Comparative Analysis of Techniques for Source Radiation in Cylindrical EBG with and without Periodic Discontinuities

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1 1398 Progress In Electromagnetics Research Symposium Abstracts, St Petersburg, Russia, May 2017 Comparative Analysis of Techniques for Source Radiation in Cylindrical EBG with and without Periodic Discontinuities Guga Burduli 1, Vakhtang Jandieri 2, Kiyotoshi Yasumoto 3, and Daniel Erni 2 1 School of Electrical and Computer Engineering, Free University of Tbilisi D. Agmashenebeli ave., 240, Tbilisi 0159, Republic of Georgia 2 General and Theoretical Electrical Engineering (ATE) Faculty of Engineering, University of Duisburg-Essen and CENIDE Center for Nanointegration Duisburg-Essen, Duisburg D-47048, Germany 3 Faculty of Information Science and Electrical Eng., Kyushu University, Fukuoka , Japan Abstract We present a semi-analytical approach for two-dimensional electromagnetic radiation in cylindrical EBGs composed of cylindrical arrays of circular rods periodically distributed along concentrically and eccentrically layered circular rings [1, 2]. Cylindrically periodic and EBG structures have received a growing attention because of their potential applications to photonic crystal fibers, directive antennas or beam-switching antennas. The rods can be dielectrics, perfect conductors, air-holes or metals. The method uses the T -matrix of a circular rod in isolation, the reflection and transmission matrices of a cylindrical array based on the cylindrical harmonics expansion, and the generalized reflection and transmission matrices for a layered structure. The formulation is rigorous. The proposed approach introduces a cylindrical layer model to the array, extracts the reflection and transmission matrices of a cylindrical periodic layer, and then obtains the characteristics of the whole layered structure by using a recursive formula. The recursive formula is based on a simple matrix multiplication, which guarantees a very short computation time for arbitrary number of layers. Moreover, it should be noted that the reflection and transmission matrices for the scattered field are expressed in terms of a block circulant matrix characterizing the periodic arrangement of the rods. Their inverse matrices can be calculated using the eigenvalues and eigenvectors of the circulant matrix, which simplifies the calculation procedure. Based on the proposed formulation we analyze the radiation characteristics of a localized source located inside the cylindrical EBG structure. Various configurations of the cylindrical bandgap structures are studied and the rods are taken as perfect conductors. We give a deep physical insight into the relations between the transmission spectra of the cylindrical harmonic waves and the radiation pattern of the excited source. The source is a line source or dipole source. Discussions about the relation between the resonance and stopband characteristics of the transmission spectra and the radiation patterns of the localized source are given from the viewpoint of the flexible design and control of the multibeam and directive beam forming characteristics. In order to further increase the radiation characteristics of the localized source, we introduce the defects in the cylindrical periodic structure. The defects are introduced by removing the particular circular rods from each circular ring. The presence of the discontinuities requires a modification of the proposed formulation. The inverse matrices cannot be defined using the eigenvalues and eigenvectors of the circulant matrix, but must be numerically calculated. The effects of the defects profile and location of the source on the far-field radiation are discussed. Optimization algorithms of 2D radiation in cylindrical EBGs are considered. REFERENCES 1. Jandieri, V., K. Yasumoto, and Y. Liu, Directivity of radiation of a dipole source coupled to cylindrical electromagnetic bandgap structures, Journal of the Optical Society of America B, Vol. 29, No. 9, , Jandieri, V. and K. Yasumoto, Electromagnetic scattering by layered cylindrical arrays of circular rods, IEEE Transactions on Antennas and Propagation, Vol. 59, No. 6, , 2011.

