L1. 3D PRINTING A ANTENNA 3D Printing a Maxwell Fish Lens With Periodic Structures Valentine Lin and Tarek Sayed Hamad

Abstract—With the rise of high frequency communication frequency band if the effective of the periodic systems such as 5G, new types of antennas has to be developed structure is negative [7]. in order to meet the new requirements. In recent years, lens The goal of this project is therefore to produce a prototype antennas made of periodic structures has been shown to have desirable performance when increasing operational frequency of a , more specifically a Maxwell fish-eye (MFE) without increasing the size of the antennas. One way of man- lens. The MFE lens is a type of lens antenna that can be ufacturing the for the antennas are with 3D printers utilised for wireless communication. loaded with dielectrics with specified permittivity. This project A MFE lens is has a refractive index that varies over the group studied the process of designing and manufacturing a flat radius, these lenses are generally called graded index (GRIN) Maxwell fish eye lens at 5 GHz with a bandwidth of 3.5 GHz to 6 GHz. The resulting design is a lens based on a periodic lenses. At the lens’s centre, the refractive index is largest and configuration of cuboid unit cells made from dielectrics which gradually decreases to a minimum at the circumference of the consisted of a hole. By choosing the ratio of dielectric and holes lens [8]. The refractive index has a direct correlation to the in the unit cells, each part of the lens could be tuned to achieve permittivity of the material [9]. Previous studies have shown a specific effective refractive index required for realising the that a MFE lens can provide perfect imaging as long as the Maxwell fish eye lens. location of the source is known and within a certain frequency Index Terms—Maxwell fish eye lens, dielectric lens antenna, band [4]. Furthermore, with the rise of 3D printers as a cheap metamaterial, periodic structure, 3D printed lenses. and reliable way of manufacturing, previous studies has also TRITA number: TRITA-EECS-EX-2019:149 shown that a GRIN lens can be created by using a 3D printer with different materials [10]. If holess are introduced in the I.INTRODUCTION lens, the transition between materials can be much smoother In the world of wireless communication systems, lens and the effective permittivity can be tailored. This can be built antennas are gaining popularity since communication systems by a 3D printer in a single process without any complicated are developing towards higher operational frequency with machinery and with only a few different materials. technologies such as 5G [1]. Since higher frequency signals require greater directivity, lenses can be utilised to focus the II.TECHNICAL BACKGROUND signals in a desired direction as shown by [2]. In the case of In order to manufacture and realise a MFE lens, some 5G, many highly directional transmitters has to be placed in theory needs to be established. Therefore, knowledge about an area to provide full coverage. There need to be line of sight lenses, electromagnetic waves, periodic structures, dispersion between the sender and receiver because the high frequency diagrams and S-parameters are essential and will be presented signals gets blocked by most objects. One way of solving in this section. this problem is by designing integrated lens antennas that can provide several targeted beams from on single transmitter as shown by [3]. A. Lenses Traditional lens antennas has typically been constructed Lenses are most popular at optic frequencies. They transmit with a homogeneous material by combining different dielectric and refracts in a way that disperses or focuses light to a materials as the bandwidth for dielectric lenses can be very certain point, a so-called focal point. An image can take form wide while the losses are low [4]. The traditional method has if light is focused on a focal point, which depends on the been very expensive and complex in order to manufacture a lens’ geometry and the material’s refractive index [9]. Optical high performance lens. As an alternative method, 3D printing lenses are usually made out of transparent materials such as has been studied as a method of producing periodic structures glass or plastic. However, they can also be made out of non- for use in lens antennas. Studies have shown that these periodic transparent materials in order to transmit, disperse and refract structures can be utilised to design a structure with the same other electromagnetic waves, not necessarily light. desired properties for a lower manufacturing cost [5]. Periodic structures could also be designed to perform functions that natural materials cannot, such as giving a structure a negative B. Maxwell Fish Eye Lens refractive index. In the case of optics, periodic structures could A Maxwell Fish-eye lens is a type of lens antenna with a be used to circumvent the physical limitations of traditional gradient refractive index. More specifically, it has a refractive optics [6]. For example cloaking an object with a layer of index profile that is spherically symmetric, which gives it a periodic structures could render object invisible for a narrow certain property. L1. 3D PRINTING A LENS ANTENNA

Figure 1. Ray trajectories for the MFE lens. As seen in the figure, the MFE lens focuses the rays from on side of the lens to the opposite side of the lens.

