line-current model for deriving the wavelength scaling of ... · sidering the wire antenna as a...

Line-current model for deriving the wavelengthscaling of linear and nonlinear optical propertiesof thin elongated metallic rod antennasM. L. NESTEROV,1 M. SCHÄFERLING,1 K. WEBER,1 F. NEUBRECH,1,2 H. GIESSEN,1 AND T. WEISS1,*14th Physics Institute and Research Center SCoPE, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany2Kirchhoff-Institute for Physics, University of Heidelberg, Im Neuheimer Feld 227, D-69120 Heidelberg, Germany*Corresponding author: [email protected]

Received 16 January 2018; revised 27 February 2018; accepted 16 April 2018; posted 19 April 2018 (Doc. ID 319861); published 4 June 2018

Thin elongated rod antennas with a diameter smaller than the skin depth exhibit surface plasmon polariton modesthat can propagate along the antenna while being reflected at the antenna ends. In the line-current model, acurrent is associated with these modes in order to approximate the optical properties of the antennas. We findthat it is crucial to correctly derive the reflection of the surface plasmon polariton modes at the antenna ends forpredicting the resonance position and shape accurately. Thus, the line-current model allows for deriving thewavelength scaling behavior of plasmonic near fields as well as the emitted third-harmonic intensity efficiently.Neglecting the frequency dependence of the nonlinear susceptibility, we find that the third-harmonic intensity ofsuch metallic rod antennas scales as the fourth power of the frequency, whereas it decreases with the twelfth powerwithin the limit of the generalized Miller’s rule. © 2018 Optical Society of America

OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (190.2620) Harmonic generation and mixing.

https://doi.org/10.1364/JOSAB.35.001482

1. INTRODUCTION

Plasmonic resonances in metallic antennas concentrate electro-magnetic fields in small volumes and, thus, amplify and controllight–matter interaction. This includes the spontaneousemission rate and the directivity of quantum emitters [1–6],the enhancement of circular dichroism signals [7–10] andFaraday rotation [11–13], second- and third-harmonic gener-ation [14–18], advanced concepts for optical refractive indexsensors [19–24], and antenna-assisted surface-enhanced infra-red absorption spectroscopy [23,25,26].

As shown in [27], plasmonic resonances in wire antennascan be understood in analogy to Fabry–Perot modes by con-sidering the wire antenna as a cavity for surface plasmon polar-itons propagating along the wire. At the wire ends, the surfaceplasmon polariton is partially reflected, propagates in theopposite direction, and is reflected at the other end. Resonancesoccur for constructive interference of the surface plasmonpolariton after one round trip. The difference to classicalantennas is that the wavelength of the surface plasmon polar-iton significantly differs from the free-space wavelength and thefield can penetrate into the antenna for the plasmon modes, aswe require working in a regime in which the diameter is smallerthan the skin depth of the metal. Based on these considerations,Dorfmüller et al. have developed a line-current model for pre-dicting the near field of the plasmonic resonances in the vicinity

of thin elongated wire antennas for illumination by far-fieldincidence [28]. The approach has then been adapted tonear-field excitation by quantum emitters in [29], where alsothe influence of the radiation damping due to the reflection atthe antenna ends is taken into account by a simple approxima-tion. More sophisticated models for the reflection at theantenna ends rely on analytical calculations restricted to flatantenna ends [30] as well as numerical simulations for morecomplex geometries [31,32].

In this paper, we combine the line-current model for far-field incidence [28] with the near-field formulation describedin [29]. Furthermore, we replace the approximate considerationof the radiation damping in [29] by numerical simulations ofthe reflection of a surface plasmon polariton at the antennaend. Comparison with full numerical simulation of the three-dimensional rod antenna geometries reveals that this approachprovides a significantly better estimation of the antenna proper-ties compared with the aforementioned approaches.

Our numerical simulations of the surface plasmon polaritonand its reflection at the antenna end as well as the referencesimulations of the optical properties of the entire antennageometry are carried out using the finite element method.While the numerical simulation of an entire three-dimensionalantenna geometry can be rather time- and memory-consuming,the simulations of the surface plasmon polariton and its

1482 Vol. 35, No. 7 / July 2018 / Journal of the Optical Society of America B Research Article

0740-3224/18/071482-08 Journal © 2018 Optical Society of America

https://orcid.org/0000-0002-9836-8457https://orcid.org/0000-0002-9836-8457https://orcid.org/0000-0002-9836-8457https://orcid.org/0000-0002-4991-6779https://orcid.org/0000-0002-4991-6779https://orcid.org/0000-0002-4991-6779mailto:[email protected]:[email protected]:[email protected]:[email protected]://doi.org/10.1364/JOSAB.35.001482https://crossmark.crossref.org/dialog/?doi=10.1364/JOSAB.35.001482&domain=pdf&date_stamp=2018-06-04

reflection at the antenna end can be carried out very efficiently,as they are effectively two-dimensional. Most importantly, thesetwo-dimensional simulations are independent of the antennalength, so that the results of the line-current model can be givenas an analytical function of the antenna length. Thus, it is pos-sible to derive the wavelength scaling behavior of the opticalresponse of thin elongated wire antennas in a fast and efficientmanner.

We demonstrate here how to use the line-current model forpredicting the wavelength scaling of the electromagnetic nearfields, which is highly relevant for resonantly enhanced refrac-tive index sensing [22,24] as well as antenna-assisted surface-enhanced infrared absorption spectroscopy [23]. Furthermore,we derive the wavelength scaling of the resonantly enhancedthird-harmonic signal from thin wire antennas.

