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Abstract
This paper presents a novel two-dimensional (2-D) partial Maxwell fish-eye (PMFE) lens with the capability of wide-angle beam scanning inspired by the Gutman lens and Eaton lens, which is obtained by cutting a part from the 2-D Maxwell fish-eye (MFE) lens along a straight line. In terms of the refractive index profile, the MFE lens is similar to the Gutman lens near the center and the Eaton lens near the edge, respectively. We demonstrate the potential of the PMFE lens in wide-angle beam scanning based on its Gutman-like focusing and Eaton-like rotating characteristics corresponding to different feed points. As an example, a fully metallic PMFE lens antenna in the Ka-band composed of a bed of nails and a series of linearly arranged waveguide feeders is designed and experimentally verified. The measured results reveal wide-angle scanning ranges, especially about ±90° at 36 GHz, low reflections and low mutual couplings. The frequency scanning due to the dispersion of the lens is also discussed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Gradient index (GRIN) lenses [1] such as Luneburg lens [2], Maxwell fish-eye (MFE) lens [3], Gutman lens [4] and Eaton lens [5] have been applied in many microwave and optical devices. Considering the difficulty of the realization of ideal gradient index profile, the researchers have tried various means such as multiple dielectric structure [6–8], parallel plate waveguide (PPW) partially filled with dielectric [9,10], PPW with gradual height [11] and curved surface [12]. Moreover, due to the flexibility of design, the artificial dielectrics or metamaterials based on the subwavelength periodic structures are attracting increasing attention in the designs of GRIN lenses such as Luneburg lenses [13–22], MFE lenses [23,24], Gutman lenses [25–27], Eaton lenses [28–30] and bifunctional lenses [31,32]. Because of the characteristics of the GRIN lens in light focusing and rotating, it has been widely applied in the microwave antennas and functional devices for beam scanning or beam control.
As known, the Luneburg lens is popular in the antenna design as it can convert a point source to a plane wave. In principle, the Luneburg lens can support the omnidirectional beam scanning. While in practice, caused by the blocking of the necessary feeding structures such as waveguides, horns or other antennas, the large-angle scanning performances of the Luneburg lens antennas are faced with the problems of gain drop and mutual coupling. It should be noted that there have been some excellent works that analyzed the wide-angle-scanning performances of the Luneburg lens antennas. In [11] the reported Luneburg lens antenna provides a beam coverage of ±80° with a scan loss of 3 dB. In [7] the Luneburg lens antenna which is fed by multiple planar log periodic dipole antennas (PLPDAs) can cover a scanning range of ±85° with a scan loss of 3 dB, showing a mutual coupling level near −10 dB corresponding to the ports for large-angle beams.
In addition, some other types of GRIN lenses have been applied in antenna design to get an improvement in several aspects. For example, the volume reduction is significant. As an effective attempt, the half-MFE (HMFE) lenses have been used in the multi-beam antennas [24]. Compared with that of the Luneburg lens antenna, however, the scanning range of HMFE lens antenna is smaller because of the plane wave deterioration as the scanning angle increases, which can be attributed to the refractive index mismatch in the radiation aperture. Another example is the Gutman lens which can be regarded as a modified Luneburg lens with a focal arc inside the lens. It has been used as a kind of beam-scanning device [25–27]. In [25], a fully metallic Gutman lens based on artificial dielectrics is designed and applied in a multi-beam antenna which is fed by a series of single ridge waveguides along the focal arc. The lens volume is obviously reduced compared with the corresponding Luneburg lens.
Meanwhile, the arrangement of feeders related to application scenarios is also concerned by many researchers. It should be noted that the feeders of the above lens antennas are all placed along the focal arc. Although the focal arc can meet most requirements of application scenarios, the lens with feed points along a straight line or a plane has attracted increasing attention because of its convenience in the integration with planar circuit. On the basis of transformation-optics (TO) theory, the feed points of Luneburg lens can be placed along a straight line [33] or on a plane [14], though the scanning ranges are limited. In addition, without TO method, a partial Gutman lens with a feeder plane has been researched on its beam-scanning performances [26].
The partial Gutman lens brings a simplified solution of lens multi-beam antenna with reduced volume and feed points placed in a plane or a straight line. However, the reported results show a limited beam scanning range, which is needed to be further improved. As known, the Eaton lens has been found with the function of rotating the beam by an expected angle such as 90° [8,28–30], but it requires very high refractive index and cannot support the small-angle beam. To solve the problem, a lens with linearly arranged feed points which can support wider beam scanning range is worth studying.
