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Abstract
Optical beams carrying orbital angular momentum (OAM) have received much attention due to the prospects of their use in terahertz communications, biomedical engineering, and imaging. Here we propose an antenna design for the generation of multiple beams carrying OAM with different topological states at the same frequency. The proposed OAM generator is based on a compact set of microstrip ring-shaped resonators. An analytical solution for the radiated field of a single circular ring resonator antenna is derived involving the cavity model and the magnetic current approach. To verify our theoretical description, the numerical full-wave simulation is performed for an actual size OAM generator with the use of the ANSYS HFSS electromagnetic solver, and an antenna prototype operating in the microwave band is fabricated and tested. Conditions of the antenna operation in the combined OAM and mode-division multiplexing (OAM-MDM) regimes are discussed. Obtained results prove that the proposed antenna can be used as a compact and low-cost generator of multiple beams with different OAM states.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM), which characterizes the helical phase pattern (vortex) of optical beams, has recently attracted enormous interest due to its potential application for high-resolution imaging [1], nanoparticle manipulation [2,3], quantum optics [4], and biomedical engineering [5]. In this regard, significant efforts of researchers have been directed toward designing specific devices and planar artificial structures (metasurfaces) suitable for the OAM beam generation. In particular, in the microwave range, antennas based on dielectric resonators [6,7], circular slots [8], patches [9–12] circular ring resonators [12,13], and microwave chiral cavities [14] are widely used to generate electromagnetic waves carrying OAM. In the terahertz and visible parts of the spectrum, many designs utilizing computer-generated holograms, spiral phase plates, subwavelength gratings, whispering gallery mode resonators, plasmonic and dielectric nanoantennas are also proposed to emit OAM beams [15–26]. Another common approach for generating OAM beams is to use antenna arrays with a successive phase shift [27].
Moreover, there is a significant interest in the use of OAM beams for wireless and optical communications [28–30]. In this field, increasing the transmission capacity of communication channels is one of the priority tasks that arise in the development of new systems. In such systems, wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and space-division multiplexing (SDM) are the most common techniques used to increase the transmission capacity [31]. Multiple independent data streams are located at different wavelengths, polarizations, or spatial channels when WDM, PDM, and SDM techniques are applied. A special case of the SDM technique is the mode-division multiplexing (MDM), in which a set of mutually orthogonal spatially overlapping and co-propagated modes can carry independent data channels at the same carrying frequency [32,33]. Several different types of orthogonal modal basis sets can be considered as potential candidates in this case. One of them is a set of OAM modes [34]. Compared to other MDM implementation methods, the combined OAM and MDM technique (OAM-MDM) has some advantages due to its circular mode symmetry, making it suitable for many communication technologies. The OAM-MDM is independent on the wavelength and polarization, so it can be additionally combined with either WDM or PDM technique to increase the capacity in both wireless [28,33] and optical [32,34] communication systems.
In wireless communication systems, microstrip resonator antennas (patch antennas) are commonly used. The convenience of these antennas comes from their easy fabrication, low cost, compact size, and low radiation loss. Because of their small dimensions and narrow bandwidth, single patch elements often are projected in large scanning arrays which makes such antennas very effective. Compared with the resonators having simpler geometries (e.g., rectangular or elliptical patches), antennas based on the microstrip ring-shaped resonators are more attractive when implementing complex systems. Such resonators combined with other elements can be a part of a compact multi-band antenna system with a fixed aperture size. This can be realized by combining a ring-shaped antenna with a higher frequency radiating element within its aperture [35,36]. Several designs of such combined emitters used for the dual-band operation [35] as well as dual-OAM modes generation [11,37] have been proposed.
In this paper, we utilize the above-mentioned features of microstrip ring-shaped resonators to design an OAM generating antenna. In particular, our antenna is composed of a set of microstrip ring-shaped resonators with different mean radii placed on the top of a dielectric substrate with a ground plane on the bottom side. For the first time to the best of our knowledge, we implement the OAM-MDM regime of the antenna to emit multiple OAM modes with the different topological charges at several carrying frequencies without changing the antenna geometry. We construct an analytical theory of such an antenna, carry out numerical simulations, and experimentally measure its performance.
The rest of our paper is organized as follows: In Sec. 2, an analytical solution for the radiated field of a single microstrip ring-shaped resonator antenna is derived using the cavity model and the magnetic current approach. In Sec. 3, a particular design of the OAM-MDM antenna is proposed, and its practical implementation for operating in the microwave range is described. Here we also provide information on our measurement method and setup. Sec. 4. discusses the obtained theoretical and experimental results. Finally, a brief conclusion is drawn in Sec. 5.
