Efficiency and Scalability of Dielectric Resonator Antennas at Optical Frequencies Volume 6, Number 4, August 2014 Longfang Zou, Member, IEEE Withawat Withayachumnankul Charan M. Shah Arnan Mitchell, Member, IEEE Maciej Klemm, Member, IEEE Madhu Bhaskaran, Member, IEEE Sharath Sriram, Member, IEEE Christophe Fumeaux, Senior Member, IEEE
DOI: 10.1109/JPHOT.2014.2337891 1943-0655 Ó 2014 IEEE
IEEE Photonics Journal
Efficiency of Dielectric Resonators
Efficiency and Scalability of Dielectric Resonator Antennas at Optical Frequencies Longfang Zou,1,2 Member, IEEE, Withawat Withayachumnankul,1 Charan M. Shah,3 Arnan Mitchell,3,4 Member, IEEE, Maciej Klemm,2 Member, IEEE, Madhu Bhaskaran,3 Member, IEEE, Sharath Sriram,3 Member, IEEE, and Christophe Fumeaux,1 Senior Member, IEEE 1
2
School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, S.A. 5005, Australia Department of Electrical and Electronic Engineering, The University of Bristol, Bristol BS8 1TH, U.K. 3 Functional Materials and Microsystems Research Group, School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology (RMIT) University, Melbourne, Vic. 3001, Australia 4 Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology (RMIT) University, Melbourne, Vic. 3001, Australia
DOI: 10.1109/JPHOT.2014.2337891 1943-0655 Ó 2014 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Manuscript received May 6, 2014; revised June 30, 2014; accepted July 1, 2014. Date of publication July 10, 2014; date of current version July 28, 2014. The work of M. Bhaskaran and S. Sriram was supported by the Australian Postdoctoral Fellowships from the Australian Research Council (ARC) through Discovery Projects DP1092717 and DP110100262, respectively. The work of S. Sriram was also supported by the ARC through Grant LE100100215. The work of C. Fumeaux was supported by the ARC Future Fellowship funding scheme under Grant FT100100585. The work of M. Klemm was supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC), CrossDisciplinary Interfaces Programme (C-DIP) Fellowship Fund, under Grant EP/I017852/1. Corresponding author: L. Zou (e-mail:
[email protected]).
Abstract: Dielectric resonators have been foreseen as a pathway for the realization of highly efficient nanoantennas and metamaterials at optical frequencies. In this paper, we study the resonant behavior of dielectric nanocylinders located on a metal plane, which in combination create dielectric resonator antennas operating in reflection mode. By implementing appropriate resonator models, the field distributions, the scaling behavior, and the efficiency of dielectric resonator antennas are studied across the spectrum from the microwave toward visible frequency bands. Numerical results confirm that a radiation efficiency above 80% can be retained up to the near-infrared with metal-backed dielectric resonators. This paper establishes fundamental knowledge toward development of high efficiency dielectric resonator antennas and reflection metasurfaces at optical frequencies. These dielectric resonators can be incorporated as basic elements in emerging applications, e.g., flat optical components, quantum dot emitters, and subwavelength sensors. Index Terms: Dielectric resonator, resonance, nanoantenna, efficiency, dielectric metasurfaces.
