Using Huygens Multipole Arrays to Realize Unidirectional Needle-Like Radiation

Using Huygens Multipole Arrays to Realize Unidirectional Needle-Like Radiation PHYSICAL REVIEW X 7, 031017 (2017) Richard W. Ziolkowski University of Technology Sydney, Ultimo NSW 2007, Australia The University of Arizona, Tucson, Arizona 85721, USA (Received 19 December 2016; published 26 July 2017) For nearly a century, the concept of needle radiation has captured the attention of the electromagnetics communities in both physics and engineering, with various types of contributions reoccurring every decade. With the near-term needs for highly directive, electrically small radiators and scatterers for a variety of communications and sensor applications, superdirectivity has again become a topic of interest. While it is well known that superdirective solutions exist and suffer ill-posedness issues in principle, a detailed needle solution has not been reported previously. We demonstrate explicitly, for the first time, how needle radiation can be obtained theoretically from currents driven on an arbitrary spherical surface, and we explain why such a result can only be attained in practice with electrically large spheres. On the other hand, we also demonstrate, more practically, how broadside radiating Huygens source multipoles can be combined into an end-fire array configuration to achieve needle-like radiation performance without suffering the traditional problems that have previously plagued superdirectivity. DOI: 10.1103/PhysRevX.7.031017 Subject Areas: Industrial Physics, Interdisciplinary Physics, Plasmonics I. INTRODUCTION soon after Oseen’s publication. Both end-fire (maximum radiated power is oriented along the array direction) [8] The concept of superdirectivity has permeated the areas and broadside (maximum radiated power is oriented of physics and applied physics repeatedly since Oseen perpendicular to the plane of the array) [9] pattern enhance- discussed the concept of “needle radiation” almost a ments from different array configurations were considered century ago [1]. While Oseen was keenly interested in initially. La Paz and Miller [10] purported to show that the how a tiny atom might absorb a large electromagnetic wave maximum directivity from an aperture of a given size was as an equivalent photon (and, consequently, the alternate fixed, but Bouwkamp and De Bruijn [11] correctly dem- translation of his paper’s title as “pinprick” radiation might onstrated that there was no theoretical limit on the direc- make more historical sense), the reciprocal problem of tivity from an aperture of any size. Dolph realized that one transmitting a needle-like radiation pattern from a small could control the sidelobe levels of the pattern by properly source has stimulated many physics and, possibly more, weighting (Chebyshev polynomial tapering) the amplitudes engineering discussions. The role of superdirectivity in of the element excitations [12]. Riblet [13,14] illustrated radio astronomy and in particle physics was discussed by that such amplitude tapering has an associated cost of Casimir [2] and Wheeler [3]. They, too, emphasized the widening the mainlobe of the pattern. However, it was possibility that the effective receiving cross section of a quickly shown by Yaru [15] that the current distribution radio telescope or an atom could be extremely large in solutions that produce superdirective beams from arrays are comparison with its physical size. This concept has been generally ill-posed [16]; i.e., small variations of the large demonstrated more recently with plasmonic particles positive and negative variations of the excitation amplitudes whose strong reactive scattering components extend to required to achieve the effect lead to its disappearance in large distances and redirect the power passing through a practice. In fact, Casimir [2] and Wheeler [3] noted this large area of an incoming plane wave and force it to flow practical difficulty and believed that one would never go towards the scatterer [4–7]. beyond combining a dipole and a quadrupole mode Engineering the emission of electromagnetic fields from together in practice. Nonetheless, this goal has been finite sources was intensely studied in the 1940s and 1950s achieved with subwavelength dielectric and plasmonic particles [17–22]. There have been and continue to be many examples of optimizing the directivity from an Richard.Ziolkowski@uts.edu.au, ziolkowski@ece.arizona.edu antenna system with constraints on its various other Published by the American Physical Society under the terms of performance characteristics to circumvent the ill-posedness the Creative Commons Attribution 4.0 International license. of the “super” outcome [23–27]. Further distribution of this work must maintain attribution to A useful operational definition of superdirectivity, e.g., the author(s) and the published article’s title, journal citation, and DOI. as emphasized by Hansen [28,29], is to achieve a directivity 2160-3308=17=7(3)=031017(13) 031017-1 Published by the American Physical Society RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) greater than that obtained with the same antenna configu- modes N in a sphere of radius r would be limited to ration being uniformly excited (constant amplitude and N ¼ kr ≡ 2πr =λ is no longer justifiable in general. 0 0 phase). Let the radiating system be either an aperture The concept of a transmitting antenna realizing a far- antenna (continuous current distribution) whose effective field needle radiation pattern is also intimately connected area is A or an array of radiating elements (set of discrete to subwavelength imaging, i.e., being able to resolve two eff currents) distributed in A . If the total efficiency (i.e., small objects separated by subwavelength distances eff taking into account the material losses, mismatch losses, [46,47]. Moreover, superdirectivity has been shown to lead to enhanced channel capacity in multiple-input–multiple- polarization mismatch,…) of the system is e , then its total output (MIMO) systems [48–51]. Thus, superdirectivity maximum gain G is related to its maximum directivity max concepts yet again become important as nanotechnology D as G ¼ e × D . Thus, if there are no losses, max max total max applications flourish and the Internet of things (IoT) comes then the maximum directivity of the antenna system to fruition. We would like to have electrically small, highly uniformly driven at the excitation wavelength λ is funda- directive, receiving or transmitting antennas (whether they mentally related to its effective area A as [30] eff are macro, micro, or even nano) for numerous wireless applications. 4π D ¼ A : ð1Þ max eff 2 There has been a significant increase in higher-directivity scattering approaches reported, but little on corresponding radiating systems. This article is timely in that it addresses two Consequently, a larger effective aperture will provide a fundamental questions: Despite nearly a century of inves- higher directivity. tigation, what would it really take to achieve actual needle- As has been shown by a number of authors, e.g., like radiation? Can one design an array that would eliminate Refs. [31–33], the fields in a region of free space outside sidelobes, has a high front-to-back ratio, has superdirective of a spherical surface that encloses all the currents can be properties, and is not plagued by ill-posedness? expanded in a series of electric and magnetic multipole fields represented by (vector) spherical harmonics (see II. FAR-FIELD RADIATED FROM A SET OF Ref. [34] for more details). This approach has proved to be ELECTRIC AND MAGNETIC SOURCES very successful for the analysis of the far-field behavior of an antenna system [30]. By taking into account both the As explained in Ref. [34], the electric field radiated into transverse electric and magnetic modes, Harrington dem- the far field of a combination of electric J and magnetic onstrated that the maximum directivity from a source currents K excited at the frequency f ¼ ω=2π can be region as a function of the number of multipole modes, written as N,is [35,36] ZZZ ikr ff −ikr ˆ·r⃗ 0 0 3 0 2 ⃗ ⃗ ⃗ E ¼ iωμ e fJ ðr⃗ Þ − r ˆ½r ˆ · J ðr⃗ Þgd r⃗ D ¼ N þ 2N: ð2Þ ω ω J;ω max 4πr ð3Þ Therefore, by exciting higher order modes, one can achieve very high directivities from a fixed source region. ZZZ ikr ff These antenna results are immediately connected to the −ikr ˆ·r⃗ 0 3 0 ⃗ ⃗ E ¼ iωε e fr ˆ × ½K ðr⃗ Þgd r⃗ : ð4Þ K;ω 4πr upper bounds on the total cross section associated with scattering from particles [37]. The concept of subwave- The primed (unprimed) coordinates are the observation length superscattering [38,39] arises from maximizing the (source) point coordinates, and the wave number is contributions from a sufficiently large number of channels; pffiffiffiffiffi k ¼ ω=v, with v ¼ 1= εμ being the wave speed in the i.e., by aligning the frequencies of higher-order resonant medium. The far-field magnetic field is simply multipole modes, arbitrarily large total cross sections can ff ff ⃗ ⃗ H ¼ r ˆ ×E =η, where the free-space impedance be achieved with subwavelength structures. In fact, the ω ω pffiffiffiffiffiffiffiffi ability to create highly subwavelength (electrically small) η ¼ μ=ε. These expressions represent the known facts radiators and scatterers has been one of the success stories that the far fields are transverse electromagnetic (TEM) and associated with metamaterials [40–42]. Moreover, recent that they are related to the Fourier transform of the current passive and active nanoparticle studies associated with the components orthogonal to the observation direction. optical theorem [43], Kerker conditions [21], and Huygens Now consider the currents to be confined to the surface source effects [44] illustrate that combining sets of electric of a small sphere of radius a. Being as general as possible, and magnetic multipoles leads to enhanced directivities. the current densities then take the form These effects have been demonstrated with numerous ˆ ˆ configurations [17–20,22,45]. The recent subwavelength ⃗ J ðr⃗ Þ¼½J ðθ; ϕ; ωÞθ þ J ðθ; ϕ; ωÞϕδðr − aÞ; ω θ ϕ superdirective results have shown that Harrington’s original ˆ ˆ K ðr⃗ Þ¼½K ðθ; ϕ; ωÞθ þ K ðθ; ϕ; ωÞϕδðr − aÞ: ð5Þ ω θ ϕ estimate that the maximum number of usable higher-order 031017-2 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) ˆ ˆ ˆ 0 0 With the standard unit vector cross products θ×ϕ¼r, 0 −ikar ˆðθ;ϕÞ·r ˆðθ ;ϕ Þ 0 0 FJ ðθ;ϕ;ωÞ¼∯ dΩ e Π ðθ ;ϕ ;ωÞ: ð11Þ θ θ ˆ ˆ ˆ ˆ r ˆ × θ ¼ ϕ, ϕ × r ˆ ¼ θ—the far-field expressions become Z Z The fact that the source points are on the sphere of radius a ikr 2π π ff 2 0 0 0 −ikr ˆ·r⃗ 0 0 0 has allowed us to write r⃗ ¼ ar ˆðθ ; ϕ Þ, while emphasizing E ðr⃗ Þ¼ iωμ a sin θ dθ dϕ e 4πr 0 0 that the observation direction r ˆ is given by the coordinates ðθ; ϕÞ. The far-field pattern function is now more clearly 0 0 0 0 × J ðθ ; ϕ ; ωÞþ K ðθ ; ϕ ; ωÞ θ θ ϕ η related to the 2D Fourier transform of the current density pattern Π over the unit sphere. 