I. Introduction

Commonly, Frequency Selective Surfaces (FSSs) consist of a two-dimensional periodic array of a metal plane aperture with openings (metal grid), or an arrangement of printed conductive elements on a dielectric substrate and they act as a band-pass or band-stop filter [^{1}],[^{2}]. For the last few decades, FSS are one of the important planar periodic structures, which are investigated and characterized in several important applications domain such as electromagnetic engineering and microwaves. Due to large demands for multi-functionality of FSS in electrical engineering and telecommunications, many researchers studied and developed the FSS for designing multiband responses[^{3}]-[^{15}].

In this paper, we propose a structure having five resonant frequencies, three resonant when the FSS is x-polarized and two resonant when the FSS is y-polarized. To perform this investigation, the proposed FSS was simulated using the wave concept iterative procedure WCIP [^{16}].

In order to validate the results of the WCIP method, the commercial finite-element software package COMSOL Multiphysics RF Module is employed, and one FSS design is fabricated and tested. The obtained WCIP and COMSOL Multiphysics results are in good agreement where compared with the experimental ones.

Moreover, for such FSS structure, a synthesis approach based on the FSS parametric study allows the determination of the resonant frequencies. The use of the least mean square method allowed the determination of the resonant frequencies equations based on polynomials and plotting charts. For complex FSS patterns characterized with a difficult resonant frequencies behavior comprehension, the synthesis approach is extended to FSS structures based on non-coupled parallel metallic strips to provide complex FSS equivalents structures. When the FSS applications are limited to the use of an FSS of a specific pattern, abacuses are implemented to get a direct determination of the FSS resonant frequencies ones the FSS dimensions are fixed without a need to run the software each time. The presented FSS synthesis approach results are compared to the WCIP results, COMSOL Multiphysics and measurements and good agreement is obtained.

II. L-Notched Rectangular metallic ring FSS design

The unit cell of the L-notched rectangular metallic ring FSS is shown in fig. 1 where the metallic ring is printed on a fiberglass (FR-4) substrate with dielectric constant ε_{r}=4.4 and a thickness h=1 mm.

The transmission characteristic of the proposed multiband FSS is determined by the WCIP method when the interface of the FSS is described by 200×200 pixels, and the iterative procedure is stopped after 600 iterations. The FSS is excited with a normal incident plane wave source.

III. Results and discussions

A. WCIP results and validation

In order to confirm the results obtained by WCIP method, a simulation with COMSOL Multiphysics software was investigated and one FSS prototype of 10×10 unit cells, shown in Fig. 2, is fabricated and experimentally characterized by using the data acquisition system that consists of an Agilent two ports microwave network analyzer N5230A and two standard horn antennas. The FSS overall size is 20 cm × 20 cm, and the metallic rings are printed on a substrate of FR-4 with thickness h=1 mm and dielectric constant ε_{r}=4.4. All the other dimensions of the fabricated FSS are the same as those of the unit cell shown in Fig. 1 when L_{h}=7.5 mm and L_{v}=4 mm.

Figs.3 (a) and (b) present the FSS transmission coefficient obtained by the WCIP method, COMSOL Multiphysics software, and measurements when a normally incident plane wave is x- polarized and y-polarized respectively.

As the incident plane wave is x-polarized, dual resonant frequencies are observed, from both simulated WCIP, COMSOL Multiphysics and measured results. The first resonance occurs at 8.7 GHz in the WCIP results, 8.9 GHz in the COMSOL results and 8.5 GHz for measurements while the second resonance takes place at 10.4GHz, 10.3 GHz and 10.1 GHz for WCIP, COMSOL and measured results respectively. In the case of y-polarization, only one resonance frequency is observed at 8.1 GHz as a WCIP result, 8.1 GHz as a COMSOL result and 7.8 GHz in measurements. A good agreement is noted when comparing the WCIP results with those of COMSOL Multiphysics and Measurement.