2. Microstrip ring-shaped resonator antenna: theoretical description
Let us consider an antenna composed of a microstrip ring-shaped perfectly conducting resonator with a width $w$ placed on the upper plane of a dielectric substrate with a ground plane (GP) on the bottom side, as shown in Fig. 1. The mean radius of the resonator is $R=\left (a+b \right )/2$, where $a$ and $b$ are radii of the inner and outer edges of the ring, respectively. The substrate has thickness $h$ and is made of a material with relative permittivity $\varepsilon _r$. Due to the azimuthal symmetry of the antenna under study, we use a cylindrical polar coordinate system ($\rho$, $\varphi$, $z$), assuming the symmetry axis of the antenna coincides with the $z$-axis of the coordinate system.
Fig. 1. Schematic representation of a microstrip ring-shaped resonator antenna on the Cartesian ($x$, $y$, $z$) and polar ($\rho$, $\varphi$, $z$) coordinate systems, where $r$ is the distance from the coordinate system origin to the observation point $P$ and the angle $\theta$ is measured from the $z$-axis, $w$ is the microstrip width, $a$ and $b$ are radii of the inner and outer edges of the ring, $R=\left (a+b \right )/2$, and $h$ and $\varepsilon _r$ are the thickness and relative permittivity of the substrate, respectively.
For an optically thin substrate ($h \ll \lambda$, where $\lambda$ is the wavelength inside the substrate), the field variations along the $z$-axis can be completely neglected [36,38,39] which corresponds to the TM$_{mn0}$ modes manifestation on the ring (here $E_\rho =E_\varphi =H_z=0$, and $m$ and $n$ are the azimuthal and radial mode indices, respectively). For brevity, we use the shortened notation TM$_{mn}$ instead of TM$_{mn0}$ throughout the paper.
A solution of the Maxwell equations for the TM$_{mn}$ modes can be written as:
The radial component of the surface current $I_{\rho }$ must vanish along the edges ($I_{\rho }\mid _{\rho =b}=H_{\varphi }\mid _{\rho =b}=0$). Therefore, the characteristic equation for the resonant TM$_{mn}$ modes resulting from the boundary conditions is:
In the zeroth-order approximation, the resonant frequency $k_{mn}$ is obtained by setting: where $\nu _{mn}$ is the root of the transcend equation $J'_m(\nu _{mn})Y'_m(\nu _{mn}b/a) = J'_m(\nu _{mn}b/a)Y'_m(\nu _{mn})$. Therefore, Eq. (5) can be used to determine the corresponding radius of the circular ring resonator for a chosen resonant (operating) frequency $f_r$.The analytical relations for the radiation fields of a microstrip ring-shaped resonator antenna are obtained from the magnetic current approach as follows (the treatment is omitted here and can be found elsewhere [36]):
The effect of substrate on the radiated fields can be taken into consideration through the correction factors $F_E(\theta )$ and $F_H(\theta )$ for both E-plane (at $\varphi =0^{\circ }$) and H-plane (at $\varphi =90^{\circ }$) patterns, respectively. These factors are:
The total radiated electric field from a two-feed excited microstrip ring-shaped resonator antenna can be obtained as follows [40]:
Finally, the $E_z$ component of the total radiated electric field in the Cartesian coordinates can be obtained from Eq. (10) performing standard coordinate transformations [41]:
In particular, in the $x$-$y$ plane (when $\theta =90^{\circ }$), we have $E_z(x,y)=-E_\theta (\varphi, \theta )$.
3. OAM-MDM antenna: prototype and experimental setup
Based on the principle of generating a beam carrying OAM by a microstrip ring-shaped resonator described above, our goal here is to design an antenna operating under MDM conditions, simultaneously generating several beams with different topological states $l$ at the same frequency. In accordance with the experimental means at our disposal, we chose the microwave part of the spectrum ($1-10$ GHz) to characterize the antenna and study its performance. In particular, we propose a design of the antenna that can simultaneously generate four beams carrying different topological states at the same resonant frequency $f_r$. At the first stage of our study, we fix the resonant frequency $f_r=4.0$ GHz and determine analytically all geometric parameters of the antenna. Then, in the second stage, we numerically and experimentally characterize the antenna at the resonant frequency chosen. For all our subsequent numerical calculations we use the commercial ANSYS High Frequency Structure Simulator (HFSS). Finally, we additionally test our antenna at the multiple resonant frequencies $f_r = 2.0$ GHz, $6.0$ GHz, and $8.0$ GHz to reveal the OAM-MDM operation conditions.
