Comparative Study of Photonic Defective Comb-Like Structure by the Green Function and the Transfer Matrix Method

. In this paper, we investigate the propagation of electromagnetic waves in periodic comblike waveguides structure made of dangling side branches (resonators) and finite waveguides. We introduce two defective resonators inserted in two di ff erent sites inside the perfect periodic system. This kind of system has been studied theoretically using Green’s function method (GFM), the main objective of this work is to make a comparative study between the results found in the literature and the results found by the Transfer Matrix Method (TMM) for the first time in such photonic defective comb-like system. Noted that the calculations made by the transfer matrix method (based on Maxwell’s equation and the continuity conditions) are simpler compared to the calculations when applying the Green function method. The results founding in this paper either at the level of the transmission and reflection rates are in agreement with the results found by the Green function method, which gives an idea of the e ff ectiveness of this simple method and their agreement with the experience.


Introduction
The propagation of electromagnetic waves has attracted increasing attention in recent years by their interaction with materials, this interaction is very important in the photonic field which is composed of the periodicity of materials.The main property of photonics is the ability to manipulate the flow of electromagnetic waves in the structure.The important property of electromagnetic wave propagation in periodic structures is the presence of photonic band gaps and pass bands.The pass band corresponds to the frequency regions the electromagnetic wave can propagate through the periodic system.A band gap corresponds to a range of frequencies where the propagation of electromagnetic waves is prohibited inside the periodic system.This periodical system has applications in telecommunication fiber [1], sensors [2], waveguides, and cavities [3].The comb-like waveguides structure (CWGs) is composed by the periodicity of segment and one grafted resonator [4].This periodic comblike waveguide structure presents photonic band gaps separated by very narrow pass bands.The insertion of defects in this CWGs periodic structure leads to obtain defect modes in the band gaps, these defect modes are very sensitive to the geometrical and material parameters of the system [5].This structure can use as an accordable photonic filter [6].Using Green's Function Method (GFM), Y. Ben-Ali et al. studied the propagation of electromagnetic waves in defective comb-like waveguides structure for double frequencies filters [7].Dobrzynski et al. investigated theoretically (using GFM) and experimentally the existence of photonic band gaps separated by passbands in CWGs structure in the frequency range up to 500MHz [8].Other teams have studied the propagation of the photonic, acoustic, and electronic waves using the Green function and transfer matrix methods [9][10][11][12], and found the presence of defect modes in the photonic, acoustic, or electronic band gaps.
Our objective is the comparison between two methods (Green's function and the transfer matrix), and we study the propagation of electromagnetic waves through periodic CWGs structure containing two defective resonators located at two different sites.The boundary conditions at the ends of the resonators are of type H = 0 (the magnetic field is equal to zero at the end of the resonators).

Dispersion Relation through comb-like Waveguides
In the configuration described here, the structure is formed by segments of length d 1 in the x-direction (Fig. 1a).Every connection point, termed as "site," is denoted by an integer n.At each of these sites, a single resonator with a length d 2 is attached.The location in a cell, delimited between sites n and n + 1, can be defined by the coordinate pair (n, x).Here, x is a local coordinate in the limits 0 ≤ x ≤ d 1 , and d 1 is the period of the structure.To compute the Green's Function for this infinite structure, it's essential to determine the Green's Function for each constituent segment of length d 1 as well as for the resonator of length d 2 .The matrix inverse of the Green function for the segment within the interface space M 1 = {0; d 1 } is presented as: depends on the choice of boundary conditions at the end of the resonator.For the case where H = 0, the transfer matrix of the resonator of de length d 2 is given by: With: In the interface space of the infinite comb-like waveguides structure, the inverse of the Green's Function matrix ) is an infinite tridiagonal matrix formed by the superposition of the elements . The matrix can be expressed as: With: C 2 Using Bloch's theorem, and after simple calculations, we deduce the dispersion relation as: Given that the system exhibits periodicity in the x direction, we can apply the Fourier transform to the infinite tridiagonal matrix within a segment of length d 1 .This process yields ← → g −1 [(k; M; M)], which is expressed as follows: We inverse the Eq.(5a) and finally we find: Or: The bulk bands characterizing the comb-like waveguides structure are determined by poles analysis of the Green's function.This relation is represented as follows: The inverse Fourier transform of ← → g [(k; M; M)] is given by: In this equation, the integers n and n ′ denote the sites on the infinite com-like waveguide, with their values ranging from −∞ ≤ n, n ′ ≤ +∞.The parameter t defined by: t = e jkd 1 (10)

