The discovery of the so-called extraordinary lighttransmission through subwavelength hole arrays1
triggered an avalanche of studies with interest ex-tending far beyond the physics community. Recentbiophysical applications used single apertures asnanocavities in trapping single biomolecules and intheir investigation. Field enhancement inside thenanostructures is aimed at further reducing the in-vestigated volume, as well as for use in near-fieldmicroscopy applications, which may require using acoaxial intrusion inside the aperture2 or structuringits neighborhood by surface corrugation.3,4
There exist many numerical studies of periodichole arrays, based mainly on grating theories, butonly few have been devoted to isolated single (orstructured) apertures in real metals. The precision offinite-difference time-domain methods is restrictedby the small dimensions of the structures comparedwith the wavelength, requiring special approaches.This method has been used with success for elliptical
apertures by application of local grid refinement.5Recently a differential theory of diffraction in cylin-drical geometry that can be used to model cylindricalobjects of finite length was developed.6 Its applicationto single-hole diffraction7 demonstrated the key roleplayed by surface plasmons in a near-field distribu-tion, even in the absence of periodicity.
Our aims in the present paper are to study in detailthe specific character of a plasmon-type homogeneoussolution of the diffraction problem in cylindrical sym-metry, to develop a perturbation analysis of singleand structured apertures, to compare the analyticalconclusions with rigorous electromagnetic computa-tions, and to use them for maximizing the plasmonexcitation. The structure under consideration is sche-matically represented in Fig. 1. It consists of a singlecircular aperture surrounded by concentric lamellargrooves of circular form. The grooves are assumed tomatch the phase of the incident wave to the plasmonsurface wave to enhance it, an expectation well mo-tivated by numerous effects related to resonanceanomalies in diffraction gratings. However, it is notclear what geometry is necessary for this phasematching when cylindrical symmetry exists. Using adirect analogy with grating structures, one would ex-pect that the grooves should be equidistant. However,such is not the case, as we show below.
Figure 2 illustrates the spectral dependence of thez component of the plasmon wave on a 230 nm thicksilver screen, calculated outside the corrugated re-gion (at x � 4 �m) for four 30 nm deep channelssituated about a circular aperture with a 50 nm ra-dius (the second curve, representing the spectral be-havior of coupling integral F0, is discussed further in
E. Popov ([email protected]), M. Nevière, and A.-L. Fehrem-bach are with the Institut Fresnel, Unité Mixte de RechercheAssociée au Centre National de la Recherche Scientifique 6133,Université de Provence, Faculté des Sciences et Techniques deSt. Jérôme, Avenue Escadrille Normandie Niémen, 13397 Mar-seille Cedex 20, France. N. Bonod is with Commissariat àl’Energie Atomique, Centre d’Etudes Scientifiques et Techniquesd’Aquitaine, B.P. 2, 33114 Le Barp, France.
Received 22 March 2005; revised manuscript received 9 May2005; accepted 9 May 2005.
the text). The channel walls are equidistant and haveradii Rj � �j � 1� � 235 nm �j � 2, . . . 9�, whichcorresponds to a quasi-period of 470 nm. These val-ues are chosen in order that the aperture will have aradius approximately ten times smaller than the op-timum wavelength, a ratio that is of practical impor-tance in biomolecular studies and near-fieldmicroscopy. According to expectations from the one-dimensional grating theories, this geometry has toresonantly (in-phase) excite a plasmon surface wavethat has propagation constant �p � ��d. When onetakes into account the refractive index of silver, thespectral maximum has to be close to 500 nm. How-ever, instead of a maximum, one finds a minimum. Inwhat follows we shall try first to explain this phe-nomenon and second to present a simple physicalmethod that will enable us to optimize plasmon ex-citation in cylindrical geometry. To this end we needfirst to discuss the diffraction of light and plasmonpropagation in cylindrical geometry. Although these
topics are not new,8–11 we find this necessary in orderto use the results in formulating a perturbative ap-proach to a physical explanation of numerical resultspresented in Section 5 below.
2. Propagation Equations and Their Solutions inCylindrical Coordinates
Because of the natural 2� periodicity with respect to�, the electric field vectorial components can be rep-resented by Fourier series in �:
Ej(r, �, z) � �n��
Ej, n(r, z)exp(in�), j � r, �, z,
(1)
and a similar expression could be written for mag-netic field H. In what follows, we assume that dielec-tric permittivity ε is piecewise constant in z andindependent of �. This independence ensures thatMaxwell equations and boundary conditions are de-coupled in �, i.e., that the propagation and scatteringof each field component Ej, n are carried out separatelyand independently for each value of n. Thus thethree-dimensional electromagnetic problem is re-duced to Nn two-dimensional problems, where Nn isthe number of terms retained when the series in Eq.(1) is truncated during the calculations.
Helmholtz equations in cylindrical coordinates (seeAppendix A) lead to coupling between Er, n and E�, n:
�E�, n �E�, n
r2 2in
r2 Er, n k2E�, n � 0,
�Er, n �Er, n
r2 �2in
r2 E�, n k2Er, n � 0,
where k � ���0 is the wave number and the Lapla-cian takes the form
� ��2
�r2 1r
�
�r 1
r2
�2
��2 �2
�z2. (3)
In what follows, k0 will denote the free-space wavenumber.