A sketch and photo of the prototype of our OAM-MDM antenna are presented in Figs. 2(a) and 2(b), respectively. The antenna is composed of a set of microstrip ring-shaped resonators having the same width $w$ and different mean radii $R_m$ ($m=1,2,..,N$). The corresponding mean radii $R_m$ of the resonators are defined with the use of Eq. (5) providing operation conditions of the TM$_{mn}$ mode. On the back side of the substrate, several ground rings are deposited which are separated by concentric gaps. In this way, the individual ring-shaped resonators appear to be electrically isolated from each other by these gaps at the ground plane (for the discussion on the design parameters, see Appendix).
Fig. 2. (a) Sketch of an antenna made in the form of $N=4$ microstrip concentric circular ring resonators, (b) photos of the antenna prototype, and (c) power divider providing $\pi /2$ phase shift between its output ports. The coaxial probes (CPs) positions are marked by points. Here, the operating frequency $f_r$ is $4.0$ GHz, and geometrical parameters of the antenna are: $A = 43$ mm, $R_1=8.71$ mm, $R_2=17.42$ mm, $R_3=26.13$ mm, and $R_4=34.84$ mm. The width of the strips on the upper side and the distance between the metallization on the bottom side of the antenna are $w=2.16$ mm and $w_g=0.4$ mm, respectively. This width of the strips provides the characteristic impedance $Z_w=50$ Ohm of a microstrip line. The substrate is made of Rogers RT/duroid 5880 plate with relative permittivity $\varepsilon _r=2.2$ and thickness $h=0.787$ mm.
The values of the mean radii can be specified more precisely by introducing effective permittivity for the substrate [36,42,43]:
Accounting for the modal dispersion $\varepsilon _\textrm {eff}(f)$ can be defined from the empirical relation which enables us to calculate $\varepsilon _\textrm {eff}(f)$ with high accuracy (error is less than $1\%$) in the frequency range from $2.0$ GHz up to $12.4$ GHz [44]:
To feed each of the ring-shaped resonators of the antenna we use a schema with two coaxial probes [40]. These probes are inner pins of SMA connectors that are soldered to the ring metallization as they pass through the substrate. To obtain equal amplitudes and phase differences at two coaxial probes, the $3$-dB power divider is designed. The practical realization of this power divider is presented in Fig. 2(c). In this device, the phase difference is realized by a differential line with the quarter-wavelength length between arms of the output ports [36,43].
Our experimental setup allows us to measure both amplitude and phase distributions of the beam generated by the antenna. A photo and schematic view of this setup are presented in Figs. 3(a) and 3(b), respectively. The primary signal is generated by the Rhode & Schwarz Vector Network Analyzer ZVA50 (VNA). The signal from the transmitting port (Port 1) of the VNA is fed to the input port of the power divider via $50$ Ohm coaxial cable. Actual S-parameters of the antenna are collected in Fig. 3(c). The output signals from the power divider are fed into additional ports (Port 3 and Port 4) of the VNA to validate the relative phase difference and equality of the amplitude. Then they are connected to corresponding coaxial probes of the antenna. For simultaneous supply of several ports in the multiplexing regime, a standard power divider is used.
Fig. 3. (a) Photo, (b) schematic view of the experimental setup, and (c) actual S-parameters of the antenna. The distance $d$ between the observation plane and antenna prototype is $75$ mm. The observation plane is $2B\times 2B$ square with $B=200$ mm. The scanning area is swept with $40 \times 40$ points at a step width of 10 mm.
The LINBOU near-field scanning platform [45] is used to perform measurements of the electric field characteristics of the radiated OAM modes. The scanning subwavelength electric probe is set normally to the antenna prototype surface. It is connected to receiving port (Port 2) of the VNA. During the measurements, the probe is automatically moved in the observation plane at the distance $d$ above the antenna prototype. At each probe position, the corresponding component of the radiated electric field ($E_z$) is sampled. All our measurements are carried out in an anechoic chamber to prevent unwanted noise (additional details on the experimental method and measurement setup can be found in Ref. [46]).
