Generation of Hankel-type surface plasmon polaritons in the vicinity of a metallic nanohole

S. Nerkararyan and Kh. NerkararyanDepartment of Radiophysics, Yerevan State University, 1 Alek Manoogian, Yerevan 375049, Armenia

N. Janunts and T. PertschInstitute of Applied Physics, Friedrich-Schiller-Universität Jena, Max Wien Platz 1, 07743 Jena, Germany

�Received 6 September 2010; published 7 December 2010�

It is shown that the electromagnetic fields of surface plasmon polaritons �SPPs� generated around a nano-hole, milled in a metal film, can be described by Hankel functions. These SPPs are dipole active and can beexcited by a linearly polarized electromagnetic plane wave under normal incidence �with respect to the metalsurface�. Two kinds of Hankel-type SPPs are generated simultaneously around a nanohole: inward and outwardpropagating with respect to the nanohole. The wave fields of the Hankel-type SPPs increase anomalously in theclose vicinity of the nanohole and exceed considerably that of the incident wave. It is shown analytically thatthe excitation cross section of Hankel-type SPPs, around a nanohole with excitation area limited by circularboundary, is significantly higher compared to the case of an infinite metal surface. We conclude that theunusually high transmission of light through arrays of nanoholes milled in a metal film can be ascribed to theexcitation of inward Hankel-type SPPs, which transfer the energy toward the nanohole region.

DOI: 10.1103/PhysRevB.82.245405 PACS number�s�: 73.20.Mf, 78.67.�n, 42.79.Ag, 68.47.De

I. INTRODUCTION

Optical transmission of subwavelength apertures has beenof interest in physical science for many years.1–5 The studyof these structures has greatly intensified since the pioneer-ing experimental work of Ebbesen and co-workers on en-hanced optical transmission through subwavelength holearrays.6,7

Considerable experimental and theoretical efforts havebeen devoted to understanding transmission through periodicarrays of nanoapertures or through a single nanoaperturewith surrounding periodic corrugations.8–10 The observationsshowed that transmission can be either enhanced or sup-pressed depending on the wavelength for periodic and qua-siperiodic arrays, as well as for arrays with translational androtational symmetry.11–13 The transmittance of these arrayshas been explained by coupling of incident light with SPPsthrough momentum matching provided by the �long-range�periodicity of the structures.12–14 Recently it has been dem-onstrated that significant spectral modulation of the transmis-sion can be achieved also in a specifically tailored quasiperi-odic geometry, where translational periodicity and long-range order is totally absent.14,15 In this case the SPPs areexcited not by the grating but by the individual nanoholes.

Generally speaking, both SPPs excited by periodicity ofthe apertures and by the individual apertures can contributeto an enhanced optical transmission. The experimental andnumerical studies of SPPs generated by single nanoapertureshave been stimulated by their essential role in the opticaltransmission through arrays of nanoapertures. The features ofcylindrical wave fields generated in the vicinity of nanoholesdue to the scattering of the incident wave have been investi-gated using analytical and numerical methods.16–20

The generation and focusing of SPPs by a single circularslit called plasmonic lens has been demonstrated. The focalposition of the plasmonic lens can be tuned by changing theincidence angle of the beam.21 Recently it has been shownthat excitation with radially polarized light delivers a tighter

focus and higher energy in the plasmonic lens compared tolinearly polarized light.22 Lezec and co-workers have madedetailed studies on understanding the excitation, propagation,and attenuation mechanism of SPPs excited by singlenanoslits as well as the phase shift between SPPs and theincident excitation light.14,23,24 It has been reported that acircular nanohole in a thin metal film acts as a pointlike SPPsource at normal or oblique incidence of linearly polarizedlight.25 Near-field optical measurements have shown the di-polar character of the SPPs launched by nanoholes in a thinmetal film.26,27

In this paper we study analytically the properties of SPPswhose wave fields are described by Hankel functions and thepossibilities of their excitation in the vicinity of a nanohole.In order to discriminate between the different contributionsof the SPP to the optical transmission through subwavelengthapertures, one has to answer two important questions. �1�Which kind of modes do exist around these apertures? and�2� whether they can be excited by normally incident light?A nanohole having a cylindric symmetry supports solutionsof the wave equation describing the wave fields of SPPs bycylindrical functions, in particular, by the Hankel functions.These functions characterize the behavior of a SPP generatedat the metal surface in close vicinity of a nanohole up to itsrim. The field values of Hankel-type SPPs increase ap-proaching the nanohole but remain finite at the nanohole rim.Therefore, though Hankel functions diverge at the origin pre-sented by the center of the nanohole, they give a valid pic-ture of the physical processes for the case of a nanohole offinite size. Some of the Hankel-type modes are dipole activeand can therefore interact with incident plane waves.

