Research Papers

Propagation of particle plasmons in sets of metallic nanocylinders at the exit of subwavelength slits

[+] Author Affiliations
Francisco J. Valdivia-Valero, Manuel Nieto-Vesperinas

Instituto de Ciencia de Materiales de Madrid, C.S.I.C., Campus de Cantoblanco 28049 Madrid, Spain

J. Nanophoton. 5(1), 053520 (August 11, 2011). doi:10.1117/1.3615984
History: Received April 05, 2011; Revised June 28, 2011; Accepted July 06, 2011; Published August 11, 2011; Online August 11, 2011
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We investigate the effect on transmission of p-polarized light of the excitation of plasmons of metallic nanocylinders, placed at the exit of subwavelength slits. Morphology-dependent resonaces of the aperture may then be excited, thus leading to supertransmission. The possible enhancement of transmittance is investigated by working out appropriate choices of the geometrical parameters and illumination. Additionally, we study conditions in which supertransmitted light may propagate through sets of metallic cylinders with excited plasmons, in front of either one slit or placed in front of a periodic array of slits. We find that the concentration and intensity of the transmitted field is mainly governed by both the material and geometrical configurations of the particle sets.

Figures in this Article

Plasmons of metal nanoparticles1 enjoy potential as elements of nanooptical networks. These resonant modes arise from the excitation of coherent oscillations of conduction-band electrons, localized on the surface of these particles. The strong light coupling leads to its absorption and spatial confinement to a nanometric scale, which results in large local enhancements of electromagnetic field intensities. Ensembles of particle chains have been extensively studied.2 This provides nanoscale control of the transmission, manipulation, and switching of optical signals.3 In addition, nanoparticles—either dielectric4 or metallic5—have been addressed in microdisks in connection with the mutual perturbation of their resonances, such as counterpropagating whispering gallery mode (WGM) splitting, modification and their monitoring in the cavity, or control of the radiating properties of the nanoparticle by the WGMs of the cavity.6

A somewhat related phenomenon known as enhanced optical transmission through subwavelength apertures, either alone or forming a grating,7 has received much attention in connection with its potential application for light concentration, detection, and wavefront steering. Morphology-dependent resonances (MDR) play an important role in the aforementioned phenomenon. They also influence the bandgap size and position of photonic crystals (PC).89 In this way, the Mie resonances of the particles forming the PC constitute the light propagation and enhancement vehicle in the upper bands of those that are called molecular photonic crystals.10 On the other hand, with regard to metallic PCs, there are certain advantages in fabricating them, such as reduced size and weight, easier production methods, and lower costs, as well as the fact that low-loss metal PCs have been studied.11 Furthermore, many applications are being developed from these structures, such as waveguiding,1215 waveguide mode-plasmon coupling,16 light transmission control,17 thermovoltaics and blackbody emission,1819 and lensing.20

Studies have been undertaken aiming to increase the efficiency of aperture supertransmission by enhancing the incident field energy. This has been sought in different wavelength regions by exciting a resonance in a particle or structure placed near the aperture entrance. Among several configurations, this included metamaterials, split-ring resonantors, or nanocylinder plasmons and nanojets (see Ref. 21 and references therein).

In this paper we address sets of metallic particles placed at the exit of apertures. In contrast with Ref. 21, because these nanoparticles are at the other side of the slab, they do not reinforce the field incident on the apertures. We then want to answer three questions: first: What is the effect of the presence of metallic particles, that support plasmons, in the transmission zone of a subwavelength aperture that is supertransmitting? Do these particles enhance or inhibit this supertransmission? And, how does the energy flow of light transmitted by this slit (or slits) propagate through these sets of particles?

We present a study by means of numerical simulations that show new effects in configurations of metallic nanoparticles that support plasmons near the exit of supertransmitting subwavelength slits practiced in a thick slab. Our study is carried out in 2-D, but the essential features observed with regard to enhanced transmission and coupling of resonances, and light transport are likewise obtained in 3-D.2223 Also, this 2-D geometry constitutes a good model with equivalent effective constitutive parameters for microdisks.2425 Furthermore, such a configuration is adequate to deal with structures of long parallel nanocylinders in 2-D PCs or metamaterials.26 We then address localized surface plasmons of metallic nanocylinders.27 We see the behavior of light concentration and transmitted field-enhancement efficiency in this kind of particle set, which appears maximized when the configuration of the set has emission properties similar to those of a nanoantenna.

Further calculations deal with sets such as linear and bifurcated nanocylinder chains, also addressing the natural step passing from particle chains, placed one close to another, toward a PC geometry situated in front of a metallic array of slits. In this way, we design a method of collimation and coupling of light from free space into the particles by placing them close to the slits. These numerical simulations allow one to study the effects that arise in the near field with regard to extraordinary transmission in the slits and the excitation of plasmons of these metallic particles; and because the calculations are exact, they constitute a reliable design of future experiments that can be performed in either 2-D or 3-D.

In this respect, our calculations indicate that although the nanoaperture behaves as a collimator that transmits light into the nanoparticles, the fundamental role concerning enhancement and field concentration corresponds to the excitation of the particle plasmon that couples with the MDRs of the slit. This is thoroughly studied by first displaying the simple configuration of a single metallic nanoparticle at the exit of a nanoaperture, and afterward, extending these observations to more complex sets, including particle chains and PCs. In these latter configurations, we take advantage of the excitation of dipolar plasmons.

Transmission into Metallic Nanoparticles through a Nanoslit

Numerical Procedures

From now on, all refractive indexes under the different wavelengths on use are taken from Refs. 2829. All particles in this study are considered of silver (Ag), (refractive index n = 0.188 + i1.610 at λ = 366 nm and n = 0.233 + i1.27 at λ = 346 nm) because of their rich plasmon spectrum in the near ultraviolet. However, it should be stressed here that this is done for the sake of illustrating the effects, and that other noble metal material can be chosen.

