Supplementary information. Dendritic optical antennas: scattering properties and fluorescence enhancement

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1 Supplementary information Dendritic optical antenna: cattering propertie and fluorecence enhancement Ke Guo 1, Aleandro Antoncecchi 1, Xuezhi Zheng 2, Mai Sallam 2,3, Ezzeldin A. Soliman 3, Guy A. E. andenboch 2, ictor.. Mohchalkov 4, and A. Femiu Koenderink 1 1 Center for Nanophotonic, AMOLF, Science Park 104, 1098 XG Amterdam, The Netherland 2 Department of Electrical Engineering (ESAT-TELEMIC), KU Leuven, Kateelpark Arenberg 10, BUS 2444, Leuven, B-3001, Belgium 3 Department of Phyic, American Univerity in Cairo, AUC Avenue, P. O. Box 74, New Cairo 11835, Egypt 4 Laboratory of Solid State Phyic and Magnetim, KU Leuven, Celetijnenlaan 200D, BUS 2444, Leuven, B-3001, Belgium S1. Formulating the interaction of light with a nanocatterer in the framework of a olume Integral Equation (IE) In the following, we dicu a olume Integral Equation (IE) formalim for light nanotructure interaction. Thi material can alo be found in our previou work [1]. It i reviewed here only for the ake of completene. The phyical proce governing the interaction between light and a general catterer can be decribed by the following two equation in the frequency domain, 01 E r E r E r, E r i G r, r' J r' dv '. (S1) tot inc cat cat In Eq. (S1), the firt equation imply tate that everywhere in pace the total electric field i the um of the impreed incident field and the cattered field. Thi cattered field i generated by the induced current flowing in the ource volume, which give the git of the econd equation in Eq. (S1). Here, μ 0 and μ 1 are the vacuum permeability and the relative permeability of the material filling the pace where the catterer i ituated. G (r, r ) i the electric dyadic Green' function. Pleae note that in thi article a e -iωt time convention i employed and the angular frequency ω ha been ytematically uppreed. Epecially at the patial poition of the catterer, the total field i linked with the induced current via, Jr r Etot r, r. (S2) i 1 0 in Eq. (S1) and Eq. (S2) denote the ource volume. In Eq. (S2), ε 0 and ε r (ω) repreent the vacuum permittivity and the relative permittivity of the material that contitute the catterer. Combining the above equation, we have

2 Jr r i G r, r' J r' dv ' E r, r. (S3) 0 1 i 0 1 inc In Eq. (S3), ince the incident electric field and the electric dyadic Green' function are aumed to be known in the firt place, the induced current i the main target to olve and can be numerically evaluated by, e.g. a Method of Moment (MoM) algorithm. Writing compactly, we have the following operator formalim a in the main text, Z r, r' ; J r', E inc r,. (S4) In Eq. (S4), the impedance operator Z(r, r ; ω) i 1 Z r, r' ; r r' i 01 G, dv '. i 1 r r' (S5) 0 r A Dirac delta i added in Eq. (S5) to emphaize the local approximation. S3. Group theory conideration Although the unconnected, 1 t generation and 2 nd generation Dendritic antenna have very different geometrie, the ymmetry operation underlying thee geometrie are unchanged. Thee ymmetry operation (See Fig. S1(a-b)) are the ame a the one for a 2D rectangle: the identity operation E where no tranformation i conducted, a rotation of π about the z axi, a mirroring operation m y with repect to the y axi, and a mirroring operation m x with repect to the x axi. Thee ymmetry operation can be proved to form a group which i a C 2v group. The order of the group h, i.e. the number of element in thi group, i four.

