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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 25 — Dec. 8, 2008
  • pp: 20706–20723

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Light propagation from a fluorescent particle embedded in a photonic cluster of micrometer-sized dielectric spheres

T. Fujishima, H. T. Miyazaki, H. Miyazaki, Y. Jimba, T. Kasaya, K. Sakoda, Y. Ogawa, and F. Minami  »View Author Affiliations


Optics Express, Vol. 16, Issue 25, pp. 20706-20723 (2008)
http://dx.doi.org/10.1364/OE.16.020706


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Abstract

In self-assembled multilayer arrays of micrometer-sized spheres that include small amounts of fluorescent particles, unique six-dot-triangular and seven-dot-hexagonal patterns have been known to appear in the fluorescence microscopic images. Although it has been suggested that these two types of patterns correspond to local domain structures, i.e., face centered cubic (fcc) or hexagonal closed packed (hcp), no conclusive evidence has been provided to support this claim. In this study, we systematically investigated the relationship between the propagation patterns and the arrangement of the particles. Through a cross-check between an experiment using well-defined clusters fabricated by a micromanipulation technique and a rigorous calculation based on the expansion of vector spherical harmonics, we confirmed that the six-dot-triangular and seven-dot-hexagonal patterns correspond to the fcc and hcp domains, respectively. Further, we also found that the propagation patterns depend on the size of the clusters. As a result of a quantitative discussion on the light propagation in clusters with various sizes, it was clarified that a sufficient domain size is necessary for the appearance of clear triangular or hexagonal patterns.

© 2008 Optical Society of America

1. Introduction

Photonic crystals [1

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

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S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

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S. Noda and T. Baba, Roadmap on Photonic Crystals (Kluwer Academic, Dordrecht, 2003).

] are periodic dielectric structures having a scale that is almost equal to the wavelength of light. Such structures are expected to exhibit various unique optical properties due to the strong interaction between light and matter. In addition to the inhibition of the penetration of light by photonic bandgaps (PBG’s) [4

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of Two-Dimensional Photonic Band Structure with Near-Infrared Band Gap,” Jpn. J. Appl. Phys. 33, 1463–1465 (1994). [CrossRef]

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T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996). [CrossRef]

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S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 251–253 (1998). [CrossRef]

] and the localization of light at a point defect [7

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003). [CrossRef]

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A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005). [CrossRef] [PubMed]

], light propagation in photonic crystals has also been investigated. For example, light propagation in line-defect waveguides in PBG crystals has been extensively studied [9

T. Baba, N. Fukaya, and J. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999). [CrossRef]

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N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. S. Jensen, O. Sigmund, P. I. Borel, M. Kristensen, and K. Asakawa, “Topology optimized photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Electron. Lett. 42, 1031–1033 (2006). [CrossRef]

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R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nature Physics 3, 401–405 (2007). [CrossRef]

]. The dispersion relations in the transparent frequency region outside the PBG have also been experimentally measured. The band structures of two-dimensional crystals have been determined using oblique incidence methods [12

V. N. Astratov, M. S. Skolnick, S. Brand, T. F. Krauss, O. Z. Karimov, R. M. Stevenson, D. M. Whittaker, I. Culshaw, and R. M. De la Rue, “Experimental technique to determine the band structure of two-dimensional photonic lattices,” IEE Proc.: Optoelectron. 145, 398–402 (1998). [CrossRef]

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T. Yamasaki and T. Tsutsui, “Fabrication and Optical Properties of Two-Dimensional Ordered Arrays of Silica Microspheres,” Jpn. J. Appl. Phys. 38, 5916–5921 (1999). [CrossRef]

, 14

K. Ohtaka, Y. Suda, S. Nagano, T. Ueta, A. Imada, T. Koda, J. S. Bae, K. Mizuno, S. Yano, and Y. Segawa, “Photonic band effects in a two-dimensional array of dielectric spheres in the millimeter-wave region,” Phys. Rev. B 61, 5267–5279 (2000). [CrossRef]

, 15

H. T. Miyazaki, H. Miyazaki, K. Ohtaka, and T. Sato, “Photonic band in two-dimensional lattices of micrometer-sized spheres mechanically arranged under a scanning electron microscope,” J. Appl. Phys. 87, 7152–7158 (2000). [CrossRef]

]. The band structures of three-dimensional colloidal crystals have been observed by white-light interferometry [16

İ. İ. Tarhan, M. P. Zinkin, and G. H. Watson, “Interferometric technique for the measurement of photonic band structure in colloidal crystals,” Opt. Lett. 20, 1571–1573 (1995). [CrossRef] [PubMed]

] and by a frequency-resolved optical gating method based on femto second pulses [17

J. Nakagawa, H. Kitano, F. Minami, T. Sawada, S. Yamaguchi, and K. Ohtaka, “Large pulse distortion in a 3D photonic crystal,” J. Lumin. 108, 255–258 (2004). [CrossRef]

]. The discovery of interesting behaviors of light such as the superprism effect in quasi-three-dimensional crystals [18

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998). [CrossRef]

] triggered the birth of the concept of the photonic band engineering. Light propagation in one-dimensional wires [19

T. Mitsui, Y. Wakayama, T. Onodera, Y. Takaya, and H. Oikawa, “Light Propagation within Colloidal Crystal Wire Fabricated by a Dewetting Process,” Nano Lett. 8, 853–858 (2008). [CrossRef] [PubMed]

] and in two-dimensional crystals comprising microspheres has been investigated by the use of near-field scanning optical microscopy and attenuated total reflection techniques [20

T. Fujimura, K. Edamatsu, T. Itoh, R. Shimada, A. Imada, T. Koda, N. Chiba, H. Muramatsu, and T. Ataka, “Scanning near-field optical images of ordered polystyrene particle layers in transmission and luminescence excitation modes,” Opt. Lett. 22, 489–491 (1997). [CrossRef] [PubMed]

, 21

M. Haraguchi, T. Nakai, A. Shinya, T. Okamoto, M. Fukui, T. Koda, R. Shimada, K. Ohtaka, and K. Takeda, “Optical Modes in Two-dimensionally Ordered Dielectric Spheres,” Jpn. J. Appl. Phys. 39, 1747–1751 (2000). [CrossRef]

]. In addition, Matsushita et al. [22

S. I. Matsushita, Y. Yagi, T. Miwa, D. A. Tryk, T. Koda, and A. Fujishima, “Light Propagation in Composite Two-Dimensional Arrays of Polystyrene Spherical Particles,” Langmuir 16, 636–642 (2000). [CrossRef]

] have reported unique light propagation that exhibits characteristic geometrical patterns in self-assembled multilayer crystals; our work is an extension of their report.

