# Sub-THz Fully-Metallic Geodesic Luneburg Lens Antenna

Pilar Castillo-Tapia, S. Yang, Ángel Palomares Caballero, Jean-Paul Guillet, N. J. G. Fonseca, Oscar Quevedo-Teruel

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Pilar Castillo-Tapia, S. Yang, Ángel Palomares Caballero, Jean-Paul Guillet, N. J. G. Fonseca, et al. Sub-THz Fully-Metallic Geodesic Luneburg Lens Antenna. IEEE Transactions on Terahertz Science and Technology, 2025, 15 (3), pp.514-518. 10.1109/tthz.2025.3548452. hal-04987665

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# Sub-THz Fully-Metallic Geodesic Luneburg Lens Antenna

P. Castillo-Tapia, Student member, IEEE, S. Yang, Student member, IEEE, Á. Palomares-Caballero, J.-P. Guillet Member, IEEE, N. J. G. Fonseca, Senior Member, IEEE, and O. Quevedo-Teruel, Fellow, IEEE

**Abstract**—We propose and validate experimentally a fully metallic geodesic Luneburg lens antenna operating in the sub-THz band. The antenna produces three beams pointing at 0, 40 and -40 degrees. To facilitate the integration, the geodesic lens is folded to reduce its height to approximately 38.7% of the original Rinehart-Luneburg lens. To reduce potential leakage resulting from manufacturing and assembly tolerances at sub-THz frequencies, the waveguide feeding structure has a deliberate small air gap alongside electromagnetic bandgap structures. This enhancement aims to bolster the robustness of the antenna, ensuring stable performance even in the presence of misalignments. The results demonstrate the robustness of geodesic lenses in the sub-THz regime; showing their suitability for applications that require multi-beam antennas at these high frequencies. The successful performance of geodesic lenses in the sub-THz regime confirms its potential for operation at higher frequencies above 300 GHz.

**Index Terms**—Electromagnetic bandgap structure, fully metallic antenna, geodesic lens, Luneburg lens, sub-THz.

## I. INTRODUCTION

Sub-terahertz (sub-THz) radar systems are interesting for many applications since they can achieve higher resolutions [1]. This technology has garnered significant interest specially in non-destructive testing. Multi-beam antennas are considered for these applications to accelerate data acquisition. A common solution is to employ phased array antennas [2]; yet, at these high frequencies, the reduced scale of the structures and intricate manufacturing processes pose significant hurdles. Additionally, losses may occur due to factors such as complex feeding systems and the use of substrate materials [3].

An alternative approach involves the utilization of quasi-optical systems [4], [5]. Lens antennas have shown potential for offering promising solutions even at lower frequencies. For instance, gradient index (GRIN) Luneburg lens antennas have demonstrated the ability to achieve promising beam scanning capabilities with highly directive beams, coupled with high

<figure><img src="image_2.png" /><figcaption>Fig. 1: Ray trajectories in (a) GRIN Luneburg lens and (b) geodesic Luneburg Lens.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Profiles of the original Rinehart-Luneburg lens (grey) and the designed modulated geodesic lens (blue). Both $\rho$ and $z$ are normalized to the lens radius. The modulated lens profile with ray trajectories is illustrated as inset.</figcaption></figure>

This work has been supported by ESA, Open Discovery Ideas Chanel project N° 4000134617/21/NL/GLC/ov.

P. Castillo-Tapia, S. Yang and O. Quevedo-Teruel are with the Division of Electromagnetic Engineering and Fusion Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mails: shiyyang@kth.se; pilarct@kth.se; oscarqt@kth.se).

Á. Palomares-Caballero is with the Institut d'Electronique et des Technologies du numéRique (IETR), UMR CNRS 6164, INSA Rennes, Rennes, 35700, (e-mail: Angel.Palomares-Caballero2@insa-rennes.fr).

J.-P. Guillet is with the IMS Laboratory, UMR CNRS 5218, University of Bordeaux, 351 cours de la Liberation, 33405 Talence, France (e-mail: jean-paul.guillet@u-bordeaux.fr).

N. J. G. Fonseca was with the Antenna and Sub-Millimetre Waves Section, European Space Agency, 2200 AG, Noordwijk, The Netherlands. He is now with the Technical Department, Anywaves, 31000 Toulouse, France (e-mail: n.fonseca@anywaves.com).

