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“Metamaterial Loaded Compact Cavity Resonators” E k H Thomas Henry Hand Duke University Department of Electrical and Computer Engineering Ph.D. Qualifying Exam Presentation Friday, October 27th, 2006 Dr. Steven Cummer Dr. William Joines Dr. David Smith Dr. Qing Liu Overview of Presentation Part I: Metamaterials Overview History and Theory of Operation Part II: Applying Metamaterials to Create Thin Subwavelength Cavity Resonators Engheta’s Idea Hrabar’s Study Kong’s Study Part III: My Work on the Metamaterial Loaded Cavity Resonator Motivation Design Approach Simulation Results Experimental Results Part VI: Conclusions Project Summary Upcoming Publications Part I: Metamaterials Overview Metamaterial = “Meta” + “Material” Meta: Greek prefix meaning “Beyond” “Metamaterials” are synthetic structures that possess electromagnetic properties “beyond” conventional materials They gain electromagnetic properties from their structure as opposed to their intrinsic material property The goal is to give a structure an “effective” permittivity and permeability by providing electric and magnetic responses using artificial metallic inclusions. These effective parameters are the result of averaging the spatial fields across the material. The Effective Medium Picture Metamaterials are useful because they are designed to function as continuous effective media to electromagnetic radiation We want to be able to characterize a sample with effective material parameters εeff and μeff , which result from spatial averaging of the electric and magnetic fields Since the metamaterial is composed of tiny metallic inclusions, we want to be sure these elements are significantly smaller than the free space wavelength λo as to prevent diffraction effects that would ruin this effective medium picture. Size Restrictions: (Pendry, et. al. “Magnetism from Conductors and a << λo = 2πcoω-1 Enhanced Nonlinear Phenomena): In practice, we like to keep the unit cell dimensions around λo/10, although cell sizes on the order of λo/6 have proved to keep the effective medium picture intact. (Smith, et. al., “Electromagnetic parameter retrieval from inhomogeneous metamaterials): What are Negative Index Metamaterials (NIMs)? Metamaterials that provide a structure with an effective negative index of refraction. First conceptualized by V.G Veselago in 1968 Pendry proposed physical structures in 1996 and 1999 that lead to the their physical realization First physically realized by Smith, et. al. in 2000. Since metamaterials were first physically realized in 2000, many research groups have exploited these synthetic structures to create novel devices and components. Timeline: 1968 1996 1999 2000 Time Veselago first studies Pendry Pendry proposes Smith is the first in the the effect a negative proposes wire Split Ring world to physically permittivity and structures to Resonators (SRR’s) realize a medium with permeability has on realize a to realize a negative an effective negative wave propagation negative permeability index of refraction permittivity Negative Index Metamaterial Features Negative Permittivity and Permeability will cause the phase velocity and power flow to be anti-parallel NIM Slab Phase velocity Power Flow (Borrowed from physics.ucsd.edu/~drs/left_home.htm) Negative ε and μ allow for a broader electromagnetic palette Example: No Cut-Off Waveguide: Dispersion Relation in Rectangular WG loaded with anisotropic NIM: y 2 2 As can be seen, by choosing εx <0, μy < 0 kz x y 2 2 ky and μz > 0 , kz will always be positive and c z there will be no lower cutoff frequency. …Negative Refraction Continued Snell’s Law at the interface between a negative index material and a positive index material: n>0 n<0 n>0 http://sagar.physics.neu.edu/wavepacket_refraction.htm http://www.utexas.edu/research/cemd/nim/Intro.html Light Bending the Wrong Way? n1 sin i n2 sin t , and for n1 > 0 and n2 < 0, n1 t sin sin i n 1 Refracted beam will be opposite to the normal as shown in the animation above. 