The Pennsylvania State University. The Graduate School. Electrical Engineering Department ANALYSIS AND DESIGN OF MICROWAVE AND OPTICAL

Transcription

1 The Pennsylvania State University The Graduate School Electrical Engineering Department ANALYSIS AND DESIGN OF MICROWAVE AND OPTICAL PLASMONIC ANTENNAS A Thesis in Electrical Engineering by Bingqian Lu 216 Bingqian Lu Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 216

2 ii The thesis of Bingqian Lu was reviewed and approved* by the following: Douglas H. Werner John L. and Genevieve H. McGain Chair Professor of Electrical Engineering Thesis Advisor Ping Werner Professor of Electrical Engineering John Roe Professor of Mathematics Kultegin Aydin Professor of Electrical Engineering and Department Head *Signatures are on file in the Graduate School

3 iii ABSTRACT The plasmonic antenna has become a popular research topic, largely as a result of the development of nano-technology. This thesis covers the design and analysis of these devices in both the microwave and optical frequency regions. First, we will present a compact methodology for the design of periodic leaky wave antennas. These antennas enable spoof plasmon type surface waves to radiate at microwave frequencies. This approach is based on structurally modifying a corrugated reactance surface. In particular, a properly engineered periodic perturbation is introduced to enable the excitation of the n = -1 spatial Floquet mode. This mode is characterized by a complex wavenumber, the real part of which is less than that of free space. As a result, the guided spoof plasmons are efficiently coupled to the radiating modes. Numerical simulation software (HFSS) has been used to validate the proposed design methodology. Second, we will present closed form expressions for the radiated fields, directivity and gain for a single nanoloop antenna. In the terahertz, infrared and optical regimes nanoloops show great promise for a variety of applications, such as solar cells and optical sensors. However, due to the complex behavior of metals at these frequencies, prior studies had not yet completed a theoretical derivation for the radiation parameters of a nanoloop. We propose a solution based on the extension of the formulation for a thin-wire Perfect-Electric Conductor (PEC) loop to include the effects of loss and dispersion. These proposed expressions contain integrals of Bessel and Lommel-Weber functions as well as Q-type integrals. Various series representations for these integrals will be presented along with guidelines for solving them. We will validate these equations through the comparison of results from full-wave solvers. These simulations typically require hours of processing time, whereas our analytical expressions can be evaluated within seconds.

4 iv Finally, we will present an extended study on the derivation of coupling between nanoloops. Due to their properties of highly directive transmission and reception, nanoloop arrays have many practical applications, such as energy harvesting. Specific equations involving numerical integrals are proposed in Chapter 4. More importantly, far-zone approximations are employed to generate simplified closed form analytical solutions. The induced current on a passive loop has been derived for the general case, where the passive loop can be located anywhere with respect to the active loop. The induced current equations have also been derived for two special cases, where the passive loop is coplanar to the active loop and where the passive loop is stacked above the active loop. Both the numerical and closed form analytical results were verified with FEKO (a full-wave Method of Moments solver). A significant reduction in computing time was observed.

5 v TABLE OF CONTENTS List of Figures... v List of Tables... vi Acknowledgements... vii Chapter 1 Introduction... 1 Chapter 2 Microwave Regime Plasmonic Antenna Design Introduction The Baseline of Surface Plasmon Waveguide In Planar Leaky Wave Antenna Design Planar Leaky Wave Antenna Design Numerical Results Cylindrical Spoof SPP LWA Design Numerical Results for Cylindrical LWA Overview Chapter 3 Optical Frequency Plasmonic Nanoloop Antenna Analysis Introduction Theoretical Formulation Input Impedance, Radiated Power, Radiation Resistance and Efficiency Radiation Intensity, Directivity and Gain Results Comparing to Simulations Conclusion Chapter 4 Optical Frequency Plasmonic Nanoloop Antenna Coupling Analysis Introduction Formulation of the Coupling Current Equation for Two Coplanar Nanoloop Antennas Analytical Solution Compared to Numerical Simulation for Coplanar Nanoloops Formulation of the Coupling Current Equation for General Configuration of Two Nanoloop Antennas Comparison of Analytical Solutions to Numerical Simulations for General Nanoloop Configurations Formulation of the Coupling Current Equation for the r Component Comparison of Analytical Solutions with Numerical Simulations for General Nanoloop Configurations Including the r Component Formulation of the Coupling Current Equation for Stacked Two Nanoloop Antennas Analytical Solution Compared to Numerical Simulation for Stacked Configuration Nanoloops... 74

6 vi Chapter 5 Conclusions and Future Work Bibliography Appendix A Calculation of the Integral of the Bessel Function Appendix B Calculation of the Integral of the Lommel-Weber Function Appendix C Calculation of the Q-Type Function Appendix D Induced Current Derivation for Coplanar Nanoloops... 9 Appendix E Induced Current Derivation for Generalized Configurations of Nanoloops Appendix F Induced Current Derivation for Stacked Configurations of Nanoloops

7 vii LIST OF FIGURES Figure 2-1. Corrugated surface geometry. T=.5mm, G+T = 2mm and l=5mm Figure 2-2. Dispersion properties of the corrugated surface shown in Figure 2-1. At 9GHz, the guided wavenumber is k x = 1.437k Figure 2-3. Side view of (a) the unperturbed structure and (b) the perturbed structure Figure 2-4.The tapering structure of the proposed leaky wave antenna. Amplitudes of the first 75 elements (5 Unit cells) increase in a cubed polynomial fashion. The 76 th to 325 th (2 Unit cells) have a constant amplitude of.5mm for the sinusoid perturbation structure. Amplitudes of the last 75 elements (5 Unit cells) decrease in a cubed polynomial fashion Figure 2-5.The near field of Leaky Wave Antenna with a flat SPP wave launching structure (no tapering). A strong 2 nd Floquet mode (n = 2) that radiates in the negative θ direction is observed... 1 Figure 2-6.The near field of the proposed Leaky Wave Antenna. Far less 2 nd Floquet mode (n = 2) raidation is observed in this near field plot Figure 2-7.The realized gain comparison of the proposed Leaky Wave Antenna with a flat SPP wave launching structure versus a tapered SPP wave launching structure. The main beam is the same while the 2 nd Floquet mode realized gain is 8.657dB for the flat SPP wave launch and.77db for the tapered SPP wave launch Figure 2-8.The realized gain of the proposed Leaky Wave Antenna and the beam steering property over frequency Figure 2-9.The normalized directivity comparison of the proposed tapered periodic Leaky Wave Antenna structure for different amplitudes of the sinusoidal perturbation. A positive correlation observed between increasing sinusoidal amplitudes, the half power beamwidths, and the directivity of the 2 nd Floquet mode Figure 2-1. S parameters from the proposed tapered structure Figure (a) Leakage rate vs. the perturbation amplitude. (b) Radiation Efficiency vs. frequency for the proposed tapered structure Figure 2-12.The cylindrical corrugation geometry. G = 1.5mm and T=.5mm. R in = 5mm and R out = 1mm Figure Dispersion diagram of the light line, planar structure (discussed in the previous section) and cylindrical groove structure with the same R in = 5mm and different R out, where (a) R out = 1mm and (b) R out = 1mm

8 Figure (a) The R in and R out pair for the desired surface impedance. (b) The radius difference between R out and R in (c) The real and imaginary components of the corresponding surface impedance for each pair of R in and R out Figure 2-15.The near field of the proposed cylindrical Leaky Wave Antenna Figure The normalized directivity for cylindrical LWA for different sinusoidal amplitudes Figure (a) Beam Steering Property of the proposed cylindrical LWA. Realized gain for 8.8GHz, 9GHz and 9.2GHz are plotted. (b) The S parameters for the cylindrical LWA for a sweep of frequencies Figure 3-1.Geometry of the thin circular loop (where the wire radius is far smaller than the loop radius, i.e., a 2 b 2 ). An infinitesimal voltage source is placed at φ = with a constant value of V Figure 3-2. Efficiency vs. k b for difference number of modes considered for a circumference b = 6nm and wire radius a = b/64.21 nm (Ω = 12) Figure 3-3. Comparison of the total power radiated vs. k b evaluated using three different simulation techniques for the (a) 6nm circumference and (b) 3 nm circumference gold nanoloops Figure 3-4. Comparison of efficiency vs. k b evaluated using four different simulation techniques for the (a) 6nm circumference and (b) 3 nm circumference gold nanoloops Figure 3-5. Input radiation and loss resistance vs. k b for (a) 6nm circumference and (b) 3nm circumference gold nanoloops Figure 3-6. Radiation resistance vs. k b for the (a) 6nm circumference and (b) 3nm circumference gold nanoloop Figure 3-7. (a) Directivity of the 6nm loop evaluated at (θ, φ) = (9, ). (b) Directivity of the 3nm loop evaluated at (θ, φ) = (, ) Figure 3-8. Directivity corresponding to a 3nm loop for k b values of (a).1 (1THz), (b).5 (5THz), (c) 1.1 (11THz) and (d) 2.5 (25 THz) Figure 3-9. Directivity of the (a) 6nm loop and (b) 3nm loop corresponding angles of (θ, φ) = (, ), (θ, φ) = (9, ) and (θ, φ) = (9, 18 ) Figure 3-1. Gain of the (a) 6nm loop and (b) 3nm loop corresponding to angles of (θ, φ) = (, ), (θ, φ) = (9, ) and (θ, φ) = (9, 18 ) Figure 4-1. Two nanoloop antennas on the same plane, θ = π 2. The left loop (blue) is the active loop with a voltage source and the loop on the right (green) is the passive loop without voltage sources. The distance between the centers of the two loops is viii

9 ix defined as r. The active loop coordinate can be expressed as (r, π/2, φ) and the passive loop coordinate can be expreseed as (r, π/2, φ ) Figure 4-2. Two nanoloop antennas on the same plane. The unit vector φ from the nonprimed coordinate and the tangential unit vector t are illustrated Figure 4-3. Two nanoloop antennas on the same plane. The left loop (blue) is the active loop with a voltage source place at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the two loops is defined as r, with an x-component of ξ and y-component of η. (x, y) denotes any arbitrary point on the passive loop Figure 4-4. Passive loop. The tangential vector t is illustrated Figure 4-5. Induced current comparison between FEKO result, numerical integration and closed form analytical solution on the passive PEC loop located at φ = and (a) 7 times and (b) 14 times the radius away from the center of the active loop. Good agreement is observed Figure 4-6. Induced current on the passive PEC loop located at φ = 3 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are all plotted. Fairly good agreement is observed Figure 4-7. Induced current on the passive PEC loop located at φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are all plotted. The second peak was not captured by both numerical and analytical solutions and is due to inaccuracies in the radial component of the electric field Figure 4-8. Induced current on the passive PEC loop located at φ = 9 (a) 7 times, (b) 14 times, (c) 28 times and (d) 56 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. The first peak was not captured by both numerical and analytical solutions and is a result of inaccuracies in the radial component of the field Figure 4-9. Two nanoloop antennas in a general configuration. The left loop (blue) is the active loop with a voltage source at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the active loop and the projection of the passive loop is defined as r, which x-component is ξ and y-component is η. (x, y) here denotes an arbitrary point on the passive loop Figure 4-1. Induced current component on the passive PEC loop located at θ = 3, φ = 2 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are each plotted Figure Induced current on the PEC passive loop located at θ = 45, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO

10 x result, numerical integration and closed form analytical solutions are all plotted. Good agreements are observed Figure Two nanoloop antennas in the most general configuration. The left loop (blue) is the active loop with a voltage source located at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the active loop and the projection of the passive loop is defined as r. The x-component of the projected loop center is denoted as ξ and the y-component of the projected loop center is denoted as η. (x, y) represents any arbitrary point on the passive loop Figure Induced current comparison (including the E r component) on the passive PEC loop located at θ = 45, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Near perfect agreement has been observed Figure Induced current (including the E r component) on the passive PEC loop located at θ = 9, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Good agreements are observed Figure Induced current including the E r component on the passive PEC loop located at θ = 3, φ = 2 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are plotted. Good agreements are observed Figure Induced current including the E r component on the passive PEC loop located at θ = 9, φ = 9 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are plotted. Good agreements are observed Figure Induced current including the E r component on the passive PEC loop located at (a) θ = 3, φ = 2, (b)θ = 45, φ = 45, (c) θ = 9, φ = 45 and (d) θ = 9, φ = 9 The center of the passive loop is 4 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted Figure Comparison between FEKO results, numerical integration and closed form analytical solutions for the induced current (including the E r component) on the passive gold loop located at θ = 9, φ =. The loops have a separation distance of (a) 7 times and (b) 14 times the radius from the center of the active loop Figure Induced current including the E r component on the passive gold loop located at θ = 9, φ = 9 that is (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted.... 7

11 xi Figure Induced current on the PEC passive loop located at θ = (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Near perfect agreement between different methods are observed for both separation distances Figure Comparison between FEKO, numerical integration and close form analytical solutions for the induced current on the passive PEC loop located at θ = and r = 4b. Analytical solution magnitude at the peak is slightly off because the far field approximation θ = θ might not hold for this short separation distance Figure Induced current comparison between FEKO result, numerical integration and closed form analytical solutions for two 6nm gold loops in the stacked configuration located at θ = (a) 7 times and (b) 14 times the radius away from the center of the active loop Figure Induced current comparison between FEKO result, numerical integration and closed form analytical solutions for 3nm gold loops located at θ = 9, φ = (a) 7 times and (b) 14 times the radius away from the center of the active loop. Good agreements between all three methods are shown Figure A-1. Relative error vs. number of terms for the integral of the Bessel function with m=1. Using (a) power series (Equation (A.1)) and (b) Bessel series (Equation (A.2)) representations. The error is computed relative to (Equation (A.8)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red Figure A-2. Relative error vs. number of terms for the integral of the Bessel function with m=35. Using (a) power series (Equation (A.1)) and (b) Bessel series (Equation (A.2)) representations. The error is computed relative to (Equation (A.8)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red Figure B-1. Relative error vs. number of terms for the integral of the Lommel-Weber function given in Equation (B.1) with (a) m=1 and (b) m=35. The error is computed relative to (Equation (B.5)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red Figure C-1. Relative error vs. number of terms for the Q-type integral with m=1 using (a) power series representation (Equation (C.1)) and (b) the Bessel series (Equation (C.4)). The error is computed relative to numerical quadrature with an extremely small tolerance. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red Figure C-2. Relative error vs. number of terms for the Q-type integral with m=35, using (a) the power series representation (Equation (C.1)) and (b) the Bessel series (Equation (C.4)). The error is computed relative to numerical quadrature with an extremely small tolerance. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red

12 xii LIST OF TABLES Table 3-1. Parameters for refractive index of gold [46]-[47] Table 3-2. Timing comparison for three different methods Table 3-3. Timing comparison for three different methods for coupling nanoloops... 71

13 xiii ACKNOWLEDGEMENTS I wish to express my deepest appreciation to my advisor, Professor Douglas H. Werner of The Pennsylvania State University, for his continual guidance and encouragement during the course of my research. I am also appreciative of the many educational opportunities that were made possible by his generosity. Professor Werner s enthusiastic devotion to research and dedication to teaching have been an inspiration to me. I wish to thank Professor Ping Werner and Professor John Roe for serving on my Master committee and for their helpful suggestions. I am especially grateful to Dr. Anastasios H. Panaretos, Dr. Mario F. Pantoja and Dr. Zhihao Jiang for their technical assistance during my study. I acknowledge and thank Joe Nagar, for his support on the nanoloop project. I also want to thank Kyle Casterline for spending his time to format equations and prepare geometry figures in this thesis. Lastly, I owe special thanks to Joe Nagar, Kyle Casterline and Allan Casterline for proofreading this thesis.

