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We describe theoretical and experimental results on near-field interaction of two-dimensionally (2D) arrayed, high-permittivity spherical particles on a substrate in the Mie resonance scattering domain for surface nano-patterning processing. When a touching particle pair of Mie resonance particles on the substrate is considered, an electromagnetic mode different from the single particle mode is excited inside the particles, resulting in an intensity enhancement in a gap between two hotspots at particle-substrate contact points. As for 2D hexagonal close-packed particle arrays on the substrate, the refractive index of particle exhibiting a maximal enhancement factor for the 2D particle arrays is found to be shifted from the Mie resonance conditions for the single particle system.
©2010 Optical Society of America
1. Introduction
Nanotechnology and advanced laser technology have been merged in various research fields. In recent years, applied researches utilizing enhanced optical near-field mediated with the light scattering by nanostructures have a growing interest [1,2]. Nanofabrication, nano-imaging and high sensitivity molecule identification have been extensively investigated [3–13]. Scattered optical field by nanoparticles was numerically analyzed using Maxwell’s equations and it is known as Mie scattering theory [14]. The theory can be applied for detailed explanation of near-field properties in homogeneous medium [15,16]. However, if the surrounding medium around a particle (scatterer) is not homogeneous (e.g., a system that the particle is placed on substrate), near-field properties are not fully understood with the theory due to the influence of capacitive coupling and multiple scattering [17–19]. In particular, a near-field analysis of a many-particle system is complicated and has not fully been done, although far-field properties such as transmission, reflection and absorption spectrum have been well investigated.
Metallic particle provides a strong near-field enhancement even with small size parameter (in Mie scattering theory) due to the effect of surface plasmon which is collective free electron oscillation. Several researches revealed that near-field properties of a many-particle system are different from those of a single particle system because of plasmon coupling among neighboring particles [20–22]. A particle pair provides approximately four orders of magnitude enhanced near-field intensity near a particle-particle touching point [23,24]. A touching particle pair and one-dimensional particle chain can behave as a single nanowire, causing a red-shift of peak resonant wavelength [25,26]. When placing the multiple particles on substrate, near-field enhancement on the substrate is governed by two competitive plasmon couplings between particle and particle, and between particle and substrate. As for a hexagonal close-packed metallic particle array on substrate fabricated by a wet self-assembling chemical process, the enhancement factor on substrate beneath the particles is much smaller than that of a single isolated particle and hotspots which has strongly enhanced intensity are observed not in the center of the 2D particle array but in the edge area of the 2D array [27]. 2D periodic nanohole arrays could not be fabricated even by the irradiation of femtosecond laser with high laser fluence onto a close-packed gold particle array on substrate [27]. Theoretical papers on the particle array with adequate interparticle distance reported that far-field radiative dipole coupling has an enhancing effect on near-field intensity, resulting in a larger enhancement factor of near-field intensity on substrate than a single particle system [28,29].
High-permittivity dielectric particle (dielectric particle with high refractive index) with small size parameter also provides a strong near-field enhancement mediated with the Mie resonance scattering. Recent papers show some benefits of high-permittivity particle system compared to the metallic particle system for near-field ablation of substrate surface with various refractive indexes and quenching-free fluorescence enhancement [9,30]. In terms of a many-particle system consisting of dielectric particles, near-field properties are affected by interaction not only with surface charge induced in the particles but also with electromagnetic field inside particles [31,32]. Theoretical results on 2D silica particle array system clarified that the interaction with neighboring particles affect a near-field distribution on substrate more for smaller size particles [31]. We expected that fabrication of high-density 2D periodic nanohole arrays with a clear boundary can be realized by selecting optimal high-permittivity particles for hexagonally close-packed particle arrays even in the small size parameter domain. In this paper, we present results on detailed theoretical and experimental analyses of near-field properties of 2D arrayed high-permittivity dielectric particles on substrate. This paper focuses on near-field interaction both among particles, and between particle and substrate.
2. Theoretical and experimental procedure
We performed the numerical analysis based on Mie scattering theory and three-dimensional (3D) Finite-Domain Time-Difference (FDTD) simulation. The simulation model consists of nanoparticle arrays placed on Si substrate, and a plane wave with circular polarization in fundamental wavelength (λ = 800 nm) or second harmonic wavelength (λ = 400 nm) of Ti:Al2O3 femtosecond laser is irradiated to the particle system. Circular point-symmetric nanoholes are obtained under particles with circular polarized light, while elongated nanohole is obtained with linearly polarized light. Optical constants of materials used in the simulation were taken from Refs [33–37].
