Electromagnetic Scattering by Networks of High-Permittivity Thin Wires
EElectromagnetic Scattering by Networks of High-Permittivity Thin Wires
Carlo Forestiere and Giovanni Miano
Department of Electrical Engineering and Information Technology,Universit`a degli Studi di Napoli Federico II, via Claudio 21, Napoli, 80125, Italy
Bruno Miranda
Department of Electrical Engineering and Information Technology,Universit`a degli Studi di Napoli Federico II, via Claudio 21, Napoli, 80125, Italy andInstitute of Applied Sciences and Intelligent Systems - Unit of Naples,National Research Council, via P. Castellino 111, Naples, 80131 Italy.
The electromagnetic scattering from interconnections of high-permittivity dielectric thin wires with sizessmaller than (or almost equal to) the operating wavelength is investigated. A simple lumped element modelfor the polarization current intensities induced in the wires is proposed. The circuit elements are capacitancesand inductances between the wires. An analytical expression for the induced polarization currents in terms ofthe magneto-quasistatic current modes is obtained. The connection between the spectral properties of the loopinductance matrix and the network’s resonances is established. The number of the allowed current modes andresonances is deduced from the topology of the circuit’s digraph. The coupling to radiation is also included, andthe radiative frequency shifts and the quality factors are derived. The introduced concept and methods may findapplications both at the microwaves and in nanophotonics.
I. INTRODUCTION
High-permittivity dielectric objects [1] are currently in-tensively studied both at the microwave and in the visible,promising diverse applications including microwave anten-nas [2], nanoscale biosensors [3], wireless mid-range en-ergy transfer [4], non-linear optics [5], and metamaterials [6].Resonances of small high-permittivity dielectric objects arisefrom the interplay between the polarization energy of the di-electric and the energy stored in the magnetic field. They canbe described within the magneto-quasistatic approximation ofthe Maxwell’s equation [7], extended by radiative corrections[8]. In the magneto-quasistatic approximation the induced po-larization density field is solenoidal in the object and its nor-mal component to the surface of the object is equal to zero[9]. These features stimulate the search of relevant geome-tries for which high-permittivity dielectric resonators behaveas lumped networks when their size is smaller or comparableto the operating wavelength. A possible candidate is an arbi-trary interconnection of high-permittivity dielectric thin wires,that we denote in the following as high-permittivity dielectricnetwork . A wire is thin if the linear dimension of its crosssection is much smaller than its length.In this paper, we propose a lumped element model to de-scribe the electromagnetic scattering from high-permittivitynetworks, when the size is smaller than (or at almost equal to)the operating wavelength. We develop the model in the frame-work of Maxwell’s equation, by expanding the polarizationcurrents in terms of magneto-quasistatic modes, and by tak-ing into account the radiation perturbatively [7, 8]. We carriedout the lumped element model in terms of capacitances, self-and mutual inductances between the wires. This simplifiedmodel leads to the analytical prediction of: the polarizationcurrents induced in the dielectric network by an external elec-tromagnetic field; the magneto-quasistatic current modes andresonances of the dielectric network; their radiative frequencyshifts and quality factors. Specifically, we found that the spec- tral properties of the circuit’s loop inductance matrix deter-mine the electromagnetic scattering resonances and modes ofthe dielectric network. Thus, instead of solving Maxwell’sequation, that would be unpractical for intricate connectionsof multiple branches, we import concepts and formulas thathave been produced by scientists and engineers working onelectric inductive network [10], in particular the analytic for-mulas produced in the first twenty years of the XX centuryand summarized in the manuscripts of Rosa [11], Grover [12],and Weber [13]. These concepts are now applied to the polar-ization current densities, rather than to the electric currents,consistently with the framework proposed by Engheta and co-workers in Refs. [14–17]. In particular, one of the mainstrengths of the proposed model lies in its simplicity: whenthe network consists of only a few loops, the calculation ofthe magneto-quasistatic current modes and the associated res-onances can be carried out with just paper and pencil, togetherwith their radiative frequency shifts and quality factors. It alsoenucleates the connection between the current modes, reso-nances, radiative corrections, and the topology of the under-lying graph. It may also help the comprehension of lasing incomplex photonic graphs and networks [18–21].The paper is organized as follows. After a brief summary ofthe properties of the magneto-quasistatic current modes, res-onances and radiative corrections, we introduce - through ex-amples of increasing complexity - the steps required to assem-bly a lumped-element model for complex high-permittivitydielectric networks. First, in Sec. III we consider an isolatedhigh-permittivity loop, and we link its self-inductance with theresonance frequency and the radiative quality factor. Then, inSec. IV, we extend the lumped element model to two interact-ing high-permittivity loops. Eventually, we consider in Sec.V an arbitrary interconnection of high permittivity thin wires. a r X i v : . [ phy s i c s . c l a ss - ph ] F e b II. ELECTROMAGNETIC SCATTERING FROM AHIGH-PERMITTIVITY DIELECTRIC OBJECT