The most important parameter that characterizes the light propagation in a system comprising spheres is the size parameter S=πD/λ, where D is the diameter of the sphere and λ, the wavelength of light in vacuum [23

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, 1983).

]. Matsushita et al. fabricated self-assembled three-layer close-packed arrays of micrometer-sized polystyrene (PSt) spheres, in which a small number of fluorescent PSt particles having the same diameter are mixed, and investigated how the light from the fluorescent particles is propagated in the arrays. They prepared three types of samples having S values of 3.02, 5.41, and 17.8. In the arrays of S=5.41 and 17.8, they observed unique triangular patterns comprising six intense dots and hexagonal patterns comprising seven intense dots. Their arrays were polycrystalline lattices comprising many domains. In a close-packed structure, the arrangement of the third layer may either be face centered cubic (fcc) or hexagonal closed packed (hcp). Based on the fluorescence and phase-contrast microscopies, they concluded that the six-dot-triangular pattern and the seven-dot-hexagonal pattern correspond to the fcc and hcp domains, respectively.

However, the resolution of optical microscopes is not sufficiently high for the exact determination of wavelength-sized structures. Further, they presented no theoretical evidence that supports the existence of such propagation modes. To clarify the explicit correspondence between the propagation modes and the structures, we assembled controlled fcc and hcp clusters by use of a micromanipulation technique within a scanning electron microscope (SEM) and observed light propagation in these clusters. We also reproduced the light propagation with a rigorous calculation based on the expansion of vector spherical harmonics (VSH). We focused on the size region of S~5. Reference [22

S. I. Matsushita, Y. Yagi, T. Miwa, D. A. Tryk, T. Koda, and A. Fujishima, “Light Propagation in Composite Two-Dimensional Arrays of Polystyrene Spherical Particles,” Langmuir 16, 636–642 (2000). [CrossRef]

] has suggested that a drastic change occurs in the propagation modes within the range of S=3.0-5.4. Thus, S~5 is an important region in which characteristic patterns appear. In this study, silica microspheres were used as non-fluorescent particles [24

At an early stage of our research, we investigated light propagation in clusters solely composed of polystyrene (PSt) particles as in the paper by Matsushita et al. However, we found that non-fluorescent PSt particles become fluorescent upon electron beam (EB) irradiation during the micromanipulation. It was difficult to distinguish the luminescence from the dye-doped and undoped PSt particles. Since silica particles were found to be unaffected by the EB irradiation, we employed silica as non-fluorescent particles.

]. Silica has a slightly smaller refractive index than that of PSt.

This paper is organized as follows. In Sec. 2, we repeat the experiment of Matsushita et al. and observe the propagation patterns in a self-assembled photonic crystal that includes fluorescent particles. This is because the emergence of similar propagation patterns in arrays of silica microspheres having a refractive index different from that of PSt spheres has not yet been confirmed. Section 3 presents the results of the fabrication of fcc and hcp photonic clusters with various sizes and the observation of light propagation in them. In Sec. 4, we reproduce the propagation modes using a calculation based on the expansion of VSH and compare them with the experimental results. Then, we discuss the light propagation in significantly larger clusters by using this calculation in Sec. 5, and discuss the behavior of light in photonic clusters comprising microspheres on the basis of all the obtained results in Sec. 6. Finally, we summarize our study in Sec. 7.

2. Light propagation in a self-assembled photonic crystal comprising silica spheres

To confirm that the characteristic propagation patterns reported by Matsushita et al. are not unique to PSt particles (refractive index n=1.59) but common to dielectric spheres with a different refractive index, we first fabricated a self-assembled array of silica spheres (n=1.4), and carried out a similar experiment. We used non-fluorescent silica spheres having a diameter of 1.0 µm and fluorescent PSt particles having the same diameter. The excitation and emission wavelengths of the fluorescent PSt particles are 542 nm and 614 nm, respectively. The dye molecules in the fluorescent particles are mainly distributed near the surface of the particle.

The imaging scheme is shown in Fig. 1. We used an objective lens (numerical aperture: 0.95, focal depth: 0.68 µm) optimized for observation through a glass substrate. A Nd:YVO 4-based green laser (wavelength: 532 nm) was used as the excitation source and the luminescence from the fluorescent particles was observed through a band-pass filter (λ=610 nm±10 nm). The size parameter in this system is S=5.15 (D=1.0 µm,λ=0.61 µm). A composite self-assembled array including silica and fluorescent PSt particles in a ratio of 100:1 was fabricated by a ring technique [25

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, I. B. Ivanov, and H. Yoshimura, and K. Nagayama, “Mechanism of Formation of Two-Dimensional Crystals from Latex Particles on Substrates,” Langmuir 8, 3183–3190 (1992). [CrossRef]

, 26

F. Juillerat, P. Bowen, and H. Hofmann, “Formation and Drying of Colloidal Crystals Using Nanosized Silica Particles,” Langmuir 22, 2249–2257 (2006). [CrossRef] [PubMed]