Figure 2. A graphical representation of how the refractive index in a MFE A source in any excited point of the circumference will lens varies with the radius. r is the distance from the centre to a point on the converge exactly at the opposite side of the circumference lens and a is the radius of the lens. (its focal point), as demonstrated in Fig. 1. Due to the lens’ varying refractive index within the lens itself, the optical path is affected and bends in a way which gives the MFE lens A unit cell’s dimensions and geometry directly affects this certain property. Theoretically, a perfect image can be the effective refractive index. By combining unit cells with produced on the other side [4]. different properties, a varying refractive index can be achieved. The refractive index of the lens is greatest at the centre One important aspect to consider is the fact that the unit cells’ of the lens, and gradually decreases to its lowest value at the dimensions needs to be much smaller than the wavelength of circumference. This can be seen in Fig. 2, where the refractive the transmitted electromagnetic waves that are being transmit- index has a value of n = 2 in the middle, and n = 1 at the ted through the lens. This is because the unit cells can then circumference. The refractive index of a MFE lens can be be considered as a homogeneous media with one refractive described by equation (1). index.

√ 2n0 n = r = r 2 (1) 1 + ( a ) D. Dispersion Diagrams Where n is the refractive index, n0 is the refractive index at A dispersion diagram represents a wave’s propagation char- the border of the lens, r is the distance from the centre and a acteristics through a structure. To understand these diagrams, is the radius of the lens. In normal cases, n0 = 1 since the lens some theory needs to be established first. is surrounded by air and there needs to be a smooth transition 1) Propagation Constant: The propagation constant de- between the uttermost layer of the lens and the surrounding scribes how the wave’s amplitude and phase varies along the medium, in this case air, that has a refractive index of n = 1. If propagation direction. In a lossy medium, the propagation the discrepancy between the refractive index of two mediums constant is complex. It is described with equation (2). are too big, it would cause reflections on the surface between the mediums. γ = α + jβ (2)

C. Periodic Structures Here α is the attenuation constant and β is the phase Periodic structure are structures that are made up out of a constant [12]. In lossless mediums, α = 0 and β can be repeated pattern of unit cells, in one, two or three dimensions. calculated using equation (3). A great example of this type of structure are honeycombs, √ β(ω) = ω µ which is made up of hexagonal wax cells and has a repeated (3) pattern of these cells periodically in two directions. 2) Effective Refractive Index: Dispersion diagrams de- A periodic structure can have similar properties to a material scribes the relation between the phase constant β and angular with a specific permittivity, depending on the unit cells. These frequency ω of an electromagnetic wave travelling through a unit cells are designed to achieve a certain effective permittiv- structure [12]. The slope of the curve in a dispersion diagram ity which which is needed to realise a lens. A surface made at one frequency is the group velocity, which can be calculated out of a periodic pattern of unit cells, that produces unusual using equation (4). reflection patterns or properties, is called a metasurface [11]. With periodic structures, a GRIN lens is possible, even if it is dω v = (4) flat [12]. g dβ L1. 3D PRINTING A LENS ANTENNA

When the slope changes, it means that the effective refract- ive index of the structure has changed. The effective refractive index can be calculated with equation (5).

c neff = (5) vg By analysing the dispersion diagrams of infinitely repeating unit cells, the effective refractive index can be determined. The relation between the phase constant of the wave in vacuum and in a unit cell provides the effective refractive index for a specific frequency, see equation (6).

βunitcell(ω) neff (ω) = (6) βvacuum(ω)