In this paper, inspired by the focusing characteristic of Gutman lens and the rotating characteristic of Eaton lens along with the similarities between them and the MFE lens in terms of refractive index profile, a novel application of the MFE lens in the wide-angle beam scanning is developed. After the comparison of the proposed partial MFE (PMFE) lens and the reference Gutman and Eaton lenses, the potential of the PMFE lens in wide angle beam scanning is discussed and further verified by experiment. Specifically, a PMFE lens which is obtained by cutting the full lens in an appropriate section can support a wide-angle beam scanning close to ±90° with low scan loss. The feeders can be placed along a straight line and maintain relatively small aberrations for the offset beams. A Ka-band PMFE lens multi-beam antenna is designed, manufactured and experimentally verified for the first time. The PMFE lens is realized by a bed of nails, which is a kind of fully metallic artificial dielectric suitable for millimeter-wave (mmW) applications [13,34–36]. In addition, the frequency-dependent radiation characteristics due to the lens dispersion are also discussed.
This paper is organized as follows. The inspiration for the PMFE lens is discussed in Section 2. The full-wave analysis of the fully metallic PMFE lens is introduced in Section 3. The experimental verification of a multi-beam antenna example is presented in Section 4. The conclusion is given in Section 5.
2. PMFE lens inspired by the Gutman lens and Eaton lens
2.1 Comparative analysis of MFE’s profile with Gutman’s profile and Eaton’s profile
The modification of MFE lens is inspired by the comparative study with the Gutman lens and Eaton lens on the refractive index profile. The MFE’s profile, Gutman’s profile and Eaton’s profile can be expressed as Eqs. (1), (2) and (3), respectively, where n is the refractive index, R is the radius of the lens, r is the position relative to the center and f is the focal distance of Gutman lens.
Here, in order to make the maximum refractive index of the reference Gutman lens equal that of the MFE lens, f is determined to be R/$\sqrt 3 $. Besides, the reference Eaton lens is the version of 90° wave-rotating. The refractive index profile curves of the MFE, Gutman and Eaton lenses are plotted in Fig. 1(a). As shown, although the refractive indices of the reference Gutman lens are higher than those of the MFE lens in almost all positions, the curves are at the same level in terms of the value. For the Eaton lens, the refractive indices corresponding to the positions close to the center are much higher than those of the MFE lens, but the index curve crosses that of the MFE lens near the position r = 0.25R and the two curves tend to be closer as r increases.
Fig. 1. (a) Refractive index profiles of MFE, Gutman and Eaton lenses. (b) First (up) and second (down) derivatives of n with respect to r/R for MFE, Gutman and Eaton lenses.
In order to further analyze the similarities and differences of the three kinds of refractive index profiles, the first and second derivatives of n with respect to r/R are calculated and plotted in Fig. 1(b). The results of the first derivative indicate the different gradients of the three refractive index profiles. Obviously, for all of the MFE, Gutman and Eaton lenses, the first derivatives less than zero reveal that the indices decrease with the increase of r. However, it should be noted that the refractive indices of Gutman lens and Eaton lens change more sharply than that of the MFE lens near the edge and the center, respectively. In addition, the results of the second derivative show the variations of the refractive index gradient. As seen, when r/R < 0.6, the second derivative of the MFE lens is less than zero like the Gutman lens, while when r/R > 0.6, the second derivative of the MFE lens is higher than zero like the Eaton lens.
The analysis of the refractive index profiles for the three lenses shows that the MFE lens is similar to the Gutman lens in the area near the center and similar to the Eaton lens in the area near the edge. Thus, the MFE lens can be approximately considered to be composed of two parts, Gutman-like part and Eaton-like part, as marked in Fig. 1(b). As shown in Fig. 2, it can be inferred that the MFE lens may perform like both of the Gutman lens and Eaton lens corresponding to different feed points, which is worth proving by further investigation.
2.2 Gutman-like characteristic
The Gutman lens is a perfect focusing lens which can focus a parallel light on a point in the position of r = f, which is shown by the ray-tracing results in Fig. 2. In order to evaluate the focusing capability of MFE lens, the light propagation in the MFE lens illuminated by a parallel light is analyzed. The results in Fig. 3 show an obvious focal spot area in the MFE lens, although there is a little defocusing. It can be observed that the focusing relies more on the refraction near the center area, so the MFE lens performs like the Gutman lens although there is a little error. The ray-tracing analysis is operated with the help of COMSOL Multiphysics.