]. An SEM image of the array is shown in Fig. 2(a), which reveals that the array is ordered over a wide range. Figure 2(b) shows a magnified image of Fig. 2(a). Since the spheres in the lower layer are observed at the interstitial points of the spheres in the top layer, we can confirm the formation of an ordered multilayer array. However, the number of layers in this array is not clear. Although it was difficult to determine the exact z position of the focal plane in the array from the optical image, triangular and hexagonal patterns similar to those in the PSt array were observed at a certain focal position. Figure 2(c) shows the fluorescence micrograph, in which circles indicate the typical triangular and hexagonal patterns. Figures 2(d) and (e) show magnified images of these patterns. Hence, the emergence of two types of propagation patterns is not unique to the PSt particles but common to the arrays of S~5 and n~1.4-1.6. In our experiment, however, we were unable to identify the domain structures corresponding to the triangular and hexagonal patterns on the basis of the transmission optical

Fig. 1. The observation of light propagation in photonic clusters. (a) The setup of the photonic cluster and the objective lens. The fluorescent particle in the photonic cluster is excited by a 532-nm laser and the fluorescence is observed through the glass substrate. The z-axis is set as the direction from the cluster to the objective lens. (b) A magnified image around the cluster. The propagation was systematically observed by moving the focal plane at various z positions.
Fig. 2. The self-assembled array of the mixture of non-fluorescent silica particles and a small amount of fluorescent PSt particles. In all images, the bar indicates 3 µm. (a) An SEM image of the array. (b) A magnified image of (a); spheres of the lower layer can be observed at the interstitial points of the top-layer array. (c) A fluorescence micrograph of the self-assembled array shown in (a). The plane of the focus is presumably positioned slightly inside the surface of the array. The circles indicate the triangular and hexagonal patterns. (d) and (e) show magnified images of the circles shown in (c). images.

3. Fabrication of photonic clusters and observation of light propagation

With the aim of investigating the relationship between the domain structures (fcc or hcp) and the propagation patterns, a micromanipulation technique [27

H. Miyazaki and T. Sato, “Mechanical Assembly of Three-Dimensional Microstructures from Fine Particles,” Adv. Robotics 11, 169–185 (1997). [CrossRef]

, 28

H. T. Miyazaki, H. Miyazaki, K. Ohtaka, and T. Sato, “Photonic band in two-dimensional lattices of micrometersized spheres mechanically arranged under a scanning electron microscope,” J. Appl. Phys. 87, 7152 (2000). [CrossRef]

, 29

F. Garcia-Santamaria, H. T. Miyazaki, A. Urquia, M. Ibisate, M. Belmonte, N. Shinya, F. Meseguer, and C. Lopez, “Nanorobotic manipulation of microspheres for on-chip diamond architectures,” Adv. Mater. 14, 1144–1147, (2002). [CrossRef]

, 30

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, “Microassembly of semiconductor three-dimensional photonic crystals,” Nat. Mater. 2, 117–121 (2003). [CrossRef] [PubMed]

] was employed for the assembly of controlled three-layer fcc and hcp clusters.

Non-fluorescent silica and fluorescent PSt particles were sprinkled on a glass substrate coated with a conductive indium tin oxide layer, and we constructed clusters by picking up and placing individual particles using a gold-coated glass probe [31

H. T. Miyazaki, H. Miyazaki, N. Shinya, and K. Miyano, “Enhanced light diffraction from a double-layer microsphere lattice,” Appl. Phys. Lett. 83, 3662–3664 (2003). [CrossRef]

, 32

H. T. Miyazaki, H. Miyazaki, Y. Jimba, Y. Kurokawa, N. Shinya, and K. Miyano, “Light diffraction from a bilayer lattice of microspheres enhanced by specular resonance,” J. Appl. Phys. 95, 793–805 (2004). [CrossRef]

]. The particles were not coated with conductive materials. For assembling accurate clusters, spheres within a diameter range of ±0.6% were selected according to the diameter measurement in the SEM image, because the size distribution of the obtained particles of ±2% (standard distribution) is not sufficient for constructing precise clusters. In addition, the spheres were positioned according to a graphical template image superimposed on the SEM monitor to form exact arrangements. A graphical templete is a computer generated outline image of the completed cluster [33

T. Kasaya and H. T. Miyazaki, “Graphical Templates for Accurate Micromanipulation in a Scanning Electron Microscope,” Rev. Sci. Instrum. , submitted. [PubMed]

]. We embedded one fluorescent particle in each cluster.

For a systematic study, fcc and hcp clusters with various sizes were fabricated. In this paper, the size of a cluster implies the number of spheres that constitute the cluster, and not the diameter of each sphere. The layer in contact with the substrate is called the first layer. The third layer is arranged in the shape of a regular hexagon in every cluster. The second layer contains only the minimum number of spheres that is necessary for supporting the third layer, and it is slightly larger than the third layer. Similarly, the first layer contains the minimum number of spheres required to support the second layer. However, the arrangement of the spheres in the first layer is different for the fcc and hcp clusters. A cluster in which one edge of the regular hexagon of the third layer comprises m particles is defined as a cluster of class m. A fluorescent particle is placed at the center of the third layer. The smallest, i.e. class 1, clusters have only one fluorescent particle and no non-fluorescent particle in the third layer. The axis perpendicular to the substrate is defined as the z-axis. As shown in Fig. 1(a), the direction from the cluster to the objective lens is the positive direction of the z-axis. The origin of the z-axis is set at the center of the first layer, as shown in Fig. 1(b). Thus, the position of the surface of the substrate is z=D/2. H=(2/3)1/2 D is the distance between the layers and is common to fcc and hcp structures. Unlike the case of the self-assembled array, the position of the focal plane could be easily identified by a comparison with the surrounding isolated particles. We observed the fluorescence images at various z positions. Figures 3(a) and (b) show the fabricated class-1 clusters. The light propagation patterns in the fcc and hcp clusters are shown in Figs. 3(c) and (d), in which six-dot-triangular and seven-dot-hexagonal patterns similar to those shown in Figs. 2(d) and (e) are seen, respectively. The corresponding patterns observed in the self-assembled array are also shown in the right-hand side panels of Figs. 3(c) and (d). Since the class-1 clusters and the self-assembled array exhibited very similar transitions in the fluorescence patterns as the focal plane moves, the z position in the self-assembled array shown in the right-hand side panel is adjusted so that good correspondence with the images of the class-1 clusters shown in the center panel is obtained. The good agreement between the center panels and the right-hand side panels in Figs. 3 (c) and (d) appears to suggest that the triangular and hexagonal patterns seen in the self-assembled array correspond to the fcc and hcp domains, respectively. However, we cannot confirm this. The results of the class-1 clusters only indicate that the light is distributed to all the spheres in the first layer. For confirming the relationship between the lattice structure and the propagation patterns, it is necessary to investigate much larger clusters, and to confirm that the six-dot-triangular and seven-dot-hexagonal patterns are still maintained and light is not propagated toward the surrounding particles.

