On the Existence of Superdirective Radiation Modes in Thin-Wire Nanoloops

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1 SUPPORTING INFORMATION. On the Existence of Superdirective Radiation Modes in Thin-Wire Nanoloops Mario F. Pantoja, Jogender Nagar 2, Bingqian Lu 2, and Douglas H. Werner 2 Department of Electromagnetism and Physics Matter, University of Granada, Granada (Spain) 2 Electrical Engineering Department, The Pennsylvania State University, University Park, PA (USA) dhw@psu.edu Appendix. Directivity of a nanoloop antenna. According to [], the surface current on a thin-wire nanoloop comprised of a lossy material can be decomposed into a Fourier sum of modal currents: where = + cos () =η +/ =η /2+/ /2 (2) are the modal admittances, " is the characteristic impedance of free space, V 0 is the amplitude of the voltage source feeding the antenna (located without loss of generality at φ = 0º), b is the radius of the loop, and a is the wire radius (see Figure A.). If we define # $ = &$, then the terms am are determined by [2] [3]: where, for, N m are auxiliary functions defined as: ' = =# $ ( ) *+) + ) 2 # (3) $ ) =) = / #+ 2 : Ω <+= <>< (4) In (4), the terms Ω, =, and 0 are the Lommel-Weber functions, Bessel functions of the first kind, and modified Bessel functions of the first and second kind, respectively. Also, γ is the Euler-Mascheroni constant. For the case where =0, we ) = ln(8 + 2 : Ω <+= <>< For a lossy cylindrical wire, the characteristic impedance can be expressed in terms of the refractive index of the material " as [4]: (5)

2 = 7 = 7 F= 7 = " " = 7 " = 7 The refractive index, from the microwave to optical regimes, can be represented by a Drude-like model extended to include critical points of the band transitions as Lorentzian resonances [5]-[6]: " = G H I H ( + H 2Γ S K ++ G H I M + R H 2βΓ 2H H H+Γ H +H Γ N O & P Q N O & P Q where M is the number of critical points, H I is the plasma frequency, G are the quantum probabilities of transition, H are the critical points, Γ are the Lorentz broadening terms, and the coefficients K,T are chosen to fit experimental data based on the DC conductivity. (6) (7) Figure A.. Thin-wire nanoloop antenna geometry. Once the source currents are known, the directivity can be expressed in terms of far-zone electric fields and radiated power as: UV,W= 4XV,W Y Z # where the far-fields are given by: = 2 " Y Z # [\]^V,W\ + ] W V,W ` (8) \]^V,W\ = " cot V 4 5 b *b b sinwsin5w= # $ sinv= b # $ sinv b (9) \] e V,W\ = " # $ b *b 4 b coswcos5w= # $ sinv= b # $ sinv b (0) and the radiated power is: Y f # $ = " # $ 4 g =0 g 2 i 2 j where the Q-type integrals [7]-[8] are defined as: # + 2 j # # j # k ()

3 <=: = <mn5vmn5v>v= < j & = *?*? 2< (2) A special case of interest for this paper is the directivity in the end-fire direction (θ = 90º, φ = 80º), which is given by: 2 U90,80 *b b b = # $ = b # $ = q 2 j # $ + 2 j # * * $ # j # $ r $ (3) Appendix 2. Maximum directivity of a linear array of two dipole antennas. The maximum directivity of two parallel linear dipoles of length L separated by a distance d, such as those shown in Figure A.2, can be derived according to the procedure outlined in [9]. The directivity in the V,W direction UV,W of this two-element array, neglecting the mutual coupling between elements, is formally expressed as: UV,W = \s +s N O?tub^vube v \ G w V,W x yz { ~z { z g x} ~ ƒ ƒ g ~ \ˆ, \ ~ v v ƒ where A and A 2 are the excitation coefficients of the array, k is the wavenumber and f L accounts for the spatial dependence of the far-zone electric field of a dipole of length L, which for orientation along the z-axis and for a particular length of L=3 /2 is: (4) G w V,W =G Œ'/ V,W =?w cos Ž mv cos?w mn5v = cos Œ& Ž mv mn5v (5) Figure A.2. Array of two thin-wire 3λ/2 dipole antennas. Now we define a column-matrix: V,W =( V,W V,W += s G Œ'/ V,W, (6) s G Œ'/ V,W and an auxiliary matrix to account for the radiated power: where: V,W =( V,W V,W V,W V,W + (7)