Manuscript received Month XX, 202X; revised Month XX, 202X.

efficiency [6], [7]. However, such antennas are implemented on metasurfaces, which consist of unit cells featuring sub-wavelength size structures. This makes manufacturing at terahertz frequencies challenging and costly [8]. Another solution is to use geodesic lens antennas. The working principle of geodesic lenses is to map the gradient index distribution of the lens into equivalent path lengths by changing its height profile [9]. In the microwave regime, a geodesic lens can be implemented using a smooth parallel plate waveguide structure and manufactured with CNC milling technology [10]–[13]. Thanks to its fully-metallic implementation and smooth surfaces, it is relatively straightforward to produce it at higher frequencies while maintaining high efficiency. Although geodesic lens antennas have demonstrated promising results from the Ka-band up to 70 GHz, their application in the sub-THz regime remains unexplored.

In this paper, we present the design, simulation, and measurement of a geodesic Luneburg lens antenna operating in the sub-THz band. The lens profile has been modulated to reduce the antenna height, resulting in a more compact design conducive to easier integration. Three waveguide feeds have been added to produce beams at 0° and ±40° angles. To mitigate leakage, which is a common issue in waveguide components at high frequencies, electromagnetic bandgap (EBG) structures have been integrated in the feeding system, as proposed in [13] for the V-band. Simulation and experimental results demon-
<figure><img src="image_4.png" /><figcaption>Fig. 3: Simulated E-field distribution propagating through the designed lens at 130 GHz. The radius of the lens is 25 mm.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: (a) Dispersion diagram of the EBG hole, with phase shift ($\beta p/\pi$) and attenuation constant ($\alpha/k_0$) represented by blue and red lines respectively. Relative parameters are: $H_{hole}$ = 1 mm, $G_{gap}$ = 0.03 mm, $R_{hole}$ = 0.632 mm, $p$ = 1.58 mm. (b) Simulated S-parameters for Waveguide with Gold Material with a conductivity of $4.561 \times 10^7$ S/m and a root-mean-square surface roughness of 0.35 $\mu$m integrated with EBG holes. The length of the waveguide is 20 mm, and $G_{gap}$ = 0.03 mm. E-field distribution at 130 GHz is illustrated as inset.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Proposed modulated geodesic lens antenna working in the sub-THz band. The top figures display the central waveguide feed in a top view and the flare in a side view. The bottom figure offers an exploded view of a segment of the feeding structure, emphasizing the air gap by highlighting it in blue. Variables are denoted by blue letters. $W_{WG}$ represents the width of the waveguide, while $L_{WG\_center}$ denotes the length of the central waveguide. $L_{flare}$ and $H_{flare}$ stand for the length and height of the flare. $R_{lens}$ indicates the radius of the modulated Luneburg lens. $G_{gap}$ specifies the height of the designed air gap.</figcaption></figure>

strate the robustness to manufacturing tolerances of geodesic lens antennas and holey EBG structures for applications in the sub-THz regime.

## II. MODULATED GEODESIC LUNEBURG LENS ANTENNA

A Luneburg lens is defined as a graded index optical device, exhibiting a refractive index profile that smoothly transitions from $\sqrt{2}$ at its center to 1 at its periphery, as defined by the equation $n = \sqrt{2} - r^2$ [14]. For the geodesic Luneburg lens, the vertical profile is approximated as $z(\rho) = h_0(1 - \rho^p)^{1/q}$, where $\rho$ is the normalized radius of the lens. Here, $h_0 \approx 0.63$ represents the normalized height of the reference Rinehart-Luneburg lens, while the superellipse exponents are $p = 2$ and $q = 1.8$ [11]. The GRIN Luneburg lens and geodesic Luneburg lens, with rays illustrating their wave propagation, are depicted in Fig. 1. This figure demonstrates that the lens can collimate the wave from a source on its surface, which translates into a highly-directive beam when radiating.

In this section, we show the design procedure of a modulated geodesic lens antenna working in the band 128-132 GHz.

To demonstrate the performance of the antenna, three waveguide feeds are integrated to produce three independent beams. Moreover, an EBG structure is added at both sides of the waveguides to avoid leakage resulting from manufacturing and assembly tolerances.