3 Fig.S1. Illutration of the ymmetry operation for an unconnected tructure. (a) and (b) demontrate the ymmetry operation for an unconnected tructure and a rectangle. Thee ymmetry operation form a group and the irreducible repreentation of the group are hown in Fig. 1(c). Since ymmetry operation are alway applied baed on coordinate, we hould be able to find a correponding et of matrice that repreent thee operation. Here, we epecially focu on the matrice with the lowet dimenionalitie, that i, the irreducible repreentation. Since the group under dicuion i an Abelian group, we have four irreducible repreentation and they are hown in Fig. S1(c). Moreover, in contrat to thee tranformation operating on coordinate, we follow Wigner convention [7-9] and define tranformation operator which operate on function, 1 1 R P f r f R r, P f r R f R r. (S6) R In Eq. (S6), thi definition i illutrated for both calar function (uch a charge, etc.) and vector function (uch a current, electromagnetic field, etc.). Thee tranformation operator are commutative with the impedance operator defined in Eq. 3 of the main text (i.e., Eq. S4,S5) Combining the group irreducible repreentation and it tranformation operator, we can further contruct a et of projection operator [7-9] for the group under dicuion, l P (S7) R * P. j j j R h R In Eq. (S7), a projection operator i characterized by the ubcript j which mark an irreducible repreentation. Here, j may run from one to four. l j i the dimenionality for an irreducible repreentation and ince every irreducible repreentation ha a dimenionality of one, l j i equal to one. Then, the ummation i carried out with repect to all the ymmetry operation.

4 S3. On the commutative relation between an impedance operator and tranformation operator In thi ection, we prove the fact that the impedance operator defined in Eq. (S4) i indeed commutative with a tranformation operator, P P Z r, r' ; J r' Z r, r' ; J r'. (S8) R R A we conider a pecific frequency, in the following we will ytematically uppre the frequency variable appearing in Eq. (S8). We aume that the targeted tructure hold ome ymmetry operation R. In accordance with thi ymmetry operation we can define a tranformation operator which work on function, for example, the current in Eq. (S1-S4), 1 P J r' R J R r'. (S9) R In thi work the tructure i put on top of a gla ubtrate occupying the lower half pace. Conequently, (1) the ymmetry operation and it correponding tranformation operator are actually confined to the x-y plane. For example, the ymmetry operation can be repreented by a matrix, Rxx Rxy 0 R Ryx Ryy 0. (S10) Epecially, here we focu on two type of elementary tranformation in the x-y plane: rotation about the origin by an angle θ and reflection about a line which make an angle θ with the x axi, R rot co in 0 co 2 in 2 0 in co 0, Rrefl in 2 co 2 0. (S11) In Eq. (S11), the matrice are orthogonal matrice with a determinant either +1 or -1. Other more complex tranformation, for example, inverion, can be contructed by combining the above two elementary operation. Moreover, due to the preence of the lower half pace, we can plit the dyadic Green function in Eq. (S5) into two part, that i, a direct wave part and a reflected wave part, 0 G r, r' G r, r' Gr r, r'. (S12) For the direct field part, a cloed form expreion in the patial domain i ikr r, r' e,. k r r' k r (S13) G0 I g I 2 2 It can be een from the lat expreion in Eq. (S13) that g(r, r ) i only dependent on the ditance between obervation and ource point and thu can be replaced by g( r r ). Correpondingly, we

5 apply the impedance operator that only regard the direct field interaction to an arbitrary current ditribution that i operated on by a tranformation operator P R, r, r' J r' ' r r' J r' dv '. (S14) k 1 R dv 2 G0 P I g R R To tackle Eq. (S14), we perform a change of variable to the original ource coordinate ytem, i.e. x = R 1 r, I g R R R I g R R 2 2 k k I g R 2 k r x' J x' det d ' r x' J x' d ' 1 r x' R J x' d ' (S15) In the above derivation, we ue the following fact: 1) Since we have changed variable, a Jacobian mut appear in combination with the infiniteimal element, that i, dv = det(r) dτ. However, the ymmetry operation R i repreented by an orthogonal matrix. Since the abolute value of the determinant of an orthogonal matrix i 1, thi term i dropped in the econd expreion. 2) The integral limit in Eq. (S15) are unchanged only becaue we aume that the tructure i invariant under the ymmetry operation R. 3) To reach the lat expreion, we notice that any orthogonal tranformation doe not affect the ditance between two point. Notice that the gradient operator in Eq. (S15) i taken with repect to the original obervation coordinate ytem. In the tranformed coordinate, the gradient operator read, f r r r r x y z R f x R f y R f z R X Y Z f x x f x x f x x X x Y x Z x X Y Z f y x f y x f y x X y Y y Z y X Y Z f z x f z x f z x X z Y z Z z 1 R f x. x X Y Z f x f x x x x x X 1 X Y Z f R r f f x R f y x x x y y y Y X Y Z f f x x z z z z Z. (S16) (S17) Combining Eq. (S16) and Eq. (S17) with Eq. (S15), the lat expreion in Eq. (S15) become

