Radio Science

Semianalytical formulation on the scattering of proximity equilibration cell closed ring photonic band gap structures

Authors

  • Yunhong Liu,

    1. Integrated Nanosystems Research Facility, Department of Electrical Engineering and Computer Science, Henry Samueli School of Engineering, University of California, Irvine, California, USA
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  • Nicolaos G. Alexopoulos

    1. Integrated Nanosystems Research Facility, Department of Electrical Engineering and Computer Science, Henry Samueli School of Engineering, University of California, Irvine, California, USA
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Errata

This article is corrected by:

  1. Errata: Correction to “Semianalytical formulation on the scattering of proximity equilibration cell closed ring photonic band gap structures” Volume 43, Issue 2, Article first published online: 4 April 2008

Abstract

[1] A novel semianalytical methodology is used to analyze a periodic array of printed metallic closed ring elements in a multilayered dielectric structure. This approach is unique in that it is the first methodology capable in modeling structures with resonant implants and interelement dimensions well beyond the effective medium theory. In addition, it yields computational efficiency by 2 orders of magnitude over standard computational methods in computing the scattering parameters for proximity equilibration cell (PEC) closed ring multilayered (electromagnetic band gap and photonic band gap (PBG)) structures. Moreover, it provides physical insight in the implementation of metallic implants for practical applications. This methodology satisfies the Kramers-Kronig relations and causality, and therefore it allows for the development of semianalytical expressions for the composite's wave impedance, index of refraction, as well as the permittivity and permeability parameters accounting for full dispersion. For general artificial multilayered structures (PBG metamaterials) with centrosymmetric scattering matrices, the composite may be replaced by an equivalent homogeneous dispersive magneto-dielectric material and may be used for the design of integrated circuits, filters, and antennas using standard methods. Otherwise, use of the scattering matrix approach to obtain the effective parameters is valid only for semi-infinite structures. The upper band edge is determined by the host material uniquely and the bandwidth is determined by the shunt susceptance for different PEC ring inclusions.

1. Introduction

[2] Early applications of one dimensional periodicity in substrate and substrate-superstrate geometries for microstrip elements produced very useful results for microstrip antenna gain enhancement, radiation pattern shaping, as well as antenna efficiency and bandwidth improvement [Alexopoulos and Jackson, 1984; Alexopoulos et al., 1985; Yang and Alexopoulos, 1987]. More recently, progress has been obtained with the introduction of periodic arrays of material blocks [Yang et al., 1997a; Yang, 1999] or metallic implants [Yang et al., 1997b; Sievenpiper et al., 1999; Kyriazidou, 1999; Contopanagos et al., 1999; Kyriazidou et al., 2000, 2001] within multilayered dielectric structures, these structures being referred to as photonic band gap (PBG) or electromagnetic band gap (EBG) materials. These materials have enabled some key and novel contributions such as the realization of high impedance surfaces or magnetic walls [Yang et al., 1997b; Sievenpiper et al., 1999] and a variety of other applications [Contopanagos et al., 1999; Kyriazidou et al., 2000, 2001].

[3] Numerical methods and commercial software have provided the backbone for the modeling and characterization of these complex geometries, such as the method of moments [Tsao and Mittra, 1987], finite elements [Zhang et al., 1999; Coccioli et al., 1997], the combined FEM/MOM approach [Gedney et al., 1992], the plane wave expansion techniques [Leung and Liu, 1990] and the mode-matching technique [Chen, 1970; Huang et al., 1994]. A distinct semianalytical methodology was developed however which not only provides a better insight on the electromagnetic characteristics of these structures but it also results in a 2 orders of magnitude faster computational methodology [Kyriazidou, 1999; Kyriazidou et al., 2000]. This is a significant approach for the characterization of the electromagnetic properties of general media with ordered metallic inclusions well beyond the use of effective medium theory. This method creates the fundamental analogies between natural and artificial (PBG) crystals through the development of the Lorentzian generator function p, henceforth referred to as the Kyriazidou-Cantopanagos-Alexopoulos (KCA) Lorentzian generator [Alexopoulos et al., 2004]. Furthermore, it satisfies the Kramers-Kronig relations and Causality and it provides therefore a correct mechanism for the derivation of the effective wave impedance, index of refraction, and permittivity and permeability of the equivalent homogeneous bulk materials. In fact, this method provided the first analytical means to prove that a purely dielectric-metal structure yields a composite with magnetic as well as dielectric properties [Contopanagos et al., 1999; Kyriazidou et al., 2000]. The first step of this semianalytical procedure is the derivation of the polarizability of the implant elements. This was first demonstrated for metallic disk implants [Contopanagos et al., 1999; Kyriazidou et al., 2000].