Hence, if the MFE lens is cut and fed by a point source near the focal spot, the output light will be an approximate parallel light which is beneficial to antenna design. The MFE lens which is cut by a straight line through the focal spot can be called a PMFE lens, as shown in Fig. 4, where point O is the center of the lens and points E and A determine the cutting straight line. In order to optimize its performance in the antenna design, an approximate focal point A need to be determined. When the PMFE lens is fed at the point A, the variation of lOA will influence the parallelism of output rays. In Fig. 4, the ray-tracing results corresponding to four different feed points with lOA = 0.4R, 0.45R, 0.5R and 0.55R reveal that when lOA = 0.45R, a satisfying parallel light can be observed.
2.3 Eaton-like characteristic
Then we evaluate the Eaton-like beam-rotating capability of the MFE lens illuminated by a parallel light beam. The ray-tracing results are shown in Fig. 5. In order to explain the similarity between the MFE lens and Eaton lens and the difference between the MFE lens and Gutman lens simultaneously, the reference Gutman lens as well as the reference Eaton lens mentioned in Section 2.1 is simulated. Here, as a typical example, the parallel light beam with a width of win = 0.1R and a feeding position of sin = 0.7R is selected as the excitation. As seen, the MFE lens can rotate the incident light by almost 90° although there are some aberrations. In contrast, the beam rotation angle of the reference Gutman lens is obviously smaller.
Furthermore, based on the determined cutting line in Section 2.2 corresponding to lOA = 0.45R, we evaluate the beam-rotating capability of the PMFE lens when the light is input near the edge point E. Considering that the actual feeders such as waveguide generally do not provide a parallel light excitation, a light beam with a divergence angle of 20° is applied as the input, as well as a parallel light. In the simulation, the distance between the feed point P and the center point A, lPA, is selected to be 0.7R. The width of the parallel light beam is 0.1R. The results in Fig. 6 show that the PMFE lens can rotate the incident light by almost 90° with some aberrations. It should be noted that the output light corresponding to the beam with a divergence angle shows the better parallelism than that corresponding to the parallel light source obviously, which is beneficial to the actual antenna design.
The above results show that although PMFE lens is not a perfect focusing and rotating lens, it can simultaneously perform like both the Gutman lens and Eaton lens within an acceptable error level, showing the potential for wide-angle beam scanning.
3. Full-wave analysis of fully metallic PMFE lens
In order to verify the performance of the PMFE lens in actual design of mmW devices, a Ka-band fully metallic PMFE lens is analyzed by full-wave simulation with the help of ANSYS HFSS. The required equivalent refractive indices are realized based on the periodic structures which consist of a bed of nails which can guide TM surface waves. In this work, a metallic plate loaded with a square nail is selected as the unit cell as shown in Fig. 7(a). As the theory in [13], when the period $p \ll {\lambda _0}$ and the width of nail $a \le {p / 2}$, for the TM surface wave, the height of nail h can be expressed as Eq. (4), where n is the corresponding refractive index and k0 is the propagation constant.
By substituting Eq. (1) into Eq. (4), the relationship between h and r for the MFE lens can be obtained as Eq. (5).
Fig. 7. (a) Unit cell of bed of nails. (b) Profile of h versus (r/R) at 36 GHz. (c) Schematic diagram of the PMFE lens with a radius of 25 mm.
In this work, a and p are determined to be 0.5 mm and 1.2 mm respectively and k0 corresponds to the frequency of 36 GHz. Then the profile of h versus (r/R) at 36 GHz can be calculated and plotted along with the MFE’s profile in Fig. 7(b). The schematic diagram of the MFE lens with a radius R = 25 mm is given in Fig. 7(c). According to the analysis in Section 2.2, the PMFE lens can be obtained by cutting the full lens along the line EA as shown in Fig. 7(c), where lOA = 0.45R = 11.25 mm.
In addition, the dispersion of this lens should be noticed. Equation (6) shows that as the profile of h has been determined according to 36 GHz, the refractive index profile for the other frequencies nother is related to the corresponding propagation constants kother.