6 1 1 d 0, R I g R ' R G R d'. x x 2 k r x' J x' r x' J x' (S18) In the derivation of Eq. (S18), it i noticed that I i imply an identity operator. A a reult, the commutative relation between the direct wave impedance part and the tranformation operator i proved,, dv ', G r r' P J r' P G r r' J r' dv '. (S19) 0 R R 0 For the reflected wave part, we can expre it correponding Green function in the Carteian coordinate ytem a well a in the cylindrical coordinate ytem, that i, p r / p / r G r, r' G,, z z'. (S20) In Eq. (S20), it i emphaized that the reflected wave i dependent on a relative angle φ, a tranvere ditance ρ and the um of vertical ditance z + z between the obervation point and the ource point, tan y y' 2 2, ' '. x x' x x y y (S21) The upercript in Eq. (S20) refer to the -polarized and the p-polarized part for the reflected wave. A cloed form expreion in the patial domain read, r, r' r,, ' r,, G G z z i co2 in 2 i 1 0 F1, z z ' F2, z z ', 4 in 2 co p r, r' r,, ' p r,, G G z z i co2 in 2 p i 1 0 p F1, z z ' F2, z z '. 4 in 2 co (S22) (S23) In Eq. (S22,S23), the function F 1, F 2, F p p 1 and F 2 are only dependent on the tranvere and vertical ditance and the detailed functional form of thee function are not of interet in thi work. Note that Eq. (S22,S23) ignore the horizontal vertical, vertical horizontal, and vertical vertical coupling, ince they are not affected by the ymmetry operation given in Eq. (S10). Next, we apply the reflected wave Green function to an operated current, p d r, R / p / r R G r, r' P J r' v' G Rx x' R J x' dv '. (S24) In Eq. (S24) the ame change of variable a in Eq. (S15) i employed. To find a relation between the

7 original and tranformed coordinate ytem, it i noticed that the rotated ource coordinate x and the rotated obervation coordinate x have no effect on the vertical ditance z + z and the tranvere ditance ρ, that i, ρ(r, r ) = ρ(x, x ). However, the relative angle φ i altered, ' for rotation, (S25) 2 ' for reflection. (S26) Combining the above obervation and applying to the relative angle, the tranvere ditance and the vertical ditance, we can re-write Eq. (S22) and Eq. (S23). Take Eq. (S22) a an example, Rx, Rx' r ',, ' i r,, G G z z co2 ' in 2 ' i 1 0 F1, z z ' F2, z z', 4 in 2 ' co2 ' (S27) Rx, Rx' r 2 ',, ' i r,, G G z z co2 2 ' in 2 2 ' i 1 0 F1, z z ' F2, z z '. 4 in 2 2 ' co2 2 ' Eq. (S27) and (S28) are repectively for rotation and reflection operation. A uggeted by Eq. (S24), we right-multiply Eq. (S26) and (S27) by R. It can be proven that (S28) co 2 ' in 2 ' co in co in co 2 ' in 2 '. in 2 ' co 2 ' in co in co in 2 ' co 2 ' (S29) co2 2 ' in 2 2 ' co2 in 2 co2 in 2 co2 ' in 2 '. in 2 2 ' co2 2 ' in 2 co2 in 2 co2 in 2 ' co2 ' (S30) Subequently, we have R R r, r,, G x, x' R R G x x'. (S31) Similar proof can be contructed for the p-polarized light a well. Subtituting Eq. (S31) into Eq. (S24) immediately give, dv R d / p / p / G p r PR Gr R Gr r, r' J r' ' x, x' J x' v' P r, r' J r' dv '. (S32) Hence, the commutative relation between the dyadic Green function ued in thi work and the rotation and reflection ymmetry operation i proved.

