Evaluation of Losses in 3-D-Printed Geodesic Lenses Using a Ray-Tracing Model

This article applies an in-house generalized ray-tracing (RT) model to efficiently compute both the radiation pattern and the efficiency of geodesic lenses with nonrotationally symmetric shapes. Losses due to ohmic effects and surface roughness are included in the model. These losses are very relevant for monolithic geodesic lens antennas as postprocessing techniques cannot be applied to reduce the surface roughness of internal part of the metallic plates. The model is validated by comparison with full-wave simulations for three different lenses: a circular flat parallel-plate waveguide (PPW), an elliptically compressed geodesic lens, and a water-drop lens. These results show a reduction in computational time by a factor of 600 using the RT model. A non-rotationally symmetric water drop lens has been manufactured in a monolithic piece using the laser powder-bed fusion (LPBF) technique with successful experimental results.

Abstract-This article applies an in-house generalized raytracing (RT) model to efficiently compute both the radiation pattern and the efficiency of geodesic lenses with nonrotationally symmetric shapes.Losses due to ohmic effects and surface roughness are included in the model.These losses are very relevant for monolithic geodesic lens antennas as postprocessing techniques cannot be applied to reduce the surface roughness of internal part of the metallic plates.The model is validated by comparison with full-wave simulations for three different lenses: a circular flat parallel-plate waveguide (PPW), an elliptically compressed geodesic lens, and a water-drop lens.These results show a reduction in computational time by a factor of 600 using the RT model.A non-rotationally symmetric water drop lens has been manufactured in a monolithic piece using the laser powderbed fusion (LPBF) technique with successful experimental results.

I. INTRODUCTION
A NTENNAS for wireless communication and radar sys- tems operating in the millimeter wave (mm-wave) regime are required to be highly directive to compensate for high path losses and to be able to steer the beam [1].Due to their simplicity and passive nature, quasi-optical solutions have been considered for these applications.Some examples include Rotman lenses [2], [3], the pillbox antenna [4], [5] and homogeneous lenses [6], [7].A prominent quasi-optical solution is the Luneburg lens antenna.Due to its rotational symmetry, this lens antenna can scan in a wide angular range with very low scan losses [8].The Luneburg lens is a graded-index lens with two focal points: one at the contour of the lens and the other at infinity.This means that it can transform a cylindrical wave excited at the contour of the lens into a planar wave on its opposite side [9].Although at low frequencies the design of a Luneburg lens antenna can be bulky, at mm-wave frequencies its physical size is reduced enough to become a compact solution.An alternative is planar Luneburg lens antennas, which can be made with discretized layers of dielectrics [10] or metasurfaces [11], [12], [13].To avoid losses due to propagation in dielectric materials in the mm-wave range, fully metallic metasurfaces are preferred [14], [15], [16].However, at high frequencies, the subwavelength size of these metasurfaces makes their manufacture challenging.An option that eases manufacturing restrictions is the use of geodesic lenses [17].
A geodesic lens consists of two parallel curved metallic plates with a homogeneous refractive index profile between these plates.Thus, the properties of equivalent planar gradedindex lenses are replicated by varying the height profile of the parallel plates [18], [19].Rinehart [18] introduced the geodesic version of the graded-index Luneburg lens.The drawback of the Rinehart-Luneburg geodesic lens is that its height is about 63% of the radius, which can be bulky for certain applications.Consequently, to obtain a more compact design, folding the vertical profile was proposed in [20] and has been implemented in antenna designs in [21] and [22].Further research has recently been reported to make geodesic lens antennas more suitable for wireless communication systems, for example, by increasing the beam crossover level [23].However, this solution considerably reduces the directivity of the lens.To avoid this reduction in directivity, a geodesic lens antenna was proposed in [24] with the focal point placed further away from the lens contour.Furthermore, in [25] a compact dual polarization system was proposed by stacking two geodesic lens antennas with perpendicular polarization.This solution was implemented by integrating two polarizers into the aperture of the lens antenna.Other 2-D solutions for beam steering have been reported using linear arrays of stacked [26] or double-layer geodesic lens antennas [27].
Lens design by means of full-wave computational electromagnetic techniques is time-consuming because of the large electrical size of the lenses.This has motivated the search for other approximate and efficient procedures, among which the ray-tracing (RT) model is one of the preferred techniques.RT models have been widely used in optics [28] and microwaves for propagation studies [29], [30].In [31] and [32] RT models have also been used to speed up lens antenna design procedures.In [33] and [34], an RT model is used to estimate the amplitude and phase distribution needed in an array with a radome on top to avoid grating lobes in a wide scanning range.Geometrical optics techniques have been widely used for the design of dielectric [35], [36] and parallel-plate waveguide (PPW) [37] lens antennas.
Although the combination of RT techniques with the principles of physical optics can be considered a mature topic (currently implemented in commercial software packages such as [38] and [39]), simple and efficient implementations of this technique can be very advantageous in some specific scenarios.When considering geodesic lenses, an approach of this kind was proposed in [22].First, the geodesic curve was mapped to its equivalent refractive index by means of transformation optics.Then, the equivalent lens was used to calculate the ray paths [23].However, these two models are restricted to rotationally symmetric lenses, since rotational symmetry is a key assumption in the derivation of the geodesic height profile.To overcome this limitation, a generalized RT code was proposed in [40] for the design of nonrotationally symmetric geodesic lenses.
In this contribution, we extend the procedure proposed in [40] to include the possibility of calculating the efficiency of the lens.This new feature is then used to estimate the losses of nonrotationally symmetric geodesic lenses for materials with different surface roughness and conductivity.These losses are very relevant for highly directive antennas where postprocessing techniques cannot be applied to increase their efficiency.Furthermore, a prototype of an elliptical water-drop lens antenna was 3-D-printed using a laser powder-bed fusion (LPBF) technique in a monolithic piece (i.e., as a single piece), and experimentally validated by means of radiation patterns and radiation efficiency.