[4] In this paper we consider a generalization of this method to metallic ring elements and the time dependance et is adopted. Metallic ring implants provide a variety of flexibilities, including metallic disks as a limiting case, and moreover the possibility of a gap variable impedance load in each ring implant which will result in reconfigurable materials, filters etc [Alexopoulos et al., 2004] as well as in metamorphic structures [Liu and Alexopoulos, 2006; Kyriazidou et al., 2006, 2007]. The latter initiated the concept of artificial magnetic conductors [Alexopoulos et al., 1974a, 1974b; Alexopoulos and Katehi, 1979] or high impedance walls [Sievenpiper et al., 1999]. A negative index of refraction has been reported in an ordered variable impedance loaded PEC ring media [Liu and Alexopoulos, 2006]. We will discuss the validity of deriving the effective parameters of layered artificial materials from the reflection and transmission coefficients. We will start from a general case and a more universal conclusion will be established for symmetric and nonsymmetric structures. We will begin the derivation of the electric and magnetic polarizabilities for a narrow planar annular ring which requires knowledge of the current distribution on the ring. For a narrow annular ring, the current distribution is obtained from that of a wire ring with a wire radius being a fourth of the width of the annular ring [Balanis, 1982, and references therein]. With this background we derive closed form expressions for the electric and magnetic polarizabilities and the corresponding shunt susceptances for different incident wave polarizations. We also obtain the composite medium's effective parameters such as effective wave impedance, refractive index, permittivity and permeability. Furthermore, we discuss the general ring band gap characteristics and we provide comparisons to metallic disk implants. A comparison of this methodology to the use of a commercial code such as Ansoft Designer is provided. Further we show that the effective parameters of the metamaterial composite can be obtained from S11 and S21 for centrosymmetric structures. Otherwise, it is possible to derive effective parameters for noncentrosymmetric composites only if the structure is semi-infinite.

2. Retrieval of Effective Parameters From Transmission and Reflection Coefficients

[5] In this section, we focus on the validity of deriving the effective parameters of the PBG structure from transmission and reflection coefficients as shown in Figure 1. We consider a reciprocal artificial material, i.e., a composite with nonactive, unbiased host of implant constituents.

Figure 1.

Illustration of the relationship between an artificial medium and the equivalent effective medium.

2.1. Transmission and Reflection Coefficients and Effective Parameters

[6] The transmission and reflection coefficients for a planar homogeneous material can be expressed as

equation image

where Γ is the reflection coefficient from a semi-infinite structure, T = exp(−jkc cos θd) is the transmittance for the planar homogeneous material, k is the wave number inside the material, c is the thickness and θd is the angle of refraction. Using the transmission and reflection coefficients, we can define the effective impedance ze = Re(ze) + jIm(ze) and refractive index ne = Re(ne) − jIm(ne) from the bulk reflection coefficient and transmittance as

equation image

and

equation image

respectively. The retrieval of the bulk reflectance Γ and transmittance T from the reflection and transmission coefficients was described by Nicholson and Ross [1970] and Alexopoulos et al. [2007]. We must choose the proper branch for ln(T), which is difficult for the material whose thickness is comparable to or larger than the wavelength in vacuum.

2.2. Artificial Materials and Effective Homogeneous Media

[7] The electromagnetic scattering from an arbitrary number of N layers of planar artificial material with ordered metallic implants may be characterized by the transfer matrix which links the incident and scattered fields as

equation image

Here Rin, Lin and Rout represent the incident, reflected and transmitted wave, respectively. Lout is the incident wave from the opposite and it is zero (Lout = 0). U is the transfer matrix for a single layer with the property

equation image

Therefore the scattering matrix is defined as

equation image

[8] Following the procedure of Kyriazidou [1999] and Contopanagos et al. [1999], the transfer matrix equation image for an arbitrary number N of identical layers can be evaluated by using the Pauli spin matrix and it turns out to be

equation image

where equation image is the transfer matrix for a single layer and the definitions of z0, κ and ξ are [Kyriazidou, 1999; Contopanagos et al., 1999]

equation image

The transmission and reflection coefficients for the composite material are given by

equation image

with

equation image

The functions F and G are expressed in more general forms than those which were derived for the disk medium by Kyriazidou [1999]. The transmission and reflection coefficients of a composite are equivalent to those of a homogeneous material, only when

equation image

holds.

