Genetic algorithms calculation and optimization of the radiation pattern for periodic structure have recently been developed in multiple searches [1] [2] [3]. This paper presents a new optimization method based on Floquet modal analysis with Genetic algorithm to optimize the radiation of coupled periodic antenna [4]. This algorithm simulates biological evolutions, and it is based on the principle of genetic and natural selection. The coupling between the radiating elements increases with the dimension of the antenna arrays, making it one of the most important factors in the choice of the analysis method. This approach forces us to do a parametric study to provide an optimum radiation pattern for the coupled antenna. In this work, Floquet modal analysis is used and combined with MoM-GEC method. Floquet analysis [5] is used to reduce the formulation of the global periodic structure to one reference cell with periodic walls.

Moment MoM-GEC can be applied to this reference cell used to formulate the electric and the magnetic fields. This Floquet approach [6] [7] shows that the electromagnetic field distribution in periodic structures changes only by multiplication of a complex constant for a translation by one period in the global structure. The electromagnetic compilation of antenna arrays with an N pattern is then reduced to a calculation on one reference cell with periodic walls in a new modal base. The Floquet modal analysis introduces all possible Floquet states [8] [9] [10] and groups the coupling information of the overall structure.

This new method is applied to improve the formulation of high-density antennas. To obtain the optimized radiation pattern of coupled cell the Genetic algorithm must be adapted and applied with Floquet analysis.

This paper is organized as follows. In section two, the analytical details of the Floquet analysis formulation for periodic structure are described and 1-D antenna array example is given. In section three, the Genetic algorithm process is described and adapted to optimize radiation pattern based on Floquet phases and amplitudes. In section four, numerical tests are performed to assess the accuracy of proposed formulas. Finally in section five, conclusions have presented the results of this paper.

This section presents the formulation of electromagnetic problem for the periodic antenna. A Floquet theory is proposed to reduce the periodic domain to a single cell with periodic walls. The Floquet approach announces that electromagnetic field distribution in periodic structure changes only by multiplication of complex constant for a translation by one period in the global structure. Then the electromagnetic compilation of N periodic antenna is reduced to a one reference cell with periodic wall in a new modal base. These periodic walls are an artificial wall, and they are implemented to group all phases coming from other cells. Floquet modal analysis is to bring back all spatial calculation to a new modal calculation. An electrical field is then formulated and solved through a MoM-GEC approach [11] [12] [13] in a spectral domain [14].

The structure under analysis is shown in Figure 1. The excitation is given by an E₀ voltage source [15] [16] placed in the middle of a metallic patch. The width and the length of patches are w and L. The spatial period along the x direction is d_x. The height of dielectric substance is h, and its relative permittivity ɛris mounted on aground plane.

Periodic antenna arrays presentation using Floquet theory and MoM-GEC method. This structure is taken as finite in ±X and periodic with a period d_x. Floquet theorem can be used with this geometric periodicity, so the study of global structure is reduced to one cell with Floquet phases. The distribution of

the electric field differs only by one phase term compared to the adjacent cell. For details, see [17]. E ˜ ( x ) represents an electric field reacting with this periodicity.

{ E ˜ ( x + d x ) = exp ( j α d x ) ∗ E ˜ ( x ) E ˜ ( x + 2 d x ) = exp ( j α 2 d x ) ∗ E ˜ ( x ) E ˜ ( x + N d x ) = exp ( j α N d x ) ∗ E ˜ ( x ) (1)

Each Floquet phase corresponds to a Floquet state, and the function F α m characterizes all possible states.

F α m = 1 d x exp ( j α x ) ∗ exp ( j 2 π m x d x ) (2)

where α and m correspond respectively to Floquet mode and spectral domain mode. The α values are in Brillouin domain:

[ − π d x , π d x ] (3)

And for N discrete values of α, α_p are given by:

α p = 2 π p L (4)

where L = N ∗ d x and: − N 2 ≤ p ≤ N 2

The electric field of the central cell in spatial domain is E ˜ m . The electric field E ˜ α was associated in spectral domain, which models all waves emitted from other cells of periodic structure.

E ˜ m + 1 = E ˜ m exp ( j α d x ) (5)

Then

E ˜ m = d x 2 π ∫ − π d x π d x E ˜ α exp ( j α m d x ) d α (6)

In the spectral domain, MoM-GEC technique [10] [18] [19] can be applied for this single cell with periodic walls to extract the electromagnetic parameter. The pertinent problem of the use an electric field integral equation can be solved by applying the GEC method. It can replace the integral equation by a simple equivalent circuit in the discontinuity surface and applies the laws of tension and current to extract the relation between electric and current field by using an admittance operator [20] [21]. The discontinuity surface contains metallic and dielectric parties. The equivalent circuit of the unit cell is shown in Figure 2.

The virtual electric field is defined on the metallic surface and is null on the dielectric part (E_eα is its dual). Similar examples are found in [22]. From this circuit, this system was deduced:

{ J ˜ e α = J ˜ α E ˜ e α = − E ˜ 0 α + 1 Y ^ α e q ∗ J ˜ e α (7)

The equivalent admittance [23] operator is:

Y ^ α e q = Y ^ α u p p e r + Y ^ α d o w n (8)

Y ^ α u p p e r is the upper admittance operator of the infinite empty wave guide with periodic walls, and Y ^ α d o w n is the down admittance operator of the short-circuited dielectric wave guide of height h with periodic walls.

