Analysis on the Radiation Property of the Bounded Modes of Periodic Leaky-Wave Structure with Finite-Length Using a Hybrid Method

In this paper the radiation property of the one-dimensional periodic leaky-wave structure is analysed using a new hybrid method, which involves the mode expansion method for expanding the periodic aperture field in terms of spatial harmonics and the method of effective radiation sections for transforming the expanded fields into far fields. Using this method, the radiation of each spatial harmonic can be achieved, and the contributions of the harmonics (especially the bounded modes) to the total radiation of the periodic leaky-wave structure can be calculated. The main findings in this paper demonstrate that the bounded modes in a finite length structure have obvious contribution to the far-field radiation, which was considered to be non-radiative and always ignored in the conventional researches.

In order to extract the propagation constant of each spatial harmonic from the periodic aperture field, mode expansion must be implemented. In 14 , a mode expansion method was presented in cylinder coordinate which is suitable for far field calculation of leaky coaxial cable. In this paper, a new mode expansion method in Cartesian coordinate is presented. Combining this method with the ERS method, a hybrid approach is achieved, which is efficient for studying the radiation property of one-dimensional periodic leaky-wave structure. Using this method we can find the fact that the bounded waves also have contributions to the total radiation field, which was considered to be non-radiative in the conventional research.

Results
Mode Expansion Method for the Aperture Field of One-Dimensional Periodic Structure. In this paper, only the one-dimensional periodic leaky-wave structures are considered. The leaky rectangular waveguide (LRWG) with periodic transverse slots is a typical type of periodic leaky-wave structure, and can be regarded as one-dimensional periodic structure. Although it is not an ideal one-dimensional structure due to the finite width and field variation in the transverse direction, the proposed method will prove effective for this structure.
The LRWG, which is widely used in the communication-based train control (CBTC) system 15 , is shown in Fig. 1. Its radiation property in yoz plane (vertical plane passing through the axis of waveguide) will be studied (one-dimension approximation). An artificial boundary line in yoz plane is set above the top of LRWG with height of y 0 , which separates the upper half space of yoz plane into two regions (named region I and region II). The waveguide is considered to be made by perfect electric conductor (PEC), and the leakage loss is small due to small slot and large period, so the attenuation of aperture field along the artificial line is very small, which is ignored in the preliminary study for simplicity.
The fields in region I denoted by E I is calculated by the full-wave method. The fields in region II can be expressed by z components of the electric and magnetic Hertzian vector potentials Π e and Π m as Since the field in region II is periodic in z direction, Π ez and Π mz can be expanded into Floquet's modes according to the Floquet theorem as following where A n and B n are unknown coefficients to be determined, and β zn and β yn are the propagation constants of the nth spatial harmonic in z and y directions respectively. β zn and β yn can be expressed as where β 0 is the propagation constant of the fundamental mode (n = 0), and k 0 is the propagation constant in free space. Substituting (2) into (1), the fields in region II can be obtained as following  The tangential components of the fields must be continuous on the artificial boundary. Since the component E x is zero along the boundary line (due to symmetry), only the component E z needs to be considered, that is The length in one period along the artificial boundary line is discretized into (2N + 1) nodes uniformly in the z direction, and the sequence number of each node is denoted by i, as shown in Fig. 2.
Using equation (6) at each node (point matching), we obtain equation (7) as In equation (7), the summation has been confined within n∈ [− N, N − 1]. Thus, 2N equations can be obtained from (7), and the number of unknowns is also 2N. Therefore, a unique solution can be achieved. The truncation error due to the limited summation in (7) can be neglected if N is large enough. Equation (7) can be written in matrix forms as Then the coefficient matrix [A] can be derived, and hence each spatial harmonic is determined. It is worth noting that this method is suitable for arbitrary multi-harmonic radiation. However, the LRWG usually works under the condition of mono-harmonic radiation. So in this work the mono-harmonic radiation condition is required to ensure the existence of only one radiating mode. Also, this makes it simple to demonstrate the contributions of the bounded modes to the far field radiation, since all the other spatial harmonics (except the sole radiating mode) will be the bounded modes. The mono-harmonic radiation condition requires that: (10), we get the mono-harmonic radiation condition:  (11), the mono-harmonic radiation condition is P < 68.62 mm, so the current period length of P = 61 mm meets the requirement. The computational accuracy of the proposed method depends on the value of N in equation (7). In order to find a proper N that ensures enough accuracy, an evaluating function is defined as where E a is the simulated aperture field in one period, E h is the summation of fields of the harmonics in one period, and Q is the total number of the simulated field points in one period. The convergent curves of this function for different height y 0 are illustrated in Fig. 3. It is found that for larger y 0 , convergence of (12) becomes faster, and smaller N can give enough accuracy. This is because that the field closer to the LRWG surface contains more contributions from the higher order harmonics. If the number of terms in equation (7) is insufficient, the contributions of higher order harmonics are not sufficiently involved. Therefore, according to Fig. 3, the value of N is set to be 50 for the case of y 0 = 0.5 mm in the following study. The simulated result of the aperture field in one period is given in Fig. 4, and it is expanded into different spatial harmonics using the mode expansion method (equations (8) and (9)). Figure 5 illustrates the magnitudes of the spatial harmonics. It can be seen that the magnitude of n = −1 harmonic is greater than that of the n = 0 harmonic. This is caused by the periodic perturbation along the structure, yielding that the n = −1 harmonic possesses more power than the n = 0 harmonic. When N = 50, the summation of all the spatial harmonics obtained using equation (7) is also given in Fig. 4 for comparison, which shows a good match with the simulated result.