8 S3. The Rank of the Impedance Matrix and the Projected Matrice We combine the above dicuion on the eigenvalue problem with the group repreentation theoretical approach. The defined projection operator can be applied to all the three element in Eq. (3, main manucript), i.e. the impedance matrix, the excitation and the full olution. For the impedance matrix, thi i equivalent to a rank decompoition where the original matrix i plit into a et of rankdeficient matrice (ee explanation in the upporting information). Mathematically, the original matrix i the direct um of the projected matrice, 4 Z Z, Z P Z. (S33) j1 j j j Intead of eigen-decompoing the original impedance operator a in Eq. Error! Reference ource not found., we can apply the eigenvalue decompoition to the projected matric Z j. In thi way, we can find the eigenmode and eigenvalue that belong to a certain irreducible repreentation. To further analyze the rank of the impedance matrix and projectd matrice, uppoe that we have an impedance matrix with dimenion N by N and the eigenvalue problem for an impedance matrix i defined in Eq. (4) in the main text. Baed on all the eigenmode, we can recontruct the impedance matrix, N T Z vv. (S34) i i i i 1 The ummation in Eq. (S34) i conducted with repect to all N eigenmode. Epecially, for a pecific T i, v i v i i a matrix with a dimenion N by N but with rank one. Since the projection operator defined in Eq. (S7) actually divide the eigenpace into ubpace, we may collect all the eigenmode that belong to a pecific projection operator, that i, a pecific (row of the) irreducible repreentation, N j T Z Z, Z vv. (S35) j j j i i i i1 In Eq. (S35), the ummation indexed by j i carried out over all the (row of the) irreducible repreentation. For each j, the ummation i taken up to N j. Thi lead to the fact that Z j i an N by N matrix with rank N j. Hence, Z j i a rank-deficient matrix. S4. Example In Fig. S2 we demontrate ome exemplary eigenmode for all three tructure. For each of the tructure conidered in our work, we plot out of each irreducible repreentation one eigenmode, at a fixed choen frequency of f=50 THz. It can be readily een from thi figure that the exemplary eigenmode in the ame column indeed belong to the ame irreducible repreentation. Take the firt irreducible repreentation (the firt column in Fig. S2) a an example. Indeed, following Eq. (S6,S7) the projection operator aociated with the firt irreducible repreentation i imply the um of all the ymmetry operation,

9 j E C m m 2 x y 4 1 P P P P P. (S36) Hence, the reult of thi projection operator i a charge ditribution which obey all the ymmetrie. For the excitation, the conequence of the projection operator i to receive the projected excitation and thi excitation can only generate the current aociated with the ame irreducible repreentation, P ZJ P E ZP J P E ZJ E. (S37) j j j j j j In Eq. (S37), the econd tep i etablihed becaue the tranformation operator which are the key element in the contruction of the projection operator (a in Eq. S6) are commutative with the impedance operator. J j and E j are projected full olution current and excitation due to the j th irreducible repreentation. Latly, we combine the eigenmode of the projected matrice with the projected excitation to evaluate the coupling coefficient defined in Eq. (5, main text). Fig. S2. Illutration of exemplary eigenmode for the Unconnected, the 1 t Generation and the 2 nd Generation tructure. In the column, we find eigenmode that belong to the ame irreducible repreentation. In all the plot, the top urface charge i hown and coded by the blue color and the yellow color to repreent the negative and poitive charge accumulation. To contruct thi plot, we conidered a fixed frequency f=50 THz, and out of each irreducible repreentation we plot jut one of the eigenmode current ditribution by way of example. To olve an actual cattering problem, one would focu on one irreducible repreentation commenurate with the excitation ymmetry (4 in our work) and conider all the eigenvector in that repreentation. The above theoretical dicuion can be immediately confirmed by numerical calculation (See

10 Fig. S3). The numerical calculation have been conducted by uing our in-houe olumetric Method of Moment (-MoM) tool [1-6]. For the unconnected tructure, we project the excitation according to different irreducible repreentation (See Fig. S1(c)). It can be readily een in Fig. S3(a) that only the projection onto the 4 th irreducible repreentation i not zero. Accordingly, two mode, the L1 and L2 mode, aociated with thi irreducible repreentation are excited (See Fig. 3(d)). The mode belonging to the irreducible repreentation are not necearily orthogonal in an inner product ene [7] and the L1 and L2 mode interfere with each other and thu generate a Fano-type line hape in the extinction pectrum in Fig. S3(a). Notice that the dicontinuity in phae (Fig.S3(d)) i due to the fact that the eigenvector ha one degree of freedom. Fig. S3. Illutration of the extinction (a), the norm of the projected excitation (b), the abolute value of the coupling coefficient (c) and the phae of the coupling coefficient (d) for the Unconnected tructure. In (b), the norm of the excitation projected according to different irreducible repreentation are marked by the blue, red, light yellow and purple color, repectively. In (c) and (d), the abolute value and the phae of the coupling coefficient for the firt and econd excited mode (L1 and L2) are denoted by the red and blue color repectively. In the inet of (d), the top urface charge for the L1 and L2 mode are hown and coded by the blue and yellow color to denote the negative and poitive charge ditribution.

