Optimum Antenna Characteristics for Maximally Harvesting Ambient RF Energy

This paper investigates the best antenna characteristics to harvest ambient RF waves maximally. The proposed approach analyzes and compares key design aspects, including multi-polarization, multi-port, multi-band, omnidirectional, quasi-isotropic, and non-quasi-isotropic pattern options, over the same antenna size defined by the enclosing sphere radius. First, the ideal isotropic and the Dirac-directive antennas are theoretically analyzed and compared within a general Gaussian channel to derive the most suitable design options. Then, several test antennas, covering the design options, are designed, fabricated, characterized, and compared within a full-wave antenna simulator under spherical incidences and polarizations. Further experiments in real ambient environments complete the tests. Two optimum antenna solutions emerge from the results: the quasi-isotropic antenna and the collocated LP-port diversity system with a power combiner. The crucial feature is optimizing antenna solutions for lower frequency bands with significant power densities that compensate for the available power ${\lambda }^{2}$ -factor.

I. INTRODUCTION
A STANDARD ambient radio-frequency (RF) harvesting system, as presented in Fig. 1, includes an antenna, a matching circuit, a rectifying circuit, and a power management circuit (PMC). The antenna captures the surrounding electromagnetic (EM) waves, driven to the antenna port, and transferred to the rectifying circuit via an impedance matching circuit, maximizing power transfer. It is worth mentioning that the antenna may cover the matching circuit functionality for simplification, loss reduction, and increased performance [1], [2], [3], [4]. The PMC boosts and compensates for the continuously varying level of the rectified power (direct current, DC) to feed the terminal low-power device, such as a sensor, a detector, etc. Therefore, each block plays a key role in the system's operation and performance. However, incident waves are the most critical factors shaping the system performance within the ambient environment. Indeed, i) incident waves behave randomly over time and space, including polarization, amplitude, and phase distributions [5, pp. [13][14][15][16], [6], [7]. This raises the question of the appropriate antenna that can optimally harvest such changing EM waves. Within a deterministic propagation medium, the use of a PMC is questionable unless it serves as a power booster. ii) Incident waves often carry low RF power densities spreading over a large frequency spectrum, including frequency modulation (FM) up to wireless fidelity (Wi-Fi) bands [8], [9]. This imposes the receiving antenna and the other RF blocks to cope with wideband or multi-band options. iii) The rectifying circuit's efficiency increases with the input RF power level [10], which means that low harvested RF power produces poor DC power, unable to switch on the PMC to feed the device.
Therefore, one of the most significant challenges in ambient RF energy harvesting is optimizing the entry block, typically the receiving antenna, to maximize the harvested RF power and increase the system's capability to feed the terminal device.
Several approaches emphasize design aspects, such as polarization, number of ports, frequency bands or tones, FIGURE 1. Architecture of a standard ambient RF energy harvesting system. The antenna may include the matching circuit for simplification, loss reduction, and better performance, as designated as antenna matching [1], [2], [3], [4]. and spatial arrangement [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Indeed, recently, a two-port dual linearly-polarized (LP) bowtie-shaped meander dipole harvested up to 1.6 times more DC power than the equivalent LP dipole [11]. This work considered a fixed transmitting horn antenna and rotated the receiving antenna to include various polarization mismatch angles. Another example considered a four-port patch antenna with two integrated T-shaped slots. This antenna had two LP ports to receive and transmit simultaneously (full-duplex) while harvesting wireless power on two other LP ports [12]. Besides, other works focused on circular polarization (CP), trying to receive right-hand (RH) and left-hand (LH) CP signals on two ports separately over a single design. These studies used a patch antenna with two-coupled crossed slots [13] or a star-like slot coupled to a dielectric resonator antenna [14]. However, even though these works showed dual-polarization benefits [11], [12], [13], [14], they omitted to compare with a reference single-polarized antenna within the "real" operating environment carrying multiple incidences simultaneously.
On the other hand, some works focused on multi-band, wideband, and multi-tone aspects [15], [16]. Indeed, the work in [15] employed a wideband single-port dual LP bowtie dipole with a ground reflector, harvesting almost four times higher voltage with four simultaneous tones than a single tone. Another example in [16], based on a single-port dual CP log-periodic crossed bowtie dipole, covered up to six bands. This antenna exhibited 26 μW more DC power than [15], attributed to the additional covered bands and possible CP benefits over LP.
Finally, some approaches use electrically small antennas in array sets to increase the effective area and boost the harvested power through the number of harvesters over the same array size [20], [21]. Besides, other approaches combined wideband or multi-band, dual-polarization, multi-port, and array sets to significantly increase the harvested power. Indeed, the work in [17] considered a two-dimensional (2D) dual-band array of 16 orthogonal LP inverted-F antennas (IFA) to harvest 13 dB (20 times) more DC power than a single-band IFA. Other similar approaches scale the array size to 64 elements based on a CP equiangular spiral antenna [18], [19]. In addition, different approaches consider several multi-beam arrays to realize a quasi-isotropic coverage sequentially with high directivity [1], [22], [23], [24]. This technique shows up to 3.4 or 4.6 times more DC power than an equivalent 3-dipole linear array fitting in the same size [22].
Despite the variety of the design options addressed, typically dual-polarization, wideband, multi-band, multi-port, multi-beam, small antenna, and array set, most works do not include the isotropic pattern option on a single antenna element. This is essential for harvesting incident waves from any direction. In addition, the testing conditions often cover a single incidence with two principal polarizations, neglecting the complete spherical coverage of the antenna pattern and its complex interaction with the incident wavefield. Table 1 summarizes the most relevant works' features and limitations.
This paper fulfills these gaps by deriving the best antenna characteristics to harvest ambient waves maximally. The proposed approach analyzes and compares key design aspects, such as multi-polarization, multi-port, multi-band, omnidirectional, quasi-isotropic, and non-quasi-isotropic pattern options on the same antenna size. First, the ideal isotropic and the Dirac-directive antennas are theoretically analyzed and compared within a general Gaussian propagation channel to derive the most suitable design option. Next, various test antennas, covering the design options, are compared within a full-wave antenna simulator under spherical incidences and polarizations. Finally, the antennas are fabricated, characterized, and tested within the ambient environment in various locations. This paper is organized as follows. First, Section II theoretically analyzes and compares the ideal isotropic antenna with the Dirac-directive one to derive the most suitable for receiving ambient waves. Then, Section III presents a set of test antennas covering various design aspects. Finally, Sections IV and V validate the optimum antenna characteristics for maximally receiving ambient waves through numerical tests and experiments in actual conditions.