Fig. 3. SEM micrographs of the class-1 photonic clusters assembled by micromanipulation. The fluorescent particles are indicated by red circles. (a) The fcc cluster and (b) the hcp cluster. (c) (Media 1), (d) (Media 2) Propagation images observed at various z positions are shown at the center, along with the patterns in the self-assembled array at the right-hand side. The illustrations presenting the position of the focal plane are shown on the left-hand side. The bars indicate 1 µm in (a) and (b) and 3 µm in (c) and (d).

Therefore, we conducted a similar experiment on the class-2 clusters. Figures 4 (a) and (b) show the fabricated class-2 clusters, and the results of the optical observation of the fcc and hcp clusters are shown in Figs. 4(c) and (d), respectively. The results of the self-assembled array are also shown in the right-hand side panels, as shown in Figs. 3 (c) and (d). As can be seen in Figs. 4(c) and (d), the propagation images in the class-2 clusters exhibited considerable differences from those in the self-assembled array; light spreads over the surrounding particles and exhibits complicated patterns. Hence, the propagation patterns are not directly correlated with the lattice structures. However, we can interpret these results in the case of the class-2 clusters as a simple combination of the central patterns common to the self-assembled arrays and the propagation toward the surrounding spheres.

4. Calculation of light propagation and comparison with the experiment

In this section, we attempt to reproduce the unique light propagation in the photonic clusters by numerical calculation. The propagation of the electromagnetic fields radiated from monochromatic oscillating dipoles (S=5.15) placed on the surface of the fluorescent particle was obtained by solving the Maxwell equations. The positions of the dipoles on the surface of the sphere were determined at 16 points by the method described in the next paragraph. At each point, three orthogonal dipoles with the same oscillation amplitude were placed. The intensity distribution of the propagating light observed at each focal position was represented by mapping the z component of the time-averaged Poynting vector at the z position of the focal plane. The calculated quantity is not the same as the observed intensity since the latter is distorted by the spheres or the fractions of the spheres between the focal plane and the objective lens (indicated by the green-colored portions in the left-hand side panels of Figs. 6–9 and 11–14).

Fig. 4. (a), (b) SEM micrographs of class-2 photonic clusters, and (c) (Media 3) and (d) (Media 4) propagation images. Refer to the caption of Fig. 3 for details.

Nevertheless, at the positions close to the objective lens (z≿0), it is expected that the difference between both quantities is not considerable. Since no coherency is expected among the dye molecules, the total Poynting vector was obtained by simply summing up the independent Poynting vectors generated by each single dipole. The numerical solution of the Maxwell equations was obtained by use of the expansion of VSH up to l max =16 [34

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62, 7976–7997 (2000). [CrossRef]

], which is sufficient for convergence. We did not take account of the contribution of the substrates because it has already been shown that the substrate has no significant effect in a similar system [32

H. T. Miyazaki, H. Miyazaki, Y. Jimba, Y. Kurokawa, N. Shinya, and K. Miyano, “Light diffraction from a bilayer lattice of microspheres enhanced by specular resonance,” J. Appl. Phys. 95, 793–805 (2004). [CrossRef]

].

In order to represent the dye molecules homogeneously distributed over the entire surface of the fluorescent particle by a small number of dipoles, the positions of the dipoles were determined in view of the symmetry of the clusters and the three-dimensional isotropy. One option might be the random placement of finite number of dipoles. However, a great number of dipoles are required to express the homogeneous distribution. All clusters discussed in this study have a three-fold rotational symmetry and three planes of symmetry including the z-axis; i.e., the clusters are formed by repeating one-sixth element around the z-axis. Therefore, arrangement of the dipoles with six-fold symmetry about the z-axis is preferable. Here, we incorporate the Brillouin zone of an fcc lattice into the discussion. In this study, the positions of the dipoles were determined as follows. In the Brillouin zone of an fcc infinite lattice, obtained by extending the cluster of Fig. 3(a), the [111] direction agrees with the z-axis, as shown in Fig. 5(a). Since the eight 〈111〉 directions are homogeneously distributed in three dimensions, the positions of the dipoles were selected at the eight points on the surface of the fluorescent particle in the 〈111〉 directions [Fig. 5(b)]. However, this arrangement has only a three-fold symmetry about the z-axis. Hence, another eight points that are positioned in the 〈±1±1±1〉 directions of the Brillouin zone rotated by 180° are also considered [Fig. 5(c)][35

The directions of the eight inverted points are [±1±1±1], [±1±1∓5], [±1∓5±1], and [∓5±1±1].

]. Two points [111] and [-1-1-1] common to both the Brillouin zones were counted twice. In this manner, the dipoles were distributed on the surface of the sphere so that they have high isotropy and sixfold symmetry. Here, the Brillouin zone of the fcc lattice was employed only for its isotropy, but not because some of our clusters have fcc structures. Therefore, the same positions of the dipoles were also adopted in the calculations of the hcp clusters.

Fig. 5. (a) The Brillouin zone of an fcc lattice utilized for the reasonable determination of the positions of the oscillating dipoles. The z-axis of the cluster is aligned to agree with one of the 〈111〉 directions. The one-sixth piece of the cluster is shown in green. (b) The points in the eight 〈111〉 directions on the surface of the fluorescent particle are indicated in blue. (c) The eight 〈111〉 directions after 180° rotation are indicated in blue.