2 Realizing a Negative Permittivity 1968 1996 1999 2000 Time Veselago first Pendry Pendry Smith is the first studies the proposes proposes Split in the world to effect a negative wire Ring realize a medium permittivity and structures Resonators with an effective permeability has to realize a (SRR’s) to negative index of on wave negative realize a refraction propagation permittivity negative permeability The Drude Model of Permittivity: 2 2 2co 2 r 1 2 Ne p 2 p 2 me o a Ln ( a / r ) We want εr to be small and negative since a large and negative εr could shrink λeff to the point where the effective medium picture disappears. In 1996, Pendry proposed a way to reduce the plasma frequency using a periodic wire lattice structure (Pendry, et. al. “Extremely Low Freq. Plasmons in Metallic Mesostructures.) Realizing a Negative Permeability 1968 1996 1999 2000 Time The Lorentz Model of Permeability: Veselago first Pendry Pendry Smith is the first r 1 F 2 studies the proposes proposes Split in the world to effect a negative wire Ring realize a medium o permittivity and structures Resonators with an effective permeability has to realize a (SRR’s) to negative index of on wave propagation negative permittivity realize a negative refraction o 2 2 j permeability Q Pendry proposed split ring resonators (SRR’s) to achieve the necessary resonant magnetic response (Pendry, et. al. “Magnetism from Conductors and Enhanced Nonlinear Phenomena) Any LC resonant particle will realize the negative permeability, such as the single ring particle I employ in practice. o H inc A / L r Is related to this I ind o 2 current 1 2 Part II: Applying Metamaterials to Create Thin Subwavelength Cavity Resonators My Work is based primarily on Three Research Papers Devoted to the Thin Cavity Resonator Concept: Engheta: Originally proposed metamaterial loaded cavity resonator Hrabar: Kong: Short Paper that summarized Engheta’s More in depth experimental theory. They measured the spatial phase investigation behind Engheta’s variation in the cavity. cavity resonator. Hand, Cummer, Engheta: Expanded upon these ideas in addition to measuring the spatial electric field distribution for further clarification of the physics inside the cavity Engheta’s Study Engheta proposed theoretically that negative index metamaterials could be used to create thin subwavelength cavity resonators In his paper, he analyzed a 1D cavity loaded with a bilayer composed of dielectric and negative index slabs His aim was to show that by loading a cavity with dielectric and negative index slabs, the resonance depends on the ratio of slab thicknesses, and not their sum k2 k1 tan(k1d1 ) tan(k2 d 2 ) 0 2 1 When the metamaterial d1 2 slab has an effective negative permeability as d2 1 well as being electrically thin, we are left with the ( 1 1 , 2 2 ) constraint: The electric fields in both slabs can be expressed as: Ex1 Eo sin(k2 d 2 )sin(k1 z ) Ex 2 Eo sin(k1d1 )sin(k2 (d1 d 2 z )) So for slab thicknesses d1 = d2 = d = 1cm, and f = 2.5 GHz, the electric field distribution inside the cavity is: * Note the change in electric field slope at the interface between slabs. This discontinuity in dE/dz arises due to the discontinuity in effective permeability And it is this triangular field distribution that I seek to measure experimentally! But what if we had used two conventional RH slabs instead? Would have remained unchanged, and since μ1 and μ2 would be positive, if one tangent term is positive, then the other would have to be negative! Thus, if d1< 2k , Then to satisfy the dispersion relation, d2 must be > 2k 1 2 This constrains And makes the resonant cavity dependent on d1+d2 ! So clearly using the metamaterial slab allows us to build a more compact resonator than if we had used a RH bilayer. Hrabar’s Experiment First experimental validation of Engheta’s metamaterial loaded resonator Made a resonant ring structure to function as the LH layer This resonant ring structure was then placed inside an evanescent waveguide to realized a LH wave! Coupling loops were used to excite the loop and measure the phase of