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This article presents a procedure for electromagnetic field and polarization control with antennas. The concept previously introduced by the authors for spatially distributed three-dimensional electromagnetic polarization (as time varies) is discussed and extended also to include non-ideal antennas and the control of electromagnetic field distributions (at a given instant of time). These polarizations and fields are herein referred to as “3D”, although time is also inherent to them. Even that the main objective is to introduce a mathematically/numerically consistent synthesis technique for controlling the 3D electromagnetic fields and polarizations, an effort is made to present and discuss possible applications, including but not limited to torus-knotted distributions and spatial multiplexing for transmission of information in wireless digital communication systems.

The increasing usage of electromagnetic polarization in telecommunication systems is well known and documented, ranging from applications such as diversity and frequency reuse to electromagnetic compatibility and future 4 k/8 k resolution Digital Television Broadcasting [

This work presents a way of generating 3D polarizations and field distributions, where the electric field intensity vector can be oriented in any direction in space. Therefore, it is possible to generate 3D polarizations and/or fields with different shapes, as time varies, such as cylinders, spheres, and others. In addition, the orthogonality between 3D polarizations is also discussed. Antennas naturally generate a polarization that is orthogonal to the desired one, normally referred to as cross polarization, but the idea here is to generate the orthogonal polariza- tions in a controlled manner, similarly to the use of orthogonal polarizations in frequency reuse systems. Cross polarization is normally undesired, with the exception of the application introduced in [

It was previously demonstrated analytically and numerically by the authors in [

It was demonstrated in [

A brief summary of the concept discussed in [

The antenna array shown in

antenna produces fields only in the radial direction. This was done to simplify the equations discussed in [

The electric field generated by one antenna is frequency dependent, however the figure formed by the fields generated in the array of ideal dipole are frequency independent if the same frequency is used for the all antennas. This occurs because the proportionality between the electric fields generated by individual antennas is kept constant when the frequency is changed. Thus, for illustration purposes, without loss of generality, we use f = 100 MHz, I = 1 mA, h = λ/100 (the length of all 3 antennas placed at a distance equal to λ from point O). It is important to note that for the scenarios considered in this study there is no propagation, so the 3D polarization needs to be visualized as the polarization of the resulting electric field, not the wave. Alternate configurations that allow the 3D polarization to move within a desired path are currently being investigated (by determining different sets of feed currents for each desired location along the trajectory, for example), but the theory does not lose its importance nor generality as herein presented. By varying continuously the phase of the ideal dipole P_{2} in _{2} in the interval 0 to 2π, with ψ_{1} = ψ_{3} = 0˚ respectively for the other two dipoles, the resulting polarization is shown in

In order to show that it is also possible to generate 3D polarizations with antennas other than the ideal dipoles, we employed the array of pyramidal horns shown in

The electric field of each horn can be approximated by (far-field) [

where E_{0} is the absolute value of the intensity of the electric field at the aperture (assuming the mode TE_{01}). The electric field at the aperture is given by

where_{1} and I_{2} are given by

where C(x) and S(x) are the Fresnel functions, and

where ζ is a variable controlled by the amplitude variation of the electric field in the aperture of each antenna, it can be shown that the total electrical field (far-field analysis) at the observation point O is given by

which in the time domain is

where C_{0} and φ_{c} are respectively the absolute value and phase of (valid for O at the origin)

The method introduced by the authors in [_{1} = 19.94 cm, R_{2} = 18.2 cm, A = 18.46 cm, B = 14.19 cm). The antennas are away 10^{3}λ from the origin (far-field), and E_{0} = 1 V/m. The spherical polarization shown in

In the previous sections the field control was done in only one point in the space; the reader is referred to [

The number of users in a given region can be increased with the new method, which is based on space division multiplexing, using the same frequency and time resources, with the least possible interference between users. Thus, the channel capacity can be increased. The interference can be further reduced between nearby users, increasing the signal to noise ratio (S/N) with the control of orthogonal polarizations.

When m antennas are used for the control of n points in space, individual electric fields generated by each antenna are added to obtain the resulting field. The

most direct method for this is to decompose the spherical components a_{r} and a_{θ} directions in cartesian coordinates a_{x}, a_{y} and a_{z}:

where a_{rx}, a_{ry} and a_{rz} respectively represent the decomposition of the components of the unit vector a_{r} in the directions a_{x}, a_{y} and a_{z}. Also, a_{θx}, a_{θy} and a_{θz} respectively represent the decomposition of the components of the unit vector a_{θ} in the directions a_{x}, a_{y} e a_{z}.

