456 J. Opt. Soc. Am. A/Vol. 21, No. 3 /March 2004 Vallius et al.

Transmission through single subwavelengthapertures in thin metal films

and effects of surface plasmons

Tuomas Vallius and Jari Turunen

Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland

Masud Mansuripur and Seppo Honkanen

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received August 5, 2003; revised manuscript received October 7, 2003; accepted October 14, 2003

The existing analyses on extraordinary optical transmission through apertures on a metal screen have beencarried out assuming perfect conductivity or by examining arrays of closely spaced holes with subwavelengthdimensions. We present an electromagnetic analysis of a single hole (modeled by use of an array of distantholes) in a finitely conducting metal membrane, applying no approximations. We demonstrate that finite con-ductivity is not of remarkable importance with small hole-diameter-to-wavelength ratios in the absence ofstrong resonances. However, if the angle of incidence of a plane wave is such that surface plasmons are ex-cited, substantial enhancement of the transmittance can be observed, and the effect of finite conductivity willno longer be negligible. Our analysis also reveals that transmission of small apertures in highly conductingmembranes can be described by approximate analytical formulas if surface waves are not excited, but withpoor conductors the full electromagnetic analysis should be applied. © 2004 Optical Society of America

OCIS codes: 240.6680, 050.1960, 050.1220.

1. INTRODUCTIONSince Ebbesen et al.1 reported on extraordinary transmis-sion through arrays of subwavelength-sized holes, thesubject has been under intense examination, resulting inseveral publications and giving rise to discussions aboutthe physical origin of the phenomenon. Various argu-ments on the mechanism of the transmission, whether itis attributable to surface plasmons or cavity resonances,have been made for subwavelength arrays of both slitsand holes.2–14 However, recent research has mainly beenconcerned with periodic configurations, which are of sub-stantial interest in, e.g., filtering applications,15–17 butonly a few papers have been dedicated to the behavior of asingle aperture in finitely conducting films, and theymainly consider the topic experimentally.18,19

Optical behavior of a single subwavelength aperture ina metal membrane is of utmost interest in high-definitionoptical data storage,20 optical probing, and near-field mi-croscopy, which require focusing of light into small areas.Such apertures are relatively easy to fabricate with mod-ern lithographic processing.21 Nevertheless, in many pa-pers the theoretical analysis has been limited to the use ofthe famous formula derived by Bethe,22 which states thatthe optical energy h transmitted through a subwave-length aperture in an opaque film follows the relationh/D2 ; (D/l)4, where D and l are the diameter of theaperture and the wavelength, respectively. Here h is theratio of the integrals over the z component of the Poyntingvector of the transmitted and incident fields (the film isassumed to be in the xy plane). For a square aperture

1084-7529/2004/030456-08$15.00 ©

with dimensions D 3 D illuminated by a plane wave atnormal incidence, Bethe’s formula assumes the form

h

D25

64

27p

~kD !4

26, (1)

where k 5 2p/l. Unfortunately, Eq. (1) contains severalapproximations, such as infinite conductivity, infinitelythin film, and the assumption l @ D, and thereby its ap-plicability is doubtful in realistic configurations.Bouwkamp23 improved Bethe’s result by deriving addi-tional terms for the expansion, whose three first termsmay be written as

h

D25

64

27p F ~kD !4

261

22

25

~kD !6

281

7312

18,375

~kD !8

210 G .

(2)

Nevertheless, formula (2) remains an approximate model,and thus it must be assumed that it illustrates onlyroughly the transmission of apertures in finitely conduct-ing metal membranes of finite thickness.

Since exact analytical solutions are rare for electromag-netic scattering problems,24,25 numerical approaches ofrigorous diffraction theory must usually be applied.26 Al-though numerical analysis of structures invariant alongone coordinate axis is straightforward,27,28 three-dimensional structures, such as a circular or square-shaped hole in a film, still result in computational prob-lems. Therefore the structures are often assumed to beperfectly conducting.15,29 This assumption holds well formany metals in radio-frequency or microwave regions but

2004 Optical Society of America

Vallius et al. Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. A 457

not within the optical part of the spectrum because of thelow conductivity.16 In addition, resonance phenomenasuch as surface plasmons and waveguide resonances,which may play a major role at certain incidence angles,cannot be modeled when perfect conductivity is assumed.Consequently, analyses based on the electromagnetic dif-fraction theory of finitely conducting materials30,31 andcomparisons between exact numerical and approximateanalytical models are needed.