By making two linear combinations of Er, n and E�, n
it is possible to decouple Eqs. (2). Defining
E�, n � E�, n � iEr, n (4)
yields for Eqs. (2)
� �2
�r2 1r
�
�r �(n � 1)2
r2 k2 �2
�z2�E�, n(r, z) � 0.
(5)
Separating variables r and z makes it easy to identifyhere the differential equations that generate Besselfunctions of order n � 1, which are denoted hence-
Fig. 1. Schematic representation of a structured aperture: (a)general view, (b) cross section with the notation used in the text.
Fig. 2. Spectral dependence of |Ez| calculated at a point�4 �m, 0, 0� for a silver film with t � 230 nm thickness, R1
� 50 nm, four equidistant channels with h � 30 nm, and �
forth by �n�1. An infinite unperturbed plane interfacerequires a nondiverging solution for each r, whichlimits the choice to first-kind Bessel functionsJn�1�krr�, where kr is a free parameter serving to sep-arate the variables in Eq. (5). It will appear later thatkr represents the radial component of the wave vec-tor.
As we discuss below, a surface perturbation couldact as a localized source for a plasmon surface wavesuch that other Bessel functions could serve as asolution and, in particular, the Hankel functionHn�1
�krr�, which represents a wave propagating in aradial direction away from the source toward r → .Because of the completeness of the Bessel-functionbasis, the general solution of Eq. (5) can be written asa linear combination of particular solutions obtainedfor each value of kr:
E�, n(r, z) �0
e�, n(kr, z)�n�1(krr)krdkr. (6)
Equation (5) determines the z variation of the fieldcomponents:
0
�n�1(krr)krdkrkr2 � k2 �
�2
�z2�e�, n(kr, z) � 0.
(7)
When one considers a homogeneous region with per-mittivity independent of r, too, and uses the orthog-onality of the Bessel functions of the same order withrespect to their arguments, Eq. (5) is further de-coupled with respect to kr. To observe this it is nec-essary to multiply both sides of Eq. (7) by �n�1�krr�rand to integrate with respect to r from zero to infinity.Recalling that
0
�n�1(k�rr)�n�1(krr)rdr ��(k�r � kr)
k�r, (8)
with � representing the Dirac function, we obtain
��2
�z2 e�, n(kr, z) � (k2 � kr2)e�, n(kr, z). (9)
Obviously, the solutions represent a combination ofwaves propagating upward and downward along thez axis:
e, n� (kr, z) � bn
E, �(kr)exp(�ikzz),
e�, n� (kr, z) � cn
E, �(kr)exp(�ikzz), (10)
with
kz � �k2 � kr2. (11)
Equations (10) and (11) establish that kr is the radialcomponent of the wave vector. In what follows, thesuperscripts � and � refer to the upgoing and thedowngoing elementary waves, respectively.
The same formulas can be written for the magneticfield components by use of the notation h instead of eand H instead of E. Thus our new set of parametersfor each value of kr will consist of eight coefficients,bn
E, �, bnH, �, cn
E, �, and cnH, �. They can serve to express
the remaining z components of the electromagneticfield by using Maxwell equations:
i��0Hz � (rot E→
)z,
i� Ez � �(rot H→
)z. (12)
After some trivial calculations, one can obtain ex-pressions similar to Eqs. (6) and (10):
Hz, n(r, z) �1
i��0
0
�n(krr)krdkr
� �bnE, �(kr) � cn
E, �(kr) exp(�ikzz),
Ez, n(r, z) � �1
i� 0
�n(krr)krdkr
� �bnH, �(kr) � cn
H, �(kr) exp(�ikzz). (13)
The form of these expressions indicates that it isuseful to consider two different polarizations withrespect to the z axis: (i) transverse electric (TE)polarization with respect to the z axis, for whichEz � 0, which implies that bn
H, � � cnH, �, and (ii)
transverse magnetic (TM) polarization with respectto the z axis, for which Hz � 0. This implies thatbn
E, � � cnE, �.
Maxwell’s equations (Appendix A) lead to furtherlinks between the field components, so each polariza-tion is characterized by
TE polarization:
bnH, � � cn
H, � ��ikz
��0bn
E, �,
bnE, � � �cn
E, �; (14)
TM polarization:
bnE, � � cn
E, � ��ikz
� bn
H, �,
bnH, � � �cn
H, �. (15)
Thus for each polarization a single independent cou-ple of parameters represents upgoing and downgo-ing waves.
3. Reflection on a Plane Interface and SurfacePlasmon Wave in Cylindrical Coordinates
Let us consider plane interface(s) perpendicular tothe z axis between different media characterized by
different values of ε. It is well known that the tan-gential component of the incident wave vector is pre-served when the interfaces are crossed. In cylindricalcoordinates, this is radial component kr. Whereas thisrule in Cartesian coordinates is quite familiar fromuniversity courses, one can easily get convinced of itsvalidity by applying the procedure of Eq. (8) to thefield expansions in cylindrical coordinates at eachside of the interface and by using the orthogonality ofBessel functions.
