Null-filling effects over measurement distances on circular Taylor patterns with optimal transition integer $\overline{n}$ and with reduced inner sidelobes

Abstract

The following paper presents the results of a study carried on the impact of null-filling on measurement distances in circular Taylor patterns. Achieving the desired ripple control over these patterns requires the introduction of M complex roots, resulting in $2^{M}$ distinct continuous aperture distributions, each corresponding to a potentially different measurement distance requirement. However, this null-filling process is associated with efficiency losses and complex distribution. To evaluate these effects, we compared the optimal -25 dB SLL distribution with $\overline{n}$=5, known to offer maximal efficiency, to the same sidelobe level and optimal transition integer, but affected by null-filling. Additionally, we analyzed the influence of null-filling on the $\overline{n}$=5, -25 dB SLL pattern with the first sidelobe depressed at -40 dB, by controlling the ripple surrounding the depressed lobe and the next four sidelobes, as well as by maintaining deep nulls around the depressed lobe, while managing the ripple of the remaining four.

Introduction

Over the last decades, significant advancements in antenna design have increased the need to create radiating patterns capable of functioning as far-field over specific distances while preserving high efficiency, that is, the ratio of the maximum directivity of an antenna aperture to an uniformly excited distribution of the same size¹. According to IEEE Standard for Definitions of Terms for Antennas¹, far-field region is understood as that region of the field of an antenna where the angular field distribution is essentially independent of the distance from a specified point in the antenna’s region.

The majority of the studies concerning this subject have concentrated on line source antennas that produce sum²$^{,}$³ and difference diagrams⁴, as well as the characteristics of shaped patterns⁵. Research related to circular aperture antennas have primarily focused on two areas: firstly, examining the commonly used $\phi$-symmetric circular Taylor tapered distributions⁶, which gradually diminish towards the aperture’s edge, and secondly, analyzing circular continuous shaped patterns⁷. On our previous research⁸ we studied the so-called $\overline{n}$ transition integer peaked distributions- characterized by a spike at the edge and the ability to deliver maximal efficiency, in contradistinction to the tapered distribution, as well as patterns with relatively low inner sidelobes.

The Elliott-Stern method⁹ enables to obtain patterns in which certain infinitely deep nulls are replaced by shallower nulls, by introducing complex roots in the expression of the generalized circular Taylor radiation expression, and obtaining, for M filled-in nulls, a multiplicity of $2^{M}$ solutions for the aperture distribution. The multiplicity of solutions characteristic of the Elliot-Stern method is due to the ± sign in the imaginary part of the root; each complex-conjugate root pair introduces two possible configurations, doubling the number of distinct designs per filled-in null.

This method was used over linear arrays, linear apertures, as well as on circular apertures¹⁰$^{,}$¹¹, focusing on the obtainment of improvements in the distribution, and alleviating the mutual coupling problem under a small loss in efficiency.

In this study, we analysed for the first time to the best knowledge of the authors how, within the multiplicity of solutions obtained by applying this method to circular $\phi$-symmetric Taylor patterns, it is possible to identify solutions that require different distances for the generated field to be classified as far-field. We examined a range of ripple levels, taking into account that achieving more precise ripple control results in greater efficiency losses, across three distinct cases: (i) over the $\overline{n}$=5 transition integer for an optimal peaked distribution, (ii) modified circular Taylor $\overline{n}$ distributions with synthesized patterns featuring a depressed inner sidelobe surrounded by deep nulls, and (iii) modified circular Taylor $\overline{n}$ distributions with a depressed inner sidelobe surrounded by nulls with specified ripple level.