Y ^ α u p p e r = ∑ m n | f m n 〉 y m n α u p p e r 〈 f m n | Y ^ α d o w n = ∑ m n | f m n 〉 y m n α d o w n 〈 f m n | (9)

f_mn are the base propagation mode functions.

Next, the Galerkin method [24] was applied, where we project the excitation mode f_mn and the test function g_pq on the previous equation. The following system

was deduced:

( I 0 0 0 ) = ( 0 〈 f 0 , g 2 〉 ⋯ 〈 f 0 , g n 〉 〈 g 1 , f 0 〉 〈 g 1 , Y e q α − 1 , g 1 〉 〈 g 2 , f 0 〉 ⋱ ⋮ 〈 g 3 , f 0 〉 ⋯ 〈 g n , Y e q α − 1 , g n 〉 ) ( V 0 α x 1 x 2 x 3 ) (10)

The matrix form of the former equation can be developed as following:

( I 0 α 0 ) = ( 0 A t − A t B ) ( V 0 α X ) (11)

where A is the excitation vector andB is the coupling matrix. The test courant functions in metallic part are g_pq. The resolution of the previous system consequently helps to calculate the virtual electric field E_eα and the electric far field E_Rad of the coupled structure.

The fact of system optimization function is to search some parameters to inhabit optimal result. The resolution of optimization problems is based necessarily on optimization algorithm. For example, genetic algorithm, gradient method, network method and quasi-Newton method [25] [26] [27] are used in various research. In this study, the Genetic algorithm has been selected. It simulates the genetic evolution. The change of the system parameters follows an evolution process based on the genetic rules that modify chromosomes. In the optimization context, the variables define each chromosomes gene. These chromosomes evolve with genetic law to an optimal chromosome. In the reproduction phase, the individuals are selected and then their structures are modified to generate new individuals for the next generation. The optimization process using genetic algorithm is clarified in Figure 3.

The initial population present a set of chromosomes that includes all the variables of this problem X 1 , ⋯ , X N . The research area is defined between

[ X i min , X i max ] , i ∈ [ 1 , N ] , where X_i is the variable to optimize and N is the number of system parameter. The coding choice of chromosomes depends on the length of search area. In our case, the binary coding has been chosen. In the reproduction loop, selection function allows to choose the individuals on which ones apply the algorithm to create the next generation. Many selections method have been found such as proportional method, tournament method and ranking method.

The proportional selection is the most used method. The crossing is applied on two different individual and the result is a chromosome formed from his parent genes.

The crossing can be done in one or many points. The mutation is applied on one individual by the modification of one or many parent genes chosen randomly. The percentage of mutation is fixed, and a single child is provided. From generation to generation, the system will converge, and the population will involve towards the optimum. The optimization of pattern radiation [28] [29] form is done using Genetic algorithm and implemented over MATLAB software. The evolution of each pattern radiation is calculated through fitness function. This function represents the standard deviation between the antenna pattern and the desired template. The usual fitness function is the sum of absolute errors between the calculated values and the reference radiation pattern template. The expression of the fitness function is given by:

Fitness = | R c ( X i , θ ) − R Template ( θ ) | (12)

where N is the iteration number of genetic algorithm. The calculation of the radiation pattern for each iteration of the algorithm has been done based on the formulation of Floquet analysis. However, at each iteration, new parameters are produced by the Genetic algorithm. Finally, an evaluation of the function Fitness will be applied to validate the optimization result. To achieve good optimization, the fitness function tends to minimal value near to zero. This research of minimum will be effected by generating a sequence of vectors X 1 , X 2 , ⋯ , X N . In a convergent iterative process, an acceptable Fitness value can be reach after N iterations. At this point, the X_N value is used to generate a diagram that meets the requirements. To simulate periodic antenna array, it is necessary to consider a periodic structure with uniform spacing between elements and uniform excitation to apply Floquet theory reduction. So, its radiation pattern can be described by all Floquet mode in spectral domain.

In the context of optimization by the genetic algorithm, the parameters to be optimized are the variables which characterize a coupled periodic antenna. First, we start by encoding the system parameters. In our case the encoding binary type is used. The production mechanism [30] [31] of the initial population must be capable of producing non-homogeneous individuals of a population. In our case, the randomly model have been chosen. At this stage, we will identify the three phases of the Genetic algorithm. So, the tournament selection model have been chosen a, a crossover model at a single point characterized by a cross rate c and a mutation model with a mutation rate m. Table 1 presents all the simulation parameters of the Genetic algorithm.

In this contribution, we have presented a theoretical analysis of 1-D coupled periodic antennas with Folquet theory and MoM-GEC method. Then we have optimized the gain and the directivity of coupled periodic structure with Genetic algorithm. To enhance the radiation characteristics, we propose to choose the optimum parameters generated by the optimization algorithm (the periodicity d_x = 120 mm, the length of the metal part L = 45 mm, and the width of the metal part w = 3 mm). Floquet analysis with MoM-GEC is more useful and more beneficial and allows us to save 96.66% of computation time for each iteration of the optimization process. For a future work, we suggest applying different optimization techniques with Floquet analysis for a periodic structure to predict the optimal radiation pattern that can be adopted to decrease the side lobe level.

The authors declare no conflicts of interest regarding the publication of this paper.

Latifa, N.B. and Aguili, T. (2022) Optimization of Coupled Periodic Antenna Using Genetic Algorithm with Floquet Modal Analysis and MoM-GEC. Open Journal of Antennas and Propagation, 10, 1-15. https://doi.org/10.4236/ojapr.2022.101001