2. LINE-CURRENT MODEL

The fundamental idea of the line-current model is sketched inFig. 1. An incident field excites a current in the antenna. Thecurrent propagates to the antenna end, where it is partiallyreflected and partially transmitted. In the limit of a thin wire,the current predominantly stems from the fundamental surfaceplasmon polariton mode, i.e., a mode that propagates along thelong wire axis and is bound in the other directions. The forwardand backward propagating surface plasmon polariton results inan oscillating current distribution, from which we can calculatethe radiation to the far field.

The material surrounding the antenna is modeled via its di-electric permittivity ε1; the permittivity of the metallic antennais ε2. In our examples, we are using ε1 � 1 and ε2 defined byexperimental permittivity data of gold [33]. The correspondingwavenumbers are k1 � ffiffiffiffiffiε1p k0 and k2 � ffiffiffiffiffiε2p k0, respectively,with k0 � ω∕c. The long axis of the antenna is oriented alongthe z direction, and the antenna length is denoted by L0. Thecross section normal to the z direction has a circular shape withradius R; the antenna ends are semispheres. When we considerproperties of the full antenna, such as scattering and absorptioncross section, we will henceforth use spherical coordinatesρ, ϑ, and φ with unit vectors ρ̂, ϑ̂, and φ̂. The origin of thecoordinate system is located at the center of the antenna.

The incident angle is defined as θ � π∕2 − ϑinc. For the cur-rent propagating along the antenna in the z direction, we willconsider a cylindrical coordinate system with coordinates ρ, ϕ,and z, with unit vectors ρ̂, ϕ̂, and ẑ.

The electric field of the fundamental surface plasmon polar-iton propagating along an infinitely long metal cylinder (radiusR) has a vanishing ϕ component. It is given in the exteriorregion (ρ > R) by

E �1�ρ � −iE1kpκ1

H �1�1 �κ1ρ�eikpz , E �1�z � E1H �1�0 �κ1ρ�eikpz ,

(1)

where H �1�l denotes Hankel functions of first kind and order l.The time dependence exp�−iωt� has been omitted for the sakeof simplicity. Furthermore, κ21 � k21 − k2p. In the interior region(ρ ≤ R), we obtain

E �2�ρ � −iE2kpκ2

J1�κ2ρ�eikpz , E �2�z � E2J0�κ2ρ�eikpz :

(2)

Here, J l denotes Bessel functions of the first kind and orderl , and κ22 � k22 − k2p. The propagation constant kp can be de-termined from the following transcendental equation [27]:

ε2�ω�κ2R

J1�κ2R�J0�κ2R�

−ε1κ1R

H �1�1 �κ1R�H �1�0 �κ1R�

� 0: (3)

Figure 2 depicts the calculated dispersion relation of thefundamental surface plasmon polariton mode propagating inan infinitely long gold cylinder of radius R � 10 nm invacuum.

A. Derivation of Current and Far-Field Properties

The incident field Einc induces a current j inside the rod an-tenna that results in a scattered field Escat outside the antenna.The total field in the exterior is given by Etot � Einc � Escat andis related to the internal field Eint inside the antenna by theboundary conditions for Maxwell’s equations. The inducedcurrent obeys

incident field

radiatedfield

I+

I||

I-

z0

1

22R

-L /20 L /20

rr

inc

Fig. 1. Schematic of the line-current model for a rod antenna (per-mittivity ε2) in homogeneous space (permittivity ε1) with length L0and a cylindrical cross section of radius R. The incident field excites asurface plasmon polariton in the rod antenna, which propagates to theantenna ends, where it is partially reflected (reflection coefficient r).The forward (backward) propagating surface plasmon polariton is de-noted by the current I� (I −), and the driving current is I∥. In thismodel, the field radiated by the antenna originates from the emissionof the line current to the far field.

0 0.5 1 1.5 2 2.5

Decay length (µm)

0 5 10 15 20 250

2

4

6

8

kp (1/µm)

k 0)

mµ/1(

light

con

e

Fig. 2. Calculated dispersion relation (solid black line) for a surfaceplasmon polariton mode propagating in an infinitely long gold cylin-der of radius R � 10 nm in vacuum. The blue solid curve denotes thecorresponding decay length. The black dashed line is the light cone ofvacuum.

Research Article Vol. 35, No. 7 / July 2018 / Journal of the Optical Society of America B 1483

j � −iωε0�ε2�ω� − ε1�|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}≡Δε

Eint: (4)

For wire antennas thinner than the skin depth of the metal,we may assume that the current is approximately constantacross the plane perpendicular to the wire. Furthermore, theelectromagnetic field of the fundamental surface plasmonpolariton propagating in such a thin wire does not dependon the azimuthal angle ϕ. It exhibits a dominant z componentand a negligible ρ component, which follows from evaluatingEq. (2) for small arguments. Hence, the current j is predomi-nantly z directed, with

j�r� ≈ I�z�Θ�R − ρ�ẑ : (5)Here, Θ denotes the Heaviside function. For the current

I�z�, we might make the following ansatz [28] in order toaccount for the contributions of the incident field as well asthe surface plasmon polariton propagating with kp alongthe wire:

