We study the excitation of whispering gallery modes (WGM) in dielectric nanocylinders by light transmitted through a subwavelength slit in a metallic slab. Calculations are done both by the finite elements method and using FDTD simulations. We discuss the effect of that excitation on extraordinary transmission by the slit. In this way, we show the dominant role of the WGMs over the aperture enhanced transmission as regards the resulting transmitted intensity and its concentration inside the cylinders. When sets of these particles are placed in front of the slit, like linear or bifurcated chains, with or without bends, the concentration of WGMs is controlled by designing the geometry parameters, so that these surface waves are coupled by both waveguiding of the nanocylinder eigenmodes and by scattered propagating waves. Also, the choice of the wavelength and polarization of the illumination, allows to select the excitation of either bonding or antibonding states of the field transmitted through the aperture into the particles. These resonances are further enhanced when a beam emerges from the slit due to adding a periodic corrugation in the slab.
© 2010 Optical Society of America
Microcylinders and microspheres , , , ,  have been a subject of active research since these microcavities offer a variety of functionalities as devices in photonics ; also their arrangements in groups as photonic molecules and waveguiding chains present new effects as regards enhancement, confinement, spectroscopic splitting and transport of optical energy , , , , , . On the other hand, near field optics and nanooptics have shown the possibilities of some of these effects at subwavelength scale, by means of nanojets  or resonant states  such as whispering gallery modes (WGM) in sets of dielectric nanoparticles, and localized surface plasmons (LSP) in metallic ones . In addition, extraordinary, or enhanced, optical transmission through subwavelength apertures , , , ,  is a resonant effect that has received much attention in connection with its potential application for light concentration, detection and wavefront steering. It seems then natural to inquire the connection and interplay between the transmission characteristics of nanoapertures and its coupling by excitation of morphology dependent resonances (MDR) in nanoparticles in front of them, either WGMs or LSPs.
In this paper we report numerical simulations that show new effects in sets of nanoparticles in front of subwavelength slits. We address dielectric particles with high refractive index, which were shown to hold high Q whispering gallery modes . The configurations of sets of these objects are considered in the form of either photonic molecules or chains, also including bends and bifurcations. In this way, several of the aforementioned effects observed in microparticles are now studied at the nanoscale level. On the other hand, by considering nanoparticles close to nanoapertures, we take into account a geometry of collimation and coupling of light from free space into the particle via the aperture; this allows us to study the effect that resonant particles have on the transmission properties of the apertures, and in particular on extraordinary transmission, a question not yet addressed. Conversely, we observe the effects of the transmission characteristics on the excitation of MDRs of these particles and on the near field intensity distribution. In this respect, our simulations indicate that whereas the nanoaperture behaves as a device that couples light into the nanoparticle, the latter acts as a switch that either enhances or not the transmission through the former. This last effect is studied in detail by first considering a single nanoparticle in front of a nanoaperture, and by further extending the observations to sets of particles. We shall see that, as a matter of fact, the enhancement of light transmitted into the particle, or particles, behind the aperture, is mainly governed by the excitation of their morphological resonances rather than by whether or not the aperture produces enhanced transmission. This is seen by studying transmission of p-waves either in or out resonances of the aperture, and thus in or out of extraordinary transmission; or of s-waves in conditions where the cut-off of the aperture does not produce propagation of a transmitted wave into the far zone.
We shall address 2D configurations. First, the essential features observed as regards enhanced transmission, coupling and resonance excitation are likewise obtained in 3D  . Second, 2D resonances have been shown ,  to constitute a good model with equivalent effective constitutive parameters for microdisks. Third, a 2D model is adequate to deal with structures of long parallel nanocylinders with arbitrary spatial distribution of their transversal section uniformly illuminated normal to their axis. This has been also used for the observation of either transport or concentration phenomena in photonic crystal ordered distributions, or other resonant effects like localization in disordered distributions , or wavefront steering by negative refraction in cylinder composite metamaterials . The light propagation between nanoparticles in this configuration is thus not only due to the transport of MDRs between adjacent nanocylinders, but also due to the coupling of these MDRs with the diffracted propagating waves emerging from the subwavelength aperture where extraordinary transmission occurs. In addition there is conversion of waves scattered by some particles into MDR surface waves. For p-polarization and dielectric particles, most of the resulting transmitted intensity is concentrated in the nanoparticles in the form of WGMs. Finally, we shall address the effects of aperture beaming by slab periodic corrugation on the transmitted intensity, inside the resonant particles.
