Superdirectivity from arrays of strongly coupled meta-atoms

We explore the possibility of achieving superdirectivity in metamaterial-inspired endﬁre antenna arrays relying on the good services of magnetoinductive waves. These are short-wavelength slow waves propagating by virtue of coupling between resonant meta-atoms. Magnetoinductive waves are capable of providing a rapidly varying current distribution on the scale of the free space wavelength. Using dimers and trimers of magnetically coupled split ring resonators with only one element driven by an external source, we introduce an analytical condition for realising superdirective current distributions. Although those current distributions have been known theoretically for a good 60 years, this is the ﬁrst time that a recipe is given to realise them in practice. Our key parameters are the size of the array, the resonant frequency and quality factor of the elements, and their coupling constant. We compare our analytical results for coupled magnetic dipoles with numerical results from CST simulations for meta-atoms of various shapes. The calculated bandwidth of 5 MHz for a dimer operating at 150 MHz indicates that, contrary to popular belief, superdirective antennas exist not only in theory but may have practical applications. V C 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY)


I. INTRODUCTION
Superdirectivity is a topic that reappears at regular intervals in the science and art of electrical engineering starting with the 1922 paper by Oseen 1 in which he refers to "Einsteinian needle radiation." Contrary to phased arrays providing directive radiation by means of constructive interference, superdirectivity is based on the principle that there can be destructive interference in all directions including the direction of the main lobe but that happens to be the direction in which the destructive interference is minimum. The result is somewhat controversial, against common sense. It maintains that arbitrarily high directivity could be achieved with an array of finite size. If this is true, there is no need for enormous apertures: A linear array would do. Let us see an example. At a wavelength of 1 m for a directivity of, say, 10 000, an aperture aerial would require a surface area of about 800 m 2 , whereas a superdirective linear array in the endfire configuration could be only a few meters long and have a radial dimension of, say, 10 cm. This is fine in principle. The difficulty is in realisation: how to realise the required current distribution. Apart from the difficulty of producing the right currents, superdirective antennas have of course other limitations: narrow bandwidth, high tolerance sensitivity, and low efficiency are the main ones. For those disadvantages, see, e.g., Ref. 2. In the present paper, we shall concentrate only on the realisation of the required currents for some very limited scenarios. Our primary aim is to maximise directivity without worrying about the necessarily reduced gain and about other limitations. We are interested in applications in which a narrow beam is needed in scenarios when space is limited, e.g., in CubeSats, nanosatellites 10 Â 10 Â 10 cm 3 in size. 3 Then what matters is accuracy not power loss or narrow bandwidth.
It has been known for a long time (see, e.g., Ref. 4) that in order to have superdirectivity, the field (or current) should exhibit fast spatial variations. Since most antennas work in free space, the variation has to be fast on the scale of the free space wavelength. To do that is difficult. There have been ad hoc solutions (see, e.g., Ref. 5) but no general theory, no general approach. The novel idea in this paper is to use the progress in the theory and practice of metamaterials, in particular that of magnetoinductive (MI) waves, 6,7 to realise the current distribution for 2 and 3-element endfire arrays with elements in the form of split ring resonators. 8,9 In general, MI waves can be employed for guiding and manipulating electromagnetic waves with potential applications ranging from the detection of nuclear magnetic resonance 10,11 to subwavelength imaging 12,13 and wireless power transfer. 14,15 For our purpose, in this paper, what is important is that MI waves are slow waves and consequently have small wavelengths compared to free space signal at equivalent frequencies. Thus, slow change over the wavelength of the magnetoinductive wave means fast change over the free space wavelength and that is exactly what is needed for superdirectivity. Our solution is to realise a superdirective current distribution by launching a magnetoinductive wave. To make that realisation easier and the analogy with magnetoinductive waves even more relevant, we shall restrict our attention to the case when only one of the elements of the array is driven by an applied voltage.