0 0 0 0 þ J ðθ ; ϕ ; ωÞ − K ðθ ; ϕ ; ωÞ ϕ ; ð6Þ The far-field expression (10) indicates that there are no ϕ θ dc components of the source excitation radiated into the far field and that the radial dependence is that of a spherical 1 wave. The Fourier transform integral determines if there are ff ff ⃗ ⃗ H ðr⃗ Þ¼ r ˆ × E ðr⃗ Þ: ð7Þ ω ω any preferred directions into which the fields are radiated. Since we desire needle radiation along the z axis, it would follow that the pattern function yields As explained in Ref. [34], it is clear that to achieve a Huygens source behavior, one can simply consider the FJ ðθ; ϕ; ωÞ¼ δðcos θ − 1ÞδðϕÞ ≡ δðr ˆ − z ˆÞ: ð12Þ contributions from either the orthogonal pair J and K or θ ϕ from its dual, J and K . Electing the former, one has ϕ θ From this relation, it is immediately apparent that the currents on the sphere will have to be azimuthally sym- Z Z ikr 2π π ff 2 0 0 0 −ikr ˆ·r⃗ metric with respect to the z axis to achieve the desired E ðr⃗ Þ¼ iωμ a sin θ dθ dϕ e 4πr 0 0 outcome. To proceed, several spherical harmonic relations, as 0 0 0 0 × J ðθ ; ϕ ; ωÞþ K ðθ ; ϕ ; ωÞ θ ; θ ϕ reviewed briefly in Ref. [34], are employed. Since the spherical harmonics are a complete basis, one can expand ff ff ⃗ ⃗ H ðr⃗ Þ¼ r ˆ × E ðr⃗ Þ: ð8Þ the angular behavior of the theta component of the electric ω ω current density pattern on the sphere as By properly selecting those currents, we will show that the ∞ l X X Π ðθ; ϕÞ¼ c Y ðθ; ϕÞ; ð13Þ desired needle radiation can be achieved. θ lm lm l¼0 m¼−l III. NEEDLE RADIATION FROM CURRENTS where Y is the spherical harmonic of degree l and order lm ON A SMALL SPHERE m and the coefficients are given explicitly as First, consider the electric far-field component produced by only the J source: c ¼∯ Π ðθ; ϕÞY ðθ; ϕÞ sin θdθdϕ; ð14Þ lm θ lm Z Z ikr 2π π J;ff 2 0 0 0 with the asterisk denoting the complex conjugate operation. E ðr;⃗ ωÞ¼ iωμ a sin θ dθ dϕ 4πr 0 0 Combining these expressions with the spherical harmonic −ikr ˆ·r⃗ 0 0 expansion of the exponential term in the pattern function × e J ðθ ; ϕ ; ωÞ; ð9Þ integral, the pattern function itself becomes which can be written immediately as FJ ðθ;ϕ;ωÞ ∞ l ikr X X J;ff 0 l  0 0 ¼∯ dΩ 4π ð−iÞ j ðkaÞY ðθ;ϕÞY ðθ ;ϕ Þ E ðr; θ; ϕ; ωÞ¼ iωμ J a FJ ðθ; ϕ; ωÞ; ð10Þ l lm 0 θ lm 4πr S l¼0 m¼−l ∞ l X X where the Fourier transform of the normalized current 0 0 0 0 0 0 × c Y ðθ ;ϕ Þ: l m l m density component, FJ ðθ; ϕ; ωÞ, has been introduced. It 0 0 0 l ¼0m ¼−l defines the angular distribution or pattern of the far field, and in this article, it will be denoted as the pattern function. Recombining terms to take advantage of spherical har- Setting J ðθ; ϕ; ωÞ¼ J Π ðθ; ϕ; ωÞ, FJ is given by an monic identities and orthogonality properties and resetting θ 0 θ θ integral over the unit sphere S , whose differential surface the indices to simplify the notations, the pattern function area dΩ ¼ sin θdθdϕ,as becomes 031017-3 RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) 2 2 ∞ l X X 2ð1 þ cos θÞ × P ðθÞ Dðθ; ϕÞ¼ : ð18Þ FJ ðθ; ϕ; ωÞ¼ 4π c 0 0 π 2 2 θ l m ð1 þ cos θÞ × P ðθÞ sin θdθ 0 0 0 l ¼0 m ¼−l ∞ l X X The directivity patterns (in dB) for several numbers of modes × ð−iÞ j ðkaÞY ðθ; ϕÞ l lm are given in Fig. 1. A comparison of the maximum l¼0 m¼−l directivity of the needle radiation for the Huygens current 0  0 0 0 0 ×∯ dΩ Y ðθ ; ϕ ÞY 0 0ðθ ; ϕ Þ case and the electric-only current case as functions of the l m lm number of modes N is presented in Fig. 2. Both of these ∞ l X X cases are then compared to the Harrington limit (2) in Fig. 3. ¼ 4π ð−iÞ c j ðkaÞY ðθ; ϕÞ: lm l lm One finds from Fig. 3 that when N modes of the electric l¼0 m¼−l currents are considered, i.e., without the Huygens factor ð1 þ cos θÞ included, the maximum directivity grows as Consequently, taking into account the need for azimuthal N 2 predicted by Eq. (2), i.e., as ð2l þ 1Þ¼ N þ 2N, symmetry in the currents from Eq. (12) and the spherical l¼1 with the N ¼ 0 term not being included since it is a leftover harmonic completeness relation given in Ref. [34], one sets from the potential formulation used to obtain the far-field rffiffiffiffiffiffiffiffiffiffiffiffiffiffi representations (see Ref. [34]) and is actually not radiated 1 2l þ 1 i c ¼ δ ð15Þ lm m0 4π 4π j ðkaÞ to obtain the desired explicit needle radiation result: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ l X X 2l þ 1 FJ ðθ; ϕ; ωÞ¼ Y ðθ; ϕÞδ 0 θ lm m0 4π l¼0 m¼−l 2l þ 1 ¼ P ðcos θÞδðϕÞ 4π l¼0 -50 ≡ δðr ˆ − z ˆÞ; ð16Þ where P is the Legendre function of degree l.Itis emphasized that the process leading to Eq. (16) is not the -100 usual invocation that ka is small, with the subsequent -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) expansion of the exponential to generate the standard multipole expansion. Rather, the desired needle radiation FIG. 1. Directivity pattern (dB) of the Huygens current needle has been obtained directly by using the exact multimode radiation limited to N modes. spherical harmonic expansions and summing over all of the azimuthally symmetric modes on the sphere. An interesting outcome of this result is the fact that the needle radiation was obtained only with electric currents Huygens (or by duality, the same outcome is obtained with only Electric magnetic currents). One would then automatically obtain a factor of 2 in amplitude of the needle peak value in the far field if the magnetic (electric) currents were also included (see Ref. [34]) with K ¼ ηJ (J ¼ K =η). On the other 0 0 0 0 hand, how does the directivity behave as a function of the number of modes? 4000 In particular, consider the pattern function for N modes, 2l þ 1 P ðθÞ¼ P ðcos θÞ: ð17Þ N l 4π l¼0 0 1020 30 40 506070 80 90 100 Number of modes The directivity for the Huygens current needle radiation approximation with N pairs of modes, i.e., when N modes FIG. 2. Comparison of the maximum directivity of the needle of both the electric and magnetic currents are present on the radiation for the Huygens current and the electric-only current sphere, is then cases limited to N modes. 031017-4 Maximum directivity Directivity (dB) USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) 2.6 80 2.4 Huygens Electric 2.2 1.8 1.6 1.4 1.2 0.8 0 10 2030405060708090 100 0 102030405060708090 100 Number of modes Number of modes FIG. 3. Comparison of the ratio of the maximum directivity of FIG. 4. FWHM values of the directivity pattern of the Huygens the needle radiation for the Huygens current and the electric-only current needle radiation limited to N modes. current cases, limited to N modes, to the Harrington limit (2). the coefficients Eq. (15) as N increases, one finds problems as an electromagnetic wave (the dipole term, l ¼ 1,isthe with these very-high-order modes. In fact, the same ill- lowest-order radiated mode). On the other hand, while posedness problems encountered with the planar arrays the directivities are apparently the same in Fig. 2 when the arise but in a slightly different manner. Huygens factor is present, Fig. 3 also shows that there is a One finds that the original considerations by Harrington very noticeable difference when only the first few modes about the sphere size and the number of modes [35,36] actually play a significant and related role in this case. If are taken into account. The Huygens case value is much one restricts the number of modes to N, i.e., to the pattern larger. This outcome was confirmed by hand for the first function P ðθÞ, and selects the electrical size of the sphere few modes. Nevertheless, the Huygens case’s maximum to be ka ¼ N, then the coefficients in the current expansion directivity eventually converges to Harrington’s limit (2) are manageable. On the other hand, if one tries to realize the as N becomes quite large. This was a totally unexpected needle radiation result from an electrically small sphere, result. It was originally anticipated that the hole in the i.e., from a sphere with ka ≤ 1, ka ¼ 1 being the Wheeler directivity in the back direction would give the Huygens radiansphere whose radius is a ¼ λ=2π [52,53], the usually source the advantage for all N. However, as shown in restrictive large current amplitudes occur quite quickly as Fig. 1, the large increase in the number of sidelobes as N the index n of the coefficient 1=j ðkaÞ increases. This is increases basically fills in the back-direction hole, and the n