B. L-notched FSS parametric study

To determine the dimensions responsible of tuning the resonant frequencies for the dual polarized FSS, a parametric study should take place. Figs. 4 and 5 illustrate the transmission coefficient of the FSS when the parallel strip length L_{h} varies from 4.5 mm to 7.5 mm. With the increase in L_{h} lengths, the highest and the lowest resonant frequencies decrease while the central resonant frequency remains constant not affected by the variation of the L_{h} values in the considered interval when the x-polarized source is used.

Concerning the FSS *y*-polarized source case, when L_{h} increases the lowest resonant frequency is shifted down, while the highest one remains constant.

The effect of the vertical strip L_{v} of the proposed FSS on the resonant frequencies is shown in Figs. 6 and 7 for *x*-polarization and *y*-polarization respectively. In the *x*-polarization case, when L_{v} changes from 0.5 mm to 2.5 mm, the lowest resonant frequency is decreased from about 8 GHz to about 7GHz. The centered resonant frequency slightly decreases while the highest resonant frequencies remain constant. The results in Fig. 7 concern the y-polarization case and shows that the lowest resonant frequency decreases with the increase of L_{v}, while the highest one remains constant.

The variation of L_{v} can serve as a large tuning way for the lowest resonant frequency for both polarizations as it can serve also as a fine tuning way for the centered resonant frequency in the *x* polarization.

Another important parameter, which affects the resonant frequencies of the proposed FSS, is the length L of the notch. Figs. 8 and 9 show the transmission coefficient due to the increase in L for *x*- polarization and *y*-polarization respectively.

As the values of L increase from 0 to 3 mm, the central resonant frequency decreases from about 9.8 GHz to 8.1 GHz while both the lowest and the highest resonant frequencies do not change. In the case of *y*-polarization source plane wave, when the height L increases the highest resonant frequency decreases, while the lowest one remains constant.

IV. Bandwidth enhancement

The adjustment of different resonance frequencies can result in widening the FSS band by fusing two adjacent resonant frequencies bandwidth. In the case of L_{h} =12 mm and the x-polarized exciting source, the highest resonant frequency decreases and merges with the central resonant frequency resulting in only one resonant frequency as the lowest resonant frequency disappears and an improvement of the FSS bandwidth has shown in Fig. 10. The enhanced bandwidth is 6.95 GHz at – 3 dB and 3.25 GHz at −10 GHz concerning the WCIP results.

V. Resonant frequency estimation

An approximate procedure can be used to estimate the resonant frequencies of the proposed geometry without using neither analytical nor numerical methods. The presented procedure is based on determining the length of the linear metallic strip shown in Fig. 11 equivalent to the complex pattern FSS for a given complex FSS dimension for the structure of Fig.1 starting from the approximate equation providing an approximate evaluation of the resonant frequency of a linear microstrip structure of length L_{e} given by [^{17}]:

c is the speed of light in free space given by c=3×10^{8} m/s, L is the length of the metallic strip, and ε_{reff} is the effective dielectric constant of the structure and can be calculated from the equation of microstrip lines as[^{18}]:

Where W is the metallic strips width (W=2 mm), h is the thickness of the substrate taken here as h=1mm due to their availability in the laboratory. Therefore, the value of the effective dielectric constant, in this case, is given as ε_{reff} =3.3425.

First, the resonant frequencies for different values of the FSS dimension L_{h} or L shown in Fig. 1 are determined by the WCIP method and the COMSOL Multiphysics software but maintaining the other FSS dimensions constant. Then, the different equivalent strips lengths called effective lengths L_{e} are calculated using:

At this stage, the real metallic strips length that if manufactured gives the same result is not yet determined since it is related to the effective length L_{e} by:

Where k is a polynomial to be determined from the curve relating the complex FSS dimension L_{h} or L and the effective length L_{e} calculated in (3). Thus, it can be noted as k(L_{h}) or k(_{L}). Once determined, the resonant frequency of the FSS of Fig. 1is directly given by only inserting the concerned dimension L_{h} or L.

