### I. Introduction

*TE*

_{10}) only. It is noteworthy that research for the optimization of the air hole design to a target value has not yet been carried out.

### II. Analysis of the Air Holes in SIW

### 1. Structure Approximation

*a*

*, air hole diameter*

_{SIW}*d*

*with the centric position*

_{h,k}*x*

_{0}

*unit cell length*

_{k,}*l*

*, number of air holes*

_{uc}*N*, and height of SIW

*b*The shorting via hole diameter

*d*

*and the separation*

_{v}*s*

*are given by the condition 0.05 <*

_{v}*s*

*/*

_{v}*g*< 0.25,

*s*

*>*

_{v}*d*

*[12], and*

_{v}*λ*

*is the guided wave length.*

_{g}*a*

*is the effective width of the equivalent RW and is given by [26].*

_{eff}### 2. Recursive Equation for SIW with Multiple Air Holes

*β*of the structure in Fig. 2 has been derived and proposed by the author [27]. The equation was successfully derived from the wave equation for the case of the partially loaded waveguide [28], which is simple but accurate. As a result, the recursive equation for the phase constant

*β*can be represented by Eqs. (4)–(7).

##### (7)

*k*

*and*

_{d}*k*

*are the cut-off wave numbers for the dielectric and air regions. The phase constant*

_{a}*β*in

*TE*

_{m}_{0}modes can be numerically obtained using Eqs. (4)–(7). The effective dielectric constant of the structure in Fig. 2 can easily be obtained from the phase constant

*β*.

### 3. Closed Form Equation for the Effective Dielectric Constant

*ɛ*

*) in the SIW with air holes is mainly subject to E-field distribution in transverse mode of wave propagation. The theory suggests that the combination of the air and dielectric media in the SIW will distort the E-field distribution profile to satisfy the boundary and phase matching condition at the interface of two different media, which will result in an increase in complexity for exact analysis. In this study, however, an approximation was used, neglecting the boundary and phase matching condition, because the E-field profile is not so significant a distortion when the operating frequency approaches the cut-off region. In this section, therefore, we proposed the closed form equation for the*

_{r, eff}*ɛ*

*from the E-field energy equivalence concept, on the assumption that the wave number and the E-field amplitude is equal to the conventional SIW filled with uniform dielectric material. We can finally obtain the closed form equation of*

_{r, eff}*ɛ*

*for Fig. 2(b) in Eq. (10), combining Eqs. (8) and (9).*

_{r, eff}##### (8)

##### (10)

*W*

*,*

_{h}*W*

*,*

_{d}*W*

*, and*

_{a}*W*

*represent the total time-average stored electric energy, the energy in dielectric, air, and equivalent dielectric media, respectively, and*

_{eff}*k*

*=*

_{c}*π*/

*a*

*is the cut-off wavenumber of the SIW filled with uniform dielectric material.*

_{eff}### 4. Efficient Method for Solving the Recursive Equation

*k*

*and*

_{a}*k*

*can be expressed in terms of*

_{d}*β*using Eqs. (4) and (5). The root of Eq. (11) can be solved numerically for

*β*using the Newton-Raphson method.

*f*

*(*

_{s}*β*)] for solving the phase constant, and the initial value

*β*

_{0}is needed for the finding of minima. We need an initial value as close as possible to the root for the fast convergence of Eq. (11), and the closed form Eq. (10) was therefore used to obtain the initial value

*β*

_{0}.

### 5. Calculation Results

*ɛ*

_{r}_{,}

*extracted at cut-off frequency by means of the proposed approaches. The results were compared with simulated results from the eigenmode solver in the HFSS. We assumed Taconic/RF30-7H substrate properties with*

_{eff}*ɛ*

*= 2.97, height*

_{r}*h*= 0.762 mm and

*a*

*= 28.87 mm,*

_{eff}*b*= 0.762 mm,

*d*

*= 1 mm,*

_{v}*s*

*= 1.4 mm varying the air hole diameter*

_{v}*d*

_{h}_{,}

*from 0.6 mm to 3.6 mm and the unit cell length*

_{k}*l*

*=*

_{uc}*d*

_{h}_{,}

*+ 0.6 mm. The air hole position for each number of holes is shown in Table 1. All*

_{k}*d*

_{h}_{,}

*are identical, and the gap between the air hole in unit cell is set to 0.3 mm.*

_{k}*ɛ*

_{r}_{,}

*using the three methods along with the air hole diameter variation, which shows excellent agreement with the proposed approaches (within 2% error for the analysis scope).*

_{eff}### III. Optimization Using Genetic Algorithm

*d*

_{h}_{,}

*, number of holes*

_{k}*N*, and number of cells

*N*

*were the variables. We show how the GA is applied when the number of holes is fixed, and then we extend this to the case where the number of holes is variable.*

_{cell}### 1. Restricted Tournament Selection

From the population, pick two parents,

*A*and*B*.Cross over and mutate

*A*and*B*to produce offspring,*A*′ and*B*′.Pick a subpopulation of

*w*chromosomes from the population.Find the chromosome that most resembles

*A*′ from the subpopulation and let it compete for a place in the population.Repeat steps 3 and 4 for

*B*′.

