PIERS ONLINE, VOL. 5, NO. 5, 2009 471 Eigenvalue Analysis of Waveguides and Planar Transmission Lines Loaded with Full Tensor Anisotropic Materials C. S. Lavranos, D. G. Drogoudis, and G. A. Kyriacou Department of Electrical and Computer Engineering, Microwaves Lab. Democritus University of Thrace, Xanthi, Greece Abstract An eigenvalue analysis of curved waveguides and planar transmission lines loaded with full tensor anisotropic materials is presented. This analysis is based on our previously established two-dimensional Finite Difference Frequency Domain eigenvalue method formulated in orthogonal curvilinear coordinates. This is properly extended herein in order to accurately handle arbitrarily shaped curved geometries filled or partially loaded with full tensor anisotropic materials. Numerous investigations are carried out involving all type of forward or backward propagating modes. 1. INTRODUCTION The eigenvalue analysis, providing the propagation constants and the corresponding fields distributions, of waveguiding structures comprise a significant key tool in the design of several microwave devices, e.g., filters, dividers and couplers. Due to the interest of the subject, a variety of such techniques have been proposed in the literature [1 3. But, to the authors knowledge, none of them can handle curved geometries or planar bend waveguides. These can be found in a plethora of microwave circuits and systems, such as airborne platforms, modern phased arrays and Radar systems. Moreover, the recent adoption of smart skin ideas demands the whole RF front end to be conformal to the host objects surface. Thus, the accurate design of conformal systems demands the knowledge of curved waveguides and printed transmission lines characteristics. In parallel, the dispersion characteristics of microwave structures involving anisotropic materials (e.g., printed phase shifters in magnetized ferrite s substrate have sparked a growing interest in the field of applications for nonreciprocal devices. Such devices could also follow a curved surface since they can be parts of a conformal system. Also these could be used for specific applications, e.g., magnetic surface wave ring interferometers, curved dielectric cylinders with ferrite sleeves, printed phase shifters on curved anisotropic substrates [4. Hence, the eigenanalysis of wave propagation along curved anisotropic surfaces becomes essential. Moreover, curved or bend transmission lines may offer additional degrees of freedom in the design of non reciprocal devices. Our research effort is based on a two-dimensional Finite Difference Frequency Domain (2-D FDFD eigenvalue method formulated in orthogonal curvilinear coordinates. The theoretical basics and a variety of applications have been presented in our previous works, e.g., [5, 6. An inherent feature of the elaborated FDFD method is the well-known Finite Difference ability of handling anisotropic materials. In particular, its implementation in matrix-form in conjunction with its formulation with complex mathematics enables the introduction of anisotropic materials, including their losses, through a simple modification of the corresponding permittivity or permeability matrices. However, until now, our work was mainly focused on isotropic or diagonally anisotropic materials. Thus, in this paper our method is extended in order to accurately handle arbitrarily shaped curved geometries full or partially loaded with full tensor anisotropic materials, like ferroelectrics or magnetized ferrites, or even artificial metamaterials. Besides, the operating modes characterization, the present effort aims at revealing any new design features offered by the curved or bend geometry. 2. FDFD METHOD FOR CURVILINEAR COORDINATES As mentioned above, our research effort is based on a two-dimensional Finite Difference Frequency Domain (2-D FDFD eigenvalue method formulated in orthogonal curvilinear coordinates [5, 6. The finite difference discretization is applied by means of an orthogonal curvilinear grid (u 1, u 2, u 3, which leads to an eigenvalue problem formulated for the complex propagation constants and the corresponding fields distributions of curved structures. Following a 2-D scheme this analysis is restricted to structures uniform along the propagation direction, while the cross section of the
