Novel Design Methodology for 3D-Printed Lenses for Travelling Wave Antennas

A novel methodology is introduced for designing bespoke homogeneous and graded index lenses for enhancing the gain of travelling wave antenna arrays (TWAs). 3D-printed lenses in the literature are majorly explored with standing wave antennas (SWAs) such as a microstrip or a horn antenna for gain enhancement. As there is progressively less power radiated from each slot in a TWA, as well as the successive phase delay between slots, the existing lens design approaches used for SWAs is not optimal for TWAs. Accordingly, we present a new approach of introducing a curvature to the lens that is derived from studying the power radiation profile of each slot of the TWA. This new methodology is demonstrated on a dielectric filled waveguide (DFW) slot array antenna operating at 26 – 30 GHz band. An optimized dielectric graded lens and an optimized homogeneous lens have both been designed, fabricated, and measured with the DFW slot array. The new lens demonstrated a gain enhancement of more than 7 dB compared to less than 4 dB with conventional dielectric lenses. The proposed lens theory has been further verified with a bespoke optimized lens for a periodic stub-loaded microstrip leaky-wave antenna with a beam-scanning of 65°. Design rules are included that can be applied for any TWA.


I. INTRODUCTION
D IELECTRIC lenses are well-known components in antenna systems that have been used for more than a decade for beam-shaping and gain enhancement, providing highly directive antenna systems. In the literature, various dielectric lenses have been described based on transformation optics [1], wave optics [2], [3], and field transformation [4], [5].
With the need for more directive beam-steering antenna systems for fifth generation mobile communications [18], [19], [20], the use of TWAs has become more popular. TWAs produce an extremely narrow beam in one plane with a wide beam in the other. A 1-D dielectric lens could be a valuable addition for such TWAs to further enhance directivity.
Conventional lenses in the literature have been used for TWAs but have a smaller gain enhancement than a similar lens when used with a SWA. A 3-D printed dielectric lens antenna was presented in [21] for improving the performance of a DFW slot array antenna for applications in 5G, and demonstrated a gain improvement of only 3 dB. A similar gain enhancement of ∼4 dB for a DFW slot array antenna was shown with the help of a metallic grating cover behaving as a phase correcting lens in [22]. Another concave-convex lens structure was demonstrated in [23] to produce a 3 dB increase in gain for a phased-array antenna. Other similar lenses have been presented with various forms of TWAs such as a leaky-wave antenna [23] and a phased array antenna [24] providing a 3 to 4.5 dB increase in antenna gain. Several other lens-based LWAs have been discussed in the literature [24], [25], [26] but they do not treat the lens as a separate entity, and hence they cannot be considered as a guide to design optimized lenses for TWAs.
Transformation optics has also been demonstrated in the literature to design bespoke lenses for different antennas [27], [28]. DFW slot array antennas have been presented with a Rotman lens [29], [30], [31], which is a reflective type of lens. Reflective type dielectric lenses are used between the feeding network and the antenna array to introduce phase delay and/or redirect the wave to specific radiating elements. Other similar reflective type lens antennas [20], [32] have been reported to operate with TWAs. Those lenses act as beamforming networks and do not have any impact on improving the antenna's directivity.
Radiating elements in a TWA are arranged in a linear sequential fashion, so each element is fed after the previous element has already radiated. Hence, unlike SWAs, radiation from each element in a TWA undergoes a series of phase delays (PD) and amplitude reductions (AR) along the length of the antenna. This continuous change affects the electrical properties of the generated wavelet, which in turn reduces the effectiveness of a conventional dielectric lens positioned in front of it.
In this paper, a detailed study of PD and AR between radiating elements of a TWA and its effect on the performance of a dielectric lens is presented with a standard DFW slot array antenna. Section II presents a comparison between the radiation characteristics for a GFL positioned in front of both a microstrip patch and an DFW slot array antenna to show the need for a different design process for dielectric lenses to make them suitable for TWAs. Section III describes the effects of PD and AR from the different radiating elements of a DFW slot array antenna on the radiated wavelet and proposes an optimized graded lens for a TWA. Section IV presents the fabricated optimized homogeneous lens (OHL) and optimized graded lens (OGL). Measured gain for two different antennas with these lenses are compared with that of a conventional plano-convex flat lens (CPFL) to verify the proposed theory. A set of design rules for optimizing a dielectric lens for any arbitrary TWA is given in Section V. These design rules are then further verified by designing and producing an optimized lens for a different antenna: a stub loaded microstrip leaky-wave antenna in Section VI. This is followed by conclusions in Section VII.