A. PRELIMINARY DEFINITIONS
Let us consider the general receiving antenna within the ambient environment, supporting non-line-of-sight (NLOS) propagation waves. This is a Rayleigh-fading propagation environment [5, pp. 13-16], [6], demonstrated in indoor and outdoor propagation media over most mobile frequency bands [6], [9]. Therefore, this propagation medium is assumed throughout the paper, referred to as ambient waves in the following sections.
The time-averaged power received by the antenna is a key performance indicator, expressed with the average available power P ref (reference power) as [5, p. 138]: (1) where is the solid angle defined at the observation point, X represents the cross-polarization ratio, that is, the ratio of the reference θ -polarized power to the φ-polarized one, The angular functions p θ and p φ represent the incident power spatial distributions in vertical and horizontal polarizations, respectively. The corresponding normalization expressions are: In (1), G θ and G φ represent the antenna realized gain in vertical and horizontal polarizations, respectively. The total realized gain G = G θ + G φ = η t D, is linked to the effective area A eff by [16]: where D is the antenna directivity, η t = τ η r , the total efficiency, η r , the radiation efficiency, and τ is a transmission coefficient between the antenna port and the remaining circuit, including a transmission line and a load. Please refer to [25] for the complete expression of τ . However, for simplicity, τ = 1 is assumed throughout the paper. This indicates a "perfect" impedance matching between the antenna port and the remaining circuit (no mismatch loss).

B. OPTIMUM RECEIVING ANTENNA UNDER AMBIENT WAVES
This section analyzes the average received power, P rec , of several special antennas within ambient waves. All antennas are matched and 100%-efficient, that is, η t = 1.

1) IDEAL ISOTROPIC ANTENNA
In the particular case of an ideal θ -polarized isotropic antenna, with G θ = D θ = 1 and G φ = D φ = 0, (1) gives: Similarly, the ideal φ-polarized isotropic antenna receives: Therefore, the system of two ideal isotropic antennas, perfectly matched and 100%-efficient, receives the entire available power in both θ -and φ-polarizations on a time average. Moreover, this result stands regardless of the incident waves' spatial distributions (p θ , p φ ).

2) IDEAL DIRAC-DIRECTIVE ANTENNA
Let us consider an ideal directive antenna, which is θ -polarized and whose gain is maximum in a particular direction = 0θ , that is, G = G 0θ δ( -0θ ), where δ is the Dirac distribution. According to (1), this Dirac-directive antenna receives where p 0θ = p θ ( 0θ ) is the θ -polarized channel value in the maximum gain's direction. However, invoking the normalization condition (2) in Gd = 4πη t , leads to: where δ( -0θ )d = 1. Therefore, the condition for the Dirac-directive antenna to receive equal or more power than the ideal isotropic antenna in θ -polarization is P rec−Diracθ ≥P ref θ , that is, Let us now analyze (8) by plotting p θ for a general propagation channel which is Gaussian-type with elevation angle θ and isotropic with azimuth φ. That is: where 0 ≤ θ ≤ π , and 0 ≤ φ ≤ 2π . Invoking (2) provides the value of A θ as: where θ cθ is the elevation angle of the channel's maximum value, while the standard deviation of the channel is σ θ . From (9), it is straightforward to see that A θ is the channel's maximum value. It depends on σ θ , but not on θ cθ . Therefore, Fig. 2a plots A θ as a function of σ θ for a given value of θ cθ = 90 • -m cθ = 70 • , where m cθ = 20 • is a standard value for a θ -polarized Gaussian channel in indoor and outdoor environments on the Global System for Mobile communication band (GSM 900: 880-960 MHz) [6], [9], [26]. In addition, Fig. 2b plots p θ as a function of θ for several values of σ θ ranging from 20 to 200 step 20, with θ cθ = 70 • . Fig. 2a indicates that the lower bound of A θ is 1/4π , the isotropic channel distribution value. Therefore, Inequality (11) means that (8) is always true whenever the Dirac-directive antenna focuses on the channel's maximum value direction c (θ cθ , φ cθ ). In this case, it systematically receives more power than the isotropic antenna. However, this result collapses when there is misalignment. Indeed, Fig. 2b indicates that not so far from the channel's maximum value direction, the Gaussian channel values drop rapidly and get below the isotropic channel value of p θ = 1/4π . For instance, this occurs for θ ≤ 42 • and θ ≥ 97 • for the case of σ θ = 20 • . This range in which the isotropic antenna outperforms the Dirac-directive antenna increases as σ θ decreases, corresponding to an increasingly directive channel. Beyond that, it is worth mentioning that the particular case of the φ-polarized Dirac-directive antenna leads to the same results and analyses.

