Antenna/Body Coupling in the Near-Field at 60 GHz: Impact on the Absorbed Power Density

: Wireless devices, such as smartphones, tablets, and laptops, are intended to be used in the vicinity of the human body. When an antenna is placed close to a lossy medium, near-ﬁeld interactions may modify the electromagnetic ﬁeld distribution. Here, we analyze analytically and numerically the impact of antenna / human body interactions on the transmitted power density (TPD) at 60 GHz using a skin-equivalent model. To this end, several scenarios of increasing complexity are considered: plane-wave illumination, equivalent source, and patch antenna arrays. Our results demonstrate that, for all considered scenarios, the presence of the body in the vicinity of a source results in an increase in the average TPD. The local TPD enhancement due to the body presence close to a patch antenna array reaches 95.5% for an adult (dry skin). The variations are higher for wet skin (up to 98.25%) and for children (up to 103.3%). Both absolute value and spatial distribution of TPD are altered by the antenna / body coupling. These results suggest that the exact distribution of TPD cannot be retrieved from measurements of the incident power density in free-space in absence of the body. Therefore, for accurate measurements of the absorbed and epithelial power density (metrics used as the main dosimetric quantities at frequencies > 6 GHz), it is important to perform measurements under conditions where the wireless device under test is perturbed in the same way as by the presence of the human body in realistic use case scenarios.


Materials and Methods
We define here the exposure scenarios considered in this study. Then, the analytical and numerical methods used for exposure assessment are presented.

Exposure Scenarios
To analyze variations of TPD in the skin-equivalent model due to the presence of a radiating structure, seven scenarios of increasing complexity are considered ( Figure 1). Scenarios 1-4 are considered to determine the fundamental limits of TPD variations for a plane-wave excitation. In scenarios 5 and 6, we model the radiation pattern of antennas neglecting the impact of the phantom and a perfect electric conductor (PEC) on the antenna performances. Finally, antennas placed close to the skin-equivalent model are considered in scenario 7. These scenarios are detailed hereafter.
• Scenario 1: Plane-wave incident from free space onto a semi-infinite flat skin-equivalent model ( Figure 1a).
Normal incidence is considered to represent the worst-case exposure scenario with maximum TPD [20]. Due to a shallow penetration depth at mmWs (<1 mm), the interaction with the human body is mainly limited to skin. As a consequence, a homogenous skin-equivalent layer is used as a model [21,22]. The dielectric properties of skin-equivalent model are those of dry skin at 60 GHz (ε = 7.98 − j10.90); they were extracted from [23]. For completeness, we also provide in the paper the main results for a wet skin model (ε = 10.22 − j11.83 [23]).
• Scenario 2: Scenario 1 adding a perfect electric conductor (PEC) parallel to the skin model ( Figure 1b).
Normal incidence is considered to represent the worst-case exposure scenario with maximum TPD [20]. Due to a shallow penetration depth at mmWs (<1 mm), the interaction with the human body is mainly limited to skin. As a consequence, a homogenous skin-equivalent layer is used as a model [21,22]. The dielectric properties of skin-equivalent model are those of dry skin at 60 GHz (ɛ = 7.98 − j10.90); they were extracted from [23]. For completeness, we also provide in the paper the main results for a wet skin model (ɛ = 10.22 − j11.83 [23]).

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Scenario 2: Scenario 1 adding a perfect electric conductor (PEC) parallel to the skin model ( Figure  1b).
The total transmitted field ( , TPD ) is equal to the superposition of the transmitted field from direct incidence ( , TPD ) and the scattered field resulting from multiple reflections at the PEC/skin-model interfaces ( , TPD ). • Scenario 3: Scenario 1 with free-space losses (i.e., the amplitude of the plane-wave is attenuated in free space) ( Figure 1c).
The amplitude of the electric field radiated by an infinitesimal dipole decreases as 1/d in the farfield, where d is the distance between the source and the observation point. Hence, we assume in this scenario that the amplitude of the incident E-field decreases with an attenuation function f(d) = 1/d.
The antenna equivalent source is defined as a combination of equivalent electric and magnetic currents flowing on a closed surface surrounding the antenna (dashed line in Figure 1e) generating the same electromagnetic field as the antenna in free space.