I�z� � I∥eik∥z � I�eikpz � I−e−ikpz : (6)In this case, k∥ ≡ k1 sin θ denotes the component of the

incident wave vector parallel to the wire antenna.Let us now relate the different contributions in Eq. (6) to

the incident field. The vector potential associated with thecurrent in Eq. (6) is [34]

A�r� � μ04π

ZdV 0j�r 0� e

ik1jr−r 0 j

jr − r 0j : (7)

Using Eq. (5) in Eq. (7) as well as jr − r 0j≈ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 � �z − z 0�2

p, we obtain

A�r� ≈ μ04π

SRẑZ

L0∕2

−L0∕2dz 0I�z 0� e

ik1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2��z−z 0�2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 � �z − z 0�2

p : (8)If we assume that ρ ≪ λ1 with λ1 � 2π∕k1, the second fac-

tor in the integrand in Eq. (8) becomes localized around z.Hence, we may change variables from z 0 to ζ � z and extractI�ζ � 0� � I�z� from the integral, resulting in

A�r� ≈ μ04π

SRẑI�z�Z

z�L0∕2

z−L0∕2dζ

eik1ffiffiffiffiffiffiffiffiffiρ2�ζ2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 � ζ2

p|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

≡Z̃

, (9)

with SR � 2πR2 being the area of the rod cross section.According to [28], Z̃ takes the role of a characteristic impedance.

The z component of the scattered field at the wire surfacecan be calculated from Eq. (9) as

E scatz �z,R� �iωk21

�k21 � ∂2z �Az �iωk21

μ04π

SRZ̃ �k21 � ∂2z �I�z�:

(10)

The boundary condition for the z component of the electricfields at the surface of the metal wire is

E scatz �z,R� � E incz �z,R� � E intz �z,R�: (11)The incident field can be assumed constant across the wire,

i.e., E incz � E inc0 cos θ exp�ik∥z�. Together with Eqs. (4) and(5) as well as Eq. (10), this results in

E inc0 cos θeik∥z � iω

k21

μ04π

SRZ̃ �k21�∂2z �I�z��iI�z�ωε0Δε

: (12)

Using Eq. (6) and the fact that the resulting equations haveto be fulfilled independently for all values of k∥ and kp, weobtain according to [28] that

I∥ � −iωε0ΔεΔkE inc0 cos θ, (13)with

Δk ≡k21 − k

2p

k2∥ − k2p

: (14)

Furthermore, we assume in our model that only the currentwith amplitude I� is transmitted at z � �L∕2, so that thecurrent reflected at the antenna ends obeys

z � L0∕2: I −e−ikpL∕2 � −I∥eik∥L∕2 − rI�eikpL∕2, (15)

z � −L0∕2: I�e−ikpL∕2 � −I∥e−ik∥L∕2 − rI −eikpL∕2: (16)Here, r denotes the reflection of the surface plasmon polar-

iton at the antenna ends and is in general a complex number.However, following [29], we can redefine r to be real by intro-ducing an effective antenna length L � L0 � ΔL. The relationbetween reflection phase ϕr and antenna length increase ΔL isthen given by ΔL � ϕr∕kp. It has to be emphasized that the line-current model with effective antenna length is not fully equivalentto the one with complex reflection coefficient. This is due to theimaginary part of kp, which results in additional artificial ab-sorption when replacing the real antenna length by the effectivelength. In the following, we will consider real r and effectiveantenna length L, as it provides a better agreement with thenumerically exact results. The reason might be that theline-current model does not properly account for radiativelosses at the antenna ends, which is counterbalanced by theoverestimation of the Ohmic losses when using the effectiveantenna length.

Equations (15) and (16) straightforwardly result in

I� � I∥reik

�p L∕2 − e−ik

�p L∕2

1 − r2e2ikpLeikpL, (17)

with k�p ≡ kp � k∥. Note that kp as well as L and r are func-tions of frequency ω. The roots of the denominator at frequen-cies ωm provide the resonance condition for the wire antenna inthe same way as in a standard Fabry–Perot cavity [27,35]: thereal part of ωm is the resonance frequency, while −2 Im�ωm�defines the linewidth of the resonance.

In summary, the current I�z� inside the rod antenna isgiven by

I�z;ω; k∥� � −iωε0Δε�ω�Δk�ω; k∥�Î�z;ω; k∥�E inc0 cos θ:(18)

For brevity of notations, we have introduced here thenormalized current:

Î�z;ω; k∥� � eik∥z � Î��z;ω; k∥� � Î−�z;ω; k∥�, (19)with Î� � exp��ikpz�I�∕I∥.

In the far-field limit, the scattered field is polarized trans-verse to the outgoing wave vector, which means that the onlyrelevant component for our resonance geometry is the ϑcomponent, for which we obtain [29]


Eϑ�ρ,ϑ�≈ iZ 1k1SReik1ρ

4πρsin ϑ

ZL∕2

−L∕2dzI�z�e−ik1 cos ϑz

� k20ΔεΔkeik1ρ

4πρsin ϑV A

�sinc

��k∥ − k1 cos ϑ�

L2

�

� Î��0�sinc��kp − k1 cos ϑ�

L2

�

� Î−�0�sinc��kp � k1 cos ϑ�

L2

��E inc0 cos θ: (20)

Here, we have introduced V A � LSR as the effectivevolume of the antenna, sinc�x� � sin�x�∕x, and Z 1 denotesthe impedance in medium 1. According to Eqs. (4)–(6) as wellas Eq. (18), the z component of the electric field inside theantenna is

