A power-distribution-ratio real-time tunable power divider based on active plasmonic waveguide

Abstract

In this paper, a dynamic controllable unequal power divider based on spoof plasmonic waveguide loaded with varactors is proposed, which is mainly composed of a double-sided corrugated metal strip coupled with two branch single-sided plasmonic strips. In order to allocate electromagnetic (EM) energy flexibly, varactors are loaded in the coupling gap to adjust the coupling coefficient. Both simulation and measurement results prove that the proposed plasmonic power divider can adjust and control the power distribution ratio in a real time manner in the range of 4.5–6 GHz frequency band with a maximum power allocation ratio of 2.7 at 5.77 GHz. The plasmonic power divider possesses good frequency selectivity, especially the dynamical response to varactor voltages. Therefore, it can be expected that the proposed unequal power divider will stimulate further research on spoof surface plasmon polaritons (SSPPs) for the design of new planar active microwave components, circuits and systems in the future.

Introduction

Surface plasmon polaritons (SPPs) are collective oscillations of free electrons, which are surface EM waves bound at the interface between metal and dielectric. Because SPPs have strong surface field localization characteristics, which makes them extremely sensitive to surface conditions and can break through the diffraction limit, they have brought a huge boost to the development of plasma devices^{1,2,3,4,5,6,7,8}. However, the rapid progress in microwave communications poses the fundamental question about the limited frequency band in nature SPPs, hence it encourages researchers to break such dilemma. Pendry et al.⁹ proposed a new paradigm of SSPPs or designed SPPs (DSPPs), which has revolutionized the traditional optical SPPs by enabling analogous SPPs working at microwave and terahertz band, enriching flexible EM devices and simultaneously numerous compact EM functionalities.

Since then, SSPPs transmission lines and related devices have been widely studied due to their unique advantages, such as wide bandwidth and high integration. Based on this, the power dividers using SSPPs structure has been developed since it benefits for the miniaturization, flexible adjust and broadband properties. However, in the past decade, SSPPs power dividers have been only designed to achieve equal division of EM power^{10,11,12,13,14,15,16,17,18,19,20}. For example, Gao et al.¹⁰ proposed an ultra-wideband Y-type power divider based on planar terahertz plasma metamaterials. Later, Zhou et al.¹⁹ also proposed a four-way SSPPs power divider with elliptical ring periodic structure, it can divide the EM wave into four paths on average, and realize four equal divisions of SSPPs EM wave.

Nevertheless, the above SSPPs power dividers are mainly designed for static power splitting, which cannot adapt the development of the fast modern EM devices. Yong et al.²¹ first proposed a new type of unequal SSPPs power divider, by changing the geometric configuration of the structure, it is easy to design and realize different broadband power distribution ratios in SSPPs waveguides. However, it works in a static mode which can not split EM power dynamically and arbitrarily. However, in modern EM communication, large amount of dynamic power dividers are urgently demanded since they can be utilized as feeding ports for continuous beam scanning antennas, which can save power consumption compared with the traditional feeding strategy, where equal power splitting loaded with attenuator is employed. Yet, as far as our knowledge, there have not been any report about the real time adjustable SSPPs power divider for unequal division of EM power.

In the present paper, we propose a dynamically controllable power divider with unequal power division. The schematic structure of the power divider is illustrated in Fig. 1. The parameters of this specific structure can be accurately retrieved as: $\hbox{T} = 98.8$ mm, $\hbox{H} = 104$ mm, $\hbox{L} = 0.5$ mm, $\hbox{a} = 5$ mm, $\hbox{b}= 22.5$ mm, $\hbox{c} = 21$ mm, $\hbox{d} = 34.7$ mm, $\hbox{f} = 9.5$ mm, $\hbox{e} = \hbox{g} = 5$ mm, $\hbox{h} = 30$ mm, $\hbox{k} = 10$ mm, $\hbox{p} = 4.5$ mm, $\hbox{m} = 0.2$ mm, $\hbox{n} = 0.2$ mm, $\hbox{o} = 0.5$ mm, $\hbox{q} = 1.7$ mm, $\hbox{v} = 3$ mm, $\hbox{gap} = 0.2$ mm, $\hbox{s} = 6$ mm, $\hbox{u} = 2.3$ mm, $\hbox{w} = 1.55$ mm. The simulation and measurement results show that the power distribution ratio of the two output ports can be adjusted over a wide range of 1–2.7 and it can be changed in real time by tuning the capacitance value of the varactor loaded into the coupling gap.