4 & & V,W = V,W = 4\G Œ'/ V,W \ : : \G Œ'/V,W\ mn5v>v>w V,W = V,W = & & 4\G Œ'/ V,W \ : : \G Œ'/V,W\ N O?tub^ube mn5v>v>w We also define another auxiliary column-matrix related to the position of the elements in the array: (88) N V,W =( N V,W N V,W += N O?tub^vube v 2. (20) Using these auxiliary matrices, the maximum directivity in any direction V,W can be determined by solving the following matrix equation [0]: V,W = V,W N V,W (2) which leads to the desired excitation coefficients A and A 2. The maximum directivity is then: U š V,W = s +s N O?tub^vube v G Œ'/ V,W (22) For the case of L = 3λ/2 dipoles, the diagonal elements of can be simplified to: V,W = V,W = 4 G Œ' V,W œ ub6 sin V = 4cos Œ& Ž mv œ ub6 (23) š where œ ub < is the cosine integral [4] defined as œ ub <={ žÿ >. The off-diagonal elements can be written as: V,W = V,W = & 2\G Œ'/ V,W \ : \G Œ'/V,W\ = #>mn5vmn5v>v = \G Œ'/ V,W \! (#> 2 + =: cos Œ& > After a series of algebraic manipulations, an exact expression can be found for that involves hypergeometric functions of the type [5], such that: (24) V,W = V,W = #> 4\G Œ'/ V,W \ œ ub6+ i+ 2! 2 +, 9 4 k (25) This series converges rapidly for values of?t 2<, which is a required condition for superdirectivity. In this case, it was found through numerical experiment that terms after =2 are negligible. This leads to the following convenient closed-form expression: V,W = V,W 4\G Œ'/ V,W \ iœ ub6 2( #> 2 + At the angle of interest (V =90,W =90 ) these expressions reduce to: + 3 (+ 3 +(#> 2 + k (26)

5 V =90,W =90 = V =90,W =90 = 4 œ ub6π V =90,W =90 = V =90,W =90 4 iœ ub6 2( #> (+ 3 +(#> 2 + k (27) Alternatively, if?t 2 the series representation involving hypergeometric functions given in (25) will be slowly converging. Therefore, in this case, a more efficient methodology for evaluating the series representation in (24) can be developed by deriving a recurrence relation for. To begin this development, we recognize that the integral can be represented as: = 2 : > + 2 : cos3 > (28) A closed-form solution to the first integral in (28) can be found in terms of gamma functions: =: > Next, we define an auxiliary variable = to represent the second integral in (28): = Γ 2Γ 2 +2 (29) = =: cos3 > (30) Combining (28)-(30) we arrive at = 2 += (3) Now, using the fact that Γ<+=<Γ< (32) we may derive the following recurrence relation for the integral defined in (29): = 2 2 (33) Next, after applying integration by parts twice on = we arrive at the following recurrence relation: = = = +2 4= (34) For <3 we have:

6 = 4 œ ub6π = 2 = (35) = 2 3 = == =0 = = 2 9 For 3 we can efficiently evaluate and = using (33) and (34), respectively, as recurrence relations with the initial values given in (35). Finally, can be calculated using (3) based on the results obtained from the application of (33) and (34). The maximum directivity in the direction (V =90,W =90 ) for inter-element distances varying between d = 0.0λ and 0.5λ are shown in Fig. A.3, left vertical axis. In addition, the phase difference required to achieve these maxima are shown in Fig. A.3, right vertical axis. Note that both elements are fed with the same current excitation amplitude. It can be seen that for a two-element array of isotropic radiators, a phase delay near 80 degrees leads to the maximum directivity. In particular, a directivity of 5.2 occurs when the elements are extremely close together and this value decreases as the inter-element spacing increases. This reduction is often tolerated because of the sensitivity and manufacturing issues which arise in feeding two radiators which are extremely close together [9]. In addition, mutual coupling can have a major impact when the elements are closely spaced, and in this case the magnitude of the current in the second dipole has to be reduced to compensate for the effects of the coupled fields [9]. Figure A.3. Maximum directivity and required phase delay in an array of two thin-wire 3λ/2 dipoles vs. inter-element spacing. Appendix 3. The lossy nanoloop acting as a two-element linear array of dipoles. The objective of this appendix is to establish a relationship between superdirective radiation in a thin-wire gold nanoloop and a two-element array of finite-length linear dipoles. Figure A.4 shows the circumference of both the 3000 nm and 600 nm gold nanoloops in terms of the effective wavelength []. As shown in the main text,

7 superdirectivity occurs for the 3000 nm loop around k b =.8, which corresponds to an effective wavelength of 3. Figure A.4. Effective wavelength for thin-wire gold nanoloops of circumference C. At this frequency, the nanoloop can be thought of as a two-element array of semicircular dipoles, as shown in Figure A.5. Figure A.5. Analogy between a thin-wire nanoloop and a two-element nanodipole array. Note that this interpretation only holds when the circumference of the loop is an integer multiple of λ eff. The separation distance d between the elements is considered roughly to be the same as the radius of the nanoloop b, which corresponds to 0.9λ at the frequency of interest. Since C = 3λ eff for the loop, our simplified model will consist of two dipoles of length L= 3λ/2. Using these values, Figure A.3 predicts a maximum directivity of approximately 4.5. While there are multiple assumptions in this simple model, which limit the accuracy of the prediction, the analogy provides physical insight into the superdirective phenomenon and provides a rough estimate of the value of the directivity that could be achieved by varying the physical dimensions of the nanoloop. This model will now be employed to analyze the case of the gold nanoloop with C = 600 nm, where the superdirective effect does not appear. Figure A.6 shows the directivity of the 600 nm nanoloop in the endfire direction, where a directivity peak of 2.3 is achieved when k b = 0.8. According to Figure A.4, that particular frequency corresponds to a length of 2λ eff, from which our simplified model can be constructed. An examination

8 of the magnitude and phase of the currents of the nanoloop (Figures A.7 and A.8) for C = 2λ eff confirms that although the magnitudes are approximately the proper values to achieve superdirectivity, the phase difference is not the required 80 degrees, and consequently the directivity is lower than the maximum achievable value. However, an examination of the phases of Figure A.8 shows that the phase difference of 80 degrees is achieved at C = λ eff and C = 3λ eff. However, the magnitude of the currents at those frequencies are either too high or too low to yield the superdirective effect. Therefore, we can conclude that the gold nanoloop with this radius is not able to achieve the superdirective effect. This simplified model gives us insight into the underlying physical mechanism that causes superdirectivity, namely the specific combination of lossy material and loop radius that leads to a current profile which closely resembles a two-element superdirective dipole array. Figure A.6. Directivity for a thin-wire 600 nm circumference gold nanoloop. Figure A.7. Currents (magnitude) at different frequencies for a thin-wire 600 nm circumference gold nanoloop.