4. Results and discussion
Since geometric parameters of our antenna satisfy the following conditions: $h \ll 2R_m$ and $w<0.5c/f_r$, the fields in the individual ring resonator can be considered in the single mode approximation, when the resonator operates on a particular TM$_{m1}$ mode (i.e., the higher-order modes with $n>1$ are excluded from our consideration). To verify the applicability of this approximation, Fig. 4 shows analytically derived and numerically simulated results for the electric field patterns of the TM$_{11}$, TM$_{21}$, TM$_{31}$, and TM$_{41}$ modes arising at the same frequency in resonators with different radii. It can be seen that the electric field patterns obtained analytically and numerically coincide well which confirms the correctness of the single mode approximation used.
Fig. 4. Analytical (upper row) and simulated (bottom row) electric field patterns ($E_z$-component) for (a) TM$_{11}$, (b) TM$_{21}$, (c) TM$_{31}$, and (d) TM$_{41}$ mode of the microstrip ring resonators operating at the same resonant frequency $f_r=4.0$ GHz. The heavy dots on the panels in the bottom row correspond to the coaxial probes (CPs) positions taken into account in the ANSYS HFSS project. The radii and all other geometric and material parameters of the antenna correspond to those indicated in the caption to Fig. 2.
It was shown earlier [8] that OAM waves can be generated based on circular traveling waves. In particular, a microstrip ring-shaped resonator antenna operating on the TM$_{mn}$ mode can generate such kinds of waves [47]. It emits a circularly polarized wave when two mutually orthogonal degenerated TM$_{mn}$ modes with relative phase shift $\pi /2$ are excited on the ring. In the two coaxial probes scheme [40], the probes are supplied with the same amplitude and relative phase shift of $\pi /2$ to excite two mutually orthogonal TM$_{m1}$ modes on the ring-shaped resonator [10,13,37]. When two mutually orthogonal TM$_{m1}$ modes are excited, then the surface current on the ring produces a circular traveling electromagnetic wave that can carry an OAM state. The correct angular distance $\alpha _m$ between two coaxial probes located on the ring can be determined from the following condition [40]: one coaxial probe is always situated in the null field region of the other probe, thus, causing insignificant mutual coupling between the probes. Such spacing is individual for each operating mode where the condition $\alpha _m=\pi /(2m)$ holds. Moreover, if $m$ is an odd number, the same angle $\alpha =\pi /2$ can be set. One can check the correctness of defined positions of coaxial probes suitable for the OAM generation considering the results of our full-wave numerical simulation for the modes of several individual ring-shaped resonators presented in Fig. 4. In this simulation, we fix the following angular distances between the respective coaxial probes: $\alpha _1=\alpha _3=\pi /2$, $\alpha _2=\pi /4$, and $\alpha _4=\pi /8$.
In what follows, we analyze the characteristics of particular OAM states generated by our antenna which is composed of a set of microstrip ring-shaped resonators. We collect in Fig. 5 the results of our analytical treatment for several OAM modes of the electromagnetic field radiated by our antenna. These data are presented in the form of the instantaneous phase $\textrm {arg}(\mathbf {E})$ and amplitude $|\mathbf {E}|$ distributions of the electric field plotted in the $x$-$y$ plane of the chosen coordinate frame. It is noteworthy that the amplitude distribution of the $E_z$-component of the OAM waves has a petal structure. This feature of the $E_z$-component, in addition to the phase distribution, can be used to classify the OAM modes.
Fig. 5. Analytical data for the phase (upper box) and amplitude (bottom box) distributions of the electric field of the radiated OAM waves with different topological charge numbers $l$: (a) $l=1$ ($m=1$), (b) $l=2$ ($m=2$), (c) $l=-3$ ($m=3$), and (d) $l=4$ ($m=4$), and (e) multiplex state presented by a superposition of three OAM modes with $l=2$, $l=-3$, and $l=4$ for the antenna operating at the fixed frequency $f_r=4.0$ GHz. All geometrical and material parameters of the antenna are listed in the captions to Figs. 2 and 3.