C. SUMMARY
According to previous Sections II-B1 and II-B2, the following conclusions are made: i) The ideal isotropic antenna receives the entire timeaveraged available power in θ -and φ-polarizations, regardless of the channel's spatial distributions. ii) The ideal directive antenna receives more power than the ideal isotropic antenna as long as the former precisely focuses on the channel's maximum value direction. Besides, like any arbitrary antenna, both antennas capture the same power within an isotropic channel.
iii) Any misalignment between the ideal directive antenna and the channel's maximum value direction leads to power loss, leading the isotropic antenna to outperform the ideal directive antenna within a dedicated elevation angle range. This range increases as the channel tends to be more directive. iv) Realizing the above antenna-channel alignment in ii) is tricky within a Rayleigh-fading environment (NLOS) since this suggests a precise preconception of the channel's space-time behavior for any frequency band of interest. Indeed, this means knowing the RF source characteristics (number, position, radiation pattern, transmitting power, etc.) and mainly the propagation loss factors, models, etc. Unfortunately, anticipating these parameters with accuracy is almost impossible for any location, considering standard wireless applications [27, ch. 4-5]. v) Based on i), iii), and iv), the ideal isotropic antenna is the most suitable receiving antenna within the ambient environment since it is channel-insensitive to capture the available power. However, the directive antenna remains the best option within a predefined or deterministic propagation environment. vi) The above conclusions also stand for any directive or quasi-isotropic antennas which underperform the ideal antennas with the same behavior. For this reason, the following sections focus on quasi-isotropic antennas. vii) Finally, even though the proposed formulation stands for Gaussian and isotropic channels within a Rayleightype fading environment, a similar approach may be followed to deal with other channel distributions.
Beyond that, the above conclusions also cover a Ricetype fading environment, including a deterministic channel component and a random one [7]. The dominant channel component will decide the most suitable antenna based on v).

D. TIME-AVERAGED AVAILABLE POWER
This section formulates the time-averaged available power (P ref ) with the space-time-averaged power density W inc . According to previous Section II-B1, the θ -polarized (resp. φ-polarized) isotropic antenna captures . Therefore, invoking (3) allows expressing the elementary time-averaged power per unit solid angle d with the time-averaged power density w incθ in direction as: where w incθ is linked to p θ by: Now, let us consider the following approximation: where W incθ is the spatial average of w incθ . It is straightforward to derive the relative error of (14) as: where p θ represents the spatial average of p θ . Fig. 3 plots ε θ for the θ -polarized Gaussian channel as a function of σ θ . Fig. 3 indicates that the error rapidly decreases with σ θ . Therefore, the error reaches 33% for highly directive channels (low σ θ ) and vanishes toward isotropic channels (high σ θ ). In practice, the θ -polarized channel in indoor and outdoor environments over most wireless communications, including GSM 900 up to Wi-Fi 2.45 GHz, can be assumed as isotropic in a first-order approximation. In this viewpoint, the error made is negligible. However, in a more precise approach, the channel is often assumed Gaussian with a standard deviation σ θ between 20 and 100 • [6], [9], [26]. This case sets ε θ between 5 and 28%, which is acceptable.
Integrating (12) over the 3D space while invoking (13) - (14) leads to: Since the exact derivation applies to φ-polarized waves, (16) still stands for P ref φ by just replacing θ with φ. Therefore, the total available power reads: where the cross-polarization ratio also reads as: From (16) - (17), one can make the following analyses: i) The time-averaged available power in each polarization increases with λ 2 , and the space-time-averaged power density W inc . Fig. 4 illustrates this result by plotting (16) up to 6 GHz for several values of W inc ranging from 0.01 to 10 nW/cm 2 . ii) Given two frequency bands around f 1 and f 2 ( with the same power density W inc , .75 dBm, leading to a power decrease factor of almost 6 dB. iii) Considering a higher-frequency receiving antenna suggests compensating for the λ 2 -power factor, like [17] increasing the number of antennas accordingly. iv) Therefore, the optimum single-band or multi-band receiving antenna within the ambient environment is the one targetting lower frequencies or those for which the power density compensates for the λ 2 -factor.

E. TIME-AVERAGED HARVESTED DC POWER
Even though the focus is on the antenna part, this section analyzes the time-averaged harvested DC power. The harvested DC power entering the PMC is linked to the conversion efficiency (η RF−DC ) and the RF power received by the antenna (P RF ) by: According to [28], under the diode's breakdown point, η RF−DC is linearly dependent on P RF , meaning that P DC is proportional to the square of P RF . Therefore, maximizing P RF maximizes P DC systematically. Moreover, this behavior extends to the time-averaged powers, P rec and P DCavg . Thus, based on (1) and (17) and assuming the diode's junction resistance (R j ) is much greater than the series resistance (R s ), R j >> R s , the time-averaged DC power under matching conditions is formulated as: where λ j = 2π cC j √ (R s R j ) is a characteristic wavelength linked to the diode's cutoff frequency, f j = c/λ j ; c is light's speed in free space, while C j and R I denote the diode's junction capacitance and current responsivity, respectively. MEG is the receiving antenna's mean effective gain [6], [7], defined in (1) in the curly brackets.
In (20), factor F represents the impact of the operating frequency or wavelength, while D includes the diode performance impact. Besides, MEG considers the interaction between the receiving antenna and the incident waves. Beyond that, (20) indicates that the most decisive factor for boosting P DCavg is the operating frequency or wavelength, acting as a power-four term (or higher). In contrast, the other factors (D, MEG, W inc ) have a power-two impact. Indeed, F behaves as λ 4 at low frequencies (λ >> λ j ) but follows λ 8 for higher frequencies (λ << λ j ).
Overall, maximizing P DCavg means optimizing the antenna with the propagation medium to get the highest MEG and choosing the most suitable diode with the highest D. Prior to that, it is crucial anticipating the operating frequency and the available power densities to maximize F*(W incθ + W incφ ) 2 .
The following sections focus on the antenna part only as the DC power follows the received RF power's behavior. The analysis will include full-wave simulations and real ambient measurements based on several test antennas.

