Enhanced transmission with coaxial nanoapertures: Role of cylindrical surface plasmons

Transcription

1 Enhance transmission with coaxial nanoapertures: Role of cylinrical surface plasmons Michael I. Haftel, Carl Schlockermann, an G. Blumberg 3 Center for Computational Materials Science, Naval Research Laboratory, Washington, DC , USA Munich University of Applie Sciences, Munich, Germany, an 3 Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974, USA Receive 8 August 006; revise manuscript receive October 006; publishe 4 December 006 We simulate the optical fiels an optical transmission through nanoarrays of silica rings embee in thin gol films using the finite-ifference-time-omain metho. By examining the optical transmission spectra for varying ring geometries we uncover large enhancements in the transmission at wavelengths much longer than the usual cutoffs for cylinrical apertures or where the usual planar surface plasmons or other perioic effects from the array coul play a role. We attribute these enhancements to closely couple cylinrical surface plasmons on the inner an outer surfaces of the rings, an this coupling is more efficient as the inner an outer ring raii approach each other. We confirm this hypothesis by comparing the transmission peaks of the simulation with cylinrical surface plasmon CSP ispersion curves calculate for the geometries of interest. One important result is that a transmission peak appears in the simulations close to the frequency where the longituinal wave number k z in the ring satisfies k z =m /L, where m is an integer an L the length of the aperture, for a normal CSP TE or TM moe. The behavior of the CSP ispersion is such that propagating moes can be sent through the rings for ever longer wavelengths as the ring raii approach, whereas the transmission ecreases only in proportion to the ring area. DOI: 0.03/PhysRevB I. INTRODUCTION The phenomenon of enhance optical transmission EOT through nanoarrays of apertures has attracte a great eal of attention ever since the publication of Ebbesen et al. s work in 998. This work, while having far-reaching implications for optical evice evelopment as well as for evices in the RF an microwave regime, has generate a lot of controversy concerning the origin of the enhance transmission. While originally thought to be attributable to surface plasmons SP s 3 on the upper an lower surfaces of the metal film in which the apertures were embee, other investigators have shown that other effects incluing shape resonances 4,5 an resonant coupling between the elements of the array,,6,7 present even for perfect conuctors where SP s are not present, 6,7 an other iffractive effects from evanescent waves, 8 can account for EOT. The explicit role of SP s has not been completely resolve since the other enhancements often occur at frequencies very close to those where surface plasmons woul be launche. Apart from the role of SP s, EOT is very sensitive to the shape of apertures. For example, the aspect ratio of rectangular apertures can have a large influence on EOT. 4 In a series of papers 8 consiering arrays of coaxial ring CR apertures, Baia an co-workers have shown in finiteifference-time-omain FDTD simulations that the EOT from coaxial ring apertures can be much larger than from cylinrical apertures. Furthermore, the transmission peaks for coaxial gol rings are consierably reshifte from those of either perfectly conucting coaxial rings or cylinrical holes. Such coaxial structures have recently been fabricate by Salvi et al., 3 who fin goo agreement between their measure transmission intensities an FDTD simulations, at least for wavelengths up to the maximum measure of 900 nm. These finings emonstrate that one can excee the iffraction limit to a much greater egree when these CR structures are use compare to apertures with a cylinrical PACS number s : n, 73.0.Mf, 4.79.Ag shape or other open structures. This coul have far-reaching consequences for optical evices in sensing, etection, an communication applications. The main purpose of this paper is to analyze the unerlying physics of the coaxial geometry an to show that cylinrical surface plasmon CSP resonances are primarily responsible for this behavior. We have previously sketche the role of these resonances for such CR structures. 4 The present paper gives a etaile analysis of their optical properties. The previous investigators attribute the enhancements at long wavelengths to the moe structure of the iniviual coaxial rings CR, 0,,3 presumably a TE moe, an not to the perioic structure. How the behavior of these moes is etermine by the properties of the gol film an gol core of the rings has not been elineate. Some of the normal moes for a CR in a perfect conuctor o iffer consierably from those of a cylinrical aperture.,5 For example, the lowest frequency TE moe has a cutoff R +R,,5 where R an R are the inner an outer ring raii. This gives the somewhat nonintuitive property that, for a constant outer raius R, the cutoff approaches a constant value R as the ring closes R R, which is the maximum value of the cutoff wavelength for a fixe R. Of course, the transmission vanishes in this limit as the ring closes up. Also, the CR supports a TEM 0 moe without cutoff. This moe, however, has raial symmetry an cannot be reaily excite by an incoming plane wave of linear or circular polarization, an thus woul not be seen in a FDTD simulation or typical experiments. In this work we show that the cutoff wavelength increases inefinitely in the above limit, an this is a consequence of the negative ielectric constant of metals an the ensuing cylinrical surface plasmons. The above work on CR structures 9 motivate us to carry out FDTD simulations to guie possible evice fabrication processing. These simulations strongly suggest that the source of the unusual EOT for the coaxial ring apertures 098-0/006/74 3 / The American Physical Society