To be specific, there are two alternative methods of the OAM modes classification. The difference between these methods lies in the fact that the calculation of phase variations for the OAM modes can be carried out by considering either the transverse ($E_{x,y}$) [6,48] or longitudinal ($E_z$) [10,13] component of the electric field (from Fig. 5 it can be easily seen that for these two classifications, the mode order will differ by one according to the number of phase variations for the $E_{x,y}$ and $E_z$ components). Since in our experimental setup we have the ability to measure both the phase and amplitude of the $E_z$ component of the electric field, we perform our classification of the OAM modes, assuming that the azimuthal mode index $m$ is equal to the absolute value of the topological charge number $|l|$ of the corresponding OAM state. Thereby, from the TM$_{m1}$ mode of the ring-shaped resonator, a beam carrying OAM with topological state $l = \pm m$ appears.
In Fig. 6, the simulated and measured patterns of the instantaneous phase and amplitude of the $z$-component of the electric field for the radiated OAM waves with states $l=1$, $2$, $-3$, and $4$ are shown. The occurrence of such beams in the field of the given antenna is consistent with the results presented earlier [10,11] where several designs of the OAM generators based on the circular patch antennas have been considered. In particular, our antenna produces OAM waves with the left-handed ($l=+m$) spiral phase structure [in accordance with the accepted terminology [49], when $l=+m$ ($l=-m$), a wave is said to have positive (negative) helicity] when the signal passing through the first coaxial probe (CP 1) is at $\pi /2$ behind the signal passing through the second probe (CP 2) [see, Figs. 6(a), 6(b), and 6(d)], and OAM waves with the right-handed spiral phase structure ($l=-m$) when the signal passing through the first coaxial probe (CP 1) is at $\pi /2$ ahead the signal passing through the second probe (CP 2) [see, Fig. 6 (c)]. In all cases, the phase change is $2\pi l$ after one period of rotation.
Fig. 6. Simulated and measured data for the phase (upper box) and amplitude (bottom box) distributions of the $E_z$-component of the OAM waves presented in Fig. 5.
Two-dimensional diagrams of the phase and amplitude distributions of the $z$-component of the electric field of the OAM waves radiated by our antenna are shown in Fig. 6. These data are collected from numerical calculations, as well as from direct experimental measurements. One can readily see that there is a reasonable agreement between the results of our analytical model (see, Fig. 5), numerical simulations, and measurement. Some unbalance in the electric field distributions as well as possible asymmetry in the far-field radiation patterns are related to our use of a feeding system with two coaxial probes for each resonator. One way to overcome this issue is to use two additional coaxial probes for feeding [40], which should be placed strictly opposite to the corresponding original feeds (see Appendix). Nevertheless, this alternative method significantly complicates the design and workability tuning of the antenna and therefore rarely used in practice.
Our obtained results suggest that for the radiated modes, the center of the vortex is exactly aligned along the direction of the electromagnetic wave propagation ($+z$-direction), where the amplitude distribution in the observation plane is hollow. The area of the hollow region for the OAM modes (vortex core) increases as the number of the topological charge rises, which is a well-known feature of the vortex beams [50]. The two-dimensional diagrams of the electric field are further supplemented by the three-dimensional far-field radiation patterns and their cross-sections plotted in the E-plane and H-plane as presented in Fig. 7 (here the analytical results for the far-field cross-sections are given only for the E-plane, so as not to overload the pictures). Although the cavity model makes it possible to evaluate the resonant frequency of the ring resonator very precisely and calculate the field distributions of the corresponding resonant TM$_{m1}$ mode, some discrepancy between the analytical and simulated far-field radiation patterns is found in the far-field cross-section diagrams. This discrepancy can be explained by the fact that the cavity model gives only a rough approximation of the ratio of the aperture fields since these fields are calculated from a standing wave distribution in which all points are considered to be in phase [36,51]. This leads to a subsequent error in the definition of both the lobes angular direction and their levels. However, this error does not greatly affect the appearance of radiation patterns, and the error decreases as the value of $l$ rises. Thus, this additionally confirms that our analytical description based on the cavity model and the magnetic current approach can be used to adequately predict both the near-field and far-field characteristics of the antenna under study.
Fig. 7. Far-field radiation patterns (upper row) and their cross-sections (bottom row) in the E-plane and H-plane for the radiated OAM waves with topological charge numbers corresponding to those given in Fig. 5.
However, there are some peculiarities in the patterns presented. For the OAM mode with $l = 1$, the radiation maximum exactly coincides with the $+z$-axis direction [Fig. 7(a)], while the hollow at the beam axis does not exist. It means that the field does not have OAM properties which is consistent with the results reported earlier [10]. It can also be seen that when several resonators are simultaneously excited [Fig. 7(e)] the helical structure of the released beam is preserved which can be used to implement a multiplex state.