II. RT MODEL
The generalized RT model for geodesic lenses reported in [40] is briefly described here for completeness.Three steps can be distinguished in this model: 1) geometric optics to find the path trajectories; 2) ray-tube theory to find the amplitude of the field in the lens aperture; and 3) Kirchhoff's diffraction formula to calculate the radiation pattern.

A. General Description of the Model
Geometric Optics is generally used to describe wave propagation using rays with an approximation of zero wavelength [42], [43].Consequently, lenses with a large electrical size are to be considered (in our experience, larger than 3-4 wavelengths).Since a TEM mode is assumed to propagate in the PPW, Fermat's principle states that the waves propagate through the shortest optical path.For example, in our case study shown in Fig. 1, the source point (in blue) connects to every target point (in pink) through the shortest geometrical path.Here, we use the Python library "potpourri3d" [44] to compute the ray trajectory on a surface with an arbitrary shape (the mean surface defined by the curved PPW).This library efficiently computes the shortest distance between two points on a surface that is here meshed using the 2-D Delaunay triangulation [41] available in the Python library "pyvista" [45].An example of this mesh is illustrated in Fig. 1.Calculating the path of the rays to the kth target point gives us a path length σ k , which is used to evaluate the phase distribution in the lens aperture, as shown in Fig. 2. It also retrieves information on the width of the ray tube close to the source dL ′ k and close to the target points dL k .This information is needed to compute the amplitude distribution with the help of the theory of power conservation of ray tubes [42].Following the notation in Fig. 2 and assuming a constant height of the PPW, the amplitude A k of the kth ray in the aperture is calculated as where A ′ k is the amplitude close to the source.The specific value of this parameter depends on the nature of the source (dipole/waveguide), as discussed in [40].The parameter dL k is given by dL k = dc k cos θ k .Finally, Kirchhoff's diffraction formula is used to numerically compute the far-field radiation pattern by considering each target point as an elementary source with amplitude A k and phase ϕ k = k 0 σ k .Following the notation in Fig. 3 and assuming R obs ≫ λ, the far-field is calculated as where r k = r k rk is a vector from the target point to the observation point.Remarkably, the simple addition of the exponential factor exp(−ασ k ) accounts well for the attenuation of energy along the path trajectory due to losses.These losses were not previously taken into account in previous work [40].

B. Calculation of Radiation Efficiency
Geodesic lens antennas are highly efficient solutions due to their fully-metallic nature.However, their efficiency depends on the conductivity and roughness of the metallic plates.Therefore, assessing the losses of the lens antenna prior to manufacturing is crucial since small antenna losses can cause an important increase in the power consumption of the system or a degradation in the equivalent isotropic radiated power (EIRP).For example, losses of 1dB mean that the antenna needs a 20% larger aperture or that the amplifiers need to provide 20% more power to the system to achieve the required EIRP.
Since Kirchhoff's diffraction formula only provides the radiation pattern in the H-plane, another procedure should be used for the calculation of the directivity.However, it is possible to calculate lens efficiency by comparing the H-plane radiation patterns with and without losses; that is, using the following ratio: where E 0 is the far-field when no losses are considered.
As proposed in Section II-A, the losses in the radiation pattern are taken into account by the exponential factor exp(−ασ k ) in (1).The attenuation constant α of the rays propagating in a PPW with a distance between plates h ppw (see Fig. 4), assuming a conductivity σ and a root mean square (rms) surface roughness in metal plates, is given by the following expression [46]: where δ s = √ 2/(ωσ µ 0 ) is the skin depth (with ω being the angular frequency) and α c is the attenuation due to perfectly smooth conductors given by [47] From the asymptotic property of the arctan function in (3) [lim x→∞ arctan(x) = π/2], it can be concluded that increasing the rms surface roughness influences losses only if

