Low Profile GRIN Lenses With Integrated Matching Using 3-D Printed Ceramic

In this paper, we investigate a shortened horn antenna with high gain that is enabled by a 3D-printed gradient index (GRIN) lens composed of high permittivity zirconia <inline-formula> <tex-math notation="LaTeX">$(ZrO_{2})$ </tex-math></inline-formula>. The baseline H-plane sectoral horn antenna is designed with length that is 1/3 of the optimal horn antenna and exhibits a low gain due to the high flaring rate of the horn. Increased gain is achieved by adding a flat GRIN lens at the horn aperture. High permittivity <inline-formula> <tex-math notation="LaTeX">$ZrO_{2}\,\,(\varepsilon _{r} = 23)$ </tex-math></inline-formula> enables lens miniaturization; however, when interfaced with air, reflections at the air interface increase the impedance mismatch. Two different methods for mitigating the reflections are studied. One is a simple <inline-formula> <tex-math notation="LaTeX">$\lambda /4$ </tex-math></inline-formula> matching layer that matches the bulk permittivity of <inline-formula> <tex-math notation="LaTeX">$ZrO_{2}$ </tex-math></inline-formula> to air. As expected, the <inline-formula> <tex-math notation="LaTeX">$\lambda /4$ </tex-math></inline-formula> layers reduce the reflections in part of the 13–18 GHz band but produce high reflection in other parts. The second approach is a GRIN lens with integrated tapered matching layer to match phase and impedance simultaneously. Three tapering methods are studied (exponential, Klopfenstein, linear) for impedance matching. Analytical expressions of the minimum thickness and permittivity distribution are derived. The lens is discretized for print and three types of unit cells are proposed to create a wide range of permittivities ranging from bulk ceramic to air. A <inline-formula> <tex-math notation="LaTeX">$ZrO_{2}$ </tex-math></inline-formula> lens prototype printed with an XJet Carmel 1400 is measured and results show good agreement with simulations, including gain performance equivalent to a horn of 2.4x longer length. The measured gain and beamwidth of the lens are 5.4dB higher and 52° narrower than those of the shortened horn alone at 15 GHz, respectively.


I. INTRODUCTION
H ORNS are one of the most widely used microwave antennas due to their simple structure, high gain, and versatile design. Horns are used as antennas for 5G and millimeter-wave applications [1], [2], [3], antennas for electronic warfare (EW) [4], [5], and feeds for lenses [6], [7]. To attain high gain with a conventional horn, wide aperture size and low flaring rate are required making a relatively long focal distance from the phase center to the aperture. In contrast, a short horn antenna with a high flaring rate can cause phase error that reduces antenna gain [8].
A gradient index (GRIN) lens with refractive index that continuously varies in space can enhance the gain of a short horn when placed in the aperture. The spatially-varying refractive index enables wave collimation in the aperture and produces a directive beam [9], [10], [11], [12], [13]. Unlike homogeneous lenses that must exhibit a specified curvature, GRIN lenses can have diverse shapes and be integrated with various types of antennas [14], [15].
It is difficult to fabricate a GRIN lens using conventional subtractive manufacturing due to its complex internal structure. While 3D printing allows the realization of complex structures, most printed materials are polymers that have low permittivity and moderate-to-high dielectric loss [16], [17], [18]. Reference [16] fabricated a lens using a printed polymer (ε r = 2.7, tan δ = 0.02). The measured results showed that the material loss decreased the antenna gain by 1.5 at higher frequencies.
Ceramic materials exhibit a wider range of permittivity with better mechanical strength and lower loss tangent than most polymers [19]. 3D printed ceramics are widely used for antenna applications, e.g., dielectric resonator antenna (DRA) [20], [21], [22], [23], GRIN lens [24], and dielectric waveguide-based antennas [25], [26]. Zirconia (ZrO 2 ), the primary material used in this paper, is a technical ceramic with high permittivity that has recently been printed with near-full density using a nanoparticle jetting (NPJ) technique [27], [28], [29]. Although high permittivity reduces the dimensions of a GRIN lens, when interfaced with air, a larger impedance mismatch occurs. In order to achieve high radiation efficiency, a matching layer (ML) can be added to the GRIN lens. Reference [6] added a GRIN lens with MLs to a shortened horn. The lens shows similar gain compared to the optimal horn even though the lens and shortened horn length is shorter than the optimal length. Reference [9] introduced a GRIN lens with a similar phase matching approach, employing a linearly tapered ML to match phase and impedance simultaneously. However, both lenses are relatively thick (1.3λ, 1.9λ) and use permittivity around ε r = 7. As we demonstrate in this paper, a higher permittivity enables a significantly thinner lens.
In this paper, we study novel GRIN lenses for a shortened H-plane horn using 3D-printed high permittivity ceramic ZrO 2 . The higher permittivity of ZrO 2 results in a thinner lens than prior work, and we provide a study of several matching techniques to assess the tradeoff between thickness and performance. In Section II, a shortened horn with a high flaring rate is designed as a baseline. Then permittivity profiles of GRIN lenses using printable materials are investigated to correct the phase error. However, when interfaced with air, impedance mismatch occurs. To mitigate the reflections, we study two different matching methods in Section III. First, a simple λ/4 matching layer is added to a GRIN lens. The other method is a tapered matching layer that combines the lens and ML into a single design stage. The tapered lens has gradient permittivity in the H-plane and the propagation direction to match phase and impedance simultaneously. The minimum thickness and 2D permittivity distribution of the lens according to the horn dimensions and permittivity is introduced by analytical expression. The three tapered lenses and quarter wave matching lens are investigated and compared to the optimal horn and simpler matching technique. In Section IV, unit cells to generate a range of effective permittivities are generated by varying the fill ratio of unit cells. Due to the range of permittivities required in the design, different types of unit cells are used in different regions. We introduce three different types of unit cell models to realize a wide range of effective permittivity. A lens is 3D printed using a nanoparticle jetting technique with ZrO 2 . In Section V the performance of the printed lens is measured and analyzed.

