Light propagation is simulated through coupled-resonator optical waveguides (CROWs) composed of seven transparent polystyrene microspheres, including micro-joints formed between the spheres. In nanojet-induced mode (NIM) light propagation, the micro-joints increased the optical coupling between microspheres drastically, and the light confinement by individual microspheres weakened as the micro-joint diameter increases. These results suggest that we can control NIM light propagation by changing the micro-joint diameter; this amounts to a nanojet throttle valve.
©2011 Optical Society of America
A dielectric sphere with a diameter of a few micrometers confines light going around its circumference and acts as an optical resonator [1–4]. By connecting optical resonators, we can induce resonance among them and can expect to guide light in arbitrary shapes with a wavelength-scale curvature. These connected optical resonators, such as microrings, microdisks, or microspheres are called coupled-resonator optical waveguides (CROWs) [3,5,6]. In CROW, because the coupling between resonators is “weak”, the light propagation along CROW is suppressed and the group velocity of light slows. This concept can be applied to optical buffer memory [7,8], rotation detection and measurement , and group velocity compensation . Moreover, some resonator-based optical devices, such as those for signal modulation, switching, and memory functions, have already been demonstrated [11,12]. Supposing that we can increase or decrease the coupling rate in a wider range, we will be able to realize better switching devices.
By using microspheres doped with CdSe nanocrystals, Möller and co-authors showed that the diameter accuracy of a microsphere should be better than < 0.05% in order to induce resonance and coupling among the microspheres . Hara and co-authors found that their numerical model is consistent with the experimental results of transparent polystyrene-microsphere chains fabricated in air or in a vacuum . On the other hand, the polystyrene-microsphere chains fabricated by a dewetting process in colloidal suspension show long-range propagation of about 20 microspheres, even though the deviation of the diameters exceeds 1.1% [6,13–16]. The difference between these two results should be attributed to differences between fabrication methods; especially, whether or not the chains are fabricated in water is essential for propagation with high efficiency. Recently, by using high-resolution scanning electron microscopy (HR-SEM), Mitsui and co-authors found that the neighboring polystyrene microspheres of CROWs fabricated in water are connected with partly dissolved polystyrene . They called this connection a micro-joint and explained the light propagation of whispering gallery mode (WGM) components by a finite difference time domain (FDTD) simulation including the influences of the micro-joints.
In the case of straight chains, the long-range propagation can be explained by the concept of a photonic nanojet consisting of focused spots with elongated shapes by spherical or cylindrical resonators. Chen and co-authors showed that the nanojets can be periodically reproduced along a chain of microspheres [18,19]. This quasiperiodic pattern of coupled nanojets, termed nanojet-induced modes (NIMs), has been observed in chains of polystyrene microspheres [14,16]. The NIMs are rather tolerant of the presence of disorder and show broad peaks. However, the influences of the micro-joints have never been included in a discussion of optical properties of NIM propagation, and the influences have not been understood sufficiently.
Though some theoretical [19–22] and experimental  studies have been reported based on the WGM concept in the case of bended or branched chains, the concept of micro-joints has also never been introduced in a discussion of WGM propagation. Benyoucef and co-authors investigated the Q-factors in coupled WGMs consisting of two GaAs microdisks spaced about 90 nm apart; they controllably increased the refractive index of a microdisk by local laser heating, and finally, they caused a significant enhancement (reduction) of the participating Q values . However, the possibility of micro-joint formation by laser heating has not been understood sufficiently.
In the present study, we first observe the detailed structure of micro-joints by HR-SEM and discuss the formation mechanism of micro-joints. We then perform FDTD simulations in straight-chain models of microspheres, including the micro-joints, and investigate the optical properties of propagating light through the chain when we change the micro-joint diameter.