14 To my dearly beloved parents Honglan Ling and Nanchun Lu xiv

15 1 Chapter 1 Introduction The general properties of the antenna have been studied and well-established in the radio and microwave frequency regimes for nearly a century. Theories have been derived from the notion that a perfect electric conductor (PEC) will have the characteristic of only carrying surface current. These theories have guided engineers to optimize designs and make significant technological impacts. However, with the fast development of nano-technology, antenna engineers are shifting their focus from radio and microwave frequency regimes toward optical and infrared frequency regimes. Many optical antenna geometries have evolved from RF antenna designs, such as dipole antennas [1, 2], patch antennas [3], loop antennas [4], etc. Optical antennas have gained the attention of many researchers due to their ability to enhance [5, 6] and direct [7, 8] emission as well as their potential for use in sensing applications and small volume spectroscopy [9, 1]. In the future, wireless communications are expected to play a key role in the development of practical nanotechnology-enabled devices. Those devices will likely be able to perform a wide variety of tasks ranging from energy harvesting to medical implantation [1]. The design of nanoantennas for these applications is particularly challenging because the materials they are made of are lossy and exhibit dispersion properties in the optical to terahertz spectra [2]. Despite such great promise, we still lack the analytical means for describing the properties of optical antennas. Presently, the only way to study these properties is through computationally intensive full-wave simulations. Noble metals show negative dielectric permittivity at optical or infrared frequencies. When a noble metal surface is illuminated, a specific type of Surface Wave (SW) is excited and travels along the interface between two dielectrics with opposite permittivity signs [11]. Such SWs are

16 2 known as Surface Plasmon Polaritons (SPPs) and can be explained by both charge motion in the metal and electromagnetic waves in dielectrics. The motion of charge in the metal is known as surface plasmons (SPs) and the electromagnetic waves in dielectrics are known as polaritons [12]. It has been previously established that the resonant excitation of surface plasmons will create a large electric field on the surface and, in turn, force the light waves through holes in the material. This will result in a high transmission coefficient. While a PEC itself cannot support surface plasmons due to the absence of an electromagnetic field inside the conductor, the addition of structured metal surface arrays enable surface plasmon generation in the microwave to terahertz frequencies. Such SWs on a PEC are called Spoof SPPs, a topic currently generating much research interest. There are several reported ways to decorate a PEC surface and create Spoof SPPs. These include one-dimensional (1D) arrays of subwavelength grooves [13, 14], two-dimensional (2D) arrays of subwavelength holes or dimples [15-17], and three-dimensional (3D) metal wires with periodic arrays of radial grooves [18]. Such decorated PECs are known as plasmonic metamaterials and they open a new area of research to control the surface electromagnetic (EM) waves on a subwavelength scale. The dispersion properties and spatial confinement of the spoof SPPs can be controlled by adjusting the geometry of the individual array unit cells. Consequently, plasmonic metamaterials provide a new method of creating spoof SPPs at terahertz and microwave frequencies. These spoof SPPs exhibit similar characteristics to natural SPPs occurring at optical frequencies. Previously, researchers have focused on creating spoof SPPs in the microwave frequency range and guiding them along specially designed SPP waveguides. However, it is also important to convert localized energy into spatially propagating waves. Such energy conversion plays a critical role in electromagnetic wireless communication. The goal of efficient energy transfer is an important field of interest to many researchers. Hence, one of the topics of this thesis is to analyze and design a leaky wave antenna (LWA) using Spoof SPPs operating at RF frequencies.

17 3 Furthermore, with recent developments in the field of nano-technology, the design and optimization of optical antennas has emerged as a popular research topic. These designs are mostly done with numerical software solvers. The more complex the investigated structures are, the greater the possibility to reveal new discoveries. However, this comes at the expense of large computational costs for the research performed. For this reason, accurate analytical models and equations have great appeal in the research community for predicting structure behavior and device performance. Unfortunately, because the phase velocity of optical frequency currents in metals is much less than the speed of light, standard radio frequency (RF) antenna theory does not directly apply. As a result, the metals no longer behave as PECs at these frequencies. Instead, they exhibit substantial dispersion and loss [23]. This has a dramatic impact on the radiation properties of nanoantennas, including the directivity, efficiency, and total radiated power [24]. This thesis also provides work on analytical representations of these antenna radiation properties. Analytical solutions for mutual coupling between these loop antennas are also derived and discussed in this thesis. These analytical solutions have considerably decreased the computational time required for simulations of two or more loops.

18 4 Chapter 2 Microwave Regime Plasmonic Antenna Design 2.1 Introduction Spoof plasmons (SPs) are a class of surface waves (SWs) which are excited along structured metallic reactance surfaces (RSs). Such types of SWs are considered the microwave and THz equivalents of the well-known optical surface plasmon polaritons (SPPs); the latter are typically excited at the interface between two dielectrics with opposite permittivity signs [1]. The excitation of optical SPPs is a direct consequence of noble metals exhibiting values of negative dielectric permittivity at these frequencies. Surface plasmon polaritons (SPPs) are highly localized electromagnetic (EM) waves at optical frequencies exhibiting exponential field decay away from the interface of metal and dielectric materials. Much of the initial work in the field of plasmonics has been focused on theories and applications in the optical frequency regime. Noble metals such as silver, copper and gold are normally used at these frequencies. As discussed in [18], when the operating frequency decreases, the localization of the fields on the dielectric side of the surface also decreases. The field confinement generally decreases with increasing conductivity of the conductor. Perfect electrical conductors (PECs), as an extreme example, will support neither surface EM waves on planar interfaces nor on wires. There has been an increase in research interest regarding the plasmonic properties of perfect electric conductors (PECs). A PEC itself cannot support SPPs; however, it can support SPPs if decorated with one of the following: - One-dimensional (1D) arrays of subwavelength grooves [13, 14] - Two-dimensional (2D) arrays of subwavelength holes or dimples [15-17] - Three-dimensional (3D) metal wires with periodic arrays of radial grooves [18].

19 5 The properties of the spoof SPPs can be tuned by altering the geometry of the surface corrugations. More importantly, these decorated surface layers can be viewed as a dielectric permittivity of the Drude form [18]. The exciting field of plasmonics is now firmly established not only within the photonics community, but also with researchers and scientists from a variety of disciplines wanting to take advantage of subwavelength light localization effects in the near zone of conductors. In the microwave frequency, numerous wave-guiding structures have been proposed that exploit the guiding properties of SPs, but it is also crucial to convert the localized energy into spatially propagating and radiating waves. In this chapter, we contribute to the existing knowledge of SP wave guiding techniques by proposing a simple yet efficient methodology to couple SPs into radiating modes. In particular, we begin with the design of a baseline planar corrugated surface to support SPs. Next, we structurally modify the baseline waveguide by introducing a properly engineered periodic perturbation. This way, we allow for the excitation of spatial Floquet modes that naturally convert the guiding structure into a periodic leaky wave antenna (LWA). Finally, we use a similar methodology on arrays of periodic radial grooves to design a cylindrical LWA. 2.2 The Baseline of Surface Plasmon Waveguide In Planar Leaky Wave Antenna Design In this section, we establish dispersion properties of a baseline SP waveguide. These properties will determine the radiation characteristics of the final LWA design. Suppose we consider the planar corrugated surface shown in Figure 2-1. The structure consists of a groundplane with a periodic arrangement of transversely infinite vertical grooves.

20 6 Figure 2-1. Corrugated surface geometry. T=.5mm, G+T = 2mm and l=5mm. The length of a single corrugation is equal to G + T, where the thickness of the grooves is equal to G. The depth of each groove is represented by the parameter l, which has a value of 5 mm. We assume the structure to be infinite along the y-axis. It is important to understand that the dispersion relation between the frequency and wavenumber is controlled by the groove geometry. To solve for the dispersion relation, we first determine the wavenumber k x of the SPs that can be supported by such a structure. The Transverse Resonance Method (TRM) is utilized by setting the sum of the input impedance at the interface of free space (Z + ) and that of the corrugated surface (Z ) to zero. With respect to Figure 2-1, this approach yields the result: Z + + Z = By applying transverse resonance, Equation (2.1) can be represented as: (2.1) Z G G + T tan(k l) + k x 2 2 k =. ωε (2.2) In the above expression Z, k, and ε are the characteristic impedance, the wavenumber and the dielectric permittivity of free space, respectively. Given that k x is purely real, Equation (2.1) can be numerically solved with the resulting dispersion diagram shown in Figure 2-2. From the dispersion diagram, a wavenumber k x supported by the corrugation is greater than that of free space k. Consequently, these SPPs will not radiate. In this study, the operating frequency is 9 GHz which corresponds to k x = 1.437k.

21 7 Figure 2-2. Dispersion properties of the corrugated surface shown in Figure 2-1. At 9GHz, the guided wavenumber is k x = 1.437k. 2.3 Planar Leaky Wave Antenna Design Leaky Wave Antenna (LWA) design requires that Floquet s theorem be invoked [3]. The design principle of a periodic LWA is most similar to that of a uniform leaky wave antenna. The main difference is that the periodic LWA s slow wave (k x > k ) can be converted into a fast wave (k x < k ) characterized by an infinite number of Floquet space harmonic modes [3, 2]. For the purposes of this study, we assume a reactance surface (a corrugated surface in this case) that supports slow SWs. Normally, a waveguide that supports slow waves has the following slow wavenumber: k x > k. When a longitudinally-periodic discontinuous structure is added to such a waveguide, (i.e., an array of metal strips or notches) an EM field can be created and the wavenumber of the perturbed RS then becomes: k k x + n p jα. (2.3) In the expression (2.3), p is the period of the perturbation, n is an integer number that indicates the corresponding Floquet mode, and α is the leakage per unit length (leakage rate) of the

22 8 LWA. The modified wavenumber offers the advantage that for some n, the condition Re{ } k x 2 n k p may be satisfied and consequently, the fast wave can be created. This condition holds for all negative n values if the length of the periodic antenna structure is properly chosen. Usually, the n = 1 space harmonic (Floquet mode) is chosen to design the periodic LWA for single beam radiation. The radiation angle θ M of the LWA s main lobe is determined by k sin(θ M ) = k x p. (2.4) Figure 2-3. Side view of (a) the unperturbed structure and (b) the perturbed structure. In this study the periodic perturbation is introduced by simply converting the flat corrugated surface (Figure 2-3(a)) into a sinusoidally varying structure. This is illustrated in Figure 2-3(b). It should be noted here that the l of each of the structure s teeth remains constant, while only the PEC bed structure height changes. This keeps the wavenumber of the SP the same as the un-perturbed base structure, and forces the SW to follow a serrated path. The goal is to expose the SW to periodic structural fluctuations.

23 9 Figure 2-4.The tapering structure of the proposed leaky wave antenna. Amplitudes of the first 75 elements (5 Unit cells) increase in a cubed polynomial fashion. The 76 th to 325 th (2 Unit cells) have a constant amplitude of.5mm for the sinusoid perturbation structure. Amplitudes of the last 75 elements (5 Unit cells) decrease in a cubed polynomial fashion. Furthermore, we also introduce a taper to the PEC bed height in order to optimize the design to have maximum leakage rate and minimum energy-transfer to the receiving end. The optimum design consists of a 5 unit cell wave guiding structure on each end, paired with a tapered perturbation structure, the amplitude of which is shown in Figure 2-4. Prior studies have shown that the leakage rate (α) is directly linked to the half power beamwidth (Δθ) of the radiation pattern [25]. We discovered that by changing the amplitude of the sinusoids, we can control the leakage rate and essentially gain control over the half power beamwidth. Δθ a/( L λ cos θ M ) α/k.183 cos θ M (2.5) The proportionality factor a is dependent on aperture distribution (a.88 for a constant aperture distribution). Radiation efficiency can also be calculated using S11 and S12 values from the proposed antenna using: η = (1 S11 2 S12 2 ) 1%. (2.6)

24 1 2.4 Numerical Results For the periodic LWA under study, we set p = 3 mm. The antenna consists of 4 periods/unit cells. TEM waveguides are located at both ends of the LWA. Only one of them is excited and the other one serves as a load of the antenna. The number of unit cells, as an important geometric parameter for LWA design, was determined through extensive numerical experimentation. The length of the LWA plays a critical role in antenna performance because an effective aperture should be able to provide a long enough path for guided SP waves to leak most of the energy out before reaching the other end. Figure 2-5 shows the near field of the LWA antenna with a sinusoidal perturbation but without a tapering structure. As shown in this figure, the second Floquet mode radiating in the negative θ direction is strong. Figure 2-5.The near field of Leaky Wave Antenna with a flat SPP wave launching structure (no tapering). A strong 2 nd Floquet mode (n = 2) that radiates in the θ direction is observed. When the tapered structure (Figure 2-4) is introduced, a large reduction of the 2 nd Floquet mode is observed. Figure 2-6 shows the electric near field distribution from the proposed LWA. It can be seen that the SW has been successfully converted into a radiating mode that gradually detaches from the corrugated surface. The majority of the observed radiating field is due to effects of the n = -1 Floquet mode previously discussed.