clearly illustrated in Fig. 5. One observes the oscillations of initial advantage is lost. While the peak values of the directivity increase quad- ratically as N increases, the corresponding full-width-at- half-maximum (FWHM) values of the main beam are given ka = 10 ka = 20 in Fig. 4. One can clearly see that the needle-like behavior ka = 30 is emerging as N increases. The width of the main beam decreases rapidly as the peak directivity increases. In contrast to the known 2D planar aperture or array 4 approaches to high directivity, some of which are discussed in Ref. [34], Eq. (16) demonstrates that currents on a 3D sphere can achieve true needle radiation in theory. Nevertheless, again examining the directivity patterns in Fig. 1, the increasing numbers of sidelobes illuminate yet another issue. While the sidelobe levels are decreasing as N increases and the outcome would be the eventual achieve- -1 0 5 10 15 20 25 30 35 40 45 50 ment of the true needle result, they are impressively present Multipole number in large numbers for a finite number of modes. This sidelobe behavior is very undesirable for many applica- FIG. 5. Needle radiation expansion coefficients for various tions, especially if only a low number of modes were electrical sizes of the sphere, ka, as functions of the number of excited. Moreover, when one examines the amplitudes of azimuthally symmetric modes, N. 031017-5 Directivity ratio Coefficient magnitude Power Pattern FWHM (degrees) RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) the amplitudes for small mode numbers and the exponential Jðx; y; zÞ¼ I lδðxÞδðyÞ½Hðz − l=2Þ − Hðz þ l=2Þx; ˆ growth of the coefficients once the mode number exceeds ð19Þ the electrical size of the sphere. Thus, in practice, one could only hope to approach a “true” needle effect from an Kðx; y; zÞ¼ I lδðxÞδðyÞ½Hðz − l=2Þ − Hðz þ l=2Þy; ˆ increasingly larger number of tailored currents on an m increasingly larger sphere, basically in agreement with ð20Þ Eq. (1). where the Heaviside function HðuÞ¼ 1 if u> 0, and ¼ 0 if u< 0. Recall that dipole antennas radiate in their IV. NEEDLE-LIKE RADIATION FROM AN broadside directions. With these current direction choices, ARRAY OF DIPOLE-CONSTRUCTED HUYGENS MULTIPOLES radiated fields along the z axis are thus possible. As shown in Ref. [34], if the current amplitudes are weighted properly Obtaining sets of specified current distributions on an so that I ¼ I ¼ I =η, one then has e m electrically small 3D object with curvature to realize an approximate needle radiation even for a few modes is ikr ff ˆ ˆ also, unfortunately, a nontrivial task. Moreover, one would ⃗ E ðr⃗ Þ¼iωμIl ð1þcosθÞ½cosϕθ−sinϕϕ; 4πr nevertheless desire a planar or conformal array or at least a ikr Ile thin stack of planar-radiating elements on a mobile plat- ff ⃗ ˆ ˆ H ðr⃗ Þ¼iωμ ð1þcosθÞ½sinϕθ þcosϕϕ: ð21Þ form for any practical application. Thus, a discrete radiating η 4πr aperture associated with a simple set of radiators to achieve It is immediately apparent that the cardioid pattern asso- high directivity, like what is obtained with now commonly ciated with a Huygens source is attained and that the field is used phased arrays, remains desirable. Can something null along the negative z axis as expected, where θ ¼ π, useful be achieved in practice? ϕ ¼ 0. The directivity is then straightforwardly calculated As the historical discussion indicated, it is well known to be that there are generally fundamental trade-offs in the patterns generated by a discrete array between the main 4πr Sðr⃗ Þ · r ˆ 3 beam width and the sidelobe levels [30]. One knows that a Dðθ; ϕÞ¼ ¼ ð1 þ cos θÞ : ð22Þ rad uniformly driven aperture generally produces the maximum P 4 tot directivity, amplitude tapering of the array elements pro- Therefore, the maximum directivity of the electric-magnetic vides control of the sidelobe levels, and phasing between dipole pair (N ¼ 1), which is along the positive z axis, where the elements yields the capability to steer the direction of θ ¼ 0, is 3, twice the value of either dipole alone, confirming the main beam. Examples of the directivity obtained from the Harrington result (2): D ¼ 1 þ 2 × 1 ¼ 3. distributions of electric and magnetic currents on a planar max Huygens source antennas have been achieved in practice disk and from a circular array of Huygens dipole sources in both electrically small [54–57] and larger [58,59] are given in Ref. [34]. packages. They have been recognized as an important One finds that in contrast to the currents on a small research direction for IoT applications [60]. Huygens sphere, those 2D current distributions do not yield the metasurfaces have already played a significant role in desired needle-like radiation pattern unless the disk radius antenna and scattering configurations [61–63]. How, then, becomes extremely large or hard-to-realize current distri- does one achieve a Huygens behavior with yet higher butions are employed. What can one then do to achieve directivity? needle-like radiation from a potentially realizable current distribution? Here, we explore how superdirectivity can be B. Multipole-based Huygens sources obtained with an end-fire array of broadside radiating In an extension of Uzkov’s results [64], it has been Huygens multipoles. demonstrated in a series of articles on end-fire arrays, e.g., Refs. [65–72], that an array of electric elements achieves A. Dipole-based Huygens sources its maximum directivity in its end-fire direction when the To understand more completely the Huygens source separation distance between the element pairs goes to zero. concept, a combination of elemental electric and magnetic These dense packing and end-fire concepts have been dipole sources is considered first. With an emphasis on the demonstrated experimentally for moderately small separa- z axis as the preferred direction, the elemental electric and tion distances as well. Moreover, it has been shown that a magnetic current densities of amplitude, respectively, I dense end-fire array of dipole Huygens sources, i.e., and I , imposed on a pair of orthogonally oriented electric electric and magnetic dipole pairs in an end-fire configu- and magnetic Hertzian dipole antennas of length l will be ration, will produce the highest possible directivity asso- taken as ciated with dipole radiating elements [70]. As an extension 031017-6 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) dipoles, it has a length 2 × NΔ but with all of the resulting multipoles being centered on the zeroth element, i.e., the simple electric (magnetic) dipole. Finally, as was done with the sphere-based currents, adding the resulting electric (magnetic) multipole fields together with the simple coefficient weightings, 1=fn!½ð−2iÞkΔ g, one obtains the pattern function Nþ1 1 − cos θ FJ ðθ; ϕ; ωÞ ≈ ðcos θÞ ¼ zx 1 − cos θ n¼0 ¼ P ðθÞð23Þ for N multipole elements. Using L’Hopital’s rule, the maximum of Eq. (24) occurs along the z axis at θ ¼ 0, max i.e., P ∼ ðN þ 1Þ. Then, arranging the electric and FIG. 6. Assemblage of the broadside radiating multipoles. magnetic current moments so that they are balanced with I ¼ ηI ¼ I, one obtains, for the far fields of N electric and m e magnetic multipole pairs, of those results, let us consider a compact end-fire array of broadside radiating Huygens multipole sources. ikr ff As noted by Harrington [73], one can use alternating ˆ ˆ E ðr⃗ Þ¼ iωμIl ð1 þ cos θÞP ðθÞ½cos ϕ θ − sin ϕ ϕ; ω N 4πr pairs of dipole current elements, as one does with alter- ikr Il e nating sets of charges to achieve electrostatic or magneto- ff ⃗ ˆ ˆ H ðr⃗ Þ¼ iωμ ð1 þ cos θÞP ðθÞ½sin ϕ θ þ cos ϕ ϕ: ω N static multipoles, to produce higher-order electromagnetic η 4πr multipoles. Again, consider the electric (magnetic) multi- ð24Þ poles to be oriented along the x axis (y axis). As discussed in Ref. [34] and as depicted in Fig. 6, the electric (magnetic) Thus, the directivity again takes the form multipoles are obtained by properly arranging combina- tions of electric (magnetic) dipoles to be compactly spaced 2 2 2ð1 þ cos θÞ × P ðθÞ along the z axis, i.e., to have an electrically small distance Δ R Dðθ; ϕÞ¼ ; ð25Þ π 2 2 ð1 þ cos θÞ × P ðθÞ sin θdθ 0 N between each of them, and to have the appropriate orienta- tions with respect to each other. The resulting electric and where the factor of 2 appears from the equal contributions of magnetic multipoles are then combined together, as depicted the balanced electric and magnetic Huygens multipoles. in Fig. 7, to form the Huygens multipole end-fire array. The directivity patterns (in dB) for several number of Accounting for all of the constituent electric (magnetic) Huygens multipoles arranged compactly along the z axis are given in Fig. 8. A polar plot of the N ¼ 1000 case is shown in Fig. 9. From both figures, one clearly sees that the Huygens source behavior has been obtained and, as the number of higher-order modes is increased, that the directivity approaches a needle-like Huygens behavior in which the sidelobes have been completely eliminated. Thus, the desired needle-like radiation from an array has been demonstrated. This behavior is further confirmed in Figs. 10 and 11. While they illustrate the needle-like behavior, they also indicate that this desirable performance, in contrast to the sphere result, is slow to evolve. Referring to Fig. 10,it is confirmed that while the peaks of the power patterns increase as 2ðN þ 1Þ, the maximum directivity only increases linearly as N becomes large, approximately as 1.5N. Moreover, the decrease of the angular FWHM of the power pattern begins to slow noticeably as N becomes quite large. This is emphasized further in Fig. 12, which presents FIG. 7. End-fire array of the assembled