Figure 1 shows a schematic of a simulation system with x-y-z coordinate. Si substrate surface has xy plane at z = 0 and the wave vector of the laser light has -z direction. In the experiment, nanoholes were fabricated on the Si substrate surface by illuminating a circularly polarized second harmonic wave (400 nm) of Ti:Al2O3 femtosecond laser to the nanoparticles placed on Si substrate. We applied loose-focus lens with focal length of 25 mm and spot diameter of 50 μm for obtaining uniform nanohole geometry over the large area. 200 nm amorphous TiO2 particles were used in experiment, and the 2D arrays were prepared by falling-drop method. Fabricated nanoholes were observed by using Field Emission Scanning Electron Microscopy (FE-SEM). We analyzed near-field properties of a single particle, a touching particle pair and 2D hexagonally close-packed particle array system. As for the theoretical simulation of the 2D array system, the infinite particle monolayer is used by applying the periodic boundary condition in calculation area.
Fig. 1 (Color online) Schematic of a simulation model and definition of x-y-z coordinate system for FDTD simulation. We analyzed three systems of a single particle, a touching particle pair and 2D hexagonal close-packed particle array.
3. Results and discussion
3.1 Single isolated particle on substrate
Figure 2 shows the near-field efficiency of a particle with various diameters (D = λ, λ/2, λ/4) in air as a function of refractive index of particle. Near-field efficiency was derived from the Mie scattering theory and is expressed by the following equation [38]:
where, an and bn are Mie coefficients for electric mode and magnetic mode, respectively, where n is mode number. h (2)(x) is spherical Hankel function of the second kind, ko is wave number outside particle, and a is radius of particle. According to the figure, when the particle diameter is smaller than the illumination wavelength, the efficiency is quite low for low-refractive-index particle (n = 1.45~1.8). However, when the refractive index of particle is high enough the efficiency becomes high even with the small size parameter. High refractive index particle means high polarizability of particle, resulting in a strong light scattering with small particle. Several peaks appear in the efficiency curve based on the Mie resonance scattering with specific refractive indexes for a given particle diameter. The first peaks appear at refractive index of 3.82 for particle with diameter of a quarter wavelength due to the magnetic dipole mode (TE1 mode) resonance, and 2.72 for particle with diameter of a half wavelength owing to the magnetic quadrupole mode (TE2 mode) resonance.Fig. 2 (Color online) Near-field efficiency of a particle with various diameters (D = λ, λ/2, λ/4) in air as a function of refractive index of particle.
Figure 3 shows dependence of the enhancement factor of optical near-field intensity on the refractive index of the particle when a particle with 200 nm diameter placed on Si substrate is irradiated by wavelength of 800 nm (D = λ/4) or 400 nm (D = λ/2). The figure evaluates enhancement factors on substrate surface (z = 2.5 nm; xy plane between particle and substrate) and inside substrate (z = −2.5 nm; xy plane inside substrate). When the irradiation wavelength is 800 nm, the peak appears at refractive index of 3.7 for the factor on surface and at refractive index of 3.4 for the factor inside substrate. With the irradiation wavelength of 400 nm, the peak appears at 2.8 for the factor on surface and at 2.7 for the factor inside substrate. Although the slight shifts of peak refractive indexes are seen due to the existence of substrate, the maximal enhancement factors among the same size dielectric particle are obtainable near the refractive index for the resonance scattering modes. Our recent results revealed that the enhancement factor becomes low with the increase of the extinction coefficients (imaginary part of complex refractive index) because of optical absorption by particle [9,39]. However, the real part of refractive index corresponding to the resonance mode excitation is not changed. Thus, we will focus on the near-field properties of high-permittivity particles with the refractive indexes for peak enhancement factor with small extinction coefficients.
Fig. 3 (Color online) Simulated enhancement factor of optical intensity when a particle with 200 nm diameter placed on Si substrate is irradiated by incident wavelength of 800 nm (D = λ/4) or 400 nm (D = λ/2), as a function of refractive index of particle. Enhancement factors on substrate surface (z = 2.5 nm; xy plane between particle and substrate) and inside substrate (z = −2.5 nm; xy plane beneath particle inside substrate) are shown.