In particular, OAM modes with different topological charge numbers $l$ are mutually orthogonal, which allows them to be multiplexed together and transmitted along the same spatial axis and then demultiplexed with low crosstalk [28,32]. Thus, by simultaneously exciting several resonators, the given antenna can generate multiple OAM modes with different states $l$ at the same carrier frequency $f_c$ providing the conditions for operating in the OAM-MDM regime (see also Refs. [52–54]).
To obtain qualitative characteristics of the generated field, the modal analysis can be applied (see a tutorial in Ref. [55]). In particular, the vortex nature and the possibility of implementing a multiplex state can be verified by comparing the OAM content (mode purity) carried by the experimental fields with the expected analytical and numerically simulated ones. For this purpose, we applied the spiral spectrum algorithm [56,57] which performs the projection of the electromagnetic field on spiral harmonics given in the $\exp (il\varphi )$ terms, similarly to the Fourier transform. Corresponding results are collected in Fig. 8. There is a correspondence between the obtained results, although one can notice the appearance of some crosstalk between the modes of the same topological charge number with different helicity. This crosstalk is most pronounced for the experimental data. It can be explained by the fact that our calculation of the mode purity is carried out only from the $z$-component of the electric field, and, apparently, for a more accurate calculation, other components of the field should also be taken into account. The increased crosstalk in the experimental data is due to the near-field perturbation that our dipole probe introduces when making measurements. However, even with this shortcoming, the desired modes can be uniquely identified in both separate and multiplex states.
For the antenna under consideration, the ratio between the width $w$ of each microstrip ring to its mean radius $R_m$ lies within the range $0.055 \leq w/R_m \leq 0.24$. For such narrow rings, Eq. (5), that determines the resonant frequencies of the TM$_{m1}$ modes, can be reduced to [58]:
and the mean radii of the ring-shaped resonators at the corresponding carrier frequency $f_c$ can be determined from Eq. (14) as follows:This condition is already taken into account in our design, which allows us to use the antenna for operating in the OAM-MDM regime at the multiple carrier frequencies $f_c=pf_r$ without changing its geometry [here, $p=q/2$ ($q \in \mathbb {Z}$), $f_r=4.0$ GHz is the reference resonant frequency at which $p = 1$, and the antenna generates OAM waves with $l=\pm m(f_r)$].
From Eq. (15), it follows that $R_m(f_c)=R_m(f_r)/p$ and the topological charges of the OAM waves at the different carrier frequencies are $l=\pm pm(f_r)$. Considering that $l$ must be an integer, the total number of the emitted OAM modes at the carrier frequency $f_c=pf_r$ can be defined as:
Therefore, the proposed antenna is able to simultaneously generate four or two OAM modes at the carrier frequencies since corresponding pairs of circular ring resonators are well matched [see, Fig. 3(c)]. In Table 1, we summarize all possible topological states $l$ of the OAM waves which can be generated by our antenna at the carrier frequencies $f_c=2.0$ GHz, $4.0$ GHz, $6.0$ GHz, and $8.0$ GHz when the reference resonant frequency is $f_r=4.0$ GHz.
Table 1. Topological states $l$ of the OAM waves at different carrier frequencies $f_c$
To demonstrate this behavior in more detail, we present in Fig. 9 the simulated and measured phase and amplitude distributions of the $z$-component of the electric field radiated by the OAM-MDM antenna operating at the carrier frequencies $f_c=2.0$ GHz ($p=1/2$) and $6.0$ GHz ($p=3/2$). They are supplemented by Fig. 10 where the mode purity conditions are shown. One can conclude that, in the experiment, our antenna exhibits performances that correspond to the numerical data obtained. This confirms the ability of the antenna to operate in the OAM-MDM regime.
Fig. 9. Same as in Fig. 6 but for the antenna operating at the carrier frequencies (a, b) $2.0$ GHz and (c, d) $6.0$ GHz. Here: (a) $l=1$, $p=1/2$, (b) $l=2$, $p=1/2$, (c) $l=3$, $p=3/2$, and (d) $l=6$, $p=3/2$.
However, for this regime, as Eq. (13) suggests, the effect of modal dispersion should be evaluated. For this evaluation, the corresponding dispersion characteristics are plotted in Fig. 11.
Fig. 11. (a) Dispersion of effective permittivity of the substrate $\varepsilon _\textrm {eff}(f)$ and (b) relative errors $\Delta \varepsilon _\textrm {eff}$ and $\Delta R$.