Results

Equations (1), (2) and (3), presented in the Methods section, were used to describe the field pattern generated by a peaked $\phi$-symmetric circular Taylor distribution, at a representative sidelobe level (from now on referred to as SLL) of -25 dB, over an optimal $\overline{n}$=5, for which the main characteristics have already been studied in our previous research⁸. The iterative Elliott-Stern method was subsequently employed to compute the real roots $u_{n}$ of equation 3, enabling precise control over the level of each sidelobe to achieve radiation patterns with a suppressed first sidelobe. We employed this same approach to fill specific nulls and analyze the characteristics of the resulting modified patterns, now incorporating complex roots of the form $u_{n}+jv_{n}$ and therefore studying the multiplicity of $2^{M}$ solutions. Six gradual ripple levels were analyzed throughout the study; considering that narrower ripple control leads to greater efficiency losses, we focused on the highest precision level that ensured efficiency losses remained below 5%.

Over the course of our study, we analyzed the measurement distance requirements based on two distinct error acceptance criteria: a strict error acceptance of 0.5 dB, which ensures high measurement accuracy, and a more permissive criterion of 1.0 dB, which allows for greater tolerance and can be employed in scenarios where high precision is not required.

Throughout this section we applied the aperture radius criterion established by R. C. Rudduck, D. C. F. Wu, and Hyneman¹², namely $a > \overline{n} \cdot \lambda$, to ensure that the entire region influenced by $\overline{n}$ is considered.

Null-filling effects over an optimal -25 dB SLL and $\overline{n}=5$ peaked distribution

Table 1 displays the efficiency (the ratio of the maximum directivity of an antenna aperture to an uniformly excited distribution of the same size) reductions encountered when the initial four nulls in the radiating pattern, with an optimal -25 dB SLL and $\overline{n}$=5, were filled. For achieving this specific ripple configuration, we employed 4 specific complex roots over Eq. (3), obtained by implementing the Elliott-Stern method⁹. This procedure provided $2^{4}$=16 solutions for each case. It should be noted that every solution’s radiating pattern behaves similarly over the far-field scenario, leading to a single efficiency measurement for every set of solutions. This efficiency was compared to the optimal peaked distribution with no null-filling, which it is known to provide maximal efficiency⁸.

Table 1 Efficiency loss due to ripple level for an optimal -25 dB SLL $\overline{n}$=5 distribution.

Naturally, achieving more precise ripple control results in increased efficiency losses. It can be noted that an optimal peaked distribution demonstrates considerable sensitivity when there are alterations in the ripple level; maintaining an exceptionally precise ripple level of ± 0.50 dB results in a substantial efficiency reduction of 13.40%, in comparison to a pattern that lacks any filling. In practical terms, such a significant efficiency loss is unlikely to be manageable in a real-world situation, due to the considerable impact on the emission pattern. Lower ripple control results in reduced efficiency losses, albeit with the attenuation of the potential effects of interest on the pattern: We observed that shorter measurement distances are required to achieve far-field measurements in patterns with more precise ripple control, as shown in Table 2. It will therefore be essential to establish a threshold for acceptable efficiency losses, ensuring that the selected ripple level remains within this limit. We imposed a maximum efficiency loss of 5%, as this value is generally considered affordable in most practical scenarios. This approach allows for controlled attenuation of ripple effects while preserving overall system efficiency within reasonable margins.

Without altering the antenna’s diameter, it is possible to achieve circular radiation patterns that necessitate different $\gamma$ values to re-establish the far-field region, while preserving the same efficiency. This adaptive far-field configurations can be useful in certain applications, such as wireless networks and point-to-point communications, as patterns can be optimized to direct more signal towards certain areas or reducing interference in others.

The ability to significantly modify the distance requirements for far-field measurements could potentially substitute the need for a compact range in antenna testing. These measurement systems consist on parabolic reflectors and range antennas that create an area where the antenna radiation can be approximated to a plane wave¹³, and are voluminous devices.

The use of extremely large-scale antenna arrays (ELAA) is expected to be a common feature of sixth generation (6G) mobile networks. These wide arrays necessitate great distances in order to operate under plane wave radiation, which may be larger than a typical cell’s radius¹⁴, and future research is expected to aim at near-field communications. Consequently, ELAA, along with future 6G networks, could significantly benefit from accurate antenna excitations that decrease and adjust distance requirements for plane wave radiation.