3.2 Particle pair on substrate
Figure 4 shows optical intensity distribution on xz plane when a single particle or a touching particle pair of different material particle is placed on Si substrate. Figure 4(a) and 4(e) show the results for 200 nm gold particle (n = 0.16 + 5.083i, λ = 820 nm) whose size corresponds to TM1 resonance mode with the excitation wavelength of 800 nm. Figure 4(b) and 4(f) show the results for 200 nm Si particle (n = 3.68 + 0.005i, λ = 800 nm) whose refractive index is close to the index at TE1 resonance scattering mode with the irradiation wavelength of 800 nm (D = λ/4). Figure 4(c) and 4(g) show the results for 200 nm amorphous TiO2 particle (n = 2.66 + 0.024i, λ = 400 nm) whose refractive index is close to that at TE2 resonance scattering mode with the wavelength of 400 nm (D = λ/2). Figure 4(d) and 4(h) show the results for 800 nm polystyrene particle (n = 1.58, λ = 800 nm) which is commonly used at the irradiation wavelength of 800 nm (D = λ). In all cases, intensity distributions of particle pairs are affected by the near-field coupling with the neighboring particle. A strong intensity enhancement takes place both at particle-substrate contact point and at particle-particle touching point. The central symmetry of the intensity distribution under each particle on substrate surface is broken and intensity enhancement inside substrate is also seen in a gap between two hotspots obtained beneath the contact points of particles and substrate. In case of (f) Si particle pair, intensity at the particle-particle touching point is larger than that at the particle-substrate contact point due to the strong capacitive coupling effect between induced charge in particles, which is similar to the case of gold particle pair [23]. On the contrary, intensity at the particle-substrate contact point is several times larger than that at the particle-particle touching point in the cases of (g) amorphous TiO2 particle pair and (h) polystyrene particle pair. These particles induce near-field light via higher multipole scattering mode, resulting in the forward scattered intensity pattern. We will later focus on the system that 200 nm particle on Si substrate is irradiated by 400 nm femtosecond laser.
Fig. 4 (Color online) Optical intensity distribution on xz plane when a single particle or a touching particle pair of different material particle is placed on Si substrate. Figure 4(a) and 4(e) show the cases of 200 nm gold particle (n = 0.16 + 5.083i, λ = 820 nm) whose size corresponds to the TM1 resonance mode with the excitation wavelength of 800 nm. Figure 4(b) and 4(f) show the cases of 200 nm Si particle (n = 3.68 + 0.005i, λ = 800 nm) whose refractive index is close to the index of TE1 resonance scattering mode with irradiation wavelength of 800 nm (D = λ/4). Figure 4(c) and 4(g) show the cases of 200 nm amorphous TiO2 particle (n = 2.66 + 0.024i, λ = 400 nm) whose refractive index is close to that of TE2 resonance scattering mode with irradiation wavelength of 400 nm (D = λ/2). Figure 4(d) and 4(h) show the cases of 800 nm polystyrene particle (n = 1.58, λ = 800 nm) which are commonly used at irradiation wavelength of 800 nm (D = λ).
Figure 5 shows Poynting vector distribution on xz plane when (a) single particle or (b) touching particle pair of 200 nm amorphous TiO2 particles placed on Si substrate is irradiated by 400 nm laser. When using a particle pair, the vector direction and position and number of phase singularity inside each particle are changed from those of single particle system. The optical power flow is observed not only to the particle-substrate contact point (x = 100 nm, z = 0 nm in Fig. 5) but also to the substrate beneath particle-particle touching point (x = 0 nm, z = 0 nm in Fig. 5). Thus, the particle pair also induces electromagnetic mode of a single ellipsoidal particle. Optical intensity distribution on xy plane inside substrate surface (z = −2.5 nm) and SEM images of fabricated nanoholes by using a single particle or a touching particle pair of 200 nm amorphous TiO2 particle with the irradiation of 400 nm femtosecond laser are shown in Fig. 6. Figure 6(a) and 6(b) are intensity distribution and fabricated nanohole (40 mJ/cm2) with a single particle. Figure 6(c) is intensity distribution using a particle pair, and Fig. 6(d) and 6(e) are fabricated nanoholes using a particle pair with different laser fluences (72 mJ/cm2 and 108 mJ/cm2). In the case of a single particle, a single nanohole with diameter less than 100 nm is fabricated with laser fluence of a half ablation threshold of bare substrate (82 mJ/cm2). According to the intensity distribution with a particle pair, the intensity enhancement also occurs at the gap of two hotspots created under particle (x = y = 0 nm) and its factor is 28% of hotspots. A pair of separated nanoholes is fabricated in the low laser fluence regime (72 mJ/cm2), and a combined nanohole pair is obtained in the high laser fluence regime (108 mJ/cm2), corresponding to the intensity distribution.
Fig. 5 (Color online) Poynting vector distribution on xz plane when a single particle or a touching particle pair of 200 nm amorphous TiO2 particle placed on Si substrate is irradiated by 400 nm laser. (a) single particle, (b) particle pair.