E intz �z� ≈ ΔkÎ�z�E inc0 cos θ: (21)Based on Eqs. (20) and (21), we can calculate the extinction,

scattering, and absorption cross sections as

σext � σscat � σabs, (22)

σscat �PradS0

� 2πρ2

jE inc0 j2Z

π

0

dϑjEϑj2 sin ϑ, (23)

σabs �PabsS0

≈k1 Im�ε2�ε1jE inc0 j2

V AL

ZL∕2

−L∕2dzjE intz j2, (24)

where S0 � jE inc0 j2∕2Z 1 is the time-averaged Poynting vectoramplitude of the incident plane wave. The integrals in Eqs. (23)and (24) can be carried out numerically. Note that Eq. (20)implies that the scattering cross section σscat is proportionalto k40V

2A, whereas Eqs. (5) and (6) result in σabs ∝ k0V A, as

in the case of small spherical antennas, because

1

L

ZL∕2

L∕2dzjE intz j2

∝1

L

ZL∕2

L∕2dzjÎ�z�j2 � 1� jÎ��0�j2 � jÎ −�0�j2

� 2Re�Î��0�sinc

�k−p

L2

� Î −�0�sinc

�k�p

L2

�

� 2Re�Î��0�Î−�0��sinc�Re�kp�L�: (25)

B. Discussion

In the above derivations, we have made the followingapproximations:

• Thin-wire approximation: Equation (5) is based on theassumption that the wire antenna has a radius smaller than theskin depth of the metal, so that we can assume a current insidethe antenna that does not depend on ρ. Furthermore, the thin-wire approximation allows for the assumption that the incidentfield is constant in the plane normal to the long wire axis.However, the exact spatial dependence of the fields might havesome impact on the excitation efficiency, because the zcomponent of the electric field at the antenna surface mightbe smaller or larger than in the center of the antenna. This alsoaffects the accuracy of the estimated near-field distributionaround the antenna.

• Single-mode approximation:Only the fundamental sur-face plasmon polariton is considered as a possible mode propa-gating along the thin wire. Higher-order surface plasmonpolaritons with field nodes in the radial or azimuthal directionas well as leaky modes are neglected. These modes provideadditional radiative and nonradiative loss channels that are nottaken into account in the line-current model. Similarly, theincident field not only excites currents in the form ofEq. (6), but should contain the excitation of the other modesas well. Consequently, the single-mode approximation resultsin an incorrect estimation of the current amplitudes I� andI∥. However, for elongated thin wires, the single-mode approxi-mation is justified, because the fundamental surface plasmonpolariton is in this case the only bound mode with a consid-erably large decay length compared to the antenna length. Forshort antennas, however, higher-order modes become morerelevant [32,36].

• Disregard of excitation and radiation at antenna ends:The main excitation of the surface plasmon polariton is as-sumed to stem from the incident electric field at the surfaceof the elongated rod antenna. Similarly, the radiation to thefar field is assumed to take place through the current oscillatingin the whole antenna. The accumulated charges at the antennaends as well as the possibility to excite the surface plasmonpolariton from these ends are neglected. This approximationas well as all other approximations in the line-current modelfail for larger diameters, for which it will result in an inaccurateestimation of the charge amplitude I�z� and its radiation to thefar field.

In addition, we have not yet specified the reflection ampli-tude r and the effective antenna length L � L0 � ΔL. In thework of Dorfmüller et al. [28], the magnitude of the reflectioncoefficient is assumed to equal unity (r � 1), while ΔL is ob-tained from fitting the line-current model to full numericalresults.

It has to be emphasized that r � 1 means neglecting theradiative decay, i.e., the radiation damping. This is of coursea very rough approximation, since the oscillating current doesradiate to the far field. In [29], it is therefore suggested to firstcalculate the so-called radiation resistance Rrad for r � 1, with

Rrad �2PI2max

, (26)

where P is the total emitted power and Imax is the maximum ofjI�z�j. This value can be used in order to derive a betterestimation for the reflection coefficient:

r�L0� ≈Re�Z � − Rrad∕2Re�Z � � Rrad∕2

≡ rapp: (27)

In this case, Z denotes the antenna wave impedance calcu-lated from the surface plasmon polariton mode in the infinitelylong wire as

Z �RdxdyExHy − EyHxj R dxdyI�z�j2 : (28)

For the effective antenna length increase, a simple approxi-mation is given by ΔLapp � 2R [27]. Alternatively, we havecalculated the reflection coefficient r and the effective antennalength increase ΔL of the fundamental surface plasmon polar-iton mode at the rod antenna end using full numerical


simulations based on the finite element method (see Fig. 3).For that purpose, we have set up a model, in which we launchthe surface plasmon polariton at a port at the antenna end.Perfectly matched layers ensure the appropriate boundary con-ditions for the scattered light, and the reflection into the surfaceplasmon polariton mode is derived in a second port. Theresulting reflection amplitude and effective antenna lengthincrease are denoted by rFEM and ΔLFEM, respectively.

Figure 4 depicts the scattering cross section of gold rod an-tennas with a diameter of R � 10 nm, where the antennalength is varied between 150 nm and 500 nm. The thin blackline has been calculated using the finite element method for thefull antenna, while the thick lines have been derived from thevarious approximations in the line-current model. In panel (a),

we have used r � 1 in combination with the approximateeffective antenna length increase ΔLapp (magenta lines) as wellas the value ΔLFEM (green lines) extracted from the two-dimensional finite element simulation of the reflection atthe antenna end. Panel (b) depicts the results calculated fromthe approximate reflection amplitude rapp with the approximateeffective antenna length increase ΔLapp (orange lines) as well asresults that are solely based on the finite element simulation ofthe reflection at the antenna end (blue lines).

