This dielectric on the opposite side of the

This report analyses the application of plasmonics in the nanoantenna space. It also includes the designand analysis of a graphene based bow-tie antenna.Introduction:Inside a metal, electric field cannot exist due to polarization. Above the plasma frequency, the electronsare too slow to oscillate at the speed of the incoming radiation due to their intertia, and the metal losesreflectivity. Plasmonics is the study of the interaction of light with materials when the frequency of theincident EM wave crosses the plasma frequency of the material. Plasma frequency is defined for a materialas the frequency above which light is no longer reflected and the material becomes transparent to theincident radiation.Surface Plasmon Resonance:Electrons in a metal are weakly bound and are free to roam. Surface plasmon resonance is the collectiveoscillation of free electrons in response to an electric field. The oscillation of the electron gas has a certainfrequency and shows resonant behavior at the plasma frequency of the metal. Surface plasmon is asurface wave that travels along the interface between a metal and a dielectric. The wavelength of asurface plasmon is the distance over which the fields due to the charge distribution undergoes oneoscillation and the distance over which the charge density undergoes one oscillation.The relationship between the frequency of radiation and the spatial wave number is given below:ko is the free space wave number, ?o is the permittivity of free space, and ?m is the frequency-dependentpermittivity of the metal or conductor.Figure 1: Dispersion relationAirMetal-Dielectric InterfaceThe amplitude of the surface plasmon waves decay exponentially as the distance from the interfaceincreases. With the presence of a dielectric on the opposite side of the metal, the plasmon wavespenetrate the dielectric and since there are no electrons to collide with and lose energy to, the plasmonwaves can travel much farther along the interface. This is visualized in the figure below:Figure 2: Propagation of surface plasmons along the interfaceSince the refractive index of the material is higher than that of air, the wavelength of the propagatingsurface plasmons is shorter, and this leads to localization of electric fields in sub-wavelength space.Electric Field EnhancementElectric Field enhancement is a combined effect of two phenomena:1. Scattered Fields:The constructive interference of multiple scattered fields can cause local field enhancement inplasmonic structures. Thus, to design nanoantennas which are characterized by the factor bywhich electric fields are enhanced, it is desirable to design structures that maximize the scatteringcross-section of the antenna.Scattering cross-section (?scat) is defined as the ratio of the scattered power to the incident power.Absorption cross-section (?abs) is defined as the ratio of the absorbed power to the incidentpower. The sum of the scattering and absorption cross-sections is defined as the extinction cross-section (?ext). It quantifies the degree of extinction of the incident light that is caused by theantenna.?ext = ?scat + ?absScattering and absorption are competing phenomena and designing structures for plasmonicapplications involve deciding which one to optimize. For example, for detecting biomolecules, itis desirable to optimize absorption cross-section and analyzing the frequency of dip in lightintensity which changes with the nature of biomolecules. For surface enhanced ramanspectroscopy (SERS), it is desirable to increase scattering cross-section as the intensified electricfield is necessary to magnify the light scattered by the bio-molecules. For cancer therapy usingplasmonic nanostructures, it is desirable to enhance absorption cross-section and scatteringcross-section as this would localize heat as well as the incoming radiation’s high energy and enablekilling cancer cells. For plasmonic antennas in photovoltaics, scattering efficiency could beincreased so that the incoming light is intensified. This increases the light to electricity conversionefficiency of solar cells.2. Field Localization:Since the wavelength of the surface plasmons is localized with respect to the incident radiation,the energy that is contained in the impinging wave is transferred to a smaller space, thusincreasing the electric field intensity in that region.Field Enhancement in Isolated StructuresSurface plasmons on closed surface like a metallic sphere exhibit discrete resonances. These modescorrespond to the different distributions of charge separation along surface of the sphere. For example,a rod has two axes of oscillation as shown:Vertical AxisHorizontal AxisFigure 3: Axes of a gold nanorodThe charge distribution can occur along the vertical or horizontal axis and these constitute two differentmodes of the electron oscillations in the nanorod. Different frequencies of incident radiation couple withdifferent plasmon oscillations. The figures below illustrate the same for a spherical nanostructure:Figure 4: Charge distribution in a metal nanosphereHigher modes of plasmon oscillations do exist, but the incident radiation doesn’t couple to these modes.These modes do not radiate and on excitation, are lost to resistive currents in the nanosphere. To harnessthe higher order modes, the symmetry of the sphere is broken so that the higher order modes can coupletogether to increase electric field enhancement. This is visualized in the figures below:Figure 5: Coupling higher order modes to increase electric field enhancementAs shown in the figures above, as the gap size decreases the higher order modes of the individual spherescouple together leading to highly localized fields. Such interaction also shifts the resonant frequency andchanges field characteristics.The change in resonant frequency with change