Scatterers in the Rx Near Field for RF Energy Harvesting Efficiency Enhancement

Abstract

In this paper, we investigate the enhancement of RF–RF energy harvesting efficiency (e_rf–rf) in multipath environments in the context of wireless power transfer (WPT). For this, we used a retrodirective transmitting (Tx) antenna array resonating at 2.4 GHz and a receiving (Rx) antenna surrounded by scatterers placed in the Rx near field. The Rx resides in the Tx far field. We showed that in a medium made of a random distribution of scatterers, a time-reversed wave field interacts with the random medium to regenerate not only the propagating waves but also the evanescent waves required to refocus the energy at the receiver location. The system was enclosed inside a 3 m³ cubical reverberating room to create a strong multipath environment. The study was done for homogeneous (free space) and heterogeneous (multipath environment) media. Different WPT scenarios were considered for different applications: 4 × 1, 4 × 2 and 4 × 4 multi input-multi output (MIMO) systems. The simulation results show that using near-field scattering generates signal focusing at the source location, which increases the RF–RF energy harvesting efficiency, especially in a multipath environment. The average received power in the frequency band 2.4–2.5 GHz was greatly enhanced in the presence of the scatterers. The investigated WPT approach showed encouraging results for charging/powering-up sensors, IoT and smart devices in indoor environments.

1. Introduction

Today, IoT is no longer an emerging trend. It has become one of the most important technologies of the current century with applicability in many industries such as transportation, energy, civil infrastructure, smart buildings, environment monitoring, healthcare, defense, manufacturing, and production. Small Internet of Things (IoT) devices, such as sensors, smart watches, medical implants, fitness trackers, and many other devices, are now being used by a growing number of people around the world. These small devices can only be equipped with low-capacity batteries, which drain faster. IoT is evolving toward creating smart environments, where a multitude of sensors and devices interact to deliver an abundance of useful information. Therefore, energy efficiency and a long battery life are of crucial importance in the design and development of wireless sensors networks (IoT devices). An effective power supply to a large number of sensors or devices is a critical issue in achieving this IoT. The integration of energy harvesting (EH) technologies with the IoTs leads to the automation of building and homes. In addition, wireless RF power transfer appears to be an alternative to providing these sensors and devices with self-sustaining operations [1], allowing for battery-free, long-term wireless networks. As a result, as long as there is wireless coverage, IoT devices can achieve long-term communication even in adverse conditions. More importantly, due to the broadcast nature of wireless channels, multiple devices can be charged simultaneously. Low energy transfer efficiency due to path loss and channel fading during wireless energy signal transmission is a challenging issue in wireless RF energy transfer.

A conventional RF WPT system is shown in Figure 1. The transmitting (Tx) antenna transmits the RF energy to the EH (energy harvester) and the receiving (Rx) antenna receives the incoming RF energy, and the rectifier converts it to DC energy. The Federal Communication Commission (FCC) limits equivalent isotopically radiated power (EIRP) in the 2.4 GHz ISM band to 36 dBm, and typical radio frequency energy harvester (RFEH) sources exhibit lower transmit power [2].

Microwaves are scattered by objects such as walls, desks, humans, etc., in many real-world environments (buildings, homes, offices, etc.), creating a multitude of paths from the transmitter to the receiver. As a result, the effective distance for wireless energy transfer is too short to achieve the broad coverage required for mass access. Another challenging issue while using the far-field WPT approach is how to increase the harvested RF power at the receiver input (receiver antenna) without increasing the transmit power; thus, increasing the DC power level at the output of the rectenna, and charging/powering-up devices located meters away from the transmitter.