1. Introduction Resonant nanostructures have aroused great interest in the last decade due to their capacity of controlling light, either through field enhancement and localization at an interface or through
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scattered far-field radiation. They have great impact on diverse applications, including biomedical spectroscopy, sensing, communications, photovoltaic power generation, near-field optical spectroscopy and photodetection [1]–[6]. Although plasmonic nanostructures are widely recognized for their ability to manipulate light at subwavelength scale, the ohmic loss intrinsic to metals adversely affects their efficiency [7], [8]. As an alternative solution to metal structure, recent research has theoretically and experimentally demonstrated electric and magnetic resonances in high-index dielectric resonators [7], [9]– [12]. Low-loss dielectric resonators, driven by displacement current, are capable of attaining high efficiency in various applications. For example, an array of silicon nanodisks embedded in low index dielectric has been shown in [7] to suppress backward scattering and enhance forward scattering as transmission metasurfaces in the infrared. In [10], a nonuniform arrangement of cylindrical dielectric resonators made of TiO2 was designed to create a periodic gradient metasurface with the goal of deflecting an incident beam toward a given deflection angle off the specular reflection at visible frequencies. Later, a similar arrangement of silicon dielectric resonators on a silver surface was demonstrated for optical vortex beam generation in the shortwavelength infrared band [13]. In the typical reflection-mode configuration as implemented in these two last references [10], [13], the dielectric resonators are mounted on a metallic plane, which leads to coupling between the dielectric resonators and the metallic surface. As the frequency increases in the optical range, the properties of the metal as imperfect conductor begin to affect the response of the resonators. This becomes significant in the infrared and visible ranges when waves couple with plasmonic modes. To characterize this behavior, the response of dielectric resonators is investigated in this paper in terms of resonant dimensions, field distribution and radiation efficiency across six frequency decades, from radio frequencies to the visible light range. Since the dielectric resonator shape significantly affects the field distribution and hence the resonance frequency, empirical models are usually employed to obtain an initial geometry of the resonators. These models are reasonably accurate where we can assume a perfect electric conductor (PEC) for the ground plane [14]. As the operation frequency approaches the infrared and optical ranges, the empirical model becomes too complicated, and cannot reproduce the results accurately. Thus, electromagnetic simulations are employed for investigation of the behavior of dielectric resonators. This numerical investigation uses appropriate frequency-specific metal models for the ground plane. Since various suitable low-loss dielectric material are available the resonator materials is set in our study to representative material properties, which remain (hypothetically) constant across the spectrum. The results of this study are consistent with previous experimental findings, and quantitatively reveals how the plasmonic effects significantly change the field distribution in the dielectric resonators and prevent a straightforward scaling of dielectric resonators. Importantly, the study of the efficiency shows that optical dielectric resonators offer a promising alternative to conventional metallic resonators in the optical range.
2. Scalable Dielectric Resonator Antenna Model Because of the scalability properties of Maxwell's equations, all electromagnetic devices can be scaled with respect to wavelength for operation at different frequencies, provided that the material properties are constant. For the scaling of dielectric resonators with a given permittivity on a metal plane, as the operation frequency approaches infrared and visible frequencies, a straightforward scaling will fail because the penetration of electromagnetic fields into the metal ground plane can no longer be neglected and collective electron oscillations must be considered. The standard engineering formulas for calculating the resonance frequency of dielectric resonators [14] are not valid in this case as they consider the metal base as a perfect symmetry plane. Thus, a more sophisticated numerical analysis is required to estimate the resonance frequency, study the scaling behavior and predict antenna efficiency from the microwave to the visible spectrum. This is investigated in the following with a generic dielectric resonator model.
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Fig. 1. Three-dimensional schematic of the proposed scalable single element dielectric resonator antenna model.
Fig. 2. (a) E-field and (b) H-field distributions on resonance in the cylindrical dielectric resonator on silver excited by an incident plane wave at 5 GHz ð0 ¼ 60 mmÞ.
The proposed scalable dielectric resonator model is illustrated in Fig. 1. A cylindrical dielectric resonator is mounted on a silver block with a size of 1:50 1:50 and a thickness of 0 =3, where 0 is the free space wavelength. The height of the dielectric resonator is fixed to 0 =10 and the resonance is achieved by varying the diameter D of the dielectric resonator. The scaling behavior is characterized by considering the ratio of the dielectric resonator diameter to the resonance wavelength (i.e., D=0 ). All the dimensions of the dielectric resonator and silver plane in this investigation are expressed as a function of the wavelength to ease the description of the scaling behavior. The lateral size of the silver plane is chosen to be larger than one wavelength to minimize edge diffraction effects while keeping the computational cost reasonable. Considering the manufacturability of optical antennas, the dielectric resonator material is set to have the same permittivity as TiO2 (titanium dioxide) at 633 nm wavelength [10]. The dielectric resonator material is then assumed with a frequency-independent anisotropic dielectric permittivity of 8.29 (in the x and y directions) and 6.71 (in the z direction) and an estimated loss tangent of tan ¼ 0:01 [15]. This TiO2 permittivity value is assumed to be unrealistically constant across the spectrum for the sake of comparison across different frequency bands. Despite that, other dielectric materials with a comparable permittivity are typically available as a substitute at a given target frequency range. There are three fundamental resonant modes in a cylindrical dielectric resonator, denoted as HEM11 (horizontal magnetic dipole mode), TM01 (vertical electric dipole mode) and TE01 mode (vertical magnetic dipole mode) [14]. The HEM11 is emphasized in this study because it can be readily accessed via a plane-wave excitation, typical for optical experiments and applications. Furthermore, this mode is of technological importance since it can create a magnetic dipole, rarely obtainable from natural materials at optical frequencies. For the scaling study, the TiO2 DRA model on a silver block is simulated by using ANSYS HFSS, employing the finite element method (FEM) in the frequency domain. In the simulation, the DRA is excited with a normally incident plane wave, with polarization in the x direction. The TiO2 -silver DRA model is first simulated at a microwave frequency of 5 GHz ð0 ¼ 60 mmÞ. At this wavelength, only a negligible fraction of the electromagnetic wave can penetrate into the metal. The silver block can be approximately treated as a nearly perfect electric conductor and simulated by using the classical skin-effect model (Appendix A). The resulting E and H field distributions of the HEM11 mode are shown in Fig. 2. According to the theoretical field distribution of this mode [14], the maximum magnetic field is located at the bottom and center of
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Fig. 3. (a) E-field and (b) H-field distributions on resonance in the cylindrical dielectric resonator on silver excited by an incident plane wave at 500 THz ð0 ¼ 600 nm).
