## Abstract

Wireless power transfer (WPT) has triggered immense research interest in a range of practical applications, including mobile phones, logistic robots, medical-implanted devices and electric vehicles. With the development of WPT devices, efficient long-range and robust WPT is highly desirable but also challenging. In addition, it is also very important to actively control the transmission direction of long-range WPT. Recently, the rise of topological photonics provides a powerful tool for near-field robust control of WPT. Considering the technical requirements of robustness, long-range and directionality, in this work we design and fabricate a one-dimensional quasiperiodic Harper chain and realize the robust directional WPT using asymmetric topological edge states. Specially, by further introducing a power source into the system, we selectively light up two Chinese characters, which are composed of LED lamps at both ends of the chain, to intuitively show the long-range directional WPT. Moreover, by adding variable capacitance diodes into the topological quasiperiodic chain, we present an experimental demonstration of the actively controlled directional WPT based on electrically controllable coil resonators. With the increase in voltage, we measure the transmission at two ends of the chain and observe the change of transmission direction. The realization of an actively tuned topological edge states in the topological quasiperiodic chain will open up a new avenue in the dynamical control of robust long-range WPT.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Coil resonators play an important role in magnetic field control; they can confine the magnetic field and greatly improve the efficiency of wireless power transfer (WPT) [1–3]. For a standard WPT system, it is mainly composed of two coupled coil resonators, which are placed on the source and receiver sides, respectively. WPT has been widely used in various fields, such as mobile phones and medical-implanted devices. However, there are some aspects of WPT applications that should be noted. Firstly, although the transmission distance based on the coupled coil resonators is larger than that of the traditional magnetic induction scheme [1–3], it is still difficult to meet the needs of long-range WPT because of the limitation of the coupling of evanescent wave [4]. Secondly, most WPT schemes have inherent sensitivity to transmission distance and structural disturbance, which is an essential challenge for various application scenarios [5–7]. Thirdly, in order to realize electromagnetic compatibility, ferrite is often added to WPT device for unidirectional magnetic shielding. However, the ferrite not only increases the cost, but also seriously increases the size and weight of the device [8–10]. Therefore, the fundamental challenges for WPT in radio frequency (RF) regime based on the coupled-resonant-coil is the long-range [11], directional [12] and robustness [13–16].

As a new generation of WPT technology, the topological edge states (TESs) in topological chains have been proposed to be able to achieve robust long-range WPT [17–21]. At present, topological photonics involving multiple topological models by mapping condensed matter physics have been carefully studied based on artificial microstructures [22–24]. These photonic topological structures can not only conveniently uncover some interesting topological phases and the TESs in experiments, but also provide powerful means for electromagnetic wave manipulation [25,26]. The TESs in classical wave systems can explore much interesting physics which involves the nonlinear [27,28], non-Hermitian properties [29–37] and quantum optics [38,39], which enable some unique applications including high-harmonic light [28], sensors [34], lasers [36,37], and filters [40]. Specially, one-dimensional (1D) topological chains have opened up exciting directions in topological photonics. Although the structure of 1D chains is simple, they have rich topological properties, such as topological phase transition, band inversion, robust TESs and so on. One of the simplest 1D topological chains is the dimer chain, i.e., Su-Schrieffer-Heeger model [41], in which a pair of TESs are localized symmetrically at two ends of the chain. It is the TESs that leads to 1D topological chain being used for robust long-range WPT [17–21]. It should be noted that although the topological dimer chain is similar to the transitional Domino structure composed of the coupled resonators for long-range WPT [11], the TESs in topological structures are topological protected. The topological array composed of coil resonators for WPT inherits the physical properties of TESs and has good electromagnetic compatibility because it is immune to impurities and perturbations [42]. In addition, this robust WPT inherits the topology protection of the TESs rather than resort to the stable mode of the nonlinear oscillatory circuit [43], metamaterials [44] and third-odd *PT* symmetric platform [45,46]. Therefore, the proposed WPT devices not only provides a deeper understanding for the long-range WPT, but also is very useful in various application scenarios, such as the robots whose structure will often deform.

