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# Planar array with bidirectional elements for tunnel environments

## Abstract

The growing demands for wireless communications in tunnel environments are driven by requests for maintaining uninterrupted internet access for users in tunnels as well as wireless connections for wireless sensors, security, and control networks. Nevertheless, wireless networks in tunnel environments are associated with technical challenges related to elongated wireless coverage in two opposite near-end-fire directions. Here, we introduce a low-profile bidirectional antenna that can be mounted on a large ground plane and that has a dual-magnetic-current mode exhibiting radiation patterns with 3-dB beamwidth coverage at near-end-fire angles. Furthermore, we realized a planar array with eight such bidirectional antennas that were configured as a sparse array in order to reduce the commonplace issues of mutual coupling and grating lobes. The radiation beams of the proposed antenna array can scan in the end-fire area (60° < φ < 120°, 45° < θ < 80°), with scanning gains of 11~15 dB in the near-end-fire directions. In addition, we demonstrate that the proposed array can adaptively generate a directional single beam or multiple beams according to the positions of users, which is suitable for intelligent communication systems with low power consumptions and high communication bandwidths in tunnel environments.

## Introduction

Although the above-mentioned antennas have superior bidirectional radiation performance in free space, they are less appropriate for on-wall or on-vehicle applications because they lack a large back ground plane to eliminate the influence of back structures and materials on radiation. As a result, a linear reconfigurable array with two isolated ports and four L-shaped monopoles vertical to a metallic ground was proposed; the array had two opposite end-fire radiation patterns that could be switched by switching the feeding signals at two ports25. However, the monopole elements have a high profile and cannot be easily fabricated using standard techniques compared to printed patch antennas. Printed leaky-wave antennas have low profiles, can be effortlessly-fabricated to generate near-end-fire radiation beams and can be backed by larger ground planes. Consequently, coplanar waveguide26 and substrate integrated waveguide27 approaches were proposed to design end-fire leaky-wave antennas. By using tapered substrate-integrated waveguides, leaky-wave antennas can also be designed for conformal applications28. A fixed-frequency beam-steering leaky-wave antenna was proposed that can scan its beams in near-end-fire directions using binary switches29. The combination of two groups of end-fire leaky-wave radiators can generate bidirectional radiation patterns. As a special kind of leaky-wave antenna, surface-wave antennas have recently been widely investigated to produce end-fire bidirectional radiation patterns30,31,32 using large ground planes33 or high impedance surfaces34,35. In addition, microstrip magnetic dipoles with a large ground plane are another method for generating near-end-fire radiation patterns with vertical polarizations36,37,38,39.

The above antennas and arrays that can be mounted on a large ground plane can produce near-end-fire radiation beams or bidirectional radiation beams, which are highly needed for tunnel environments. Nevertheless, it is much more energy-efficient to automatically direct radiation beams to users in modern intelligent communication systems. Therefore, phased arrays capable of beam steering are highly needed in communication systems40,41,42. When we focus on tunnel communication environments, a new challenge arises. Generally, phased arrays can scan their beams in broadside directions, and scanning beams in near-end-fire directions remains a technological challenge43,44,45. To solve that problem, some methods have been proposed to increase element beamwidth, including the high impedance surface method46, image-theory method47, magnetic current method48, and reconfigurable element method49,50 for open space applications. In addition to approaches for widening element beamwidths, metallic wall51,52,53,54,55,56 and wide-angle impedance matching (WAIM) approaches57,58,59 were proposed to match the active impedances of elements at different scanning angles. However, those widening element beamwidth approaches are not adapted to tunnel environments because scanning range in broadside directions is redundant in tunnel environments and the arrays can only scan beams in a certain plane rather than over bidirectional areas.

In this paper, a low-profile, easily fabricated dual-mode bidirectional antenna and a planar phased array with bidirectional elements are proposed for tunnel environments. The design methodology is first analysed considering three important key factors that are introduced to guide the design of an array for use in a tunnel: designing bidirectional elements, dealing with the strong coupling, and avoiding grating lobes. Furthermore, a dual-magnetic-current antenna with two bidirectional modes is proposed based on the operational principle of phase-inversed binary arrays. By using a patch with opposite short edges and opposite open edges, a dual-magnetic-current antenna is realized. The 3-dB beam-width can cover bidirectional near-end-fire directions from 40° to 85° and from −40° to −85°. A planar array with eight bidirectional elements is then presented. Decoupling cavities and a sparse array configuration are used to achieve weak mutual coupling and avoid grating lobes in the array. Finally, the time reversal synthesis method is applied to adaptively synthesize single beam and multiple beams according to the users, which is suitable for intelligent communication systems used in tunnel environments.