Table 2 Comparison of the $\gamma$ required to regain the far-field condition throughout different ripple level and error tolerance criteria.

The results in Table 2 refer to the shortest and greatest $\gamma$ requirement among the $2^{4}$ different solutions, and can be compared to the optimal -25 dB SLL case with $\overline{n}$=5 and no null-filling. For this optimal no-filled reference, the shortest distance required to return to the far-field, computed with a tolerance of 0.5 dB, was $\gamma = 1.7$, and for a tolerance of 1.0 dB it was $\gamma = 1.20$, as previously studied⁸.

As the degree of ripple increases, there is a noticeable reduction in the range between the minimum and maximum requirements for the parameter $\gamma$, and maximum distance requirements reduce more significantly than minimum distances increase. Each of the various ripple levels examined, which fall within the efficiency threshold, might enable a decrease in the necessary measurement distances, or a significantly increase, provided the antenna is excited in the appropriate manner.

High precision control of ripples enables operations under far-field conditions at reduced distances, which can be particularly advantageous when utilizing antennas with large diameters. On the other hand, thicker ripple control ranges, such as ± 4.25 dB or greater, result in only slight adjustments, although the associated efficiency losses remain minimal. Certain antenna excitations will require substantially greater distances in order to obtain far-field measurements; for a ripple control of ± 0.50 dB, it is possible to obtain antenna excitations which require 5.4$\gamma$ above the original pattern, and even for a wide ripple control of ± 4.25 dB, there is a difference of 3.2$\gamma$ compared to the same pattern. For an error tolerance of 1.0 dB, these differences decrease, although remain substantial, requiring 2.6$\gamma$ above when applying a ripple control of ± 0.50 dB, and 1.5$\gamma$ over a ripple of ± 4.25 dB.

In order to obtain the ± 2.50 dB ripple level with an efficiency loss of 4.57%, we utilized the complex roots introduced in Table 3:

Table 3 Root values for a null-filling of ± 2.50 dB over the optimal 25 dB SLL distribution.

The necessity of four roots arises due to working under the context of $\overline{n}$=5, which entails to control the main lobe along with the first four sidelobes; nulls between the main lobe and this four sidelobes is affected by the ripple control. Table 4 presents notable variations in the behavior concerning the multiplicity of $2^{4}$ solutions, specifically analyzing the distance needed to recover the far-field characteristics. This specific root configurations, which can be obtained by precisely exiting the antenna’s field, require highly significantly different measurement distances depending on the excitation of the antenna; over an error acceptance of 0.5 dB, this differences can vary up to 6.70$\gamma$. When working with null-filled patterns, it is crucial to consider which antenna excitation is implemented, as different configurations can substantially impact the measurement distance needed to regain the far-field condition. Some of these configurations demand considerably greater distances compared to the non-filled pattern. Notably, the root configurations $-v_{1},+v_{2},+v_{3},+v_{4}$ and $+v_{1},-v_{2},-v_{3},-v_{4}$, identified as the ones requiring the greatest measurement distances, require a significant increase of 4.2$\gamma$ and 4.1$\gamma$, respectively, over the pattern with no filled nulls, for an error tolerance of 0.5 dB, and an increase of 1.8$\gamma$ and 2.0$\gamma$ over the 1.0 dB criterion. On the other hand, it it possible to obtain shorter measurement distances; both configurations $-v_{1},-v_{2},+v_{3},+v_{4}$ and $-v_{1},-_{2},+v_{3},-v_{4}$ require 0.5$\gamma$ less than the no-filled scenario over an error of 0.5 dB, and 0.3$\gamma$ over the 1.0 dB error acceptance.