In this paper we shall provide electromagnetic analysesof rectangular and circular apertures in finitely conduct-ing screens. The perforated film will be examined notonly under normal incidence but also in a tilted angle thatexcites surface plasmon resonance within the unperfo-rated surface. The dependence of the transmittance onthe size of the hole and the influence of surface plasmonswill be studied.

2. NUMERICAL METHODThe numerical analysis is carried out by use of a rigorousFourier modal method for crossed gratings30,31 with theS-matrix algorithm,32 which is currently the most reliablemethod for three-dimensional structures.33 To enhancethe computational efficiency of the method, symmetryrules of the Fourier coefficients34,35 are applied under nor-mal incidence. Readers not familiar with the approachmay consult Ref. 31, which contains a full description ofthe method with a complete list of the original publica-tions.

Since the Fourier modal method is derived for periodicstructures, we have to include a large nonmodulated areabetween the holes to eliminate the effects of periodicity inthe near field. On the other hand, the larger the period,the longer the computation time, and thus we have tocompromise between computational efficiency and nu-merical accuracy. From the studies on one-dimensionalgratings, we know that when dielectric structures areseparated by a nonmodulated distance of 10l the couplingbetween isolated segments becomes negligible, and eachsegment can then be analyzed separately.36,37 In thecase of conducting materials, that separation distance canbe reduced because of the small skin depth of metals38; weuse the period d 5 4.9l (see Fig. 1 for geometry), which

Fig. 1. Metal membrane perforated by square apertures of thesize D 3 D. The thickness of the film is denoted by h. The in-cident field propagates in the xz plane at an angle u with respectto the z axis; the electric field vector of the incident field is in theplane of incidence. The grating period used in the calculationsis d 3 d.

yields rather reliable results within reasonable computa-tion times. A noninteger value was chosen to avoid Ray-leigh anomalies.39

3. RESULTSLet us restrict ourselves, for the time being, to a squareaperture in a metal membrane. We denote the refractiveindices on the side of incidence, on the transmission side,and in the metal by n1 , n3 , and n2 , respectively. In ad-dition, u, h, and D denote the angle of incidence with re-spect to the z axis, the thickness of the metal film, and thelength of the side of the square aperture.

A. Normal IncidenceAs the first example, let us consider a metal membrane(n2 5 0.1 1 3i) perforated by a square aperture illumi-nated by a plane wave under normal incidence. The re-fractive indices of the incident and transmission mediaare n1 5 n3 5 1. The result of interest is the ratio of theenergy transmitted through the film to the incident en-ergy on the aperture. It follows that we must normalizethe transmittance by taking into account the proportionof the period covered by the hole when the transmittanceassumes the form hd2/D2, which is depicted in Fig. 2 fortwo different values of h, namely, 0.3l and 0.4l. Theseare both chosen to be greater that 0.25l to avoid trans-mission through the film; i.e., opaque films are assumedas in the Bethe and Bouwkamp models.22,23

Even though some numerical noise attributable to in-complete convergence appears (we used 27 3 27 diffrac-tion orders in the calculations), we can clearly observe thepeak of transmittance at around D 5 0.4l and, in addi-tion, that Bethe’s formula [Eq. (1)] (dashed–dotted curve)produces the characteristic form of the curve within therange D , 0.3l but Bouwkamp’s approach [Eq. (2)] (dot-ted curve) clearly outperforms Bethe’s after that.Bouwkamp’s formula contains more terms, and thus itshould be more accurate than Bethe’s; it is evidentlycloser to the curve corresponding to the thinner mem-brane because both approximate formulas assume an in-finitely thin film. When D . 0.3l, both approximatemodels diverge away from the rigorous curves. In viewof these results, it can be concluded that neither Bethe’snor Bouwkamp’s formula is very accurate in the visibleregion, although each gives reasonable approximationswhen D , 0.3l.

The dependence of the transmittance on the thicknessof the film is illustrated in Fig. 3, where the case of theunperforated film is also included. In contrast to the pa-pers examining subwavelength periodic slits or holes,9,12

no anomalous behavior manifests itself in the form ofsharp peaks in the transmittance because at smooth sur-faces under normal incidence strong coupled resonancesbetween the holes are not excited.