Because the slab is thicker than usual in supertransmission experiments, in order that the slit walls present high reflection and as small as possible skin depth and losses, the metal of the slab is assumed to be tungsten (W) (refractive index n = 3.40 + i2.65 at λ = 366 nm and n = 3.15 + i2.68 at λ = 346 nm). It should be remarked in this connection that, ideally, a quasi-perfectly conducting slab would exhibit the most pronounced supertransmission effects under study; but if experiments are done with thinner slabs, other materials, such as noble metals Al, Au, or Ag, may be employed. Also, contrary to noble metal films, the W slab is dielectric at the frequencies employed here and, hence, it does not support surface plasmon polaritons (SPPs). Nevetheless, note that, in contrast to initial interpretations, it has been proven7,3031 that the phenomenon of subwavelength aperture supertransmission does not require slab SPPs, but other type of resonances such as MDRs of the aperture, Fabry–Perot resonances, or some kind of wood anomalies in periodic arrays of such apertures, such as those appearing at Rayleigh wavelengths (i.e., at the onset of a grating propagating mode becoming evanescent3032).

Depolarization is prevented by launching on the aperture linearly polarized light beams of rectangular profile, their widths being that of the simulation window (the latter always coincides with the slab width D; thus, these beams behave as plane waves in the domain of the calculations). The direction of propagation of such beams is normally incident to the axis of the infinite cylinders and points upward in all windows of the calculated magnitudes to be shown in this work. A schematic diagram of the calculation geometry is shown in Fig. 1.

Grahic Jump LocationF1 :

Schematic of the geometry of the calculations: An incident p-polarized plane wave with magnetic vector Hz and Poynting vector Sy impinges a W slab of width D and thickness h, containing and aperture of width d. The transmitted intensity is evaluated at a rectangular monitor ax × ay. However, when a cylinder of radius r is placed, the transmitted intensity is evaluated at an annulus of exterior radius re = (9/8)r.

At the illuminating wavelengths, the slit geometrical parameters have been chosen on or near those values that maximize their light transmission. The same applies to the distances between all particle sets and the exit plane of the slit.

Maxwell equations are solved by using a finite element (FE) method (FEMLAB of Comsol).33 The solution domain is meshed with the element growth rate, 1.55, and meshing curvature factor, 0.65. The geometrical resolution parameters consist of 25 points per boundary segment to take into account curved geometries in order to adapt the finite elements to the geometry and optimize the convergence of the solution. The final mesh contains 104 elements. To solve the Helmholtz equation, the UMFPACK direct is employed. The boundary conditions of the simulation space are properly set both to keep the calculations from undesired window reflections and to avoid possible geometrical discontinuities. We hence ensure that neither inconsistencies due to property discontinuities of the objets under study nor possible systematic errors because of the simulation window interfere with the field calculation.

We always select p-polarized incident waves to seek extraordinary transmission in the 2-D slit (s-polarized waves do not produce such a phenomenon in 2-D subwavelength slits30,34). The results are thus expressed in terms of the magnetic vector H(r), which is along the cylinder OZ-axis, the electric vector E(r), and the time-averaged energy flow ⟨S(r)⟩; these last two are both transversal, namely, in the plane of the images to show next (see Fig. 1). The incident field is a plane wave H(x, y) = H0exp {[(2π/λ)y − ωt]} of unit amplitude H0 = 1 A/m (SI), which corresponds to ⟨S⟩ ≈ 190 W/(m2).

Finally, the nomenclature followed to classify the plasmon resonances of the cylinders will use the subscripts (i, j), i and j standing for their angular i'th and radial j'th orders, respectively.

One Metallic Particle in Front of a Nanoslit

The phenomenon of field-transmission enhancement through a subwavelength aperture by excitation of plasmon resonances in nearby metallic particles is studied by comparing it to the transmission through a slit alone, practiced in a metallic slab. We choose λ = 364.7 nm for the incident wave, where the aperture alone is supertransmitting. It should be noted, however, that at this fixed wavelength, the presence of the particle, or sets of particles, would slightly alter the geometrical parameters at which the slit alone yields a transmittance peak. For this reason, we have chosen near, but not exactly equal, values of these parameters when the cylinders are present.

Figure 2 shows a diagram of the p-polarized transmitted intensity by the aperture alone in terms of its width d for several thicknesses h. It is seen that there is an optimum value of these parameters for slit supertransmission at the chosen wavelength λ = 364.7 nm. As shown next, this kind of peak corresponds to modes in the aperture that represent MDRs of the slit and that, due to its rather large thickness, is also mixed with its waveguide modes.

Grahic Jump LocationF2 :

Variation of the averaged energy flow |⟨S(r)⟩| (in units of electron volts per nanometer squared, per second), transmitted by the slit in the W slab (n = 3.39 + i2.66; λ = 364.7 nm) versus its width, illuminated by a p-polarized plane wave near its TE51 MDR. Each curve with symbols corresponds to a different slab thickness. The highest peak (out of the scale this graph) occurs at the slit width d = 55 nm, thickness h = 259.4 nm, and has a value of its ordinate equal to 3.5 × 105 eV/(nm2 s). Averages have been performed over the rectangular monitor at the slit exit (see Fig. 1), of area: S = 1.76 d (nm) × 54.09 nm = 95.09 d (nm2) at the slit exit.

Figures 3 show the magnetic field magnitude |H(r)| and the time-averaged energy flow ⟨S(r)⟩ distributions in the slit, which produce the supertransmission peak of Fig. 2, namely, that with d = 55 nm and h = 259.4 nm. These distributions show two interesting features: (i) the waveguide and MDR mixtured eigenmode inside the aperture and (ii) the change of direction of the energy flow, from the lower region of the aperture at which the energy is partly reflected to the upper region of the slit where the energy manifests transmission upward.

Grahic Jump LocationF3 :

(a) Magnetic field magnitude |H(r)| (in amperes per meter), in a W slab aperture (refractive index n = 3.39 + i2.66, slab width D = 2850 nm, slab thickness h = 259.40 nm, slit width d = 55 nm). The incident radiation at (λ = 364.7 nm) is p-polarized, of unit amplitude, and impinges on the slab from below. (b) Time-average energy flow ⟨S(r)⟩ (in Joules per meters square per second), maximum arrow length ≈1.05 MeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)), both magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown under the same conditions as in Fig. 3. (c) Detail of the saddle point evidenced by ⟨S(r)⟩ inside the slit (the arrows here appear normalized to their magnitude which is shown according to the color bar). (d) Detail of the electric field E(r) (in volts per meter), both magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown under the same conditions as in Fig. 3.

