A Shape Generation Method for 3D Printed Antennas With Unintuitive Geometries

Additive manufacturing used in combination with versatile shape generation methods can enable designers to realize unintuitive antenna designs with bespoke electromagnetic behaviors that would normally be extremely difficult or even impossible to manufacture using conventional techniques. In this paper, we present a new custom algorithm that produces arbitrary three-dimensional (3D) meander line antennas. This algorithm is used in conjunction with multi-objective optimization to create two quadrifilar helix antennas (QHAs) and one monopole antenna, all with unique electromagnetic performances. One QHA has a wide bandwidth, high broadside gain, and compact size, while the other has a dual-band nature with different radiation patterns in each band. Similarly, the monopole example is a dual-wideband design which targets Wi-Fi applications. These structures possess a meander radius that varies with height as well as conductor thickness that varies along the meander path which allows for improved antenna performance over conventional meander line antennas. An example design was fabricated and tested in order to validate the performance of the optimized virtual antenna model.


I. INTRODUCTION
All antennas, no matter what their design, have inherent performance constraints based on their geometry [1]. Generally, fewer constraints result in a larger design space which can be exploited to realize antennas with unintuitive and complex geometries that have desirable properties such as wide bandwidth, high gain, strong circular polarization, and compact size [2], [3], [4]. However, some level of constraints are needed as completely unrestricted antennas may not be fabricable with traditional methods such as CNC machining. Fortunately, additive manufacturing techniques such as 3D printing provide a good solution to realizing freeform antenna structures since most structurally stable designs that can be modeled in CAD tools can be 3D printed [5], [6], [7], [8]. However, since the design space for truly three-dimensional The associate editor coordinating the review of this manuscript and approving it for publication was Chan Hwang See. antennas is potentially much larger than those using traditional highly constrained methods, new modeling techniques must be used to take full advantage of this increased dimensionality while making optimization of these 3D designs tractable. [9], [10], [11], [12].
Many of these unintuitive geometries are based on the helix antenna, originally introduced by John Kraus in the late 1940s [13], [14] with two primary modes of operation: normal and axial. Helix antennas are widely used for many applications, including global positioning systems, cellular communication [15], weather satellites [16], satellite tracking [17] and television broadcasting [18], but they tend to be limited in bandwidth [19].
Four individual helical antenna elements can be combined to form a quadrifilar helix antenna (QHA). This improves performance in multiple ways such as reduced sidelobes and better axial ratio albeit with increased manufacturing difficulty [20], [21]. One alternative design with improved bandwidth is the conical spiral antenna (CSA). It was invented by John Dyson in the late 1950s and is based on Rumsey's principles of frequency independent antennas [22], [23]. Different sections of the antenna are excited by different frequencies, creating active regions. This allows the structure to operate effectively over a wide frequency range [24], [25], [26]. There are also examples of optimization being employed in the design of these types of antennas [27].
Another important concept in antenna design is tapering [28]. An electromagnetic structure could make use of conductors of varying width to modify its impedance and improve matching. This commonly occurs in the vicinity of the feed or can sometimes be found in the antenna body itself, especially in planar designs [29]. Tapers are usually limited to simple geometries and rarely appear on volumetric wire antennas [30].
A technique for generating freeform helical-like antennas with spatially-varying conductor thickness and arbitrary cross-section could improve the performance of conventional helical antennas while being compatible with state-of-the-art multi-objective optimization procedures. This would enable a straightforward methodology for inverse-design of antennas with complex and unintuitive geometries which meet a set of user-defined performance objectives. To this end, we introduce our approach.
This manuscript presents a new freeform design method for wire antennas that combines properties of QHAs, CSAs, and tapering in order to overcome each of their limitations. As mentioned previously, QHAs traditionally have narrow bandwidth. This can be improved slightly by using multiple concentric helices, but the difference is minor given the increase in complexity [31]. CSAs have better bandwidth but do not typically operate in normal mode and can have asymmetries in their gain pattern. In addition, tapering can be difficult to achieve using traditional subtractive fabrication methods.