III. TEST ANTENNAS
This section presents several test antenna geometries, design steps, and characteristics. This includes reflection coefficient, efficiency, directivity, and realized gain. In addition, results include simulations and measurements. The multiband option focuses on GSM 900, around 940 MHz, and DCS 1800, about 1840 MHz, corresponding to the center frequencies of the related band downlinks. Table 2 details the test antennas' design options.
The dipoles cover single LP antennas with omnidirectional radiation patterns. While the in-phase-fed dipoles (crossed dipole, tripole) cover non-quasi-isotropic antennas with multiple LP and nulls in both θ -and φ-polarizations. The optimally-fed dipoles, typically the turnstile (2D) and the optimum tripole (3D), are quasi-isotropic antennas with  isotropic patterns in φ-polarization. In θ -polarization, the former has the entire horizontal plane as the null domain, while the latter has "just" a null pair. They radiate CP along the z-axis and primarily elliptical polarization everywhere, based on [29].
For comparison needs, all the antennas have the exact occupied area or volume, fitting in the same sphere radius of r aut ≈ L PCB /2, where L PCB = 161.4 mm is the size of the square printed circuit board (PCB). Furthermore, the printing substrate is FR4, with claimed permittivity of ε r = 4.2, and loss tangent of tanδ = 0.014, specified at 2 GHz [30].
Straight dipole-based antenna geometries are used for their design simplicity, allowing antenna collocation with minimum impedance and pattern distortions. Another benefit is their potential to cover several design options, such as quasi-isotropic, non-quasi-isotropic, and omnidirectional patterns, by "just" controlling the feed and the number of elements. Besides, even though more compact antennas may fit the specified size, like in [20], [21], this small antenna design option is not addressed in this study. Therefore, fixed geometries (here, half-wavelength) are suitable to cover the envisioned design options.

A. DIPOLES 1) THE X-AXIS DIPOLE
This is a standard half-wavelength dipole aligned along the x-axis of a Cartesian coordinate system. The dipole length is L D , width, W, and operates around 940 MHz.
A microstrip line feeds the dipole through the top conductor, directly connected to one dipole's arm, while the other arm connects to the ground plane through a via. Fig. 5a gives the schematic view of the dipole, while Fig. 5b presents the overview. In addition, Fig. 5c shows the fabricated prototype's photograph, while Table 3 provides the dimensions of the fabricated prototype.

2) THE Y-AXIS AND Z-AXIS DIPOLES
Rotating the x-axis dipole by 90 • around the z-axis leads to the y-axis dipole. Meanwhile, the same rotation around the yaxis gives the z-axis dipole. Therefore, these rotations do not affect the "intrinsic" performance data, such as the reflection coefficient (S 11 ), efficiency, maximum directivity, and gain. However, rotations change the null domains. Indeed, the y-axis dipole's radiation patterns consist in rotating those of the x-axis dipole by 90 • along the z-axis. However, the special case of the z-axis dipole shows a null in the entire φ-polarization. Please refer to [29] and [37, pp. 182-184] for more details on these dipoles' null domains.

3) THE 3-PORT TRIPOLE
This antenna is a collocated version of the previous three dipoles aligned along the x-, y-, and z-axes. A simplified version of this antenna considered in simulations includes 3-lumped ports for simplification. Simulations, not detailed for brevity, indicate low coupling levels between the dipoles, lower than -40 dB. Therefore, each dipole in the lattice behaves almost the same as it operates lonely. Full-wave simulations confirm this behavior in the radiation patterns. Consequently, there is no motivation to fabricate this complex design. Instead, rotating the single dipole (Fig. 5c) over three orthogonal positions will sequentially provide this 3-port tripole's experimental results. However, simulations consider the three individual dipoles and the 3-port tripole separately for comparison and validation purposes.

4) THE 2-PORT CROSSED DIPOLE
This antenna is a 2-port version of the previous tripole without the z-axis dipole (port #3). Therefore, the previous results of the x-and y-axis dipoles also represent the 2-port crossed dipole.

