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Salandrino et al. Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. B 3007

Parallel, series, and intermediate interconnectionsof optical nanocircuit elements.

1. Analytical solution

Alessandro Salandrino, Andrea Alù,* and Nader Engheta

Department of Electrical and Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia,Pennsylvania 19104, USA

*Corresponding author: [email protected]

Received June 11, 2007; revised September 18, 2007; accepted October 7, 2007;posted October 11, 2007 (Doc. ID 84012); published November 19, 2007

Following our recent development of a paradigm for extending the classic concepts of circuit elements to theinfrared and optical frequencies [Phys. Rev. Lett. 95, 095504 (2005)], in this paper we investigate the possibil-ity of connecting nanoparticles in series and in parallel configurations, acting as nanocircuit elements. In par-ticular, here we analyze a pair of conjoined half-cylinders whose relatively simple geometry may be studied andanalyzed analytically. In this first part of this work, we derive a novel closed-form quasi-static analytical so-lution of the boundary-value problem associated with this geometry, which will be applied in Part 2 for a nano-circuit and physical interpretation of these results. © 2007 Optical Society of America

OCIS codes: 350.4600, 240.6680, 290.5850.

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. INTRODUCTIONdvances in the science and engineering of nanofabrica-

ion and characterization techniques over the past few de-ades have provided the possibility for exploring opticalnd electronic phenomena at the nanometer scale withrecision previously unimagined [1]. Among the opticalhenomena characterizing the nanoscale domain, surfacelasmons have attracted a great deal of attention in thecience, applied science, and engineering communities.

Plasmonic materials indeed offer interesting possibili-ies in manipulation of optical fields, ranging from scat-ering enhancement to subdiffraction confinement anduidance. These effects are indeed striking and are not or-inarily observed with normal dielectrics, so they are of-en called “anomalous,” even though they may be pre-icted and described within the framework of the classicalcattering theories [2,3] and solid-state physics [4]. It ismportant to point out that in this case the fabricationechniques are leading ahead of the design techniques,nd therefore new design ideas are required in order toully exploit many possibilities provided by the currentechnologies. Many groups all over the world have concen-rated their efforts in the area of nanotechnology [5–15],ith an ever-increasing number of suggestions and ideas

or applications and devices based on control of plasmonicesonances.

We have recently conducted some studies for develop-ng and extending the concept of circuit elements to theptical domain [16] that implied a partial redefinition ofhe relevant electrical quantities commonly involved inhe low frequency designs, shifting the focus from the con-uction currents to the displacement currents flowing inn optical nanocircuit. The operation through displace-ent rather than conduction currents has major conse-

uences in the way that the circuit elements may be con-

0740-3224/07/123007-7/$15.00 © 2

ected. As a matter of fact, the nature of the displacementurrents makes it more difficult to confine them throughpecific paths and directions, as is done by the physicaloundary of a conductor when an actual flow of chargearriers is concerned. To this end, an important extensionf our nanocircuit theory may reside in the possibility ofnterconnecting multiple nanocircuit elements together toorm a complex nanocircuit system.

The present work is focused on a first step toward suchgoal, i.e., the analysis of the series and parallel inter-

onnections between two nanocircuit elements, in theense defined in the framework of our optical nanocircuitheory [16]. As we describe in detail in the second part ofhis paper [17], in which we focus on the physical aspectsehind this interconnection and its role in the nanocircuitramework, one relevant geometry in this sense consistsf a pair of two conjoined half-cylinders that, dependingn the orientation of the impinging electric field, may acts a parallel or a series combination of nanocircuit ele-ents.In this first part, we present a detailed electromagnetic

olution for the quasi-static scattering problem associatedith this geometry. In particular, here we derive a novel

losed-form solution for this specific geometry, which hassimple physical interpretation in terms of the reso-

ances associated with the coupling of the two nanocir-uit elements. Moreover, the field distributions obtainedhrough this analysis fully confirm the nanocircuit inter-retation of the geometry under analysis. The results ofhis first part will therefore be of importance and use tohe second part of this work [17], showing how such aimple geometry may act as a basic parallel or series in-erconnection between two nanocircuit elements, and pro-iding a first step toward the design of a more-complex in-erconnected nanocircuit system.

