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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 17351–17361
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Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres

Vasily N. Astratov and Shashanka P. Ashili  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 17351-17361 (2007)
http://dx.doi.org/10.1364/OE.15.017351


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Abstract

Using techniques of flow-assisted self-assembly we synthesized three-dimensional (3D) lattices of dye-doped fluorescent (FL) 5 µm polystyrene spheres with 3% size dispersion with well controlled thickness from one monolayer up to 43 monolayers. In FL transmission spectra of such lattices we observed signatures of coupling between multiple spheres with nearly resonant whispering gallery modes (WGMs). These include (i) splitting of the WGM-related peaks with the magnitude 4.0–5.3 nm at the average wavelength 535 nm, (ii) pump dependence of FL transmission showing that the splitting is seen only above the threshold for lasing WGMs, and (iii) anomalously high transmission at the WGM peak wavelengths compared to the background for samples with thickness around 25 µm. We propose a qualitative interpretation of the observed WGM transport based on an analogy with percolation theory where the sites of the lattice (spheres) are connected with optical “bonds” which are present with probability depending on the spheres’ size dispersion. We predict that the WGM percolation threshold should be achievable in close packed 3D lattices formed by cavities with~103 quality factors of WGMs and with ~1% size dispersion. Such systems can be used for developing next generation of resonant sensors and arrayed-resonator light emitting devices.

© 2007 Optical Society of America

1. Introduction

In the case of microspheres, the cavities can be micromanipulated and sorted individually which opens a possibility to select much more uniform resonators [10

10. 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]

,12

12. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Photonic molecule lasing,” Opt. Lett. 28, 2437–2439 (2003). [CrossRef] [PubMed]

,15

15. Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, “Fine structure of coupled optical modes in photonic molecules,” Phys. Rev. A 70, 051801(R) (2004). [CrossRef]

17

17. B. M. Möller, U. Woggon, and M. V. Artemyev, “Coupled-resonator optical waveguides doped with nanocrystals,” Opt. Lett. 30, 2116–2118 (2005). [CrossRef] [PubMed]

] on the basis of spectroscopic characterization of their WGM peak positions. These techniques allow selecting spheres with the size uniformity ~0.03%. The band structure effects due to tight-binding model [16

16. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005). [CrossRef] [PubMed]

] as well as normal mode splitting effects [10

10. 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]

,15

15. Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, “Fine structure of coupled optical modes in photonic molecules,” Phys. Rev. A 70, 051801(R) (2004). [CrossRef]

,17

17. B. M. Möller, U. Woggon, and M. V. Artemyev, “Coupled-resonator optical waveguides doped with nanocrystals,” Opt. Lett. 30, 2116–2118 (2005). [CrossRef] [PubMed]

] have been observed in supermonodispersive spheres. However the sorting of large numbers of microspheres (Q>104) with overlapping positions of WGM peaks still remains a challenging problem.

The optical transport in disordered mesoscopic systems of coupled cavities can be compared with the case of random waveguides [3

3. M. Stoytchev and A. Z. Genack, “Measurement of the probability distribution of total transmission in random waveguides,” Phys. Rev. Lett. 79, 309–312 (1997). [CrossRef]

,4

4. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998). [CrossRef]

] formed by wavelength scale scatterers. In random waveguides the importance of interference phenomena such as strong fluctuations in the transmitted intensity through a disordered sample is determined by the ratio Nl/d, where N is the number of transverse propagating channels in the waveguide, d its length and l is the elastic mean free path. To observe strong fluctuations, it is important to achieve as low values of Nl/d as possible [4

4. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998). [CrossRef]

]. In coupled cavities, the light effectively propagates along the lines connecting close neighbors forming a network. Changing direction or “scattering” effectively takes place at the touching points between the cavities, so the elastic mean free path l can be associated with the size of the cavities. In the case of large scale 2D or 3D networks of coupled resonators, the number of transverse paths N can be extremely high meaning that the interference phenomena can be averaged.

It is interesting to note that the optical transport in systems of coupled cavities can be considered by analogy with the “bond percolation” problem [31

31. R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). [CrossRef]

,32

32. C. D. Lorenz and R. M. Ziff, “Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices,” Phys. Rev. E 57, 230–236 (1998). [CrossRef]

] in percolation theory. The cavities arranged as a lattice are connected at the WGM wavelengths with optical “bonds” which are present with probability p depending on the cavities’ size dispersion (assuming p≈1 in the resonant case). At small p only a few bonds are present, thus only small clusters of sites (spheres) can form, but at a critical probability p c, called the percolation threshold [31

31. R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). [CrossRef]

,32

32. C. D. Lorenz and R. M. Ziff, “Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices,” Phys. Rev. E 57, 230–236 (1998). [CrossRef]