III. NUMERICAL RESULTS
In this section, three lenses are considered to numerically validate the model: a circular flat lens, an elliptically compressed geodesic lens, and an elliptically compressed waterdrop lens.In all cases, the lenses are fed by a rectangular waveguide, whose amplitude is approximated by the Gaussian function A ′ k = 10 −(3ξ 2 /20) [40].The waveguide source has a width of 8.64 mm and height 2 mm, the same height as the PPW that makes up the lens.The numerical results obtained with the proposed RT method are compared with the full-wave simulation results from HFSS.

A. Circular Flat Lens
Fig. 5 shows the results of a circular flat lens of radius 6λ at 30 GHz (λ is the wavelength of free space at the operating frequency).Although the calculation of the ray trajectories in this lens is straightforward, it is considered a good benchmark to validate the radiation patterns provided by our model.In Fig. 5(a), the geodesics computed by the RT model are plotted, which in this case are straight lines from the source to the different target points.Fig. 5(b) shows the vertical component of the electric field, E z , simulated using HFSS.The calculated radiation patterns are plotted in Fig. 5(c) and (d) for two values of the rms surface roughness.The results are normalized to the maximum value of the radiation pattern when no losses are considered.In both figures, good agreement is found between the RT results and those of HFSS, thus validating the ability of our proposed method to calculate the efficiency of the lens.The dashed lines in Fig. 5(c) are the results of a lens with perfectly smooth conducting plates ( = 0) and a conductivity of σ = 3.56 × 10 5 S/m.Its efficiency is −0.7 dB in the broadside direction for both RT and full-wave simulation.The results in dashed lines in Fig. 5(d) correspond to a lens with = 6 µm and conductivity σ = 3.56 × 10 5 S/m, showing an efficiency of −1.5 dB in both RT and full-wave simulation.This result agrees with the condition (5), which for this conductivity and frequency states that a value of rms surface roughness ≲ 50 µm will have a noticeable effect on the losses.

B. Elliptically-Compressed Geodesic Lens
In this section, we study an elliptically-compressed geodesic lens.The lens height profile before compression follows the superellipse function z/R = h 0 [1 − (ρ/R) p ] 1/q [22], where z is the height, ρ the radial position, R the lens radius, and h 0 , p, and q are the design parameters.A toroidal bend with Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.radius R b is added at the end of the profile to avoid reflections due to discontinuities.In this example, the radius of the lens is R = 7.5λ at 30 GHz and is compressed in the x-direction by a factor a = 0.5.Here, the lens is fed at two different positions: φ s = 0 • and φ s = 45 • .Fig. 6(a) shows the ray paths of the generalized RT model for these two different source positions.Fig. 6(b) shows the electric field, E z , calculated with HFSS.In both cases, a planar wave is achieved on the opposite side of the feeding waveguide.Normalized radiation patterns are shown in Fig. 6(c).The results with losses consider a conductivity of σ = 3.56 × 10 5 S/m and roughness = 6 µm.For the source located at φ s = 0 • , the efficiency has a value of −1.7 dB in RT and −1.5 dB in HFSS.For the