S21 S21 phase was then measured, showing that the metamaterial behaves as a phase compensator Kong’s Experiment Expands upon Hrabar’s experiment Shows that the resonant frequency is invariant for various slab thicknesses, as long as Engheta’s dispersion relation is satisfied Experiment: Metamaterial was fabricated and transmission properties were measured to verify LH behavior The phase difference across the bilayer cavity was measured to see whether or not it approached zero as predicted by Engheta The slab thicknesses (d1 and d2) were simultaneously varied to see if the resonant frequency in the cavity remained unchanged. Part III: My Work on the Metamaterial Cavity Resonator To help make more lucid the relationship between the properties of the metamaterial slab and the field structure inside the cavity, I decided to measure the spatial electric field magnitude distribution inside the cavity. This study provides a more in depth look at the behavior of the fields inside the cavity. It allows us to see some interesting physical effects, such as how well defined the boundary is between the air and metamaterial layers, and how the effective permeability changes as frequency is shifted. Field Analysis Inside the Cavity Using a 1-D Cavity Topology, we assume a metamaterial slab of thickness d and effective material parameters ε2 and μ2. The goal is to show that the metamaterial permittivity ε2 has little effect on the electric field in the air region. After applying boundary conditions at –d and 0, the electric field in the air region is expressed as: E x1 Eo e jko ( z d ) 1 e jko ( z d ) 2 k 1 j o 2 tan (k 2 d ) k2 o Electrically Thin Layers 1 jko ( z d ) E x1 2 Eo 1 1 jk d o r2 For there to exist a null in Ex1, we require 1 jko ( z d ) 1 jko r 2 d And since in our domain z < -d, equality of imaginary parts tells us that μr2 must be < 0 as well! zo d ( r 2 1) (Null Location, where μr2 < 0) From this requirement, the null location will be dependent on the metamaterial’s permeability and thickness Thus, we can vary the null position zo by controlling the properties of the metamaterial slab! Varying the Material Parameters of the Metamaterial Slab Holding εr2 constant at +1 Holding μr2 constant at -1 Showing how variations in the metamaterial’s permeability will affect the electric field. In this plot, f = 2.5 GHz, d2 = 5 mm (k2d < 0.3) It is evident that this field structure is a strong function of the metamaterial’s effective permeability. How do Metamaterial Losses Affect the Field Structure? As we add losses to the metamaterial slab, the electric field becomes unable to reach a true null. This is understood from the analytical electric field magnitude in the air: adding an imaginary component to μr2 makes it essentially impossible for Ex1 to reach a null. Designing the Metamaterial Slab Used resonant rings to realize the effective negative permeability No need for wire structures (negative ε) if the phase variation across the cavity is small. Extracted effective permeability using method discussed in Smith, et. al. “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients.” Where should we probe the fields? Because the metamaterial is composed of discrete metallic inclusions, we must be careful where we place our thin wire probe when making field measurements. We do not want to probe along (line 1) where the quasi-static fields associated with it dominate over the effective medium fields. In experiment, we will probe somewhere between lines 2 and 3, where the effective medium fields dominate the response. This idea was explored in the paper Cummer, Popa, “Wave Fields measured inside a negative refractive index metamaterial.” Fabricating the Rings The optimal ring design from HFSS was then realized on FR4. I utilized standard photolithography methods to etch the ring design. Because the gap g = 0.3 mm was at the limit of the resolution of this process, the actual fabricated rings were resonant at a slightly higher frequency (2.7 GHz). Above: Dimensions of the ring designed to Above: Fabricated rings on FR4. In