], a giant cluster appears spanning the entire network. The bond percolation threshold for 2D triangular lattice is p c=0.3472963…[33

33. M. F. Sykes and J. W. Essam, “Exact critical percolation probabilities for site and bond problems in two dimensions,” J. of Math. Phys. (N.Y.) 5, 1117–1127 (1964). [CrossRef]

], whereas for 3D face-centered-cubic (fcc) lattice it is only p c=0.1201635 [32

32. C. D. Lorenz and R. M. Ziff, “Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices,” Phys. Rev. E 57, 230–236 (1998). [CrossRef]

]. It should be noted that in the case of WGM coupling in 3D case the multiple spheres can be preferably connected in the “atomic” plans of such fcc lattice that should result in anisotropic properties and higher than p c=0.1201635 thresholds of such percolative transport, however still the WGM transport in a 3D lattice of spheres is expected

to be much more robust to the presence of disorder compared to that in 1D chains or 2D arrays of cavities.

Fig. 1. (a) Sketch of the cell for the hydrodynamic flow-assisted self-assembly of microspheres, (b) SEM image of the top surface of the sample showing its polycrystalline structure, (c) experimental set up, (d) single dye-doped sphere transmission spectrum.

2. Structures and experimental setup

The 3D close-packed structures formed by 5 µm dye-doped (Green FL, Duke Scientific Corp.) polystyrene microspheres with ~3% size dispersion were synthesized by the technique [34

34. B. Gates, D. Qin, and Y. Xia, “Assembly of nanoparticles into opaline structures over large areas,” Adv. Mater. 11, 466–469 (1999). [CrossRef]

] of hydrodynamic flow-assisted self-assembly. As shown in Fig. 1(a) the suspension of spheres was injected into a cell fabricated by sandwiching a mylar film with a rectangular hole between two glass substrates. Submicron scratches fabricated on the surface of the mylar film allowed the liquid to leak out whereas the spheres were trapped inside the cell. The growth of the close-packed structure with ~1 cm2 area was accelerated under continuous sonication. The thickness (d) of this structure was controlled by the mylar films in 5–177 µm range.

Although polycrystalline, such samples have fcc domains with typical dimensions greater than 50 µm, as illustrated in Fig. 1(b). The triangular packing of spheres in Fig. 1(b) represents [35

35. V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, “Interplay of order and disorder in the optical properties of opal photonic crystals,” Phys. Rev. B 66, 165215 (2002). [CrossRef]

] domains with (111) planes parallel to the surface. About 90% of the total area of the sample was shown to contain domains with the (111) planes parallel to its surface.

White light illumination was used for imaging different areas of the samples and for taking transmission spectra of individual spheres, as illustrated in Figs. 1(c,d). The pronounced dip, around 467 nm, in Fig. 1(d) is due to the absorption of dye molecules doped throughout the entire volume of microspheres. It should be noted that the illumination of the microspheres with plane waves leads to the formation of “photonic nanojets” [36

36. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12,1214–1220 (2004). [CrossRef] [PubMed]

] at the back side of spheres and to the formation of nanojet-induced modes [37

37. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32, 409–411 (2007). [CrossRef] [PubMed]

] in cavity chains. However the plane wave illumination is poorly coupled to WGMs in spheres.

Fig. 2. (a) Single 5µm sphere emission spectrum below the threshold for lasing WGMs and (b-f) emission spectra of a single monolayer of 5µm spheres collected as a function of average excitation intensity from 0.03 W/cm2 to 100 W/cm2. Inset illustrates geometry of excitation and collection of FL emission.

In order to study the transport of WGMs we created a localized FL source at one side of the sample by illuminating it with the focused beam from an optical parametric oscillator (OPO) system with 5ns pulses and repetition rate of 20Hz, tuned to the center of the absorption band of the dye (467 nm). The focused spot had a Gaussian intensity distribution at the sample with the half width a=30 µm leading to an excitation of about 30 spheres in the first monolayer. The attenuation length of the pump, due to absorption of the dye, can be estimated as l a ~13 µm. Thus, in thick lattices (d>l a) the FL source was confined near the illuminated surface of the structure. As shown in Fig. 1(c) the optical transport properties were studied by detecting FL transmission spectra from the opposite side of the sample. We used a 100×objective (NA=0.5) coupled to the spectrometer through a spatial filter selecting a ~10 µm circular area at the surface of the sample located opposite to the center of the excitation spot.

3. Experimental results and discussion

3.1. Pumping dependence of emission of a single monolayer

We first present in Fig. 2 the results of characterization of a single monolayer of microspheres. In this case the spheres are coupled laterally, and the effects of transport between the layers are not involved.