The relative errors arising in the definition of the effective permittivity of the substrate and mean radii of the ring-shaped resonators, neglecting the effect of modal dispersion, can be written as:
where $G$ takes the values $\varepsilon _\textrm {eff}$ and $R$, respectively.From our evaluation, it follows that the relative error in the mean radii calculation is $\Delta R \sim (\Delta \varepsilon _\textrm {eff})^{1/2}$ which does not exceed $2.5\%$. Thus, the correctness of the introduced value of $\varepsilon _\textrm {eff}(f_r)$ is justified over the entire frequency band of interest. If one consider the operation frequency of the antenna in the band above $10$ GHz, the corresponding estimations should be accounted for: $\varepsilon _\textrm {eff}(f) \to \varepsilon _r$ and, consequently, $\Delta R \to 0$.
5. Conclusions
In conclusion, we have demonstrated a strategy of the multi-band OAM-MDM operation of the microwave antenna based on a compact set of microstrip ring-shaped resonators. An analytical solution for the radiated field of a single resonator antenna is derived involving the cavity model and the magnetic current approach. To verify our theoretical description, full-wave numerical simulations are performed for an actual size OAM-MDM generator using the ANSYS HFSS electromagnetic solver, and an antenna prototype operating in the microwave range is fabricated and tested. The measured phase patterns and amplitude profiles of the antenna have been shown, confirming that the antenna can generate OAM waves with different order and helicity. The spiral spectrum algorithm is applied to derive the OAM content carried by the actual antenna operating in the OAM-MDM regime.
The results obtained indicate the possibility of using the proposed antenna as a small-sized and inexpensive generator capable to operate in the OAM-MDM regime at several carrier frequencies. By further modifying the design, an antenna for generating composite OAM beams [59,60] can be realized. Moreover, the identified parameters of generation of OAM beams can be applied in the designs of reconfigurable antennas made of novel 2D materials such as graphene for operating in the terahertz (THz) range [61,62].
Appendix: design parameters
In this section, we briefly discuss the effect of a feed scheme selection as well as a ground plate design on the resulting far-field radiation pattern of the antenna. For this purpose, we study the properties of a ring-shaped resonator antenna operating on the TM$_{21}$ mode.
In the proposed antenna, to reduce the mutual coupling of resonators, the concentric circular slits with equal width $w_g$ have been applied on the ground plate. The variation in the radiation pattern of the antenna for two different slits width ($w_g=0.2$ mm and $w_g=6.55$ mm) is presented in Fig. 12. For clarity of discussion, the bottom view of the antenna is also shown. Note, that the second case corresponds to the design in which the width of the rings (metallization) on both sides of the antenna are the same. From here one can conclude that increasing the width of the slits on the ground plate leads to increasing the level of the backward lobes whereas in the range of slits widths $w \in \left [0.1\,\textrm {mm},0.5\,\textrm {mm}\right ]$ the far-field radiation pattern remains almost invariable.
Fig. 12. Effect of the slits width $w_g$ in the ground plate of the ring-shaped resonator antenna on its far-field radiation pattern where (a) $w_g=0.2$ mm and (b) $w_g=6.55$ mm.
Figure 13 suggests that when the two-probe feed scheme is used, the asymmetry in the far-field radiation pattern can appear. To more clearly demonstrate this effect, we compare the far-field radiation patterns generated by the ring-shaped resonator antenna fed by either two-probe or four-probe scheme. In both schemes, the antenna is placed on a substrate with a ground plate without slits. One can conclude that the asymmetry in the far-field radiation pattern of the antenna with the two-probe feed scheme can be minimized (or completely suppressed) with the use of the four-probe feed scheme where the coaxial probes should have the following phase arrangement: $\left [0^{\circ }, 90^{\circ }, 180^{\circ }, 270^{\circ } \right ]$ or $\left [0^{\circ }, 90^{\circ }, 0^{\circ }, 90^{\circ }\right ]$ for odd and even order modes of the resonator, respectively [40].
Funding
Project of Jilin Province Development and Reform Commission (2022C047-6); Project of Science and Technology Development Program of Changchun City (21ZY23); Project of the Education Department of Jilin Province (JJKH20211093KJ); National Natural Science Foundation of China (62271229).
Acknowledgments
V.R.T. acknowledges Jilin University’s hospitality and financial support.
Disclosures
The authors declare no conflicts of interest.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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