## Design Procedures and Results

### Design procedures

The arrays used for tunnel environments in modern intelligent communications systems should have three key performance traits: (1) large back ground planes for on-wall or on-vehicle applications, (2) the ability to generate radiation that can cover near-end-fire directions, (3) ideally, the radiation beams that are automatically directed to users automatically. The schematic diagram of the planar phased array for use in tunnel environments is shown in Fig. 1.

To design an array for use in tunnels, there are three challenges: (1) designing low-profile bidirectional elements with a large back ground plane, (2) dealing with the strong coupling between bidirectional elements, especially in the along-tunnel direction and (3) avoiding grating lobes and large sidelobes. To homogenize the radiation distribution in a tunnel, the radiation pattern of each antenna array should cover its near-end-fire directions, and the directions along the broadside of one antenna array are covered by another antenna array, which means that the broadside radiation patterns of the array elements are redundant and that the bidirectional element is suitable for tunnel applications. To design the bidirectional antennas with large ground planes, a dual-magnetic-current structure based on an inversed-phase binary array is proposed in this paper. The radiation pattern of the bidirectional element is mainly focused in the two near-end-fire directions along the tunnel; therefore, the coupling between the elements arranged along the tunnel direction may be very strong. In this paper, two techniques are used to decrease coupling between elements. First, a grounded decoupling cavity is placed around the elements, and second, the distance between the elements along the tunnel direction is approximately one wavelength. However, according to the product principle of array patterns, grating lobes may appear when the distance between array elements exceeds a half wavelength. Therefore, decreasing coupling is contradictory to avoiding grating lobes. To solve these problems, we propose that the elements along the vertical direction to the tunnel are arranged as stair steps, such that the element distance in the tunnel direction decreases and grating lobes can be avoided.

### Dual-mode dual-magnetic-current bidirectional antenna

The geometry of the bidirectional antenna is shown in Fig. 2, and its optimum dimensions are indicated in the figure. A patch with two slots is printed on a substrate with a thickness of 4 mm and a relative permittivity of 2.65. Two edges of the patch are shorted and the other two edges are partially shorted at the ends. The open area of two opposite edges can support two inversed equivalent magnetic currents, which compose an inversed-phase binary array and produce a bidirectional radiation pattern. In this antenna, there are two close resonant modes, which are denoted TM12 and TM14. The two slots on the patch are used to excite the TM14 mode, which has a resonance frequency close to that of TM12. The feed position is at the centre of the patch in the y direction but not at the centre in the x direction. The operational principle will be analysed step by step below.

To clearly display the operational principle of the proposed bidirectional antenna, the evolutionary process of magnetic-current antennas is shown in Fig. 3. Antenna 1 is a fundamental magnetic-current antenna and consists of an open edge and three short edges. The open edge can be equivalent to a magnetic current and generate a dipole-like radiation pattern. The fields on the open edge of the patch can be equivalent to a magnetic current by analogy to the radiation principle of slot antennas:

$${J}_{m}=-\,{n}_{y}\times {E}_{t}$$
(1)

where, $${n}_{y}$$ is a unit vector in the z direction and $${E}_{t}$$ is the electric field in the slots. According to the operational principle of inversed-phase binary arrays, two single-magnetic-current antennas are arranged back-to-back and are labeled Antenna 2 in the figure. When the two single-magnetic-current antennas of Antenna 2 are fed with equal magnitude and phase, the equivalent magnetic currents on the two open edges will have inversed phase because of the opposite directions of the open edges. As we know, inversed-phase binary array can produce a bidirectional radiation pattern, and Antenna 2 therefore emits a bidirectional radiation pattern. The structure of Antenna 3, which has two open edges and two short edges, is a simplification of the structure of Antenna 2. Antenna 3 has only one off-centre feed, the position of which controls impedance matching. Antenna 4 is an example of the practical antenna shown in Fig. 2. The sheets on the short edges of Antenna 3 are replaced with grounded holes, and the two slots that are etched on the top patch excite two binary magnetic-current modes and widen the operational frequency band.