For comparison purposes Fig. 2(a) shows the low-intensity FL emission spectrum of a single isolated sphere demonstrating WGM peaks with 11–12 nm separations which is expected given the free spectral range of a 5 µm sphere. This spectrum was obtained by selecting the emission from a central area on the sphere equator using a spatial filter. Generally, the WGM resonances in spherical cavities [28

28. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes — part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006). [CrossRef]

,29

29. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes — part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). [CrossRef]

] are characterized by their polarization (TE/TM) and by radial (n), angular (l), and azimuthal (m) numbers. In perfect spheres the WGM modes are 2l+1 fold degenerate in m. This degeneracy can be removed by deformations from the spherical shape [38

38. V. S. Ilchenko, P. S. Volkov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Comm. 145, 86–90 (1998). [CrossRef]

]. The modes with n=1 (one antinode of the electromagnetic field in the radial direction) are most closely confined to the surface of the sphere, and they have highest Q-factors of their WGM resonances. The spectral resolution in Fig. 2 is limited by the spectrometer at ~0.2 nm level. Our measurements with higher spectral resolution showed that the sharpest peaks in Fig. 2(a) (corresponding to n=1) are characterized with Q=4×103. It has been demonstrated [39

39. N. Le Thomas, U. Woggon, W. Langbein, and M. V. Artemyev, “Effect of a dielectric substrate on whispering-gallery-mode sensors,” J. Opt. Soc. Am. B 23, 2361–2365 (2006). [CrossRef]

] that such values of Q-factors are much smaller than that predicted by the Mie theory as a result of an inhomogeneous broadening due to the spheres shape deformations and as a consequence of a homogeneous broadening for modes with small azimuthal numbers m due to the tunneling to the substrate. The latter factor leads to highest Q factors for modes with |m|~l located in the vicinity of equatorial plane of spheres.

These maxima are weak since up to a threshold of ~0.3 W/cm2 corresponding to Fig. 2(d) only a few percent of the total spontaneous emission intensity is coupled to WGMs [12

12. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Photonic molecule lasing,” Opt. Lett. 28, 2437–2439 (2003). [CrossRef] [PubMed]

], and most of the FL intensity is emitted into radiative modes with a broad (510–570nm) spectrum. Above this threshold, however, the individual spheres start to operate as WGM microlasers. Lateral coupling of ~30 size-disordered WGM microlasers located within the excitation spot gives rise to very strong inhomogeneously broadened peaks in the emission spectrum of single monolayer, as illustrated in Fig. 2(f).

3.2. Pumping dependence of FL transmission of several monolayers thick structures

The spectra of FL transmission through the six monolayer thick structure with d=25.4 µm are presented in Fig. 3. Since the structure is strongly absorbing at the pump wavelength (467nm), and nearly transparent at the emission wavelengths at 510–570 nm, the transport of light from the illuminated side of the sample to the area of collection plays a key role in formation of FL transmission spectra.

There are two major mechanisms which can be involved in such transport. One is connected with nonresonant diffusive propagation of radiative modes emitted by the source spheres. Using traditional scattering terminology [40

40. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

] this transport can be referred to as a scattering by large spherical particles (x=2πa/λ>>1, where a=2.5 µm is the radius of microsphere) with relatively small (0.59) index contrast. In this limit most of the light is transmitted through each sphere after two refractions without a significant inner reflection. In the case of plane wave illumination this leads to formation of photonic nanojets [36

36. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12,1214–1220 (2004). [CrossRef] [PubMed]

,37

37. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32, 409–411 (2007). [CrossRef] [PubMed]

] due to the focusing effect produced by individual spheres. The second mechanism is connected with evanescent coupling [18

18. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006). [CrossRef]

] of WGMs. High pumping intensities strongly favor the latter mechanism since at highest powers almost 100% of the emission of the source spheres is provided into the WGM 1 modes as opposed to the radiative modes.

A series of the FL transmission spectra measured as a function of I av is presented in Fig. 3. Up to a threshold of I av~0.3 W/cm2 corresponding to Fig. 3(c), the spectra display a set of nearly equidistant broad maxima similar to that in Figs. 2(b-c). However, at higher I av each maximum is observed to transform into a double-peak structure with 4.0-5.3 nm splitting, as indicated by dashed lines in Fig. 3. Similar splitting at I av>0.3 W/cm2 was observed in all samples with thicknesses ranging from 9.1 µm (2 monolayers) up to 177 µm (43 monolayers). It is seen in Fig. 3 that the magnitude of splitting does not depend on the pumping intensity.

