Propagating plasmonic mode in nanoscale apertures and its implications for extraordinary transmission

Transcription

1 Journal of Nanohotonics, Vol. 2, 2179 (12 February 28) Proagating lasmonic mode in nanoscale aertures and its imlications for extraordinary transmission Peter B. Catrysse and Shanhui Fan Edward L. Ginzton Laboratory and Deartment of Electrical Engineering Stanford University, Stanford, CA Abstract: We studied the interaction of different athways by which extraordinary transmission through nanoscale aerture arrays arises and obtained a comlete hysical icture that incororates both roagating lasmonic and surface lasmon modes. The transmission behavior is qualitatively different deending on the number of transmission athways resent in the regime of oeration. If only one athway is resent, it can give rise to high transmission. When multile athways are resent simultaneously, their interlay must be studied in order to understand the rich and comlex transmission behavior. The frequency range of these athways can be controlled by varying the structures and, in articular, by coating the surface of the arrays or by filling the aertures with dielectrics that differ from the surrounding medium. Keywords: aertures, aerture arrays, extraordinary transmission, Fano interference, nanoscale aertures, lasmon-olaritonic material, roagating lasmonic modes, surface lasmon modes, transmission athways, transmission suression 1 INTRODUCTION The otical roerties of nanoscale aertures in otically thick metallic films have been intensely researched in the ast several years due to their fundamental imortance in nearfield otics and nano-hotonics, as well as their ractical significance for hotonic devices and alications, including filters, near-field robes, otical data storage, and nanolithograhy [1,2]. It is well-established that the transmission characteristics of aertures strongly vary deending on whether they allow or rohibit roagating modes [3,4]. For examle, enhanced transmission in metallic nano-slits is attributed to roagating transverse magnetic (TM) modes that are suorted by slits, indeendent of their width [3,5-9]. Cylindrical holes with a circular cross-section in a erfect metal, on the other hand, do not suort roagating modes when the hole diameter is smaller than λ 2n, where λ is the vacuum wavelength of incident light and n is the refractive index of the dielectric inside the hole [1,]. Following Ebbesen et al. s ioneering exeriments [12], extraordinary otical transmission has therefore commonly been associated with the excitation of surface wave resonances on the front and back surfaces of the metallic film and an evanescent tunneling rocess through the subwavelength holes [-16]. More recently, localized surface lasmon modes or shae resonances inside subwavelength holes have been identified as an alternative athway for extraordinary transmission [17-22]. In either case, the use of resonances results in transmission eaks of relatively narrow line width. In recent work of our own, we numerically demonstrated that subwavelength holes in a metal always suort a roagating lasmonic HE mode near the surface lasmon frequency, regardless of how small the holes are, and that this mode can lead to near-comlete otical transmission through a subwavelength hole array [23,24]. The imortance of such roagating modes is that, under aroriate conditions, they rovide a transmission window with a relatively broad bandwidth. Hence, it has become clear that there are multile mechanisms by which 28 Society of Photo-Otical Instrumentation Engineers [DOI: 1.17/ ] Received 4 Se 27; acceted 2 Dec 27; ublished 12 Feb 28 [CCC: /28/$25.] Journal of Nanohotonics, Vol. 2, 2179 (28) Page 1

2 extraordinary transmission can occur. The interaction between these mechanisms, however, has not received much attention. In this aer, we study the interference of different mechanisms or athways by which extraordinary transmission through subwavelength hole arrays arises. We rovide a comlete hysical icture that incororates all reviously reorted athways and unifies them in a comrehensive framework based on the analysis of their resective disersion relations. We show that the transmission behavior is qualitatively different deending on the number of transmission athways resent in the regime of oeration. If only one mechanism is resent, it gives rise to extraordinary transmission. When multile mechanisms are resent simultaneously, their interlay must be studied in order to understand the rich and comlex transmission behavior. We further demonstrate that the frequency range of these athways can be controlled by varying the structures and, in articular, by coating the surface of the hole arrays or by filling the holes with dielectrics that differ from the surrounding medium. This aer is organized as follows: we start by revisiting the regime exlored in exeriments insired by Ebbesen et al. and we show that our aroach exlains all the salient features in the transmission sectrum, including the ones ascribed to surface waves. In addition, we rovide evidence that shae resonances or localized surface lasmons are, in fact, due to the behavior near cut-off of roagating lasmonic modes inside the holes. Next, we design subwavelength hole arrays that robe other regimes of oeration in which either one or multile athways are resent. In each case, a disersion analysis exlains all the details of the transmission sectrum. 2 BACKGROUND AND NUMERICAL APPROACH Following Ebbesen et al. s ioneering exeriments [12], the most-studied athway for extraordinary otical transmission through subwavelength hole arrays has been surface waves. As a starting oint, we analyze the transmission roerties of a subwavelength hole array with a 75-nm eriod (a) square lattice in a 32-nm thick (h) metallic film (Fig. 1 inset). This structure was reviously investigated exerimentally and analyzed theoretically by Martin- Moreno et al. [14]. In our analysis, for simlicity and without loss of generality, we assume cylindrical holes with a circular cross-section and a 12-nm radius (r ). We describe the otical roerties of the metal using a Drude free-electron or lasmonic model: 2 ( ) 1 ω ε1 ω =, (1) ω( ω iωτ ) where ω reresents (angular) frequency, ω is the lasma frequency and ω τ is the collision frequency. The dielectric function in this model takes into account the contribution of free electrons only and dislays a lasma-like disersion. Desite its aarent simlicity, the lasmonic model has been a valuable source of insights into the behavior of real metals. Its regime of validity extends dee into the visible wavelength regime for aluminum and most alkali metals [25]. It has roven to be accurate in the near- and far-infrared wavelength regimes in describing the otical behavior of noble metals (e.g., silver, gold, coer) [26,27], while being a reasonable aroximation in the visible wavelength range above 5 nm. In 16 this aer, for examle, we model silver with arameter values ω = rad/s and 13 ω τ = rad/s. By allowing for additional Lorentzian resonance terms, the use of the lasmonic model can be easily extended to the entire visible wavelength range, i.e., below 5 nm, where inter-band transitions often contribute to the dielectric function [28]. While a model extension of this tye might be more realistic and result in alicability to a larger grou of metals in a wider wavelength range, its study is beyond the scoe of this aer. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 2