broadside radiating multipoles. the directivity patterns of the needle and the multipole 031017-7 RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) 40 35 20 100 -20 -40 -60 -80 -100 -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) 0 100 200 300 400 500 600 700 800 900 1000 Number of multipoles FIG. 8. Directivity pattern (dB) attained with N Huygens multipoles arranged compactly along the z axis. FIG. 10. Maximum directivity (dB) attained with N Huygens multipoles arranged compactly along the z axis. 330 30 300 60 270 90 240 120 0 100 200 300 400 500 600 700 800 900 1000 Number of multipoles 210 150 FIG. 11. FWHM values of the power patterns generated by N Huygens multipoles arranged compactly along the z axis. FIG. 9. Polar plot of the directivity produced by 1000 Huygens multipoles arranged compactly along the z axis. because the nth multipole amplitude is the inverse of the antennas for a finite number of terms adjusted for the larger product of the factorial terms n! and ðkΔÞ . This means the number of multipoles needed to recover the maximum coefficient magnitudes first increase algebraically with n, obtained with a much smaller number of needle terms. but they then reach a tractable maximum (as long as the While it does take more terms, the multipole antenna does multipole index is not exceedingly large, which it would recover the needle behavior without the sidelobe issues. not be in a realistic antenna) and start to decrease as the Furthermore, considering the coefficient weightings, one factorial term becomes dominant. Thus, one could hope to does not encounter the exponential blowup associated with generate needle-like behavior in practice. These results the sphere results. This outcome is clearly illustrated in suggest that advancing to a few Huygens multipole pairs Fig. 13, where the coefficient amplitudes for different from the simple Huygens dipole pair can significantly electrical spacings (i.e., kΔ, where Δ is the physical improve the directivity associated with a compact system. distance between the elements) between the multipoles In fact, it proves that one is not bounded by the simple are compared for variable numbers of multipoles. The dipole pair, which disproves previous conclusions (see, results are different from the conventional arguments e.g., Ref. [74]). 031017-8 Directivity (dB) Maximum directivity (dB) Power Pattern FWHM (degrees) USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) Needle, 5 modes 25 Multipoles -20 -40 -60 -80 -100 -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) FIG. 12. Comparison of the directivity patterns (dB) of the needle and the multipole antennas. k = 0.2 3 k = 0.1 k = 0.05 2.5 FIG. 14. Dielectric resonator stack approach to realize the Huygens multipole array. 1.5 desired broadside radiating fields. These elements would 0.5 then be assembled into the end-fire array configuration as depicted. This dielectric resonator antenna (DRA)-based -0.5 system is currently being explored for experimental vali- dation of the Huygens multipole end-fire array concept. -1 Another approach, particularly suited for nano-antennas, -1.5 would be to have a multilayered, subwavelength-size, 0 5 10 15 20 25 Multipole number resonant core-shell particle configuration in which the electric and magnetic multipole modes were simultane- FIG. 13. Needle radiation expansion coefficients for various ously excited and their resonance frequencies adjusted by electrical sizes of the sphere, ka, as functions of the number of the geometry and material values to be coincident. While azimuthally symmetric modes, N. this was accomplished at the dipole level (e.g., Ref. [44]), discussions about the relationship between the quality factor and directivity [27] remind us that the higher-order Even achieving directivities that are an order of magni- multipoles will have narrower bandwidths and, hence, may tude larger than an electrically small dipole from a similar be quite sensitive to their design parameters. Nonetheless, a footprint holds many potential benefits for future IoT two-dimensional version of this multilayer concept, as wireless and mobile platform-based communication and sensor devices. Moreover, the amplitudes needed to realize depicted in Fig. 15, has been verified [75]. Yet another the outcome are well-posed and reasonable. In practice, the technique would be to employ a similar number of (single) desired needle-like pattern outcome could be realized by resonant core-shell particles, each producing one of the developing a compact antenna constructed as a stack of thin requisite multipole fields and then aligned and excited in an layers, each layer having the proper elements (dipoles or end-fire configuration. other structures) to realize the requisite higher-order multi- All of these arrangements are intimately related to and pole. The thickness of the layers would be subwavelength supported by the near-field resonant parasitic (NFRP) to ensure that kΔ is small. metamaterial-inspired engineering of electrically small As shown in Fig. 14, a parasitic stack of high-permittivity antennas paradigm [76], which has successfully produced annular dielectric resonator elements could be tuned to a large variety of multifunctional compact systems. For instance, with one of the Huygens dipole sources produce the desired electric and magnetic multipoles appro- priate for each layer at the same frequency and with the already realized (see, e.g., Ref. [56]), any one of the 031017-9 Coefficient magnitude Directivity (dB) RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) array, which combines the first six Huygens multipoles, N ¼ 0 to N ¼ 5, are shown in Figs. 16–18, respectively. The Huygens behavior in all of these cases is immediately observed. The Huygens multipole array result in Fig. 18 illustrates the absence of any sidelobes and the beginning of the needle-like behavior. FIG. 15. Two-dimensional version of a coated nanoparticle realization of the multipole effects. aforementioned “arrays” of NFRP electric and magnetic multipole elements could be driven at RF and micro- FIG. 16. NewFasant predicted 3D directivity pattern for the wave frequencies. This system is highlighted in Fig. 14. Huygens dipole (N ¼ 0) antenna. Moreover, as noted in the earlier discussions, the particle approaches have been demonstrated for various individual multipole orders at optical frequencies; hence, their combi- nations are also very realizable. Therefore, these compact Huygens multipole end-fire arrays of NFRP elements would overcome the usual argument that superdirectivity cannot be achieved because of the ill-posedness encoun- tered with the currents typically associated with its synthesis. V. SIMULATED HUYGENS MULTIPOLE END-FIRE ARRAY RESULTS To confirm these analytical superdirectivity results and to set the stage for future experimental efforts, simulations of FIG. 17. NewFasant predicted 3D directivity pattern for the the Huygens multipole end-fire array were performed with Huygens N ¼ 2 multipole antenna. the commercial method of moments code NewFasant (see Ref. [34]) [77]. It is the only commercial code (to the best of our knowledge) that has multipole sources built into it. The simulation parameters used to obtain the results were a 3.0 MHz excitation source (λ ¼ 100.0 m), 10.0 cm long dipoles, with Δ ¼ 0.1 cm spacings between them. The electric and magnetic dipole Huygens source was first created, and the Huygens quadrupole source was con- structed with it. The result was then used to create the Huygens hexapole source and so on until the Huygens duodeca-multipole was created. Finally, these six multi- poles, n ¼ 0; 1; …; 5, were combined together in the desired end-fire configuration as illustrated in Fig. 7. The overall length of this N ¼ 5 Huygens multipole −4 end-fire array is 1.0 cm (i.e., 2 × NΔ ¼ 10 λ ). The simulated 3D directivity patterns for the Huygens FIG. 18. NewFasant predicted 3D directivity pattern for the dipole (N ¼ 0 multipole) and hexapole (N ¼ 2 multipole) Huygens multipole end-fire array, which combines the first six antennas and for the composite Huygens multipole end-fire multipoles: N ¼ 0 to N ¼ 5. 031017-10 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) Analytical or from the multiple sidelobes produced by a mode-limited Numerical needle source. The results disprove many previous state- ments that the directivity from a Huygens dipole source was the best one could accomplish from an electrically small -20 source region. Moreover, given the many recent multipolar nano-antenna predictions and initial macroscopic-sized -40 Huygens source realizations, an electrically small, NFRP- based Huygens multipole end-fire array should be demon- -60 strable in the near future. ACKNOWLEDGMENTS -80 The author expresses his profound gratitude to the CEO -100 of NewFasant, Professor Felipe Catedra of the University -200 -150 -100 -50 0 50 100 150 200 of Alcala, Alcala de Henares, Spain, for his kindness in Angle (degrees) providing access to the NewFasant code, for writing a FIG. 19. Comparison of the analytically and numerically couple of “How to Use NewFasant for Dummies” tutorial calculated directivities for the Huygens multipole end-fire array, documents, and for actually running some initial cases including the dipole through the N ¼ 5 multipoles. for my education and benefit. This work was supported in part by the Australian Research Council Grant No. DP160102219. The 2D directivity pattern for the Huygens multipole end-fire array predicted numerically by NewFasant is compared with the analytical result (23) for the first six multipoles in Fig. 19. No coefficient weights were applied [1] C. W. 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Using Huygens Multipole Arrays to Realize Unidirectional Needle-Like Radiation

