Periodic Structures Eigenanalysis Incorporating the Floquet Field Expansion

S. J. Lavdas1 C. S. Lavranos1 G. A. Kyriacou1

1Department of Electrical and Comuter Engineering, Microwaves Lab,Democritus University of Thrace, Xanthi 67100, Greece e-mail: gkyriac@ee.duth.gr

Abstract − The current work elaborates on the study of periodic structures loaded either with anisotropic or isotropic media. An eigenanalysis methodology is adopted using Finite Difference in Frequency Domain (FDFD) in order to evaluate the Floquet wavenumbers. An eigenvalue problem is addressed and solved with Arnoldi iterative Algorithm. The periodicity of the structure is accounted in two alternative approaches. Initially Periodic Boundary Conditions (PBCs) are imposed on the periodic surfaces whose results found to be in a very good agreement with analytical ones. However, there is a deviation when the phase difference between periodic surfaces rise above 150 degrees. In order to get more accurate results, a Floquet Field Expansion is incorporated into the FDFD formulation. Also, adaptive meshing is employed for the accurate study of very fine discontinuities. In turn certain periodic structures loaded with anisotropic media are simulated in order to reveal the so-called Frozen Modes.

1 INTRODUCTION

Periodic structures have received a great deal of attention in the last decade since they exhibit many interesting and potentially useful features such as band gap and modes localized around the discontinuities of the loading media. These phenomena arise from coherent (constructive) or destructive multiple scattering and interference due to the existence of periodically located obstacles and are similar to Bragg diffraction. In this sense, a lot of applications based on periodic structures have been deployed such as frequency selective surfaces, phased arrays antennas and band gap structures. The basic properties of periodic structures are well studied back in 1970s by Brilluine [1]. The interest in periodic structures is also motivated by the recent development of metamaterials with their extraordinary properties. One of the most known phenomenon is the so called “frozen mode”. Namely, electromagnetic waves presenting a group velocity which tends to zero, as extensively described by Figgotin [2]. Moreover, the dispersion diagrams of such structures are not symmetrical while the corresponding ones of isotropic media are symmetrical. The published theory of periodic structures has a strong formal analysis of the quantum theory of electrons in crystals and thus makes heavy use of the concepts of Bloch waves. Hence, most of researchers deal with periodic structures at photonic frequencies. Even though there are a lot of published articles, mainly in optics, most

of them are focused toward deterministic approaches. Namely, electromagnetic structures excited by specific sources are analysed therein. An obvious limitation of this analysis is that it cannot offer any physical insight into the structure. On the contrary, an eigenanalysis reveal the physical behavior of the structure. Regarding the eigenanalysis methodology most of the published articles are restricted to simple geometries using analytical or approximate techniques. The essential contribution of the proposed method is the eigenanalysis of structures that have arbitrary shape filled with anisotropic or inhomogeneous media.

2 FDFD-FORMULATION

This paper is focused on the development of an eigenanalysis methodology for infinite periodic 3-D structures. Following our previous work [3] an eigenvalue problem is formulated in the form :

2e eωΑ = (1)

Where A matrix is equal to : 1 1

m eE R M R− −Α = (2)

E is the discretised permittivity tensor for the unit cell likewise M is the discretised permeability tensor. Also, Rm is a matrix representing the discretised magnetic rotation H∇ × and E∇ × is the corresponding electric field rotation.

2.1 Periodic Boundary Conditions

The periodicity of structures can be expessed by two approaches based on Floquet Theorem which states that in a periodic system, for a given mode of propagation at a given steady-state frequency, the fields at one cross section differ from those one period (or an integer multiple periods) away by only a complex exponential constant [4]. Thus, the first approach is the replacement of an infinite periodic structure shown in Figure.1 by a unit cell. Secondly Periodic Boundary Conditions (PBCs) are imposed on periodic surfaces of the unit cell. PBCs ,in turn, are applied to rotation matrices of electric and

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magnetic field. Moreover, appropriate boundary conditions for the remaining surfaces which are either Perfect Electric Conductor (PEC) or Perfect Magnetic Conductor (PMC) are imposed on the previous matrices (Re, Rm).

Figure 1: The analysis of an infinite periodic structure is replaced by that of a unit cell.

