Introduction
Circularly polarized antennas can accept arbitrary polarization of the coming wave and have a wide range of applications in wireless communications [Reference Gao, Luo and Zhu1], such as global positioning system (GPS), radio frequency identification (RFID), satellite communications, wireless local area networks (WLAN), etc. Multi-band operation has become a hot topic in the current research of the new electronic devices, which can work flexibly in multiple communication states. Therefore, various types of multi-band antennas have emerged and how to achieve better communication applications is a popular research question in current multi-band antenna design [Reference Verma and Kumar2–Reference Hoang and Park7 ].
Various types of multi-band antennas have been designed in wireless communication systems. Sharma et al. [Reference Sharma, Das, Gupta and Gangwar8] presented a quad-band circularly polarized dielectric resonator antenna for GPS/CNSS/WLAN/WiMAX applications. In [Reference Ta, Park and Ziolkowski9], a crossed dipole on a high impedance surface was proposed for triple-band WLAN applications. Slot antennas are more often chosen in multi-band antenna design. However, slot antennas are more complicated to implement, such as comb slot [Reference Abdalrazik, Gomaa and Kishk10], multi-branch slot [Reference Baek and Hwang5], non-concentric circles slot and so on. The multi-band patch antenna is another option for researchers [Reference Lu and Huang11–Reference Qian, Chen, Xiang and Chu14 ].
Characteristic mode theory (CMT) was first proposed by Garbacz and Turpin as a set of modes diagonalizing a scattering matrix in 1965 [Reference Garbacz15] and refined by Harrington and Mautz for perfectly electrically conducting (PEC) object in 1971 [Reference Harrington, Mautz and Chang16, Reference Chang and Harrington17 ] that the current distribution can be considered as a linear superposition of the modal currents. CMT was developed in dielectric and magnetic bodies [Reference Chen and Wang18, Reference Harrington and Mautz19 ], and then generalized for antenna design [Reference Harrington and Mautz20].
The patch antennas are designed for a wide range of applications using characteristic mode analysis (CMA), such as multi-mode MIMO antenna [Reference Sharma, Das, Gupta and Gangwar21] and orbital angular momentum (OAM) [Reference Parvin, Ahmadi-Shokouh and Dashti22]. The circular polarization patch antennas are achieved by making two characteristic modes orthogonal and the character angle difference is 90∘ and the broadband characteristic is realized by the resonate frequency points of multiple modes [Reference Li, Li, Zhu, Zhang and Liu23–Reference Zeng, Zhang, Lin and Guan26 ]. In [Reference Xu, Nan, Liu and Agostino27], the multi-band characteristics were achieved by five modes separations, but only two bands were obtained. Therefore, using CMA to design multi-band patch antennas is worth studying.
This paper presents a multi-band circularly polarized patch antenna for WLAN and WiMAX applications based on the characteristic mode theory. In section “Multi-band circularly polarized antenna,” the antenna design process is completed in the order of characteristic mode analysis, feed design, and the influence of important geometric parameters. The simulated and measured results of the antenna are presented and discussed in section “Results of simulation and measurement,” and the full-wave simulation and measured results verify the CMT. Conclusion is presented in last section.
Multi-band circularly polarized antenna
Characteristic mode theory
The total current J on the antenna surface can be written as the superposition of multiple characteristic modal currents as (1) according to CMT [Reference Garbacz15]:
\begin{equation}
J = \sum_{1}^{N} \alpha_{n}J_{n},
\end{equation}where Jn is the nth characteristic current and αn is a complex modal weighting coefficient (MWC), which can be written as (2):
\begin{equation}
MWC= \alpha_{n}= \frac{\left\langle J_{n},E^{i}\right\rangle}{\left(1+j\lambda_{n}\right)},
\end{equation}λn is the real eigenvalue solved by the eigenvalue equation and E i is the incident electric field [Reference Deng28].