Different from the symmetrical distribution of the TESs in the topological dimer chain, researchers have uncovered the interesting asymmetric topological TESs in 1D trimer [47] and quasiperiodic chains [18,48–56]. The TESs in the quasiperiodic Harper chain are localized at left or right end of the chain [18,50], which may be used for the directional WPT [57,58]. In this work, we theoretically design and experimentally fabricate a finite quasiperiodic Harper chain based on the ultra-subwavelength coil resonators for long-range WPT. By using the near-field measurement technology, we can obtain the density of states (DOS) spectrum of the topological Harper chain. In addition, the distribution of the asymmetric TESs in the Harper chain is observed from the local density of states (LDOS) spectrum. Specially, using the asymmetric TESs, we selectively light up two Chinese characters composed of LED lamps at both ends of the chain, which intuitively show the directional WPT in the topological Harper chain. Moreover, given the robustness of TESs, the designed WPT device will be robust to the disorder perturbation inside the structure. The TESs for directional WPT not only extend previous research work on long-range WPT, but they have a circuit structure which is easier to integrate and for active control by considering the variable capacitance diodes (VCDs) into the resonators [59]. As a result, by adding electrical VCDs into the system, we experimentally observe the actively tuned transmission direction by modulating the external voltages applied in the VCDs.

This paper is organized as follows: Sec. 2 covers the design of the topological Harper chain and the demonstration of the robust asymmetric TEMs with topological protection; in Sec. 3, the experiments are carried out to verify the long-range directional WPT based on the topological Harper chain; in Sec. 4, an active directional WPT system is constructed based on the coil resonators with VCDs. Finally, Sec. 5 summarizes the conclusions of this work.

## 2. Characterization of the asymmetric TEMs in a topological quasiperiodic Harper chain

In the experiment, the 1-D Harper chain is constructed by ultra-subwavelength coil resonators. The coil resonator is placed on a polymethyl methacrylate (PMMA) substrate with thickness of *h* = 1 cm as shown in Fig. 1(a). All the coil resonators in the Harper chain are identical with the resonant frequency of *f*_{0} = 5.62 MHz, which is determined by the loaded lumped capacitor *C* = 100 pF and the geometric parameters including the inner diameter *D*_{1} = 5.2 cm and outer diameter *D*_{2} = 7 cm. Here, considering the miniaturization of the device, we study the topological WPT in MHz regime; however, similar results can be well extended to kHz regime. For the deep sub-wavelength (*D*_{2} < λ/100) coil resonator, its radiation ability is very weak. Because the coil resonator can't radiate electromagnetic wave like antenna, the composition of far-field can be ignored. In addition, the magnetic field of the coil resonator can be well localized in the inner of the structure, and its intensity will decay exponentially with the distance from the coil. Therefore, the coupling between different coil resonators only considers the near-field coupling [1,11,14]. In particular, two Chinese characters composed of LED lamps connected to non-resonant coils are loaded on the back layer of the left end and right end of the Harper chain, respectively. The non-resonant coil is shown in Fig. 1(b) and the ports ‘*A*’ and ‘*B*’ are connected with the LED lamps. Once the magnetic field in the top coil is strong enough, the LED lamps loaded in the back layer of the both ends resonators can be lighted up. Figure 1(c) shows the schematic of a Harper chain with 16 same resonators. In this tight binding model, the quasiperiodic modulation is controlled by tuning the coupling strength. Specially, by breaking the translation symmetry rather than the rotational symmetry, the topological protected asymmetric edge states are achieved in the quasiperiodic chain.