in gap size is visualized in figure 6. Here ?sp is thewavelength of the surface plasmon and ? is the wavelength of incident radiation.Figure 6: Variation of resonant frequency with gap size and sphere diameterBow-tie Antenna DesignDue to the coupling effect of nanosphere dimers as discussed above, bow-tie antennas with pointed tipsexhibit very high electric field enhancements in the nanogap. A bow-tie antenna was designed withgraphene to demonstrate field enhancement.Graphene:Graphene has been rising in popularity over the past couple of years since its discovery in 2004. Gold andsilver are popular metals currently to be used in plasmonic antennas due to their very high conductivity.Graphene was chosen in the design because of its tunability and bandgap-less structure. Thus, grapheneexhibits strong light-matter interaction and supports SPP propagation over a broad frequency band.Further, the response of a graphene nanoantenna can be tuned by chemical or electrostatic doping.The specifications of the design are shown below. These parameters were chosen on a trial and error basisby examining the change in electric field enhancement by varying the parameters.Lumerical:The simulations were performed on the photonics design software called Lumerical. Lumerical offers avery easy-to-use graphical user interface and provides a wide variety of in-built structures to experimentwith. The simulation set-up in lumerical is shown below. The electric field oscillation is shown as bluearrows and the direction of propagation is shown by the pink arrow(normal incidence).Figure 7: Simulation structure: Bowtie graphene antennax 103Results:The scattering cross-section was plotted as a function of frequency and the resulting plot is shown below:Figure 8: Scattering cross-section as a function of frequencyIt can be seen that the antenna shows a resonant behavior at around 7.4 THz. The electric fieldenhancement was visualized by taking the ratio of the electric field intensity to the incidentelectric field of 1V/m at 7.4 THz and an enhancement of 5000 was obtained. This is shown below:Figure 9: Field Enhancement in the XY planex 103x 103Figure 10: Field enhancement in the XZ planeFigure 11: Field enhancement in the YZ planeMaximum field enhancement over all wavelengths (|E|^2/|Einc|^2) in y-z plane is: 6862.79Maximum field enhancement over all wavelengths (|E|^2/|Einc|^2) in x-z plane is: 6862.79Maximum field enhancement over all wavelengths (|E|^2/|Einc|^2) in x-y plane is: 6862.79The results agree with published work analyzing the electric field enhancements of graphene patchnanoantennas.Conclusion and Future Work:The goal of the project is to efficiently model the plasmonic effects in optical nano-antennas usingconcepts in circuit theory. For example, the kinetic inductance of an optical nanoantenna can bemodelled as an inductor and analyzed using circuit theory. Efficient modelling of opticalnanoantennas can enable easy designing of structures for applications. The electric fieldenhancement factor can also be easily extracted by converting the quantities so that they beeasily analyzed by using concepts in circuit theory. The modelling of nanoantennas is shown inthe two figures below:b)Figure a) shows how elements in a nanoantenna like resistance, capacitance and inductance can bemodelled using circuit theory. CA is the antenna capacitance. ?CA is the capacitance due to fringing fieldsthat is expressed as a fraction of the antenna capacitance. Cgap is the gap capacitance. Rrad and R?are the radiation and ohmic resistances respectively. L is the kinetic inductance.Figure b) shows the elements of the quantum emitter where I is the quantum mechanical current and Rradis the radiation resistance associated with the quantum emitter.The above diagrams do not include plasmonic effects and are illustrated here just to give a sense of thefuture work that needs to be accomplished. The Purcell factor of enhancement can be found by takingthe ratio of the power radiation in Rrad in figure b) to the power radiated in Rrad in figure a). The challengeis to extract the above parameters using lumerical (or any other software), run circuit simulations andestablish a clear transition from quantum mechanical model of optical antennas to a classical model byverifying the equivalence of the enhancement factor from the simulations and circuit analysis.References:1. Michael S. Eggleston, Sujay Desai, Kevin Messer, Surabhi Madhvapathy, Jun Xiao3, Xiang Zhang,Eli Yablonovitch, Ali Javey, Ming C. Wu, “Enhanced Spontaneous Emission from an OpticalAntenna Coupled WSe2 Monolayer “2. Shinji Hayashi and Takayuki Okamoto 2012 J. Phys. D: Appl. Phys. 45 4330013. Ignacio Llatser Mart√≠, Christian Kremers, Albert Cabellos-Aparicio, Josep Miquel Jornet, EduardAlarc√≥n and Dmitry N. Chigrin, “Scattering of terahertz radiation on a graphene-based nano-antenna”4. Josep Miquel Jornet and Ian F. Akyildiz , “Graphene-Based Nano-Antennas for ElectromagneticNanocommunications in the Terahertz Band”5. The Promise of Plasmonics: Scientific American6. Krasnok, A. E. et al. An antenna model for the Purcell effect. Sci. Rep. 5, 12956; doi:10.1038/srep12956 (2015)7. Anil Kumar, “Optical Nano-antennas: Fabrication, Characterization and Applications”8. Jean-Jacques Greffet, Marine Laroche, Fran ?cois Marquier. Impedance of a nanoantenna and asingle quantum emitter. Physical Review Letters, American Physical Society, 2010, 105 (11),pp.1177019. Shaloo Rakheja and Parijat Sengupta J. Phys. D: Appl. Phys. 49 (2016) 115106 (14pp)10. BedirB.Yousif and Ahmed S.Samra, “Modeling of Optical Nanoantennas”11. Lukas novotny and Niek Van Hulst, “Antennas for light”12. Journal of Applied Physics 112, 114915 (2012); doi: 10.1063/1.476884013. http://people.ee.duke.edu/~drsmith/plasmonics/enhancement.htm14. Scientific Reports | 5:12956 | DOi: 10.1038/srep12956