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On the Existence of Superdirective Radiation Modes in Thin-Wire Nanoloops Mario F. Pantoja, Jogender Nagar, Bingqian Lu, and Douglas H Werner ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00486 • Publication Date (Web): 07 Feb 2017 Downloaded from http://pubs.acs.org on February 12, 2017

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On the Existence of Superdirective Radiation Modes in Thin-Wire Nanoloops Mario F. Pantoja‡1, Jogender Nagar‡2, Bingqian Lu2, and Douglas H. Werner2* 1

Department of Electromagnetism and Physics Matter, University of Granada, Granada (Spain)

2

Electrical Engineering Department, The Pennsylvania State University, University Park, PA (USA)

KEYWORDS: Superdirectivity, nanoantennas, nanoloops, electromagnetic radiation.

ABSTRACT:

The problem of obtaining broadband superdirective radiation from an electrically small, easy

to manufacture antenna is among the most challenging and elusive problems in electromagnetics. Superdirective arrays tend to be narrowband and sensitive to tolerancing, while a single superdirective radiator typically requires a very complicated and difficult to manufacture design. In this Article, we report on a new and transformative discovery, the fact that broadband superdirectivity naturally occurs for a single thin-wire nanoloop of the appropriate material composition and size. Full-wave simulations have revealed end-fire directivity of above 4.0 (6 dBi) for a nanoloop with radius less than 0.2 wavelengths. This surprising phenomenon is explained in two ways: by comparison with two-element superdirective arrays and via a spherical multipole decomposition. This finding offers a solution to the problem of the inherently short-range communication of nanodevices, and thus has the potential to strongly impact the fields of sensors, electronics, and wireless communications.

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The creation of superdirective antennas has been the Holy Grail for physicists and engineers ever since the concept was first postulated nearly a hundred years ago1. While pioneering work in the field has shown that unlimited directivity can theoretically be achieved with arrays of ideal perfectly conducting elements2-4, this superdirectivity comes at the cost of extreme sensitivity to electrical and mechanical errors, low radiation efficiency and a very narrow bandwidth5-7. The first experimental demonstration of the phenomenon was reported in 1969 for antenna arrays operating at radio frequencies8. It was later shown that the concept of superdirectivity could be applied to the optical regime9. A brief overview of the history and theory of superdirective arrays can be found in the literature10. More recent research has focused on achieving superdirectivity in the RF regime using an electrically small antenna (i.e., with dimensions less than a wavelength)11-13. Recently, an interest in the phenomenon has been rekindled with the advent of metamaterials14. In this context, superdirectivity has been achieved using meta-arrays15-16, metasurfaces17, or through coupling between metaparticles18.

In the optical regime, remarkable

contributions to achieving directive radiation patterns with single-element radiators have been reported in recent years, with structures such as dielectric nanospheres19 or metallic nanorods and nanodisks20 being among the most popular. Along with these single-element radiators, the use of coupling between two elements to create directional radiation in the optical regime has also been achieved by utilizing plasmonic nanoparticles21 and magnetic resonators22. In addition, arrays of a higher number of elements23 can lead to efficient, highly directive designs in the optical regime, such as a superdirective Yagi-Uda array of dielectric nanoparticles24. However, these nanoantenna array designs suffer from the same drawbacks that have plagued more traditional superdirective arrays in the RF regime: namely, a high directivity can be achieved only at the expense of narrow bandwidth and an extreme sensitivity to the physical parameters5. The main contribution of this Article is to show that, remarkably, thin-wire metallic nanoloops with the appropriate dimensions and material composition are able to naturally produce the superdirectivity phenomenon with a large bandwidth in the optical regime, thus overcoming the, until now, persistent 2