25 11 Figure 2-6.The near field of the proposed Leaky Wave Antenna. Far less 2 nd Floquet mode (n = 2) raidation is observed in this near field plot. We can conclude that the tapering structure is more efficient than the flat launch LWA design by comparing the realized gain of the two aforementioned LWA antennas in Figure 2-7. The flat launch LWA has an un-desired 2 nd Floquet mode gain of 8.657dB, whereas the tapering structure LWA shows a gain of only.77db. As a result, all the discussion henceforth is based on the tapered structure.

26 12 Figure 2-7.The realized gain comparison of the proposed Leaky Wave Antenna with a flat SPP wave launching structure versus a tapered SPP wave launching structure. The main beam is the same while the 2 nd Floquet mode realized gain is 8.657dB for the flat SPP wave launch and.77db for the tapered SPP wave launch. Given the design specifications established in Section 2.2, we can now utilize Equation (2.4) to calculate an angle for the main radiating lobe. This calculated angle is predicted to occur at 19. The realized gain of the LWA, shown in Figure 2-8, is computed at three different frequencies: 8.8 GHz, 9 GHz and 9.2 GHz. First, the numerically predicted maximum radiation at 9 GHz occurs at 21. This agrees with the theoretically expected value. This is the result that validates our proposed design methodology. Second, it can be seen that as the operating frequency sweeps from 8.8 GHz to 9.2 GHz, the major lobe of the antenna scans from 18 to 22. For each of the main beams, minor lobes are observed at approximately -52. It can be shown from Equation (2.4) that these minor lobes result from the n = -2 Floquet mode.

27 13 Figure 2-8.The realized gain of the proposed Leaky Wave Antenna and the beam steering property over frequency. Furthermore, as mentioned in Section 2.3, we can tune the leakage rate by changing the geometric sinusoidal amplitude of the perturbation structure. Because leakage rate is directly linked to the half power beamwidth, we can control this value by altering the amplitude of the sinusoids in the perturbed structure. Figure 2-9 shows a comparison of the normalized directivities (in db) for sinusoids of three different amplitudes:.3mm, 1.5mm and 3mm. The resulting half power beamwidth values are 2.7, 3.1 and 8, respectively. This trend is not only limited to the (n = 1) mode; the second mode (n = 2) exhibits the same trend with increasing directivity.

28 14 Figure 2-9.The normalized directivity comparison of the proposed tapered periodic Leaky Wave Antenna structure for different amplitudes of the sinusoidal perturbation. A positive correlation observed between increasing sinusoidal amplitudes, the half power beamwidths, and the directivity of the 2 nd Floquet mode. Figure 2-1 shows the S11 and S12 of the proposed tapered LWA, where both values are used to calculate the radiation efficiency shown in Figure 2-11(b). At this target frequency, the efficiency is calculated to be 84%. Leakage rate is also calculated (Figure 2-11(a)) based on the sinusoidal amplitude of the structure. Figure 2-11(a) shows that the leakage rate increases as the amplitude increases.

29 15 Figure 2-1. S parameters from the proposed tapered structure. Figure (a) Leakage rate vs. the perturbation amplitude. (b) Radiation Efficiency vs. frequency for the proposed tapered structure.

30 Cylindrical Spoof SPP LWA Design In this section, we first establish the dispersion properties of the cylindrical baseline SP waveguide. These will determine the radiation properties of the final cylindrical LWA design. Consider the cylindrical corrugated surface shown in Figure The structure consists of alternating large and small cylindrical plates creating cylindrical corrugations. Figure 2-12.The cylindrical corrugation geometry. G = 1.5mm and T=.5mm. R in = 5mm and R out = 1mm. Similar to the planar LWA design, cylindrical corrugations are also being used to launch spoof SPP waves. The length of a single corrugation is equal to G + T. The large cylindrical plate has a radius R out and thickness T, where the small cylindrical plate has a radius R in and thickness G. The wavenumber k z of the SPs supportable by this type of structure needs to be determined in order to solve for the dispersion relation between wavenumber and frequency. The transverse resonance method (TRM) is applied at the interface between free space and the corrugated surface. With respect to Figure 2-12, this approach yields the resulting equations: Z ρ + + Z ρ = (2.7)

31 17 Assuming the TM 1 mode is sufficient in describing the electromagnetic field created by the cylindrical corrugation structure (Figure 2-12), the impedance expressions in Equation (2.7) can be expressed by [2] Z + ρ = j k (2) ρ H (kρ R out ) ωε (2) H 1 (kρ R out ) (2.8) and Z G J (k R out )Y (k R in ) J (k R in )Y (k R out ) ρ = jz G + T J 1 (k R out )Y (k R in ) J (k R in )Y 1 (k R out ) (2.9) In the equation shown above, H (2) n ( ) is the n-th order Hankel function of the second kind, J n ( ) is a Bessel function, and Y n ( ) is a modified Bessel function of the n-th order. In addition, k ρ is the radial wavenumber with a purely imaginary value. This is because k ρ is associated with the guided surface wave along the axis of the rod. k ρ = j k z 2 k 2 (2.1) Equation (2.1) describes k ρ, where k z is real and k z > k. Given that k z is purely real, Equation (2.7) can be numerically solved. Figure 2-1 shows the resulting dispersion diagram based on this solution. It is important to understand that the dispersion relation between the frequency and wave number is controlled by the radial groove geometry. The planar structure can be viewed as a type of cylindrical corrugation when the large cylindrical plates have infinite radii. Figure 2-13 verifies this claim. We see that for a fixed R in, the dispersion diagram of the cylindrical corrugation approaches that of the planar corrugation as R out increases.

32 18 Figure Dispersion diagram of the light line, planar structure (discussed in the previous section) and cylindrical groove structure with the same R in = 5mm and different R out, where (a) R out = 1mm and (b) R out = 1mm. We assume a target radiation angle of 3 with respect to the normal vector of the antenna s length. In this scenario, the required k z will be 1.611k (calculated using Equation (2.4)). The surface impedance for R in = 5mm and R out = 1mm pair is j1.413 (from Equation (2.9)). For a given R in = 5mm, the average corresponding R out needs to be.92 mm to maintain the same surface impedance. We can find each pair of R in and R out values yielding the desired surface impedance for a fixed range of sinusoidal amplitudes. This essentially provides the target radiation angle. Figure 2-14(a) shows the R in and R out pair for a desired surface impedance shown in Figure 2-14(c). It is important to note that the difference between R in and R out is also shown as a sinusoidally varying function (Figure 2-14(b)).

33 Figure (a) The R in and R out pair for the desired surface impedance. (b) The radius difference between R out and R in (c) The real and imaginary components of the corresponding surface impedance for each pair of R in and R out. 19

34 2 2.6 Numerical Results for Cylindrical LWA Similar to the periodic planar LWA under study, we set p = 3 mm for a cylindrical LWA. There are 4 periods (unit cells) in the proposed cylindrical LWA. Two TEM waveguides are placed on both ends and the one located at the origin excites the antenna. Extensive numerical experimentation has been performed to obtain the number of unit cells. This value is important because the antenna should be long enough to allow most of the guided SP energy to leak away as radiation before reaching the opposite end of the LWA. Figure 2-15 shows the cylindrical periodic LWA antenna with sinusoidal perturbations and without a tapering structure. The guiding structure used in a planar LWA has been verified as not necessary for a cylindrical LWA. Figure 2-15.The near field of the proposed cylindrical Leaky Wave Antenna. Figure 2-16 shows the normalized directivity for three cylindrical LWAs with different sinusoidal structure amplitudes. It is important to note that the target radiation angle is 3 in the +z direction away from the radial axis (perpendicular to the z-axis). Therefore, the main beam angle is complimentary when shown as a value of θ. It will be denoted as θ = 6. Similar to the claim for planar LWAs, changing the sinusoidal amplitudes will also change the beamwidth. A comparison of the normalized directivity (in db) for different sinusoidal amplitudes with values of

35 .15mm,.25mm and.35mm result in the half power beamwidths of 2.2, 2.8 and 5.4, respectively. 21 Figure The normalized directivity for cylindrical LWA for different sinusoidal amplitudes. As previously discussed, the target main beam radiation angle is θ = 6. The realized gain of the cylindrical LWA is shown in Figure 2-17(a) computed at three different frequencies: 8.8 GHz, 9 GHz and 9.2 GHz. First, the numerically predicted maximum radiation at 9 GHz occurs at θ = 61.2 which is in excellent agreement with the theoretically expected value. This confirms that the proposed design methodology also works for cylindrical LWA. Furthermore, it can be seen that as the operating frequency sweeps from 8.8 GHz to 9.2 GHz, the major lobe of the antenna scans from 57.5 to 64. Both S11 and S12 are shown in Figure 2-17(b), where both values are less than -1dB over the range of interest.

36 22 Figure (a) Beam Steering Property of the proposed cylindrical LWA. Realized gain for 8.8GHz, 9GHz and 9.2GHz are plotted. (b) The S parameters for the cylindrical LWA for a sweep of frequencies. 2.7 Overview To date, researchers have focused on studying the guiding properties of SPs along engineered surfaces. However, the effective conversion of localized SPPs into radiating waves is a research topic that needs to be fully explored. In this paper, we have described a simple and compact method for the design of SP based LWAs. Our proposed approach requires only minimal and cost effective structural modifications for creating a corrugated surface, which yield highly predictable and successful results. This is primarily a result of our design process being strongly rooted in the physics of electromagnetic antenna theory. It is expected that the development of spoof SPP based LWAs will have a large impact in modern antenna engineering. They are low profile, very simple from a manufacturing standpoint, conformal (since the SW is always attached to the corrugated surface) and they offer all the advantages of conventional LWAs: high gain and major lobe scanning as a function of frequency.

37 23 Chapter 3 Optical Frequency Plasmonic Nanoloop Antenna Analysis Loop antennas have been thoroughly analyzed in the microwave/rf regimes due to their simplicity and versatility. On the other end of the Electromagnetic spectrum (i.e., terahertz, infrared, and optical regimes), nanoloops show considerable promise for use in a variety of applications, including solar cells and optical sensors. However, a complete theoretical derivation of the radiation parameters of a nanoloop operating at these frequencies has not yet been performed because noble metals behave differently at these frequencies. This chapter will extend the formulation of thin-wire Perfect-Electric Conductor (PEC) loops to include the effects of loss and dispersion. Closed form expressions for the radiation fields, directivity and gain will be presented. These expressions can be evaluated within an order of seconds, whereas other simulations take on the order of hours to complete. 3.1 Introduction The theory of perfectly conducting antennas in the microwave and RF frequency ranges has been thoroughly studied and is well understood. There is a well-established literature base to aid in the design and physical understanding of these antennas. In particular, the conducting loop has been studied for its simplicity and versatility [21]. Recently there has been much interest in utilizing antennas at higher frequencies (terahertz, infrared and optical frequencies) for use in applications such as energy harvesting and wireless sensing [22]. Metals no longer behave as PECs when operating within the terahertz and optical frequency spectra. Rather, they exhibit dispersion and loss. Therefore, simply scaling an antenna such as a loop to work in these frequency range may yield unpredictable performance. A complete theoretical description of the nanoloop structure will

38 24 lead to a better understanding of the physical phenomena governing their radiation properties. This chapter will provide a set of closed-form expressions for various parameters of the nanoloop structure. These characteristics we discuss are the radiated fields, total radiated power, radiation resistance, directivity, and gain. The primary contribution of this work is to extend the previously derived integration procedures for the far field parameters of a PEC loop [23] to include the effects of dispersion and loss. These properties are exhibited by antennas in the optical regime [24]. In addition, we will present the results in a simple closed form. The closed form expressions for the aforementioned parameters will be compared with results obtained via full-wave numerical simulations. Several series representations for the integrals involved will be presented. 3.2 Theoretical Formulation Figure 3-1 depicts the geometry of a thin circular loop with wire having a diameter of 2a and a loop radius of b. Figure 3-1.Geometry of the thin circular loop (where the wire radius is far smaller than the loop radius, i.e., a 2 b 2 ). An infinitesimal voltage source is placed at φ = with a constant value of V. The derivation of the electric current and input impedance can be considered an extension of the Perfect Electric Conductor (PEC) case, which accounts for the lossy properties of noble