2 Comparative Analysis of Techniques for Source Radiation in Cylindrical EBG with and without Periodic Discontinuities Guga Burduli 1, Vakhtang Jandieri 2, Kiyotoshi Yasumoto 3 and Daniel Erni 2 1 School of Electrical and Computer Engineering, Free University of Tbilisi, Tbilisi, Republic of Georgia 2 General and Theoretical Electrical Engineering (ATE) and CENIDE Center Nanointegration Duisburg-Essen, Duisburg, Germany 3 Faculty of Information Science and Electrical Eng., Kyushu University, Fukuoka, Japan PIERS 17 Saint-Petersburg, Russia

3 Photonic Crystals 1D 2D 3D PBG (MIT) (JRCAT) ( Sandia) 1μm 1μm (NECI)

4 3D Photonic Crystals Diamond Si Woodpile (Sandia) BANDGAP BANDGAP

5 A periodic array of circular cylinders is typical of a discrete periodic structure. When the periodic arrays are multilayered, it constitutes a electromagnetic bandgap (EBG) structure. Various analytical and numerical techniques have been developed to formulate the electromagnetic interactions with EBG structures. However, the previous pertinent efforts have been mainly concerned with the planar EBG structures. Recently, cylindrical structures, which are formed by circular rods periodically distributed on layered concentric or eccentric circular rings, have received a growing attention because of their use in novel modern devices. V. Jandieri, K. Yasumoto and Y. Liu, Journal of the Optical Society of America B, vol.29, no.9, pp , V. Jandieri and K. Yasumoto, IEEE Transaction on Antennas and Propagation, vol.59, no.6, pp , V. Jandieri, K. Yasumoto and Young-Ki Cho, Progress in Electromagnetics Research (PIER), vol.121, pp , 2011.

6 Photonic Crystal Fiber 1. High refractive index core features: Greater freedom in fiber design. Single mode guidance over wide wavelength regions. 2. Low refractive index core (air) features: Core is air and nonlinear effect is small. No need for ultra low-loss material.

7 Antenna Application After H. Boutayeb and T. Denidni, IEEE TAP, vol.55, no.11, pp , 2007

8 We have developed a semi-analytical method, which can be applied to the guiding, scattering and radiation problems in cylindrical periodic (or EBG) structures. The approach uses: T-matrix of a circular rod; Translation matrices for cylindrical waves; Reflection and transmission matrices based on cylindrical waves for each layer; Generalized reflection and transmission matrices for multilayered structure.

9 Structure of layered cylindrical arrays (a) (b) Fig.1. (a) Cross-sectional view of N-layered cylindrical arrays formed by M ν circular rods distributed on each of N-concentric circular rings with radii Rν ( ν = 1, 2,, N) and (b) Schematic view of ( ν ) scattering process through the ν -th layer of the cylindrical array where b denotes the amplitude ( ν ) vector of incoming standing cylindrical waves and c is that of the outgoing cylindrical waves.

10 ( ) For the incidence of incoming wave with b ν Scattered Field: ψ s( ν ) M ν = Ψ j= 1 ( ν) T ( ν) j a j Ψ ( ν) (1) j ( ν) ( ν) j = m ( κρ ν, j ) ], a j = jm, imϕν, [ H e [ a ] Boundary condition is applied on the i -th circular rod (i=1,2 M). Scattered field components of the local coordinates ν ν ( ρj, ϕj)( j i) are transformed into the local coordinate for i -th rod (Graf Theorem).

11 The linear system of equations for unknown scattering amplitude is obtained: a 1 ( ν) ( ν) ( ν ) ( ν) = Γ Τ+ b ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) I T1 K12 T1 K13 T1 K 1Mν ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) T2 K21 I T2 K23 T2 K2M ν ( ν ) Γ ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) = T3 K31 T3 K32 I T3 K3M ν ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) TM K ν Mν1 TM K ν Mν 2 TM K ν Mν3 I M ν M ν M matrix ν Number of scatterers ( ν ) (1) m i( m+ n)( π+ θ, ij )/2 ij, mn = m n (, ij )( 1) e ν ν K H kd d = 2R 1 cos θ, θ = θ θ ν, ij ν ν, ij ν, ij ν, i ν, j

12 Solution of the Linear System In case of pure periodic cylindrical structure: θ = θ = 2 π / M ν, ij ν ν d = d = 2R 1 cosθ ν, ij ν ν ν ( ν ) Γ - Block Circulant Matrix Solution is given using eigenvalues and eigenvectors of the circulant matrix.