the dielectric resonator, just above the silver plane. Thus, a field probe is placed at this location and the magnitude of the magnetic field along the y axis is monitored with the maximum value clearly identifying the resonance frequency. The resonance of 5 GHz ð0 ¼ 60 mmÞ is achieved for a diameter D ¼ 27:2 mm ðD 0:450 Þ in the case of the mentioned fixed dielectric resonator height of 0 =10 ¼ 6 mm. The skin effect model neglects relaxation effects and remains valid up to the THz range. Above around 1 THz, the more advanced modified relaxation-effect model (Appendix B) is required for accurate description of the surface impedance of the metal. The model provides more accurate results by taking into account electron-phonon collisions. The field penetration into the metal becomes significant in the near-infrared and visible parts of the spectrum. Thus, the metallic structures cannot be treated by using the surface impedance model and the Drude model is usually employed to describe the dispersion of a metal. The formulation of the Drude model and the silver parameters used in the simulation are explicitly given in the Appendix C. Fig. 3 shows the simulated E-field (a) and H-field (b) distributions of a DRA operating at 500 THz ð0 ¼ 600 nmÞ. The field distributions confirm the excitation of a magnetic dipole mode qualitatively similar to the fundamental HEM11 mode in the cylindrical dielectric resonator. However, compared to the field distributions in Fig. 2, the field extends significantly into the silver block (Fig. 3) since the resonator couples strongly with the surface-plasmon polaritons (SPPs). It can thus be clearly observed from Figs. 2 and 3 that field penetration and mode conversion prevent a direct scaling of the dielectric resonator behavior, and that a significant antenna size reduction is required to compensate the change. At a wavelength of 600 nm and for a dielectric resonator height of 0 =10 ¼ 60 nm, the desired HEM11 mode is excited for a resonant diameter of 128 nm ðD 0:210 Þ. This stands in contrast to a value of 272 nm which would be expected by direct scaling from 5 GHz. The scaling behavior of DRA from microwave to visible frequencies is studied further in Section 3.
3. DRA Scaling Behavior Using the scalable model described in the previous section, the scaling behavior of resonant DRAs has been investigated further in detail. Fig. 4 shows the computed resonant diameter (expressed in D=0 ) of the DRA as a function of the wavelength of operation, for a spectrum covering a range extending from microwave to the optical region. At frequencies below 1 THz ð0 9 300 mÞ, the classical skin-effect model is usually utilized for most purposes because of its simplicity and accuracy. In Fig. 4, the black line with circle markers indicates the expected constant ratio of the DR diameter to wavelength ðD=0 0:45Þ for the DRAs operating in this low frequency range. Beyond 1 THz ð0 G 300 mÞ, the frequency-dependent complex surface impedance model for silver (calculated using Eq. (2) in the Appendix B) must be employed to account for the metallic loss, and the blue curve with star markers represents the resonant DR diameters obtained using this method. With an increase of the electromagnetic wave penetration depth in the silver, the resonant DR diameter progressively decreases by a few percents. For even higher frequencies, as discussed in Section 2, the Drude model is employed to describe the dispersion of silver and the red curve with diamond markers shows the obtained resonant DR diameters. In this frequency range, the plasmonic effects significantly reduce the resonant DR diameters down to 1/3 of the values that would be expected from direct scaling. The analysis using the Drude model can only be conducted in the visible and near-infrared
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Fig. 4. Scaling behavior of a cylindrical DRA from microwave to visible frequencies. The markers indicate the simulated wavelengths, whereas the connecting lines are merely to assist visual perception. (Inset: Zoom in Y axis shows the overlap and divergence of the skin-effect model and surface impedance model.)