To investigate the influence of the refractive index ontransmittance, we next consider an identical case exceptfor the refractive index of the metal n2 5 1.2 1 2i. Ow-ing to poor conductivity, the film transmits more light,and thus we have to use larger values for the thickness h;now we analyze with h 5 0.5l and h 5 0.6l. The be-havior of the hole is plotted in Fig. 4, illustrating the ex-

458 J. Opt. Soc. Am. A/Vol. 21, No. 3 /March 2004 Vallius et al.

tent to which we may trust Bethe–Bouwkamp formulas[Eqs. (1) and (2)] with poor conductors. Even with smalldiameter-to-wavelength ratios (D/l , 0.2), the approxi-mate formulas are untrustworthy although they yieldedreliable results in the previous case, in which the metalwas a better conductor. From Fig. 5 we notice that thetransmittances of the perforated and intact films arenearly identical, implying that a remarkable proportion oflight impinging on the hole is absorbed. In fact, morethan 50% of the incident light is absorbed in the case ofthe unperforated film.

B. Surface Plasmon ExcitationNext we will consider a case in which light is absorbed bythe unperforated film because of surface plasmon excita-tion within the metal layer.40 In principle, there are twoways of generating surface plasmons. On the one hand,we may use the grating period to invoke surface wavespropagating along the grating. On the other hand, theangle of incidence can be applied to excite surface plas-mons at the critical angle of total reflection. In this pa-per we consider single apertures, and thus we resort tothe latter option. The refractive indices before and after

Fig. 2. Dependence of the normalized transmittance on the holesize D for two different film thicknesses h. Also, the results ofBethe’s and Bouwkamp’s formulas are plotted. The refractiveindex of the metal is n2 5 0.1 1 3i.

Fig. 3. Transmittance of both the perforated (D 5 0.3l) andthe unperforated membranes as functions of the thickness withrefractive index n2 5 0.1 1 3i.

the film are n1 5 1.5 and n3 5 1, respectively. Themetal membrane (n2 5 0.1 1 3i) is perforated by asquare-shaped aperture having a refractive index n35 1. A plane wave arrives at the metal surface at anangle 45.1° with respect to the z axis, and the electric fieldvector is in the plane of incidence corresponding to TM po-larization. We analyze this configuration by using twodifferent values of film thickness, h 5 0.1l and h5 0.15l; the transmittance is zero without the aperturebecause the angle of incidence is greater than the criticalangle of total internal reflection. The effect of the aper-ture size is illustrated in Fig. 6, which depicts the depen-dence of the normalized transmittance on the hole size.

In contrast to the case of normal incidence, in whichthe efficiency was larger than 1 only within a narrowrange in the neighborhood of D 5 0.4l, we may achievetransmittance with high efficiency over a wide range ofhole sizes, 0.1l , D , 0.8l with an efficiency over 1.5.This is attributable to longitudinal surface waves thatpropagate in the metal layer and couple to diffraction or-ders, resulting in the transmission anomalies for sub-wavelength periodic structures reported by Ebbesen et al.in Ref. 1. This demonstrates that the same phenomenoncan be obtained with single holes as well, even withoutthe surface corrugations used by Thio et al. in Ref. 19.

The dependence of the transmittance on the incidentangle u as well as the absorption of the unperforated

Fig. 4. Same as Fig. 2 but with n2 5 1.2 1 2i.

Fig. 5. Same as Fig. 3 but with n2 5 1.2 1 2i and D 5 0.4l.

Vallius et al. Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. A 459

metal film are plotted in Fig. 7. The trends of bothcurves are similar, implying that the transmittance de-pends on the amount of light coupled into surface waves.Consequently, surface plasmons play a major role in thetransmission, which is not necessarily correct with infi-nite slits.11 The angular width of the resonance in Fig. 7is approximately 1°, and thus it may be realized also ex-perimentally without great difficulties, since such angu-lar divergences can easily be obtained with common laserbeams.

C. Near-Field ProfileHaving analyzed efficiencies of transmission through sub-wavelength apertures, we now proceed to consider energydistributions immediately after the aperture. A questionarises as to whether the transmitted energy density canbe concentrated into a small spot, which would be useful,for instance, for optical writing purposes. Instead of

Fig. 6. Dependence of normalized transmittance on the holesize D for different film thicknesses at the surface plasmon exci-tation angle (u 5 45.1°) with n1 5 1.5, n2 5 0.1 1 3i, and n35 1.

Fig. 7. Normalized transmittance of an aperture (n2 5 0.11 3i, D 5 0.3l, h 5 0.1l) as a function of the incident angle u.