13 Reflection Matrix M ( ν) ( ν) νν, 1 = j +, j j= 1 R β T Reflection Matrix β [ im j π M ] m n κ ν ( ν ) ( 1)2 / j = J ( d )e ( translation matrix) M ( ν) ( ν) ν 1, ν = + j +, j j= 1 F I η T Transmission Matrix [ im j π M H κd ] ν ( ν ) (1) ( 1)2 / η j = m n( )e ( translation matrix)

14 Incident Field Ψ c T ( ν 1) Boundary condition is applied on the i -th circular rod (i=1,2 M). Scattered field components of the local coordinates ν ν ( ρj, ϕj)( j i) are transformed into the local coordinate for i -th rod (Graf Theorem). Schematic view of Scattering process

15 Reflection Matrix M ( ν) ( ν) ν 1, ν = j, j j= 1 R η T Reflection Matrix [ im j π M H κd ] ν ( ν ) (1) ( 1)2 / η j = m n( )e ( translation matrix) M ( ν) ( ν) νν, 1 = + j, j j= 1 F I β T Transmission Matrix β [ im j π M ] m n κ ν ( ν ) ( 1)2 / j = J ( d )e ( translation matrix)

16 Generalized Reflection and Transmission Matrices R = R + F R Γ ν 1, ν ν 1, ν ν 1, ν ν, ν+ 1 ν, ν 1 Γ = ( I R R F 1 νν, 1 νν, 1 νν, 1 ) νν, 1 y ( N ) F R = 0 R NN, + 1 ν : 1, ν Generalized reflection matrix viewed from region ( ν 1) to the whole outer regions from ( ν ) to ( N ) R N R 2 #2 R #1 1 O (0) (1) (2) ( ε 0, µ 0) ( N ) # N x ( N ) F = Γ NN, 1ΓN 1, N 2 Γ2,1Γ1,0

17 Some Physical Explanation About Difference Between Planar and Cylindrical Configurations 1. For isotropic scatterer the reflection matrix of single planar layer always satisfies the following equality: (Reciprocity Relation). R = R νν, 1 ν 1, ν 1. Since the reflection matrices are expressed in terms of the cylindrical waves, even for isotropic scatterer: R R νν, 1 ν 1, ν 2. Each Floquet mode is excited only in some particular frequency range. 2. All orders of cylindrical harmonic waves are excited. However, due to the resonances and stopbands nature some of them are enhanced, whereas others are strongly suppressed.

18 Our formulation is general and straightforward. Various configurations of Cylindrical EBG with different types and locations of the excitation sources can be rigorously studied. We take into account all cylindrical modes and the interactions between the modes. Method is not time-consuming. Recursive relation for the layered structure is based on a simple matrix multiplication. V. Jandieri, K. Yasumoto and Y. Liu, Journal of the Optical Society of America B, vol.29, no.9, pp , V. Jandieri and K. Yasumoto, IEEE Transaction on Antennas and Propagation, vol.59, no.6, pp , V. Jandieri, K. Yasumoto and Young-Ki Cho, Progress in Electromagnetics Research (PIER), vol.121, pp , 2011.

19 (a) (b) (c) Cross-sectional view of three different configurations of three-layered cylindrical EBG structure of metallic circular rods with periodically located on the concentric circular layers. (a) twelve circular rods are symmetrically distributed along the 1 st layer, 2 nd layer and 3 rd layer and the source is located at the global origin; (b) one additional metallic circular rod is additionally placed at the origin of cylindrical structure (a); (c) 12 circular rods are symmetrically distributed along the 1 st layer, 24 circular rods along the 2 nd layer and 36 circular rods along the 3 rd layer. The source is located at the global origin.