It is important to mention that both configurations $-v_{1},-v_{2},+v_{3},+v_{4}$ and $-v_{1},-_{2},+v_{3},-v_{4}$ consistently offer lesser distance requirements throughout each of the six ripple levels studied, whilst $+v_{1},-v_{2},-v_{3},-v_{4}$ requires the greatest distances. Independently of the ripple level being used, there seems to be a tendency for certain antenna excitations to produce specific characteristics in the radiating pattern. Even with similar $\gamma$ requirements, different aperture distributions can be achieved due to phase variations; for instance, the root configuration $-v_{1}, -v_{2}, +v_{3}, +v_{4}$ exhibits a dynamic range of 2.08, whereas $-v_{1}, -v_{2}, +v_{3}, -v_{4}$ results in a higher dynamic range of 2.33. These differences highlight the impact of phase adjustments on the overall excitation and radiation characteristics of the antenna pattern, and has already been studied¹⁰.

Table 4 $\gamma$ requirement across the different combinations of complex roots.

Figure 1 allows to examine the effects of ± 2.50 dB null-filling on the far-field radiation pattern. As previously discussed, each combination of solutions behaves similarly when analyzed under far-field conditions. In this figure, the -25 dB SLL peaked distribution pattern for $\overline{n} = 5$ is represented with a dotted line, while the ripple-controlled pattern is shown using a continuous line. It can be clearly observed the effects of the null-filling over the original pattern, obtaining a peak difference of 5 dB between the maximum and minimum around the rippled area. The peak level of each lobe remains roughly the same after applying ripple control over the aperture distribution, even on the sidelobes uncontrolled by the roots previously presented.

Fig. 1

Comparison of the far-field expression for -25 dB SLL and $\overline{n}=5$ for a pattern with, and without, null-filling.

The introduction of controlled null-filling altered the energy distribution across the pattern, leading to a redistribution of radiated power that may impact directivity as well as dynamic range, and will require specific distances to regain the far-field condition. While highly precise ripple control results in significant losses of efficiency, it is possible to synthesize patterns with a multiplicity of solutions, with different continuous aperture distributions, dynamic range and distance requirements; this diversity of solutions is particularly relevant in applications where the antenna can be precisely excited to benefit from certain specific characteristic.

Null-filling effects over depressed patterns with the first sidelobe affected

In order to generate patterns with a single depressed sidelobe, we followed the method of Elliott-Stern to obtain the root values for implementation in Eq. (3), which are presented in Table 5. This patterns require significantly greater distances to regain the far-field, and also carry a considerable reduction in efficiency, relative to the optimal transition integer for a peaked distribution at -25 dB SLL, with no suppression of sidelobes. However, in situations where it is preferred to focus solely on the region containing suppressed sidelobes, such as for interference elimination, the suppressed patterns proved to be significantly more effective⁸. This specific pattern required 8.1$\gamma$ to regain the far-field over a error acceptance of 0.5 dB, and 5.6$\gamma$ for a more permissive 1.0 dB acceptance, whilst its efficiency calculated was 0.8680.

Table 5 Root values of $u_{n}$ for one depressed sidelobe at -40 dB, for -25 dB SLL and $\overline{n}$=5.

To control the ripple level of patterns with depressed lobes, one approach is to fill the first four nulls, similarly to how we described in the previous section, resulting in $2^{4}$ solutions. Alternatively, it is possible to maintain the first two nulls, corresponding to the depressed sidelobe, at a deep level by using real roots, while filling the remaining two nulls of the controlled lobes to the desired ripple level, and thereby obtaining $2^{2}$ solutions.

Depressed sidelobe with deep-nulls

In Table 6 it is shown the reduction in efficiency experienced when applying null-filling to a pattern characterized by having its first sidelobe depressed. By comparing this to Table 1, it can be noticed that this second pattern has substantially higher resilience to ripple control. Specifically, for a null-filling specification at -25 dB SLL $\overline{n}$=5 without depressed sidelobes, the efficiency loss stands at 13.40% for a ripple control of ± 0.50 dB. In contrast, the same level of ripple control applied to a pattern with a depressed sidelobe results in a more manageable efficiency loss of 5.64%, compared to the depressed pattern without null-filling. It should be noticed that patterns with a single depressed sidelobe demonstrate lower efficiency than those with no sidelobe depression⁸.