Considering an antenna located at point P_{1}, and an observation point in O_{1}, the equations below are obtained in cartesian coordinates:

where P_{1} is the position vector of antenna 1, S_{1} is the vector direction of antenna 1, and O_{1} is the position vector of the observation point 1. Superscript T in the equations was used for simplification, and indicates the transpose of the row vector. All these vectors are referenced to the origin of the cartesian system. P_{1} and O_{1} start from the origin and go to the specified points, and the distances represent the amplitudes. S_{1} starts from point P_{1} and is defined here as a unit vector, and its amplitude equals 1. The distance vector D_{11} between O_{1} and P_{1} is_{11} and S_{1} is given by

The distance vector D_{11} has the same direction of the vector a_{r}_{11}. In general, establishing the variable u as being equal to x, y or z, a_{ru}_{11} can be determined as:

With a_{rx}_{11}, a_{ry}_{11}, and a_{rz}_{11} values obtained through (17), a_{θ}_{x}_{11}, a_{θ}_{y}_{11}, and a_{θ}_{z}_{11} can be determined as:

Using

where φ_{Eu}_{11} is the argument of E_{u}_{11}. The coefficient C_{u}_{11} has the unit Ω/m. C_{u}_{11} and φ_{Eu}_{11} are calculated as

with

The phase and amplitude can be grouped in a single complex variable σ_{u}_{11} as shown below

Once the values of the length of the antenna and the frequency of the excitation source are determined in the vacuum, the coefficient σ_{u}_{11} will vary only with changes in the position and direction of the antenna, as well as the distance of this to the observation point. With m antennas and n observation points (m = 3n), Equations (25) and (26) can be generalized to

where i and l are integers number, with

Calling s the matrix of coefficients, I the current vector and E the vector fields, the values of the excitation sources of the antennas can be determined by:

where

In this section the volumetric antennas array and fields control can be employed for the transmission of digital signals as well as to minimize the interference by spatial multiplexing. For the simulation, the antenna array presented in

For the observation point O_{1}, the main antennas are 1, 2 and 3 because they contribute more significantly to the resulting electric field at this point, since the value of the coefficient C_{u} has a higher value. Similarly, for point O_{2}, the main antennas are 4, 5 and 6. In the configuration of

Assuming

Note that the mutual couplings between the transmitting antenna and the reception points are negligible, since the distances between the points of observations and the antennas are in far-fields. The amplitudes of the mutual impedances of the antennas in transmission and reception are considerably reduced if the distances between them are large [

It was considered for the simulation digital transmissions ASK with binary signals with m_{a} < 1. Although other modulations can be employed, we adopted this modulation for being simpler than the others, given that the focus of the simulation is to illustrate the transmission of signals to two different receiving points, sustaining isolation between the points at a same frequency and polarization.

For the simulation, it was also considered as bit 0 an amplitude of electric field equals to 2 × 10^{−3} V/m, and bit 1 an amplitude equals to 4 × 10^{−3} V/m. In the absence of transmission, the amplitude must be zero. A circular polarization was

used in the plane

to generate the desired electric field.

The simplest condition happens when there is no transmission at both observation points. In this configuration, the desired fields E_{0} and E_{1} are zero, and current feeds of the antennas are zero. For other conditions, Equation (30) was used to obtain the amplitudes and phases of the currents.

In the transition times of the desired signals, the currents must be changed in these moments. According to the table, all antennas must be fed, even when there is transmission at only one point. This is due to the fact that all antennas should work together to nullify the field in one of the reception points. It is also noted that the electric current amplitudes of main antennas for the point where

Desired Signal | Antenna Current Phasors | ||||||
---|---|---|---|---|---|---|---|

O_{1 } | O_{2 } | I_{1} (mA) | I_{2} (mA) | I_{3} (mA) | I_{4} (mA) | I_{5} (mA) | I_{6} (mA) |

null | null | 0 | 0 | 0 | 0 | 0 | 0 |

null | bit 0 | ||||||

null | bit 1 | ||||||

bit 0 | null | ||||||

bit 1 | null | ||||||

bit 0 | bit 0 | ||||||

bit 0 | bit 1 | ||||||

bit 1 | bit 0 | ||||||

bit 1 | bit 1 |

there is no transmission is small compared with the current amplitudes of the main antennas for the point where there is transmission.

Thus, all antennas work together and are responsible for the transmission of all signals. As each antenna carries a small portion of the information, the fading can be mitigated, due to the path diversity.

For illustration,

signals, that is, a circular polarization in the plane

amplitude equals to 4 × 10^{−3} V/m for the point of observation O_{1} (a), and an amplitude equals zero for the point of observation O_{2} (b).

In this way, any electric field is generated at the observation point, regardless of the desired field at another point, resulting in a transmission isolated by spatial separation or spatial multiplexing.