PIERS ONLINE, VOL. 5, NO. 5, 2009 472 waveguide structure can be of arbitrary shape loaded with inhomogeneous and in general anisotropic materials. Its direct implementation in orthogonal curvilinear coordinates leads to an accurate description of curved or bend geometries with a coarse enough grid but free of the well known stair case effect. The waveguiding structure can be curved in all directions (obeying some limitations with respect to the curvature along the propagation axis and this constitutes its main advantage. The main contribution of this work refers to the analysis of general anisotropic media, where complex tensors constitutive parameters (dielectric permittivity ɛ and magnetic permeability µ are used. The dielectric permittivity tensor ɛ constitutes the relation between the electric flux density D and the electric field intensity Ē as: [ D1 D 2 D 3 = [ ɛ11 ɛ 12 ɛ 13 ɛ 21 ɛ 22 ɛ 23 ɛ 31 ɛ 32 ɛ 33 [ E1 E 2 E 3 Similarly, magnetic permeability tensor µ constitutes the relation between the magnetic flux density B and the magnetic field intensity H as: [ B1 B 2 B 3 = [ µ11 µ 12 µ 13 µ 21 µ 22 µ 23 µ 31 µ 32 µ 33 [ H1 H 2 H 3 For each unit cell, the dielectric or magnetic material is assigned at its four transverse and four longitudinal components, as shown in Fig. 1. Particularly, the dielectric permittivities (ɛ 11, ɛ 21, ɛ 31 e 1 are defined on the same points as the E 1 transverse electric components, the dielectric permittivities (ɛ 12, ɛ 22, ɛ 32 e 2 are defined on the same points as the E 2 transverse electric components and the dielectric permittivities (ɛ 13, ɛ 23, ɛ 33 e 3 are defined on the same points as the E 3 longitudinal electric components. Analogously happens for the magnetic permeability tensors. Thus, half cell homogeneity is imposed. This is accurate in the case of dielectric (or magnetic materials, where the electric (or magnetic grid can be properly designed in order to precisely follow the material boundaries. However, in the case of a lossy inhomogeneous or anisotropic material with both ɛ 1 and µ 1, the half-cell misalignment is inevitable. This is due to the electric and magnetic grid half cell shifting and can be reduced by using a denser grid near the material boundaries, or some advanced techniques such as the condensed nodes technique [7. 3. NUMERICAL RESULTS The first geometry analyzed is a vertically curved rectangular waveguide, filled with a full permittivity tensor ɛ r but scalar permeability (µ r material, as shown in Fig. 2. Retaining their tensor (or matrix representation they read: ɛ = ɛ 0 ( 18.5875 j2.57 3.8841 + j1.029 0 3.8841 + j1.029 14.1025 j1.39 0 0 0 11.86 j0.80, µ = µ 0 ( 1 0 0 0 1 0 0 0 1 (1 (2 (3 Figure 1: Discretization of dielectric and/or magnetic material tensors according to the electric or magnetic curvilinear grid.
PIERS ONLINE, VOL. 5, NO. 5, 2009 473 Figure 2: A downward curved rectangular waveguide filled with full tensor dielectrically anisotropic material, (a = 47.746 mm, b = 21.26 mm, Curvature Ratio: R/b = 2.25. (a (b Figure 3: Normalized propagation constants for the curved waveguide of Fig. 2, compared against those of the straight waveguide, Nuno [3: (a phase constants, (b attenuation constants. This waveguide is curved downward, but, due to its symmetry, the same behavior occurs when it is curved upward. The orthogonal curvilinear coordinates system and the corresponding metric coefficients for this case are given by [8 as: u 1 = x, u 2 = y, u 3 = s and h 1 = 1, h 2 = 1, h 3 = 1 + y R (4 The phase constants for the first two modes are presented in Fig. 3(a, along with those for the corresponding straight waveguide (of the same cross section. Moreover, in Fig. 3(b the attenuation constants for the same complex modes are also compared against again those of the corresponding straight waveguide. The results for the straight waveguide are obtained by the proposed FDFD method and compared with those given by Nuno [3, while those for the curved waveguide are obtained exclusively by the proposed FDFD method. As shown in Fig. 3, the phase and attenuation constants for the straight case are very close to those given by Nuno [3, since a maximum deviation about to 0.1% is observed. Figure 3(a shows a continuous increase in the normalized phase constants versus frequency for both modes, as compared to the straight case. This increase leads to +11% for the first and +10.7% for the second mode at 10 GHz, for a curvature ratio equal to R/b = 2.25. In parallel, the attenuation constants for both modes are also shifted upwards comparing to the straight case, but this increase starts well beyond the cutoff frequency. Both shiftings rise to +15% at 10 GHz. The second examined case is a vertically (downward and sideways (bend right curved rectangular waveguide partially loaded with a ferrite slab. The coordinate system and the corresponding metric coefficients are given by (4 for the downward curvature, while for the sideways (bend right curvature, only the h 3 metric coefficient is changed as: h 3 = 1 + x R (5