The reflection coefficients and electric field distributions of the bidirectional antenna in Fig. 2 with/without slots on the patch are shown in Fig. 4. When there are two slots on the top patch, which is the structure shown in Fig. 2, we can see two resonant valleys at 5.61 GHz (Position B) and 5.78 GHz (Position C), and when there are no slots on the patch, only one resonant valley at 5.48 GHz (Position A) can be seen. At Position A, the electric field distributions are from the ground to the patch in the central area and from the patch to the ground at the open edges, which indicates a TM12 mode. At Position B, the electric field distributions are similar to those at Position A. At Position C, the electric field distributions in the central area and at the open edges are from the ground to the patch, and the electric fields between the open edges and slots are from the patch to the ground, which indicates a TM14 mode. Therefore, the modes at Positions A, B, and C are the TM12 mode, the TM12 mode, and the TM14 mode, respectively. The slots on the patch result in a higher-order mode close to the resonant frequency of TM12 mode, and the two resonant modes induce a relatively wide operational frequency band.

Aided by the slots on the top patch, the two modes resonate at two frequencies that are close to each other. To increase the operational frequency bandwidth, the feed position and grounded holes at the ends of the open edges are also important for impedance matching. From Fig. 2, we can see that there are 3 grounded holes (noted as side additional grounded holes) at each end of the two open edges, in addition to the grounded holes at the two short edges on the patch. The relation between the reflection coefficient and the number of additional side grounded holes, n, is shown in Fig. 5(a). When n increases from 1 to 3, the resonant depth of the TM14 mode gradually increases, and when n further increases from 3 to 5, the resonant depths of the two modes gradually decrease. The relation between the reflection coefficient and feed position is shown in Fig. 5(b). With increasing distance D, the two resonant points in the map gradually become closer. For D ~ 2 mm, the antenna is matched to the TM12 mode, for D ~ 3.3 mm, the antenna is matched to the TM14 mode, and for intermediate values of D, the antenna has a compromised impedance matching between the two resonant modes. When D = 2.9 mm, a relatively wide operational frequency bandwidth can be achieved, as the vertical line shows.

The simulated normalized patterns of the proposed dual-mode bidirectional antenna with infinite ground plane are shown in Fig. 6. For this antenna, the yz and xy planes are the E and H planes, respectively. In the yz plane, the vertically polarized components are shaped like the letter “m” at both 5.6 GHz and 5.8 GHz. The horizontally polarized component at 5.6 GHz is less than −30 dB, and that at 5.8 GHz is less than −25 dB. In the xy plane, the vertically polarized components are shaped like the number “8” at both 5.6 GHz and 5.8 GHz, and the horizontally polarized components are very small around the two end-fire directions. Figure 6 indicates that the radiation is bidirectional over a relatively wide frequency range, and the horizontally polarized component in the near-end-fire directions is much smaller than the vertically polarized component.

From the above analysis, we can see that the significant challenge in designing low-profile bidirectional elements for use in tunnel arrays can be overcome by using the proposed dual-magnetic-current method. In the next section, a planar array with bidirectional elements will be proposed for use in tunnel.

### Array with bidirectional elements for use in tunnels

The eight elements shown in Fig. 7(a) were arranged as a planar array, as shown in Fig. 8. From the last section, we know that two main beams of the bidirectional elements are located in the yz plane and are directed close to ±y directions. Therefore, the coupling between elements arranged along the y axis may be very strong, even with a decoupling cavity. To decrease the coupling, the element distance is set to 52 mm, which is approximately one wavelength at 5.8 GHz. However, on the basis of the product principle of array patterns, grating lobes may appear when the distance between array elements exceeds a half wavelength. To solve this problem, the elements along the x direction are arranged as stair-steps. Therefore, the element distance in the y direction is smaller than half wavelength. This sparse distribution allows the grating lobes to be avoided and makes the coupling between the elements in x the direction to decrease. The details of the array arrangement will be analysed below. The measured active S parameters of the array are shown in Fig. 8(c). All the reflection coefficients are smaller than −10 dB from 5.75–5.86 GHz, and the couplings between elements are smaller than −15 dB. The weak coupling between the elements indicates that the decoupling cavity and sparse arrangement are effective, which can also be verified by the magnetic field distribution plots shown in Fig. 8(d).