frequencies (i.e., from 0.5 to 6 m), constrained by the resolution of the thin penetration layer. The transition between the three models and their overlap at the boundary of their validity range provide a consistency validation for this practical method of characterizing metal properties from microwave to optical frequencies. It is noted that the experimental results obtained for resonant DRA operating at microwave [14] and millimeter-wave frequencies [16], [17], as well as the reduced dimensions of nano-DRAs resonant for 633 nm red light as presented in [10] are consistent with the results presented in this section, providing confidence in the reliability of the presented computations.
4. Optical DRA Radiation Efficiency The scalable DRA model provides guidelines for the design of DRAs across the spectrum. The reduced diameter at optical frequencies clearly originates from higher field penetration into the metal plane. This increased interaction with the lossy metal is expected to be detrimental to the radiation efficiency of the antenna. To investigate this effect, the scalable DRA model has also been employed to study the radiation efficiency of DRAs from microwave to optical frequencies. The efficiency study has been performed in CST Microwave Studio using the time-domain solver, which is based on the Finite-Integration Technique (FIT). When the DRA is illuminated by a plane wave, the angle of incidence influences the coupling of the wave to the DRA and thus the excitation of the mode in the resonator. Since the HEM11 resonant mode is equivalent to a horizontal magnetic dipole on a lossy ground plane, its amplitude when excited by a plane wave will generally decrease as a function of the incidence angle. Furthermore, the finite size of the underlying silver block can introduce parasitic resonant effects. For these reasons, the plane wave illumination is not adopted for characterizing the intrinsic radiation efficiency of the excited HEM11 resonant mode, which is independent of the illumination conditions. Therefore, an inner excitation port is defined to optimally excite the resonant mode. This port takes the form of a short narrow slot, which approximates an infinitesimal magnetic current source along the y axis. The excitation port is located at the bottom center of the DR, at a height of 5 nm above the silver top surface. The radiation efficiency is defined as the ratio of the total power radiated by an antenna to the net power accepted by the antenna from the excitation port. This definition eliminates the mismatching loss between excitation port and antenna and thus is suitable for studying the net material losses (i.e. dissipation as heat) inside the antenna. Similarly to in the scaling behavior study, the height of the DR is fixed to 0 =10 and the resonance is achieved by varying the diameter D of the DR. It is worth noting that the resonant DR
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Fig. 5. Simulated DRA efficiency from microwave to optical frequencies. The solid curves indicate the efficiency for a DR made of lossless dielectric material, whereas the dashed lines are obtained with a lossy dielectric with loss tangent of 0.01. (Inset: Zoom in Y axis shows the overlap and divergence of the skin-effect model and surface impedance model.)
diameters calculated in CST Microwave Studio have excellent agreement with these calculated in ANSYS HFSS (shown in Fig. 4). The use of the two different software tools has been on the one hand motivated by practical considerations (availability and minor software features), but on the other hand, the cross-check between these two software tools, based on different numerical methods and excitations, offers an additional validation of the proposed method. As before, the skin-effect model is employed at wavelengths above 300 m; the modified relaxation–effect model is employed between 3 mm to 3 m; and the Drude model is employed below 6 m. As shown in Fig. 5, the combination of these three models allows coverage of the full frequency range and their partial overlap in the ranges of common validity provides confidence in the accuracy of the results. At low frequencies, the results of the skin-effect model (solid black line with circle markers) predicts a DRA efficiency close to 100% with a lossless DR (i.e., tan ¼ 0) and hence the loss in the silver can be neglected. Introducing the relatively high dielectric loss tangent of tan ¼ 0:01 decreases the DRA efficiency to around 91%. When the operating frequency is increased and approaches terahertz and far infrared frequencies, the loss in the silver begins to become apparent, which results in a slight decrease in the DRA efficiency (solid blue curve with star markers). Nevertheless, the antenna efficiency remains well above 90% for a lossless DR, indicating a nearly negligible influence of metallic losses on the performance. The efficiency of