Using our method, we can create antennas which have significantly enhanced performance over comparable designs found in the literature. To demonstrate this capability, we realize a compact QHA with over 100% fractional bandwidth. Furthermore, another benefit of our method is the ability to create innovative antennas with disruptive behavior not common in literature, such as a QHA that possesses distinct and tailored radiation patterns in different frequency bands. In addition, to demonstrate that the proposed method is not limited to helix-like geometries, we realize a dual-wideband volumetric monopole for use in Wi-Fi applications. We then demonstrate that such antennas can be 3D printed using Verowhite plastic and then metallized with conductive silver paint [32], [33]. This approach allows for cheap and rapid prototyping.
An overview of the proposed method is shown in Fig. 1. We start by replicating a standard wideband monopole, similar to [34]. Then, we proceed to make various modifications such as changing the base shape, varying the monopole's profile along the height, introducing twisting, and altering Starting with a standard wideband monopole, various types of adjustments can be made to achieve many different results. the meander path. There is a large potential for disruptive and unintuitive antenna geometries through the use of this method, and the three examples presented in this work only represent a small portion of this new and considerably expanded design space. The remainder of this work is organized as follows: Section 2 outlines the process for designing and evaluating antennas using this method, starting with the definition of an arbitrary meander line and ending with a simulated model of the antenna. Section 3 presents three example designs, which consist of two QHAs and one unconventional monopole. It then discusses how they compare to current work in this area, and demonstrates the benefits of searching for antenna designs that go beyond the constraints of traditional geometries. Section 4 will discuss fabrication and measurement results for one of the example designs. Finally, Section 5 provides a summary and conclusions of the paper.

II. ANTENNA DESIGN
In order to effectively generate and simulate an arbitrability shaped 3D meander with variable cross-sectional area, a new design methodology is required to allow the designer to easily generate geometrically diverse meander-line antennas with only a few parameters. Furthermore, converting the shape into a form that can be imported and simulated with full-wave electromagnetic solvers must be an efficient and straightforward process. Once these steps are in place, then optimization can be applied to discover antenna designs which meet certain user-defined performance criteria. Subsection A discusses the shape generation and simulation methods, while subsection B presents the optimization process.

A. SHAPE GENERATION
The first step of the mesh generation process is to define a meander line in 3D space to serve as the path that the antenna must follow. This can be as simple as a straight line in one direction or could be something more complex such as a helical or a knot structure [35], [36], [37]. The path can be created in any number of ways, such as by manually defining points, using a parametric curve, or implementing a spline. Then, a ''base shape'' must be defined, which corresponds to the 2D cross-section of the meander line that follows the 3D path to create the geometry. While a circular crosssection is used as the base shape for the helical designs going forward, any closed planar geometry can be implemented including those based on Bezier or spline surfaces [38] or the superformula [39]. Fig. 2 shows some of the many possible meander cross sections that can be implemented. These include common shapes, such as a circle ( Fig. 2(a)) and square ( Fig. 2(b)) as well as more unintuitive shapes, such as those in Fig. 2(c) through Fig. 2(f). To the author's knowledge, the ability to generate antenna geometries that are based on an arbitrary 3D meander that also possesses an arbitrary cross section has yet to be reported in the literature. The next step requires the user to define a number of points and rings which set the element size of the resulting mesh. The number of points specifies how many vertices