2.2 Floquet Field Expansion in FDFD

A more accurate simulation of infinite periodic structures is expected when the Floquet Field Expansion is incorporated into the FDFD formulation. Floquet Theorem is a discrete spatial Fourier transform (series) since the electric/magnetic field of lossless periodic structures is a spatial periodic functions. Hence, the electric and magnetic field is expressed as:

( , , )n

E

nj z

x y z nn

E C e β=+∞

−

=−∞

= ∑ (3)

( , , )n

H

nj z

x y z nn

H C e β=+∞

−

=−∞

= ∑ (4)

Where ,E HnC are called Floquet coefficients

( , , )0

1n

E

pj z

n x y zC E e dzp

β= ∫ (5)

( , , )0

1n

H

pj z

n x y zC H e dzp

β= ∫ (6)

Also nβ is the Floquet Wavenumber which is equal to :

02

nnpπβ β= + (7)

and n is the number of Spatial Harmonics of Floquet series. The evaluation of the integrals of Equations (5),(6) requires a linear interpolation between the adjacent nodes of the electric and magnetic field, discretized according to Yee’s cell. Linear interpolation is being applied only in the direction of the structure’s periodicity as depicted in Figure 2 . It is important to mention that the discretized Floquet Coefficients via Finite Differences, lead to equations (8-9) similar to Bragg Diffraction as it will be proved in the next

section. The discretized Floquet coefficients are the same either for electric or magnetic field. There are two types of Floquet Coefficients depending on the position of the electric/magnetic nodes according to Yee cell.

XYnC is related to the nodes 1 2( , )i ie e that are placed in XY-Plane and

ZnC is related to the

corresponding ones 3( )ie that are placed to Z-Plane as shown in Figure 2.

1 1( 1)2

1 20 0

1 1( 1) ( ) ( )n o

n n n

XY

j z iN Nj z ji z j i zi i

ni in n

e eC e e e eep j z j

ββ β β

β β

− −Δ Δ + Δ

= =

⎧ ⎫= − − +⎨ ⎬Δ⎩ ⎭

∑ ∑ (8)

13

0

1 1( )n

n

j zj zi

nZn

eC e ep j

βιβ

ι β

ΔΝΔ+

=

−= ∑ (9)

Figure 2: A unit cell with periodicity in z-direction. Linear interpolation is being applied between the corresponding pairs of successive nodes of xy-Plane_i and Z-Plane_i.

3 BRAGG DIFFRACTION

A short review of Bragg diffraction is presented first in order to reveal the same phenomenon as appears in our FDFD formulation. Let us consider a wave to impinge perpendicularly on a periodic structure as shown in Figure 3. As the wave propagates, at each obstacle (discontinuity) there will be a transmitted and a reflected wave. In this manner a forward and a backward travelling wave will be generated within each waveguide section as shown in Figure 3. Observing the evolution of this phenomenon in time at each point there will be multiple waves returning from the 1st, 2nd ….Nth discontinuity. At certain frequencies these waves will be coherent (in phase) resulting to high field intensity, while other frequencies the non-zero phase difference yields a null field or a band gap. Assuming a lossless waveguide, the total reflected electric field at the input (N=0) of Figure 3, reads :

0

1nji pi

rTotal riE E T ep

β

ι

Κ

=

= ∑ (10)

Which is a geometrically decreasing progression of the following form :

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2 2 4 4{1 ....}j p j prTotal inE E R T e T eβ β− −= + + + (11)

Where R is the reflection coefficient and T is the transmission coefficient at each dielectric slab. The ratio between two successive reflected waves is given by the angular frequency of the following term : 2 2j pe βω −= Τ (12)

When the phase of (2 )pβ equals an integer multiple of 2π (rad), the reflected waves will add in phase resulting to a standing wave with the transmitted ones. In this sense equation (12) shows a selective strong reflection when :

2( )

2g

T p

n T p p m

ϕ βλ

π β

Δ = ∠ − ⇔

⇔ = ∠ − ⇔ = (13)

Considering the power conservation law, when strong reflection occurs then the transmitted power becomes minimum. Hence, equation (13) represents the so-called band gap condition.Even if the previous analysis look like an optical approach, the results of the next section will verify the some condition for periodically loaded waveguides.

Figure 3: A periodic waveguide loaded with dielectric slabs of infinitesimal width.