In the linear regime, resonant electromagnetic scatteringfrom nonmagnetic and small objects, occurs according to twodifferent mechanisms (e.g., [7], [8]). If the real part of thepermittivity is negative (e.g., as in metals) resonances arisefrom the interplay between the electric field energy of theelectro-quasistatic current modes and the polarization energy.Instead, if the real part of the permittivity is positive andvery high (e.g., as in dielectrics) resonances arise from theinterplay between the magnetic field energy of the magneto-quasistatic current modes and the polarization energy. Thequasistatic current density modes are solutions of the source-free Maxwell equations in the quasistatic limits. In particular,the magneto-quasistatic current density modes are solenoidalin the object and their normal component to the boundary ofthe object is equal to zero. In the following, we briefly re-sume the principal features of the magneto-quasistatic currentmodes together with the radiative corrections (for details seein [7], [8]).Let us consider an isotropic and homogeneous dielectricsurrounded by vacuum and occupying a volume Ω ; ∂ Ω is theboundary of Ω with outward-pointing normal ˆ n . The object isilluminated by a time harmonic electromagnetic field incom-ing from infinity Re (cid:8) E ext ( r ) e i ω t (cid:9) , where ω is the angularfrequency. We denote with ε R the relative permittivity of theobject, with χ = ε R − ε the vac-uum permittivity.We now introduce the dimensionless size parameter of theobject x = ω / ω c where ω c = c / l c , l c is the radius of the mini-mum sphere circumscribing the dielectric object, and c is thelight velocity in vacuum; x is equal to the ratio between l c andthe operating wavelength in vacuum. A. Magneto-quasistatic approximation
In objects with high relative permittivity, ε R (cid:29)
1, and withsizes much smaller than the operating wavelength, i.e., x (cid:28)
1, the contribution of the magneto-quasistatic current densitymodes { j h } to the induced polarization current density field isgiven by [7]: J p ( ˜ r ) = i ω χε ∑ h − x χ / κ ⊥ h (cid:104) j h , E ext (cid:105) j h ( ˜ r ) (1)where j h ( ˜ r ) are solutions of the eigenvalue problem14 π ˆ ˜ Ω j h ( ˜ r (cid:48) ) | ˜ r − ˜ r (cid:48) | d ˜ V (cid:48) = κ ⊥ h j h ( ˜ r ) , ∀ ˜ r ∈ ˜ Ω (2)with j h ( ˜ r ) · ˆ n ( ˜ r ) = ∀ ˜ r ∈ ∂ ˜ Ω , (3)and κ ⊥ h is the eigenvalue associated to the eigenfunction j h ( ˜ r ) ; (cid:104) A , B (cid:105) = ´ ˜ Ω A · B dV is the standard inner product. In Equa- tions 1-3 the position vector r have been normalized by l c , i.e.,˜ r = r / l c , and ˜ Ω is the corresponding scaled domain and ∂ ˜ Ω its boundary.Apart from the factor µ , the integral operator on the lefthand side of equation 2 gives the static magnetic vector poten-tial in the Coulomb gauge as function of the current densityfield (here we call it the magneto-quasistatic integral opera-tor). Equation 2 holds in weak form in the functional spaceequipped with the inner product (cid:104) A , B (cid:105) , and constituted bythe vector fields that are solenoidal in ˜ Ω and have zero normalcomponent to ∂ ˜ Ω . The spectrum of the magneto-quasistaticintegral operator with the boundary conditions 3 is discrete (cid:8) κ ⊥ h (cid:9) h ∈ N , and the eigenvalues are real and positive. The cur-rent density modes { j h } h ∈ N are solenoidal in ˜ Ω , have zero nor-mal component to ∂ ˜ Ω , and are orthonormal according to thescalar product (cid:104) A , B (cid:105) . The modes are ordered in such a way κ ⊥ < κ ⊥ < κ ⊥ < ... . The expression 1 has been obtained bysolving the full wave electromagnetic scattering problem inthe quasistatic limit (for detail see in [7]). Here we have disre-garded the contribution of the electro-quasistatic current den-sity modes because they give nonresonant terms in the elec-tromagnetic scattering from a high-index dielectric.The resonance frequency ω ⊥ h of the h -th mode j h is obtainedby maximizing its amplitude in the expression 1. It is solutionof the equation (cid:32) ω ⊥ h ω c (cid:33) = κ ⊥ h Re (cid:8) χ ( ω ⊥ h ) (cid:9) . (4) B. Radiative corrections
When the dimensionless size parameter of the object be-comes almost equal to one, x (cid:39)
1, the set of modes { j h } still approximates well the modes of the objects, neverthelessthe magneto-quasistatic approximation cannot provide theirradiative shift and quality factors. The results obtained bymagneto-quasistatic approximation have to be supplementedby radiative corrections [8]. Indeed, expression 1 is still avalid approximation for the induced polarization current den-sity when x (cid:39) κ ⊥ h is replaced with κ h = κ ⊥ h + κ ( ) h x + κ ( ) h x + . . . (5)where κ ( n ) h is the n -th coefficient in the expansion of κ h ina power series of x (for details see in [8]). For any objectshape it results κ ( ) h =
0. Taking into account the second order(real) correction κ ( ) h , we obtain that the resonance frequencyis solution of the equation (cid:18) ω h ω c (cid:19) = κ ⊥ h Re { χ ( ω h ) } − κ ( ) h . (6)The coefficient κ ( ) h is negative, therefore the second order ra-diative correction introduces a negative frequency shift withrespect to the quasistatic resonance given by 4. The first non-vanishing imaginary correction of order n i ≥