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narrowband limitation as well as tolerance and sensitivity issues. While a superdirective thin-wire nanoloop has a low radiation efficiency, it will be shown that by increasing the wire thickness, extremely high directivities over a broad bandwidth can be achieved at a higher frequency range with an efficiency of about 98%. Furthermore, this finding paves the way for the implementation of more complex designs (i.e., designs involving arrays with a large number of elements and/or combining the advantages of both dielectric and metallic nanoantennas), and brings the original concept of superdirectivity much closer to reality. Although superdirective antennas are traditionally used in space communications and radio astronomy, the nanoloop design considered here is a subwavelength structure and has potential applications in areas which include, among others, optical filters, medical devices, sensors, and lenses for high-gain nanoantennas. From a physical point of view, the superdirectivity phenomenon occurs when a significant enhancement of the directionality of an electrically small antenna is achieved, in comparison to the directionality of a Hertzian or ideal point dipole. A more quantitative definition, in the context of arrays of antennas, is any directivity higher than that obtained with an array of the same length where the elements are excited by currents with uniform amplitude and phase25. Of particular interest for this Article is the special case of the two-element array, which was presented originally in Reference 3 and later numerically studied for isotropic elements26. If we consider elements with an azimuthally symmetric radiation pattern, then a maximum end-fire directivity is achieved when both elements are fed with the same current amplitude and a phase difference of about 150-200 degrees. Exact values depend on the distance between elements, and as the spacing is decreased the end-fire directivity increases, ranging from a value of 2 (3 dBi) for a spacing of 0.5

(where

is the free-space wavelength) up to nearly 4 (6 dBi) for distances close to zero.

This value of 4 (N2 for an array containing N elements) is usually taken as the reference value when considering superdirective radiation from a two-element antenna array. From a practical point of view, the matching network required to feed elements of the array is highly sensitive to the distance between 3


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them. Moreover, for very close non-isotropic elements (e.g., linear dipoles) the mutual coupling has a strong influence on the magnitude and phase of the currents, thus making robust implementation of the superdirective effect with a two-element array extremely difficult to achieve in practice, even over a narrow bandwidth. With all this in mind, the theoretical analysis of the currents on thin-wire nanoloops in the optical and infrared regimes has been presented27. In essence, the idea was to extend the Fourier expansion of the currents in the thin-wire PEC loop, originally introduced in Reference 28, to include the losses in the constituent material that occur at high-frequency regimes29. From this analysis, far-field radiation parameters, including directivity and gain, can be derived. Closed form expressions for the currents and the directivity in the end-fire direction are included for completeness as extra material in Appendix 1 and a full derivation can be found in Reference 30.

Figure 1. Directivity of a thin-wire gold nanoloop of circumference C = 2πb = 3000 nm located in the XY plane. (Left) Directivity vs. frequency along three directions of interest. (Right) 3D radiation pattern at 118 THz ( = 2540 nm) or at 2 / = 1.18. For validation purposes, results include both full-wave simulations provided by CST31 and analytical expressions based on the theoretical development presented in Appendix 1 of the Supplemental Materials.

The nanoloops under study in this paper have loop radius b and a thickness measure22 Ω = 2 ln(2πb/a), where a is the wire radius. Primed spherical coordinates (r’ = b, θ’ = 90°, φ’) designate source points around the perimeter of the nanoloop, while unprimed spherical coordinates (r, θ, φ) correspond to field 4


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points. Note that in the far-zone of the nanoloop, the r-dependence is suppressed and the field point coordinates are designated simply by (θ, φ). The supplemental materials contain figures depicting these coordinate systems for clarity. Figure 1 shows the directivity versus frequency for a thin-wire gold nanoloop with a circumference of 3000 nm and Ω = 12 along three different directions of interest. For values of