Fig. 1

Schematic diagram of the proposed plasmonic power divider.

Coupling principle

In order to reveal the unequal powder divider theoretically, the dispersion curve and electric field distribution of T-shaped and rectangular unit structures are investigated firstly. The dispersion curve is obtained through theoretical calculation and numerical simulation of the eigenmodes. The dispersion curve is carried out by CST EM simulation software. The substrate is Rogers 5880, with the dielectric constant of 2.2 and the dielectric loss angle tangent being 0.0009. The periodic length is $\hbox{p} = 4.5$ mm, and the metal is annealed copper. By placing the proposed slot unit in the air box, the dispersion characteristics of the periodic arrangement of T-shaped and rectangular units (marked as “+” structures) are studied. The boundary in the x direction is set as the periodic boundary, and the other boundaries in the y and z directions are set as the electric field boundary with an open space being 60 mm. When the phase difference between the two periodic boundaries is swept from $0^{\circ }$ to $180^{\circ }$, all eigen-frequencies are calculated.

The input port starts from a single-mode waveguide structure, and it couples to two closely packed plasmonic waveguides with mirror symmetry arrangement (see Fig. 1). The coupling is realized by a separation gap of 0.2 mm, which is about 0.01 wavelengths at 5 GHz. Since a key goal in this paper is to evaluate as well as enhance the coupling between the two plasmonic waveguide, it is reasonable for us to consider whether the propagation constant and the mode pattern of the two waveguides are similar or not for coupling. The simulated dispersion curve is illustrated in Fig. 2, where k represents the propagation constant on the x axis. It can be seen that the dispersion curves of T-shaped and “+” structures are obviously deviated from the light, indicating that these two kinds of structure can both constrict the surface EM wave. Also, it can be seen from Fig. 2b, due to the mirror symmetry of the structure, it can support two orthogonal eigenmodes with either symmetric (Es (x, y)) or antisymmetric (Ea (x, y)) electric field distribution with respect to the y axis. We present electric-field distributions on the plasmonic waveguide on the x − y plane, which is 1.0 mm above the structure surface at 5 GHz, in which the fields are excited from the left of the SSPP waveguide. From the electric field distribution of T-shaped branch and rectangular metal, it can be observed that the electric field energy is symmetrically (Es (x, y)) concentrated on the branch and the two EM modes are in high similarity. therefore, the phase matching can be automatically satisfied between the two coupled structures and a good transmission performance can be expected.

Fig. 2

(a) The dispersion characteristics of the master and slave unit cell on the waveguide, it should be noted that the “+” structure represents the element structure of the main waveguide, and the T-shaped represents the element structure of the slave waveguide. (b) Electric field distribution of main waveguide and slave waveguide at 5 GHz.

Based on the concept of microwave network, the coupling part can be described by a four-ports network, whose transmission matrix can be obtained as refer to (1):

$$\begin{aligned}&\left[ \begin{array}{cc} b_1 & b_2 \\ b_4 & b_3\\ \end{array}\right] =e^{-j\beta x}\left[ \begin{array}{cc} \cos (kx) & -i\sin (kx) \\ -i\sin (kx) & \cos (kx)\\ \end{array}\right] \cdot \left[ \begin{array}{cc} a_2 & a_1 \\ a_3 & a_4\\ \end{array}\right] \end{aligned}$$