2. Transmission into nanoparticles through a nanoslit
2.1. Numerical simulations
Dielectric particles in this study are Si (refractive index n = 3.670 + i0.005 at λ = 919nm) . The metal for either the particles or the slab, is assumed to have refractive index n = 0.135 + i10.275. The 2D geometries constitute transversal sections of 3D infinite cylinders. The incidence being normal to the cylinder axis, implies that the field component along this axis will keep its polarization. Maxwell equations are solved by using either a finite element method (FE) (FEMLAB of COMSOL, http://www.comsol.com) or an FDTD procedure (Fullwave, http://www.rsoft.com) . In the former, the solution domain is meshed with element growth rate: 1.55, meshing curvature factor: 0.65, approximately; the geometrical resolution parameters consist of 25 points per boundary segment to take into account curved geometries. This configuration is adapted to the geometry and optimizes the convergence of the solution. The final mesh contains about 104 elements. To solve Helmholtz equation, both the the UMFPACK direct and the Good Broyden iterative solver are employed. Their results were in agreement with each other. The determination of convergence of these methods is global, however the local errors are distinguishable by changing the number of elements in the mesh like for instance as reported in . Low reflection boundary conditions are set at the boundaries of the simulation space, except those that coincide with the exterior limit of the metallic slab that contains the aperture, for which a conductor condition is selected. With the former boundary condition, reflections in the window are as low as possible so that one deals with almost free space; the latter boundary condition, on the other hand, is employed whenever a metal slab was displayed, its refractive index n having a high imaginary component(n = 0.135 + i10.275). The interior boundaries of the calculation space were treated as continuous. Field computations are done with waves in their stationary regime of propagation. The results are thus expressed in terms of E(r)[V/m] for the electric vector in the case of s-polarization, or TE, (namely, E(r) along the cylinder axis) and H(r)[A/m] for the magnetic vector in the case of p-polarization, or TM, (namely, H(r) along the cylinder axis). We select p or s waves according to whether we seek enhanced transmission or its suppression in the 2D aperture , .
To observe mode time propagation, we employ the FDTD software. The boundary conditions selected in this method were similar to those chosen in the finite elements one. The FDTD simulations were run until a stationary state was reached, which involved times ct = 50μm (where c stands for the light velocity in the vacuum). Time steps Δt = ct/104 where run with the space discretized into cells of side length 1.5 Δt × 1.5 Δt. Stability of the solution was tested for different discretization sizes. Also, we checked that the two methods provide similar results in the stationary state. In this regime, the field norm, ∣E(r) ∣ or ∣H(r) ∣, is the detectable physical quantity on illumination with a time-harmonic wave. On the other hand, the field, when light starts interaction with the objects, is expressed in terms of Re[E(r,t)] or Re[H(r,t)]. Unless explicitly stated, all calculations are done with the FE method.
2.2. One particle in front of a nanoslit
We first consider a nanoslit of width d = 117.5nm, thickness h = 705nm, practiced in a metallic slab of refractive index n = 0.135 + 10.275i. The system is illuminated by a linearly polarized plane wave at normal incidence whose electric or magnetic field modulus, depending on the polarization, is normalized to unity. For p-polarization such a slit presents morphological resonances that yield extraordinary transmission ,,. We shall choose these slit parameters so that its resonant transmission wavelengths may coincide with those of resonance of nanoparticles that eventually will be placed in front of it.