A. GEOMETRY OF SWA AND TWA
A conventional graded flat lens (GFL) was simulated with both a SWA and a TWA to demonstrate that these designs are not optimal for TWAs. The study used a microstrip patch  antenna (representing a standard SWA) operating at 28 GHz and a dielectric filled waveguide (DFW) slot array antenna (representing a TWA) operating in the 26 -30 GHz band. The microstrip patch antenna had a width of w m = 4 mm and length, l m = 3.6 mm, and was fed with a 50 microstrip line (see Figure 1(a)). The DFW slot array antenna was designed using general guidelines specified in [31] and [32]. It consisted of 16 equally spaced bow-tie shaped slots of length 7.5 mm and spacing of 6 mm in two rows, with an offset of 1 mm from the center (see Figure 1(b)). The DFW slot array antenna was fed from one end and the other end was closed with a copper wall. Both the patch and the DFW slot array antennas were designed on a Taconic TLP-5 substrate (ε r = 2.2, loss tangent = 0.0009, thickness, h = 0.787 mm).

B. SIMULATED RESULTS AND DISCUSSION
Radiating elements in a TWA are arranged linearly, resulting in a wider beamwidth in the plane perpendicular to the array compared to the plane parallel to the array. Therefore, a GFL with focal length of 10 mm and a uniform relative permittivity distribution in the direction of the radiating elements was designed based on the wave optics theory described in [3]. Each layer of the lens is shown in Figure 2 and their dimensions are calculated as follows: width w = 8 mm and thickness t = 20 mm. The antenna is placed parallel to the lens at a distance equal to the focal length of 10 mm. The length l of the lens is determined by the required focal length, size of the antenna and its beamwidth. The GFL length l = 130 mm was chosen to cover the full length of the DFW slot array antenna. Note, the performance of the patch antenna was the same for l = 30 mm and l = 130 mm, however, the same sized lens was used for both antennas. Simulated S11 results for both antennas mounted with lenses are given in Figure 1(c). Figure 3 shows that the lens improved the gain of the microstrip antenna from 7.6 dBi to 14.6 dBi, demonstrating an overall increase of 7 dB. This agrees with the general improvements seen in the literature when the lens relative permittivity variation is only in one direction. However, the same lens showed an improvement of only 3.9 dB when tested with a TWA where the gain increased from 13.1 dBi to 17.0 dBi. To understand if the tilt in the beam was limiting the gain improvement, the lens was then placed perpendicular to the beam angle. Figure 3(a) shows the radiated beam is tilted by 24 • at 28 GHz.
The tilt in the radiated beam is because of the travelling wave characteristics of the slot array and can be estimated using the general theory of leaky wave antennas [35], [36]. The tilt angle can be further confirmed with the simulated E-field plot for the DFW slot array antenna shown in Figure 4(a). Note, a sidelobe is formed in Figure 3(a) at an approximate angle of 50 • . This is because the radiated wavefront from the TWA is not perpendicular to the lens and hence, the transmission of the wavefront through the lens is not collimated within a single beam, leading to the formation an additional sidelobe.
When the lens is physically tilted at 24 • with respect to the DFW antenna (see Figure 4(e)), the E-field plot shows the wavelet arriving on the lens cross-section with the same phase and hence avoiding any distortion of the overall wavelet generated by the GFL (see Figure 4(c)). This tilted  flat lens (TFL) further improved the gain of the DFW slot array antenna by 1.2 dB compared to the GFL, but still short of the performance achieved when using the patch antenna. To confirm 24 • tilt was the optimum angle for this antenna, simulations were conducted with the lens tilt angle changing from 18 • to 32 • . The gain was found to be highest at 24 • , with a slight but acceptable reduction of 0.6 dB in antenna gain performance for a tilt from 22 • to 26 • .
For completeness, the E-fields in the YZ-plane for each scenario are shown in Figure 4 (b, d, and f). The 2D and 3D simulated radiation patterns for the DFW slot array antenna with the tilted lens are shown in Figure 5. Figure 4 and Figure 5 show the conventional dielectric lens is more effective for a standard SWA as compared to a TWA, even when the lens has been tilted. This difference in performance is hypothesized to be because of amplitude reduction effects that exist between the radiating elements of a TWA. It is also apparent, when titled, the focal plane of the lens is no longer coincident with the phase center of each slot. Hence, there is a need to design an optimized lens for TWAs, which is addressed in Section III.

