of σ(θ) for each optimization result. (c) Plot of each optimization result in terms of material parameters. (d) Distribution of modal coefficients for each result.

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We then performed a more detailed analysis of the [ε₁, ε₂] = [483.83, 25.96] solution (design 3 from Fig. 3). Figure 4(a) shows the modal distribution, where modes n = -2 through n = 2 have magnitudes ∼1. Figure 4(b) presents an evaluation of each modal coefficient as well as σ_sca across a frequency sweep centered at 2.2 GHz. The five different modes add up to an σ_sca of ∼5, and the dispersion of σ_sca indicates that this design has a high quality (Q-) factor. The σ(θ) at different frequencies is illustrated in Fig. 4(c). It is notable that even though the optimization procedure does not consider the angle of maximum scattering, this design is highly directive. Lastly, Fig. 4(d) presents the scattered electric field outside and inside the cylinder.

 

Fig. 4. Two-layer “pure” result. (a) Cylinder configuration and modal coefficients. (b) Frequency sweep showing the values of each modal coefficient as well as the scattering cross section. (c) Polar plot of σ(θ) for the cylinder design evaluated at different frequencies. (d) Scattered near field outside the cylinder at 2.2 GHz. Inset: Scattered field inside the cylinder.

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Figure 5 displays an example of a three-layer cylinder with a “pure” solution, and the corresponding material parameters are [ε₁, ε_2, ε₃] = [537.4, 49.5, 516.2]. It behaves in a similar manner to the “pure” two-layer structure, yet Fig. 5(b) shows the bandwidth of this solution is narrower. Accordingly, when considering the far field plot in Fig. 5(c), the σ(θ) patterns quickly degrade when moving away from the central frequency. Lastly, for Fig. 5(d), it is interesting how the interior electric field differs greatly from the two-layer case, but the exterior scattered field looks nearly identical. This observation again reveals that the exterior scattered field is determined by the resonances excited in all layers of the cylinder. It should be noted that, given the high permittivity nature of our dielectric scatterers, they can support multiple resonances if a broader frequency range is considered. A multi-frequency superscattering effect can possibly be achieved by maximizing scattering at multiple different frequencies using multi-objective optimization [36].

 

Fig. 5. Three-layer “pure” result. (a) Cylinder configuration and modal coefficients. (b) Frequency sweep showing the values of each modal coefficient as well as the scattering cross section. (c) Polar plot of σ(θ) for the cylinder design evaluated at different frequencies. (d) Scattered near field inside and outside the cylinder at 2.2 GHz.

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After finding two- and three-layer “pure” solutions for σ_sca= 5, the next goal was to identify “pure” solutions involving more modes. However, our study shows that within the allowed permittivity range (1 < ε_r < 2000), “pure” solutions for σ_sca = 7 utilizing all modes from n = -3 to n = 3 cannot be supported by the proposed cylinder systems. We then looked for the highest possible σ_sca that could be obtained without aiming for a particular modal distribution. Optimizations that started at mode n = 4 tended to be the most successful. The results are shown in Fig. 6 and will be referred to as the “complex” designs. The “complex” two-layer design with permittivity values of [ε₁, ε₂] = [178.8251, 52.6046] and its modal distribution are displayed in Fig. 6(a). Next, Fig. 6(b) shows the dispersion of the modal coefficients and σ_sca as well as σ(θ) at 2.2 GHz. The maximum σ_sca is 6.46, which corresponds to the sharp peak at 2.2 GHz. This result outclasses the “pure” solutions shown in Fig. 4(c). The corresponding scattered electric field on the interior of the cylinder is shown in Fig. 6(c). Moreover, Fig. 6(d), (e), and (f) show the modal distribution, frequency sweep study, and near fields for a three-layer “complex” solution where [ε₁, ε₂, ε₃] = [1244.2, 1796.8, 80.42]. Compared with the “complex” design of the two-layer cylinder, similar scattering responses are observed. The modes n = ± 2 of the three-layer system show a narrower peak around 2.2 GHz, indicating the two solutions exist in regimes with distinct mode structures in the permittivity space.

 

Fig. 6. Summary of maximum scattering “complex” designs. (a) Configuration and modal distribution of two-layer case. (b) Frequency sweep of “complex” two-layer cylinder, showing each mode and total σ_sca. The inset displays σ(θ) at 2.2 GHz. (c) Scattered electric field distribution inside the two-layer cylinder. (d) Configuration and modal distribution of three-layer, maximum scattering design. (e) Frequency sweep of maximum scattering three-layer design, showing each mode and total σ_sca. The inset displays σ(θ) at 2.2 GHz. (f) Scattered electric field distribution inside the three-layer cylinder.