To further elucidate the nature of the Hankel modes let usconsider a pointlike dipole source situated at the center of astructure and its radiation is described by the functions withan infinite value at the origin. Since electrodynamic pro-cesses are reversible, the existence of waves propagating notonly from the center but also toward it can be reasonablypresumed. Given that, a strong light localization at the center

PHYSICAL REVIEW B 82, 245405 �2010�

1098-0121/2010/82�24�/245405�6� ©2010 The American Physical Society245405-1

is expected. The realization of such strong localization inthree dimensions is very difficult due to a generally complexphase-amplitude distribution of localizing waves whereas inthe two-dimensional case the realization of such a process isquite permissible. We note that the field description in theregion of the nanohole itself is beyond the scope of our con-siderations.

The SPPs described by Hankel functions have been pre-viously discussed in the context of light diffraction by thenanohole, where the nanohole was considered as a secondarysource of SPPs.8,28 Here we focus on the analytical descrip-tion of the generation of Hankel-type SPPs transferring theenergy toward the nanohole. We believe these are the wavesstrongly contributing in the field enhancement in the imme-diate vicinity of the nanohole.

II. HANKEL-TYPE SPP

In this chapter we will show that a metal surface contain-ing a nanohole supports Hankel-type SPPs propagating in-ward and outward with respect to the nanohole. We considerthe problem in a cylindrical coordinate system �� ,� ,z� withthe origin placed at the center of the nanohole, the coordinatez oriented perpendicular to the surface and z=0 being thedielectric-metal interface. The dielectric occupies thehalf-space associated with z�0 and is described by thedielectric permittivity �d. The metal in the half-spacecorresponding to z�0 has the dielectric permittivity�m ��m=�m� + i�m� , �m� �0� and the nanohole is characterizedby the radius r.

We consider the surface waves of TM type with the zcomponent of the magnetic field Hz=0. The z components ofthe electric field of a SPP can be defined by the followingwave equation:

�2Ez

�z2 +�2Ez

��2 +1

�

�Ez

��+

1

�2

�2Ez

��2 −�d,m

c2

�2Ez

�t2 = 0. �1�

Here c is the speed of the light. In the region of the dielectric�z�0 and ��r�, this equation has two linearly independentsolutions, which are described by Hankel functions of thefirst and second kinds

Ez,n�1,2� = DHankn

�1,2����kd2 + �d

2�e�−�dz+it� cos n� , �2�

Where Hankn�1��x�=Jn�x�+ iYn�x�, Hankn

�2��x�=Jn�x�− iYn�x�,where Jn�x� and Yn�x� are integer-order Bessel functions ofthe first and second kinds, respectively, kd=�d2 /c2 and isthe wave frequency. The constants �d and D are to be deter-mined from the boundary conditions. The other componentsof the wave fields in this region can be calculated via Max-well equations and the expressions in Eq. �2�. In the sameway the electric fields in the metal region �z�0,��r� canalso be described by Hankel functions

Ez,n�1,2� = CHankn

�1,2����km2 + �m

2 �e�+�mz+it� cos n� , �3�

where km=�m2 /c2 and the constants �m and C are deter-mined from the boundary conditions.

A typical three-dimensional surface plot of the real part ofthe electric field Re�Ez� around the nanohole is shown in Fig.

1, where the z=0 plane coincides with the metal-dielectricinterface. The Hankel-type SPP has a dipole character andpropagates along the polarization direction of the incidentbeam shown by the arrow in the picture.