C. Elliptically-Compressed Water-Drop Lens
The geodesic surface of this lens before compression follows the spline profile described in [22] with the optimized design parameters of [40].This profile is shown in Fig. 4 in dashed lines.The lens has a radius of R = 7.5λ and a compression ratio of a = 0.7.Fig. 7 shows the ray paths from the RT model and the electric field from the full-wave simulation at three different feeding positions φ s = 0 • , 30  = 20 µm.The first material will later be used in the manufacturing of prototype.Normalized radiation patterns without losses are omitted here to simplify the plots.The beam shape and SLL agree well between the RT model and HFSS.However, for the beam at φ s = 60 • , there is some disagreement in the SLL.As explained in [40], at these extreme angles, some reflections increase the side lobes, which are not taken into account in the RT model.Our aim was to find a balance between SLLs and scan losses for this design.We could have achieved lower SLLs; however, this would have come at the expense of increased scan losses.Fig. 8 shows the radiation efficiency of both lenses in the frequency band of interest.In the low-loss case (aluminum, = 8 µm), the efficiency is around 0.27 dB throughout the band.On the other hand, in the lossy case (steel, = 20 µm), the efficiency goes from −1 to −1.3 dB until φ s = 50 • .However, for the port scanned at φ s = 60 • , the efficiency in HFSS is −1.5 dB.Again, this disagreement is attributed to some reflections that are not taken into account in the RT model.
These results show the importance of calculating the efficiency of these antennas before choosing the manufacturing technique and the metallic materials.In terms of computational time, to calculate the results of the seven ports the full-wave IV.ADDITIVE-MANUFACTURED COMPRESSED WATER-DROP LENS ANTENNA Here, we present the design and measurements of a compressed water-drop lens antenna optimized for manufacturing using LPBF.In this technique, a sequence of layers of metal powder, selectively bonded together by the application of a laser source, generates the desired structure.This manufacturing technique has been used to manufacture waveguide components [50], [51], horn arrays [52], periodic structures [53] and polarizers [54].Recently, it has been used to build a Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.half-Luneburg geodesic lens antenna in a single piece [55].The prototypes made with this 3-D-printed technique are lightweight and solely made in metal, which is of interest for on-board space applications.

A. Lens Antenna Design
The RT model was used to efficiently optimize the profile of the geodesic lens and to select the manufacturing technique and material.After that, final numerical and experimental validations are performed using full-wave simulations.
A model of the lens antenna designed is shown in Fig. 9.We used the lens presented in Section III-C with a targeted operation frequency band of 24-32 GHz.Thirteen feedings are added to steer the beam ±60 • with steps of 10 • .On the opposite side of the lens, an exponential flare with a length of 10 mm and a height that smoothly increases from 2 to 15 mm is added to match the impedance of the free space.Furthermore, the feeds must match the impedance of the PPW with commercial coaxial connectors, which are inserted into the waveguide to excite the TE 10 mode.Thus, a stepped waveguide transition is added, similar to the one proposed in [21].The transition has four steps of 8.64 mm width and height from 2 to 4.3 mm.The total length of the feeds is 11 mm, which maintains the compactness of the design.

B. Additive-Manufactured Prototype
In previous works, geodesic lenses have been typically manufactured in multiple pieces with CNC milling [21], [22],  [23], [25], [26], [27].This can cause misalignments, leakage between layers, and an unnecessary increase in the prototype weight.Recently, a geodesic half-Luneburg lens antenna was manufactured in [55] as a single piece using LPBF, which demonstrated the potential of this technique.The metallic part of the lens was optimized to create a more compact and lightweight prototype while maintaining its mechanical properties.Here again, this technique and optimization are carried out with a final design as shown in Fig. 9.The metallic walls are conformal to the geodesic shape with a thickness of 2 mm.In light of the results presented in Fig. 8(c), the material chosen for manufacturing is aluminum alloy AlSi10Mg (with σ = 1.68 × 10 7 S/m), with an expected roughness of 8 µm.This roughness is estimated from previous experience taking into account the elliptical-compressed shape and printing direction, and it is achieved by configuring the height of the printed layer to be as small as possible for the 3-D printer.The manufactured prototype is shown in Fig. 10.Because the prototype is manufactured in one single piece, it is not possible to use post-processing techniques, such as polishing, to reduce the roughness in the inner surface of the metal.

C. Experimental Results
The simulated and measured reflection coefficients of the first seven ports are shown in Fig. 11(a) and (b).Simulations have been carried out using HFSS.The results of the remaining six ports are omitted as they are symmetric to the first  The prototype was measured in the anechoic chamber of KTH using the setup shown in Fig. 12.The simulated and measured radiation patterns at 24, 28, and 32 GHz of the first seven ports are plotted in Fig. 13.There is good agreement between the simulations and the measurements.The maximum scan losses at 28 GHz are 3.5 and 2.7 dB for simulations and measurements.
The measured radiation patterns at 24, 28, and 32 GHz for ports 4 and 7 are depicted in Fig. 14.The results demonstrate that a fan beam is generated at each scanning angle.
Fig. 15 shows the simulated and measured realized gain for ports 1-7.Simulations were carried out considering aluminum material (σ = 1.6835016 × 10 7 S/m) and = 8 µm.The realized gain goes from 15 to 20 dB in simulations and from 13.6 to 19 dB in measurements, showing a good agreement.
The prototype's radiation efficiency, measured and averaged for the seven ports and frequencies ranging from 24 to 32 GHz, was found to be −1 dB, while simulations indicated a value of −0.27 dB.These discrepancies are within the cumulative effect of the tolerances of the manufacturing and the test uncertainties in the anechoic chamber at KTH (±0.25 dB).
V. CONCLUSION In this work, we have presented a generalized RT model to efficiently compute the radiation pattern and efficiency Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
of geodesic lenses with nonrotationally symmetric shapes.Different from previous implementations of RT for geodesic lenses, Kirchhoff's diffraction formula has a new term to account for the losses in the parallel plate waveguide due to finite conductivity and roughness.The calculation of these losses is key when monolithic antennas are manufactured since the surface roughness of the internal part of the metallic plates cannot be reduced in post-processing.This model was validated by comparison with full-wave simulations for three different lenses: a circular flat lens, an elliptically-compressed geodesic lens, and an elliptical water-drop lens.The latter lens was chosen for experimental validation.A prototype was manufactured using the LPBF technique with aluminum alloy.The prototype has been tested and its experimental results are in good agreement with the simulations.The proposed RT approach has sped up the design process by a factor of 600.Furthermore, this method helped to choose the material and the targeted roughness of the additive-manufactured prototype.