the base shape contains while the number of rings specifies how many cross-sectional base shapes are created along the 3D path. The shape generator places these cross-sections centered along the meander line and performs a 3D rotation using quaternions to align them with the slope of the line at that point. Each point on a given ring is then connected to two corresponding points on the next ring to create the triangular elements that define the mesh. The total number of mesh triangles is the product of the number of rings and the number of points. Lastly, points are added at start and end of the meander line to serves as caps for the shape and close the mesh. For the studies considered in this paper, we constrained the bottommost ring to be perpendicular to the z-axis to simplify connecting of the antenna feeds. Fig. 3 displays the main steps involved in mesh creation. An example 3D path which is differentiable at all points is given in Fig. 3(a) while Fig. 3(b) shows the vertices of a mesh which have been generated by rotating and translating a circular base shape according to the position and slope of the line. Fig. 3(c) illustrates how each of the vertices are connected to form a mesh. Each point is connected to the two adjacent ones in the same ring as well as the corresponding point in the next and previous rings. Fig. 3(d) presents the Once the mesh is created, it can be imported into an electromagnetics simulation tool to assess its performance as an antenna. For the purposes of the studies presented in this paper, FEKO, a commercial method-of-moments (MoM) solver, was chosen for multiple reasons [40]. Firstly, MoM is well suited for analyzing the behavior of a perfect electrical conducting (PEC) body in free space. Secondly, FEKO's built-in scripting language (i.e., EditFEKO) makes it straightforward to import the mesh points and triangle data from the shape generation code. A MATLAB script was created in which the user can input all the necessary parameters to define a geometry. Then it generates an EditFEKO script based on those parameters. It is possible in future work that the shape generation code could be converted to another language such as Python and a different electromagnetic solver package such as HFSS or CST could be utilized.
This method can be further exploited to create designs consisting of more than one meander. In these cases, the shape generator can be run multiple times and a translation or rotation can be applied to each meander. Then, each meander can be individually fed and the properties of each feed, such as input voltage phase and magnitude, can be varied as needed.
In the case of a QHA, four separate meanders are generated and then rotated around the z-axis by a 90-degree rotation angle. Note that the relative spacing of each element can be parameterized and tuned during the optimization process. A feed is placed on each arm and the phase of input voltages are offset by π/2 radians in an order opposed to the helix winding sense. This approach could also be extended further to realize octafilar helical antennas and even designs with an arbitrary number of meanders.
While our shape generation procedure can vary typical parameters such as meander height, radius, and number of turns, there are many other parameters that can be adjusted. Firstly, the meander radius can vary as a function of height. This allows for a meander whose radial position can vary as a function of the height above the ground plane (or virtual reference surface) while being continuously differentiable at all points. Secondly, the conductor width (i.e., area) of the meander cross-section can be varied along the path of the meander.
For the helical-like antennas discussed in this paper, their meanders can be defined using a simple set of equations. Equation (1) defines the parameter t where h is the height and n is the number of turns. The x and y coordinates are calculated in (2) and (3), respectively, where r(z) is the height dependent meander radius. The z coordinate is bounded between zero and the total height. While only a few parameters are needed to produce geometrically diverse meander antennas, by connecting the shape generation procedure to an appropriate optimization algorithm we can tune the shape parameters in order to realize a custom set of performance criteria.