Superdirectivity has a vast literature. A very influential paper was that by Schelkunoff 16 who showed that arbitrarily narrow beams can be obtained with the right design. A rather surprising conclusion was reached by Uzkov 17 who showed that as the length of the linear endfire array tends to zero, the directivity increases as the square of the number of elements in the array. Gain maximisation for a finite number of elements of any linear array was performed by Bloch et al. 18 and by Uzsoky and Solymar 19 who also introduced auxiliary conditions defining both an array-Q and tolerance sensitivity. Directivity maximisation for a linear array of dipoles was done by Tai. 20 Generalisation to any dipole distribution (linear, plane, or 3D arrays) was due to Shamonina and Solymar. 21 There were a number of optimisations of Yagi antennas (see, e.g., Refs. 22 and 23) which resulted in much improved directivity but they were still far from superdirectivity. Further progress in the theory of superdirectivity was made by Lo et al. 24 who introduced the signal-to-noise ratio as an auxiliary condition when maximising directivity. For some more recent directivity maximisations, see Azevedo 25,26 and Smierzchalski et al. 27 The experimental work gained new momentum with the publication of the paper by Newman and Schrote 28 who built a four-element array. More recently, practical superdirective arrays consisting of two elements, driven independently, or parasitic, were reported in Refs. 29-37. The authors stressed the significance of using electrically small resonant elements that eases the problems of impedance matching. Work on metamaterial-inspired superdirective antennas was published in Refs. 38-44. The aim of this paper is to formulate the design rules for constructing superdirective endfire arrays comprising strongly coupled meta-atoms. The structure of the present paper is as follows. In Sec. II, we treat theoretically the two-element case based on the coupling between resonant elements approximated as magnetic dipoles. "Superdirective conditions" are derived which tells us how two coupled circuits can realise the required superdirective current distribution. The model is tested in Sec. III by comparison to numerical simulations for realistic elements resonant in the MHz frequency range. In Sec. IV, we discuss the issue of the bandwidth, the extension of the model to three-element arrays, and potential applications. Details of analytical derivations are delegated to two Appendixes in order not to interrupt the flow of argument.

II. SUPERDIRECTIVE CONDITIONS FOR A DIPOLE DIMER
A schematic representation of the two-element array is shown in Fig. 1(a). The elements are shown here as split ring resonators 7 widely used for metamaterials, although capacitively loaded rings and squares will also be considered. We shall model the elements arranged in the xy plane at a distance d as two vertical magnetic dipole radiators oriented normally, i.e., along the z axis as shown in Fig. 1(b). The current distribution required for realising superdirectivity is derived in Ref. 21 and given in Appendix A for the general case of N dipole radiators making up a linear array. We shall evaluate them here for the case of two elements and radiation in the endfire (x) direction. The optimum current distribution for two elements presented in the form of a vector with two components I 1 and I 2 has the form k ¼ x/c is the wave number, x is the frequency at which the antenna is radiating, and c is the velocity of light. We are interested in the kd ( 1 condition when the distance between the elements is small relative to the free space wavelength. We can therefore expand Eq. (1) in terms of kd. We obtain for the ratio of the two currents In the limit of kd ! 0, it is sufficient to take into account only the first two terms.
We shall now return to our proposed technique of realising this current distribution. Our goal is to realise the required current distribution by driving only one of the elements by external voltage and by choosing the right coupling constant and the right quality factors of the two coupled circuits. In this section, we assume that the coupling constant is negative and that element 2 is driven to enable superdirective radiation in the positive x direction. By modelling the metaatoms as coupled LCR circuits as shown in Fig. 1(c), the ratio of currents for two identical resonant circuits, coupled by a mutual inductance M, is derived in Appendix B as where are the quality factor of the elements and the coupling constant between them, respectively.