MS in (3) is the modal significance, defined as the normalized current amplitude. MS is an inherent property of the mode and is independent of the external source. Characteristic angle (CA) is another important parameter, defined as shown in (4). CA characterizes the phase difference between the characteristic current Jn and the associated electric field En. The mode is considered significant when MS is close to 1 or CA is close to 180∘:
\begin{equation}
MS_{n}=\left| \frac{1}{1+j\lambda_{n}}\right|,
\end{equation}
\begin{equation}
CA_{n}=180^\circ-\tan^{-1}\lambda_{n}.
\end{equation}The modes used to achieve the circular polarization should satisfy the following conditions:
(1) MS
The MS curves of the two modes intersect means that the MS values are equal.
(2) CA
CA difference between the two modes is around 90∘.
(3) Characteristic Current
The current of the two modes should be excited simultaneously and the direction should be perpendicular.
(4) Directivity
The directivity of the two modal radiation patterns is consistent.
CMA of the multi-band antenna without excitation
Although the characteristic mode analysis of antennas with substrate has been studied [Reference Chen and Wang18], the influence of substrate on the mode of the proposed antenna is limited. In this section, FEKO is utilized to do the CMA of the radiating patch in Fig. 1, which is derived from a combination of multiple rectangles. The first 15 modes of the radiating patch without feeding structure are computed. The final structure is obtained by optimizing the parameters of the antenna with substrate. Figure 2 shows the MS curves, it can be seen that a small number of modes and simple distribution in the low frequency band and a large number of modes and complex distribution in the high frequency band. It is worth noting that the significance curves of different modes have intersection points, such as modes 1 and 3 intersects at 2.5 GHz, modes 3 and 4 intersects at 4.5 GHz, modes 4 and 6 intersects at 5.3 GHz and so on. According to the CMT, modes are called significant modes when MS 
$gt$ 0.707, so the main significant modes around 7 GHz are modes 4, 8, and 10.
Figure 1. Antenna structure without feeding network and metal ground in FEKO.
Figure 2. Modal significance curves without feeding structure.
Besides, the circular polarization also requires attention to the difference of the CA between two orthogonal modes with the same current amplitude. As shown in Fig. 3, the CA phase difference of the intersecting modes are 99∘, 81∘, 78∘, and 96∘ at 2.5, 3.5, 5.3, and 6.5 GHz where the MS curves intersect, respectively. It can be observed that the CA curves change slowly around 6.5 GHz, which indicates that there is potential to form a broadband circular polarization.
Figure 3. CA curves without feeding structure.
The characteristic currents and the radiation patterns of each mode at the corresponding frequencies are shown in Figs. 4 and 5. Fig. 4(a, b) shows the current distribution of modes 1 and 3 at 2.5 GHz, both with vertical directions, and in turn (c, d) show the current distribution of modes 3 and 4 at 3.5 GHz, (e, f) show the current distribution of modes 4 and 6 at 5.3 GHz, (g, h) show the current distribution of mode 4 and mode 8 at 6.5 GHz, (i, j) show the current distribution of mode 4 and mode 10 at 6.5 GHz. As shown in Fig. 5,the radiation direction of each orthogonal modes at 3.5, 5.3, and 6.5 GHz is consistent.
Figure 4. Characteristic currents of the antenna without feeding structure: (a) mode 1 at 2.5 GHz, (b) mode 3 at 2.5 GHz, (c) mode 3 at 3.5 GHz, (d) mode 4 at 3.5 GHz, (e) mode 4 at 5.3 GHz, (f) mode 6 at 5.3 GHz, (g) mode 4 at 6.5 GHz, (h) mode 8 at 6.5 GHz, (i) mode 4 at 6.5 GHz, and (j) mode 10 at 6.5 GHz.
Figure 5. Radiation patterns of the antenna without feeding structure: (a) mode 1 at 2.5 GHz, (b) mode 3 at 2.5 GHz, (c) mode 3 at 3.5 GHz, (d) mode 4 at 3.5 GHz, (e) mode 4 at 5.3 GHz, (f) mode 6 at 5.3 GHz, (g) mode 4 at 6.5 GHz, (h) mode 8 at 6.5 GHz, (i) mode 4 at 6.5 GHz, and (j) mode 10 at 6.5 GHz.