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Scenario 6: Scenario 2 with the antenna equivalent source replacing the plane-wave illumination ( Figure 1f). • Scenario 7: Realistic antennas placed in the vicinity of the skin model ( Figure 1g).
The source main beam is directed towards the phantom representing the worst-case exposure scenario. Several sources have been considered: single patch antenna (SPA) and patch antenna array (PAA) with 4, 8, or 16 (2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively) radiating elements, inspired from [10,24] (Figures 2 and 3) and matched to 50 Ω in free-space at 60 GHz. All results are provided for an antenna input power of 10 mW. The total transmitted field (E Total , TPD Total ) is equal to the superposition of the transmitted field from direct incidence (E Direct , TPD Direct ) and the scattered field resulting from multiple reflections at the PEC/skin-model interface es (E R , TPD R ).

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Scenario 3: Scenario 1 with free-space losses (i.e., the amplitude of the plane-wave is attenuated in free space) (Figure 1c).
The amplitude of the electric field radiated by an infinitesimal dipole decreases as 1/d in the far-field, where d is the distance between the source and the observation point. Hence, we assume in this scenario that the amplitude of the incident E-field decreases with an attenuation function f(d) = 1/d.
The antenna equivalent source is defined as a combination of equivalent electric and magnetic currents flowing on a closed surface surrounding the antenna (dashed line in Figure 1e) generating the same electromagnetic field as the antenna in free space.

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Scenario 6: Scenario 2 with the antenna equivalent source replacing the plane-wave illumination ( Figure 1f).
• Scenario 7: Realistic antennas placed in the vicinity of the skin model ( Figure 1g).
The source main beam is directed towards the phantom representing the worst-case exposure scenario. Several sources have been considered: single patch antenna (SPA) and patch antenna array (PAA) with 4, 8, or 16 (2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively) radiating elements, inspired from [10,24] (Figures 2 and 3) and matched to 50 Ω in free-space at 60 GHz. All results are provided for an antenna input power of 10 mW.

Analytical Method: Plane Wave Illumination
The problem of a normally-incident plane wave (scenarios 1-4) at a planar interface between free space and a lossy region representing skin was solved analytically in [25]. Without loss, the electric field is given by: where E is the amplitude of the electric field, k is the free space wavenumber, and d is the normal distance between the plane-wave source and skin model interface. Assuming that the plane wave is attenuated in free space, its E-field vector is given by: where f(d) is an attenuation function. The transmitted E-field vector ( ) at the phantom interface is therefore expressed as [25] (Appendix A): Scenario 2: Scenario 3:

Analytical Method: Plane Wave Illumination
The problem of a normally-incident plane wave (scenarios 1-4) at a planar interface between free space and a lossy region representing skin was solved analytically in [25]. Without loss, the electric field is given by: where E is the amplitude of the electric field, k is the free space wavenumber, and d is the normal distance between the plane-wave source and skin model interface. Assuming that the plane wave is attenuated in free space, its E-field vector is given by: where f(d) is an attenuation function. The transmitted E-field vector ( ) at the phantom interface is therefore expressed as [25] (Appendix A): Scenario 2:

Analytical Method: Plane Wave Illumination
The problem of a normally-incident plane wave (scenarios 1-4) at a planar interface between free space and a lossy region representing skin was solved analytically in [25]. Without loss, the electric field is given by: where E 0 is the amplitude of the electric field, k 0 is the free space wavenumber, and d is the normal distance between the plane-wave source and skin model interface. Assuming that the plane wave is attenuated in free space, its E-field vector is given by: Appl. Sci. 2020, 10, 7392 where f(d) is an attenuation function. The transmitted E-field vector (E tr ) at the phantom interface is therefore expressed as [25] (Appendix A): Scenario 1 : Scenario 2: Scenario 3: Scenario 4: where T 1 and R 1 are the transmission and reflection coefficients, respectively, calculated using Fresnel coefficients at the free space/skin model interface [25]. The TPD for a plane-wave can be computed as: where η is the complex intrinsic impedance of the skin.