II. DESIGN APPROACH OF SHORTENED HORN AND GRIN LENS
An H-plane sectoral horn has an optimal length that depends on its aperture width as [8] where a is the aperture width in the H-plane, λ is operating wavelength, l is the distance from the focal point to aperture. We consider a specific baseline horn design with aperture width a = 3 √ 2λ at the operating frequency of 15 GHz. The horn has an optimal length of length l 1 = 6λ as shown in Fig. 1. Note that these dimensions are the same as those studied in [6]. The goal of this study is to introduce a gradient index lens that is placed in front of the horn allowing reduction of its electrical length while maintaining comparable electrical characteristics. In the specific example studied here, we reduce the horn's length by a factor of 3 and replace it with a shortened horn of length l 2 = 2λ as shown in Fig. 1. As the maximum directivity is related only to the aperture size, in principle, the shortened horn can achieve equal directivity to the longer optimal horn, if the phase error is corrected by the lens.

A. DESIGN OF PERMITTIVITY PROFILE OF GRIN LENS
The shortened H-plane sectoral horn antenna is designed with length 2λ at 15 GHz, which is 1/3 of the optimal horn antenna and exhibits 3 lower gain due to the high flaring rate of the horn. To correct the phase error in the aperture, a GRIN lens must be added as illustrated in Fig. 2 to selectively delay the fields in the center relative to the edge. Since the lens is flat, the phase error is corrected by varying the effective permittivity of the material from the center to the edges along the x-axis.
To find the required permittivity distribution, we assume that cylindrical waves are radiated from the focal point toward the horn aperture at z = 0. The phase difference between waves arriving at the center of the aperture x = 0 and at other points on the x-axis varies as [8] φ where R is the distance from the focal point, P, to the lens interface (z = 0) and β 0 is the phase constant in free space. The lens is designed to apply a variable phase shift to waves passing through the lens so that they are co-phasal in at the front of the lens aperture z = t. Neglecting reflections for now, the required phase shift as a function of position is [10] φ where t is the thickness of the lens, β(x) = β 0 √ ε r (x) is the phase constant in the material, which has spatially varying permittivity. From (3), the permittivity along the x-axis is given by where ε r,max = ε r (x = 0) is the largest permittivity that can be obtained with the composing material (i.e., its bulk permittivity). Rays inside of the lens are assumed straight lines as shown in Fig. (2). Recently, [30] derived a new equation that is a solution of curved lines in a lens and demonstrated that the lens reduced the phase errors in simulation compared to [10] and [31]. However, the permittivity distributions along the x-axis are almost identical between (3) and the modified equation in [30] since the shortened horn in this paper has a small ratio of t/a. Therefore, we use (3) in this paper. Fig. 3(a) represents the permittivity distribution along the x-axis computed by (4). Four widely used 3D printable materials are considered for the lens-two ceramics, zirconia [27] and alumina [32], a photocurable polymer, Formlabs Clear [33], and a composite ABS/SrTiO 3 material [6]. These materials have permittivities of 23, 9.8, 2.8, According to (4), the thickness, t, of the lens alone is defined as ε r,min is the permittivity value at the edge (x = a/2). Therefore, t is determined by the bulk permittivity of the material and the minimum permittivity at the edge, and lens thickness can be adjusted by the bulk permittivity and set of the minimum permittivity at the edge. The minimum thickness, t min , of the lens is achieved by assuming ε r,min = 1, i.e., the permittivity approaches air at the edge of the aperture so that the permittivity profile ranges as 1 ≤ r ≤ r,max . Thus, t min is determined by the bulk permittivity of the material, and lens thickness can be reduced by using high permittivity materials as seen in Fig. 3(b). The required thickness of ZrO 2 lens is 4.8 mm, is 5.5× thinner than a Formlabs Clear lens. While the higher permittivity of a printable ceramic can produce a thinner lens, higher permittivity materials cause larger reflections when interfaced with air due to the impedance mismatch. This increases mismatch at the feed of the horn antenna, resulting in a poor S 11 and realized gain. Fig. 3(c) shows the S 11 results of four lenses. The lenses that have continuously varying permittivity values are placed on the aperture of the shortened horn fed by WR-62 waveguide and simulated using HFSS full-wave electromagnetic simulation software. The polymer lens shows small reflections across the frequency range since the permittivity is small. However, the other three lenses present high reflection, resulting in low realized gain as shown in Fig. 3(d).
While the gain and realized gain of the polymer lens are almost the same, the realized gain of the ZrO 2 lens shows noticeable drops due to impedance mismatch. Therefore, a matching layer is needed to reduce reflections.