This change of orientation of the energy flow is evidenced in Fig. 3 as a potential saddle point due to a change of sign of the magnetic field H(r) inside the slit. On the other hand, Fig. 3 exhibits the electric field E(r) which shows a strong charge concentration, and a resulting a dipolar pattern configuration at the corners of both the entrance and exit of the slit [see the two upper and lower vertices of the aperture in Fig. 3]. In this respect, note that these charge peaks in the W slab aperture are due to polarization charges (i.e., to bounded charges which are known to play an important role in field enhancement35).

Incidentally, we believe that this high-intensity concentration at the edges of the aperture exit is responsible for the existence of gradient optical forces on dielectric nanoparticles placed in its proximity and the corresponding creation of two potential wells, each of them in front of each corner of the aperture exit [cf. Fig. 3], as observed in Ref. 36. Note in Fig. 3 that the supertransmitted intensity is already seen in the spatial distribution of |H(r)|, E(r), and |⟨S(r)⟩| in the neighborhood of the slit exit, [we recall that incident |H| = 1 A/m, |E| = 380 V/m, and |⟨S⟩| = 190 W/(m2)].

As a first example of placing a metallic particle near the aperture exit, Fig. 4 deals with an Ag nanocylinder placed in front of the exit of a subwavelength slit in a W slab, which presents a large transmission near the chosen wavelength. As shown, at λ = 366 nm and at a distance between the cylinder and the slab d = 549 nm, a strong-intensity enhancement of |H(r)| appears on the cylinder surface, manifested by the plasmon stationary interference pattern surrounding it, and a standing wave pattern due to reflections between the slab/aperture and the particle is observed. It should be noted that this cylinder alone has this TE51 mode at λ = 359.7 nm, which is red-shifted as seen in Figs. 4 in presence of the slab and aperture. A well-known phenomenon explained on the basis of the driven oscillator model.37

Grahic Jump LocationF4 :

(a) Magnetic field magnitude |H(r)| (colors in amperes per meter) and time-averaged energy flow ⟨S(r)⟩ [arrows in Joules per meters squared per second), maximum arrow length ≈80.02 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)), localized on the surface of an Ag cylinder (radius r = 200 nm, refractive index n = 0.186 + i1.61) in front of a slit in a W slab at the same illumination as in Fig. 3. The distance between the cylinder surface and the exit plane of the slit is dlc = 3λ/2 = 549 nm (λ = 366 nm); (b) Variation of the concentration of |⟨S(r)⟩| (in electron volts per nanometer squared per second), on the cylinder surface versus its vertical separation (in nanometers) from the slit exit, illuminated at λ = 366 nm (near the plasmon TE51 resonance). (c) The same quantity versus wavelength at the distance seen in Fig. 4, dlc = 549 nm. The curves with squares and upward triangles stand for the response of the slab alone, and those with circles and downward triangles correspond to the slab with cylinder, respectively. The ordinate of the curves with both upward and downward triangles are plotted with a reduction factor of 6.24. (d) Electric field E(r) (in volts per meter); both its magnitude (online version: colors, printed version: gray levels) and directions (arrows), under the same conditions as in Fig. 4. Black and red curves in Figs. 4 have been calculated by averaging the quantity in an annulus of area A = π{[(9/8)r]2r2} [see the two concentric circles in Fig. 4, which are drawn according to the scheme of Fig. 1], whose internal circle coincides with either the cylinder section [case of the slit with the particle in Figs. 4] or an imaginary circle coincident with that cylinder section [case of the slit alone in Fig. 4]. The green and blue curves in Fig. 4 have been calculated by averaging the quantity in a rectangular monitor of area S = 76 nm × 58.46 nm = 4442.96 nm2 at the slit exit [see the rectangle drawn in Fig. 1].

Also, both vortices and a saddle point are exhibited by ⟨S(r)⟩ in the standing wave pattern between the nanocylinder and the slab. The vertical separation between the cylinder and the slit exit has been chosen in order to optimize this field enhancement. Figure 4 shows this intensity concentration in terms of the energy-flow magnitude |⟨S⟩| as the nanocylinder is gradually moved away from the slit at λ = 366 nm. In order to quantify it, we have averaged this quantity in an annulus surrounding the cylinder, whose internal circle coincides with the cylinder section.

The image of Fig. 4 corresponds to the third peak of Fig. 4 from the left. The spectrum of this enhancement versus the wavelength with the cylinder at d = 549 nm, can be seen in Fig. 4, which shows a comparison between this enhancement to the cylinder at the aforementioned distance d = 549 nm with the one obtained in the same region with the slit alone (red and black curves, respectively), as well as in the region immediately outside the slit exit (blue and green curves). In addition, Fig. 4 shows the pattern of E(r) for the same configuration as Fig. 4. This illustrates the enhancement of the electric field on the particle surface, as well as the interesting vortices described by its wavevector around it, due to multiple reflections with the upper surface of the slab. This is in contrast with the case in which the particle is dielectric and the excited MDR is a whispering gallery mode (WGM), in which case the field is mainly confined inside the particle and exponentially decays outside.38 In this latter case, the cylinder reflectivity is much lower and the multiple reflections are much weaker.

This high reflectivity of the metallic cylinder produces not only a high concentration of transmitted energy around its surface, but also outside it. In particular, at the slit exit [compare Figs. 34]. This is related to a relatively lower reflectivity at the slit entrance, and in fact, already this lower reflectivity constitutes another signature of a larger transmission into the space where the particle is. Hence, this cylinder acts as an extractor of transmittance upon the slit [see Figs. 4].

Figure 5 shows the behavior of both the magnetic field and the energy flow with a lateral separation of the nanoparticle from the slit exit. Now, the asymmetry of its location suppresses the standing plasmon pattern on the upper side of the cylinder surface, showing the energy circulation around the particle, [see Fig. 5]. Note that the spatial distribution seen in Fig. 5 corresponds to the highest peak of Fig. 5. Again, the transmittance of the slit is now much larger than when it is alone, also, a standing wave is now formed by the coupling between the light emerging from the slit and the cylinder plasmon, and is confined by the high reflectivity of the slab, also projecting energy into the upper space.

Grahic Jump LocationF5 :

(a) Variation of the magnetic field concentration |H(r)| on the cylinder surface versus its horizontal separation from the slit exit. The particle is illuminated at λ = 366 nm (i.e., near the nanocylinder TE51 resonance). The black, red, green, and blue curves stand for vertical distances between the particle surface and the slab equal to λ/8, λ/4, 3λ/8, and λ/2, respectively. (b) Map of |H(r)| (online version: colors, printed version: gray levels) and ⟨S(r)⟩ (arrows) when the cylinder is at a vertical and horizontal distance from the slit exit: λ/8 and 730 nm, respectively, and illuminated at λ = 364.7 nm (p-polarization, near the TE51). Maximum arrow length ≈26.09 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s). Calculations in Fig. 5 have been made as in Figs. 4.

