3-D Printed All-Dielectric GRIN Lens Antenna With an Integrated Feeder

In this paper we present the design, fabrication, and experimental verification of a new type of Graded-index (GRIN) lens antenna with an integrated feeder. The continuously varying refractive index distribution is chosen appropriately to offer the rays collimation at the lens aperture. It is practically implemented by varying the material density in a host medium, thus realizing a new type of all-dielectric high gain antenna, entirely using 3D printing. This solution can find application to high gain wireless communication and measurement systems. This GRIN lens antenna is printed in one monolithic process and does not require the feeder to be placed at a focal distance, thus complying with more strict space requirements. It accepts interchangeable feeds that can cover a wide frequency range. The directivity and gain are evaluated using near-field measurements in the Ku-band. A 40% measured aperture efficiency is achieved at 14GHz. The challenges and performance limitations that come with 3D printing, as compared to the design of idealized continuous distribution GRIN lenses are discussed.

antennas. Despite the fact that the majority of the antenna was realized by 3D printing, the associated system still relies on an off-the-self waveguide adapter that suffers from unavoidable mismatch due to reflections in the air-dielectric interface that can reduce the realized gain of the antenna.
In this study, a GRIN lens antenna with an embedded feed is realized for the first time. Such an arrangement represents an all-dielectric alternative with no metallic walls. Also, it demonstrates the application of a zero focal length GRIN lens. In our recent study [1], we have presented the derivation of new formulas based on Geometrical Optics for a compact GRIN lens with embedded feeder (i.e., placed at a null focal length F=0). By assuming that the focus is inside the medium, the radially varying refractive index profile is derived in analytical form via the inverse truncated Abel transform. The refractive index profile is given as a function of the lens geometrical parameters, namely the lens radius and thickness. The GRIN solution for infinite cylinder and radial graded index already existed in the literature [18], [19], named by the author in [19] "Mikaelian Lens". However, it was mostly used in the framework of fiber optics and, to the authors' knowledge, this is the first time an antenna with an embedded feed is realized.
In this work, we proceed to the practical realization, by using 3D printing technology, and to the experimental verification of the GRIN lens with an integrated feeder proposed in [1]. All the challenges related to the unavoidable discretization of the refractive index, required for the practical realization, are presented and discussed. To validate the 3D printed GRIN lens performance, the full 3D far-field radiation pattern is measured using our in-house spherical near-field facility in a wide frequency range between [12][13][14][15][16][17][18][19][20][21][22] GHz. This is accomplished by swapping the feeders operating in different frequency bands. It should be noted that the bandwidth of the antenna system is bounded to both the frequency response of the feeder, as well as to selected discretization of the perforated 3D printed GRIN lens towards the higher frequencies. The directivity and the gain are extracted.
The article is organized as follows. Section II presents the design methodology of the GRIN lens together with the discretization and unit cell design suitable for 3D printing. Section III shows numerical simulations of the designed GRIN lens using a discrete number of permittivity layers. In Section IV, the 3D printing of the GRIN lens is presented together with the characterization of the 3D printed filament. In Section V, the measurement results are provided and in Section VI conclusions are drawn.

II. GRIN LENS DESIGN AND DISCRETIZATION
Let us first consider a cylindrical GRIN lens with a continuous refractive index, varying with respect to the radial coordinate ρ, as depicted in Fig. 1 (a), (b). According to [1], the radially varying refractive index profile, allowing for the rays collimation, is given by the following equation (plotted in Fig. 1 (c)): where the maximum refractive index as a function of the thickness-to-radius ratio d/a is given by: Here we select the thickness-to-radius ratio to be d/a = 1.2 and as a result the maximum refractive index is n 0 = 2 (or permittivity r = 4). For these lens dimensions, from [1], the calculated total aperture efficiency is 73% for a feed with U(θ ) = cos 3 (θ ) pattern. This is translated to an estimated directivity of 28.6 dB at 20 GHz. In order to use common thermoplastic materials for the lens fabrication such as ABS or PLA with a maximum permittivity of r = 2.5 − 2.7 respectively, the lens thickness-to-radius ratio d/a should be properly tuned in (2) to be d/a = 1.5.