It can be seen in Fig. 4(a) that the simple approximation ofr � 1 overestimates the magnitude of the scattering cross sec-tion. Furthermore, using ΔLapp, the resonance wavelength ofthe line-current model (magenta lines) is shifted comparedto the result of the full numerical simulation. The shift ofthe resonance wavelength originates in the simple approxima-tion for the effective antenna length. The mismatch in reso-nance wavelength almost vanishes when using ΔLFEM (greenlines). In order to obtain a better agreement between the mag-nitude of the scattering cross section from the line-currentmodel and the full numerical simulations, one has to replacer � 1 by rapp from Eq. (27) or rFEM from the finite elementsimulation for the reflection at the antenna end. When com-bining rapp with ΔLapp [orange lines in Fig. 4(b)], the magni-tude of the scattering cross section is reduced compared toFig. 4(a), while the resonance wavelength is shifted. This mis-match in resonance wavelength vanishes when combining rappwith ΔLFEM (not shown here). However, taking the maxima ofthe scattering cross sections as a guide to the eye, one can seethat rapp underestimates the magnitude of the scattering crosssection. In contrast, the numerically derived magnitude rFEM ofthe reflection coefficient and the resulting effective antennalength LFEM yield an excellent agreement between the line-current model and full numerical simulations, as seen forthe blue lines in Fig. 4(b).

Note that the dimensions of our elongated rod antennaswith a radius of R � 10 nm are in a regime where the thin-wire approximation and the single-mode approximation canbe applied. For radii larger than the skin depth, the agreementbetween the line-current model and full numerical simulationsbecomes worse, so that the line-current model can be used onlyfor qualitative predictions, as shown in [23].

3. APPLICATIONS

A. Wavelength Scaling

Whenever it is desired to predict the wavelength scaling behav-ior of antenna properties such as maximum scattering cross sec-tion or enhancement of the near fields, numerical simulationsrely on solving Maxwell’s equations repeatedly. Hence, full-fieldsimulations become rather inefficient, while the line-currentmodel provides the required results in significantly less calcu-lation time. As an example, we derive here the wavelength scal-ing of the magnitude of the electric field in the vicinity of thewire antenna at the fundamental resonance, which is importantfor sensing applications, such as antenna-assisted surface-enhanced infrared absorption spectroscopy [23,25].

Near resonances, jÎ�j ≫ 1, so that Eq. (19) can bewritten as

0

0.1

0.3

)mµ

(L

0 2 4 6 8 100.97

0.98

0.99

Wavelength (µm)

rM

EF

0.22R

FEM

Fig. 3. Magnitude of the reflection coefficient rFEM (black solidline) and effective antenna length increase ΔLFEM (blue dashed line)for the reflection of a surface plasmon polariton mode at a roundedantenna end with antenna radius R � 10 nm calculated by the finiteelement method (FEM). The blue dotted line denotes the simpleapproximation ΔLapp � 2R of Ref. [27].

0

0.02

0.04

0.06

0.08

0.10

mµ(noitces-ssorc

gnir et tacS

2 )

1.0 1.5 2.0 2.5 3.00

0.02

0.04

0.06

0.08

Wavelength (µm)

(a)

(b) L =500 nm0400

300

200150

FEM

r=1, Lappr=1, LFEM

rapp, Lappr , LFEM FEM

Fig. 4. Scattering cross section for antenna length between 150 mmand 500 nm and a radius of 10 nm. Thin black lines denote fullnumerical simulations based on the finite element method (FEM),whereas thick lines are derived from the line-current model using dif-ferent approximations: (a) approximate effective antenna length in-crease ΔLapp (magenta lines) and numerically calculated effectiveantenna length increase ΔLFEM (green lines) at the antenna ends with-out radiation damping (r � 1); (b) approximate effective antennalength increase ΔLapp and radiation damping based on the reflectionmagnitude rapp (orange lines) as well as numerically calculated effectiveantenna length increase ΔLFEM and radiation damping based on rFEM(blue lines).


Î�z;ω; k∥� ≈ Î��z;ω; k∥� � Î −�z;ω; k∥� ≡ Î p�ζ;ω; k∥�,(29)

which is a function of ζ ≡ kpz. This functional behavior in thez direction due to the dominating surface plasmon polariton isvalid even beyond the limits of the line-current model whenbeing close to a resonance. Therefore, we may approximatethe electric field inside the layer including the weak radialdependence as

E intz �ρ, z� ≈ E int0 J0�κ2ρ�f �kpz�, (30)

E intρ �ρ, z� �1

κ22

∂E intz∂ρ∂z

� −E int0kpκ2

J1�κ2ρ�∂f∂ζ

, (31)

where f specifies the z dependence of the fields. For smallvalues of ρ, the z component is roughly constant while theρ component is much smaller for kpρ ≪ 1, thus revealingthe basic assumption of the line-current model:

E intz �ρ, z� ≈ E int0 f �kpz�, (32)

E intρ �ρ, z� ≈ −E int0kpρ2

∂f∂ζ

: (33)

Using Eqs. (21) and (29) in Eq. (33) in combination withthe boundary condition for the radial field component, the ρcomponent in the exterior yields

E extρ �ρ, z� ≈ε2ε1

kpR2

∂I p∂ζ

H �1�1 �κ1ρ�H �1�1 �κ1R�

ΔkE inc0 cos θ: (34)

The z component in the exterior is approximately given by

E extz �ρ, z� ≈ −Î p�ζ;ω; k∥�H �1�0 �κ1ρ�H �1�0 �κ1R�

ΔkE inc0 cos θ: (35)

It turns out that j∂Îp∕∂ζj ≫ j2ε1Î p∕kpRε2j, so that thedominant field component in the exterior is the ρ component.