Thus each elementary solution with a given valueof kr is independent of the solutions for other valuesof kr. The plane interface couples only the upgoingand downgoing component for each kr. However, suchis not the case for an interface with apertures, whichwill couple all kr components, thus leading to diffrac-tion phenomena.
Equations (13) show that the fundamental TE andTM polarizations are preserved at each interface, be-cause of the continuity of �0Hz and Ez. Thus, as isdone in Cartesian coordinates, an arbitrary polarizedincident field can be decomposed into two fundamen-tal polarizations, and the reflected and transmittedfields will be the sum of the fields reflected and trans-mitted in each polarization, independently of theother polarization. Let us consider the two cases, tak-ing for simplicity a single interface at z � 0 thatseparates the upper medium with permittivity fromthe lower medium with permittivity ε.
We denote by bnE, � and bn
H, � the TE and TM wavesincident from the upper medium; the continuity ofthe tangential electric and magnetic field componentsimplies, from Eqs. (6) and (10) and the analogousequations for H, that
Thus the Fresnel coefficients attain the same form8
as in Cartesian coordinates:
rTE �bn
E,
bnE, �
�kz � kz
kz kz
, rTM �bn
H,
bnH, �
�kz� � kz�
kz� kz� ,
tTE �bn
E, �
bnE, �
�2kz
kz kz
, tTM �bn
H, �
bnH, �
�2kz�
kz� kz� .
(18)
As one can observe from Eqs. (18), nonmagnetic me-dia cannot exhibit TE polarized surface waves,10,11 afact that is well known when one is working in Car-
tesian coordinates. The only possible surface wavesexist in TM polarization when the interface sepa-rates a metal from a dielectric. There is nothingsurprising in obtaining the same results when one isworking in cylindrical coordinates, for the followingreasons: A plane-wave plasmon solution in Carte-sian coordinates is characterized by its TM naturewith respect to the plane of propagation; i.e., themagnetic field vector is parallel to the interface, in-dependently of the direction of propagation alongthis interface. Thus any combination of plasmonspropagating in different directions will have Hz
� 0. They all will have a unique propagation constantkr
p along the radial direction, equal to the pole of theTM reflection and transmission Fresnel coefficients, asolution of the equation
[k2 � (krp)2]1�2
[k2 � (krp)2]1�2
� 0, (19)
where k2 � �2�0 . The corresponding z components ofthe wave vector are given by
kzp � �k2 � (kr
p)2 1�2,
kzp � �k2 � (kr
p)2 1�2, (20)
with nonnegative imaginary parts.From Eqs. (4), (6), (10), (15), and (16), the electric
field of the elementary plasmon surface wave in cy-lindrical coordinates can be derived:
In addition, Eq. (19) ensures the continuity of Er, E�,and Ez. In the lower medium the field has the sameform, except that we have to suppress the circumflexand replace kz
p with �kzp. Let us consider the field
forms that correspond to several values of n and havedifferent degrees of symmetry.
A. n � 0
The elementary field that corresponds to n � 0 is �independent. It has no � electric field component�J�1 � �J1� and represents a radial plasmon thatpropagates from the origin in all radial directions.The r dependence is given by J0�kr
pr� for the z compo-nent and by J1�kr
pr� for the r component. Such a sur-face wave is quite difficult to excite, because itrequires a radially polarized source field.
B. n � �1
The most important elementary solution regardingexcitation with a linearly polarized incident field,the most common experimental situation when oneis using laser sources, is n � �1. From this point ofview it is more important to consider two indepen-dent linear combinations that include n � �1 andn � 1, although they could be treated separately.Let us consider two combinations, the sum �p� andthe difference �p��, between the components for n� �1 and n � 1 that have the following properties(it is necessary to remember that each solution ofthe homogeneous problem is determined within amultiplicative constant):
1. b�1H, � b1
H,
Er, p, krp
p (r, �, z) �ikz
p
� b1
H, (krp)exp(ikz
pz)
� [�0(krpr) � �2(kr
pr)]sin(�),
E�, p, krp
p (r, �, z) �ikz
p
� b1
H, (krp)exp(ikz
pz)
� [�0(krpr) �2(kr
pr)0]cos(�),
Ez, p, krp
p (r, �, z) ��4kr
p
� b1
H, (krp)exp(ikz
pz)�1(krpr)sin(�).
(22)
These equations can be further simplified by use ofthe relations between Bessel functions:
Er, p, krp
p (r, �, z) �2ikz
p
� b1
H, (krp)exp(ikz
pz)��1(krpr)sin(�),
E�, p, krp
p (r, �, z) �ikz
p
� b1
H, (krp)
� exp(ikzpz)
2
krpr
�1(krpr)cos(�),
Ez, p, krp
p (r, �, z) ��4kr
p
� b1
H, (krp)
� exp(ikzpz)�1(kr
pr)sin(�), (23)
which form gives a better understanding of thefield behavior. Recalling the asymptotic behaviorof Bessel functions [see relations (24) below], onecan observe first that the � component of the fielddeclines as exp�Im kr
pr��r3�2, much faster than the rand z components, which behave asymptotically asexp�Im kr
pr��r1�2.Second, the r and z components are zero at �
� 0, �; thus the electric field declines much morerapidly in directions close to � � 0, � and much moreslowly in directions � � ����2�, as if the plasmonwere able to propagate more easily along the y axisthan along the x axis. In addition, along the y axis,component E� � �Ex is null, as happens for a planeplasmon wave propagating along y.