Analytical Method: Equivalent Source
For scenarios 5 and 6, the electric field was modeled analytically using the plane-wave spectrum theory [26][27][28]. It represents the spatial distribution of each field component over a transverse plane as a superposition of the plane waves propagating along different directions defined by the couplet K = k xx + k yŷ , also called the plane wave spectrum (PWS). The PWS of an electric field phasor component E(R, z 0 ) over a plane Ψ identified by z = z 0 and R = xx + yŷ is expressed as: The strength of this approach is its ability to represent the propagation of a complex field topography through space. The PWS over any plane parallel to Ψ located at distance l in a homogenous medium is computed by multiplying the PWS at z = z 0 by the propagator P(K, l) = e −jk z l : where k z is the longitudinal propagation constant given as k z = k 2 − |K| 2 , k is the propagation constant. For exposure scenario 5, the tangential spectrum components of the incident and transmitted fields at the air/phantom interface are related as: where Π 1 is the spectral transmission operator given in [26]. The normal field spectrum component is obtained from the tangential field spectraÊ || using the Gauss law: Appl. Sci. 2020, 10, 7392 6 of 16 For exposure case 6, the total transmitted field spectrum is given as (refer to Appendix A for more details): where I is the identity matrix, d is the PEC-phantom separation distance, Γ 1 is the spectral reflection coefficients at the phantom interface given in [26]. The H-field spectrum is calculated as [28]: The spatial field components (E and H) are retrieved using the inverse Fourier transform of the field spectra. The TPD is calculated as [6]: where ds is the integral variable vector with the normal direction to the integral area A on the body surface. All results are provided for an averaging area A of 1 cm 2 (except 2-dimensinoal TPD distributions provided in Sections 3.2 and 3.3). Note that the TPD is identical to the absorbed power density as defined in [6] and to the epithelial power density as defined by [7].

Numerical Method: Patch Antenna Arrays
Scenario 7 was analyzed numerically using the finite integration technique (FIT) implemented in CST Studio Suite 2019. The convergence is reached by setting a finer mesh around the air/phantom interface (i.e., 1 µm) and larger beyond (i.e., 0.356 mm corresponding to λ g /50, where λ g is the guided wavelength in the phantom). Open boundaries are used representing the free-space conditions (i.e., no reflected field at the boundaries of the computational volume). The number of mesh cells varies from 26 to 80 million with the antenna/phantom separation distance. Typical duration of single simulation varies from 35 to 75 min using high-performance workstations with accelerators (Xeon Gold 6140, 768 Go RAM, NVIDIA Quadro GV100; Dell, TX, USA).

Results
To analyze the TPD variations due to the antenna/body coupling, the following figure of merit is defined: where TPD m and TPD n are the TPD from exposure scenarios m and n, respectively, with m ∈ {2, 4, 6, 7} and n ∈ {1, 3, 5}. In practice, the separation distance between a wireless device and its user may vary.
To account for this variation during exposure, we also calculated the floating average Υ (m,n) over the range of distances ∆d ∈ {1, 3, 5} mm.