A. CONTINUOUS AND DISCRETIZED MODEL
In this section we present the GRIN lens design together with the unavoidable discretization of the continuous permittivity distribution into a staircase distribution. Indeed, the 3D printing process requires the lens to be printed with a VOLUME 4, 2023 529 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. finite number of layers of different effective permittivity. The effective permittivity required in each layer can be realized by varying the infill percentage of the dielectric material.
The specific permittivity values are assigned by approximating the continuous permittivity profile given by (1) with a staircase permittivity distribution. The staircase distribution and the continuous one are plotted in Fig. 1 (c). The discretization process produces two kinds of errors in the permittivity distribution: i) an error due to the discretization of the lens in a finite number of rings, each one with a constant permittivity ii) an error in the homogenization process due to the approximation of the constant permittivity in the i-th ring to a holey unit cell. Due to the selection of 11 layers, the first error is negligible.
As it is noted in [12], [13], the GRIN lens performance is affected by the number of discrete permittivity layers and the maximum unit cell (UC) size. We decide to use 11 layers for achieving a smooth transition of permittivity and to avoid phase aberrations at the lens output aperture due to fragmentation.
As a first step, we compare the effect of the discretization of the cylindrical GRIN lens via electromagnetic (EM) simulations (see Fig. 2). For this purpose, a lens with radius a=75mm (corresponding to 2.5λ at 10 GHz and 5λ at 20 GHz) and thickness d=90mm has been chosen. In the discretized GRIN lens ( Fig. 2 (b)), each ring width is rw=0.1 a=7.5mm, except from the first and last which have rw=0.05 a=3.75mm. The different infill percentages assigned to the 3D printer, and corresponding to the different permittivity levels, are given in Table 1. The lowest permittivity for the lens is set to ε r,min = 1.5 due to fabrication limitations. The unit cell size used in the simulations of the perforated GRIN lens prior to 3D printing is also listed in the last column of Table 1.
In order to link the required infill percentage F with the specific effective permittivity from the staircase model, a permittivity measurement is done for different infill values using a rectangular sample. More details are given in Section IV. The direct measurements of the permittivity allow to reduce the homogenization error.
From the EM simulation we derive a directivity of 29 dB at 20 GHz for the continuous model of Fig. 2a, while for the discretized model of Fig. 2b, we find a directivity of 28.8 dB, which is only 0.2 dB reduced. This result confirms the absence of phase errors due to the lens discretization.

FIGURE 3. Shaped GRIN lens (a) continuous and (b) discretized into 10 layers.
A further modification of the cylindrical GRIN lens is considered as an attempt to reduce weight and volume. In [1], assuming the GO approximation, the analytical expression for the ray-trajectories inside the GRIN lens has been derived (3).
With the support of an in-house code implementing a raytracing algorithm, we noticed that only a few rays exceed the angle θ max (see Fig. 1 (b)) hence contributing to negligible spill-over. Thus, a possible improvement of the cylindrical GRIN lens with the embedded feeder is a shaped GRIN lens following the extreme ray-path ((3) for θ = θ max ), This opens the possibility of the realization of an all-dielectric lens-antenna (see Fig. 3).
The shaped GRIN lens in [1] ('cup lens' in the following) has negligible performance deterioration with respect to the cylindrical shape and it allows for weight and cost reduction since less material is used. Indeed, we have seen from simulations that the directivity of the continuous cup lens is 27.8 dB, −1.2 dB less than the cylindrical GRIN lens. This concept has been already discussed in [1], however has not been implemented yet. We then decide to realize the 'cup lens' instead of the cylindrical lens by using 3D printing.

B. UC DESIGN
Before proceeding to the lens fabrication, we will estimate the perforated 3D printed model performance via EM simulations by modelling the same unit cell used for 3D printing. As it is shown in Fig. 4, the unit cell is a rectangular hollow dielectric cell with thickness equal to the measured extrusion width of the 3D printer. It is measured to be 0.7 mm, expanded by 20%, from the nominal nozzle diameter of 0.6mm due to heat expansion and extrusion pressure. So, in the analysis below for the unit cell we set w = 0.7 mm.
The relationship between the unit cell size and the ε r,eff is plotted in Fig. 4. It is extracted numerically by using CST Studio Suite, using a method first used for a metamaterial unit cell in [20]. To achieve the effective permittivity gradient, the volume of the air gaps in the grid type 2D pattern in tuned in respect to the full solid host dielectric material volume. The size of the UC is 2w+b.
To guarantee that a similar performance is achieved of the perforated model as compared to the homogeneous discretization one (satisfying the effective medium approximation), the following rules should apply at any frequency point: • the ring width should be greater than two unit-cells [13].
• the size of the unit-cell should be at least 5 times smaller than the wavelength [16]. A smaller extrusion width w allows for a smaller unit cell size for a given infill ratio. This means that more unit cells can fit in the ring width making the effective permittivity approximation valid towards even higher frequencies and the GRIN lens bandwidth larger.

