light transmission through a subwavelength hole

Optics Communications 255 (2005) 338–348

www.elsevier.com/locate/optcom

Light transmission through a subwavelength hole

Evgeny Popov a,*, Michel Neviere a, Philippe Boyer a, Nicolas Bonod b

a Institut Fresnel, Case 161, Unite Mixte de Recherche Associee au Centre National de la Recherche Scientifique (UMR 6133),

Universite de Provence, Faculte des Sciences et Techniques de St. Jerome, Avenue Escadrille Normandie-Niemen,

13397 Marseille Cedex 20, Franceb Commissariat a l’Energie Atomique, Centre d’Etudes Scientifiques et Techniques d�Aquitaine, BP No. 2, 33114 Le Barp, France

Received 7 April 2005; received in revised form 1 June 2005; accepted 3 June 2005

Abstract

We use a recently developed electromagnetic theory to get some insight of the physics that governs the extraordinarytransmission of light by a hole pierced in a metallic screen. When the film thickness increases, a comparison is madebetween the decay of the total field and the decay of the field of the evanescent mode in the corresponding infinite hol-low metallic waveguide. The decay behavior and the comparison between their field maps demonstrates that the trans-mission through the hole is determined by this guided mode. The near field maps below the hole shows small oscillationsaround a decaying mean value, when the observation points moves away from the hole ridge. The study of the plasmonexcited at the bottom interface of the screen allows interpreting all the fine phenomena in terms of simple concepts.� 2005 Elsevier B.V. All rights reserved.

Keywords: Single aperture; Surface plasmon; Evanescent modes; Enhanced transmission

1. Introduction

Despite the great amount of theoretical workpublished since the discovery of the extraordinarylight transmission by subwavelength hole arrays[1], the physics of the process still remains not fullyunderstood. Concerning periodic arrays, the keyrole of surface plasmons has been invoked from

0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2005.06.010

* Corresponding author.E-mail address: [email protected] (E. Popov).

the very beginning [2–5]. The plasmon excitationat bottom interface was shown to occur thanksto a special channel [6,7] provided by periodic holearrays. Then the coupling between the plasmonsexcited at both interfaces was shown both byexperiments [8] and theory [9,10] to increase againthe transmission. It was even found that such anextraordinary transmission could be obtained withmodulated films without holes [11–13], providedthey are thin enough to allow a coupling betweenthe plasmons at the top and bottom interface.All the theoretical studies took advantage of the

ed.

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E. Popov et al. / Optics Communications 255 (2005) 338–348 339

bi-periodicity of the device to express the fieldcomponents in terms of Fourier series.

A single aperture with subwavelength diameterinside a metallic screen is a key component fornanophotonic applications. Despite the fact thatthe standard aperture theory [14] has been devel-oped since 60 years, analyzing a nano hole piercedinside a real metallic sheet which cannot be as-sumed as perfectly conducting turns out to be adifficult task. But due to the interest of the prob-lem, a recent effort succeeded in developing a rig-orous electromagnetic theory [15,16] of thatdiffraction problem. The analysis makes use of aBessel–Fourier basis to represent the various cylin-drical components of the electromagnetic field andreduces Maxwell equations into a first order differ-ential set to integrate. In the case of a cylindricalhole, the integration is made via an eigenvaluetechnique which leads to short computation time.It is then possible to construct field maps and,more generally to study the field evolution in termsof any parameter in order to extract the physicalprocess which governs the extraordinarytransmission.

2. Fringe structure of the transmitted field map

The nano hole under study is illustrated inFig. 1, which states the notations. The radius R

(typically equal to 50 nm) is assumed to be muchsmaller than the wavelength k = 500 nm. The

x

z

t

o

θ

incidentE→

incidentk→

Fig. 1. Illustration of a nanohole inside

thickness t will be varied between 10 and 500nm. The device is lighted under normal incidenceby a plane wave with vector ~E parallel to the x-axiswith a unit amplitude. We first choose R = 50 nmand t = 150 nm. We consider an observation pointM located just below the bottom interface atz = �1 nm and calculate the square modulus ofthe total electric field j~EðMÞj2 when the x and y