### 2. Chromosome and Fitness Function

*N*holes, a vector of

*N+*1 double-precision floating-point numbers represents a single chromosome. The first number in the vector represents the number of cells. The rest of the number represents the hole diameters. The (

*i*+ 1)

*gene represents the diameter of the*

^{th}*i*

*hole. The population is simply a 2-D array of size (*

^{th}*N+*1)×

*N*

*, and*

_{p}*N*

*is the size of population. Note that the first gene of the chromosome does not need to be a floating-point number; this choice was made for the sake of simplicity.*

_{p}*U*{

*N*

_{cell}_{,}

*,*

_{min}*N*

_{cell}_{,}

*} and the rest of the genes to*

_{max}*U*(

*d*

*,*

_{h,k,min}*d*

*), where*

_{h,k,max}*U*{

*a*,

*b*} and

*U*(

*a*,

*b*) are discrete and continuous uniform random variables in the ranges [

*a*,

*b*] and (

*a*,

*b*)

*,*respectively,

*n*

_{cell}_{,}

*is the minimum number of cells,*

_{min}*n*

_{cell}_{,}

*is the maximum number of cells,*

_{max}*d*

*is the minimum diameter of a hole, and*

_{h,k,min}*d*

*is the maximum diameter. The fitness function, which assigns a value of fitness to a chromosome, is defined as follows:*

_{h,k,max}*x*

*is the*

_{i}*i*

*chromosome,*

^{th}*T*is the target value, and we can freely assign this value (i.e., the phase constant or effective dielectric constant etc. can be selected), and

*g*(

*x*

*) is a computer function that computes the wavenumber for*

_{i}*x*

*. The chromosomes that have wavenumbers closer to the target have higher fitness values. This fitness function directs the evolving population toward the configurations that have the target wavenumber. The population stops evolving when a given computation time is exhausted or the solutions have converged.*

_{i}### 3. Variable Number of Holes

### 4. Genetic Operators

*x*

*and*

_{i}*y*

*are the*

_{i}*i*

*genes of two parents, assuming*

^{th}*x*

*<*

_{i}*y*

_{i}*, z*

*is the*

_{i}*i*

*gene of the offspring, and α is a factor that controls the expansion. α = 0.5 was used.*

^{th}*p*

*, and its operation is given by*

_{m}*x*

*is the*

_{i}*i*

*gene before mutation, and*

^{th}*N*(0,

*σ*

^{2}) is a Gaussian random variable with mean 0 and variance

*σ*

^{2}. The extent of the disruptiveness of mutation can be precisely controlled with the probability and variance.

*p*

*=1 and*

_{m}*σ*

^{2}= 8.33×10

^{−4}(

*d*

*−*

_{h,k,max}*d*

*) were used in this study.*

_{h,k,min}### 5. Optimization Results

*β*, in which the air hole diameters

*d*

_{h}_{,}

*and number of holes*

_{k}*N*were the variables. The number of cells

*N*

*was fixed to 1 for convenience. We used Taconic/RF30-7H substrate properties with*

_{cell}*ɛ*

*= 2.97, height*

_{r}*h*= 0.762 mm,

*a*

*= 28.87 mm,*

_{eff}*l*

*=*

_{uc}*d*

_{h}_{,}

*+ 0.6 mm and a gap between the air holes in the unit cell of 0.3 mm.*

_{k}*β*is 80 rad/m and the errors in percentage indicate the difference between the optimized and target values of

*β*. Several types of solution are suggested in Table 2, and they show good agreement (less than 2.1e-8% optimization errors). The optimization process took about a minute using a typical laptop computer.

### IV. Validation by Bandpass Filter Design

*d*

_{h}_{,}

*, number of holes*

_{k}*N*, and number of cells

*N*

*. In this paper, a very accurate approach is applied to find the dimension of the filter by means of the recursive equation and optimization by GA.*

_{cell}*K*

_{n}_{−1,}

*/*

_{n}*Z*

_{0}was obtained from [34]. We used Taconic/CER-10 substrate properties with

*ɛ*

*= 10, height*

_{r}*h*= 0.635 mm and

*a*

*= 15.8 mm,*

_{eff}*a*

*= 16.34*

_{eff}*b*= 0.635 mm,

*d*

*= 0.7 mm,*

_{v}*s*

*= 0.95 mm*

_{v}*l*

*=*

_{uc}*d*

_{h}_{,}

*+0.25 mm, and the gap between the air hole in unit cell was set to 0.25 mm. The number of air holes*

_{k}*N*was fixed to 7, and the diameters of seven air holes

*d*

_{h}_{,}

*were set to equal values for design convenience. Table 3 shows the desired values for the propagation constant*

_{k}*β*

*and the effective dielectric constant*

_{n}*ɛ*

_{r}_{,}

_{eff}_{,}

*corresponding to the*

_{n}*K*

_{n}_{−1,}

*/*

_{n}*Z*

_{0}values. The optimization results from the GA for the number of unit cell

*N*

*, air hole diameter*

_{cell}*d*

_{h}_{,}

*, half-wave resonator length*

_{k}*l*

_{r}_{,}

*, and coupling length*

_{n}*l*

_{c}_{,}

*are also shown in Table 3.*

_{n}*TE*

_{10}mode [35].