Applying the boundary conditions �the continuity of thewave fields at the metal-dielectric interface at z=0� leads tothe following expressions:

D�d = − C�m,

D�d = + C�m,

kd2 + �d

2 = km2 + �m

2 . �4�

We can obtain the dispersion relation for the Hankel-typeSPP by substituting the expressions for km,d in Eq. �4�

�d

�d=

�m

��m� �,

�d =

c

�d

���m� � − �d

,

�m =

c

�m�

���m� � − �d

. �5�

Note that kd,m2 +�d,m

2 = 2

c2

�d�m

�d+�m= �kSPP�2, where kSPP is the

wave vector of the planar SPP.In the wave zone, where ��1 and ��=�kSPP, the Hankel

functions can be approximated as29

Hankn�1����� �� 2

���exp�+ i��� −

n�

2−

�

4� , �6�

Hankn�2����� �� 2

���exp�− i��� −

n�

2−

�

4� . �7�

From Eqs. �6� and �7� it can be concluded that the wavefields containing the functions Hankn

�1� describe inwardpropagating SPPs while the fields containing Hankn

�2� de-scribe outward propagating SPPs. In the following we willcall them converging and diverging Hankel-type SPPs, re-spectively.

Z YX

E

FIG. 1. �Color online� Three-dimensional plot of the z compo-nent of the electrical field �Ez� of a Hankel SPP in the vicinity of ananohole plotted at the gold/air interface calculated for the param-eters �=532 nm and r=50 nm. The arrow shows the polarizationdirection of the incident beam which is parallel to the x axis. Theamplitude of the electrical field of the Hankel SPP significantlyincreases in the immediate vicinity of the nanohole.

NERKARARYAN et al. PHYSICAL REVIEW B 82, 245405 �2010�

245405-2

The converging Hankel-type SPPs transfer wave energyto the nanohole region. Therefore, in the area of the nanoholethe wave fields increase substantially. This fact also followsfrom the field expression in the immediate vicinity of thenanohole which corresponds to the limiting case of �� 1

Hankn�1����� � −

i

���n����

2�−n

�n � 0� , �8�

where ��n� is the gamma function. In the next section it willbe shown that the mode associated with n=1 is the onlymode excited in the case of a normally incident and linearlypolarized plane wave.

III. EXCITATION EFFICIENCY OF THEHANKEL-TYPE SPP

We have shown in the previous section the existence of anentire class of converging and diverging Hankel-type SPPs atthe metal surface around a nanohole. Now we will focus oncalculating the cross section and the efficiency of the con-verging Hankel-type SPP excitation via external currentsproduced by a linearly polarized electromagnetic wave illu-minating the metal surface containing a nanohole. For sim-plicity we will consider the case of normal incidence withthe polarization direction parallel to the x axis. The electricfields of the incident beam then have the following form:

Ex = Ae−i�t−kdz� + Be+i�t−kdz� �z � 0� , �9�

Ex = Qe−it+qz with q2 = kx2 −

2

c2 �d �z � 0� . �10�

Relationships between the amplitudes are found from theboundary conditions:

Q = −2i��d

���m� � − i��d

A ,

B = −���m� � + i��d

���m� � − i��d

A . �11�

The components of the current density in the metal can beobtained from Ohm’s law

j� = +�m�

4�Qe−it+qz cos��� ,

j� = −�m�

4�Qe−it+qz sin��� . �12�

Hankel-type SPPs are excited by the external currents gen-erated in the metal by a linearly polarized incident planewave. Formally the procedure to determine the SPP wavefields excited by the external currents is identical to the de-termination of the fields in metallic waveguides. This proce-dure is described in detail in Ref. 30 and here we will onlyshow the final formulas. Inside the metal and in the vicinityof the nanohole, the electric and magnetic components of the

total field of converging Hankel SPPs can be decomposedinto the sum of all harmonics of the converging Hankel SPPs

E� = n=0

�

cn�1�E� n

�1�, H� = n=0

�

cn�1�H� n

�1�, �13�

where

cn�1� =

1

N�

V

j�E� n�2�dV �14�

and

N =c

4�� ��E� n

�2�H� n�1� − �E� n

�1�H� n�2� ���̂dS . �15�

Here the components E� n�1,2� and H� n

�1,2� are determined fromEqs. �2� and �3� via Maxwell equations. In Eq. �14� the in-tegration is carried out over the volume r����,0���2�, and z�0. The factors cn

�1� are proportional to theexcitation cross section of the Hankel SPPs and represent theoverlap integral of the generated current and the HankelSPPs. In Eq. �15� N is a normalizing coefficient and theintegration is done over any surface �=const, where ��̂ is theunit vector.