The polynomials are found by the resolutions of linear equations as a form of matrix systems A*X=b. When the number of points used for determining the polynomial is equal to the order of the polynomial, the matrix X is square in this case the solution is unique. In the literature, there are many methods for optimization like the genetic algorithm, neural network, gradient method, least square method … In the following, the optimization with the least square method of one variable on the resonance frequencies by using the Scilab software [^{19}] is used. Therefore, knowing the two polynomials concerning the FSS dimensions L_{h} and L, the resonant frequency of the FSS structure of Figs. 1 and 2 can be easily anticipated only by knowing the values of the influencing dimensions L_{h} and L, the dielectric without using neither the WCIP method nor the COMSOL Multiphysics or HFSS or other commercial software. The precise equation giving the FSS resonant frequency is then given by:

where L_{h} is the varied dimension FSS from the standard FSS dimensions shown in Fig. 1 in addition to the substrate thickness h=1mm and its dielectric constant ε_{r}=4.4.

where L is the varied dimension FSS from the standard FSS dimensions shown in Fig. 1 in addition to the substrate thickness h=1mm and its dielectric constant ε_{r}=4.4, and *c* is the speed of light c=3×10^{8} m/s.

Fig.12 shows the five resonant frequencies in function of the length affecting the FSS resonance obtained by the WCIP method, COMSOL Multiphysics software and equation 5. By comparison, a good agreement is observed.

VI. L-notched rectangular metallic ring fss equivalent structure determination approach

The dimensions affecting each resonant frequency of a complex metallic pattern FSS such as the notched quasi-square FSS are determined after achieving the parametric study of the structure. A deep comprehension of the FSS behavior leads to a determination of a simple equivalent circuit based on series LC components branches which are not all the time possible for all FSS structures. To overcome this drawback, equivalent FSS structures based on non-coupled parallel metallic strips are introduced which necessitate the presentation of an approximate approach determining the real or the physical length L_{r}, Fig. 1, of the metallic strip FSS to end with an FSS having the desired resonant frequency. First, the curve describing the relationship between the resonant frequency and the actual length of the strip L_{r} is plotted in Fig. 13, by recording the resonant frequency calculated by the WCIP method for different strip lengths L_{r} varying from 5.25 mm until 15.75 mm with a step equal to 0.25 mm, so that to cover the interval of L_{r} values possible by the standard metallic strip FSS being W=1 mm, the substrate dielectric constant ε_{r} = 4.4, the substrate thickness h = 1 mm, the dimensions of the basic unit cell are a × b = 20 mm × 20 mm as shown in Fig. 11.

A comparative study is made to estimate the actual strip's length L_{r} having a specific resonant frequency and this by determining the curve that describes the relationship between the resonant frequency and the actual length as shown in Fig. 13. The blue circles concern the resonant frequency obtained by the WCIP method for a given strip's length L_{r}. The red curve is obtained by an interpolation process of the WCIP results.

From Fig. 13 the length L_{r} can be determined for any value of the desired frequency varying from 6.29 to 12.60 GHz. To validate the approximate proposed approach, the measured resonant frequencies of three proposed FSS are used.

The first structure presents the unit cell of the notched rectangular metallic ring FSS shown in Fig. 3, where the resonant frequencies given by the system of measurement and considered as desired resonant frequencies for the presented approach are 8.4 GHz and 10.1 GHz when the incident wave is *x*-polarized and 7.8 GHz when the incident wave is *y*-polarized. An FSS structure equivalent to the structure of Fig. 3 and based on two non-coupled metallic strips according to *x*-polarization with lengths L* _{1x}* and

*L*and a single metallic strip according to y with length

_{2x},*L*is shown in Fig. 14.a.