A. EFFECTS OF AMPLITUDE REDUCTION
In this section, a DFW slot array antenna has been used to study how amplitude reduction affects the effectiveness of the lens. The design being a leaky-wave architecture on a waveguide-based guiding structure, means that it reduces leakage losses. Further, at 28 GHz DFWs are easier and cheaper to manufacture compared to standard air-filled waveguides.
The decrease in the wave amplitude in a DFW slot array antenna is illustrated in Figure 6. Neglecting all the losses within the DFW (dielectric and conductive), as the wave reaches the first slot, a fraction of the power is radiated out, some of the power is reflected and the rest continues propagating to the next slot.
To quantify the amplitude reduction, simulations were conducted for 16 different DFW slot array antennas of fixed length (l = 130 mm) but with different number of slots, the first antenna having only one slot and the sixteenth DFW antenna having 16 equidistant slots. Waveguide ports were defined at both ends of all the DFWs. All sixteen antennas have the same width of 5.2 mm, height of 0.787 mm, and relative permittivity of 2.2.
The simulated and calculated S11 of the single slot antenna shows a good match and only 4% of the incident power is reflected. It is, therefore assumed that the reflection from the i th slot has negligible effects on the (i − 1) th slot. The power reflected, R 1 , and the power continuing forward, F 1 , for the single slot in the first DFW antenna can be calculated from the S11 and S21 simulated results if the input power, P in is normalized to 1 W, as, The total power radiated from the first slot, P 1 can simply be calculated by subtracting from the input power P in the reflected and forwarded powers. The same can be repeated for subsequent slots to calculate the power radiated by i th slot, using F i−1 as the input power. The average relative radiated power (ARRP) by each slot, γ can be calculated as the ratio of power radiated by i th slot, i.e., P i to the power received by that slot, i.e., F i−1 , where F 0 is equal to the normalized power 1 W. Therefore, the ARRP can be written as, Using the S-Parameters from the 16 simulated antennas, the ARRP was found to be approximately constant at 17% (ranging from 16.9% to 17.2%). This means i th slot on the DFW receives 83% of the power present at (i − 1) th slot, assuming negligible losses. Hence, the power radiated by i th slot can be defined as, This can be confirmed from Figure 7(a) and (b) which compare the simulated and calculated power radiated by individual slots and the overall antenna at 28 and 29 GHz. To incorporate the effects of this amplitude reduction on the wavelet, the TFL was further modified to follow the same logarithmic curve defined by eq. (3) and is described in detail in Sections III-B and III-C.

B. SIMULATED RESULTS FOR OPTIMIZED GRADED LENS
The optimized graded lens (OGL) was produced by implementing eq. (3) to calculate distance between lens and antenna at different points in x-axis, so that the OGL has the same curvature profile as seen in Figure 7(b). The X and Z-coordinates defining the lens curvature are given by, where, i is the slot number, l 1 is the distance between the DFW feed and first slot, l s is the slot length, n is the total number of slots and g is the distance between each slot. The parameter d is the distance from the first slot to the lens and is used to replace P 1 in eq. (3). The lens was simulated with a parametric sweep of d values between 10 and 30 mm to find the separation distance resulting in the maximum gain, and this was achieved for d = 20 mm. The location of the origin O (0, 0) for X and Z is shown in Figure 8(a). The 16 points created from eq. (4) and (5) on a  Cartesian surface with the antenna at z = 0 defines the logarithmic curvature of the lens and this cross-sectional view is shown in Figure 8 with the logarithmic curve presented as a dotted line.
With the logarithmic shape of the OGL, the antenna gain is further improved to 19.2 dBi at 28 GHz, giving an average increase of 7.6 dB compared to the gain of the DFW slot array antenna, and an average increase of 3.6 dB compared to the DFW antenna with the GFL (see Table 1).
The HPBW of the antenna system reduced from 68 • without any lens to 36 • with the GFL to 28 • with the OGL in the elevation plane, demonstrating a significant reduction by a factor of 2.4. The 2D gain patterns demonstrating this are shown in Figure 9. The reshaping of the lens with the proposed OGL allows us to reduce the oblique incident wave from the TWA, as confirmed by the smaller side lobe levels seen in Figure 9(a). A detailed discussion on the analysis of the proposed lens design is presented in Section III-C.