As mentioned above, the strength of the near field is impor-tant for antenna-assisted surface-enhanced infrared absorptionspectroscopy. In this case, a thin layer of molecules is placeddirectly on the antenna surface. If the layer is thinner thanthe decay length of the surface plasmon polariton outsidethe antenna, the ρ component in Eq. (34) is roughly constantin the radial direction. Defining V m as the volume of the mol-ecules, we thus obtain for the normalized near-field intensity inthe volume of the moleculesZ

V mdV

Eρ�ρ, z�E inc0 cos θ

2 � jε2kpRj

2

4ε21jΔkj2SR

ZL∕2

−L∕2dz

∂Îp∂ζ

2

:

(36)

In Fig. 5, we are displaying the result of Eq. (36) at theresonance wavelength for varying antenna lengths and a fixedradius of 10 nm (blue line). The prediction of the line-currentmodel agrees well with finite element results of the correspond-ing antenna geometries (orange dots). For such thin rod anten-nas, the overall scaling follows a λ1.2 scaling (thin black dashedline), while there is a λ3 scaling for thicker antennas (for details,see [23]) with a radius of 100 nm that is well above theskin depth of our metal (25–27 nm in the wavelength rangeof interest). Interestingly, the line-current model allows for

predicting the λ3 scaling for thicker antennas, but it can nolonger predict the magnitude of the field enhancement (notshown here).

B. Third-Harmonic Generation

The full numerical simulation of third-harmonic generation israther time-consuming, even in the undepleted pump approxi-mation [37], where any back-action of the field at the thirdharmonic to the fundamental wavelength is neglected. The rea-son is that the wavelength at the third harmonic is one third ofthe incident wavelength, requiring a very fine mesh to resolveall features in the electromagnetic near fields. In the line-current model, all numerical simulations are effectively two-dimensional, thus reducing the numerical effort significantly.The third-harmonic intensity can be calculated using the reci-procity principle [38]. In a first step, the electric fields are cal-culated at the fundamental wavelength, i.e., at frequency ω.This provides the nonlinear polarization PTH at the third har-monic 3ω, which acts as a source for the emitted electric fieldETH at the third harmonic. Given another source j 0 thatgenerates a field E 0, the reciprocity theorem statesZ

dV j 0 · ETH � −i3ωZ

dV PTH · E 0: (37)

Assuming j 0 � ĵδ�r − r 0� with jĵj � 1 and r 0 being locatedin the homogenous surrounding of the wire antenna, theanalytic solution for the Green’s dyadic in homogeneous spaceresults in

E 0ext�r� � ik1Z 1�ĵ�1� ik1jRj − 1

k21jRj2

� R�R · ĵ� 3 − 3ik1jRj − k21jRj2

k21jRj4�eik1jRj

4πjRj

≈ ik1Z 1 ĵeik1r

0

4πr 0e−ik1 cos ϑ

0z ≡ ik1Z 1 ĵeik1r

0

4πr 0e−ik

0∥z , (38)

where R � r − r 0, and we have used the thin-wire approxima-tion as well as r 0 ≫ r. The field in Eq. (38) acts as an incidentfield at frequency 3ω, which can be put into the line-currentmodel in order to derive the internal field occurring in Eq. (37).

analytical model1.2

FEM

2 4 6 8 10 120

2

4

6

8

10

Wavelength (µm)

|E|

detargetnI2

). u.a(

Fig. 5. Wavelength scaling of the near-field intensity around anelongated rod antenna with a radius of 10 nm. The blue curve hasbeen obtained from the line-current model by evaluating Eq. (36)at the resonance wavelength for varying antenna lengths. The resultsare in good agreement with the numerical results derived by finiteelement modeling of the corresponding antenna geometries (orangedots). The thin black dashed line depicts a λ1.2 scaling as a guideto the eye.


In particular, we have to replace E inc0 cos θ in Eq. (21) byE 0ext · ẑ exp�ik 0∥z�, which provides for ĵ⊥r 0 the transverse fieldcomponents emitted at the third harmonic:

ĵ · ETH�r 0� ≈ �3k0�2eik1�3ω�r

0∕c

4πr 0χ�3�

× �Δk�ω�E inc0 cos θ�3Δk�3ω� sin ϑ 0

×V AL

ZL∕2

−L∕2dz�Î�z;ω; k∥��3Î�z; 3ω; −k 0∥�: (39)

The integral in Eq. (39) can be easily evaluated. The third-harmonic intensity emitted in a certain direction and polariza-tion is then proportional to jĵ · ETHj2.