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PHYSICAL REVIEW X 7, 031017 (2017) Richard W. Ziolkowski University of Technology Sydney, Ultimo NSW 2007, Australia The University of Arizona, Tucson, Arizona 85721, USA (Received 19 December 2016; published 26 July 2017) For nearly a century, the concept of needle radiation has captured the attention of the electromagnetics communities in both physics and engineering, with various types of contributions reoccurring every decade. With the near-term needs for highly directive, electrically small radiators and scatterers for a variety of communications and sensor applications, superdirectivity has again become a topic of interest. While it is well known that superdirective solutions exist and suffer ill-posedness issues in principle, a detailed needle solution has not been reported previously. We demonstrate explicitly, for the first time, how needle radiation can be obtained theoretically from currents driven on an arbitrary spherical surface, and we explain why such a result can only be attained in practice with electrically large spheres. On the other hand, we also demonstrate, more practically, how broadside radiating Huygens source multipoles can be combined into an end-fire array configuration to achieve needle-like radiation performance without suffering the traditional problems that have previously plagued superdirectivity. DOI: 10.1103/PhysRevX.7.031017 Subject Areas: Industrial Physics, Interdisciplinary Physics, Plasmonics I. INTRODUCTION soon after Oseen’s publication. Both end-fire (maximum radiated power is oriented along the array direction) [8] The concept of superdirectivity has permeated the areas and broadside (maximum radiated power is oriented of physics and applied physics repeatedly since Oseen perpendicular to the plane of the array) [9] pattern enhance- discussed the concept of “needle radiation” almost a ments from different array configurations were considered century ago [1]. While Oseen was keenly interested in initially. La Paz and Miller [10] purported to show that the how a tiny atom might absorb a large electromagnetic wave maximum directivity from an aperture of a given size was as an equivalent photon (and, consequently, the alternate fixed, but Bouwkamp and De Bruijn [11] correctly dem- translation of his paper’s title as “pinprick” radiation might onstrated that there was no theoretical limit on the direc- make more historical sense), the reciprocal problem of tivity from an aperture of any size. Dolph realized that one transmitting a needle-like radiation pattern from a small could control the sidelobe levels of the pattern by properly source has stimulated many physics and, possibly more, weighting (Chebyshev polynomial tapering) the amplitudes engineering discussions. The role of superdirectivity in of the element excitations [12]. Riblet [13,14] illustrated radio astronomy and in particle physics was discussed by that such amplitude tapering has an associated cost of Casimir [2] and Wheeler [3]. They, too, emphasized the widening the mainlobe of the pattern. However, it was possibility that the effective receiving cross section of a quickly shown by Yaru [15] that the current distribution radio telescope or an atom could be extremely large in solutions that produce superdirective beams from arrays are comparison with its physical size. This concept has been generally ill-posed [16]; i.e., small variations of the large demonstrated more recently with plasmonic particles positive and negative variations of the excitation amplitudes whose strong reactive scattering components extend to required to achieve the effect lead to its disappearance in large distances and redirect the power passing through a practice. In fact, Casimir [2] and Wheeler [3] noted this large area of an incoming plane wave and force it to flow practical difficulty and believed that one would never go towards the scatterer [4–7]. beyond combining a dipole and a quadrupole mode Engineering the emission of electromagnetic fields from together in practice. Nonetheless, this goal has been finite sources was intensely studied in the 1940s and 1950s achieved with subwavelength dielectric and plasmonic particles [17–22]. There have been and continue to be many examples of optimizing the directivity from an Richard.Ziolkowski@uts.edu.au, ziolkowski@ece.arizona.edu antenna system with constraints on its various other Published by the American Physical Society under the terms of performance characteristics to circumvent the ill-posedness the Creative Commons Attribution 4.0 International license. of the “super” outcome [23–27]. Further distribution of this work must maintain attribution to A useful operational definition of superdirectivity, e.g., the author(s) and the published article’s title, journal citation, and DOI. as emphasized by Hansen [28,29], is to achieve a directivity 2160-3308=17=7(3)=031017(13) 031017-1 Published by the American Physical Society RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) greater than that obtained with the same antenna configu- modes N in a sphere of radius r would be limited to ration being uniformly excited (constant amplitude and N ¼ kr ≡ 2πr =λ is no longer justifiable in general. 0 0 phase). Let the radiating system be either an aperture The concept of a transmitting antenna realizing a far- antenna (continuous current distribution) whose effective field needle radiation pattern is also intimately connected area is A or an array of radiating elements (set of discrete to subwavelength imaging, i.e., being able to resolve two eff currents) distributed in A . If the total efficiency (i.e., small objects separated by subwavelength distances eff taking into account the material losses, mismatch losses, [46,47]. Moreover, superdirectivity has been shown to lead to enhanced channel capacity in multiple-input–multiple- polarization mismatch,…) of the system is e , then its total output (MIMO) systems [48–51]. Thus, superdirectivity maximum gain G is related to its maximum directivity max concepts yet again become important as nanotechnology D as G ¼ e × D . Thus, if there are no losses, max max total max applications flourish and the Internet of things (IoT) comes then the maximum directivity of the antenna system to fruition. We would like to have electrically small, highly uniformly driven at the excitation wavelength λ is funda- directive, receiving or transmitting antennas (whether they mentally related to its effective area A as [30] eff are macro, micro, or even nano) for numerous wireless applications. 4π D ¼ A : ð1Þ max eff 2 There has been a significant increase in higher-directivity scattering approaches reported, but little on corresponding radiating systems. This article is timely in that it addresses two Consequently, a larger effective aperture will provide a fundamental questions: Despite nearly a century of inves- higher directivity. tigation, what would it really take to achieve actual needle- As has been shown by a number of authors, e.g., like radiation? Can one design an array that would eliminate Refs. [31–33], the fields in a region of free space outside sidelobes, has a high front-to-back ratio, has superdirective of a spherical surface that encloses all the currents can be properties, and is not plagued by ill-posedness? expanded in a series of electric and magnetic multipole fields represented by (vector) spherical harmonics (see II. FAR-FIELD RADIATED FROM A SET OF Ref. [34] for more details). This approach has proved to be ELECTRIC AND MAGNETIC SOURCES very successful for the analysis of the far-field behavior of an antenna system [30]. By taking into account both the As explained in Ref. [34], the electric field radiated into transverse electric and magnetic modes, Harrington dem- the far field of a combination of electric J and magnetic onstrated that the maximum directivity from a source currents K excited at the frequency f ¼ ω=2π can be region as a function of the number of multipole modes, written as N,is [35,36] ZZZ ikr ff −ikr ˆ·r⃗ 0 0 3 0 2 ⃗ ⃗ ⃗ E ¼ iωμ e fJ ðr⃗ Þ − r ˆ½r ˆ · J ðr⃗ Þgd r⃗ D ¼ N þ 2N: ð2Þ ω ω J;ω max 4πr ð3Þ Therefore, by exciting higher order modes, one can achieve very high directivities from a fixed source region. ZZZ ikr ff These antenna results are immediately connected to the −ikr ˆ·r⃗ 0 3 0 ⃗ ⃗ E ¼ iωε e fr ˆ × ½K ðr⃗ Þgd r⃗ : ð4Þ K;ω 4πr upper bounds on the total cross section associated with scattering from particles [37]. The concept of subwave- The primed (unprimed) coordinates are the observation length superscattering [38,39] arises from maximizing the (source) point coordinates, and the wave number is contributions from a sufficiently large number of channels; pffiffiffiffiffi k ¼ ω=v, with v ¼ 1= εμ being the wave speed in the i.e., by aligning the frequencies of higher-order resonant medium. The far-field magnetic field is simply multipole modes, arbitrarily large total cross sections can ff ff ⃗ ⃗ H ¼ r ˆ ×E =η, where the free-space impedance be achieved with subwavelength structures. In fact, the ω ω pffiffiffiffiffiffiffiffi ability to create highly subwavelength (electrically small) η ¼ μ=ε. These expressions represent the known facts radiators and scatterers has been one of the success stories that the far fields are transverse electromagnetic (TEM) and associated with metamaterials [40–42]. Moreover, recent that they are related to the Fourier transform of the current passive and active nanoparticle studies associated with the components orthogonal to the observation direction. optical theorem [43], Kerker conditions [21], and Huygens Now consider the currents to be confined to the surface source effects [44] illustrate that combining sets of electric of a small sphere of radius a. Being as general as possible, and magnetic multipoles leads to enhanced directivities. the current densities then take the form These effects have been demonstrated with numerous ˆ ˆ configurations [17–20,22,45]. The recent subwavelength ⃗ J ðr⃗ Þ¼½J ðθ; ϕ; ωÞθ þ J ðθ; ϕ; ωÞϕδðr − aÞ; ω θ ϕ superdirective results have shown that Harrington’s original ˆ ˆ K ðr⃗ Þ¼½K ðθ; ϕ; ωÞθ þ K ðθ; ϕ; ωÞϕδðr − aÞ: ð5Þ ω θ ϕ estimate that the maximum number of usable higher-order 031017-2 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) ˆ ˆ ˆ 0 0 With the standard unit vector cross products θ×ϕ¼r, 0 −ikar ˆðθ;ϕÞ·r ˆðθ ;ϕ Þ 0 0 FJ ðθ;ϕ;ωÞ¼∯ dΩ e Π ðθ ;ϕ ;ωÞ: ð11Þ θ θ ˆ ˆ ˆ ˆ r ˆ × θ ¼ ϕ, ϕ × r ˆ ¼ θ—the far-field expressions become Z Z The fact that the source points are on the sphere of radius a ikr 2π π ff 2 0 0 0 −ikr ˆ·r⃗ 0 0 0 has allowed us to write r⃗ ¼ ar ˆðθ ; ϕ Þ, while emphasizing E ðr⃗ Þ¼ iωμ a sin θ dθ dϕ e 4πr 0 0 that the observation direction r ˆ is given by the coordinates ðθ; ϕÞ. The far-field pattern function is now more clearly 0 0 0 0 × J ðθ ; ϕ ; ωÞþ K ðθ ; ϕ ; ωÞ θ θ ϕ η related to the 2D Fourier transform of the current density pattern Π over the unit sphere. 