007 Optical Society of America

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3008 J. Opt. Soc. Am. B/Vol. 24, No. 12 /December 2007 Salandrino et al.

Throughout the following analysis an e−i�t time depen-ence is assumed.

. GEOMETRY OF THE PROBLEM: AIELECTRIC CONJOINED CYLINDERhe geometry we consider in the following is shown inig. 1 and consists of two half-circular cylinders of radius, cut perpendicularly to their circular cross section alongdiameter and conjoined along the cut. Under the as-

umption that the size of the structure is sufficientlymaller than the operating wavelength, which is a validssumption for the typical size of the nanocircuit ele-ents of interest in this work, the following analysis may

e performed in the quasi-static approximation. It may benderlined that this assumption does not affect the nec-ssary time variation of the involved phenomena, re-uired for the definition of displacement current and foraving plasmonic materials [18], but it rather limits thepatial variation of the fields across each nanoparticle,wing to its small electrical size.

The structure is excited by a uniform electric field ofmplitude E0 incident at an angle � with respect to thenternal interface of the cylinder.

The goal of this part of our manuscript is to derive theistribution of the electric potential �� ,�� inside andround the structure of Fig. 1. Such a potential is the so-ution of the Laplace equation in a suitable cylindrical ref-rence system �� ,��:

�2�2��,��

��2 + ��,��

��+

�2��,��

��2 = 0. �1�

ere we have used the translational symmetry of the 2Droblem at hand, for which the potential is uniform alonghe z axis, i.e., the axis of the cylinder.

. FORMULATION OF THE DUAL PROBLEMIA KELVIN TRANSFORMATIONhe boundary-value problem shown in Fig. 1 cannot beasily handled in its original form, because a set of or-hogonal eigenfunctions conforming to such geometry isot readily available. For this reason our approach is toperate a conformal mapping of the original geometrynto a new form that may be more easily solved.

In the following, we solve the boundary-value problemf Fig. 1 by applying a conformal Kelvin transformation19]:

ig. 1. Geometry of the problem: two conjoined half-cylinders ofifferent permittivities.

�K = �2/�, �2�

hich maps the geometry of interest into a related dualroblem. This transformation belongs to the inversionsith respect to an analytic curve and specifically consistsf an inversion with respect to the circle of radius �,alled the circle of inversion and centered at the origin ofur cylindrical reference system. In general, the mappingetween the original geometry and the transformed geom-try is depicted in Fig. 2, where a circle (passing throughoints 1 and 2) is mapped into another circle (passinghrough points 1K and 2K) after applying Eq. (2).

The circle of inversion, represented by the dotted blackircle, and centered at the origin, maps any analytic curvenside its area into another analytic curve outside of it. Inarticular, the mapping is invariant for circular shapes,s Fig. 2(a) shows, and a circle of radius R centered at theartesian coordinates �0,r0� inside the (dotted) circle of

nversion is mapped into a circle of radius RK=�2R / �r02

R2� and center �0,�2r0 / �r02−R2�� positioned outside of it.

e note how in general the transformed circles are scaledn size with the quantity �2 / �r0

2−R2� and reversed horizon-ally with respect to the original circles [the points 1 andon the original circle are, respectively, mapped into the

oints 1K and 2K on the transformed circle].In the special case where the original circumference

asses through the center of the circle of inversion, i.e.,hen r0=R, the mapped circle [top circle in Fig. 2(a)] de-enerates into the line y=�2 / �2R�, as shown in Fig. 2(b).his particular case allows us to map any circumference

nto a straight line, which may simplify the originaloundary-value problem of Fig. 1 into the dielectric wedgeroblem of Fig. 3.

ig. 2. (Color online) Schematic diagram of the mapping repre-ented by Eq. (2). (a) The smallest circle, located inside the circlef inversion (dotted), is mapped into the circle above it. (b) De-eneration of the mapped circle in the case in which the originalircumference passes through the center of the circle ofnversion.