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2.4 Tolerance studies

Fabrication and measurement of the devices presented in this paper is of potential future interest. Thus, it is paramount to evaluate the robustness of these solutions when subject to real world constraints, such as manufacturing tolerances. To address this issue, we performed two tolerance studies of the two-layer design from Fig. 3 by executing various parametric sweeps. The first sweep evaluated 16 < ε₂< 36, and the results are presented in Fig. 7(e) and (h). The modal coefficients and σ(θ) patterns for assorted designs are displayed, which demonstrates how performance varies. Figure 7(f) provides a zoomed-in perspective on the permittivity sweep as well as showing how introducing dielectric loss affects performance. An ε₂ variation of ±1 results in designs with 3.3 < σ_sca< 5. The system still performs well under realistic loss conditions, maintaining σ_sca> 4 for tan(δ) = 10⁻³ at the optimized value of ε₂= 25.96.

 

Fig. 7. Tolerance study. (a),(b),(c) Two-dimensional parametric sweeps of mode n = 1, 2, and 3, respectively, versus inner and outer permittivities. (d) Two-dimensional parametric sweep of σ_sca versus [ε₁, ε₂]. (e) Effect of modifying outer permittivity on σ_sca, with σ(θ) patterns included for specific designs. (f) Tolerance study of ± 1 variation of outer permittivity as well as the use of lossy dielectrics on σ_sca. (g) Tolerance study for ± 0.8 mm variation in inner radius of the two-cylinder system (r₂ is fixed). (h) Modal distribution for various cylinder designs corresponding to the marked points in (e).

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The second tolerance study concerned variations in the cylinder’s dimensions, specifically 8.2 mm < r₁< 9.8 mm, whereas r₂ (the total width of the cylinder system) was fixed. The results are displayed in Fig. 7(g), where σ_sca> 4 is maintained for changes in r₁ of approximately ± 0.2 mm. The parameter ε₁ was not assessed in this fashion because σ_sca has loose dependence on ε₁, as can be seen in Fig. 7(d). Therefore, it is concluded that the cylinder has reasonable robustness and is not overly sensitive to changes in material parameters or dimensions.

As a contrast, designs utilizing modes up to n = 4, such as the “complex” solutions from Fig. 6, may have poor robustness which would make them difficult to fabricate. Likewise, there may be a desire to create a cylinder with four or more layers with the hope it would achieve even higher σ_sca compared to our “complex” designs. Nonetheless, it would need to utilize higher order modes which are sensitive to changes in material parameters, making the corresponding design impractical. Therefore, designs with greater than three layers were not considered in our study. Figures 7(a), (b), and (c) demonstrate this tolerance concept by showing how higher order modes become increasingly sensitive to changes in permittivity.

2.5 Material considerations

We have presented how our “pure” solutions are reasonable from a robustness point of view, but there is also the concern of the availability of materials with the necessary permittivities. When reexamining the two-layer structure from Fig. 4, it has [ε₁, ε₂] = [483.83, 25.96]. Ceramics with high permittivities can be realized through a variety of techniques, such as doping or sintering [19,20,37]. This flexibility enables the fabrication of cylinders with permittivities from a few hundred to a few thousand. For example, materials with permittivities in this range have been attained through using magnesium titanate [38]. Hence, based on the experimentally measured permittivity of MgTiO₃-CaTiO₃ ceramics [21], we study a two-layer cylinder with the ε₂ = 23.6. For the same operating frequency of 2.2 GHz, the cylinder dimensions were slightly modified to [r₁, r₂] = [9.4, 18.8] mm to account for the change in material parameters. Additionally, to better visualize the design space, we executed a parametric sweep of ε₁, and Fig. 8 shows the outcome. The design holds up well compared to the two-layer cylinder discussed in Fig. 4, achieving σ_sca ∼5.

 

Fig. 8. Two-layer cylinder design based on a realistic case (ε₂ = 23.6). A sweep of inner permittivity versus the magnitude of each mode and the σ_sca was performed to demonstrate robustness.

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The relationship between the core’s permittivity and scattering in this cylinder is of note because it can be taken advantage of in practical applications. As previously mentioned, the permittivities of certain ceramics can vary due to environmental factors [30–33], so this device has potential use as a temperature sensor based on scattering properties. It could be configured to precisely detect a particular temperature value, which is a capability that is useful in countless real-world scenarios.

3. Conclusion

In summary, through leveraging mode analysis and an optimization approach based on gradient descent, we have numerically demonstrated superscattering from individual multilayered dielectric cylinders. Our mode optimization strategy enables the identification of the “pure” mode configuration, i.e., degenerate resonances with individual channels approaching the single-channel limit, as well as the “complex” mode configuration, i.e., the scenario with the maximized value of scattering cross section. The strength of the observed scattering is primarily limited by the maximum allowed permittivity in the cylinder structure. Well-developed microwave and sub-THz dielectric materials (such as ceramics) can offer a relative dielectric constant ranging up to a few thousand, indicating the feasibility of the proposed system. Given the recent advances in additive manufacturing techniques, we envision that a combination of the mode analysis approach and our customized optimization methods can be exploited for the creation of complex dielectric subwavelength structures exhibiting exotic scattering behaviors.