(1)

where k is the coupling coefficient and x is the length of the coupling part. Suppose the EM wave of the SSPP waveguide propagates along the x direction, the transmission coefficient $S_{21}$ of the coupling part can be derived as:

$$\begin{aligned} S_{21} =\frac{2}{A+B/Z_{0}+CZ_{0}+D} =-\frac{2i\sin (kx)e^{j\beta x} }{e^{2j\beta x}+\sin ^{2}(kx)} \end{aligned}$$

(2)

where $\beta$ is the propagation constant. From Eq. (2), it can be noticed that $S_{21}$ is decided by the coupling coefficient k as well as coupling length x. The $S_{21}$ curves have distinct dips in the transmission spectra of SPPP waveguides, which suggests that when sin(kx) equals zero, the E-field energy can not propagate through the coupling part of the SSPPs waveguide and the null point in the $S_{21}$ item can be manipulated through coupling in this case.

According to the coupled mode theory, the transmission power $P_1(\textit{x})$ transmitted through the propagation waveguide and the coupling power $P_2(\textit{x})$ of adjacent SSPPs waveguides can be written as refer to²²:

$$\begin{aligned} P_{1} (x)&=P_{0} \cos ^{2} (kx)e^{-2ax} \\ P_{2} (x)&=P_{0} \sin ^{2} (kx)e^{-2ax} \end{aligned}$$

(3)

where $P_0$ is the initial incident power, a is the attenuation constant. The frequency-dependent coupling coefficient k can be expressed as:

$$\begin{aligned} k=\frac{\omega \varepsilon _{0}}{4}\iint \left( n^{2}-n_{0}^{2}\right) \left[ E_{1x}^{*}\cdot E_{2x}+\left( \frac{n_{0}^{2}}{n^{2}}\right) E_{1y}^{*}\cdot E_{2y} \right] \end{aligned}$$

(4)

where n and $n_0$ are the refractive index distribution of the surrounding medium and waveguide respectively, $E_{jx}$ and $E_{jy}$ ($\hbox{j} = 1$, 2) are the normalized transverse and longitudinal electric field distribution of the plasma waveguide respectively.

From Eqs. (3) and (4), it can be known that the EM energy of the two waveguides can be converted to each other, which mainly depends on parameter k and x. Here we only consider the variation of k, and the coupling length x is regarded as a constant. In this case, the coupling energy between the two waveguides depends on the electric field distribution ($E_{jx}$ and $E_{jy}$), in terms of Eq. (4). Therefore, we can control the coupling energy between SSPPs by regulating the field distribution. Generally, for passive coupling devices, the electric field distribution can be controlled by changing the gap between SSPPs waveguides²³. This is mainly because the energy becomes more prominent with the decrease of the gap (the capacitance becomes larger). On the contrary, with the increase of gap, the coupling energy become weaker since the capacitance becomes smaller^24,25. Thus, the design of an appropriate separation interval is very important.

We simulate the S-parameters curve of the power divider structure with different coupling gap values, and the results are shown in Fig. 3. Taking $\hbox{gap} = 0.05$ mm as an example, it be seen that the EM wave input from one port can be equally transmitted into two single-sided metal structure branches through coupling, and the insertion loss in the wide range of 4.5–6 GHz is less than $-4$ dB. The S₂₁ parameter rapidly decreases at a frequency of approximately 5.4 GHz, which means that the transmission zero has isolated the working frequency band. Then, it rises again at a frequency of approximately 5.6 GHz. The presence of the transmission zero should be due to the coupling gap of the SSPPs waveguide and the null point can be manipulated through different gap value, or different coupling coefficient. From Eq. (2), we know that when 2isin(kx)$e^{j\beta x}$ is taken as infinite, the transmission zero will be introduced. As the gap value decreases, the coupling energy increases. As indicated by Eq. (4), as the coupling coefficient increases, sin(kx) is an increasing function in the range of (0, $\pi$), and an increase in the (kx) production will cause the increase of function. Then, to ensure that 2isin(kx)$e^{j\beta x}$ remains constant, the $e^{j\beta x}$ item will decrease, which guarantees the reduced value of $\beta$ since $e^{j\beta x}$ is an increasing function. This confirms the drift of the transmission zero, as shown in Fig. 3.