Fundamental Limits: Plane-Wave Illumination
First, we assess the TPD changes due to presence of a PEC layer in front of the skin model for plane-wave illumination (scenarios 1 and 2). To this end, TPD , TPD , and Υ ( , ) are calculated using (3), (4), (7), and (15) for E = 10 V/m and R = −0.59 + j0.16 (Figure 4).   Figure 4 shows that the TPD at the surface of the phantom is strongly altered by the presence of PEC (increase up to 574% and decrease down to 61.7%). Υ ( , ) can be expressed as: The positions of Υ ( , ) maxima and minima (d ( , ) and d ( , ) ) depend on the phase of R and given by:  Figure 4 shows that the TPD at the surface of the phantom is strongly altered by the presence of PEC (increase up to 574% and decrease down to 61.7%). Υ (2,1) can be expressed as: The positions of Υ (2,1) maxima and minima (d Υ (2,1) max and d Υ (2,1) min ) depend on the phase of R 1 and given by: where n is an integer number, R 1 is the phase shift introduced by the phantom interface to the reflected plane-wave. The fundamental limits of Υ (2,1) can be found by replacing (17) and (18) in (16): Equation (19) shows that the higher the magnitude of the phantom reflection coefficient, the higher the TPD variations. For example, for a wet skin model, the variations of TPD are more pronounced (increase up to 629% and decrease down to 62.3%, respectively). Note that these variations are also age-dependent as the tissue properties evolve with age [29]. In particular, for 5 year old children the enhancement increases to 640%. For the sake of brevity, in the rest of the paper, the analysis for wet skin and age-dependent effects will be omitted (except Section 3.3). Table 1 provides the maximum and minimum Υ (m,n) . The results show that the average TPD increases due to the presence of PEC (roughly a 60% increase for ∆d = 5 mm). Next, the free-space losses are taken into account in the analysis (scenarios 3 and 4). TPD 3 , TPD 4 , and Υ (4,3) are calculated using (5), (6), (7), and (15) ( Figure 5).

Fundamental Limits: Equivalent Sources
Here, we consider the equivalent sources corresponding to the patch antenna arrays radiating in free space ( Figure 2). This allows us to model the case where the free-space antenna matching, efficiency, and radiated field are preserved and not modified by the phantom (scenarios 5 and 6). TPD 4 demonstrates a damped oscillatory behavior around TPD 3 (increase up to 80% and decrease down to 28%). Due to the free-space loss, the oscillation amplitude of Υ (4, 3) is lower compared to Υ (2,1) . The averaged TPD over distance increases due to the presence of PEC (i.e., maximum increase of 41%, 15%, 5% for ∆d = 1 mm, 3 mm, 5 mm, respectively) ( Table 2).