Other Sets of Metallic Nanocylinders

In order to test light transmission and concentration by means of aperture MDR particle plasmon interaction and through adjacent cylinder plasmon coupling, Figs. 6 show |H(r)|, ⟨S(r)⟩ and E(r) for a bifurcated chain of Ag cylinders near a slit. The distance between the slit and the cylinders has been chosen to optimize the extraction of energy by the particles through the aperture.

Grahic Jump LocationF6 :

(a) Magnetic field magnitude |H(r)| spatial distribution for five Ag cylinders (radius r = 200 nm, refractive index n = 0.186 + i1.610), disposed in a bifurcated chain (bifurcation angle θ = 45 deg, distance between cylinder surfaces dcc = 100 nm), in front of the slit of Figs. 3 illuminated at λ = 364.7 nm (p-polarization, near the TE51 resonance). The vertical distance between the bottom cylinder surface and the exit plane of the slit is λ00 = 366 nm near the TE51 resonance of an isolated cylinder). (b) Time-averaged energy flow ⟨S(r)⟩ [maximum arrow length ≈15.60 KeV/(nm2s), minimum arrow length ≈0 eV/(nm2 s)] in the same configuration as in Fig. 6. (c) Electric field E(r) distribution, both the magnitude in volts per meter (online version: colors, printed version: gray levels) and directions (arrows), are displayed in the same configuration as in Fig. 6.

Light transport through the chain up to the upper particles is now obtained as a concentration of energy spread in the region of the four upper cylinders as shown in Figs. 6 with, of course, a lower intensity concentration in the aperture exit zone. Compare |⟨S(r)⟩| = 766.86 eV/(nm2 s) in the area inside a rectangular monitor appropriately scaled and equivalent to that shown in Fig. 4, over the slit exit when no particles are present against |⟨S(r)⟩| = 10698.50 eV/(nm2 s) averaged over the same monitor in the configuration of Figs. 6. Note that now, as before, the optimum resonant wavelength λ = 364.7 nm for this combined system of particles and slit is again redshifted with respect to the individual elements (λ = 359.4 nm).

The field transmission through the particle chains is accomplished by both the propagating waves surrounding the set and a field hopping process between neighbor particles. An appropriate choice of set parameters and illumination allows one to select the particle of the set with the most enhanced plasmon intensity on its surface.

Chains of smaller size nanocylinders at distance dcc from each other, show dipolar plasmon coupling between neighbors [see |H(r)| in Figs. 7], both near the exit plane of the slit and at a larger distance from it. A detail of the transmission in the first three cylinders of such a chain can be seen in Figs. 7, where the energy flow and electric field are shown. The latter quantity exhibits on the surface of each cylinder a typical dipolar plasmon distribution [see Fig. 7]. This transmitted light through the particles does not take place by tunneling, but by a hopping mechanism, as in dielectric molecular photonic crystal rows,10 due to a

dcc2
dipolar interaction (where dcc is the distance between cylinder surfaces) with frequency splitting of the single metallic cylinder spectral line and a redshift of its extinction peak. This coupling diminishes the energy of the ensemble in the bonding state for this configuration (parallel dipoles).2

Grahic Jump LocationF7 :

(a) Magnetic field magnitude |H(r)| distribution for five Ag cylinders (radius r = 30 nm, refractive index n = 0.173 + i1.95) of a linear chain, (distance between cylinder surfaces dcc = 100 nm), in front of a slit in a W slab, (slit width d = 39.59 nm, slab width D = 2610 nm, slab thickness h = 237.55 nm, refractive index n = 3.39 + i2.41) illuminated at λ = 400 nm (p-polarization). The vertical distance between the bottom cylinder surface and the exit plane of the slit is λ0/8 (λ0 = 346 nm is a wavelength near the TE11 of an isolated cylinder); (b) This is same configuration as in Fig. 7, changing the distance from the slit to 3λ0/2 (λ0 = 346 nm), illuminated at λ = 349.3 nm (p-polarization), (c) Detail of the time-averaged energy flow ⟨S(r)⟩ in the first three cylinders of the chain for the arrangement of Fig. 7, both the magnitude (colors) and directions are shown [maximum arrow length ≈1.75 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2s)]. (d) Detail of the electric field E(r) in the first three cylinders of the chain for the case of Fig. 7, both the magnitude in volts per meter (online version: colors, printed version: gray levels) and directions (arrows) are shown.

The field confinement along the chain line and between cylinders in the case in which they are close to the slit [cf. Figs. 7], contrasts with the large diffraction occurring when their distance to the slab is large [cf. Fig. 7], this latter case also showing a strong standing wave below the chain. The energy flow |⟨S⟩| in the area immediately outside the slit exit is reduced with respect to the case of the slit alone in the case of Fig. 7 [compare for Fig. 7 192.48 eV/(nm2 s) against 1536.35 eV/(nm2 s) for the same slit alone]; however, it is enhanced in the case of Fig. 7 (24674.90 eV/(nm2 s) compared to 6531.39 eV/(nm2s) for the same slit alone. These numbers were obtained by averaging in a suitable rectangular monitor at the slit exit. Nevertheless, both linear configurations render a good response regarding to the averaged energy flow concentrated around the cylinders [see the corresponding low reflected energy in Fig. 7]. The linear chain near the slit of Fig. 7 achieves an averaged |⟨S⟩| of 1100 eV/(nm2 s) against 188.87 eV/(nm2 s) in the same area when the slit transmits without the cylinders.

Other distributions considered include those of bifurcations with elbows of nanoparticles at subwavelength distance dcc from each other, as illustrated in Fig. 8, which shows (|H(r)| in color and ⟨S(r)⟩ in arrows. Now the dipolar interaction is of order

dcc3
because it takes place in the near field. The enhancement of transmitted light on top of the set is now quite sharp as the cylinders approach each other (not shown). This set presents the bottom cylinder placed at a distance from the slit that optimizes the energy concentration both on the top particles and at the aperture exit.

