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Lab boratory for Applied Electromagnetics and Commun nications Electromagnetic Scattering by an Impedance p Sheet with a 1-D Inhomogeneity g y in a Rectangular Waveguide Keith W. Whites† and Brian B. Glover‡ †Laboratory for Applied Electromagnetics and Communications, South Dakota School of Mines and Technology, USA ‡Los Alamos National Laboratory, USA Metamaterials 2009: 3rd International Congress on Advanced Electromagnetic Materials in Microwaves and Optics London, UK, Aug. 30 – Sept. 4, 2009 Sponsored by the National Science Foundation through an EPSCoR Research Infrastructure Improvement (RII) program grant (EPS0554609) and an Integrative, Hybrid & Complex Systems (IHCS) program grant (ECCS-0824034). Keith W. Whites Metamaterials 2009 September 2, 2009 1 Adjustable and Tapered R card Lab boratory for Applied Electromagnetics and Commun nications One aspect of this work is to produce lossy films with adjustable sheet impedance: – Perhaps isotropic and uniform – Perhaps spatially varying – Perhaps anisotropic; etc. As will be discussed shortly, this will be accomplished by physically altering commercially available resistive films. Applications include: – Reduce backscattering (Tapered R card: Senior and Liepa, 1984; Haupt and Liepa Liepa, 1987) – Adaptive reflector antenna (Haupt, 2006) – Wu-King taper for ultrawideband antenna: • Radially inhomogeneous sheet impedance • Printed conductor onto 370 OPS Kapton XC (Glover, Kirschenmann, and Whites, “Engineering R-Card Surface Resistivity with Printed Metallic Patterns, Metamaterials’2007, Oct. 2007.) Keith W. Whites Metamaterials 2009 September 2, 2009 2 1 Outline Lab boratory for Applied Electromagnetics and Commun nications This work requires a method for characterizing the scattering by films that are spatially varying, which is the topic of this talk. Eventually would like to measure the effective sheet impedance . z Creating effective sheet impedance films z Mode matching solution o scattering by spatially varying sheet impedance films in a rectangular waveguide z Using perforations to create spatial variation z Properties of DuPont Kapton 370 XC z Measurements for two region perforated Kapton 370 XC z Conclusions o accuracy of the Maxwell/Maxwell Garnett mixing rule Keith W. Whites Metamaterials 2009 September 2, 2009 3 Lab boratory for Applied Electromagnetics and Commun nications Creating Effective Films Methods to create tapered sheet impedance films include: Printing electrically small, metallic patterns on commercially available films to reduce the effective surface resistivity. [Films include DuPont's Kapton XC® or Ohmega Technologies OHMEGA-PLY® OHMEGA PLY® or 200 Ω/sq stainless steel coated PET (polyethylene terephthalate), for example.] Open loops print quicker and use less conductive “ink.” 1 mm Perforating commercially available films to increase the effective surface resistivity. i i i Laser cut square holes Kapton 370 XC Directly manufacture. Expensive, difficult to realize spatially varying. Keith W. Whites Metamaterials 2009 September 2, 2009 4 2 Lab boratory for Applied Electromagnetics and Commun nications Outline z Creating effective sheet impedance films z Mode matching solution o z scattering g by y spatially p y varying y g sheet impedance p films in a rectangular g waveguide Using perforations to create spatial variation o accuracy of the Maxwell/Maxwell Garnett mixing rule z Properties of DuPont Kapton 370 XC z Measurements for two region perforated Kapton 370 XC z Conclusions Keith W. Whites Metamaterials 2009 September 2, 2009 5 Waveguide Characterization Lab boratory for Applied Electromagnetics and Commun nications Quest is to develop a method for characterizing