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One of the important properties of the Kelvin transfor-ation is that it is invariant with Laplace’s equation [19]:

�2��,�� = 0 ⇔ �2��K��,�� = 0. �3�

his ensures that a solution of Laplace’s equation in theapped geometry is still a harmonic function in the origi-

al geometry.The electromagnetic equivalence of the mapped prob-

em is obtained by further showing that there is a one-to-ne correspondence between the boundary conditions inhe original domain and in the mapped domain. Theelvin transformation, like most of the conformal map-ings, is generally applied to problems involving only me-allic boundaries or, in other words, Dirichlet boundaryonditions where the consistency of the solutions in thewo domains is guaranteed by the continuity of the map-ing. Our geometry, on the other hand, involves interfacesetween different dielectrics and, to the best of our knowl-dge, the Kelvin transform has not been applied in theast to this kind of problems.Again, the continuity of the potential at any interface is

uaranteed by the continuity of the mapping, but the con-inuity of the normal component of the electric displace-ent D at the interface between two different media isot as immediately evident. For this reason, in order torove the complete correspondence of the original and theual problem, in Section 4 we will prove the consistency ofhe boundary conditions between the two geometries.

. ELECTROMAGNETIC CONSISTENCY OFHE KELVIN TRANSFORMATIONOR THE PRESENT PROBLEMhe continuity of the potential is clearly maintained afterhe Kelvin transformation, since imposing that the solu-ion is continuous along the three interfaces separatinghe three materials in Fig. 3 automatically implies thathe conformal solution is continuous along the corre-ponding interfaces, also in the original geometry of Fig..More intricate is the boundary condition on the normal

omponent of the electric displacement vector D. First weote that the vertical line at the interface between thewo half-cylinders of Fig. 1 does not change its orientationfter the mapping into Fig. 3 and that the Kelvin trans-

ig. 3. Mapping of the geometry of Fig. 1, after applying theelvin transformation [Eq. (2)] in the specific case that is illus-

rated in Fig. 2(b), in order to obtain the double-wedge rectangu-ar problem.

ormation involves only the radial variable �. This implieshat the normal to the interface is parallel to � in botheometries and that �� ,�� /��=��K�� ,�� /��. Theoundary condition at this interface therefore remainsatisfied after the mapping.

Figure 2(b) shows the orientation of the unit vector normal to the original circumference and of the trans-ormed unit vector y. In particular along the circumfer-nce and along the line y=�2 / �2R� we get

n = � sin � − � cos �,

y = � sin � + � cos �. �4�

e have to prove that if

�in � �in��K��,�� · y = �0 � �0��K��,�� · y �5�

n the boundary y=�2 / �2R� in the mapped geometry,here in and �in are, respectively, the permittivity and

he potential in one of the two materials and �0 is the po-ential in the outer region, then

�in � �in��,�� · n = �0 � �0��,�� · n �6�

n the original circumference of Fig. 2(b).To this end, Eq. (5) can be rewritten as

�in� ��in��K��,��

��sin � +

1

�

��in��K��,��

��cos �

= �0� ��0��K��,��

��sin � +

1

�

��0��K��,��

��cos � .

�7�

pplying transformation (2) we obtain

�in� ��in��,��

��sin � +

1

�

��in��,��

��cos �

= �0� ��0��,��

��sin � +

1

�

��0��,��

��cos � , �8�

hich coincides with Eq. (6), confirming that theoundary-value problem in Fig. 3 is consistent with thene depicted in Fig. 1. This shows that it is sufficient toolve for the potential in one of the two geometries in or-er to determine the electromagnetic solutions in thether mapped geometry.