3 1.8 a 2 r h HE T.6 (2,) or (,2) (1,1) (1,) or (,1) λ (nm) Fig. 1. Normal-incidence transmission sectrum of a cylindrical hole array with 12-nm radius (r ) holes on a 75-nm eriod (a) square lattice in a 32-nm (h) thick metallic film surrounded by air. The inset shows the geometry. The surface lasmon-olariton resonances are indicated by solid red and cyan vertical arrows and the corresonding Rayleigh-Wood anomalies by dashed vertical arrows of the same color. The blue horizontal arrow shows the range of the roagating lasmonic HE mode in the hole. For the transmission calculations, we use a three-dimensional (3D) total-field finite-difference time-domain (FDTD) imlementation [29]. The simulation domain includes a single unit cell of the square-lattice array. On the to and bottom surfaces of the comutational domain, we imose Perfectly Matched Layer (PML) absorbing boundary conditions [3]. For the remaining four surfaces that are erendicular to the metallic film, we imose Bloch eriodic boundary conditions [31]. Furthermore, we aly a normally incident ulsed lane-wave excitation to obtain the resonse in the visible and near-infrared wavelength range from a single simulation. The incidence lane is chosen above the metallic film and the field data for determining the sectral transort roerties of the waveguide, through direct integration of the Poynting vector, are collected in an observation lane laced behind the metallic film. Such calculation measures the total amount of ower that can ass through the hole array. The transmittance is defined as the ratio of the ower assing through the subwavelength hole array to the incident ower. Figure 1 shows the calculated transmission sectrum at normal incidence for the aforementioned structure. The sectrum agrees very well with ublished exerimental transmission measurements (Fig. 1 in Ref. 14). Moreover, it matches almost erfectly with reorted theoretical transmission calculations for a structure with 24-nm wide square holes (Figs. 2 and 3 in Ref 14). In that aer, the authors rovided the first three-dimensional theoretical study of extraordinary transmission through subwavelength hole arrays and showed close corresondence between their numerical results and exeriments. They also develoed a simlified analytic model to account for the longest-wavelength feature. In their analytic model, they attribute the highest eaks in the sectrum at 8 nm to the excitation of a surface wave resonance on the to and bottom metal-dielectric interfaces and the twin eaks to the formation of an "SP molecule" [14]. Their model, however, does not account for the remaining features in the transmission sectrum. In what follows, we show that a descrition Journal of Nanohotonics, Vol. 2, 2179 (28) Page 3

4 based on surface lasmon-olariton resonances alone, in fact, does not cature the full hysics involved in the transort roerties of light through subwavelength hole arrays. To suort our claims, we rovide an analysis based on disersion for all the modes suorted by the structure. We show that our model indeed catures all the hysics of light transort through subwavelength hole arrays and that it quantitatively exlains all the features in the transmission sectrum, including those that can not be exlained by surface lasmonolariton modes alone. 3 DISPERSION ANALYSIS We seek to understand the roerties of extraordinary transmission, by erforming a disersion analysis of the surface waves on the flat metal-dielectric interfaces, and the roagating modes inside the subwavelength holes. Both the interfaces and the holes are integral comonents in systems that feature extraordinary transmission. When the metal is assumed to exhibit a lasmonic resonse [Eq. (1)], neither of these comonents is inert; rather, each comonent lays an active and distinctive role in the transmission behavior. 3.1 Surface waves on flat metal-dielectric interfaces A flat interface between a dielectric, with a ositive dielectric function ε 2 >, and a metal, for which the real art of the dielectric function ε 1 ( ω ) is negative, suorts surface waves rovided that ε 1 ( ω) > ε 2 (Fig. 2 inset). Such surface lasmon-olaritons (SPPs) have wavevectors k ( ω ) related to the wave frequency ω by the following disersion relation, SPP k SPP ( ω) ω = c ( ) ( ) ε1 ω ε2. (2) ε ω + ε 1 2 Figure 2 shows the disersion relation for surface lasmon-olaritons on a single flat metaldielectric interface (solid blue curve). While it covers the entire frequency range from zero to the surface lasmon frequency ωs = ω 1+ ε2, we identify two regimes with qualitatively different roerties. At small frequencies, the SPP disersion relation lies below the light line in the dielectric (dashed gray curve), but does not substantially deviate from it. This is the "hoton" regime, in which the excited surface waves are highly delocalized and their fields extend far away from the metal film. The SPP wave extends significantly into the dielectric and the field concentration is largely diminished. At frequencies closer to the surface lasmon frequency, the disersion relation lies far to the right of the light line in the dielectric and the SPPs become dee-subwavelength. We call this the "lasmon" regime, where the surface waves are very localized and their fields are highly concentrated near the interface. In general, SPPs on a flat metal-dielectric interface are confined to the surface; they can not coule to far-field incident light. The resence of a eriodic array, however, rovides a hase-matching mechanism that allows surface waves to coule to normally incident light. The frequencies ω mn at which surface waves are resonantly excited by a square lattice of eriod a are estimated using 2 2 2π kspp ( ω mn ) = m + n, (3) a where mn, are integers [12]. The use of such surface resonances to enhance transmission is well documented. In ast studies, the lowest-order surface wave resonance (1,) or (,1), which usually rovides the strongest signature, has received most attention [14]. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 4