Third, Ez and Er are antisymmetric with respect tothe directions of propagation, i.e., when � changes itssign.
Fourth, depending on the type of solution (Jn orHn
), the modulus of each component decreases with rbut either varies rapidly, representing a standing-wave feature (for Jn), or represents a wave propagat-ing toward r → (for Hn
), which could be observed intheir asymptotic behavior:
quite similar to Eqs. (23); the only difference is as if� → � ���2�. The same conclusions are valid aswhen b�1
H, � b1H, with axes x and y interchanged.
C. n � �2, �3, . . .
Higher-order solutions can easily be obtained fromthe general form written in Eqs. (21). Similarly to thecase with n � �1, the solutions can be combined to
form field maps that exhibit higher-order symmetry,which behave as sin�n�� or cos�n��. For n � 2 the fieldmap is doubly periodic with respect to �; for n � 3 themap is triply periodic, etc. Whereas this is interestingfrom a theoretical point of view, we do not go furtherin detail, because these solutions (as well as forn � 0) cannot be excited with a linearly polarizednormally incident wave that is singly periodic in �(Appendix A) and that is the subject of our numericalstudy, which follows.
4. Perturbation Analysis of Surface-Wave ExcitationOwing to Surface Defects
Let us now consider a metal–dielectric interface sup-porting a plasmon surface wave that is slightly per-turbed with a perturbation geometry that isindependent of �. Let us assume that the surfaceequation is expressed in the form
zS � hf(r), (26)
where h is a small parameter that represents theheight of the perturbation and has a dimension oflength. If h → 0, we can assume that the boundaryconditions require the continuity of E�, n�r, zS�. Be-cause of surface perturbation, the component of thewave vector in the �x, y� plane is no longer preserved,and an incident wave characterized by a single valueof kr � kr
i of the form
E�, ni (r, z) � b�, n
i (kri)exp(�ikz
iz)Jn�1(krir) (27)
will generate a continuum of coupled componentswith 0 � kr � . In Eq. (27) it is necessary to use thefirst-kind Bessel function, which is the only nondi-verging Bessel function for any value of r. From thegeneral form given in Eq. (6), one can conclude that
bnH, �(kr) � k0
2bnH, i
�(kr � kri)
kri . (28)
Let us recall that we are interested in TM polariza-tion because a plasmon wave does not exist in TEpolarization. Because of the coupling between thediffracted components, the simple equation (16) isreplaced by a more complicated form:
Within the first-order approximation in h, the expo-nential terms become
exp(�ikzzS) � 1 � ihkzf(r). (30)
The zeroth-order terms correspond exactly to Eq.(16) for bn
H�kri�, which are the only field components
that are present when h � 0. The other components,characterized by kr � kr
i, are of higher order in h. Wecan further simplify the first-order approximation ofEqs. (29) by multiplying by ��k�rr�r and integratingwith respect to r:
kz�
bn
H, (k�r) ikz
i
bn
H, iFn1(k�r, kri) � �
k�z
bn
H, �(k�r),
bnH, (k�r) � ibn
H, iFn1(k�r, kri) � bn
H, �(k�r), (31)
with coupling coefficient Fn1�kr, kri� equal to
Fn1(kr, kri) � k0
2kzih
0
f(r)�n1(krr)Jn1(krir)rdr.
(32)
On comparison of Eq. (31) with Eq. (16), it is evidentthat the diffracted-field amplitudes are expressedthrough the incident-field amplitude with similarFresnel coefficients, multiplied by the coupling inte-gral, Eqs. (32):
It has to be stressed that these expressions are validonly in a first-order approximation in h. However,they permit us to draw three important conclusions:First, Eqs. (33) have the same poles as Eqs. (18), andthus the condition for the existence of surface plas-mons given for a flat interface by Eq. (19) remainsthe same for a perturbed surface. Second, to maxi-mize certain diffracted components by a suitablechoice of the surface perturbation it is necessary tomaximize coupling integral Fn1�kr, kr
i�. Third, forvalues of kr close to the pole kr
p, the field amplitudescan be expressed in series of kr � kr
p. However, be-cause according to Eq. (11) kz � kz�kr
2�, the depen-dence becomes
bnH, (kr), bn
H, �(kr) �1
kz� kz� Fn1(kr, kr
i) bnH, i
�C
kr2 � (kr
p)2 Fn1(kr, kri) bn
H, i.
(34)
The validity of this formula is illustrated in Fig. 3,where the kr dependence of b1
H, , expanded on thebasis of J1�krr�,6 is represented for linearly polarizedlight normally incident upon an air–silver interface,where the silver film is pierced with a 50 nm radiushole and the wavelength is � � 500 nm. As can beobserved, in the vicinity of the pole the behavior isclearly expressed by relation (34), which is known ingrating theories as a phenomenological formula.