2 HAFTEL, SCHLOCKERMANN, AND BLUMBERG FIG.. Schematic iagram of problem geometry for the basic case. A square array of coaxial cyliners with gol cores an silica rings are embee in a gol layer itself embee in a silica meium. The ring geometry, film thickness, an perioicity are inicate. Light is incient normally at the bottom of the film. are cylinrical surface plasmon CSP resonances, with the propagating CSP s on each of the metal-ielectric interfaces becoming strongly couple as the ring narrows. In the remainer of this paper we present the evience for such an interpretation. Section II briefly escribes our moel system an the basics of our FDTD simulation, incluing the parameters of an extene Drue moel for the ielectric constant of gol Au. Section III examines the analytic features of the normal moe problem, as a function of the inner an outer ring ratio, for a coaxial cyliner with a metal core an a ielectric ring embee in a metal film. By iagonalizing the appropriate 8 8 matrix we erive ispersion curves for CSP normal moes an from these erive the resonant conitions, which allow us to preict the peak positions as a function of ring geometry. We show that the cutoff for propagating moes increases inefinitely as the ring narrows with a resultant reshifting of the transmission peaks. We assess the role of losses in limiting this behavior. In Sec. IV we carry out FDTD simulations for ifferent CR geometries. These simulations confirm the analytic trens establishe in Sec. III. While the transmission peaks increasingly reshift with narrower rings, the intensity of transmission ecreases no faster than the expose ring area, an the transmitte energy is two to four times that incient on the ring. In Sec. V we summarize our results an present conclusions. II. MODEL SYSTEM FIG.. The ielectric constant of gol as calculate from Eq. an the experimental measurements of Johnson an Christy Ref. an Lynch an Hunter Ref. 3. Our basic moel consists of an infinite array of coaxial silica inex of refraction=.46 rings space in a square pattern at =555 nm intervals, with inner an outer raii R =50 nm an R =00 nm, imbee in an L=90 nm thick Au film. The metal film is itself imbee in a uniform silica meium. The incient light is a linearly polarize plane wave traveling in the silica meium incient normally on the Au film from the bottom. Figure illustrates our moel system. We consier variations of the CR geometry, namely changes in R, L, an, as well as consiering our basic geometry with Au replace by a perfect conuctor PC. We simulate the fiels with the NRL High Accuracy Scattering an Propagation HASP coe. 6,7 This coe employs a nonstanar finite ifference NSFD algorithm that yiels much smaller amplitue an ispersion errors than the stanar Yee algorithm when use on a coarse /0 gri. 8,9 We employ perioic bounary conitions in the x an y irections parallel to the film an Mur absorbing bounary conitions in z. The spatial step is nm in all irections, an the typical time step is 0.0 fs. The incient pulse is a plane wave of perio 3.0 fs =900 nm in vacuum, 63 nm in silica with a Gaussian envelope with a full with half maximum FWHM of fs. The frequency epenence of the fiels at any point is calculate from the time epenence obtaine from the simulation using stanar fast Fourier transform FFT techniques. We compute the transmission intensity at a given frequency by comparing the real part of the z component of the Poynting vector integrate over a plane 700 nm above the film with the corresponing quantity in the incient beam. This intensity is stable as the height above the film is varie from nm. We have verifie the HASP coe a number of ifferent ways. Simulations of Mie scattering from cyliners 0 an spheres reprouce the analytic results to about % even on a coarse /0 gri. Here we are using about a /00 gri. The HASP coe also gives goo agreement with the near-surface fiels in the surface plasmon jet observations of Egorov et al. 7 HASP also reprouces the positions of the transmission peaks observe by Martin-Moreno et al. for a free staning silver film with cylinrical apertures although the magnitue of the peaks iffers from the experiment. We moel the ielectric constant of Au with an extene Drue moel, i.e., m = i / i +4 i /, where,,, an are fit to the experimental ielectric constant. This form is a generalization from the Drue moel that allows us to fit the ielectric constant over a much larger range of wavelengths than the pure Drue moel. Thus we are not require to carry out numerous simulations over limite ranges of wavelengths to cover the full range of interest. Values of =7.99, = fs we use Gaussian units, = an =9.00 fs yiel a goo fit to the experimental ata of Johnson an Christy for = nm. Figure illustrates this fit for 450 nm. The real part fits the experimental ata well for 50 nm, while the imaginary part fits well for 630 nm. We inclue more recent experimental ata of Lynch an Hunter 3 up to nm, an our fit agrees reasonably well at least up to