III. EM SIMULATIONS OF THE SHAPED CUP GRIN LENS
In this section, we compare the shaped GRIN lens model constructed by homogeneous layers and the model constructed by perforations. In the literature, most of the times, a discretized model of homogeneous permittivity levels is used. However, when the GRIN lens is realized via 3D printing using a single host material, this assumption of homogeneous permittivity is not yet valid. The frequency dependence of  the perforated 3D printed pattern is predicted via full wave analysis. In the perforated model of the GRIN lens (Fig. 5b) each of the ten layers is filled up with a different unit cell with size d associated to the respective permittivity value, given in Table 1. In the literature, only in a few cases, such as in [21], [22], there has been an investigation via EM simulations on the performance of the fully realistic 3D printable GRIN lens model (including the perforations using the unit cell) prior to the 3D printing. This is probably because this simulation model is computationally intensive requiring high computational resources. The simulation model of the perforated cup GRIN lens is shown in Fig. 5b, realized using a single material with r = 4.6, which is the measured permittivity of the printed filament.
The effective permittivity remains valid as long as the unit-cell dimensions remain small as compared to the wavelength (effective medium approximation). As the frequency is increased, this stops to be valid and the bandwidth of the lens is limited. In Table 2, the unit cell size in terms of the wavelength is listed.

A. INTEGRATED FEED DESIGN
The integrated feed design is based on a simple circular waveguide. The selection of the waveguide as compared to other type of antennas is the easy manufacturability and the rotational symmetry, same as for the GRIN lens. The circular waveguide is loaded with the same 3d printed material with effective permittivity as the one used for the first central layer of the lens. By doing so, we avoid any unwanted reflections at the interface between the lens and the feeder. We use metallic flanges of various diameters connected with a miniaturized SSMA connector to generate the TE11 mode. In this work, two circular loaded waveguides are realized, one with ø8 and the other with ø10mm diameter, to be embedded to the shaped GRIN lens to cover the Ku (12-18 GHz)  frequency band. CST Microwave Studio simulations are carried out to design the feeding of the loaded waveguides using an SSMA connector.
To investigate the frequency response of the perforated shaped GRIN lens we used four different waveguide feeds in EM simulations ( Table 3). The waveguide operating range is between the first and second cutoff mode. In [1] it had been shown that aU(θ ) = cos m θ pattern, with m=3 achieves the best aperture efficiency and lens illumination. All the feeds approximate this cosine pattern. Hence, they guarantee a low spillover.
In Fig. 7. the directivity of the shaped GRIN lens is plotted versus frequency, for both the discrete homogeneous and perforated models. For the homogeneous model, we observe that the trend of the directivity is increasing, meaning that there is no bandwidth limitation. As for the perforated model, we see that only at the low frequencies matches with the homogeneous model while as the frequency increases above 15 GHz, the directivity is reduced. This means that the effective medium approximation with the specific UC is no longer valid. The highest achievable directivity is spotted between 12-17 GHz at the range of 22dB.
The phase distribution at the output surface of the cup lens is derived from EM simulations. As it is observed in Fig. 6, the phase starts to be distorted at the outer rings from 15 GHz and higher.

IV. 3D PRINTING
The GRIN lens is realized using Additive manufacturing, and more specifically, Fused filament fabrication (FFF) technique. A nozzle of 0.6 mm diameter is used. The bigger nozzle diameter (standard is 0.4mm) allows for a faster printing and smaller probability of clogging, which is possible to happen in big models. The nozzle diameter determines the minimum layer thickness which should not be less than 25%. So, we selected a layer height of 0.2mm.
In the 3D printer software, or "slicer", we tune the infill percentage that describes the volume fraction of the  thermoplastic filament material to the total volume, thus the permittivity is tuned. Normally, the infill percentage is varied from 15% to 100%. As a host material, we use a custom filament PREPERM [2] based on ABS material loaded with ceramic particles to achieve a permittivity as high as 5. Due to the difficulty of reaching εr=1 using a 3D printing material, we have truncated the permittivity of the outmost ring (Ring 10) to be 1.5 instead of 1.3 that can be realized with an infill percentage of 15%.

