1. Introduction

When a radiated field is distributed in space, there are three regions that identify the process ^[1]: Rayleigh region, Fresnel region and Fraunhofer region; Rayleigh region refers to the individual point field, which means that the radiation generated at each point of the structure has not been mixed with the other. In Fresnel region, the generated individual fields begin to interact with each other, but their phase differences are significant, so the radiation pattern does not have the shape that must adopt in far-field. In Fraunhofer region, fields are added almost in phase and as they move away from the source, the differences begin to be further reduced, so this region is referred as the far-field region. Figure 1 describes the three regions of the field distribution process in space. Table I shows conditions to determine the field region distances, where D is the antenna diameter, R1 and R2 are Rayleigh and Fresnel regions and λ is the wavelength ^[2].

FIGURA 1 Field distribution regions. 

According with Fig. 1 and Table I, the minimum distance to obtain the far-field of a 1.5 m diameter structure, working at 5 GHz, is at least 75 m, which represents a problem if we try to measure far-field inside a controlled environment (as in an anechoic chamber, making nearly unmanageable to address the problem). Our method proposes to perform a field modeling at the Rayleigh region, define virtual point currents over the surface through discrete Pocklington equation, and then obtain far-field using array theory; essentially a near-field to far-field transformation is carried out. Comparing with commercial software, we reduce computational resources and the proposed method has been used to transform near-field into far-field from real measurements ^[1].

The method requires knowing the electric field near to reflector surface; this data can be obtained from measurements as ^[3-^6] or by field modeling in the Rayleigh region as in ^[7], for this purpose, we implement a model based on FDTD method in order to obtain the magnitude and phase of field in a small region, near to the reflector surface. Descriptions of FDTD, discrete Pocklington equation and parabolic array factor (PAF) are presented in following Subsecs. 1.1 and 1.2.

1.2 Discrete Pocklington equation and parabolic array factor

The method considers that the generalized Pocklington equation ^[11] is valid not only on the antenna surface, but also outside very near to it, as near as the Rayleigh region defines it. Hence, it is possible to perform a field modeling by FDTD or by measurement as in ^[1] to acquire field points near the structure.

The current can be determined by transforming the generalized Pocklington in a discrete form as Eq. (6) calculating virtual currents In along the structure.

Emi=-1jωε∑n=1N[R2(k2R2-1-jkR)Sm⋅sn'+(3+3jkR-k2R2)(R⋅Sm)(R⋅sn')]e-jkR4πR5In. (6)

Where:

Rmn=|r-r'|=[x(s)-x'(s')]2+[y(s)-y'(s')]2+[z(s)-z'(s')]2

k=Wavenumber

Solving Eq. (6) by the structure geometry, it can be constructed a matrix as (7), where ZMN represent impedances, IN represents a virtual current to be calculated and EM represents the modeled electric field. The matrix is solved by multiplying the electric field vector, obtained by the FDTD modeling, by the inverse of the impedance matrix, so vector current is obtained.

*20cZ11⋯Z1N⋮⋱⋮ZM1⋯ZMN*20cI1⋮IN=*20cE1⋮EM (7)

Finally, far-field is calculated using the array theory ^[12,^13] under the assumption that calculated point currents IN in (7), represent an array of specific elements that generate the far-field as is shown in Fig. 2. Field radiated by a parabolic array of point sources is given by (8):

PAF(θ)=E0∑n=1N|IN|ej(kRncos(θ-ϕ)+αn) (8)

FIGURE 2 Current in a parabolic profile. 

In (8) E0 represents the reference field of the array point sources with unitary magnitude; the sum of the individual fields from each point antenna considers the phase current αn, due the distance and the position of each current point Rn defined by ϕ while θ represents the angle between each point and the far-field position, Fig. 2 describes Eq. (8).

2. Near-field analysis and far-field transformation

Table II shows the modeling parameters, where physical dimensions are defined in meters and cells, in order to apply our computational technique based on Pocklington equation and FDTD.

