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We investigate the optical behaviors of metallic nanoparticle (MNP) chains supporting localized surface plasmon (LSP) for different distances between particles. MNPs are excited through the fundamental TE mode of a silicon waveguide. Finite difference time domain (FDTD) calculations and optical power transmission measurements reveal three different behaviors. For short distances between particles, dipolar coupling occurs, and the MNP chain behaves as a waveguide. For the longest distances, nanoparticles are uncoupled, and the MNP chain acts as a LSP Bragg grating. Finally, for intermediate distances, we observe one behavior at a time, i.e. dipolar coupling or LSP Bragg reflection. There is only a small range of wavelengths within which both behaviors can coexist.
© 2013 Optical Society of America
1. Introduction
Metallic nanostructures are of great interest in photonics since they can provide subwavelength confinement of light through the excitation of surface plasmon-polaritons (SPP) or localized surface plasmons (LSP). Very compact periodic structures have been proposed such as metallic nanoparticle chains [1–8], photonic crystals [9, 10] metallic gratings [11–14]. It has recently been shown that metallic nanoparticle (MNP) periodic chains integrated on a dielectric (silicon) waveguide could be used either as individual waveguides with a true capacity of transmitting light by dipolar coupling between particles [15, 16] or as resonant metallic gratings operated either in the refractive index modulation regime or in the loss modulation regime [17, 18]. For given gold MNP sizes and shapes allowing LSP resonance at 1.55µm, typical periods were d~150 nm to obtain a collective plasmonic oscillation of particles [15], while they were larger than 1µm to strongly attenuate coupling between LSP resonators and achieve a Bragg grating behavior of high order at 1.55µm [18]. However, it is possible to dimension MNP chains so as to simultaneously favor near-field dipolar coupling between MNPs and first order Bragg grating effects. For instance, using a period of 300 nm, conditions are fulfilled for each of the two behaviors of the MNP chain at 1.55µm.
The aim of this paper is to evaluate the coexistence of both effects (collective plasmon oscillations and Bragg grating effects): does one dominate? Can they combine each other to give a new behavior? In what follows, we first separately investigate configurations where only one effect is dominant. Then, we focus on a configuration where both effects are present, i.e. a first order Bragg grating supporting collective LSP oscillations. Experimental investigations are carried out from transmission measurements on the different waveguide devices fabricated in this work. Theoretical investigations are based on the Fourier analysis of propagative modes.
2. Description of devices and simulation methods
The basic structure consists of a dielectric waveguide loaded by MNP chain. This allows an easy comparison between all configurations by analyzing the properties of the guided light.
Fabricated devices are made of a TE monomode silicon (SOI) waveguide of 500 nm × 220 nm cross section, loaded with gold ellipsoids whose long axis and small axis are D1≈210 nm and D2≈80 nm, respectively [Fig. 1]. The thickness of gold is fixed at 30 nm while the center-to-center distance between particles (or the chain period), d, depends on the chosen configuration. The device fabrication was made in two steps. Firstly, the silicon waveguide was fabricated by using deep UV lithography followed by an etching process. Secondly, gold nanoparticles were realized by using e-beam lithography and a lift off process after deposition of a 1 nm titanium adhesion layer and a 30 nm gold layer by e-beam evaporation.
Fig. 1 (a) Schematic view of the investigated structure. (b) Picture showing the ellipsoidal shape of gold nanoparticles with D1 (respectively D2) the long (respectively small) axis size and d the center-to-center distance between particles.
All the transmission spectra of the MNP-loaded SOI waveguides were measured by injecting a wavelength-tunable TE polarized light at the waveguide entrance. A lensed polarization maintaining fiber was used to couple the laser TE light to the entrance facet of the SOI waveguide. A reference waveguide without MNP was used on the same chip for transmission normalization. The input light was delivered by a tunable laser scanned by steps of 1nm over the 1260-1630 nm range. The light at the sample output was collected by an objective and was focused onto a power meter.