III. MATCHING LAYER-EMBEDDED GRIN LENS A. QUARTER-WAVE MATCHING LENS (QWML)
Clearly the primary reflection occurs due to the high permittivity material in the center of the lens as shown in Fig. 3(a). As one simple matching network, we applied the λ c /4 matching rule, where λ c is the wavelength in the center of the lens at the operating frequency of 15 GHz. The matching layers (MLs) are placed both top and bottom of the lens where they contact the air, as shown in Fig. 4(b). The permittivity and thickness of the MLs are ε r = 4.8 and λ c /4 = 2.28 mm, respectively. Fig. 4(c) shows the permittivity distribution in the lens. While the permittivity values are constant along the z-axis and in MLs, they gradually decrease along the x-axis to collimate the radiated field. The thickness of the lens including MLs is 0.47λ (9.36 mm). The total thickness of the lens and the shortened horn is 2.47λ (49.36 mm), which is less than half of the optimal horn.
We also designed λ c /4 MLs for the alumina, ABS/SrTiO 3 , and Formlabs Clear lenses. The thicknesses of the lenses are 0.7λ (14.1 mm), 0.84λ (16.8 mm), and 1.73λ (34.5 mm), including MLs of 5.6 mm, 6.1 mm, 7.7 mm, and the ML permittivities are 3.1, 2.7, and 1.7, respectively. Then, the MLs with lenses were simulated in HFSS. Equation (4) is applied to material properties in HFSS without tangent loss for a continuously varying permittivity model.  range. Fig. 5(b) presents comparison of the realized gain. While alumina, polymer, and ABS/SrTiO 3 lenses show that the realized gain with MLs is lower than without ML in some frequency ranges due to high side lobes in H-plane, the ZrO 2 lens with ML constantly enhances the realized gain in a majority of the frequency range. The realized gain of the ZrO 2 lens is 5.1dB higher than that of the shortened horn at 15 GHz.
We selected ZrO 2 as a component material for the GRIN lens. Since ZrO 2 is a high permittivity ceramic (ε r = 23), it reduces thickness of the lens according to (5) and Fig. 3(b). It is also mechanically stable and has low loss tangent (0.0013) [27]. In addition, the ZrO 2 lens with QWML shows consistent realized gain in the frequency range.