Grahic Jump LocationF8 :

Magnetic field magnitude |H(r)| in amperes per meter (online version: colors, printed version: gray levels), and time-averaged energy flow ⟨S(r)⟩ (maximum arrow length ≈1.52 105 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)) concentrated on the surfaces of seven Ag cylinders (radius r = 30 nm, refractive index n = 0.186 + i1.61) in bifurcation (bifurcation angle θ1 = 45 deg, distance between cylinder surfaces dcc = 0 nm), with elbow (elbow angle θ2 = 45 deg), in front of a slit in a W slab (slit width d = 39.59 nm, slab width D = 2610 nm, slab thickness h = 237.55 nm, refractive index n = 3.39 + i2.66), illuminated at λ = 364.7 nm (p-polarization), which is near the TE11 of each cylinder. The vertical distance between the bottom cylinder surface and the exit plane of the slit is dlc = λ0 = 346 nm, near the TE11 of the isolated cylinder.

Most light passes through the aperture and reaches the cylinder set, although the radiation pattern reveals coupling between cylinders and slab. Note the low value of |H(r)| in the reflection region below the slab in Fig. 8. The standing wave between the slab and the cylinders now embraces the set in one of the maxima and makes it to strongly emit upward along three main directions, hence, exhibiting a nanoantenna-like behavior. (Radiation directions can be controlled by changing the configuration at the chosen wavelength.)

Again, this effect is more pronounced with the cylinders at a certain distance from the slit, as shown in Fig. 8, than when they are very close to it. In either case, however, this intensity, strongly transmits even into the aperture exit [42216.07 eV/(nm2 s) for Fig. 8 against 2783.01 eV/(nm2 s) for the same slit alone], these values obtained by an average over the area of a suitable rectangular monitor at the slit exit, although most energy is now transmitted to the cylinders and associated with a very small reflection below the slab. This process of transmission up to the top of the set is much more efficient than that of transmission by the excitation of the morphological resonance of the slit alone, which was shown in Fig. 4.

A Metallic Photonic Crystal in Front of an Array of Slits

Aperture supertransmission is further enhanced by periodically repeating it in the slab.7,30 Accordingly, we next arrange chains of metallic particles, each placed in front of a slit of such an array. Introducing some distance between cylinders in each chain, we build in this form a metallic PC.

Figures 9, corresponding to the PC alone and to the PC in front of the slit array, respectively, show the first step of our approach. At the chosen illumination wavelength λ = 400 nm, the slit grating is supertransmitting and does the array because this wavelength is both close to a Rayleigh anomaly of the slit array and near the TE11 of the PC cylinders.

Grahic Jump LocationF9 :

(a) Magnetic field magnitude |H(r)| distribution in a photonic crystal (horizontal period ax = 275.50 nm, vertical period ay = 75 nm) formed with 68 Ag cylinders (radius r = 30 nm, refractive index n = 0.173 + i1.95) and illuminated at λ = 400 nm (p-polarization, the TE11 excited is near that of the isolated particle which would appear at λ ≈ 346 nm); (b) Magnetic field magnitude |H(r)| distribution in the Ag photonic crystal of Fig. 9, this time in front of a grating of slits practiced in a W slab (period P = 275.50 nm, slit width d = 39.36 nm, slab width D = 8P, slab thickness h = 94.46 nm, refractive index n = 3.39 + i2.41), illuminated at λ = 400 nm (i.e., near the TE11 of an isolated particle), (p-polarization). The distance between the cylinder surfaces of the first row and the exit plane of the slits is 22.5 nm (fitted to get the best response).

As a result, a strong concentration of field in the vertical rows of the PC appears. This is again due to a dipolar interaction between neighbor particles of the same vertical row and between adjacent rows as discussed in connection with Figs. 7. The enhancement of the field is alternating between particle gaps in each row, at difference with the case of a single chain, shown in Figs. 7. The qualitative aspect of the distribution of Figs. 9 is similar, except for the collimation effect produced by the slit grating and the slight weakening of the energy enhancement between particles due to intensity concentration within the slits, shown in Fig. 9. These results in Fig. 9 are in contrast to those obtained when the incident wavelength is out of resonance of the cylinders in this latter case, light passes through the PC suffering very little interaction with the cylinders (this is not shown here for the sake of brevity).

In the second step of our analysis, the distance between cylinders in each vertical chain is incremented to a value comparable to that of the horizontal distance between chains. Figures 10 correspond to the responses of the grating alone, this new PC alone and the combination of both, respectively. Figure 10 shows an effective bandgap in the XM direction of transmission in PC reciprocal space [upward direction in Figs. 9 and Figs. 11]. Nevertheless, an enhancement of transmission along this direction is achieved in the whole range studied [compare the values of Fig. 10 to those of Fig. 10]. Furthermore, a transmission peak rises for the PC/grating arrangement, in the range where the gap was for the PC alone [see Figs. 10], the two last effects being due to the presence of the grating. Note the match in wavelength between the transmission peak of the grating alone in Fig. 10 and that mentioned above for the combination of PC and grating in Fig. 10.

Grahic Jump LocationF10 :

(a) Magnitude |⟨S⟩| in electron volts per nanometer squared per second versus wavelength λ transmitted by the W array of slits alone of Fig. 9. (b) Average energy flow magnitude |⟨S⟩| units vs. wavelength passing through the PC of Ag cylinders of radius r = 30 nm, [horizontal period ax = 275.50 nm, vertical period ay = 160 nm, see also Figs. 11]. (c) Magnitude (|⟨S⟩|) versus wavelength for the same PC, now in front of the W grating of Fig. 10. The square and circle curves (black and red in the online version) stand for the magnitude of energy flow averaged over each square (120 × 120 nm2) circumscribed to each cylinder section and over each rectangular strip (120 × 920 nm2) circumscribed to each PC vertical row, respectively. In the case of Fig. 10, these circles are imaginary and coincide with the cylinders sections of Figs. 10 [see also Figs. 11].

Grahic Jump LocationF11 :

(a) Map of the magnetic field magnitude |H(r)| for the Ag cylinder photonic crystal, studied in Fig. 10, placed in front of the exit of the W array of Figs. 910, illuminated at λ = 450 nm [see Fig. 10]. The distance between the first horizontal row lower edges and the slab is 22.5 nm. (b) Detail of the time-averaged energy flow S(r) [showing both its magnitude (online version: colors, printed version: gray levels) and directions (arrows), maximum arrow length ≈11.23 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)]. (c) Detail of the electric field E(r), both its magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown.