20 Cylindrical EBG Structure Configuration 1 (a)

21 1-layer Transmission Spectra F m,0 1,0 ( m = 0, ± 12 ) Radiation Pattern in H-plane (blue line) and E-plane (red line)

22 3-layers F Transmission Spectra m,0 3,0 ( m = 0, ± 12, ± 24 ) Radiation Pattern in H-plane (blue line) and E-plane (red line)

23 Cylindrical EBG Structure Configuration 2 (b)

24 R1 ω / c= 0.37 R1 ω / c= Radiation Pattern in H-plane (blue line) and E-plane (red line)

25 Cylindrical EBG Structure Configuration 3 (c)

26 F Transmission Spectra m,0 3,0 ( m = 0, ± 12, ± 24, ± 36 )

27 R1 ω / c= (a) (b) R1 ω / c= 1.34 (c) (d) Radiation Pattern in H-plane (blue line) and E-plane (red line)

28 Source Source Cross-sectional view of two different configurations of the three-layered cylindrical EBG structure with defects composed of perfect conductor circular rods periodically located on the concentric circular rings. Radii of the circular rods are the same along the different rings and equal to. The defects are introduced by removing: (a) two, three and four circular rods from the 1st, 2nd and 3rd circular rings; (b) three, five and seven circular rods from the 1st, 2nd and 3rd circular rings. The hollow circles indicate the missing circular rods. 1. Improvement of Radiation Characteristics Comparison to Cylindrical EBG without Defects. 2. Practical Application in Design of Directive EBG Based Antennas.

29 ( ) For the incidence of incoming wave with b ν Scattered Field: ψ s( ν ) M ν = Ψ j= 1 ( ν) T ( ν) j a j Ψ ( ν) (1) j ( ν) ( ν) j = m ( κρ ν, j ) ], a j = jm, imϕν, [ H e [ a ] Boundary condition is applied on the i -th circular rod (i=1,2 M). Scattered field components of the local coordinates ν ν ( ρj, ϕj)( j i) are transformed into the local coordinate for i -th rod (Graf Theorem).

30 The linear system of equations for unknown scattering amplitude is obtained: a 1 ( ν) ( ν) ( ν ) ( ν) = Γ Τ+ b ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) I T1 K12 T1 K13 T1 K 1Mν ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) T2 K21 I T2 K23 T2 K2M ν ( ν ) Γ ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) = T3 K31 T3 K32 I T3 K3M ν ( ν) ( ν) ( ν) ( ν) ( ν) ( ν) TM K ν Mν1 TM K ν Mν 2 TM K ν Mν3 I M ν M ν M matrix ν Number of scatterers ( ν ) (1) m i( m+ n)( π+ θ, ij )/2 ij, mn = m n (, ij )( 1) e ν ν K H kd d = 2R 1 cos θ, θ = θ θ ν, ij ν ν, ij ν, ij ν, i ν, j

31 Solution of the Linear System 2. When the periodic cylindrical structure does introduce defects: a 1 ( ν) ( ν) ( ν ) ( ν) = Γ Τ+ b Solution is given using the direct numerical calculations of the inverse matrix. V. Jandieri, K. Yasumoto and Y. Liu, Journal of the Optical Society of America B, vol.29, no.9, pp , V. Jandieri and K. Yasumoto, IEEE Transaction on Antennas and Propagation, vol.59, no.6, pp , V. Jandieri, K. Yasumoto and Young-Ki Cho, Progress in Electromagnetics Research (PIER), vol.121, pp , 2011.

32 Numerical Analysis and Discussions Source R1/ λ 0 = 0.20 R1/ λ 0 = 0.35 R1/ λ 0 = 0.50 d / R = 0.0 s 1 ϕ θ H-plane E-plane

33 Numerical Analysis and Discussions Source R1/ λ 0 = 0.20 R1/ λ 0 = 0.35 R1/ λ 0 = 0.50 d / R = 0.0 s 1 ϕ θ H-plane E-plane

34 Optimization Smajic, Hafner, Erni, Optimization of photonic crystal structures, JOSA A.

35 Optimization Application of the developed optimization method to the cylindrical EBGs with defects to further increase the radiation characteristics. How many rods and which rods should be removed in order to improve the radiation? Accurate and fast calculation of the reflection and transmission spectra developed in this work allows us to efficiently apply the optimization method to cylindrical EBGs. Source

36 Thank You for your kind Attention.