B. TAPERED MATCHING LAYERS
The λ c /4 ML of ZrO 2 reduces reflections in part of the 13-18 GHz band but presents S 11 > −10 in some frequencies. The QWML has constant permittivity values in the propagation direction and matches only the bulk permittivity of ZrO 2 at the center of the lens. Alternatively, an integrated matching layer with continuously varying permittivity in two dimensions can be designed to ensure the impedance matching and phase collimation at every point of the lens.
The basic assumption for the analytical solution of tapered lenses is the same as the quarter wavelength matching layers: as waves pass through the lens, both the center and an arbitrary point on the x-axis, require the same phase to become a plane wave as shown in Fig. 6. The permittivity value is tapered from 1 (z = 0) to the peak (z = t 1 ) in half of the lens, and the other half is its mirrored version. The analytical solution of the permittivity distribution for the lens is defined as [34]: (6) where t 1 is the half of the lens thickness. Equation (6) is solved to find ε r (x, z) using three tapering methods (exponential [35], klopfenstein [36], linear) in the propagation direction for impedance matching. This section presents analytical expression for the permittivity satisfying (6) using two methods (exponential, linear), and computes the minimum thickness for the lenses. Also, the advantages and disadvantages of these three types of tapered lenses will be compared with each other and the QWML.

1) EXPONENTIAL TAPER
An exponential impedance taper for half of the lens in the z-axis is expressed as [37]; where Z 0 is the impedance in free space. α is a constant. The peak permittivities are at z = t 1 where Z(t 1 ) = Z t1 = Z 0 e αt 1 , which determines the constant α. Since ZrO 2 is a non-magnetic material, μ r = 1, the impedance is inversely proportional to the square root of the permittivity. Therefore 2D permittivity on the xz-plane is written as where X(x) is the peak permittivity at z = t 1 and it can vary along the x-axis. Equation (8) is substituted into (6) to calculate the minimum thickness, t min = 2t 1 as where X(0) = ε r,max and X( a 2 ) ≈ 1. The computed t min is 12.7 mm, when a is 84 mm, R is 2λ 0 at 15 GHz, and ε r,max is 23. X(x) is defined by solving (6) as where W is the Lambert W function [38] and ψ varies by changing x-coordinates. Fig. 7(a) and (b) show the permittivity and impedance profiles at x = 0, both of which are tapered exponentially. Fig. 7(c) represents the 2D permittivity distribution of the lens, which matches impedance and phase simultaneously.

2) LINEAR TAPER
Similarly, a linear impedance taper for half of the lens in the z-axis is expressed as; Like the exponential taper, (12) can be written as a 2D permittivity equation as where Y(x) is the peak permittivity at z = t 1 . Equation (13) is also applied to (6). The minimum thickness, t min , is calculated as the same way as the exponential taper.
The t min of the linear taper is 18.4 mm, where the dimensions, a, R, and the peak ε r are the same as the exponential case. Y(x) is also defined by solving (6) as where ζ also varies by changing x-coordinates. The linear taper has thicker t min since the permittivity varies more slowly than that of the permittivity in the lower z values as shown in Fig. 7(a). Fig. 7(b) and (c) represent linear variation in impedance and 2D permittivity variation, respectively.

3) KLOPFENSTEIN TAPER
Unlike the previous two tapering methods, the Klopfenstein taper can be optimized for minimum reflection. However, the tapered lens has continuously varying electrical length since the permittivities keep changing. The lens has to have a constant electrical length along the propagation direction to achieve the desired impedance matching. Firstly, a Klopfenstein taper of unit length (z = 1) is calculated from the following equation [39] where ε 0 is the free space permittivity and ε s is the peak permittivity at z = t 1 . The function (ξ, A) is defined as where I 1 (ξ ) is the modified Bessel function. The initial value of reflection coefficient ( 0 ), the maximum ripple in the reflection ( m ), and A are given by are used for the calculation. The thickness of the QWML is lens + MLs thickness.
ε r (z) at x = 0 for half of the lens in the z-axis is calculated by (17) where the m is set to -30. Then, the permittivity values are put into circuit simulation in AWR software. A cascaded transmission line consisting of circuit elements with different normalized impedances (Z(z) = 1/ √ ε r (z)) and constant electrical length is used to find the appropriate total electrical length and thus lens thickness. The minimum electrical length that produces S 11 < −10 across the entire band is determined from AWR simulation.
The lens thickness is then the aggregate thickness of the transmission line lengths and is found from the chosen electrical length and permittivity values. Permittivity values along other x-axis cuts are selected to keep the lens a fixed physical length in z while meeting the phase requirement in (6) with a taper of the form (17). The values are determined by trial and error in MATLAB. The resulting lens thickness is 18.6 mm where ε r = 1 at z = a/2. This tapered lens has constant electrical length in the propagation direction and satisfies both phases matching in the H-plane and impedance matching in the propagation direction. Fig. 7 shows the taper profile and 2D permittivity distribution.