is the resonant frequency of the circuit, L is its self-inductance, R the resistance, C the capacitance, and is the reciprocal frequency of radiation, normalised to the resonant frequency. If we want to realise the superdirective current distribution, the ratio of currents in the coupled circuits should agree with the required current distribution for superdirectivity, i.e., Equating the real and imaginary parts, we find two important conditions and The first condition [Eq. (9)] may be rewritten as that unsurprisingly happens to be the same as the resonance frequency of the antisymmetric mode in coupled circuits [Eq. (B9)]. This is very reasonable as superdirectivity for two elements requires the elements to be nearly in antiphase. The second condition [Eq. (10)] may be rewritten by substituting Eq. (11) into Eq. (10) as which for small coupling constant j ( 1 can be approximated by This relationship is valid when the elements are composed of elementary dipoles. For isotropic radiators, the right-handside of Eq. (12) modifies to 6/kd, and, as we will see later, for elements of different shapes, the coefficient on the right hand side will again be different. In this section, we continue considering dipoles but when we come to the CST results, we shall have to modify Eq. (12). Note that there are five variables x, x 0 , j, Q, and d, and there are two equations to satisfy. A possible way to proceed is to choose first the frequency, x, the coupling constant, j, and the quality factor, Q, and then find the resonant frequency from Eq. (11) and d from Eq. (12). An alternative is to start with x and kd because the value of x is dictated by the system which needs the antenna and d must be chosen so that kd ( 1. Once the physical realisation of the element is chosen, we know the range of values which Q might take. Then, the required coupling constant could be found from Eq. (12).
Equations (11) and (12) have fundamental importance. The first one defines the frequency at which, for a given system of coupled resonators, superdirectivity can occur. The second one is the condition for the existence of superdirectivity, prescribing, for a given geometry (given kd), the required values of the coupling constant j and the quality factor Q. We shall refer to Eqs. (11) and (12) as the SD 1 and SD 2 conditions. Provided we satisfy SD 2 , we have, in the vicinity of the antisymmetric resonance, the current ratio needed for superdirectivity and we can be sure that we have a superdirective radiation pattern. It has been known for a good 60 years what the desired current distribution should be but this is the first time that a recipe has been given how to achieve it. In the present section, the analysis is done for only two elements but this metamaterial-inspired method can also be used for multi-element cases as discussed later.
A superdirective current distribution will yield the maximum achievable for a given frequency and geometry. For a small array with kd ( 1, it is of the form found by expanding Eq. (A10) from Appendix A. Taking just the first two terms of the expansion offers a wide range of validity of up to kd ' 2 as illustrated by Fig. 2. The maximum as kd ! 0 is D ¼ 5.25. It is a reasonable value considering that the maximum directivity of an array of N elements is equal to N 2 when kd ! 0 17 and the directivity of an elementary dipole (electric or magnetic) is 1.5. (Note here that for finding the resultant radiation pattern, we need to multiply the element radiation pattern with that of the array. This simple method of multiplication does not apply to the respective directivities. The resultant directivity is always less than the product of the individual directivities.) Let us see now a few examples. In our first example, we intend to show that it is worth adhering to the SD 2 condition and that it will indeed give a superdirective current distribution and the corresponding high directivity. We shall choose the resonant frequency as f 0 ¼ 150 MHz, the inter-element distance as d ¼ 64 mm, and the coupling constant as j ¼ À0.1. The choice of the coupling constant corresponds to a possible realisation in the form of two closely spaced circular loops, each 62 mm in diameter, in a planar arrangement side by side with the centreto-centre distance of 64 mm. 45 We shall choose three different values of the quality factor: Q ¼ 26 (a), 2600 (b), and 2625 (c) of which the second one satisfies the SD 2 condition and the two others do not, with Q being 10 times smaller than needed in case (a) and 10 times larger than needed in case (c). The modulus of the current ratio, its phase, and the corresponding directivity are shown in Fig. 3 as a function of the normalised frequency for the parameters given above. The maximum directivities obtained in the vicinity of the antisymmetric resonance are 2.28, 5.24, and 4.03 for the three cases, respectively. Insets show the corresponding radiation patterns at the optimum frequency. Although in each of the three cases, the optimum directivity is found in the vicinity of the antisymmetric resonance, i.e., when the SD 1 condition is satisfied, the highest directivity is obtained in case (b), when the SD 2 condition is satisfied as well. The directivity of 5.24 in case (b) does correspond to the theoretical maximum expected for two dipoles with the specified inter-element distance and wavelength, see Eq. (14) and Fig. 2.