By contrast to Fig. 9, when the slit grating is added to this PC and illumination is out of both the plasmon resonance and the PC effective gap [see Fig. 10], one again observes an enhancement of transmission into the PC, a strong intensity concentration in the slits, and a relatively low reflection below the slab [as example, that shown in Figs. 11], even though now this reflection is larger, relative to the transmittivity of the slits, than that shown for resonantly excited nanoparticle sets in front of one slit [compare to Figs. 78]. On the other hand, the presence of the slits modifies the gap of the PC which, as shown in Fig. 10, has a transmission peak near λ = 360 nm. This is a new feature of the effect of the slits on the transmittivity of the PC.

We have presented a simulation study that shows how the excitation of plasmons of metallic nanoparticles at the exit of subwavelength apertures (Ag cylinders and W slabs have been used in this work), under polarization that excites the slit propagating modes, (p-polarization in 2D), may enhance the transmission. Supertransmission is not essential for this effect, providing the particle-excited resonance couples with the wave transmitted by the slit. However, such enhancement also occurs and is quite interesting, when the slit alone already yields supertransmission through the excitation of its MDRs. Even though the presence of the slab slightly redshifts the particle plasmon resonance and reciprocally does the cylinder with respect to the aperture MDR associated to its supertransmission.

When more than one cylinder is placed in front of the aperture, the light extracted by this set of particles through the slit can couple and propagate in these particles when one does an appropriate design of their geometrical configuration. There is a specific distance for each of such cylinder–slit configuration that optimizes the transmitted energy passing into the nanoparticles. This suggests a control of light transmission via plasmon excitation of metallic particles. Also, we observe that it is possible to fitting nanoparticle set parameters and illumination in such a way that the transmitted intensity is concentrated in certain cylinders when the stationary regime of propagation is reached.

The case of a metallic photonic crystal (e.g., an Ag PC), in front of a metallic slit array shows the effects of effective bandgaps on the interparticle plasmon transmission and the enhancement of the transmittivity of the grating due to the excitation of plasmons in the metallic PC.

All these results are also expected with 3-D particles at the exit of apertures with any geometry, in particular, the latter being subwavelength holes, and have a potential for controlling transmitted near fields at the nanoscale. We hope that they will stimulate experiments both in 2D and 3D. The nanoparticle, or the sets of nanoparticles, with excited plasmons may constitute efficient coupling devices to extract highly localized energy through nanoapertures and transport it in, e.g., chains of these particles, or in other sorts of nanowaveguides (or even filtering such transmitted intensities if a PC configuration is chosen). Alternatively, this supertransmitted energy may be directly delivered into free space or transported elswhere through another coupling device.

Our research is supported by the Spanish MiCINN through Grant No. UNSPECIFIED FIS2009-13430-C02-C01 and Consolider NanoLight (Grant No. UNSPECIFIED CSD2007-00046) research contracts. The latter grant supports the work of FJVV.

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Lezec  H. J., and Thio  T., “ Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelenght hole arrays. ,” Opt. Express. 12, , 3629–3651  ((2004)).
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Qiu  M., and He  S., “ Guided modes in a two-dimensional metallic photonic crystal wave-guide. ,” Phys. Lett. A. 266, , 425–429  ((2000)).
Zhao  Y., and Grischkowsky  D. R., “ 2-D terahertz metallic photonic crystals in parallel-plate waveguides. ,” IEEE Trans. Microwave Theory Tech.. 55, , 656–663  ((2007)).
Wang  S., , Lu  W., , Chen  X., , Li  Z., , Shen  X., , and Wen  W., “ Two-dimensional photonic crystal at THz frequencies constructed by metal-coated cylinders. ,” J. Appl. Phys.. 93, , 9401–9403  ((2003)).
Baida  F. I., , van Labeke  D., , Pagani  Y., , Guizal  B., , and Al Naboulsi  M., “ Waveguiding through a two-dimensional metallic photonic crystal. ,” J. Microsc.. 213, , 144–148  ((2004)).
Christ  A., , Tikhodeev  S. G., , Gippius  N. A., , Kuhl  J., , and Giessen  H., “ Plasmon polaritons in a metallic photonic crystal slab. ,” Phys. Stat. Sol. (c). 0, (5 ), 1393–1396  ((2003)).
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Hossain  Md. M., , Chen  G., , Jia  B., , Wang  X., , and Gu  M., “ Optimization of enhanced absorption in 3D-woodpile metallic photonic crystals. ,” Opt. Express. 18, , 9048–9054  ((2010)).
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Biographies and photographs of the authors not available.

© 2011 Society of Photo-Optical Instrumentation Engineers (SPIE)

Citation

Francisco J. Valdivia-Valero and Manuel Nieto-Vesperinas
"Propagation of particle plasmons in sets of metallic nanocylinders at the exit of subwavelength slits", J. Nanophoton. 5(1), 053520 (August 11, 2011). ; http://dx.doi.org/10.1117/1.3615984


Figures

Grahic Jump LocationF1 :

Schematic of the geometry of the calculations: An incident p-polarized plane wave with magnetic vector Hz and Poynting vector Sy impinges a W slab of width D and thickness h, containing and aperture of width d. The transmitted intensity is evaluated at a rectangular monitor ax × ay. However, when a cylinder of radius r is placed, the transmitted intensity is evaluated at an annulus of exterior radius re = (9/8)r.

Grahic Jump LocationF2 :

Variation of the averaged energy flow |⟨S(r)⟩| (in units of electron volts per nanometer squared, per second), transmitted by the slit in the W slab (n = 3.39 + i2.66; λ = 364.7 nm) versus its width, illuminated by a p-polarized plane wave near its TE51 MDR. Each curve with symbols corresponds to a different slab thickness. The highest peak (out of the scale this graph) occurs at the slit width d = 55 nm, thickness h = 259.4 nm, and has a value of its ordinate equal to 3.5 × 105 eV/(nm2 s). Averages have been performed over the rectangular monitor at the slit exit (see Fig. 1), of area: S = 1.76 d (nm) × 54.09 nm = 95.09 d (nm2) at the slit exit.

