Bio-Inspired Dielectric Resonator Antenna for Wideband Sub-6 GHz Range

: Through the years, inspiration from nature has taken the lead for technological development and improvement. This concept ﬁrmly applies to the design of the antennas, whose performances receive a relevant boost due to the implementation of bio-inspired geometries. In particular, this idea holds in the present scenario, where antennas working in the higher frequency range (5G and mm-wave), require wide bandwidth and high gain; nonetheless, ease of fabrication and rapid production still have their importance. To this aim, polymer-based 3D antennas, such as Dielectric Resonator Antennas (DRAs) have been considered as suitable for fulﬁlling antenna performance and fabrication requirements. Di ﬀ erently from numerous works related to planar-metal-based antenna development, bio-inspired DRAs for 5G and mm-wave applications are at their beginning. In this scenario, the present paper proposes the analysis and optimization of a bio-inspired Spiral shell DRA (SsDRA) implemented by means of Gielis’ superformula, with the goal of boosting the antenna bandwidth. The optimized SsDRA geometrical parameters were also determined and discussed based on its fabrication feasibility exploiting Additive Manufacturing technologies. The results proved that the SsDRA provides relevant bandwidth, about 2 GHz wide, and satisfactory gain (3.7 dBi and 5 dBi, respectively) at two di ﬀ erent frequencies, 3.5 GHz and 5.5 GHz.


Introduction
The use of peculiar geometries and shapes inspired by nature has demonstrated, through the years, to be an intriguing and successful option capable of supporting the development of advanced electronic devices. In particular, these bio-inspired designs are able to cope with the main criticalities related to metal-based antenna systems aimed at WLNA, WiMAX, 5G and mm-wave communication applications: wide bandwidth and high radiation efficiency. In this viewpoint, Panda et al. [1] presented the design of a microstrip antenna resembling a flower shape, conceived on a defective ground structure (DGS); the numerical results showed that a dramatic improvement of the bandwidth and return loss was achieved at the operating frequency (15 GHz). Abolade et al. [2] exploited the shape of Carica Papaya leaf to realize a microstrip antenna which exhibited large bandwidth and high gain for three different bands, centered at 4.3 GHz, at 7.2 GHz and 9.3 GHz, respectively, providing a compact and suitable solution for a variety of applications (GSM1900, UMTS, WLAN, LTE2300 and of using bio-inspired geometry was presented by Sankaranarayanan et al. [33]: in this work, snow-flake fractal and half split fractal rings were implemented on the classic cylindrical DRAs, thus achieving 90% of fractional bandwidth and gain enhancement in the frequency range between 4.7 and 12.4 GHz. Fractal segmented patterns were also employed by Gupta et al. [34] to design FR4-based DRAs with ultra-wideband performance (122.5% and 94.89%) from 4.8 GHz to 20 GHz and characterized by high compactness and gain at 8.5 GHz (8.75 and 15.65 dBi, respectively). Actual 3D star-super-shaped DRAs (S-DRAs) based on Gielis' superformula were reported by Simeoni et al. [35], Petrignani et al. [36] and Basile et al. [37]. In [35], the super-shaped DRAs (S-DRA) were designed to operate at 7.65 GHz and realized by considering polyvinyl chloride (PVC), a thermoplastic material having low permittivity (ε r~2 .8) and negligible tangent loss. The results showed that in the single feed case study, a fractional bandwidth of 74% and gain of~8 dBi were accomplished between 6 and 13 GHz. Conversely, in [36,37], twisted star-based S-DRAs were designed to operate in the sub-6 GHz range, at 3.5 GHz. The S-DRAs were realized by means of a photopolymer resin material having ε r = 2.7 and low tangent loss, suitable for stereolithography. The experimental results showed that this kind of DRAs was able to provide high gain (around 3 dBi) and radiation efficiency (~90%) at 3.5 GHz, while keeping their volume significantly small.
However, although the use of bio-inspired geometry and DRAs proved to lead to an overall improvement of the antenna performance, especially in terms of bandwidth enhancement, works about the actual combination of these two strategies for a further boost in the antenna development are still at their beginning. Indeed, very few Authors attempted to design and realize bio-inspired DRAs and, in particular, none of them considers complex and thin 3D shapes, like Button snail Modulus modulus (spiral seashell) for the DRA applications.
Therefore, in this research, the design and optimization of a bio-inspired super-shaped DRA, resembling a Spiral shell of a sea mollusk, generated by means of Gielis' superformula and suitable for sub-6 GHz applications, is presented. The numerical analyses were performed by fixing Gielis' parameters and varying the geometrical sizes of the DRA, with the aim of accomplishing wideband behavior in the frequency range of interest. In particular, the influence of the coaxial cable pin height, DRA thickness and height on the scattering parameter S 11 and antenna gain are evaluated. The impact of the exponential coefficient of Gielis' superformula on the antenna design was also investigated in terms of antenna design and prototype fabrication, envisaged to be realized by means of Additive Manufacturing processes. The 3D radiation patterns, electric and magnetic field distributions and radiation efficiency of the optimized Spiral shell DRA are also reported. The overall performance of the SsDRA was then compared to other DRA geometries, conceived and realized by using the same low-permittivity material. Finally, a detailed discussion about fabrication feasibility and issues emerged for the realization of the Ss-DRA prototype is reported.