9 Figure A.8. Currents (phase) at different frequencies for a thin-wire gold 600 nm circumference nanoloop. Appendix 4. Derivation of electric and magnetic multipole coefficients. According to [2], the radiated fields can be expanded in spherical coordinates in terms of multipole coefficients as: ± ] Z,V,W=] ± ²24+q # ³4, ± #Z ± + S 4, ± #Z ± r ± ± ± Z,V,W= ] ± ²24+q ³ 4, ± #Z ± + # S4, ± #Z ± r ± ± where ± are the vector spherical harmonics, ± are spherical Hankel functions of the first kind, # is the wavenumber, is the impedance of free space, and ³ and S are the electric and magnetic mutipole coefficients, respectively. Following the derivation presented in [3], we can calculate these coefficients by assuming knowledge either of the electric fields surrounding the antenna: or a knowledge of the magnetic fields: (36) (37) S 4,= ] ± #Z ± ²24+ : ± ] >Ω (38) #Z ³ 4,= ] ± #Z ±* ²44+²24+ : ± ] f >Ω (39) ³ 4,= ] ± #Z ± ²24+ : ± >Ω (40) #Z S 4,= ] ± #Z ± ²44+²24+ : ± f>ω (4) or any combination of the above equations (38)-(4) if both the electric and magnetic fields are known. In the above expressions, ± represent the scalar spherical harmonics. For the spherical multipole analysis performed in the main text, (9)-(0) are used to compute the electric fields, the far-field identities =]^/ and ^ = ] / are used to compute the magnetic fields, and (38) and (40) are used to compute the electric and magnetic multipole coefficients.

To provide more insight, a brief parametric study will now be performed to show the sensitivity of the phenomenon as the geometrical dimensions and material properties are varied. Fig. A.

10 Appendix 5. A parametric study of Al/Au superdirective nanoloops. The main text of this contribution was devoted to the discovery of superdirectivity in thin-wire nanoloops exhibited in the optical and terahertz regimes. To provide more insight, a brief parametric study will now be performed to show the sensitivity of the phenomenon as the geometrical dimensions and material properties are varied. Fig. A.9 presents contour plots of the directivity of nanoloops comprised of gold, aluminum and an ideal perfect electric conductor for a thickness factor Ω=245 ~z@ =2 where b is the radius of the nanoloop and a ¹ is the radius of the wire. The dependent variables for the graphs are the loop circumference and the parameter # $ = &$. The red regions mark the superdirective radiation regime, which can be reached only at those ' frequencies where constructive interference is achieved between the two halves of the loop (as explained in Appendixes 2 and 3). From these graphs, it can be seen that the existence and frequency range in which superdirectivity occurs is extremely complex and highly dependent on geometry and material choice. At low frequencies the electrical size of the nanoloop is small and there is not enough length to create the superdirective condition. At very high frequencies the electrical size of the nanoloop is too large and the attenuation of the current leads to a current magnitude too small to yield superdirectivity. Only an intermediate part of the frequency spectrum is suitable to create superdirective radiation, which is not produced in any case by the PEC material due to the absence of losses. As expected, the directivity for the thin-wire PEC loop with a fixed Ω=2 is dependent on frequency and not on the specific loop circumference. At higher frequencies the directivity of a PEC loop increases and the radiation pattern is symmetric. Regarding the constitutive material, we can see the influence of the particular material through the refractive index and extinction coefficients of Fig. A.9(d). Although more quantitative conclusions would require a deeper study, we can infer that the lower refractive index of gold enables superdirectivity at lower frequencies because of the lower phase velocity. The role of the extinction coefficient is more related to the bandwidth of the superdirective regime, which is higher for aluminum because the extinction coefficient has a shorter dynamic range compared to that of gold. (a) (b)

$(d) Refractive index versus frequency for gold and aluminum. Finally, we also show some preliminary results on the influence of the thickness of the wire on the directivity and the efficiency. Fig. A.$ 0(a)-(c) show the directivity and gain of nanoloops of circumference 3000 nm for gold with thickness factors of Ω = 2, 0 and 8, respectively, while Fig. A.

0(a)-(c) show the directivity and gain of nanoloops of circumference 3000 nm for gold with thickness factors of Ω = 2, 0 and 8, respectively, while Fig. A.