It is known that the two orthogonal modes have equal MS. When their CA difference is around 90∘, circular polarization can be achieved. All of the characteristic modes have orthogonal current direction and the consistent radiation directivity at the same frequency, respectively. The above performances meet the four conditions for circular polarization under the CMA, so the antenna structure can theoretically achieve multi-band circular polarization performance.
CMA of the multi-band antenna with feeding structure
The configuration of the proposed antenna is built in the full wave simulation software HFSS as shown in Fig. 6. Rogers 5880 is chosen as the dielectric substrate with a thickness of 1.575 mm and a relative permittivity of 2.2. The upper layer of the substrate is the radiating part of the antenna, and the apex of the largest square of the radiating part is used as the feeding position. Microstrip structure is used for antenna feeding and the microstrip transition is used to adjust the impedance matching. A metal ground is placed at 33.5 mm below the antenna as a reflector. The design of antennas with substrate will have a certain impact on the characteristic mode analysis [Reference Harrington and Mautz20], so the optimized geometric parameters are shown in Fig. 6.
Figure 6. Configuration of the structure.(a) Antenna, (b) top view, (c) bottom view. w1 = 18 mm, w2 = 8.1 mm, w3 = 9.3 mm, w4 = 5.4 mm, w5 = 3.8 mm, w6 = 6 mm, l1 = 52 mm, l2 = 19 mm.
Whether the characteristic mode can be excited as desired depends on the selection of the feeding position. In Fig. 7, four points are selected on the radiation structure for performance comparison through microstrip feeding. From the performance of impedance matching and axial ratio in Figs. 8 and 9, feeding-1 can excite all the modes in second section), in which the impedance bandwidth covers 2.45, 3.5, 5.8, and 6.5 GHz, and the resonance frequencies of AR are at 2.45, 3.5, 5.3, 5.8, and 6.6 GHz. While the impedance bandwidth of the other three feeding points cannot cover all the desired frequency bands. Therefore, the feeding-1 point is selected as the feeding position.
Figure 7. Configuration of the structure for feeding position.
Figure 8. The S-parameter for different feeding positions.
Figure 9. The AR for different feeding positions.
Figures 10 and 11 show the characteristic currents and the radiation patterns of the radiation part when feeding at the feeding-1 position, in which characteristic modes have vertical current direction and the consistent radiation directivity at the resonant frequency. Figures 12 and 13 show the MS and CA curves, in which the feeding structure has a significant impact on modes 5 and 8, and the MS curve intersection frequency point of modes 4 and 6 has moved from 5.3 to 4.9 GHz, but the circular polarization performance is basically not affected, so that satisfying the circular polarization condition. The changes in CMA before and after adding the feed structure are shown in Table 1.
Figure 10. Characteristic currents of the antenna with feeding structure: (a) mode 1 at 2.5 GHz, (b) mode 3 at 2.5 GHz, (c) mode 3 at 3.5 GHz, (d) mode 4 at 3.5 GHz, (e) mode 4 at 4.9 GHz, (f) mode 6 at 4.9 GHz, (g) mode 4 at 5.8 GHz, (h) mode 8 at 5.8 GHz, (i) mode 4 at 6.5 GHz, and (j) mode 10 at 6.5 GHz.
Figure 11. Radiation patterns of the antenna with feeding structure: (a) mode 1 at 2.5 GHz, (b) mode 3 at 2.5 GHz, (c) mode 3 at 3.5 GHz, (d) mode 4 at 3.5 GHz, (e) mode 4 at 4.9 GHz, (f) mode 6 at 1.9 GHz, (g) mode 4 at 5.8 GHz, (h) mode 8 at 5.8 GHz, (i) mode 4 at 6.5 GHz, and (j) mode 10 at 6.5 GHz.
Figure 12. Modal significance curves with feeding structure.
Figure 13. CA curves with feeding structure.