[9] Substituting (9) into (10), we have

equation image

Condition (11) is simplified by using (4). The final result is

equation image

[10] It is interesting to note that the formulas of the transmission and reflection coefficients for both natural and artificial materials are the same, when the scattering matrix of an artificial structure is centrosymmetric (S11 = S22 and S21 = S12). The derivation of effective parameters from transmission and reflection coefficients is valid only for a homogeneous material. The transmission and reflection coefficients of an artificial material without centrosymmetric scattering matrix have a different form from (1). Therefore the retrieval method of effective parameters which was described above is not valid for an artificial material without a centrosymmetric scattering matrix [Smith et al., 2005]. For structures without a centrosymmetric scattering matrix, the definition of effective parameters is only available for a semi-infinite structure [Collin, 1991, and references therein].

3. Formulation of the Problem

[11] We consider a material constructed by stacking an arbitrary number N of planar arrays of thickness c. Each planar array consists of an orthogonal lattice of PEC annular rings embedded in the middle of a homogeneous dielectric of complex permittivity εd = εdr(1 − j tan (δ)), as shown in Figure 2. The size of a unit cell is a × b × c. The propagation of an electromagnetic wave within the material is equivalent to that of a transmission line loaded periodically by the shunt admittance Y of a planar array of scatterers.

Figure 2.

Metalodielectric planar array and corresponding PBG material.

[12] The reflection and transmission coefficients of N layers were given in (8), where F(i), G(i) and ξ(i) are defined as

equation image

ηa(i) and ηd(i) are the relative impedances for the air and dielectrics, respectively. kF(i) is the Floquet propagation constant. The superscript (i) corresponds to different incident polarizations (⊥ for TE and ∥ for TM). p(i) is the KCA Lorentzian generator function, which is defined by Kyriazidou [1999] and Contopanagos et al. [1999]

equation image

and Y(i) is the shunt admittance for two-dimensional (2-D) periodic array of implants.

4. Current Distribution

[13] In this section, the induced current distribution on a printed PEC annular ring illuminated by a plane electromagnetic wave is determined. The effective radius of the equivalent wire is assumed to be one fourth of the width of the printed strip ring [Balanis, 1982, and references therein].

4.1. Induced Current on the Wire Circular Loop

[14] A wire loop is placed in the x-y plane. A plane wave equation imagei = E0iequation imageejequation image illuminates the loop, where equation image = k0(cos equation imageequation image − sin θequation image) as shown in Figure 2. The mixed potential integral equation (MPIE) method is used to determine the induced current distribution on a wire circular loop. The loop radius is rc and the radius of the wire is rw. The integral equation is given as

equation image

where Eϕi represents the ϕ component of the incident electric field.

[15] Taking the Fourier expansion on (15), we obtain the current distribution

equation image

where Z0 = 377Ω,

equation image

and

equation image

Kn, the Fourier coefficient of the kernel of integral (15), is evaluated by Wu [1962].

[16] It is easy to show that fn = fn. The current distribution is derived to be

equation image

4.2. Effective Electric and Magnetic Dipoles

[17] The current density distribution on a printed ring is

equation image

where w = rori is the width, ri, ro are the inner and outer radii of the printed ring, respectively.

[18] The effective electric and magnetic dipoles are defined by

equation image

Obviously, only the first and second terms in (19) contribute to the final electric and magnetic dipoles, respectively. a0, a1 are given in (18).

[19] The final expressions for the effective electric and magnetic dipoles are

equation image

from which the electric and magnetic polarizabilities are obtained as

equation image

where θd is the refractive angle within the dielectric host.

4.3. Shunt Susceptance for Planar Printed Ring Array

[20] A 2-D infinitely thin array with canonical shapes may be modelled as a shunt susceptance jB [Collin, 1991, and references therein]. The relative shunt admittance for the 2-D annular ring array is

equation image

and the effective shunt susceptances for different polarizations are given by the expressions

equation image

for TE incidence, and

equation image

for TM incidence.

[21] Taking the interaction field among the nearest-neighbor elements into account, the lattice interaction constants for electric and magnetic dipoles are added. Thus equation (24) becomes [Collin, 1991, and references therein]

equation image

where Ce and Cm are the lattice interaction constants for electric and magnetic dipoles, respectively,

equation image

where K0 is the modified Bessel function of second kind.

5. Accuracy

[22] The accuracy range is determined primarily by two factors: First, the predominant evanescent mode along the longitudinal direction gives the range as [Kyriazidou, 1999]

equation image

Second, the precision in approximating the current distribution on the PEC annular ring sets the second factor.