Table 6 Efficiency loss due to ripple level for one depressed sidelobe with deep-nulls.

To achieve high narrow ripple control while maintaining efficiency losses under 5%, a precise ripple adjustment of ± 1.00 dB can be applied to the system pattern, resulting in an efficiency reduction of 4.15%. The root values that allow this precise null-filling are shown in Table 7.

Table 7 Root values for a null-filling of ± 1.00 dB for one depressed sidelobe with deep-nulls at -40 dB, for -25 dB SLL and $\overline{n}$=5.

The presence of only two complex roots results in a substantially lower multiplicity of solutions in comparison to cases where four roots are controlled. This reduction in the number of solutions−by precisely twelve−directly impacts the systems adaptability capacity in meeting the specific operational requirements of the antenna. This constraint suggests that increasing the number of controlled roots could enhance the design flexibility, allowing for a more precise adjustment of radiation patterns and overall performance optimization. Table 8 shows the different distance requirements over the four combination of roots regarding the synthesized pattern with an efficiency loss of 4.15%. It can be observed that this pattern exhibits lesser variability over the required $\gamma$ to regain far-field, presenting a maximal difference of 0.2 $\gamma$ for an error acceptance of 0.5 dB, as well as for an error of 1.0 dB. Root configuration $+v_{3},-v_{4}$ requires the least distance to regain far-field condition over both the criteria of 0.5 dB and 1.0 dB. Configuration $+v_{3},+v_{4}$ requires the greatest distance over an error of 0.5 dB, although this different is not highly significative; and over an error of 0.1 dB the root configuration $-v_{3},-v_{4}$ requires further positioning. The observed variation in root configuration concerning error acceptance criteria suggests that the root configuration $-v_{3},-v_{4}$ demonstrates a slightly more stable behavior over extended distances when compared to the configuration $+v_{3},+v_{4}$; however, the $-v_{3},-v_{4}$ configuration exhibits a quicker degeneration as the distances are reduced. Pattern behaviour can vary depending on the specific aperture distribution of the antenna, not necessary evolving in a linear manner.

Table 8 $\gamma$ requirement across the different combinations of complex roots over the ± 1.0 dB filled pattern with one depressed sidelobe unaffected by filling.

The shortest distance requirement we obtained throughout the multiplicity of solutions presented in Table 8 offers a unit of $\gamma$ less than the pattern generated without filling the controlled sidelobes, at an error tolerance of 0.5 dB, and a decrease of 0.5$\gamma$ over a more permissive error acceptance of 1.0 dB. Over wide error tolerance criteria, differences between the one lobe depressed pattern with, and without null-filling in the controlled undepressed lobes decreases; stricter tolerance carries distance differences which can become significative, especially when working with large diameter antennas.

Fig. 2

Comparison of the far-field expression for -25 dB SLL and $\overline{n}=5$ over a single lobe depressed pattern with ± 1.00 dB null-filling, and without filling.

The pattern generated after applying a ripple control of ± 1.00 dB, as well as without null-filling, is introduced in Fig. 2. Nulls between the second, third and fourth sidelobes are clearly filled up, leaving a difference between each relative maximum and minimum of 2 dB. It can be observed in this figure that the controlled sidelobes clearly maintain their lobe level similarly to the pattern without filling; on the other hand, the remaining lobes present a minor reduction in their peak amplitude. This slight variation observed in the remaining lobes implies a slight redistribution of energy across the pattern, leading to reduced far sidelobe radiation, which can be beneficial in applications where minimizing interference is crucial, as less energy is radiated at larger angles.