Fig. 1 .Fig. 2 .
Fig. 1.Mean curve of a non-rotationally symmetric lens with source and targeted points.The mesh has been computed using Delaunay triangulation [41].

Fig. 3 .
Fig. 3. Lens parameters involved in the calculation of the far-field pattern.φ s is the angle at which the source is placed.

Fig. 4 .
Fig. 4. Lens profile along the y-axis of the conducting plates (solid lines) and the mean path (dashed line).

Fig. 7 .
Fig. 7. (a), (c), and (e) RT model ray paths and (b), (d), and (f) HFSS electric field of the compressed water-drop lens for feed positions (a) and (b) φ s = 0 • , (c) and (d) φ s = 30 • , and (e) and (f) φ s = 60 • .Fig. 8(a) and (b) shows normalized radiation patterns for seven feed positions evenly distributed from 0 • to 60 • at 30 GHz.In this case, two real materials are considered: aluminum (σ = 1.6835016 × 10 7 S/m [48]) with = 8 µm and steel (σ = 1.020408 × 10 6 S/m [49]) with= 20 µm.The first material will later be used in the manufacturing of prototype.Normalized radiation patterns without losses are omitted here to simplify the plots.The beam shape and SLL agree well between the RT model and HFSS.However, for the beam at φ s = 60 • , there is some disagreement in the SLL.As explained in[40], at these extreme angles, some reflections increase the side lobes, which are not taken into account in the RT model.Our aim was to find a balance between SLLs and scan losses for this design.We could have achieved lower SLLs; however, this would have come at the expense of increased scan losses.Fig.8shows the radiation efficiency of both lenses in the frequency band of interest.In the low-loss case (aluminum, = 8 µm), the efficiency is around 0.27 dB throughout the band.On the other hand, in the lossy case (steel, = 20 µm), the efficiency goes from −1 to −1.3 dB until φ s = 50 • .However, for the port scanned at φ s = 60 • , the efficiency in HFSS is −1.5 dB.Again, this disagreement is attributed to some reflections that are not taken into account in the RT model.These results show the importance of calculating the efficiency of these antennas before choosing the manufacturing technique and the metallic materials.In terms of computational time, to calculate the results of the seven ports the full-wave

Fig. 15 .
Fig.15.Simulated (solid and measured (dashed line) realized gain of the lens antenna prototype.six ports.The simulated results are below −15 dB in the band from 24 to 32 GHz, while the measured results are below −12 dB.The prototype was measured in the anechoic chamber of KTH using the setup shown in Fig.12.The simulated and measured radiation patterns at 24, 28, and 32 GHz of the first seven ports are plotted in Fig.13.There is good agreement between the simulations and the measurements.The maximum scan losses at 28 GHz are 3.5 and 2.7 dB for simulations and measurements.The measured radiation patterns at 24, 28, and 32 GHz for ports 4 and 7 are depicted in Fig.14.The results demonstrate that a fan beam is generated at each scanning angle.Fig.15shows the simulated and measured realized gain for ports 1-7.Simulations were carried out considering aluminum material (σ = 1.6835016 × 10 7 S/m) and = 8 µm.The realized gain goes from 15 to 20 dB in simulations and from 13.6 to 19 dB in measurements, showing a good agreement.The prototype's radiation efficiency, measured and averaged for the seven ports and frequencies ranging from 24 to 32 GHz, was found to be −1 dB, while simulations indicated a value of −0.27 dB.These discrepancies are within the cumulative effect of the tolerances of the manufacturing and the test uncertainties in the anechoic chamber at KTH (±0.25 dB).