total, there were 10 achieve a negative permeability at about 2.5 rings in the transverse section of the waveguide, and two GHz. rings in the vertical direction. Creating the Parallel Plate Waveguide for Field Measurements Because all of the simulations and analysis thus far assumed the unit cell(s) to be illuminated by a uniform plane wave, it would be inappropriate to make measurements in a closed waveguide. The parallel plate waveguide allows TEM propagation at 0 Hz, and the higher order modes propagate according to: Where c is the speed of light in a mc fc vacuum, h is the height of the waveguide (hmax in the bottom 2h diagram), and m = 1, 2,3,… Slit for probing the Waveguide Unit Cells (10 in transverse direction) Copper Wall Because only the TEM mode is propagating the waveguide, the slit running down the propagation axis will not disturb the field patterns, because the current induced in the top and bottom plate of the waveguide to support this mode will run parallel to this slit. Experimental Setup Left: The experimental setup showing the parallel plate waveguide with slot for field measurements. Slot used for Field Measurements Copper plate to act as PEC Unit Cell Structures 1.5cm 3 cm Above: Top view of waveguide with Above: Side view of experimental setup. The white center slot. Markings are spaced 1mm outlined box shows where the other copper slab can be apart from each other allowing for placed to realize the thin cavity resonator, and the green highly resolved electric field dashed line shows the theoretical electric field amplitude magnitude measurements. distribution inside the cavity Experimental Results Above: Overlay plot showing Analytical, Simulated, and Measured Spatial Electric Field Magnitude distributions inside the cavity resonator. It is clear from the measurements that the electric field forms a triangular shape, a characteristic predicted by Engheta. It is clear that Ex1 reaches a null close to the interface, thus allowing us to form a cavity that is 3 cm in thickness (At this frequency, an unloaded cavity would require a thickness of λo/2 = 5.4 cm) The effect of frequency variation on the field structure It is seen that as frequency is increased to where μr2 approaches zero, the field null pushes closer and closer to the interface as predicted in my analysis. Loss tangent tan δ = μr2’’/μr2’ becomes more sensitive to subtle changes in μr2’ as μr2 approaches zero. This explains why the loss seems to increase as frequency is increased towards the μr2= 0 frequency. Notice as frequency increases, quasi-static fields inside the metamaterial slab tend to dominate over the effective medium fields. Effective Medium Field Behavior as Permeability approaches zero As μr2 approaches zero, the effective medium fields in the slab must vanish: Ex 2 E e 2x jk2 z e jk2 z 0 Since k2 c r 2 r 2 0 Fields Plotted for various permeabilities r 2 1 Permittivity of the slab is held r 2 0.5 constant at +1. r 2 0.1 Notice how effective medium fields tend to vanish as μr2 tends to zero ? Coupling to S11 No slab present Resonance due to the shrunken 1.5 cm cavity (no air layer present) Resonance due to 3 cm cavity (d1 = d2 = 1.5 cm) Unloaded 3 cm cavity Loaded 3 cm cavity So the S-Parameter data proves we have coupled We have successfully made the to a resonant mode at f = 2.776 GHz for a 3 cm cavity 55% its unloaded size! cavity Project Summary Verified theoretically that electrically thin slabs can realize the thin resonator, where the resonance depends on the metamaterial’s effective permeability, as well as the ratio of the two slab thicknesses. Field magnitude measurements inside the cavity suggest that we can reduce the size of the resonator significantly Showed that the thin cavity functions as desired by measuring both spatial electric field magnitude, as well as the coupling to S11 to verify resonant characteristics. Can better understand the effect that the slab parameters has on the electric field structure inside the cavity. Reduced the cavity to 55% its unloaded size! Upcoming Publications and