spatially varying impedance films. In this work, will assume variation in only one Cartesian dimension. Will develop a waveguide technique hoping for higher accuracy in a confined measurement system. In actuality, the film is a thin slab of material. Maybe multilayered in the case of printed film (though not the case with perforated). In inhomogeneous materials, difficult to obtain mode expansion of electromagnetic fields. However, here we have very thin, high contrast films. Model these with sheet impedance boundary condition. Effect sheet impedance Zs(x) will vary across the surface. Keith W. Whites Metamaterials 2009 September 2, 2009 6 3 Mode match solution - 1 Lab boratory for Applied Electromagnetics and Commun nications Solve with mode match approach. Incident TE10 mode: Top View ⎛ π x ⎞ − jβ z ,10 z E = sin ⎜ ⎟e ⎝ a ⎠ β ⎛ π x ⎞ − jβ z ,10 z H xi = − z ,10 sin ⎜ ⎟e ωμ ⎝ a ⎠ i y ⎛ mπ ⎞ β z2,m 0 = ω 2 με − ⎜ ⎟ ⎝ a ⎠ n̂ 2 With no y variation in the incident fields or the specimen, then only TEm0 modes (and no TM modes) will be scattered by the specimen. S Subsequently, non-zero components off the reflected f fields f include β z ,m 0 ⎛ mπ x ⎞ jβ Am sin ⎜ ⎟e ⎝ a ⎠ m=1 ωμ ∞ ⎛ mπ x ⎞ jβ z ,m 0 z E yr = ∑ Am sin ⎜ ⎟e ⎝ a ⎠ m=1 ∞ H xr = ∑ z ,m 0 z Non-zero components of the transmitted fields include β z ,m 0 ⎛ mπ x ⎞ − jβ Bm sin ⎜ ⎟e ⎝ a ⎠ m=1 ωμ ∞ ⎛ mπ x ⎞ − jβ z ,m 0 z E yt = ∑ Bm sin ⎜ ⎟e ⎝ a ⎠ m=1 ∞ H xt = −∑ Keith W. Whites z ,m 0 z September 2, 2009 Metamaterials 2009 7 Mode match solution - 2 Lab boratory for Applied Electromagnetics and Commun nications z Employing the impedance sheet boundary condition at z = 0 in which the tangential electric field is continuous across the sheet while the tangential magnetic field is discontinuous: nˆ × E = Z s ( x ) nˆ × ⎡⎣ nˆ × ( H + − H − ) ⎤⎦ z Leads to a matrix equation for the vector of transmitted field amplitudes Bi Q⋅B = P where 2β a Qij = δ ij + z , j 0 I R ,ij 2 ωμ Pi = 2β z ,10 10 ωμ I R ,i1 ⎛ iπ x ⎞ ⎛ jπ x ⎞ I R ,ij ≡ ∫ Z s ( x ) sin ⎜ ⎟ sin ⎜ ⎟ dx ⎝ a ⎠ ⎝ a ⎠ 0 a and δij is the Kronecker delta function. Keith W. Whites Metamaterials 2009 September 2, 2009 8 4 Compare with CST MWS Simulation Lab boratory for Applied Electromagnetics and Commun nications To help validate the accuracy of this mode matching solution, comparisons were made with the frequency domain solver of CST Microwave Studio. Considering a two region resistive film: 333.33 OPS 833.33 OPS Within the frequency domain solver, CST MWS provides for an ideal impedance sheet boundary condition, with infinitesimal thickness, in an integral equation solution. No such boundary condition is available with the time domain solver. Keith W. Whites September 2, 2009 Metamaterials 2009 9 Two Region Specimen Results Used 20 modes in the mode matching solution (both transmitted and reflected), and ~70k mesh cells in the CST MWS solution. 0.8 S parameter magnitude 0.7 178.8 0.6 CST: Mag S11 CST: Mag S21 0.5 Mode match: Mag S11 Mode match: Mag S21 0.4 0.3 1.4 178.6 178.4 CST: Ang S11 178.2 Mode match: Ang S11 1.3 0.2 8 9 10 11 12 13 1.2 f (GHz) 178.0 Angle of S21 Angle of S11 Lab boratory for Applied Electromagnetics and Commun nications 177.8 177.6 177.4 177.2 1.1 1.0 CST: Ang S21 Mode match: Ang S21 0.9 177.0 0.8 176.8 176.6 8 9 10 11 12 0.7 13 8 f (GHz) 9 10 11 12 13 f (GHz) Effects of higher order modes directly observed in phase angle data: would be 180º or 0º for uniform resistive sheets. Excellent agreement: good confidence in mode match solution. Keith W. Whites Metamaterials 2009 September 2, 2009 10 5 Lab boratory for Applied Electromagnetics and Commun nications Outline z Creating effective sheet impedance films z Mode matching solution o z scattering g by y spatially p y varying y g