11 (c) (d) Figure A.9. Contour plots of directivity versus loop circumference and normalized frequency for (a) gold, (b) aluminum, and (c) PEC material. (d) Refractive index versus frequency for gold and aluminum. Finally, we also show some preliminary results on the influence of the thickness of the wire on the directivity and the efficiency. Fig. A.0(a)-(c) show the directivity and gain of nanoloops of circumference 3000 nm for gold with thickness factors of Ω = 2, 0 and 8, respectively, while Fig. A.0(d)-(f) show the corresponding results for aluminum with Ω = 2, 0 and 8, respectively. Interestingly, the region of high directivity shifts to higher frequencies as Ω decreases (i.e., as the wire thickness increases), with a moderate variation in the peak values of the directivity. According to the Harrington limit [7]- [8], the theoretical maximum value of the directivity increases for higher frequencies as well, so we can conclude that the increase of the wire thickness is not efficient in terms of getting close to the theoretical maximum. However, this theoretical drawback has a practical advantage because the efficiency improves significantly as the loop thickness increases. This effect is most pronounced in gold for Ω = 8, where efficiencies of above 90 over a large bandwidth are seen in the region of high directivity. To further study this effect we present in Fig. A. the current on the surface of the gold nanoloop for Ω = 2 and 8, at the frequencies corresponding to maximum peaks of directivity (8 THz and 350 THz, respectively). For the Ω = 2 case the losses of the material at 8 THz are appreciable and the relative phase of 80º is achieved between the source point and the opposite point of the nanoloop. For the Ω = 8 case, the two-element dipole model is no longer valid, which requires that the directivity be analyzed in a different manner. Another interesting observation is that the number of maxima of the nanoloop is related to the effective wavelength [] for which the directive radiation is achieved: for the thinner wire (Ω = 2) the nanoloop length is around 3λ eff, while for the Ω = 8 case the directive pattern arises at 5λ eff. In summary, the thick wire nanoloop is able to produce directive radiation patterns with significantly higher efficiencies than its thin wire counterparts, but it is not technically operating in the superdirective mode since for higher frequencies the number of spherical modes which can be excited in a given volume increases according to the Harrington and Geyi limits of superdirectivity [7]-[8].

12 (a) (b) (c) (d) (e) (f) Figure A.0. Directivity and gain for 3000 nm circumference gold nanoloops of Ω = (a) 2, (b) 0, (c) 8 and aluminum nanoloops of Ω = (d) 2, (e) 0, (f) 8.

(a) (b) Figure A.. Current distributions (magnitude and phase) for 3000 nm gold nanoloops of (a) Ω = 2 and (b) Ω = 8. Appendix 6. Full-wave simulation of nanoloops.

This allows closed-form expressions for the far-field properties which can be computed extremely efficiently.

In addition to the theoretical method described in Appendix, the performance of a thin-wire nanoloop can be also explored by using full-wave numerical electromagnetic solvers which evaluate the

13 (a) (b) Figure A.. Current distributions (magnitude and phase) for 3000 nm gold nanoloops of (a) Ω = 2 and (b) Ω = 8. Appendix 6. Full-wave simulation of nanoloops. The main text has analyzed the superdirectivity of nanoloops considering the simplest scenario: the thin-wire nanoloop in vacuum fed with a delta-gap source. This allows closed-form expressions for the far-field properties which can be computed extremely efficiently. Appendix 5 presented a parametric study to show the sensitivity of the superdirective phenomenon to different loop geometries and material composition. In addition to the theoretical method described in Appendix, the performance of a thin-wire nanoloop can be also explored by using full-wave numerical electromagnetic solvers which evaluate the integral or differential forms of Maxwell's equations either in the frequency or time domain [9]. The numerical solution of these equations is usually achieved by discretizing the model through a geometrical mesh and accounting for the different materials by considering their constitutive parameters [20]. As shown in Fig. of the main text and Fig. A.2 of this Appendix, both theoretical and full-wave simulations give very similar results for the radiation properties, but the computational burden of the full-wave solvers is significantly higher than the theoretical method. When more

complex scenarios are considered, the theoretical model presented here needs to be extended or full-wave solutions must be employed. (a) (b) Figure A.3.