After some mathematical calculations, using formulas �2�,�3�, and �14�, we obtain for the factors cn

�1� the followingexpression:

cn�1� =

�

2

�m� �����m��� + i��d�

����m��� + �d�2�����m��� + ����m��� − �d�

A

D�n1Hankn

�1�

��kSPPr�kSPPr , �16�

where �n1 is the Kronecker symbol. Therefore, under normalincidence with respect to the metal surface the Hankel SPPwith the order number n=1 is the only mode excited.

The cross section of the excitation is the ratio of thepower of the converging Hankel SPP and the intensity of theincident beam

���r,�� =�2

4

c2

2

��m� �2��d����m��� −��d�

���m�������m��� + �d�2�����m��� + ����m��� − �d�2�Hank1

�1�

��kSPPr�kSPPr�2. �17�

This formula shows the excitation cross section in the case ofa homogeneous and infinite illumination of the metal surface.In the case of longer wavelengths, when �d ��m� � and smallnanohole radius �rkSPP 1�, formula �17� can be simplifiedsubstantially to

�� =c2

42

�m�2 · ��d

��m� �7/2 . �18�

The cross section decreases strongly with increasing ��m� ��corresponding to increasing wavelength� due to two rea-sons. First, according to Eqs. �10� and �11� the amplitude ofthe electric field decreases in the metal with increasing ��m� �.Second, the q and �m parameters increase with increasing��m� � �see Eqs. �5� and �10� which decreases the volume of

GENERATION OF HANKEL-TYPE SURFACE PLASMON… PHYSICAL REVIEW B 82, 245405 �2010�

245405-3

interaction �overlap� of the incident wave and the HankelSPP.

The excitation cross section can be increased significantlyby limiting the SPP excitation area, which can be achievedeither by limiting the diameter of the incident beam or bylimiting the metal surface around the nanohole. The latter iseasier to realize experimentally. Hence we will consider thestructure shown in Fig. 2, which is a metallic disklike struc-ture with a nanohole in the center. The Hankel SPPs areexcited in the area enclosed by the nanohole radius r and thedisk radius R. In order to study the excitation of Hankel SPPssolely two assumptions are made. At first, the disk height isassumed to be large enough to provide a good separationfrom the metal substrate in order to neglect the influence ofSPPs excited in the substrate region. At second, the rims ofthe nanohole and the disk are assumed to be blunt in order toneglect the excitation of the SPP at the disk surface due tolight scattering at the rims. In the frame of these assumptionswe obtain an expression for the cross section of the HankelSPP excitation

��R,r,�� = ���r,��

��Hank1

�1��kSPPR�kSPPR − Hank1�1��kSPPr�kSPPr�2

�Hank1�1��kSPPr�kSPPr�2

.

�19�

The dependence of the normalized cross sections �̃�R ,r ,��=��R ,r ,�� /���r ,�� on the disk radius and wavelength isshown in Fig. 3 for the case of homogeneous illumination.Gold is used as a metal and air as a dielectric in thecalculations.31 �̃�R ,r ,�� represents the enhancement factorof the Hankel SPP excitation cross section in a metallic diskwith radius R compared to the case of an infinite metal sur-face. At short wavelengths �̃�R ,r ,�� does not benefit muchfrom the increasing disk radius because of strong damping ofthe SPP while for large disk radii the longer wavelengthslead to a larger enhancement factor. This fact is attributed tothe larger propagation length at longer wavelengths. Al-though the excitation efficiency increases with increasingdisk radius it exhibits a maximum around the value equal tothe propagation length of the planar SPP represented by theorange curve in Fig. 3. The characteristic behavior of thenormalized cross section depending on the disk radius at awavelength of �=700 nm and a nanhole radius ofr=50 nm is shown in Fig. 4. �̃�R ,r ,�� exhibits fast oscilla-tions with the period of the wavelength of the planar SPP and