E. Waveguides A rectangular waveguide made from a perfect electric conductor (PEC) is designed for feeding the lens. PEC is chosen since it reflects electromagnetic waves fully. A rectan- gular waveguide can propagate TM and TE modes which are Figure 3. 3D model of a cuboid unit cell with a hole, along with its different field distributions. In a rectangular waveguide, with dimensions. Its dimensions can be seen in table I. the width W and height h, and W > h, the dominant mode is called TE10. This is due to the fact that this mode has the lowest attenuation of all TE-modes in a rectangular waveguide. A. Design of Unit Cells These types of rectangular waveguides has a cutoff frequency, which indicates what frequencies of electromagnetic waves To design appropriate unit cells, there are several parameters can propagate through it. The cutoff frequency can be calcu- that need to be considered. For this project, the main para- meters are: the effective refractive index n , dimension and lated using equation (7), for cases where W > h [13]. eff geometry of the unit cells. All of these parameters affect the 1 lens’ structure, its electromagnetic properties, and whether the (f ) = √ (7) c 10 2W µ lens is possible to manufacture with 3D printer. For example, if the walls of the unit cells are too thin, its structural integrity This equation also gives some design tips for a waveguide. can be compromised and may break easily. The 3D printer For example, to make sure that electromagnetic waves of that was used to manufacture the lens had a nozzle width of 5 GHz can propagate through the waveguide, a cutoff fre- 0.4 mm which means that our unit cells were required to have quency should be lower than that. a minimum thickness of 0.4 mm. The geometry that was chosen for the unit cells were cuboids containing cuboid holes. This was chosen to simplify F. S-parameters the manufacturing process since the lens was printed in two The performance of the antenna can be measured numeric- separate parts. A visual representation of the unit cells can be ally by analysing the S-parameters. The S-parameters represent found in Fig. 3. However, it is worth noting that the unit cells the gain of the signal between two ports and are denoted as can also take on several different shapes, such as a cuboid containing cylindrical air pockets etc., but the dimensions of Sij where j is the source port and i is the receiving port. If i and j are the same port, then the corresponding S-parameter the unit cells would have to be different. represents the gain of the reflected electromagnetic waves Simulations were made using CST Studio’s (reflection coefficient) [14]. If i and j are not the same port, Eigenmode Solver to calculate the dispersion diagrams of then Sij corresponds to the power transfer between port i respective unit cell in an infinite periodic structure. A table and port j. Worth noting is the fact that S-parameters can be of the unit cell’s dimensions can be seen in table I and the displayed in either a linear or decibel scale. dimensions of the holes can be seen in table II.

Table I III.DESIGN DIMENSIONSOFTHEUNITCELLS With the theory established, one can design a MFE lens Dimension Term Magnitude Length of unit cell d 8 mm together with a feeding structure. First, the unit cells will be Length of hole ` Varies designed. Secondly, they will be used to design the entire lens. Height of unit cell hU 5 mm Lastly, the feeding will be designed. Height of hole hA 5 mm L1. 3D PRINTING A LENS ANTENNA

Figure 4. Dispersion diagram of unit cell (r = 3) with different sizes of Figure 5. Dispersion diagram of unit cell (r = 4.4) with different sizes of holes. Each line that is dashed corresponds to a unit cell which has a hole holes. Each line that is dashed corresponds to a unit cell which has a hole with dimension `. Other dimensions of the unit cells are fixed and can be with dimension `. Other dimensions of the unit cells are fixed and can be seen in table I. seen in table I.

Table II DIMENSIONSOFTHEHOLES.

The hole’s side Magnitude `1 1.4 mm `2 2 mm `3 2.6 mm `4 3.2 mm `5 3.8 mm

B. Unit Cells Results Each simulation of the unit cells resulted in a dispersion diagram. The dispersion diagrams were further analysed in order to acquire the effective refractive index for each unit cell and its hole, using equation (6). The dispersion diagrams were compiled to one plot which can be seen in Fig. 4 for unit cells with r = 3 and in Fig. 5 for unit cells with r = 4. This in turn gives the effective refractive index which can be plotted as a function of the hole’s side, as seen in Fig. 6. Figure 6. A graphical representation of the refractive index of a unit cell as Since only 5 dispersion diagrams were analysed, the rest of a function of the hole’s size. Each curve corresponds to a unit cell with a the data to create a smooth graph were generated with spline specific r. Other dimensions of the unit cells are fixed and can be seen in interpolation using MATLAB. table I.

C. Design of Lens part of the lens was constructed with a material with r = 3. Since a 3D printer was going to be used to manufacture the This can be seen in Fig. 7, where the pink part corresponds lens, a flat MFE lens was designed since it takes less time to to a material with r = 4.4 and the green part corresponds to 3D print than a three dimensional one. r = 3. The reason for why these materials were used, was The final lens consisted of 34 unit cells in diameter, res- because those materials were available for us to use in the ulting in a diameter of 27.2 cm. The thickness of the lens lab. Since the unit cells are not infinitesimal the dimensions was 5.4 mm. With simulations, the lens was simulated with of each unit cell where chosen based on the refractive index different thicknesses, resulting in not so different results. The the centre of the cell should have according to 1. Note that lens’ thickness was therefore chosen depending on how long the lens was covered by PEC parallel plates over and under it would take to print, since a smaller lens takes less time it. than a thicker lens. The thickness also depended upon if the lens would be able to hold itself up, hence you cannot have a D. Design of Feeding lens that is to thin. The unit cells in the inner part of the lens Waveguides needed to be designed to be able to test the was constructed with a material with r = 4.4 and the outer performance of the lens. A basic cuboid waveguide made L1. 3D PRINTING A LENS ANTENNA

Table III DIMENSIONS WAVEGUIDES.