The comparison between the calculation and the experiment at each z position is shown in Figs. 6–9 for class-1 fcc and hcp and class-2 fcc and hcp clusters, respectively. Note that the position of the observation plane z=H/2 [(e) in Figs. 6 and 7 and (b) in Figs. 8 and 9] is slightly inside the cluster, and that of z=H [(f) in Figs. 6 and 7 and (c) in Figs. 8 and 9] is slightly outside the clusters (inside the substrate in the experiment). The six-dot-triangular patterns are clearly discerned in Figs. 6(e) and (f). Digitized images, obtained by casting the calculation results into three-value maps, reflect the major features of the observed images. Figure 7(f) shows the seven-dot-hexagonal pattern. The appearance of the characteristic wing-shaped patterns in Fig. 7(d) also corresponds well to the observation in Ref. [22

S. I. Matsushita, Y. Yagi, T. Miwa, D. A. Tryk, T. Koda, and A. Fujishima, “Light Propagation in Composite Two-Dimensional Arrays of Polystyrene Spherical Particles,” Langmuir 16, 636–642 (2000). [CrossRef]

]. However, the existence of the intense spot at the center of the three wings is a different feature from the case of PSt spheres. In (a)–(c) of both Figs. 6 and 7, the agreement between the observation and the calculation is poor. This may be because the distortion induced by the structure between the focal plane and the objective lens (indicated by the green-colored part in the left-hand side panels of Figs. 6 and 7) is not negligible and then, the approximation of the Poynting vector to the observed image is no longer valid. On the other hand, in (d)–(f) (z≥0), the agreement between the calculation and the experiment is remarkable. In the following, we discuss cross sections only at z≥0.

Next, we discuss the class-2 clusters. The calculation agrees well with the experiment in the results of the fcc cluster shown in Fig. 8. On the other hand, the agreement in the hcp cluster shown in Fig. 9 is poor. However, if we focus on the one-third part at the upper left of the images, the agreement between both results is fair even in hcp. It is assumed that these facts are related to the disorder in the fabricated clusters; this is because we can find some imperfections in the arrangement in the class-2 clusters in Figs. 4(a) and (b). It appears that the light propagation in the fcc cluster was hardly affected by the lattice disorder (Fig. 8). In contrast, the propagation in the hcp cluster appears to be sensitive to the accuracy of the arrangement (Fig. 9). [36

The dye molecules doped in the PSt particles gradually degrade upon electron beam irradiation. Therefore, the time allowed for the manipulation is limited. Since the top layer of the class-2 or larger clusters is composed of a large number of particles, it was difficult to assemble sufficiently accurate lattices within the limited time.

].

Fig. 6. Comparison of the light propagation images in the class-1 fcc cluster obtained in the experiment and that obtained in the calculation. From left to right: the illustrations presenting the position of the focal plane, experimental results, calculation results (blue indicates zero and red indicates the maximum), and the three-value map obtained by digitizing the calculation result. The distribution of the light intensity was obtained in the range of z=-3H/2-+H at a step of H/2. Black dashed lines in the calculation images of (b) and (d) indicate the positions of the spheres of the second and first layers. The portions in the cluster between the focal plane and the objective lens are indicated in green in the illustrations on the left-hand side. The triangular patterns are seen in both the experiment and calculation at z=H/2 and H. The color scales in the experiment and calculation are common for (a)–(f).
Fig. 7. Comparison of the light propagation images in the class-1 hcp cluster obtained in the experiment and that obtained in the calculation. Refer to the caption of Fig. 6 for details. The hexagonal pattern and the wing-shaped patterns (yellow lines) are observed in both of the experiment and calculation at z=H and 0, respectively. The appearance of an intense spot at the center at z=0 (green circle) is a characteristic feature for the clusters comprising non-fluorescent silica particles.
Fig. 8. Comparison of the light propagation images in the class-2 fcc cluster obtained in the experiment and that obtained in the calculation. From left to right: the illustrations presenting the position of the focal plane, experimental results, calculation results (blue indicates zero and red indicates the maximum), and the three-value map obtained by digitizing the calculation result. The distribution of the light intensity was obtained in the range of z=0-+H at a step of H/2. Black dashed lines in the calculation image of (a) indicates the position of the spheres of the first layer. The portions in the cluster between the focal plane and the objective lens are indicated in green in the illustrations on the left-hand side. The color scales in the experiment and calculation are common for (a)–(c) and also for Figs. 13 (a)–(c).

Because of the imperfections in the fabricated class-2 clusters, a clear agreement between the experiment and the calculation was not obtained. Consequently, we fabricated intermediate clusters that are larger than the trivial class-1 structures but smaller than the class 2 ones in order to realize precise clusters. The intermediate clusters are different from the class 2 ones in that the former only have fluorescent particles and no non-fluorescent ones in the third layer. We call such intermediate clusters class 1.5. The fabricated clusters are shown in Fig. 10; these have no discernible distortion. The experimental and calculation results are shown in Figs. 11 and 12, respectively. Both the experimental and the calculated results exhibit very similar features. This excellent agreement implies that some approximations in the calculation, e.g., neglecting the substrate, discrete distribution of the dipoles, and use of the Poynting vector as the observed quantity, are reasonable. Moreover, the agreement also supports the suggestion that the disagreement in the class-2 hcp cluster originated from the lattice disorder.

Fig. 9. Comparison of the light propagation images in the class-2 hcp cluster obtained in the experiment and that obtained in the calculation. Refer to the caption of Fig. 8 for details. The one-third part at the upper left of the experimental image is indicated by the white lines in (c). The color scales in the experiment and calculation are common for (a)–(c) and also for Figs. 14 (a)–(c).
Fig. 10. Scanning micrographs of the clusters in which all the non-fluorescent particles in the third layer are removed from the class-2 clusters; these clusters are referred to as class-1.5 clusters. The fluorescent particles are indicated by red circles. The bars indicate 1 µm. (a) fcc and (b) hcp. The arrows in (b) show the possible propagation paths that dominate the emergence of the three bright spots shown in Fig. 12.
Fig. 11. Comparison of the light propagation images in the class-1.5 fcc cluster obtained in the experiment and that obtained in the calculation. Refer to the caption of Fig. 8 for details. The color scales in the experiment and calculation are common for (a)–(c).
Fig. 12. Comparison of the light propagation images in the class-1.5 hcp cluster obtained in the experiment and that obtained in the calculation. Refer to the caption of Fig. 8 for details. The color scales in the experiment and calculation are common for (a)–(c).