B. SINGLE-PORT SINGLE-BAND TEST ANTENNAS 1) THE OPTIMALLY-FED CROSSED DIPOLE AND TRIPOLE
Figs. 6a -6d shows the general design's views with the tripole. It is worth mentioning that the tripole includes three printed circuit boards (PCBs), while the crossed dipole design holds on the horizontal PCB only in the xy-plane. The two vertical PCBs above and beneath the horizontal PCB include the z-axis dipole's arms. The horizontal PCB holds the x-and y-axis dipoles and includes the feed line, consisting of a primary suspended line (L S , W S ), a microstrip impedance transformer (L T , W T ), a matching capacitor (C M ), and a 50 -microstrip line (L M , W M ) connected to an SMA connector. Fig. 7 presents the design steps, summarized as follows: Step #1. Set the initial design parameters according to the following relationships: where λ g is the guided wavelength. The extra length L on the y-axis dipole implements the 90 • -phase shift condition [32], while the extra length L z on the z-axis dipole compensates for the x-axis and y-axis dipoles' offsets due to lines L O and L S , introduced to free some space for the feed.
Choosing L O and L S as low as possible helps minimize the dipoles' offsets from the phase center. Thin or moderately wide dipole arms (W) may be used. Besides, the offset line widths may be identical, as W O ≈ W S ≈ W.
Step #2. Run a full-wave simulation to evaluate the gain or directivity variation, D irφ (dB) = D irφmax − D irφmin , performed over the entire 3D space on the φ-component since only this component can shape an isotropic behavior [29]. If D irφ( dB) is below the target value, often set at 6 dB or lower, this pattern design process stops.
Step #3. Whenever D irφ( dB) is not good, the pattern optimization process continues by resetting L and L and checking D irφ( dB) until getting satisfactory results.  Step #4. This step improves the impedance matching level by adjusting the transformer length and characteristic impedance through L T and W T . A supplemental series matching capacitor (C M ) may be necessary to complete the impedance matching process by compensating for the inductive part of the obtained impedance.
The entire design process, involving steps #1 through #4, is carried out at the fundamental resonance of the tripole and the crossed dipole to yield the corresponding optimally-fed structures.

2) THE IN-PHASE-FED CROSSED DIPOLE AND TRIPOLE
These structures have the exact dimensions as the optimallyfed crossed dipole and tripole, except for L = 0, setting the in-phase-fed condition. Beyond that, these structures may require a dedicated impedance matching process (step #4) due to the dipole lengths' change. Table 4 gives the dimensions of the fabricated crossed dipoles and tripoles based on the previous design procedure, while Figs. 8a -8b shows the prototypes' photographs.

C. DUAL-BAND QUASI-ISOTROPIC ANTENNA
This antenna includes two collocated quasi-isotropic dipoles operating at two separate frequencies around 900 and 1800 MHz, as seen in Fig. 9a. The concept has been recently proposed in [33] and is scalable to more than two isotropic bands. One of the design challenges is setting up the singleport feed network. The reader is advised to refer to [33] for more details. Note that optimizing this antenna means  going through steps #2 to #3 in Fig. 7 for each frequency band of interest. Then, an impedance-matching process with a dedicated feed will complete the design.
As seen in Fig. 9b, the proposed feed is hybrid, with direct connections to the lower-band dipole and a coupling feed for the higher-band dipole. Please refer to [33] for more details. Table 5 shows the dimensions of the optimized prototype, while Fig. 9c presents a photograph of the fabricated prototype. Overall, all the test antennas cover the GSM 900 (880 -960 MHz) at S 11 ≤ -10 dB or S 11 ≤ -6 dB, which is acceptable for mobile communications. Beyond that, the dual-band quasi-isotropic antenna covers the upper-band wireless standards, including DCS 1800 (1710-1880 MHz), PCS1900 (1850-1990 MHz), and UMTS (1920-2170 MHz).  Table 6 summarizes the measured and simulated performances over the covered frequency bands. This includes efficiency, directivity, and realized gain results.

D. MEASURED PERFORMANCES
It is seen that the test antennas' measured efficiencies range between 70 and 95%, while their realized gains are within 1.08 -3 dBi. These values correlate with simulation results at 84 -95% and 0.8 -1.9 dBi, respectively.
The noted differences between the measured and simulated performances are due to several factors: i) the substrate loss, which is underestimated in simulation, ii) the efficiency measurement errors inherent to the Wheeler cap approach through the cavity and antenna modes' overlappings [35], and finally iii) the measuring cable's parasitic radiation, which affects directivity and radiation patterns, not detailed for brevity. These parasitic effects increase the directivity variation ( D ir ), which remains acceptable compared with simulations (Table 6).
Measurements confirm that the optimally-fed prototypes have a near-isotropic pattern in φ-polarization and a null in the entire horizontal plane for θ -polarization, except for the optimum tripole showing a null pair. Besides, the inphase-fed prototypes exhibit larger null domains in both polarizations. These radiation characteristics are present over the covered frequency bands.
The following section evaluates the performances of the test antennas within a controlled propagation environment.

IV. NUMERICAL TESTS
This section evaluates the performance of the previous test antennas within a dedicated propagation environment while controlling the incident power density under spherical incidences and polarizations. Such a controlled environment is synthesized inside a full-wave antenna simulator, typically CST [34]. One of the key features of this study is including most electromagnetic interaction phenomena between the receiving antenna and the complex incident wavefield simultaneously. Another option is testing the antennas within the exact propagation characteristics, which are not evident within the ambient environment.

A. INCIDENT WAVEFIELD SYNTHESIS
The simulation starts by synthesizing the incident wavefield of interest in spatial distribution and power density at a specified frequency, typically 940 and 1840 MHz. Let us consider an isotropic wavefield in both θ -and φpolarizations, with a unit power density, typically W incθ = W incφ = 1nW/cm 2 , providing X = 1. These channel characteristics are standard for most wireless applications, typically over GSM 900 and DCS 1800 bands in indoor and outdoor environments [5], [6], [9].
According to [36], synthesizing an isotropic propagation environment requires an array of three electrically small dipoles with thin or moderately thick radii, L d /d ≥ 3, where L d and d represent the dipoles' length and diameter, respectively. Two array sets are required to cover the two polarizations while setting the dipoles' centers at the elevation angles of θ d = {-90 • , 0, +90 • } over a radius of R d , in a Cartesian coordinate system (Fig. 11).
For the θ -polarized wavefield, the dipoles are tangential to the positioning arc, while in φ-polarization, they lie orthogonally to the arc area. The reader is advised to refer to [36] for more details.
The dipoles are fed in phase with a feed current amplitude coefficient of I 0 for adjusting the power density target value. Table 7 gives the simulation setup parameters.