The simulations in this paper were made by a 3D FDTD method (Lumerical FDTD Solutions) with a uniform mesh of 3 nm × 3 nm × 3 nm. Accurate dispersion data were introduced for deposited gold after fitting a Drude model to experimental ellipsometric measurements. The presence of a thin layer of native oxide between Si and gold was also accounted for in calculations.
In a previous work [15], coupling mechanisms between the MNP chain and the silicon waveguide have been clearly highlighted in the case where the two structures form a coupled waveguide system. The behavior of this coupled system is characterized by (i) a deep transmission minimum with low reflections, (ii) a quasi-total and periodic energy transfer from one guide to the other, and (iii) finally an anticrossing in the dispersion relation diagram (coupled modes).
3. Coupled vs uncoupled particle systems (10 particles)
Let us first compare the case of 10 particles separated by d = 150nm to the one with 10 particles separated by 1µm. Figures 2(a) and 2(b) show SEM pictures of the two fabricated structures. The calculated and measured transmission spectra are depicted for each structure in Figs. 2(c) and 2(d). In both cases the agreement obtained between simulations and experiments is very good.
Fig. 2 Top: scanning electron microscope image of gold nanoparticles deposited on top of SOI waveguide with d = 150 nm (a) and d = 1µm (b). Middle: normalized transmission spectra of the SOI waveguide with gold nanoparticles deposited on top for d = 150 nm (c) and d = 1µm (d), respectively. Blue lines are for measurements, blue dash curves are for FDTD calculations. Bottom: calculated re〉ection (red) and transmission (blue) spectra for d = 150 nm (e) and d = 1µm (f), respectively.
As it is seen, transmission curves are radically different for the two cases. For d = 150 nm, the transmission curve exhibits a dip between 1250 and 1400 nm, which is characteristic of the interaction between the TE waveguide mode and the MNP chain at LSP resonance [16]. The whole system behaves as a two-coupled-waveguide system. For d = 1µm, the transmission curve presents an abrupt transition near 1300 nm followed by a slow decrease, a peak around 1550 nm and finally a dip at the longest wavelengths. This different behavior stems from the fact that the system is operated as a single Si waveguide with a Bragg grating formed on top by the periodic MNP chain [18].
The reflection curves also display different shapes for the two structures [Figs. 2(e) and 2(f)]. For d = 150 nm, wide oscillations at low level are caused by the finite size of the MNP chain. For d = 1µm, two reflections peaks due to the MNP based Bragg grating are clearly visible at 1335 nm and 1572 nm, respectively. By using the simple formula for Bragg gratings: mλ = 2neffd, where m is the order of the Bragg resonance, λ the wavelength in free space, and neff the effective index of the silicon waveguide mode, we can identify the reflection peaks as the fourth order (1350 nm) and third order (1572 nm) of the Bragg resonance, respectively. Because of the high orders of the Bragg grating, the incident light is in fact re-directed into several directions, among which is the backward (reflection) direction. The LSP Bragg grating behavior is indeed confirmed by the dip in the transmission curve around 1550 nm as described previously [15]. The small wavelength shift between simulations and measurements in Fig. 2(d) is readily explained by the slightly different grating periods in the simulated (1 µm) and fabricated (997 nm) structures [Fig. 2(b)].
Let us now show the electric field distributions calculated for the two configurations [Figs. 3(a)-3(d) and Figs. 4(a)-4(d)]. When d = 150 nm [Fig. 3(a)-3(d)], three different behaviors can be observed. At 1250 nm, the light propagates in the Si waveguide, but excites individual MNPs without the occurrence of collective oscillations. There is no waveguide coupling behavior. The guided light follows an exponential decrease along the guide. At 1321 nm, the chain starts behaving like a waveguide, but the energy transfer between the two waveguides is not complete (partial phase matching). The energy transfer becomes more efficient at 1370 and 1450 nm (better phase matching). The electric field distribution is radically different when the spacing between particles is 1 µm [Figs. 4(a)-4(d)]. At 1337 nm (first reflection peak in Fig. 2(f)), four maxima can be seen between particles. This confirms that the Bragg grating is operated in its fourth order. At the second reflection peak (1572 nm), three maxima are now observed between particles. The Bragg grating is working in the third order.