2. Detail mechanism of micro-joints formation between polystyrene microspheres
To fabricate the micro-joints, we aligned and connected microspheres by using a self-assembly technique in colloidal suspension within pure water on a lithographically patterned substrate [25–30]. Figure 1(a) is a schematic of the self-assembly technique. First, we fabricate a patterned substrate that has a lot of dimples with tetragonal symmetry. The dimples have an inverse shape of a frustum of a tetragonal pyramid; they are arranged for the template of CROWs and are spaced 2.0 μm apart from center to center. Next, we placed a glass coverslip on the patterned substrate with a small gradient (1–2 °). Next, a colloidal suspension of polystyrene or borosilicate glass microspheres was put inside the gap and then slowly evaporated. The microspheres in the meniscus were then dragged by the capillary force and definitely trapped in the dimples. Since the distance between dimples and the diameter of the microspheres were the same, the microspheres trapped in the nearest neighbors were in close contact with each other. This technique was described in detail in Ref .
Figures 1(b) and 1(c) are the HR-SEM images of the CROWs neighboring points, which consist of polystyrene and borosilicate glass microspheres, respectively. In Fig. 1(b), we note that the neighboring polystyrene microspheres are connected by micro-joints with a diameter of about 300 nm. In addition, the polystyrene microspheres keep a nearly globular shape, and the micro-joints look like extra viscous liquid. On the other hand, Fig. 1(c) shows that the neighboring borosilicate glass microspheres are perfectly separated. In both CROWs, we did not use any bonding agents for curing, unlike the case in Ref , except for the polystyrene microspheres dispersed in pure water. Moreover, since Figs. 1(b) and 1(c) were taken in a vacuum, water as a simple substance should have evaporated.
Here, we notice the difference in chemical properties between polystyrene and borosilicate glass. Figure 2 is a schematic of the micro-joint formation mechanism. In the case of polystyrene microspheres, hydrophilic initiator residues at their terminals are exposed to water phase and protect the hydrophobic polystyrene. Therefore, the surface layer of polystyrene microspheres is strongly hydrated and slightly swollen. If two microspheres are close to each other, the neighbors should have a large contact area due to surface tension. During the drying process, polystyrene microspheres should shrink slightly, and the contact area should be left as a micro-joint.
There is a possibility that the micro-joints shrink and break as a consequence of dehydration. However, observations by conventional optical microscopy have revealed that the micro-joint rarely breaks by dehydration. This is because the chain polymers, once entangled, are not easily broken.
3. Model of FDTD simulation and spectra of propagated light
The micro-joints might be able to drastically increase the optical coupling between microspheres. Therefore, the influence of micro-joints should be included in the model of FDTD simulation. However, to investigate the optical properties of propagating light through the CROW that has micro-joints, we first simulated the spectra of propagated light without and with micro-joints.
A schematic of the model for FDTD simulation is shown in Fig. 3(a) . The simulations were performed in 3 dimensional models. We noted the electric intensity mapping in X-Y plane, and depicted the results in 2 dimensional mapping. We performed the simulations by using the FDTD Solutions system of Lumerical Solutions, Inc. Figure 3(a) indicates a plan-view schematic illustration of a CROW of a straight chain of seven microspheres having a diameter of 2 μm as a model for our FDTD simulation. In this model, the refractive indices of inside and outside of microspheres are 1.59 (polystyrene) and 1.00 (air), respectively, and the boundary condition assumed at the boundary of the simulation domain is a perfectly matched layer (PML). For a light source, an oscillating point-dipole parallel to the X- and Y-axes is set at a point about 0.1 μm inwards from the end of the CROW on the X-axis. The reason why we set a light source at this point about 0.1 μm inwards is the following; we expect that the light causes total reflection along the circumference of the microsphere, and the light excites WGMs within it. If the light source is set at more inward point, the intensities of WGMs should be smaller. The reason why we set a light source on the X-axis is to simplify the NIM light propagation. When the light source is set at a point deviated from the X-axis, we have already found that the light propagates with rather complicated zigzagged path. Therefore, in this article, we only note the case that the light source is just on the X-axis. The results of zigzagged propagation will be shown at any other place in future. On the other hand, the measuring point in which the spectra of propagated light are calculated is set at about 0.1 μm outwards from the other end of the CROW on the X-axis. The spectra of propagated light without and with 0.6 μm-diameter micro-joints are shown in Figs. 3(b) and 3(c), respectively. The blue and red lines in Figs. 3(b) and 3(c) represent the spectra of propagated light from point-dipole polarized parallel to the X- and Y-axes, respectively.