5. Optimization of Surface Perturbation
Let us go back to the second conclusion of Section 4and try to optimize the coupling integral to enhancethe excitation of the plasmon surface wave. For agiven perturbation form, the coupling is assumed toincrease with the corrugation amplitude, at least upto certain values, because the linear dependence in
Eqs. (32) is limited to the first-order approximation inh that is used. Higher-order terms will change thislinear dependence, and they are automatically takeninto account in the numerical result. In what follows,we limit ourselves to normally incident linearly po-larized monochromatic light containing a single kr
component, kri � 0. The coupling integral becomes
F0(kr, kri) � k0
2kzih
0
f(r)�0(krr)rdr. (35)
It is thus evident that the optimal surface profile thatpresents the maximum coupling between the nor-mally incident wave and the kr diffracted componentwill be obtained when the profile is represented bythe zeroth-order Bessel function:
f(r) � J0(k�rr) ) F0(kr, kri) � k0
2kzih
�(kr � k�r)k�r
. (36)
The coupling is then infinitely strong. However,higher-order diffraction limits the coupling becauseof the radiation, as reciprocity requires that the cou-pling between the incident and the kr components beequal to the inverse coupling. Indeed, the couplingintegral is symmetric with respect to kr and kr
i.Further on in this section we analyze numerically
such strong singular coupling predicted by expres-sions (36), but it is quite difficult to fabricate thisstructure, so we deal first with more realisticlamellar-form corrugation.
A. Single Aperture
The validity of the approach is tested first on a singlecircular hole without surface corrugation. The plas-mon wave is excited at the aperture and propagatesoutward; thus its field is expressed in the formHn
�krpr� rather than in the standing-wave form,
Jn�krpr�. There is no contradiction of the fact that in
relation (34) we used the basis of J1�krr� because ofthe relations between Bessel functions of differentkinds. In particular, there is the following link be-tween J1 and H1
for complex values of krp:
0
J1(krr)krdkr
kr2 � (kr
p)2 � i�
2 H1(kr
pr). (37)
Thus, by using relation (34) and working with a J1basis with real values of kr, one can obtain an outgo-ing propagating solution with complex propagatingconstant kr
p corresponding to a pole of Eqs. (18). Thisfact can be observed from Fig. 4, where the modulusof Ez exactly follows the function H1
�krpr� outside the
aperture.However, inside the hole, the field components
must be represented in the basis of Jn, which does notdiverge at origin. Then the coupling integral is ex-tended from 0 to R and takes a simple form:
Fig. 3. Comparison of the computed kr dependence of |b1H, | and
its approximation given by the phenomenological formula [relation(34)] in the vicinity of the plasmon propagation constant for a silverscreen with 50 nm radius of aperture and � � 500 nm.
At � � 500 nm, the plasmon propagation constant onan air–silver interface is �p � kr
p�k0 � 1.068. Theincident wave has no z component of the field, and thedependence of |Ez| along the surface is determinedmainly by the plasmon surface wave. Its value farfrom the aperture �x � 4 �m� is plotted in Fig. 5 as afunction of the aperture radius, together with thedependence on the perturbative approach given byEq. (38). One can observe a very good correspondencebetween the rigorous numerical results and the per-turbation analysis. Figure 6 presents a comparison ofthe two sets of results, given as a function of thewavelength for a fixed aperture dimension, R� 50 nm. Here it is necessary to rigorously take intoaccount the dispersion of �p.
B. Lamellar Corrugation
In analogy with the diffraction grating behavior, oneexpects to resonantly enhance the surface wave byusing a suitably positioned corrugation to excite thesurface wave in phase along a surface much largerthan the single aperture. As is obvious from the anal-ysis of expressions (36), singular resonant excitationcan be expected if the corrugation follows the Bessel-function quasi-periodicity rather than using equidis-tantly corrugated channels. To investigate thisassumption it is necessary first to calculate the cou-pling integral for the multichannel corrugation thatwas schematically presented in Fig. 1. The disconti-nuity of the radial electric field components acrossthe channel walls requires that the coupling integraltake different forms in the channels and betweenthem:
F0(krp, 0) � k0
2�kzih
0
R1
J0(krpr)rdr
kzih
R1
R2
J0(krpr)rdr
kzih
R2
R3
J0(krpr)rdr . . .�
� k02(kz
i � kzi)
h
krp �
j�1
jmax
(�1)jRjJ1(krpRj). (39)
As observed further, sometimes it is better to use analternative form of this expression because the radialcomponent of the plasmon field is represented by theexpression �0 � �2 [Eqs. (22)] rather than simply by�0. Another form of Eq. (39) is then easily obtained:
F0(krp, 0) � k0
2(kzi � kz
i)h
krp �
j�1
jmax
(�1)jRj
� [J1(krpRj) � J3(kr
pRj)]. (40)
Fig. 4. Field map of a surface plasmon wave excited by a circularaperture of 50 nm radius in a silver film by an x-polarized normallyincident plane wave with wavelength equal to 500 nm. Rigorousresults, solid curve; corresponding Hankel function, squares.
Fig. 5. Comparison of values of |Ez| at point �4 �m, 0, 0� com-puted from the rigorous theory and from the perturbative ap-proach. Aperture radius R is varied; other parameters are thesame as in Fig. 4.
Fig. 6. Same as in Fig. 5 but with the wavelength varied and R� 50 nm.