Dimension Magnitude [mm] Width W 50 Length L 80 Height H 5.6 Thickness t 0.3 Gap height gh 5 Focal point fp 25 Focal height fh 5

Figure 7. Top view of the simulated Maxwell fish eye lens. Pink represents a material with r = 4.4 and green represents a material with r = 3. Its diameter is 27.2 cm and the thickness is 5 mm.

Figure 9. Simulated field distribution over the lens. Figure 8. Model of feeding structure, top view. Its dimensions can be seen in table III. its structure. The S-parameters were also determined and can be seen in Fig. 10. out of a PEC material is sufficient to check if the lens is working as intended. Each waveguide is fed by a coaxial cable V. TEST SETUP that is placed inside the waveguide. The dimensions of the waveguides was determined through simulations in CST with A. The 3D Printed Lens the help of Equation (7) and the final dimensions can be seen The prototype lens was successfully manufactured using two in table III. A model of the feeding structure can be seen in different materials with different relative permittivities, r = Fig. 8. It is worth to mention the fact that in simulation, the 3 and r = 4.4 respectively. The diameter of the lens was thickness of the waveguide do not really matter. This is due to 27.2 mm and the thickness was 5.4 mm. As can be seen in the material being PEC, which reflects waves fully and do not Fig 11, the lens is consisting of two parts and the hole varies depend on the thickness. Because of this, the thickness of the with the radius. waveguides were only 0.3 mm in simulations. However, the physical waveguide needs to have a bit more thickness to it in B. Lens with 8 Waveguides order to support the structure, hence the physical waveguide’s thickness is 1 mm. The physical prototype was constructed The S-parameters were measured one at a time with a with thin copper plates that were cut to the dimensions in passive 50 Ω resistor loaded in each port that was not tested to table III simulate the load of a coaxial cable. The bottom was covered with a copper plate and the top was covered with a copper net. A copper barrier was also added around the lens. The setup IV. SIMULATION RESULTS for measuring the lens can be found in Fig. 12. The structure consisted of the MFE lens and eight wave- guides symmetrically placed around the lens. All the wave- VI.RESULTS guides where placed 8 mm from the circumference of the lens, The result of the measurements, found in Fig. 13, show since that is the same length as a unit cell. A plot of the field that the lens has a S7.1 varying between 0.18 and 0.6 in distribution over the lens can be seen in Fig. 9, together with the frequencies between 3 GHz and 6 GHz. The S1.1 of the L1. 3D PRINTING A LENS ANTENNA

Figure 12. Testing setup for the lens with 8 waveguides numbered 1 to 8. Waveguide 1 is the source port while waveguide 7 is the target port. Figure 10. S-parameters (linear scale) for lens with 8 waveguides. S1.1 is equal to the reflection coefficient, and S7.1 is equal to the power transfer to the target port.