5 1.5 1 ω/ω.5 z x ε 2 ε k x /k Fig. 2. Disersion relation for surface lasmon-olaritons (SPPs) on a single flat metal-dielectric interface (blue curve). The inset shows the geometry. The red curve indicates the regimes of extraordinary transmission robed in exeriments insired by Ebbesen et al. The black curve deicts the bulk metal disersive roerties. Shaded areas reresent the continuum of modes extended in the resective materials. Dashed gray line reresents the light line in the dielectric (air). In such a case, the surface resonance is in the "hoton" regime, where the surface lasmonolariton exhibits low losses (solid red curve). Meanwhile, higher-order modes also show u as distinct features in the transmission sectrum. As can be seen from Eq. (3), the transmission roerties for any eriodic structure that relies on this athway will be highly deendent on the lattice constant a of the array (and the dielectric constant of the material interfacing with the metal). Conversely, the lattice constant a is an imortant design arameter for any extraordinary transmission based on this mechanism. 3.2 Proagating modes inside subwavelength holes The roagating modes inside a subwavelength hole are calculated by considering a z - invariant cylindrical waveguide with a circular cross-section of radius r in the transverse xy -lane (Fig. 3 inset). For the dielectric inside the hole, we use a real, frequencyindeendent dielectric function ε 3. For the surrounding metal, we assume a frequencydeendent dielectric function ε 1 ( ω ). The roagating modes ( mn, ) of the waveguide are found by solving Maxwell s equations in cylindrical coordinates for electric and magnetic fields of the form ψ( r, φ, z, t) = ψn( r)ex( jmφ)ex[ j( ωt kzz)], where m is an integer denoting angular momentum, n is related to the number of nodes in the radial direction, and k z is the roagation constant of the mode. The disersion relation is derived after solving a transcendental equation as obtained by matching boundary conditions [23,32,33]; i.e., (1) (1) 2 2 ε3 J m ε1 H m 1 J m 1 H m 2 ck z 1 1 = m (1) (1) , (4) kt,3 Jm kt,1 Hm kt,3 Jm kt,1 Hm ω r kt,3 k T,1 2 Journal of Nanohotonics, Vol. 2, 2179 (28) Page 5

6 where Jm( kt,3r ) and first kind, and H ( k r ) reresent m -th order Bessel and Hankel functions of the (1) m T,1 2 ω 2 Ti, = ε i z = 1, 3 k k i. (5) c The rime above these functions denotes differentiation with resect to their argument. The disersion relation Eq. (4) differs qualitatively from that of a cylindrical erfect electrical conducting (PEC) waveguide with radius r, for which we readily obtain J m( kt,3r ) = for transverse electric (TE) and Jm( kt,3r ) = for TM modes [34]. The roagating modes inside the cylindrical hole are calculated by solving for the roots of the disersion equation, i.e., we numerically determine the ( ω, ) airs that satisfy Eq. (4). The rocedure involves k z first a coarse scanning of the ( ω, k z ) sace to determine the aroximate location of the disersion relation. The solution is then refined using Newton s method. In what follows, we assume that ε 1 ( ω ) takes on the form of Eq. (1) with the collision frequency ω τ set to zero. This amounts to a lossless lasmonic model. Without loss of generality, we set ε 3 = 4 (e.g., Si 3 N 4 ). For a hole radius r =.36λ, where λ = 2πc ω is the lasma wavelength, the resulting disersion relations are shown in Fig. 3. We distinguish two tyes of lasmonic modes: bulk modes, which extend into the metal region and lie above the line defined by ω = ω + ckz; and roagating waveguide modes, which are confined to the hole. The latter exhibit two discrete bands labeled HE and EH. The field vector lots of these modes are shown in Fig. 3. These modes with angular momentum m = 1 have the roer diole symmetry to coule to a normally incident lane wave. The disersion diagram for the roagating lasmonic modes exhibits three distinct features that have a rofound imact on the transmission of incident light through subwavelength cylindrical holes [23]. First, such holes always suort roagating modes near the surface lasmon frequency, regardless of how small the holes are. Even when material losses are included as art of the lasmonic model, the modes still roagate over several microns when the radius of the holes is much smaller than λ 2n3. Second, the fundamental (lowest-frequency) mode has an HE signature, which enables it to coule to a normally incident lane wave. The HE mode is located comletely below the surface lasmon frequency of the metal. Thus, a cylindrical hole in a lasmonic metal always exhibits a ass-band below the surface lasmon frequency. Third, when the radius of the hole is small, there exists a sto-band for a normally incident lane wave, where the light does not transmit. This sto band occurs when the cut-off at k z = for the EH mode lies above the surface lasmon frequency. Hence, a single hole or a hole array may behave as a band-ass filter and allow longer wavelengths to ass through while rejecting shorter ones. All of these features are fundamentally different from the behavior of a PEC metal, including a commonly-used modified PEC model where an effective hole radius derived from a skin deth calculation is emloyed instead [23]. The transmission sectrum of an individual subwavelength hole can be comletely exlained by the roerties of these roagating lasmonic modes. In the case of subwavelength hole arrays, the disersion relation of a single hole remains very relevant rovided that the hole searation is such that the roagating modes inside nearest-neighbor holes do not interact. In fact, this is shown to hold for thickness of searating metal walls as small as 8 nm [23]. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 6

