Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2007 Investigation of the Reflective Properties of a Left-Handed Metamaterial Amanda Durham Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Durham, Amanda, "Investigation of the Reflective Properties of a Left-Handed Metamaterial" (2007). Browse all Theses and Dissertations. 92. https://corescholar.libraries.wright.edu/etd_all/92 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. INVESTIGATION OF THE REFLECTIVE PROPERTIES OF A LEFT-HANDED METAMATERIAL A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science By AMANDA DURHAM B.S., Embry-Riddle Aeronautical University, 2004 2007 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES March 29, 2007 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Amanda Durham ENTITLED Investigation of the Reflective Properties of a Left-handed Metamaterial BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science. _______________________ Lok Lew Yan Voon, Ph.D. Thesis Advisor _______________________ Lok Lew Yan Voon, Ph.D. Department Chair Committee on Final Examination ____________________________ Lok Lew Yan Voon, Ph.D. ____________________________ Gregory Kozlowski, Ph.D. ____________________________ Douglas Petkie, Ph.D. ____________________________ Joseph F. Thomas, Jr., Ph.D. Dean, School of Graduate Studies Abstract Durham, Amanda, M.S., Department of Physics, Wright State University, 2007. Investigation of the Reflective Properties of a Left-handed Metamaterial. The purpose of this project is to investigate the reflective properties of a left-handed metamaterial (LHM) through the use of a finite element analysis software called FEMLAB. In the 1960’s, V. Veselago theorized that a material with negative permeability and negative permittivity has a negative index of refraction. In 2000, such a metamaterial was built and demonstrated at microwave frequencies. Previous work had focused on the transmission properties of the metamaterial. In our work, the reflected wave was examined for a LHM subject to an incident transverse electric wave. The different generalizations, first proposed by Veselago, of the Fresnel and Snell’s equations for LHM’s were rederived. We show that the reflectance does not distinguish between normal materials and metamaterials, and, through computational results, that FEMLAB can be used for LHM’s. iii Contents Page 1 INTRODUCTION.......................................................................................................... 1 1.1 Definition of a Left-Handed Metamaterial ......................................................................................1 1.2 A Different Backwards Wave Source..............................................................................................4 1.3 Report Explanation ..........................................................................................................................5 2 LITERATURE REVIEW ............................................................................................. 7 2.1 Examples of LHM ...........................................................................................................................7 2.1.1 Split-ring resonators (SRR)...........................................................................................................8 2.1.2 Thin rod array ...............................................................................................................................8 2.2 Recent Research...............................................................................................................................9 3 BACKGROUND .......................................................................................................... 13 3.1 Maxwell’s Equations .....................................................................................................................13 3.2 Physical Property Choices for this Report .....................................................................................14 3.2.1 LHM vs. photonic crystal ...........................................................................................................14 3.2.2 Transverse electric vs. transverse magnetic wave.......................................................................15 3.2.3 Fresnel diffraction vs. fraunhofer diffraction..............................................................................16 3.2 FEMLAB .......................................................................................................................................16 3.2.1 FEMLAB’s computational method compared to more popular methods ...................................17 3.2.2 Why FEMLAB ...........................................................................................................................19 3.2.3 Deriving single equation for FEMLAB ......................................................................................19 3.2.4 Perfectly matched layers.............................................................................................................21 3.3 Fresnel’s Equations........................................................................................................................23 3.4 Fabry-Perot Equations ...................................................................................................................29 3.4.1 Electrodynamic approach............................................................................................................29 3.4.2 Airy’s formulation ......................................................................................................................32 3.4.3 Preparing equations for LHM .....................................................................................................35 4 MODELING PROCESS ............................................................................................. 38 5 RESULTS ..................................................................................................................... 47 5.1 FEMLAB Solution Visualization Verification ..............................................................................48 5.2 FEMLAB Results Verification ......................................................................................................50 5.2.1 Expected values ..........................................................................................................................50 5.2.2 FEMLAB values.........................................................................................................................51 5.3 LHM Results..................................................................................................................................55 5.3.1 FEMLAB LHM results...............................................................................................................55 5.4 Veselago’s Theory: n vs Z .............................................................................................................64 6 CONCLUSION ............................................................................................................ 70 A: PROOF OF NO REFLECTION AT INTERFACE .............................................. 72 B: AVERAGE PEAK VALUE JUSTIFICATION ..................................................... 76 C: DETERMINING PERMITTIVITY FROM COMPELX PERMEABILITY AND INDEX OF REFRACTION ........................................................................................... 83 iv D: MATLAB ROUTINE ................................................................................................ 86 E: EXTRA NOTES AND HELPFUL HINTS.............................................................. 88 WORKS CITED.............................................................................................................. 90 v Figures Page Figure 1: Ray diagram of interface between n>0 and n<0 media....................................... 2 Figure 2: All possible combinations for ε and μ................................................................. 3 Figure 3: Example of group and phase velocities moving in opposite directions .............. 4 Figure 4: Diagram of geometry used for paper................................................................... 6 Figure 5: Coils of SRR (left) and representation current flow of a SRR (right)................. 8 Figure 6: Array of thin rods ................................................................................................ 9 Figure 7: Ray diagram of LHM ........................................................................................ 10 Figure 8: Setup used by Schurig and coworkers during cloaking experiment ................. 11 Figure 9: Diagram of cloak and electromagnetic waves................................................... 11 Figure 10: Symmetric lattice structure for a photonic crystal........................................... 14 Figure 11: Original and meshed geometry comparison.................................................... 18 Figure 12: PML used in FEMLAB runs ..........................................................................