5. Light propagation in larger clusters

Each domain in a self-assembled array can be considered as a cluster with a much larger size. Since our calculation has been confirmed to be sufficiently reliable in the previous section, much larger clusters will be discussed on the basis of this calculation. In this section, lmax is limited to 9 due to the limitation of the computing power. Nevertheless, it has been confirmed that no significant errors appear.

Figures 13 and 14 show the calculation results for class-3 and class-4 clusters, respectively, along with the propagation patterns observed in the self-assembled array. In both results of class 3, the propagation intensity toward the surrounding spheres is remarkable. However, as the number of surrounding spheres increases (class 4), the propagation modes become closer to that of the self-assembled array. Therefore, the domains in the self-assembled array in which the clear triangular or hexagonal patterns were exhibited were presumed to be larger than class 4.

Fig. 13. Comparison of the six-dot-triangular patterns in the experimental images for the self-assembled array and in the calculated light propagation images for the fcc clusters of class 3 and class 4. From left to right: the illustrations presenting the position of the focal plane, triangular patterns in the self-assembled array, and calculation results (blue indicates zero and red indicates the maximum) of class 3 and class 4. The z values in the self-assembled array are determined in the same manner as that shown in Fig. 3. The distribution of the light intensity was obtained in the range of z=0-+H at a step of H/2. Black dashed lines in the calculation image of (a) indicate the positions of the spheres of the first layer. The portions in the cluster between the focal plane and the objective lens are indicated in green in the illustrations on the left-hand side. The color scales in the experiment and calculation are common for (a)–(c) and also for Figs. 8 (a)–(c).
Fig. 14. Comparison of the seven-dot-hexagonal patterns in the experimental images for the self-assembled array and in the calculated light propagation images for the fcc clusters of class 3 and class 4. Refer to the caption of Fig. 13 for details. The color scales in the experiment and calculation are common for (a)–(c) and also for Figs. 9 (a)–(c).

6. Discussion

In contrast to the class-1 clusters, in the class-1.5 and class-2 clusters shown in Figs. 8–12, light propagates toward the surrounding spheres and exhibits complex patterns. This implies that the outward propagation in the first and second layers is considerable. In addition, the propagation in the class-1.5 clusters is also different from that of the class-2 clusters, which indicates that the structure of the third layer also affects the propagation toward the first layer. One of the most significant discrepancies between class-1.5 and class-2 clusters is the three bright spots at the outer edges in the class-1.5 hcp cluster [Fig. 12 (a)]. These spots are positioned on the lines from the fluorescent sphere to the second nearest spheres, as shown by the arrows in Fig. 10(b). This propagation could be due to the contribution of the specular resonance in bispheres; this is known to occur in spheres with a similar size parameter [37

H. T. Miyazaki, H. Miyazaki, and K. Miyano, “Anomalous scattering from dielectric bispheres in the specular direction,” Opt. Lett. 27, 1208–1210 (2002). [CrossRef]

, 38

H. T. Miyazaki, H. Miyazaki, and K. Miyano, “Analysis on specular resonance in dielectric bispheres using rigorous and geometrical-optics theories,” J. Opt. Soc. Am. A 20, 1771–1784 (2003). [CrossRef]

]. The presence of the non-fluorescent spheres around the fluorescent particle in the third layer of the class-2 clusters would disturb these propagation paths; thus, the three bright spots are no longer apparent in the class-2 clusters.

From the results of much larger clusters (Figs. 13 and 14), the propagation can be regarded as the superposition of two propagation modes with different characteristics, i.e., one gives unique geometrical patterns at the center and the other that is diffused outwards. Here, the former is referred to as the central propagation and the latter as the outward propagation. We discuss these modes quantitatively. Through the discussion, the intensity of the central propagation is defined as the summation of the intensity of the light that propagates through the six or seven spheres at the center of the fcc or hcp clusters, respectively, while the outward propagation intensity is defined as the summation of the intensity through other spheres. These intensities were calculated at the cross section of z=0, which is the center of the first layer. Figure 15(a) shows the relationship between these intensities and the size of the clusters. The central propagation intensity decreases until class 2 and remains constant in larger classes. This result indicates that the central propagation is fundamentally independent of the cluster size; however, the class-1 and class-1.5 clusters are special exceptions. The enhanced central propagation in the class-1 clusters can be attributed to light confinement to the central (six or seven) spheres due to the absence of the surrounding spheres. Moreover, the comparison between the class-1.5 and class-2 clusters suggests that the non-fluorescent particles surrounding the fluorescent particle in the third layer contribute to the distribution of light to the outward propagation. The intensity of the central propagation in the fcc clusters agrees very well with that in the hcp clusters. This implies that the central propagation intensity is determined by the upper two layers since the arrangement of the spheres in the fcc and hcp clusters are common in the upper second and third layers but different only in the first layer. On the other hand, the outward propagation intensity increases in accordance with the cluster size. This indicates that a greater amount of light from the fluorescent particle can be trapped by the entire cluster as the cluster size increases. The good agreement in the outward propagation intensity between the fcc and hcp clusters shows that the diffusion of light is not significantly affected by the lattice structure.

Fig. 15. (a) The relationship between the integrated intensities of the two propagation modes (central and outward propagations) and the cluster sizes. (b) Similar to (a), except that the vertical axis shows the intensities per sphere. The red and blue lines correspond to the fcc and hcp clusters, respectively. The solid lines indicate the central propagation and the dashed lines indicate the outward propagation. The intensity ratio of the central propagation to the outward propagation in (b) gives the contrast of the triangular or hexagonal patterns to the background intensity in the propagation images.