coordinates of M are varied. Fig. 2(a) shows theglobal field map, while Fig. 2(b) is a zoom in thecentral part corresponding to the hole. It is clearthat j~Ej2 decreases when x and/or y are increased,but the slope of the surface is not the same in thex- and y-direction. Clearly, the polarization of theincident field introduces an anisotropy. Moreover,the zoom in the central region clearly shows thatthe maximum of j~Ej2 is not located at x = y = 0.A sharp maximum occurs at the point M withx = 50 nm and y = 0, i.e., on the edge of the metal,while nothing is observed at y = 50 nm. This isconsistent with the results that we obtained in aprevious study [17] and is due to the fact that atthe edges of the holes, where the incident electricvector is perpendicular to the metal-air borders,there is a charge accumulation caused by the dis-continuity of the induced current [18], which re-sults in a sharp increase of the electric fieldamplitude (edge effect). Recent experimental re-sults of Degiron (see Fig. 2(a) of [19]) fully con-firms the existence of such field enhancement onthe aperture boundary perpendicular to the inci-dent polarization direction.

y

a metallic screen and notations.

Fig. 2. (a) Square modulus of the electric field map obtained at z = �1 nm. (b) Detail of the central region (hole). R = 50 nm, k = 500nm, t = 150 nm.

340 E. Popov et al. / Optics Communications 255 (2005) 338–348

01E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

100

1000

R = 50nm= 500 nm

t (nm) 50100150200250300350400450500

|E|2 (x

,0,-

1)

x (nm)

50001000 2000 3000 4000

01E-12

1E-7

0.01

1000

|E|2

t (nm)

σ = infinite

x = 0x = Rx = 2R

100 200 300

a

b

Fig. 3. (a) Variation of the square modulus of the total electric field at a point M just below the bottom interface (z = �1 nm) as afunction of abscissa x, for various thicknesses of the screen noted in the figure. (b) Variation of |E|2 as a function of t, at a pointM(x, 0,1 nm) for a hole pierced in a perfectly conducting screen (full line) and in a silver screen for three values of x noted in the figure;R = 50 nm.


Now, allowing the thickness t to take differentvalues ranging between 50 and 500 nm, we studythe evolution of j~EðMÞj2 when the pointM(x, 0,�1) is moved parallel to the x-axis. Fig. 3shows the results. Whatever the thickness t maybe, the function j~EðxÞj2 presents a maximum whenx � 50 nm and decays from this value when x is in-creased. As soon as the thickness t is greater thanor equal to 200 nm, the field map remains the samemodulo a vertical translation. On the other hand,when t is reduced below 150 nm, a quasiperiodicityin x-direction appears, with an amplitude modula-tion which increases when t decreases. Of course,the thinner the screen, the larger the mean trans-

mitted square field modulus. This behavior of lightdiffracted by thin screens pierced by a nanoholehas already been predicted by a different theoreti-cal approach using a multiple multipole technique,as shown in Fig. 7(b) of [20]. Our first aim is, bycomparing the field map and decay constant withthe properties of the mode guided by an infinitelylong hollow metallic waveguide, to demonstratethat the transmission through the hole is deter-mined by the behavior of this mode, propagatingor evanescent, depending on the hole radius.

The quasiperiodic fringe structure has alreadybeen reported in an experimental work [21], andthe authors have used a FDTD simulation to


account for their observations. They have con-cluded that the fringes result from an interferenceprocess between a surface plasmon radiated bythe hole edge and a wave directly transmittedthrough the metal film. The second aim of the pa-per is to confirm this explanation.

In order to address the first point, Fig. 3(b)shows the total electric field intensity as a functionof the film thickness at a point M(x, 0,�1 nm) sit-uated immediately below a nano hole withR = 50 nm pierced inside a perfectly conductingscreen (full line at x = 0) and a silver screen. Con-cerning the perfectly conducting screen, thestraight line shows a fast exponential decrease,since the transmitted field may only come from aguided mode inside the nano hole, which is far be-low the cut-off. In that case, the field has been nor-malized to unity for t = 0. With the finiteconductivity, the curves present two different re-gions. For thickness greater than 100 nm, the fieldintensity for the three different positions decreasesexponentially with a slope much weaker than theone observed with infinite conductivity. This re-gion is discussed in detail in the next section andits properties are due to the existence of an evanes-cent mode in the hollow metallic waveguide.