The black curve in Fig. 1 shows the transmission of such a subwavelength slit. The parameters correspond to those of one of the resonances found in ,  re-scaled in such a way that the wavelength of enhanced transmission will match that of the nanoparticle resonance when placed in front of the aperture exit. This transmission, represented in A·m, is evaluated by integrating the magnetic field magnitude ∣H z∣[A/m] in a 130nm × 100nm rectangular section whose bottom side coincides with the slit exit plane. There is a transmission maximum of the slit at 945nm, which also corresponds to the wavelength excitation of the WGH 21 of a cylinder of refractive index n = 3.670 + i0.005 and radius r = 200nm. The peaks of the red curve corresponding to the magnitude of ∣H∣ (we omit the subindex z from now on) transmitted into the cylinder when placed in front of the slit, represent the estimated excitation of different WGMs, by integrating ∣H∣ in a circular area that coincides with the cylinder transversal section. On comparing the red and black curves, we notice that the enhancement of the field magnitude confined inside the cylinder at λ = 945nm rises from 1A·m to 3.5A·m (× 10-13) whereas that due to the slit alone rises from 9A·m to 15A·m (× 10-14) (we wish to point out the aforementioned different surfaces of integration leading to these values). Also, the peaks due to WGMs are narrower than those due to localized plasmons of the metallic aperture which have larger damping, as expected . This result indicates that the resonance of the particle reinforces the slit transmission whether it is extraordinary or not, and it is a dominant effect upon the latter. Thus selecting the transmission maximum of the slit is not determinant of enhanced intensity concentration in the resonant nanoparticle unless coupling to its MDRs appear. The cylinder acts like a piston that extracts radiation through the slit when some of its WGMs are excited. It should be remarked that the proximity between the particle and the slit slightly alters the value of their respective resonant wavelengths, however, as seen in what follows, this shift is more critical for the particle than for the slit, due to the larger influence of the former on the transmitted intensity and its narrower lightline.
To reinforce this observation, in Fig. 2 the concentration of ∣H∣ in the cylinder is evaluated versus the real part nreal of the cylinder refractive index while maintaining all the other parameters of the slab-slit-cylinder configuration. Two wavelengths are chosen: one in a peak of the slit resonant transmission, λ = 945nm, and one out of it: λ = 750nm. Both curves present peaks of total field intensity concentrated inside the particle at values of nreal that correspond to the excitation of the particle WGMs. The nature of this excitation dominates these intensity enhancements (see e.g. the large red peak near nreal = 4.45), irrespective of whether or not the wavelength is that of the slit resonant extraordinary transmission, and confirms the dominance of the WGM excitation in the particle upon that of the localized plasmons in the slit leading to extraordinary transmission, as regards the resulting enhancement and concentration of transmitted intensity in the particle.
To compare the above with a situation of a slit with almost no transmission, a slab with an aperture of width 440nm which renders a cutoff wavelength of 880nm illuminated under s-polarization, at the wavelength of 919nm, transmits very little radiation. When a resonant particle is placed in front of it, there is coupling of both the evanescent and weak propagating components of the wavefront at the aperture exit with the cylinder eigenmodes, and the change in the resulting transmission is dramatic as shown in Fig. 3(a), where we see the “extraction” of intensity and its confinement in the nanoparticle by excitation of its WGE 31 mode. The field modulus in the cylinder reaches values four times higher than that of the incident field. In general, due to the large cylinder refractive index, there is a tendency of field concentration inside it, even when there is no WGM excitation. In particular, this configuration of slit with a larger particle also yields a nanojet inside the cylinder; this is not shown here for the sake of brevity.