39 metals in the terahertz, infrared and optical frequency regions. Current on a PEC loop can be represented as a Fourier series [35] in terms of resonant modes as [37] 25 I(φ) = I m e jmφ = V [Y + Y m cos(mφ) ] m= m=1 (3.1) where the input impedance for each mode is Y = Z 1 = [jπη a ] 1 Y m = Z m 1 = [jπη (a m /2)] 1 (3.2) η in the preceding expression is the free space characteristic impedance, and a m are given by [35]- [36] The term k b = b λ a m = a m = k b ( N m+1 + N m 1 ) m2 N 2 k m (3.3) b (shown above) is unit-less. Its value depends on the loop radius and the operating frequency. For m 1, the functions N m are defined as N m = N m = 1 π [K ( ma b ) I ( ma b ) + C m] 1 2 2k b [Ω 2m(x) + jj 2m (x)]dx (3.4) m 1 C m = ln(4m) (2k + 1) k= where Ω m, J m, I and K are the Lommel-Weber, Bessel, and modified Bessel functions of the first and second kind, respectively. When m =, (3.4) can be simplified to N = 1 π ln (8 b a ) 1 2k b 2 [Ω (x) + jj (x)]dx. (3.5) The characteristic impedance of the wire Z s must be considered when working with an imperfect conductor. Z s for a cylindrical wire can be written as follows [44]-[45] Z s = γ J (γa) σ J 1 (γa) (3.6) where the propagation constant γ and the conductivity σ are both functions of the material

40 26 refractive index, η: γ = ω c η (3.7) σ = jωε (η 2 1) (3.8) The refractive index within frequencies ranging from microwave to optical regimes can be represented by a Drude-like model [3]. Hence, the refractive index expression may be expressed in a convenient analytical formulation [19] η 2 = 1 f 2 ω p ω ( 1 + ω j2γ M π j e γ m α ) ω j2βγ π j e γ m + f 2 mω p ( + ) 2ω m ω m ω + jγ m ω m + ω jγ m m=1 (3.9) where M is the number of critical points, ω p is the plasma frequency, f m are the quantum probabilities of transition, ω m are the critical points, Γ m are the Lorentz broadening terms, and the coefficients α, β are chosen to fit experimental data based on the DC conductivity. This equation describes an important property of an imperfect conductor when it is composed of noble metals in the terahertz, infrared and optical frequency ranges. When considering the imperfect conductors, the surface current on the loop Equation (3.1) can be extended and modified to include the characteristic impedance of the wire I(φ) = V [Y + Y m cos(mφ) ] m=1 (3.1) where Y = [jπη a + (b/a)z s ] 1 (3.11) Y m = [jπη (a m /2) + (b/a)(z s /2)] 1 The source currents in the Fourier series form enable the derivation of expressions for the far-zone electromagnetic fields. As a result, the associated far-field antenna parameters can be expressed conveniently. The far-zone electric field may be shown in spherical coordinates (θ, φ) as [33]-[34]

41 27 E θ = η e jk r cot θ 2r E φ = η e jk r k b 2r E r m=1 m= mj m I m sin(mφ) J m (k b sin θ) j m I m cos(mφ) J m (k b sin θ) (3.12) where J m is the derivative of the m-th order Bessel function, and the modal currents I m are derived from Equation (3.1) and (3.11) as D I m = m V jπη a m + Z = Y m V = (Z m ) 1 V s D = 1, D m = 2 ( for m = 1,2, ), (3.13) where Y m and Z m are the modal input admittance and impedance, respectively. The computational implementation of the current equation, and consequently the electric field equations, require solving the integrals of Lommel-Weber and Bessel functions. These are found in Equation (3.4) and (3.5). A detailed discussion of solution for solving such integral functions is provided in Appendices A and B. 3.3 Input Impedance, Radiated Power, Radiation Resistance and Efficiency In this section, we present several analytical representations for parameters that characterize the radiation properties of a loop antenna, namely the input impedance, radiated power, radiation resistance and efficiency. The input impedance of the thin-wire loop can be obtained from (3.14) Z in = V = [Y I + Y m in m=1 ] 1 (3.14) Next, the total radiated power is given by (3.12) π P r (k b ) = E 2 r 2 sin θ dθdφ, 2η (3.15) where E 2 = E θ E θ + E φ E φ. Using the orthogonality of sinusoidal functions shown below:

42 28 sin(mφ) sin(nφ) dφ cos(mφ) cos(nφ) dφ, m n = { π, m = n, m n = { π, m = n (3.16) the integral Equation (3.15) can be simplified as E θ E θ dφ = πη 2 cot 2 θ 4r 2 m 2 I m 2 J m 2 (k b sinθ) m=1 E φ E φ dφ = πη 2 2 k b 4r 2 I m 2 J 2 m (k b sinθ) m= (3.17) Hence, by substituting Equation (3.17) into (3.15), the total radiated power representation yields P r (k b ) = η 2 π πk b I 8 m 2 [ sin θ J 2 m (k b sin θ)dθ + m2 π m= 2 k cos2 θ sin 1 θ J 2 m (k b sinθ)dθ] b (3.18) A more compact way to represent Equation (3.18) is to invoke the Q-type integrals (see [2]-[23] for thin-wire PEC loops), which are defined as (p) (x) = J m (xsinθ)j n (xsinθ) sin p θ dθ Q mn π 2 (3.19) Because the integrant in Equation (3.18) contains multiplications of identical derivative Bessel functions and Bessel functions of the same order, a useful recurrence relation can be derived for the case where n = m: Q ( 1) mm (x) = x2 (1) 4m 2 [Q m 1 m 1 (1) (x) + 2Q m 1 m+1 (1) (x) + Q m+1 m+1 (x)] (3.2) Next, the expression for the total radiated power given in (3.18) can be simplified by employing Equation (3.19) and (3.2): P r (k b ) = η 2 πk b V 4 2 T(k b ) (3.21) where the modal current has been replaced by the multiplication of the source voltage V and modal

43 admittances Y m (see Equation (3.13)). The function T represents the following Q-type integrals: 29 T(k b ) = Y m 2 [ 1 2 Q (1) (k m 1 m 1 b ) Q (1) (k m+1 m+1 b ) m2 2 k Q (1) mm(k b )] b m= (3.22) A detailed derivation for the above expression can be found in Appendix C. The input radiation resistance given in [5] as a function of total radiated power and the input current is shown as R rad,in = 2P r(k b ) I in 2 = 2 Z in 2 P r (k b ) V 2 (3.23) Similarly, this expression can also be written in a more compact fashion using Q-type integrals and modal admittances through employing Equation (3.21): R rad,in = k b 2 πη Z 2 in 2 T(k b ) (3.24) Another possible way to express the radiation resistance is to include the maximum current [5] given by R rad = 2P r(k b ) I max 2 (3.25) The maximum current often occurs at the input terminals placed at φ = or at φ = 18. When the maximum current is at the input terminals, the two definitions for radiation resistance (Equation (3.24) and (3.25)) yield identical results. For the case of φ = 18, R rad can be shown in the following convenient form R rad = k b 2 πη 2 Next, the loss resistance is defined as [25] T(k b ) Y + ( 1) m m=1 Y m 2 (3.26) R loss = b Re(Z s ) 1 a I in 2 I(φ) 2 dφ = Re(Z s ) b Z in 2 [2 Y a Y m 2 ] m=1 (3.27)

44 Then, the radiation efficiency expression can now be derived by substituting Equation (3.24) and (3.27) into its well-known form 3 R rad,in e = = [1 + b Re(Z s ) R rad,in + R loss a k 2 b πη T(k b ) (2 Y 2 + Y m 2 )] m=1 1 (3.28) Because the expressions presented in this section include infinite series, in a practical implementation truncation is required. In general, the number of modes needed for the series to converge and achieve reasonable accuracy increases as the value of k b increases. Figure 3-2 shows the efficiency of a gold nanoloop with a computed circumference of 6nm versus a variety of different number of modes. Figure 3-2. Efficiency vs. k b for difference number of modes considered for a circumference b = 6nm and wire radius a = b nm (Ω = 12). 3.4 Radiation Intensity, Directivity and Gain The radiation intensity at an arbitrary point of interest can be defined using normalized farzone electric fields E θ = re jk r E θ and E φ = re jk r E φ. The radiation intensity is given in the form

45 31 U(θ, φ) = 1 2η [ E θ 2 + E φ 2 ] (3.29) where the squared magnitude of both non-zero electric far-field can be obtained using Equation (3.12): E θ 2 = V 2 η 2 cot 2 θ 4 [ mn( 1)n j m+n Y m Y n sin(mφ) sin(nφ) J m (k b sin θ)j n (k b sin θ) ] m=1 n=1 E φ 2 = V 2 η 2 2 k b ( 1) n j m+n Y [ m Y n cos(mφ) 4 cos(nφ) J m (k b sin θ)j n (k b sin θ) ] m= n= (3.3) Next, an expression for the directivity at the same point of interest can be found from this wellknown formula: D(θ, φ) = 4πU(θ, φ) P r (k b ) (3.31) by using expressions (3.21), (3.22), and (3.29). Expressions useful to describe the directivity for some special cases of interest can be obtained from (3.31) as D(⁰, ⁰) = Y 1 2 2T(k b ) D(9⁰, ⁰) = 2 T(k b ) j m+n Y [( 1)n m Y n J m (k b )J ] n (k b ) m= n= D(9⁰, 18⁰) = 2 T(k b ) j m+n Y m Y n [( 1)m J m (k b )J n (k b ) ] m= n= (3.32a) (3.32b) (3.332c) Finally, the gain of the nanoloop antenna can be expressed by multiplying the radiation efficiency and the directivity. This is shown in the following convenient form G(θ, φ) = ed(θ, φ) = D(θ, φ) [1 + b Re(Z s ) a k 2 b πη T(k b ) (2 Y 2 + Y m 2 )] m=1 1 (3.33)

46 Results Comparing to Simulations Gold loops with two different circumferences expressed as b (namely 6 nm and 3 mm) are considered when carrying out the validation of analytical solutions presented in the previous section. Equation Ω = 2 ln ( b ) = 12 (see [36]) is used to define the wire radius of both a loop models. The resulting wire radii are 9.3 nm and 46.7 nm for the 6 nm and 3 µm loop, respectively. The parameters of gold are modeled using the values listed in Table 3-1. Table 3-1. Parameters for refractive index of gold [46]-[47]. Parameters Value Parameters Value α 1.54 f 2.36 β ω f.37 γ 2 4. Ω.5 Γ f 1.2 f 3.6 ω ω 3 7. γ 1 4. γ 3 4. Γ 1.6 Γ ω p 9. The equations derived in Section 3.2 can be implemented through the use of a symbolic mathematical engine (i.e., Mathematica), or by using appropriate numerical methods (i.e., MATLAB). When implemented in MATLAB, the power series representations provided in the Appendices are used to replace the integrals containing Bessel, Lommel-Weber functions and the Q-type integrals. To further verify the accuracy of the analytical results, three full-wave frequencydomain tools are used to provide simulation results: a) The thin-wire Integral Equation (IE) solver [47]

47 33 b) FEKO solver (the full-wave Method of Moments) [48]; c) CST Microwave Studio (the Finite Element Method solver) [49]. When creating the model in both FEKO and CST, a voltage gap having negligible capacitance was created for the excitation. The expression for input impedance in Equation (3.14) was implemented in MATLAB, where the integrals involving Bessel and Lommel-Weber functions were evaluated using exact series representations discussed in Appendices A and B, respectively. Table II illustrates both the computational time and memory comparison results for each simulation method considered. Conducted tests were run on the same machine (a windows-based desktop containing dual Intel Xeon Processors with 1 cores). FEKO was run using all 1 cores in parallel, whereas the other simulation methods used a single core. In conclusion, the full-wave simulation methods required over 2 hours to complete, while the analytical solution in MATLAB completed within approximately 2 seconds. There is also a significant difference in memory usage; the fullwave solvers require 2-16 GB whereas the analytical solution requires only 16.7 MB. Table 3-2. Timing comparison for three different methods

48 34 Figure 3-3. Comparison of the total power radiated vs. k b evaluated using three different simulation techniques for the (a) 6nm circumference and (b) 3 nm circumference gold nanoloops. Radiation parameter results were considered for two gold loop antennas with two circumferences of 6 nm and 3 µm. Figure 3-3 and 3-4 show plots of the total radiated power and efficiency, respectively. The analytical results were obtained by using Equation (3.21) and (3.22) and compared to full-wave simulations obtained from FEKO and CST. As shown in Figure 3-2, approximately 35 modes were sufficient to accurately determine the efficiency for all k b values up to 5. Hence, 35 modes were used for all analytical results presented in this section. The analytical and full-wave methods are in near perfect agreement. Noticeable discrepancy in the integral equation (IE) based solution has also been observed. This is primarily because this method employs a piecewise approximation to describe the loop curvature. This approximation resulted in slightly higher radiated power and efficiency when comparing with the actual values [51].

49 35 Figure 3-4. Comparison of efficiency vs. k b evaluated using four different simulation techniques for the (a) 6nm circumference and (b) 3 nm circumference gold nanoloops. To further analyze these results, we must consider the effective wavelength concept presented in [3]. The effective wavelength essentially accounts for the propagation of the plasmonic mode (i.e., the TM mode). Therefore, the physical phenomena in the infrared/optical frequency regions can be described by using the effective wavelength. The maximum radiated power and efficiency occur at the antenna s resonant frequency. Multiple resonances are observed within the frequency range of interest. This result explains the 6 peaks observed in the k b value range from to 2.5 for the 6nm loop (Figure 3-4(a)), and 7 peaks are seen in the k b value range from to.5 for the 3nm loop (Figure 3-4(b)). The maximum total radiated power for each loop occurs at the first resonant mode, followed by successively decreasing increments as frequencies increase. This behavior is similar to that of microwave loop antennas [5]. Additionally, higher values of the radiated power and efficiency are observed as the loop size increases. Figure 3-5 shows a MATLAB implementation of both the input radiation resistance and loss resistance given in (3.24), (3.26) and (3.27), respectively. The maximum input radiation resistance of the 6 nm circumferences loop is only 4 Ohms. This value, in an addition to low input admittance and driving current, leads to the low total radiated power indicated in Figure 3-3 (a). Moreover, the loss

50 36 resistance for this 6nm circumference loop is very high, resulting in an efficiency of only.7% (shown in Figure 3-4 (a)). However, the maximum input radiation resistance is around 8 Ohms, with a corresponding efficiency of above 6% in the 3 nm circumference case. Figure 3-5. Input radiation and loss resistance vs. k b for (a) 6nm circumference and (b) 3nm circumference gold nanoloops. Figure 3-6 shows an implementation of the radiation resistance in terms of the maximum current given in (3.25). The curves in Figure 3-6 have the same shape as those of the input radiation resistance in Figure 3-5. However, the maximum value for the 6 nm loop is now about 2 Ohms and the max for the 3 nm loop is about 4 Ohms. These values are about half of what we saw from the input resistances for both loops.