According to Eq. (6), the curves of nother versus (r/R) at 32, 33, 34, 35, 37 and 38 GHz are all given in Fig. 8. As shown, the dispersion characteristics are obvious. In this work, the frequency scanning beams caused by the dispersion will be also discussed.
The structure of the lens with a WR-28 rectangular waveguide feeder is given in Fig. 9. The beam scanning can be obtained by moving the feeder from the center with different shifts. To provide a transition between the lens and the free space, an extended ground plane with the width e = 10 mm is added around the lens. It should be noted that the distance between the waveguide feeder and the ground plane t obviously affects the impedance matching between the feeder and the lens. According to the simulated results of reflection coefficients varying with t in Fig. 10(a), the t is determined to be 1 mm to meet the desired frequency band from 33 to 36 GHz.
Fig. 10. (a) Reflection coefficients changing with t. (b) Reflection coefficient curves corresponding to different feed positions
To evaluate the beam scanning capability, we choose five different feeding positions with shifts of s = 0 mm, 4.5 mm, 9 mm, 13.5 mm and 18 mm as examples. At first, the reflection coefficient curves corresponding to different feed positions are observed. As shown in Fig. 10(b), the reflection coefficients below −15 dB are obtained at most frequencies in Ka-band, indicating the good impedance matching. Especially in the main operating band from 33 to 36 GHz, the reflection coefficients are all lower than −18.5 dB. To further study the wave propagation in the PMFE lens, the electric fields are simulated at a series of frequencies. Figure 11(a) gives the distributions of the three electric field components, Ex, Ey and Ez, in x-y plane at 36 GHz, where s = 0mm. The results show an obvious propagation mode of TM in the lens and indicate that the main component for radiation is Ez. Thus, the z-component is focused on in the following analysis. As shown in Fig. 11(b), when s = 0 mm, there is no beam rotation in the propagation directions at all the frequencies. While, when s = 9 mm and 18 mm, the frequency scanning can be observed clearly as well as the beam rotation phenomena. Especially, the rotation angles at 35 and 36 GHz are close to and even more than 90° when the feeder moves to the position of s = 18 mm. These full-wave simulated results agree well with the ray-tracing analysis and show the application potential of the PMFE lens in wide-angle beam scanning. It should be pointed out that as discussed in [37], because of the asymmetry structure of the surface wave lens and the finite extended ground plane, the beams show an upward deflection near 20° in the elevation plane, which can be observed by the electric field distributions in the x-z plane in Fig. 11(c). The beam deflection also occurs in the other surface wave antennas such as the works in [38] and [39].
Fig. 11. (a) Distributions of Ex, Ey and Ez in x-y plane at 36 GHz, where s = 0 mm. (b) Distributions of Ez in x-y plane corresponding to different feeding positions and different frequencies. (c) Ez distributions in x-z plane at different frequencies.
As a modification, another version is designed to eliminate the upward beam deflection by adding a cover to version 1. The side view of the version 2 is shown in Fig. 12. It should be noted that a flare whose size is determined by l and q is added around the edge of the lens to maintain the impedance matching with the air. Here, l = 10 mm, q = 11.556 mm, and the spacing of the two plates, g = t + t + 3.556 mm = 5.556 mm. Because of the addition of the cover plate, the refractive indices increase slightly. In Fig. 13, compared with those of version 1, the refractive index profiles of version 2 are given based on the eigen mode simulation with the help of HFSS. As seen, the lens formed by a bed of nails with a cover plate in version 2 can also approximately meet the MFE’s profile with some dispersion.
The simulated electric fields in Fig. 14 show that the lens of version 2 can also support the wide-angle beam scanning by the movement of feed position. Meanwhile, the frequency scanning can also be observed. As seen from the results at 36 GHz, TM mode is still the main propagation mode. Especially, differing from the version 1, the version 2 can limit the radiation wave to 0° elevation angle as shown in Fig. 14(b).
Fig. 14. (a) Distributions of Ez in x-y plane corresponding to different feeding positions and different frequencies, and (b) Ez distributions in x-z plane at different frequencies for version 2.
The results of full-wave simulation verify the conclusion of ray-tracing analysis, showing that the PMFE lens can be used in the design of wide-angle scanning multi-beam antenna.