In order to increase the RF–RF energy harvesting efficiency, the multi-antenna beamforming technique has been proposed to steer RF signals in the desired direction towards the intended receivers in order to combat the severe signal propagation loss over distance, thus improving the energy transfer efficiency [3,4]. Retrodirective [5] and time-modulated arrays [6] and time-reversal (TR) [7] are alternative techniques to multi-antenna beamforming that also enable directional/energy focusing. Although these techniques may be used for both near- and far-field regions, the retroreflective technique is suitable for a variety of short- and mid-range MPT applications for wireless sensors or mobile devices [8,9]. An enhancement in WPT system efficiency can be achieved by using time-reversal focusing because the re-radiated field focuses in the incoming field direction due to the phase conjugation relationship, regardless of the placement of the devices to be charged. In addition, the experimental results in [10] show that time reversal is able to compensate for multiple reverberation and recreate a short electromagnetic pulse at the source.

From the receiving point of view, the vast majority of the technical efforts in the literature have been devoted to the design of efficient rectenna [11,12] as another promising approach to increase the input RF power and increasing the output DC power level as a result. Different rectenna architectures were considered to increase the RF power received from a multiple-antenna array in a rectenna array in order to maximize the received power. The first method involves a RF-combining structure that uses coherent reception of the incoming waves [13,14,15]. The second method uses DC-combining structures [15,16]. Thus, the end-to-end efficiency of a wirelessly-powered system is dependent on the performance of the individual components such as the transmitter (Tx), the receiving antenna (Rx), the rectifier and power management circuit. However, Rx antenna design for radio frequency energy harvesting (RFEH) has not been considered as a key parameter.

In our work we have focused on the Rx antenna design in order to increase the RF energy harvesting efficiency at the receiver input.

Based on the work presented in [17], we studied the effect of the presence of randomly distributed scatterers in the Rx antenna near-field on the end-to-end RF power transfer efficiency (e_rf–rf) in a multipath environment.

The authors of [17] showed that in a medium composed of a random distribution of sub-wavelength scatterers, a time-reversed wave field interacts with the random medium to regenerate not only the propagating waves, but also the evanescent waves required to refocus below the diffraction limit. For this, they used a retrodirective Tx array made of eight commercial dipolar antennas placed in the far-field, ten wavelengths apart from a receiving array. The remarkable results of their study showed that two adjacent antennas can be addressed independently, and focusing spots created around them as small as λ/30 were demonstrated. Thus, a large increase in the information transfer rate by time reversal in such disordered media was achieved.

Our goal was to investigate this technique in the context of WPT, with the aim of maximizing the RF received power from a retrodirective transmitting array (Tx) at the rectenna input with an efficient, compact and simple receiving antenna (Rx) design. As per our knowledge, this technique has never been studied in the context of RF–RF energy harvesting or WPT approaches. We focused our study on multitarget (multi-users) scenarios.

In this paper, we investigated the RF energy harvesting efficiency enhancement in a multipath environment in the presence of scatterers in the receiving antenna near field. For this, we used a linear transmitting retrodirective antenna array (Tx) composed of four printed dipole antennas, and a printed dipole receiving antenna (Rx). Simulations were performed using CST microwave studio.

This paper is organized as follows: Section 2 presents the retrodirective WPT approach. In Section 3, we present the WPT system model. In Section 4, we study the RF energy harvesting scenario in homogeneous and heterogeneous media. Section 5 concludes the paper.

2. Retrodirective WPT Approach

The reciprocity approach treats the receiver as the source of the RF power and then by finding the fraction of this power which is absorbed by the transmitter, we can obtain all the information regarding the spatial channel.

2.1. Time-Reversal Approach

The time-reversal process usually consists of three steps, as illustrated in Figure 1. First, the signals (pilot signal) emitted by a source (Rx) are recorded by a time-reversal array (Tx); second, the recorded signals are time-reversed in either the time domain or the frequency domain (a complex conjugation in the frequency domain is equivalent to time-reversal in the time domain); third, the time-reversed signals are re-radiated by the retrodirective array as a focused beam. In our work, we used the phase conjugation approach (Figure 2).

2.2. Evanescent Wave Reconstruction

In addition, another approach to increase the focusing efficiency is to time-reverse the evanescent waves, which contain the subwavelength information about the source.