Grahic Jump LocationF3 :

(a) Magnetic field magnitude |H(r)| (in amperes per meter), in a W slab aperture (refractive index n = 3.39 + i2.66, slab width D = 2850 nm, slab thickness h = 259.40 nm, slit width d = 55 nm). The incident radiation at (λ = 364.7 nm) is p-polarized, of unit amplitude, and impinges on the slab from below. (b) Time-average energy flow ⟨S(r)⟩ (in Joules per meters square per second), maximum arrow length ≈1.05 MeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)), both magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown under the same conditions as in Fig. 3. (c) Detail of the saddle point evidenced by ⟨S(r)⟩ inside the slit (the arrows here appear normalized to their magnitude which is shown according to the color bar). (d) Detail of the electric field E(r) (in volts per meter), both magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown under the same conditions as in Fig. 3.

Grahic Jump LocationF4 :

(a) Magnetic field magnitude |H(r)| (colors in amperes per meter) and time-averaged energy flow ⟨S(r)⟩ [arrows in Joules per meters squared per second), maximum arrow length ≈80.02 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)), localized on the surface of an Ag cylinder (radius r = 200 nm, refractive index n = 0.186 + i1.61) in front of a slit in a W slab at the same illumination as in Fig. 3. The distance between the cylinder surface and the exit plane of the slit is dlc = 3λ/2 = 549 nm (λ = 366 nm); (b) Variation of the concentration of |⟨S(r)⟩| (in electron volts per nanometer squared per second), on the cylinder surface versus its vertical separation (in nanometers) from the slit exit, illuminated at λ = 366 nm (near the plasmon TE51 resonance). (c) The same quantity versus wavelength at the distance seen in Fig. 4, dlc = 549 nm. The curves with squares and upward triangles stand for the response of the slab alone, and those with circles and downward triangles correspond to the slab with cylinder, respectively. The ordinate of the curves with both upward and downward triangles are plotted with a reduction factor of 6.24. (d) Electric field E(r) (in volts per meter); both its magnitude (online version: colors, printed version: gray levels) and directions (arrows), under the same conditions as in Fig. 4. Black and red curves in Figs. 4 have been calculated by averaging the quantity in an annulus of area A = π{[(9/8)r]2r2} [see the two concentric circles in Fig. 4, which are drawn according to the scheme of Fig. 1], whose internal circle coincides with either the cylinder section [case of the slit with the particle in Figs. 4] or an imaginary circle coincident with that cylinder section [case of the slit alone in Fig. 4]. The green and blue curves in Fig. 4 have been calculated by averaging the quantity in a rectangular monitor of area S = 76 nm × 58.46 nm = 4442.96 nm2 at the slit exit [see the rectangle drawn in Fig. 1].

Grahic Jump LocationF5 :

(a) Variation of the magnetic field concentration |H(r)| on the cylinder surface versus its horizontal separation from the slit exit. The particle is illuminated at λ = 366 nm (i.e., near the nanocylinder TE51 resonance). The black, red, green, and blue curves stand for vertical distances between the particle surface and the slab equal to λ/8, λ/4, 3λ/8, and λ/2, respectively. (b) Map of |H(r)| (online version: colors, printed version: gray levels) and ⟨S(r)⟩ (arrows) when the cylinder is at a vertical and horizontal distance from the slit exit: λ/8 and 730 nm, respectively, and illuminated at λ = 364.7 nm (p-polarization, near the TE51). Maximum arrow length ≈26.09 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s). Calculations in Fig. 5 have been made as in Figs. 4.

Grahic Jump LocationF6 :

(a) Magnetic field magnitude |H(r)| spatial distribution for five Ag cylinders (radius r = 200 nm, refractive index n = 0.186 + i1.610), disposed in a bifurcated chain (bifurcation angle θ = 45 deg, distance between cylinder surfaces dcc = 100 nm), in front of the slit of Figs. 3 illuminated at λ = 364.7 nm (p-polarization, near the TE51 resonance). The vertical distance between the bottom cylinder surface and the exit plane of the slit is λ00 = 366 nm near the TE51 resonance of an isolated cylinder). (b) Time-averaged energy flow ⟨S(r)⟩ [maximum arrow length ≈15.60 KeV/(nm2s), minimum arrow length ≈0 eV/(nm2 s)] in the same configuration as in Fig. 6. (c) Electric field E(r) distribution, both the magnitude in volts per meter (online version: colors, printed version: gray levels) and directions (arrows), are displayed in the same configuration as in Fig. 6.

Grahic Jump LocationF7 :

(a) Magnetic field magnitude |H(r)| distribution for five Ag cylinders (radius r = 30 nm, refractive index n = 0.173 + i1.95) of a linear chain, (distance between cylinder surfaces dcc = 100 nm), in front of a slit in a W slab, (slit width d = 39.59 nm, slab width D = 2610 nm, slab thickness h = 237.55 nm, refractive index n = 3.39 + i2.41) illuminated at λ = 400 nm (p-polarization). The vertical distance between the bottom cylinder surface and the exit plane of the slit is λ0/8 (λ0 = 346 nm is a wavelength near the TE11 of an isolated cylinder); (b) This is same configuration as in Fig. 7, changing the distance from the slit to 3λ0/2 (λ0 = 346 nm), illuminated at λ = 349.3 nm (p-polarization), (c) Detail of the time-averaged energy flow ⟨S(r)⟩ in the first three cylinders of the chain for the arrangement of Fig. 7, both the magnitude (colors) and directions are shown [maximum arrow length ≈1.75 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2s)]. (d) Detail of the electric field E(r) in the first three cylinders of the chain for the case of Fig. 7, both the magnitude in volts per meter (online version: colors, printed version: gray levels) and directions (arrows) are shown.

Grahic Jump LocationF8 :

Magnetic field magnitude |H(r)| in amperes per meter (online version: colors, printed version: gray levels), and time-averaged energy flow ⟨S(r)⟩ (maximum arrow length ≈1.52 105 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)) concentrated on the surfaces of seven Ag cylinders (radius r = 30 nm, refractive index n = 0.186 + i1.61) in bifurcation (bifurcation angle θ1 = 45 deg, distance between cylinder surfaces dcc = 0 nm), with elbow (elbow angle θ2 = 45 deg), in front of a slit in a W slab (slit width d = 39.59 nm, slab width D = 2610 nm, slab thickness h = 237.55 nm, refractive index n = 3.39 + i2.66), illuminated at λ = 364.7 nm (p-polarization), which is near the TE11 of each cylinder. The vertical distance between the bottom cylinder surface and the exit plane of the slit is dlc = λ0 = 346 nm, near the TE11 of the isolated cylinder.