2

/

(where b is the radius of the loop) less than 0.25, the radiation pattern exhibits a

directive response along (θ, φ) = (90°, 0°). For values of kb larger than 1.5, the nanoloop has similar radiation properties to a nanodipole oriented parallel to the direction of the feed gap (i.e., along the Yaxis), a result consistent with previous findings32. Surprisingly, however, the (θ, φ) = (90°, 180°) direction yields unusually high values of directivity (near 4.5 or 6.53 dBi) in the far-infrared regime (i.e., at 118 THz or kb = 1.18). This does not occur for the nanoloop studied in Reference 32 due to the particular combination of size, material composition and frequency range under consideration. The 3D radiation pattern at 118 THz shown in Figure 1 explicitly exhibits superdirectivity, a phenomenon that is not present in perfect-electric conductor (PEC) loops or in thin-wire gold nanoloops where the circumference is either too large or too small. It is important to note that superdirectivity (where the directivity is larger than 4) occurs over a relatively large fractional bandwidth (absolute bandwidth normalized by the center frequency) of 10 percent. The fractional bandwidth for directivity larger than 3 is 25 percent. These values are much greater than any previously reported fractional bandwidths that have been obtained for two-element arrays, which are typically on the order of only 1-2 percent33, as well as other single-element nanoantennas such as dielectric nanodisks and nanospheres which yield typical fractional bandwidths for directivities larger than 3 of approximately 9% and 14%, respectively19-20. To expand on the claim that superdirectivity only occurs for a nanoloop with a very specific combination of geometrical parameters and material composition, the supplementary material includes: a) videos presenting a comparison of the radiation patterns for gold nanoloops and their PEC counterparts, and b) additional analysis of the 600 nm gold nanoloop, in which superdirective radiation does not appear. As quantum dots34 and optical 5


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waveguides35 are the most common feeding techniques at the nanoscale, it is important to note that efficient feeding of a nanoantenna is still an area of intense research due to the difficulties involved in impedance matching quantum systems35. Here we consider idealized computational models of quantum dots and simplified delta-gap numerical models to feed the nanoantennas (details and comparisons of the quantum dot and delta-gap model are provided in Appendix 6 of the Supplementary Material).

Figure 2. Current distribution (magnitude and phase) for the nanoloop of Figure 1 operating at 118 THz ( = 2.54 µm) where the coordinate system is shown in Figure 1.

At the frequency of maximum directivity, the thin-wire gold nanoloop operates as a resonant loop of length 3 times the effective wavelength of propagation37. Figure 2 shows the magnitude and phase of the current distribution at this frequency as a function of azimuthal angle φ’. The most important observations are that there is a near 180° phase difference between the feed point (φ’ = 0°) and the opposite point on the nanoloop (φ’ = 180°) and the magnitude of the current at φ’ = 180° is about 30% lower than it is at φ’ = 0°. These properties of the current, as well as the fact that superdirectivity occurs along the (θ, φ) = (90°, 180°) direction, suggests that there is a relationship between the superdirective two-element array of parallel linear dipoles and the thin-wire nanoloop. The supplementary material included in Appendix 2 provides a derivation for the maximum directivity of a linear array of two dipole antennas.

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To provide a way to explain this anomalous effect, we will consider the thin-wire nanoloop as a superposition of two dipoles of length 3 /2 separated by a distance equal to the radius of the nanoloop b. The rationale for this is further explained in Appendix 3. At the frequency kb = 1.18, the gold nanoloop with a circumference C = 2πb = 3000 nm can be approximated by two dipoles with a separation distance of d = 0.6 /π = 0.19 . Neglecting mutual coupling, the elements should have a uniform amplitude and phase difference of 180 degrees for maximum directivity. Taking into account the mutual coupling between two closely-spaced dipoles requires a slight decrease in the excitation current amplitude for one of the elements. The superdirective model26 predicts a maximum directivity of 4.5 (6.53 dBi), which is very close to the value of 4.25 (6.28 dBi) achieved by the nanoloop. There are multiple assumptions in this simple model which limit the accuracy of the prediction but it provides an instructive and intuitive model to explain the superdirectivity effect. Analogous to the amplitudes and phases of the two-element dipole array, the thin-wire nanoloop can be viewed as two sources: one located at φ’ = 0° and a second at φ’ = 180° with a slightly lower current excitation magnitude and a phase difference of approximately 180 degrees, as indicated by the plots shown in Figure 2.