We start with the Harper model based on the tight-binding mechanism. The topological Harper chain with finite resonators in a quasiperiodic arrangement is controlled by tuning the coupling strength as [18]:

*n*

^{th}and (

*n*+1)

^{th}resonators, ${\kappa _0} = 1$ is a scaling constant, $\varepsilon = 0.5$ is the coefficient that controls the strength of the modulation, and $\tau$= ${{\left( {\sqrt 5 + 1} \right)} / 2}$ is the golden ratio. $\phi$ denotes the topological parameter. We vary the value of $\phi$ from 0 to $2\pi$ and obtain the topological band diagram. For the finite-size Harper chain with 16 resonators, we calculate the projected band structure, as shown in Fig. 2(a). The left and right TESs exist in two bandgaps, which are marked by green and red curves, respectively. Here, the value of $\phi$ is 4 marked by the black dotted line, to facilitate viewing. The values of the coupling strengths from ${\kappa _1}$ to ${\kappa _{15}}$ are calculated from Eq. (1) and shown by the pink dots in Fig. 2(b). Table 1 shows the corresponding relation between the distances of adjacent resonators and the coupling coefficients. In general, the definition of long-range under near-field WPT regime is based on the ratio of transmission distance

*d*and the radius of the coils

*r*[11]. Here, the topological chain arranged in straight coaxial form with a

*d/r*ratio equal to 13.1 (i.e., 45.8/3.5), which is termed ‘long-range’ WPT for a ratio of

*d/r*> 10.

Based on the near-field detection method, we measure the DOS spectrum of the finite 1D topological Harper chain. The near-field magnetic probe is a loop antenna, which is connected to the port of the vector network analyzer (Agilent PNA Network Analyzer E5071C). The radius of the loop probe is 2 cm. It can be taken as a non-resonant antenna with high impedance. This small loop antenna acts as a source to excite the sample and then measure the reflection. And the LDOS of each site is obtained from the reflection by putting the probe into the center of the corresponding resonator. The DOS spectrum is obtained by averaging the LDOS spectral over all 16 sites. Figure 3 shows the measured DOS spectrum of the topological Harper chain. It should be noted that for the topological Harper chains, the left and right edge states exist in two bandgaps. Here, considering that the low-frequency modes are less affected by the loss [21], we observe the topological edge modes in the low-frequency bandgap, and use them to realize the directional WPT. The left edge state (*f* = 5.26 MHz) and right edge state (*f* = 5.45 MHz) exist in the bandgap of the spectrum.

Figure 4 shows the calculated and measured LDOS distributions of the left edge state and right edge state, which are marked by the green and red arrows in Fig. 3(a), respectively. For the left edge state (*f* = 5.26 MHz), its LDOS is mainly distributed in the left end of the chain, as shown in Figs. 4(a) and 4(c). And Figs. 4(b) and 4(d) shows the LDOS of the right edge state (*f* = 5.45 MHz) is mainly distributed in the right end of the chain. It can be clearly seen that the calculated results are in good agreement with the experimental results. The deviation mainly come from the errors of sample construction and experimental measurement.

## 3. Directional long-range WPT implemented by quasiperiodic Harper chain with asymmetric TESs

After determining the asymmetric TESs in the Harper chain, we further observe the efficient transmission of TEMs based on the standard multi-coil WPT system, as shown in Fig. 5. The source coil is placed at the center of the chain as transmitter and two non-resonant receiving coils are placed at the left and right ends of dimer chain as receiver.

The WPT system of magnetic resonance has been demonstrated to be the powerful tools to improve the functionalities and obtain new performance beyond the WPT systems of magnetic induction [1–3]. In particular, magnetic resonance WPT can achieve long-range and efficient transmission [11]. However, directional long-range WPT with topological protection has not been observed. Figure 6 shows the transmission ratio of the two TESs in topological Harper chain, in which the transmission ratio of the left edge state and right edge state (*S _{L}/S_{R}*) and the transmission ratio of the right edge state and left edge state (

*S*) are marked by green line and purple line, respectively. It can be clearly seen that the