. TRANSFORMATION OF THE INCIDENTIELDSo complete the formulation of the dual problem, the in-ident field also needs to be properly transformed accord-ng to the Kelvin mapping. If we assume that the con-oined half-cylinders are exposed to a uniform electriceld linearly polarized along the line forming a generalngle � with respect to the normal to the interface be-ween the two half-cylinders, then the distribution of im-ressed (i.e., “incident”) potential in the original geometrys given by �inc�� ,��=−E0� cos��−��. As we will show andiscuss in detail in Part 2, the conjoined half-cylinders ofig. 1 may be seen as interconnected in series or in par-

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llel as a function of the orientation of the impinging field,nd in particular the two special cases of �=0 and � /2 correspond, respectively, to the series and the par-llel connection.After the Kelvin mapping, the impressed potential is

ransformed into

�inc��K��,�� = −2�0E0�2

2�0

cos�� − ��

�K. �9�

his expression coincides with the potential of a dipoleriented along the angle � with respect to the axis xK,ith dipole moment p=20E0�2 and located at the origin

18]. Not surprisingly, this is physically related to theroperties of the Kelvin transformation, which mapsoints placed at infinity in the original reference systemnto the origin of the conformal reference system, since aniform electric field may be considered to be generatedy two equal and opposite charges located at two points atnfinity, located symmetrically with respect to the origin,nd the Kelvin transformation does not affect the angularistribution. This explains how these two charges collapsento a localized dipole at the origin in the conformal ref-rence system, and oriented along the same angle with re-pect to the xK axis as the original field forms with the xxis (normal to the interface between the half-cylinders).

. SOLUTION OF THE DUAL PROBLEM:HE DOUBLE DIELECTRIC WEDGEhe geometry and the excitation of the dual problem haveeen completely described in Section 5. We now proceed tots formal solution. First of all it is convenient to translatehe double-wedge boundary shown in Fig. 3 on the xKxis, with a new transformation rule given by

Y� = yK −�2

2R

X� = xK�, �� = �X�2 + Y�2

�� = arctan Y�

X�� . �10�

he boundary-value problem of Fig. 1 has been reduced tohat of a double dielectric wedge excited by an electric di-ole placed at the distance r0=�2 / �2R� from the interfacef the double wedge with free space, as depicted in Fig. 4.

ig. 4. Transformed boundary-value problem after the Kelvinransform [Eq. (2)] and the translation [Eq. (10)]; a double wedgexcited by an electric dipole forming an angle � with the X� axisqual to the angle formed by the uniform electric field with theormal to the interface between the two 2D half-cylinders in theriginal geometry (Fig. 1).

his problem may be solved for the translated potential�� ,�� under this quasi-static assumption by applying

he Mellin transformation, as suggested in [20].After having solved the rectangular problem of Fig. 4

or the potential �� ,��, we apply the inverse transla-ion [Eq. (10)] and the inverse Kelvin transform [Eq. (2)]o return to the original problem and evaluate the poten-ial distribution �� ,�� for the conjoined half-cylinders ofig. 1. In particular, this distribution assumes a closed-

orm expression when the permittivities of the two half-ylinders are equal and opposite in sign, which, as we dis-uss in Section 7, corresponds to a resonant configuration.

Applying the Mellin integral transform [20], defined as

f�s� =�0

�

��s−1f��d��,

f�� =1

2i�c−i�

c+i�

��−sf�s�ds, �11�

e can expand the integral �� ,�� of the Laplace equa-ion in the double-wedge problem of Fig. 4 into a set ofnalytical functions ��s ,��, depending on the Mellin ra-ial frequency s and the angular coordinate ��. In par-icular, in the four regions of Fig. 4 we get

�0+� �s,�� = A0+�s�cos�s�� + B0+�s�sin�s��,

�0−� �s,�� = A0−�s�cos�s�� + B0−�s�sin�s��,

�1��s,�� = A1�s�cos�s�� + B1�s�sin�s��,

�2��s,�� = A2�s�cos�s�� + B2�s�sin�s��. �12�

he unknown coefficients A and B in each region are de-ermined by imposing the boundary conditions at thehree interfaces of Fig. 4. This allows the determination ofhe potential ��s ,�� in the Mellin domain that is relatedo the potential distribution �� ,�� in the double-edge problem through Eq. (11), by applying an inverseellin transform with a residue evaluation of the integral

20]. The convergence of the integrals in Eq. (11) for theresent configuration is guaranteed by the form of excita-ion, i.e., an electric dipole, whose radial dependence sat-sfies the convergence requirements [20].