4 NUMERICAL RESULTS The proposed eigenanalysis methodology is validated by comparing results with the commercially available simulator Comsol Multiphysics [5] and the results found to be in very good agreement with analytical ones. However, there is a significant deviation between the results of the current work and the commercially simulator when the phase difference between periodic surfaces exceeds 150 degrees, especially when the periodic structures are loaded with different dielectric media. Therefore, the next step of this work is oriented in clarifying the previous deviation even if there is great agreement with analytical solutions that have been derived from a classical Bragg diffraction in optics. Firstly, an empty waveguide, is considered as a “periodic structure”, is simulated as shown in Figure 4. The results of the proposed analysis are identical with the analytical waveguide propagation constant. In contrary results from a commercial simulator and our previous work imposing PBC shows significant deviation from the analytical solution as shown in Figure 5.

Figure 4: A cross section of an empty waveguide that is mentioned as periodic structure.

Figure 5: Dispersion diagram of an empty waveguide of Figure 6 (a=0.05m,b=0.03m) .

In turn, the same closed periodic waveguide is loaded with two different dielectric media ,εr1=30 and εr2=25, as shown in Figure 6 with two different periods p1=0.019m and p2=0.15m while a=0.05m, b=0.03m are the same.

Figure 6: A cross section of an infinite periodic waveguide loaded with two different dielectric media. It is obvious from the dispersion diagrams of Figures 7-8, that when the length of the period exceeds λg/2 , where λg is related to the fundamental mode, a band gap appears since destructive interference of the waves takes place. Computing the above band gap by the analytical solution [6], which refers to infinitely extended dielectric films in xy-plane (similar to Figure 3) which have a periodicity along the z-axis, is equal to 31MHz, according to :

sin( / )

md pf f ε π

ε πΔΔ = (14)

Where fm is the centered frequency of the band gap and d is the width of dielectric media which is supposed to be the same for both of them as well as the term Δε/ε is less than unity. Furthemore, simulating the periodic structure of Figure 6 with different values and widths (w1=0.0213m, w2=0.0123m) of two dielectric media (εr1=3, εr2=9) a band gap of 346MHz appears.

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Figure 7: A dispersion diagram for a periodic structure (Figure 6) loaded with two different dielectric slabs(w1=w2=9.5mm and p1=λg/5).

Figure 8: A periodic structure (Figure6) loaded with two different dielectric slabs(w1=w2=9.5mm and p2=2λg/2).

The analytical solution [6], for arbitrary Δε/ε, of equation (15) is 344MHz. Both results are found to be in very good agreement.

1 21

1 2

4 sin ( )mf fε ε

π ε ε−

−Δ =

+ (15)

The strengths of the present method are actually revealed when a very demanding periodic structure such as that of Figure 10 is simulated. The major difficulty stems from the very thin ferrite media for which adaptive meshing (very dense inside the ferrite) is employed.

Figure 9: A periodic structure loaded with two different dielectric, (εr=3, εr=9), (p=5λg/2).

Figure 10: A periodic structure loaded with two misaligned Dielectric media (A1, A2) and one gyrotropic Ferite (F).

Figure 11: Dispersion Diagram of structure of figure 10. The TE20 mode is identified as a frozen mode since at 100 degrees the curve is flat causing a zero group velocity (Ug=0) as well as the curve is asymmetric unlike the rest ones of the diagram.

5 CONCLUSION The incorporation of Floquet Field Expansion into the Finite Difference of Frequency Domain formulation is successfully implemented and verified for closed periodic waveguides loaded with isotropic and anisotropic media. The next step of this work is the study of open periodic structures by employing a global radiation boundary condition. References [1] Leon Brilluine, Wave Propagation in Periodic Structures, first edition,1946. [2] Alex Figgotin, Peter Kuchment, Band-Gap Structure of Spectra of Periodic Dielectric AND acoustic Media I&II,SIAM J. APPL. MATH,Vol.56No.1, pp. 68-88, February 1996. [3] Lavranos, C. S. and G. Kyriacou (2009), Eigenvalue analysis of curved waveguides employing an orthogonal curvilinear Frequency Domain Finite Difference method, IEEE Microwave Theory and Techniques, Vol. 57, 594-611. [4] Keqian Zhang,Dejie Li,Electromagnetic Theory for Microwaves and Optoelectronics, second edition, Springer,1998. [5] Comsol_Multiphysics©, ed. available at http://www.comsol.com/: COMSOL AB, (1998-2010). [6] John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, Rober D. Meade, Photonic Crystals, Molding the Flow of Light,2008. [7] Ryan A. Chilton, Kyung-Young Jung, Robert Lee, Fernado L. Teixeira, Frozen Modes in Parallel-Plate Waveguides With Magnetic Photonic Crystals,IEEE Transactions on Microwave Theory and Techniques,Vol 55,No. 12, December 2007

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A2

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