7 EH ω/ω.6 HE.4 HE r =.36λ y.2 ε 3 z x ε k z /k EH Fig. 3. Disersion diagram in normalized frequency and wavevector coordinates for a cylindrical hole with radius.36λ, where λ is the lasma wavelength, and the hole is filled with a dielectric ( ε 3 = 4 ). Shown are the two lowest-order waveguide modes with angular momentum m = 1 for a lossless lasmonic model. The solid red line corresonds to the disersion of the HE mode, while the dashed blue line shows the disersion of the EH mode. The side anels show vector lots for the electric fields of the HE and the EH mode, resectively. 3.3 Disersion-based interretation of extraordinary transmission We now interret the transmission sectrum for the subwavelength hole array in Fig. 1 in terms of the disersion relations for the surface lasmon waves and the roagating lasmonic modes suorted by the structure. Figure 4 (left anel) shows disersion diagram ( ω, k x ) for the surface waves, where ω reresents (angular) frequency and k x denotes the roagation constant arallel to the front and back interfaces of the metallic film. It is obtained using the disersion model for a single, flat metal-dielectric interface (Section 3.1). We show the lowest-order modes (red and cyan curves) and higher-order modes (gray curves) in the first Brillouin zone of a reduced-zone scheme assuming a square lattice. In this reresentation, a surface wave will be resonantly excited by normal incident light when the curves intersect the k x = axis. The labeled arrows indicate these locations. For a eriodicity a = 5.43λ = 75 nm, the lowest-order (1,) or (,1) resonances are excited at ω (1,) =.18ω, the (1,1) resonance occurs at ω(1,1) =.25ω, and the (2,) or (,2) resonances are located at ω(2,) =.34ω, as determined by Eq. (3). Henceforth, we simlify the notation and refer to these resonances as (1,), (1,1) and (2,) SPP resonances, resectively. Figure 4 (right anel) deicts the disersion diagram ( ω, k z ) of the roagating modes, where k z denotes the roagation constant along a single subwavelength hole of circular cross-section with radius r = 12nm in a lasmonic metal (blue curves). It is obtained using the method described in Section 3.2. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 7

8 ω/ω.4.3 (2,) or (,2) (1,1).2 (1,) or (,1).1.5 k /k x k /k z Fig. 4. Disersion diagram for surface lasmon-olariton modes on a single flat metal-dielectric interface (left anel) and roagating lasmonic modes inside a cylindrical subwavelength hole with circular cross-section (right anel). For the surface waves (red, cyan and gray curves), we assumed a square lattice with eriodicity a = 75 nm. For the roagating modes (blue curves), we defined a circular hole with radius r = 12 nm ( r =.864λ ). The metal is modeled as a 16 lossless lasmonic material with λ = 138nm ( ω = rad/s ) and the dielectric at the interfaces and inside the hole is air ( ε 2 = ε 3 = 1). We highlight the following features in the disersion diagram: ωhe =.21ω is the cutoff frequency for the fundamental HE mode, ω =.46ω is the cutoff frequency for the EH mode, and ωs = ω 2 =.71ω is the surface lasmon frequency inside the hole. Next, we overlay the disersion features on to of the transmission sectrum (Fig. 1). Several SPP resonances occur (solid red and cyan vertical labeled arrows) at 762 nm, 548 nm, and 42 nm. The sectral bandwidth of the roagating HE mode (solid blue horizontal arrow) ranges from 651 nm (cutoff wavelength) to 39 nm (surface lasmon wavelength inside the hole). This clearly shows that in the wavelength range for which the transmission is shown, both SPP resonances and a roagating HE mode are resent. If only surface waves rovide a athway, as is the case above 651 nm, and normally incident light is couled into SPP resonances through the momentum added by the hole array, it can give rise to extraordinary transmission. This is evidenced by the transmission eaks at 8 nm, near the fundamental (1,) SPP resonance at 762 nm. This feature is angle and eriodicity deendent. The shar resonance with an asymmetric Fano-interference shae is due to the SPP resonance sitting on a low-transmission (evanescent tunneling inside the holes) background [35-37]. Without material losses, the eaks can reach the 1% transmission level. The double eaks are due to couling of SPPs on front and back interfaces. The broad high-transmission lateau below 651 nm arises from the roagating HE mode inside the holes. Within this range of high transmission, the SPP resonances show u as a di in the transmission sectrum. In general, when both mechanisms (surface waves and roagating modes) are resent simultaneously, their interlay must be studied in order to understand the rich and comlex transmission behavior that results. Indeed, the (1,1) and (2,) SPP resonances reside inside the HE transmission band. Instead of giving rise to a EH Journal of Nanohotonics, Vol. 2, 2179 (28) Page 8