3, has a positiveimaginary part that we denote with κ ( n i ) h . It determines theradiative broadening of the mode, related to the inverse of thequality factor, that is given by [8] Q h = κ ⊥ h κ ( n i ) h (cid:18) x h (cid:19) n i , (7)where x h is the size parameter evaluated at the resonance fre-quency given by 6. The order of the first non-vanishing imag-inary correction n i returns the multipolar scattering order [8].For instance, the modes with n i = n i = E2 moment [22] (also called toroidaldipole) different from zero, etc.. The general expressions forthe radiation corrections of magneto-quasistatic current den-sity modes are given in Ref. [8]. III. ISOLATED HIGH-PERMITTIVITY THIN WIRE LOOP
We now consider a high-permittivity thin wire dielectricloop with uniform cross section Σ (the transverse linear di-mensions are much smaller compared to the length). The wireaxis is represented by the closed curve Γ with tangent unitvector ˆ t . With abuse of notation we also indicate with Σ thecross-sectional area, with Γ the loop length, and with Ω theloop volume, which is given by Ω = Γ × Σ . A. Fundamental magneto-quasistatic current mode
In a loop with small cross section, the lowest eigenvalue κ ⊥ is well separated from the remaining eigenvalues, as shown in[7] for a torus of finite cross section. This separation increasesas the wire cross section reduces. This fact suggests that for x smaller than (or almost comparable to) one in a thin wiredielectric loop only the mode j ( ˜ r ) associated to the small-est eigenvalue κ ⊥ (fundamental magneto-quasistatic mode) isexcited. Therefore, in this case the expression of the inducedpolarization current density 1 reduces to J p ( ˜ r ) = i ω χε − x χ / κ ⊥ (cid:104) j , E ext (cid:105) j ( ˜ r ) . (8)The fundamental mode j ( ˜ r ) is directed along the wire axis ˆ t ,and its module is uniform in ˜ Ω (as shown in [7] for a torus offinite cross section), j ( ˜ r ) = (cid:26) ˆ t ( ˜ r ) / √ ˜ Ω in ˜ Ω , otherwise. (9)The mode is normalized in such a way (cid:104) j , j (cid:105) =
1. The corre-sponding eigenvalue κ ⊥ is given by Eq. 2, i.e.˜ Ω κ ⊥ = π ˆ ˜ Ω ˆ ˜ Ω ˆ t ( ˜ r ) · ˆ t ( ˜ r (cid:48) ) | ˜ r − ˜ r (cid:48) | d ˜ V (cid:48) d ˜ V , (10) where ˜ Ω on the left hand side denotes the scaled volume of theobject ( ˜ Ω = Γ × Σ / l c ). Apart from the factor ( l c / Σ ) µ , theintegral of the right hand side is equal to the self-inductanceof the loop L , therefore κ ⊥ = l c ΓΣ l c µ L . (11) B. Lumped element model for a single loop
The expression 8 has a very simple physical explanation.The polarization current density field induced in the loop isgiven by J p = i ωε χ ( E + E ext ) (12)where E is the induced electric field. By performing the lineintegral along Γ on both sides of equation 12 we obtain1 χ i ω C I = E ext + E (13)where I is the intensity of the induced polarization current inthe wire, C = ε ΣΓ , (14) E is the induced voltage along the loop E = ˛ Γ E · ˆ t dl , (15)and E ext is the applied voltage E ext = ˛ Γ E ext · ˆ t dl . (16)On the other hand, from the Neumann-Faraday law we obtain: E = − i ω LI (17)where L is the self-inductance of the loop. Using Eq. 17 inEq. 13 we have: (cid:18) i ω L + χ i ω C (cid:19) I = E ext , (18)which is the equation governing the equivalent circuit in Fig.1. Since 1 ω c κ ⊥ = LC , (19)the solution of equation 20 is expressed as I = i ω C χ − ( ω / ω c ) χ / κ ⊥ E ext . (20)This expression coincides with the expression of the polariza-tion current intensity obtained from 8.The quasi-static resonance frequency, solution of equation4, is the value of ω for which (cid:12)(cid:12)(cid:12) − ( ω / ω c ) χ / κ ⊥ (cid:12)(cid:12)(cid:12) is mini-mum. C. Radiative corrections
In this case the radiative corrections κ ( ) h and κ ( ) h have thefollowing expressions [8]: κ ( ) = (cid:0) κ ⊥ (cid:1) π ˆ ˜ Ω ˆ ˜ Ω (cid:12)(cid:12) ˜ r − ˜ r (cid:48) (cid:12)(cid:12) j ( ˜ r ) · j (cid:0) ˜ r (cid:48) (cid:1) d ˜ V (cid:48) d ˜ V , (21) κ ( ) = − i (cid:0) κ ⊥ (cid:1) π ˆ ˜ Ω ˆ ˜ Ω (cid:12)(cid:12) ˜ r − ˜ r (cid:48) (cid:12)(cid:12) j ( ˜ r ) · j (cid:0) ˜ r (cid:48) (cid:1) d ˜ V (cid:48) d ˜ V . (22)By using 9, expression 21 becomes κ ( ) = − π ΣΓ (cid:16) κ ⊥ (cid:17) ∆ (23)where ∆ = − π ˛ ˜ Γ ˛ ˜ Γ ˆ t ( ˜ r ) · ˆ t (cid:0) ˜ r (cid:48) (cid:1) (cid:12)(cid:12) ˜ r − ˜ r (cid:48) (cid:12)(cid:12) d ˜ l (cid:48) d ˜ l >
0; (24)we have chosen l c = Γ / π . By using 9 expression 22 be-comes: κ ( ) = i (cid:16) κ ⊥ (cid:17) π (cid:107) P M (cid:107) , (25)where P M is the magnetic dipole moment of the mode in thescaled object, P M = ˆ ˜ Ω ˜ r × j d ˜ V . (26)The radiation quality factor is given by Q = πκ ⊥ (cid:107) P M (cid:107) x r , (27)where x r = ω r / ω c and ω r is the resonance frequency; it is so-lution of equation 6 and takes into account the radiative shift.The expression of the induced polarization current in theloop with the radiative corrections is obtained by substituting κ ⊥ with κ = κ ⊥ + κ ( ) x + κ ( ) x into expression 20. D. Circular loop
To illustrate the use of the formulas derived above, wenow consider a circular loop (ring) with circular cross section(torus), minor radius r w and major radius a . Hollow nanodiskshave been recently used to tailor the magnetic dipole emission[23]. The circular loop was the platform where dielectric res-onances have been first proposed [1], and its modes were stud- χ CI L − + E ext FIG. 1. Lumped element circuit for a high-permittivity loop a r w .