Maps of the overall dependence of the optimum directivity for dimer configurations with various values of j and Q are shown in Fig. 4 as contour plots D max (j, Q) for the parameters: (a) f 0 ¼ 50 MHz and d ¼ 16 mm, (b) f 0 ¼ 50 MHz and d ¼ 64 mm, and (c) f 0 ¼ 150 MHz and d ¼ 64 mm. Every point on each map corresponds to a dimer with specified j and Q and the frequency yielding the best directivity. The SD 2 condition is also shown as a red dotted line. It may be seen that configurations which satisfy the SD 2 condition for j and Q indeed show the maximum directivity, whereas for configurations for which the jQ product is too low or too high, the maximum achievable directivity is lower. It is worth mentioning that as long as we keep kd ( 1, the dependence of the optimum directivity on kd is weak, with D gradually reducing from 5.25 at kd ! 0 to 5.17 at kd ¼ 0.5 [see Eq. (14) and Fig. 2]. Therefore, there will also be very little change in the shape of the optimum radiation pattern for any optimum configuration shown by dashed lines in Fig. 4 in comparison to the radiation pattern already shown in the inset to Fig. 3(b).
It can also be seen from Fig. 4 that, for a sufficiently small coupling constant, the SD 2 condition appears as a straight line on the double logarithmic scale of the graph, yielding evidence that Eq. (13) would be a good approximation of the SD 2 condition. Comparing cases (a)-(c), one can also notice that as kd increases [kd is the smallest in case (a) and the largest in case (c)], the optimum value of the jQ product reduces. This indicates that the choice of j and Q depends strongly on the size of the array and there might very well be situations when superdirectivity is not achievable, e.g., due to unrealistic values for j and Q required.

III. SIMULATIONS
To verify our analytical approach enabling us to predict the values of j and Q that would yield superdirectivity for a chosen geometry, numerical simulations were performed using the time domain solver of CST Microwave Studio. We looked at two types of copper split pipe resonators, of circular and square shape, shown schematically in Fig. 5. To enable a direct comparison between arrays of different shapes, we chose the diameter of circular elements, 2R, and the side length of the square element, a, both to be equal to 23 mm. Other geometric parameters of the elements were the same for both circular and square elements, namely, the wall thickness w ¼ 1 mm, the height h ¼ 5 mm, and the gap g ¼ 2 mm. By adjusting the value of an external lossless capacitor inserted into the gap of an element, the resonant frequency could be tuned. In the simulations, we excite one of the split pipes by a voltage source and find the current distribution that optimises the directivity by varying the available parameters, namely, the distance between the elements and the resonant frequency.
The capacitively loaded split-pipe elements, although small relative to the wavelength, cannot be regarded simple magnetic dipoles due to their extended height, and hence, we need to generalise the dipole-model SD 2 condition [Eq. (12)] to where a is a coefficient dependent on the shape of the metaatom. As mentioned in Sec. II, for isotropic elements, a ¼ 1.2, and for dipoles, a ¼ 1. We find that for the split pipe elements of extended height h ¼ 5 mm used in numerical simulations, for both the circular and square varieties, we need to use a ¼ 0.8.
In the first series of simulations, summarised in Fig. 6, we keep the dimer geometry unchanged, with the centre-tocentre distance d ¼ 24 mm, i.e., with the separation between the walls of the neighbouring meta-atoms being 1 mm. The coupling constant for the circular-element dimer for this geometry is found to be equal to j ¼ À0.1, and for the square-element dimer, it is À0. 19. These values of the coupling constant are calculated from the split of the resonance into the symmetric and the antisymmetric mode, see Eq. (B9), by following the procedure detailed in Ref. 9. By adjusting the value of the capacitors, the resonant frequency varies in the range between 50 and 150 MHz. The left column, Fig. 6(a), shows the results for the circular elements and the right column, (b), is for the square elements. The top row in Fig. 6 shows the numerically calculated variation of the quality factor of a single element with resonant frequency f 0 . For elements of either shape, the quality factor increases with the resonant frequency as expected for a copper resonator of the chosen size in the chosen frequency range.