9 transmission eak, they are resonsible for the suression of transmission at 548 nm and 42 nm. (As a side note, the same dis in the sectrum were reviously interreted as Rayleigh- Woods anomalies [14]. In this articular structure, the SPP disersion relation is very similar to the vacuum disersion relation. The comlex line shaes at 75 and 53 nm in the transmission might be due to the interaction between these two effects [22]. On the other hand, when the SPP and the Rayleigh-Wood anomaly are sufficiently far from each other, both show a di in the transmission sectra, as seen for examle for the two dis in the vicinity of 4 nm. We will show, in Section 4.2, a case where surface lasmon-olariton resonances are well searated from the Rayleigh-Wood anomaly.) We also note that, in recent studies of metallic systems, localized surface lasmon modes or shae resonances inside the subwavelength holes have been identified as a otential athway for extraordinary transmission [17-19,22]. These modes are, in fact, a feature of the roagating lasmonic mode near its cutoff wavelength. In order to describe the entire transmission sectrum, in most cases, one really needs to take into account the entire disersion diagram of the roagating modes and not just their behavior near cutoff. To illustrate this, we refer to the localized waveguide resonance identified in metallic systems featuring subwavelength holes with a rectangular cross-section [21]. For a PEC metal, the resonance is related to a cavity mode at the cut-off frequency of the lowest-order roagating TE mode. Due to the large imedance mismatch between the hole in a PEC metal and free sace, this resonance manifests itself as a single eak in the transmission sectrum surrounded by regions of near-zero transmission (see for examle Fig. 2 of Ref. 21). In the case of a lasmonic metal, the lowest-order roagating mode is the HE mode. It also gives rise to a cavity mode or localized waveguide resonance near its cutoff wavelength. Here, however, the imedance mismatch is smaller and the transmission does not dro abrutly around the highest wavelength eak (see for examle Fig. 4 of Ref. 23) and, hence, the entire ass-band of the HE mode has to be taken into account to exlain all the features in the sectrum. In summary, we find that all salient features in the transmission sectrum (Fig. 1), including dis and eaks, match the highlighted characteristics obtained from the disersion relations in Fig. 4. Our model not only accounts for the extraordinary transmission eaks at 8 nm due to the (1,) SPP resonance, but also for the region of background transmission u to 651 nm due to the HE roagating mode, and for the transmission suression coinciding with the (1,1) and (2,) SPP resonances. Hence, the entire extraordinary transmission sectrum can be exlained by studying the disersion of the surface waves and roagating modes suorted by the structure. The tyical structure considered in Fig. 1 reresents, in fact, only one ossible scenario for effects related to extraordinary transmission. In this articular case, the lowest-frequency (1,) SPP resonance is located below the cut-off frequency of the lowest-order HE mode, i.e., ω(1,) < ωhe. While this regime is interesting from a theoretical oint of view, its usefulness for ractical alications merits some discussion. Holes in these designs tyically "oerate in cutoff", i.e., the cutoff frequency of the roagating lasmonic modes in the hole is above the frequency regime of oeration ( ω < ω HE ). Simultaneously, the eriodicity of the array is increased to manage losses by bringing the frequency of the (1,) SPP resonance in the visible or near-infrared range, i.e., away from the surface lasmon frequency of the metal-dielectric interface. Such structures comrise a sarse array (large eriodicity) of very small holes and tyically feature relatively low transmittance (< 1%), which makes filters based on this aroach less attractive. Alications that can benefit from this design should be able to accommodate the lower eak transmittance with a narrow bandwidth that results from the SPP resonance henomenon. Hence, this regime is more suitable for sectral filtering alications where absolute transmittance is not the main secification. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 9

10 4 REGIMES OF SUBWAVELENGTH OPERATION Based on the disersion analysis for surface waves and roagating modes, we can define several additional regimes of oeration for subwavelength aertures. Each regime features subwavelength oeration, yet differs in the nature of the athway(s) that are resonsible for transmission. The behavior of subwavelength hole arrays is therefore qualitatively different in each regime, nevertheless the transmission features in each case remain in very good agreement with the disersion characteristics from surface waves and roagating modes suorted by the structures. We identify four characteristic frequencies in the disersion analysis of the roagating modes and the surface waves: the cutoff frequency of the HE mode in the subwavelength holes ω HE, the surface lasmon frequency in the hole ω s, h, the frequency of the lowestorder (1,) SPP resonance ω (1,), and the surface lasmon frequency of the flat metaldielectric interface ω s, i. The roagating lasmonic HE mode in the hole is imortant in the frequency range ωhe < ω < ω s, h, whereas the SPP resonances aear in the frequency range ω(1,) < ω < ω si,. We define the regimes of oeration for subwavelength aerture arrays based on the resective order of these characteristic frequencies. This order establishes the location and otential sectral overla of the different mechanisms or athways. It rofoundly imacts the transmission sectrum, as evidenced in the "Ebbesen" regime (Section 3.3) where the order is ω(1,) < ωhe < ω s, h = ωs, i. In what follows, we assume again, for simlicity and without loss of generality, a circular hole with radius r =.36λ and a square lattice with eriodicity a = 18 nm. The metal is 16 described by the lasmonic model Eq. (1) with lasma frequency ω = rad/s and 13 collision frequency ω τ = rad/s. When a lossless lasmonic model is used, the collision frequency ω τ is set to zero. Using a combination of disersion analysis and 3D FDTD simulations, we investigate the transmission of subwavelength hole array structures for each of the different regimes and identify which modes lay the dominant role in the transort roerties for incident light. 4.1 Regime 1: ω HE < ω s,h < ω (1,) In the structure of Fig. 1, the SPP resonances occur at discrete locations in the frequency range between ω(1,) =.18ω and ωs = ω 2, while the hole suorts roagating modes in the entire range between ωhe =.21ω and ωs = ω 2. Hence, the two frequency ranges largely overla. By lacing a dielectric inside the hole, however, it is ossible to shift the range of roagating modes to lower frequencies, thereby avoiding sectral overla with the SPP resonance range. In this regime, we have ωhe < ω s, h < ω(1,) and each athway should searately lead to large transmission and no destructive interference (transmission suression) should arise. We recently identified this regime of oeration for subwavelength holes and found that the resence of a lasmonic HE mode alone, indeed, leads to nearcomlete otical transmission [23]. To the best of our knowledge, this regime has not been robed yet exerimentally. Figure 5 shows the disersion diagram ( ω, k x ) for the surface waves (left anel; red, cyan and gray curves) and the disersion diagram ( ω, k z ) for the roagating modes (right anel; blue and magenta curves). For eriodicity a = 1.3λ, the lowest-order (1,) SPP resonance Journal of Nanohotonics, Vol. 2, 2179 (28) Page 1