_{R}/S_{L}*S*is significantly higher than

_{L}/S_{R}*S*for the left edge state (

_{R}/S_{L}*f*= 5.26 MHz), and the

*S*is significantly higher than

_{R}/S_{L}*S*for the right edge state (

_{L}/S_{R}*f*= 5.45 MHz), which clearly illustrate that TESs are selectively localized at the left or right ends of the quasiperiodic chain at different frequencies. Because Harper chain is an asymmetric structure, it not only leads to the different distribution of left and right edge states, but also leads to the different transmission efficiency of left and right edge states, so that there will be different transmission ratios

*S*and

_{L}/S_{R}*S*in Fig. 6. Therefore, based on the asymmetric TESs in the Harper chain, the directional WPT is realized.

_{R}/S_{L}Then, a demonstration experiment is carried out to exhibit a high-power and directional WPT control. The high-power signal source (AG Series Amplifier, T&C Power Conversion) instead of the vector network analyzer, is used to excite the TESs. A source non-resonant coil is placed at the center of the structure. In particularly, to show the directional WPT intuitively, the resonator on left (right) end of the chain is added with the non-resonant coil with the Chinese characters for ‘tong’ (‘ji’) composed of LED lamps. At the working frequency of the left edge state (*f* = 5.26 MHz), the Chinese character ‘tong’ is lighted up on the left end of the chain whereas the Chinese character ‘ji’ loaded on the right end of the chain remains dark, as shown in Fig. 7(a). When the Harper chain with perturbation, the Chinese character ‘tong’ is still lighted up and ‘ji’ is still dark in Fig. 7(b), just as the case without perturbation in Fig. 7(a). Then, at the working frequency of the right edge state (*f* = 5.45 MHz), whether there is disorder or not, the Chinese character ‘tong’ is always dark on the left end of the chain whereas the Chinese character ‘ji’ loaded on the right end of the chain remains lighted up, as shown in Figs. 7(c) and 7(d). However, different from the two TESs, Figs. 7(e) and 7(f) show that both the Chinese characters ‘tong’ and ‘ji’ are lighted up weakly at the working frequency of the bulk state (*f* = 5.08 MHz) without the perturbation. Then both ‘tong’ and ‘ji’ are extinguished when perturbation is introduced into the interior of the chain. These results in Fig. 7 indicate that when perturbation is added into the interior of the chain, the TESs will nearly not be affected. As a result, the transmission efficiency of the asymmetric TESs is much higher than the bulk state, which is highly significant for robust directional WPT. Moreover, our results may be extended to other physical platforms such as sound [60] and heat [61] transfer.

## 4. Actively controlled directional long-range WPT using external voltage in topological Harper chain

The technologies require the actively controlled long-range WPT, which will greatly improve the flexibility of WPT devices. However, the experimental realization of the actively tunable robust WPT is still a great challenge. In this section, we introduce an active WPT platform based on the circuit-based coil resonators to experimentally demonstrate the actively controlled long-range robust WPT. Figure 8(a) shows the details of the circuit-based coil resonator in the actively controlled topological WPT. The corresponding full circuit model is shown in Fig. 8(b). The circuit-based coil resonator is composed of a fundamental LC resonator, a VCD component (Philips BB181), and the protection elements. In the experiment, a direct voltage source is connected to the sample from the top. The signal is input from the source non-resonant coil and the protection component is added to avoid the interaction between direct current (DC) source and the signal source.

The experimental data on the relationship between the used voltage and the resonant frequency of the active coil resonator is displayed by the green dots in Fig. 9(a). For a clearer view, we also made a fitted line, which corresponds to the red line. From Fig. 9(a), We can clearly see that as the applied voltage increases, the frequency of the coil resonator increase, which means the capacitance of the VCD decreases. Then we study the property of coupling between two active coil resonators in Fig. 9(b). Take the case U = 0 V for example, the coupling strength of coil resonators decreases exponentially with the increase of distance, which is marked by the purple dots in Fig. 9(b). This property is also applicable to the case U = 4 V [the purple line in Fig. 9(b)]. Specially, coupling strengthen between the coil resonators is nearly independent of the applied external voltage. Therefore, we can easily move the TESs by tunning the applied voltage without changing the structure.