_{1y}The resonant frequencies given by the measurements and corresponding to the structure of Fig. 3 are replaced in Fig.13 as desired resonant frequencies giving the real lengths of the strips as *L _{1x}*= 10.98 mm,

*L*= 8.41 mm,

_{2x}*L*= 7.8 mm. The basic FSS unit cell has dimensions of

_{1y}*a*=20 mm and

*b*=20 mm, substrate dielectric constant of

*ε*=4.4 and a thickness of

_{r}*h*=1 mm.

The resonant frequencies obtained by the WCIP method when characterizing the equivalent FSS resulting from the presented synthesis approach are 8.2 GHz and 10 GHz when the incident wave is *x*-polarized and 7.8 GHz when the incident wave is *y*-polarized as shown in Fig. 14.b. The relative error between the measured results of the notched quasi-square FSS of Fig. 1 and the results of the proposed equivalent FSS structure of Fig. 14.a is 2.38% and 0.99% respectively in *x*-polarization and 0 % in *y*-polarization.

The same approach's steps are repeated to find the equivalent FSS structures of the open notched quasi-square metallic ring FSS proposed in [^{9}] and shown in Fig. 15. The manufactured FSS are printed on a fiber glass (FR-4) substrate with dielectric constant ε_{r}=4.4 and a thickness of h=1 mm [^{9}].

Table I summarizes the different obtained parallel metallic strips lengths of the equivalent FSS structure corresponding to the measured resonant frequencies of the open notched quasi-square metallic ring FSS for Lh=9.5 mm [^{9}].

Measured resonant frequency (GHz) of the open notched quasi square metallic ring FSS | real metallic strip length Lr(mm) | Equivalent metallic strips FSS resonant frequency (GHz) | Error % | |
---|---|---|---|---|

x-polarized source |
fr_{1x}=9.1 |
L_{1x}=9.6 |
fr_{1x}=9 |
1.10 |

fr_{2x}=9.9 |
L_{2x}=8.67 |
fr_{2x}=10.2 |
3 | |

fr_{3x}=11.7 |
L_{3x}=6.49 |
fr_{3x}= 11.6 |
0.86 | |

y-polarized source |
fr_{1y}=6.7 |
L_{1y}= 14.76 |
fr_{1y}=6.6 |
1.49 |

The resulting equivalent FSS structure to the structure presented in Fig. 15 is based on three uncoupled parallel metallic strips, according to *x*-polanzation with lengths *L _{1x}*,

*L*and

_{2x}*L*and a single metallic strip according to y polarization as shown in Fig. 16.a.

_{3x}When L_{h}=11.5 mm, the parallel metallic strips FSS structure equivalent to the open notched quasisquare metallic ring FSS of Fig. 15 is shown in Fig. 17.a. This structure is based on two metallic strips according to x-polarization with lengths *L _{1x}* and

*L*and two metallic strips according to

_{2x}*y-*polarization with length

*L*and

_{1y}*L*.

_{2y}
Table II summarizes the different obtained results for the strips lengths corresponding to the measured frequencies of the open notched quasi-square metallic ring FSS show in Fig. 15 for L_{h}=11.5mm.

Measured resonant frequency (GHz) of the open notched quasi square metallic ring FSS | real metallic strip length L_{r} (mm) |
Equivalent metallic strips FSS resonant frequency (GHz) | Error % | |
---|---|---|---|---|

x-polarized source |
fr_{1x}=8.8 |
L_{1x}=10.28 |
fr_{1xe}=8.8 |
0 |

fr_{2x}=11.7 |
L_{2x}=6.49 |
fr_{2xe}=11.6 |
0.86 | |

y-polarized source |
fr_{1y}=6.5 |
L_{1y}= 15.3 |
fr_{1ye}=6.4 |
1.54 |

fr_{2y}=8.8 |
L_{2y}=10.28 |
fr_{1x}=8.8 |
0 |

The transmission coefficients of the equivalent FSS structure in addition to measured ones [^{9}] of the open quasi-square metallic ring FSS for the *x* and *y* polarized excitations are given in Figs.16.b and 17.b for L_{h}=9.5 mm and L_{h}=11.5 mm respectively. The equivalent FSS structures results show a good agreement.