Projects Hand, Cummer, Engheta, “The Measured Electric Field Spatial Distribution Within A Metamaterial Subwavelength Cavity Resonator” (Submitted to IEEE Transactions on Antennas and Propagation) Cummer, Popa, Hand “Accurate Q-Based Design Constraints for Resonant Metamaterials and Experimental Validation” Hand, et. al., “Accurate Method for Determining Q, F and effective Material Parameters of Magnetically Resonant Materials” (Journal and Submission Date TBD) Laboratory Manual for EE 53 – Steady State and Transients on Transmission Lines, Crosstalk, and Antenna Experiments. References  V. G. Veselago, “The electrodynamics of substances with simultaneously  D.R. Smith, S. Schultz, P. Markos, C.M. Soukoulis, “Determination of effective negative values of epsilon and mu," (also in Russian, Usp. Fiz. Nauk., vol. 92, pp. permittivity and permeability of metamaterials from reflection and transmission co- 517- 526, 1967), Soviet Phys. Uspekhi, vol. 10, no. 4, pp. 509- 514, 1968. efficients," Phys. Rev. B, vol. 65, No. 195104, pp. 1-5, 2002.  N. Engheta, “An Idea for Thin Subwavelength Cavity Resonators Using Meta-  Koschny, et. al., “Resonant and Anti-resonant frequency dependence of the ef- materials With Negative Permittivity and Permeability," IEEE Trans. Antennas fective parameters of metamaterials," Physical Review E, vol. 68, no. 065602, pp. Propagation, vol. 1, no. 1, pp. 10-12, 2002. 1-4, 2003.  A.E. Centeno and P.S. Excell, “High-Q Dielectrically Loaded Electrically Small  Bogdan-Ioan Popa, Steven A. Cummer, “Determining the effective electromag- Cavity Resonators," IEEE Microwave and Guided Wave Letters, vol. 3, no. 6, netic properties of negative-refractive-index metamaterials from internal Fields," 1993, Phys. pp. 173-174. Rev. B., vol. 72, no. 165102, pp. 1-5, 2005.  Z.M. Hejazi, P.S. Excell, “Miniature HTS spiral cavity resonator for mobile tele-  B. J. Justice, J. J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, phone bands," Int. J. Electronics, vol. 86, no. 1, pp. 117-126. “Spatial mapping of the internal and external electromagnetic ¯elds of negative in- dex metamaterials," Opt. Express 14, 8694-8705 (2006).  James K. Plourde, Chung-Li-Ren, “Application of Dielectric Resonators in Mi- crowave Components," IEEE Transactions on MTT, vol. MTT-29, no. 8, 1981,  J.B. Pendry, et.al.,”Extremely Low-Frequency Plasmons in Metallic Mesostruc- pp.745-770. tures," Phys. Rev. Lett., vol. 76, no. 25, 1996, pp. 4773-4776.  Jin Au Kong, et. Al. “Experimental Realization of a One-Dimensional LHM-  Popa, Cummer, \Derivation of effective parameters of magnetic metamaterials RHM Resonator," IEEE Transactions on MTT, vol. 53, no. 4, 2005, pp.1522-1526. composed of passive resonant LC inclusions," Physics Archives, 2006.  Hand, Cummer, Engheta, “The Measured Spatial Electric Field Distribution  Steven A. Cummer, Bogdan-Ioan Popa, “Wave Fields measured inside a Within a Metamaterial Subwavelength Cavity Resonator," submitted to IEEE negative AWPL, refractive index metamaterial," App. Phys Lett., vol. 85, no. 20, pp. 4564-4566, October 2006. 2004.  C.A. Balanis, “Advanced Engineering Electromagnetics," John Wiley and  S.A. Tretyakov, S.I. Maslovski, I.S. Nefedov, M.K. Krkkinen, “Evanescent modes Sons, stored in cavity resonators with backward-wave slabs", Microwave and Optical Inc., 1989. Tech- nology Letters, vol. 38, no. 2, pp. 153-157, 2003.  “Characterization of Left-Handed Materials”, 6.635 Lecture Notes, Massachusetts  Hrabar, S., Bartolic, J., Sipus, Z., ”Experimental investigation of subwavelength Institute of Technology., 2006. resonator based on backward-wave meta-material," Antennas and Propagation Soci-  Uspekhi, “Electrodynamics of materials with negative index of refraction," ety International Symposium, 2004. IEEE , vol.3, no.pp. 2568- 2571 Vol.3, 20-25 Physics June 2004. Conferences “nd Symposia.  J.B. Pendry, et. al. “Magnetism from Conductors and Enhanced Nonlinear Phe- nomena," IEEE Transactions on MTT, vol. 47, No. 11, 1999, pp. 2075-2084.
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