Radiations of the Harmonics Calculated by ERSs.
The method and whole procedure to calculate the radiation of periodic aperture field by using ERSs are stated as following: Each spatial harmonic extracted from  the periodic field can be viewed as a traveling wave, and the radiation characteristic can be analysed using ERS method. The determination of ERSs and the formulae for calculating far field by ERSs were given in the supplementary information according to the study in 13 . So the far fields either from radiating harmonics or from the bounded harmonics can be calculated. Then the far-field radiation of the whole periodic structure can be expressed as the summation of radiation fields from all the spatial harmonics as following where L e0 and L e are the location and length of the ERS. r is the distance from observation point to origin. J Mx is the equivalent magnetic current for each spatial harmonic. The two terms in (13) represents the magnetic current integrations along the two sections of ERSs, respectively. So firstly, we calculate the ERSs of each spatial harmonic by supplementary formulas (S4) and (S5) in the supplementary information, and the results are shown in Fig. 6. It can be seen that for the fundamental mode (n = 0), the ERSs corresponding to the main beam direction (around θ = 34.8°) cover the whole length of the LRWG, demonstrating that the far field in the main beam direction is generated by the whole structure. For the n = −1 spatial harmonic, the ERSs are very short at first, but increases rapidly as θ increases. When θ enters the angular range corresponding to the last several lobes close to θ = 180°, the ERSs of the n = −1 spatial harmonic is found to be much larger than that of the n = 0 spatial harmonic. Predictably, this will lead to a much stronger radiation in this angular range generated by the n = −1 spatial harmonic, though it is actually a bounded mode. For the n = + 1 and − 2 spatial harmonics, for any angle in space, the corresponding ERSs are very short, which means a weak radiation. It is worth noting that different spatial harmonics have different locations and lengths of ERSs, due to different propagation constants.  Checking each term of the summation in equation (13) individually, the radiation pattern of each spatial harmonic can be calculated by the corresponding ERSs. Figure 7 shows the normalized radiation patterns of each spatial harmonic and the summation of all harmonics, respectively, and the radiation pattern calculated from the simulated aperture field is also given for comparison. It is observed that the radiation patterns generated by the fundamental mode (n = 0, the radiating mode, blue curve with triangles) and the simulated aperture field (red curve) match well with each other within θ∈ [0°, 65°]. However, as θ increases, the difference between the two curves becomes significant, and reaches 18.52 dB on the last lobe. Contrarily, when θ increases, the radiation pattern of n = −1 harmonic (green curve) approaches to the radiation pattern calculated by the aperture field gradually, and agrees with it on the last several lobes. This demonstrates that the last several lobes generated by the aperture field are mainly contributed by the radiation of n = − 1 harmonic. Although the n = −1 spatial harmonic is a bounded mode, its effect on the total radiation cannot be ignored, which can be also inferred from the larger ERSs in Fig. 6. In addition, in Fig. 7 the radiation strengths of n = − 2 and n = +1 spatial harmonics are relatively weak, corresponding to the smaller ERSs in Fig. 6. In Fig. 7, the black curve with dots (the summation of all the spatial harmonics) is in good agreement with the red curve (direct radiation of the aperture field), because the contributions of higher order harmonics (bounded waves) have been sufficiently involved when N = 50. In addition, the radiation patterns of the whole LRWG obtained by the three-dimensional full-wave simulation (CST) is also given in Fig. 7 (grey curve), from which we find that the radiation pattern calculated directly from the aperture field of one-dimension model is very close to that of three-dimensional full-wave method. This means that the proposed method is valid for the analysis of this type of periodic leaky-wave structure. Figure 8 illustrates the difference between the last lobe levels obtained by the fundamental mode (n = 0) and the total aperture field, respectively. It can be seen that the difference becomes larger as P increases, and reaches 32 dB when P is 68.6 mm (this value of P is the critical value of the mono-harmonic condition). Therefore, the radiation property of this periodic structure is dominated by the fundamental mode only when the period of the slots is small, even when the mono-harmonic condition is satisfied. In addition, the period of the slots is the determination factor for the main beam direction, which is very important in practical applications of waveguide slot array. Therefore, proper period of slots should be chosen through a comprehensive analysis.

Discussion
In this paper, taking the LRWG as an example, the radiation property of one-dimensional periodic leaky-wave structure is studied with a new hybrid method, which involves the mode expansion method and the ERSs method. Using this method, the periodic aperture field of the LRWG can be efficiently expanded into spatial harmonics. Then the radiating mode and bounded modes can be easily extracted from the total radiation field, and the contribution of each mode can be calculated using the ERSs of each spatial harmonics. The results show that the n = − 1 spatial harmonic, which was usually supposed to be the non-radiating mode (slow wave) in the conventional research, has great influence on the total far-field radiation, yielding an obvious difference between the radiation patterns of the fundamental mode (n = 0) and the total periodic aperture field. The maximum difference of the last lobe levels can reach 32 dB even under the condition of mono-harmonic radiation. Therefore, the contribution of the bounded waves to the far-field radiation should not be ignored if high accuracy is demanded. Although the bounded waves contribute mainly to the higher order side lobes, the discovery in this paper changes the conventional opinion that the bounded mode (slow wave) can only be guided along the waveguide and do not have any contribution to the far fields. This is the most important discovery in this paper. From another perspective, this work presents a new insight into the mechanism of high side lobe levels generated by bounded-wave radiation, and this might be instructive for further research on the suppression of bounded-wave radiation.