Table 1. Comparison between with and without feeding structure in CMA
Effect of antenna geometric parameters
The effect of the geometric parameters on antenna performance is introduced in more detail in this section. Figure 14(a) shows the effect of the width l2 of the bottom metal slab at radiation layer on 
$| S_{11} |$. Notably, as l2 increases, the impedance matching at low band shifts to better, but first widens and then deteriorates at high band. So l2 of 19 mm is the most suitable, which not only reduces the downward radiation of the microstrip feeding line, but also reduces the influence on the field distribution and impedance of the radiation structure. In Fig. 14(b), the AR is mainly affected by w5 at low band. As w5 increases, the curve at 2.5 GHz shifts to low frequency, and the bandwidth at 3.5 GHz increases. In Fig. 14(c), w3 has the opposite effect on the axial ratio at 6 and 7.5 GHz. So compromise w5 = 3.8 mm, w3 = 9.3 mm.
Figure 14. Effect of parameters on performance. (a) The effect of l2 on S-parameter. (b) The effect of w5 on AR. (c) The effect of w3 on AR. (d) The effect of h on AR. (e) The effect of h on gain. (f) The effect of h on S-parameter.
h is an important value because it satisfies the reflections of multiple bands. Due to the higher frequency, the shorter the wavelength, making it more sensitive to the effect on high band. From Fig. 14(d)–(f), it can be seen that a decrease in h is beneficial for improving the performance of axial ratio and cross polarization level. However, considering the possible machining errors, 
$| S_{11} |$ may be higher than −10 dB at 3.5 GHz, so h = 33.5 mm is chosen.
Results of simulation and measurement
In order to verify the proposed multi-band circularly polarized performance, the antenna is fabricated and measured as shown in Fig. 15. Figure 16 shows the S-parameter simulated and measured results of the multi-band circularly polarized antenna. From the simulation results, it can be seen that there are two resonance points around 2.6 and 4 GHz, which are known from second section to be the series resonance points caused by modes 1 and 4, respectively. It is known from the CMA that the number of resonant modes increase around 6.5 GHz, so that all impedance matching is achieved above 5.2 GHz. The simulated impedance bandwidth (
$| S_{11} |$ ≤ −10 dB) is 53.53% and 47.28% (2.4–4.15, 5.25–8.5 GHz). The measured impedance bandwidth is 52.43% and 42.8% (2.43–4.2, 5.5–8.5 GHz). The minor discrimination between the simulation and measurement may be due to the assembly and fabrication tolerance.
Figure 15. Photograph of fabricated antenna and measurement setup.
Figure 16. The S-parameter of the antenna.
Axis ratio (AR) is an important parameter for circularly polarized antennas, which directly reflects the purity of circular polarization. Figure 17 shows simulated results of the AR of the multi-band antenna, which is less than 3 dB in the bands of 2.38–2.7, 3.2–3.7, 4.6–4.8, and 6.38–7.02 GHz. The wider AR at 6.5 GHz also confirms the conclusion of the CMA that a constant and stable CA difference in this band will provide a broad AR bandwidth. The above bandwidths are in general agreement with the results of the CMA at 2.4, 3.5, 5.8, and 6.7 GHz. Due to the MS curve after adding the feeding structure shifted from 5.3 GHz to around 4.9 GHz, which is slightly different from the initial expected results. Table 2 shows the comparison between the results of full wave simulation and CMA with the feeding structure. It can be seen that the antenna operating frequency is basically the same.
Figure 17. The axis ratio of the antenna.
Table 2. Comparison between the results of characteristic mode analysis and full wave simulations
The gain of the multi-band circularly polarized antenna is shown in Fig. 18 with left-hand circular polarization at 2.5 and 3.5 GHz, and right-hand circular polarization at 5.8 and 6.5 GHz. The maximum gain of the antenna can reach 6.42 and 10.0 dBic respectively. From the conclusions of the CMA in second section, there are multiple pairs of orthogonal modes around 6.5 GHz, which are linearly superimposed on the right-hand circularly polarized mode, resulting in a significant gain increase. The simulated and measured radiation patterns in 0∘ and 90∘ planes at 2.45, 3.5, 5.3, and 5.8 GHz are shown in Fig. 19. The measured results match well with the simulated results at the frequency bands.
Figure 18. The gain of the antenna.
Figure 19. The simulated and measured radiation patterns of the proposed antenna. Xoz-plane: (a) 2.45 GHz, (b) 3.5 GHz, (c) 5.3 GHz, (d) 5.8 GHz; Yoz-plane: (e) 2.45 GHz, (f) 3.5 GHz, (g) 5.3 GHz, and (h) 5.8 GHz.