3 ENHANCED TRANSMISSION WITH COAXIAL FIG. 3. The transmission spectra for various inner raii R.All other parameters are the same as the basic moel. this wavelength. The c conuctivity in our moel is 7 fs, whereas the observe c conuctivity for Au is about 440 fs Ref. 4. Thus our fit must eventually lose its accuracy in the long wavelength limit, but this must be well beyon 0 m. We aapt the algorithm evelope by Gray an Kupka 5 originally for a pure Drue moel to hanle this frequency epenence for the Au regions, whereas we use the NSFD algorithm of the HASP coe to upate the fiels in the uniform silica meium. III. CYLINDRICAL SURFACE PLASMON DISPERSION Before consiering an analytic treatment of the CR problem, we illustrate in Fig. 3 the simulate transmission spectrum for our basic case an for variations in the inner raius R. A significant feature is that the prominent peak at the longest wavelength is progressively reshifting as the ring narrows R R =00 nm. Actually it appears that a pair of peaks is reshifting. This pair correspons to the peaks at 770 an 000 nm for R =50 nm, but correspons to the two longest wavelength peaks for larger R. For our moel for the ielectric constant Eq. the planar surface plasmon PSP i.e., the surface plasmon propagating on the flat upper an lower surfaces of the film resonances shoul be at 843 nm for a 0, moe, an at 63 nm for a, moe. These are the vacuum wavelengths. All of these simulations prouce a peak at 880 to 960 nm, which correspons to the usual PSP resonance, which is typically reshifte somewhat from the theoretical position. 3,6 These peaks shift a little with R, but not ramatically as o the aforementione peaks. The long wavelength pairs of peaks appear unrelate to the planar surface plasmons, an their epenence on R implies they are relate to the moes of the coaxial rings themselves. Our simulation results closely resemble those of Baia et al. 9 as oes the conclusion that some of the peaks are not ue to planar surface plasmons. Allowing for the slightly ifferent perioicity theirs is 600 nm an ielectric meium they use glass, =.34 the peak positions an features of our Fig. 3 for R =0 are very similar to those of Fig. 4 of Ref. 0, an the features in Fig. 3 for R =50 nm are very similar to those of Fig. 8 of Ref. 0. For example, for R =50 nm an R =00 nm see Fig. 8, Ref. 0, both our results an those of Baia et al. 0 exhibit a pair of peaks at 000 nm an 903 nm in Fig. 3 at long wavelengths separate by 00 nm, followe by a tria of peaks at wavelengths nm shorter at 654, 74, an 763 nm in Fig. 3, an a rather small bump at a somewhat shorter wavelength 598 nm in Fig. 3, at about 680 nm in Fig. 8, Ref. 0. An even smaller bump 40 nm blueshifte from the latter that is har to iscern in either figure. The relative magnitues of these features in our simulations an in Ref. 0 are comparable in magnitue. In aition to the aforementione SP moes, corresponing Woo s anomalies WA, in our case theoretically at 80 an 57 nm, coul play a role in accounting for some of these simulate features. The higher orer features at the shorter wavelengths 700 nm in our simulations an those in Ref. 0 o not show up as clearly as in some other simulations, such as those of Chang et al. 7 who consier 00 nm open square apertures. Here the ecrease expose area of our coaxial ring apertures compare, e.g., to open circular or square apertures may partially explain the repression of these features. Inee, these higher orer features are further represse in the R =75 nm an R =85 nm results in Fig. 3. There are some peaks, however, e.g., the peaks at 000 nm an 763 nm in Fig. 3 with corresponing peaks at 50 an 900 nm in Fig. 8 of Ref. 0 that cannot be explaine by PSP or WA effects. We have also simulate the system stuie by Salvi et al. 3 We obtain peaks at 660 nm an 300 nm versus their results of 700 an 30 nm. The ifference is likely ue to the ifferent treatment of the ielectric constant of Au. They use ifferent pure Drue moel fits over limite wavelength intervals. How o the moes of the iniviual coaxial rings account for the behavior of the transmission spectra shown in Fig. 3? To investigate this question we first carry out a normal moe analysis of the geometry of interest Fig. in cylinrical coorinates to obtain CSP ispersion curves for a single infinitely long CR completely surroune by Au. We seek the relation between the longituinal wave number k z in the CR, the azimuthal state n, an the frequency, enote by n k z or the corresponing wavelength n k z. We follow the proceure escribe by Schröter an Dereux, 8 who erive the CSP ispersion curves for a metal ring embee in a ielectric. Kushawa et al. 9,30 have evelope Green s function techniques for calculating ispersion curves for coaxial 9 an multiaxial 30 structures with arbitrary ielectric constants with applications to quantum wire an nanotubelike structures. We straightforwarly aapt the methos of Ref. 7, except now with a ielectric ring embee in a metal. We concentrate on n = as this is the azimuthal moe most reaily excite by a linearly polarize plane wave. A normal moe exists whenever the appropriate bounary conitions on E, E z, H, an H z at the two cylinrical metal-ielectric interfaces can be satisfie, which amounts to fining the zeroes of the eterminant of an 8 8 matrix involving TE an TM coefficients for the various cylinrical Bessel or Neumann or Hankel functions in the raial wave function for the three raial regions. For n=0 or k z =0 pure TE or TM moes can be foun; otherwise, the CSP moes are TE-TM combinations. We will show that these moes have a surface