C. ANALYSIS AND DISCUSSION OF OPTIMIZED LENS
The improvement in gain with the introduction of the OGL can be explained by two observations indicated in Figure 10(a). The region highlighted by 'i' shows the radiation from the 8 th slot combining with the radiation from previous slots to enhance the main beam. This phenomenon is clearly absent in the GFL in Figure 4(c) and is solely due to the curvature of the OGL. Secondly, and more importantly, region 'ii' shows very little radiation from slot 1 missing the lens, compared to the significant wastage seen with the TFL in Figure 4(e). The curving of the OGL closer to slot 1, which radiates the most power, ensures more of the antenna's radiated power falls on the lens and so constructively contributes to the main beam. The high level of collimation seen in Figure 1(b) confirms the enhance performance of the OGL.
Due to the curving of the lens to produce the OGL, its focal line is also expected to curve. To establish its shape, a series of focal points are obtained through examining the E-field in simulations, which can be joined to form a focal line. First, the antenna is removed and the OGL is illuminated from the front with a uniform plane wave at an angle of 24 • . The focal point distance in the YZ plane behind the lens is recorded (see Figure 11(a)) at 10 mm intervals across the length of the lens in X-axis (from 10 to 120 mm). In Figure 11(b), which is a reproduction of the E-field plot shown in Figure 10(a), the recorded focal points are superimposed. It is clear the focal line is not incident on the antenna surface. Although the radiated waves originate from the 16 slots, due to the amplitude reduction, phase delay, constructive and destructive interference, the resultant cylindrical wave appears, at least to the OGL, to originate from a virtual radiating line parallel to the wavefront, which we suggest is close to the focal line of the OGL.
To verify this was in fact the correct shape, separation distance, tilt angle for maximum gain enhancement, a parametric study was conducted. Figure 12 shows the variation in gain when the (b) curvature (relating to ARRP), (c) distance between the lens and antenna, and (d) tilt angle of the lens, are changed in simulation. It is clear the gain decreased as the design deviates from the calculated optimum parameters.
As the direction of the main beam changes with frequency, there is a small deviation in phase center. However, this only causes a minimal decrease in gain, which makes the OGL effective across the whole frequency bandwidth. We note, the focal line (composed of focal points) of the TFL would be perfectly parallel to the cylindrical wavefront, but because of its large distance from slot 1, the gain is decreased. Curving the OGL not only brings the lens closer to slot 1, but it also ensures its focal line is approximately parallel to the cylindrical wavefront. It is worth noting that the phase center of the overall antenna is at X = 40 mm in Figure 11(b), which coincidentally intersects the OGL's focal line. Furthermore, parameter d, which was optimized to be 20 mm, is also equal to the focal length of the original GFL introduced in  Section II-B. This was further verified by testing the lens with different LWAs and is demonstrated in Section VI.
It is important to note that there is a slight difference between the S21 for 28 and 29 GHz, as seen in Figure 7. This should theoretically affect the shape of the lens with change in operating frequency, making the lens frequency sensitive. However, when the average relative radiated power by each slot is calculated for all the frequencies, it is 0.17 for 28 GHz and 0.16 for 29 GHz, and hence, the relative difference is very low and can be neglected. This can further be verified by studying the consistent improvement in the gain of DFW slot array antenna described in Table 1.