11 Fig. 4. Illutration of the firt five eigenmode ( L1 L5 ) and their correponding eigenvalue for the 1 t Generation tructure. The firt, econd and third column cover the frequency range: 50 THz 150 THz, 150THz 300 THz and 300 THz 500 THz, repectively. (a) (c) demontrate the extinction power of the tructure. (d) (f) how the real (olid line) and imaginary (dahed line) part of the eigenvalue. (g) (i) plot the eigenmode coupling coefficient. Throughout the figure, the eigenmode are marked by the red, green, blue, cyan and purple color. Their top urface charge ditribution are illutrated on the top of the figure. A imilar analyi can be carried out for the 1 t Generation tructure and the 2 nd Generation. In Fig. S4 we demontrate the five reonant eigenmode (ee the inet in Fig. S4 for the top urface charge ditribution) for the 1 t Generation tructure. A uggeted by Eq. (5, main text), when the abolute value of an eigenvalue become minimum, the correponding eigenmode reache it reonance. Thi obervation i clearly reconfirmed by comparing the frequency point where the imaginary part of an eigenvalue croe zero (ee Fig. S4. (d) (f)) yielding the reonant frequency for the coupling coefficient (ee Fig. S4. (g) (i)) and the reonant frequency for extinction (ee Fig. S4. (a) (c) ). S5. Illutration of Full Solution and Dominant Eigenmode at Reonant Frequencie for the 2 nd Generation Structure

12 Fig. S1. Illutration of Full Solution and Dominant Eigenmode for the 2 nd Generation Structure. The reonance are marked by the red, green, blue and cyan circle. The top urface charge ditribution of the full olution and the dominant eigenmode at thee reonance are hown on the left hand ide, while the extinction pectrum i hown on the right. Reference: [1] X. Zheng,. K. alev, N. erellen, Y. Jeyaram, A.. Silhanek,. Metluhko, M. Ameloot, G. A.E. andenboch,.. Mohchalkov, olumetric Method of Moment and conceptual multi-level building block for nano topologie, IEEE Photonic J. 4, (2012). [2] G. A. E. andenboch and A. R. an de Capelle, Mixed-potential integral expreion formulation of the electric field in a tratified dielectric medium-application to the cae of a probe current ource, IEEE Tran. Antenna Propag. 40, 806 (1992). [3] F. J. Demuynck, G. A. E. andenboch, and A. R. an de Capelle, The expanion wave concept Part I: Efficient calculation of patial Green' function in a tratified dielectric medium, IEEE Tran. Antenna Propag. 46, 397 (1998). [4] M. rancken, and G. A. E. andenboch, Hybrid dyadic-mixed-potential and combined pectralpace domain integral-equation analyi of quai-3-d tructure in tratified media, IEEE Tran. Microw. Theory Techn, 55, (2003). [5] Y. Schol and G. A. E. andenboch, Separation of horizontal and vertical dependencie in a urface/volume integral equation approach to model quai 3-D tructure in multilayered media, IEEE Tran. Antenna Propag. 55, 1086 (2007). [6] G. A. E. andenboch,. olkiy, N. erellen, and.. Mohchalkov, On the ue of the method of moment in plamonic application, Radio Sci. 46, RS0E02 (2011). [7] X. Zheng, N. erellen, D. ercruye,. olkiy, P. an Dorpe, G. A. E. andenboch, and.. Mohchalkov, On the ue of group theory in undertanding the optical repone of a nanoantenna, IEEE Tran. Antenna Propag. 63, (2015). [8] M. Tinkham, Group theory and quantum mechanic, Dover publication (2003). [9] E. P. Wigner, Group theory and it application to the quantum mechanic of atomic pectra, Academic Pre (1959).