51 37 Figure 3-6. Radiation resistance vs. k b for the (a) 6nm circumference and (b) 3nm circumference gold nanoloop. To validate the expressions given in (3.31)-(3.32c) for directivity, both MATLAB and Mathematica implementations are performed and their results are compared with full-wave solvers. The directivity for the 6 nm loop evaluated at (θ, φ) = (9, ) and the 3 nm loop evaluated at (θ, φ) = (, ) are shown in Figure 3-7 to illustrate the comparison. Both MATLAB and Mathematica yield nearly equivalent values, which are in reasonable agreement with the full-wave solvers. 3D directivity plots are generated using FEKO for the 3 nm loop antenna for k b =.1,.5, 1.1 and 2.5 in Figure 3-8. Similar radiation/directivity patterns are observed for the 6 nm loop. For k b =.1, PEC loops show an omnidirectional far-field pattern in the loop plane with a null in the normal direction (similar to that of a magnetic dipole). However, a directivity pattern with a peak of 6 dbi is exhibited by the nanoloop for k b =.1, shown in Figure 3-9 (a). This directivity pattern is caused by the non-symmetric current distribution resulting from the losses of the gold material. The non-symmetric current distribution results in constructive interference for the case where (θ, φ) = (9, ). For the PEC case, the peak directivity gradually shifts to the

52 38 normal direction (θ, φ) = (, 18 ) from k b = to 1. [5]. Peak directivity shifts to the normal direction for the nanoloop at around k b =.5, as shown in Figure 3-8 (b). However, at around k b = 1.1, the peak shifts to (θ, φ) = (9, 18 ) with a directivity of 7.5 dbi, as shown in Figure 3-8 (c). This is again due to the loss of the gold material, where extremely strong constructive interference occurs at this frequency. Lastly, the directivity pattern becomes omnidirectional in the x-z plane at around k b = 2.5 and stabilizes for larger k b, (see Figure 3-8 (d)). In this frequency range, only the source and its surrounding loop area are radiating. This leads to a radiation pattern similar to nanodipoles, which are quite different from the behavior of PEC loops. The far field patterns of PEC loops begin to exhibit multiple lobes at around k b = 2., increasing more when k b > 2., subsequently resulting in more complex patterns. These far field pattern trends are valid for all PEC loops despite circumference values. However, unlike PEC loops, the behavior of nanoloops relies considerably upon the circumference and material properties. Figure 3-7. (a) Directivity of the 6nm loop evaluated at (θ, φ) = (9, ). (b) Directivity of the 3nm loop evaluated at (θ, φ) = (, ).

53 39 Figure 3-8. Directivity corresponding to a 3nm loop for k b values of (a).1 (1THz), (b).5 (5THz), (c) 1.1 (11THz) and (d) 2.5 (25 THz). To fully understand these trends, the directivity expressions given in (3.32a)-(3.32c) were implemented in MATLAB. This generates the results shown in Figure 3-9 for the three different angles of interest. In Figure 3-9(a), the two largest directivities occur at (θ, φ) = (9, ) around k b =.1, and at (θ, φ) = (9, 18 ) near k b =.175 for the 6 nm loop. As k b exceeds this value, the pattern becomes omnidirectional in the xz-plane and remains that way as the frequency continues to increase. Because the current has low magnitudes and is symmetric about the yz-plane, the directivity remains low for all frequencies. Similar directivity versus frequency behavior has been observed for the 3nm circumference loop, but with more ripples due to the existence of higher order modes in the currents.

54 4 Figure 3-9. Directivity of the (a) 6nm loop and (b) 3nm loop corresponding angles of (θ, φ) = (, ), (θ, φ) = (9, ) and (θ, φ) = (9, 18 ). Remarkably, superdirectivity is attained along the (θ, φ) = (9, 18 ) direction over a narrow band for the 6 nm loop and over a broad band for the 3 nm loop. It is well-documented that superdirectivity along the end-fire direction of an antenna can occur when two PEC monopoles are excited with the same current magnitude but opposite phases. This phenomenon does not occur for PEC loops because it is a direct consequence of the material properties of gold, which result in attenuation of the current on the nanoloop. An analysis of the current distribution indicates that this phenomenon occurs naturally on the nanoloop over a certain frequency range. This surprising behavior can be effectively and efficiently predicted using the directivity equation (3.32c). Finally, the gain defined in Equation (3.33) is implemented for the three directions under consideration, shown in Figure 3-1. As expected, the gain is much lower for the 6 nm loop since it corresponds to an electrically smaller antenna having both a lower efficiency and directivity than those of the 3nm loop. Gain decreases as the frequency increases due to the decreasing efficiency described in Figure 3-4. In addition, both the gain and directivity maxima for the 3

55 41 nm circumference case appear when k b 1 at (θ, φ) = (9, 18 ). This frequency, k b 1, corresponds to L/λ eff 2.5 (i.e., around the third resonance), as indicated by the equation in [1]. In microwave applications, most designs operate at the first resonance (L/λ eff.5). In contrast, these results imply that nanoloops can be used at higher-order resonances in the terahertz, infrared and optical frequency ranges where the pattern is more directive and the gain is larger. Figure 3-1. Gain of the (a) 6nm loop and (b) 3nm loop corresponding to angles of (θ, φ) = (, ), (θ, φ) = (9, ) and (θ, φ) = (9, 18 ) 3.6 Conclusion Efficient and compact closed-form expressions for the radiation properties of lossy nanoloops in the terahertz, infrared and optical regimes have been presented in this paper. In particular, analytically described closed-form far-zone radiated electric fields, total radiated power, radiation resistance, directivity, and gain expressions are given. These expressions are derived by extending the known formulation for thin-wire perfect-electric conductor (PEC) loops to include the effects of dispersion and loss from the imperfect conductors. Closed form expressions were shown to be in agreement with the full-wave simulations of gold nanoloops based on different

56 42 computational electromagnetics techniques. The full-wave solvers required more time and memory as compared to the analytical solutions implemented in MATLAB. Lastly, we have contributed guidelines for computing the required integrals of the Bessel and Lommel-Weber functions, as well as the Q-type integrals, described in Appendices A, B and C.

57 43 Chapter 4 Optical Frequency Plasmonic Nanoloop Antenna Coupling Analysis In this chapter, we present a closed form analytical solution for the problem of mutual coupling between nanoloops. This is an extension of the single nanoloop concept discussed in Chapter 3. The mutual coupling between two nanoloop antennas is analyzed by formulating the induced current equation on the passive loop. In 198, Abul-Kassem and Chang [59] derived moment functions associated with the mutual coupling between two loops through using a double Gaussian quadrature scheme. In this chapter, we provide another approach using only the electric field from the active loop and a coordinate transformation. Results of the closed form analytical method have been verified with Method of Moments (i.e., FEKO) and numerical integration methods for a wide range of angles and distances. Furthermore, results for special cases of the coplanar and stacked nanoloops are simplified and verified. 4.1 Introduction Loop antennas have long been considered the basic antennas for transmitting and receiving signals due to their practical applications as probe antennas for geophysical explorations, submarine antennas, etc. These antennas also have applications in direction finding, and as a result, their RF properties have been studied in depth [5, 63]. With the development of nanotechnology and metamaterials, the nanoloop antenna has re-gained attention among researchers. Presently, not only are the properties of the single nanoloop of interest, but the mutual coupling characteristics are also useful in the design process. To the best of our knowledge, the first approach to analyze element admittances of a pair of staggered loop antennas was performed by Bhattacharya and Bandyopadhya [6]. A staggered configuration has the advantage of achieving two distinctive

58 44 amplitudes and phases. In this chapter, we will analyze the mutual coupling between two nanoloop antennas, and provide analytical expressions for the induced current on the passive loop using the electric field expression provided in [33]. A thorough understanding of the mutual coupling between nanoloop antennas is important because it can be applied to antenna array synthesis. There are several properties of nano-scale antenna arrays that have interested researchers and scientists since their discovery. These arrays exhibit properties that give them high directivity and consequently enable them to steer light. These properties allow for the use of these arrays in several valuable applications such as solar harvesting [65] and directive scattering. It was not until 213 that researchers had completely developed closed-form representations of these antenna s radiation resistance, radiation efficiency, and input impedance properties. A complete analysis of these antenna array types has yet to be performed. One specific property of interest to be later addressed in this chapter is the mutual coupling parameter. However, performing a full wave analysis on these types of structures require large computational resources. The development of a closed-form analytical representation of the mutual coupling between array elements would mitigate the required computational time, and will potentially contribute substantially to the understanding the physics of nanoloop antenna array coupling. Representing the current distribution on a PEC thin-wire circular loop by a Fourier series [33] enables the efficient calculation of radiation properties [25, 37]. It simplifies the derivation for closed-form representations of vector potential. Consequently, exact near-zone and far-zone electric and magnetic field representations can be derived [33]. By including the imperfect conductor material properties (i.e., dispersion and loss), these equations for a PEC loop can be extended to represent nanoloops in the optical frequency region. This chapter provides a complete closed-form induced current solution for nanoloop arrays, where the mutual coupling equation integrals can be simplified using far-zone approximations. The proposed closed-form analytical representations can be written as summations of Bessel functions and derivative Bessel functions.

59 45 The full-wave Method of Moments (i.e., FEKO) solver requires approximately ten hours to compute a result, whereas the analytical solution and numerical quadrature solver can produce a result in under a minute. Some useful guidelines for deriving efficient analytical solutions are also provided in this chapter. 4.2 Formulation of the Coupling Current Equation for Two Coplanar Nanoloop Antennas Consider two thin circular loop antennas with radii of b, and wire radius of a arranged in the co-planar configuration shown in Figure 4-1. The electric current and input impedance can be derived by considering the loss properties of noble metals (i.e., gold and silver) in terahertz, optical and infrared frequency ranges. We denote the active loop using non-primed coordinates, and the passive loop using primed coordinates. Figure 4-1. Two nanoloop antennas on the same plane, θ = π 2. The left loop (blue) is the active loop with a voltage source and the loop on the right (green) is the passive loop without voltage sources. The distance between the centers of the two loops is defined as r. The active loop coordinate can be expressed as (r, π 2. φ) and the passive loop coordinate can be expreseed as (r, π 2, φ ).

60 46 We first use far-field expressions when deriving the induced current on a passive loop located in the far-field region of the active loop. Similar to Chapter 3, we assume the loop has an arbitrary current distribution represented by the following Fourier decomposition [33]. I(φ) = I n cos(nφ) n= (4.1) This form of current distribution is very general and has been applied for the analysis of loop antennas in a wide variety of environments [5, 25, 63]. The simplified general far-zone Electric field for arbitrary loop current distributions (4.1) can be shown as [33] E r E θ η cot θ e jk r n(j) n I 2 r n sin(nφ)j n (k b sin θ) n=1 (4.2) E φ = η k b e jk r 2 r (j) n I n cos(nφ) J n (k b sin θ) n where sin θ = 1 for the co-planar case. Equation (4.2) can then be simplified to E r E θ = (4.3) E φ = η k b e jk r 2 r (j) n I n cos(nφ) J n (k b ) The induced current on the passive loop can be expressed as n I φ (Φ) = V Y m cos(m(φ φ )) m= (4.4) where Φ denotes the location on the passive loop, Y m is the input impedance for each mode (shown in Equation (3.1) and (3.2)), and V is the induced voltage shown as

61 47 V = E φ dl = E φ bt dφ (4.5) where t is the tangential vector on the second loop. Figure 4-2. Two nanoloop antennas on the same plane. The unit vector φ from the non-primed coordinate and the tangential unit vector t are illustrated. Note that E φ is in terms of non-primed coordinates and t is in terms of primed coordinates. In order to perform the dot product, we first perform a coordinate transformation to transform all the non-primed coordinate parameters into the primed coordinates through using a Cartesian coordinate system. From Figure 4-3, we create three vectors: v 1 and v 2 v 1 = (b cos φ, b sin φ ) (4.6) v 2 = r = (ξ, η) By using vector addition: v 3 = v 2 + v, 1 we have where (x, y) = (ξ + b cos φ, η + b sin φ ) (4.7) η = r sin φ (4.8) ξ = r cos φ (4.9) Therefore, we can express x = ξ + b cos φ and y = η + b sin φ.