VII. the L-notched rectangular metallic ring FSS abacuses

When the applications are restricted to a specific FSS pattern as that shown in Fig. 1, abacus can provide rapid precise results without a need to use neither software nor experimental measurements. In this part abacuses are proposed and used to extract the resonant frequencies of the L- notched rectangular metallic ring FSS structure but without doing any simulation.

So according to the parametric study of the FSS of Fig. 2, the resonance frequencies fr_{1x}, fr_{3x} and fr_{1y} depends on the length L_{h} while fr_{2x} and fr_{2y} are varied according to L.

Figs.18 and 19 show the effect of increasing the lengths L_{h} and L on L-notched rectangular metallic ring FSS resonant frequencies respectively.

Figs. 18 and 19 can be used as abacus for the choice of the dimensions of the L- notched rectangular metallic ring FSS of Fig. 2 providing desired resonant frequencies. According to Fig.18, the values of the resonant frequencies fr_{1x}, fr_{2x}, fr_{3x}, fr_{1y} and fr_{2y} for a fixed value of the length L_{h}=4.8 mm are respectively: 6.5 GHz, 9 GHz, 11.3 GHz, 6 GHz et 8.2 GHz.

The confirmation of this abacus approach results is done by simulating the resulting L- notched rectangular metallic ring FSS of Fig. 2 with L_{h}=4.8 mm using COMSOL Multiphysics. The obtained results are illustrated in Fig. 20 for both the *x* and *y* polarizations of the normally incident plane wave source. The simulated resonant frequencies fr_{1x}, fr_{2x}, fr_{3x}, fr_{1y} and fr_{2y} are respectively 6.5 GHz, 8.8 GHz, 11.2 GHz, 6 GHz and 8.2 GHz with errors 0%, 2.22 GHz, 0.89%, 0% and 0%.

For L fixed at 1.7 mm with L_{h}=4.5 mm and L_{v}=4 mm, Fig. 19 is used to extract the corresponding resonant frequencies fr_{1x}, fr2x, fr3x, fr_{1y} and fr_{2y}. They are respectively 6.6 GHz, 9.2 GHz, 11.4 GHz, 6 GHz and 8.6 GHz. Then the FSS structure of Fig. 2 is simulated with COMSOL Multiphysics for the dimensions L=1.7 mm, L_{h}=4.5 mm and L_{v}=4 mm and the results of the scattering coefficients are shown in Fig. 21. The simulated resonant frequencies fr_{1x}, fr_{2x}, fr_{3x}, fr_{1y} and fr_{2y} are respectively 6.6 GHz, 9 GHz 11.4 GHz 6.1 GHz and 8.6 GHz with errors of 0%, 2.22%, 0%, 1.63% and 0%.

VIII. CONCLUSION

In this paper, the L- notched rectangular metallic ring FSS is analyzed using WCIP method. The proposed FSS presents a frequency behavior that may be suitable in the design of multiband structures where the FSS can be used simultaneously for dual and three-band applications when the two perpendicular polarizations are considered. One FSS has been fabricated and characterized. The FSS resonant frequencies can be adjusted by changing the dimensions of the FSS determined by the parametric study. To determine the approximate equations of the different FSS resonant frequencies an approximate procedure is presented. In complex pattern FSS such as the open notched quasisquare metallic ring FSS the determination of an equivalent circuit based on resonant LC circuit offering a complete FSS behavior comprehension is quite difficult. Thus the approximate procedure is extended to determine the complex patterns FSS equivalent structures based on non-coupled parallel metallic strips that can replace the complex pattern FSS for given resonant frequencies. When the applications are limited to a specific FSS such as L- notched rectangular metallic ring FSS due to its availability in terms of cost and manufacturing, abacuses can be used to save time without a need to software. Two abacuses are plot and verified.