Bio-Inspired Spiral Shell DRA
The Button snail Modulus modulus (spiral seashell) presented in Figure 1a was the bio-inspiration for the device of the SsDRA shown in Figure 1b. The SsDRA is placed in the center of a ground plane, a squared substrate with side L SUB , in the origin of the reference system; the dielectric thickness t and the height h DRA of the antenna are also indicated. The considered dielectric material has a permittivity of 2.7 and negligible loss tangent (0.008) in the frequency range of interest [36,37]. The DRA is fed by a coaxial cable (Z 0 = 50 Ω, conductor diameter Ø 0.48 mm, insulation diameter Ø 1.52 mm) placed in the antenna origin: the pin height is indicated by h PIN .

Gielis Parameters: Impact on the SsDRA Design
The design initiator was defined by considering that the variation of the "a" and "b" parameters affects the sizes and volume of the DRA, and so the footprint it can actually occupy. The angular position of the shell wave ridges remains constant, while the shell wave amplitudes increase proportionally as "a" and "b" increase. The "m1" and "m2" parameters influence the frequency of the shell wave over the base spiral: when these coefficients are increased, the shell wave number increases along the spiral. The parameters "n1", "n2" and "n3" affect the shell wave amplitude on each loop of the shell: in particular, when n1 = n2 = n3 = 2, the output geometry becomes the well-known Archimedean spiral. For values of "n1", "n2" and "n3" higher than 2, alternate waves are overlapped on the spiral. When these parameters are lower than 2, the waves resemble a rectified sine (or cosine) with an outward concavity, thus orienting the wave ridges inward. The constant scaling factor K regulates the overall amplitude of the geometry and, as such, it affects the DRA radius and volume. The planar exponential function f θ , expressed by the following Equation (2): acts on the closeness of the wavy spiral loops, thus on the space actually filled by the DRA. In particular, the parameter ctheta (which is a rational number) has a sharp influence on the radial expansion of the spiral loops, i.e., the aperture velocity ΔR/Δθ. On the basis of these considerations, we fixed the starting initiator values reported in Table 1 where the angle parameter θ is varied in the range [−3π, 3π] (i.e., three complete rounds). The defined parameters lead to a SsDRA footprint smaller than the ground plane whose dimensions are 100 × 100 mm 2 , thus with LSUB = 100 mm.

Evaluation of SsDRA Design Based on the Fabrication Feasibility
The proposed SsDRA has a typical 2.5D geometry, since it develops as a straight protrusion of a thin planar section. The main fabrication issue displayed along the z-axis direction concerns the

