Reconfigurable High-Efficiency Power Dividers Using Waveguide Epsilon-Near-Zero Media for On-Demand Splitting

Abstract

Although epsilon-near-zero (ENZ) media have emerged as a promising platform for power dividers, the majority of existing designs are confined to fixed power splitting. In this work, two dynamically tunable power dividers using waveguide ENZ media are proposed by precisely modulating the internal magnetic field and the widths of the output waveguides. The first approach features a mechanically reconfigurable ring-shaped ENZ waveguide. By continuously re-distributing the magnetic field within the ENZ tunneling channels utilizing rotatable copper plates, arbitrary power division among multiple output ports is constructed. The second design integrates a rectangular-loop ENZ cavity into a substrate-integrated waveguide, with four positive–intrinsic–negative diodes embedded to dynamically activate specific output ports. This configuration steers electromagnetic energy toward output ports with varying cross-sectional areas, enabling on-demand control over both the power division and the number of output ports. Both analytical and full-wave simulation results confirm dynamic power division, with transmission efficiencies exceeding 93%. Despite differences in structure and actuation mechanisms, both designs exhibit flexible field control, high reconfigurability, and excellent transmission performance, highlighting their potential in advanced applications such as real-time wireless communications, multi-input–multi-output systems, and reconfigurable antennas.

1. Introduction

Owing to their ability to divide a single input signal into multiple output signals with a balanced or imbalanced power ratio, power dividers have been widely explored in radio-frequency (RF), microwave, and optical circuits [1,2,3]. For example, in reflectarrays and transmitarrays, power dividers are essential parts of the feeding network and excite every element [4]. Conventional microwave power dividers, typically implemented using microstrip technology such as the Wilkinson type [5], provide equal power division and port isolation through quarter-wavelength transmission lines and matching resistors. Despite their widespread use, high energy loss and structural constraints may limit their further applications in high-quality electromagnetic (EM) wave manipulation and compact system integration [6]. In contrast, metamaterial-based power dividers exploit engineered EM responses to achieve functionalities beyond those of conventional designs. For instance, a 1:4 series power divider based on zero-degree metamaterial structures delivers equal power to all four output ports over a broad frequency range [7]. Furthermore, PT-symmetric non-Hermitian photonic systems enable control of the near-field routing of hyperbolic polaritons, breaking intrinsic symmetry constraints and allowing subwavelength-scale energy guidance [8].

Epsilon-near-zero (ENZ) media [9,10,11,12,13] are elaborately engineered materials with a near-zero dielectric constant and wavenumber, resulting in an extremely small optical index and a nearly uniform phase distribution. Due to their natural temporal and spatial decoupling properties [14,15,16,17,18,19], ENZ media have emerged as promising platforms for high-efficiency signal transmission [20,21,22,23,24,25], division [26], and combination [27], exhibiting great potential in massive multi-input–multi-output (MIMO) systems, the Internet of Things (IoT), etc. As one of the most effective construction strategies, waveguides operating at the cut-off frequency can be equivalently considered ENZ media [28,29,30,31,32]. It has been experimentally demonstrated in waveguides that their equivalent dielectric constant near the cut-off frequency can be considered near zero, resulting in a near-infinity wavelength and longitudinal propagation constant. Therefore, featuring merits of low loss and geometry insensitivity, many recent efforts have been devoted to ENZ-based power dividers. For example, an arbitrarily shaped multi-port power divider is proposed by connecting multiple waveguides through an ENZ host medium, which is embedded with a dielectric impurity to facilitate impedance matching [33]. Arbitrary power allocation could be realized with pre-designed output waveguide widths. Additionally, an acoustic power divider and multiplexer have been developed utilizing an index-near-zero medium [34], capable of tunneling sound to an arbitrary number of output ports with phase shifts of 0° or 180°. Precise power and phase control are implemented by adjusting each output channel’s cross-sectional area and arranging the output port positions. Despite significant advancements, the above approaches are restricted to fixed power division after fabrication, which may constrain their possibilities in dynamic application scenarios, for example, real-time wireless communications.