Also shown is absorption of the unperforated film versus the in-cident angle u for a film having h 5 0.1l.

square apertures, we shall consider circular holes hence-forth, since they can produce more symmetric field distri-butions after the aperture.

The electric energy density immediately after a circularaperture is shown in Fig. 8(a), assuming illumination by aTM-polarized plane wave propagating in the xz plane to-ward the positive x direction at the surface plasmon angle(u 5 45.1° with respect to the z axis). The incident fieldexcites a surface plasmon that propagates along the xaxis. We note that, even though electric fields propagat-ing within the metal film generate substantially strongenergy distribution throughout the membrane, a remark-able peak can be observed near the upper edge of the hole,on which the propagating surface waves impinge. Thewidth of the peak is w/l ' 1/4, indicating that light canbe confined into a very small spot of high energy. Notethat the center of the spot does not coincide with the cen-ter of the hole. However, considering Fig. 8(b), whichshows the time average of the z component of the Poyn-ting vector, we discover that the propagating part of thefield is concentrated in the vicinity of the edge; it actuallyhas negative values within the aperture, which is not an

Fig. 8. (a) Distribution of the electric energy density at the exitplane of a circular hole with diameter D 5 0.2l and film thick-ness h 5 0.1l at the surface plasmon angle (n1 5 1.5, n3 5 1,u 5 45.1°) with n2 5 0.1 1 3i. (b) Same for the time average ofthe z component of the Poynting vector whose maximum is nor-malized to unity. The hole is centered at the origo of the coor-dinates.

460 J. Opt. Soc. Am. A/Vol. 21, No. 3 /March 2004 Vallius et al.

Fig. 9. (a) Diagram of the simulated system. The annular beam at the entrance pupil of the objective lens (l 5 650 nm) has a Gauss-ian cross section between the inner and the outer radii rmin 5 1.75 mm, rmax 5 1.96 mm; this beam is linearly polarized in the radialdirection. The angle of incidence of the various rays at the metal film ranges from 42.13° to 48.56°, centered around the resonance angle(ures 5 45.1°) for surface plasmon excitation in the thin metal film. The film thickness is h 5 65 nm, and its complex refractive indexis n2 5 0.1 1 3i. (b) Intensity distribution of the annular beam at the entrance pupil of the objective lens. (c) Localized intensitydistribution, uExu2 1 uEyu2, at the focal plane of the objective. Like the incident beam, the focused spot has radial polarization; that is,at all points within the focal plane of the objective, the in-plane component of the electric field is oriented radially relative to the focalpoint (x, y) 5 (0, 0).

unusual phenomenon with rapidly oscillating fielddistributions.41

Using plane-wave illumination, we can get only a smallfraction of the incident energy inside the hole; a focusedspot would thus be more beneficial. Also tunnelingthrough the metal might induce problems as the evanes-cent field has a considerable amplitude outside the aper-ture. It follows that light would transmit also throughthe unperforated film, which is undesirable in several op-tical applications. If we try to form a small focal spot byusing a focused Gaussian beam that covers only the aper-ture and its neighborhood, the spread of the angular spec-trum of the incident beam is so large that only a smallfraction of light satisfies the surface plasmon resonancecondition. Therefore an intuitive way of solving theproblem would be to employ an electromagnetic Bessel

field.42 The angular spectrum of a propagation-invariantBessel field is confined to a cone. Thus all plane-wavecomponents are incident on the xy plane at the sameangle u with respect to the z axis; only the azimuthalangle varies. If this angle is chosen equal to the surfaceplasmon resonance angle, the light-excitation efficiencyis, in principle, optimized (the efficiency optimization re-quires further investigation, since each sidelobe of theBessel beam contains approximately equal energy, andthus a larger portion of incident light falls outside the ap-erture than is the case with a Gaussian beam).