Grahic Jump LocationF9 :

(a) Magnetic field magnitude |H(r)| distribution in a photonic crystal (horizontal period ax = 275.50 nm, vertical period ay = 75 nm) formed with 68 Ag cylinders (radius r = 30 nm, refractive index n = 0.173 + i1.95) and illuminated at λ = 400 nm (p-polarization, the TE11 excited is near that of the isolated particle which would appear at λ ≈ 346 nm); (b) Magnetic field magnitude |H(r)| distribution in the Ag photonic crystal of Fig. 9, this time in front of a grating of slits practiced in a W slab (period P = 275.50 nm, slit width d = 39.36 nm, slab width D = 8P, slab thickness h = 94.46 nm, refractive index n = 3.39 + i2.41), illuminated at λ = 400 nm (i.e., near the TE11 of an isolated particle), (p-polarization). The distance between the cylinder surfaces of the first row and the exit plane of the slits is 22.5 nm (fitted to get the best response).

Grahic Jump LocationF10 :

(a) Magnitude |⟨S⟩| in electron volts per nanometer squared per second versus wavelength λ transmitted by the W array of slits alone of Fig. 9. (b) Average energy flow magnitude |⟨S⟩| units vs. wavelength passing through the PC of Ag cylinders of radius r = 30 nm, [horizontal period ax = 275.50 nm, vertical period ay = 160 nm, see also Figs. 11]. (c) Magnitude (|⟨S⟩|) versus wavelength for the same PC, now in front of the W grating of Fig. 10. The square and circle curves (black and red in the online version) stand for the magnitude of energy flow averaged over each square (120 × 120 nm2) circumscribed to each cylinder section and over each rectangular strip (120 × 920 nm2) circumscribed to each PC vertical row, respectively. In the case of Fig. 10, these circles are imaginary and coincide with the cylinders sections of Figs. 10 [see also Figs. 11].

Grahic Jump LocationF11 :

(a) Map of the magnetic field magnitude |H(r)| for the Ag cylinder photonic crystal, studied in Fig. 10, placed in front of the exit of the W array of Figs. 910, illuminated at λ = 450 nm [see Fig. 10]. The distance between the first horizontal row lower edges and the slab is 22.5 nm. (b) Detail of the time-averaged energy flow S(r) [showing both its magnitude (online version: colors, printed version: gray levels) and directions (arrows), maximum arrow length ≈11.23 KeV/(nm2 s), minimum arrow length ≈0 eV/(nm2 s)]. (c) Detail of the electric field E(r), both its magnitude (online version: colors, printed version: gray levels) and directions (arrows) are shown.

Tables

References

Pelton  M., , Aizpurua  J., , and Bryant  G., “ Metal-nanoparticle plasmonics. ,” Laser Photon. Rev.. 2, , 136–59  ((2008)).
Maier  S. A.,  Plasmonics: Fundamentals and Applications. ,  Springer Science + Business Media LLC ,  New York  ((2007)).
Maier  S. A., and Atwater  H. A., “ Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures. ,” J. Appl. Phys.. 98, , 011101  ((2005)).
Hiremath  K. R., and Astratov  V. N., “ Perturbation of hispering gallery modes by nanoparticles embedded in microcavities. ,” Opt. Express. 16, , 5421–5426  ((2008)).
Elston  S., “ Finite-difference time-domain simulations of metallic nanoparticles in whispering gallery mode resonators. ,” PhD Thesis, Brown University ((2009)).
Mazzei  A., , Gotzinger  S., , Menezes  L. de S., , Zumofen  G., , Benson  O., , and Sandoghdar  V., “ Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light. ,” Phys. Rev. Lett.. 99, , 173603  ((2007)).
Lezec  H. J., and Thio  T., “ Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelenght hole arrays. ,” Opt. Express. 12, , 3629–3651  ((2004)).
Lidorikis  E., , Sigalas  M. M., , Economou  E. N., , and Soukoulis  C. M., “ Tight-binding parametrization for photonic band gap materials. ,” Phys. Rev. Lett.. 81, , 1405–1408  ((1998)).
Moroz  A., and Tip  A., “ Resonance-induced effects in photonic crystals. ,” J. Phys.: Cond. Matter. 11, , 2503–2512  ((1999)).
Vanderbem  C., and Vigneron  J. P., “ Mie resonances of dielectric spheres in face-centered cubic photonic crystals. ,” Opt. Soc. Am. A. 22, , 1042–1047  ((2005)).
El-Kady  I., , Sigalas  M. M., , Biswas  R., , Ho  K. M., , and Soukoulis  C. M., “ Metallic photonic crystals at optical wavelengths. ,” Phys. Rev. B. 62, , 15299–15302  ((2000)).
Qiu  M., and He  S., “ Guided modes in a two-dimensional metallic photonic crystal wave-guide. ,” Phys. Lett. A. 266, , 425–429  ((2000)).
Zhao  Y., and Grischkowsky  D. R., “ 2-D terahertz metallic photonic crystals in parallel-plate waveguides. ,” IEEE Trans. Microwave Theory Tech.. 55, , 656–663  ((2007)).
Wang  S., , Lu  W., , Chen  X., , Li  Z., , Shen  X., , and Wen  W., “ Two-dimensional photonic crystal at THz frequencies constructed by metal-coated cylinders. ,” J. Appl. Phys.. 93, , 9401–9403  ((2003)).
Baida  F. I., , van Labeke  D., , Pagani  Y., , Guizal  B., , and Al Naboulsi  M., “ Waveguiding through a two-dimensional metallic photonic crystal. ,” J. Microsc.. 213, , 144–148  ((2004)).
Christ  A., , Tikhodeev  S. G., , Gippius  N. A., , Kuhl  J., , and Giessen  H., “ Plasmon polaritons in a metallic photonic crystal slab. ,” Phys. Stat. Sol. (c). 0, (5 ), 1393–1396  ((2003)).
Veronis  G., , Dutton  R. W., , and Fan  S., “ Metallic photonic crystals with strong broadband absorption at optical frequencies over wide angular range. ,” J. Appl. Phys.. 97, , 093104  ((2005)).
Fleming  J. G., , Lin  S. Y., , El-Kady  I., , Biswas  R., , and Ho  K. M., “ All-metallic three-dimensional photonic crystals with a large infrared bandgap. ,” Nature (London). 417, , 52–55  ((2002)).
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