Fig. 6 (Color online) Optical intensity distribution on xy plane inside substrate surface (z = −2.5 nm) and SEM images of fabricated nanoholes by using a single particle or a touching particle pair of 200 nm amorphous TiO2 particle with the irradiation of 400 nm femtosecond laser. (a) intensity distribution on xy plane with a single particle, (b) SEM image of fabricated nanohole with a single particle (laser fluence 40 mJ/cm2), (c) intensity distribution on xy plane with a particle pair, (d) SEM image of fabricated nanohole pair with a particle pair (laser fluence 72 mJ/cm2), (e) SEM image of fabricated nanohole pair with a particle pair (laser fluence 108 mJ/cm2).
3.3 2D particle array on substrate
Figure 7 shows (a) intensity distribution and (b) Poynting vector distribution on xz plane when applying 2D hexagonally close-packed amorphous TiO2 particles with diameter of 200 nm on Si substrate with the excitation wavelength of 400 nm. According to Fig. 7(a), a strong enhancement of intensity at particle-particle touching point is not seen and hotspot sites differ from those of the particle pair system. Moreover, near-field intensity inside the substrate near the particle-substrate contact point becomes quite weak. As for the Poynting vector distribution seen in Fig. 7(b), the power convergence to the particle-substrate contact point is not seen inside particle, which shows totally different distribution from the case of a particle pair. Figure 8 shows SEM images of fabricated nanohole arrays on Si substrate using 2D particle array of 200 nm amorphous TiO2 particles with 400 nm femtosecond laser irradiation at different laser fluences. Irradiated laser fluences are (a) 61 mJ/cm2, (b)118 mJ/cm2 and (c) 133 mJ/cm2. 2D array of 50~100 nm nanoholes was fabricated at a low laser fluence of 61 mJ/cm2. Ablated area was observed in the gap of the 200 nm period nanoholes at laser fluence of 118 mJ/cm2. Arrayed nanoholes were combined with further increase of laser fluence (133 mJ/cm2).
Fig. 7 (Color online) Intensity distribution and Poynting vector distribution on xz plane when applying 2D hexagonally close-packed amorphous TiO2 particles with diameter of 200 nm on Si substrate at the excitation wavelength of 400 nm. (a) intensity distribution, (b) Poynting vector distribution.
Fig. 8 (Color online) SEM images of fabricated nanohole array on Si substrate using 2D particle array of 200 nm amorphous TiO2 particles at 400 nm femtosecond laser irradiation of different laser fluences. (a) laser fluence: 61 mJ/cm2, (b) laser fluence: 118 mJ/cm2, (c) laser fluence: 133 mJ/cm2.
Figure 9 represents evaluated factors for intensity enhancement when the hexagonally close-packed particle array with the diameter of 200 nm is placed on Si substrate, as a function of particle’s refractive index (λ = 400 nm). Enhancement factor of the intensity inside substrate beneath particle-substrate contact point [x = 0 nm, z = −2.5 nm in Fig. 7(a)] is shown in the blue line. Enhancement ratio of the intensity beneath particle-substrate contact points [x = 0 nm, z = −2.5 nm in Fig. 7(a)] divided by that at the gap of the contact points [x = 100 nm, z = −2.5 nm in Fig. 7(a)] which is evaluated as a guidepost for avoiding the joining of neighboring nanoholes is shown by the green line. Enhancement factor of maximal intensity around particle array above substrate which is commonly evaluated for the research of surface-enhanced Raman scattering (SERS) is shown in the red line. Enhancement factor of the intensity inside substrate beneath particle-substrate contact point for the single particle system is also shown in the gray dashed line. Each factor of 2D arrays takes a peak value with refractive indexes around 2.1 and 3.1, shifted from the optimal refractive index of 2.7 for the single particle system. Thus, the optimal refractive index for nanohole array fabrication is determined by the near-field resonance excitation condition for 2D particle array, different from that for the isolated single particle system. When refractive index of particle is 3.1, the intensity enhancement factor under particles is larger than twice the single particle system. Based on photonic band theory, it was theoretically analyzed that the frequencies of the peak enhancement factor of near-field intensity and the frequencies of peak reflectivity from the monolayer of hexagonally close-packed particle array are consistent [40,41]. Thus, photonic band effect of the hexagonally close-packed particle system may act at the peak refractive index. When reflectivity of the system is high, ablation influence in the area other than near-field intensity hotspots seems to be small, because propagating light intensity through the particle monolayer is reduced. Consequently, the selection of the optimal particle materials for 2D array enables one to fabricate 2D periodic nanohole arrays with a clear boundary. The spacing distance between particles also plays an important role for near-field distributions around particle arrays. Optimal refractive index of particles seems moving to that of isolated particle case with the increase of the spacing due to the decrease of near-field coupling between particles.