The experimental setup of actively controlled topological Harper chain for directional long-range WPT is shown in Fig. 10. The experimental sample based on near-field coupling mechanism is constructed according to the scheme in Fig. 5. However, it should be noted that in order to achieve actively controlled WPT, all the coil resonators are connected in parallel to a DC source. The *d/r* ratio in this actively controlled Harper chain is equal to 13.6 (i.e., 43.5/3.2). It can be seen from Fig. 9(a) that the frequency of all resonators will change with the change of applied voltage. The signals are generated by a vector network analyzer and then input to the sample based on the non-resonant coil, which functions as the source for the system. In addition, another non-resonant coil is placed at two ends of the chain to measure the transmission on the left and right, respectively.

The asymmetric distribution of the TESs can be used for the selective directional WPT. When a source is put at the center of the chain, for the left and right edge states, the energy mainly propagates to the left end and right end of the chain, respectively. The active tuning of Harper chain is very interesting. In the previous active tuning of periodic dimer chains, one can only tune the state from a bulk state to an edge state or vice versa [62]. However, in the quasiperiodic chain, we can actively tune the state from one edge state to the other edge state with inversion field distribution. In our structure, loading VCDs into the coil resonators can actively control the resonance frequency. So, by applying an external voltage on the diode, we can actively control the topological Harper chain. For example, when the external voltage is a large value U = 4 V, the resonant frequency of the coil resonator is large. In this case, the left edge mode corresponds to *f* = 38.4 MHz and the energy will propagate to the left end when a source is put at the center of the chain, which is shown in Figs. 11(a) and 11(b). Specially, the contrast ratio $|{({S_L} - {S_B})/({S_R} + {S_L})} |$ is $|{(0.32 - 0.08)/(0.32 + 0.08)} |= 0.6$. However, when the external voltage increases to a large value U = 0 V, the resonant frequency of the coil resonator is small. In this case, at *f* = 38.4 MHz, the previous left edge mode becomes the right edge mode and the energy will propagate to the right end of the chain which is shown in Figs. 11(c) and 11(d). Similarly, the contrast ratio $|{({S_L} - {S_B})/({S_R} + {S_L})} |$ is $|{(0.26 - 0.01)/(0.26 + 0.01)} |\approx 0.93$. In addition to the left and right edge states, the other modes in Fig. 11 are bulk states. However, the transmission ratio of *S _{L}/S_{R}* and

*S*are not obvious for the bulk states, and the bulk states are easily affected by the system disturbance. So, by applying external voltages, one can actively tune the state from one edge state to the other edge state at the same frequency. This property can be used to reverse the propagating direction of energy and switch on/off the WPT device in one certain direction.

_{R}/S_{L}## 5. Conclusion

In summary, based on the 1D topological Harper chain composed of ultra-subwavelength coil resonators, we theoretically and experimentally verify that the asymmetric TESs can be used for directional WPT. The asymmetric TESs are further used to selectively light up two Chinese characters at two ends of the chain. Specially, this directional WPT is robust to the disorder perturbation inside the structure. The parameters used in this work are all readily accessible for WPT applications. Moreover, we realize tunable TESs by actively controlling Harper chains that are composed of the circuit-based coil resonators. The results in this work not only provide the directional WPT, but also may facilitate to explore the topological properties in higher-order or more complex setup for long-range WPT, such as the topological corner states.

## Funding

National Natural Science Foundation of China (11774261, 12004284, 61621001); National Key Research and Development Program of China (2016YFA0301101); Natural Science Foundation of Shanghai (18JC1410900); China Postdoctoral Science Foundation (2019M661605, 2019TQ0232); Shanghai Super Postdoctoral Incentive Program.

## Disclosures

The authors declare no conflicts of interest.

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