the DRA is still above 80% by considering loss in dielectric ðtan ¼ 0:01Þ. As the operating wavelength is further deceased to the infrared and visible spectrum, the loss in the silver dramatically increases, which significantly reduces the DRA efficiency (solid red curve with diamond markers). In this case, the loss in the silver is larger than that of the dielectric, as clearly seen from the dashed curved obtained with the lossy dielectric. Taking the wavelength of 600 nm as an example, the efficiency of the DRAs is predicted to be around 67% for a lossless dielectric material, and 58% for realistic material with loss tangent of tan ¼ 0:01. In other words, about 33% of the power loss is attributed to dissipation in the silver and only about 9% power is consumed in the dielectric. Importantly, high efficiencies of above 80% are predicted to be maintained well into the near-infrared regime, for wavelength longer than 1 m. The result coincides with the measured efficiency of dielectric meta-reflectarray in the infrared [13] and explains the lower value experimentally observed in the visible range [10]. Since the loss is mainly in the metal due to SPPs at high frequencies, the overall efficiency can be enhanced by considering multi-layer DRAs [18], where a dielectric layer with a low permittivity would act as a spacer between the metal and the high permittivity dielectric resonators. This would decrease interaction, and consequently, the loss in the metal could be reduced. Another option to avoid loss in metal could be replacing the metal layer with a full dielectric mirror. As an example, a highly efficient
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Fig. 6. Optical patch antenna efficiency study. (a) The sketch of a optical patch antenna model ð0 ¼ 600 nmÞ. (b) Simulated patch antenna resonant length and efficiency at the resonant wavelength of 600 nm (500 THz) as a function of the substrate height.
dielectric mirror composed of several dielectric layers has been proposed in [19]. This is one of the potential solutions for realizing full dielectric resonator antennas and metasurfaces with very high efficiency at optical frequencies.
5. Comparison to Metallic Patch Antenna in the Optical Range A patch antenna model is built to compare its efficiency with the DRA at the resonance wavelength of 600 nm (500 THz). The optical patch antenna consists of a square silver patch with a length of Lp mounted on a SiO2 spacer with refractive index n of 1.45, extinction coefficient of 108 (equivalent to the loss tangent of tan ¼ 2:9 108 ) and thickness of Hs , as shown in Fig. 6(a). The silver patch thickness of 50 nm is selected as two times the skin depth in silver at this wavelength. For a fair comparison, the size of the substrate and bottom silver layer are chosen the same as for the DRA model, i.e. 1:50 ¼ 900 nm. The thickness of the bottom silver is 200 nm to block all incident radiation. Interestingly, a similar metallic patch configuration has been utilized for perfect absorbers at near-infrared and visible frequency ranges, where the absorption is caused by localized surface plasmon resonances [20], [21]. As in the DRA study and for optimal excitation of the patch resonant mode, the patch antenna model is excited by a port located in the bottom center of the spacer, at a height 5 nm above the bottom silver block. Since the patch antenna resonance is influenced by the thickness of the spacer [22], the efficiency of the antenna is studied for different spacer thickness Hs . The resonance is obtained by varying the patch length Lp , as shown in Fig. 6(b). By increasing the substrate thickness, the efficiency can be raised from 20% to 45%. However, the efficiency cannot be further improved by increasing the substrate thickness because the resonance between the top silver patch and bottom silver layer diminishes beyond Hs ¼ 125 nm. Thus, the efficiency of the patch antenna is convergent to about 45% at the resonant wavelength of 600 nm, about 13% lower than that of DRA. It is worth mentioning that the efficiency of the DRA is obtained with DR height of 0 =10, where this denominator 10 is chosen as typical value appropriate for operation in the fundamental HEM11 mode.
6. Conclusion In this paper, the scaling behavior and efficiency of DRAs on a metal plane have been studied from the microwave to the visible part of the spectrum (0 from 0.3 m to 500 nm). The study is a fundamental investigation for bringing the concept of conventional microwave DRAs to optical frequencies. Two commercial electromagnetic simulation tools have been employed for optical antenna design, with particular care taken in the use of three appropriate material models across the different frequency scales. The transition and overlap of the scaling curves for the three models validate this practical method of characterizing metal properties from microwave to optical frequencies.