Gielis Parameters: Impact on the SsDRA Design
The design initiator was defined by considering that the variation of the "a" and "b" parameters affects the sizes and volume of the DRA, and so the footprint it can actually occupy. The angular position of the shell wave ridges remains constant, while the shell wave amplitudes increase proportionally as "a" and "b" increase. The "m 1 " and "m 2 " parameters influence the frequency of the shell wave over the base spiral: when these coefficients are increased, the shell wave number increases along the spiral. The parameters "n 1 ", "n 2 " and "n 3 " affect the shell wave amplitude on each loop of the shell: in particular, when n 1 = n 2 = n 3 = 2, the output geometry becomes the well-known Archimedean spiral. For values of "n 1 ", "n 2 " and "n 3 " higher than 2, alternate waves are overlapped on the spiral. When these parameters are lower than 2, the waves resemble a rectified sine (or cosine) with an outward concavity, thus orienting the wave ridges inward. The constant scaling factor K regulates the overall amplitude of the geometry and, as such, it affects the DRA radius and volume. The planar exponential function f(θ), expressed by the following Equation (2): acts on the closeness of the wavy spiral loops, thus on the space actually filled by the DRA. In particular, the parameter c theta (which is a rational number) has a sharp influence on the radial expansion of the spiral loops, i.e., the aperture velocity ∆R/∆θ. On the basis of these considerations, we fixed the starting initiator values reported in Table 1 where the angle parameter θ is varied in the range [−3π, 3π] (i.e., three complete rounds). The defined parameters lead to a SsDRA footprint smaller than the ground plane whose dimensions are 100 × 100 mm 2 , thus with L SUB = 100 mm.

Evaluation of SsDRA Design Based on the Fabrication Feasibility
The proposed SsDRA has a typical 2.5D geometry, since it develops as a straight protrusion of a thin planar section. The main fabrication issue displayed along the z-axis direction concerns the aspect ratio (AR) of the geometry, defined as h DRA /t. In order to guarantee the successful fabrication of a stiff structure, the AR should be higher than a critical threshold, which depends on the mechanical properties of the material and the chosen fabrication technology. Therefore, in the antenna optimization, the values of t and h DRA were set by taking into account this limitation.
The spiral section evolving along the X-Y plane represents the main challenge for the fabrication of the proposed geometry, and, as such, it dictates the actual selection of the most suitable Additive Manufacturing process. In order to come up with feasible solutions for the further SsDRA fabrication, preliminary considerations were made, basing on the SsDRA prototype drawings reported in Figure 2, as follows: By fixing the parameter "m" equal to 10, the geometry has 10 peaks and valleys and the step-angle between them is kept constant and equal to 36 degrees (360/m), independently of the position along the spiral curve. Ridges and valleys are rounded, but the lower the cylindrical coordinate θ (and distance from the origin GT(θ)) is, the lower the radius of the circle which approximates locally the curve on ridges or valleys. In addition, the amplitude of the oscillation along the spiral is proportional to the values of parameters n 1 , n 2 and n 3 of the Gielis formulation (1): in this case, they are all equal to 5. This setting allows us to have smaller amplitude close to the origin, i.e., micro-features which requires an accurate fabrication assessment. b.
Since the radius of the spiral increases monotonically from the origin to the end of the spiral, the smaller features of the geometry are in close proximity to the origin, thus generating micro-scale features.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13 aspect ratio (AR) of the geometry, defined as hDRA/t. In order to guarantee the successful fabrication of a stiff structure, the AR should be higher than a critical threshold, which depends on the mechanical properties of the material and the chosen fabrication technology. Therefore, in the antenna optimization, the values of t and hDRA were set by taking into account this limitation. The spiral section evolving along the X-Y plane represents the main challenge for the fabrication of the proposed geometry, and, as such, it dictates the actual selection of the most suitable Additive Manufacturing process. In order to come up with feasible solutions for the further SsDRA fabrication, preliminary considerations were made, basing on the SsDRA prototype drawings reported in Figure  2, as follows: a. By fixing the parameter "m" equal to 10, the geometry has 10 peaks and valleys and the stepangle between them is kept constant and equal to 36 degrees (360/m), independently of the position along the spiral curve. Ridges and valleys are rounded, but the lower the cylindrical coordinate θ (and distance from the origin GT(θ)) is, the lower the radius of the circle which approximates locally the curve on ridges or valleys. In addition, the amplitude of the oscillation along the spiral is proportional to the values of parameters n1, n2 and n3 of the Gielis formulation (1): in this case, they are all equal to 5. This setting allows us to have smaller amplitude close to the origin, i.e., micro-features which requires an accurate fabrication assessment. b. Since the radius of the spiral increases monotonically from the origin to the end of the spiral, the smaller features of the geometry are in close proximity to the origin, thus generating micro-scale features. The measurement of circle approximations of the curve is important to evaluate the feasibility of 3Dprinting technologies in the SsDRA fabrication. Indeed, the majority of the Additive Manufacturing technologies build the objects using a layer-by-layer strategy, accomplished by moving a circular end-effector: in particular, the end-effector is an extrusion nozzle in the Fused Deposition Modelling (FDM), and a laser spot in the stereolithography (SLA) [39,40]. Therefore, the previous considerations lead to select an AM process that should be capable of reproducing accurately the micro-features and ensuring the target value of roundness for the ridges and valleys. Detailed discussions about the evaluation of the best candidate among AM processes along with fabrication challenges and solutions about the optimized SsDRA fabrication are reported in Section 4. The measurement of circle approximations of the curve is important to evaluate the feasibility of 3Dprinting technologies in the SsDRA fabrication. Indeed, the majority of the Additive Manufacturing technologies build the objects using a layer-by-layer strategy, accomplished by moving a circular end-effector: in particular, the end-effector is an extrusion nozzle in the Fused Deposition Modelling (FDM), and a laser spot in the stereolithography (SLA) [39,40]. Therefore, the previous considerations lead to select an AM process that should be capable of reproducing accurately the micro-features and ensuring the target value of roundness for the ridges and valleys. Detailed discussions about the evaluation of the best candidate among AM processes along with fabrication challenges and solutions about the optimized SsDRA fabrication are reported in Section 4.