However, these waves decay exponentially with the distance from the source and therefore they cannot be detected in the far field. To reconstruct these waves, the authors of [18] placed subwavelength scatterers in the near-field of the source. By diffracting off the scatterers, evanescent waves can convert into propagating waves, and are captured by the time-reversal Tx array in the far-field region. When the time-reversed fields are transmitted back, they are converted back into evanescent waves originating from the source and a super-resolution can be obtained. Combining the retrodirective approach with the evanescent wave reconstruction approach by placing scatterers in the receiving antenna (Rx) near field, we investigated the enhancement in the RF energy harvesting efficiency in a multipath environment (Figure 3). Based on the presented approach, if the target receivers broadcast pilot signals, the wireless power beam is able to follow any receiver motions in time, thus, allowing charging/power-up multiple stationary or moving devices in a dynamic environment.

3. WPT System Model

We investigated the retrodirective WPT system behavior in the presence of scatterers in the Rx antenna near field in free space and in multipath medium. A printed dipole antenna was considered for both the retrodirective Tx array and Rx antenna (antenna array).

3.1. Antenna Design

Figure 4 shows the printed dipole antenna used in our study. The dipole is printed on FR-4 (epoxy) substrate with εr = 4.3, h = 1.5 mm, and tanδ = 0.025.

The length and position of the scatterers were optimized in order to keep the antenna impedance unaffected by the presence of the scatterers.

3.2. Simulation Results

The simulation results show that the presence of the scatterers does not affect the antenna behavior in terms of impedance matching (Figure 5a) and radiation pattern (Figure 5b,c).

We studied the RF–RF efficiency enhancement in the presence of the scatterers. The study was done in homogenous (free space) and heterogenous medium (reverberating cavity to create a multipath scheme).

4. RF Energy Harvesting in Homogenous and Inhomogeneous Medium

4.1. Transmitting (Tx) Antenna Array and Rx Configurations

For the wireless power transmitter Tx, we considered a linear uniform antenna array with elements arranged in a linear fashion. The distance between the two adjacent radiators was constant and greater than λ/2 (6 cm) to reduce the mutual coupling, with λ denoting the wavelength in free space. The transmitting array includes four uniformly distributed Tx antenna elements. Each element is a printed dipole vertically polarized and oriented along the z direction. The four-element array is deployed along the x axis. The wireless power receiver includes a z-oriented printed dipole denoted as Rx and assumed to reside in the far-zone of the wireless power transmitter. The investigated system operates in the frequency band of 2.2–2.5 GHz with a central frequency of 2.45 GHz. The overall system was enclosed inside a metal cavity from PEC (perfect conductor) to create a strong reverberating medium.

We investigated different WPT scenarios in the presence of scatterers in the Rx near field: single Rx, 2 × 1 and 2 × 2 Rx arrays for MIMO systems and RF power combining systems. Furthermore, we investigated the randomly distributed multi-user (multi device) scenario, for which we considered four Rx randomly distributed in a room for IoT applications. These configurations could be used for charging/powering-up multi loads (devices) or to increase the RF–RF energy harvesting efficiency, and thus increasing the DC power level to be delivered to the load.

We applied the reciprocity theorem to all configurations; first the Rx antenna was excited (a pilot signal was sent), the received signals at the Tx antenna ports were collected, and the Tx antennas were then re-excited by the respective conjugate phases (retrodirective approach) of the received signals for which the Rx was the source of power. The total transmitted power from the Tx array was 2 W (0.5 W per element).

4.2. Rx Configurations

Single Rx

Figure 6 shows a WPT system with four Tx antenna array and one Rx located in the far field of the Tx.

Table 1. Received power at the Rx port.

Received Power (1 Rx)	Without Scatterers	With Scatterers
Homogenous medium	6.82 × 10−4	2.42 × 10−4
Heterogenous medium	4.8 × 10−3	5.1 × 10−3

summarizes the simulation results in the homogenous and heterogenous media without and with scatterers. The results show that the same amount of power was received without and with scatterers in both media.