sheet impedance p films in a rectangular g waveguide Using perforations to create spatial variation o accuracy of the Maxwell/Maxwell Garnett mixing rule z Properties of DuPont Kapton 370 XC z Measurements for two region perforated Kapton 370 XC z Conclusions Keith W. Whites Metamaterials 2009 September 2, 2009 11 Spatially Varying Sheet Impedance Lab boratory for Applied Electromagnetics and Commun nications z z z Ideal resistive sheets with constant sheet resistance were used in the previous simulations. How can one create physically realizable films with varying sheet impedance? One approach is to use the concept of “effective film,” as mentioned at the beginning of this presentation, and print metallic particles onto a resistive film, or else perforate the film: Resistive sheet Constant effective Zs Perforations Increasing effective Zs z Perforations in a host resistive film more closely matches an effective film than printing particles on top of the film. Keith W. Whites Metamaterials 2009 September 2, 2009 12 6 Accuracy of Maxwell Garnett The effective sheet resistance of square holes in a SC lattice are extremely well predicted by the simple Maxwell Garnett formula: Rs ,eff = Rs ,0 1 +ν 1 −ν ν = perforation area fraction Rs,eff extracted for 228 OPS ideal resistive film in X band waveguide with uniform square perforations on a SC lattice at 10 GHz. Using CST MWS (frequency domain solver) and averaging Rs,eff from S11 and S21, though no difference. 10000 Effective Rs (OPS) Lab boratory for Applied Electromagnetics and Commun nications Extracted from CST MWS simulation Maxwell Garnett No more than 2 2.4% 4% difference! MG slightly over-predicts Rs (because it’s a lower bound on σ). 1000 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Area fraction of square perforations We first observed this behavior with effective quasistatic permittivity (conductivity) of SC lattice of cubes (Whites and Wu, JAP, 2000). Keith W. Whites September 2, 2009 Metamaterials 2009 13 Linearly Tapered Resistive Sheet Comparing mode match solution to CST MWS (frequency domain) with ideal resistive sheet: Linear taper Rs from 228 OPS to 684 OPS. Four modes in mode match solution: has converged to four decimal places in magnitude and two decimal places in phase (compared to 20 modes). 5 0.70 180 0.44 0.68 0.42 4 179 0.66 0.36 177 |S21| 0.38 0.64 Ang(S11) 178 3 0.62 CST MWS Mode match 2 Ang(S21) 0.40 |S11| Lab boratory for Applied Electromagnetics and Commun nications 0.60 0.34 CST MWS Mode match 0.32 176 0.30 Hole sizes NOT linearlyy tapered! p 8 9 10 11 1 0.56 0 8 12 9 10 11 12 Frequency (GHz) Frequency (GHz) 0.58 175 From these results, conclude: – – – – Have created effective linear tapering in the effective sheet resistance. Maxwell Garnett formula accurately predicting effective sheet resistance. This spatial rate of change in the effective sheet resistance is acceptable. While geometry changing in vertical direction, it is electromagnetically uniform. Keith W. Whites Metamaterials 2009 September 2, 2009 14 7 Lab boratory for Applied Electromagnetics and Commun nications Outline z Creating effective sheet impedance films z Mode matching solution o z scattering g by y spatially p y varying y g sheet impedance p films in a rectangular g waveguide Using perforations to create spatial variation o accuracy of the Maxwell/Maxwell Garnett mixing rule z Properties of DuPont Kapton 370 XC z Measurements for two region perforated Kapton 370 XC z Conclusions Keith W. Whites Metamaterials 2009 September 2, 2009 15 Two Region Perforated Specimen Lab boratory for Applied Electromagnetics and Commun nications z z z z z The last set of data we will show involves the measurement of a two region specimen in a WR-90 rectangular