62 48 Figure 4-3. Two nanoloop antennas on the same plane. The left loop (blue) is the active loop with a voltage source place at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the two loops is defined as r, with an x-component of ξ and y-component of η. (x, y) denotes any arbitrary point on the passive loop. We use the following coordinate conversion tensor to convert active loop spherical coordinates to passive loop Cartesian coordinates, φ = [ sin φ cos φ ] (4.1) where sin φ = x r, (4.11) cos φ = y r, (4.12) and r = x 2 + y 2. = b 2 + r 2 + 2r b cos(φ φ ) (4.13)

63 49 Figure 4-4. Passive loop. The tangential vector t is illustrated. As shown in Figure 4-4, the tangential vector on the passive loop can be expressed as t = [ sin φ cos φ ] (4.14) and the dot product between φ and t in terms of the primed coordinates can be shown as y sin φ φ t = r x cos φ + r (4.15) = 1 r [r cos(φ φ ) + b 2 ] Hence, the expression for the induced current can be written as I(Φ) = η k b b 2r 2 e jkr (b + r cos(φ φ )) (j) n I n T n ( x r ) J n (k b ) 1 cos(m(φ φ )) dφ Z m n= m= (4.16) where T n ( ) is the Chebyshev polynomial defined in (4.17) T n ( x r ) = T n(cos φ) = cos(nφ) (4.17) The distance from the center of the active loop to any point on the passive loop can be simplified by the employing the following assumptions

64 5 r = x 2 + y 2 = (r cos φ + b cos φ ) 2 + (r sin φ + b sin φ ) 2 (4.18) = r 2 + b r b cos(φ φ ) If we assume the passive loop is in the far-zone of the active loop, then r b and Equation (4.17) can be expressed as r r 1 + 2b r cos(φ φ ) (4.19) This can be represented as the binomial expansion shown in (4.2). r r [ (2b r cos(φ φ )) 1 8 (2b r cos(φ φ )) + ] (4.2) Without loss of generality, we can use the far field approximation in (4.21) and (4.22) to simplify the current equation in (4.23): r r for amplitude terms { r r (1 + b cos(φ r φ )) for phase terms (4.21) In addition, we can assume that φ φ if the passive loop is located in the far zone of the active loop, then T n ( x r ) = cos(nφ) = cos(nφ ) (4.22) This approach leads to the following simplified current integral I(Φ) = η k b b 2r 2 (j)n I n cos(nφ ) J n (k b )e jk r n= e jk b cos(φ φ ) (b + r cos(φ φ )) Y m cos(m(φ φ )) dφ m= (4.23) By using the following generating Bessel function,

65 51 e x(t 1 t )/2 = J k (x)t k, t k= (4.24) The exponential term in the integrant can be shown as e jk b cos(φ φ ) = J k ( k b) (j) k e jk(φ φ ) k= (4.25) We apply Euler s identity along with the Equation in (4.26) to solve the integral shown in (4.23). e jpφ dφ, p = {, p = (4.26) The induced current on the passive loop can be fully simplified with the derivative Bessel function identity J v (z) = 1 (J 2 v 1(z) J v+1 (z)) and then solved as a closed-form analytical solution (4.27). I(Φ) = η k b bπ e jk r (j) n I r n J n (k b ) cos(nφ ) n= Y m (((j) m 1 J m ( k b) + b r (j) m J m ( k b))) cos(m(φ φ )) m= (4.27) Appendix D includes detailed induced current derivations for the coplanar nanoloops. It is important to note that the current equation in (4.27) can be written as a Fourier decomposition. Therefore, we can write the far-zone electric field caused by the induced current by following a procedure in [33]. Calculations of the induced far zone electric field are beyond the scope of this thesis. However, the far field of any nanoloop array can be derived through using the induced farzone electric field equation with an array factor. This will likely reduce computation time considerably when computing induced far fields and other radiation properties of nanoloop arrays.

66 Analytical Solution Compared to Numerical Simulation for Coplanar Nanoloops In this section we compare three methods of finding the induced current on the passive loop located on the same plane as the active loop. The methods compared are as follows: - The FEKO (method of moment) simulation - A numerical integration method (shown in Equation (4.16)) that does not incorporate any far field assumptions. - A closed form analytical solution derived from solving Equations (4.16) and using far field assumptions. Figure 4-5. Induced current comparison between FEKO result, numerical integration and closed form analytical solution on the passive PEC loop located at φ = and (a) 7 times and (b) 14 times the radius away from the center of the active loop. Good agreement is observed. Figure 4-5 and Figure 4-6 show a near perfect agreement between all three methods when the passive PEC loop is located at φ = and at φ = 3 and distances 7 times and 14 times the radius away from the center of the active loop.

67 53 Figure 4-6. Induced current on the passive PEC loop located at φ = 3 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are all plotted. Fairly good agreement is observed. Figure 4-7 shows that the numerical integration and closed form analytical methods begin to break down when the passive loop is located at φ 45. This is because the assumption of the far zone electric field E r = no longer holds inside this range of angles. Figure 4-7. Induced current on the passive PEC loop located at φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are all plotted. The second peak was not captured by both numerical and analytical solutions and is due to inaccuracies in the radial component of the electric field.

68 54 When φ is within the range φ < 45, the results agrees well, however the E r component becomes more prominent outside this range of angles. This was not observed in Figure 4-3 and Figure 4-4 where φ is outside the range. When using the E r = assumption, both the distance and the angle φ must be considered. Figure 4-8 further reinforces our understanding of this phenomena by showing that the discrepancy between the FEKO and both numerical and analytical methods diminish as the distance between the loops increase. Figure 4-8. Induced current on the passive PEC loop located at φ = 9 (a) 7 times, (b) 14 times, (c) 28 times and (d) 56 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. The first peak was not captured by both numerical and analytical solutions and is a result of inaccuracies in the radial component of the field.

69 Formulation of the Coupling Current Equation for General Configuration of Two Nanoloop Antennas Configurations of coplanar nanoloops can comprise antenna arrays for use in practical applications, but other nanoloop configurations are also possible in array design. It is important to have a generalized induced current equation for any of these configurations. In this section, we present a generalized induced current equation on the passive loop located anywhere relative to the first loop. There will be two induced current components, I θ and I φ. This is due to E θ and E φ being non-zero (i.e., E r = ). Consider the configuration in Figure 4-5, we can convert both primed and non-primed spherical coordinates into primed Cartesian coordinates for the passive loop. Since φ is always in an x-y plane, we make a projection of the passive loop into the same plane where the active loop is located. Let the passive loop be centered at (ξ, η, z ). In addition, the distance between the center of the active loop and the center of the projection is denoted as r, and the distance between the center of the active loop and the center of the passive loop is denoted as R. Similarly, we denote the distance between the center of the active loop and any arbitrary point on the passive loop as R, and the distance between the center of the active loop and the projection of any arbitrary point on the passive loop as r. These values are represented as: R = x 2 + y 2 + z 2 r = x 2 + y 2 R = (ξ + b 2 cos φ ) 2 + (η + b 2 sin φ ) 2 + z 2 = r 2 + z 2 (4.28)

70 56 Figure 4-9. Two nanoloop antennas in a general configuration. The left loop (blue) is the active loop with a voltage source at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the active loop and the projection of the passive loop is defined as r, which x-component is ξ and y-component is η. (x, y) here denotes an arbitrary point on the passive loop. Similar to the co-planar case, x = ξ + b cos φ and y = η + b sin φ. The conversion from non-primed Spherical coordinates to non-primed Cartesian coordinates is defined in (4.29) r = [sin θ cos φ sin θ sin φ cos θ] { θ = [cos θ cosφ cos θ sin φ sin θ] φ = [ sin φ cos φ ] (4.29) where each of the sine and cosine terms can be expressed by primed coordinates parameters x and y. cos θ = z R = z x 2 + y 2 + z 2 sin θ = 1 cos 2 θ = x2 + y 2 x 2 + y 2 + z 2 = r R (4.3)

71 57 cot θ = z r x cos φ = x 2 + y = x 2 r y sin φ = x 2 + y = y 2 r First, we solve for the current θ component induced by the far-zone electric field E θ shown in Equation (4.2). The conversion from non-primed Spherical to primed Cartesian coordinates for unit vector θ can be shown as: θ = [ z x R r z y R r r R ] (4.31) The tangential unit vector t on the passive loop is shown in Equation (4.14). The dot product of θ and t becomes: θ t = z x Rr ( sin φ ) + z y (cos φ ) Rr = z Rr (η cos φ ξ sin φ ) (4.32) The voltage θ component can be expressed by: V θ = E θ ( θ bt )dφ = η z b 2 cot θ 2 e jk R R 2 r (4.33) (r sin(φ φ )) n(j) n I n J n (k b sinθ) sin(nφ) dφ n=1 All terms in Equation (4.34) with the exception of the sin(nφ) term can be written in with respect to φ by substituting equations (4.3) and (4.32) into (4.33)

72 58 I θ (Φ) = η z 2 b 2 r 2 n(j) n I n Y m n=1 m= J n ( k br R ) e jk R R 2 r 2 (4.34) (sin(φ φ )) sin(nφ) cos(m(φ φ )) dφ Because the passive loop is located in the far-zone of the active loop, we assume the distance between loop centers to be considerably larger than the loop radius (i.e., r b). The following approximation then holds: r r for amplitude terms { r r (1 + b 2 cos(φ r φ )) for phase terms (4.35) and: R R for amplitude terms { R r 2 + z 2 (1 + b 2r cos(φ φ )) for phase terms R 2 (4.36) In addition, we can also assume that φ φ, then the current equation may be re-written as I θ (Φ) = η 2 b 2 z 2 2r R e jk R n(j) n I n J n ( k br ) sin(nφ R ) Y m e jk b 2 r cos(φ R φ ) sin(φ φ ) cos(m(φ φ )) dφ n=1 m= (4.37) Similarly, by using generating Bessel function (4.24) and applying (4.26), the current equation can be solved and expressed as a closed form analytical equation, I θ (Φ) = η z 2 π r 2 R n(j) n I n J n ( k br ) sin(nφ R ) e jk R ( (j) m 1 Y m m sin(m(φ φ )) J m ( k b 2 r n=1 )) [ R ] m= (4.38)

73 Secondly, we perform the same procedure for I φ (Φ). The φ Spherical to Cartesian coordinate conversion was shown in Equation (4.29). This can be re-written as 59 φ = [ y r x r ] (4.39) The tangential component t was shown in Equation (4.14). The dot product of φ and t then becomes φ t = 1 r [r cos(φ φ ) + b 2 ]. (4.4) The φ component of the induced current can be written with the far-zone E φ expression (4.2) as I φ (Φ) = η k b b 2 2 e jk R rr (j) n I n cos(nφ) J n ( k br R ) dφ (4.41) n= (b 2 + r cos(φ φ )) Y m cos(m(φ φ )) [ m= ] We substitute Equation (4.38), (4.39) and φ = φ into (4.41) to make a far-zone approximation. The result of which is shown in (4.42). I ϕ (Φ) = η k b b 2 e jk R (j) n I 2r R n cos(nφ ) J n ( k br ) R n= (4.42) Y m e jk r m= R cos(φ φ ) (b 2 + r cos(φ φ )) cos(m(φ φ )) dφ Similarly, by using the generating Bessel function in (4.24) and applying (4.26), the current equation can be solved and expressed as a closed form analytical equation. I φ (Φ) = m= η (j) n I n J n ( k br ) cos(nφ k b b 2 π R ) (j) m 1 Y m e jk R R ( b 2 (j)j [ [ r m ( k b 2 r ) + J R m ( k cos(m(φ φ b 2 r )) m= )) R ] ] (4.43)

74 6 Finally, the total induced current on the passive loop centered at (r, θ, φ ) in the far-zone of the active loop is now in a closed-form analytical representation. This representation assumes E r is approximately zero. I(Φ) = I θ (Φ) + I φ (Φ) (4.44) Appendix E includes a detailed derivation procedure of all steps in Section Comparison of Analytical Solutions to Numerical Simulations for General Nanoloop Configurations In this section we compare three methods of finding the induced current on the passive loop located anywhere in free space relative to the active loop. In this section, we still assume the far-field approximation E r = holds. The three methods compared are as follows: - The FEKO (Method of Moments) simulation - A numerical integration method (I total = I θ + I φ, where I θ can be found in Equation (4.38) and I φ can be found in Equation (4.43)) that does not incorporate any far field assumptions. - A closed form analytical solution using far field assumptions derived from solving I total = I θ + I φ, where I θ can be found in equation (4.38) and I φ can be found in equation (4.43).

75 61 Figure 4-1. Induced current component on the passive PEC loop located at θ = 3, φ = 2 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are each plotted. Figure 4-1 and Figure 4-11 show a near perfect agreement between all three methods when the passive loop is located at θ = 3, φ = 2 and at θ = 45, φ = 45 for two PEC loops located distances 7 times the radius away and 14 times the radius away from the center of the active loop. However, discrepancies are observed when testing other angles, such as θ = 8, φ = 8. This is also because the E r = far-zone approximation is less true when the observation point is along directions where φ > 45. Modification of this approximation needs to be performed so that the analytical solution will account for all angles.

76 62 Figure Induced current on the PEC passive loop located at θ = 45, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Good agreements are observed. 4.6 Formulation of the Coupling Current Equation for the r Component We have discovered that the E r term is necessary to be included in the induced current equation because it slowly decreases in the φ = 9 direction relative to the φ = direction. In this section we will apply the far-zone approximation on the E r near-field expression [33].