4.3. Rx Arrays for MIMO and Power Combining Applications

4.3.1. 2 × 1. Rx Array

For this configuration, we considered a linear antenna array Tx and a 2 × 1 Rx antenna array with distance inter-antennas λ/4 (Figure 7). For this configuration, the Tx and Rx arrays are in line of site (LOS) and the distance between arrays is 1 m. Simulations were done for homogenous (free space, a 3 m³ room with vacuum walls) and heterogeneous (3 m³ reverberating room) media, without and with scatterers in the Rx antenna near field.

Table 2. Average received power in watts.

Pavg	P(Rx1)	P(Rx2)
Homogeneous Medium
Without scatterers	6 × 10−4	7.4 × 10−4
With scatterers	2.4 × 10−4	2.7 × 10−4
Heterogeneous Medium
Without scatterers	8.5 × 10−4	3.4 × 10−4
With scatterers	3.6 × 10−3	4.3 × 10−3

summarizes the average received power (Pavg) in the frequency band of 2.4–2.5 GHz.

The simulation results show that in homogeneous medium, the presence of the scatterers had no effect on the RF–RF efficiency enhencement, while in heterogenous medium the e_rf–rf was greatly enhanced. The average received power was enhanced by a factor of 4 (P_{1 with scatterers} = 4∗P_{1 without scatterers}) and 12 (P_{2 with scatterers} = 12∗P_{2 without scatterers}), respectively.

The RF energy harvesting enhancement in the presence of the scatterers is due to the electromagnetic reciprocity theorem [18,19]. By reciprocity, the signal at the receiver when we apply a current at the transmitter should be proportional to the signal at the transmitter, when we apply a current at the receiver. Thus, in the case with the scatterers, larger fields could be found at the transmitter when we excite from the receiver. Consequently, the amount of received RF power increases at the Rx port when the retrodirective Tx array sends the conjugated signal. Furthermore, the enhancement of the Rf received power in heterogenous medium is due to the fact that the scatterers allow waves coming from different directions to add constructively at the receiver. As a result, the received power is increased.

4.3.2. 2 × 2. Rx Array

For this configuration, we considered a 2 × 2 Rx array. The inter-antennas distance was set to be λ/4 (Figure 8). The same simulation conditions used in the previous case were used in regard to the Tx array, transmitted power and media.

Table 3. Average received power in watts.

Pavg	P(Rx1)	P(Rx2)	P(Rx3)	P(Rx4)
Homogeneous Medium
Without scatterers	1.9 × 10−3	2.2 × 10−3	1.5 × 10−3	1.7 × 10−3
With scatterers	1.8 × 10−3	2 × 10−3	1.4 × 10−3	1.6 × 10−3
Heterogeneous Medium
Without scatterers	1.8 × 10−3	1.7 × 10−3	2.3 × 10−3	2.4 × 10−3
With scatterers	2.8 × 10−3	3 × 10−3	3.4 × 10−3	3.9 × 10−3

summarizes the simulation results for this configuration.

The same results were found as for the previous configuration; the presence of the scatterers in the Rx antenna near field enhanced the average received power in a multipath environment [20].

4.3.3. Multi-User Scenario

For this configuration, we considered a real-life scenario where a Tx provides wireless power for multiple devices in an indoor environment, where the wireless power transmitters can be embedded in walls, on the ceiling, etc. We investigated the WPT scenario for four Rx randomly distributed inside a 3 m³ cubic room (Figure 9). For the multi-user scenario, two Tx configurations were considered: Tx on the walls and Tx on the ceiling.

Table 4. Average received power in watts for multi-user WPT scenario.