Fig. 3

Transmission coefficient S₂₁ with different values of the coupling gap. The inset shows the E-field distribution with coupling $\hbox{gap} = 0.05$ mm @5.5 GHz (isolation band) and 5.8 GHz (pass band).

It can be deduced from Fig. 3, the transmission and coupling power ($P_{2}$ and $P_{3}$) of the SSPP waveguide can be adjusted by changing the gap. And the smaller the gap, the greater the coupling strength. In terms of the exponential attenuation characteristics of SSPPs waveguides, the E-field strength rapidly weakens in the normal direction at the coupling interface of the waveguide. Therefore, the coupling performance will decrease with increasing distance, thus, we choose the coupling gap of 0.15 mm as the optimum value for the following investigation. Also, it is worthy mentioning that the power divider can be extended to working in multi-band characteristic by compound corrugation engineering^26,27.

Design of plasmonic unequal power divider

However, it is still a static working mode through controlling the coupling strength via the gap. In order to further dynamically control the propagation and suppression of SSPPs wave in the propagation band, we load a row of varactor diodes at the coupling gap and change the coupling coefficient of the power divider by changing the capacitance of the varactor diodes, as Fig. 4a. For ease of study, the row of capacitors near port 2 is labeled C2, and the row of capacitors near port 3 is labeled C3, coupling distance $\hbox{gap} = 0.15$ mm. The capacitance regulation of the varactor diode is achieved by connecting the branches of the main waveguide and the slave waveguide to the biasing circuit for applying bias voltage, as depicted in Fig. 4b. The sector structure of the slave waveguide is utilized to separate the direct biased voltage and the alternating signal as to prevent it from influencing on the measuring instrument.

Fig. 4

(a) Enlarged view of plasmonic waveguide loaded with varactors. (b) Photograph of the experimental system.

The S-parameters are measured using the Agilent N5230C network analyzer. the model of varactor diode is chosen as SMV2202-040LF, which is a silicon hyperabrupt tuning varactor diode with its maximum capacitance value being 3.14 pF under no biased voltage, and with the parasitic resistance of $2 \Omega$ inductance of $4.5 \times 10^{-10}H,$ and its capacitance decreases with the increase of the bias voltage.

We refer to the data given by the manufacturer (Skyworks Solutions Inc.) that the capacitance value of the varactor varies from 3.14 pF, 1.34 pF, 0.76 pF to 0.45 pF as the biased voltage varies from 0 V, 4 V, 8 V to 12 V respectively. The corresponding simulation and measurement results are carried out according to the above data, as can be seen from Fig. 5, the experimental results are basically consistent with the simulation results, Fig. 5a–d respectively illustrate the variation of S-parameters as the capacitance of the varactor diode changes. As shown in Fig. 5a, when the capacitance values of the varactor diodes are equal, the output power at the output port is the same. However, as the difference in capacitance values of the varactor diodes increases, the gap between S₂₁ and S₃₁ also widens. As depicted in Fig. 5d, when the capacitance of varactor diode C3 is 3.14 pF and C2 is 0.45 pF, the ratio of S₂₁ to S₃₁ can reach 2.77 at 5.77 GHz. However, it can be observed that regardless of whether it is simulation or actual measurement, the reflection is relatively large. The reason for this occurrence is due to the mismatch of the impedance of the structure itself and the varactor diode, which can be further improved by optimizing the geometric parameters of the structure.

Fig. 5

The measured and simulated S-parameters of the proposed plasmonic divider. (a) The capacitance value is set as $\hbox{C2} = \hbox{C3} = 3.14$ pF during simulation, and the voltage is set to 0 V during measurement. (b) The capacitance value is set as $\hbox{C3} = 3.14$ pF and $\hbox{C2} = 1.34$ pF during simulation, and the voltage is set to 4 V during measurement. (c) The capacitance value is set as $\hbox{C3} = 3.14$ pF and $\hbox{C2} = 0.76$ pF during simulation, and the voltage is set to 8 V during measurement. (d) The capacitance value is set as $\hbox{C3} = 3.14$ pF, $\hbox{C2} = 0.45$ pF during simulation, and the voltage is set to be 12 V during measurement. (e) The reflection performances under different capacitance during simulation. (f) The reflection behaviors under different voltages during actual measurement.