Transmission rate
In this part, we focus on calculating the transmission rate of finite comb-like waveguides system containing the defects at the resonators.As shown in Figure 1b, the structure is constructed as follows: We start with an infinite periodic system as shown in Figure 1a, and extract a finite segment containing N uniformly spaced resonator sites.This finite structure is then bordered by two semi-infinite segments at its extremities.Each segment of this finite structure is made up of a resonator ("medium 2") of the length d 2 , and these resonators are periodically spaced at a spacing of d 1 .For simplicity, we've assumed that both the semiinfinite segments and the resonator share identical physical properties.The introduction of defects is realized by altering the resonators at sites J and J ′ .Specifically, the first defect is introduced by replacing the resonator at site J with one of length d 02 , which differs from d 2 .Similarly, the second defect involves substituting the resonator at site J ′ with a new one of length d 03 , distinct from d 2 .This modification in the system leads to a set of disturbed states, defined as ), manifests as a 6 × 6 matrix.This matrix operates within the interface domain, encompassing sites With: E3S Web of Conferences 469, 00044 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900044 By understanding the elements of the response function situated in the interface space of the infinite comb-like waveguide structure, denoted as , along with the elements of the cleavage operator ← → V (M S M S ), we can obtain the elements of the response function for the finite structure.Accordingly, the interface response operator, represented by ← → A (M S M S ), is as follows The function ← → g (M s M s ) is calculated from Eq. (7a) The parameter t is given in Eq. (7b).The operator ← → ∆ (M S M S ) is given by the following relation: After calculating the operator ← → ∆ (M s M s ), we write this operator in the interface space M 0 = {0, J, J ′ , N}.
The Green's function ← → d (M 0 M 0 ) for an infinite comb-like waveguides structure is defined in the interface space M 0 by the following equation: Finally, the Green's function of finite photonic comb-like waveguides From where: The transmission rate through the structure is given by the following relation: The symbol 's' represents the interface connecting the first input waveguide (Medium 1) to the comb-like waveguide.Conversely, 'e' denotes the interface where the second input waveguide (Medium 1) meets the comb-like waveguide.

Transfer Matrix Method
The transfer matrix method is a useful technique to calculate the dispersion relation, the transmission and reflection rates.The expressions of the electric fields for a cell constituted by a waveguide of length d 1 and a grafted resonator of length d 2 situated between two semiinfinite segments is given by [12]: for: The first term of the electric field represents the transmitted wave and the second term indicates the reflected wave, α i is the wave number in medium i (i=0, 1, 2, s), and the coefficients A p , B p , C p and D p are constants (p = 0, 1, s).
The transfer matrix of segment and resonator located between two semi-infinite segments, is written as follows: With: M cell is the matrix of a unit cell.The set of 2 × 2 complex matrix of the form M cell satisfying the constraint det (M cell ) = 1.
The dispersion relation of the infinite system is given by: With k is the Bloch vector.Moreover, the system considered is formed of N cells (see Fig. 1), the matrix corresponding to this periodic structure can be obtained by calculating the product of each in matrix: The unit matrix of order N can be simplified by the following identity matrix: The reflection and transmission rates of the perfect structure are given by: New, we are interested in creating two distinct geometric defects located at sites J and J ′ in the periodic comb waveguide structure.
The total transfer matrix of our proposed system is given by: M D (J) and M D (J ′ ) represent the defect transfer matrix at two different sites J and J ′ .Here, N represents the total number of cells in the structure.
After a detailed calculation, we find the transfer matrix in the form: The elements of the matrix M T of the periodic comb-like waveguides structure containing two defects are calculated numerically.We write the reflection and transmission rates of this defective structure as follows:

Results and Discussions
In this section, we discuss our numerical results by plotting the band structure and the transmission rate.For our structure, the materials used of the segments and resonators are to be: assumed homogeneous, nonmagnetic (µ 1 = µ 2 = 1), and of identical relative permittivity, i.e.
which is a dimensionless quantity, with c represents the speed of electromagnetic waves in a vacuum and ω represents the pulsation.The frequency domain depends strongly on the size of lengths.Furthermore, we utilized the Fortran programming language to visualize our results.

Perfect Comb-Like Waveguides Structure.
In this study, the propagation of electromagnetic waves through a perfectly periodic CWGs structure is explored using the transfer matrix method.Fig. 2a represents the evolution of the transmission rate as a function of the reduced frequency with N = 8, d 1 = 1d and E3S Web of Conferences 469, 00044 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900044d 2 = 0.5d.We observe the existence of two pass band separated by a large band gap in which the electromagnetic waves cannot propagate (evanescent waves), these bands are caused by the periodicity of the system as well as by the resonance mode of the grafted resonator.Our results are similar to those found by Y. Ben-Ali et al. when studied defect modes in onedimensional photonic comb-like waveguides structure using the Green function method (Fig. 2b represents the result found by Y. Ben Ali et al. [13].The electromagnetic waves can be controlled for a wide range of applications, including filtering and multiplexing devices.Furthermore, this perfect periodic system functions as a low pass filter.

Effect of the Defect Lengths
In this analysis, we explore the behavior of electromagnetic waves propagating through a periodic comb-like waveguide structure.This structure encompasses two defective resonators located at sites J and J ′ .Fig. 3a  Hence, the location of the defect modes in the band gap is influenced by the variation of the defect lengths of both defective resonators located in two different sites.Specifically, the variation of two defect lengths can result a shift of the defect mode frequencies in the band gap.This shift can be either toward the limit of the band gap or toward the center of the band gap, depending on the direction of the defect length variation.In addition, the shape of the band gap can also be changed by the variation of defect lengths.We note that the location of the defects depend on the resonant mode frequencies and the lengths of defective resonators.This result is similar to those found by Y. BenAli et al. when studied periodic photonic comb-like waveguides system contains two defects using the Green's function method (Fig. 3b) [13].

