INTRODUCTION

Leaky wave antennas can radiate at the zero, minus the first, and other spatial harmonics [1–3]. In waveguide-slot arrays, radiation occurs, as a rule, at minus the first spatial harmonic; radiation at the zero harmonic is leaky. To suppress radiation at the zero harmonic, waveguides with paired longitudinal slots located along the waveguide with a shift by half the length of the waveguide mode at the center frequency of the range are used. In this case, beam scanning occurs, as a rule, in the rear half-space with respect to the direction of waveguide mode propagation. With an increase in frequency, the direction of radiation moves into the front half-space; however, leaky radiation can occur at the minus second spatial harmonic. In addition, when the beam passes through the normal to the array, the value of the reflection coefficient sharply increases due to the in-phase addition of reflection from the slots.

A single-beam radiation mode is possible at minus the first harmonic, at which, as the frequency increases, the beam scans throughout the angle sector from –90° to + 90° with respect to the normal to the radiating aperture. However, this mode is achieved for a material filling a waveguide with a high dielectric constant (ε > 9) [4]; at the same time, the heat loss significantly increases.

A wide frequency scanning sector in a leaky wave antenna can also be obtained with a rectangular metal-dielectric waveguide in integral form in a dielectric substrate with the use of SIW technology [5]. The waveguide is loaded on slots frequently located in the upper metallization, forming a quasi-homogeneous radiating structure in which the main mode H10 is leaky. With an appropriate choice of waveguide parameters and slots, it is possible to achieve propagation modes for direct (in the upper part of the operating range) and return (in the lower part of the range) leaky waves. In [6], when using an array of slots in the form of a meander, the scanning sector of such an antenna for a change in frequency from 8.6 to 12.8 GHz was from –60° to + 60°, including the direction of radiation normal to the frequency of the transition between direct and backward waves.

The aim of this work is to implement a 180° frequency scanning sector of a waveguide-slot antenna array using zero spatial harmonic radiation in the low-frequency part of the single-mode range and minus the first spatial harmonic in the high-frequency part of a rectangular metal waveguide filled with a dielectric with a relatively small dielectric constant.

1. ARRAY SYNTHESIS

Consider a waveguide-slot array in the form of a metal waveguide with a rectangular cross section, filled with a dielectric with a relative dielectric constant ε, the size of the wide wall of which is a, and the narrow wall b. In the upper wide wall, paired narrow transverse slots are cut along the waveguide with a period p and distance d between their centers (Fig. 1).

Fig. 1.
figure 1

Linear waveguide-slot array with pair of transverse slots in wide wall; 1 and 2, input numbers.

Full size image

Suppose that the slots do not affect the propagation constant β0= [(k2ε – (π /a)2]1/2 of the main mode (H10) of the rectangular waveguide, where k = 2π/λ is the wavenumber of the free space, λ is the wavelength at the operating frequency f, and ε is the dielectric constant. The critical waveguide frequency fc= cε/λc where λc= 2a is the critical wavelength and cε= c0/(ε)1/2 is the the speed of light in the material filling the waveguide. With an increase in frequency, the direction of radiation changes from the normal to the critical frequency fc to the direction of waveguide mode propagation at frequency f1. It is easy to show that the frequency f1 corresponds to the wavenumber of the free space k1= 2π/λ1= kc/(ε – 1)1/2 where kc= 2π/λc.

Let us etermine the dielectric constant ε and period p based on two conditions. The first is that at the same frequency f1 where the direct radiation of the zero harmonic ceases, the radiation of minus the first spatial harmonic occurred. Note that at this frequency, minus the first harmonic radiates along the axis waveguide in the direction opposite to the direction of leaky mode propagation. The propagation constant of minus the first harmonic in a periodic structure with period p is β–1 = β0– 2π/p [1–3]. Its radiation at frequency f1 occurs if β–1 in absolute value becomes equal to the wavenumber of the free space at this frequency: β‒1(f1) = –k1 under the condition β0(f1) = k1. Hence, we find that in order for the first condition to be fulfilled, the period of the structure must satisfy the condition p = λ1/2. Or, using the above expression for k1p = a(ε – 1)1/2.