Figure 3(b), which represents the field norm confined in the cylinder versus the wavelength of the electromagnetic radiation for s-polarization, is a proof of both field localization in the dielectric cylinder and slit-slab effects on it. These curves show that there is a tendency for the confined field modulus to decrease due to the presence of the metallic layer and to increase as the cylinder separates beyond a distance of the order of the cylinder diameter. The less affected resonance is that corresponding to λ = 733nm for the isolated cylinder (WGE 41). The peak does not appreciably shift when the particle moves around the aperture, but its Q-factor, evaluated as in , diminishes from 216 to 144 (corresponding to the isolated and close-to-aperture cylinder, respectively). The other resonance at λ = 919nm and Q-factor = 107 for the isolated cylinder (WGE 31) is more affected by the slab-slit configuration since the maximum change occurs, as before, when the cylinder is tangent to the aperture exit plane, now being λ = 914nm and the Q-factor = 89.
In a similar way, a resonant dielectric particle switches on and enhances the transmitted radiation by a subwavelength slit that already is near its resonant transmission. Figure 4(a) shows the case of the slit of Fig. 1 illuminated at a wavelength (λ = 750nm), which is near but not exactly at the peak of extraordinary transmission under p-polarization. When the dielectric cylinder is placed in front of the aperture, the transmission and intensity concentration inside it, is again much larger than without that object. The WGH 31 is excited exhibiting a typical enhanced stationary wave distribution, whose intensity peaks inside the cylinder are about 100 times larger than that of the incident field (see Fig. 4(b) which also shows that the mode inside the aperture is present, like in Fig. 4(a), but its intensity now remains similar to that in absence of particle). Another interesting fact of Fig. 4(b) is that the WGH 31 intensity is much larger than that of the field reflected below the slab, contrary to what happens with the transmitted intensity in absence of particles, as seen in Fig. 4(a).
Figure 4(c) represents the field norm integrated in the cylinder versus the wavelength for p-polarization. We prove with these curves both field concentration in the dielectric cylinder and slit effects on it, the latter being more dramatic than in the case of s-polarization (cf. Fig. 3(b)). These curves show the same tendency for the confined field in the particle to decrease due to the presence of the slab. The low-wavelength resonance is that one corresponding to λ = 748nm for the isolated cylinder (WGH 31). The peak slightly shifts to λ = 742nm when moving the particle around the aperture, but its Q-factor diminishes from 97 to 24 (isolated and close-to-aperture cylinder, respectively). The other resonance, at λ = 944nm and Q-factor = 20 for the isolated cylinder (WGH 21), seems to be more affected by the slit configuration since its peak change occurs, as before, when the cylinder is tangent to the aperture, then shifting to λ = 914nm. The Q-factor in this case does not change much as the particle moves around the slit, but it slightly oscillates around a value of 19.
2.3. Coupling between two or more particles
Figure 5 shows the coupling between two cylinders at p-polarization, vertically arranged in front of the subwavelength aperture, manifested as peak resonance splittings. The state degeneration is broken into two new modes, one at higher wavelength with even symmetry of the field intensity spatial distribution and another at lower wavelength with odd symmetry. This arises when the cylinders are placed near enough each other. The presence of the slab reduces not only the peak amplitudes but the Q-factor too. The separation between cylinders plays a different role for each type of resonance, i.e. whether it is bonding (higher wavelength and even symmetry of field distribution) or antibonding (lower wavelength and odd symmetry of field distribution). In the first case the Q-factor decreases as the separation increases, while in the second case it increases instead, which is in agreement with  about the existence of a high-Q bonding mode and a low-Q antibonding mode under s-polarization only when the cylinders are in contact. This is the opposite behavior, even though more attenuated, as the cylinders go off the slit a distance of the order of their constant separation from each other. The Q-factor either decreases or increases due to the proximity of the slit, depending on the bonding or antibonding nature of the resonance: compare the peaks at λ = 983nm with Q-factor = 45 and at λ = 38nm with Q-factor = 28 (black curve, isolated cylinders together) with the peaks at λ = 973nm with Q-factor = 34 and at λ = 909nm with Q-factor = 29 (green curve, cylinders together close to the slab). These values correspond to the WGH21 resonance, but the other resonance (WGH 31) follows the same behavior. Now, this effect relates this configuration to that of photonic molecules . Indeed, a cylinder chain, or even a more complex structure, may be interpreted as a photonic molecule, as we next show.