To clearly show the effect of the decoupling cavity and element arrangement on array coupling, two elements arranged side by side and two elements arranged end to end were analysed, as shown in Figs 9 and 10, respectively. From Fig. 9, we can observe that the mutual coupling decreases with distance Dy in both two cases, with and without the decoupling cavities, and S21 in the case with the decoupling cavity is approximately 5 dB weaker than that in the case without the decoupling cavity, which means that both the decoupling cavity and the large element distance can weaken mutual coupling. From Fig. 10, we can see that S21 in the case with the decoupling cavity is approximately 3 dB weaker than that in the case without the decoupling cavity, which is similar to Fig. 9. However, the trend in the variation in mutual coupling with Dx differs from that with Dy. In the operational frequency band, the mutual coupling decreases when the distance Dx changes from 0 to 10 mm and then increases in both two cases (with and without decoupling cavities). When Dx is near 0, the distance between the two elements is very small, and the coupling is therefore strong; as Dx increases, the distance increases, and the coupling decreases. However, when the right element moves further in the –y direction, the direction of the radiation emitted by the proposed bidirectional element becomes significant. We know that the radiation of the element is near the ± y direction. In the near field, the electric fields are strong around the open edges and weak around the short edges, as shown in Fig. 4. When Dx increases above 10 mm, the two elements approach the direction of strong radiation and enter the strong field area, where the mutual coupling increases. Therefore, our proposed stair-step distribution for the antenna array is able to not only significantly avoid scanning grating lobes but also efficiently decreases mutual coupling.

The active far-field radiation patterns of each element in the proposed array are shown in Fig. 11. All the elements have bidirectional radiation patterns in both the simulated and measured cases shown in the figures, and the vertically polarized component is 15 dB stronger than the horizontally polarized component. For each element, the two near-end-fire beams are asymmetrical in both the simulated and measured results because of the effect of the other elements in the array.

Subsequently, the time reversal synthesis method was used to adaptively determine the feeding amplitude and phase corresponding to the desired radiation pattern. Due to its ability to automatically focus, the time reversal technique has been widely used in super-resolution imaging60,61,62, high-capacity communications63,64,65,66, and phased array synthesis47,67. The time reversal synthesis method consists of the following characteristics47,67: (1) The excitation amplitude and phase of each element can be adaptively determined regardless of array arrangement and mutual coupling. (2) The excitation signals of all the elements can be determined by a time reversal operation both in simulations and in practical communication systems; therefore, time reversal is a solution for intelligent communication systems. It should be noted that the radiation efficiency of the antenna array can be reduced by the mutual coupling, although the main radiation beam of the array can be adaptively pointed to the desired directions using the time-reversal approach in situations of strong mutual coupling. Therefore, the strategies for reducing mutual coupling discussed in the previous paragraphs remain necessary. The basic process of the time reversal synthesis method is described as follows: In Step 1, the phased array is illuminated by detecting signals with temporal waveforms that cover the operating frequency bandwidth of the array from the desired steering directions. In Step 2, each element of the array receives the signal at the same time, and the time-reversal operation is applied. In Step 3, each element of the array transmits the time-reversed signal. In the frequency domain, the time reversal operation is equivalent to phase conjugation. Thus, the process can be finished in the frequency domain. The amplitude $${{\rm{A}}}_{n}$$ and phase $${{\rm{\phi }}}_{n}$$ at operational frequency $${f}_{0}$$ of the received signal of Element n can be obtained by Fourier transform and the amplitude and phase of the corresponding exciting signal in Step 3 can be determined as $${{\rm{A}}}_{n}$$ and $$-{{\rm{\phi }}}_{n}$$. It is difficult to control amplitude excitations in practical applications. It has been proven that the amplitude $${{\rm{A}}}_{n}$$ of each element can be set uniformly to 1 in the time reversal synthesis process, with a weak effect on the radiation pattern of the array47. Therefore, only the phase is controlled using the time reversal synthesis method in this paper. In other words, in Step 3, each element of the array transmits a signal with an amplitude of 1 and a phase of $$-{{\rm{\phi }}}_{n}$$. In Step 4, the signal is focused in the directions of the detecting signals, so that the desired main beams will be obtained. The process of the time reversal synthesis method is shown in supplementary Movie 1.