Pavg	P(Rx1)	P(Rx2)	P(Rx3)	P(Rx4)
Homogeneous Medium
Without scatterers	8.3 × 10−6	5.6 × 10−5	3.4 × 10−5	2 × 10−4
With scatterers	8.3 × 10−6	6 × 10−5	3.6 × 10−5	2.2 × 10−4
Heterogeneous Medium
Without scatterers	2.5 × 10−3	2.6 × 10−3	2 × 10−3	2.5 × 10−3
With scatterers	3.4 × 10−3	3.3 × 10−3	3 × 10−3	4 × 10−3

summarizes the simulation results for the average received power on the Rx ports. It is worth noting that similar results were found for the configuration where the Tx antenna array was placed on the ceiling. The obtained results show that the microwave power transfer in multipath environment is more efficient in the presence of the scatterers in the Rx antenna near field. Furthermore, in our real-life scenario, microwaves are scattered by objects such as walls, desks, humans, etc., which produces a multitude of paths from the transmitter to the receiver. The presence of the scatterers in the Rx antenna near field showed the ability to compensate for the path loss during the wave propagation in a multipath environment.

4.4. Results Analysis

If the medium is time-invariant and reciprocal, the time-reversed signals will add up constructively at the original source location, in both space and time [20].

When time-reversal is performed in a homogeneous medium with no multipath effect, the size of the spatial focusing spot is limited by the half-wavelength (usually termed as diffraction limit). However, when the medium is heterogeneous or there is multipath reflection, the focusing spot is smaller than in a homogeneous medium. This enhanced resolution is called “super-resolution” (Figure 10).

The results obtained with scatterers in homogenous and heterogeneous media and presented in Table 1, Table 2, Table 3 and Table 4 for different Rx configurations correspond well to this theory. More energy focus could be obtained in multipath environments in the presence of the scatterers. As a result, the amount of received power increases and the RF–RF efficiency could be enhanced greatly.

In addition, the electromagnetic properties of arrays of scatterers can be analytically investigated in several different ways. The key problem is the calculation of the field that excites every particle in the array, the so-called local field. The local field acting on an inclusion is the sum of the external field of the incident wave and the field created by all the other particles of the system at the point where the considered inclusion is located (so-called interaction field) [21]. The interaction field can be evaluated by direct summation of the scatterers’ fields. In our study, we have made no simplifications, and have induced all the terms of the dipole field: the near-field contribution (quasi-static field), which is proportional to 1/r³, the intermediate-zone field 1/r², and the wave field 1/r, where r is the distance from an inclusion to the point where the field is evaluated. Magnetic field is not considered here.

4.4.1. Calculation of the Local Field

The electric field generated by an electrical dipole is (e.g., [22]):

$${\mathit{E}}_{\mathit{p}}=\frac{1}{4\pi {\epsilon }_0}\left\{k^2\right.\left(\mathit{n}\times \mathit{p}\right)\times n\frac{e^{-jkr}}{r}+\left[3\mathit{n}\left(n.p\right)-\mathit{p}\right]\left(\frac{1}{r^3}+\frac{jk}{r^2}\right)\left.e^{-jkr}\right\}$$

(1)

where $\mathit{p}$ is the electric dipole moment, $\mathit{n}=\frac{\mathit{r}}{r}\mathrm{is}\mathrm{the}\mathrm{unit}\mathrm{vector}$directed from the source to the observation point, $r=\left|\mathit{r}\right|$, vector r points from the observation point to the dipole, and $k=\omega \sqrt{{\epsilon }_0{\mu }_{00}}$ is the wave number.