In Fig. 3(b), the highest-intensity peak of the red line appears at a wavelength of about 500 nm, and most of the intensity peaks of blue and red lines do not overlap. On the other hand, in Fig. 3(c), most of the intensity peaks of two lines become broader, and cause overlapping. This result suggests the micro-joints have an influence on the optical properties of propagating light. Especially, at a wavelength of 505 nm, we note that the intensity of the polarized light parallel to the Y-axis with micro-joints is rather large though the polarized light hardly reaches the measuring point without micro-joints. Here, with micro-joints, since the intensities of the X- and Y-polarized light are relatively approximate to each other at a wavelength of 505 nm, we simulate light propagation at a wavelength of 505.248 nm including the influence of micro-joints. In addition, Figs. 3(b) and 3(c) also show the change in relative intensities among intensity peaks, and show the shift in wavelength of intensity peaks. However, because these results are so complicated, we only note the light propagation at this wavelength of 505.248 nm.
4. Influences of micro-joints in NIM light propagation
Figure 4(a) is a schematic of the model for FDTD simulation, CROWs being straight chains of seven microspheres with micro-joints. The light source is set at the same point as shown in Fig. 3(a), and the polarization direction is parallel to the Y-axis. The simulation was performed at a 505.248 nm wavelength. In this model, the micro-joints are shaped like disks that fill the gaps between neighboring microspheres. We have performed FDTD simulations in cases where the micro-joint diameters are 0 μm, 0.2 μm, 0.4 μm, and 0.6 μm. Figures 4(b) to 4(e) show the intensity mappings of propagating light simulated using the model in which the micro-joint diameters are 0 μm, 0.2 μm, 0.4 μm, and 0.6 μm, respectively. In addition, the intensity line profiles of propagating light on the X-axis in Figs. 4(b) to 4(e) are indicated in Fig. 4(f).
Figures 4(b) to 4(d) show a large electric field appears across the centers of the microspheres, and show rather weak resonance in the WGM appears around the circumferences of the microspheres. Because it is slightly difficult to cause resonance with WGM for the polarized light emitted from the light source directly, the origin of the WGM resonance should be the scattered components at the contacted points of microspheres. On the other hand, the large electric fields across the centers of the microspheres should indicate light propagation with NIMs in which the microspheres act as periodically coupled spherical lenses . Moreover, these large electric fields show local field enhancements at the contacting points of microspheres or in the micro-joints. Then, we note that the local field enhancement is the largest when the micro-joint diameter is 0.2 μm as indicated in Fig. 4(c). Because this value is smaller than the half of wavelength, the micro-joints should act as sub-wavelength scale aperture. Therefore, the local field enhancement should be attributed to the micro-joint and evanescent-field around it. Supposing that a case of closed aperture, i.e., there is no micro-joint as indicated in Fig. 4(b), NIM light propagation should be blocked and should be attenuated strongly. When the micro-joint diameter becomes larger than the wavelength, the shape of electric field shift from that of NIM light propagation to that of far-field type light propagation as shown in Fig. 4(e).
In Fig. 4(b), we note that the intensity of propagated light at the point of (X, Y) = (11.1, 0) is very weak. Since the simulation model of Fig. 4(b), whose micro-joints have zero diameter, means that the simulation model is perfectly equal to that in Fig. 3(a), the weak intensity at the point of (X, Y) = (11.1, 0) is consistent with the spectrum of the red line in Fig. 3(b). However, in the case of micro-joints with a 0.6-μm diameter indicated in Fig. 4(d), the intensity of propagated light is rather strong. These results suggest that the micro-joints increase the optical coupling between microspheres drastically; this is also indicated in Fig. 3(c). Therefore, if we can change the micro-joint diameter arbitrarily—that is, if we can create nanojet throttle valves—we can control NIM light propagation. Moreover, the intensity line profiles indicated in Fig. 4(f) show that the position of maximum intensity, except for the contacting points between microspheres, shifts to the positive direction of the X-axis in proportion to the increase in the micro-joint diameter. The positions of the maximum intensity except for the contacting points in Fig. 4(f) are about X = 2.0 μm for 0 μm-diameter micro-joints case, about X = 6.0 μm for 0.2 μm-diameter micro-joints case, and about X = 9.5 μm for 0.4 μm-diameter micro-joints case. This result suggests that the light confinement by individual microspheres weakens as the micro-joint diameter increases, and also suggests that the light can reach a farther point with NIM light propagation. In other words, by increasing the micro-joint diameter, the coupling coefficient among individual resonators increases, and individual resonators shift to long-chain coupled resonators, which finally become a simple waveguide ultimately when the micro-joint diameter is equal to that of the microspheres.