4 HAFTEL, SCHLOCKERMANN, AND BLUMBERG plasmonlike character, i.e., the fiels are concentrate near the metal-ielectric cylinrical interfaces. In general, all the components of E an H can be expresse in terms of the TM an TE wave functions, TM an TE, an their first an secon erivatives with respect to the cylinrical coorinates,, an z, where, in a raial region i for azimuthal moe n, TM = A im F m n k i exp ik z z exp in, a m TE = B im F m n k i exp ik z z exp in, b m where F m n z is a cylinrical Bessel-type function whose type Bessel, Neumann, or Hankel is labele by m, an k i is the raial wave number that satisfies k i + k z = i i /c, where i is the ielectric constant in region i, an i is the magnetic permeability usually one. The longituinal wave number k z an the azimuthal moe n are common to all raial regions. In the inner metallic region m= an F is a Bessel function F n z =J n z. In the ielectric ring m= or with F n m z a Bessel function J n z for m=, or a Neumann function N n z for m=. In the outer metallic region m=, an F is a Hankel function of the first kin H n z. The eight coefficients A im, B im are gotten from matching the continuity of E, E z, H, an H z at the metal-ielectric interfaces at inner raius R an outer raius R. This involves fining the zeroes of the eterminant of the matrix M, 3 J J N i J /k c i J /k c i N /k c 0 nk z J /k R nk z J /k R nk z N /k R 0 0 J 3 N 3 H M = 0 i J 3 /k c i N 3 /k c i 3 H 4 /k 3 c 0 nk z J 3 /k R nk z N 3 /k R nk z H 4 /k R J J N 0 nk z J /k R nk z J /k R nk z N /k R 0 i J /k c i J /k c i N /k c J 3 N 3 H 4 0 nk z J 3 /k R nk z N 3 /k R nk z H 4 /k R 0 i J 3 /k c i N 3 /k c i H 4 /k 3 c, where J =J n k R, J =J n k R, J 3 =J n k R, with similar labeling for the N j, an H 4 =H n k 3 R. In our case, where the core of the coaxial rings an the film are both Au, = 3 = Au, = silica =.3, an k =k 3. We have assume that = = 3 =. The continuity conitions are summarize in the matrix equation Ma=0, where a is the coefficient vector whose components are A,A,A,A 3,B,B,B,B 3 of Eq., an the eight components of the vector Ma, when equate to zero, give the continuity conitions on E z R, H R, E z R, H R, H z R, E R, H z R, an E R, respectively. If the metal is perfectly conucting, then pure TE an TM solutions are possible. In the case of a perfectly conucting metal with R 0, a cylinrically symmetric TEM 0 moe is also possible, an this moe has no cutoff. For a real metal finite pure TE or TM solutions are couple unless n=0 or k z =0 Eq. 4, an except for these cases the solutions are amixtures of TE an TM. The TEM 0 moe is still possible, but the cylinrical symmetry of this moe means that it cannot be excite by an incient linearly or circularly polarize plane wave. The evience to ate,3 inicates that the long wavelength moes are preominantly TE, which is most easily excite by incient plane waves. Figure 4 gives the n = ispersion curves for Au, expresse in terms of the wavelength k z for six values of the inner raius R R =00 nm. To simplify matters we have use only the real part of the ielectric constant of Au. Incluing the imaginary part harly alters these curves for the real part of k z, but of course a finite with will be introuce usually wiening to 0 30 nm in. The vertical lines correspon to values of k z that satisfy the relation k z L= values for various L. As the conition k z L = m is the general conition for a longituinal staning wave in the cyliner, the intersection of any of these lines the vertical axis with a given curve will give the wavelength for m = m=0 at which the cylinrical moe is in resonance with the longituinal moe. Eq. 5 is the usual longituinal resonance conition for a cavity with perfectly conucting walls; we assume it is approximately true for the real metal, which our later simulations bear out. Physically, we tentatively ientify these as CSP resonances, as our following analysis will confirm that these solutions have the characteristics of surface plasmons propagating on the metal-ielectric interfaces because of the negative ielectric constant of the metal. Thus a picture of CSP assiste extraorinary transmission takes form. As previously note,,3 the EOT is primarily riven by TE wave guie moes k z 0, which iffers from the EOT prouce by planar surface plasmons, where the

5 ENHANCED TRANSMISSION WITH COAXIAL FIG. 4. The n= CSP ispersion curves for silica CR s embee in a gol film for the basic case but with various R values. The soli vertical line inicates the values of k z corresponing to k z = /L for L=90 nm, while the otte vertical lines inicate corresponing values of k z for L=40 an 000 nm. latter involves evanescent moes propagating through the aperture. We fin a very close corresponence between the resonant frequencies in the simulations Fig. 3 with those obtaine with the ispersion curves with the resonance conition of Eq. 5 see Fig. 4. For example, for n= with our basic moel R =50 nm, L=90 nm the ispersion analysis preicts m = 0, peaks at 979 nm an 789 nm, respectively, while the simulation gives these peaks at 000 nm an 757 nm. For R =75 nm the ispersion analysis gives 376 nm an 035 nm, while the simulations in Fig. 3 give 406 nm an 0 nm. The simulation results closely follow the trens of the ispersion analysis, but ten to be reshifte somewhat from the