Here, ${\mathit{E}}_{\mathit{p}}$ and $\mathit{p}$ are the vector representation of the electric field induced and the dipole moment on each inclusion in a 3D Cartesian coordinate system:

$$\begin{gathered} \mathit{E}\mathit{p}=E_x+E_y+E_z \\ \mathit{p}=p_x+p_y+p_z \end{gathered}$$

Solving Equation (1), we get Equation (2):

$${\mathit{E}}_{\mathit{p}}=\frac{1}{4\pi {\epsilon }_0\left|r^3\right|}\left\{k^2{e}^{-jkr}\left[\begin{array}{c}\left(\left({\left|r\right|}^2-x^2\right)\widehat{x}-xy\widehat{y}-xz\widehat{z}+\left(\left(3x^2-{\left|r\right|}^2\right)\widehat{x}+3xy\widehat{y}+3xz\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)px\\ \left(-xy\widehat{x}+\left({\left|r\right|}^2-y^2\right)\widehat{y}-yz\widehat{z}+\left(3xy\widehat{x}+\left(3y^2-{\left|r\right|}^2\right)\widehat{y}+3yz\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)py\\ \left(-xz\widehat{x}-zy\widehat{y}+\left({\left|r\right|}^2-z^2\right)\widehat{z}+\left(3xz\widehat{x}+3zy\widehat{y}+\left(3z^2-{\left|r\right|}^2\right)\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)pz\end{array}\right]\right\}$$

(2)

The local field ${\mathit{E}}^{\mathit{l}\mathit{o}\mathit{c}}$ acting on every particle is the sum of the external field: ${\mathit{E}}^{\mathit{e}\mathit{x}\mathit{t}}=E_0{e}^{-jkr}$ and the interaction field${\mathit{E}}^{\mathit{i}\mathit{n}\mathit{t}}$. To calculate the interaction field, we remove one dipole and sum up the fields generated at its location by the other dipoles:

$$\begin{gathered} {\mathit{E}}_{\mathit{i}}^{\mathit{i}\mathit{n}\mathit{t}}={\displaystyle \sum }_{\begin{array}{c}p\\ p\ne i\end{array}}^N{\mathit{E}}_{\mathit{p}} \\ {\mathit{E}}_{\mathit{i}}^{\mathit{l}\mathit{o}\mathit{c}}={\mathit{E}}^{\mathit{e}\mathit{x}\mathit{t}}+{\mathit{E}}_{\mathit{i}}^{\mathit{i}\mathit{n}\mathit{t}} \\ {\mathit{E}}_{\mathit{i}}^{\mathit{l}\mathit{o}\mathit{c}}={\mathit{E}}^{\mathit{e}\mathit{x}\mathit{t}}+{{\displaystyle \sum }}_{\begin{array}{c}p\\ p\ne i\end{array}}^N{\mathit{E}}_{\mathit{p}} \end{gathered}$$

(3)

On substituting (2) in (3), we get Equation (4):

$${\mathit{E}}_{\mathit{i}}^{\mathit{l}\mathit{o}\mathit{c}}={\mathit{E}}^{\mathit{e}\mathit{x}\mathit{t}}+{{\displaystyle \sum }}_{\begin{array}{c}p\\ p\ne i\end{array}}^N\frac{1}{4\pi {\epsilon }_0\left|r^3\right|}\left\{k^2{e}^{-jkr}\left[\begin{array}{c}\left(\left({\left|r\right|}^2-x^2\right)\widehat{x}-xy\widehat{y}-xz\widehat{z}+\left(\left(3x^2-{\left|r\right|}^2\right)\widehat{x}+3xy\widehat{y}+3xz\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)px\\ \left(-xy\widehat{x}+\left({\left|r\right|}^2-y^2\right)\widehat{y}-yz\widehat{z}+\left(3xy\widehat{x}+\left(3y^2-{\left|r\right|}^2\right)\widehat{y}+3yz\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)py\\ \left(-xz\widehat{x}-zy\widehat{y}+\left({\left|r\right|}^2-z^2\right)\widehat{z}+\left(3xz\widehat{x}+3zy\widehat{y}+\left(3z^2-{\left|r\right|}^2\right)\widehat{z}\right)\left(\frac{1+jkr}{\left|r^2\right|}\right)\right)pz\end{array}\right]\right\}$$

(4)

where $={\mathit{r}}_{\mathit{p}}-{\mathit{r}}_{\mathit{i}}$, $r=\left|\mathit{r}\right|$ is the distance between the dipole’s centers.