The two expressions differ significantly for a single-channel corrugation [jmax � 3; Fig. 7(a)], and theirdifference is reduced when the number of channelsincreases, as shown in Fig. 7(b). As is evident fromEq. (39), to maximize the coupling integral for a givenvalue of the corrugation depth it is necessary to in-troduce the channel walls (i.e., Rj) consecutively, atthe maxima and minima of the function rJ1�kr
pr�.Then, increasing the total number of channels willincrease the coupling integral and change its spectralbehavior significantly (Fig. 8), the maxima becominghigher and narrower. The curves are drawn with val-ues of Rj optimized to obtain the maximum of thecoupling integral at � � 500 nm. With the value ofthe plasmon propagating constant equal to 1.068, thecorresponding optimal values for Rj are as given inTable 1. We keep the same the hole radius, R1� 50 nm. These results explain why in Fig. 2 the spec-tral maximum positions differs significantly from500 nm.
The expectations from the approximate formulasare fully confirmed by rigorous numerical results. Aswe observe below, for a given configuration the max-imum field enhancement is obtained for shallow cor-
rugation depth, of the order of 30–50 nm (Fig. 9, four-channel corrugation). This is explained by the factthat deeper corrugations lead to stronger radiation ofthe plasmon and perturb its propagation by cuttingthe surface. Similar behavior exists when a plasmonsurface wave is excited on the surface of sinusoidal orlamellar metallic diffraction gratings. In what fol-lows, we have chosen h � 30 nm.
Figure 10 shows the dependence of the field en-hancement on the number of corrugation channels,compared with a single-aperture configuration. Theplasmon field is calculated at x � 4 �m away from thecorrugation, because at the channel walls one ob-serves strong local field fluctuations. Both the electricfield and the total transmission by the aperture arestrongly enhanced, as expected from Fig. 8. The spec-tral behavior of the plasmon amplitude is quite wellpredicted by Eq. (40) or (39) for one or four channels,respectively, as seen from Fig. 11, although a smallspectral shift to a longer wavelength is observed forthe rigorous results.
Figure 12 presents the spectral dependence of theplasmon field for a single corrugated channel withwalls situated at 235 and 470 nm, a part of the equi-distant structure discussed in Fig. 2. A sharp mini-mum is observed close to 480 nm, the spectralposition close to which one would expect a maximum,judging from the one-dimensional grating theory�500 nm�. However, this geometry outcome is not sur-prising because the correct phenomenological for-mula from Eq. (40) matches the numerical results inboth Figs. 2 and 12. The field amplitude is similar invalues when an equidistant corrugation is used com-pared with the optimized corrugation from the tableabove; however, the spectral position of the maxi-mum is significantly changed.
C. Bessel-Function Corrugation
As was already discussed, the optimal structure isassumed to have a geometry that varies according tothe zeroth-order Bessel function [expressions (36)].Because of numerical problems, we are not yet able to
Fig. 7. Comparison of two expressions for the coupling integralgiven in Eqs. (39) and (40) as a function of wavelength for a Bessel-function-type corrugation with radii given in Table 1 and situatedabout a 50 nm radius aperture.
Fig. 8. Spectral dependence of the coupling integral given by Eq.(39) and its evolution when the number of channels is increased forthe same structure as in Fig. 7.
model rigorously a continuous-profile surface corru-gation in cylindrical geometry. However, the code candeal with multilevel corrugation; thus we investi-gated a surface that has the profile given in Fig. 13.In fact, the figure presents a staircase approximationof the zeroth-order Bessel function, limited in the rdirection. The phenomenological curves calculatedfrom Eq. (39) for two values of Rmax are shown in Fig.14. If the corrugation as well as the incident wavewere extended to infinity, the response would be highand narrow, as we observe in Subsection 5.D below.However, this geometry is of little practical interestbecause the energy density of the incident wave de-creases with the beam width. Anyway, the main dif-ference between the curves in Fig. 14 compared withthe curves in Fig. 8 is the lower value of the secondary
maxima of the former, which is a natural conse-quence of the smoother profile.
The numerical results (Fig. 15) fully confirm thephenomenological predictions concerning the spec-tral width and position of the main maximum and thereduction of the secondary maxima. Again, a rela-tively small shift toward a longer wavelength is ob-served in the rigorous results.
D. Phase Modulation
The optimal corrugation given by expressions (36) isquite difficult to fabricate. However, instead of a sur-face corrugation, it is possible to use a phase modu-lation of an adjacent dielectric layer, in analogy withthe phase gratings used for mode coupling in inte-grated optics. Such a phase structure can be manu-factured by a holographic technique. Let us assumethat it consists of a plane dielectric layer deposited
Fig. 9. Field enhancement as a function of channel depth for afour-channel structure surrounding a single 50 nm radius aper-ture at � � 500 nm. Field calculated on the upper and lower filmsurfaces compared with the enhancement of |b1
H, |.
Fig. 10. Near-, transmitted-, and plasmon-field enhancement as afunction of the number of channels for h � 50 nm and �
� 500 nm.