0 0 0 0 þ J ðθ ; ϕ ; ωÞ − K ðθ ; ϕ ; ωÞ ϕ ; ð6Þ The far-field expression (10) indicates that there are no ϕ θ dc components of the source excitation radiated into the far field and that the radial dependence is that of a spherical 1 wave. The Fourier transform integral determines if there are ff ff ⃗ ⃗ H ðr⃗ Þ¼ r ˆ × E ðr⃗ Þ: ð7Þ ω ω any preferred directions into which the fields are radiated. Since we desire needle radiation along the z axis, it would follow that the pattern function yields As explained in Ref. [34], it is clear that to achieve a Huygens source behavior, one can simply consider the FJ ðθ; ϕ; ωÞ¼ δðcos θ − 1ÞδðϕÞ ≡ δðr ˆ − z ˆÞ: ð12Þ contributions from either the orthogonal pair J and K or θ ϕ from its dual, J and K . Electing the former, one has ϕ θ From this relation, it is immediately apparent that the currents on the sphere will have to be azimuthally sym- Z Z ikr 2π π ff 2 0 0 0 −ikr ˆ·r⃗ metric with respect to the z axis to achieve the desired E ðr⃗ Þ¼ iωμ a sin θ dθ dϕ e 4πr 0 0 outcome. To proceed, several spherical harmonic relations, as 0 0 0 0 × J ðθ ; ϕ ; ωÞþ K ðθ ; ϕ ; ωÞ θ ; θ ϕ reviewed briefly in Ref. [34], are employed. Since the spherical harmonics are a complete basis, one can expand ff ff ⃗ ⃗ H ðr⃗ Þ¼ r ˆ × E ðr⃗ Þ: ð8Þ the angular behavior of the theta component of the electric ω ω current density pattern on the sphere as By properly selecting those currents, we will show that the ∞ l X X Π ðθ; ϕÞ¼ c Y ðθ; ϕÞ; ð13Þ desired needle radiation can be achieved. θ lm lm l¼0 m¼−l III. NEEDLE RADIATION FROM CURRENTS where Y is the spherical harmonic of degree l and order lm ON A SMALL SPHERE m and the coefficients are given explicitly as First, consider the electric far-field component produced by only the J source: c ¼∯ Π ðθ; ϕÞY ðθ; ϕÞ sin θdθdϕ; ð14Þ lm θ lm Z Z ikr 2π π J;ff 2 0 0 0 with the asterisk denoting the complex conjugate operation. E ðr;⃗ ωÞ¼ iωμ a sin θ dθ dϕ 4πr 0 0 Combining these expressions with the spherical harmonic −ikr ˆ·r⃗ 0 0 expansion of the exponential term in the pattern function × e J ðθ ; ϕ ; ωÞ; ð9Þ integral, the pattern function itself becomes which can be written immediately as FJ ðθ;ϕ;ωÞ ∞ l ikr X X J;ff 0 l  0 0 ¼∯ dΩ 4π ð−iÞ j ðkaÞY ðθ;ϕÞY ðθ ;ϕ Þ E ðr; θ; ϕ; ωÞ¼ iωμ J a FJ ðθ; ϕ; ωÞ; ð10Þ l lm 0 θ lm 4πr S l¼0 m¼−l ∞ l X X where the Fourier transform of the normalized current 0 0 0 0 0 0 × c Y ðθ ;ϕ Þ: l m l m density component, FJ ðθ; ϕ; ωÞ, has been introduced. It 0 0 0 l ¼0m ¼−l defines the angular distribution or pattern of the far field, and in this article, it will be denoted as the pattern function. Recombining terms to take advantage of spherical har- Setting J ðθ; ϕ; ωÞ¼ J Π ðθ; ϕ; ωÞ, FJ is given by an monic identities and orthogonality properties and resetting θ 0 θ θ integral over the unit sphere S , whose differential surface the indices to simplify the notations, the pattern function area dΩ ¼ sin θdθdϕ,as becomes 031017-3 RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) 2 2 ∞ l X X 2ð1 þ cos θÞ × P ðθÞ Dðθ; ϕÞ¼ : ð18Þ FJ ðθ; ϕ; ωÞ¼ 4π c 0 0 π 2 2 θ l m ð1 þ cos θÞ × P ðθÞ sin θdθ 0 0 0 l ¼0 m ¼−l ∞ l X X The directivity patterns (in dB) for several numbers of modes × ð−iÞ j ðkaÞY ðθ; ϕÞ l lm are given in Fig. 1. A comparison of the maximum l¼0 m¼−l directivity of the needle radiation for the Huygens current 0  0 0 0 0 ×∯ dΩ Y ðθ ; ϕ ÞY 0 0ðθ ; ϕ Þ case and the electric-only current case as functions of the l m lm number of modes N is presented in Fig. 2. Both of these ∞ l X X cases are then compared to the Harrington limit (2) in Fig. 3. ¼ 4π ð−iÞ c j ðkaÞY ðθ; ϕÞ: lm l lm One finds from Fig. 3 that when N modes of the electric l¼0 m¼−l currents are considered, i.e., without the Huygens factor ð1 þ cos θÞ included, the maximum directivity grows as Consequently, taking into account the need for azimuthal N 2 predicted by Eq. (2), i.e., as ð2l þ 1Þ¼ N þ 2N, symmetry in the currents from Eq. (12) and the spherical l¼1 with the N ¼ 0 term not being included since it is a leftover harmonic completeness relation given in Ref. [34], one sets from the potential formulation used to obtain the far-field rffiffiffiffiffiffiffiffiffiffiffiffiffiffi representations (see Ref. [34]) and is actually not radiated 1 2l þ 1 i c ¼ δ ð15Þ lm m0 4π 4π j ðkaÞ to obtain the desired explicit needle radiation result: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ l X X 2l þ 1 FJ ðθ; ϕ; ωÞ¼ Y ðθ; ϕÞδ 0 θ lm m0 4π l¼0 m¼−l 2l þ 1 ¼ P ðcos θÞδðϕÞ 4π l¼0 -50 ≡ δðr ˆ − z ˆÞ; ð16Þ where P is the Legendre function of degree l.Itis emphasized that the process leading to Eq. (16) is not the -100 usual invocation that ka is small, with the subsequent -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) expansion of the exponential to generate the standard multipole expansion. Rather, the desired needle radiation FIG. 1. Directivity pattern (dB) of the Huygens current needle has been obtained directly by using the exact multimode radiation limited to N modes. spherical harmonic expansions and summing over all of the azimuthally symmetric modes on the sphere. An interesting outcome of this result is the fact that the needle radiation was obtained only with electric currents Huygens (or by duality, the same outcome is obtained with only Electric magnetic currents). One would then automatically obtain a factor of 2 in amplitude of the needle peak value in the far field if the magnetic (electric) currents were also included (see Ref. [34]) with K ¼ ηJ (J ¼ K =η). On the other 0 0 0 0 hand, how does the directivity behave as a function of the number of modes? 4000 In particular, consider the pattern function for N modes, 2l þ 1 P ðθÞ¼ P ðcos θÞ: ð17Þ N l 4π l¼0 0 1020 30 40 506070 80 90 100 Number of modes The directivity for the Huygens current needle radiation approximation with N pairs of modes, i.e., when N modes FIG. 2. Comparison of the maximum directivity of the needle of both the electric and magnetic currents are present on the radiation for the Huygens current and the electric-only current sphere, is then cases limited to N modes. 031017-4 Maximum directivity Directivity (dB) USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) 2.6 80 2.4 Huygens Electric 2.2 1.8 1.6 1.4 1.2 0.8 0 10 2030405060708090 100 0 102030405060708090 100 Number of modes Number of modes FIG. 3. Comparison of the ratio of the maximum directivity of FIG. 4. FWHM values of the directivity pattern of the Huygens the needle radiation for the Huygens current and the electric-only current needle radiation limited to N modes. current cases, limited to N modes, to the Harrington limit (2). the coefficients Eq. (15) as N increases, one finds problems as an electromagnetic wave (the dipole term, l ¼ 1,isthe with these very-high-order modes. In fact, the same ill- lowest-order radiated mode). On the other hand, while posedness problems encountered with the planar arrays the directivities are apparently the same in Fig. 2 when the arise but in a slightly different manner. Huygens factor is present, Fig. 3 also shows that there is a One finds that the original considerations by Harrington very noticeable difference when only the first few modes about the sphere size and the number of modes [35,36] actually play a significant and related role in this case. If are taken into account. The Huygens case value is much one restricts the number of modes to N, i.e., to the pattern larger. This outcome was confirmed by hand for the first function P ðθÞ, and selects the electrical size of the sphere few modes. Nevertheless, the Huygens case’s maximum to be ka ¼ N, then the coefficients in the current expansion directivity eventually converges to Harrington’s limit (2) are manageable. On the other hand, if one tries to realize the as N becomes quite large. This was a totally unexpected needle radiation result from an electrically small sphere, result. It was originally anticipated that the hole in the i.e., from a sphere with ka ≤ 1, ka ¼ 1 being the Wheeler directivity in the back direction would give the Huygens radiansphere whose radius is a ¼ λ=2π [52,53], the usually source the advantage for all N. However, as shown in restrictive large current amplitudes occur quite quickly as Fig. 1, the large increase in the number of sidelobes as N the index n of the coefficient 1=j ðkaÞ increases. This is increases basically fills in the back-direction hole, and the n clearly illustrated