77 63 Figure Two nanoloop antennas in the most general configuration. The left loop (blue) is the active loop with a voltage source located at φ =. The loop on the right (green) is the passive loop with no voltage source. The distance between the centers of the active loop and the projection of the passive loop is defined as r. The x-component of the projected loop center is denoted as ξ and the y-component of the projected loop center is denoted as η. (x, y) represents any arbitrary point on the passive loop. The near field E r expression is given in [33]. E r (r, θ, φ) = ηk 2 b 1 sin θ 4 (k b ) 2 C 3 mn sin(nφ) [ k 2 b 1 r 1 sin θ ] 2 [ m=1 m n=1 m n=2k k=,1, m 1 (2) hm+1 (k R 1 ) (k R 1 ) m+1 (4.45a) m C 4 mn sin(nφ) [ k 2 m 1 b 1 r 1 sin θ (2) hm (k R 1 ) ] 2 (k R 1 ) m, m=1 n=1 m n=2k k=,1, ]

78 64 where n 3 = I n m n + n [ 2 ]! [m 2 ]! C mn (m + 1)n 4 3 C mn = (m + 1)C mn = I n m n + n [ 2 ]! [m 2 ]! (4.45b) (4.45c) Substituting the following far-field approximations into (4.45a) yields R for amplitude term r 1 = R 1 = R = { R (1 + b 2r cos(φ φ )) for phase term R 2 sin θ = sin θ = r R φ = φ lim h (2) ejx x l (x) = (j) l+1 x (4.46a) (4.466b) (4.46c) (4.46d) We then factor the common terms shown in (4.47). E r (r, θ, φ) = ηk 2 b 1 sin θ 4 m sin(nφ ) m=1 n=1 m n=2k k=,1,2.. [ k 2 m 1 b 1 sin θ ] 2 (4.47) k 2 2 R [jk b C mn (j)m+1 R C 4 mn ] e jk R We incorporate the far-field approximation when solving the dot product of r and t. r b 2 t = x R ( sin φ ) + y R cos φ r b 2 sin(φ R φ ) (4.48) We can now write the current term as

79 65 I r (Φ) = E r (r b 2 t ) Y p cos(p(φ φ )) dφ p= = ηb 2 1b 2 r 4 sin(nφ 4R ) m=1 m n=1 m n=2k k=,1,2.. [ k 2 m 1 b 1 sin θ ] (j) m+1 e jk R 2 (4.49) [ jk b C mn R C 4 mn ] Y p T p= where T = e jk b 2 r R cos(φ φ ) sin(φ φ ) cos(p(φ φ )) dφ (4.5) Let us evaluate the integral T using the generating Bessel function, T = 1 4j J k ( k b 2 r ) (j) k k= R +pφ) ej((k+1)φ e j(k+1+p)φ + e j((k+1)φ pφ) e j(k+1 p)φ dφ e j((k 1)φ +pφ) e j(k 1+p)φ e j((k 1)φ pφ) e j(k 1 p)φ (4.51) The integral has non-zero terms when k = p 1, p 1, p + 1 and p + 1. We simplify each non-zero term using the Bessel function identity J m (x) = ( 1) m J m (x). The integral can then be written as T = π sin(p(φ φ )) (j) p 1 [J p 1 ( k b 2 r R ) + J p+1 ( k b 2 r R )] (4.52) We apply a second Bessel function identity shown in (4.53) 2p x J p(x) = J p 1 (x) + J p+1 (x) (4.47) this leads to a simplified closed form solution for the integral T T = pr r k b 2 sin(p(φ φ )) (j) p 1 J p ( k b 2 r R ) (4.48) When substituting (4.54) into (4.49), the current can be expressed as

80 66 I r (Φ) = πηb 1r 2R 3 sin(nφ ) k [ jk b C mn R m=1 m n=1 m n=2k k=,1,2.. [ k 2 m 1 b 1 sin θ ] (j) m+1 e jk R 2 C 4 mn ] Y p p sin(p(φ φ )) (j) p 1 J p ( k b 2 r ) R p= (4.49) 4.7 Comparison of Analytical Solutions with Numerical Simulations for General Nanoloop Configurations Including the r Component The results illustrated in this section for PEC loops are similar to those of Sections 4.3 and 4.5 with the exception that the previous assumption E r = is no longer used. Detailed derivations including the far zone E r component are established in Section Formulation of the Coupling Current Equation for the r Component. In addition, the results of simulations using two gold loops are also compared in this section. Figure Induced current comparison (including the E r component) on the passive PEC loop located at θ = 45, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Near perfect agreement has been observed.

81 67 Figure 4-13, 4-14, 4-15 and 4-16 show perfect agreement between all three methods when the passive loop is located at (θ = 45, φ = 45 ), (θ = 9 φ = 45 ), (θ = 3 φ = 2 ), (θ = 9 φ = 9 ), respectively. These results are for two PEC loops located at distances 7 times and 14 times the radius away from the center of the active loop. Figure Induced current (including the E r component) on the passive PEC loop located at θ = 9, φ = 45 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Good agreements are observed. Figure Induced current including the E r component on the passive PEC loop located at θ = 3, φ = 2 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are plotted. Good agreements are observed.

82 68 Figure Induced current including the E r component on the passive PEC loop located at θ = 9, φ = 9 (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solution are plotted. Good agreements are observed. By including the r-component, the distance needed for the far-zone approximation to be valid will be decreased. Figure 4-16 shows all four aforementioned angles with a shorter distance of separation, i.e., r = 4b. Near-perfect agreement between the numerical and FEKO methods was observed. Additionally, relatively good agreement between the numerical, FEKO and the solved analytical solutions was also seen.

83 69 Figure Induced current including the E r component on the passive PEC loop located at (a) θ = 3, φ = 2, (b)θ = 45, φ = 45, (c) θ = 9, φ = 45 and (d) θ = 9, φ = 9 The center of the passive loop is 4 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. The above results are performed on PEC loops. Figure 4-17 and 4-18 show comparison results for gold loops at θ = 9, φ = and θ = 9, φ = 9, respectively. Thus, for both the numerical and analytical gold loop solutions to match the FEKO results, the separation distance must be greater than for PEC loops. We believe this is because gold becomes very lossy at those frequencies and the resulting current no longer has constant amplitude. Therefore, the far-field approximation begins to fall apart for shorter distances. Agreement improves when the two gold loops are 14 times the radius away.

84 7 Figure Comparison between FEKO results, numerical integration and closed form analytical solutions for the induced current (including the E r component) on the passive gold loop located at θ = 9, φ =. The loops have a separation distance of (a) 7 times and (b) 14 times the radius from the center of the active loop. Figure Induced current including the E r component on the passive gold loop located at θ = 9, φ = 9 that is (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. As discussed before and shown in Table 3-3, both closed form analytical methods and numerical integration methods are considerably faster than the FEKO simulation. Thus, the closed

85 form analytical method shows significant improvement, especially for stacked configurations. In addition to speed, a significant reduction in computational memory has also been observed. 71 Table 3-3. Timing comparison for three different methods for coupling nanoloops TABLE I COMPARISON OF REQUIRED COMPUTATIONAL RESOURCES Method Time Memory FEKO 5.9 hours 8.2 GB MATLAB - Numerical 32.5 seconds 1 GB + 14 MB MATLAB Analytical (general) 1.9 seconds 1 GB + 7 MB MATLAB Analytical (stacked).1 seconds 1GB + 2 MB 4.8 Formulation of the Coupling Current Equation for Stacked Two Nanoloop Antennas Many Yagi-Uda loop antennas are in a stacked configuration. Here, we present the induced current on the passive loop located directly above or below the active loop. Figure 4-2. Two nanoloop antennas in the stacking configuration. The bottom loop (blue) is the active loop with a voltage source and the loop on the top (green) is the passive loop without voltage sources. The distance between the centers of two loops is defined as R.

86 When compared to the general configurations (Section 4.4), stacked cases result in considerably simpler geometric parameters 72 ξ = η = x = b 2 cos φ y = b 2 sin φ r = b 2 2 cos 2 φ + b 2 2 sin 2 φ = b 2 (4.56a) (4.56b) (4.56c) (4.56d) (4.56e) R = b 2 2 cos 2 φ + b 2 2 sin 2 φ + z 2 = b z 2 (4.56f) The Spherical to Cartesian conversion can then be shown as r = [ cos φ z R ] θ = [ z R cos z φ R sin φ b R ] { φ = [ sin φ cos φ ] (4.57) The tangential vector on the passive loop is denoted as t = [ sin φ cos φ ] (4.58) In this configuration, both of the dot products θ t and r t are zero. The dot product between φ and t is shown in (4.59) φ t = 1 (4.5) Therefore, the total induced current is only due to E φ component. The current equation now becomes

87 73 I φ (Φ) = η (j) n I n J n (k b sin θ) k b b 2R ejkr [ ( Y m cos(nφ ) cos(m(φ φ )) dφ ) ] n= m= (4.51) Applying the far-field approximation in equations (4.35) and (4.36) as well as substituting θ = θ = results in a much simpler current expression. By using the generating Bessel function along with Euler s identity, we can write the current equation as I φ (Φ) = η k b b 8R ejkr (j) n I n J n (k b sin θ) Y m [e jmφ n= m= e j(n m)φ dφ + e jmφ e j(n+m)φ + e jmφ e j(n m)φ dφ + e jmφ dφ ] e j(n+m)φ dφ (4.52) I φ has non-zero components when n = m and n = m =. The current equation may then be simplified to I φ (Φ) = η k b bπ 2R e jkr (j) n I n J n ( k bb R ) Y n(cos(mφ)) n= + I Y J ( k bb R ) (4.53) Because I r and I θ are both zero, the induced current on the second loop in the stacked configuration can be shown as I total (Φ) = I φ (θ = π 2, Φ) (4.54) = η k b bπ 2R + I Y J ( k bb R ), e jkr (j) n I n J n ( k bb R ) Y n(cos(mφ)) n= Appendix F includes detailed derivation procedures for Sections 4.8.

88 Analytical Solution Compared to Numerical Simulation for Stacked Configuration Nanoloops In this section we compare three methods of finding the induced current on the passive loop located directly above the active loop. The methods compared are as follows: - The FEKO (method of moment) simulation - A numerical integration method (shown in Equation (4.6)) that does not incorporate any far field assumptions. - A closed form analytical solution (4.63) derived from solving Equation (4.6) and using far field assumptions. Figure Induced current on the PEC passive loop located at θ = (a) 7 times and (b) 14 times the radius away from the center of the active loop. FEKO result, numerical integration and closed form analytical solutions are plotted. Near perfect agreement between different methods are observed for both separation distances. Near perfect agreement has been observed for the stacked case in Figure 4-19 when the PEC passive loop is located directly above the active loop 7 and 14 times loop radius apart. A relatively good agreement can also be seen in Figure 4-2 when the two loops are 4 times loop radius apart.

89 75 Figure Comparison between FEKO, numerical integration and close form analytical solutions for the induced current on the passive PEC loop located at θ = and r = 4b. Analytical solution magnitude at the peak is slightly off because the far field approximation θ = θ might not hold for this short separation distance. The above results are performed on PEC loops. Figure 4-23 shows comparison results for two stacked gold loops. Similar to the conclusion we arrived in Section 4.7, the separation distance for gold loops needs to be farther than that for PEC loops for the both numerical and analytical solution to match FEKO result. Good agreement is observed when the two gold loops are 14 times the radius away. Figure Induced current comparison between FEKO result, numerical integration and closed form analytical solutions for two 6nm gold loops in the stacked configuration located at θ = (a) 7 times and (b) 14 times the radius away from the center of the active loop.

90 Figure Induced current comparison between FEKO result, numerical integration and closed form analytical solutions for 3nm gold loops located at θ = 9, φ = (a) 7 times and (b) 14 times the radius away from the center of the active loop. Good agreements between all three methods are shown. 76

91 77 Chapter 5 Conclusions and Future Work Considering the designs and equations developed in this thesis, we hope this research will open opportunities to further explore more antenna designs using the spoof SPPs approach as well as other complex leaky-wave antenna designs. The advantage of using spoof SPPs in the RF frequency spectrum is that it has low loss, which can make the design more efficient. The proposed LWA was not implemented due to time constraints, however using 3D printing technologies, it should be fairly easy to implement by using extruded polymers with a metal coating. The analytical equations developed for single nanoloops should be a solid stepping-stone for other researchers in the study of nanoloops. Furthermore, the mutual coupling analysis in this thesis between two nanoloops will contribute to the practicality and flexibility of future research. In our research, the closed form analytical solution has reduced the computational time from 6 hours (using FEKO) to 1.9 seconds (in MATLAB) for gold loops. Because the induced currents in our study is in the form of a Fourier series decomposition, far-zone electric fields can be later derived with minimal effort. Nanoloop arrays can be easily characterized with the far-zone electric field expression and array theory. Nanoloop applications such as solar panels for energy harvesting or high directivity nanoloop Yagi-Uda antennas can potentially use such an approach. It is also worth noting that lossy nanoloop antennas with a certain loop radius exhibit properties of superdirectivity. These properties are of interest to researchers and are worth studying for use in other nanoloop applications.

92 1 Bibliography [1] L. Tang, S. E. Kocabas, S. Latif, A. F. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, Naanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna, Nat. Photon, vol.2, no.4, pp , 29. [2] M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, Controlling the near-field oscillations of loaded plasmonic nanoantennas, Nature Photonics, vol. 3, no. 5, pp , 29. [3] R. Esteban, T. V. Teperik, and J. J. Greffect, Optical patch antennas for single photon emission using surface plasmon resonances, Phys. Rev. Lett., vol. 14, no. 2, p. 2682(1)-(4), 21. [4] A. F. McKinley, T. P. White, and K. R. Catchpole, Theory of the circular closed loop antenna in the terahertz, infrared and optical regions, Journal of Applied Physics, vol. 114, pp (1)-(1), 213. [5] P. Anger, P. Bharadwaj and L. Novotny, Enhancement and quenching of single-molecule fluorescence, Phys. Rev. Lett., vol. 96, no. 11, pp. 1132(1)-(4), 26 [6] T. H. Taminiau, F. D. Stefani, F. B. Segerink, N. F. van Hulst, Optical antennas direct singlemolecule emission, Nat. Photonics, vol.2, no.4, pp , 28 [7] T. H. Taminiau, F. D. Stefani, N. F. van Hulst, Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna, Opt. Express, vol. 16, no.14, pp , 28 [8] K. Li, M. I. Stockman, and D. J. Bergman, Self-Similar chain of metal nano-spheres as an efficient nano-lens, Phys. Rev. Lett., vol. 91, no. 22, pp (1)-(4), 23 [9] S. Li, M. L. Pedano, S. H. Chang, C. A. Mirkin, G. C. Schatz, Gap structure effects on surfaceenhanced Raman scattering intensities for gold gapped rods, Nano. Lett., vol. 1, no. 5, pp , 21 [1] N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. V. Dorpe, P. Nordlander and S. A. Maier, Fano resonances in individual coherence plasmonic nanocavities, Nano Lett., vol. 9, no.4, pp , 29 [11] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York, 27 [12] S. Zeng, D. Baillargeat, H. Ho and K. Yong, Nanomaterials enhanced surface plasmon resonance for biological and chemical sensing applications, Chem. Soc. Rev. vol. 43, pp , 214 [13] F. J. Garcia-Vidal, L. Martin-Moreno and J. B. Pendry, Surface with holes in them: new plasmonic metamaterials, J. Opt. A-Pure Appl. Opt., vol. 7, no. 2, pp.s97-s11, 25 [14] Q. Q. Gan, Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings, P. Natl. Acad., Sci. USA, vol. 18, no. 13, pp , 211 [15] J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vida, Mimicking surface plasmons with structured metal surfaces, Science, vol. 35, pp , 24 [16] C. R. Williams, Highly confined guiding of terahertz surface plasmon polaritons on structured metal surface, Nat. Photonics, vol. 2, pp , 28