To observe how the signal is degraded when users move away from the observation point used in the design, consider other example using the same conditions, except for_{1} e O_{2}, with the new current values obtained by (30), electric fields were simulated for different points

The time-average amplitude of the resulting field at the observation point i is calculated as follows:

In possession of the

Instead of calculating the difference of electric field amplitude at each instant of time, (32) was used in order not to consider the propagation delays between observations points. As the polarizations are in different planes, and the observa- tions points are separated by multiple wavelengths, the Equation (32) becomes more appropriate to the intended results.

In

Despite that the O_{1n} and O_{2n} points are approximately 300 m from the points O_{1}

Desired Signal | Time-average amplitude of the resulting field | Error | |||||
---|---|---|---|---|---|---|---|

O_{1 } | O_{2 } | ||||||

null | bit 0 | 0 | 2 × 10^{−3} | 0.143 × 10^{−3} | 1.932 × 10^{−3} | 0.143 × 10^{−3} | −0.068 × 10^{−3} |

null | bit 1 | 0 | 4 × 10^{−3} | 0.286 × 10^{−3} | 3.865 × 10^{−3} | 0.286 × 10^{−3} | −0.135 × 10^{−3} |

bit 0 | null | 2 × 10^{−3} | 0 | 1.932 × 10^{−3} | 0.143 × 10^{−3} | −0.068 × 10^{−3} | 0.143 × 10^{−3} |

bit 1 | null | 4 × 10^{−3} | 0 | 3.865 × 10^{−3} | 0.286 × 10^{−3} | −0.135 × 10^{−3} | 0.286 × 10^{−3} |

bit 0 | bit 0 | 2 × 10^{−3} | 2 × 10^{−3} | 1.942 × 10^{−3} | 1.942 × 10^{−3} | −0.058 × 10^{−3} | −0.058 × 10^{−3} |

bit 0 | bit 1 | 2 × 10^{−3} | 4 × 10^{−3} | 1.962 × 10^{−3} | 3.871 × 10^{−3} | −0.038 × 10^{−3} | −0.129 × 10^{−3} |

bit 1 | bit 0 | 4 × 10^{−3} | 2 × 10^{−3} | 3.871 × 10^{−3} | 1.962 × 10^{−3} | −0.129 × 10^{−3} | −0.038 × 10^{−3} |

bit 1 | bit 1 | 4 × 10^{−3} | 4 × 10^{−3} | 3.883 × 10^{−3} | 3.883 × 10^{−3} | −0.117 × 10^{−3} | −0.117 × 10^{−3} |

and O_{2}, the amplitudes of the electric fields generated and the polarization planes almost did not suffer degradation.

Considering the small decoupling between the receiving antennas due to the difference between the planes of polarization, the received power will be slightly smaller. Consequently, the largest error obtained in the example will have a small reduction.

In fact, spatial regions whose deviations are small are formed around the observation point O_{i}, and the isolation is preserved. Thus, the reception point could move within that region without suffer interference. The region depends on the threshold detection to identify the signals.

Although the distance between the new observations points and those used in the design are large, it represents only a percentage of the distance from the observation points O_{1} and O_{2} to the origin, which equals 3.000 m. Thus, as these points move away from their main antennas, the greater will be the region of reception, where users can move without any significant errors. This is because the amplitude of the electric fields suffers smooth transitions between the two observation points.

Ignoring the effects of fading, the theoretical perfect isolation can be maintained if the system knows the locations of all the reception points. So with the new position of the observation point, the matrix of complex coefficients s could be updated, if necessary and new values of currents are obtained using (30). The sensitivity of the signals with respect to various distances from the target points, as well as case examples with pyramidal horn antennas, is discussed in depth in [

The theory and possible applications related to torus-knotted electromagnetic fields is discussed in [

This work extended substantially a procedure introduced by the authors in [

vector can be oriented in any direction in space. These polarizations were referred to as “3D” due to their distribution in space, although time is also inherent. It was shown analytically and numerically that it is possible to generate 3D polarizations with different shapes, as time varies, such as cylinders, spheres, and others. In contrast to [

Although the main purpose was to introduce a mathematically consistent synthesis procedure for controlling the three-dimensionally distributed electromagnetic fields and polarizations, with applications yet to be adapted or discovered, an effort was made to illustrate that the technique can be used for digital transmission of information in wireless communication systems with spatial multiplexing. For each observation point, three antennas will be necessary. If more antennas are used, the fading will be smaller and the results will be closer to the desired. Volumetric regions are created with this array, and isolation could be preserved if the reception points are within these regions. The interference for spatial multiplexing can be further reduced if the method of electric field control is combined with the use of orthogonal 3D polarizations between close users, as in nano and phantom cellular systems [

Pereira, L.P.S. and Terada, M.A.B. (2017) Synthesis of Antennas for Field and Polarization Control. Journal of Electromagnetic Analysis and Applications, 9, 97-112. https://doi.org/10.4236/jemaa.2017.97009