It may be shown that the poles of the coefficients in Eq.12), which determine the location of the residues in theeneral case and therefore the distribution of the trans-ormed potential, are given by the following dispersionquation in the general case:

��s� = �4�0�1�2 + �02��1 + �2� + �1�2��1 + �2�

+ ��0 + �1��0 + �2��1 + �2�cos�s��sin2�s/2� = 0.

�13�

t must be noticed how in the special case of 1=2=0 theispersion equation yields the set of solutions sN=N, withbeing any integer, which corresponds to the poles of the

riginal potential distribution, given by an electric dipole,onsistent with the fact that in this case the conjoined

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alf-cylinders and the corresponding double wedge areot present. When 1=−2, instead, independent of theirelative value with respect to 0, Eq. (13) admits the set ofolutions sN=2N, which corresponds to the potentialsrom two electric dipoles, one placed in the original sourceosition, again corresponding to the impressed potential,he other placed and oriented symmetrically with respecto the origin, which is an image of the original source. Ase discuss further in detail in the second part of thisanuscript, this special condition corresponds to a special

esonant configuration for the conjoined geometry undernalysis.

ig. 5. (Color online) Potential distributions for a double wedgeith permittivities 1=−2=−2. The orientation of the excitingipole is (a) �=0, (b) �= /2, and (c) �= /4.

. INVERSION OF THE MELLINRANSFORM IN THE RESONANT CASE

n the resonant case 2=−1, the coefficients appearing inhe Eqs. (12) have the following simple expressions:

A0+ = A1 =pr0

s−1 sin��

2�0 sin�s/2�,

B0+ = −pr0

s−1 cos��

2�0 sin�s/2�,

A0− = −p

�0r0

s−1 cos��cos�s/2�

+pr0

s−1 sin��cos�s�

2�0 sin�s/2�,

B0− = −pr0

s−1 cos��cos�s�

2�0 sin�s/2�

−p

�0r0

s−1 sin��cos�s/2�,

B1 = −pr0

s−1 cos��

2�1 sin�s/2�,

A2 = −pr0

s−1 cos��cos�s/2�

�1

+pr0

s−1 sin��cos�s�

2�0 sin�s/2�,

B2 =pr0

s−1 cos��cos�s�

2�1 sin�s/2�

+p

�0r0

s−1 sin��cos�s/2�. �14�

n the inversion of the Mellin transform, only the termsith poles give contributions to the potential through

heir residues. These poles, consistent with Eq. (13), areocated at s=2N, and the transformed potential distribu-ion may be written in compact form as

�i��s,�� = r0s−1

p

�0

�i sin��cos�s�� − �0 cos��sin�s��

2�i sin�s/2�,

�15�

ith i=0,1,2 in the different regions of space.The inversion of the Mellin transform may be per-

ormed by applying the residue theorem, once a properromwich contour is chosen [20], resulting in

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�i��,�� = sgn�� − r0�p

�N=0

� �i sin��cos�2N�� − �0 cos��sin�2N��

�0�ir0 cos�N�

�min�r0,��

max�r0,��2n�1 −1

2��N� . �16�

quation (16) may be easily evaluated as a geometric se-ies in this configuration, yielding the expression

�i��,�� =p��4 − r0

4�sin��

2�0r0��4 + r04 + 2��2r0

2 cos�2��

+�ip��2r0 cos��sin�2��

�02��4 + r0

4 + 2��2r02 cos�2��

, �17�

hich corresponds to the potential distribution of the ex-iting dipole in Fig. 4 superimposed to a dipole-like singu-arity placed at X�=0, Y�=�2 /2R. This can be seen in theistribution of potential shown in Fig. 5 for several orien-ations of the impressed electric dipole.