When one plots the field spatial distribution, one sees a detail of the splitting of modes in the two cylinders close to the slit. The lowest wavelength state is that in which field maxima of one cylinder match to minima in the other cylinder then showing an odd symmetry distribution. Correspondingly, the highest wavelength state makes field maxima of both cylinders to appear in front of each other, thus the field spatial distribution presents an even symmetry.
2.4. Transmission in particle chains.
Figures 6(a)-6(c) show p-wave illumination of a chain of dielectric cylinders with bifurcation in front of the aperture of Fig. 4(a). All the elements and incident wave parameters are the same as in the case of a single cylinder, shown in Fig. 4(b). Namely, we are near the extraordinary transmission peak of the slit. The cylinder in the vertex is tangent to the exit plane of the aperture. Again, this cylinder ”extracts” more light intensity through the slit, when its WGM is excited, than the aperture alone resonantly transmits (cf. Fig. 4(a)); and the light is transported to the next cylinders even though the stationary state regime shown here exhibits maximum light concentration and enhancement in the first particle that now acts as a source for the others of the chain. This structure, on the other hand, represents an example of a complex photonic molecule, not only because of the anti-bonding and bonding states (cf. Fig. 6(a) and Fig. 6(b)), but due to the existence of an hybrid state (see Fig. 6(c)) in which several bonding and anti-bonding areas now appear.
We see this kind of enhanced transmission also in presence of bends as shown in the FE calculation of Fig. 7. In agreement with the conditions put forward in  we have observed optimum configurations for transmission and WGM concentration in the particles, given a bifurcation angle, for certain bending angle and air gaps between cylinders, and they are independent of whether the slit supertransmits or not. We should also remark that, although not shown here for brevity, when one cylinder is placed tangent to the exit plane of the slit, the intensity of the WGH 31 mode is smaller than when placed at a certain distance from it. However, when other cylinders are also present, like in chains, they contribute to enhance and extract more light through the slit that eventually will become localized as WGMs in these cylinders. We have also observed that in general the intensity does not always tend to concentrate in the first cylinder in front of the slit, but its distribution in the particles much depends on the geometry. This point is important because it means that with the appropriate design, the field can be manipulated and thus obtained more intensely concentrated in certain cylinders of the chains.
At this stage, it should be reminded, however, that all field moduli ∣E(r)∣ (or ∣H(r)∣) (or its equivalent: intensity) shown so far represent the time average: (< (Re[E(r,t)])2 >)1/2 (or (< (Re[H(r,t)])2 >)1/2) and hence correspond to the stationary state of the field. On the other hand, the time evolution from the beginning of the interaction of either the field or its intensity, may be displayed by using FDTD calculations. When this last procedure is employed, one observes the following: when the slit is illuminated either with a wavelength in cut-off, e.g. with s-polarization such that the field emerging from its exit is evanescent, like in the case of Fig. 3(a) and lower ratios d/λ, or with p-polarization in supertransmission regime like in Fig. 4(a), this field couples with the WGM of the particle in front of the slit, like e.g. in Fig. 3(a) and Fig. 4(b), and also with propagating waves that are scattered into free space by the cylinder. When more cylinders exist, like in a chain, either linear, or bifurcated with or without bends, there is a coupling with the WGMs of the next particle both from the previously formed WGMs and from the scattered waves propagating in free space. So that when the stationary state is reached, most of the light intensity is concentrated in the cylinders as WGMs. In this connection, if one seeks WGM pure waveguiding from the first particle up to the last of the chain, with minimum scattering into free space, the optimum configuration, as shown in previous works (see ) ), is coupling of evanescent waves created by TIR at an interface close to the first particle of the chain.