in Fig. 5. One observes the oscillations of initial advantage is lost. While the peak values of the directivity increase quad- ratically as N increases, the corresponding full-width-at- half-maximum (FWHM) values of the main beam are given ka = 10 ka = 20 in Fig. 4. One can clearly see that the needle-like behavior ka = 30 is emerging as N increases. The width of the main beam decreases rapidly as the peak directivity increases. In contrast to the known 2D planar aperture or array 4 approaches to high directivity, some of which are discussed in Ref. [34], Eq. (16) demonstrates that currents on a 3D sphere can achieve true needle radiation in theory. Nevertheless, again examining the directivity patterns in Fig. 1, the increasing numbers of sidelobes illuminate yet another issue. While the sidelobe levels are decreasing as N increases and the outcome would be the eventual achieve- -1 0 5 10 15 20 25 30 35 40 45 50 ment of the true needle result, they are impressively present Multipole number in large numbers for a finite number of modes. This sidelobe behavior is very undesirable for many applica- FIG. 5. Needle radiation expansion coefficients for various tions, especially if only a low number of modes were electrical sizes of the sphere, ka, as functions of the number of excited. Moreover, when one examines the amplitudes of azimuthally symmetric modes, N. 031017-5 Directivity ratio Coefficient magnitude Power Pattern FWHM (degrees) RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) the amplitudes for small mode numbers and the exponential Jðx; y; zÞ¼ I lδðxÞδðyÞ½Hðz − l=2Þ − Hðz þ l=2Þx; ˆ growth of the coefficients once the mode number exceeds ð19Þ the electrical size of the sphere. Thus, in practice, one could only hope to approach a “true” needle effect from an Kðx; y; zÞ¼ I lδðxÞδðyÞ½Hðz − l=2Þ − Hðz þ l=2Þy; ˆ increasingly larger number of tailored currents on an m increasingly larger sphere, basically in agreement with ð20Þ Eq. (1). where the Heaviside function HðuÞ¼ 1 if u> 0, and ¼ 0 if u< 0. Recall that dipole antennas radiate in their IV. NEEDLE-LIKE RADIATION FROM AN broadside directions. With these current direction choices, ARRAY OF DIPOLE-CONSTRUCTED HUYGENS MULTIPOLES radiated fields along the z axis are thus possible. As shown in Ref. [34], if the current amplitudes are weighted properly Obtaining sets of specified current distributions on an so that I ¼ I ¼ I =η, one then has e m electrically small 3D object with curvature to realize an approximate needle radiation even for a few modes is ikr ff ˆ ˆ also, unfortunately, a nontrivial task. Moreover, one would ⃗ E ðr⃗ Þ¼iωμIl ð1þcosθÞ½cosϕθ−sinϕϕ; 4πr nevertheless desire a planar or conformal array or at least a ikr Ile thin stack of planar-radiating elements on a mobile plat- ff ⃗ ˆ ˆ H ðr⃗ Þ¼iωμ ð1þcosθÞ½sinϕθ þcosϕϕ: ð21Þ form for any practical application. Thus, a discrete radiating η 4πr aperture associated with a simple set of radiators to achieve It is immediately apparent that the cardioid pattern asso- high directivity, like what is obtained with now commonly ciated with a Huygens source is attained and that the field is used phased arrays, remains desirable. Can something null along the negative z axis as expected, where θ ¼ π, useful be achieved in practice? ϕ ¼ 0. The directivity is then straightforwardly calculated As the historical discussion indicated, it is well known to be that there are generally fundamental trade-offs in the patterns generated by a discrete array between the main 4πr Sðr⃗ Þ · r ˆ 3 beam width and the sidelobe levels [30]. One knows that a Dðθ; ϕÞ¼ ¼ ð1 þ cos θÞ : ð22Þ rad uniformly driven aperture generally produces the maximum P 4 tot directivity, amplitude tapering of the array elements pro- Therefore, the maximum directivity of the electric-magnetic vides control of the sidelobe levels, and phasing between dipole pair (N ¼ 1), which is along the positive z axis, where the elements yields the capability to steer the direction of θ ¼ 0, is 3, twice the value of either dipole alone, confirming the main beam. Examples of the directivity obtained from the Harrington result (2): D ¼ 1 þ 2 × 1 ¼ 3. distributions of electric and magnetic currents on a planar max Huygens source antennas have been achieved in practice disk and from a circular array of Huygens dipole sources in both electrically small [54–57] and larger [58,59] are given in Ref. [34]. packages. They have been recognized as an important One finds that in contrast to the currents on a small research direction for IoT applications [60]. Huygens sphere, those 2D current distributions do not yield the metasurfaces have already played a significant role in desired needle-like radiation pattern unless the disk radius antenna and scattering configurations [61–63]. How, then, becomes extremely large or hard-to-realize current distri- does one achieve a Huygens behavior with yet higher butions are employed. What can one then do to achieve directivity? needle-like radiation from a potentially realizable current distribution? Here, we explore how superdirectivity can be B. Multipole-based Huygens sources obtained with an end-fire array of broadside radiating In an extension of Uzkov’s results [64], it has been Huygens multipoles. demonstrated in a series of articles on end-fire arrays, e.g., Refs. [65–72], that an array of electric elements achieves A. Dipole-based Huygens sources its maximum directivity in its end-fire direction when the To understand more completely the Huygens source separation distance between the element pairs goes to zero. concept, a combination of elemental electric and magnetic These dense packing and end-fire concepts have been dipole sources is considered first. With an emphasis on the demonstrated experimentally for moderately small separa- z axis as the preferred direction, the elemental electric and tion distances as well. Moreover, it has been shown that a magnetic current densities of amplitude, respectively, I dense end-fire array of dipole Huygens sources, i.e., and I , imposed on a pair of orthogonally oriented electric electric and magnetic dipole pairs in an end-fire configu- and magnetic Hertzian dipole antennas of length l will be ration, will produce the highest possible directivity asso- taken as ciated with dipole radiating elements [70]. As an extension 031017-6 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) dipoles, it has a length 2 × NΔ but with all of the resulting multipoles being centered on the zeroth element, i.e., the simple electric (magnetic) dipole. Finally, as was done with the sphere-based currents, adding the resulting electric (magnetic) multipole fields together with the simple coefficient weightings, 1=fn!½ð−2iÞkΔ g, one obtains the pattern function Nþ1 1 − cos θ FJ ðθ; ϕ; ωÞ ≈ ðcos θÞ ¼ zx 1 − cos θ n¼0 ¼ P ðθÞð23Þ for N multipole elements. Using L’Hopital’s rule, the maximum of Eq. (24) occurs along the z axis at θ ¼ 0, max i.e., P ∼ ðN þ 1Þ. Then, arranging the electric and FIG. 6. Assemblage of the broadside radiating multipoles. magnetic current moments so that they are balanced with I ¼ ηI ¼ I, one obtains, for the far fields of N electric and m e magnetic multipole pairs, of those results, let us consider a compact end-fire array of broadside radiating Huygens multipole sources. ikr ff As noted by Harrington [73], one can use alternating ˆ ˆ E ðr⃗ Þ¼ iωμIl ð1 þ cos θÞP ðθÞ½cos ϕ θ − sin ϕ ϕ; ω N 4πr pairs of dipole current elements, as one does with alter- ikr Il e nating sets of charges to achieve electrostatic or magneto- ff ⃗ ˆ ˆ H ðr⃗ Þ¼ iωμ ð1 þ cos θÞP ðθÞ½sin ϕ θ þ cos ϕ ϕ: ω N static multipoles, to produce higher-order electromagnetic η 4πr multipoles. Again, consider the electric (magnetic) multi- ð24Þ poles to be oriented along the x axis (y axis). As discussed in Ref. [34] and as depicted in Fig. 6, the electric (magnetic) Thus, the directivity again takes the form multipoles are obtained by properly arranging combina- tions of electric (magnetic) dipoles to be compactly spaced 2 2 2ð1 þ cos θÞ × P ðθÞ along the z axis, i.e., to have an electrically small distance Δ R Dðθ; ϕÞ¼ ; ð25Þ π 2 2 ð1 þ cos θÞ × P ðθÞ sin θdθ 0 N between each of them, and to have the appropriate orienta- tions with respect to each other. The resulting electric and where the factor of 2 appears from the equal contributions of magnetic multipoles are then combined together, as depicted the balanced electric and magnetic Huygens multipoles. in Fig. 7, to form the Huygens multipole end-fire array. The directivity patterns (in dB) for several number of Accounting for all of the constituent electric (magnetic) Huygens multipoles arranged compactly along the z axis are given in Fig. 8. A polar plot of the N ¼ 1000 case is shown in Fig. 9. From both figures, one clearly sees that the Huygens source behavior has been obtained and, as the number of higher-order modes is increased, that the directivity approaches a needle-like Huygens behavior in which the sidelobes have been completely eliminated. Thus, the desired needle-like radiation from an array has been demonstrated. This behavior is further confirmed in Figs. 10 and 11. While they illustrate the needle-like behavior, they also indicate that this desirable performance, in contrast to the sphere result, is slow to evolve. Referring to Fig. 10,it is confirmed that while the peaks of the power patterns increase as 2ðN þ 1Þ, the maximum directivity only increases linearly as N becomes large, approximately as 1.5N. Moreover, the decrease of the angular FWHM of the power pattern begins to slow noticeably as N becomes quite large. This is emphasized further in Fig. 12, which presents FIG. 7. End-fire array of the assembled broadside radiating