Figure 9 shows a radially polarized ring of coherent,monochromatic light (l 5 650 nm) brought to focus atthe focal plane of an aberration-free objective lens. Asmall, hemispherical glass lens of refractive index n15 1.5 is placed immediately before the focal plane of the

Vallius et al. Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. A 461

objective, and a thin metal film is coated on the flat facetof the hemisphere. The angular spectrum is assumed tobe radially polarized, i.e., locally linearly polarized in theradial direction. The central cone angle is chosen to co-incide with the surface plasmon resonance angle at themetal film. The intensity profile of the angular spectrumis shown in Fig. 9(b), and the resulting intensity distribu-tion at the focal plane is shown in Fig. 9(c). Like thering, the focused spot in the xy plane is linearly polarizedin the radial direction. The full diameter of the brightcentral ring of the focused spot (namely, the diameter ofthe first dark ring) is 0.7 mm, which happens to coincidewith the diameter of the first dark ring of an Airy patternproduced by the same objective–hemisphere combination,i.e., 1.22l/(nNA) 5 0.7 mm (NA, numerical aperture).As already mentioned, the sidelobes of the focused spot inFig. 9(c) are brighter than those of the Airy pattern anddepend on the width of the angular spectrum, whichshould be optimized to obtain the highest efficiency.

Since all the plane waves contained in the cone are inthe immediate neighborhood of the resonance, the focusedspot is strongly absorbed in the metal. The glass hemi-sphere helps to reduce the size of the focused spot (by afactor n 5 1.5 relative to a similar spot formed in the air),but it is also essential for the excitation of surface plas-mons. The transmission through the unperforated metalfilm is exactly zero because all the rays incident at thebottom of the glass hemisphere have incidence anglesgreater than the critical angle for total internal reflection.The small fraction of the optical energy that is not ab-sorbed by the metal film is thus reflected back toward theobjective.

At first sight, the intensity distribution in Fig. 9(c) doesnot appear to be optimal because of the central zero (dueto a field singularity). However, the radial polarizationensures that all plane waves are TM polarized. Hencethe charge carriers within the metal film oscillate collec-tively to and from the dark spot at the center of the firstbright ring of the focused beam. If a small circular holehappens to exist in the region of this dark spot, the E-fieldoscillations of the bright ring should then produce uni-formly charged walls on the cylindrical inner surface ofthe hole. These charges will then radiate into the holeand beyond, as they oscillate between positive and nega-tive values at the frequency of the incident radiation.

We illustrate the configuration described in Fig. 9 by aradially polarized conical wave with u 5 45.1°. Eventhough the radial component of the incident field is zeroover the aperture, the z component has its peak value inthat region. The field is confined around the aperture,thus reducing the amplitude of the field within the unper-forated region. The distribution of electric energy den-sity of the transmitted field at the exit plane of the aper-ture is illustrated in Fig. 10(a). One observes that theconfiguration differs substantially from the case of plane-wave illumination because the ratio between the energydensity over the aperture and after the unperforated filmis substantially greater than in the case of plane-wave il-lumination. Although the incident field is confined to afairly small spot, each plane-wave component is coupledinto surface plasmons and results in the sharp peak afterthe aperture. Since the incident energy is confined

around the aperture, we may now obtain substantiallyhigher efficiencies.

Nevertheless, Fig. 10(b) reveals that even though theelectric energy density is confined within the aperture,the z component of the Poynting vector has positive val-ues only at the edges of the hole, indicating that the elec-tric energy peak consists of evanescent fields. Over theaperture the values are negative, which is quite a com-mon feature of electromagnetic conical waves.41 Al-though the diameter of the hole is D 5 0.2l, the size ofthe region where the energy flow differs from zero is of theorder of 1 wavelength. Thus the size of the aperture in asemitransparent film in the Kretschmann configurationdoes not necessarily define the spot size of the transmit-ted field.

4. CONCLUSIONSIn view of the aforementioned results, we may concludethat, by applying the Bethe–Bouwkamp formulas to well-conducting metals, one may obtain good approximationson the transmittance of the hole with small D/l ratiosproviding the right magnitude of the transmission, but,on the other hand, with poor conductors or larger hole di-ameters the analytical results are not trustworthy.Therefore Bethe’s or Bouwkamp’s results should be used

Fig. 10. Same as Fig. 8 but the radial component of the incidentfield is the first-order Bessel function.

462 J. Opt. Soc. Am. A/Vol. 21, No. 3 /March 2004 Vallius et al.

with care within the optical region where metals cannotbe assumed to be perfectly conducting.

Our analysis reveals that the transmittance of a singleaperture in a metal membrane can be enhanced simply byuse of an incident angle that excites a surface plasmonwave at the smooth dielectric–metal interface. This pro-cedure allowed enhanced transmission with high effi-ciency and was not very sensitive to changes in the size ofthe aperture or the angle of incidence. Angular depen-dence of the transmission was similar to that of absorp-tion, implying that the effect arises from surface plasmonresonances. Near-field analysis on the transmission sideshowed that under plane-wave illumination a sharp peakin the electric energy density is observed at the edge ofthe aperture. However, the electric energy density wasalso high after the unperforated film. That was avoidedby use of a finite incident beam with a radially polarizedangular spectrum, in which the electric energy densitywas confined to the neighborhood of the aperture.