Since Eq. (17) enters in Î�z;ω; k∥� and Î�z; 3ω; −k 0∥�, wededuce that the third-harmonic intensity becomes significantlyincreased due to a resonance at the fundamental harmonic, asÎ�z;ω; k∥� contributes with the third power in Eq. (39). Aresonance at the third harmonic increases the third-harmonicintensity as well, but it only enters with first power inÎ�z; 3ω; −k 0∥�. These findings are confirmed in Fig. 6, wherewe calculated the third-harmonic intensity emitted in the xdirection and z polarization using Eq. (39) for different antennalengths (blue lines). Black arrows indicate one third of the res-onance wavelength of the fundamental resonance at ω1,whereas red arrows depict the position of the third-order res-onance at ω3. For longer antennas, these resonances becomemore and more matched, i.e., 3ω1 ≈ ω3.

Neglecting the frequency dependence of χ�3�, the overallscaling of the third-harmonic intensity follows a λ−4 behavior,as shown by the orange line in Fig. 6. The reason is that theintegral in Eq. (39) is roughly constant over frequency. Thus,the only frequency scaling of the field comes from the prefactorω2, so that the third-harmonic intensity scales with ω4 ∝ λ−4.

In its range of applicability, the generalized Miller’s rule [39]provides

χ�3��3ω� ∝ χ�1��3ω��χ�1��ω��3: (40)For a Drude metal with χ�1��ω� ∝ ω−2, we thus obtain that

χ�3��3ω� ∝ ω−8. The third-harmonic intensity of the gold rodantenna is proportional to ω4 times the square value of χ�3�, sothat it scales as ω−12 ∝ λ12. Hence, the third-harmonic inten-sity grows with antenna length, until Miller’s rule breaks down.

4. SUMMARY

We have shown that, within the single-mode approximation,the line-current model is capable of predicting the optical prop-erties of thin elongated metal antennas. This requires, however,to accurately know the reflection coefficient of the propagatingsurface plasmon mode at the antenna ends. Simple models forthe reflection coefficient fail, while we can calculate it efficientlyby numerical approaches, such as the finite element method. Ithas to be emphasized that the resulting numerical effort is sig-nificantly lower than that required to simulate the optical prop-erties of the full three-dimensional antenna, because therequired numerical calculations for the line-current modelare effectively two-dimensional. Most importantly, the line-current model allows for covering a large range of antennalength with negligible computational effort, which can be usedfor predicting the wavelength scaling behavior of certain reso-nant properties, such as the near-field enhancement as well asthe third-harmonic intensity. For the latter, we obtain a λ−4

scaling from the line-current model when neglecting the fre-quency dependence of χ�3�. Combining this result with thegeneralized Miller’s rule, we obtain an overall λ12 scaling ofthe third-harmonic intensity.

Funding. Bundesministerium für Bildung und Forschung(BMBF) (13N9048, 13N10146); H2020 EuropeanResearch Council (ERC) (COMPLEXPLAS); Baden-Württemberg Stiftung (Internationale Spitzenforschung II);Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg (MWK) (Zukunftsoffensive IV); VolkswagenFoundation; Deutsche Forschungsgemeinschaft (DFG)(SPP1839); Alexander von Humboldt Foundation.

REFERENCES

1. E. M. Purcell, “Spontaneous emission probabilities at radio frequen-cies,” Phys. Rev. 69, 681 (1946).

2. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory ofthe spontaneous optical emission of nanosize photonic and plasmonresonators,” Phys. Rev. Lett. 110, 237401 (2013).

3. G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J.Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanismsof large Purcell enhancement in plasmonic nanoantennas,” Nat.Photonics 8, 835–840 (2014).

4. E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell fac-tor of open optical systems,” Phys. Rev. B 94, 235438 (2016).

5. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, andN. F. van Hulst, “Unidirectional emission of a quantum dot coupled to ananoantenna,” Science 329, 930–933 (2010).

6. D. Dregely, K. Lindfors, M. Lippitz, N. Engheta, M. Totzeck, and H.Giessen, “Imaging and steering an optical wireless nanoantenna link,”Nat. Commun. 5, 4354 (2014).

7. A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik,“Theory of circular dichroism of nanomaterials comprising chiral mol-ecules and nanocrystals: plasmon enhancement, dipole interactions,and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).

8. E. Hendry, T. Carpy, J. Johnston, M. Popland, R. V. Mikhaylovskiy,A. J. Lapthorn, S. M. Kelly, L. D. Barron, N. Gadegaard, and M.Kadodwala, “Ultrasensitive detection and characterization of biomole-cules using superchiral fields,” Nat. Nanotechnol. 5, 783–787 (2010).

9. M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “Therole of plasmon-generated near fields for enhanced circular dichroismspectroscopy,” ACS Photonics 3, 578–583 (2016).

10-8

10-610-410-2

Fundamental wavelength (µm)

ytisnetnicino

mr ah-drihT

-4

L =0.40 µm 0.5 0.6 0.8 11.5 2 2.5

3

Fig. 6. z-polarized third-harmonic intensity (blue lines) emitted inthe x direction by rod antennas of 10 nm radius and varying lengthsbetween 400 nm and 3 μm. The results have been obtained from theline-current model and the reciprocity theorem for χ�3� � 1. The thickorange line is a guide to the eye for the asymptotic scaling of the maxi-mum intensity. Black arrows indicate one third of the fundamentalresonance wavelength, whereas red arrows denote the resonance wave-length of the third-order resonance.


10. W. Zhang, T. Wu, R. Wang, and X. Zhang, “Surface-enhanced circulardichroism of oriented chiral molecules by plasmonic nanostructures,”J. Phys. Chem. C 121, 666–675 (2016).