The second condition for selecting the dielectric constant of the filling material is to set the radiation direction of minus the first harmonic at the upper frequency of the operating range. We choose this frequency equal to the critical frequency of the mode of the nearest higher type (H20). Since the radiation direction of minus the first harmonic at this frequency can be set rather arbitrarily, the choice of ε is not strictly defined. We denote the radiation direction of minus the first harmonic at frequency f2 = 2fc via the angle θ0 with respect to the normal to the wide wall of the waveguide. Then,

$${{\beta }_{{ - 1}}}({{f}_{2}}) = {{\beta }_{0}}({{f}_{2}}) - {{2\pi } \mathord{\left/ {\vphantom {{2\pi } p}} \right. \kern-0em} p} = {{k}_{2}}{\text{sin}}\left( {{{\theta }_{0}}} \right).$$

If the angle θ0 positive, then minus the first harmonic at frequency f2 radiates in the front half-space with respect to the propagation direction of the leaky mode. When both conditions are fulfilled, we obtain a transcendental equation that should be satisfied for ε at a given angle θ0

$$\begin{gathered} \frac{{\sqrt \varepsilon }}{{\sqrt {\varepsilon - 1~} }}\frac{{\sin {{\theta }_{0}}}}{{\left[ {\varepsilon - {{{(\sin {{\theta }_{0}})}}^{2}}} \right]}} + \frac{{\sqrt \varepsilon }}{{2\sqrt {\varepsilon - 1~} }} \\ \times \,\,\frac{1}{{\sqrt {\varepsilon - {{{(\sin {{\theta }_{0}})}}^{2}}} }}\sqrt {\left[ {\frac{{4{{{(\sin {{\theta }_{0}})}}^{2}}}}{{\varepsilon - {{{(\sin {{\theta }_{0}})}}^{2}}}}} \right] + \varepsilon + 3} = 1. \\ \end{gathered} $$
((1))

Its solution is presented in Fig. 2 as the dependence ε(θ0). For a change in θ0 from 0° to +30,° the corresponding value of ε varies from 2.33 to 3.71.

Fig. 2.
figure 2

Dependence of relative dielectric constant of material filling waveguide on direction of radiation of minus first harmonic at upper frequency of range.

Full size image

Thus, for simultaneous fulfillment of both of the above conditions, we have the following radiation pattern in the frequency range. In the lower part of the range (at frequencies from fc to f1 = fc [ε/(ε – 1)]1/2), a zero harmonic is emitted, the radiation direction of which with respect to the normal varies from 0 to +90°. As well, at frequency f1 , radiation of minus the first harmonic occurs at an angle of –90°. In the frequency range from f1 to f2 = 2fc its radiation direction changes from –90° to θ0. Radiation along the normal to the array plane takes place at frequency fn= fc[(ε + 3)/(ε – 1)]1/2. Thus, with θ0 > 0 in the frequency band of a single-mode waveguide, we have the complete (180°) frequency scan sector.

The possibility of array radiation at minus the second spatial harmonic has not yet been considered. Analysis using the same approximate model shows that the frequency at which it begins to radiate is determined by the following expression:

$$\begin{gathered} f\left( { - 2} \right) = {{f}_{{\text{c}}}}\frac{{4\sqrt \varepsilon }}{{\sqrt {\varepsilon - 1~} }} \\ \times \,\,\left[ {\sqrt {\frac{1}{{{{{\left( {\varepsilon - 1} \right)}}^{{2~}}}}} + \frac{1}{{\varepsilon - 1}} + \frac{1}{{16}}} - \frac{1}{{\varepsilon - 1}}} \right], \\ \end{gathered} $$
((2))

where fc is the critical waveguide frequency. In deriving this formula, we assume that the period of the structure, depending on ε determined by the above-mentioned formula p = a(ε – 1)1/2. Figure 3 shows the dependence of the relative frequency of radiation of minus the second harmonic f(–2) /fc from the value of ε. In the same figure, the dotted line shows the ratio of the critical frequency of the nearest higher mode H20 to the critical waveguide frequency.