FIGURE 12. Synthesized isotropic wavefields at 940 and 1840 MHz in CST.
According to [36], R d should be a few wavelengths to maximize the size of the test zone (r test ) around onetenth wavelength. Let us consider R d = 750 mm, that is, 2.35λ and 4.6λ at 940 and 1840 MHz, respectively, with r test ≈ 0.16λ. Fig. 12 presents the calibrated wavefields at 940 and 1840 MHz.
As expected, Fig. 12 shows isotropic wavefields in both polarizations with almost a unit power density over the full incidence range of θ ∈ [-90 • , 90 • ]. From these results, one can compute the total available power as [36]: where subscript p indicates either θ or φ-polarizations. Table 8 compares P incp computed from (22) with the predicted (16), where a "1/2" factor is added to only consider the upper half hemisphere incidence.   Table 8 shows that the predicted available power correlates with the simulated one over the two frequencies, confirming the λ 2 -factor, typically 6 dB decrease from 940 to 1840 MHz.

B. TEST ANTENNAS
The test antennas presented in Section III are now placed in the test zone, each as a receiving antenna. However, the large PCB size with FR4 (L PCB = 161.4 mm) sets the enclosing sphere radius at r aut ≈ L PCB /2 ≈ 80 mm, much larger than the test zone radius of r test ≈ 51 mm at 940 MHz. Therefore, the antennas are redesigned with a thicker lossloss substrate to decrease the PCB size accordingly. The new substrate is Rogers TMM 13i, with a permittivity of ε r = 12.2, a thickness of h = 1.905 mm, and a loss tangent of tanδ = 0.0019, stated at 10 GHz. The new PCB size is L PCB = 105 mm for all antennas, setting r aut ≈ 52.5 mm, almost equal to r test ≈ 51 mm. Simulations show that the antennas with Rogers TMM 13i perform similarly to FR4.

1) CASE OF 2D-INCIDENT WAVEFIELDS
Each receiving antenna is sequentially rotated every φ i = 10 • in the azimuth plane to cover various incidence angles φ i from 0 to 180 • under a given illumination frequency. The received power is recorded for every incidence plane successively.
Figs. 13a -13d illustrates some test antennas' received powers, typically the x-axis dipole, the turnstile, the optimum tripole, and the dual-band quasi-isotropic dipole. Fig. 13a shows that the x-axis dipole receives the maximum power in φ-polarization at φ i = 90 • , whereas the power received in θ -polarization almost vanishes. These observations cover the two frequencies. Indeed, the maximum power reception corresponds to a "perfect" alignment of the x-axis dipole's polarization (x-axis) and maximum gain with the incident wavefield's polarization and spatial distribution. In contrast, the minimum reception corresponds to a complete polarization misalignment between the channel and the x-axis dipole.
In addition, Fig. 13a shows a lower received power at 1840 MHz than at 940 MHz, consistent with performance decrease (gain, efficiency) with frequency.
Overall, the received powers' profiles follow the test antennas' 3D-radiation patterns, as noted in Figs. 13a -13d. Besides, one notes a more steady received power profile at 940 MHz for the three quasi-isotropic antennas (turnstile, optimum tripole, dual-band dipole) than the dipole. Furthermore, this stability extends at 1840 MHz for the dualband dipole, except for a "rapid" deep around φ i = 130 • , corresponding to destructive interference from the feed network's parasitic radiation. Except for the dipole, negating a feed network in simulation (lumped port), this phenomenon predominantly occurs at 1840 MHz for all the test antennas around the feed network's location.

2) CASE OF 3D-INCIDENT WAVEFIELDS
The power received by the test antenna under the 3D-incident wavefield is the collection of powers received from each incidence plane, P rec (φ i ). Therefore, the total power received in p-polarization is: Applying (23) for each test antenna in both θ -and φpolarizations under the two illumination frequencies provides the results in Table 9, analyzed in the following points, first focusing on 940 MHz results: i) Except for the z-axis dipole, all the antennas receive almost the same total power, P recθ + P recφ , consistent with the well-known result in [7] setting the mean effective gains of any arbitrary antennas identical within an isotropic channel. ii) The z-axis dipole receives the lowest powers for two reasons: first, the incident wavefields illuminate the test antennas only from the upper half hemisphere, avoiding the lower half-dipole. Secondly, regardless of the incidence plane, the θ -polarized incident wavefield is not well aligned with the dipole's polarization (zaxis). At the same time, the dipole poorly receives the φ-polarized incident wavefield. iii) The powers separately received by the x-axis, yaxis, and z-axis dipoles are almost the same as those received by the 2-port crossed dipole and the 3-port tripole. This confirms the forecasts in Sections III-A3-III-A4, claiming that the 3-port tripole and 2-port crossed dipole can be viewed as separate orthogonal dipoles. iv) Among single-port dipole associations (crossed dipole, tripole), the optimally-fed structures receive up to 1 dB more power than the in-phase-fed structures. v) Two optimum designs emerge, as presented in Figs. 14a -14b. The first options are quasi-isotropic antennas: single and dual-band designs. While the second option consists in adding an N-way power combiner to either the 3-port tripole or the 2-port crossed dipole to combine the power received by each dipole. This collocated LP-port diversity system will outperform quasi-isotropic antennas as long as the insertion loss IL(dB) in the power combiner is lower than the combining gain (10logn), that is, 10logn -IL > 0, where n is the number of antenna ports receiving significant and comparable powers (n ≤ N). According to Table 9, it is clear that n ≈ 2 for both the 3-port tripole and the 2-port crossed dipole. vi) Finally, the dual-tone received power (1840 + 940 MHz) by the dual-band quasi-isotropic antenna is almost equal to its lower frequency received power (940 MHz). The λ 2 -factor on the available power explains this negligible contribution from 1840 MHz under the same W inc . This also stands for the other antennas, even though they are not necessarily well-matched and efficient on this higher frequency. Therefore, as mentioned in Section II-D, it is worth emphasizing that the multi-band design option benefits lower frequencies or those with significant power densities to compensate for the λ 2 -factor. The following section deals with real ambient tests.