4. Experimental realization of multi-beam PMFE lens antenna
To experimentally verify the performance of the PMFE in wide-angle beam scanning multi-beam antenna, we take the lens of version 2 in Section 3 as an example for further study. To make a multi-beam antenna, we add six waveguide feeders along the cutting line of the PMFE lens with a distance of d = 7 mm, as shown in Fig. 15(a). Considering the matching of the feeder size and the lens size, the width of the waveguides, w, is determined to be 6 mm which can meet the operating frequencies. The prototype of the fully metallic PMFE lens multi-beam antenna has been manufactured by machining technology and the photos are shown in Fig. 15(b). For the convenience of the experiment, the transitions between the waveguide with the cross section of 6 mm × 3.556 mm to the WR28 waveguide are employed with the necessary bends, which can hardly affect the performances of the antenna.
The S-parameters were measured by the vector network analyzer and the results are shown in Fig. 16. The reflection coefficients in Fig. 16(a) are all below −12.5 dB from 33 to 36 GHz. The coincidence of the results corresponding to the symmetric ports (port 1 vs port 6, port 2 vs port 5 and port 3 vs port 4) shows the good fabrication precision. The low reflection coefficients show that the feed efficiencies of all ports are better than 94%. The mutual couplings are plotted in Fig. 16(b). As shown, the couplings below −20 dB are obtained for all the ports in the operating band, which reveal the good independence of the channels corresponding to the multiple beams. The measured results are in agreement with the simulation.
Figure 17 gives the measured normalized radiation patterns at 33, 34, 35 and 36 GHz in the azimuth plane when the elevation angle is 0°. As seen, the maximum beam scanning angles at the four frequencies are respectively 60°, 65°, 73.5° and 85°. The low cross polarization levels with a typical value of −20 dB show the obvious linear polarization characteristics. Furthermore, Fig. 18 plots the measured elevation radiation patterns for port 3 at 33, 34, 35 and 36 GHz, showing the beams without upward deflection in the elevation plane.
Fig. 17. Measured and simulated azimuth radiation patterns at (a) 33, (b) 34, (c) 35 and (d) 36 GHz, when elevation angle is 0°.
Fig. 18. Measured and simulated elevation radiation patterns, when the antenna is fed by port 3 and azimuth angle is 15°.
The peak gains and radiation efficiencies for port 1, port 2 and port 3 are plotted in Fig. 19(a), showing low scan loss and high radiation efficiency. As seen, the scan losses are lower than 3 dB. After taking into account the conductor loss of aluminum and the typical surface roughness of 800 nm, the simulated radiation efficiencies are higher than 84.6%. Besides, as given in Fig. 19(b), the frequency scanning of the azimuthal beams can be observed. It should be noted that at 36 GHz, the beam corresponding to port 1 shows a realized gain above 12.5 dBi at the scanning angle of 90°, which demonstrate the acceptable wide-angle scanning performance. The measured and simulated radiation results agree well.
Fig. 19. (a) Measured and simulated gains, simulated radiation efficiencies and (b) measured azimuthal frequency scanning.
In order to further evaluate the contribution of this work, a comparison between the proposed PMFE lens and several reported GRIN lenses for beam scanning is given in Table 1. As seen, although the scan loss of 3 dB cannot be avoided, the PMFE lens antenna supports a beam scanning coverage of ±90°, which is an obvious improvement compared with the previous works. Meanwhile, the prototype realized by fully metallic structure yields high radiation efficiency better than 84% and good port isolation higher than 20 dB.
Table 1. Comparison With Reported GRIN Lenses for Beam Scanning
5. Conclusion
A novel PMFE lens and its application in fully metallic mmW wide-angle scanning multi-beam antenna have been proposed. The inspirations from the Gutman lens and Eaton lens have been presented and the Gutman-like and Eaton-like characteristics have been analyzed by ray-tracing and full-wave simulation. A prototype in Ka-band consisting of a bed of nails and six linearly arranged waveguide feeders has been experimentally realized. In the band from 33 to 36 GHz, the antenna can achieve wide-angle beam scanning ranges. Especially at 36 GHz, the beam can scan to ±90° with a gain above 12 dBi. The reflection coefficients and mutual couplings of the prototype have been verified to be lower than −12 dB and −20 dB, respectively, in the operating band. The radiation efficiencies of the antennas are higher than 84.6%. The results indicate that the PMFE lens can be used in the multi-beam antenna with a wide-angle scanning range.
Funding
National Natural Science Foundation of China (61901040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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