These patterns reached the far-field region at shorter distances, and exhibited lower far-sidelobe radiation. However, this comes at the expense of a reduced number of possible solutions, limiting the flexibility in pattern synthesis. The excitation of the antenna field results in less variation compared to patterns with a greater number of null-controlled lobes ($2^{2}$ over $2^{4}$). This trade-off suggests that while this specific sidelobe control may improve energy distribution, it also imposes constraints on the degrees of freedom available for optimizing other performance metrics, such as $\gamma$ requirements.

Depressed sidelobe with filled nulls

Patterns with their depressed sidelobe affected by null-filling are more susceptible to reductions in efficiency, compared to patterns that sustain their depressed lobe surrounded by deep-nulls. As indicated in Table 9, for a ripple control level of ± 0.50 dB, these particular patterns exhibit a reduction in efficiency of an additional 1.27%, and a ripple control level of ± 1.75 dB is required to ensure that the efficiency does not exceed the previously defined threshold of a 5% decrease. Nevertheless, this pattern allows for significantly greater ripple control over the pattern with no depressed lobes, previously introduced.

Table 9 Efficiency loss due to ripple level for one depressed sidelobe with ripple level.

Table 10 presents the required four complex roots values that synthesize this desired pattern. The presence of four complex roots implies that there are 16 configurations that can be achieved through antenna excitation; this configuration offers greater adaptability compared to the scenario in which the depressed lobe remains unaffected by null-filling.

Table 10 Root values for a null-filling of ± 1.75 dB affecting the first sidelobe depressed.

The different configurations in which the antenna can be excited lead to notably diverse distance requirements for achieving far-field patterns, as detailed in Table 11. Over an error of tolerance of 0.5 dB, specific antenna excitation can result in distances requirements that vary from 7.2$\gamma$ to 29.2$\gamma$, allowing to obtain drastically different results. A variation of 22.0$\gamma$ implies a specially broad range of distance requirements which can be adjusted to meet the specific needs of the application. When considering an error threshold of 1.0 dB, the necessary distances varies from a minimum requirement of 5$\gamma$ with the root configuration of $-v_{1},-v_{2},+v_{3},+v_{4}$, to a maximum requirement encountered with both configurations of $+v_{1},-v_{2},-v_{3},-v_{4}$ and $+v_{1},-v_{2},-v_{3},+v_{4}$ of 15.3$\gamma$, that is, the possible range in distance requirements amounts to a significant variance reaching up to 10.3$\gamma$. For an error acceptance criterion of 0.5 dB, $-v_{1},-v_{2},+v_{3},+v_{4}$ configuration requires 0.9$\gamma$ less to obtain far-field emissions than the pattern generated with no null-filling, and 0.6$\gamma$ less under the threshold of 1.0 dB acceptance, being the same root configuration which offered the least distance requirements on the study of patterns with no depressed lobes.

Throughout the range of solutions available by filling the nulls surrounding the depressed lobe, significantly greater distance requirements can be achieved compared to those obtained when maintaining deep-nulls, even while the minimal value of $\gamma$ remains similar.

Table 11 $\gamma$ requirement across the different combinations of complex roots for a null-filling of ± 1.75 dB and one depressed lobe affected.

Figure 3 presents a comparison between the pattern obtained after filling the nulls of the depressed lobe as well as the next three, and the original -25 dB SLL pattern with $\overline{n}$= 5, where the first lobe was depressed without null-filling. The first two nulls, which in Fig. 2 were previously deep-nulls, now exhibit a ripple of 3.5 dB, obtained by working with the first two roots having their imaginary part distinct to zero, and indicating a partial energy redistribution into these regions.

A stronger resemblance to the pattern without null-filling can be observed, particularly in the behaviour of the lateral sidelobes; peak amplitude of lateral sidelobes, even though slightly inferior to the patter without null-filling, is notably closer to it than in the previous case, where the depressed lobe remained with deep-nulls.

Fig. 3

Comparison of the far-field expression for -25 dB SLL and $\overline{n}=5$ over a single lobe depressed pattern with ± 1.00 dB null-filling also affected.