Here, we generalize the concept of dynamic high-efficiency on-demand power division with arbitrarily shaped waveguide ENZ media by proposing two distinct types of reconfigurable ENZ waveguide-based power dividers. With the inherent magnetic field homogeneity, tunable power division ratios can be achieved by adjusting the internal magnetic field and/or the widths of the output waveguides. A mechanically reconfigurable power divider is presented by coupling rectangular waveguides via a ring-shaped air-filled narrow ENZ tunnel. Rotating the embedded copper plates continuously tunes the magnetic field in the sub-channels, directly controlling the power delivered to the two output waveguides and enabling arbitrary split ratios. The transmission efficiency remains above 93% across all power divisions. As another approach, an electrically reconfigurable power divider is proposed with output waveguides of different cross-sectional areas. It consists of a substrate-integrated waveguide (SIW) and a rectangular-loop air-filled ENZ cavity with four embedded positive–intrinsic–negative (PIN) diodes. Selective switching of these diodes dynamically reconfigures both the number and cross-sectional areas of the output SIWs, enabling flexible output port selection and on-demand power division. Regardless of the output power split ratios, the transmission efficiencies remain above 93%. Analytical and numerical models are conducted to characterize the EM performances of the proposed reconfigurable ENZ-based power dividers, validating their desired EM regulation with high transmission efficiency, real-time tunability, on-demand power allocation, and flexible geometry parameters. The presented methods may provide new strategies for advanced reconfigurable power dividers (RPDs).

2. Concept and Theoretical Analysis

As conceptually illustrated in Figure 1, we first consider a two-dimensional (2-D) RPD comprising a narrow ENZ channel with an arbitrary geometry, which further interconnects three identical standard waveguides. The waveguides are terminated by ports labeled 1 to 3, with Port 1 designated as the input and Ports 2–3 serving as the output. The external boundaries of the structure, except for at the waveguide ports, are enclosed by a perfect electric conductor (PEC). The incident wave is polarized with the magnetic field along the out-of-plane axis.

Transmission efficiency is an essential index of power dividers, which is quantified as $\eta ={\left|{\mathit{S}}_{21}\right|}^2+{\left|{\mathit{S}}_{31}\right|}^2=1-{\left|{\mathit{S}}_{11}\right|}^2-{\alpha }_{\mathrm{loss}}$ [35], where ${\alpha }_{\mathrm{loss}}$ denotes the material loss factor. To maximize signal transmission efficiency, reflected waves and energy loss should be suppressed as low as possible. Due to the low-loss characteristic of the ENZ waveguide media, ${\alpha }_{\mathrm{loss}}$ approaches zero. The reflection coefficient $\mathit{R}$ at Port 1 can be expressed as [36,37,38]

$$R={\displaystyle \frac{(a_1-a_2-\dots -a_n)+i{k}_0{\mu }_{r,p}{A}_p}{(a_1+a_2+\dots +a_n)-i{k}_0{\mu }_{r,p}{A}_p}},$$

(1)

which can be further written as

$$\mathit{R}={\displaystyle \frac{1+i{\mathit{k}}_0{\mu }_{r,p}{\mathit{A}}_p/a_1-{\sum }_{n=2}^N{a}_n/a_1}{1-i{\mathit{k}}_0{\mu }_{r,p}{\mathit{A}}_p/a_1+{\sum }_{n=2}^N{a}_n/a_1}},$$

(2)

where ${\mathit{k}}_0$ is the free-space wavenumber, ${\mu }_{r,p}$ denotes the relative permeability of the medium inside the ENZ channel, ${\mathit{A}}_p$ is the cross-sectional area of the channel, $a_1$ is the width of the input waveguide, and the widths of the output waveguides are $a_2,a_3,\dots a_n$ (with $n=3$ in the conceptual illustration). When the output waveguide widths satisfy ${\sum }_{n=2}^N{a}_n=a_1$ and the ENZ channel area approaches zero ($k_0{\mu }_{r,p}{\mathit{A}}_p\to 0$), the reflection coefficient $\mathit{R}$ tends toward zero, allowing EM waves to be efficiently squeezed and transmitted through a narrow, arbitrarily shaped ENZ channel.