Figure 3. Near-zone electric field (left) magnitude and (right) phase in the XY plane (i.e. the plane containing the nanoloop as shown in Figure 1) for the 3000 nm thin-wire gold nanoloop at 118 THz.

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Another interpretation of the phenomenon can be offered by performing a multipole decomposition of the near fields. According to Reference 38 the multipole expansion of the near field surrounding a localized charge-current distribution leads to an angular spectrum representation which completely describes the radiated electromagnetic field. In this way, properties related to the radiation pattern such as the directivity can be expressed simply in terms of coefficients in a multipole expansion of vector spherical harmonics (see Appendix 4 for a summary of the equations and procedures to compute the multipole coefficients). First, the near fields of the nanoloop will be examined, then a more rigorous multipole decomposition will be performed. Figure 3 depicts the near-zone electric fields of the gold nanoloop at 118 THz. The magnitude plot shows intense electric fields around the periphery of the nanoloop, with 6 peaks corresponding to the fact that the loop has a circumference of C = 3λeff, where λeff is the effective wavelength38. It also shows the directive pattern in the end-fire direction, with a difference of 40 dB between the forward (X, Y) = (1 µm, 0) and backward (X, Y) = (-1 µm, 0) direction. The angular width can also be determined by examining the phase of the electric field, as shown in Figure 3 (right). It can be seen that a constant phase front forms over the effective radiation aperture, and the diameter of this region dictates the angular width (beamwidth) of the main beam. At 118 THz the diameter of the nanoloop is 0.37 , and the rapid spatial oscillations of the current in this electrically small region leads to the excitation of high-order multipole moments which results in superdirectivity39. The effective aperture area39 S = D /(4π) as shown in Figure 3 (right) at 118 THz is 2.31 µm2. Normalizing this value to the geometric aperture of a spherical antenna of radius b leads to a normalized effective aperture39 Sn = D /(4π2b2) of 3.23. For a conventional spherical particle, Sn > 4 is considered to be the superdirective limit. While the normalized effective aperture doesn’t quite achieve superdirectivity based on this metric, it also occupies significantly less volume than the sphere.

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Figure 4. Amplitude of the (a) electric and (b) magnetic multipole coefficients and phase differences of the (c) electric and (d) magnetic multipole coefficients corresponding to the radiated fields for the 3000 nm thin-wire gold nanoloop.

An analysis of the magnitudes and phases of the specific electric and magnetic modes, similar to the study performed in Reference 39 for a superdirective dielectric nanoantenna, can provide insight into the physical mechanisms which result in superdirectivity. Typically, superdirective behavior is achieved when a specific set of multipole coefficients are out of phase, resulting in strong constructive interference in a single direction39,40. Figure 4 shows the amplitude and phase of the dominant electric and magnetic multipole coefficients41 up to order l = 3 for the 3000 nm gold nanoloop. A comparison of the magnetic multipole coefficients shown in Figure 4 (b) and the directivity plot of Figure 1 indicates that the l = 1 magnetic mode peaks when superdirectivity occurs. The superposition of this mode and the l = 1 electric mode leads to a directive pattern along (θ, φ) = (90°, 0°) and (θ, φ) = (90°, 180°). However, the magnitude 9