PIERS ONLINE, VOL. 5, NO. 5, 2009 474 The ferrite bias DC magnetic field H 0 is vertical, parallel to the waveguide s short dimension. Thus, the direction of the DC bias for the curved structures must be always transverse to the waveguide s cross section, perpendicular to the propagation direction curved arc. The ferrite slab has an isotropic (scalar dielectric permittivity and an anisotropic-gyrotropic magnetic permeability equal to: ɛ = ɛ 0 ( 10 0 0 0 10 0 0 0 10 ( 0.875 0 j0.375, µ = µ 0 0 1 0 j0.375 0 0.875 Due to the ferrite transverse bias, the wave propagation is non reciprocal, theoretically expected to yield different positive β + and negative β phase constants. Specifically, the forward wave propagation (toward positive ŝ should have different phase constants as compared to the backward wave propagation (toward negative ŝ. This is indeed observed in the numerical results of Fig. 5, where the normalized dispersion curves for the forward and backward propagating dominant mode of the curved waveguides are presented. These are also compared against those for the corresponding straight waveguide (of the same cross section. Note, that even not shown in Fig. 5, the forward (β + > 0 and backward (β < 0 phase constants have different signs, with their fields proportional to e jβ+s and e +jβ s (with β < 0 respectively. For better presentation and direct comparison reasons, all phase constants are depicted as absolute values, β ± /K 0. Again, the results for the straight waveguide are obtained by the proposed FDFD method and compared with those given by Nuno [3, while those for the curved waveguide are obtained exclusively by the proposed FDFD method. As shown in Figs. 5(a and 5(b, the phase constants for the straight case are very close to those given by Nuno [3, since a maximum deviation about to 0.5% is observed. When the waveguide is curved downward, a continuous increase in the normalized phase constants for both propagating modes, as compared to the straight case, is observed, which leads to (6 (a (b Figure 4: Curved rectangular waveguide (a = 47.746 mm partially loaded with a transversely magnetized ferrite slab: (a Vertical (downwards curvature (R/a = 2, (b sideways (bend right curvature (R/2a = 2. (a (b Figure 5: Normalized phase constants for the forward and backward propagating dominant mode of the curved waveguides shown in Fig. 4, compared against those of the straight waveguide, Nuno [3: (a Downward curvature, (b sideways curvature.
PIERS ONLINE, VOL. 5, NO. 5, 2009 475 +12% for both modes at 3 GHz. On the other hand, the sideways curvature leads to a continuous decrease in the normalized phase constants for both propagating modes comparing to straight waveguide, which leads to 24% for backward and 28% for forward propagating mode at 3 GHz. The different behavior can be explained by the cross section s asymmetry due to the asymmetric location of the ferrite slab, as clearly explained in [5. 4. CONCLUSION An eigenvalue analysis of curved waveguides loaded with full tensor anisotropic materials was validated herein. A variety of simulations for different curved anisotropic structures (practical microwave ferrite or ferroelectric devices will be presented at the conference. Our next task refers to eigenanalysis of open curved waveguiding structures. ACKNOWLEDGMENT This work is implemented in the framework of Measure 8.3 through the O.P. Competitiveness 3rd Community Support Programme and is co-funded by: 75% of the Public Expenditure from the European Union European Social Fund, 25% of the Public Expenditure from the Hellenic State Ministry of Development General Secretariat for Research and Technology, and Private Sector (INTRACOM SA. REFERENCES 1. Hwang, J.-N., A compact 2-D FDFD method for modeling microstrip structures with nonuniform grids and perfectly matched layer, IEEE Transactions on Microwave Theory and Techniques, Vol. 53, No. 2, 653 659, 2005. 2. Valor, L. and J. Zapata, Efficient finite element analysis of waveguides with lossy inhomogoeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors, IEEE Transactions on Microwave Theory and Techniques, Vol. 43, No. 10, 2452 2459, 1995. 3. Nuno, L., J. V. Balbastre, and H. Castane, Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite element method (FEM using edge elements, IEEE Transactions on Microwave Theory and Technique, Vol. 45, No. 3, 446 449, 1997. 4. Srivastava, N. Propagation of magnetostatic waves along curved ferrite surfaces, IEEE Transactions on Microwave Theory and Techniques, Vol. 26, No. 4, 252 256, 1978. 5. Lavranos, C. S. and G. A. Kyriacou, Eigenvalue analysis of curved waveguides employing an orthogonal curvilinear frequency domain finite difference method, IEEE Transactions on Microwave Theory and Techniques, Vol. 57, No. 3, 594 611, 2009. 6. Lavranos, C. S. and G. A. Kyriacou, Eigenvalue analysis of curved waveguides employing FDFD method in orthogonal curvilinear coordinates, Electronics Letters, Vol. 42, No. 12, 702 704, 2006. 7. Afande, M. M., K. Wu, M. Giroux, and R. G. Bosisio, A finite-difference frequency domain method that introduces condensed nodes and image principle, IEEE Transactions on Microwave Theory and Techniques, Vol. 43, No. 4, 838 846, 1995. 8. Lewin, L., D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures, Peregrinus, London, UK, 1977.