11 is excited at ω(1,) =.56ω (left anel). The disersion diagram for the roagating modes in the hole, on the other hand, features three discrete bands (right anel). The first band (lower solid blue curve), located between ωhe =.24ω and ωs = ω 5, corresonds to the lowest-order HE mode. A second band (uer solid blue curve) features frequencies larger than ωeh =.63ω and deicts the disersion of the higher-order EH mode. Both modes have angular momentum m = 1 and coule to normally incident light. A third mode (solid magenta curve) is located between ωm= 3 =.38ω and ωs = ω 5, and has angular momentum m = 3. In a single hole, the m = 3 mode does not have the right diole symmetry to coule to a lane wave. As we will show below, the square lattice breaks the continuous rotational symmetry of a single hole and allows the lane wave to actually coule to this mode as well. Between the discrete bands, from ωs = ω 5 to ωeh =.63ω, is a stoband where no roagating modes exist in a subwavelength hole with radius r =.36λ. Figure 6 shows the transmission sectrum at normal incidence for a subwavelength hole array oerating in this regime. The dominant feature is a high-transmission ass-band from 39 nm to 573 nm. The ass-band features a bandwidth (blue horizontal arrow) that agrees erfectly with the disersion of the HE roagating mode (lower solid blue curve in the right anel of Fig. 5). We note that, in contrast to the dominant feature in the "Ebbesen" regime (Fig. 1), the ass-band is due to the roagating mode inside the subwavelength hole. Hence, it is largely indeendent of incidence angle or eriodicity of the array. It is also the longest-wavelength feature in the transmission sectrum of the structure, since ωhe =.24ω is smaller than ω(1,) =.56ω. A second distinct feature in the transmission sectrum is the low-transmission region between 22 nm and 39 nm. This is the sto-band between the roagating HE and EH modes and once again the range agrees very well with the disersion analysis (Fig. 5 right anel). A third sectral feature is a shar, narrow 1% transmission eak in the sto-band at 245 nm (vertical red arrow). This wavelength corresonds with the location of the (1,) SPP resonance in the disersion analysis (Fig. 5 left anel). Since the SPP resonance is located in the sto-band between roagating lasmonic modes in the hole, it shows u as a resonant transmission eak. Finally, notice the shar asymmetric eaks between 39 nm and 413 nm. This range agrees with sectral bandwidth of the lasmonic roagating mode with angular momentum m = 3 (magenta curve in right anel of Fig. 5). The eaks result from Fano interference between multile athways. In this case, the athways are formed by the fundamental HE mode (blue horizontal arrow), which coules very efficiently to the incident light, and the higher-order m = 3 mode (magenta horizontal arrow). In a single hole, this mode does not have the roer (diole) symmetry to coule to an incident lane wave. The resence of a square lattice, however, may result in couling of light into this mode, since 3-fold rotation is no longer the symmetry of the structure. To verify this interretation, we lot the H z - comonent of the magnetic field as an inset for the Fano interference-eak at 48 nm. The field exhibits the six nodes that are characteristics of modes with m = 3 symmetry. The frequency of this mode lies outside the sectral band of the m = 3 mode for a single subwavelength hole (magenta horizontal arrow). Since the symmetry-breaking arises from the lattice and, hence, the couling between the holes, we seculate that the couling between the holes may also lead to a shift in the sectral band in the hole array. These interference eaks have not yet been reorted exerimentally. Indeed, we have verified that the features in the band due to the higher-order m = 3 roagating mode, as well as the (1,) surface resonance eak, all vanish when losses are included in the metal model (Fig. 7). In what follows, we roceed with a lossy lasmonic model. Journal of Nanohotonics, Vol. 2, 2179 (28) Page

12 (1,) or (,1) ω/ω k /k x k /k z Fig. 5. Disersion diagram for the surface lasmon-olariton modes (red, cyan and gray curves) and the roagating lasmonic modes in Regime 1. Shown are the HE and EH modes with angular momentum m = 1 (blue curves) and the first roagating mode with m = 3 (magenta curve). The metal is modeled as a lossless 16 lasmonic material with λ = 138nm ( ω = rad/s ), the film is surrounded by air ( ε 2 = 1 ) and the hole is filled with a dielectric ( ε 3 = 4 ). 1.8 T.6.4 -H z,max H z,max λ (nm) Fig. 6. Normal-incidence transmission sectrum of a 5-nm radius ( r =.36λ ) cylindrical hole array with 18-nm eriod in a 25-nm thick metal film. The hole, surround, and metal roerties are the same as in Fig. 5. The surface resonance is indicated by the red vertical arrow. The blue and magenta horizontal arrows show the range of the lowest-order roagating modes in the hole with m = 1 and m = 3, resectively. Inset shows the H z -comonent of the magnetic field for the Fanointerference eak at 48 nm. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 12

13 1.8 T λ (nm) Fig. 7. Normal-incidence transmission sectrum of a 5-nm radius ( r =.36λ ) cylindrical hole array with 18-nm eriod in a 25-nm thick metal film. The metal 16 is modeled as a lossy lasmonic material with λ = 138nm ( ω = rad/s ) 13 and ω τ = rad/s. The surface resonance is indicated by the red vertical arrow. The blue horizontal arrows show the range of the lowest-order roagating modes in the hole with m = 1. Hole array designs based on this aroach tyically feature a high acking density and exhibit diffraction-less behavior. These structures have a high-transmission ass-band with a finite bandwidth that is non-resonant and that can be tailored at both the low and high-frequency ends. This regime is ideal for alications that require a broad-band filter such as solar cells, color imaging, olarimetry, or color dislays. 4.2 Regime 2: ω HE < ω (1,) < ω s,h Another class of subwavelength hole array designs consists of selecting the eriodicity of the array and the hole size such that the resence of the lowest-order (1,) SPP resonance falls within the sectral range of the roagating HE mode. In this regime ωhe < ω (1,) < ωs, h and both athways are resent simultaneously in the regime of oeration and lay a crucial role in the transmission behavior. Figure 8 shows the disersion diagram ( ω, k x ) for the surface waves (left anel; red, cyan and gray curves) and the disersion diagram ( ω, k z ) for the roagating modes (right anel; blue curve). For a lattice constant a = 1.3λ, the lowest-order (1,) SPP resonance is excited at ω(1,) =.56ω (left anel). The disersion diagram for the roagating modes in the hole (right anel), on the other hand, features the lowest-order HE mode ( m = 1 ) that exists between ωhe =.46ω and ω = ω 2. Hence, the (1,) SPP resonance is located within the sectral band of the roagating HE mode. Figure 9 shows the transmission sectrum for this design obtained with 3D FDTD calculations. The agreement with the disersion characteristics is indicated, as before, by the blue horizontal arrow and the vertical red arrow. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 13