For thickness lower than 100 nm, the exponen-tial behavior is first slightly modified, due to theinfluence of higher order evanescent modes in thenano hole. Then, for t 6 40 nm, a more complex

0

1E-3

0.01

0.1

1

10

cut-off: Ag

Re(

γ),Im

(γ)

R(nm)50 100

Fig. 4. Real and imaginary part of the propagation constant of the fuwith silver or with a perfectly conducting metal, as function of the ra

behavior occurs, which can be interpreted in thefollowing manner. For t = 0, the transmitted fieldis unity. As soon as a small thickness (10 nm) ofsilver is introduced, a charge accumulation occursat top and bottom edge, leading to a field enhance-ment. But for small t, Coulomb repulsion limitsthe charge accumulation on the two edges on theside of the hole. This repulsion decreases when t

is increased, and we first observe an increase ofj~Ej2, with a maximum value for t = 20 nm, fol-lowed by a decrease due to the exponential de-crease of the transmitted field. One can see that,depending on the observation point, the maximumof the field intensity enhancement obtained fort = 20 nm ranges between 13 and 900, occurs nearthe ridge (x = R) in agreement with the results pre-sented in Fig. 2.

3. Guided mode inside a nano circular metallic

waveguide

In view of explaining the origin of the slow de-cay shown in Fig. 3(b), we first calculated the effec-tive index c of the fundamental mode of an infinitemetallic circular waveguide made with a perfectlyconducting metal or with silver. Fig. 4 shows thereal and imaginary part of c as function of the ra-dius R. For a perfectly conducting metal, a sharpcut-off phenomenon occurs below R = 146.5 nm,

cut-off: infinite conductivity

λ = 500 nmσ = inf: Re(γ)σ = inf: Im(γ)Ag: Re(γ)Ag: Im(γ)

150 200

ndamental mode of an infinite circular metallic waveguide madedius.


the propagation constant moving from pure realvalues to pure imaginary ones. The transition issmoother in a real metal like silver, for which areal and an imaginary part always cohabitate.Anyway, below about R = 103 nm, the fundamen-tal mode becomes predominantly evanescent. It isworth noticing that the finite conductivity of silvergreatly reduces the cut-off radius (the field pene-trates in the metallic walls and the effective radiusis increased) and that below the cut-off, Im(c) is al-ways smaller for the silver waveguide than for theperfectly conducting one, which explains the re-sults shown in Fig. 3(b).

Having determined the cut-off radius, close to100 nm, we plot the transmission factor of a nano-hole inside a silver screen, as a function of thethickness, below, on, and above the cut-off radius(Fig. 5). The calculations are done twice, the firsttime by computing the total field transmittedthrough the hole, the second time by consideringthe fundamental waveguide mode only. Abovethe cut-off, for R = 125 nm, the transmissionof the single mode as a function of t is almost con-stant, with a small slope linked with the smallimaginary part of c, close to 10�2. The total fieldpresents small oscillations, due to a Fabry–Perotinterference effect produced by the reflection ofthe propagation field at the top and bottom ofthe hole. Below the cut-off, the slope of the curvesis much steeper and agree with the value of

01E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

Tra

nsm

issi

on

t(nm)500 1000

Fig. 5. Transmission of three nanoholes with different radii pie

Im(c) � 1, obtained with R = 75 nm in Fig. 4, tak-ing into account the fact that the transmission isproportional to the square of the field. This slopeis weaker than the one observed in Fig. 3(b), whichis quite natural since in the latter R = 50 nm in-stead of 75 nm, that leads to a higher Im(c). Ofcourse, for R = 100 nm, the slope shows an inter-mediate steepness, as could be deduced fromFig. 4. For all values of R, the decrease of the totalfield is governed by the decrease of the fundamen-tal mode, since the slope of the decay of the totalfield and of the mode is the same when staying be-low the cut-off. Moreover, the maxima of the smallfluctuations observed above the cut-off (R = 125nm) follow the straight line of the correspondingsingle mode,which is now propagating.

In view of pointing out the link between theguided mode inside the hole and the transmissionthrough the nanohole, Fig. 6 presents the fieldmaps of the modulus of the r, h and z componentsof the electric field, for both the fundamentalguided mode and for the real total field calculatedinside the hole at z = t/2. It is evident that the totalfield inside the hole has the spatial structure of thefundamental guided mode. Moreover, as far as thedecay constant of this waveguide mode is alwayssmaller than the attenuation through the film (ob-tained in Fig. 4 for R = 0), the former plays a keyrole in the extraordinary transmission process (asviewed in Fig. 5).