B. OPTIMIZATION
Multi-objective optimization is a powerful tool for electromagnetic device inverse-design since it can simultaneously minimize the values of multiple competing cost functions (e.g., impedance matching, gain, and other performance metrics) [41]. In this paper, Borg, a multi-objective evolutionary algorithm (MOEA), was chosen due to its ability to self-tune its optimization parameters and robust performance on a wide range of electromagnetics problems [42], [43]. Furthermore, Borg is well suited for parallel processing and can even be run asynchronously for improved multicore performance. The input parameters for optimization include the meander height, number of turns, N points for meander radius, and M points for conductor width, where N and M can be any positive integer. The points for meander radius are evenly distributed along the height of the structure. They are then fit to a polynomial so the transitions between different radii are smooth. The points for conductor width are distributed in a similar manner, except they are fitted to a spline instead. The values for N and M must be less than or equal to the number of mesh rings. It was found that, for the helix example designs presented later in this work, a value of N = M = 4 provided a good balance of optimization time and design space of antenna geometries. With the additional parameters of meander height and number of turns, this results in a total of 10 optimization parameters. , and 5(f) portray increasing, decreasing, and oscillating conductor widths respectively. Moreover, designs can have both varying meander radius and varying conductor thickness simultaneously. The minimum meander radius can be zero, which represents a straight wire in the z-direction. However, the conductor width must have a nonzero lower limit or else the antenna would become arbitrarily thin and fabrication would be impractical. The choice of upper limits on each of these parameters are at the designer's discretion, but the max conductor width must be small enough compared to the minimum meander radius as to avoid self-intersection. VOLUME 10, 2022

III. EXAMPLES
In this section, three example meander antennas were optimized to demonstrate the versatility of the proposed design method. The first example is a dual-band QHA with unique radiation characteristics in each band, the second is a broadband QHA with high gain and over 100% fractional bandwidth, and the third is a dual-wideband monopole for use in Wi-Fi applications. With the dual-band example, our goal was to demonstrate that the proposed method is capable of creating innovative antennas that achieve performance profiles which are not feasible using conventional design techniques. Next, with the broadband example, our goal was to demonstrate that the proposed technique can also be utilized to construct more conventional designs albeit ones that realize enhanced performance over antennas created using traditional methods. Lastly, the monopole example demonstrates the versatility of the method, in that it is not limited to only designing helix-like geometries.

A. DUAL-BAND HELIX
Dual-band QHAs that possess different radiation characteristics in each band typically require a complex mechanical system with moving parts [44] or multiple concentric antennas [32] in order to realize their multi-functionality. Using the method presented in this work, it is possible to create a single antenna with dual-band, dual-mode performance without the need for moving parts. The targeted design is a QHA that exhibits an omnidirectional, linearly polarized gain pattern in one band and a broadside, circularly polarized pattern in the second band. An antenna such as this would be useful in applications where communication to both earth and space-based receivers is desired. For demonstration purposes, the lower band was chosen to be centered at 2 GHz, while the upper was set an octave higher at 4 GHz. However, all dimensions can easily be scaled based on wavelength, so the separation between the two bands as well as the targeted operating frequencies can be chosen arbitrarily. To the author's knowledge, a single QHA with no moving parts and with this performance profile has yet to be presented in the literature.
The following cost functions were utilized to optimize the dual-band QHA Equations (4) and (5) measure the impedance matching in the lower and upper bands, respectively. Equation (6) concerns the omnidirectionality of the antenna's radiation pattern in the lower band by maximizing the minimum value of the Earth mode gain. Equation (7) seeks to maximize the broadside gain in the upper band. Finally, (8) considers the antenna height with the goal of keeping its footprint small. Note that for (6) and (7), gain values are considered on a linear scale, not dB. In future work, additional cost functions could be implemented to further lessen the antenna's footprint. It could be optimized for properties such as minimal volume and low weight.
The lower and upper bounds on the total height are set to 15 mm and 150 mm, respectively. At 4 GHz, this corresponds to a range from 0.2 to 2.0 wavelengths. The number of turns is bounded between 0.5 and 6. The values for conductor width fall between 0.75 mm and 3 mm, which is equivalent to a span from 0.01 to 0.04 wavelengths at 4 GHz. The meander radius was bounded between 6 mm and 75 mm, which corresponds to a range of 0.08 to 1.0 wavelengths in the upper band. Meanwhile, the antenna is fixed to the ground plane with a 1.86 mm gap between the two.
After optimization, a Pareto front of over 100 designs was obtained and a design was selected that met the S 11 requirement and had the best Earth mode gain, which is the most difficult metric in which to achieve good performance. The selected design has a height of about 70.3 mm with each arm making only half a turn. The meander radius is widest at the base with a value of about 46.7 mm. It then decreases sharply until a quarter of the way up the structure, reaching a minimum radius of 17.7 mm. After that it gradually increases for the remaining height, reaching a value of 32.5 mm at the top. In a similar fashion, the conductor radius is largest at the bottom of the antenna at around 2.7 mm. It then decreases to around 1.4 mm partway up the height and gradually grows for the remaining distance, reaching a final value of 2 mm. The FEKO model showing the geometry of this example antenna is provided in Fig. 6 and a plot of the S 11 versus frequency is shown in Fig. 7.