The alternative proposed here is that of controlling the distribution of WGMs in the different particles of the set in the stationary state by means of the transmission characteristics of the aperture. In this respect, when the aperture is illuminated at an extraordinary transmission wavelength under p-polarization, as shown in the previous examples, the field concentration once reached the stationary state, will depend on the parameters chosen for the set and will be built by both the WGMs created from the first particle in front of the aperture and by a cylindrical wave that, after coupling with the first particle, propagates in free space further illuminating the other particles of the chain and contributing to the build-up of their corresponding WGMs (see the FDTD calculation shown in Fig. 8(a)). Once again, the result in the stationary state is a strong distribution of field as WGMs in the particles of the chain with higher concentration in certain cylinders of the set and less intensity distribution outside them, depending on the chosen parameters, as shown in Fig. 8(b) at a time instant near the stationary state. This intensity in the particles therefore results in an enhancement of the extraordinary transmission that would present the aperture alone.
Further, it is worth remarking that when sets of dielectric particles over an aperture are illuminated under s-polarization, like in Fig. 3(a), and although not shown here for brevity, one observes that the coupling with the most intense WGM at a given particle has a similar dependence on the geometry and parameters chosen. But like in Fig. 3(a) the electric field distribution exhibits a larger intensity in the near region of the exterior space emanating from the most intensely illuminated particles.
2.5. Effects of beaming by periodic corrugation in the slab surface
It was shown in  that introducing a chosen periodic corrugation in the surface of the slab in which the aperture is practiced, produces at the appropriate wavelength a peak of extraordinary transmission with a concentrated angular distribution, similar to that of a beam emerging from the aperture. To see the effect of this phenomenon in our configuration of particles, we show in Fig. 9 the magnetic field modulus for a set of 8 dielectric nanocylinders in rhombus formation under p-wave illumination. The corrugation is now introduced in the slab with the same grating structure as that of the slit of Fig. 2(A) and Fig. 2(D) of , but with the period and depth of the grooves of that reference now rescaled to our geometry. The effect of beaming by this system in the transmitted intensity is to produce a field spatial distribution almost similar to that in absence of corrugation, but with much higher intensity inside the particles (c.f. a peak of 35 units in ∣H∣ in Fig. 9 versus 8 units in absence of corrugation). Therefore, the effect of beaming in the wave transmitted by the slit due to the slab grating corrugation, is to further enhance the intensity transmitted into the resonant particles. Again, one can design the parameters so that there is maximum intensity concentration in a different particle; for example in the structure of Fig. 9 the other supertransmission wavelength λ = 750nm produces the highest enhancement on the top vertex cylinder with WGH 31 modes.
We have shown by means of both the finite elements method and FDTD simulations, that the excitation of morphological resonances of nanocylinders placed in front of a subwavelength slit dramatically enhances the extraordinary transmission that the aperture would produce alone. Conversely, the transmitted intensity which appears mainly concentrated in the nanoparticles, does not strongly depend on whether the aperture is on or out of its enhanced transmission regime. Also, it is possible to fitting particle set parameters and illumination such that the transmitted intensity is concentrated in certain cylinders when the stationary regime of propagation has been reached.
When an appropriate periodic corrugation is introduced in the slab, the slit supertransmits producing an emerging beam. Then, the excitation of the MDR is further enhanced, even though they are in the near field zone of the aperture. Thus the amplitude and phase of the emerging wavefront that will eventually give rise to a beam at larger distances, is determinant as regards the WGMs or LSPs excited on the nanoparticles.
All these results should be reproducible for 3D particles in front of apertures with any geometry, in particular circular nanoholes, and would offer new possibilities for controlling transmitted near fields in the nanoscale region.
We thank Drs. J.L. García-Pomar and R. Gómez-Medina for help with the computations in the beginning of this study. Work supported by the Spanish MEC through grants FIS2009-13430-C02-01 and Consolider Nanolight. F.J.Valdivia-Valero acknowledges a scholarship financed by the last quoted grant.
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