analytical results. Differences can come from several sources incluing losses, presence of other moes, interference of CSP an PSP moes, an the staircase representation of the CR on a finite rectangular mesh, as well as the conition of Eq. 5 being approximate. Figures 5 a an 5 b show the raial epenence of the ominant fiel components for a typical solution corresponing to k z =.008 nm for inner raius equal to 50 nm an 85 nm, respectively. The magnitue of the fiels is maximum at the inner metal-ielectric interface with cusplike curvature at both interfaces like that of ecaying exponentials away from the interfaces on either sie. For some components the fiel is actually ecreasing or nearly constant as it approaches the outer raius, but with positive curvature. This is the expecte behavior of a cylinrical surface plasmon, FIG. 5. Raial epenence of the fiels an electromagnetic energy ensity arbitrary units for n=, k z =0.008 nm an R =50, 85 nm, obtaine from the CR normal moe analysis. i.e., surface states on the inner an outer raius of the ring. As the ring narrows see the 85 nm case in Fig. 5 the fiels o not have a chance to substantially fall off away from the inner raius, an the average fiel intensity in the ielectric ring is higher than for a wier ring. In a sense the CSP wave functions from the two interfaces are overlapping more. Another possible physical picture is that the surface plasmons from the two surfaces are more coherent, i.e., that while the surface plasmons have the same angular momentum the same azimuthal quantum number n they also approach having the same angular velocity as R R. From the ispersion curves in Fig. 4 it is evient that the wavelengths at the peaks for m=0 an m= increase rapily as R R. What is the limit for such behavior, i.e., oes the peak reshift inefinitely? To etermine this epenence on the with of the ring we consier the conition for et M =0 in the limit R=R R 0. For n= an k z =0 the TE an TM moes ecouple an the TE moes are etermine bya4 4 matrix J k m R J k R N k R 0 i J k m R i J k R i N k R 0 k m c k c k c M = 0 J k R N k R H k m R 0 i J k R k c i N k R k c i H k m R. k m c 6 The eterminant of M is straightforwar to evaluate. Since, for 600 nm m, thus k m k, we express the eterminant in powers of k m /k,

6 HAFTEL, SCHLOCKERMANN, AND BLUMBERG k m k J k m R H k m R N k R J k R J k R N k R et M = k m c + k m k J k m R H k m R J k R N k R N k R J k R + J k m R H k m R J k R N k R N k R J k R, 7 + J k m R H k m R N k R J k R J k R N k R where H z is shorthan for H z. We assume that k R R =k R, an expan the Bessel functions whose arguments are k R in terms of those evaluate at k R an their erivatives. The first square bracket in Eq. 7 is of first orer an reuces to N k R J k R J k R N k R k R, which can be further reuce, by application of the Bessel function ifferential equation to w z = w z /z /z w z, /k R N k R J k R J k R N k R k R. The first orer contribution to the first square bracket in the k m /k term in of Eq. 7 vanishes leaving only a zero orer contribution. The secon square bracket in the k m /k term becomes, upon expansion to first orer J k R N k R N k R J k R + k R J k R N k R N k R J k R, which reuces to, upon application of the Bessel function ifferential equation J k R N k R N k R J k R R/R J k R N k R N k R J k R. The last square bracket in Eq. 7 vanishes in zero orer an only the first orer contribution N k R J k R J k R N k R k R remains. Upon application of all of these relations, a common factor W k R = N k R J k R J k R N k R 9 appears in Eq. 7. Equation 7 now becomes et M = W k R k m k J k m R H k m R k R k R k m c k. m 0 J k m R H k k m R J k m R H k m R R/R + J k m R H k m R k R 8 Further simplification occurs because of the ientity W z = / z. Further assuming that k m R to be justifie latter, Eq. 0 becomes, ignoring the prefactor since we are only intereste in the zero of the eterminant, et M k m k J k m R H k m R k R k R k m J k m R H k k m R J k m R H k m R R/R + J k m R H k m R k R + k m k J k m R H k m R J k m R H k m R k m R. We further utilize the ientity J k m R N k m R N k m R J k m R =/ k m R to obtain J k m R H k m R J k m R H k m R =i/ k m R an the ifferential equation for H z to eliminate H k m R an simplify Eq. further to first orer in R,

7 ENHANCED TRANSMISSION WITH COAXIAL et M k m k J k m R H k m R k R i k R k R k m J k k m R H k m R R/R + J k m R H k m R k R + k m k k m R H k m R J k m R H k m R J k m R R + k m R J k m R H k R k m R. 3 There are two regimes where our approximations are vali. The first is, for most metals, the optical an near infrare. For these frequencies in the extene Drue moel Eq., but k R, an in this case m / an k m = / /c. For wavelengths nm k m is approximately the constant value 0.04i for Au. The secon regime is the extreme 0 limit which correspons to the extreme R 0 limit where m i/. With an R as small parameters, in either of the above cases the leaing terms proportional to R 0 an R are the thir an first terms of Eq. 3, respectively. Keeping these leaing terms an equating the eterminant to zero yiels the following conition for a CSP resonance: or k = / i k m J k m R H k m R R/R, 4 res = c / i k m J k m R H k m R R/R / /, 5 where k m may epen on. In the regime where m is real an negative, k m =i m, where m is real an positive, res is real an positive an can be expresse res = c m I m R K m R R/R / /, 6 where I z an K z are moifie Bessel function of the first an