Fig. 3 Maps of the electric field (|Ey|) calculated in the vertical symmetry plane of the structure with d = 150 nm for four wavelengths: 1250 nm (a), 1321 nm (b), 1370 nm (c) and 1450 nm (d).
Fig. 4 Maps of the electric field (|Ey|) calculated in the vertical symmetry plane of the structure with d = 1µm for four wavelengths: 1337 nm (a), 1500 nm (b), 1572 nm (c) and 1600 nm (d).
Let us finally focus on calculated dispersion relations. For d = 150 nm [Fig. 5(a)], the dispersion diagram is limited to the first Brillouin zone which shows an anti-crossing associated to the mode coupling between the MNP chain and the Si waveguide. This is in agreement with results reported in [15]. For d = 1µm, the diagram is presented in two figures, Figs. 5(b) and 5(c), for clarity. Figure 5(b) corresponds to wavelengths from 1250 nm to 1450 nm (i.e. for frequencies from 1.51 × 1015 rad/s to 1.3 × 1015 rad/s), while Fig. 5(c) is for wavelengths from 1450 to 1650 nm (i.e. for frequencies from 1.3 × 1015 rad/s to 1.14 × 1015 rad/s). The Fourier transform of the electrical field along the 1µm period chain is calculated for the first five Brillouin zones so as to account for the contribution of the fourth order of the Bragg grating [10, 11]. In both Figs. 5(b) and 5(c), the fundamental component of the guided Bloch mode around k//~4π/d appears to be the strongest component.
Fig. 5 Dispersion curves of supermodes extracted from the spatial Fourier transform of the complex electric field (|Ey|) calculated in the mid-plane of the MNP chain along the z axis. Black lines delimit the light cone in air. Black dashed curves are light lines for Si and SiO2. White dashed lines are the dispersion curves of the fundamental TE mode of the Si waveguide. Figure (a) is for d = 150 nm, figures (b) and (c) are for d = 1µm.
As is seen in Fig. 5(b), the stand-alone Si waveguide dispersion relation (white dots) intersects harmonic −4 (H-4) of the Bloch mode in the fourth Brillouin zone near ω = 1.43 × 1015 rad/s (λ = 1321 nm), i.e. for k//~- 4π/d. This induces a portion of negative group velocity in the dispersion curve. In other words, the propagating light is reflected at this wavelength. The diagram also shows that harmonics H-3 and H-1 are between the light lines in air and in silica and are thus leaky in the silica. Besides, another harmonic exist, H-2 above the light line in air. This non-guided harmonic corresponds to a vertical emission (90°) in free space.
For wavelengths in the range from 1450 to 1650 nm [Fig. 5(c)], the dispersion relation of the Si waveguide follows that of the loaded waveguide. In the diagram half plane where k// is negative, the dispersion relation of the stand-alone waveguide intersects harmonic H-3 of the Bloch mode at around ω≈1.2 × 1015 rad/s (λ = 1571 nm). The group velocity becomes negative around this frequency, and the field intensity is more intense, thus confirming the high reflection level calculated in Fig. 2(f). Harmonics H-1 and H-2, which are visible in the diagram, are not guided.