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of the electric multipoles shown in Figure 4 (a) are dominant over most of the frequency range and dictate whether or not high directivity occurs specifically at (θ, φ) = (90°, 180°). Figure 4 (c) shows that the difference between the phases of the (l, m) = (2, ±2) modes undergoes a jump from +π to –π at around kb = 0.72, which corresponds to the frequency point where superdirectivity first starts to appear. When the phase difference is + π, these modes can only add up constructively along φ = 0°, but when the difference is -π these modes add up constructively along φ = 180°. A peak in the magnitude of the (l, m) = (2, ±2) mode occurs at kb = 0.82, a point at which there is also a peak in the directivity along (θ, φ) = (90°, 180°). Beyond this frequency point, the magnitude of this mode decreases while the magnitude of the (l, m) = (3, ±3) mode increases until a maximum is reached at kb = 1.18, a point at which the directivity along (θ, φ) = (90°, 180°) also peaks. The in-phase superposition of the (l, m) = (3, ±3) modes coupled with the out-of-phase superposition of the (l, m) = (2, ±2) modes leads to extremely high directivity along (θ, φ) = (90°, 180°). It is interesting to note that this same phase jump occurs for the l = 2 magnetic multipole modes, as shown in Figure 4 (d), but this contribution is negligible. Beyond about kb = 1.5 the electric and magnetic multipole coefficients have approximately the same magnitude and superdirectivity no longer occurs. It is important to note that the physical mechanism leading to superdirectivity in the thinwire metallic nanoloop is fundamentally different than that of dielectric nanospheres and nanodisks, such as the structure presented in Reference 29. While dielectric structures have a large spectrum of high-order magnetic multipoles which contribute to a high directivity over a narrow bandwidth, the thin-wire metallic nanoloop considered in this paper is dominated by low-order electric multipoles which results in a very large frequency bandwidth at the expense of more loss. This poor efficiency can be circumvented by increasing the wire thickness, which also shifts the highly directive band up in frequency.

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Figure 5. Directivity in the forward end-fire direction vs. normalized wavelength for 3000 nm loops comprised of gold, aluminum and perfect electric conductor materials.

Now that the physics leading to the superdirective effect for nanoloops has been explained, several questions arise from this analysis: a) whether the phenomenon can be achieved with materials other than gold, b) the dependence of the radiation properties on the geometrical parameters, and c) how this metallic structure compares to the performance of superdirective dielectric nanoantennas. To address the first issue, Figure 5 shows the directivity of the 3000 nm and Ω = 12 loop when it is comprised of gold, aluminum and PEC materials. It can be seen that the superdirective effect also appears for aluminum, but at a higher frequency and with an even wider bandwidth when compared to gold. As expected, the PEC loop never achieves superdirectivity. The difference in radiation properties between Al and Au can be explained by studying the material parameters (plots of these parameters and a more detailed explanation are given in Appendix 5 of the Supplemental Materials). To summarize, the Al presents a lower dynamic range of the extinction coefficient over the band of frequencies considered, which allows a more stable distribution of currents over a wider bandwidth42. Next we study the dependence of the directivity on the geometrical parameters of the nanoloop. Appendix 5 of the Supplemental Materials presents a comprehensive parametric study of the directivity 11


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of nanoloops comprised of Au and Al. The fundamental takeaway is that electrically small nanoloops cannot support the current distributions required for superdirectivity while electrically large nanoloops exhibit too much attenuation. Only an intermediate portion of the frequency spectrum is suitable for superdirectivity, and this frequency range increases for larger loop circumferences and wire thicknesses. More details related to the directivity and efficiency are provided in Appendix 5 of the Supplemental Material.

Figure 6. Comparison of directivity (left) and gain in dB (right) for Ag and Au thin-wire nanoloops with b = 477.5 mm and thickness factors Ω = 8 and Ω = 12, a Si (ε r = 11.9) nanosphere of radius 477.5 nm, and a Si nanodisk of radius 477.5 nm and height h = 15 nm.

