Comparative Study of Surface Waves on High-Impedance Surfaces With and Without Vias O. Luukkonen, A. B. Yakovlev, C. R. Simovski, and S. A. Tretyakov AP-S International Symposium URSI National Radio Science Meeting San Diego, California 5 – 11 July, 2008 Outline Introduction and Motivation  Surface Waves on HIS Structures  Without Vias  Dynamic Models  Patch and Jerusalem Cross Arrays EBG Properties of HIS Structures With  Vias  Wire Media Slab and Mushroom Array  Mushroom-like Jerusalem Cross Array Conclusion  2 Introduction Analytical modeling of dense FSS grids and HIS structures with and without vias  Homogenization of impedance surface in terms of effective circuit parameters Dynamic model obtained from full-wave scattering problem via the averaged impedance boundary condition  Homogenization of wire media slab and mushroom-like HIS structures – ENG approximation 3 Surface Waves on HIS Structures Without Vias Dynamic Models 4 Model 1 – Impedance Surface Transmission Line Model Z Z g d η Z Z Z o g d s Z Z g d Z s Impedance Surface Model TEz TMz H E y y E H x x H E Z z z s h z No fields beyond the impedance surface 5 S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003 Model 1 – Impedance Surface Impedance Surface Model Impedance Boundary Condition at y=0: TEz TMz H E y y E Z yˆ H E H x H x E Z s z z s z No fields beyond the impedance surface TEz TMz E ZTEH E E e jkzTEz kTyEy E ZTM H H H e jkzTMz kTyM y x s z x 0 z s x x 0 2 2 j ZTM kTE 0 kTE k 1 0 kTM j ZTM kTM k 1 s y ZTE z 0 ZTE y 0 s z 0 s s 0 6 S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003 Grounded Dielectric Slab with Grid Model 2 - Impedance on Air-Dielectric Interface TEz y H y E x H air z 1 1 grid Z TMz g E h slab y 2 2 H x E z z PEC Two-sided impedance boundary condition at y=h E E Z yˆ H H 1 2 g 1 2 7 Dispersion Equations Two-sided impedance boundary condition at y = h TEz-odd TMz-even TE TM E E Z H H E E Z H H x1 x2 g z1 z2 z1 z2 g x1 x2 Dispersion equations j j k 2 k k coth(k h) 2 k tanh(k h) ZTM 2 y1 y1 y2 y2 ZTE y2 y2 g j ZTM k 1 g 1 g y1 8 Complex Wavenumber Plane Branch points in the complex k -plane at k k z z 1 Re{k } 0 - proper modes on the top Riemann sheet y 1 Re{k } 0 - improper modes on the bottom Riemann sheet y 1 Re{k } 0 - branch cuts condition y 1 Im{k }Re{k } Im{k } 1 1 Hyperbolic k -plane branch cuts: z Re{k } z z Re{k } Re{k } z 1 Im{k / k } z 1 k k2 k2 y z i i 2 -1 1 Re{k / k } k2 n2 z 1 i i c 1/2 c 9 0 0 Transmission-Line Network Analysis k θ η Z Z o g d Z s HIS Reflection coefficient Z Z g d Z Z cos s Z Z TE , s 0 g d Z cos s 0 Parallel resonance Z cos X X 0 TM , s 0 g d Z cos s 0 10 S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003