This examination also showed that confining light intosmall spots by use of subwavelength apertures is not astrivial as one may expect, since the distribution of the zcomponent of the Poynting vector after the aperture mayvary greatly depending on the illuminating field. There-fore further investigations are needed to define the opti-mal illumination to confine the transmitted field within asubwavelength area. For any practical application, themedium with which light passing through the aperturewould interact must also explicitly be taken into account.

ACKNOWLEDGMENTSThis paper was written when Tuomas Vallius was visitingthe Optical Sciences Center of the University of Arizona.We thank Philippe Lalanne from Laboratoire CharlesFabry de l’Institut d’Optique for helpful comments. Wealso gratefully acknowledge discussions about electro-magnetic Bessel fields with Pertti Paakkonen and aboutenhanced transmission with Jani Tervo from the TeamJoensuu.

Tuomas Vallius, the corresponding author, can bereached by e-mail at Tuomas.Vallius@joensuu.fi.

REFERENCES1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A.

Wolff, ‘‘Extraordinary optical transmission through sub-wavelength hole arrays,’’ Nature 391, 667–669 (1998).

2. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H.J. Lezec, ‘‘Surface plasmons enhance optical transmissionthrough subwavelength holes,’’ Phys. Rev. B 58, 6779–6782(1998).

3. U. Schroter and D. Heitmann, ‘‘Surface-plasmon-enhancedtransmission through metallic gratings,’’ Phys. Rev. B 58,15419–15421 (1998).

4. M. M. J. Treacy, ‘‘Dynamical diffraction in metallic opticalgratings,’’ Appl. Phys. Lett. 75, 606–608 (1999).

5. J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, ‘‘Transmis-sion resonances on metallic gratings with very narrowslits,’’ Phys. Rev. Lett. 83, 2845–2848 (1999).

6. T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W.Ebbesen, ‘‘Surface-plasmon-enhanced transmissionthrough hole arrays in Cr films,’’ J. Opt. Soc. Am. B 16,1743–1748 (1999).

7. I. Avrutsky, Y. Zhao, and V. Kochergin, ‘‘Surface-plasmon-

assisted resonant tunneling of light through a periodicallycorrugated thin metal film,’’ Opt. Lett. 25, 595–597 (2000).

8. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, ‘‘Theory oflight transmission through subwavelength periodic hole ar-rays,’’ Phys. Rev. B 62, 16100–16108 (2000).

9. S. Astilean, P. Lalanne, and M. Palamaru, ‘‘Light transmis-sion through metallic channels much smaller than thewavelength,’’ Opt. Commun. 175, 265–273 (2000).

10. L. Martın-Moreno, F. J. Garcıa-Vidal, H. J. Lezec, K. M.Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, ‘‘Theoryof extraordinary optical transmission through subwave-length hole arrays,’’ Phys. Rev. Lett. 86, 1114–1117 (2001).

11. Q. Cao and P. Lalanne, ‘‘Negative role of surface plasmonsin the transmission of metallic gratings with very narrowslits,’’ Phys. Rev. Lett. 88, 0574031–0574034 (2002).

12. P. Lalanne, C. Sauvan, J.-P. Hugonin, J. C. Rodier, and P.Chavel, ‘‘Perturbative approach for surface plasmon effectson flat interfaces periodically corrugated by subwavelengthapertures,’’ Phys. Rev. B 68, 125404 (2003).

13. N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, ‘‘Reso-nant optical transmission through thin metallic films withand without holes,’’ Opt. Express 11, 482–490 (2003).

14. A. Moreau, G. Granet, F. I. Baida, and D. Van Labeke,‘‘Light transmission by subwavelength square coaxial aper-ture arrays in metallic films,’’ Opt. Express 11, 1131–1136(2003).

15. R. C. McPhedran, G. H. Derrick, and L. C. Botten, ‘‘Theoryof crossed gratings,’’ in Electromagnetic Theory of Gratings,R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

16. V. Kettunen, M. Kuittinen, J. Turunen, and P. Vahimaa,‘‘Spectral filtering with finitely conducting inductive grids,’’J. Opt. Soc. Am. A 15, 2783–2785 (1998).