Spiral Shell DRA Analysis and Optimization
In order to evaluate the SsDRA performance and proceed with the antenna optimization, numerical analyses have been carried out on the SsDRA geometrical parameters by means of the CST Microwave Studio.
3.1. Influence of SsDRA Thickness, Pin Height and SsDRA Height on the Scattering Parameter S 11 : Evaluation of the Bandwidth In order to optimize the SsDRA design in terms of bandwidth, the scattering parameter S 11 was evaluated by varying the central pin height, h PIN , and the SsDRA shell thickness t, by fixing the SsDRA height h DRA = 29 mm. The pin height h PIN is changed between 14.5 and 18 mm, while keeping the DRA thickness equal to t = 2 mm. The trend of the scattering parameter S 11 as a function of h PIN is reported in Figure 3a, where it can be noticed that when the pin height increases, the impedance matching becomes wider. In particular, for a pin height h PIN = 17 mm (cyan curve), a good impedance matching can be achieved at 3.5 GHz and 5.5 GHz, leading to a bandwidth of about 2 GHz. Figure 3b depicts the trend of the scattering parameter S 11 when h PIN is fixed at 17 mm and the SsDRA thickness t is varied (t = 1, 1.5 and 2 mm). The plots show that as t increases, the scattering parameter S 11 values improve, especially at 5.2 GHz, and, when t = 2 mm (yellow curve), the bandwidth at −10 dB is equal to about 2 GHz.

Spiral Shell DRA Analysis and Optimization
In order to evaluate the SsDRA performance and proceed with the antenna optimization, numerical analyses have been carried out on the SsDRA geometrical parameters by means of the CST Microwave Studio.