Fig. 9 (Color online) Evaluated factors for intensity enhancement when the hexagonally close-packed particle array with the diameter of 200 nm is placed on Si substrate, as a function of particle’s refractive index (λ = 400 nm). The blue line shows the enhancement factor of intensity inside substrate beneath particle-substrate contact point [x = 0 nm, z = −2.5 nm in Fig. 7(a)]. The green line show the enhancement ratio of intensity beneath particle-substrate contact points [x = 0 nm, z = −2.5 nm in Fig. 7(a)] divided by that at the gap of the contact points [x = 100 nm, z = −2.5 nm in Fig. 7(a)] which is evaluated as a guidepost for avoiding combination of nanoholes. The red line shows the enhancement factor of maximal intensity around particle array above substrate. The gray dashed line shows the enhancement factor of intensity inside substrate beneath particle-substrate contact point for single particle case.
Figure 10 shows intensity distribution inside Si substrate on xz plane when applying a single particle of 200 nm amorphous TiO2 particle and 2D particle array of 200 nm amorphous TiO2 particles (n = 2.66 + 0.024i), 200 nm ZnO particles (n = 2.18) and 200 nm polycrystalline rutile TiO2 particles (n = 3.182 + 0.00145i) at the irradiation wavelength of 400 nm. ZnO particle and poly-rutile TiO2 particle have refractive indexes near the peak refractive indexes for particle array shown in Fig. 9. In the case of amorphous TiO2 particle, the intensity enhancement in the 2D array system is much lower than that in the single particle system. Intensity hotspots with a clear boundary are obtained under the particles-substrate contact points with the ZnO particle array and poly-rutile TiO2 particle array, but not obtained with the amorphous TiO2 particle array. The fabrication ability of 2D nanohole array with a clear boundary is done by smart selection of appropriate particle materials. Figure 11 shows Poynting vector distribution on xz plane when applying 2D particle array of (a) ZnO particles and (b) poly-rutile TiO2 particles on Si substrate (λ = 400 nm). Near-field focusing to the particle-substrate contact points is observed outside particles with ZnO particles and inside particles with poly-rutile TiO2 particles, resulting in the strongly enhanced hotspots with a clear boundary under particles.
Fig. 10 (Color online) Intensity distribution inside Si substrate on xz plane when applying a single particle of 200 nm amorphous TiO2 particle and 2D particle array of 200 nm amorphous TiO2 particles, 200 nm ZnO particles and 200 nm poly-rutile TiO2 particles with irradiation wavelength of 400 nm. (a) single particle of 200 nm amorphous TiO2 particle (n = 2.66 + 0.024i) on Si substrate, (b) 2D particle array of 200 nm amorphous TiO2 particles (n = 2.66 + 0.024i) on Si substrate, (c) 2D particle array of 200 nm ZnO particles (n = 2.18) on Si substrate, (d) 2D particle array of 200 nm poly-rutile TiO2 particles (n = 3.182 + 0.00145i) on Si substrate.
Fig. 11 (Color online) Poynting vector distribution on xz plane when applying 2D particle array of ZnO particles and poly-rutile TiO2 particles on Si substrate (λ = 400 nm). (a) ZnO particle array on Si substrate and (b) poly-rutile TiO2 particle array on Si substrate.
4. Conclusion
In conclusion, we have presented experimental and theoretical results on near-field properties of 2D arrayed high permittivity particles on substrate. For a single particle system, a particle with the refractive index of Mie resonance scattering provides the maximal enhancement factor of near-field intensity on substrate and inside substrate among the same size dielectric particles. When a touching particle pair is applied, near-field around a particle interacts with that of adjacent particle and the mode of electromagnetic field inside particles are changed. It results in the intensity enhancement on substrate surface not only at the particle-substrate contact points but also in the gap between the contact points. A combined nanohole pair is fabricated in the high laser fluence regime. In terms of the 2D hexagonally close-packed particle array, the optimal refractive index of particle is changed from the Mie resonance condition for a single particle system. Theoretical results revealed that the 2D particle array with optimal refractive index can fabricate 2D periodic nanohole array with a clear boundary.
Acknowledgement
This study was supported in part by a Grant-in-Aid for Challenging Exploratory Research (22656019), and was supported in part by a Grant-in-Aid for the Global Center of Excellence for High-Level Global Cooperation for Leading-Edge Platform on Access Spaces from the Ministry of Education, Culture, Sport, Science and Technology of Japan. Y. Tanaka is grateful for the JSPS Fellowship for Young Scientists.
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