### A. Modulated geodesic lens design

To achieve a more compact design, the lens profile can be folded, as suggested by Kunz [15]. In our configuration, the modulated profile is designed using splines, due to its capability to yield smoother transitions and, consequently, to minimize reflections [11]. An in-house ray-tracing tool serves to optimize the modulation of the profile, offering an efficient means to design the lens, as detailed in [16]. As depicted in Fig. 2, This modulation achieves a lens height of only 38.7% compared to the original height, resulting in a more compact design that is easier to integrate.

The lens is implemented in a parallel plate waveguide (PPW) of height 0.5 mm, which is less than a quarter of the wavelength at 132 GHz, ensuring quasi-TEM mode wave propagation within the PPW [17]. The lens radius is 25 mm, approximately equivalent to 10.8 wavelengths at 130 GHz. To showcase the characteristics of the modulated Luneburg lens, a full-wave model is simulated in CST Microwave Studio to observe electromagnetic field propagation. The lens is fed by a discrete port producing a cylindrical wave at the contour of the lens, and open boundary conditions were applied to the PPW.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Simulated S-parameters of the antenna.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7: Efficiency of the antenna.</figcaption></figure>

environment to minimize reflections. Fig. 3 depicts the electric field distribution within the lens when excited at 130 GHz. It can be seen that the wavefront is planar at the opposite side of the port. This means that a geodesic lens antenna utilizing this modulated profile exhibits the same focusing properties as the reference geodesic lens design and is capable of generating directive beams.

## B. Feeding structure design

The feeding system is configured as a three-waveguide system. Although more feeds can be added to have a finer scanning progression, three scanning angles were found enough for this proof of concept. In sub-THz systems, small gaps or misalignments, due to manufacturing inaccuracies, can lead to pronounced wave leakage, resulting in performance degradation of the antenna [13]. Conventionally, this is addressed by applying increased pressure on the structure, typically using clamps or additional screws. However, this solution is not ideal as non-uniform pressure distribution can produce impedance mismatch and compromise structural performance. This not only escalates the complexity of ensuring consistent performance across mass production but also restricts the range of potential applications [18]–[21].

Consequently, an EBG structure is added around the waveguide feeds of the antenna to avoid leakage and reduce complexity. More concretely, we use glide symmetric holes, as they have shown high robustness in previous designs at lower frequencies [22]–[24]. This structure has been designed to optimize their performance within the operational frequency band. Figure 4(a) shows the dispersion diagram, both phase constant and attenuation constant, of the glide-symmetric holes. An illustration of the unit cell is also shown as an inset.

<figure><img src="image_9.png" /><figcaption>Fig. 8: Measurement setup in an anechoic chamber.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9: Measured reflection coefficients of the antenna.</figcaption></figure>

The band of operation of the designed antenna is highlighted in grey. It can be seen that the designed holey structure has a stopband in that frequency range with an attenuation constant of around $\alpha = 0.54k_0$.

A simulation of a waveguide with a uniform gap of 0.03 mm and surrounded by the holey EBG structure was performed. The waveguide has a length of 20 cm and the metallic plates have a conductivity of $4.561 \times 10^7$ S/m and a root-mean-square surface roughness of 0.35 µm (provided by the manufacturer). In Fig. 4(b) the S parameters together with the electric field distribution are shown. The transmission coefficient is over -1 dB across the band, meaning that over 80% of the energy is transmitted. These results affirm the effective prevention of leakage by the structure.

## C. Prototype design

The final design of the modulated geodesic lens antenna is illustrated in Fig. 5. The structure is divided into two pieces joined using screws. This is done to allow for milling manufacturing and a post processing of the metallic plates to reduce surface roughness. Moreover, the prototype is also coated in gold to further reduce the losses.

The antenna comprises a modulated geodesic lens and a feeding network. A linear flare structure is added to the lens, to match it with free-space. This flare increases the height from 0.5 mm to 3 mm and it has a length of 4 mm. At the opposite side, the feeding network is placed, which is composed by three waveguide feeds. These waveguides feed the antenna at angles of -40°, 0°, and 40°, thereby generating three distinct beams. A small intended gap, $G_{gap} = 0.03$ mm, is added to the feeding part, remaining within the bounds of manufacturing limitations. The highlighted gap is depicted at the bottom of Fig. 5, where air is represented in blue. At the two ends of the waveguide, stages were implemented to accommodate screws (see the top left of Fig. 5). When the screws are tightened, a good contact between the upper and lower stages is achieved
<figure><img src="image_11.png" /><figcaption>Fig. 10: Simulated (solid lines) and measured (dashed lines) normalized radiation patterns of the antenna at (a) 128 GHz, (b) 130 GHz, (c) 132 GHz.</figcaption></figure>

thanks to the intended gap. The EBG holes are strategically placed along the waveguides to avoid leakage where the gap is. This robust feature helps to mitigate the effects of potential manufacturing and assembly errors.