A. PERMITTIVITY CHARACTERIZATION OF 3D PRINTED SAMPLES
The effective permittivity as a function of the infill percentage is measured using a waveguide transmission/ reflection technique to extract the permittivity and loss tangent of 3D printed rectangular slabs. These slabs of various infill percentages are placed inside a rectangular waveguide WR159 in the range between 5-7 GHz. The S-parameters of the rectangular test samples were recorded inside the waveguide and the dielectric properties were extracted by using the Nicolson-Ross-Weir method [23], [24]. In Fig. 10a, the setup is presented and in Fig. 10b, the measured samples of effective permittivity are fitted with a polynomial model. The Wiener max model is also plotted based on the nominal permittivity of the filament, which has been used in [25] to approximate different filaments permittivity response to the filling factor. As it is observed from the measured samples in Fig. 10b, the permittivity of the full solid 3D printed sample (for material filling factor 100%), is found to be r = 4.6 in respect to the nominal permittivity of the filament r = 5. This difference is explained from the non-uniformity created by the air gaps formed between the layers and is taken into account in the lens fabrication following the permittivity model of Fig. 10b extracted from the measurements.
A second order polynomial model is used for fitting the measured samples data of the effective permittivity versus

B. LENS FABRICATION CONSIDERATIONS
The optimization of the 3D printing parameters is an important step in the GRIN lens antenna design. In the following table (Table 4) the most important parameters together with the optimized values are listed. The fabrication of the samples and the GRIN lens prototype is done by using our in-house 3D printer (Raise 3D Pro 3) with a dual nozzle capability. The two filaments used, one for the lens mounting and one for the lens, are a standard and a custom ABS filament loaded with ceramic particles to achieve a nominal permittivity of 4 needed for the lens maximum permittivity. Using a 0.6mm nozzle, a trace width of 0.7mm was obtained and a height of 0.2mm was used to achieve good printing quality.

V. FAR-FIELD RADIATION PATTERN MEASUREMENTS
In this section, we present the far-field radiation patterns of the proposed GRIN lens antenna. The antenna measurements have been done in the spherical near-field facility of University of Siena. This part of the work is particularly important to verify the performance of the 3D printed GRIN lens in terms of the directivity and gain. The polarization of the feed is y-polarized. A standard gain horn (MVG QR2000) is used as a reference gain antenna for estimating the gain and dielectric losses of the lens antenna. The placement of the GRIN lens antenna in the anechoic chamber is shown in Fig. 11. The E-and H-plane radiation patterns are plotted in Fig. 12. A close matching of the main beam between simulations and measurements is observed. However, the sidelobe levels (SLL) are higher in the measured patterns, especially at 12 and 15 GHz.
The measured and simulated directivity and aperture efficiency are listed in Table 5. The difference between simulation and measurement is also noted. The maximum difference is found at 12 GHz, to be 1.7 dB. The maximum measured aperture efficiency is found to be 40% at 14 GHz, with the directivity to be 22.9 dB. The average losses across the frequency range are found to be 2.7 dB.
Comparison with the State-of-the-art: Here we compare our findings with the current state-of-the-art in the topic of 3D printed dielectric GRIN lens antennas. In [5], a dielectric dome Lens array is 3D printed, which is used in wide angle scanning applications. Even though it is used in a different application from increasing the broadside directivity, a perforated 3D printed lens design is adopted. A similar concept to the one presented here, of the zero focal GRIN lens is presented in [28]. An H-plane horn antenna with enhanced directivity using conformal transformation optics is presented. Although, the realization is done with a parallel plate waveguide (PPW) and a fan beam pattern is provided. In Table 6, the most relevant and recent works on the activity of 3D printed GRIN lenses antennas are compared.
As compared to the reported works, our 3D printed GRIN lens appears to be more compact, low-cost and achieves an aperture efficiency of 40%. In a future work, a different UC will be designed and fabricated to increase the aperture efficiency and gain bandwidth of the proposed cup GRIN lens. To enlarge the BW of the proposed antenna even at higher frequencies, the extrusion width in the 3D printing should be reduced. The model can become even more compact when using a higher permittivity material. Finally, a multiple feeds design will be investigated for multiple beam capability.

VI. CONCLUSION
For the first time a zero focal graded index lens antenna is realized with 3D printing, allowing the feed to be embedded within the lens and not at a finite focal distance, thus making the system more compact. Different frequency bands can be covered using interchangeable feeds. A 40% measured aperture efficiency is achieved while a directivity of 20 dB or higher is guaranteed at the Ku-band. This study shows the challenges and performance limitations that come with 3D printing and the frequency dependence of the selected 3D printing unit cell. This work opens the possibility to simply replace traditional metallic horn antennas with this new type of all-dielectric 3D printed GRIN lens antenna.