17. J. R. Sambles, ‘‘More than transparent,’’ Nature 391, 641–642 (1998).

18. T. Thio, H. J. Lezec, and T. W. Ebbesen, ‘‘Strongly enhancedoptical transmission through subwavelength holes in metalfilms,’’ Physica B 279, 90–93 (2000).

19. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W.Ebbesen, ‘‘Enhanced light transmission through a singlesubwavelength aperture,’’ Opt. Lett. 26, 1972–1974 (2001).

20. M. Mansuripur, The Physical Principles of Magneto-OpticalRecording (Cambridge U. Press, Cambridge, 1995).

21. J. Turunen and F. Wyrowski, eds., Diffractive Optics for In-dustrial and Commercial Applications (Wiley-VCH, Berlin,1997).

22. H. A. Bethe, ‘‘Theory of diffraction by small holes,’’ Phys.Rev. 66, 163–182 (1944).

23. C. J. Bouwkamp, ‘‘Diffraction theory,’’ Rep. Prog. Phys. 18,35–100 (1954).

24. A. Sommerfeld, ‘‘Mathematische theorie der diffraction,’’Math. Ann. 47, 317–374 (1896).

25. G. Mie, ‘‘Beitrage zur optik truber median, speziell kolloi-daler metallosungen,’’ Ann. Phys. (Leipzig) 25, 377–445(1908).

26. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

27. K. Knop, ‘‘Rigorous diffraction theory for transmissionphase gratings with deep rectangular grooves,’’ J. Opt. Soc.Am. 68, 1206–1210 (1978).

28. L. Li, ‘‘Use of Fourier series in the analysis of discontinuousperiodic structures,’’ J. Opt. Soc. Am. A 13, 1870–1876(1996).

29. F. J. Garcıa de Abajo, ‘‘Light transmission through a singlecylindrical hole in a metallic film,’’ Opt. Express 10, 1475–1484 (2002).

30. E. Noponen and J. Turunen, ‘‘Eigenmode method for elec-tromagnetic synthesis of diffractive elements with three-dimensional profiles,’’ J. Opt. Soc. Am. A 11, 2494–2502(1994).

31. L. Li, ‘‘New formulation of the Fourier modal method forcrossed surface-relief gratings,’’ J. Opt. Soc. Am. A 14,2758–2767 (1997).

32. L. Li, ‘‘Formulation and comparison of two recursive matrixalgorithms for modeling layered diffraction gratings,’’ J.Opt. Soc. Am. A 13, 1024–1035 (1996).

Vallius et al. Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. A 463

33. P. Dansas and N. Paraire, ‘‘Fast modeling of photonic band-gap structures by use of a diffraction-grating approach,’’ J.Opt. Soc. Am. A 15, 1586–1598 (1998).

34. P. Lalanne and D. Lemercier-Lalanne, ‘‘On the effective me-dium theory of subwavelength periodic structures,’’ J. Mod.Opt. 43, 2063–2085 (1996).

35. P. Lalanne, ‘‘Improved formulation of the coupled-wavemethod for two-dimensional gratings,’’ J. Opt. Soc. Am. A14, 1592–1598 (1997).

36. B. Layet and M. R. Taghizadeh, ‘‘Analysis of gratings withlarge periods and small features by stitching the electro-magnetic field,’’ Opt. Lett. 21, 1508–1510 (1996).

37. T. Vallius, V. Kettunen, M. Kuittinen, and J. Turunen,

‘‘Step-discontinuity approach for non-paraxial diffractiveoptics,’’ J. Mod. Opt. 48, 1195–1210 (2001).

38. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, ‘‘Useof grating theories in integrated optics,’’ J. Opt. Soc. Am. A18, 2865–2875 (2001).

39. Lord Rayleigh, ‘‘On the dynamical theory of gratings,’’ Proc.R. Soc. London Ser. A 79, 399–416 (1907).

40. H. Raether, Surface Plasmons on Smooth and Rough Sur-faces and on Gratings (Springer, New York, 1988).

41. J. Turunen and A. T. Friberg, ‘‘Self-imaging andpropagation-invariance in electromagnetic fields,’’ PureAppl. Opt. 2, 51–60 (1993).

42. J. Durnin, ‘‘Exact solutions for nondiffracting beams. I.The scalar theory,’’ J. Opt. Soc. Am. A 4, 651–654 (1987).