Fundamental Limits: Equivalent Sources
Here, we consider the equivalent sources corresponding to the patch antenna arrays radiating in free space ( Figure 2). This allows us to model the case where the free-space antenna matching, efficiency, and radiated field are preserved and not modified by the phantom (scenarios 5 and 6). TPD 5 , TPD 6 , and Υ (6,5) are calculated from Equations (8)- (15) (Figure 6a-e).
Significant differences in Υ (6,5) maxima and, to a smaller extent, minima between the antenna-equivalent sources are noted for d < 25 mm (Figure 6e). The TPD increases (decreases) up to (down to) 174% (39%), 342% (54%), 421% (54.7%), and 497% (54.7%) for the SPA, 2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively. This is due to the differences in the attenuation rate of the peak power density in free-space PD fs of the antenna-equivalent sources (Figure 6f). Indeed, when the PD fs attenuation rate is higher, Υ (6,5) is lower. At d = 35mm, Υ(6,5) of all antenna-equivalent sources converges to the same oscillatory function (with relative difference < 10%) (Figure 6e). Figure 7 shows that as d increases, Υ (6,5) converges to Υ (4,3) as the power density in free-space decreases as 1/d 2 in the far-field. Maximum values of Υ (6,5) are obtained for 4 × 4 PAA (i.e., the maximum increase of 247%, 124%, 76% for ∆d = 1mm, 3mm, 5mm, respectively) ( Table 3). Note that for the antenna-equivalent sources maxΥ is up to 2.5 times higher compared to the plane wave with free-space losses (compare Tables 2 and 3). Significant differences in Υ ( , ) maxima and, to a smaller extent, minima between the antennaequivalent sources are noted for d < 25 mm (Figure 6e). The TPD increases (decreases) up to (down to) 174% (39%), 342% (54%), 421% (54.7%), and 497% (54.7%) for the SPA, 2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively. This is due to the differences in the attenuation rate of the peak power density in free-space PD of the antenna-equivalent sources (Figure 6f). Indeed, when the PD attenuation rate is higher, Υ ( , ) is lower. At d = 35mm, Υ(6,5) of all antenna-equivalent sources converges to the same oscillatory function (with relative difference < 10%) (Figure 6e). Figure 7 shows that as d increases, Υ ( , ) converges to Υ ( , ) as the power density in free-space decreases as 1/d in the farfield. Maximum values of Υ ( , ) are obtained for 4 × 4 PAA (i.e., the maximum increase of 247%, 124%, 76% for Δd = 1mm, 3mm, 5mm, respectively) ( Table 3). Note that for the antenna-equivalent sources maxΥ is up to 2.5 times higher compared to the plane wave with free-space losses (compare Tables 2 and 3).  To obtain a deeper insight into the TPD variations, we analyzed the changes in the spatial distribution of TPD for the SPA equivalent source due to the presence of PEC (Figure 8). The distribution of TPD is affected by the presence of PEC and evolves with d. For d corresponding to the maximum TPD (i.e., 4.75 mm, 7.25 mm, 17.25 mm), the absorbed power density is concentrated around its maximum. It extends progressively over a larger surface when d approaches the value corresponding to TPD minima (i.e., 6.5 mm, 9.0 mm, 18.75 mm). When the spatial distribution of TPD is concentrated around its maxima, the spatial averaging area has a stronger impact on the mean TPD, which rapidly decreases with the averaging area (e.g., the ratio between TPD averaged over 1 cm and 4 cm equals to 3.26 and 1.86 for d = 4.75 and 6.5 mm, respectively).  To obtain a deeper insight into the TPD variations, we analyzed the changes in the spatial distribution of TPD for the SPA equivalent source due to the presence of PEC (Figure 8). The distribution of TPD 6 is affected by the presence of PEC and evolves with d. For d corresponding to the maximum TPD (i.e., 4.75 mm, 7.25 mm, 17.25 mm), the absorbed power density is concentrated around its maximum. It extends progressively over a larger surface when d approaches the value corresponding to TPD minima (i.e., 6.5 mm, 9.0 mm, 18.75 mm). When the spatial distribution of TPD is concentrated around its maxima, the spatial averaging area has a stronger impact on the mean TPD, which rapidly decreases with the averaging area (e.g., the ratio between TPD averaged over 1 cm 2 and 4 cm 2 equals to 3.26 and 1.86 for d = 4.75 and 6.5 mm, respectively). distribution of TPD is affected by the presence of PEC and evolves with d. For d corresponding to the maximum TPD (i.e., 4.75 mm, 7.25 mm, 17.25 mm), the absorbed power density is concentrated around its maximum. It extends progressively over a larger surface when d approaches the value corresponding to TPD minima (i.e., 6.5 mm, 9.0 mm, 18.75 mm). When the spatial distribution of TPD is concentrated around its maxima, the spatial averaging area has a stronger impact on the mean TPD, which rapidly decreases with the averaging area (e.g., the ratio between TPD averaged over 1 cm and 4 cm equals to 3.26 and 1.86 for d = 4.75 and 6.5 mm, respectively).  The cross-section distributions along x-axis at y = 0 mm of the x component of E Direct , E Total , and E R are plotted in Figure 9. The amplitude of E Total is directly related to the phase difference between E Direct and E R . For d = 7.25 mm, E Direct is in phase with E R around x = 0 mm. This null phase difference evolves periodically along the x-axis resulting in either constructive or destructive interferences. Consequently, this results in an enhancement of the E Total amplitude for x ∈ (−5.0; 5.0) mm and in a decrease for x ∈ (−10; −5) U (5; 10) mm, thus explaining the higher spatial gradient of TPD in Figure 8a,c,e (middle line, scenario 6). On the other hand, for d = 9.0 mm, E Direct and E R are out of phase at x = 0 mm. This results in a decrease in the E Total for x = (−5.0; 5.0) mm and its enhancement for x = (−10; −5) U (5; 10) mm, resulting in spread of TPD in Figure 8b,d,f (middle line, scenario 6). Note that similar observations were made for the y and z components of the field (for the sake of brevity, the data are not shown).