14 .7.6 (1,) or (,1).5 ω/ω k /k x k /k z Fig. 8. Disersion diagram for the surface lasmon-olariton modes (red, cyan and gray curves) and the roagating lasmonic modes (blue curves) in Regime 2. The metal is modeled as a lossy lasmonic material with λ = 138nm ( ω = rad/s ) and ω τ = rad/s. The film is surrounded by air ( ε 2 = 1 ) and the hole is filled with air ( ε 3 = 1). 1.8 T λ (nm) Fig. 9. Normal-incidence transmission sectrum of a 5-nm radius ( r =.36λ ) cylindrical hole array with 18-nm eriod in a 25-nm thick metal film. The hole, surround, and metal roerties are the same as in Fig. 8. The surface resonance is indicated by the red vertical arrow. The blue horizontal arrow shows the range of the roagating mode in the hole. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 14

15 Due to the simultaneous resence of two transmission athways, we observe two very striking effects. First, the sectrum features a high-transmission ass-band between 25 nm and 35 nm. This feature is rimarily due to the excitation of roagating modes inside the holes. Even though the interaction between roagating modes and surface resonances leads to a shift of the high-transmission window to the longer wavelength range, the oscillation due to Fabry-Perot effects inside the hole is reserved. Second, the transmission is comletely suressed from 225 nm to 25 nm. This feature arises from Fano interference [38] between two transmission athways, formed by roagating modes in the holes and surface resonances on the interfaces. The high-transmission eak at 3 nm, which coincides with the cutoff frequency of the roagating HE mode ( z k = ), can be interreted as a localized waveguide resonance [21]. It is evident from the transmission sectrum, however, that a descrition in terms of a localized waveguide resonance alone is not comlete. Instead, the full ass-band of the lasmonic mode has to be taken into account (This was also evidenced in Figs. 6 and 7, where only the Fabry-Perot wavelength eaks corresond to the localized waveguide resonances, but there are regions of high transmission in between as well). 4.3 Regime 3: ω (1,) < ω s,i < ω HE It is ossible, in general, to indeendently engineer the hole and the interface roerties in the aerture systems discussed in this aer. For examle, rather than lacing a dielectric inside the hole (Regime 1), we can instead aly a thin dielectric coating to the interfaces of the film. The resence of a coating shifts all the SPP resonances to lower frequencies so that they do not overla with the range in which the roagating modes occur. In this case ω(1,) < ωs, i < ωhe and the structure oerates in a regime where each athway can searately lead to large transmission without destructive interference (transmission suression). This regime looks similar to Regime 1. In comarison, however, the frequency-order of athways is reversed in this regime. The design of such a structure, secifically, is based on the idea that one can engineer the cutoff frequency of the hole ω to lie above the surface lasmon frequency of the interfaces HE ω s, i. For examle, the hole radius ( r =.36λ ) and dielectric inside the hole ( ε 3 = 1) determine the cutoff frequency of the roagating HE mode inside the subwavelength hole at ωhe =.46ω. The thin dielectric coating enforces a limiting frequency for the surface waves suorted by the interfaces. For large k x -values, the surface waves are highly confined to the interface and they see only the coating. Hence, their limiting frequency value is the surface lasmon frequency of the metal-coating interface ωs = ω 1+ εc < ωhe (irresective of the roerties of the surrounding medium), where ε c is the dielectric constant of the thin coating. In this case, it is no longer sufficient to emloy the disersion relations for a single metal-dielectric interface [Eqs. (2) and (3)] to estimate the disersion diagram. Rather, one has to calculate the disersion of the modes suorted by a three-layer system, consisting of a semi-infinite dielectric suerstrate with dielectric constant ε 2 >, a thin dielectric coating with dielectric constant ε c > and thickness h c, and a semi-infinite metal substrate with dielectric function ε 1 ( ω ) <. The resulting disersion equation, which has to be solved, is given by k k ε k k k + = j + 2z 1z c 2z 1z cz ε2 ε1 kcz ε1ε2 εc tan ( k h ) cz c, (6) Journal of Nanohotonics, Vol. 2, 2179 (28) Page 15

