Microwave backscatter enhancement using radial anisotropy in biomimetic core-shell spheres

Electromagnetic metamaterials are engineered composites that are typically composed of constituents with sub-wavelength dimensions. They can be designed to have tailored or enhanced electromagnetic properties and can even exhibit material parameters that are not found in nature. The ability to realize the often complex geometries required has been enhanced by advances in fabrication techniques such as 3D printing and nanofabrication. Moreover, the natural world itself provides instructive examples that inform electromagnetics research through the many elaborate structures that have evolved in living organisms for the control of light.¹ Natural structures that determine optical properties, including reflectivity, iridescence, and polarization control, have provided inspiration and design information for analogous biomimetic devices² as well as the translation of their geometries from optical to microwave length scales.³ One goal for engineered metamaterials and meta-atoms is the manipulation of the scattering of electromagnetic waves by particles with dimensions smaller than or close to the wavelength. It has been shown that the strength and directionality of scattering may be controlled by structuring a particle in order to adjust the frequencies and magnitudes of the Mie resonances that it supports.⁴ The term “superscattering” is used to refer to total scattering from multiple resonances that exceeds the theoretical limit for that of a single resonance.⁵ Diverse applications have been identified for tailoring of scattering, including wireless power transfer, radar signature control, and the enhancement of nonlinear optical processes,^4,6,7 while mode selection and tuning are used in the design of dielectric resonance antennas.^8,9

An important concept arising from the spectral coincidence or “stacking” of multiple Mie resonances is termed the first Kerker condition,¹⁰ as illustrated in Fig. 1.¹¹ In this case, the electric dipole (ED) and magnetic dipole (MD) scatter the incident field with the same phase in the forward direction and opposite phase in the backward direction. If the magnitudes of the two modes are matched, the consequent wave interference eliminates backscatter and enhances forward scattering. For modes arising from a dielectric particle, the contrasting electric field distributions between the electric and magnetic mode families within the particle allows their resonances to be tuned differentially through structuring of the internal geometry. For example, the introduction of a low permittivity core within a high permittivity sphere was proposed in order to preferentially move the spectral positions of the electric modes to higher frequencies due to their greater electric field flux at the center of the sphere.¹² The enhancement of backscattering was then demonstrated by using this effect to pull electric modes into overlap with selected magnetic modes having the requisite relative phase.¹³ Among other geometries and concepts that have been considered theoretically, the example of homogeneous radially anisotropic spheres has relevance to this work.^14,15

In recent years, researchers have identified materials and nano-structures that use optical scattering to enhance the sensitivity of compound eyes of certain shrimp species. The presence of the crystalline pteridine material isoxanthopterin was identified,¹⁶ and its anisotropic refractive index was found to have the value n = 1.96 for one of its principal axes. This very high index among biological materials suggests utility of isoxanthopterin in optical reflectors. The arrangement of isoxanthopterin crystals in spherical core-shell nanoparticles was characterized as featuring a radially anisotropic shell surrounding an isotropic core¹⁷ and was identified as providing efficient backscattering. The role of the scattering in enhancing the sensitivity of the compound eyes was reported, whereby the presence of particles in a retinal structure called the tapetum reflector helps to concentrate light in the light-sensitive rhabdoms of the retina as well as reducing crosstalk between adjacent rhabdoms. Beck et al. described the influence of the high refractive index for light polarized tangentially to the particle surface on the magnitude of the scattering cross section due to the alignment of the electric fields of the magnetic modes.¹⁸ Short range ordering of the nanoparticles has also been reported to contribute to the light gathering.¹⁹ 

In this work, we demonstrate a functional biomimetic component for microwave backscatter enhancement inspired by the shrimp eye structures. The component harnesses an advantageous physical configuration arrived at through evolution and showcases the use of additive manufacture to realize a microwave version requiring a complex geometry. Experimental results and simulations are presented for design variants that elucidate the role of the anisotropy in manipulating the backscatter, while providing a quantitative indication of the performance benefit introduced vs equivalent structures without anisotropy. The work is motivated by the desire to increase the palette of techniques available for the manipulation of electromagnetic radiation.