V. EXPERIMENTS IN REAL AMBIENT ENVIRONMENTS
This section deals with experiments in real ambient environments to compare the test antennas' performances, typically their time-averaged received powers.

A. AMBIENT TESTS 1) MEASUREMENT SETUP
The measurement setup includes the fabricated test antennas in Section III, a 60 cm-long 50 -coaxial cable, and a spectrum analyzer, model FieldFox N9918A from Keysight, as presented in Fig. 15.
A standard personal computer (PC) with a dedicated LabVIEW program controls the spectrum analyzer (SA) for automated power acquisitions.

2) ACQUISITION
The embedded LabVIEW software on the PC allows setting some of the key measurement parameters, defined as follows: frequency band: 0.6 -3 GHz, Number of points: 401, sweep mode: continuous, amplitude type: power (dBm), video bandwidth (VBW): 10 kHz, resolution bandwidth (RBW): 10 kHz, Attenuation: 0 dB, and Preamplifier: Off. Under these settings, the SA's noise floor decreases to around -90 dBm, which is decent for measuring low-power signals in the ambient environment. It is worth mentioning that any maximum power tracking function is deactivated during measurements to get ambient power's "real" time-varying behavior. The system stores the received power every 10 seconds to reach 50 measured samples with each test antenna. Each sample includes all the frequency points. All the antennas are measured on the same support the exact day between 10 -12 am before changing the location.

3) LOCATIONS
Three locations with different RF propagation conditions are considered, as seen in Figs. 16a -16c.
Location #2 is downtown Rouen, within a dense area with relatively high domestic and professional buildings. The closest RF transmitters to this location are 210 m, corresponding to ORANGE, operating over the same frequency bands as Location #1.
Location #3 corresponds to one of ESIGELEC's buildings in a suburban area with dense forests and sparse buildings in Saint-Etienne-du-Rouvray, another city close to Rouen in the most southern part. The closest RF transmitters to this location are from ORANGE base stations at 340 m, operating over the same frequency bands as Locations #1 and #2. Only one mobile provider operates closely around this location, while four illuminate Locations #1 and #2.

4) POST-PROCESSING
Four post-processing steps apply to the raw received power samples. First, the measured samples are time-averaged for each frequency point. The second step cancels the SA's noise floor fluctuations by setting at -90 dBm any signal equal to or lower than -87 dBm, considering 3 dB noise fluctuation. The third step averages the obtained power locally on each frequency band where power exists, typically on LTE 20, GSM 900, etc. Finally, the time-averaged power on each frequency band is added to form the total time-averaged received power. Fig. 17a shows the power received by the optimallyfed tripole over time on LTE 20 and GSM 900 bands. Furthermore, Fig. 17b presents the raw received power over the frequency domain with the post-processed steps.
One can see that the time-domain received power's profile resembles a Rayleigh fading's behavior. The cumulative distribution function (CDF), computed for the three locations over LTE 20 and GSM 900, confirms this behavior.

5) INCIDENT POWER DENSITY
A commercial field meter from Fields At Work, model ExpoM -RF 4 [38], is used to measure the available power densities in the three locations. First, the field meter measures the available incident field amplitude E inc over several center frequencies with 100 MHz bandwidth. Then, the power densities are computed as E inc 2 /120π , assuming plane wave modes. Fig. 18 presents the measured power densities over time in Location #1, indicating a pulsed behavior for Wi-Fi signals around 2.45 GHz. However, mobile communications  show regular profiles, explained by downlink signals from base stations. Table 10 summarizes the time-averaged power densities in the three locations. Table 10 shows that the time-averaged power density decreases from Locations #1 to #3, consistently with the number of RF sources and their proximity to the three locations. Besides, LTE 20 and GSM 900 show the most significant power densities in Locations #1 and #2. However, Wi-Fi power density predominates in Location #3.
Among the existing data, the work in [8] issued average power densities in London's underground at 36 and 84 nW/cm 2 over the downlinks of GSM 900 and DCS 1800 bands, respectively. These data, like those of Imperial College London (ICL) [8], are much higher than the present values. The differences may be explained by the "maxhold" function, activated during measurements, and a denser RF source distribution, as reported in [8]. Another example includes the downtown Montreal survey setting the average power over the GSM/LTE 850 band between -30 to -20 dBm [39], which is almost 40 dB higher than that of LTE 20 in Fig. 17b. Again, a denser RF source distribution may explain this gap. Beyond that, comparing ambient RF signal measurement data suggests including the full testing parameters to make a reliable analysis. These factors include the measurement setup (measuring antenna's gain/pattern, cable loss, etc.), the post-processing method, the location configuration (RF sources, distance from the measurement location, vegetation, etc.), weather conditions, etc.