In summary of this section, we have shown the different behaviors of the MNP chain depending whether the distance between particles is small or long, i.e. whether the equivalent dipoles can be near-field coupled or not. If the MNPs are very close to each other (150 nm), the MNP chain is operated as a waveguide, which couples to the Si waveguide. In contrast, the period is too short to produce a Bragg grating effect. This results in a unique dip in the transmission curve, and in an anticrossing of both waveguide modes in the dispersion diagram. When the MNPs are sufficiently distant from each other, the MNP dipoles cannot oscillate in a collective manner, and the chain does not behave as a waveguide. The MNP chain acts only as an LSP Bragg grating for the fundamental TE mode of the Si waveguide, the grating order depending on the distance between particles. This situation results in two relative maxima in the transmission curves and two peaks in the reflection spectra at the grating phase matched wavelengths.
4. Coupled particles at Bragg resonance
Both the guide-like and Bragg grating-like behaviors of the MNP chain can in principle coexist by choosing appropriate geometrical parameters for the structure. For illustration, we now theoretically investigate the case of 20 gold particles spaced by 300 nm. At this distance the dipolar coupling between each particle is possible, while the first order Bragg resonance occurs around 1500 nm. Figure 6 shows the calculated transmission and reflection spectra of such a device. A high reflection level (up to 95%) is found between λ≈1400 nm and λ≈1500 nm. Correspondingly, a wide transmission dip is visible around 1450 nm with transmission levels less than −30 dB, which is the rejection rate of the Bragg grating. This dip is followed by a transmission peak near 1500 nm.
Electric field amplitude maps in Figs. 7(a)-7(d) show four different regimes depending on wavelength. At 1350 nm, the Si waveguide mode excites each individual MNP, but is slightly modified by the weak reflections at the grating end. At the transmission minimum near the Bragg resonance (λ = 1450 nm, ω = 1.30 × 1015 rad/s), the field propagates only over a short distance (about 3µm) in the region with MNPs. At the maximum of the transmission peak (λ = 1495 nm, ω = 1.26 × 1015 rad/s), MNPs are collectively excited. The coupled waveguide regime appears, but phase matching between waveguide modes is not perfect. Reflections still occur at the end of the MNP chain. Finally, at λ = 1550 nm (ω = 1.215 × 1015 rad/s), the chain is fully excited by the SOI waveguide in such a way that all field intensity maxima are now located in the chain. However, because of the strong (exponential) decrease experienced by the field along the chain, re-coupling from the MNP chain to the SOI waveguide cannot be observed.
Fig. 7 Maps of the electric field (|Ey|) calculated in the vertical symmetry plane of the structure for four wavelengths:1350 nm (a), 1450 nm (b), 1495 (c) and 1550 nm (d).
The dispersion relation of the structure calculated in the first Brillouin zone [Fig. 8(a)] clearly shows a photonic band gap between ω ≈1.35 × 1015 rad/s (λ ≈1396 nm) and ω ≈1.275 × 1015 rad/s (λ ≈1478 nm). In the diagram half plane where k// is negative (k// = 2π neff /λ), the dispersion curve is split into two branches for frequencies higher than ω ≈1.35x1015 rad/s (λ ≈1396 nm) and lower than ω ≈1.275x1015 rad/s (λ≈1478 nm). This splitting is not visible in the diagram half plane where k// is positive: the dispersion curve follows that of the stand-alone SOI waveguide fundamental TE mode. However, for frequencies smaller than ω ≈1.275 × 1015 rad/s (λ ≈1478 nm), the slope of the dispersion curve of the Si waveguide loaded with MNPs differs from that of the stand-alone Si waveguide one.
Fig. 8 (a) Dispersion curves of supermodes extracted from the spatial Fourier transform of the complex electric field (|Ey|) calculated in the mid-plane of the MNP chain along the z axis. Black lines delimit the light cone in air. Black dashed curves are light lines for Si and SiO2. White dashed lines are the dispersion curves of the fundamental TE mode of the Si waveguide. (b), (c) and (d): Spatial distributions and effective refractive indexes of supermodes revealed by the spatial Fast Fourier Transform (FFT) of the complex electric field Ey at 1375 nm (ω = 1.37 × 1015 rad/s) (b), at 1495 nm (ω = 1.26 × 1015 rad/s) (c) and at 1550 nm (ω = 1.215 × 1015 rad/s) (d). Bottom figures in (b), (c) and (d) show exploded views of supermode field distributions for positive wavevectors (k// = 2π neff /λ ≥ 0).