16 where ( ) 2 2 k ε ω c k = is the wavevector comonent erendicular to the surface and iz i x k x is arallel to the surface. Solving this transcendental equation leads to the disersion diagram for modes suorted by a dielectric coated interface. The lowest-order mode that results is a surface wave. Figure 1 shows the disersion diagram ( ω, k x ) for the surface waves suorted by the coated surface (left anel; red, cyan and gray curves) and the disersion ( ω, k z ) of the roagating mode inside the hole (right anel; blue curve). The lowest-order (1,) SPP resonance is excited at ω(1,) =.37ω (left anel) and all higher-order surface resonance are located below ωs = ω 5. The disersion diagram for the roagating HE mode in the hole (right anel), on the other hand, starts at ωhe =.46ω. Therefore, both mechanisms are sectrally searated and do not interact. Figure features the transmission sectrum at normal incidence obtained using 3D FDTD calculations. The arrows indicate the features derived from the disersion diagram. Secifically, the (1,) SPP resonance at 368 nm due to the surface modes (red vertical arrow), agrees very well with the transmission eak at 4 nm. The transmission regime between 2 nm and 3 nm matches erfectly with the ass-band due to the roagating HE mode (blue horizontal arrow). These results identify a new regime of oeration that until now has not been analyzed theoretically and has not been exlored exerimentally either. Nevertheless, it features a clean searation of the surface lasmon and roagating lasmonic modes by design and therefore should be useful in establishing the transmission roerties that are unique to each of the tyes of modes. 5 DISCUSSION AND SUMMARY We studied the interaction of different mechanisms or athways by which extraordinary transmission through subwavelength and nanoscale aerture arrays can arise in metallic systems. We rovide a comlete hysical icture that incororates both roagating lasmonic and surface lasmon modes. Our study incororates all reviously reorted mechanisms, e.g., surface-lasmon resonances, localized lasmon resonances and roagating lasmonic modes, and unifies them in a comrehensive disersion-based framework. We revisited the regime exlored by Ebbesen et al. [12] and showed that our aroach exlains all the salient features in the transmission sectrum, including the ones ascribed to surface waves. We rovided evidence that shae resonances or localized surface lasmons are, in fact, due to the behavior of roagating lasmonic modes inside the holes near their cutoff frequency. We identified four characteristic frequencies: the cutoff frequency of the HE mode in the holes, the surface lasmon frequency in the hole, the frequency of the lowest-order SPP resonance, and the surface lasmon frequency of the interface. We defined several regimes of oeration for subwavelength aerture arrays, based on the resective order of the characteristic frequencies. We designed subwavelength hole arrays to robe these regimes, including the "Ebbesen" regime [14], a regime we reviously reorted in Refs. 23 and 24, and a newly identified regime. In each case, the disersion analysis exlains the details of the transmission sectrum. Finally, we showed that the transmission behavior is qualitatively different deending on the number of transmission athways resent in the regime of oeration. If only one mechanism is resent, it can give rise to extraordinary transmission. When both mechanisms are resent simultaneously, their interlay must be studied in order to understand the rich and comlex transmission behavior that results. The same set of interference effects and mechanisms can be found in other resonant systems as well. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 16

17 ω/ω.4.3 (1,) or (,1) k /k x k /k z Fig. 1. Disersion diagram for roagating lasmonic (blue curves) and surface lasmon-olariton modes (red, cyan and gray curves) in Regime 3. The metal is 16 modeled as a lossy lasmonic material with λ = 138nm ( ω = rad/s ) 13 and ω τ = rad/s. The hole is filled with air ( ε 3 = 1), the horizontal metal surfaces are coated with the thin dielectric layer ( ε c = 4, hc = 2nm ) and the film is surrounded by air ( ε 2 = 1 ) T λ (nm) Fig.. Normal-incidence transmission sectrum of a 5-nm radius ( r =.36λ ) cylindrical hole array with 18-nm eriod in a 25-nm thick metal film. The hole, coating, surround, and metal roerties are the same as in Fig. 1. The surface resonance is indicated by the red vertical arrow. The blue horizontal arrow shows the range of the roagating mode in the hole. Journal of Nanohotonics, Vol. 2, 2179 (28) Page 17

18 For examle, we recently reorted at mid-infrared wavelengths that honon-olaritonic thin films with a eriodic array of subwavelength holes allow near-comlete transmission in the olariton ga where a homogeneous film comletely suresses transmission and we observed similar interference effects [37]. Acknowledgments This work was suorted in art by the Stanford Global Climate and Energy Project (GCEP), and by the NSF-NIRT Program (Grant No. ECS-5731). The comutation was erformed with suort from the NSF-LRAC rogram. References [1] S. Blair and A. Nahata, "Focus issue: Extraordinary light transmission through subwavelength structured surfaces - Introduction," Ot. Ex. 12(16), (24) [doi:1.1364/opex ]. [2] C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445(7123), (27) [doi:1.138/nature535]. [3] J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83(14), (1999) [doi:1.3/physrevlett ]. [4] F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic film: Annular aerture arrays," Phys. Rev. B 67(15), (23) [doi:1.3/physrevb ]. [5] E. Poov, M. Neviere, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength eriodic hole arrays," Phys. Rev. B 62(23), (2) [doi:1.3/physrevb ]. [6] P. Lalanne, J. P. Hogonin, S. Astilean, M. Palamaru, and K. D. Moller, "One-mode model and Airy-like formulae for one-dimensional metallic gratings," J. Ot. A: Pure Al. Ot. 2, (2) [doi:1.188/ /2/1/39]. [7] S. Astilean, P. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Ot. Commun. 175(4), (2) [doi:1.6/s3-418()462-4]. [8] Y. Takakura, "Otical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86(24), (21) [doi:1.3/physrevlett ]. [9] Q. Cao and P. Lalanne, "Negative role of surface lasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88(5), 5743 (22) [doi:1.3/physrevlett ]. [1] W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface lasmon olaritons and their role in the enhanced transmission of light through eriodic arrays of subwavelength holes in a metal film.," Phys. Rev. Lett. 92(1), 1741 (24) [doi:1.3/physrevlett ]. [] W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface lasmon subwavelength otics," Nature 424(695), (23) [doi:1.138/nature1937]. [12] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary otical transmission through sub-wavelength hole arrays," Nature 391(6668), (1998) [doi:1.138/3557]. [13] A. Krishnan, T. Thio, T. J. Kima, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Evanescently couled resonance in surface lasmon enhanced transmission," Ot. Commun. 2(1), 1-7 (21) [doi:1.6/s3-418(1) ]. [14] L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary otical transmission through Journal of Nanohotonics, Vol. 2, 2179 (28) Page 18