One lobe depressed patterns affected by null filling exhibit altered energy distributions in comparison to patterns that lack null filling. Synthesized ripple controlled patterns with their first depressed lobe surrounded by deep-nulls present reduced sidelobe radiation than ripple controlled patterns with their depressed lobe also affected.

Certain antenna excitations result in reduced distance requirements, such as a reduction of 1$\gamma$ at an error margin of 0.5 dB and 0.5$\gamma$ at an error margin of 1.0 dB for a depressed sidelobe surrounded by deep nulls, while comparing it to the one lobe depressed pattern without null-filling. In contrast, by also filling the nulls around the depressed lobe there is a distance decrease of 0.9$\gamma$ for a 0.5 dB error margin and 0.6$\gamma$ for a 1.0 dB error margin. This decreased distances are especially significant when working with big diameter antennas, and can be obtained under a manageable efficiency loss of under 5%. The pattern with its depressed lobe affected by null-filling also provides a significantly thicker range of distance requirements obtainable, due to its increased multiplicity, and allows to require the greatest $\gamma$ to regain far-field.

Discussion

By implementing ripple control over a circular Taylor distribution, it becomes possible to obtain, for M complex roots, a range of $2^{M}$ distinct continuous aperture distributions, each related to different distances required for accurate far-field measurements. These solutions are inherently complex, and are associated with a reduction in efficiency. Specifically, lower ripple levels, which result in greater null-filling, tend to be associated with increased losses in efficiency than thicker ripple levels. However, this approach potentially decreases moderately the distances needed to achieve far-field measurements and offer greater adaptability, as the antenna can be excited to work under specific complex root configurations. The -25 dB SLL $\overline{n}$ = 5 distribution exhibits a notably higher sensitivity to ripple control when compared to patterns generated with depressed lobes; patterns synthesized with one depressed lobe allow for greater reduction of measurement distances in comparison to the null-filled pattern with no depressed lobes.

The potential to extend this technique to include both symmetric and asymmetric linear Taylor distributions, as well as linear and circular Bayliss distributions with arbitrary sidelobe topographies¹⁵, is currently under consideration. Numerical validation of large planar arrays with circular boundaries, composed of electromagnetic microstrip dipoles whose excitations are obtained through the sampling of the aperture distribution-while accounting for mutual coupling-is also subject of further analysis. In this scenario, a great number of elements may be required in order to ensure the sampling interval is small.

Methods

In accordance with Fig. 4, we examined a planar aperture defined by a circular boundary of radius a.

Fig. 4

Geometry of the problem.

The known general expression of a circular aperture in the Fresnel region, given by Hansen¹⁶ and Walter¹⁷, can be operated over a $\phi$-symmetric region, in order to obtain the following expression of the field⁸:

$$\begin{aligned} F(\gamma ,u)=\int _{0}^{\pi }g_{0}(p) e^{-j\frac{p^{2}}{8\pi \gamma }}J_{0}(pu) p dp \end{aligned}$$

(1)

from where $u = \frac{2a}{\lambda } \sin (\theta )$ and $\gamma = \frac{r}{8a^{2}/\lambda }$ are given in terms of wavelength $\lambda$, and p=$\frac{\pi }{a}r'$. It should be noted that g$_{0}(p)$ corresponds to:

$$\begin{aligned} g_{0}(p)=\frac{2}{\pi ^{2}}\sum _{m=0}^{\overline{n}-1} \frac{F(\gamma _{1m})}{J_{0}^{2}(\gamma _{1m}\pi )} J_{0}(\gamma _{1m}p) \end{aligned}$$

(2)

and $F(\gamma _{1m})$, the expression of the generalized circular Taylor radiation pattern, can be computed from:

$$\begin{aligned} F(u)=\frac{J_{1}(\pi u)}{\pi u} \frac{\displaystyle \prod _{n=1}^{\overline{n}-1}\left( 1-\frac{u^2}{u_{n}^2}\right) }{\displaystyle \prod _{n=1}^{\overline{n}-1}\left( 1-\frac{u^2}{\gamma _{1n}^2}\right) } \end{aligned}$$