Calculus region dimensions are 1.5×2.25 m corresponding to 500×750 cells, considering the size of each cell as λ/20, even several authors define λ/10 is an acceptable size ^[9]. Our results are compared with those of commercially available software CST, using same parameters for both experiments. Figure 3 shows the calculus region and electric field distribution for both computational methods. To have a better view of the parabolic profile, Fig. 3 (a) is shown in a smaller area defined by (90:335,95:645), where reflector vertex is located at (100,375) and its focus at (300,375). Figure 3 displays both incident and reflected fields, which visual comparison with CST shows a very good concordance.

FIGURE 3 Near-field modeling at 5 GHz, a) FDTD and b) CST. 

The electric field vector EM in Rayleigh region required in (6), is defined using field samples from 500 points in the FDTD matrix at 0.003 m, one cell away from reflector surface along the parabolic profile. Figure 4 shows EM vector in magnitude, real and imaginary parts. Field phase is shown in Fig. 5, and is obtained using distances from reflector surface to its focus, for each field point using parabola equation^[14].

FIGURE 4 Superficial electric field distribution. 

FIGURE 5 Field distribution phase. 

Next step in the methodology is to apply Eqs. (6) and (7) to obtain current along the reflector surface. Equation (6) requires the use of parabolic geometry for distace Rmn and vectors sm and sn, that is, distances between each point of the reflector axis and each current point ^[7].

Finally, the field distribution is transformed in virtual point currents along the plate surface that will be applied in the array factor equation for the parabolic profile (8). Figure 6, shows the magnitude, real and imaginary normalized parts for current distribution IN. We can notice a similar behavior with electric field displayed on Fig. 4, including the 180∘ phase change of imaginary part due the reflection. Figure 7 shows the calculated phase for each current point.

FIGURE 6 Superficial current distribution. 

FIGURE 7 Current phase. 

Once field distribution is transformed into virtual currents along the parabolic profile, magnitude and phase values are used in the parabolic array factor Eq. (8) to get the far-field pattern shown in Fig. 8. As it is shown, radiation pattern has a main lobe width of 2.5∘ which is consistent with the theory of parabolic reflectors ^[15], also the nearest sidelobes are observed at -17 dB, other lobes have amplitudes between -20dB and -55 dB.

FIGURE 8 Reflector field pattern at 5 GHz. 

3. Comparison

In previous work ^[7,^16], calculated radiation pattern was compared versus a measured radiation pattern however, actual analysis allows us to observe and compare electric field behavior with those calculated in CST to show its viability.

Figure 9 shows a comparison between near-field behavior for both simulations (FDTD and CST) in magnitude. Figures 10 and 11 show an excellent agreement of the near-field in real and imaginary parts; finally, near-field phase is shown in Fig. 12, as seen there is a clear similarity between FDTD and CST curves, specially real and imaginary parts with small differences at the beginning and at end of curves.

FIGURE 9 Magnitude Electric field comparison between FDTD and CST. 

FIGURE 10 Electric field real part comparison between FDTD and CST. 

FIGURE 11 Electric field imaginary part comparison between FDTD and CST. 

FIGURE 12 Electric field phase comparison between FDTD and CST. 

Radiation patterns comparison is shown in Fig. 13, black curve represents the pattern of our method and blue curve is the CST pattern. Similarity between both curves is evident; it can be seen that the main lobe width is about 2.5∘ for both patterns and small differences of 5 dB at the edges (0° o 30° and 150° to 180°).

FIGURE 13 Reflector field pattern comparison at 5 GHz. 

4. Conclusions

We presented a method to transform near-field (Rayleigh region) into far-field, without performing simulation in large computational regions, then reducing computing resources. Even we define the field in the Rayleigh region using the FDTD Method, an alternative proposed as future work is to measure the field near the reflector surface and obtain far-field with Pocklington equation and Array Theory. In addition, our method can be applied to other antenna geometries as lineal radiators or even linear arrays or some other type of reflectors.

Comparison between our methodology (FDTD-Pocklington-PAF) and commercial software shows that results are very similar not only for radiation pattern, but also for near-field behavior.