has a total maximum at the disk radius equal to the propaga-tion length �LSPP� of the planar SPP. In this case the excita-tion cross section of Hankel SPPs can be increased by twoorders of magnitude compared to the case of infinite illumi-nation. The inset in Fig. 4 shows the zoomed region of theplot in the vicinity of the disk radius equal to LSPP �shownwith orange line� to demonstrate the fast oscillations whichare the results of SPP interference. The blue and red circlesshow the local minima and maxima, respectively, separatedby half the SPP wavelength. The SPPs generated at the disksurface with radii resulting in local maxima are in phasewhile for radii leading to local minima the generated SPPsare out of phase. The amplitude of the fast oscillations hasthe same behavior as the normalized cross section. It in-

FIG. 2. �Color online� �a� Three-dimensional sketch and �b�two-dimensional cross section of the metallic disk structure geom-etry, which provides limited area excitation of Hankel-type SPP.The SPP is generated at the disk surface enclosed by the radii of thenanohole r and the disk R.

Disc radius R [µµµµm]

Normalized cross section ( , , )R rσ λ�

1008040 6020

1350

0

1000

900

800

700

600

500Wavelengthλ[nm]

FIG. 3. �Color online� Density plot of the normalized excitationcross section of the Hankel-type SPP versus the disk radius and thewavelength in the case of homogeneous illumination and assuminga nanohole radius of r=50 nm. The orange curve shows the propa-gation length of the planar SPP �locus of points��2�Im kSPP���� −1 ,���. �̃�R ,r ,�� represents the enhancement factorof the excitation cross section of a Hankel SPP in a metallic diskwith radius R compared to the case of an infinite metal surface,when both surfaces contain nanoholes with radius r.

0 50 100 150 200 2500

20

40

60

80

100

120

Disc radius R �Μm�

Nor

mal

ized

cros

sse

ctio

n��R

,r,Λ�

25 26 27 28

90

100

110

120

130

ΛSPP

r�50 nmΛ�700 nmLSPP�26.4 Μm

FIG. 4. �Color online� Dependence of the normalized excitationcross section of the Hankel-type SPP on the disk radius R in thecase of homogeneous illumination at the wavelength of�=700 nm and a nanohole radius of r=50 nm. The cross sectionexhibits fast oscillations as a function of the disk radius �see inset�with the period of the planar SPP wavelength and has a total maxi-mum at the propagation length of the planar SPP �orange line�. TheHankel SPPs generated at disk radii resulting in local maxima of thecross section �blue circles in the inset� are in phase. In case of radiiwith local minima of the cross section �red circles� the Hankel SPPsare out of phase.

NERKARARYAN et al. PHYSICAL REVIEW B 82, 245405 �2010�

245405-4

creases with the disk radius, reaches a maximum value whenthe radius equals the LSPP and then decreases to unity. Asimilar effect has been observed in the slit-nanoholestructure.32

The excitation efficiency � of Hankel SPPs is determinedas the ratio of the Hankel SPP power generated at the disksurface and the power of the incident beam on the disk.Since the cross section ��R ,r ,�� is proportional to the Han-kel SPP power, the excitation efficiency � then can be de-fined as

��R,r,�� =��R,r,��

��R2 − r2�. �20�

��R ,r ,�� shows the portion of the energy of the incidentbeam transferred into the Hankel SPP at the disk surface. Adensity plot of the excitation efficiency on a logarithmicscale is shown in Fig. 5�a� as a function of wavelength � anddisk radius R. The observed oscillating character and thecontinuous decrease in the excitation efficiency with increas-ing wavelength and disk radius are a result of the similarbehavior of the tangential components of Hankel SPPs. Theefficiency is the highest around the wavelength of 500 nmwhich is caused by the strong absorption of gold in thisspectral domain. Furthermore, it sharply decreases with in-creasing wavelength due to the weak absorption at longerwavelengths as shown in the Figs. 5�a� and 5�b� �see alsoexpression �18� . In other words, the efficiency is propor-tional to the factors cn