Influence of SsDRA Thickness, Pin Height and SsDRA Height on the Scattering Parameter S11: Evaluation of the Bandwidth
In order to optimize the SsDRA design in terms of bandwidth, the scattering parameter S11 was evaluated by varying the central pin height, hPIN, and the SsDRA shell thickness t, by fixing the SsDRA height hDRA = 29 mm. The pin height hPIN is changed between 14.5 and 18 mm, while keeping the DRA thickness equal to t = 2 mm. The trend of the scattering parameter S11 as a function of hPIN is reported in Figure 3a, where it can be noticed that when the pin height increases, the impedance matching becomes wider. In particular, for a pin height hPIN = 17 mm (cyan curve), a good impedance matching can be achieved at 3.5 GHz and 5.5 GHz, leading to a bandwidth of about 2 GHz. Figure 3b depicts the trend of the scattering parameter S11 when hPIN is fixed at 17 mm and the SsDRA thickness t is varied (t = 1, 1.5 and 2 mm). The plots show that as t increases, the scattering parameter S11 values improve, especially at 5.2 GHz, and, when t = 2 mm (yellow curve), the bandwidth at −10 dB is equal to about 2 GHz.  Figure 4a shows the scattering parameter S11 when both hPIN and t are varied, while hDRA is set equal to 29 mm; the behavior of the monopole antenna (blue curve) having a pin height hPIN equal to 17 mm is also reported for reference. The monopole exhibits a scattering parameter S11 value of about −14 dB at about 5.7 GHz with a bandwidth of about 1 GHz (from 4 GHz to 5 GHz). When the SsDRA is considered and its thickness t is increased, a better impedance matching is accomplished: in particular, the antenna bandwidth at −10 dB increases from 1.5 GHz for t = 1 mm and hPIN = 19 mm (red curve), up to about 2 GHz for t = 2 mm and hPIN = 17 mm (purple curve), along with lower scattering parameter S11 value (−16 dB), especially around 5.2 GHz.  Figure 4a shows the scattering parameter S 11 when both h PIN and t are varied, while h DRA is set equal to 29 mm; the behavior of the monopole antenna (blue curve) having a pin height h PIN equal to 17 mm is also reported for reference. The monopole exhibits a scattering parameter S 11 value of about −14 dB at about 5.7 GHz with a bandwidth of about 1 GHz (from 4 GHz to 5 GHz). When the SsDRA is considered and its thickness t is increased, a better impedance matching is accomplished: in particular, the antenna bandwidth at −10 dB increases from 1.5 GHz for t = 1 mm and h PIN = 19 mm (red curve), up to about 2 GHz for t = 2 mm and h PIN = 17 mm (purple curve), along with lower scattering parameter S 11 value (−16 dB), especially around 5.2 GHz. Figure 4b depicts the realized gain of the antenna that takes into account the losses due to the matching: as we can infer from the plot, as the SsDRA thickness is increased, gain values are equal to 3.3 dBi at 2.8 GHz (red curve), and increase up to 5 dBi at 5.5 GHz (purple curve). Generally, the gain values improve by about 0.2 dB up to about 2 dB with respect to the reference monopole in the overall frequency range of interest. Furthermore, the inspection of Figure 4b shows that SsDRA allows the additional coverage of the WiFi band, at 2.45 GHz.  Figure 4b depicts the realized gain of the antenna that takes into account the losses due to the matching: as we can infer from the plot, as the SsDRA thickness is increased, gain values are equal to 3.3 dBi at 2.8 GHz (red curve), and increase up to 5 dBi at 5.5 GHz (purple curve). Generally, the gain values improve by about 0.2 dB up to about 2 dB with respect to the reference monopole in the overall frequency range of interest. Furthermore, the inspection of Figure 4b shows that SsDRA allows the additional coverage of the WiFi band, at 2.45 GHz. Figure 5 reports the evaluation of the scattering parameter S11 and gain as the SsDRA height hDRA is varied between 10 mm and 30 mm, while hPIN and t are fixed to 17 mm and 2 mm, respectively. As evident from Figure 5a, when the SsDRA height is increased from 10 mm and 20 mm, the bandwidth at −10 dB increases as well, covering a range from about 1.3 GHz to 1.8 GHz (red and yellow curves). Conversely, as shown in Figure 5b, as the hDRA is increased, the SsDRA gain improves in the range between 2.5 GHz to 4.5 GHz (red and yellow curves are relevantly higher than the blue curve), while it shows negligible changes around 5.5 GHz.

