1 Introduction

Researchers studied the fifth generation cellular network technology (5G), and they found that it has a greater capacity while it uses a new spectrum [1,2,3,4,5,6,7]. 5G millimetre-wave bands use frequency bands ranging from 24–100 GHz [8,9,10,11,12], supporting a wide range of devices [13,14,15,16] and demonstrating the performance of outlining 5G service requirements [17, 18]. Leaky-wave antennas (LWAs) are highly promising millimetre and submillimetre wave directional antennas [19,20,21] due to their ability to achieve higher direct radiation [22,23,24] with single or multiple feedings [25, 26], resulting in a compact structure [27] with low energy consumption [28, 29]. Numerous studies have discussed the functionality of resonant cavities by using the LWA principle [30,31,32,33,34,35]. For the first time, an antenna of this type was proposed by Oliner [36] and was called the “leaky-wave”. A typical antenna of this type is a surface waveguide with a periodic system of irregularities [37, 38]. The presence of irregularities leads to the transformation of the surface waves into radiating waves. This is reflected in the interpretation of the wave propagation coefficient as a complex quantity \({\gamma = \alpha +j \beta }\), the real part of which is due to radiation into the surrounding space. The imaginary part characterises the phase delay of the wave propagating in the waveguide. The flexibility of periodic type LWAs for scanning over a wide angular range is one of their advantages [39]. Filling the guiding structures with dielectric is a good approach for producing a slow-wave mode [40,41,42,43]. There are different types of open waveguide structures (dielectric rods with different cross-sectional shapes). The most common are dielectric rods with a ground plane and a dielectric rod placed in a channel metal waveguide. In addition, there are various irregularities, such as longitudinal or transverse conductor strips on the waveguide surface [44], grooves [45, 46], holes in the dielectric rod [47, 48], quarter-wave pins on the walls of the channel metal waveguide [49], and fractal antennas [50,51,52,53,54,55]. Each of the presented options for dielectric waveguide irregularities is quite simple in terms of technological implementation and has characteristic advantages and disadvantages. Metal strips are easily implemented by using printing technology with foil materials. However, there is a slight increase in thermal losses in this case. Irregularities in the form of grooves do not exhibit this disadvantage. However, their execution requires mechanical processing operations, which complicates manufacturing. The remainder of this paper is organized as follows. Section 2 presents the methodology of the current distribution of LWA array model with non-identical irregularities. The contribution and advantages of the proposed work were involved. Section 3 details the synthesis of LWAs with non-identical irregularities. A new idea is put forward for how to implement the antenna element geometry based on each component’s coupling coefficient value. Section 4 presents the simulation results and is divided into three sections. The first shows the results for an antenna with identical irregularities. The second shows results for an antenna with non-identical irregularities. The third section provides a comparison between the results of the antenna with identical and non-identical irregularities. Finally, Sect. 5 concludes the paper and summarises the radiating characteristics based on a modified energy approach and work results.