Ag: λ = 500 nm

total field:R = 75 nmR = 100 nmR = 125 nm

single mode:R = 75 nmR = 100 nmR = 125 nm

1500

rced inside a silver screen as a function of the thickness t.

020406080100

0

50

100

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

020406080100

0

50

100

0.00

0.01

0.02

0.03

0

50

100

020406080

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0

50

100

020406080

0.00

0.01

0.02

0.03

0

50

100

020406080

0.000

0.005

0.010

0

50

100020406080

0.000

0.005

0.010

field in the aperturemodal field

|Ez|

|Eθ|

|Er|

y(nm

)

y(nm

)

y (nm

)

y(n

m)

y (nm

)

y(n

m)

x (nm)x (nm)

x (nm) x (nm)

x (nm) x (nm)

Fig. 6. Maps for the modulus of the components of the electric field for the fundamental guided mode and for the total field in the holeat mid depth.


4. Interference effect

In order to clarify the fringe structure shown inFig. 3(a), we study separately the x- and z-compo-nents of the field. Using the same optogeometricalparameters as in Figs. 3(a), 7(a) and (b) show |Ex|and |Ez| as functions of x for several metal thick-nesses. Since the incidence is normal, unpierced sil-ver films will produce a transmitted field withEz = 0, while Ex will be different from zero. Thus,if we follow the explanation developed in [19] and

recalled in Section 2, the field maps are expected tobe different. Comparison of Fig. 7(a) and (b) con-firms these expectations. One observes fluctuationsof |Ex| for thinner layers which are absent in |Ez|.The relative amplitude of these fluctuations in-crease with t up to t � 150 nm and rapidly de-creases when t > 250 nm.

In addition, one can observe a small variationin their quasiperiod, which becomes shorter forthinner layers. For example, when the thicknessis increased from t = 50 to 250 nm, the quasiperiod

Table 1Propagation constant of a plasmon along a silver film fordifferent film thicknesses

t (nm) a0p Imða00pÞ5 4.0181 0.1323310 2.203 0.06115 1.6627 0.0365120 1.425 0.024525 1.30215 0.017730 1.22985 0.01346250 1.0668 0.00257

Fig. 7. Field amplitude at a point M(x, 0,1 nm) below ananohole in silver screen as a function of x for variousthicknesses t (a). |Ex| (b). |Ez| (c). Modulus of the x-componentof the field below the nanohole minus the field below theunpierced silver film called Ex,film.


d increases from about 450 to 470 nm. This ten-dency is even more pronounced for thinner layersand is due to the coupling between the plasmons

propagating on the two faces of the layer. We de-vote the next section to its study.

Assuming that a surface plasmon Ep propagat-ing at the bottom interface of a silver film has an x-dependence of the following form [17]:

Ep;x ¼axffiffiffix

p exp ik0a0pxþ iux

� �exp �k0a00px

� �; ð1Þ

Ep;z ¼azffiffiffix

p exp ik0a0pxþ iuz

� �exp �k0a00px

� �; ð2Þ

where k0 = 2p/k, and ux and uz are phase factors,the phase of the plasmon varies with a x-period dprelated to a0p by a0p ¼ k

dp. Taking d = 470 nm (for

t > 150 nm) and k = 500 nm, we find a0p ¼ 1.064,which is in excellent agreement with the value ofthe plasmon propagation constant along air–silverinterface. Since the z-component of the transmit-ted field is null, |Ez| reduces to the modulus ofthe z-component of the plasmon Ep and no inter-ference effect is seen (Fig. 7(b)). On the other hand(Fig. 7(a)), Ex is the sum of a transmitted fieldEx film equal to a constant a1 for z = const., plusa plasmon field Ep,x equal to ðax=

ffiffiffix

p Þ expðik0a0pxþiuxÞ expð�k0a00pxÞ. Thus, the square modulus ofthe sum is modulated with a period dp ¼ k

a0p¼

470 nm for large thicknesses. Moreover, when t

is increased from 50 nm, a1 is decreased more rap-idly than the field transmitted through the hole, asdiscussed in detail in the previous section. Thus,the contrast of the interference figure varies witht and depends on the relative strength of a1 andax. When the film thickness is sufficiently large,a1 becomes negligible compared with ax and theinterference effect disappears. Fig. 3(a) combinesthe behaviors of Fig. 7(a) and (b). In order to con-firm this interpretation, we have plotted in