Because the magnetic fields are equal at the interfaces between the ENZ region and the input/output waveguides, the power flow densities are also identical at both interfaces under the perfect tunneling condition. Thus, the time-averaged input power can be calculated as [37,38]

$$P_{in}={\eta }_0{\left|H_{\mathrm{c}}\right|}^2(1-{\left|R\right|}^2)a_1.$$

(3)

Accordingly, the output power of each port can be written as

$$P_n={\eta }_0{\left|H_{\mathrm{c}}(1+R)\right|}^2{a}_n,$$

(4)

where ${\eta }_0$ is the free-space impedance, and $H_c$ denotes the magnetic field inside the ENZ region. By reducing $\mathit{R}$ to near zero, the output power can be simplified as

$$P_n={\eta }_0{\left|H_{\mathrm{c}}\right|}^2{a}_n.$$

(5)

This relation indicates that the output power can be modulated by tuning the magnetic field ($H_{\mathrm{c}}$) within the sub-channels and/or the widths ($a_n$) of the output waveguides.

Figure 1a illustrates the method of achieving arbitrary power division by only tuning the internal magnetic field ($H_{\mathrm{c}}$). To minimize reflection, the input waveguide width $a_1$ is set equal to the sum of the output widths, i.e., $a_1=a_2+a_3$. Equal output widths ($a_2=a_3$) are chosen to ensure identical output cross-sectional areas. To modulate the magnetic field, two copper plates are inserted into the ENZ channel. Copper plate 1 is located between the output waveguides and acts as an EM barrier to suppress inter-port crosstalk. Copper plate 2 is positioned near the entrance of each sub-channel, where it alters the effective entrance width ($b_1$ or $b_2$) and introduces localized field perturbations. Due to the quasi-static field distribution and the high sensitivity of the ENZ region to local changes, such perturbations can significantly redistribute the internal magnetic field, thereby enabling tunable power $P_n$ delivery to each output port. Therefore, the output power ratio between Port 2 and Port 3 can be estimated from the width ratio of $b_1$/$b_2$. In this manner, precise and reconfigurable power division can be achieved by tuning the internal magnetic field.

As conceptually sketched in Figure 1b, dynamic power division can also be achieved by tuning the output waveguide widths ($a_n$). Similarly, to suppress reflected waves, these widths are set to satisfy the relation $a_1=a_2^{\prime }+a_3^{\prime }$. Given the spatially uniform magnetic field profile inside the ENZ channel, the power distribution among the output ports is determined solely by their cross-sectional areas. Accordingly, arbitrary power splitting ratios between Port 2 and Port 3 are attainable with the required output waveguide width ratio of $a_2^{\prime }/a_3^{\prime }$.

Guided by the theoretical analysis, two types of highly efficient ENZ-based power dividers were developed with dynamic and on-demand power division by tuning the internal magnetic field ($H_{\mathrm{c}}$) and the widths of the output waveguides ($a_n$), respectively.

3. Practical Designs of Reconfigurable ENZ-Based Waveguide Power Dividers

3.1. Mechanically Reconfigurable ENZ-Based Power Divider Based on Magnetic Field Redistribution

A mechanically reconfigurable ENZ-based power divider is proposed to enable continuous and arbitrary tuning of power division ratios via adjustment of the internal magnetic field ($H_{\mathrm{c}}$). As displayed in Figure 2a,b, the entire structure is enclosed by metallic walls with copper, except for at the interfaces of three rectangular waveguide ports. An air-filled cylindrical waveguide with a height of $h=50$ mm and a radius of $r_1=20$ mm is integrated with a copper post (radius of $r_2=18$ mm and conductivity of $\sigma =5.998\times {10}^7$ S/m) to form a 2 mm wide narrow annular channel. Operating near the ${\mathrm{TE}}_{10}$ cut-off frequency (i.e., $f_{\mathrm{c}}=0.5\mathrm{c}/h=3$ GHz), the channel exhibits equivalent ENZ behavior, enabling efficient wave tunneling across geometric discontinuities. The ENZ channel is coupled to three rectangular waveguide ports, namely, one input port (Port 1) and two output ports (Ports 2 and 3). As indicated in the previous theoretical analysis, to minimize the reflection coefficient according to Equation (2), the input port width is set to 20 mm, while the output ports are designed with equal widths of 10 mm each. All waveguides are filled with polytetrafluoroethylene (PTFE, $\epsilon =2.1$, and tan$\delta$ = 0.0004) and are symmetrically arranged at 120° intervals.