Scattering Parameter S11 and Realized Gain as a Function of the Exponential Function f(θ) of the Optimized SsDRA
In order to show the effect of the planar exponential function f(θ) on the SsDRA performance, the coefficient ctheta was varied by considering three values: 0.16, 0.18 and 0.20. The geometrical sizes of the SsDRA were kept unchanged: t = 2 mm, hDRA = 29 mm and hPIN = 17 mm. Figure 6a reveals that, when this coefficient is reduced, the lower cut-off frequency shifts at higher frequencies (from 3.5 GHz to about 4 GHz), while the minimum of the scattering parameter shifts from 5 GHz to 5.5 GHz.  Figure 5 reports the evaluation of the scattering parameter S 11 and gain as the SsDRA height h DRA is varied between 10 mm and 30 mm, while h PIN and t are fixed to 17 mm and 2 mm, respectively. As evident from Figure 5a, when the SsDRA height is increased from 10 mm and 20 mm, the bandwidth at −10 dB increases as well, covering a range from about 1.3 GHz to 1.8 GHz (red and yellow curves). Conversely, as shown in Figure 5b, as the h DRA is increased, the SsDRA gain improves in the range between 2.5 GHz to 4.5 GHz (red and yellow curves are relevantly higher than the blue curve), while it shows negligible changes around 5.5 GHz.  Figure 4b depicts the realized gain of the antenna that takes into account the losses due to the matching: as we can infer from the plot, as the SsDRA thickness is increased, gain values are equal to 3.3 dBi at 2.8 GHz (red curve), and increase up to 5 dBi at 5.5 GHz (purple curve). Generally, the gain values improve by about 0.2 dB up to about 2 dB with respect to the reference monopole in the overall frequency range of interest. Furthermore, the inspection of Figure 4b shows that SsDRA allows the additional coverage of the WiFi band, at 2.45 GHz. Figure 5 reports the evaluation of the scattering parameter S11 and gain as the SsDRA height hDRA is varied between 10 mm and 30 mm, while hPIN and t are fixed to 17 mm and 2 mm, respectively. As evident from Figure 5a, when the SsDRA height is increased from 10 mm and 20 mm, the bandwidth at −10 dB increases as well, covering a range from about 1.3 GHz to 1.8 GHz (red and yellow curves). Conversely, as shown in Figure 5b, as the hDRA is increased, the SsDRA gain improves in the range between 2.5 GHz to 4.5 GHz (red and yellow curves are relevantly higher than the blue curve), while it shows negligible changes around 5.5 GHz.

Scattering Parameter S11 and Realized Gain as a Function of the Exponential Function f(θ) of the Optimized SsDRA
In order to show the effect of the planar exponential function f(θ) on the SsDRA performance, the coefficient ctheta was varied by considering three values: 0.16, 0.18 and 0.20. The geometrical sizes of the SsDRA were kept unchanged: t = 2 mm, hDRA = 29 mm and hPIN = 17 mm. Figure 6a reveals that, when this coefficient is reduced, the lower cut-off frequency shifts at higher frequencies (from 3.5 GHz to about 4 GHz), while the minimum of the scattering parameter shifts from 5 GHz to 5.5 GHz.

Scattering Parameter S 11 and Realized Gain as a Function of the Exponential Function f(θ) of the Optimized SsDRA
In order to show the effect of the planar exponential function f(θ) on the SsDRA performance, the coefficient c theta was varied by considering three values: 0.16, 0.18 and 0.20. The geometrical sizes of the SsDRA were kept unchanged: t = 2 mm, h DRA = 29 mm and h PIN = 17 mm. Figure 6a reveals that, when this coefficient is reduced, the lower cut-off frequency shifts at higher frequencies (from 3.5 GHz to about 4 GHz), while the minimum of the scattering parameter shifts from 5 GHz to 5.5 GHz. Figure 6b shows that when c theta is reduced, the antenna gain undergoes a reduction at 3.5 GHz (blue curve), while it exhibits a slight increase (about 0.4 dB) at 5.5 GHz (red curve).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 Figure 6b shows that when ctheta is reduced, the antenna gain undergoes a reduction at 3.5 GHz (blue curve), while it exhibits a slight increase (about 0.4 dB) at 5.5 GHz (red curve). Finally, we underline that, if θ is varied in the range 0-2π (1 complete round), while other parameters are fixed-t = 2 mm, hDRA = 29 mm, hPIN = 17 mm and ctheta = 0.2-the SsDRA bandwidth experiences a significant reduction, decreasing down to about 1.5 GHz, while the realized gain value approaches the monopole one, in the frequency interval from 4.5 GHz to 6 GHz.