2 Methods

2.1 The model of LWA with non-identical irregularities

To achieve the LWA amplitude distribution of radiating current |J(z)| (different from the exponential), the attenuation coefficient of the wave propagating along the aperture must be changed as follows: \({\alpha (|X|,z) \overset{|X|}{\rightarrow } \alpha _\mathrm{given}(z,|J_\mathrm{given}(z)|)}\). (|X|) -parameters of waveguide and irregularities, z-coordinate along the antenna axis). The implementation of this principle requires the presence of data about the quantitative dependence of the attenuation coefficient \({\alpha (z)}\) in a longitudinally inhomogeneous dielectric waveguide on the parameters of the waveguide and irregularities [56, 57]. A solution to a specified task is difficult. The use of numerical electromagnetic modelling (CST, HFSS, etc.) to determine the parameters of irregularities based on a given \({\alpha _\mathrm{given}(z,|J_\mathrm{given}(z)|)}\) is difficult due to the need to organize a large number of options that differ in waveguide size, irregularities, and interelement distance. Therefore, it is necessary to develop simple engineering procedures for the synthesis of irregularities in longitudinally inhomogeneous waveguides using a given amplitude-phase distribution of radiating currents in an antenna aperture. These can be used to obtain the first approximation when designing such antennas and refer to the methods of local optimization. This technique can be developed during the transition to an antenna model as a linear antenna array [58]. According to the indicated model, the antenna is considered an antenna array with serial excitation. The energy approach is widely used for the approximate calculation of linear radiating systems in the form of a periodic system of identical radiators with a sequential excitation scheme [57]. In [59], a modified energy approach was proposed and approved, which summarises the approach for the case of an almost periodic system of non-identical radiators. According to the specified approach, a dielectric waveguide with irregularities is represented by an antenna array model with a serial excitation of its elements. The antenna array model is shown in Fig. 1a. A section of a waveguide with irregularity is considered an element of the array. The coordinates of the radiators correspond to the centers of the sections. The interaction of the waveguide mode with the radiators is characterised by the power-coupling coefficient. Figure 1b shows the mechanism of antenna element excitation. According to the energy approach in the antenna, with sequential excitation the amplitude-phase radiating current distribution of the antenna array model can be represented as:

$$\left| {I_{n} } \right| = \sqrt {s_{n} \prod\limits_{{k = 1}}^{{n - 1}} {(1 - s_{k} )} }$$
(1)
$$\begin{aligned}&arg(I_{n}) = \sum _{k=1}^{n-1}\beta _{k}d_{k} \end{aligned}$$
(2)

where \({s_{k}}\) is the power coupling coefficient of the k-th array element (relative power fraction of the undergoing wave radiated by the n-th element) and \({\beta _{k}}\) is the wave-number in the waveguide part from k-1 to the k-th element.

Fig. 1
figure 1

Antenna array model: a current distribution of the antenna array model; b The power coupling coefficients of the k-th array element

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2.2 The contributing advantages of the proposed work are as follows:

(1) The design uses a periodic structure of a dielectric waveguide with non-identical irregular grooves in a dielectric rectangular rod, which are placed inside the channelled waveguide. One of the disadvantages of LWAs with identical irregularities is the relatively high level of side lobes (\({-}\,10 \ldots {-}\,12\) dB). However, the disadvantage is the exponential distribution of the radiating equivalent current in the antenna. The use of non-identical irregularities allows for a change in the distribution of the current and a reduction in the level of the side lobes. (2) The non-identical irregularities methodology allows us to correct the phase velocity in a dielectric waveguide. This study proposes a simple and effective method for synthesizing the parameters of irregularities depending on the required form of the current distribution.

3 LWA synthesis

The LWA under consideration consists of a dielectric rectangular rod with irregularities in the form of grooves and a channel metal waveguide (Fig. 2). Serial excitation of the antenna through the section of a rectangular waveguide with a transition horn was used.

Fig. 2
figure 2

A dielectric waveguide with non-identical irregularities

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The calculation of the antenna radiation pattern is performed using classical relations, where the amplitude-phase distribution \({I_{n} = \left| I_{n} \right| e^{j arg(I_{n})}}\) is in accordance with (1)–(2) and the coordinates of the radiators (Fig. 1):

$$\begin{aligned} F(\theta )=\sum _{n=1}^{N}g_{n}(\theta )\left| I_{n} \right| e^{jarg(I_{n})}e^{-2j\pi z_{n}(sin(\theta )/\lambda )} \end{aligned}$$
(3)

where \({g_{n}(\theta )}\) represents the n-th radiator’s radiation pattern. The n-th section of the waveguide exhibited irregularity. Radiation pattern \({g_{n}(\theta )}\) can be considered equal to \({g_{n}(\theta )\approx g(\theta )}\) because the electrical dimensions of the sections are almost the same. Then:

$$\begin{aligned} F(\theta )=g(\theta )\sum _{n=1}^{N}\left| I_{n} \right| e^{jarg(I_{n})}e^{-2j\pi z_{n}(sin(\theta )/\lambda )} \end{aligned}$$
(4)