secon kin. 3 If m is approximately constant, the resonant frequency is real, positive, an is ecreasing to zero as R /. This behavior of the resonant wavelength, R /, is roughly consistent with the extraction of resonant frequencies from the ispersion curves Fig. 4 an simulation results Fig. 3 in the optical an near infrare regimes. Equation 6 further simplifies if y= m R is very large or very small. In the problem of interest y 4. In the large an small m R limits res c m R/R / /, 7a FIG. 6. The real an imaginary parts of the m=0 k z =0 resonant frequency for the full ielectric constant of Eq.. The approximate values of Eq. 5 are inicate. an res c m / R/R / /, 7b respectively. The fact that the resonant frequency is real an ecreases as R /, or that the resonant wavelength increases as R /,as R 0, follows from m being real in Eqs. 7a an 7b, an is a consequence of the negative ielectric constant of the metal. This woul not occur for the infinite imaginary ielectric constant of an ieal conuctor. As the ring further narrows, the resonant frequency ecreases an eventually we are in the regime where the metal behaves as a goo conuctor an its ielectric constant becomes pure imaginary, m =4 i /, where in our extene Drue moel Eq. = /4. In the limit k m R, J k m R H k m R i/, an we obtain res i R/ R, 8 i.e., the resonant frequency ecreases as the ring with, but becomes entirely imaginary. The frequency in our formulation has a negative imaginary part because we have assume that the fiels have a time epenence exp i t, thus a negative imaginary part correspons to a finite lifetime. For Au the losses become significant for 4 m, whereas the conition k m R hols for 30 m. Therefore, as one procees from the optical to the THz regime the resonant wavelength increases inefinitely inversely as R, but procees off of the real axis, i.e., evelops a larger with. Figure 6 illustrates the epenence of the resonant frequency for k z =0 on R using the full extene Drue moel of Eq., incluing the imaginary part i.e., losses. We also inclue results for the approximation of Eq. 5, which are very close to the exact results for R 0 nm. The behavior res R / is confirme for.0 nm R 0 nm, which roughly correspons to.0 fs 3.0 fs or wavelengths 0.6 to 80 m. For narrower CR s i.e., lower frequencies the resonant frequency becomes imaginary an linear in R. The ominance of the imaginary part formally means that the with becomes large compare to the real part of the resonant frequency, which means that a resonance oes not really exist. The crossover point is R 0.0 nm or 0.0 fs 80 m for the extene Drue moel we employ. The c conuctivity employe in our moel is about a factor of 3.5 too small. Thus our extene Drue moel unerestimates losses in the low frequency limit. However, the losses are slightly overestimate with respect to experiment at 0 m see Fig.. Realistically, the

8 HAFTEL, SCHLOCKERMANN, AND BLUMBERG FIG. 7. The real an imaginary parts of the m= k z = nm resonant frequency for the full ielectric constant of Eq.. results in Fig. 6 shoul be accurate at least own to 0. fs with the ownturn for both curves becoming more severe at lower frequencies. Figure 7 shows the corresponing curves for the case k z = nm, which is the m = longituinal resonance conition for L = 90 nm. Here also the res R / behavior occurs over the same range of R an as in Fig. 6. IV. FDTD SIMULATION OF COAXIAL RING APERTURES The above CSP normal moe analysis largely explains the unusual epenence of the ecreasing resonant frequency or increasing resonant wavelength with the narrowing of the coaxial rings, as well as preicts this epenence for rings much narrower than can be explore with numerical simulation. Numerical simulation, however, is necessary to supplement these preictions for several reasons: i The CSP ispersion analysis oes not tell us how efficiently the CSP normal moes couple with the incient raiation, i.e., the intensity of these resonances. ii The resonant conition k z L=m is approximate. iii All resonant moes SP, CSP, n interfere naturally in a FDTD simulation, an shoul give more reliable preictions of transmission peaks, etc. iv The simulations inclue the full effect of the ielectric constants, incluing losses. In this section we confirm the theoretical preictions of the previous section an further assess the properties of the coaxial ring arrays through FDTD simulations. The previous theoretical analysis assumes that the CSP enhancements are the result of the CSP normal moes of iniviual CR apertures. This woul seem justifie because the fiels rop precipitously between apertures in the gol film see Fig. 5, i.e., a tight-bining picture seems to apply. However, perioicity effects o exist between cylinrical apertures even for perfectly conucting PC films. To see how these peaks are affecte by the perioicity of the lattice, in Fig. 8 we compare the transmission spectrum of 555 nm perioicity with that of 888 nm perioicity. The longest wavelength PSP for 888 nm perioicity shoul be at about 350 nm. A peak oes show up at 333 nm, but it is harly iscernible in Fig. 8. There is a larger peak at 43 nm, but this is harly iscernible also. The largest peak appears at 004 nm, very close to the 000 nm peak for 555 nm perioicity. The intensity is own by a factor of 3, but this mainly reflects the reuction of the aperture ensity by a factor.56. Thus the position of this peak is harly affecte by perioicity. Baia et al. 0 obtain the same