Knowing the numerical values for j, Q, and kd, we can now employ the generalised SD 2 condition to find the optimum resonant frequency. The middle row of Fig. 6 shows the plots for the frequency dependence of the left-hand side of the generalised SD 2 condition (solid line) and of the righthand side (dashed line) using the analytical SD 1 condition for the optimum frequency. The numerical data and the analytical data can be seen to agree well. Obviously, the left hand side of SD 2 , jQ, increases with increasing frequency because of the increase in Q. The right hand side, 4/kd, decreases with f 0 . Their intersection predicts at what resonant frequency the superdirective condition is fulfilled-this frequency is marked by a vertical dotted line for both types of arrays, for circles (left plot) and for squares (right plot). The optimum frequencies predicted by the SD 2 method are 100 MHz for the circular meta-atoms and 67 MHz for the square meta-atoms.
The bottom row in Fig. 6 shows plots of the directivity values found numerically as a function of the resonant frequency, clearly confirming that the SD 2 condition is indeed valid. Also, in these plots, we mark the optimum frequency by a vertical dotted line, and it can be seen that the values agree well with those predicted by the SD 2 method. The corresponding directivities are close to 5 for both the circular and square elements.
Our next series of simulations (Fig. 7) is analogous to that shown in Fig. 6, but this time, the optimisation is by inter-element distance at a resonant frequency of 150 MHz. The left column (a) is for circular elements and the right column (b) is for square elements. The variation of the coupling constant with the centre-to-centre distance is shown in the top row. As expected, for any distance between the elements, the coupling between square shaped elements is stronger than the coupling between circular elements, because the areas on the two coupled elements are closer to each other. The middle row shows how the left-hand side and the right hand side of the SD 2 condition vary with the centre-to-centre distance. The optimum distance, when the SD 2 condition is fulfilled, is marked by vertical dotted lines for both types of arrays. The directivity plot versus distance in the bottom row demonstrates clearly that our method works again-the maximum directivity corresponds to the predicted distance with good accuracy for both types of arrays. The respective optimum directivities are close to 5 in both cases. The optimum centre-to-centre distance for the circular elements is 29 mm, and for the square elements, it is 35 mm. In both cases, when elements are placed side by side, the resulting coupling is too strong, and the elements have to be moved away to reduce the coupling strength to the optimum value, with square elements requiring a larger separation.
Similar optimisation procedures can be performed for other scenarios, e.g., at any resonant frequency. It would not be a difficult exercise to predict that at a lower resonant frequency, as the right hand side of the SD 2 condition is inversely proportional to the frequency (see Fig. 6), a larger value of the coupling constant would be required for the optimum configuration than the values observed in Fig. 7. If the chosen resonant frequency would be too low, the optimisation might fail altogether for the elements of the chosen shape, and a different geometry of meta-atoms would be required that would enhance the coupling constant even further. Suitable candidates would be, e.g., elongated rectangular elements, for which the coupling constant is known to increase significantly. 46

A. Bandwidth
Our SD method provides a simple test whether or not a chosen geometry is capable of superdirectivity. By estimating the product of jQ and comparing it to 5/kd, we know what changes to the inter-element distance, the resonant frequency, or the shape of the elements are needed for a dimer of meta-atoms to satisfy the SD 2 condition. If, in an optimised configuration, the actual frequency deviates from the optimum one, then of course both the currents and the value of kd change. As a consequence, both the radiation pattern and the directivity will change, and the radiation pattern will widen and the directivity of the array will decline, as can be seen, e.g., in Fig. 3(b, bottom plot). Our measure of the bandwidth is based upon the decline of directivity. We define the array bandwidth in the usual way where Df 3 dB is the width between the points in frequency where directivity drops by 3 dB from its maximum value obtained at f opt . How would the configurations shown in Fig. 4 score in terms of the bandwidth? The bandwidth is plotted in Fig. 8 against the absolute value of the coupling constant j (solid lines) for the parameters corresponding to the optimum SD 2 curve in Fig. 4. We also plot in Fig. 8 the variation of the passband of the corresponding magnetoinductive waves with frequency (dashed lines). The two are obviously correlated although they represent different physical quantities. One shows the frequency range yielding the directivity close to the superdirective value and the other one is the (much wider) frequency range in which magnetoinductive waves can propagate. The fact that both depend on j in the same manner is a corroboration of our method for realising superdirectivity by virtue of inter-element coupling between meta-atoms.