Fig. 7(c) the modulus of the difference between Ex

and the field Ex,film transmitted by the unpiercedsilver film. No fringes appear and the curves arequite similar to those obtained for |Ez|, not only

0 1000 2000 3000-2

0

2

: Re(Ex)

: Im(Ex) t = 5 nm

Re(

Ex)

, Im

(Ex)

0 1000 2000 3000-2

0

2 t = 15 nm

Re(

Ex)

, Im

(Ex)

0 1000 2000 3000-2

0

2

4

t = 25 nm

x (nm)

Re(

Ex)

, Im

(Ex)

Fig. 8. Variation of Ex(x) for

for the shape but also for the amplitudes. The con-clusion is that the interference effect suggested bythe authors of [19] is fully observed on relativelythin films, not thick enough to completely attenu-

0 1000 2000 3000-2

0

2

t = 10 nm

0 1000 2000 3000-2

0

2 t = 20 nm

x (nm)

0 1000 2000 3000-2

0

2

4

6

t = 30 nm

several low thicknesses.


ate the field transmitted through the film whencompared to the field transmitted through theaperture.

5. Extremely thin screens

We have observed in Fig. 7(a) that when thethickness t is reduced, the plasmon period reduces.This is due to the increasing coupling between the

-1

0

1Re(Ez)

t = 5 nm

-2

0

2

Im(

-2

0

2t = 10 nm

-2

0

2

4

-2

0

2

4t = 15 nm

-2

0

2

4

-2

0

2

4

t = 20 nm

-2

0

2

4

-2

0

2

4 t = 25 nm

-2

0

2

0

-202468

d

t = 30 nm

X Axis Title

0

2

30001000 2000

Fig. 9. Variation of Ez(x) for

surface plasmons at the upper and lower interface[10,12]. It greatly modifies the plasmon propaga-tion constant ap, as is shown in Table 1, which pre-sents ap for several thicknesses. The result is areduction of the period of the interference fringesby a factor 3–5, when t is reduced from 40 to 5nm. Such a reduction is thus expected in the exactnumerical results. Using the same optogeometricalparameters, Fig. 8 shows the real and imaginarypart of Ex calculated using the electromagnetic

Ez)

0X Axis Title

30001000 2000

several low thicknesses.


theory [16] as function of x, for different values of tranging from 5 to 30 nm. Fig. 9 presents the sameresults for Ez. Comparing Figs. 8 and 9, it is clearthat Ex has a non-zero mean value, while for Ez itis not so. This comes from the fact that the deviceis lighted under normal incidence, so that the z-component of the transmitted field is zero. In allfigures, it is seen that the period of the modula-tions is the same for Ex and Ez and that it increaseswhen t is increased, as can be deduced from Table1, which establishes the plasmon origin of the phe-nomenon. Moreover, as expected from the discus-sion in Section 3, whereas the mean valuedecreases with t as expected for a transmitted field,the values of Ex(0) and Ex(R) increase with t with-in this range of its variation. An increase of Ez(R)can also be observed.

The reduction of the plasmon phase period dueto coupling phenomena is not limited to thin films.A recent work [22] demonstrates experimentallythat a significant shift in the phase period of ex-cited plasmon on a pair of apertures in Au filmswhen the distance between the apertures decreases,due to the coupling between the neighboring local-ized surface plasmons.

6. Conclusion

The extraordinary light transmission throughnano holes is linked with the finite conductivityof the screen which produces lower attenuationfor guided modes below the cut-off than in the caseof infinite conductivity. In addition, the finite con-ductivity allows for surface plasmon excitation,which field dominates the near-field distributionin the very vicinity below the film. The numericalresults obtained using a rigorous electromagneticanalysis can be fully explained using the conceptsof leaky guided mode inside the hole, and surfaceplasmon at the bottom interface, the latter being,eventually, modified by coupling with the upperinterface plasmon when the thickness of the screenis small. For thicknesses at which the transmittedfield through the layer is non-negligible, a fringestructure of the field map exists for some field com-

ponents, due to interference with the surface plas-mon field.

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light transmission through a subwavelength hole

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