Moreover, the far-field patterns in each band are also simulated. At 2 GHz, the QHA operates in normal mode with an omnidirectional radiation pattern. At 4 GHz, the antenna operates in axial mode with a broadside radiation pattern. The maximum gains in the lower and upper bands are 5.9 dB and 10.3 dB respectively. A polar plot of the farfield patterns is shown in Fig. 8. As one can see, the optimizer was able to tune the antenna geometry in order to produce the two desired and distinctly different radiation patterns. The unique performance profile of this antenna is most likely  due to its similarity to conical spiral antennas. It exploits the properties of active regions which come from Rumsey's angle and truncation principles [22], [24]. The tapering also further improves the impedance matching.

B. BROADBAND HELIX
While the goal of the previous example was to create a new type of QHA, our method can also be used to address the existing limitations of more conventional designs. While QHAs commonly suffer from narrow bandwidths, the introduction of a spatially varying meander radius and conductor width can significantly improve their performance in this regard. Thus, the goal for the second example is to optimize an arbitrary QHA that achieves the highest possible bandwidth.
Four different cost functions are used to optimize the geometry of this antenna: Equation (9) accounts for the bandwidth, which takes the worst S 11 value over the entire bandwidth and attempts to minimize it. Equation (10) evaluates for the minimum broadside gain over the bandwidth. Equation (11) is the gain span and minimizes the difference between the highest and lowest gain to enable more consistent performance. Equation (12) is meant to minimize the footprint of the antenna. Note that for costs 2 and 3, gain values are considered on a linear scale, not dB. The bounds on all the parameters are identical to the previous example design. However, the gap between the antenna and ground plane was adjusted to a value of 1.24 mm. This gap was smaller than in the dual-band design because it was scaled according to the center frequency of operation. Note that, similar to the dual-band design, additional cost functions could be added to minimize properties such as overall antenna volume and weight.
After optimization, several hundred Pareto optimal designs were found. A design was selected that demonstrated a good balance of wide bandwidth, high gain, and short height while also keeping the difference between the highest and lowest gains small. The FEKO model of this example antenna is shown in Fig. 9 with labels for height, conductor width, and meander radius. The antenna height is only 28 mm, which corresponds to about λ/2 at the center frequency, giving the antenna a small footprint.
Similar to the dual-band design, each arm makes a little over half a turn. In terms of other parameters, the meander radius is widest at the base with a value of 32.1 mm. It then oscillates between low and high values, reaching a minimum radius of 9.24 mm and ending at a radius of 19.8 mm at the top. The conductor radius gradually increases from 2.4 mm to 2.9 mm along the height, before decreasing to 2 mm once reaching the top.
The simulation results show that the antenna is able to obtain an S 11 of less than −10 dB over nearly the entire band from 2 GHz to 7 GHz (Fig. 10(a)). This corresponds to a fractional bandwidth of 111% which is higher than what conventional QHAs can normally achieve. Meanwhile, the antenna maintains a maximum gain value between 8 and 10 dB over the entire frequency range of interest ( Fig. 10(b)). Similar to the dual-band example, the design of this antenna exploits properties of active regions to achieve a wide bandwidth, and its tapered arms further improve impedance matching.   The far-field gain pattern was then calculated and is presented in Fig. 11. One can see that the antenna forms a stable broadside beam over the entire frequency range. Moreover, the gain is entirely left-hand circularly polarized (LHCP) with no coupling to the orthogonal polarization.