Two 1 mm thick copper plates are symmetrically installed within the ENZ channel and electrically connected to the surrounding metallic enclosure. Copper plate 1 is introduced to suppress coupling between the output ports, and copper plate 2 divides the annular ENZ channel into two sub-channels. By rotating copper plate 2, the magnetic field is redistributed between the two sub-channels, enabling direct modulation of the output power delivered to each output waveguide (as indicated by the dashed arrows). Accordingly, the power ratio between Port 2 and Port 3 is governed by the ratio of their effective input widths ($b_1$/$b_2$). Notably, both plates are synchronously rotated via a central copper post for practical feasibility, and the position of copper plate 1 has little influence on the EM performance. To enable manually tunable control, the central copper post is mechanically connected to a disk and base via a rotatable supporting shaft so that disk rotation simultaneously drives both the post and copper plates, allowing precise and repeatable adjustment of the plate rotation angles in real time. Overall, despite minor angular deviations and a tiny shift of the central post during rotation, the ENZ channel maintains highly efficient tunneling with negligible impact on the overall transmission.

Full-wave simulations were performed using the commercial software COMSOL (Version 6.1) to evaluate the performance of the RPD at an operating frequency of 3 GHz. As shown in Figure 2c, the output power varies continuously with the rotation angle $\theta$ ranging from −29° to 29°. When Port 1 is excited with a total power of 1 $\mathrm{W}$, the output power at Port 2 increases from 0 to 0.93 $\mathrm{W}$, while that at Port 3 simultaneously decreases from 0.93 to 0 $\mathrm{W}$. This indicates a continuous and controllable redistribution of EM power between the two output ports, with the reflected power at Port 1 remaining below approximately 0.07 $\mathrm{W}$. Owing to the matched input and output waveguide widths, the channel supports efficient tunneling of EM waves through the low-loss ENZ waveguide. While the finite channel width leads to a slight increase in the reflection coefficient $\mathit{R}$, the transmission efficiency consistently exceeds 93% across the entire tuning range.

To characterize the frequency-dependent performance, the simulated results of the proposed power divider with exemplary power ratios of 1:1, 1:2, and 0:1 between Ports 2 and 3 are presented in Figure 3a–c, respectively. As the first example of equal power splitting, copper plate 2 is aligned with the input waveguide axis with $\theta =0$° and $b_1=b_2$. As shown in Figure 3a, simulated $S_{21}$ and $S_{31}$ are identical to −3.07 dB at the ENZ frequency, which conforms to the design target and leads to a transmission efficiency of 98%. However, the reflection introduced by copper plate 2 slightly shifts the optimal operating frequency. Additionally, a balanced power flow density between the two sub-channels is observed in Figure 3d, and the elevated power flow density within the ENZ channel is mainly due to its reduced cross-sectional area. The power split ratio of 1:2 between Port 2 and Port 3 is investigated as the second example of imbalanced power division. By setting the rotation angle $\theta$ to −23.5°, simulated $S_{21}$ = −4.86 dB and $S_{31}$ = −1.78 dB are obtained at 3 GHz, as plotted in Figure 3b, and the simulated power flow densities within the sub-channels are approximately $0.65\times {10}^4$ $\mathrm{W}/{\mathrm{m}}^2$ at Port 2 and $1.3\times {10}^4$ $\mathrm{W}/{\mathrm{m}}^2$ at Port 3, as exhibited in Figure 3e, respectively. The desired power splitting is implemented with a high transmission efficiency of 99%.