�1� which are the overlap integrals of theHankel-type SPPs and the current density generated in themetal by the incident light. Therefore the stronger the ab-

sorption the stronger is the generated current and the higheris the efficiency. The comparison of Figs. 5�c� and 5�d� dem-onstrates that the peak value of the excitation efficiency at�=480 nm is about 4000 times larger than at �=1000 nm.The strong absorption will increase the excitation efficiencybut it will also damp the excited SPP. The efficiency at�=480 nm drops from its peak value of 250 to almost 0 at adisk radius of about R=2.5 �m while at �=1000 nm itdrops from 0.075 to 0.005 �15 times� at around R=8 �m.The reason of the slow decrease in ��R ,r ,�� over the diskradius at larger wavelengths is the longer propagation lengthcompared to that at shorter wavelengths. Consequently SPPsgenerated at longer distances from the nanohole contributemore efficiently to the formation of the Hankel SPPs.

However, a significant drop of the efficiency occurs al-ready at distances of about two to three times of surfacePlasmon wavelength �SPP. This suggests that the major partof the Hankel-type SPP generation occurs in close vicinity ofthe nanohole. Therefore we claim that this is the regionwhere the Hankel-type SPP originates from.

It is necessary to compare our current work with previousstudies of SPPs generated due to the light scattering from acylindrically symmetric nanoapertures. First, we investigatethe generation of SPPs which transfer the energy to the nano-hole region, while in the scattering problems, mostly the caseof SPPs escaping from the nanohole is discussed. Second, wediscuss the case of normal incidence of the beam on thesurface when generated SPPs can be described by the Hankelfunction of the first order. In the case of inclined illuminationthe excited SPPs can be described by the infinite sum of allorders of Hankel functions. In the latter case the wave fieldscan be quite different from the case discussed in our paper.18

Finally, the nanohole can generate also radiative modeswhich are beyond the scope of this work. It has been shownthat in the immediate vicinity of the nanohole the total fieldis different from the SPPs fields, and this difference de-creases quickly for larger wavelengths.17,19,32 Since the Han-kel SPPs decay rapidly other diffracted waves such as“creeping waves” or “Norton waves” can dominate in thetotal field also at large distances from the nanohole.19,32

IV. CONCLUSION

In conclusion �a� we have shown that the wave fields ofSPPs, generated at a metal surface possessing a nanohole,can be described by Hankel functions. Simultaneously twokinds of Hankel-type SPPs are excited around the nanohole.One of them converges in the nanohole while the second onediverges from the nanohole. In the close vicinity of the nano-hole, the wave fields of the SPPs increase anomalously andconsiderably surpass the values of the wave fields of theincident beam. �b� It has been found that Hankel-type SPPsare dipole active and can be excited by an incident linearlypolarized electromagnetic plane wave. They originate in theclose vicinity of the nanohole. �c� We have applied an ana-lytical method, developed in the metallic waveguide theory,to calculate the excitation cross section and to derive explicitanalytical expressions for the excitation efficiency. �d� It isfound that the efficiency can be increased by several orders

500 600 700 800 900 10000

1

2

3

4

Wavelength Λ �nm�

Exc

itat

ion

effi

cien

cy�1

04 Η�

r�50nm

R�1800nm

1 2 3 4 5 6 7 80.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Disc radius R �Μm�

Exc

itat

ion

effi

cien

cy�1

04 Η�

r�50nm

Λ�1000nm

1 2 3 4 5 6 7 80

50

100

150

200

250

Disk radius R �Μm�

Exc

itat

ion

effi

cien

cy�1

04 Η�

r�50nm

Λ�480nm

FIG. 5. �Color online� �a� Density plot of the excitation effi-ciency distribution on a logarithmic scale �log10��� versus thewavelength � and the disk radius R. �b� Dependence of the effi-ciency on wavelength in the case of a nanohole radius ofr=50 nm, disk radius R=1800 nm �along the blue line�. �c� and �d�show the efficiency dependence on the disk radius for a nanoholeradius of r=50 nm and wavelengths �c� �=1000 nm and �d��=480 nm, respectively.