multipoles. the directivity patterns of the needle and the multipole 031017-7 RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) 40 35 20 100 -20 -40 -60 -80 -100 -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) 0 100 200 300 400 500 600 700 800 900 1000 Number of multipoles FIG. 8. Directivity pattern (dB) attained with N Huygens multipoles arranged compactly along the z axis. FIG. 10. Maximum directivity (dB) attained with N Huygens multipoles arranged compactly along the z axis. 330 30 300 60 270 90 240 120 0 100 200 300 400 500 600 700 800 900 1000 Number of multipoles 210 150 FIG. 11. FWHM values of the power patterns generated by N Huygens multipoles arranged compactly along the z axis. FIG. 9. Polar plot of the directivity produced by 1000 Huygens multipoles arranged compactly along the z axis. because the nth multipole amplitude is the inverse of the antennas for a finite number of terms adjusted for the larger product of the factorial terms n! and ðkΔÞ . This means the number of multipoles needed to recover the maximum coefficient magnitudes first increase algebraically with n, obtained with a much smaller number of needle terms. but they then reach a tractable maximum (as long as the While it does take more terms, the multipole antenna does multipole index is not exceedingly large, which it would recover the needle behavior without the sidelobe issues. not be in a realistic antenna) and start to decrease as the Furthermore, considering the coefficient weightings, one factorial term becomes dominant. Thus, one could hope to does not encounter the exponential blowup associated with generate needle-like behavior in practice. These results the sphere results. This outcome is clearly illustrated in suggest that advancing to a few Huygens multipole pairs Fig. 13, where the coefficient amplitudes for different from the simple Huygens dipole pair can significantly electrical spacings (i.e., kΔ, where Δ is the physical improve the directivity associated with a compact system. distance between the elements) between the multipoles In fact, it proves that one is not bounded by the simple are compared for variable numbers of multipoles. The dipole pair, which disproves previous conclusions (see, results are different from the conventional arguments e.g., Ref. [74]). 031017-8 Directivity (dB) Maximum directivity (dB) Power Pattern FWHM (degrees) USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) Needle, 5 modes 25 Multipoles -20 -40 -60 -80 -100 -200 -150 -100 -50 0 50 100 150 200 Angle (degrees) FIG. 12. Comparison of the directivity patterns (dB) of the needle and the multipole antennas. k = 0.2 3 k = 0.1 k = 0.05 2.5 FIG. 14. Dielectric resonator stack approach to realize the Huygens multipole array. 1.5 desired broadside radiating fields. These elements would 0.5 then be assembled into the end-fire array configuration as depicted. This dielectric resonator antenna (DRA)-based -0.5 system is currently being explored for experimental vali- dation of the Huygens multipole end-fire array concept. -1 Another approach, particularly suited for nano-antennas, -1.5 would be to have a multilayered, subwavelength-size, 0 5 10 15 20 25 Multipole number resonant core-shell particle configuration in which the electric and magnetic multipole modes were simultane- FIG. 13. Needle radiation expansion coefficients for various ously excited and their resonance frequencies adjusted by electrical sizes of the sphere, ka, as functions of the number of the geometry and material values to be coincident. While azimuthally symmetric modes, N. this was accomplished at the dipole level (e.g., Ref. [44]), discussions about the relationship between the quality factor and directivity [27] remind us that the higher-order Even achieving directivities that are an order of magni- multipoles will have narrower bandwidths and, hence, may tude larger than an electrically small dipole from a similar be quite sensitive to their design parameters. Nonetheless, a footprint holds many potential benefits for future IoT two-dimensional version of this multilayer concept, as wireless and mobile platform-based communication and sensor devices. Moreover, the amplitudes needed to realize depicted in Fig. 15, has been verified [75]. Yet another the outcome are well-posed and reasonable. In practice, the technique would be to employ a similar number of (single) desired needle-like pattern outcome could be realized by resonant core-shell particles, each producing one of the developing a compact antenna constructed as a stack of thin requisite multipole fields and then aligned and excited in an layers, each layer having the proper elements (dipoles or end-fire configuration. other structures) to realize the requisite higher-order multi- All of these arrangements are intimately related to and pole. The thickness of the layers would be subwavelength supported by the near-field resonant parasitic (NFRP) to ensure that kΔ is small. metamaterial-inspired engineering of electrically small As shown in Fig. 14, a parasitic stack of high-permittivity antennas paradigm [76], which has successfully produced annular dielectric resonator elements could be tuned to a large variety of multifunctional compact systems. For instance, with one of the Huygens dipole sources produce the desired electric and magnetic multipoles appro- priate for each layer at the same frequency and with the already realized (see, e.g., Ref. [56]), any one of the 031017-9 Coefficient magnitude Directivity (dB) RICHARD W. ZIOLKOWSKI PHYS. REV. X 7, 031017 (2017) array, which combines the first six Huygens multipoles, N ¼ 0 to N ¼ 5, are shown in Figs. 16–18, respectively. The Huygens behavior in all of these cases is immediately observed. The Huygens multipole array result in Fig. 18 illustrates the absence of any sidelobes and the beginning of the needle-like behavior. FIG. 15. Two-dimensional version of a coated nanoparticle realization of the multipole effects. aforementioned “arrays” of NFRP electric and magnetic multipole elements could be driven at RF and micro- FIG. 16. NewFasant predicted 3D directivity pattern for the wave frequencies. This system is highlighted in Fig. 14. Huygens dipole (N ¼ 0) antenna. Moreover, as noted in the earlier discussions, the particle approaches have been demonstrated for various individual multipole orders at optical frequencies; hence, their combi- nations are also very realizable. Therefore, these compact Huygens multipole end-fire arrays of NFRP elements would overcome the usual argument that superdirectivity cannot be achieved because of the ill-posedness encoun- tered with the currents typically associated with its synthesis. V. SIMULATED HUYGENS MULTIPOLE END-FIRE ARRAY RESULTS To confirm these analytical superdirectivity results and to set the stage for future experimental efforts, simulations of FIG. 17. NewFasant predicted 3D directivity pattern for the the Huygens multipole end-fire array were performed with Huygens N ¼ 2 multipole antenna. the commercial method of moments code NewFasant (see Ref. [34]) [77]. It is the only commercial code (to the best of our knowledge) that has multipole sources built into it. The simulation parameters used to obtain the results were a 3.0 MHz excitation source (λ ¼ 100.0 m), 10.0 cm long dipoles, with Δ ¼ 0.1 cm spacings between them. The electric and magnetic dipole Huygens source was first created, and the Huygens quadrupole source was con- structed with it. The result was then used to create the Huygens hexapole source and so on until the Huygens duodeca-multipole was created. Finally, these six multi- poles, n ¼ 0; 1; …; 5, were combined together in the desired end-fire configuration as illustrated in Fig. 7. The overall length of this N ¼ 5 Huygens multipole −4 end-fire array is 1.0 cm (i.e., 2 × NΔ ¼ 10 λ ). The simulated 3D directivity patterns for the Huygens FIG. 18. NewFasant predicted 3D directivity pattern for the dipole (N ¼ 0 multipole) and hexapole (N ¼ 2 multipole) Huygens multipole end-fire array, which combines the first six antennas and for the composite Huygens multipole end-fire multipoles: N ¼ 0 to N ¼ 5. 031017-10 USING HUYGENS MULTIPOLE ARRAYS TO REALIZE … PHYS. REV. X 7, 031017 (2017) Analytical or from the multiple sidelobes produced by a mode-limited Numerical needle source. The results disprove many previous state- ments that the directivity from a Huygens dipole source was the best one could accomplish from an electrically small -20 source region. Moreover, given the many recent multipolar nano-antenna predictions and initial macroscopic-sized -40 Huygens source realizations, an electrically small, NFRP- based Huygens multipole end-fire array should be demon- -60 strable in the near future. ACKNOWLEDGMENTS -80 The author expresses his profound gratitude to the CEO -100 of NewFasant, Professor Felipe Catedra of the University -200 -150 -100 -50 0 50 100 150 200 of Alcala, Alcala de Henares, Spain, for his kindness in Angle (degrees) providing access to the NewFasant code, for writing a FIG. 19. Comparison of the analytically and numerically couple of “How to Use NewFasant for Dummies” tutorial calculated directivities for the Huygens multipole end-fire array, documents, and for actually running some initial cases including the dipole through the N ¼ 5 multipoles. for my education and benefit. This work was supported in part by the Australian Research Council Grant No. DP160102219. The 2D directivity pattern for the Huygens multipole end-fire array predicted numerically by NewFasant is compared with the analytical result (23) for the first six multipoles in Fig. 19. No coefficient weights were applied [1] C. W. 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