For comparison with the anisotropic shrimp-eye replica, two designs were specified with shells that were isotropic both in terms of the materials of their construction as well as omitting form anisotropy. Design 2 featured a shell permittivity of ε_r = 3.12, thereby providing an isotropic equivalent to the anisotropic sample according to Eq. (1). The purpose of this design was to investigate the difference in scattering behavior depending on the presence or absence of anisotropy in the shell. In a third design, the permittivity of the isotropic shell was tailored in order to obtain an approximate match in the backscattering magnitude and spectrum to that of the shrimp-eye analogue. The purpose of this experiment was to identify the increase in isotropic permittivity required to realize the same performance offered by a lower index anisotropic medium. All three designs featured an overall diameter of 66 mm, comprising a core of radius r = 19 mm and a shell of thickness t = 14 mm. The anisotropic shell was formed from six layers, each 2.33 mm-thick, alternating between solid dielectric and air.

The frequency-dependent backscattering from the microwave samples was measured using a quasi-monostatic configuration in an anechoic chamber, as shown in Fig. 3. The transmission and collection of radiation were performed by a pair of Flann Microwave DP240-AB dual-polarized horn antennas, which were approximately co-located, connected to an Anritsu MS46122B vector network analyzer. The sample was mounted on a fiberglass pole topped with Rohacell 31HF foam, which has a low relative permittivity (ε_r = 1.04). The horns were positioned 3 m from the sample in order to present approximately flat incident phase fronts across its spatial extent. The measurements were calibrated using a 50 mm-diameter stainless steel sphere.

In this section, results are first presented for designs 1 and 2, comparing the anisotropic shrimp-eye analogue with its isotropic equivalent with shell permittivity ε_r = 3.12. Design 3 is discussed later in the context of the effect of the use of materials with larger permittivity. The simulations and experimental results for the monostatic radar cross section (RCS), normalized to the cross-sectional area, for designs 1 and 2 are shown in Fig. 3. The experimental data [Fig. 3(b)] show quantitative agreement with the finite-element method simulations [Fig. 3(a)]. The comparison between the designs shows that the introduction of anisotropy results in an RCS enhancement of more than a factor of two for the lowest frequency scattering peak at around 3 GHz. A small downward shift in frequency is also observed. An alternative measure of backscatter is presented in finite-element simulations in the supplementary material, whereby the normalized backscatter is integrated over the backward hemisphere [Fig. S1(b)] and is compared with the total normalized scattering cross section [Fig. S1(a)]. This measure shows a similar backscatter enhancement to the monostatic RCS results. The hemispherical backscattering accounts for approximately one quarter of the total scattering in the anisotropic design at frequencies near the lowest frequency RCS peak.

Figures 3(c) and 3(d) show analytical Mie theory calculations for the same two designs of the scattering cross section (σ_sca) contributions, normalized to the sample cross-sectional area, within the frequency range of interest due to the first three modes of each of the electric and magnetic resonance families: dipoles, quadrupoles, and octupoles. Of particular note are the relative strengths of the electric dipole (ED) and magnetic dipole (MD) modes. The simulations show that neither of the designs have a match in the ED and MD strengths within the 2–3 GHz frequency range in which the relevant RCS peak occurs. Accordingly, they do not satisfy the first Kerker condition for total backscatter suppression. However, the results show that the design with radial anisotropy in the shell has a strengthened MD resonance relative to that of the isotropic design, while leaving the ED strength largely unaffected. For the isotropic component [Fig. 3(d)], the dimensionless scattering parameter for the MD mode has a peak value of 1.65 at 2.64 GHz, which is a factor of 2.06 stronger than the broad ED resonance that peaks at 2.32 GHz. For the anisotropic design [Fig. 3(c)], the MD parameter reaches 2.16 at 2.35 GHz, a factor of 3.27 stronger than the approximately coincident ED mode.

Observing the effect of introducing anisotropy on the backscattered radiation highlights the importance of the difference between the internal field geometries of the two mode families as well as their relation to the orientation of the principal axes of the shell permittivity. The electric dipole has no radial magnetic field component, while the magnetic dipole has no radial electric field.¹¹ The strengthening of the MD magnitude is, therefore, expected due to the alignment of its electric fields with the large tangential permittivity, while the electric fields of the ED mode are polarized orthogonally to that direction. By altering the relative strength of the electric and magnetic resonances in this way, the scattering performance is pulled further from the first Kerker condition by the weakening of the destructive inference in the backward direction. Consequently, the backscatter is enhanced. The observed downward shift in frequency may also be attributed to the larger tangential permittivity, which results in the resonance occurring at a longer wavelength.