Fig. 3.
figure 3

Dependence of relative radiation frequency of minus second harmonic on dielectric constant of filling material (curve 1) and ratio of critical frequency of nearest higher mode H20 to critical waveguide frequency (curve 2).

Full size image

Thus, analysis assuming no effect of the slots on the phase constant shows that radiation at minus the second spatial harmonic for ε <2.9 occurs at frequencies higher than that of the occurrence of the highest mode (H20). For ε > 2.9, the limit of the upper operating frequency of the array is not determined by the critical frequency of higher mode H20, but by the beginning of radiation of minus the second harmonic.

2. NUMERICAL SIMULATION

To check the solution of the problem found using the approximate theory and to obtain accurate quantitative reflection and radiation characteristics with frequency scanning, an electrodynamic model of a waveguide-slot antenna array was constructed in the form of a metal rectangular waveguide filled with a dielectric with pairwise transverse slots in the wide wall (see Fig. 1). For the numerical simulation, the finite element method was used. As an example, we consider a waveguide-slot array with 15 pairs of slots with a period of p = a (ε – 1)1/2 = 18.3 mm in the wide wall of the rectangular metal-dielectric waveguide with a cross section of 13.92 × 6.05 mm and length of 274.5 mm with a dielectric constant of the filling material of ε = 2.73, which corresponds to an angle θ0 = 10° (see Fig. 2). The critical frequency (fc = 6.52 GHz) coincides with that of a standard hollow waveguide 23 × 10 mm in size, while the frequency at which radiation of the zero harmonic ceases and that of minus of the first harmonic occurs, f1 = fc [ε/(ε – 1)]1/2 = 8.19 GHz. The frequency along the normal fn= fc[(ε + 3)/(ε – 1)]1/2 = 11.87 GHz, and the critical frequency of the nearest higher mode (N20) 2fc= 13.04 GHz.

To reduce the value of the reflection coefficient from the array at the radiation frequency along the normal, the slots in each pair should be offset by about a quarter of the wavelength of the mode H10 in the waveguide at this frequency. The slot parameters are length 7 mm, width 1.5 mm, distance between centers d = 4 mm. The value of d was chosen slightly less than a quarter of the wavelength of the unperturbed mode H10 in the waveguide, taking into account the effect of the slots on the propagation constant.

Figure 4 shows the results of calculating the frequency dependence of the reflection coefficient at the input of еру waveguide with one pair of slots. The waveguide output is loaded on an ideal matched load. The figure shows that for the selected parameters, reflection from one pair of slots in the vicinity of the frequency fn is quite small.

Fig. 4.
figure 4

Dependence of reflection coefficient on pair of slots in vicinity of radiation frequency along normal.

Full size image

The results of calculating the frequency dependence of the reflection coefficient S11 and passage of S21 in the range of 7–14 GHz are presented in Fig. 5, which shows the presence of three maxima of the reflection coefficient in this frequency range. The first maximum (at frequencies near f1 = 8.2 GHz) corresponds to the transition from radiation of the zero spatial harmonic to radiation of minus the first harmonic. At this frequency, the period is equal to half the wavelength in the waveguide, which in this case is equal to the wavelength in the free space, which ensures fulfillment of the condition β0(f1)p = π while reflections from pairs of slots are in phase. The second maximum near the frequency \(f_{{\text{n}}}^{'}\) = 11.2 GHz corresponds to the radiation frequency along the normal, at which the distance between pairs of slots is equal to the wavelength in the waveguide; then condition β0 is satisfied (\(f_{{\text{n}}}^{'}\))p = 2π and reflection from all slots is also summed in phase. This maximum is significantly lower than the first due to suppression of reflection from each pair of slots (see Fig. 4). The increase in reflection at frequencies of 13–14 GHz is apparently due to the combined effect of summation of reflection from slots, the appearance of the waveguide mode H20, and the onset of radiation of minus the second spatial harmonic. Additional studies are required for a more accurate description of how these effects influence reflection of the array in this frequency range.