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The numerical modeling quantitatively shows how the electromagnetic field penetration into the metal prohibits a straightforward scaling of DRAs from the microwave range. In terahertz and visible frequencies (wavelength from 300 m to 500 nm), the resonant DR diameter is gradually reduced down to one third of the value that would be expected from direct scaling. From the study of the radiation efficiency, it has been shown that from the microwave, up to the far-infrared region, the efficiency of the DRA remains as high as 90% for a dielectric material with a loss tangent of tan ¼ 0:01. In this frequency range, only a negligible fraction of the electromagnetic wave can penetrate into the silver layer and the loss mainly results from the DR. The efficiency of DRAs can be improved by employing lower loss dielectric material. As the operating wavelength decreases toward the visible spectrum, the fields begin to penetrate deeply into the metal, leading to higher ohmic losses. As a result, below 1 m, the radiation efficiency drops significantly due to an increase of loss in the silver. Despite that, in the visible range, the efficiency of DRAs is expected to be higher than that of comparable patch antennas, as demonstrated for an example at 600 nm. The findings presented in this paper provide guidelines for the design and scaling of DRAs at optical frequencies, taking into account decreasing metal conduction due to plasmonic effects.
Appendices: Approaches for Electromagnetic Modeling of Metal From the computational point of view, electromagnetic simulation software based on volume discretization generally simplifies a metal block into its outer surface mesh, when the metal can be approximated as a perfect electric conductor or characterized by a frequency-dependent complex surface impedance. This significantly reduces the computational expenditure in terms of CPU time and memory requirements. At low frequencies, in the microwave range, the skindepth model (Appendix A) is traditionally used and has been shown to provide highly accurate results. At higher frequencies, in the low THz range, the more realistic surface impedance model (Appendix B), taking into account relaxation effects, keep a low computational expenditure and has been demonstrated to yield acceptable accuracy below the near-infrared region. However, the model is too simple to describe the realistic penetration of electromagnetic fields into the silver block in the visible frequencies, where a volumetric discretization of the metal for application of the Drude model is required (Appendix C).
Appendix A Classical Skin-Effect Model
At long wavelength in the radio-frequency range (usually for 0 9 300 m), only a negligible fraction of the electromagnetic wave can penetrate into the metal. The electron scattering relaxation and displacement current terms can be ignored [23]. The classical skin effect model surface impedance Z0 is given by the well-known first-order Leontovich surface boundary condition [24] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j!0 r Z0 ¼ ¼ R0 ð1 þ jÞ 0
(1)
where ! ¼ 2f , 0 is the permeability of free space, r is the relative permeability of the medium, 0 is the intrinsic bulk conductivity at DC, and R0 is classical skin-effect surface resistance.
Appendix B Modified Relaxation–Effect Model At terahertz frequencies ð30 G 0 G 3000 mÞ, a more significant part of the field penetrates into the metal and therefore the scattering relaxation has to been taken into account. Considering
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electron-phonon collisions, the surface impedance ZSR of the modified relaxation–effect model is given by [25] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j!0 r ZSR ð!Þ ¼ R þ j!"0
(2)
where "0 is the permittivity of free space. The intrinsic bulk conductivity of the metal is R ¼ 0 =ð1 þ j!Þ, where is the phenomenological scattering relaxation time for the free electrons. The value of is typically on the order of 1014 s at room temperature. The displacement current term j!"0 can been ignored at long wavelengths (0 93000 m, ! 1), where Eq. (2) can be simplified to the classical skin-effect model in Eq. (1).
Appendix C Drude Model As the operation wavelength approaches the near-infrared and visible range, electric field penetration increases significantly, and becomes a non-negligible fraction of the free-space wavelength. An increase in the wave-electron interaction with a high scattering rate leads to high energy dissipation. Importantly, the field penetration prohibits a straightforward scaling of structures from the microwave range. The Drude model is a classical approximation that describes the optical response of noble metals. The dielectric function of the free electron gas is represented as [23] "ð!Þ ¼ "1
!2
!2p þ j !
(3)
where "1 is the sum of interband contributions (usually 1 G "1 G 10) and !p is the plasma frequency. The damping constant ¼ 1 þ !2 is composed of a frequency independent part 1 and a frequency dependent term !2 . The parameters for the Drude model found in the literature might differ between different sources, as the exact properties depend on the quality of investigated samples, i.e., the density of the film, the grain size, and the surface roughness. The density affects the strength of the interband absorption and the plasma frequency !p . The grain size and the surface roughness dominate the damping constant . The parameters for silver adopted here are "1 ¼ 4, !p ¼ 1:38 1016 s1 , 1 ¼ 2:73 1013 s1 , and ¼ 5:9 1018 s1 [26].
Acknowledgment A. Mitchell's time commitment to this project was under the ARC Centre of Excellence program (CE110001018). However, this project was not supported with funding from this center.
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IEEE Photonics Journal
Efficiency of Dielectric Resonators
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