For a more precise understanding of the structure behavior, the spatial distribution of the guided light is plotted versus the effective index, thereby giving evidence of individual modes or supermodes involved in the coupled waveguide system. This distribution was retrieved from FDTD simulations by calculating the spatial Fast Fourier Transform (FFT) of the complex electric field EY. Figures 8(b) and 8(c) represent the FFT module (color scale) versus the effective index, neff, and the Z position in the vertical symmetry plane of the structure for λ = 1375 nm (ω ≈1.37 × 1015 rad/s) and λ = 1495 nm (ω ≈1.26 × 1015 rad/s). As revealed by the two-dimensional maps, components with high effective index (≥ 7 and ≤ −7) are located in the MNP chain, and are harmonics of the MNP chain waveguide mode. At 1375 nm (ω ≈1.37 × 1015 rad/s), only one component with an effective index of 2.45 is visible in the MNP chain and Si waveguide for positive k//. This is the fundamental component of the propagating Bloch mode. In contrast, for negative k//, two components are visible with effective indexes equal to −2.49 and −2.1, respectively. These values correspond to the reflected wave and to one harmonic of the Bloch mode, respectively. At λ = 1495 nm (ω ≈1.26 × 1015 rad/s), a weak additional component appears for neff ≈2.6 just aside the main one for neff ≈2.2 (see bottom view in Fig. 8(c). The presence of these two supermodes indicates that waveguide coupling exists at frequencies below the Bragg band gap, while it is combined with reflections due to the Bragg grating. At λ = 1550 nm, the additional component is not clearly visible in the field distribution [Fig. 8(d)]. Its presence manifests itself only as a broad spot aside the main mode. Let us however recall that for this wavelength, the optical energy is fully transferred from the SOI waveguide to the MNP chain Fig. 7(d). Actually, all happens as though the coupling length was longer than the propagation length, which is shortened by optical losses. On the contrary, in the forbidden band, the Bragg effect kills the waveguide coupling regime, despite the fact that collective oscillation of the chain MNPs could occur a priori.
5. Discussion and conclusion
The behaviors of periodic MNP chains fabricated on a dielectric (silicon) waveguide have been investigated for different situations depending whether the MNPs are dipolar coupled or not. In most situations, we have observed only one behavior at a time, i.e. dipolar coupling or LSP Bragg grating reflection. There is only a small range of wavelengths (near the long wavelength boundary of the Bragg band gap) within which both behaviors can coexist. In contrast, we have shown that the field transmission along the MNP chain is totally cancelled for wavelengths within the Bragg band gap. One interest in combining both behaviors is the narrower resonance dip observed in the transmission spectrum as compared to the case where Bragg reflection does not exist. Another interest stems from the relatively high reflectivity of the LSP Bragg grating, thereby leading to a high rejection rate in transmission (>20 dB) and an abrupt transmission dip profile. Such features can be of utility for photonics integrated circuits since they are obtained with a small number of grating periods (here a number of ten). They can also find applications in highly sensitive biosensing.
Acknowledgments
The authors acknowledge Alexeï Chelnokov (CEA Leti) for providing them with SOI waveguides, Abdel Aassime and David Bouville for their help in final sample preparation. They also thank Sylvain Blaize, Rafael Salas Montiel and Aniello Apuzzo for fruitful discussions. This work has been supported by the Agence Nationale de la Recherche under contract PLACIDO N° ANR-08-BLAN-0285-01, and by French RENATECH network. The Mickaël Février grant has been funded by Region Ile-de-France.
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