The efficiency coefficient and antenna aperture efficiency are:

$$\begin{aligned}&\text {Efficiency} =1-\prod _{k=1}^{N}(1-s_{k}) \end{aligned}$$
(5)
$$\begin{aligned}&\text {Antenna Aperture Efficiency} = \frac{\left| \sum _{1}^{N}I_{n}^{2} \right| }{N\sum _{1}^{N}\left| I_{n} \right| ^{2}} \end{aligned}$$
(6)

Relations (1)–(6) can be used to solve the analysis problem (looking for radiation pattern and other basic parameters) and the synthesis problem (looking for geometric parameters) using a given radiation pattern or other criteria. Let us consider an antenna in the form of a dielectric waveguide with non-identical irregularities in the channel of a metallic waveguide, connected to a matching load. For the given waveguide parameters and irregularities, the amplitude-phase distribution for the antenna array model is determined by the relations (1)–(2). The objective of antenna array synthesis is to determine the parameters of the irregularities and their locations according to the predetermined amplitude-phase distribution in the antenna aperture. In the case of travelling wave antennas, the specified distribution is in-phase (or linear for inclined radiation). We restricted ourselves to the case of transverse radiation obtained with a given beam width and side-lobe level. Under these conditions, the specified in-phase distribution of the phase and amplitude distributions can be considered as a function of \({J_{n(\mathrm{given})}=\left| I_{n}\right| }\). The form of \({J_{n(\mathrm{given})}}\) and the electrical length of the antenna \({L_\mathrm{ant}}\) are determined according to the classical aperture theory of antennas. The number of radiators is \({N=L_\mathrm{ant}/\lambda _\mathrm{waveguide}}\). The value of \({\lambda _\mathrm{waveguide}}\) by the first approximation is determined by the specified dimensions and the parameters of the unloaded (without grooves) waveguide. The solution of the synthesis problem (looking for the irregularity parameters) is carried out in two stages, each of which is submitted in a closed form.

Stage 1: Let the specified distribution be normalised in such a way that:

$$\begin{aligned} \sum _{n=1}^{N}J_{n}^{2}=1 \end{aligned}$$
(7)

From relationship (1) directly follows:

$$\begin{aligned}&s_{1}=C_\mathrm{Norm}J_{1}^{2} \end{aligned}$$
(8)
$$\begin{aligned}&s_{2}=C_\mathrm{Norm}J_{1}^{2}/1-s_{1} \end{aligned}$$
(9)
$$\begin{aligned}&s_{n}=C_\mathrm{Norm}J_{n}^{2}/\prod _{k=1}^{n-1}(1-s_{k}) \end{aligned}$$
(10)

In relationships (8)–(10), the normalising multiplier \({ C_{Norm}<1}\) is selected such that all the found coupling coefficients do not exceed the maximum possible value \({s_{max}}\) for this type and size of waveguide.

$$\begin{aligned} \left\{ \begin{array}{l} C_\mathrm{Norm}\rightarrow \max \\ \text {When } s_{n}(C_\mathrm{Norm}) < s_{\max }, n=1\cdots N \end{array}\right. \end{aligned}$$
(11)

The dimensions of the \({w_{n}}\) and \({h_{n}}\) irregularities are determined using the inverse of function s(w, h) based on the discovered \({s_{n}}\).