[23] To validate our theory, we plot the reflection coefficient for one layer of material which is under consideration in Figure 2. The array parameters a × b × c are selected to be 5.25 × 5.25 × 15 mm3 and the host dielectric constant is εd = 2.2(1, 10−3) as what were used by Liu and Alexopoulos [2006]. We will use the same array parameters and host dielectric constant in the following theoretical discussion throughout this work. The numerical data from commercial simulator Ansoft Designer v1.1 are plotted in Figure 3 and compared with our analytical results for different sizes of annular rings. An excellent agreement between our analytical predictions and the commercial code results up to

equation image

which is a narrower range than that given in (29).

Figure 3.

Comparison between the analytical predictions (ro = 1.1 mm, ri = 0.9 mm (thin solid) and ro = 1.6 mm, ri = 1.4 mm (heavy solid)): and numerical results from Ansoft Designer (ro = 1.1 mm, ri = 0.9 mm (dashed) and ro = 1.6 mm, ri = 1.4 mm (dotted)).

6. Band Gap Structure

[24] The photonic band gaps are achieved by stacking N layers of 2-D arrays along the longitudinal direction. Taking the limit of N → ∞ in (8), we get the bulk reflection coefficient [Contopanagos et al., 1999]

equation image

where ηa(i) and ηd(i) are the relative impedances for the air and dielectrics, respectively. The forbidden band is confined within ∣τ(i)∣ > 1, where τ is defined as [Contopanagos et al., 1999]

equation image

We plot τ(i) in Figure 4 together with the bulk reflectivity ∣Γ2 at normal incidence. ∣τ(i)∣ = 1 exactly defines the band edges and leads to

equation image

where m is the order for each band gap.

Figure 4.

τ(i)∣ (dashed) and bulk reflectivity ∣Γ2 (solid) at normal incidence.

[25] Solving (33), we get the upper and lower boundaries for each band gap

equation image

and

equation image

[26] It is interesting to note that the upper boundary for the mth band gap is fixed at nk0c cos θd = and the lower boundary given in (35) is tunable by changing the value of shunt susceptance B(i). After investigating (35), we find that the larger the shunt susceptance is, a wider bandwidth is achieved. Equations (25) and (26) lead to BB for any incident angle θ, which results in the bandwidth for oblique TE incidence being wider than for oblique TM incidence. This was also observed for the disk medium [Kyriazidou et al., 2000].

[27] Figure 5 shows the bulk reflectivity for TE and TM at incident angles θ = 30° and 60°. We observe that the band gaps for TE are wider than the ones for corresponding TM polarization. The bulk reflectivity is not unity within the forbidden band, which is caused by the loss of the host material.

Figure 5.

Bulk reflectivity of PBG medium at oblique incidence for TE (solid) and TM (dashed) polarizations: (a) 30° and (b) 60°.

7. Effective Refractive Index

[28] An expression for the effective refractive index ne,z(i), which satisfies the Kramers-Kronig relations and causality, is given by Contopanagos et al. [1999], provided the KCA Lorentzian generator function p(y) is known,

equation image

The method provides an accurate and meaningful result for any analytical form of p(y). Unfortunately, p(y) is not explicitly analytical for the ring medium. This will introduce numerical error, which is comparable to the imaginary part with very small value.

[29] We propose a modified approach to derive the effective refractive index from ξ(i)(13) directly. Taking the inverse form of (13), we have the longitudinal and total effective refractive indices ne,z(i) and ne(i)

equation image

where ξ(i) can be expressed as [Contopanagos et al., 1999]

equation image

and Ψ(i) is [Contopanagos et al., 1999]

equation image

[30] We note that y = log(x) is continuous on the Riemann surface, even though the function y = ln(x) is multivalued for the imaginary part. The imaginary parts of y differ by j2π between two neighboring Riemann sheets. Given a reference point, an accurate y at a fixed frequency, we can get a complete and accurate y for the whole frequency range. A natural reference point is provided at 0 frequency.

[31] We plot ϕF = kFc in Figure 6. We observe that ϕF equals within the mth forbidden band. Equation (37) gives the effective refractive index within band gaps at normal incidence

equation image

Equation (40) explains very well the decrease of the effective refractive index ne(i) from the lower band edge to the upper band edge. It predicts that a higher effective refractive index can be achieved at the lower band edge for a wider band gap, which is shown clearly in Figure 7. In Figure 7, we plot ne(i) for different ring sizes and for disks at normal incidence.

Figure 6.

Dispersion diagrams for PBG system in a continuous normalized frequency domain.

Figure 7.

Comparison of ne between different sizes and shapes of scatterers at normal incidence: (a) real part and (b) imaginary part.

[32] It is interesting to note that the relative refractive index equation image of the host material provides a lower limit to the real part of the relative refractive index Re(ne(i)) of the composite material at the higher band edge, which is predicted in (34). The peaks of Re(ne(i)) and Im(ne(i)) increase with increasing frequency, which is significantly different from the behavior of the disk medium.