Using the time reversal synthesis method, radiation patterns of the proposed array were synthesised. The simulated scanning patterns of the proposed array are shown in Figs 12 and 13. Given the structural symmetry of the proposed array, the scanning patterns around only one end-fire direction are shown in the figures; the scanning performances around the other end-fire direction are similar. From Fig. 12, we can see that when the main beam is directed to $${\rm{\theta }}=45^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=35^\circ$$ to $$\,{\rm{\theta }}=60^\circ$$. When the main beam is directed to $${\rm{\theta }}=60^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=45^\circ$$ to $$\,{\rm{\theta }}=80^\circ$$. When the main beam is directed to $${\rm{\theta }}=70^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=55^\circ$$ to $$\,{\rm{\theta }}=90^\circ$$. From Fig. 13, we observe that the main beam of the array can be scanned from $${\rm{\theta }}=60^\circ$$ to $${\rm{\phi }}=120^\circ$$ and that the radiation beams of the proposed array can be scanned in the end-fire area. In addition, the horizontally polarized components in all the scanning cases are more than 10 dB smaller than the vertically polarized components.

The measured scanning radiation patterns are shown in Fig. 14 around only one end-fire direction of the proposed array as a result of the symmetry of the structure. When the main beam is directed to $${\rm{\theta }}=45^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB radiation beam in the yz plane approximately covers $${\rm{\theta }}=36^\circ$$ to $$\,{\rm{\theta }}=65^\circ$$. When the main beam is directed to $${\rm{\theta }}=60^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=44^\circ$$ to $$\,{\rm{\theta }}=84^\circ$$. When the main beam is directed to $${\rm{\theta }}=70^\circ ,{\rm{\phi }}=90^\circ$$, the 3-dB beam in the yz plane approximately covers $${\rm{\theta }}=50^\circ$$ to $$\,{\rm{\theta }}=90^\circ$$. When the radiation beam is scanned in the yz plane, the horizontally polarized components in all the scanning cases are approximately 15 dB smaller than the vertically polarized components. When the beam is scanned in the xy plane, the main beam of the array can be scanned from $${\rm{\phi }}=60^\circ$$ to $${\rm{\phi }}=120^\circ$$ and the horizontally polarized components in all the scanning cases are more than 6 dB smaller than the vertically polarized components. The measured scanning gains of the proposed array are shown in Fig. 15. From Fig. 15, we can see that the scanning gain decreases when the main beam direction is away from the end-fire direction. The scanning gains in the near-end-fire directions (60° < φ < 120°, 45° < θ < 80°) are within 11~15 dB, and the gain fluctuations are small, which indicates good scanning performance.

The above results show the scanning performances in the near-end-fire directions of the proposed array, which indicate that the proposed array can scan its main beam in hotspot areas of tunnel environments. In a time reversal communication system, after a user transmits a detection signal in the tunnel, the proposed array can automatically focus its main beam to the user using a tracking mechanism. In this situation, one main radiation beam can increase the transmission efficiency and avoid signal mutual interference. In other situations, for example, when users are located in two opposite directions and transmit detection signals simultaneously, the proposed array is expected to have multi-beam performance. To demonstrate the multi-beam performance, the proposed array is simultaneously illuminated by two plane waves from ±y, i.e., $${\rm{\theta }}=90^\circ ,{\rm{\phi }}=90^\circ$$ and $$\,{\rm{\theta }}=90^\circ ,{\rm{\phi }}=270^\circ$$. Based on the time reversal synthesis method introduced above, a bidirectional end-fire radiation pattern can be realized, as shown in Fig. 16. The 3-dB beamwidth of the vertically polarized component can cover the bidirectional near-end-fire directions, including $${\rm{\theta }}=90^\circ ,{\rm{\phi }}=90^\circ$$ and $$\,{\rm{\theta }}=90^\circ ,{\rm{\phi }}=270^\circ$$. The horizontally polarized component is 10 dB smaller than the vertically polarized component, and an excellent bidirectional radiation pattern is demonstrated. From the above analysis, we can see that the proposed array can be used in both single-user and multi-user situations in tunnel environments.