The antenna maintains a good S 11 from 2 GHz to 7 GHz, which corresponds to a fractional bandwidth of 111%. Additionally, in order to demonstrate the benefits of the taper, a separate simulation was performed. A duplicate of this antenna was created in FEKO but with the same conductor width along the entire antenna structure. It is observed in Fig. 12 that the taper significantly improves impedance matching in the middle of the frequency band. Also, note that the antenna was simulated using an infinite ground plane to decrease the required computational time. However, additional tests were performed which demonstrated that a 10 cm ground plane was sufficient in minimizing any effects on performance. Compared to similar antennas in the literature, this antenna surpasses most previous designs in terms of fractional bandwidth and gain, while remaining comparable in terms of size (see Table 1). While the design presented  in [49] achieves similar performance to our design, it requires a much larger footprint.

C. WI-FI MONOPOLE
While the previous examples have focused on QHAs, the proposed method is not limited to this type of geometry. In fact, it is trivial to restrict the meander path to be a straight line along the z-axis and proceed to vary the meander width, twisting, and base shape to create geometrically diverse monopoletype antennas. Our goal here is to create a multi-band 3D monopole which operates over different Wi-Fi regimes.
This antenna targets the frequencies of 2.4 GHz and 6 GHz to make it applicable for present and future Wi-Fi applications [50]. In total, 13 design parameters were used including 9 values defining the meander width along the height, and values which govern the monopole height, offset above the ground plane, amount of twisting, and a selection of one of six base shapes (Fig. 2). The following four cost functions were employed: Equation (13) attempts to achieve a good match in each of the two target bands. Next, (14) makes sure the antenna maintains a multi-band property by forcing poor matching at frequencies which are out of band. Lastly, (15) and (16) attempt to improve the omnidirectional gain within the lower and upper bands, respectively. Potential future studies could add additional cost functions which would aim to reduce the volume and weight of the antenna for further reduced footprint.
After optimization, a Pareto front of about 100 designs was generated, and one design was selected that provided a good balance of each of the performance goals. Fig. 13(a) shows the selected antenna's geometry which is modeled in FEKO. Fig. 13(b) presents the S 11 behavior as a function of frequency and compares it to a traditional quarter-wave wire monopole. Fig. 13(c) shows the omnidirectional radiation patterns over various in-band frequencies. Note that these patterns are entirely vertically polarized. Lastly, Fig. 13(d) compares the peak gain of our design to a quarter-wave monopole over various frequencies. The final design's tapered structure with narrow and wide sections contributes to its dual-wideband nature.
While there exist examples of dual-band monopoles in the literature; however, to the author's knowledge, they are either planar [51], [52] or have relatively narrow bandwidth in each band [53]. In addition, this antenna is able to achieve good performance over common Wi-Fi bands without the need for a matching network and while maintaining the same size as a conventional quarter-wave monopole.

IV. EXPERIMENTAL DEMONSTRATION
The broadband QHA example design was selected as a good candidate for fabrication and testing in order to validate the efficacy of the proposed design approach. A mount was designed in Solidworks to provide mechanical support for the antenna during operation and was produced using a Stratasys Objet260 Connex3 3D printer and VeroWhite plastic. The mount was designed to provide mechanical rigidity for the structure while having minimal impact on the electromagnetic performance.
Next, high resolution 3D models of each QHA arm were exported from FEKO as.stl files so as to be compatible with the 3D printer system. Since each arm is identical, four copies of one arm were printed. Once printed, each VeroWhite arm was sprayed with conductive silver paint and a 0.065-inch bit was used to drill a hole into the bottom of each to facilitate easy connection to the feeds. The antenna was assembled on the mount and attached to a square 10 cm copper ground plane with four holes. SMA connectors were placed on the underside of the ground so as to pass through it and slot into the holes in each antenna arm. This allows each arm to be fed individually while adding a minimal amount of conductive material above the ground plane that might interfere with performance.
The conductivity of the silver paint is 8.3 x 10 5 S/m. Through an additional Feko study, its radiation efficiency was found to be approximately 99% over the frequency band. Therefore, the metallization process is not expected to have any significant detrimental effects on performance for this design.