52 (a) ε R x r = ω r / ω c the modelComsol ε R Q the modelComsol1 FIG. 2. Comparison between the results obtained by the proposedmodel and by Comsol for a circular loop: (a) Resonance frequencynormalized to ω c , x r = ω r / ω c , and (b) radiative quality factor of ahigh-permittivity loop with major radius a and minor radius r w = . a as a function of the permittivity ε R . ied in Ref. [7, 24]. A circuit model of the dielectric circularloop has been proposed in Ref. [25], and bulk metamaterialsmade of rings have been also investigated [26].For a circular loop the expression of κ ⊥ is κ ⊥ = a r w a µ L , (28)where L is the inductance of the loop [10], L = µ (cid:112) a ( a − r w ) (cid:20)(cid:18) k − k (cid:19) K ( k ) − k E ( k ) (cid:21) + µ a , (29) k = (cid:115) a ( a − r w )( a − r w ) , (30) K ( k ) and E ( k ) are the complete elliptic integrals of the firstand second kind, where the second term in Eq. 29 representsthe internal inductance of the ring [10]. The expression of thesecond order correction is κ ( ) = −
23 2 a r w (cid:16) a µ L (cid:17) . (31)Therefore for a non dispersive material the expression of the χC I L − + E ext, I χC − + E ext, L M FIG. 3. Lumped element circuit for two mutually coupled high-permittivity loops. resonant frequency is: ω r = c a (cid:32)(cid:115) r w a (cid:18) La µ (cid:19) Re { χ } + (cid:16) a µ L (cid:17)(cid:33) − . (32)The expression of the radiative quality factor is given by: Q = π (cid:18) La µ (cid:19) x r . (33)We have validated formulas 32 and 33 against Comsol mul-tiphysics (wave optics package) considering a high permittiv-ity solid torus with r w = . a . Specifically, in Fig. 2, wecompare the resonance position obtained by Eq. 32 againstthe position of the resonance peak of the scattering spectrumcomputed in Comsol, when the torus is excited by an electricpoint dipole, located in the equatorial plane of the torus, at adistance 3 a from the center, and oriented along the toroidaldirection. We found very good agreement for very high per-mittivities, while the error slightly deteriorates for smaller val-ues of permittivity where the dimension of the object becomescomparable to the resonance wavelength. In Fig. 2 (b) wecompare the quality factor obtained by Eq. 33 against the in-verse of the full-width at half maximum of the resonance peakobtained in Comsol. Good agreement is found. IV. TWO INTERACTING HIGH-PERMITTIVITY LOOPS
In this section, we consider a pair of high-permittivity thinwire loops occupying two disjoint spatial domains Ω and Ω with cross sections Σ and Σ . The wire axes are representedby the closed curve Γ and Γ with tangent unit vector ˆ t andˆ t , respectively. The polarization current density in each loopis uniformly distributed across the wire cross-section and di-rected along its axis; I and I are, respectively, the polariza-tion current intensities of the two loops. A. Lumped element model for a loop pair
By following the procedure introduced in section III B, weobtain the following system of equations for the intensities of the induced polarization currents (the lumped element circuitis shown in Fig. 3) i ω L I + i ω MI + i χω C I = E ext , , i ω MI + i ω L I + i χω C I = E ext , ,. (34)where C q = ε Σ q Γ q q ∈ , , (35) L and L are the self-inductions of the loops, and M is themutual inductance between the two loops, M = µ π ˛ Γ ˛ Γ ˆ t · ˆ t | r − r (cid:48) | dl dl . (36)The modes of the loop pair are solutions of the generalizedeigenvalue problem L u = κ ⊥ ω c C − u (37)where L = (cid:18) L MM L (cid:19) (38)and C = (cid:18) C C (cid:19) . (39)The generalized eigenvalue problem has two eigenvalues, κ ⊥± ,and two current modes, u ± . These modes exhibit equidirectedand counter-directed currents, which are called Helmholtz and anti-Helmholtz modes, respectively. They are orthogonal ac-cording to the weighted scalar product u (cid:124) ± C − u ∓ = ε l c u (cid:124) ± C − u ± = . (40)Since the magnetic energy of each current mode is strictly def-inite positive, the matrix L is strictly definite positive, thus L L > M . The solution of the problem Eq. 34 in terms ofthe current modes isI = i ε ω χ l c ∑ h = ± u (cid:124) h E − x χ / κ ⊥ h u h , (41)where E ext = ( E , ext , E , ext ) (cid:124) . (a) (b) a Σ Σ bd ˆ t ˆ t . . . . . . d/a x r = ω r / ω c Resonances ε R = 100the model A-HComsol A-Hthe model HComsol H1 FIG. 4. (a) Two loops with major radii a and b , distance d , sections Σ and Σ . (b) magnetic field lines of Helmholtz (H) and anti-Helmholtz(A-H) current modes for a = b and d = a /
2. Comparison between the results obtained by the proposed model and by Comsol: resonancefrequency (c) normalized to ω c , x r = ω r / ω c . B. Radiative corrections
The second order corrections to κ ⊥± are given by κ ( ) ± = (cid:0) κ ⊥± (cid:1) π u (cid:124) ± ∆ ( ) u ± , (42)and the first non-vanishing imaginary corrections (of odd or-der n i ≥
3) are given by κ ( n i ) ± = i ( − ) ( n i − ) / (cid:0) κ ⊥± (cid:1) π n i ! u (cid:124) ± ∆ ( n i ) u ± , (43)where the elements of the matrix ∆ ( n ) are (cid:16) ∆ ( n i ) (cid:17) pq = ˛ ˜ Γ p ˛ ˜ Γ q ˆ t p ( ˜ r ) · ˆ t q (cid:0) ˜ r (cid:48) (cid:1) (cid:12)(cid:12) ˜ r − ˜ r (cid:48) (cid:12)(cid:12) n i − d ˜ l (cid:48) d ˜ l . (44)For loops with different shape and/or sizes, the first non-vanishing imaginary corrections for both the modes is 3. Ifthe two loops are equal the first non-vanishing imaginary cor-rections for the anti-Helmholtz mode is 5.The expression of the induced polarization currents in theloop pair with the radiative corrections is obtained by substi-tuting κ ⊥± with κ ± = κ ⊥± + κ ( ) ± x + i κ ( n i ) ± x ni into expression41. C. Two coaxial circular loops