GENERATION OF HANKEL-TYPE SURFACE PLASMON… PHYSICAL REVIEW B 82, 245405 �2010�

245405-5

of magnitude when the Hankel SPPs are excited in the lim-ited region of the surface, particularly at the surface of adisklike structure. The efficiency reaches its maximum val-ues around disk radii equal to the propagation length of aplanar SPP. It also exhibits fast oscillations with a period of�SPP caused by the interference of SPPs. �e� We claim thatthe enhanced optical transmission through arrays of nano-

holes can be attributed to the excitation of the convergingHankel-type SPP which delivers energy to the nanohole.

ACKNOWLEDGMENTS

This research was supported by the following grants:ANSEF-2202, BMBF-PhoNa, and DFG-Nano Guide.

1 H. A. Bethe, Phys. Rev. 66, 163 �1944�.2 C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 �1954�.3 A. V. Zayats, I. I. Smolyaninovb, and A. A. Maradudin, Phys.

Rep. 408, 131 �2005�.4 F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L.

Kuipers, Rev. Mod. Phys. 82, 729 �2010�.5 R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, Laser

Photonics Rev. 4, 311 �2010�.6 T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and H. J.

Wolff, Nature �London� 391, 667 �1998�.7 H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-

Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, Science 297,820 �2002�.

8 S.-H. Chang, S. K. Gray, and G. C. Schatz, Opt. Express 13,3150 �2005�.

9 F. J. García de Abajo, Rev. Mod. Phys. 79, 1267 �2007�.10 H. Liu and Ph. Lalanne, Nature �London� 452, 728 �2008�.11 F. Przybilla, C. Genet, and T. W. Ebbesen, Appl. Phys. Lett. 89,

121115 �2006�.12 M. Sun, J. Tian, Z.-Y. Li, B.-Y. Cheng, D.-Z. Zhang, A.-Z. Jin,

and H.-F. Yang, Chin. Phys. Lett. 23, 486 �2006�.13 T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, Nature

�London� 446, 517 �2007�.14 D. Pacifici, H. J. Lezec, L. A. Sweatlock, R. J. Walters, and H. A.

Atwater, Opt. Express 16, 9222 �2008�.15 C. Rockstuhl, F. Lederer, Th. Zentgraf, and H. Giessen, Appl.

Phys. Lett. 91, 151109 �2007�.16 F. de León-Pérez, G. Brucoli1, F. J. García-Vidal, and L. Martín-

Moreno, New J. Phys. 10, 105017 �2008�.17 J. Alegret, P. Johansson, and M. Käll, New J. Phys. 10, 105004

�2008�.18 A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, Phys.

Rev. Lett. 78, 4269 �1997�.19 A. Yu. Nikitin, F. J. García-Vidal, and L. Martín-Moreno, Phys.

Rev. Lett. 105, 073902 �2010�.20 H. Liu, Opt. Lett. 35, 2876 �2010�.21 Zh. Liu, J. M. Steele, H. Lee, and X. Zhang, Appl. Phys. Lett.

88, 171108 �2006�.22 G. M. Lerman, A. Yanai, and U. Levy, Nano Lett. 9, 2139

�2009�.23 G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J.

Weiner, and H. J. Lezec, Nat. Phys. 2, 262 �2006�.24 J. Weiner, Opt. Express 16, 950 �2008�.25 L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H.

Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kim-ball, Appl. Phys. Lett. 85, 467 �2004�.

26 L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U.Welp, D. E. Brown, and C. W. Kimball, Nano Lett. 5, 1399�2005�.

27 H. Gao, J. Henzie, and T. W. Odom, Nano Lett. 6, 2104 �2006�.28 J. Beermann, A. Evlyukhin, A. Boltasseva, and S. I. Bozhevol-

nyi, J. Opt. Soc. Am. B 25, 1585 �2008�.29 M. Abramowitz and I. A. Stegun, Handbook of Mathematical

Functions with Formulas and Mathematical Tables �Dover, NewYork, 1965�.

30 L. A. Vaynshteyn, Electromagnetic Waves �Sovietskoye Radio,Moscow, 1957�.

31 P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 �1972�.32 P. Lalanne and J. P. Hugonin, Nat. Phys. 2, 551 �2006�.

NERKARARYAN et al. PHYSICAL REVIEW B 82, 245405 �2010�

245405-6