Radiation Pattern, Realized Gain and Radiation Efficiency of the Optimized SsDRA
The previous analyses allowed to define the geometrical parameters of the bio-inspired SsDRA which exhibits wider bandwidth and a satisfactory gain value at two frequencies, 3.5 GHz and 5.5 GHz: t = 2 mm, hDRA = 29 mm, hPIN = 17 mm and ctheta = 0.2. Figure 7 shows the 3D radiation patterns of the optimized antenna, along with the polar cuts in the E and H planes, at 3.5 GHz and 5.5 GHz. In particular, when the frequency is equal to 3.5 GHz (Figure 7a-c), the maximum value of the gain is equal to 3.78 dBi. The main lobe magnitude in the Eplane (φ = 0 deg, Figure 7b) is equal to 0.727 dBi, the main lobe direction is equal to 53 deg and the angular width (3 dB) is equal to 82.7 deg. Conversely, when the H-plane is considered (φ = 90 deg, Figure 7c), the main lobe magnitude is equal to 1.05 dBi, while the main lobe direction is equal to 53 deg, with an angular width (3 dB) of 76.3 deg. Figure 7d-f plot the 3D radiation patterns and far-field radiation patterns of the optimized antenna at 5.5 GHz. In particular, in Figure 7e referring to E-plane, the maximum gain value is equal to 5.08 dBi, the main lobe magnitude at φ = 0 deg is equal to 3.91 dBi and the main lobe direction is equal to 61.0 deg with an angular width (3 dB) of 52.5 deg. In the H-plane (Figure 7f), for φ = 90 deg, the main lobe magnitude is equal to 4.22 dBi, the main lobe direction is equal to 61.0 deg and the angular width (3 dB) is equal to 50.4 deg.

Radiation Pattern, Realized Gain and Radiation Efficiency of the Optimized SsDRA
The previous analyses allowed to define the geometrical parameters of the bio-inspired SsDRA which exhibits wider bandwidth and a satisfactory gain value at two frequencies, 3.5 GHz and 5.5 GHz: t = 2 mm, h DRA = 29 mm, h PIN = 17 mm and c theta = 0.2. Figure 7 shows the 3D radiation patterns of the optimized antenna, along with the polar cuts in the E and H planes, at 3.5 GHz and 5.5 GHz. In particular, when the frequency is equal to 3.5 GHz (Figure 7a-c), the maximum value of the gain is equal to 3.78 dBi. The main lobe magnitude in the E-plane (ϕ = 0 deg, Figure 7b) is equal to 0.727 dBi, the main lobe direction is equal to 53 deg and the angular width (3 dB) is equal to 82.7 deg. Conversely, when the H-plane is considered (ϕ = 90 deg, Figure 7c), the main lobe magnitude is equal to 1.05 dBi, while the main lobe direction is equal to 53 deg, with an angular width (3 dB) of 76.3 deg. Figure 7d-f plot the 3D radiation patterns and far-field radiation patterns of the optimized antenna at 5.5 GHz. In particular, in Figure 7e referring to E-plane, the maximum gain value is equal to 5.08 dBi, the main lobe magnitude at ϕ = 0 deg is equal to 3.91 dBi and the main lobe direction is equal to 61.0 deg with an angular width (3 dB) of 52.5 deg.
In the H-plane (Figure 7f), for ϕ = 90 deg, the main lobe magnitude is equal to 4.22 dBi, the main lobe direction is equal to 61.0 deg and the angular width (3 dB) is equal to 50.4 deg.
The electric field and the magnetic field intensities of the optimized SsDRA at 3.5 GHz and 5.5 GHz are displayed in Figure 8. The higher intensity is recorded at the coaxial cable pin, but larger part of it is distributed throughout the whole available SsDRA volume. As noticeable, at 3.5 GHz, both the electric field (Figure 8a) and the magnetic field (Figure 8b) intensities are symmetric with respect to the pin while, at 5.5 GHz, the electric field ( Figure 8c) and the magnetic field (Figure 8d) intensities follow the dielectric resonator shape. In particular, in this last case, both intensities are located between the second and third loops of the shell. The analysis of the axial ratio reveals that the antenna shows a linear polarization as for the monopole over the frequency range of interest. The electric field and the magnetic field intensities of the optimized SsDRA at 3.5 GHz and 5.5 GHz are displayed in Figure 8. The higher intensity is recorded at the coaxial cable pin, but larger part of it is distributed throughout the whole available SsDRA volume. As noticeable, at 3.5 GHz, both the electric field ( Figure 8a) and the magnetic field (Figure 8b) intensities are symmetric with respect to the pin while, at 5.5 GHz, the electric field ( Figure 8c) and the magnetic field (Figure 8d) intensities follow the dielectric resonator shape. In particular, in this last case, both intensities are located between the second and third loops of the shell. The analysis of the axial ratio reveals that the antenna shows a linear polarization as for the monopole over the frequency range of interest. The radiation efficiency calculated for the optimized SsDRA is reported in Figure 9: the antenna exhibits an efficiency of about 45% around 3.5 GHz, while it increases up to about 85% at 5.5 GHz. The radiation efficiency calculated for the optimized SsDRA is reported in Figure 9: the antenna exhibits an efficiency of about 45% around 3.5 GHz, while it increases up to about 85% at 5.5 GHz. The radiation efficiency calculated for the optimized SsDRA is reported in Figure 9: the antenna exhibits an efficiency of about 45% around 3.5 GHz, while it increases up to about 85% at 5.5 GHz.