waveguide. Using Kapton 370 XC as the resistive sheet (~40 μm thick). Specimen was laser cut to produce a SC lattice of square holes: Perforation area fraction = 0.340. Using Maxwell Garnett, expect Zs to increase by 2.03 [=(1+0.340)/(1-0.340)] over neat R card. Some discoloration around edges of perforations. Localized heating during laser cutting? Heating will change properties of Kapton XC. Keith W. Whites Metamaterials 2009 September 2, 2009 16 8 Properties of Kapton XC Kapton XC is a fairly complex material. Carbon black dispersed in polyimide in a multilayered structure. Complex εr from 5 specimens measured in a flanged waveguide: 80 0 εr'eff 60 51.44 40 -100 20 εr"eff 0 εr"eff -50 εr'eff Lab boratory for Applied Electromagnetics and Commun nications -150 -20 -79.56/ωε0 -200 9 10 11 12 Frequency (GHz) Appreciable variation between specimens: a well-known property of Kapton XC. (Also known to be somewhat anisotropic.) A parallel RC model with εr,eff = 51.44-j79.56/ωε0 approximately describes the measured complex permittivity of Kapton XC within the X-band, though a more sophisticated model is likely necessary especially for broadband modeling. Keith W. Whites Metamaterials 2009 September 2, 2009 17 Lab boratory for Applied Electromagnetics and Commun nications Outline z Creating effective sheet impedance films z Mode matching solution o z scattering g by y spatially p y varying y g sheet impedance p films in a rectangular g waveguide Using perforations to create spatial variation o accuracy of the Maxwell/Maxwell Garnett mixing rule z Properties of DuPont Kapton 370 XC z Measurements for two region perforated Kapton 370 XC z Conclusions Keith W. Whites Metamaterials 2009 September 2, 2009 18 9 Two Region Specimen Measurement For mode match calculations, extracted Zs from waveguide measurement of uniform specimens (neat R card, then perforated specimen). i ) Used these measured Zs values at each frequency to calculate scattering for the two region specimen. Lab boratory for Applied Electromagnetics and Commun nications z z 0.80 -164 0.55 -6 -166 0.75 0.50 -168 |S21| -172 Ang(S11) |S11| Measured Mode match solution 0.40 0.70 -174 -10 0.65 Ang(S21) -8 -170 170 0.45 -12 -176 Measured Mode match solution 0.60 0.35 -178 0.30 -180 8 9 10 11 Frequency (GHz) Keith W. Whites -14 0.55 8 12 9 Measurements by T. Amert 10 11 12 Frequency (GHz) Metamaterials 2009 September 2, 2009 19 Conclusions Lab boratory for Applied Electromagnetics and Commun nications 9 Overall quest of this work is to develop a method for characterizing the scattering by spatially varying impedance films. 9 Presented a rectangular waveguide mode matched-based solution for characterizing films with a 1-D variation in sheet impedance. 9 Showed a method for spatially tailoring such films by using perforations of varying size on a uniform grid. 9 Presented evidence that the change in sheet impedance is extremely well predicted by the Maxwell Garnett formula. 9 Measured results with a two region specimen of neat and perforated Kapton XC were shown. Keith W. Whites Metamaterials 2009 September 2, 2009 20 10 Lab boratory for Applied Electromagnetics and Commun nications Thank You Keith W. Whites and Brian B. Glover Laboratory for Applied Electromagnetics and Communications Department of Electrical and Computer Engineering South Dakota School of Mines and Technology 501 East Saint i Josephh Street, Rapid id City, i SD 57701 USA †Voice: +1-605-394-6861, E-mail: [email protected] Sponsored by the National Science Foundation through an EPSCoR Research Infrastructure Improvement (RII) program grant (EPS0554609) and an Integrative, Hybrid & Complex Systems (IHCS) program grant (ECCS-0824034). Keith W. Whites Metamaterials 2009 September 2, 2009 21 11