Figure 13. Measured S-parameters of the prototype, the S1.1 corresponds to Figure 11. 3D printed MFE lens prototype. The lens’ diameter was 27.2 cm the reflection coefficient and the S7.1 is the power transfer to the target port. and the thickness was 5.4 mm. It consisted of two dielectric materials with r = 3 and r = 4.4 respectively. world results. First of all, a net of copper was used as the top layer of the construction instead of a copper plate used prototype is lowest between 5.1 GHz and 5.6 GHz. At the for the bottom layer. By using a copper net, it can act as same frequencies, the S7.1 is the largest, which means we a Faraday cage which blocks the waves from going out of have the greater power transfer between those frequencies. the lens construction. The net was installed since it gives people the ability to see the actual lens and the construction. VII.DISCUSSION With a copper plate, this would not have been possible. In order to minimise reflections along the edges of the lens, However, the copper net introduces some irregular airgaps the effective refractive index along the circumference had to between the actual lens and the net itself, which affects the be close to n = 1 which proved to be quite difficult to performance. By installing a copper plate which lies along the achieve without making the walls of the unit cells too thin lens’ upper surface instead, this problem could be fixed which to manufacture. The resulting unit cells at the edge had a would improve the performance. The net introduced unwanted wall thickness of 0.4 mm which made them very fragile. But reflections together with the waveguide ports. since the operational frequency of the lens is relatively low, at Secondly, the entire construction around the lens was hand- 5 GHz, small imperfections does not affect the performance of made out of copper plates. Although a template was used the lens. A solution to this problem would be to use a dielectric to construct the waveguides, it must be stated that their with a lower permittivity for the outermost layer which would dimensions and geometry is not as perfect as the waveguides result in thicker unit cells at the circumference. in the simulations. The same goes for the rest of the entire con- It is worth to mention sources of errors or things that struction. This problem could be used by improved methods contribute to the differences between the simulations and real of constructing metallic structures, such using machines and/or L1. 3D PRINTING A LENS ANTENNA letting a professional metallic worker construct the parts. [4] O. Quevedo-Teruel, R. C. Mitchell-Thomas, and Y. Hao, “Frequency Thirdly, the inner part of the lens needed to be rasped a dependence and passive drains in fish-eye lenses,” Phys. Rev. A, vol. 86, p. 053817, Nov 2012. [Online]. Available: https: little bit in order for it to fit inside the outer part of the lens. //link.aps.org/doi/10.1103/PhysRevA.86.053817 When the inner part was later installed, it introduced a slight [5] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of bending of the entire lens. Therefore, the geometry of the lens a negative index of refraction,” Science, vol. 292, no. 5514, pp. 77–79, 2001. [Online]. Available: https://science.sciencemag.org/content/292/ is likewise not as perfect as in the simulations. 5514/77 [6] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science, vol. 305, no. 5685, pp. 788–792, VIII.CONCLUSIONS 2004. [Online]. Available: https://science.sciencemag.org/content/305/ 5685/788 A Maxwell fish eye lens was realised. It was based on peri- [7] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, odic structures and was 3D printed using dielectric materials A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak with  = 3 and  = 4.4. Also, a feeding structure for the lens at microwave frequencies,” Science, vol. 314, no. 5801, pp. 977–980, r r 2006. [Online]. Available: https://science.sciencemag.org/content/314/ was realised. The result was a prototype lens that performed 5801/977 similarly to the simulations. The lens performed well even [8] J. C. Maxwell, “The scientific papers of : 1846- if the waveguides were not pointed towards each other. It 1862,” in The Scientific Papers of James Clerk Maxwell, W. D. Niven, M.A, and F.R.S., Eds. Toronto, Canada: CUP Archive, 1890, vol. 1,2, had acceptable power transfer and low reflections, especially ch. 17, pp. 284–285. between 5 GHz and 5.5 GHz. The results also show that the [9] H. Young, R. Freedman, A. Ford, F. Sears, and M. Zemansky, University power transfer to the other ports is higher it the measurements Physics with Modern Physics. Boston, MA: Person, vol. 14, pp. 1062, 1081. compared to the simulations. This means that the lens does [10] S. Zhang, R. K. Arya, S. Pandey, Y. Vardaxoglou, W. Whittow, and not focus the electromagnetic waves as well as it does in the R. Mittra, “3D-printed planar graded index lenses,” IET simulations and some of the power is scattered into other ports. ANTENNAS & PROPAGATION, vol. 10, no. 13, SI, pp. 1411–1419, OCT 22 2016. Differences came mainly from unwanted reflections from [11] O. Quevedo-Teruel, M. Ebrahimpouri, and M. Ng Mou Kehn, “Ultraw- both the copper net and waveguide ports, losses in the ma- ideband metasurface lenses based on off-shifted opposite layers,” IEEE terials which were not taken into account in the simulations Antennas and Wireless Propagation Letters, vol. 15, pp. 484–487, Dec 2016. and lastly the geometry of every hand-made component which [12] J. Miao, “Ka-band 2D luneburg lens design with glide-symmetric were not as perfect as in the simulations. These things can be metasurface,” p. 10, 2017. improved in the future to realise even better lenses. [13] D. Cheng, Field and wave electromagnetics, ser. The Addison-Wesley series in electrical engineering. Addison-Wesley Publishing Company, The final prototype of the lens and its construction worked, 1989, pp. 355–356, 375–376. as can been seen in the results. The VNA used to measure the [14] K. Kurokawa, “Power waves and the scattering matrix,” IEEE Transac- S-parameters was calibrated for a frequency band of 3 GHz- tions on Microwave Theory and Techniques, vol. 13, no. 2, pp. 194–202, March 1965. 6 GHz, hence why there is no measured results for other frequencies. The power transfer in the measurement were significantly lower compared to the simulations and the transfer to the other ports also appears to be higher.

ACKNOWLEDGEMENTS We would like to give a special thanks to our great super- visors: Oscar Quevedo Teruel, Oskar Dahlberg and Fatemeh Ghasemifard. They have all been supportive and pedagogic throughout the whole project. Without them, we would not have been able to understand the theory, the applications and the usefulness of this topic. Their expertise and their passion to educate and help have been inspiring and this project would not have been successful and interesting as it has been without them. We thank them greatly.

REFERENCES

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