We now apply these results to two coaxial circular loopswith the same permittivity ε R , circular cross sections, majorradii a and b , equal minor radii r w = . a , and axial distance d , as sketched in Figure 4 (a). The self-inductances L and L are given by 29 and the mutual inductance is given by [12, 27] M = µ √ ab (cid:20)(cid:18) k (cid:48) (cid:19) K ( k (cid:48) ) − k (cid:48) E ( k (cid:48) ) (cid:21) (45) where K ( k ) and E ( k ) are the complete elliptic integrals of thefirst and second kind, and k (cid:48) = (cid:115) ab ( a + b ) + d . (46)Figure 4 (b) shows the field lines of the magnetic field gen-erated by the Helmholtz u − and anti-Helmholtz u + currentmodes for a = b . Figure 4 (c) shows the resonance frequenciesof these modes normalized to ω c , x r = ω r / ω c , as a function ofthe distance between the loops d , for b = . a and ε R = a , at a distance 3 a from its center, oriented alongits toroidal direction. The agreement is very good.We now validate the expression of the quality factor 7against the full-width at half maximum obtained by Comsol.Table I shows the quality factor of the Helmholtz and anti-Helmholtz modes as function of the loop distance. We com-pare them with the inverse of the full-width at half maximumof the peak, and we found qualitative agreement. For the anti-Helmholtz mode, since for some values of the ratio d / a theoverall magnetic dipole moment of the system is vanishing(or nearly vanishing), both the imaginary corrections associ-ated to n i = n i = V. HIGH-PERMITTIVITY DIELECTRIC NETWORK
In this Section, we generalize the concepts introduced inthe previous two Sections by considering an arbitrary inter-connection of high-permittivity thin wires, e.g. Fig. 5. Thelumped element model we propose allows to study effectivelythe main properties of the electromagnetic scattering fromsuch structures: in particular, the induced polarization cur-rents expressed in terms of the magneto-quasistatic currentmodes, their resonance frequencies, their radiative frequencyHelmholtz mode d / a . . . . . . . . . d / a . . TABLE I. Comparison between the radiative quality obtained by theproposed model and by Comsol for b = . a , r w = . a , ε R =
100 asa function of the distance d / a .FIG. 5. Top view of an example of a network of dielectric wires. shifts and their radiative quality factors.In the magneto-quasistatic limit the normal component ofthe polarization current density vanishes at the object’s sur-face, hence there is not leakage of polarization current den-sity. Therefore, at each node of the network the sum of thepolarization current intensity is conserved, an analogous tothe Kirchhoff’s law for electric currents. A. Network digraph
As for electric circuits [28], it is convenient to associate tothe dielectric network a digraph G , i.e. an oriented graph, with n nodes and b branches. We denote with e h the h -th branch ofthe digraph, ˆ t h its tangent unit vector, and I h the intensity ofthe branch polarization current. The digraph associated to thenetwork of Fig. 5 is shown on the top layer of Fig. 6. Forsimplicity, we restrict our analysis to connected graphs. If thegraph is not connected, one may simply combine the follow-ing approach with the one carried out in the previous section.A loop L i of G is a connected sub-graph where exactly twobranches are incident in each node. A tree T of a connecteddigraph G is a connected subgraph that contains all the nodesof G , but no loop. For any given digraph G , many possiblechoices of trees are possible. Given a connected digraph and G r a ph G L oop L (cid:9) J L oop L (cid:9) J L oop L (cid:9) J L oop L (cid:9) J FIG. 6. Graph G associated to the high-permittivity network, wherethe chosen tree T is highlighted in red. a chosen tree, the branches of T are partitioned in two dis-joint sets: the ones belonging to T , called twigs , and the onesthat do not belong to T that are called links . The fundamentaltheorem of graphs states that, given a connected graph with n nodes and b branches and a tree T , there are n − (cid:96) = b − ( n − ) links. Every link (e.g. the p -th link) togetherwith a proper choice of twigs constitutes a unique loop, calledthe fundamental loop associated to the link. For instance, con-sidering the graph shown in the top layer of Fig. 6 we canassociate to every link of the particular tree T highlighted inred the four fundamental loops L , . . . , L , shown in the lay-ers below.A set of fundamental loops are identified through the l × b fundamental loop matrix B associated with the correspond-ing tree. The jk occurrence is defined as follows: b jk = k is in the loop j and their reference directions are thesame; b jk = − k is in the loop j and their referencedirections are opposite; b jk = k is not in the loop j . B. General lumped element model
We now use the loop analysis to formulate the electro-magnetic scattering problem from the high-permittivity net-work: the b polarization current intensities of the networkare expressed in terms of the l fundamental loop currents.This choice guaranties that the polarization currents satisfythe Kirchhoff’s law at any node of the circuit.Let be I the (column) vector representing the polarizationcurrent intensities of the branches of the circuit { I , I , ..., I b } ,and J the column vector representing the current intensi-ties of the links associated with the tree T of the circuit, { J , J , ..., J l } . The conservation of the sum of the polariza-tion currents at the nodes of the circuit implies thatI = B (cid:124) J . (47)Let be E the column vector representing the set ofinduced loop voltages { E , E , ..., E l } and E ext the col-umn vector representing the set of loop external voltages (cid:8) E ext , , E ext , , ..., E ext , l (cid:9) . The constitutive relations of the di-electric thin wires give1 χ B (cid:0) i ω C (cid:1) − I = E + E ext , (48)where C is the diagonal matrix whose elements are C h =( ε Σ h ) / Γ h with h = , , ..., b , Σ h is the cross-section of the h -th wire and Γ h is the length. On the other hand the Faraday-Neumann law gives E = − i ω L J (49)where L is the l × l inductance matrix of the set of fundamentalloops. By combining equations 47-49 we obtain the system ofequations governing the set of the link currents associated tothe tree T , i ω L J + i ω χ B C − B (cid:124) J = E ext . (50)The current modes { u h } of the network and the correspondingeigenvalues (cid:8) κ ⊥ h (cid:9) are solution of the generalized eigenvalueproblem L u = κ ⊥ ω c B C − B (cid:124) u . (51)The number of magneto-quasistatic current modes and of res-onances of a high-permittivity dielectric network is equal tothe number l of links of the digraph G of the network. Thematrices L and B C − B (cid:124) are symmetric and definite positive.As a consequence, the eigenvalues (cid:8) κ ⊥ h (cid:9) are real and positive,and the current modes satisfy a weighted orthogonality, ε l c u (cid:124) h (cid:0) B C − B (cid:124) (cid:1) u k = δ hk . (52)The solution of Eq. 50 isJ = i ε ω χ l c (cid:96) ∑ h = u (cid:124) h E ext − x χ / κ ⊥ h u h . (53) L P χ C I χ C L P I χ C L P I χ C L P I L P χ C I χ C L P I χ C L P I L P I χ C χ C L P I L P L P L P L P L P L P L P L P FIG. 7. Lumped element circuit of capacitances and partial induc-tances. The self partial inductances L P qq of each branch of the di-graph are shown. The mutual partial inductances between the firstbranch and any other branch are shown in red. The dotted conven-tion commonly employed in magnetically coupled circuits is used inthis case. C. Radiative corrections
