Wave Interactions in a 2-D D Left-Handed Structure

Transcription

1 Wave Interactions in a 2-D D eft-handed Structure Anthony ai 1, Christophe Caloz 2, and Tatsuo Itoh 1 1 Electrical Engineering Department, University of California, os Angeles, USA 2 École Polytechnique de Montréal, Québec, Canada Outline eft-handed Materials (HMs) Background Mushroom Structure as an Effective Composite ight/eft-handed (CH) Surface (PI/NI) ε, µ; n Mushroom Characteristics Fields, efractive Index, Isotropy Mushroom Design Parabolic efractor

2 Background (1) eft-handed Materials Simultaneous negative permittivity (-ε ) and permeability (-µ ). eversal of Snell s aw (negative index of refraction), Doppler Effect, and Cerenkov Effect. Electric field, Magnetic field, and Wavevector of electromagnetic wave in a HM form a eft-handed triad. HMs support backward waves: anti-parallel group and phase velocity. Background (2) ight-handed eft-handed Composite ight/eft-handed + = ω < ω ω > ω ω dω vp = vg = > vp vg < vp vg < vp vg > β dβ HMs support backward waves: anti-parallel group and phase velocity; energy travels away from the source, but wavefronts travel backwards toward the source.

3 2-D eft-handed Material (1) Z CH T C Y C z z y TEM CH PPWG H H x E = E = y β CH jb = Z ' Y ' B = ω µε d x Z o = Z Y ' ' η = µ ε Transmission line relations related to constitutive parameters of CH material. 1 µ = ω and ε = C 2 2 C ω 1 n( ) ω = c εµ 2-D eft-handed Material (2) Distributed Implementation of a 2-D eft-handed Material Unit Cell Mushroom Structure Equivalent CH Circuit 2C 2 p period of unit cell p C eft-handed Capacitance (C ): Due to mutual capacitive couplings. eft-handed Inductance ( ): Due to via to ground plane. ight-handed Capacitance (C ): Due to capacitor-like nature of unit cell. ight-handed Inductance ( ): Due to current flow on top patch. * Electromagnetic waves see the unit cell as effectively homogenous in the long λ region, i.e. p < λ /4

4 Mushroom Characteristics (1) Field Distribution (H egion) Quasi-TEM field distribution. E-field H-field H-field is locally circulating around via to provide H shunt inductance. x On average, H-field is perpendicular to E-field. S-vector (Poynting) Mushroom Characteristics (2) Dispersion Diagram Brillouin Zone efractive Index k y π a Γ π a M π a X k x negative refractive index in H region π a Isotropicality 2 2 ( ) k k β ω = + x y Mushroom is nicely isotropic close to the spectral origin.

5 Mushroom Improvement (1) Ways to Increase eft-handed Attributes of Mushroom Structure Caps to increase capacitor couplings from adjacent cells (Sanada, Caloz, & Itoh). Pros: H mode occurs at lower frequency; H-ness occurs closer to Γ- point more homogeneous. Cons: Difficult to implement. Increase substrate height. Pros: Dispersion curve shifted lower. Cons: Need to dramatically increase height to see small improvement. Thinner Via. Pros: H slope is improved; dispersion curve shifted lower. Cons: Need a very thin via (radius ~.1 mm) to obtain improvement; resistance increases lossy; smallest available drill bit has radius of.175 mm. Increase permittivity (ε ) of host substrate.* Pros: Similar affect as adding caps without the need for difficult implementation. Cons: High permittivity substrate is more expensive. * It may appear that increasing the permittivity will dramatically increase C and therefore decrease the H attributes of the mushroom structure, but it appears that C is increased significantly and dominates at lower frequency. Mushroom Improvement (2) Increase permittivity (ε ) of host substrate Dispersion Diagram Dimension of Unit Cell period : 5 mm top: 4.8 x 4.8 mm 2 via diameter:.24 mm Frequency (GHz) ~3 GHz Drop Blue height: 1.57 mm ε =2.2 Green height: 1.27 mm ε =1.2 *Further improvement can be accomplished by increasing height and thinner via. βp (degrees)

6 Parameter Extraction (1) C C,, & 2C 2 C Parameter Extraction (2) H Capacitance (C ) Z C =1/(jωB) A=1 B=Z C= D=1 H & H Values (C,, & ) Z S Z S Z 11 =Y -1 +Z s Z 21 =Y -1 Y Z 12 =Y -1 Z 22 =Y -1 +Z s C C 1 Y = ω + Y 2 jω ω Z = Z jω j Y = ω Y ω ω 1

7 Parameter Extraction (3) ADS --- (with extracted parameters) HFSS --- Measured --- db(s(1,1)) db(s(7,7)) db(s(5,5)) S freq, GHz db(s(2,1)) db(s(8,7)) db(s(6,5)) S freq, GHz Parameter Extraction (4) 1 5 S 11 ADS --- HFSS --- Measured --- S 21 db(s(1,1)) db(s(7,7)) db(s(5,5)) db(s(2,1)) db(s(8,7)) db(s(6,5)) freq, GHz freq, GHz

8 Generalized Conical Meta-Interfaces For this conic section to focus, n I > n II i.e. wave needs to be slower in n I than in n II n AB + n r = n DC + n f I II I DC = AB + ( r cos θ f )... r = f ( ni nii) n cosθ n I II II n I > n II : Hyperbola n I < n II : Ellipse n I =-n II : Parabola Paraboloidal efractor Principle eal Parabola 2 y z = f 4 f 2 f r = n + n cosθ 1 2 Plane Wave Γ= Cylindrical Wave Image Parabola 2 y z = 2 f 4 f Effective Medium Full-Wave Demonstration Mushroom Implementation N > 1,!