Figure 15(b) shows these propagation intensities per sphere. This result represents the contrast between the central propagation and the outward propagation in the images shown in Figs. 6–9 and 11–14. The two propagation intensities are most similar in the class-2 clusters. This corresponds to the remarkable outward propagation seen in Figs. 8 and 9. On the other hand, the outward propagation intensity per sphere monotonously decreases with the cluster size. This yields the feature that the central propagation gradually becomes dominant with an increase in the cluster size, as shown in Figs. 13 and 14.

Finally, we would like to remark on the relationship between the light propagation in the clusters and the photonic band structures. Yamilov et al. showed that the photonic band structures begin to be formed in a cluster composed of only five spheres [39

A. Yamilov and H. Cao, “Density of resonant states and a manifestation of photonic band structure in small clusters of spherical particles,” Phys. Rev. B 68, 085111 (2003). [CrossRef]

]. Therefore, the photonic bands are expected to contribute to the light propagation in our clusters as well. The frequency region discussed here is much higher than those in the usual discussion on photonic bands. In terms of the dimensionless frequency, our fcc system corresponds to ω=d/λ=2.31, where d=√2D is the lattice constant of a fcc lattice, while the low-frequency region of ω≲1.5 has usually been discussed [40

K. Ohtaka and Yukito Tanabe, “Photonic Band Using Vector Spherical Waves. I. Various Properties of Bloch Electric Fields and Heavy Photons,” J. Phys. Soc. Jpn. 65, 2265–2275 (1996). [CrossRef]

]. In general, a large number of bands are densely formed at ω≲1, and therefore, photons are propagated almost freely in the crystals. Light diffusion in our clusters regardless of the lattice structures can also be interpreted as the result of the densely formed band structures.

We also add a comment about the influence of the resonance of the individual spheres. In systems made of larger spheres, coupling of whispering gallery modes has been discussed [41

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-Binding Photonic Molecule Modes of Resonant Bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999). [CrossRef]

, 34

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62, 7976–7997 (2000). [CrossRef]

]. However, in our systems, the resonance of the spheres seems to be less remarkable. We have analyzed the electromagnetic energy of individual spherical modes in the clusters, and found that many low-order modes of l≲5 equally contribute to the propagation and there is no dominating resonance mode. A similar discussion can be found in an article on the light scattering by bispheres [38

H. T. Miyazaki, H. Miyazaki, and K. Miyano, “Analysis on specular resonance in dielectric bispheres using rigorous and geometrical-optics theories,” J. Opt. Soc. Am. A 20, 1771–1784 (2003). [CrossRef]

].

7. Conclusion

In summary, we have proved that the triangular and hexagonal patterns previously discovered in self-assembled multilayer arrays of micrometer-sized spheres including a small number of fluorescent particles originate from the local domain structures, i.e., fcc and hcp lattices. Although this has already been suggested by Matsushita et al., we have provided direct evidence through a cross-check between an experiment using well-defined clusters fabricated by micromanipulation and a calculation based on the expansion of vector spherical harmonics. We also clarified that the propagation patterns depend on the cluster size. While a minimum, class-1, fcc (hcp) cluster exhibits a six-dot-triangular (seven-dot-hexagonal) pattern, complex patterns appear in larger, class-2, clusters; the relationship between the patterns and the lattice structures is not straightforward. However, on the basis of a quantitative discussion based on the calculation of the light propagation in much larger clusters, clear triangular (hexagonal) patterns were found to emerge in fcc (hcp) clusters as the lattice size increased. The light from the fluorescent particle is divided into the central propagation path and the outward propagation path, and the former gives these geometrical patterns. The latter is strongly diffused around the central core and becomes less remarkable as the cluster size increases. This implies that the self-assembled multilayer array that exhibited clear triangular and hexagonal patterns has ordered domains to a sufficient extent. Although our calculation includes several approximations, the obtained results were found to be sufficiently credible judging by the excellent agreement with the experimental results.

The numerical calculation technique enables a systematic and quantitative analysis for various size parameters and refractive indices of the spheres; this will be studied in the future. Although we used silica particles as non-fluorescent spheres in this study, it is important to confirm the emergence of unique propagation patterns in systems comprising only PSt particles. Furthermore, it is necessary to investigate the intermediate range of S=3-5 because Matsushita et al. predicted a drastic change in the patterns at S=3.63 (ω=1.63, or ω=1 by their definition). In this low-frequency region, the photonic band structures could have a significant role in the appearance of the unique propagation patterns.

Acknowledgments

The authors thank S. Matsushita and S. Fujiyoshi for their helpful discussions. This study was supported by Grant-in-Aids for Scientific Research and by the Nanotechnology Network Program of the Ministry of Education, Culture, Sports, Science and Technology.

References and links

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2.

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

3.

S. Noda and T. Baba, Roadmap on Photonic Crystals (Kluwer Academic, Dordrecht, 2003).

4.

K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, “Fabrication of Two-Dimensional Photonic Band Structure with Near-Infrared Band Gap,” Jpn. J. Appl. Phys. 33, 1463–1465 (1994). [CrossRef]

5.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature (London) 383, 699–702 (1996). [CrossRef]

6.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 251–253 (1998). [CrossRef]

7.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425, 944–947 (2003). [CrossRef]

8.

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005). [CrossRef] [PubMed]

9.

T. Baba, N. Fukaya, and J. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999). [CrossRef]

10.

N. Ikeda, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, J. S. Jensen, O. Sigmund, P. I. Borel, M. Kristensen, and K. Asakawa, “Topology optimized photonic crystal waveguide intersections with high-transmittance and low crosstalk,” Electron. Lett. 42, 1031–1033 (2006). [CrossRef]

11.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nature Physics 3, 401–405 (2007). [CrossRef]

12.