Fig. 11. Spectral dependence of |Ez| at point �4 �m, 0, 0� com-puted from the rigorous theory (solid curves) and the perturbativeapproach (squares): (a) single channel and Eq. (40), (b) four chan-nels and Eq. (39). Corrugation radii follow the values in Table 1.
onto a metallic surface and characterized by a radialdependence of its refractive index. Applying the sameprocedure as when we went from Eq. (7) to Eq. (9), weobtain a coupled-wave equation:
kr2e�, n(kr, z) �
�2
�z2 e�, n(kr, z)
� �2�0 0
(r)�n�1(krr)Jn�1(krir)rdre�, n(kr, z).
(41)
The coupling integral that appears in Eq. (41) is re-sponsible for the diffraction effects and has almostthe same form as Eqs. (32), as can easily be under-stood because one can consider a lamellar structureas a phase grating with large contrast.
As has already been discussed with respect to ex-pression (36), the strongest excitation can be ex-pected if the perturbation follows the zeroth-orderBessel function. The response is expected to be dom-inated by a Dirac distribution, infinitely narrow andhigh. This is so because both the modulated regionand the incident wave extend to infinity in directspace, which results in a single-frequency excitationin kr space. This is an exact analogy of a diffractionorder of a classic diffraction grating being infinitelynarrow spectrally and angularly if the grating hasinfinite dimensions and is illuminated by a perfectplane wave. Whereas this situation represents anacademic abstraction, its analysis enables us to un-derstand the underlying physics better. Anyway, areal finite system or a finite-width incident beam orboth are assumed to exhibit responses much closer tothose shown in Figs. 14 and 15.
Taking into account the above-mentioned remarks,let us analyze a system consisting of a single circularaperture in a 200 nm thick silver screen covered by athin high-index dielectric layer with its dielectric per-mittivity modulated according to the following for-mula:
(r) � d � J0(krmodr), (42)
where krmod is a constant chosen to be equal to the
propagation constant of the plasmon surface wave
Fig. 12. Same as in Fig. 11(a) but for a single channel with R2
� 235 nm and R3 � 470 nm.
Fig. 13. Staircase approximation of smoothly varying surfacemodulation described by the zeroth-order Bessel function.
Fig. 14. Spectral dependence of the coupling integral obtained byuse of two values of radius Rmax of the modulated region.
Fig. 15. Spectral dependence of the rigorous (solid curve) andperturbative (squares) values of |Ez| at point �4 �m, 0, 0� for thestructure presented in Fig. 13.
propagating along the metal–dielectric interface for agiven wavelength. If it is thick enough, the dielectriccover layer can support waveguide modes. To preventthis from occurring it is necessary to have a suffi-ciently thin layer. In addition, the layer’s existencewill modify the plasmon propagation constant, as canbe observed from Fig. 16, where the kr decompositionof the z electric field component is represented for theupper and the lower metallic surfaces, similarly toFig. 3. There are several features, indicated in thefigure by numbers inside rectangles:
1. There is an anomaly that is due to the plasmonexcitation at the lower metal–air interface. It is sim-ilar to that in Fig. 3, except for the appearance of asmall feature with number 3. The amplitude of thefield at this interface is naturally lower than on theupper boundary.
2. The existence of this surface wave is also man-ifested in the field at the upper boundary, owing tothe coupling through the circular aperture.
3. The constant krmod � 0.02649 nm�1 in the mod-
ulation function is chosen to correspond to theupper-boundary plasmon propagation constant at� � 500 nm. The effect of the modulation is quitestrong at the upper metallic boundary but can alsobe observed at the lower boundary.
4. Feature 4 consists of a relatively wide maxi-mum representing the upper-surface plasmon excita-tion by the aperture, similar to Fig. 3. The width isdetermined by attenuation constant �p � Im��p� ofthe plasmon, equal to �0.04 at � � 500 nm. In addi-tion to this broad peak, one can observe a verticalfeature with one-pixel width; the pixel is equal to asingle step in the numerical discretization along thekr axis. This feature is due to the effect of the modu-lation of the dielectric layer and is analyzed in furtherdetail below.
5. Feature 4 is translated along the kr axis with a
Fig. 16. Dependence of |b1H, | at z � 0 (upper metallic interface)
and of |b1H, �| at z � �200 nm (lower surface) on the values of kr�k0
for a 50 nm radius aperture in a silver film covered with a 50 nmthick dielectric layer with permittivity modulated according to Eq.(42), � � 500 nm, d� 0 � 3.5, � � 0 � 0.1, and kr
mod
� 0.02649 nm�1.
Fig. 17. Spectral dependence of |b1H, | for the structure de-
scribed in Fig. 16.
Fig. 18. |b1H, | as a function of kr�k0 in the vicinity of plasmon
excitation on the interface metal-modulated dielectric layer for thestructure described in Fig. 16 and presented for four wavelengths.
Fig. 19. Spatial variation of the plasmon field for three wave-lengths for the structure given in Fig. 16.
period equal to krmod, representing a higher-order
coupling.