Fig. 5.
figure 5

Frequency dependence of reflection coefficients S11 (solid line) and passage of S21 (dashed line) of the array.

Full size image

Figure 6 shows the dependence of gain on the angle θ in the upper half-plane of the longitudinal plane YOZ at various frequencies. Here, as above, angle θ is measured from the normal to the array aperture. It can be seen from the figure that at a frequency above critical, the antenna begins to radiate at the zero harmonic at an angle θ close to sin–1(β0/k), like in any leaky wave antenna radiating at the zero harmonic [1]. Curves 1–3 correspond to this mode. With an increase in frequency, the phase constant β0 approaches the wavenumber in the free space and the radiation direction approaches the waveguide axis in the direction of mode propagation (curve 4). At a frequency higher than f1 = 8.2 GHz, zero-harmonic radiation conditions (β0<k) cease to be met, but radiation occurs at minus the first harmonic. With increasing frequency, the angle of its radiation changes from –90° to θ0. Radiation in the direction of the normal (θ = 0°) occurs at frequency \(f_{{\text{n}}}^{'}\) ≈ 11.2 GHz. According to our analysis, this frequency is determined by the formula

$${{f}_{{\text{n}}}} = {{f}_{{\text{c}}}}{{\left[ {{{\left( {\varepsilon + 3} \right)} \mathord{\left/ {\vphantom {{\left( {\varepsilon + 3} \right)} {\left( {\varepsilon - 1} \right)}}} \right. \kern-0em} {\left( {\varepsilon - 1} \right)}}} \right]}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \approx {\text{ }}11.9\,\,{\text{GHz}}.$$
Fig. 6.
figure 6

Antenna radiation patterns on frequency grid: 7 (curve 1), 7.3 (2) 7.5 (3), 8(4), 8.5 (5), 9 (6), 9.5 (7), 10 (8) 10.5 (9), 11 (10), 11.5 (11), 12 (12), 12.5 (13), 12.7 GHz (14).

Full size image

Thus, the value of this frequency obtained from numerical simulation differs from the approximate value by about 5–6%.

In Fig. 6, it can be seen that the maximum gain values of the radiation patterns at certain frequencies (curves 1, 4, 14) are less than at neighboring frequencies (adjacent curves). The first minimum is explained by the minimum electrical size of the source at the minimum frequency. Others are caused by strong reflections from the slots when the leaky mode propagates in the waveguide. This leads not only to a sharp increase in the reflection coefficient at the input of the array (see Fig. 5), but also, apparently, to strong distortions in the amplitude-phase distribution of the field along the waveguide, which in turn lead to a drop in the gain and an increase in lateral radiation.

CONCLUSIONS

Thus, the example of a rectangular metal waveguide filled with a dielectric with periodically located paired transverse slots in the wide wall was used to demonstrate the possibility of implementing a 180° frequency scanning sector, including axial and normal radiation. Such a scanning sector is provided by the implementation of two radiation modes of the leaky mode (at zero and minus the first spatial harmonics).

The use of paired slots offset by a quarter of the wavelength in the waveguide made it possible to substantially suppress the reflection coefficient maximum at the frequency corresponding to radiation along the normal. However, the reflection coefficient maxima at the radiation frequency at minus the first harmonic and at the upper limit of the operating range remained. Analysis of methods to suppress them goes beyond the scope of this work.

It should be noted that the frequency band required for implementation of a 180° frequency scanning sector is almost an octave; i.e., the waveguide-slot array is ultrawideband.