result as long as the perioicity is greater than 300 nm in silver. This strongly implies that the long wavelength enhancements epen only on the physics of the iniviual CR an the CSP moes uncouple from the other CR s. Figure 8 also inclues the spectrum for the Au film replace by a perfectly conucting film with CR apertures of the same geometry. A peak appears at 808 nm 554 nm in the silica, well beyon the cutoff of 687 nm for the TE moe. This peak is likely the result of the resonant coupling between the rings, 6 which occurs at a wavelength in the meium close to the lattice spacing, an occurs even for PC films. Our analysis oes not preclue such effects also in the real metal, but the long wavelength enhancements an the singular behavior of the resonant wavelength as R approaches R are not relate to such effects an explicitly FIG. 8. The transmission spectra for the perioicities =555 nm an =888 nm for gol films, an for =555 nm for a perfectly conucting PC film. All other parameters are the same as our basic moel

9 ENHANCED TRANSMISSION WITH COAXIAL FIG. 9. The transmission spectra for various R, L combinations. The soli arrows inicate the theoretical positions of the m=0 k z =0 resonances, an the otte arrows the positions of the m= k z = /L resonances. epens on the negative ielectric constant of the real metal. To test our preictions on the epenence of the transmission peaks on R an L, we illustrate in Fig. 9 the transmission spectra for two ifferent L values for R =50 nm an R =75 nm. The theoretical resonance positions, with the conition k z L=m assume, are also inicate. The theory an simulations are in goo agreement with respect to the movement an spacing of the peaks with R an L. The peaks in the simulation usually are slightly reshifte by an average 30 nm from the theoretical positions, a feature that is also common for peaks associate with planar surface plasmon resonances. 3,6 The peak positions at the longest wavelengths harly change with L because they correspon to m=0 an in this case k z =0 inepenent of L. However, the secon peaks the thir peak for R =50 nm, L=90 nm, since the secon peak here is the PSP peak, corresponing to m=, o significantly reshift for larger L. This is consistent with the ispersion curves of Fig. 4 since here k z = /L, an the resonant wavelength is a monotonically ecreasing function of k z Fig. 4. Aitionally, the PSP peak at 903 nm for our basic case harly changes when L increases from 90 to 000 nm, thus istinguishing PSP behavior from that of CSP s. The intensity, however, is significantly epresse for L=000 nm, which is mainly the result of losses. Moreover, the results in Fig. 9 closely confirm the theoretical preictions of Sec. III. We can also extract spatial fiels an Poynting vectors energy flow from the simulations to examine the character of the solutions. The raial behavior of the fiels in the CR we extract closely resemble those in Fig. 5. Figure 0 illustrates the real an imaginary parts of the Poynting vector S for our basic case at the PSP peak 903 nm an at the CSP peak 000 nm. The signature of a CSP resonance is a large imaginary part of the Poynting vector along the surfaces of the CR throughout the CR. This is observe in Fig. 0 b at the CSP peak even in the mile of the CR, but not in Fig. 0 a for the PSP peak, although there is a large Im S at the top an bottom surfaces. Im S is strongest at the inner metal-ielectric interface, although this is har to see in the figure. Also, as a by-prouct, the real part of S is appreciable in the CR an ecays only very slowly ue to losses in both cases. The behavior of S is inicative of guie moes rather FIG. 0. Diagram of the real black arrows an imaginary gray arrows parts of the Poynting vector in the CR for a =903 nm the PSP peak an b =000 nm the CSP peak. The real an imaginary parts have a ifferent scaling in the plots. The gri correspons to the coorinates, in nm, in the simulation. than evanescent moes in the CR, even at nonresonant frequencies. Thus the moes in the coaxial apertures influence the propagation through the CR even off-resonance an can lea to overall enhancements even for the PSP moes, by ramatically increasing the TE moe cutoff. One of our most important results is that the resonant wavelength seems to be ever increasing as the CR is narrowe, behaving as R R / at least in the optical an the near infrare. Of course in the R =R limit the transmission vanishes because the aperture vanishes, i.e., the expose ielectric area vanishes. For a cylinrical aperture the transmission ecays faster than the expose area because the waves in the aperture are evanescent. For the CR apertures the moes are propagating, thus we woul expect the transmission to fall off roughly as the expose area. Table I gives scale values of the m=0 an m= peak wavelengths scale to R R / i.e. sc = peak / R R /, an the transmission at the peak scale to the expose area T sc =Transmission FDTD Area FDTD cell /Area CR for L=90 nm. Results for R 85 nm are for FDTD simula