In order to have a direct insight into the field propagation and EM energy confinement on SSPPs power divider loaded with varactor diodes, we simulate and measure the electric field distribution of the power divider at 5.8 GHz frequency, where the capacitance values of the varactor diode are chosen as $\hbox{C2} = 3.14$ pF (0 V) and 0.45 pF (12 V), respectively, as shown in Fig. 6. The measurement result show that when $\hbox{C2} = 3.14$ pF, the allocated power P2 is equal to P3 because $\hbox{C2} = \hbox{C3}$, As shown in Fig. 6a. Meanwhile, In Fig. 6b, when $\hbox{C2} =0.45$ pF, the allocated power P2 and P3 becomes unequal due to the different capacitance between C2 and C3 at this moment, which is consistent with the conclusion drawn above in Fig. 5.

Fig. 6

The electric field distribution along the x-axis in the plane of $\hbox{z} = 1$ mm above the plasmonic waveguide at 5.8 GHz. (a) Electric field profile measured when loading the 0 V bias voltage. (b) Electric field profile measured when loading the 12 V bias voltage.

In order to clearly reveal the physical mechanism of the unequal plasmonic divider, we extract the normalized coupling strength of the SSPPs coupling structure from the simulation results, refer to Eq. (3). The results are depicted in Fig. 7. It can be look that the coupling strength of the structure increases with the increase of voltage, which is consistent with the conclusion drawn in Fig. 5.

Fig. 7

The normalized coupling strength with different coupling capacitors.

We then measure the S-parameter by changing the voltage of the varactor diode to adjust its capacitance value and reveal the relationship between the bias voltage and the power ratio P2/P3 in different frequency bands, as shown in Fig. 8. It can be clearly seen that the power ratio P2/P3 increases significantly with the increase of the bias voltage, which indicates that the varactor has an obvious regulating effect on the power ratio of the proposed power divider.

Fig. 8

Three-dimensional hologram of power distribution ratio with different voltages and working frequencies. In experiment, the voltage at both ends of the varactor C3 is constant at 0 volts, and the voltage at both ends of the varactor C2 changes in real time.

It can be deduced from Eq. (3) that the transmission power ratio of the two ports can be expressed as:

$$\begin{aligned} \sqrt{\frac{P_{21} }{P_{31} } =\frac{\tan (k_{21} x)}{\tan (k_{31} x)} } \end{aligned}$$

(5)

where $k_{21}$ and represent the coupling coefficient between the main waveguide and the slave waveguide 2 and the slave waveguide 3, respectively. Equation (5) reveals that the power distribution ratio of the structure is directly related to the ratio of the coupling coefficient. The larger the coupling coefficient is, the stronger the coupling ability becomes.

Table 1 compares the power divider performances with previously related works. Compared with other reported power dividers, as can be observed, the proposed power divider possesses controllable performance and the power divider can export unequal power ratio when loaded with varactor diodes, which provides great flexibility for designing multifunctional MW devices.

Table 1 Comparison with different plasmonic power dividers.

Conclusion

We propose a novel plasmonic power divider composed of plasmonic waveguide coupled with two slave plasmonic waveguides. The unique and beneficial features of field confinement and power splitting of the plasmonic waveguides have been verified by both S-parameter measurement and near-field mapping. Compared with traditional plasmonic dividers, the proposed power divider can work in a real time manner and the power can be converted from equal to unequal allocation with good spectral controllability, which possesses the potential to achieve multi-function. The effectively variable power distribution is very attractive for a variety of applications, such as high-speed beam scanning antennas as well as conformal high-density microwave circuits.