The bandwidth can be seen to vary between 100 kHz and 5 MHz for realistic values of the coupling constant, j ¼ À0.1 to À0.4, for frequencies in the range of 50-150 MHz and distances between the meta-atoms of 16-64 mm. For example, in case (c) with the frequency of 150 MHz, the inter-element distance of 64 mm, and the coupling constant of À0.1, the array bandwidth is 1.6 MHz, and for the coupling constant of À0.4, it is 4.8 MHz. This may be a reasonable value for wireless data transfer applications. Using 64QAM modulation, this would deliver up to 30 Mb/s sufficient for several compressed video channels. Clearly, increasing the frequency or increasing the size of the array, we can increase that number even further. Taking as another example parameters suitable for 700 MHz band UHF TV channel frequencies, choosing f 0 ¼ 744.5 MHz, d ¼ 16 mm, j ¼ À0.19, and Q ¼ 120, we obtain the 3 dB bandwidth of over 16 MHz (frequency range 703.2-719.4 MHz), sufficient to cover completely two UHF TV channels, channel 50 and channel 51. Simulation results for split pipes confirm the values of the bandwidth obtained from the dipole model supporting the conclusion that the array bandwidth depends primarily on the coupling strength rather than on the radiation properties of individual resonators.
It can be seen from Fig. 8 that for a larger-size dimer, the bandwidth curve approaches the MI passband curve. Importantly, the values of the bandwidth estimated analytically for dimers agree with those obtained numerically for split-pipe resonators of extended height, suggesting that it is the nature of the magnetoinductive waves and the coupling strength that determine the bandwidth and not so much the actual shape of the resonators. As mentioned before, in a practical realisation, the bandwidth can be increased by making the dimer structure larger by scaling up all the dimensions, while still staying within the requirement of it being small on the scale of the free wavelength. The passband of the magnetoinductive wave sets the asymptotic limit for the bandwidth; hence, realising dimers with a very strong coupling would be the way of achieving a large-bandwidth superdirective performance. Hence, equipped with our simple method of designing superdirective dimers by relying on the SD 2 condition, we can embark on the next stage of optimising superdirectivity by looking at meta-atoms of different shapes while ensuring that the coupling strength is sufficient.

B. Trimers and beyond
Can we find the superdirective condition, analogous to SD 2 for dimers, for a general case of "meta-molecules" comprising many coupled resonators? It would be a worthwhile exercise, considering that the maximum directivity of an array of N elements follows the N 2 trend for kd ! 0. 17 The recipe would be the same: by comparing the superdirective current distribution (Appendix A) to that prescribed by the generalised Ohm's law for coupled LCR circuits (Appendix B), we can identify required values of the quality factors and of the coupling constants. How can we match the superdirective current distribution for N elements? The solution is known-the distribution is close to binomial in terms of the amplitude, and the phases are such that the elements are nearly in antiphase. Our intuitive solution is to excite an endfire array of meta-atoms in the centre, the MI wave that goes in the backward direction from centre to element 1 has to be a backward wave, and hence, the inter-element coupling has to be negative. The other half of the array, from the centre to the end of the array, has to carry a forward wave and that requires positive inter-element coupling. Figure 9 illustrates the principles upon which a superdirective array, realised by magnetoinductive waves, can be built. How can the positive coupling be realised in the planar configuration? One solution would be to use differently shaped elements, e.g., figure-of-eight shaped resonators with an overlap yielding strong positive coupling. 11 Another possibility would be to rely on electric and not magnetic coupling, e.g., by using elements with splits on both sides-that would give positive electric coupling. 47,48 We test this approach taking a trimer as an example. We return to our analytical dipole model used in Sec. II but consider a trimer, an endfire array of three equidistantly spaced dipoles arranged along the x axis and pointing in the z direction. The optimum directivity of an endfire linear trimer with kd ( 1 can be written as and As before, in the limit of kd ! 