11. V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinarymagneto-optical effects and transmission through metal-dielectricplasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).

12. J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov,B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giantenhancement of thin-film faraday rotation,” Nat. Commun. 4, 1599(2013).

13. D. Floess, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Lorentz non-reciprocal model for hybrid magnetoplasmonics,” Phys. Rev. Lett.117, 063901 (2016).

14. H. Aouani, M. Navarro-Cia, M. Rahmani, T. P. H. Sidiropoulos, M.Hong, R. F. Oulton, and S. A. Maier, “Multiresonant broadband opticalantennas as efficient tunable nanosources of second harmonic light,”Nano Lett. 12, 4997–5002 (2012).

15. K. Thyagarajan, S. Rivier, A. Lovera, and O. J. Martin, “Enhancedsecond-harmonic generation from double resonant plasmonicantennae,” Opt. Express 20, 12860–12865 (2012).

16. H. Aouani, M. Rahmani, M. Navarro-Cia, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductornanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol.9, 290–294 (2014).

17. M. Hentschel, T. Utikal, H. Giessen, and M. Lippitz, “Quantitativemodeling of the third harmonic emission spectrum of plasmonic nano-antennas,” Nano Lett. 12, 3778–3782 (2012).

18. B. Metzger, M. Hentschel, T. Schumacher, M. Lippitz, X. Ye, C. B.Murray, B. Knabe, K. Buse, and H. Giessen, “Doubling the efficiencyof third harmonic generation by positioning ITO nanocrystals into thehot-spot of plasmonic gap-antennas,” Nano Lett. 14, 2867–2872(2014).

19. A. B. Dahlin, J. O. Tegenfeldt, and F. Höök, “Improving the instrumen-tal resolution of sensors based on localized surface plasmon reso-nance,” Anal. Chem. 78, 4416–4423 (2006).

20. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infraredperfect absorber and its application as plasmonic sensor,” Nano Lett.10, 2342–2348 (2010).

21. A. E. Cetin, A. F. Coskun, B. C. Galarreta, M. Huang, D. Herman, A.Ozcan, and H. Altug, “Handheld high-throughput plasmonic biosensorusing computational on-chip imaging,” Light Sci. Appl. 3, e122 (2014).

22. T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, andE. A. Muljarov, “From dark to bright: first-order perturbation theory withanalytical mode normalization for plasmonic nanoantenna arraysapplied to refractive index sensing,” Phys. Rev. Lett. 116, 237401(2016).

23. K. Weber, M. L. Nesterov, T. Weiss, M. Scherer, M. Hentschel, J.Vogt, C. Huck, W. Li, M. Dressel, H. Giessen, and F. Neubrech,

“Wavelength scaling in antenna-enhanced infrared spectroscopy:towards the far-IR and THz region,” ACS Photonics 4, 45–51 (2017).

24. T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev,W. Langbein, and E. A. Muljarov, “Analytical normalization of resonantstates in photonic crystal slabs and periodic arrays of nanoantennasat oblique incidence,” Phys. Rev. B 96, 045129 (2017).

25. F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri,and J. Aizpurua, “Resonant plasmonic and vibrational coupling in atailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101,157403 (2008).

26. F. Neubrech, C. Huck, A. Pucci, and H. Giessen, “Surface-enhancedinfrared spectroscopy using resonant nanoantennas,” Chem. Rev.117, 5110–5145 (2017).

27. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys.Rev. Lett. 98, 266802 (2007).

28. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich,and K. Kern, “Plasmonic nanowire antennas: experiment, simulation,and theory,” Nano Lett. 10, 3596–3603 (2010).

29. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorodantennas modeled as cavities for dipolar emitters: evolution of sub-and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).

30. R. Gordon, “Reflection of cylindrical surface waves,” Opt. Express 17,18621–18629 (2009).

31. S. B. Hassan, R. Filter, A. Ahmed, R. Vogelgesang, R. Gordon, C.Rockstuhl, and F. Lederer, “Relating localized nanoparticle resonan-ces to an associated antenna problem,” Phys. Rev. B 84, 195405(2011).

32. H. Jia, H. Liu, and Y. Zhong, “Role of surface plasmon polaritons andother waves in the radiation of resonant optical dipole antennas,” Sci.Rep. 5, 8456 (2015).

33. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry,“Optical properties of Au, Ni, and Pb at submillimeter wavelengths,”Appl. Opt. 26, 744–752 (1987).

34. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).35. T. Weiss, N. A. Gippius, G. Granet, S. G. Tikhodeev, R. Taubert, L. Fu,

H. Schweizer, and H. Giessen, “Strong resonant mode coupling ofFabry–Perot and grating resonances in stacked two-layer systems,”Photonics Nanostruct. Fundam. Appl. 9, 390–397 (2011).

36. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strongcoupling of single emitters to surface plasmons,” Phys. Rev. B 76,035420 (2007).

37. T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for an-alyzing higher harmonic generation in multilayer nanostructures,”J. Opt. Soc. Am. B 27, 1118–1130 (2010).

38. K. O’Brien, H. Suchowski, J. Rho, A. Salandrino, B. Kante, X. Yin, andX. Zhang, “Predicting nonlinear properties of metamaterials from thelinear response,” Nat. Mater. 14, 379–383 (2015).

39. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).


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