obtained for the unknown expansion coefficient Ciy, and Cjz. The two linear equations are expressed in matrix form, as following.

Where A is the moment matrix for all self and mutual interactions of electric fields components with electric current components, and C is an unknown electric current expansion coefficient.

Uinc represents incident electric fields. The elements of A, and Uinc matrices, obtained in the following integral form:

then the scattering parameter of S11 is obtained by following equation.

Sm

where P is the total incident power, and its value is unit according to normalization. Then, using Eq. 20 one can obtain the scattering parameters of proposed structure in a given frequency range. All of mentioned equations can be computed through MATLAB codes. The procedure that leads to the optimum pattern is performed by genetic algorithm optimization, and is explained in the next section.

2.3 Genetic algorithm optimization

The longitudinal E-plane pattern consists of m 9 n rectangular subsections, which are whether metalized, or non-metalized, as it is shown in Fig. 1b. For computation time

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consideration, based on the symmetry excitation requirements (TE10 incident wave), the whole pattern plane is chosen symmetrical with respect to the horizontal axis of waveguide. Therefore, the genetic algorithm operates on half region of the patterned plane. For finding optimum pattern of the longitudinal plane, MATLAB genetic algorithm optimization tool is used. Genetic algorithm population type is set to bit string, consisting of ‘‘0’’, and ‘‘1’’ s binary digits, correspond to the nonmetal, and metal parts on the plane, respectively (Ghajar and Ghorbaninejad 2015; Ghorbaninejad-Foumani and KhalajAmirhosseini 2011). The size of population is specified 200, and they are produced with uniform creation function. Parents are selected by a stochastic uniform, and new generation is reproduced by recombination with the crossover fraction of 0.6–0.8, and uniform mutation with rate of 0.02–0.03. Other parameters considered their defaults.

Since the number of subsections (optimization variables) is very large, optimization by commercial electromagnetic simulator software is impossible. However, since the proposed approach is concise, it acts much faster. In the proposed numerical design method, electric field integral equations are solved only one time and stored as basic matrices; so, the scattering parameters of a desired pattern can be extracted from these basic matrices. So the presented method allows to considerably decrease the computational time. For obtaining basic matrices, it is assumed that all parts of pattern are metalized part.

The designing method is based on the optimization of a suitable cost function. In the cost function, the amplitude of S21 and S11 of the proposed structure is adjusted to those of arbitrary desired filter in a desired frequency range. For this purpose, some frequency samples are selected from frequency range, so, the following cost function can be defined

for M discrete frequencies f1; f2; . . .; fM, all in the frequency domain. v

Here, the cost function is defined based on the mean square error. In Eq. 21, SMoM21 ðfmÞ and SMoM11 ðfmÞ are transmission and reflection coefficient of the structure which is derived and calculated by the method of moment, also HðfmÞ; and RðfmÞ are any desired transfer and reflection function, respectively, all at frequency fm. Also, P ? Q = M, which is total frequency numbers.

As it was noted in the design approach, in the presented method, patterned planes as resonators (or anti resonators), are longitudinally located in a waveguide so that the length between them is odd multiples of quarter wavelength. So, the mutual couplings between subsequent resonators (or anti resonators), should be considered due to effect of evanescent modes, and filter specifications. Once all resonators (or anti resonators) were designed, the numerical computation method is performed again to the whole structure for calculating their mutual coupling effects. In the proposed method due to good convergence of the double series, including the term expð cjz z0jÞ, and in far enough distances, the coupling effect would not be very much. The simulation results validate this conclusion. In general, the proposed method can be used to accurately take in account coupling effect and there is no limitation in the proposed design procedure. In such cases, one should perform a new optimization for the whole structure, assuming SMoM21 ðfmÞ and SMoM11 ðfmÞ are transmission and reflection coefficient of the whole structure. In the final optimization, the values (0 and 1 s) of each patterned planes may be considered as an initial value.

It should be noted, the Green’s function of the series form, that is slowly converging, can be accelerated by Poisson summation formula and the Kummer transform. Moreover,

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