The SEM image of the CROW in Fig. 1(b) shows that the micro-joint diameter in the actual experiment is about 0.3 μm. Therefore, in actual experiments, the optical properties of light propagation should resemble those indicated in Figs. 4(c) or 4(d). Since Figs. 4(c) and 4(d) suggest that the NIMs component of propagating light shows rather long range propagation, the influence of micro-joint should be appeared in CROW as indicated in Fig. 1(b) [15,17].
5. Influences of micro-joints in WGM light propagation
Figure 5(a) shows a schematic of the model for FDTD simulation with micro-joints in a case where the polarization direction of the light source is parallel to the X-axis. The other conditions of FDTD simulation are the same as those indicated in Fig. 4(a). Figures 5(b) to 5(e) show the intensity mappings of propagating light simulated on the model in which the micro-joint diameters are 0 μm, 0.2 μm, 0.4 μm, and 0.6 μm, respectively. The intensity line profiles of propagating light on the X-axis in Figs. 5(b) to 5(e) are indicated in Fig. 5(f).
As Figs. 5(b) to 5(e) show, a resonant mode of WGM appears around the circumference of the microspheres and the WGM component propagates to the point of (X, Y) = (11.1, 0). Moreover, as shown in Fig. 5(f), the intensity at the point of X = 11.1 μm for 0.6 μm-diameter micro-joints case is slightly larger than the intensity for 0 μm-diameter micro-joints case. This result is consistent with the spectra indicated by blue line in Figs. 3(b) and 3(c); the intensity peak of the polarized light parallel to the X-axis with micro-joints is slightly larger than that of without micro-joints in the wavelength of 505.248 nm. These two results suggest that the micro-joints increase the optical coupling between microspheres. However, the distributions indicated in the intensity mappings do not show any obvious difference among them. One plausible reason for this is that the wavelength of 505.248 nm does not meet the condition of resonance phase-matching to WGMs. Another possible reason for this is that our FDTD simulation model uses exactly the same microsphere diameter, i.e., there is no deviation in diameter. This is because the resonance between the microspheres is very sensitive to their diameter, as mentioned in the Introduction [2,3]. If a microsphere chain includes a deviation in diameter, then WGM light propagation will be stopped at the deviated microsphere. Since the deviation in diameter of microsphere was not included in the simulation, any drastic influence of micro-joints on WGM light propagation was not found.
We investigated the influence of micro-joints between neighboring microspheres on the optical properties of propagating light in microsphere CROWs by FDTD simulation. We found that the micro-joints increase the optical coupling between microspheres drastically in the NIM light propagation. In this propagation, the light confinement by individual microspheres of the CROWs in the NIM components weakens as the micro-joint diameter increases, and the light can reach a farther point. This result suggests that we can control the NIM light propagation essentially by forming nanojet throttle valves, i.e., by changing the micro-joint diameter. In this article, we did not find any drastic influence of micro-joints on WGM light propagation. The FDTD simulation, including the effect of deviation on microsphere diameter, should be examined in future.
This study was carried out as a collaborative project between the Institute of Multidisciplinary Research for Advanced Materials of Tohoku University and the National Institute for Materials Science. This study was financially supported by a Grant-in-Aid for Scientific Research (B) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of the Japanese Government (19310092). A part of this work was supported by MEXT’s “Nanotechnology Network Project”.
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