0, we shall only take into account the first two terms. Next, we shall treat the elements of the trimer as coupled LCR circuits and apply the generalised Ohm's law (see Appendix B). We assume for simplicity that only nearestneighbour coupling matters (hence mutual coupling between elements 1 and 3 is taken as zero, M 13 ¼ 0). Then, the solution for the currents can be written as with j n2 being the coupling constant between elements n and 2, Q n is the quality factor of element n, and n is the reciprocal frequency normalised to the resonant frequency x 0n of element n, see also Appendix B. We further assume that although the resonant frequencies in the three elements can be different, x 01 6 ¼ x 02 6 ¼ x 03 , their self-inductances and resistances are the same, L n ¼ L and R n ¼ R for n ¼ 1, 2, 3. The design idea is similar to that of the superdirective dimer case: to have the edge currents almost in anti-phase to the central one, we have to rely on anti-symmetric modes for both the central-left pair and the central-right pair of resonators. To achieve opposite phase shifts for the edge currents I 1 /I 2 ¼ conj(I 3 /I 2 ), we detune the resonances of the edge elements in opposite directions x 01;03 ¼ x 02 ffiffiffiffiffiffiffiffiffi 16d p and choose the coupling constants between the central element and the edge elements as of opposite sign but equal in magnitude j 12 ¼ Àj 32 ¼ Àj ¼ Àd. This enables our design frequency to be equal to the resonant frequency of the central element, x 02 , and to be simultaneously in the vicinity of the lower resonance of the left-central pair of resonators (close to the anti-symmetric mode for the pair of I 1 and I 2 currents) and in the vicinity of the upper resonance of the right-central pair (close to the anti-symmetric mode for the pair of I 2 and I 3 currents). Then, the SD 2 condition for such a trimer takes the form which, for small coupling, practically coincides with the SD 2 condition obtained for the dimers, jQ ¼ 5/kd. Figure 10 shows the frequency variation of the currents [the amplitudes (a) and the phases (b)] and the resulting directivity (c) for the optimum case for the same set of parameters as in Fig. 3, f 02 ¼ 150 MHz, d ¼ 64 mm, j ¼ 0.1, and Q 2 ¼ 236. It can be seen that at x 02 , I 1 /I 2 and I 3 /I 2 are complex conjugates of each other and have both the required magnitude and phase yielding the superdirective radiation pattern. Superdirectivity is indeed achieved at x ¼ x 02 with D ¼ 10.78 which is the maximum value for the chosen distance between the elements. The inset shows the superdirective radiation pattern at 150 MHz. The array bandwidth is 1.4 MHz, and it would further increase to 2.2 MHz if we would use j ¼ 0.4.
So far, we have relied on the ability of slow magnetoinductive waves to produce a rapidly varying current distribution required for superdirectivity. Magnetoinductive waves are of course not unique in this sense; one can rely on plasmon waves, 7 on waves traveling on electric dipole arrays, 49 or on complementary dipole arrays, 50 or indeed on other kinds of slow waves. Applications can include wireless communication, and the bandwidth achievable seems reasonable. Another potential application for superdirective metamaterial-based compact-size antennas could be in 1U CubeSats, nanosatellites where space is limited, e.g., to 10 Â 10 Â 10 cm 3 , and enhanced directivity is required. 3 X band superdirective antennas would be sufficiently small and compatible with this kind of size-limited platform although efficiency might be a problem.

V. CONCLUSIONS
For the first time to our knowledge, we formulated the design rules for constructing superdirective endfire arrays comprising coupled meta-atoms with only one element being driven by external voltage. The mechanism of imposing the required rapidly varying current distribution is the propagation of slow magnetoinductive waves. Our analytical "superdirective conditions" for the values of the quality factor and the coupling constant allow rapid design of superdirective dimers consisting of two coupled meta-atoms. The analytical model is verified numerically for dimers of capacitively loaded split pipe resonators of circular and square shape. The array bandwidth is shown to be determined by the passband of the magnetoinductive waves and increase with the coupling constant reaching significant values moving superdirectivity from the area of scientific curiosity to the area of practical applications. We conjecture that this model can be generalised to superconductive arrays of 2N þ 1 elements in which the central element is excited and the desired current distribution is obtained by a magnetoinductive forward wave in the direction of endfire radiation and by a magnetoinductive backward wave in the opposite direction.