Stage 2: After determining the irregularity parameters, the coordinates of the radiators were adjusted. The requirement is determined as follows. The initial geometry of the lattice is an equidistant system with a step equal to the wavelength of the waveguide (unloaded). The presence of irregularities leads to a change in the phase velocity of the waves in the waveguide. The phase distribution changes accordingly, which ceases to be in-phase. The coordinates of the radiators should be slightly changed to restore the in-phase distribution in the aperture. We assume that in the neighbourhood of the n-th irregularity, the value of the phase coefficient is equal to:

$$\begin{aligned} \beta _{n}=\beta (w_{n},h_{n}) \end{aligned}$$
(12)

Knowing the dependency of the wave phase velocity on the parameters of irregularities, the correction of the coordinates of the radiators is performed such that the distance between the n-th and n+1 radiators is equal to the local wavelength value in the loaded waveguide.

$$\begin{aligned} z_{n+1}-z_{n}=\frac{2\pi }{\beta _{n}}=\lambda _\mathrm{waveguide}(w_{n},h_{n}) \end{aligned}$$
(13)

The modified energy approach is distinguished by the algorithm for determining the relationships s(w, d) and \({\lambda _\mathrm{waveguide}(w,h)}\) [57]. The algorithm is based on the numerical calculation of the \({S_{21}}\) coefficient of the channel waveguide with the LWA and antenna radiation pattern using electromagnetic simulation software. In this case, the LWA model was a periodic dielectric waveguide with identical irregularities. Consequently, for different combinations of specified values of w and d, we determine the absolute values \({\left| S_{21} \right| }\) that are used to determine the relationship for the coupling coefficient:

$$\begin{aligned} s(w,h)=\frac{1-\left| S_{21} \right| ^{2}}{N} \end{aligned}$$
(14)

The radiation pattern was used to determine the \({\lambda _\mathrm{waveguide}(w,h)}\) function. The angular position \({\theta _{\max }(w,h)}\) of its maximum is determined, and the value of is

$$\lambda _{{{\text{waveguide}}}} (w,h) = \left( {sin(\theta _{{\max }} (w,h)) + \frac{{\lambda _{0} }}{d}} \right)\lambda _{0}$$
(15)

Now, we present the results of the of the transversely radiating antenna with N = 21 (as shown in Table 1 and Fig. 3 with the given amplitude distribution).

Table 1 Given amplitude distribution for \(N = 21\) transversely radiating antenna
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Fig. 3
figure 3

A predetermined amplitude distribution

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Table 2 lists the calculated values of the coupling coefficients, and Fig. 4 shows them in a graph.

Table 2 The values of distances \({d_{n}}\) and coupling coefficients \({s_{n}}\) corresponding to design geometry
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Fig. 4
figure 4

Antenna coupling coefficients

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The calculated local values of the wavelength \({\lambda _\mathrm{waveguide}(w_{n},h_{n})}\) in a loaded waveguide, where \({w_{n}}\) and \({h_{n}}\) are the dimensions of irregularities listed in Table 3.

Table 3 The dimensions of irregularities \({w_{n}}\) and \({h_{n}}\), and the calculated local values of the wavelength \({\lambda _\mathrm{waveguide}(w_{n},h_{n})}\) for individual radiating element
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The data below are the results of the electrodynamic modelling of the antenna performed on the basis of a groove dielectric waveguide. The dimensions of the groove correspond to the standard metal waveguide with a cross section of \(7.2\times 3.6\) mm, dielectric-Teflon, 2 mm thick. The excitation is carried out by a standard waveguide using the transition shown in Fig. 5.

Fig. 5
figure 5

Antenna simulation model (geometry with one-sided excitement, number of elements 21, dimensions of waveguide \(7.2 \times 3.6\) mm)

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4 Results and discussion

4.1 Antenna results with identical irregularities

We should start with the results of identical irregularities and, after showing the performance parameter, make a comparison with the new results of non-identical irregularities. The design provides \({S_{11}}\) and a radiation pattern as shown in Figs. 6, 7 and 8. \({S_{11}}\) equals − 14.97 dB at frequency 37.2 GHz.