As a result, the induced electric field on the Rx antenna is the sum of the local electric field and the sum of all electric field induced from the random array of scatters. Thus, an increase in the RF received power at the Rx port could be achieved.

4.4.2. Green’s Function

Furthermore, the Green’s function defines the physics of wave propagation from the source point r′ to the receiver point r in electromagnetics theory. The well-known scalar Green’s function in the homogeneous (free-space) scenario is defined as [17]:

$$G_0\left(r,r^{\prime },k\right)=\frac{e^{-jk\left|r^{\prime }-r\right|}}{4\pi \left|r^{\prime }-r\right|}=R_0\left(k\right)+j{X}_0\left(k\right)$$

(5)

where $k=\frac{2\pi f}{c}$ is the wavenumber corresponding to the operating frequency f, and c is the speed of light. The $R_0\left(k\right)$ and $X_0\left(k\right)$ represent real and imaginary parts of complex $G_0\left(r,r^{\prime },k\right).$

The imaginary part of the Green’s function is associated with power transfer, corresponding to the radiative field between the source and receiving points, since it relates the current at the source point to the vector potential at the receiving point. In the time domain, this means that the time-reversed Green’s function is the difference between the advanced and retarded Green’s functions [23]. By placing randomly distributed subwavelength scatterers in the near field of the source, evanescent waves can be converted into propagative waves that can be detected in the far field by diffracting off the scatterers. As a result, subwavelength coupling between a collective of subwavelength scatterers changes the spatial dependence of the imaginary part of the Green’s function, which now oscillates on scales much smaller than the wavelength.

In order to provide the time-reversed field with subwavelength confinement, the scatterers have to create a pattern $Im\left[G\left(r,r^{\prime },\omega \right)\right]p^{*}$ exhibiting subwavelength variations, where $p$ is the dipole moment of the source. As a result, the expression of the time reversed field E_TRC, produced by a Tx retrodirective array is [10]:

$$E_{TRC}\left(r,\omega \right)=-2j{\mu }_0{\omega }^{2}Im\left[\overleftrightarrow{G}\left(r,r^{\prime },\omega \right)\right]p^{*}$$

(6)

where, ${\mu }_0$ is the permeability of a vacuum, and $\overleftrightarrow{G}$the tensor stands for the dyadic green function of the medium. Equation (2) accounts for the presence of nearfield scattering in both the direct and time-reversed processes.

The results presented in Table 1, Table 2, Table 3 and Table 4 for the average received power in the presence of the scatterers in the Rx antenna near field show that the combination of the time reversal of electromagnetic waves and near-field scattering is an efficient way of generating focusing at the source location and results in an increase in the RF energy harvesting efficiency.

5. Conclusions

In this paper, the enhancement in RF energy harvesting efficiency by evanescent wave reconstruction in multipath environments was studied. The study was done in the context of wireless power transmission and in the presence of the scatterers in the Rx antenna near field. Different WPT scenarios were considered for different applications: 2 × 1 and 2 × 2 MIMO systems/RF-power combining, and a multi-user scenario for indoor wireless charging. The simulation results showed a considerable enhancement in the power transmission efficiency in the presence of scatterers in heterogenous medium. On the other hand, for the propagation in homogeneous medium (free space), we noticed that the presence of the scatterers has no effect on the RF–RF efficiency enhancement, and the same amount of power was received without and with scatterers. Thus, the presence of the scatterers in the Rx antenna near field in heterogeneous medium has the ability to reconstruct the evanescent waves and compensate for the path loss due to multipath propagation. As a conclusion, our study has shown that an enhancement in the RF energy harvesting could be obtained in a dense multipath environment when using a retrodirective Tx array and surrounding the Rx antenna by subwavelengths scatterers. The investigated concept could be used in the context of wireless power transfer for charging and or/powering-up sensors, IoT and other small mobile devices, eliminating the need to replace batteries and wires for charging.