(3)

Using the Elliott-Stern method one can identify complex roots in the form $u_{n} \rightarrow u_{n}+jv_{n}$ which produce a pattern characterized by well-defined nulls and managed sidelobe levels. This roots can be designed to precisely control each ripple’s amplitude within the filled region, modifying each sidelobes height in the unfilled region. In order to achieve this, Equation 3 can be rewritten as:

$$\begin{aligned} F(u) = f(u) \prod ^{\overline{n}-1}_{n=1} \left[ 1 - \frac{u^{2}}{(u_{n}+jv_{n})^{2}} \right] \end{aligned}$$

(4)

where

$$\begin{aligned} f(u) = \frac{J_{1}(\pi u)}{\pi u} \frac{1}{\displaystyle \prod ^{\overline{n}-1}_{n=1} \left( 1 - \frac{u^{2}}{\gamma ^{2}_{1n}} \right) } \end{aligned}$$

(5)

and therefore

$$\begin{aligned} F(u)F^*(u) = f^2(u) \prod _{n=1}^{\overline{n}-1} \frac{ u_n^4 + v_n^4 + u^4 + 2u_n^2 v_n^2 - 2u^2 u_n^2 + 2u^2 v_n^2 }{(u_n^2 + v_n^2)^2} \end{aligned}$$

(6)

As a consequence, the power pattern G(u), which can be expressed in dB as $G(u) = 10 \log _{10} S(u)S^*(u)$⁹, is given by the following equation:

$$\begin{aligned} G(u) = \frac{10}{\ln 10} \left\{ \ln f^2(u) + \sum _{n=1}^{\overline{n}-1} \ln \left( u_n^4 + v_n^4 + u^4 + 2u_n^2 v_n^2 - 2u^2 u_n^2 + 2u^2 v_n^2 \right) - 2 \sum _{n=1}^{\overline{n}-1} \ln (u_n^2 + v_n^2) \right\} \end{aligned}$$

(7)

Assuming the power pattern to be G(u) and the target pattern to be S(u), the difference G(u)-S(u) represents the gap between the existing and desired patterns. Adjustments $\delta u_{n}$ and $\delta v_{n}$ can be operated to align G(u) closer to S(u). The overall differential is expressed as:

$$\begin{aligned} d(G-S) = \sum ^{\overline{n}-1}_{n=1} \left( \frac{\delta G}{\delta u_{n}} \delta u_{n} + \frac{\delta G}{\delta v_{n}} \delta v_{n} \right) \end{aligned}$$

(8)

where the partial derivatives of Eq. (8) are to be evaluated at the points:

$$\begin{aligned} P_{m}=P(u_{1}^{0},v_{1}^{0},...u_{\overline{n}-1}^{0},v_{\overline{n}-1}^{0},u=u_{m}) \end{aligned}$$

(9)

Let $u_{m}$ represent the positions corresponding to the sidelobe and dip levels of the radiation pattern. The initial values for the inner roots $u_{n}+jv_{n}$ are given by $(u_{1}^{0},v_{1}^{0},...u_{\overline{n}-1}^{0},v_{\overline{n}-1}^{0},u=u_{m})$. By appropriately selecting $u_{m}$, one can determine key directions in the pattern, including the location of sidelobe peaks and the maxima and minima of ripples within the filled region, allowing to control the radiation characteristics of the design.

The pattern S(u) remains unchanged when $v_{n}$ is substituted by its complex conjugate, although such a substitution can lead to significant changes in the aperture distribution. With M complex roots, there are $2^{M}$ continuous aperture distributions that yield the same pattern; however, due to the system’s degeneracy, several configurations of $v_{n}$ and its conjugate may be operated while maintaining the same overall effect.