Fig. 6
figure 6

\({S_{11}}\) at frequency \(F= 37.2\) GHz for antenna with identical irregularities

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As shown at Figs. 7 and 8, the calculated radiation pattern at frequency \(F = 37.2\) GHz is 20.9 dBi, width of radiation pattern is 2.8 \(^{\circ }\), and the side lope equals − 10.5 dB.

Fig. 7
figure 7

1-D radiation pattern at frequency \(F= 37.2\) GHz for antenna with identical irregularities

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Fig. 8
figure 8

3D radiation pattern at frequency \(F= 37.2\) GHz (RP: 20.9 dBi, width of RP: 2.8 \(^{\circ }\), side lope: \({-}\) 10.5 dB) for antenna with identical irregularities

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4.2 Antenna results with non-identical irregularities

Now the proposed design provides good performance parameters as shown at Figs. 9, 10 and 11. \({S_{11}}\) equals \({-}\) 16.04 dB at frequency 37.2 GHz. As shown in Fig. 9, the calculated radiation pattern at frequency \(F = 37.2\) GHz is 22 dBi, the width of the radiation pattern is 3.2\(^{\circ }\), and the side lope equals \({-}\) 18.3 dB.

Fig. 9
figure 9

S11 at frequency \(F= 37.2\) GHz for antenna with non-identical irregularities

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Figures 10 and 11 show 1D and 3D radiation patterns according to the predetermined amplitude distribution (Fig. 3) calculated by electromagnetic numerical modelling in the CST. Figure 12 shows a comparison between the results obtained by numerical simulation in CST (black line) and a modified energy approach (purple line). It can be seen that the similarity is very close, especially in the area of the main lobe. The efficiency of the considered antenna was 83.6\(\%\).

Fig. 10
figure 10

1-D radiation pattern at frequency \(F= 37.2\) GHz for antenna with non-identical irregularities

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Fig. 11
figure 11

3D radiation pattern at frequency \(F= 37.2\) GHz (RP: 22 dBi, width of RP: 3.2 \(^{\circ }\), Side lope: − 18.3 dB) for antenna with non-identical irregularities

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Fig. 12
figure 12

Comparison of radiation patterns for antenna with non-identical irregularities obtained by numerical electromagnetic calculation and modified energy approach

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4.3 A comparison of the results of antennas with identical and non-identical irregularities

Figures 13 and 14 show comparisons of radiation patterns and \({S_{11}}\) coefficients for antennas with identical and non-identical irregularities.

Fig. 13
figure 13

Comparison of the 1-D calculated radiation pattern at frequency \(F= 37.2\) GHz for antenna with identical and non-identical irregularities

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Fig. 14
figure 14

Comparison of the S11 at the same frequency \(F= 37.2\) GHz for antenna with identical and non-identical irregularities

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An antenna with non-identical irregularities has a better side lobe level (7.8 dB better) with almost identical gain and an absolute value of \({S_{11}}\). Only a slight increase in radiation pattern width (0.4\(^{\circ }\)) is observed. Table 4 shows a comparison between the proposed work and previous works. The proposed antenna exhibits superior characteristics at mm-wave frequency bands, including low SLL with high efficiency and gain.

Table 4 Performance comparison of the presented waveguide arrays
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5 Conclusions

Antennas of the “leaky-wave” type, based on dielectric waveguide structures, are a promising class of flat antennas in the EHF band. The approach proposed in the work and synthesis of such structures is based on antenna representation as an almost periodic antenna array with a preliminary analysis of radiating characteristics based on a modified energy approach. The analysis showed the validity of the proposed approach, which ensured acceptable coordination of the calculated data and was obtained using modern means of electromagnetic modeling. In this study, we developed a combination of technical solutions to improve the technical indicators of antennas fabricated using dielectric waveguides. The solution to the task includes the following components: First, the development of the synthesis method of a linear antenna array model of LWA with non-identical irregularities Second, analysis of an antenna containing irregularities in the form of grooves in a dielectric.