A single output port configuration is explored as the third example, in which copper plate 2 is rotated to −29°. This further rotation places the plate near the input waveguide edge, effectively suppressing the EM propagation toward Port 2. It is seen in Figure 3c that the simulated $S_{21}$ drops below −170 dB, and Port 3 retains near-unity transmission with $S_{31}$ = −0.32 dB. As shown in Figure 3f, the power flow within the sub-channels diminishes at Port 2, and nearly all the transmitted energy is effectively directed to Port 3. The transmission efficiency is calculated as 93% and is slightly lower than that in previous cases. According to Equation (2), the reduced transmission efficiency is primarily attributed to increased reflections, which is caused by a reduced number of output ports and mismatches in waveguide widths. It is noted that positive rotation angles invert the magnetic field within the sub-channels, yielding a power splitting pattern exactly opposite to that with negative rotation angles.

In this design, efficient arbitrary power division is achieved by redistributing the magnetic field within the ENZ channel in a mechanically reconfigurable manner. All reconfiguration modes exhibit a reasonable bandwidth around the ENZ frequency, achieving transmission efficiencies above 93%, even when considering conductor loss, dielectric loss, and dimensional tolerances.

3.2. Electrically Reconfigurable ENZ-Based Power Divider Based on Cross-Sectional Area Modulation

An electrically reconfigurable ENZ-based power divider is further proposed to facilitate on-demand control over both power division modes and the number of output ports by dynamically tuning the cross-sectional areas of the output waveguides.

To increase the number of output ports, we propose a double-layer SIW power divider. The equivalent width $a^{\prime }$ of the SIW is calculated from the width a of the conventional rectangular waveguide as [39]

$$a^{\prime }={\displaystyle \frac{2a}{\pi }}\mathrm{arcctg}\left({\displaystyle \frac{\pi W}{4a}}\ln {\displaystyle \frac{W}{4R}}\right),$$

(6)

where $\mathit{R}$ denotes the radius of the metallic vias, and $\mathit{W}$ is the center-to-center spacing between adjacent vias. The formulation is valid under the constraint $W⩽1/20{\lambda }_g$, where ${\lambda }_g$ represents the guided wavelength. When satisfying $R=W/4$, the effective SIW width $a^{\prime }$ equals a, ensuring propagation characteristics equivalent to a conventional rectangular waveguide. Accordingly, the design adopts $\mathit{R}$ = 1.25 mm, $\mathit{W}$ = 5 mm, and $a^{\prime }$ = 50 mm, which correspond to a standard rectangular waveguide with a width of 50 mm. Owing to the magnetic field homogeneity at the cut-off frequency, the power delivered to each output port is primarily determined by the cross-sectional areas of the output waveguides. By precisely tailoring these dimensions, various power division ratios can be effectively realized, including both equal and unequal splitting among the output ports.

As illustrated in Figure 4, the proposed electrically reconfigurable SIW power divider is fully enclosed by metallic walls with a height of h = 6 mm, except for at the SIW interfaces. A rectangular copper block embedded in the center of an air-filled waveguide forms a rectangular-loop-shaped transmission channel with a width of $w_1$ = 0.2 mm and a length of l = 50 mm. This configuration yields a cut-off frequency of 3 GHz and facilitates equivalent ENZ tunneling behavior. The central ENZ channel interconnects four SIWs filled with F4B ($\epsilon =2.2$ and tan$\delta$ = 0.001), with Port 1 designated as the input and Ports 2 to 4 designated as the outputs. To suppress inter-channel coupling and ensure EM compatibility, metallic isolation layers with heights of $h_5$ = 2 mm and $h_6$ = 3 mm are incorporated between adjacent SIWs. The cross-sectional areas of the output waveguides are precisely engineered with $h_1$ = 3 mm, $h_2=h_3$ = 1 mm, and $h_4$ = 2 mm.