In Fig. 17, the measured results do not have good consistency with the simulated results, which is due to the use of the ideal lumped port for feeding in the simulation without considering the influence of the SMA connector. After the measured results are generated, an antenna structure with a SMA connector is simulated again. It can be seen that the AR curve with a SMA connector is mostly consistent with the test results, thus verifying the above conjecture. Finally, an antenna structure with SMA connector is optimized also shown in Figs. 16, 17, and 18. The modified impedance bandwidth (
$| S_{11} |$ 
$\leq$$-$10 dB) is 60.03% and 44.35% (2.46–4.57, 5.3–8.32 GHz). The 3 dB axis ratio bandwidth covers 2.4–2.65, 3.25–4, 5.18–5.35, and 5.7–6.05 GHz, respectively. The maximum gain reaches 6.5 and 10 dBic at LHCP and RHCP, respectively.
Table 3 shows the performance comparison of several multi-band antennas. Compared to [Reference Garbacz and Turpin29] and [Reference Hoang, Le, Li and Park30], the proposed antenna has similar dimensions but significant advantages in terms of gain. Although it has good multi frequency performance in [Reference Paul, Kandasamy and Sharawi31] and [Reference Abdalrazik, Gomaa and Kishk10], it does not cover frequency bands above 5 GHz, and there is no advantage in 
$| S_{11} |$ bandwidth. Overall, the proposed antenna can utilize the characteristic mode theory and highlight the advantages of high gain and multiple bands in a single radiation patch design.
Table 3. Comparison of the performance with other similar antennas
NG: not given.
Conclusion
This paper proposed a multi-band circularly polarized antenna for WLAN and WiMAX applications based on characteristic mode theory. The antenna can support 2.4, 5.8, 6.4–7 GHz WLAN bands and 3.5 GHz, 5.3 GHz WiMAX bands simultaneously. The antenna provides sufficient bandwidth in the communication band with a high gain of 10 dBic around 6.5 GHz. The antenna possesses advantages of compact size, simple feed structure and low cost as compared to other similar antennas, making it an excellent candidate for sub-6 GHz communication applications.
Acknowledgements
This work was supported by National Key R&D Program of China (No. 2022YFF0604200).
Competing interests
None declared.
Yutong Yang received the B.E. degree in electronic information science and technology from Xidian University, Xian, China, in 2021. She is currently pursuing the M.S. degree in electronic science and technology from Beijing University of Posts and Telecommunications. Her current research interests include characteristic modes theory and millimeter-wave antennas.
Zihang Qi received the B.E. degree in electronic and information engineering from China Three Gorges University, Yichang, China, in 2013, and the Ph.D. degree in electronic science and technology from the Beijing University of Posts and Telecommunications, Beijing, China, in 2019. He is currently an associate research fellow with the Beijing University of Posts and Telecommunications. His current research interests include OAM antennas, millimeter-wave/THz antennas and microwave filters.
Yongxin Chen received the B.S. degree in Electronic Information Science and Technology from Nanjing University of Aeronautics and Astronautics, Nanjing, China. And, the M.S. degree in electronic engineering from Beijing University of Posts and Telecommunications, Beijing, China. His current research interests includecharacteristic modes theory.
Xiuping Li received the B.S. degree from Shandong University in 1996, the Ph.D. degree from Beijing Institute of Technology in 2001.From 2001 to 2003, she joined in Positioning and Wireless Technology Center, Nanyang Technological University, where she was a research fellow and involved in the research and development of RFID system. In 2003, she was a research professor in Yonsei University, South Korea. Since 2004, she joined Beijing University of Posts and Telecommunications as associate professor, and promoted to professor in 2009. She has been selected into the New Century Excellent Talents Support Plan in National Ministry of Education, the Beijing Science and Technology Nova Support Plan. She won the second prize of the Progress in Science and Technology of China Institute of Communications and the Excellent Achievements in Scientific Research of Colleges and Universities. Her research interests include millimeter-wave antennas, THz antennas, RFID systems, and MMIC design.