The expression of the second order (real) radiative correc-tion for the h − th mode is κ ( ) h = (cid:0) κ ⊥ h (cid:1) π u (cid:124) h ∆ ( ) u h , (54)and the expression of the lowest order n i imaginary correction(which is an odd number n i ≥
3) is κ ( n i ) h = i ( − ) ( n i − ) / (cid:0) κ ⊥ h (cid:1) π ( n i ) ! u (cid:124) h B ∆ ( n i ) B (cid:124) u h (55)where the elements of the matrix ∆ ( n ) are (cid:16) ∆ ( n ) (cid:17) i j = ˛ e i ˆ t i ( ˜ r ) · ˛ e j ˆ t j (cid:0) ˜ r (cid:48) (cid:1) (cid:12)(cid:12) ˜ r − ˜ r (cid:48) (cid:12)(cid:12) n − d ˜ l (cid:48) d ˜ l . (56)The expression of the induced polarization currents in thelinks of the dielectric network taking into account the radiativecorrections is obtained by substituting κ ⊥ h with κ h = κ ⊥ h + κ ( ) h x + i κ ( n i ) h x ni for h = , (cid:96) into expression 53. D. Partial inductances
The direct calculation of self- and mutual- inductance offundamental loops may not be the most efficient method to as-sembly the matrix L, since different loops may share severalbranches, resulting in redundant hence inefficient computa-tions. It is instead convenient, as for electric circuits [10, 12],to preliminary assemble the partial loop inductances matrixL P . Its i j - occurrence is the partial inductance L P i j betweenthe branches e i and e j . It is defined as the ratio between themagnetic flux produced by the density current flowing in thebranch e i , through the surface between the second branch e j and infinity, and the current of the branch e i , namely: L P i j = µ π ˛ e i ˛ e j ˆ t i ( r ) · ˆ t j ( r (cid:48) ) | r − r (cid:48) | d ˜ l (cid:48) d ˜ l . (57)There are b ( b + ) / L P i j , be-cause L P i j = L P ji by reciprocity. The loop inductance matrix Lis given by L = B L P B (cid:124) (58)If the circuit is composed by an arbitrary interconnection of straight wires laying on the same plane, then we need formu-las for calculating: i) self-partial inductance of a straight wire;ii) the mutual partial inductance between wires at an angle toeach other which includes as a limit case the mutual partialinductance between parallel wires, and the mutual partial in-ductance of wires meeting in a point (given by G.A. Campbell[29]). If the circuit is not planar, then also the mutual partialinductance between skewed and displacement wires first de-rived by F. F. Martens and G. A. Campbell [10, 12, 29, 30]are needed. All these formulas have been analytically deriveda century ago, and are reported in the Appendix. In Fig. 7we show the lumped circuit for the dielectric network shownin Fig. 5, where we illustrate the self partial inductances L P qq ,and for sake of clarity the mutual partial inductance L P q . E. Sierpinski triangle
Let us analyze the dielectric network shown in Fig. 6, con-stituted by b = l w , with circular cross section with radius r w = . l w . Thewires are interconnected accordingly to a Sierpinski triangle.The minimum circle circumscribing the network is chosen ascharacteristic length l c .The graph of this network is shown in the top layer of Fig.6, where the twigs and links, associated to a chosen tree, arehighlighted in red and black, respectively. The fundamentalloops associated to each link are shown in the layers below.The lumped element circuit is shown in Fig. 7. The 45 inde-pendent partial inductances are firstly evaluated, then the 10independent elements of the 4 × κ ( ) h can be carried out with pen and paper, return-ing the four values listed in Tab. II. The four current modesare shown in Fig. 8 (the second and the third mode are degen-erate). These modes correspond to the magneto-quasistaticcurrent density modes reported in 8. Then, the matrices ∆ ( ) and ∆ ( n i ) are assembled, where n i = n i = FIG. 8. Current modes of the dielectric network arranged accord-ingly to a Sierpinsky triangle with generation number 1. Each wirehas length l w and radius r w = . l w . The modes are arranged witha lexicographic order which follows the corresponding eigenvalue.Above each mode is the order n i of the first non-vanishing imaginarycorrection. κ ⊥ h
134 181 181 379 κ ( ) h -20.1 -6.7 -6.7 -26.0 n i κ ( n i ) h TABLE II. Eigenvalues κ ⊥ h , second order (real) radiative correction κ ( ) h , imaginary correction κ ( n i ) h of the lowest order n i . corrections κ ( ) h and κ ( n i ) h are given in Tab.II.Table III gives the normalized resonance frequencies andthe quality factors of the four current modes of the Sier-pinsky network for two different values of the permittivity, ε R =
100 and ε R = .
45. For ε R =
100 the first (fundamen-tal) resonance is located at x = .
04: for l c = cm it is ε r = x r , x r , x r , x r , Q Q = Q Q the model 1 .
06 1 .
30 1 .
30 1 .
74 11.7 244 6.5Comsol 1 .
04 1 .
26 1 . −
31 280 − ε r = . x r , x r , x r , x r , Q Q = Q Q the model 1 .
94 2 .
86 2 .
86 3 .
02 1.9 23 1.2Comsol 2 .
29 2 .
89 2 . − . . − TABLE III. Normalized resonance frequencies x h = ω h / ω c and qual-ity factors of the four modes of the dielectric network of Fig. 5 for ε R =
100 and ε R = . h κ ⊥ h Square GridSierpinski1
FIG. 9. First 21 eigenvalues κ ⊥ h of a Sierpinski triangle (generationnumber 3) with b = , n = ,(cid:96) =
37, and first 21 eigenvalues κ ⊥ h of a 7 × b = , n = ,(cid:96) = ω = . GHz . The scattering peak positions and the qual-ity factors of the four current modes have been also estimatedby using Comsol Multiphysics. The dielectric network shownin Fig. 5 has been excited by an electric dipole, laying on itsequatorial plane, located at 3 l w on the left of its center, and ori-ented along the vertical in-plane direction. The quality factoris estimated as the inverse of the full-width at half maximum.The lumped circuit model exhibits a good accuracy in locat-ing the resonances, while returning the order of magnitude ofthe quality factors. The comparison is repeated for ε = . x = .