93 [17] A. P. Hibbins, B. R. Evans, and J. R. Sambles, Experimental verification of designer surface plasmons, Science, vol. 38, no. 5722, pp , 25 [18] S. Maier, S. Andrews, L. Martin-Moreno and F. J. Garcia-Vidal, Terahertz surface plasmonpolariton propagation and focusing on periodically corrugated metal wires, Phys. Rev. Lett., vol. 97, no. 17, pp (1)-(4), 26 [19] D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin- Moreno, and E. Moreno, Domino plasmons for subwavelength terahertz circuitry, Opt. Express, vol. 18, no. 2, pp , 21. [2] D. M. Pozar, Microwave Engineering, Wiley, 1998 [21] J. Schulte (Ed.), Nanotechnology: Global Strategies, Industry Trends and Applications. UK: John Wiley & Sons, 25. [22] P. Biagioni, J.-S. Huang, and B. Hecht, Nanoantennas for visible and infrared radiation, Rep. Prog. Phys., vol. 75, pp. 2442/1-4, Jan [23] P. B. Johnson, and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B, vol. 6, no. 12, pp , [24] L. Novotny, and B. Hecht, Principles of Nano-Optics. Cambridge, UK: Cambridge University Press, 212. [25] C. A. Balanis, Antenna Theory. New York: Wiley, [26] P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Resonant optical antennas, Science, vol. 38, no. 5728, pp , June 25. [27] P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas, Phys. Rev. Lett., vol. 94, 1742/1-4, Jan. 25. [28] G. S. Smith, Loop antennas, in Antenna Engineering Handbook, R. Johnson, Ed., New York: McGraw-Hill, [29] A. Ahmadi and H. Mosallaei, "Plasmonic nanoloop array antenna," Opt. Lett. vol. 35, pp , 21. [3] A. Locatelli, Peculiar properties of loop nanoantennas, IEEE Photonics Journal, vol. 3, no. 5, Oct [31] D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, Experimental verification of the concept of all-dielectric nanoantennas, Applied Physics Letters, vol. 1, 21113/1-4, 212. [32] B. R. Rao, "Far field patterns of large circular loop antennas: Theoretical and experimental results," IEEE Trans. Antennas Propagat., vol. 8, no. 4, pp , July 196. [33] D. H. Werner, An exact integration procedure for vector potentials of thin circular loop antennas, IEEE Trans. Antennas and Propagation, vol. 44, pp , [34] D. H. Werner, Lommel expansions in electromagnetics, in Frontiers in Electromagnetics, D. H. Werner and R. Mittra, E d s. N e w J e r s e y : Wiley-IEEE Press, 2. [35] J. E. Storer, Impedance of thin-wire loop antennas, Trans. AIEE, vol. 75, pp , Nov [36] T. T. Wu, Theory of the thin circular loop antenna, J. Mathematical Physics, vol. 3, no. 6, pp , Nov.-Dec

94 [37] A. F. McKinley, T. P. White, I. S. Maksymov, and K. R. Catchpole, The analytical basis for the resonances and anti-resonances of loop antennas and meta-material ring resonators, Journal of Applied Physics, vol. 112, pp /1-9, 212. [38] L. Novotny, Efficient wavelength scaling for optical antennas, Physical Review Letters, vol , pp , 27. [39] S. V. Savov, An efficient solution of a class of integrals arising in antenna theory, IEEE Antennas and Prop. Mag., vol. 44, no. 5, pp , Oct. 22. [4] J. D. Mahony, A comment on Q-type integrals and their use in expressions for radiated power, IEEE Antennas and Prop. Mag., vol. 45, no. 3, pp , June 23. [41] S. V. Savov, A comment on the radiation resistance, IEEE Antennas and Prop. Mag., vol. 45, no. 3, pp. 129, June 23. [42] J. D. Mahony, Circular microstrip-patch directivity revisited: An easily computable exact expression, IEEE Antennas and Prop. Mag., vol. 45, no. 1, pp , Feb. 23. [43] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, [44] Hanson, G.W., "Radiation efficiency of nano-radius dipole antennas in the microwave and farinfrared regimes," IEEE Antennas and Propagation Magazine, vol. 5, no. 3, pp , June 28. [45] P. G. Etchegoin, E. C. Le Ru, and M. Meyer, An analytic model for the optical properties of gold, The Journal of Chemical Physics, vol. 125, pp /1-3, 26. [46] P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: An analytic model for the optical properties of gold, The Journal of Chemical Physics, vol. 127, pp /1, 27. [47] M. F. Pantoja, M. G. Bray, D. H. Werner, P. L. Werner, and A. R. Bretones, "A computationally efficient method for simulating metal-nanowire dipole antennas at infrared and longer visible wavelengths," IEEE Transactions on Nanotechnology, vol. 11, no. 2, pp , March 212. [48] FEKO Suite 7., EM Software & Systems - S.A., Stellenbosch, South Africa, [49] CST Studio Suite 215, Computer Simulation Technology AG, Darmstadt, Germany, [5] E. K. Miller and J. A. Landt, Direct time-domain techniques for transient radiation and scattering from wires, Proc. IEEE, vol. 68, pp , 198. [51] L.F. Shampine, "Vectorized adaptive quadrature in MATLAB," Journal of Computational and Applied Mathematics, vol. 211, pp , 28. [52] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products. NY: Academic Press, [53] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1 st ed., New York: Cambridge University Press, 21. [54] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, [55] A. P. Clarke and W. Marwood, A compact mathematical function package, Australian Computer Journal, vol. 16, no. 3, pp ,

95 [56] W. F. Perger, M. Nardin, and A. Bhalla. (199, June). Solution to the generalized hypergeometric function. [Online]. Available at: [57] D. E. Amos, "A portable package for Bessel functions of a complex argument and nonnegative order," Trans. Math. Software, [58] S. Zhang and J. Jin, Computation of Special Functions, 1 st ed., New York: Wiley, [59] A. S. Abul-kassem and D. C. Chang, On Two Parallel Loop Antennas, IEEE Transactions on Antennas and Propagation, vol. AP-28, no.4, pp , 198 [6] T. Bhattacharyya and K. K. Bandyopadyay, Analysis of element admittances of a pair of staggered loop antennas, in Electrical Engineers, Proceedings of the Institution of, vol. 12, no.12, pp , 1973 [61] R. W. P. King, C. W. Jun., Harrison, and D. G. Tingley, The admittance of bare circular loop antennas in a dissipative medium, IEEE Transaction, AP-12, pp , [62] K. Iizuka, The circular loop antenna immersed in a dissipative medium, IEEE Transactions on Antennas and Propagation, vol. 13, no. 1, pp , 1965 [63] R.W. P. King, The loop antenna for transmission and reception, in Antenna Theory: Part I, R. E. Collin and F. J. Zucker, eds, McGraw, New York, pp , 1969 [64] R. W. P. King and S. Prasad, Fundamental Electromagnetic Theory and Applications, Prentice-Hall, Englewood, Cliffs, NJ, pp , 1986 [65] A. F. McKinley, T. P. White and K. R. Catchpole, Designing Nano-loop antenna arrays for light-trapping in solar cells, Photovoltaic Specialists Conference (PVSC), 213 IEEE 9 th, Tampa, FL, pp ,

96 1 Appendix A Calculation of the Integral of the Bessel Function Integration of the Bessel functions of integer order are required to solve for the coefficients of the modal impedances described by (3.4) and (3.5). This can be performed numerically by using global adaptive quadrature techniques [51]. Alternatively, the integral can be represented and accurately calculated using a power series representation. The Bessel function is expressed in the form [52] [53]: J 2m (x) = ( 1)n 2n+2m n! (2m + n)! (x 2 ) n= (A.1), which leads to the result: 2k b J 2m (x)dx = 2 ( 1)n (k b ) 2n+2m+1 n! (2m + n)! (2n + 2m + 1) n= (A.2) Another useful representation that is especially useful for larger arguments can be employed, where the definite integral of Bessel function is expressed in terms of an infinite series of Bessel functions [52]-[53]: 2k b J 2m (x)dx = 2 J 2m+2k+1 (2k b ). k= (A.3) A third formulation can be determined by using generalized hypergeometric functions [54], which are the solutions to the hypergeometric or Gauss s equation, and defined as pfq ( a 1, a 2,, a p ; z) = (a 1) k (a p ) k z k b 1, b 2,, b q (b 1 ) k (b q ) k! k k= (A.4) where ( ) k denotes the Pochhammer symbol, which is defined as

97 83 (q) n = Γ(q + n) Γ(q) = { 1 n = q(q + 1) (q + n 1) n > (A.5) Bessel functions, as well as many other special functions, can be expressed in terms of generalized hypergeometric functions. For example, J 2m (z) = z 2m z2 4 m F1 (; 2m + 1; Γ(2m + 1) 4 ) (A.6) By applying the substitution u = 1 4 z2, and the integration identity [34]: z α 1 F 1 (; b; z)dz = z α Γ(α) Γ(α + 1) 1F2 (α; b, α + 1; z), (A.7) the integral in (A.2) may be written as 2k b J 2m (z)dz = 2k b 2m+1 (2m + 1)! 1F2 (m ; 2m + 1, m ; k b 2 ) (A.8) A robust numerical technique for evaluating the generalized hypergeometric function of the form contained in (A.8) is discussed in [56] and source code can be found in [57]. Figure A-1. Relative error vs. number of terms for the integral of the Bessel function with m=1. Using (a) power series (Equation (A.1)) and (b) Bessel series (Equation (A.2)) representations. The error is computed relative to (Equation (A.8)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red.

98 84 The series representations given in (A.2) and (A.3) provide convenient alternatives to solving the integration rather than solving it numerically. The series of (A.2) converges rapidly for small arguments using lower orders of m. However, the number of terms (order m) required to achieve convergence dramatically increases for larger values of k b, because the summation is not monotonically decreasing. Therefore, the series representation involving Bessel functions (A.3) is employed for larger arguments because it converges much quicker. This is illustrated in Figure A- 1 (a) and (b). This figure also shows the relative error versus the number of terms in the series expansions for m = 1, using (A.2) and (A.3). When performing a ratio test on the series, the maximum value of k b can be computed accurately. This can be determined by relating k b to a given accuracy A, a fixed number of terms n in the power series, and the Bessel function of order m: (n + 1)(2m + 2n + 3)(2m + n + 1) k b < A (2m + 2n + 1) (A.9) This equation explains the observation that more terms are required for a given accuracy as k b increases. A comparison between the two series representations is illustrated in Figure A-2 for large values of m. Again, the case of m = 35 is considered because it has been proven that 35 modes is sufficient for the infinite series to accurately converge.

99 Figure A-2. Relative error vs. number of terms for the integral of the Bessel function with m=35. Using (a) power series (Equation (A.1)) and (b) Bessel series (Equation (A.2)) representations. The error is computed relative to (Equation (A.8)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red. 85

100 86 Appendix B Calculation of the Integral of the Lommel-Weber Function Determining solutions for integrals of the Bessel functions and, additionally, the Lommel- Weber function are both key in calculating the modal admittance coefficients. This Appendix addreses solutions to the integral of the Lommel-Weber function. A power series representation and an analytical form in terms of hypergeometric functions will be presented and compared. The following power series representation of the Lommel-Weber function is the starting point of this derivation, which is valid for integer values of m and n: ( 1) n+m Ω 2m (x) = ( x 2n+1 Γ(n + m + 3/2)Γ(n m + 3/2) 2 ) n= (B.1) The required integral of the Lommel-Weber function leads to the result using (B.1): 2k b ( 1) n+m Ω 2m (x)dx = Γ(n + m + 3/2)Γ(n m + 3/2) (n + 1) n= k b 2n+2 (B.2) provided: Next, a hypergeometric function representation of the Lommel-Weber function is Ω 2m (x) = 2 π z 1F 2[1; 1 2 ( 2m + 3), 1 2 (2m + 3); 1 4 z2 ] (1 2m)(1 + 2m) (B.3) By applying the substitution u = 1 4 z2, and the integration identity [53]: z α 1 1F 2 [1; m + 3 2, m + 3 ; z] dz = 2 (B.4) it can be shown that: z α α 2F3 [α, 1; α + 1, m + 3 2, m ; z], 2k b Ω 2m (z) dz = 4k 2 b 2F 3[1,1; 2, m + 3 2, m ; k b 2 ] π(2m 1)(2m + 1) (B.5)

101 The hypergeometric function that appears in (B.5) may be computed efficiently by using currently available libraries of built-in functions in MATLAB. 87 Figure B-1. Relative error vs. number of terms for the integral of the Lommel-Weber function given in Equation (B.1) with (a) m=1 and (b) m=35. The error is computed relative to (Equation (B.5)), which involves the confluent hypergeometric function. The parameter k b increases from.1 to.5 as the color of the curve varies from blue to red. Figure B-1 (a) and (b) shows the relative error versus the number of terms for small and large m. Similar to the case of integrating the Bessel function discussed in Appendix A, only a few terms in the series are required to achieve a given accuracy for low order m and small values of k b. Yet again, (B.5) provides a more accurate answer. This approach is also more efficient in terms of computational time than the power series representation (B.2).