We may deduce from Fig. 5 some interesting propertiesf this resonant double-wedge configuration. When thempressed dipole is parallel to the external boundary of
he double wedge [Fig. 5(a)], as far as the external field is w
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oncerned, the structure behaves as a perfect electric con-uctor, with the equipotential line parallel (and thereforehe electric field orthogonal) to this boundary. On thether hand, when the impressed dipole is normal [Fig.(b)], as far as the external field is concerned, the struc-ure behaves in a dual way, like a perfect magnetic con-uctor. In other words, the surface impedance shown byhe external interface of the resonant double wedge istrongly affected by the orientation of the impressed di-ole and may vary from zero to infinity according to theangent of the angle �. The case of Fig. 5(c) is an interme-iate situation between the two extremes, since the excit-ng dipole forms an angle of 45 deg with the boundary. Weill see how these phenomena correspond to the seriesnd parallel response of resonant conjoined cylinders asanocircuit elements in the second part of this paper [17].

. INVERSION OF THE KELVINRANSFORMhe potential distribution for the conjoined cylinder ofig. 1 may now be evaluated after proper translation andelvin inversion of Eq. (17), resulting in

�i��,�� = �iparallel��,��sin�� + �i

series��,��cos��, �18�

here

�iparallel��,�� =

E0�2R��2 + R2� − �2R cos�2�� − ��2 + 4R2�sin��

�2 + R2 − 2�R sin��

�iseries��,�� =

�0E0�2 cos��2R sin�� − ��

�i��2 + R2 − 2�R sin�� , �19�

hich is valid in the resonant configuration 1=−2. Theeaning of the superscripts series and parallel in the pre-

ious expressions follows the earlier remarks and findsroper justification and detailed discussion in Part 2 ofhis manuscript [17].

After translating the reference system in such a wayhat the center of the resonant conjoined half-cylindersoincides with the origin, we get the final simple expres-ions

�iparallel��,�� = − E0

�2� + R2

��sin ��

�iseries��,�� = − E0

�0

�i

�2� − R2

��cos �� , �20�

here �� ,�� are the cylindrical coordinates in the neweference system.

Potential distributions for this resonant configurationnd physical interpretations and nanocircuit implicationsf Eq. (20) are provided in [17].

Similar to what was noticed in the case of the rectan-ular wedge, the split cylinder behaves differently de-ending on the polarization of the incident electric field,

hich is consistent with their nanocircuit interpretation,s we discuss in Part 2. Mathematically this analogy be-ween planar and cylindrical configurations is expected,ince the Kelvin transformation is a conformal mappinghat locally preserves the angles, including those betweenhe interfaces and the equipotential lines. The interestingnomalous features of this configuration and their physi-al interpretation in terms of nanocircuits will be also dis-ussed in detail in Part 2.

. CONCLUSIONSn this first part of our manuscript, we have presented aovel closed-form quasi-static analytical solution of theoundary-value problem associated with the geometry ofwo conjoined dielectric half-cylinders, one of which maye plasmonic in nature. As we show in [17], this may beully interpreted as a series and parallel interconnectionf two nanocircuit elements, as a case study of the gener-lization of our nanocircuit paradigm to take into accountoupling and interconnections among nanocircuit ele-ents in a complex nanocircuit. In Part 2 we will apply

he results of this analytical formulation to the nanocir-

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uit paradigm introduced in [16], and we will discuss inetail the implications of these results from a physicaloint of view.

CKNOWLEDGMENTShis work is supported in part by the U.S. Air Force Officef Scientific Research (AFOSR) grant FA9550-05-1-0442.. Alù was partially supported by the 2004 SUMMAraduate Fellowship in Advanced Electromagnetics.

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