To enable effective dynamic modulation of the EM responses, four PIN diodes (${\mathrm{D}}_1$–${\mathrm{D}}_4$) are strategically positioned near the region with a concentrated electric field intensity. Specifically, ${\mathrm{D}}_2$ and ${\mathrm{D}}_4$ are mounted at the midpoints of the upper and lower channel walls, whereas ${\mathrm{D}}_1$ and ${\mathrm{D}}_3$ are mounted at the midpoints between the central isolating layer and the rectangular copper block. Each diode (SMP 1320-079LF, Skyworks) is modeled as a series resistor–inductor–capacitor (RLC) circuit, with parameters $R_d$ = 0.5 $\mathsf{\Omega }$ and $\mathit{L}$ = 0.7 nH in the ON state and $\mathit{L}$ = 0.5 nH and $\mathit{C}$ = 0.24 pF in the OFF state. The cathodes of these diodes are connected to the inner rectangular copper block, while the anodes are individually connected to the four outer walls of the ENZ channel. When the PIN diodes are switched on, they effectively behave as conductive “copper plates”, blocking or redirecting EM wave propagation within the rectangular-loop channel. Conversely, when the diodes are switched off, nearly unobstructed wave transmission through the rectangular-loop ENZ channel is enabled. By appropriately switching the diodes, both the output ports and internal EM field distributions can be dynamically reconfigured, enacting on-demand power division with varying cross-sectional areas.

As an example of equal power splitting between two output ports, State 1 is realized by switching on ${\mathrm{D}}_3$ and ${\mathrm{D}}_4$ while turning off ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$. In this configuration, Port 4 is effectively isolated and results in a two-port output of Ports 2 and 3. The power ratio between them is determined by their corresponding cross-sectional areas, which are designed to be equal and confirmed by the simulation results shown in Figure 5a. In this state, the transmission efficiency reaches 93%. In contrast, State 2 is configured by turning on ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$ while switching off ${\mathrm{D}}_3$ and ${\mathrm{D}}_4$, thereby blocking Port 2 and redirecting the energy to Ports 3 and 4. The area ratio between Ports 4 and 3 is approximately 2:1, leading to an unequal power split of 2:1, as illustrated in Figure 5b. Notably, the total output width in this case exactly matches that of the input, minimizing the reflection coefficient and achieving a high transmission efficiency of 97%. When all diodes are turned off (State 3), Ports 2, 3, and 4 all serve as output ports, resulting in a three-port power distribution. The simulated results exhibit an asymmetric power ratio close to 1:1:2, as shown in Figure 5c. This state approximately satisfies the ideal matching condition, resulting in a slightly lower transmission efficiency of 95%. Due to the ENZ tunneling effect, energy can freely propagate to the activated ports, leading to unavoidable inter-port coupling and reduced isolation. Despite the differences in output port numbers and power splitting ratios, it is observed in Figure 5d–f that the simulated magnetic fields in the output waveguides are in phase and of equal amplitude.

These simulated results collectively confirm that the proposed ENZ-based RPD exhibits stable and efficient transmission performance in multiple operation modes. Notably, delicate variation in the cross-sectional areas of the output SIWs allows for controllable and on-demand power division among the output ports. Furthermore, the number of output ports can be dynamically reconfigured, allowing flexible transitions between two-port and three-port power division configurations. The achievable power split ratios are not limited to those analyzed herein, and, in general, more possibilities could be realized by varying the polarity configurations of the PIN diodes. In addition to high reconfigurability and mode selectivity, the proposed power divider features a simple structure and practical feasibility.

4. Conclusions

In summary, this work presents versatile, high-efficiency, reconfigurable power division based on the tunneling characteristics of waveguide ENZ media. Two reconfigurable control mechanisms are introduced, supporting a flexible power distribution by tuning the magnetic field within the ENZ channel or adjusting the widths of the output waveguides. To validate these theoretical concepts, two distinct experimentally feasible implementations are presented. Compared with previous works reporting insertion losses of 0.88 dB [7] and 0.60 dB [33], the proposed design achieves a transmission efficiency of 93%, corresponding to an insertion loss of approximately 0.31 dB. It also enables dynamic tunability of power division ratios and flexible reconfigurability of output ports while maintaining a reasonable bandwidth around the ENZ frequency and a simple profile. These results confirm the feasibility of efficient ENZ-based RPDs, which may pave the way for RF front ends, real-time wireless communications, and integrated ENZ-based microwave systems.