IV. FABRICATED AND MEASURED OPTIMIZED DIELECTRIC LENS WITH DFW SLOT ARRAY ANTENNA
The optimized graded lens (OGL) presented in Figure 8 as well as an optimized homogeneous lens (OHL) shown in Figure 13, were fabricated for use with the DFW slot array antenna.
Both lenses were fabricated via FDM 3D-printing using polylactic acid (PLA) thermoplastic. PLA has a dielectric constant of up to 2.7 with 100% infill density and a loss tangent of 0.007 at the required frequency band [37], [38]. For both the OGL and OHL, triangular infill patterns are used for 3D-printing. The design model for the OHL is presented in Figure 13 and was designed based on the principles stated in [3]. Instead of varying the relative permittivity in the Yaxis, the dielectric thickness was changed to be maximum at the center (t 0 = 20 mm) and minimum at the corners (t −Y = t +Y = 8 mm), forming a plano-convex lens structure. The curvature in the X-axis was the same as for the OGL. The dielectric lens thickness distribution in the Y-axis is described through the equation, where, t y is the thickness of the lens at any point on the y-axis, t 0 is the thickness of the lens at the center (y = 0), f L is the focal length of the lens, ε r is the relative permittivity of the lens material and δ is the angle between the normal to the antenna and the edge of the lens. The lens width w h in the Y-axis is calculated from, A conventional plano-convex flat lens (CPFL) was also 3Dprinted with a focal length of 20 mm and is shown in Figure 14. Both, the OHL as well as the CPFL were fabricated with an infill density of 60% which gave the relative permittivity of the material to be ε 60% = 2.0. The different layers of the OGL were created from different infill densities of 100% (for ε 0 = 2.7), 80% (for ε 1 = 2.4), 60% (for ε 2 = 2) and 30% (for ε 3 = 1.47). The three lenses positioned in front of the DFW slot array antenna are shown in Figure 15. Besides the difference in the physical structures, the two optimized lenses (graded and homogeneous) are electrically similar and should provide similar radiation characteristics. An OHL can be fabricated easily with any generic 3D printer, however the OGL either needs to be fabricated in separate layers or needs specialized software to locally control the 3D-printed infill density in the same print.
The radiation patterns of the three fabricated lenses were measured with the 16-slot DFW slot array antenna in an anechoic chamber and are presented in Figure 16. The patterns were normalized to compare the half power beamwidth of

FIGURE 16. Measured normalized radiation patterns at 28 GHz for DFW slot array with optimized graded lens (OGL), optimized homogeneous lens (OHL) and homogeneous conventional plano-convex flat lens (CPFL) in (a) azimuth and (b) elevation planes.
the antenna system. It was found to be reduced from 68 • to 32 • with the CPFL and to 24 • with the optimized lenses in the elevation plane. The HPBW in the azimuth plane stayed almost constant at 10 • for the CPFL and reduced to 8 • for the optimized lenses.
The measured gain of the antenna system increased from 11.47 dBi to 15.5 dBi at 28 GHz for the CPFL (presenting a 3.8 dB improvement in the gain), which was further improved to 18.95 dBi with the optimized graded or homogeneous lenses, presenting a relative increase of 3.5 dB above the CPFL and gain enhancement compared to no lens of 7.3 dB for the antenna averaged over the whole bandwidth. There was no significant change in the antenna efficiency. This also agrees with the average simulated gain enhancement of 7.6 dB that was presented in Section III-B. The measured and simulated gain comparison for different lens configurations is presented in Figure 17. Note, that the small grooves that can be seen in the fabricated lens in Figure 14 do not significantly affect the lens performance, as they are located near the farthest radiating element on the DFW, where the radiated power is smallest. The measured results are very close to the simulated results, see Figure 17, which provides further evidence that these fabrication imperfections do not have a significant effect.

V. DESIGN GUIDELINES
As described in Section III, variation in the phase delay and amplitude reduction along the radiating elements of a TWA need to be taken into consideration when designing the dielectric lens for a TWA. Results presented in Sections III-C and IV can be replicated for any uniformly or non-uniformly distributed travelling wave antenna with the help of the steps described in this section.
The first step involves the study of phase delay introduced in the EM wave while travelling from the feeding point to each radiating element individually. This can be studied by calculating the equivalent electrical path length and is dependent on the operating frequency and wavelength (or guided wavelength in case of DFWs). Theory of leaky-wave antennas [35], [39] can also be used to calculate the tilt in the wavelet generated by a TWA using eq. (1). Once this tilt has been identified, the conventional flat lens should be placed parallel to the wavelet to reduce distortion in the radiated wave as it passes through the tilted lens.
The second step involves the study of radiation characteristics of each individual radiating element of the TWA. For a uniformly distributed array, the function defined in eq. (3) can be used directly, given that the averaged relative radiated power (ARRP) from each radiating element of the TWA is known, which can be calculated by the multipleantenna test process described in Section III-B. This ARRP is dependent on the electromagnetic properties of each radiating element and may vary depending on factors such as element spacing, and its dimensions. Therefore, for a generic TWA with non-uniform spacing or varying properties of radiating elements, eq. (3) can be re-written as, where, γ i is the ARRP for the i th radiating element with γ 0 = 0 and can be calculated individually through the multiple-antenna test process that has been described in Section III-B. The ARRP is found to be constant for uniformly spaced radiating elements, but for non-uniformly spaced elements, the value of ARRP varies with each radiating element. This has been further verified by simulating the DFW slot array antennas with varying element spacing which demonstrated a continuously varying ARRP for each radiating slot. Furthermore, the radiation characteristics of individual elements vary with change in frequency. This is because of variation in electrical path length with frequency, which affects the beam-angle. The ARRP shows slight deflection as we move across different frequency bands, however, this change is very small (within 3%), and hence, has a negligible effect on the performance of the lens with the change in beam-angle or frequency of operation. This can further be verified with the consistent gain enhancement as a function of frequency of the TWA shown in Figure 17. Equation (3) describes a logarithmically decreasing function, and hence, the general shape of the lens stays the same as presented in Section III-B, with the curvature redefined according to the values of ARRP in eq. (8). Therefore, eq. (8) can be used directly to define the curvature of the optimized dielectric lens for a TWA for both uniformly and non-uniformly distributed radiating elements. This shape of the lens is based on the reducing amplitude of the wavelet radiated by each element, and its shape incorporates the effects of phase delay and amplitude reduction.
This paper presents a complete lens design methodology for optimized lenses. It is worth noting that the design methodology is also applicable to SWAs where each element is identical and receives an equal amount of power. In this case, the ARRP becomes 0 and hence, eq. (8) gives a flat lens, which is identical to a conventional dielectric lens.