B. RESULTS
As described before, the total time-averaged received power is summarized in Tables 11 through 13 for the three locations for all the test antennas. In Locations #1 and #2, the total power primarily derives from the two lower bands (LTE 20, GSM 900), driving the most significant power densities. However, in Location #3, since the power density on the Wi-Fi band compensates for the λ 2 -factor typically, with (2.45/0.8) 2 ≈ 9 < 0.65/0.05 ≈ 13, most of the received power derives from the Wi-Fi band, except for the turnstile (most contribution from GSM 900).
In all locations, one of the single dipole's three positions is unfavorable, minimizing the received power compared with the two other positions. This poor reception is systematic, occurring horizontally in locations #2 or vertically in Locations #1 and #3. For comparison needs, let us consider this "negative" dipole as a reference to derive the test antennas' relative power gains ( G) within the ambient environment.
Indeed, among the single dipole's positions in the three locations, G rises to 4.37 dB (Location #1). The dipole's directivity and polarization misalignments explain this variation. Another point is the power imbalance between the dipole's three orthogonal positions within a given location. This is expected to limit the gain of the diversity system compared with isotropic antenna solutions. Besides, the optimally-fed tripole and crossed dipole outperform their equivalent in-phase-fed structures in all locations, confirming simulation results. Effectively, the turnstile receives up to 8 dB more power than the in-phase-fed crossed dipole. However, the optimum tripole captures up to 1.6 dB more power than the in-phase-fed tripole. These figures confirm simulation values around 1 dB.
Among quasi-isotropic antennas, the turnstile shows the best gain in Locations #1 and #3, receiving between 0.3 to 4.74 dB more power than the other antennas. This antenna's broad bandwidth, comfortably covering LTE 20 and GSM 900, explains this good performance. However, in Location #2, the dual-band quasi-isotropic dipole shows the highest power gain, 0.5 dB more than the others. Despite the dual-band option, this antenna only shows a limited gain compared with the other quasi-isotropic antennas. Therefore, the dual-band quasi-isotropic benefit is thin, targetting a higher frequency band (DCS 1800) on which the available power density does not compensate for the λ 2 -factor. The three locations' results confirm this limited performance, as noted from simulations.
Finally, among all the test antennas, the turnstile reaches the best power gain of 8.37 dB (Location #3), which remains closer to the highest received powers in Locations #1 (single dipole) and #2 (dual-band dipole). The turnstile's quasiisotropic pattern and broadband options, optimized for the lower frequency bands with significant power densities, explain this decent performance.
The following section precisely compares the two optimum antenna solutions, typically the best quasi-isotropic antenna (turnstile) and the diversity system.

C. OPTIMUM ANTENNAS WITHIN THE AMBIENT ENVIRONMENT
Let us compare the best-optimized quasi-isotropic antenna (turnstile) with the diversity system in Fig. 14b. Table 14 summarizes the relative power gain ( G), considering the turnstile's received power as the reference. Table 14 shows that the diversity system, which omits insertion loss, outperforms the turnstile in Locations #1 and #2. However, the turnstile shows the highest relative power gain and beats the diversity system in Location #3. Therefore, the two antenna solutions are competitive, but the following factors mitigate the diversity system's predicted performances: i) In practice, the power combiner introduces losses that increase with frequency and the number of ways to combine. Therefore, the designer should choose the appropriate combiner with minimum insertion loss (IL), verifying IL < G = 2.34 or 4.46 dB for the 2-port and 3-port dipoles, respectively. These requirements are realizable with current low-loss 2-and 3-way combiners. ii) The diversity system's overall size, including the multi-port dipole and the power combiner, is bulky, particularly at low frequencies. iii) In addition, the 2D or 3D dipole collocations are complex geometrical assemblies, particularly challenging to fabricate. Moreover, supplemental losses and parasitic radiations from the lines connecting the dipole ports to the power combiner should be anticipated. These limitations also apply to the simplified version of the diversity system, considering non-collocated LP dipoles at the expense of increased size.
The above issues may make the diversity solution impractical unless a DC-power combining approach is envisioned instead of RF summation [40]. Instead, like the designed turnstile, a well-optimized quasi-isotropic antenna provides competitive performances with a reduced size and a single port.

VI. CONCLUSION
This paper dealt with the best antenna characteristics for maximally harvesting ambient waves. The proposed approach analyzed and compared key design aspects, such as multi-polarization, multi-port, multi-band, omnidirectional, quasi-isotropic, and non-quasi-isotropic pattern options, over the same antenna size.
Several test antennas covering these design aspects have been designed, fabricated, characterized, and compared within a full-wave antenna simulator under spherical incidences and polarizations. Further comparisons were based on experiments in real ambient environments.  Two optimum antenna solutions emerged from the results: the quasi-isotropic antenna and the collocated LP-port diversity system, covering lower frequency bands with significant power densities that compensate for the available power λ 2 -factor.
Besides, the single-polarized dipole with an omnidirectional radiation pattern showed a systematic poor power reception among three orthogonal positions, leading to a power imbalance in the diversity system. However, the optimally-fed dipoles (crossed dipole, tripole) with quasiisotropic patterns systematically outperformed the in-phasefed dipoles with non-quasi-isotropic patterns.