Next, we compare the performance of nanoloops with different thickness factors to that of dielectric nanospheres and nanodisks. Figure 6 shows a comparison of the directivity (left) and gain in dB (right) for a silver nanoloop with Ω = 8, gold nanoloop with Ω = 12, a Si nanosphere and a Si nanodisk. Note that the figure shows the directivity and gain over two regions of interest where the structures are highly directive (kb between 0.5 and 1.5 for the gold nanoloop and Si nanosphere and kb between 3 and 4 for the silver nanoloop and Si nanodisk). The reason dielectric nanospheres and nanodisks are superdirective is due to inner displacement currents which, at some resonance frequencies, have a phase distribution able to cancel radiation completely on one side of the nanostructure, resulting in the directive pattern19,24. From 12


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our analysis, we’ve found that two remarkable differences exist between dielectric nanostructures and metallic thin-wire nanoloops. First, the effective wavelength of the thin-wire nanoloop changes nonlinearly for different frequencies and wire radii a37, and the current pulses will propagate in the nanoloop as a slow wave. Therefore, the directive patterns will appear in thin-wire nanoloops at lower frequencies than dielectric nanospheres of the same size, provided the nanoloop and nanosphere have similar values of electric permittivity. This is illustrated in Fig. 6 (left), where the Au thin-wire nanoloop of Ω = 12 creates a superdirective pattern with larger bandwidths and at lower frequencies than the corresponding dielectric nanospheres and nanodisks. Second, the magnitude decrease in the distribution of currents required to accomplish the unidirectional radiation presented in Appendix 2 can only be satisfied when the extinction coefficient of the material is suitably large, a condition which does not hold for non-lossy dielectric nanostructures. For practical purposes, the radiation efficiency and gain of nanoantennas need to be studied since these parameters dictate the range of communication for these devices, and as shown in Fig. 6 (right) the dielectric nanostructures present outstanding performance due to the absence of losses. This is not surprising, because, as a consequence of the large effective wavelength, the thin-wire nanoloops can be considered electrically small antennas42. This presents drawbacks in terms of the maximum achievable efficiency and gain according to the Chu-Harrington limit43, recently reformulated by Geyi44,45. A way to increase the efficiency of the nanoloops is to decrease the thickness factor (i.e. increase the wire radius). For instance, an Ag nanoloop with Ω = 8 is also presented in Fig. 6, showing that higher gains are achievable at higher frequencies. Ag was chosen as it yields radiation efficiencies of around 98% in the frequency range of interest (verification with full-wave solvers is presented in Appendix 6 of the Supplemental Materials). Fig. 6 shows that the Ag Ω = 8 loop shows extremely stable directivity over a broad frequency range, while the dielectric structures have many narrowband resonances. Note that the dielectric structures have been excited with an infinitesimal dipole source while the metallic nanoloops have been excited with a delta-gap voltage source. In reality, the dielectric 13


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structures would likely be fed with a quantum source while the nanoloops would be fed with an optical transmission line34-36. While obtaining a broadband impedance match between the quantum source and a nanoantenna could be very difficult, it is expected to be easier to achieve when coupling a transmission line to the feed-gap of a nanoloop. As can be seen in Figure 6, the Ω = 12 Au nanoloop first exhibits superdirectivity in the end-fire direction at kb = 0.8, while the nanosphere first exhibits superdirectivity at kb = 1.05. A comparison of the directivities between these two structures at the two frequencies is shown in Fig. 7. While the nanoloop is highly directive over this frequency range, the nanosphere is not. The Si nanodisk exhibits its first superdirective peak at around kb = 1.52, while after that the directivity is highly oscillatory with another peak at kb = 3. The Ω = 8 Ag nanoloop exhibits a peak at around 3.3. The directivites of these two structures at these frequencies is shown in Fig. 8. While the nanoloop remains directive with a peak at (θ, φ) = (90°, 180°), the directivity of the nanodisk switches from a peak at (θ, φ) = (90°, 180°) to a peak at (θ, φ) = (90°, 0°). As can be seen, the primary advantage of the nanoloop is its broadband superdirective behavior. In fact, the Ω = 8 Ag nanoloop has unidirectional radiation with a directivity above 3 from kb = 2.5 to kb = 4, at which point large sidelobes can be seen in the radiation pattern. We can conclude that the differences in the constitutive material of the nanostructures leads to dramatically different behavior in metallic and dielectric nanoantennas. The identification of the different mechanisms (i.e., electric and displacements current) to create superdirective patterns enables the search for a new generation of superdirective nanoantennas, which could eventually combine the advantages of both forms of currents.