[33] The real and imaginary parts of the relative refractive index are plotted in Figure 8. We observe that the real part of the refractive index for TE polarization is higher than for TM polarization, which illustrates that the scattering by the electric dipole is a leading effect in the problem.

Figure 8.

Effective refractive index at normal incidence (solid) and at 30° incidence for both TE (dashed) and TM (dotted) polarizations: (a) real part and (b) imaginary part.

8. Effective Wave Impedance

[34] The effective wave impedance ze (normalized to 120π) is defined by the bulk reflectivity (31)

equation image

[35] We give the effective wave impedance in Figure 9 for both polarizations at different incidence angles θ = 0° and 30°. We note that high effective wave impedance can be achieved when a magnetic wall occurs, and reaches zero, when an electric wall is established. The real part of the effective wave impedance is larger than that of the host material. The imaginary part of the effective wave impedance Im(ze) is capacitive within the gap where the electric wall occurs, inductive within the gap where the magnetic wall occurs. The real part of the effective wave impedance for TE polarization is higher than the one for TM polarization at θ = 30°.

Figure 9.

Effective wave impedance at normal incidence (solid) and at 30° incidence for both TE (dashed) and TM (dotted) polarizations: (a) real part and (b) imaginary part.

9. Effective Permittivity and Permeability

[36] The effective refractive index and wave impedance for the ring medium form a complete set of parameters which describe the macroscopic behavior of electromagnetic waves travelling through the material. In this section, we present an equivalent set of parameters, the effective permittivity and permeability, which govern the microscopic mechanism of electromagnetic wave interaction with the material.

[37] We adopt the universal definition for both the effective permittivity and permeability through the effective refractive index and wave impedance

equation image

where ne(i) and ze(i) are defined in equations (37) and (41), respectively.

[38] We plot the effective permittivity in Figure 10. We observe that the real part of the effective permittivity for TM incidence is larger than for TE incidence. The real part of the effective permittivity approaches zero within the band gaps and always stays above unity outside the band gaps. This provides another method to determine the boundaries for each band gap.

Figure 10.

Effective permittivity at normal incidence (solid) and at 60° incidence for both TE (dashed) and TM (dotted) polarizations: (a) real part and (b) imaginary part.

[39] We also present the effective permeability in Figure 11. We observe that the real part of the effective permeability for TE incidence is larger than that for TM incidence. Outside the band gap the artificial material swings between diamagnetic and paramagnetic materials alternatively in the pass bands, which is a significant property for an artificial material.

Figure 11.

Effective permeability at normal incidence (solid) and at 60° incidence for both TE (dashed) and TM (dotted) polarizations: (a) real part and (b) imaginary part.

10. Conclusions

[40] A universal approach has been established for the derivation of the effective parameters for PBG materials from the reflection and transmission coefficients S11, S21. It shows that the definition of effective parameters of a finite thickness artificial structure is only meaningful when the structure's scattering matrix is centrosymmetric (S11 = S22 and S21 = S12). Otherwise, the effective parameters can only be defined for the semi-infinite structure. We investigated the medium with closed ring implants which is obtained by stacking an infinite planar array of PEC rings in the middle of a dielectric layers. Semianalytical formulae for the electric and magnetic polarizabilities and effective shunt susceptance were provided for different plane wave polarizations. The key factors which determine the location and width for each band gap were given analytically for the proposed structure. The host material uniquely defines the position for the upper edge of each band gap. Wider bandwidth can be achieved for larger susceptance. The effective parameters such as effective refractive index, wave impedance, permittivity and permeability have been obtained. The refractive index shows a distinct behavior from that for the disk medium. The peaks of both real and imaginary parts of the effective impedance increase with increasing frequency. We found that the medium switches periodically between inductive and capacitive behavior due to the fact that the induced current is ahead or it lags in phase the incident field, respectively. This causes the change in the sign of imaginary part of the effective medium permittivity and permeability. Meanwhile, the whole structure remains as a passive medium as the imaginary part of the effective refractive index is positive. The ring medium provides us more design flexibilities as well as wider bandwidth compared to the disk medium. This semianalytical approach generates all results in less than 10 s which improves computational efficiency by two orders of magnitude over the numerical techniques. The semianalytical method we have presented in this paper is highly accurate for narrow printed closed ring implants for an arbitrary number of layers.

Acknowledgments

[41] The authors would like to thank Fariborz Maseeh of the Massiah Foundation for supporting in part this research. The authors would like to thank Franco De Flaviis and Harry Contopanagos for valuable discussions.

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