C. DISCUSSION
Both single-layer and continuous matching networks are studied in the previous section. Fig. 7(a) and (b) show the permittivity and impedance profiles of the three tapered lenses and QWML at x = 0 along the propagation direction. Fig. 7(c)-(f) show the 2-D permittivity distributions of the three tapered lenses and QWML. The minimum thickness of exponential, linear, and QWML are defined in (5), (9), and (14). Fig. 8 compares the lens thickness of the two tapered lenses (exponential, linear) and the QWML against the peak permittivity value. The QWML is the thinnest because it has uniform permittivity in the propagation direction unlike tapered lenses. The exponential taper lens is thinner than the other two tapered lenses. The three tapered lenses, QWML, and optimal horn were simulated with continuously varying permittivity distributions in HFSS and the results are shown in Fig. 9. Without a lens, the shortened horn has a significant phase error between the center and edges, reducing the gain. The four lenses and optimal horn reduce the phase error, though some error remains at the edges. In Fig. 9, all lenses show some frequency dependence, with the linear taper showing limited benefit around 14.5 GHz. The other three lenses with matching layers indicate performance similar to the optimal horn for most of the band, even though those horn-lens systems have less than half length compared to the optimal length horn. Specifically, the total length (shortened horn + lens) of the horn with λ c /4 ML is 49.36mm, with the three tapered lenses it is 52.7 mm, 58.4 mm, 58.6 mm, respectively, and the optimal horn is 120 mm. Table 1 gives the average S 11 and realized gain of the antenna with different lengths and matching approaches. We find that the matching is quite poor if no matching is used, whereas the matched lenses produce return loss that averages > 10 across the band. The Klopfenstein taper produces the lowest average return loss across the band. The average realized gain improves by 2.6-4.4, depending on the matching technique used.

IV. DISCRETIZATION AND 3D PRINT
The results of the prior section indicate that the QWML produces attractive return loss and realized gain with the lowest overall thickness. Therefore, we decided to 3D print the QWML. In these simulation comparisons, the minimum thickness of the lenses is achieved where ε r ≈ 1 at the edges. In our fabricated design, the minimum permittivity was selected to be 3.2 due to minimum feature size allowable while maintaining structural stability. This increased the lens thickness from the absolute minimum (assuming air at the edges) of 4.8 mm to the realized thickness of 6 mm as seen from (5). The resulting lens described in this section has a total thickness of 10.56 mm (lens 6 mm + 2× QWML 4.56 mm).
To fabricate the GRIN lens, the continuously varying permittivity distribution of the lens is discretized, and the permittivity range is from the minimum of 3.6 to the bulk permittivity of 23. Then, the lens is implemented with appropriate permittivity values using an additive manufacturing process. A single material (zirconia) is printed using an XJet Carmel 1400, and the required permittivity gradient is created by varying the fill of the printed material and air.
The design of printed unit cells to create a range of effective permittivity values has been well-established [16], [18], [40]. Most of these studies have used low permittivity polymers across a narrow range of effective permittivity values. However, it becomes difficult to create a wide range of effective permittivity values (e.g., from air to bulk zirconia) with a single high permittivity material. As the ceramic volume fill ratio of a unit cell approaches 1, the effective permittivity varies rapidly with small fill ratio and the accuracy to which the effective permittivity can be realized is limited by the 3D printer resolution. At the low end of the permittivity scale, most of the structure must be air so the effective permittivity is limited by the mechanical stability of a thin unit cell structure. Therefore, we divide the range of effective permittivity into three regions and propose different unit cell designs for each region as shown in Fig. 10(a), (c). The base unit cell is a cubic element composed of struts centered on a cube. The struts are connected to the adjacent units, and the cube size variation adjusts the filling ratio. It is used to realize mid-effective permittivities. A chamfered unit cell is used to implement high effective permittivities by chamfering four edges on the central cube to provide finer control over the effective cell permittivity at high fill ratios. A gapped unit cell has gaps in the E-field direction that act as series capacitors and dramatically lowers the effective permittivity of the cell. The gap size can be finely adjusted by the required effective permittivity without significantly compromising the mechanical stability of the structure.
Unit cell size is determined through simulation of the three unit cell models. We analyzed the frequency dependence of the unit cells and determined that unit cells smaller than λ eff /3 shows a deviation from the low frequency value of less than 5%. Also, the lens thickness should be an integer mutiple of the unit cell size. Therefore, a 1.5mm unit The 3D printed lens is composed of the three types of unit cells as illustrated in Fig. 10(b). The chamfered unit cell is used for a high permittivity range of 17.2 − 23 (red dashed rectangle), the cubic unit cell is used for the mid-permittivity range of 5.9 − 16.2 (black and blue dashed rectangles), and the gapped unit cell (pink dashed rectangle) is used for the lowest permittivity value in the lens, r = 3.6 . Fig. 11 shows a comparison of the required permittivity values for the lens and the achieved effective permittivity values with the three unit cells. The percent difference between the required and achieved permittivity values is lower than 5% in most values. Two permittivity values have 17% and 22% differences assuming the 3D printer resolution to be 50 microns.