94, thusfor l c = λ = b =
81 thin wires interconnected accordinglyto a Sierpinski triangle with generation number 3; the sec-ond one is instead composed of b =
112 thin wires, intercon-nected accordingly to a square grid. The first (smallest) 21eigenvalues of both the networks are compared in Fig. 9. Thecorresponding current modes are shown in Figs. 10 and 11.The number reported above each mode is the order n i of thefirst non-vanishing imaginary radiative correction. This num-ber returns the power dependence of the quality factor on thesize-parameter, accordingly to Eq. 7, which is related to themultipolar components of the mode. VI. CONCLUSIONS
We investigated the electromagnetic scattering from high-permittivity dielectric networks . They are interconnections ofhigh-permittivity dielectric thin wires. If the overall size ofthe network is smaller than (or at most equal to) the operatingwavelength the dielectric network can be modeled as a lumpedcircuit constituted by capacitances, self- and mutual- loop in-ductances. The resonant modes are equal in number to thelinks of the network’s digraph and are related to the spectrumof the loop-inductance matrix. Closed form expressions aregiven for the frequency shifts and quality factors due to theradiation. The inductance matrix can be assembled from the partial self- and mutual- inductance of the constituent wires,transplanting to the electromagnetic scattering theory formu-las introduced a century ago for inductive network. For net-works with sizes smaller than the incident wavelength, theerror in locating the resonance is acceptable, below 15% inthe investigated numerical experiments. This fact promotesthe proposed model as a fast computational tool for prelim-inary analysis, that could be later refined by more accuratetools that are usually associated with much higher computa-tional burden. This manuscript may also stimulate the graft-ing of several other ideas and methods from the electric andelectronic circuit onto design of high-index resonators in boththe microwaves and visible spectral range. This approach to-gether with the one proposed in Ref. [31] represent the firststeps toward the derivation of a full-wave treatment of com-plex networks of wires.
Appendix A: Partial Inductances Calculation1. Self partial inductance of a straight wire
The self partial inductance L p of the wire of Fig. 12 withradius r w and length a is L P = a µ π ln ar w + (cid:115)(cid:18) ar w (cid:19) + − (cid:114) + (cid:16) r w a (cid:17) + r w a (cid:35) + µ π a . (A1)
2. Mutual partial inductance between two unequal parallelwires that are offset
The mutual partial inductance M p between the two wiresshown in Fig. 13 of negligible cross section, length a and b .1 FIG. 10. First 21 MQS modes of a high permittivity circuit made of b =
81 thin wires interconnected accordingly to a Sierpinski triangle. Eachwire has length a and radius r w = . a . The modes are lexicographically ordered in terms of increasing MQS eigenvalues. Above each modeis the order n i of the first non-vanishing imaginary correction. FIG. 11. First 21 MQS modes of a high permittivity network made of b =
112 thin wires arranged accordingly to a square grid. Each wirehas length a and radius r w = . a . The modes are arranged with a lexicographic order which follows the increasing MQS eigenvalues. Aboveeach mode is the order n i of the first non-vanishing imaginary correction. r w a FIG. 12. Self partial inductance of a wire of length a and circularcross section of radius r w . at a distance d , with an offset δ , is given by [12, 32]: M P = µ π (cid:104) z sinh − z d − z sinh − z d − ( z − a ) sinh − z − ad + ( z − a ) sinh − z − ad − (cid:113) z + d + (cid:113) z + d + (cid:113) ( z − a ) + d − (cid:113) ( z − a ) + d (cid:21) (A2)where z = a + b + δ and z = a + δ .2 b a b δ d FIG. 13. Mutual partial inductance of two wires of lengths a and b ,equal circular cross section of radius r w , at distance d and offset z . b P bb b b P P P P α β a b r r r r FIG. 14. Mutual partial inductance of two wires of lengths a and b ,at an angle to each other.
3. Mutual partial inductance between wires at an angle to eachother
Let us consider the two wires of Fig. 14, of length a and b ,of negligible cross section (filaments). The wires are coplanar,forming an angle θ to each other [10, 12, 29]. Their mutualinductance has the following expression: M P = µ π (cid:20) ( β + b ) ln r + r + ar + r − a − β ln r + r + ar + r − a + ( a + α ) ln r + r + br + r − b − α ln r + r + br + r − b (cid:21) (A3)if the two wires are touching in P then the above equationreduces to [29]: M P = µ π cos θ (cid:18) a ln r + a + br + a − b + b ln r + a + br + b − a (cid:19) . (A4) [1] R. Richtmyer, “Dielectric resonators,” Journal of AppliedPhysics , vol. 10, no. 6, pp. 391–398, 1939. Publisher: AIP. [2] S. Long, M. McAllister, and Liang Shen, “The resonant cylin-drical dielectric cavity antenna,”
IEEE Transactions on Anten- nas and Propagation , vol. 31, no. 3, pp. 406–412, 1983.[3] O. Yavas, M. Svedendahl, P. Dobosz, V. Sanz, and R. Quidant,“On-a-chip biosensing based on all-dielectric nanoresonators,” Nano letters , vol. 17, no. 7, pp. 4421–4426, 2017. Publisher:ACS Publications.[4] A. Karalis, J. D. Joannopoulos, and M. Soljaˇci´c, “Efficientwireless non-radiative mid-range energy transfer,”
Annals ofPhysics , vol. 323, pp. 34–48, Jan. 2008.[5] K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bog-danov, H.-G. Park, and Y. Kivshar, “Subwavelength dielec-tric resonators for nonlinear nanophotonics,”
Science , vol. 367,pp. 288–292, Jan. 2020. Publisher: American Association forthe Advancement of Science Section: Report.[6] C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “Adouble negative (DNG) composite medium composed of mag-netodielectric spherical particles embedded in a matrix,”
IEEETransactions on Antennas and Propagation , vol. 51, pp. 2596–2603, Oct. 2003. Conference Name: IEEE Transactions on An-tennas and Propagation.[7] C. Forestiere, G. Miano, G. Rubinacci, M. Pascale, A. Tam-burrino, R. Tricarico, and S. Ventre, “Magnetoquasistatic reso-nances of small dielectric objects,”
Phys. Rev. Research , vol. 2,p. 013158, Feb. 2020. Publisher: American Physical Society.[8] C. Forestiere, G. Miano, and G. Rubinacci, “Resonance fre-quency and radiative Q-factor of plasmonic and dieletric modesof small objects,”
Phys. Rev. Research , vol. 2, p. 043176, Nov.2020. Publisher: American Physical Society.[9] J. V. Bladel, “On the Resonances of a Dielectric Resonator ofVery High Permittivity,”
IEEE Transactions on Microwave The-ory and Techniques , vol. 23, pp. 199–208, Feb. 1975. Con-ference Name: IEEE Transactions on Microwave Theory andTechniques.[10] C. R. Paul,
Inductance: loop and partial . John Wiley & Sons,2011.[11] E. B. Rosa,
The self and mutual inductances of linear conduc-tors . No. 80, US Department of Commerce and Labor, Bureauof Standards, 1908.[12] F. W. Grover,
Inductance calculations: working formulas andtables.