Patch Antenna Arrays
When an antenna is located in the vicinity of a scatter, its matching and radiation are altered. To exclude the effect of the antenna mismatch, TPD is normalized to (1-S / ) and TPD to (1-S / ), where S / and S / are S of the antenna in the presence of the phantom and in free space, respectively. Note that modern wireless devices are equipped with matching networks designed to compensate for the mismatch.

Patch Antenna Arrays
When an antenna is located in the vicinity of a scatter, its matching and radiation are altered. To exclude the effect of the antenna mismatch, TPD 7 is normalized to (1-S 11/Ph 2 ) and TPD 5 to (1-S 11/FS 2 ), where S 11/Ph and S 11/FS are S 11 of the antenna in the presence of the phantom and in free space, respectively. Note that modern wireless devices are equipped with matching networks designed to compensate for the mismatch.
The changes in the TPD due to the antenna/phantom coupling (scenarios 5 and 7) are shown in Figure 10. For d < d r , where d r denotes the interface between the reactive and radiating near-field regions, the changes in term of the absolute value of TPD are more pronounced (Figure 10a-d). In terms of the relative variations, for this range of d, the TPD increases up to 79.2%, 71.6%, and 43.8% and decreases down to 4.4%, 9.75%, and 9.84% for 2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively (Figure 10e-h). The results shown in Figure 10i demonstrate that there is no direct correlation between Υ (7,5) and the source directivity. Note that Υ (7,5) is lower compared to Υ (6,5) . This difference is attributed, to a smaller extent, to losses inside the antenna (15.7%, 18.6%, 27.8%, 33% in respect to the total accepted power at d = 2.25 mm and for SPA, 2 × 2 PAA, 2 × 4 PAA, and 4 × 4 PAA, respectively) and, to a larger extent, to the scattering properties of the antennas. The higher the scattering, the lower the TPD variations.
The maximum values of maxΥ (7,5) are obtained for 2 × 4 PAA (i.e., increase up to 52%, 25%, 21% for ∆d = 1mm, 3mm, 5mm, respectively) ( Table 4). It is worthy to note that the ground plane size impacts the TPD variations. For instance for the SPA, the TPD variations increase with size until the ground plane becomes large enough (e.g., for ground plane dimensions of 2.5 × 2.5 to 10 × 10 mm 2 , Υ (7,5) increases from 10% to 79%). show that the spatial distribution of TPD 7 is altered by the antenna/phantom interactions in a similar way as in scenario 6. This suggests that the exact distribution of TPD cannot be retrieved from measurements of the incident power density in free-space in absence of the body model.     7) show that the spatial distribution of TPD is altered by the antenna/phantom interactions in a similar way as in scenario 6. This suggests that the exact distribution of TPD cannot be retrieved from measurements of the incident power density in free-space in absence of the body model.

Conclusions
In this study, we analyzed the impact of the near-field antenna/body interactions on TPD at 60 GHz. To assess the variations of TPD due to presence of a skin-equivalent model, sources of increasing complexity were considered, including plane wave with and without a PEC, plane wave with free-space losses, antenna-equivalent sources with and without a PEC, and patch antenna arrays.
The spatial distribution of the TPD is impacted by the presence of a body due to constructive or destructive interferences impacting both peak and averaged TPD. Our results demonstrate that, for all scenarios considered in this study, the presence of the body in the vicinity of a source results in an increase in the average TPD. The local TPD variations depend on the source/body separation distance. The TPD enhancement due to presence of the human body reaches 574% and 80% for the plane-wave excitation with and without free-space losses, respectively. For the antenna-equivalent sources, the presence of a PEC increases (decreases) local TPD from 174% to 497% (39% to 54.7%)