Discussion
The following Table 2 reports a comparison summarizing the performances obtained by numerical results of different DRAs, such as cylindric (C-DRA), rectangular (R-DRA), star-shaped (S-DRA) and Spiral shell (SsDRA). All antennas were modelled by considering the same lowpermittivity material, Resin V04, suitable for stereolithography [36,37]. As noticeable, the optimized SsDRA displays relevant wider bandwidth behavior in the sub-6GHz range compared to other geometries, and it also covers two frequency bands, 3.5 GHz and 5.5 GHz. Additionally, good gain values are calculated at both frequencies, despite the significant reduction of dielectric material volume used for the antenna realization. In this viewpoint, the choice of bio-inspired geometry allows

Discussion
The following Table 2 reports a comparison summarizing the performances obtained by numerical results of different DRAs, such as cylindric (C-DRA), rectangular (R-DRA), star-shaped (S-DRA) and Spiral shell (SsDRA). All antennas were modelled by considering the same low-permittivity material, Resin V04, suitable for stereolithography [36,37]. As noticeable, the optimized SsDRA displays relevant wider bandwidth behavior in the sub-6GHz range compared to other geometries, and it also covers two frequency bands, 3.5 GHz and 5.5 GHz. Additionally, good gain values are calculated at both frequencies, despite the significant reduction of dielectric material volume used for the antenna realization. In this viewpoint, the choice of bio-inspired geometry allows us to have bandwidth improvement, without affecting gain, yet reducing the required material quantity. Concerning the realization of a prototype, if the antenna was thinner than 2 mm and taller than 29 mm, the antenna would be structurally weak. Therefore, the set t and h DRA values set the threshold corresponding to an aspect ratio AR = 14.5, enabling the actual realization of the antenna and ensuring its structural robustness. Additionally, referring to Figure 2, it can be noticed that, at the end of the spiral, the bigger circle, which approximates the curve, has a radius of about 9 mm, while the smaller circle in the proximity of the origin has a radius equal to 77 µm. This minimum round radius represents the actual fabrication challenges also for 3Dprinting technology. As the technology feature resolution is defined by the diameter of the circular end-effector and being the minimum required radius equal to 77 µm, this turns into a minimum end-effector diameter of D e-e_min =2 * R = 154 µm. This consideration allows us to conclude that the SsDRA geometry cannot be feasible for the FDM process, as the majority of FDM machines are typically equipped with nozzle diameters of 200 µm or 400 µm. However, the SLA technology can be successfully exploited for the fabrication of the optimized SsDRA, as the laser spot diameter of the machine allows us to accomplish a micro-feature resolution of 85 µm [37].

Conclusions
In this work, the design of a bio-inspired Spiral shell DRA (SsDRA) implemented by means of Gielis' superformula was proposed. The antenna was thoroughly optimized by properly setting Gielis' superformula parameters considering two aspects: antenna performance and fabrication feasibility via Additive Manufacturing processes. In particular, the numerical analysis shows that optimized SsDRA exhibits very wide bandwidth behavior, up to 2 GHz, comprising two main frequencies: 3.5 GHz and 5.5 GHz. This characteristic made the proposed SsDRA feasible for applications in the sub-6GHz range. Moreover, the calculated gain is satisfactory, as 3.7 dB can be found at 3.5 GHz, while a gain of 5 dB is calculated at 5.5 GHz. From the fabrication viewpoint, the evaluation of the optimized SsDRA design proved that an accurate AM process is required to realize the micro-features characterizing the spiral rounds of the antenna; in particular, as the minimum size exhibited by the design is in the order of few tens of microns, stereolithography (SLA) is the best candidate to fabricate the SsDRA prototype.