## Discussion

In this study of a planar phased array for use in tunnel environments, we implemented three key techniques for the proposed array: the bidirectional element design method, the bidirectional array arrangement method, and the adaptive array synthesis method. To design the bidirectional element, the primary concept was to manipulate dual-magnetic currents based on a phase-inversed binary array. Moreover, the dual-magnetic-current antenna was realized using a patch with opposite short edges and opposite open edges. To increase the operational frequency bandwidth, two slots were etched on the patch of the antenna to excite two bidirectional modes. In addition, the two contradictory issues of reducing strong coupling and avoiding grating lobes were solved using the decoupling cavity and the sparse array configuration in the proposed tunnel-used array arrangement process. Finally, the time reversal synthesis method was applied to synthesize array patterns. The time reversal method can be used not only to synthesize desired radiation patterns in simulations but also to manipulate the focus of radiation beams in practical intelligent communication systems. With the time reversal method, the proposed array can adaptively generate a single beam or multiple beams according to the positions of users in tunnel environments. In conclusion, excellent beam-scanning performances can be realized using the proposed array and methods. The proposed methods can be used to improve bandwidth and reduce energy consumption in demanding wireless communication applications in tunnel environments.

## Methods

### Simulation method

The proposed bidirectional antenna and planar phased array were simulated and optimized using CST Microwave Studio. The selected operating frequency band of the proposed antenna and array was centred around 5.8 GHz, which is a part of the spectrum open for academic research in China. The time reversal synthesis method was implemented in CST Microwave Studio using the following steps: (1) The proposed array is illuminated by a plane wave of 0–10 GHz from the desired steering directions. (2) Each element of the array receives the signal at the same time and the received signals are exported. (3) The amplitude $${{\rm{A}}}_{n}$$ and phase $${{\rm{\phi }}}_{n}$$ at 5.8 GHz of the received signal of Element n are calculated by Fourier transform in Matlab. (4) Array Element n transmits a signal with an amplitude of 1 and a phase of $$-{{\boldsymbol{\phi }}}_{{\boldsymbol{n}}}$$, and the desired far-field radiation pattern is obtained. When multiple beams are required, the illumination process should be performed several times because only one plane wave can be set during a calculation cycle in CST Microwave Studio. For example, two illumination calculations were performed in CST Microwave Studio to obtain Fig. 16. First, the proposed array was illuminated by a plane wave from the y direction, and the received signal at each port was exported. Subsequently, the proposed array was illuminated by a plane wave from the -y direction, and the received signal at each port was exported. Finally, the signals received in the two calculation steps at each port were added, and the Fourier transform of the added signals was calculated in Matlab to obtain the exciting phase for each port.

### Measurement method

The scattering matrices S mn (m, n = 1, 2, 3, …, 8) for the proposed planar phased array were measured using an E8361A Vector Network Analyzer. When the S matrices of the two ports were measured, the other ports were connected to matched loads. The far-field radiation patterns of the proposed antenna and array were measured in a microwave anechoic chamber equipped with a SATIMO measurement system. When the active patterns of one element were measured, the other ports were connected to matched loads. The scanning performances were measured using the feed system shown in Fig. 14(e). Eight ports of the measured array were connected to an eight-channel phase shifter, which was fed by a source through a one-to-eight power divider. The phase shifter was controlled by computer software, and each phase channel could be controlled independently. In the measurement, the feed amplitude and phase of Element n in each case were set to 1 and $$-{{\boldsymbol{\phi }}}_{{\boldsymbol{n}}}$$, respectively, based on the results simulated in CST Microwave Studio and the Matlab calculations (see the section on the simulation method).

### Data availability

The data from this paper can be obtained from the University of Southampton ePrints research repository: http://doi.org/10.5258/SOTON/D0292.

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## Change history

• ### 13 June 2018

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

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## Acknowledgements

This work was supported by funds from National Natural Science Foundation of China (NSFC Grant No. 61331007, 61361166008, and 61401065), and University of Southampton Zepler Institute Research Collaboration Stimulus Fund (Grant No. 516719104), enabled by the 2016/17 UK Higher Education Innovation Fund.

## Author information

### Affiliations

1. #### Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China

• Ren Wang
• , Bing-Zhong Wang
•  & Xiao Ding

• Jun-Yu Ou

### Contributions

B.W. initiated and supervised the project, R.W. performed the simulations and experiments, R.W. and X.D. analyzed the results and developed the theory, R.W. and J.Y.O. wrote the main manuscript text. All authors reviewed the manuscript.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to Bing-Zhong Wang.