### III. SIMULATION AND MEASUREMENT RESULTS

The antenna is simulated using CST Microwave Studio. The material of the metallic plates is gold, assuming a conductivity of $4.561 \times 10^7$ S/m and a root-mean-square surface roughness of $0.35\mu m$, as provided by the manufacturer. The simulated S-parameters of the lens antenna are presented in Fig. 6. Given the symmetric nature of the antenna structure with respect to the y-axis, the results of port 3 have been omitted for clarity. The reflection coefficients are below -15 dB and the mutual coupling below -30 dB across the band, attesting to the commendable performance of the antenna.

The simulated efficiency-frequency diagram of the lens antenna is depicted in Fig. 7 for ports 1 and 2 (port 3 is omitted due to symmetry). The aperture efficiency (effective aperture over actual aperture) is higher than 57% across the frequency range. These outcomes are calculated considering a rectangular aperture with dimensions of 2.3 mm × 58 mm. Notably, both total (radiated power over stimulated power) and radiation (radiated power over accepted power) efficiencies are at approximately 80%, which is commendable for antennas operating within the sub-THz range.

The two pieces forming the antenna were manufactured in aluminum with CNC milling technique and gold coated to reduce metal loss. The measurement of the antenna was conducted at the anechoic chamber of University of Granada, Spain. The antenna prototype mounted on the measurement setup is shown in Fig. 8. The measured reflection coefficients are shown in Fig. 9. The three ports have a reflection coefficient below -15 dB across the frequency band of operation.

The simulated and measured radiation patterns at frequencies of 128, 130, and 132 GHz are presented in Fig. 10. The radiation patterns are normalized to the maximum value at broadside direction. These patterns underscore that the antenna maintains consistent and stable beam scanning performance throughout the operational frequencies. The side-lobe level are below -14 dB and the scan losses at 40° are below 0.3 dB.

The simulated and measured peak realized gain vs. frequency is plotted in Fig. 11. The maximum gain is around 22.5 and 22 dB for the center and side ports in simulations. In measurements the three ports have a gain of around 21 dB

<figure><img src="image_12.png" /><figcaption>Fig. 11: Simulated (solid lines) and measured (dashed lines) realized gain over frequency.</figcaption></figure>

across the frequency band. This disagreement is attributed to surface roughness and measurement uncertainties.

A comparison with geodesic lens antenna designs at lower frequency bands is shown in Table I. It can be appreciated that as the frequency increases, the total efficiency of the antenna reduces due to the manufacturing tolerances. Notably, although the total efficiency of the proposed prototype is around 60%, the radius of the lens is more than double of previous works.

<table><caption>TABLE I: Comparison with previous geodesic lens antennas.</caption><thead><tr><th>Ref</th><th>Frequency band (GHz)</th><th>Radius</th><th>Material</th><th>Total Efficiency</th></tr></thead><tbody><tr><td>[9]</td><td>24-36</td><td>4λ<sub>0</sub></td><td>Aluminum</td><td>≈90%</td></tr><tr><td>[12]</td><td>56-62</td><td>4λ<sub>0</sub></td><td>Aluminum</td><td>≈75%</td></tr><tr><td>This work</td><td>128-132</td><td>10.8λ<sub>0</sub></td><td>Gold</td><td>≈60%</td></tr></tbody></table>

### IV. CONCLUSION

This work introduced a fully metallic geodesic Luneburg lens antenna operating in the sub-THz band. The antenna generates three beams directed at 0° and ±40° with its waveguide feeding system. To address potential leakage issues at these high frequencies, the feeding system incorporated an intentional air gap of 0.03 mm, considering manufacturing constraints, along with an EBG structure. A prototype has been manufactured using CNC milling and coated in gold to reduce losses. The experimental results demonstrate promising performance, with high directivity and stable scanning capabilities. This shows the potential of geodesic lens antennas for applications in the sub-THz regime. Thanks to its robustness in manufacturing, these antennas could potentially be used above 300 GHz, in the THz regime.
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