The results presented above indicate that the presence of anisotropy provides an advantage in biological optical structures, where the available dielectric material parameters are constrained. Greater flexibility is available in the microwave domain in terms of permittivity, which allows one to investigate what isotropic shell permittivity is required to achieve equivalent backscattering performance to the anisotropic case. Figure 4(a) shows finite-element method simulations of monostatic RCS and corresponding experimental measurements in the absence of anisotropy for a sample with a shell permittivity of ε_r = 4.5 (design 3). The results are comparable to that of the anisotropic sample; however, the permittivity corresponds to a refractive index of 2.12, which is larger than is typically found in biological materials.^17,23,24 Figure 4(b) shows simulations of the scattering cross sections for the modes of this design with shell permittivity ε_r = 4.5. In this case, the MD scattering parameter reaches 2.63 at 2.15 GHz, a factor of 2.18 stronger than the ED mode. While the magnitude of all of the modes is increased relative to those of the lower permittivity version shown in Fig. 3(d), the relative strengths of the ED and MD modes remain similar. Therefore, an increase in RCS is expected as a consequence of an increase in total scattering. Simulations of the backscatter integrated over the backward hemisphere [supplementary material, Fig. S2(b)] show comparable magnitude with the anisotropic design, while the total normalized scattering cross section [Fig. S2(a)] shows increased scattering at the relevant frequencies. These results demonstrate the utility of the inclusion of anisotropy as a strategy for scattering enhancement in the shrimp eye.

Advances in additive manufacturing techniques have expanded the range of dielectric materials that they offer for microwave applications. The available material properties include larger relative permittivities than those discussed above,²⁵ leading to further scope to manipulate the resonances using the same multilayered shell design concept. The consequence of increasing the permittivity of the solid layers in the shell of our anisotropic architecture is illustrated in analytical method calculations in Fig. 5. The scattering cross section contributions from the first five multipoles of each mode family are compared for electric modes [Fig. 5(a)] and magnetic modes [Fig. 5(b)]. The monostatic RCS is shown in Fig. 5(c). The effective anisotropy parameter ε_∥ – ε_⊥ is displayed on the primary ordinate axis. The isotropic permittivity of the solid layers of the shell required to obtain this anisotropy, as calculated using Eqs. (2), is shown on the secondary ordinate axis. Its magnitude covers the range 5–30, while all other design parameters are unchanged from the shrimp-eye analogue design. Each plot is normalized to the cross-sectional area of the design. The results show that the spectral positions of the electric and magnetic modes move down in frequency at differing rates as the anisotropy is increased. For larger anisotropies, the ED and MD modes can be separated in frequency, thus further preventing destructive interference in the backward direction at frequencies approaching 1 GHz. This mechanism complements the tuning of the relative strengths of the modes. Furthermore, the potential for stacking higher-order multipoles to achieve a superscattering effect²⁶ is apparent, although beyond the scope of this work. Through these mechanisms, the concentrically layered shell concept offers a plurality of processes for mode manipulation and scattering control, over and above that suggested by the nanoparticle found in the shrimp eye.

In conclusion, we have designed, fabricated, and experimentally demonstrated a metamaterial component that provides enhanced backscattering of microwave radiation. Analytical theory and numerical method simulations have been used to explain how the spherical structure achieves the enhancement by utilizing radial anisotropy to manipulate its Mie resonances. The demonstration of this mechanism using a 3D-printed structure adds to the suite of available geometries and concepts for the control of Mie scattering. Through the use of isotropic materials with form anisotropy, the need for anisotropic materials that would require a means of angular alignment in fabrication is avoided. This bio-inspired development has shown the transfer of optical concepts that benefit the visual acuity of the eye of a shrimp to a manufacturable microwave device that has applicability in the manipulation of radar return signals, and can potentially be adapted for other wavelength regimes. The component offers an all-dielectric and partially hollow construction, while providing reflectivity without the requirement for a specific orientation. In this way, the format of the structure is suited to each of the very diverse deployments of the shrimp eye and radar scatterer, while also being capable of assembly using biological processes in optical form as well as fabrication in scaled-up form by additive manufacture for operation at microwave frequencies as demonstrated in this work.