In what follows, we turn our attention to feature 4in particular, as it represents the resonant plasmonexcitation discussed in this paper. The spectral be-havior of the amplitude of the electric field of theplasmon surface wave is given in Fig. 17. As expected,the curve is much narrower than for a limited-regionmodulation (Figs. 14 and 15) but not infinitely thin,as expected from expressions (36). This is so becausethe response depends both on the coupling integraland on the plasmon spectral curve, not to forget aboutits dispersion characteristics. All these argumentsare illustrated in kr space in Fig. 18. When the wave-length is varied, the constant of modulation kr
mod doesnot move in kr space, although there is a slight driftalong the normalized kr–k0 axis. This representationwas chosen to distinguish the variation of the peakamplitudes with wavelength. As implied by relation(34), the modulation of the refractive index induces(through the coupling integral) variations of the elec-tromagnetic field that has the same modulation char-acteristics, i.e., has a single frequency representationin k space. From the point of view of plasmon excita-tion, this field modulation serves as an external res-onant source.
The plasmon propagation constant varies with thewavelength. This variation is due to the dispersion ofsilver and, mainly, to the geometric dispersion of thedielectric layer; the modification of the plasmon thatis due to its presence is strongly influenced by wave-length, as can be observed in Fig. 18 as a relativelyrapid displacement of the broad maximum towardsmaller kr�k0 values when the wavelength increases.Depending on the frequency mismatch between theexternal �kr
mod� and the proper �krp��� metallic–
dielectric interface resonance, the resonance excita-tion could be stronger or weaker. Its spectral width isdetermined mainly by the wider resonance curve;this explains why spectral curve width W in Fig. 17 isdirectly linked to the attenuation constant of theplasmon: W � ��p � 20 nm.
Another direct consequence of the existence of tworesonances (external, which is due to the layer indexmodulation, and proper, which is the plasmon surfacewave) is that the diffracted field no longer representsan outgoing (toward r → ) wave, as presented in Fig.4, but is rather a combination of two coupled waves:the outgoing plasmon field and a standing wave ex-cited by the modulation of the dielectric layer, a mod-ulation that extends to infinity. Because of thecoupling, the field exhibits a rapidly varying charac-ter [period of variation determined by the mean fre-quency �kr
mod krp��2] and large-period beating with
frequency equal to |krmod � kr
p���|, as can be observedfrom Fig. 19 for the exact frequency match (500 nm,no beating) and for two other frequencies that pro-duce smaller- or larger-period beating effects. Resultsnot reproduced here show that rigorous calculationsconfirm that the plasmon excitation is a linear func-tion of � in a large interval �0 � � � 2.5 0�, aspredicted by expressions (36) and (42).
6. Conclusions
A rigorous electromagnetic study combined with per-turbative analysis has shown that it is possible tooptimize the optogeometrical parameters of struc-tured circular apertures in metallic films for strongerexcitation of surface plasmon waves and thus of thelocal field close to the surface by using simple rulesbased on knowledge of plasmon field peculiarities incylindrical geometry. The optimal surface corruga-tion has to follow the Bessel functions rather than beequidistant. A lamellar cross-section corrugationwith constant groove depth has to be structured ac-cording to the minima and maxima of the first-orderBessel function (Table 1), whereas a smoothly vary-ing surface relief or refractive-index (phase) modula-tion has to follow the zeroth-order Bessel function[expression (36) or (42)].
Appendix A. Maxwell’s Equations in CylindricalCoordinates
Maxwell’s equations written in cylindrical coordi-nates have the form
where the set at the right is written with the � de-pendence in Eq. (1) taken explicitly into account. Thethird and the sixth of Eqs. (A1) can be simplified byuse of Eqs. (4) and (6):
i��0Hz, n �0
(e, n � e�, n)�n(krr)kr2dkr,
�i� Ez, n �0
(h, n � h�, n)�n(krr)kr2dkr. (A2)
Let us consider a homogeneous region where is con-stant. Then Eqs. (A2) immediately lead to Eqs. (13).The derivatives of Ez, n and Hz, n with respect to r inEqs. (A1) can be explicitly obtained from the relation��n � n�n�r � ��n�l, and Eqs. (A1) can be rewrittenfor the new field components E�, n and H�, n. They canbe further simplified by separate consideration of thetwo fundamental polarizations:
A. TE Polarization
For TE polarization the first two of Eqs. (A1) can becombined if one takes into account that Ez, n � 0:
�
�z E, n � ��0H, n ) �ikzbnE, � � ��0bn
H, �,
�
�z E�, n � ���0H�, n ) �ikzcnE, � � ���0cn
H, �,
(A3)
a link used in Eqs. (14).
B. TM Polarization
For TM polarization the fourth and fifth of Eqs. (A1)can be combined if one takes into account that Hz, n
� 0:
�
�z H, n � �� E, n ) �ikzbnH, � � �� bn
E, �,
�
�z H�, n � � E�, n ) �ikzcnH, � � � cn
E, �, (A4)
a link used in Eqs. (15).A linearly polarized normally incident plane wave
has a simple expression. In the numerical calcula-tions we use a monochromatic incident wave polar-ized in the x direction. In cylindrical coordinates ithas the following electric field components:
Eri � Ex
i cos �, E�i � �Ex
i sin �, Ezi � 0, (A5)
which shows that such a wave has only n � �1non-null Fourier harmonics.
The support of the EC-funded projects PHOREMOST(FP6�2003�IST�2-511616) is gratefully acknowledged.The content of this work is the sole responsibility of theauthors.
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