VI. FURTHER VERIFICATION OF THE THEORY WITH A MICROSTRIP PERIODIC STUB LOADED LWA
The lens design methodology was further verified by applying it to another TWA, a microstrip line based periodic stub loaded leaky-wave antenna (MPS-LWA). The design was taken from [40], [41] and has been modified to operate at 26 -29 GHz band and has a frequency based beam-scanning of 15 • to 80 • . The antenna and its measured S-parameters are shown in Figure 18. An OHL was designed for a focal length, f L = 20 mm using the design guidelines described in Section V and is shown in Figure 19.
The measured normalized radiation patterns for the antenna with and without the optimized lens in the azimuth and elevation planes at 26 -29 GHz are shown in Figure 20. The HPBW reduced by a factor of 3.2 for all the frequencies. An averaged gain improvement over the frequency band of 7.4 dB was observed with the OHL and is shown in Figure 21. This further confirms the effectiveness of the proposed optimized lens design for any travelling wave antenna. It is noteworthy, that the gain improvement is found to reduce as the beam moves out of the physical area covered by the lens as the lens will only be effective for the beams passing through its cross-section.

VII. CONCLUSION
Lenses are well known antenna components. Although they have been used for TWAs, the gain enhancement is not optimal. This has been highlighted through a comparison of the same lens with both a microstrip patch antenna (representing a standing wave antenna (SWA)) and a dielectric filled waveguide (DFW) slot array antenna (representing a TWA). The comparatively lower gain enhancement recorded for the TWA with the conventional lens, even when tilted appropriately to accommodate the phase delay between radiating slots, demonstrated the need to improve the lens design process for TWAs.
For a 16-slot DFW array antenna, the power radiation profile from each slot and the phase delay between each slot was investigated in detail. Subsequently, equations were derived to transform the conventional graded dielectric lens  from a flat shape to one that has a logarithmic curvature. The theory was verified by fabricating and measuring the radiation patterns of an optimized graded lens (OGL) and an optimized homogenous lens (OHL) with the DFW slot array antenna. The novel optimized lenses enhanced the gain compared to no lens by 7.6 dB (7.5 dB in simulations). The optimized lenses demonstrated a 3.8 dB improvement compared to a plano-convex flat lens (3.5 dB in simulations) and other works compared to literature, see Table 2. This magnitude of gain enhancement is equivalent to that of a conventional dielectric lens when used with a standard SWA. The optimized lens reduced the HPBW accordingly by a factor of approximately three. The total antenna efficiency was found to stay approximately constant within ± 2%.
The proposed lens theory was further verified with an OHL designed for a microstrip periodic stub loaded leaky-wave antenna operating at 26 -29 GHz. The lens demonstrated a similar average gain enhancement of 7.5 dB over the frequency band with a HPBW reduction by a factor of 3. The cross-polarization results for both the antennas were found to be below −40 dB, and hence are not included in this paper. Finally, a procedure has been proposed that can be followed to optimize dielectric lenses for all TWAs, and the theory is further verified using an MPS-LWA.