V. N. Astratov, M. S. Skolnick, S. Brand, T. F. Krauss, O. Z. Karimov, R. M. Stevenson, D. M. Whittaker, I. Culshaw, and R. M. De la Rue, “Experimental technique to determine the band structure of two-dimensional photonic lattices,” IEE Proc.: Optoelectron. 145, 398–402 (1998). [CrossRef]

13.

T. Yamasaki and T. Tsutsui, “Fabrication and Optical Properties of Two-Dimensional Ordered Arrays of Silica Microspheres,” Jpn. J. Appl. Phys. 38, 5916–5921 (1999). [CrossRef]

14.

K. Ohtaka, Y. Suda, S. Nagano, T. Ueta, A. Imada, T. Koda, J. S. Bae, K. Mizuno, S. Yano, and Y. Segawa, “Photonic band effects in a two-dimensional array of dielectric spheres in the millimeter-wave region,” Phys. Rev. B 61, 5267–5279 (2000). [CrossRef]

15.

H. T. Miyazaki, H. Miyazaki, K. Ohtaka, and T. Sato, “Photonic band in two-dimensional lattices of micrometer-sized spheres mechanically arranged under a scanning electron microscope,” J. Appl. Phys. 87, 7152–7158 (2000). [CrossRef]

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24.

At an early stage of our research, we investigated light propagation in clusters solely composed of polystyrene (PSt) particles as in the paper by Matsushita et al. However, we found that non-fluorescent PSt particles become fluorescent upon electron beam (EB) irradiation during the micromanipulation. It was difficult to distinguish the luminescence from the dye-doped and undoped PSt particles. Since silica particles were found to be unaffected by the EB irradiation, we employed silica as non-fluorescent particles.

25.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, I. B. Ivanov, and H. Yoshimura, and K. Nagayama, “Mechanism of Formation of Two-Dimensional Crystals from Latex Particles on Substrates,” Langmuir 8, 3183–3190 (1992). [CrossRef]

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28.

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29.

F. Garcia-Santamaria, H. T. Miyazaki, A. Urquia, M. Ibisate, M. Belmonte, N. Shinya, F. Meseguer, and C. Lopez, “Nanorobotic manipulation of microspheres for on-chip diamond architectures,” Adv. Mater. 14, 1144–1147, (2002). [CrossRef]

30.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, “Microassembly of semiconductor three-dimensional photonic crystals,” Nat. Mater. 2, 117–121 (2003). [CrossRef] [PubMed]

31.

H. T. Miyazaki, H. Miyazaki, N. Shinya, and K. Miyano, “Enhanced light diffraction from a double-layer microsphere lattice,” Appl. Phys. Lett. 83, 3662–3664 (2003). [CrossRef]

32.

H. T. Miyazaki, H. Miyazaki, Y. Jimba, Y. Kurokawa, N. Shinya, and K. Miyano, “Light diffraction from a bilayer lattice of microspheres enhanced by specular resonance,” J. Appl. Phys. 95, 793–805 (2004). [CrossRef]

33.

T. Kasaya and H. T. Miyazaki, “Graphical Templates for Accurate Micromanipulation in a Scanning Electron Microscope,” Rev. Sci. Instrum. , submitted. [PubMed]

34.

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62, 7976–7997 (2000). [CrossRef]

35.

The directions of the eight inverted points are [±1±1±1], [±1±1∓5], [±1∓5±1], and [∓5±1±1].

36.

The dye molecules doped in the PSt particles gradually degrade upon electron beam irradiation. Therefore, the time allowed for the manipulation is limited. Since the top layer of the class-2 or larger clusters is composed of a large number of particles, it was difficult to assemble sufficiently accurate lattices within the limited time.

37.

H. T. Miyazaki, H. Miyazaki, and K. Miyano, “Anomalous scattering from dielectric bispheres in the specular direction,” Opt. Lett. 27, 1208–1210 (2002). [CrossRef]

38.

H. T. Miyazaki, H. Miyazaki, and K. Miyano, “Analysis on specular resonance in dielectric bispheres using rigorous and geometrical-optics theories,” J. Opt. Soc. Am. A 20, 1771–1784 (2003). [CrossRef]

39.

A. Yamilov and H. Cao, “Density of resonant states and a manifestation of photonic band structure in small clusters of spherical particles,” Phys. Rev. B 68, 085111 (2003). [CrossRef]

40.

K. Ohtaka and Yukito Tanabe, “Photonic Band Using Vector Spherical Waves. I. Various Properties of Bloch Electric Fields and Heavy Photons,” J. Phys. Soc. Jpn. 65, 2265–2275 (1996). [CrossRef]

41.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-Binding Photonic Molecule Modes of Resonant Bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999). [CrossRef]

OCIS Codes
(170.2520) Medical optics and biotechnology : Fluorescence microscopy
(220.4000) Optical design and fabrication : Microstructure fabrication
(290.0290) Scattering : Scattering
(290.5850) Scattering : Scattering, particles
(350.5500) Other areas of optics : Propagation

ToC Category:
Scattering

History
Original Manuscript: October 7, 2008
Revised Manuscript: November 17, 2008
Manuscript Accepted: November 23, 2008
Published: December 1, 2008

Citation
T. Fujishima, H. T. Miyazaki, H. Miyazaki, Y. Jimba, T. Kasaya, K. Sakoda, Y. Ogawa, and F. Minami, "Light propagation from a fluorescent particle embedded in a photonic cluster of micrometer-sized dielectric spheres," Opt. Express 16, 20706-20723 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20706


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References

  1. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  2. S. John, "Strong Localization of Photons in Certain Disordered Dielectric Superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  3. S. Noda and T. Baba, Roadmap on Photonic Crystals (Kluwer Academic, Dordrecht, 2003).
  4. K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J. W. Haus, "Fabrication of Two-Dimensional Photonic Band Structure with Near-Infrared Band Gap," Jpn. J. Appl. Phys. 33, 1463-1465 (1994). [CrossRef]
  5. T. F. Krauss, R. M. De La Rue, and S. Brand, "Two-dimensional photonic-bandgap structures operating at nearinfrared wavelengths," Nature (London) 383, 699-702 (1996). [CrossRef]
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