The steps for antenna fabrication are highlighted in Fig. 14. The printed antenna arms and the mount prior to metallization are displayed in Fig. 14(a), while Fig. 14(b) shows the metallized antenna and mount attached to the copper ground plane.  Copper tape is added over the screws to prevent them having any negative impact on the performance. An image of the underside of the antenna setup with the connectors passing through the ground plane is given in Fig. 14(c). Finally, Fig. 14(d) shows a close-up of how the port and the antenna are connected.
In order for this QHA to operate properly, the phase of the input voltage on each arm must be offset with phases of 0, 90, 180, and 270 degrees and arranged in such a fashion where the direction of increasing phase opposes the helix winding sense. To ensure performance is maintained over the entire bandwidth, the phase difference between the four arms must be held constant. This can be achieved using a wideband feeding network built from commercially available phase shifters [54]. We note that future alterations of the design could make use of a more compact custom feeding network [55]. The feeding network consists of two 180-degree hybrid phase shifters and one 90-degree hybrid phase shifter, which are all rated for operation between 2 GHz to 8 GHz. The specific models in use were purchased from RF-Lambda and include the RFHB02G08GPI coaxial 30W 180-degree hybrid and the RFHB02G08GVT coaxial 50W 90-degree hybrid coupler respectively. Both are lightweight and compact.  A diagram of the phase shifter arrangement is illustrated in Fig. 15(a). A single input is connected to the 90-degree phase shifter. Then each output from that component connects to the input of one of the 180-degree phase shifters. Finally, each output of the 180-degree phase shifter connects to each antenna arm oriented counter to the helix winding sense. The isolated ports on every component are connected to 50-Ohm load.
The phase shifters are placed on a copper-clad FR4 plate underneath the antenna and ground plane. SCA49240-12 and RFC6969A-E9NL305 cables were used to connect the components. A FR4 dowel rod is also mounted on the base plate to support the antenna and the ground plane. The entire setup with the base plate, phase shifters, dowel, cables, ground plane, mount, and antenna is shown in Fig. 15(b).
The antenna and ground plane system was measured using a vector network analyzer by connecting each port to a different VNA input. The resulting S 11 versus frequency plot for each antenna arm is shown in Fig. 16. The measured results indicate good impedance matching, with the S 11 being less than −10 dB over most of the band. Also, each of the four QHA arms have near identical performance, which is expected based on the simulated results. However, there is reduced performance at low frequencies around 2.0 to 2.5 GHz, which is most likely a result of minor fabrication inaccuracies.
The antenna was then measured in an anechoic chamber. Fig. 17 shows the normalized measured LHCP far-field patterns compared to the simulation case at four different frequencies. The normalized patterns are offset by a constant of 6 degrees. This is most likely a result of misalignment when mounting in the chamber. After adjusting, the experimental data matches well to the simulated results.

V. CONCLUSION
The ability to easily model freeform 3D meanders with arbitrary cross-sections and varying conductor thicknesses represents a new paradigm in antenna design methodology. Utilizing this new method in combination with 3D printing technology allows for the fabrication of a variety of new and unintuitive antenna configurations. In addition, meanders designed using this technique are defined by a small number of parameters and can be easily optimized for any arbitrary criteria using powerful multi-objective techniques.
The three examples showcased here were a dual-band QHA with different radiation characteristics in each band, a wideband high-gain QHA, and a dual-wideband monopole. The wideband QHA outperforms similar designs in literature and has applications in areas such as global positioning, cellular communication, and radio astronomy. While the dual-band antenna could potentially be used in a situation which would usually require two separate antennas or a complex system involving moving parts. Lastly, the monopole example is capable of being employed in current and future Wi-Fi systems as well as demonstrates that the method introduced here is not limited to helix-like geometries. Fabrication is straightforward and inexpensive with feeding networks that can be easily constructed using commercially available phase shifters. Measured results for the broadband QHA compare favorably to simulations. Finally, the designs shown in this paper are only three examples, and the proposed design procedure is capable of producing a multitude of other complex structures with unique and disruptive behaviors.