V. MEASUREMENT AND DISCUSSION
The zirconia lens is 3D printed using a nanoparticle jetting (NPJ) technology and the resulting structure is shown in Fig. 10(b) and Fig. 12. Some sagging is evident in the middle of the lens, which may be due to uneven shrinking during the firing process. With printed ceramics, there is often significant shrinking of the green ceramic after firing and shrinkage in the NPJ-printed zirconia has been previously estimated at 18% [27], [41]. The measured electrical performance of the lens shown in this section suggests that the small deformations may have some effect on sidelobe level but the main beam is largely unaffected.
The lens is placed on the aperture of the shortened horn for measurement as shown in Fig. 12. Fig. 13 shows the measured S 11 of the shortened horn without the lens and the shortened horn with lens and λ/4 MLs. The S 11 of the horn with lens is similar to that of the shortened horn alone,  and remains lower than −10 from 13 to 17 GHz. Although the λ c /4 ML is designed to match the bulk permittivity of ZrO 2 at the center frequency (15 GHz), the lens has wide impedance bandwidth that covers the majority of Ku band. Fig. 14 demonstrates that the simulated and measured realized gain patterns in the H-plane and E-plane at 15 GHz have a good agreement. The wide beamwidth in the E-plane and narrow beamwidth in the H-plane are expected for a proper H-plane horn. The gain is significantly increased by the lens while the beamwidth narrows as simulated. The sidelobes of the printed lens are slightly higher than simulated, presumably due to small print variations. However, the back lobes of the lens are not significantly increased compared to those of the shortened horn. Fig. 15 shows that the lens reduces the H-plane half-power beamwidth (HPBW) over the frequency range compared to that of the shortened horn except at 13 GHz. At the center frequency of 15 GHz, the HPBW of the lens and shortened horn in the H-plane are 18 • and 70 • , respectively, indicating a beamwidth reduction of 52 • . The lens with the shortened horn generates a fan beam that is narrow in the H-plane, but wide in the E-plane.   except at 13 GHz where there is a small gain decrease. In particular, the lens enhances realized gain by 5.4 at 15 GHz. The pattern of the lens at 13 GHz shows the formation of two separate beams, which accounts for the lower gain. Across much of the band, the lens increases the gain but the measured realized gain of the lens is ∼ 1.5 lower than in simulation. These differences may be due to shrinkage in printed structure or deviations in the material properties compared to those assumed in simulation.

VI. CONCLUSION
This paper presented a shortened horn antenna with high gain that is implemented by a 3D-printed gradient index (GRIN) lens composed of high permittivity ceramic ZrO 2 . The shortened H-plane horn typically presents a low gain due to the high flaring rate of the horn, but this is improved by adding a flat GRIN lens at the aperture. Although high permittivity ZrO 2 reduces the lens size, when interfaced with air, impedance mismatch occurs. Different matching techniques are investigated including a simple λ/4 ML and several tapered MLs in which the ML and phase corrections are integrated into a single permittivity distribution that is graded in the propagation direction and the H-plane. Analytical expressions for the minimum thickness and 2D permittivity distribution of two tapered lenses are introduced. The three tapered lenses have a thicker length in the propagation direction than the QWML for all maximum permittivity values. Ultimately, the three tapered lenses and QWML show similar realized gain compared to the optimal horn even though they have a length less than half of the optimal horn. We 3D printed the QWML since it has the lowest profile and presents a high realized gain despite the narrowband matching design. The measured S 11 is lower than −10 in the majority of Ku band. The lens enhances the measured realized gain by 5.4 and reduced the beamwidth by 52 • at the target frequency (15 GHz).