14 complex scenarios are considered, the theoretical model presented here needs to be extended or full-wave solutions must be employed. (a) (b) Figure A.3. Comparison between analytical theory and full-wave solvers for the (a) directivity of a 3000 nm Ω = 2 gold nanoloop and (b) efficiency of a 3000 nm Ω = 8 silver nanoloop. There are two major unavoidable features to be taken into account before designing more realistic nanodevices: the feeding source and the substrate. For the source issue, nanoantennas are usually fed by quantum sources or waveguides [2]-[23]. Since we have presented our analysis based on emitting antennas, the quantum source will be considered here (usually, as presented in [23], waveguides are often employed as collectors of energy in receiving nanoantennas). In this case, an infinitesimal electric dipole model adequately describes the electromagnetic radiation from the quantum dot [2]-[22], but the quantitative performance of the nanoantenna may be affected by the distance between the quantum source and the nanoantenna. As an example of the effect of the quantum source, Fig. A.3(a) and (b) show the directivity of the 3000 nanometer length loop of gold and aluminum, respectively, where the feed is an ideal delta-gap source, a dipole in a gap and an offset dipole. While some minor differences in the peak values can be noticed, the curves follow each other very closely. (a) (b) Figure A.3. Directivity of a 3000 nm (a) aluminum and (b) gold nanoloop fed with different models.

15 The issue of the substrate is dealt with in the main text. References [] McKinley, A. F.; White, T. P.; Maksymov, I. S.; Catchpole, K. R. Journal of Applied Physics 202, 2, [2] Storer, J. E. Trans. AIEE 956, 75, [3] Wu, T. T. J. Mathematical Physics 962, 3, [4] McKinley, A. F.; White, T. P.; Catchpole, K. R. Journal of Applied Physics 203, 4, [5] Etchegoin, P. G.; Le Ru, E. C.; Meyer, M. The Journal of Chemical Physics 2006, 25, [6] Etchegoin, P. G.; Le Ru, E. C.; Meyer, M. The Journal of Chemical Physics 2007, 27, [7] Savov, S. V. IEEE Antennas and Prop. Mag. 2002, 44 (5), [8] Mahony, J. D. IEEE Antennas and Prop. Mag. 2003, 45 (3), [9] Altshuler, E. A.; O'Donnell, T. H.; Yaghjian, A. D.; Best, S. R. IEEE Trans. Antennas Prop. 2005, 53, [0] Collin, R. E.; Zucker, F. J. Antenna Theory; McGraw-Hill: New York, USA, 969. [] Novotny, L. Physical Review Letters 2007, 98, [2] Jackson, J. D. Classical Electrodynamics; Wiley: New York, USA, 999. [3] Grahn, P. Multipole Excitations in Optical Meta-atoms; Aalto University, Aalto, Finland, 202. [4] Gradshteyn, I.; Ryzhik, I. Table of Integrals, Series and Products; Academic Press: New York, USA, 965. [5] Olver, F. W. J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W. NIST Handbook of Mathematical Functions; Cambridge University Press: New York, USA, 200. [6] Balanis, C. A. Antenna Theory: Analysis and Design; Wiley, New Jersey, USA, [7] Harrington, R. IRE Transactions on Antennas and Propagation 958, 6, [8] Geyi, W. IEEE Transactions on Antennas and Propagation 2003, 5, [9] Jin, J.-M. Theory and Computation of Electromagnetic Fields; Wiley, New York, USA, 200. [20] Johnson, P. B.; Christy, R.W. Phys. Rev. B 972, 6, [2] Taminiau, T. H.; Stefani, F. D.; van Hulst, N. F. Optics Express 2008, 6, [22] Klemm, M. International Journal of Optics 202, /-7. [23] Olmon, R. L.; Raschke, M. B. Nanotechnology 202, 23, 44400/-28.

The Pennsylvania State University. The Graduate School. Electrical Engineering Department ANALYSIS AND DESIGN OF MICROWAVE AND OPTICAL

The Pennsylvania State University The Graduate School Electrical Engineering Department ANALYSIS AND DESIGN OF MICROWAVE AND OPTICAL PLASMONIC ANTENNAS A Thesis in Electrical Engineering by Bingqian Lu