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(a)

(b)

(c)

(d)

Figure 7. Directivity of the Si nanosphere at (a) kb = 0.8 and (b) kb = 1.05 and the Ω = 12 Au nanoloop at (c) kb = 0.8 and (d) kb = 1.05. Simulations were performed with FEKO45.

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(a)

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(b)

(c)

(d)

Figure 8. Directivity of the Si nanodisk at (a) kb = 3 and (b) kb = 3.3 and the Ω = 8 Ag nanoloop at (c) kb = 3 and (d) kb = 3.3. Simulations were performed with FEKO46.

Finally, the effect of a substrate will be considered to facilitate the design of more realistic nanodevices. The availability of the closed-form expressions derived in Appendix 1 for nanoloop antennas in free space provides the basis for extending the exact analytical formulation to include the effects of placing the nanoloop on a dielectric substrate. This would involve extending the formulation to include Sommerfeld integrals and discrete complex image theory to calculate the radiated field when the nanoloop is placed on a dielectric medium. Such a technique would be much more computationally efficient than using a full-wave solver. However, this analytical derivation has not been performed yet so instead full-wave analysis will be utilized. The presence of the substrate leads to a deviation in the direction of the main beam as a consequence of the higher permittivity of the substrate34-35. Furthermore, directivity and gain will be affected both in bandwidth and peak values, depending on the particular choice of substrate. Fig. 9 shows the near-field and directivity of the gold nanoloop of Fig. 1 when placed above

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a substrate of glass (εr = 2.6, tan δ = 0.0057). As can be seen, the radiation pattern remains qualitatively the same other than a slight deviation in the direction of maximum directivity.

Figure 9. 3000 nm gold nanoloop (placed in the XY plane) above a substrate (located at Z < 0) of glass. Near-field and radiation patterns are shown at 130.48 THz. Simulations were performed with FEKO46.

In conclusion, this paper reports the discovery that superdirective radiation naturally occurs for the thinwire nanoloop when the proper material composition and loop circumference are chosen. This finding has the potential to transform the fields of sensing and wireless communications. Analysis of the current distribution shows that the lossy nanoloop acts as a superdirective array of two closely coupled dipoles, without the added complexity and sensitivity issues associated with a two-port feeding network. A spherical multipole decomposition shows that very particular electric multipoles are excited with the proper phase difference to achieve superdirectivity. The most remarkable finding is that these specific conditions for superdirectivity occur over a relatively broad bandwidth naturally, without any additional geometrical or electrical modifications. A preliminary comparison between superdirective dielectric and metallic nanostructures shows that the advantageous features of thin-wire nanoloops in terms of electrical size and bandwidth also has a downside in the lower radiation efficiency. However, if the wire thickness 17


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is increased, it was shown that the nanoloop is capable of exhibiting an efficiency around 98% and the region of broadband unidirectional radiation shifts to a higher frequency region. Future studies will include analysis of the lossy thin-wire nanoloop on a substrate and in an array environment, as well as exploring potential designs of nanostructres that combine electric and displacement currents.

ASSOCIATED CONTENT Supporting Information. Derivations associated with the results included in the paper. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *E-mail:[email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally. Funding Sources The authors would like to acknowledge the Spanish Ministry of Education-Commission Fulbright Program “Salvador de Madariaga” (PRX14/00320) for sponsoring the joint research project. This work has also been partially financed by the Spanish and Andalusian research

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programs TEC2013-48414-C3-01 and P12-TIC-1442 as well as the Penn State MRSEC - Center for Nanoscale Science NSF DMR-1420620. Notes The authors declare no competing financial interests.

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3D radiation pattern exhibiting superdirectivity for a 3000 nm circumference gold nanoloop operating at 118 THz.


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Existence of Superdirective Radiation Modes in ... - ACS Publications

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