Courier Corporation.[13] E. Weber, “Electromagnetic Fields : Theory and Applications,”1950. Publisher: Wiley.[14] N. Engheta, A. Salandrino, and A. Alu, “Circuit elements at op-tical frequencies: nanoinductors, nanocapacitors, and nanore-sistors,”
Physical Review Letters , vol. 95, no. 9, p. 095504,2005. Publisher: APS.[15] A. Al`u and N. Engheta, “Optical ‘Shorting Wires’,”
Opt. Ex-press , vol. 15, pp. 13773–13782, Oct. 2007. Publisher: OSA.[16] A. Al`u, A. Salandrino, and N. Engheta, “Parallel, series, andintermediate interconnections of optical nanocircuit elements.2. Nanocircuit and physical interpretation,”
J. Opt. Soc. Am. B ,vol. 24, pp. 3014–3022, Dec. 2007. Publisher: OSA.[17] A. Salandrino, A. Al`u, and N. Engheta, “Parallel, series, andintermediate interconnections of optical nanocircuit elements.1. Analytical solution,”
J. Opt. Soc. Am. B , vol. 24, pp. 3007–3013, Dec. 2007. Publisher: OSA. [18] S. Lepri, C. Trono, and G. Giacomelli, “Complex Active Op-tical Networks as a New Laser Concept,”
Phys. Rev. Lett. ,vol. 118, p. 123901, Mar. 2017. Publisher: American Physi-cal Society.[19] M. Gaio, D. Saxena, J. Bertolotti, D. Pisignano, A. Camposeo,and R. Sapienza, “A nanophotonic laser on a graph,”
Naturecommunications , vol. 10, no. 1, pp. 1–7, 2019. Publisher: Na-ture Publishing Group.[20] A. Lubatsch and R. Frank, “A Self-Consistent Quantum FieldTheory for Random Lasing,”
Applied Sciences , vol. 9, no. 12,p. 2477, 2019. Publisher: Multidisciplinary Digital PublishingInstitute.[21] L. M. Massaro, S. Gentilini, A. Portone, A. Camposeo,D. Pisignano, C. Conti, and N. Ghofraniha, “HeterogeneousRandom Laser with Switching Activity Visualized by ReplicaSymmetry Breaking Maps,”
ACS Photonics , Jan. 2021. Pub-lisher: American Chemical Society.[22] J. v. Bladel, “Hierarchy of terms in a multipole expansion,”
Electronics Letters , vol. 24, pp. 492–493, Apr. 1988. Publisher:IET Digital Library.[23] T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollownanodisk for tailoring magnetic dipole emission,”
Opt. Lett. ,vol. 41, pp. 5011–5014, Nov. 2016. Publisher: OSA.[24] M. Verplanken and J. V. Bladel, “The Magnetic-Dipole Reso-nances of Ring Resonators of Very High Permittivity,”
IEEETransactions on Microwave Theory and Techniques , vol. 27,pp. 328–333, Apr. 1979. Conference Name: IEEE Transactionson Microwave Theory and Techniques.[25] L. Jelinek and R. Marqu´es, “Artificial magnetism and left-handed media from dielectric rings and rods,”
Journal ofPhysics: Condensed Matter , vol. 22, p. 025902, Dec. 2009.Publisher: IOP Publishing.[26] R. Marques, L. Jelinek, M. J. Freire, J. D. Baena, and M. Lap-ine, “Bulk Metamaterials Made of Resonant Rings,”
Proceed-ings of the IEEE , vol. 99, pp. 1660–1668, Oct. 2011. Confer-ence Name: Proceedings of the IEEE.[27] J. C. Maxwell,
A treatise on electricity and magnetism , vol. II.Oxford: Clarendon Press, 1873.[28] L. O. Chua, C. A. Desoer, and E. S. Kuh,
Linear and Non-linear Circuits . McGraw-Hill, 1987. Google-Books-ID:hl0pAAAACAAJ.[29] G. A. Campbell, “Mutual inductances of circuits composed ofstraight wires,”
Physical Review , vol. 5, no. 6, p. 452, 1915.Publisher: APS.[30] F. Martens, “ ¨Uber die gegenseitige Induktion und pon-deromotorische Kraft zwischen zwei stromdurchflossenenRechtecken,”
Annalen der Physik , vol. 334, no. 10, pp. 959–970, 1909. Publisher: Wiley Online Library.[31] C. Forestiere, G. Miano, M. Pascale, and R. Tricarico, “Electro-magnetic Scattering Resonances of Quasi-1-D Nanoribbons,”
IEEE Transactions on Antennas and Propagation , vol. 67,pp. 5497–5506, Aug. 2019. Conference Name: IEEE Trans-actions on Antennas and Propagation.[32] W. Eccles,