Effect of the Defect Position J ′
In this paragraph, we examine the variation of the transmission rate of defect modes in the band gap as a function of the reduced frequency (Fig. 4a).According the Fig. 4a d 1 = 1d, d 2 = 0.5d and N = 8.For the case J ′ = 4, we observe the appearance of four defect modes in the band gap with low transmission rates.For the case J ′ = 5, the defect modes observed in the previous case, approach each other while their transmission rates increase.For the case J ′ = 6, we notice that the four defect modes overlap each other with high transmission rates and becoming two defect modes.We see that the optimal positions of defects are located at the sites J = 3 and J ′ = 6 because they represent high transmission rates of defect modes.This result is similar to those found by Y. Ben-Ali et al. when studied multi-channel filters in 1D defective comb-like waveguides structure using the Green function method (Fig. 4b) [13].
We conclude that when we start to move the defect position J ′ away from the defect position J, the defect modes start to approach each other and they superpose when the two defects are very far from each other.The superposition phenomenon is consistent with the results of researchers when studied one-dimensional defective multilayer system in the field of phononic [14] or photonic [15].

Proper Modes of the Defect Resonator
In the last section, we examine the variation of the reduced frequency of the defect modes as a function of the defective resonators of lengths d 02 and d 03 , with N = 8, J = 3 and J ′ = 6 (Fig. 5 a).The gray areas, denoting pass bands, correspond to values of reduced frequency when |cos (kd 1 )| ≤ 1, while the white area represents the band gap.The black branches, representing defect modes, correspond to the transmission maximum of the periodic defective system.When a geometrical defect is introduced in a perfect photonic system, the properties of the band gap are modified, which generally results in the appearance of one or more defect modes in the band gap.This band structure shows that the defect modes emerge from the second pass band, then move to a lower frequency by increasing the value of d 02 = d 03 and finally merge into the first pass band.The behavior observed in Fig. 5a is agree with several works when plotting the band structure [16][17].Moreover, we observe that the two defect modes move closer together in the photonic band gap.For specific values of d 02 = d 03 , the superposition phenomenon of two defect modes can obtain and generate the defect modes.In practical applications, it is commonly desirable to design narrow selective filters capable to modulate photonic defect modes across a wider frequency range, especially in cases where the defect modes exhibit higher transmission rates and quality factors.
The result found in Fig. 5a is similar to the result obtained by Y. Ben-Ali et al. when studied the propagation of electromagnetic waves in one-dimensional defective photonic comblike waveguides structure using the Green function method (Fig. 5b) [18].

Summary
This present work is based on the Transfer Matrix Method (TMM) to calculate both the dispersion relation and the transmission rate of our proposed defective comb-like waveguides structure containing two defective resonators located at the sites J and J ′ .The obtained results are similar to that found by the Green function method.Our proposed TMM is computationally easy and programmatically fast, but the Green function method is very E3S Web of Conferences 469, 00044 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900044efficient for studying the case of complex structures.Also, we have shown that the creation of two defective resonators lead to the appearance of four defect modes in the band gap, these defect modes depend on the positions and the lengths of defective resonators.Our comb-like waveguide structure is suitable for use in telecommunications applications and selective filters capable of modulating defect modes in a wider frequency range.

Figure 1 .
Figure 1.(a) One-dimensional photonic infinite comb-like waveguides structure.(b) Finite comb-like waveguides structure containing two defective resonators, the first defect of length d 02 is located at the site J, and the second defect of length d 03 is located at the site J ′ [13].

3 .
The segment length is d 1 = 1d and the resonator length is d 2 = 0.5d with d = d 1 represents the period of the structure.The number of cells is N = 8.The lengths of the two defective resonators are noted d 02 and d 03 , these defects are located at the sites J and J ′ .The reduced frequency is given by Ω

Figure 2 .
Figure 2. Transmission rate versus the reduced frequency of the perfect periodic CWGs structure.
shows the variation of transmission rate as a function of the reduced frequency by choosing different lengths of defective resonators, with J = 3, J ′ = 6, d 1 = 1d, d 2 = 0.5d and N = 8.In the first case, for d 02 = d 03 = 0.25d, we observe the existence of two defect modes in the band gap around the reduced frequency Ω = [1.15−1.3].In the second case, for d 02 = d 03 = 0.75d, we observe the existence of four defect modes in the band gap.In the third case, for d 02 = d 03 = 1d, there are four defect modes in the middle of the band gap with a high transmission rate.

Figure 3 .
Figure 3. Evolution of the transmission rate versus the reduced frequency Ω for different values of defective resonators of lengths d 02 and d 03 .

Figure 4 .
Figure 4. Variation of the transmission rate versus the reduced frequency Ω by choosing different values of the defect position J ′

Figure 5 .
Figure 5. Variation of the reduced frequency of the defect modes as a function of the defect lengths d 02 and d 03 .