10 HAFTEL, SCHLOCKERMANN, AND BLUMBERG TABLE I. The m=0 an m= CSP transmission peak wavelengths, etermine from the ispersion curves Fig. 4, scale to R R / i.e., sc = peak / R R / an the transmitte intensity scale to the expose CR area T sc =Transmission FDTD Area FDTD cell /Expose Area CR for various R values. Results for R 85 nm are for FDTD simulations, while for R 85 nm the results are from the ispersion analysis Fig. 4. m=0 m= R nm L nm sc nm / T sc sc nm / T sc tions, while for R 85 nm the results are from the ispersion analysis Fig. 4. The R R / scaling of the peak wavelength, as iscusse in Sec. III, hols rather well. The transmission intensities at these peaks are typically 4 times that incient on the expose area. As is often escribe to be the case with PSP enhancements, there seems to be a mechanism rawing raiant flux into the apertures when CSP resonant moes are present. We are presently looking into the origin of this phenomenon. V. SUMMARY AND CONCLUSIONS We have investigate the optical transmission spectra of silica coaxial ring arrays in gol films as functions of the perioicity, ring geometry, an film thickness from analytic consierations an by FDTD simulations. Long wavelength transmission peaks occur in FDTD simulations well beyon the aperture cutoffs an beyon wavelengths where iffractive effects or planar surface plasmons play a role. Normal moe analysis inicates that transmission peaks occur when cylinrical surface plasmon n= moes TE or preominantly TE are supporte an staning waves exist along the length of the CR. The cutoffs for these moes, unlike those of cylinrical apertures or CR s in a perfectly conucting film, inefinitely increase as the ring raii approach each other, behaving as R R /. These properties result from the negative ielectric constant of a real metal an woul not ensue for a metal viewe as an ieal conuctor. The singular behavior of the cutoff wavelength hols as long as the influence of the imaginary part of the ielectric constant is small compare to the real part. For Au this correspons to 0 m. Past this wavelength the resonant wavelength continues to increase, but becomes wier an wier until the resonance peak gets completely washe out. In the long wavelength limit the magnitue of the resonant wavelength increases as R R, but becomes completely imaginary, as expecte of a perfect conuctor. The enhancements from cylinrical surface plasmons, unlike those from planar surface plasmons, involve propagating moes through the CR. The peak transmission intensities ecrease no faster than the expose CR area, enabling a super extraorinary transmission at very long wavelengths. The transmission intensities at the peaks, in fact, are typically 4 times the incient flux on the expose portion of the CR. We propose the following possible physical argument for the CSP enhancements an their epenence on ring geometry. CSP s are launche analogously to PSP s. PSP s are launche when the photon can exchange planar momentum to the SP equal to a characteristic momentum of the twoimensional D lattice. Uner perioic bounary conitions this sets up staning wave PSP s. The analogy to the perioic bounary conition azimuthally is the requirement of single-valueness, an this is characterize by the epenence exp ±in, which can be interprete as the CSP exchanging n units of angular momentum with the cyliner. If a CSP also propagates in the z irection, staning wave bounary conitions normally require k z L=m, where k z is the characteristic unit of momentum which is associate with the length of the aperture an exchange with the CSP. Together n an k z escribe the CSP motion on a cylinrical surface as k sp escribes the motion of a PSP on a flat surface. When n,k z has a characteristic value of the cyliner integral n, k z L=m a CSP is launche. When the ring is narrow, the inner an outer CSP s not only have the same angular momentum, but they have nearly the same angular velocity, meaning that the two CSP s have more coherence. A complementary view is that the wave functions of the two CSP s overlap increasingly as the ring narrows, as seen in Fig. 5. The structures we stuy are ifficult to fabricate in the optical range because of the high aspect ratios involve, an also because silver an gol, which are the metals that give the best SP response i.e., low losses, are very inert. They are ifficult to process in such imensions. Electron an ion beam lithography, liftoff, ry etch, electroplating, an ion beam milling are potential techniques that coul be use to fabricate such structures. Recently there has been significant progress in fabricating these coaxial structures. As previously mentione, Salvi et al. 3 employe electron-beam lithography an gol liftoff to form 330 nm iameter coaxial aperture arrays in 40 nm gol films on glass. However, they i not measure the structures in the near infrare range where we woul have expecte the CSP peak. Very recently Orbons et al., 3 using ion-beam lithography, fabricate similar arrays on nm thick silver films on a glass substrate an measure the transmission up to 700 nm, which inclues the region where the m=0 CSP peak shoul occur. We are presently analyzing these results. To ate both of these experiments 3,3 have been limite to rather low aspect ratios for the apertures. While the present stuy was restricte to the optical an IR regimes, beyon which losses woul ominate, one coul exploit the features in other regimes THz, RF by the evelopment of metamaterials 33 with negative effective ielectric constants an low losses containing such CR structures. The properties we foun shoul have a ramatic effect on evice esign an hopefully motivate experiments to fabricate i.e., through ion beam milling, ry etch, electroplating, or relate techniques an explore these structures, as a start in this irection has alreay occurre. 3,

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