ACKNOWLEDGMENTS
Financial support by the John Fell Fund (University of Oxford) and by the EPSRC UK (SYMETA, EP/N010493/1) is gratefully acknowledged. We would like to thank also colleagues from the OxiMeta Oxford University Network on Metamaterials for fruitful discussions.

APPENDIX A: SUPERDIRECTIVE LINEAR ENDFIRE ARRAY
The expressions in this Appendix (used in Secs. II and IV) adopt a more general case derived in Ref. 21 for an endfire array of N parallel magnetic dipoles. In the far field, the electric field produced by an N-element array of identical magnetic dipoles placed equidistantly along the x axis pointing in the z direction may be written as the inner product of two N-dimensional vectors I and F E ¼ CFI; (A1) where is the current distribution within the array, gives information about the geometry of the array, C is a constant, and h and u are the elevation and the azimuthal angle of the spherical coordinate system, with h measured from the axis of dipoles (z axis) and u measured in the xy plane from the axis of the array (x axis), d is the distance between the elements, and k is the free space wavelength. The power density is then of the form where B, an N-dimensional matrix, is expressed as the outer product of the vector F with itself, and the subscript H indicates the Hermitian transpose of the vector. The average power density over the solid angle 4p is given by with the diagonal and non-diagonal elements in the form A nm ¼ sin ðjn À mjkdÞ jn À mjkd þ cos ðjn À mjkdÞ ðjn À mjkdÞ 2 À sin ðjn À mjkdÞ ðjn À mjkdÞ 3 : The directivity is then where the power density in the numerator is taken in the endfire, h ¼ p/2, u ¼ 0, direction. The optimum current is given as 19 and the maximum available directivity is Consider a linear array of N resonant elements modelled as LCR circuits with an inductance, L n , capacitance, C n , and resistance, R n , coupled by mutual inductances, M nm . The general relationship to satisfy is the generalised Ohm's law for the column vectors of the excitation voltages and the resulting currents The impedance matrix Z contains self-impedances of the elements Z nn ¼ jxL n 1 À 2 n À j n Q n (B3) as the main diagonal terms, where n ¼ x 0n x ; Q n ¼ x 0n L n R n ; x 0n ¼ 1 ffiffiffiffiffiffiffiffiffiffi L n C n p (B4) are the reciprocal frequency normalised to the resonant frequency x 0n , the quality factor, and the resonant frequency of element n. The non-diagonal elements of the matrix Z are the mutual impedances Z mn ¼ jxM nm ; m 6 ¼ n: The current vector may be obtained as In the special case of a dimer with identical elements (selfimpedance, Z 0 , mutual impedance, Z M , with element 2 being excited by an external voltage and element 1 being passive), the currents I 1 and I 2 can be found as The resonant frequency will split due to the coupling, so there will be a lower and an upper resonance at the frequencies defined by the condition is the coupling constant widely used in the theory of MI waves. 6,7 Note that if the coupling constant is negative, then the lower resonance is antisymmetric, with the currents in antiphase, whereas the upper resonance is symmetric, with both currents in phase. The symmetric and antisymmetric resonances swap if the sign of the coupling constant changes. Note that the concept of magnetoinductive waves propagating on a chain of coupled meta-atoms is valid also in this elementary case of two coupled meta-atoms. Within the general MI wave picture, 7 a chain of N meta-atoms supports N distinct eigenmodes of magnetoinductive waves (standing MI waves with sinusoidally varying current along the structure with zero currents at the boundaries, i.e., in elements at sites zero and N þ 1) for an element n of an eigenmode l, with l varying from 1 to N and n from 0 to N þ 1. In the special case of just two coupled meta-atoms, the two eigenmodes correspond to the symmetric and antisymmetric resonances of Eq. (B9).