An inset of Fig. 3 shows the dependence of the FL transmission on the pump intensity for the peak at 529.3 nm (red line) and for the FL background at the same wavelength (blue line).

Fig. 3. FL transmission spectra of a six layer thick sample (d=25.4 µm) collected as a function of excitation intensity (I av) from (a) 0.03 W/cm2 to (h) 100 W/cm2. (c-h) For I av>0.3 W/cm2 each maximum transforms into a double peak. Inset: pump dependence of the peak at 529.3 nm (red line), background emission (blue line) and a linear fit (black line).

The observed formation of a double peak structure in the FL transmission spectra can be explained by the WGM coupling phenomena in clusters of touching cavities. Due to the 3% size disorder of cavities in our samples the probability (p) for two randomly selected cavities to have overlapping WGM resonances with Q=4×103 is of the order of 1%. In 3D close-pack lattices however each sphere has 12 nearest neighbors that significantly increase the probability of finding resonant WGMs. This probability peaks at wavelengths corresponding to the WGMs in spheres with the mean sizes, as represented by the FL transmission maxima in Figs. 3(a,b). The likely explanation of the observed double peak structure is connected with the well-known property of systems of resonant coupled cavities that form two peaks of the normalized group delay [41

41. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11, 381–391 (2003). [CrossRef] [PubMed]

43

43. M. Sumetsky, “Modelling of complicated nanometer resonant tunneling devices with quantum dots,” J. Phys.: Condens. Matter 3, 2651–2664 (1991). [CrossRef]

] at the edges of the transmission band. These peaks provide a distributed feedback for lasing thus explaining why this double peak structure is seen above the lasing threshold.

In our structures these configurations of cavities with nearly resonant WGMs occupy only a small fraction of volume of the disordered 3D lattice. For this reason these configurations are not visibly contributing to the FL transmission spectra at low pumping intensities. With increasing pumping however, their lasing emission grows stronger than the emission of individual disordered microlasers due to their higher spectral density and competition between the modes. This seems to be the reason why we can observe a spectral signature of coupling between uniform spheres in such disordered 3D lattices. The fact that such clusters of well coupled cavities can be located in different atomic planes of fcc lattice seems to be an important condition for observation of splitting perpendicular to the substrate. As an example in the case of a single monolayer presented in Fig. 2 such photonic molecular states can be formed only in its plane. Using an analogy with the previously observed directional emission of strongly coupled states in resonant bispheres [10

10. 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]

] along the axis of bisphere it can be suggested that the molecular states in a single monolayer should be observable rather in the plane of monolayer than in a perpendicular direction.

This interpretation is also consistent with the magnitude of the observed splitting. For multiple cavities it is expected [41

41. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11, 381–391 (2003). [CrossRef] [PubMed]

] to be higher (by the factor of 2) than the normal mode (WGM) splitting in two identical touching spheres. In this sense the experimentally observed splitting, indicated in Fig. 3 (4.0–5.3nm for 5 µm spheres at the average wavelength 535 nm), is found to be in reasonable agreement with the results of measurements [10

10. 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]

,15

15. Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, “Fine structure of coupled optical modes in photonic molecules,” Phys. Rev. A 70, 051801(R) (2004). [CrossRef]

,17

17. B. M. Möller, U. Woggon, and M. V. Artemyev, “Coupled-resonator optical waveguides doped with nanocrystals,” Opt. Lett. 30, 2116–2118 (2005). [CrossRef] [PubMed]

] and modeling [18

18. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006). [CrossRef]

] of normal mode splitting in polystyrene bispheres with the comparable sizes.

3.3. Thickness dependence of FL transmission

Fig. 4. (a) Geometry of excitation and collection of the FL transmission illustrated for the six monolayers thick sample. (b) Thickness dependence of the peak-to-background ratio at 529.3nm in the FL transmission spectra for two pumping intensities. (c) Thickness dependence of the intensity of the peak at 528nm (red lines) and of the FL background transmission (blue lines). Different curves correspond to excitation intensity from 0.3 W/cm2 to 30 W/cm2.

4. Conclusions

In this paper we developed an approach to understanding the optical transport properties of such systems based on the analogy with the bond percolation problem [31

31. R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). [CrossRef]

33

33. M. F. Sykes and J. W. Essam, “Exact critical percolation probabilities for site and bond problems in two dimensions,” J. of Math. Phys. (N.Y.) 5, 1117–1127 (1964). [CrossRef]

] in percolation theory. In this approach, the lattice sites (spheres) are connected with optical “bonds” that are present with probability p depending on the cavities’ size dispersion (assuming p≈1 in the case of resonance between WGMs). Due to a 3% size disorder, the structures studied in this work with Q=4×103 are characterized with p~0.01, thus only small clusters of sites connected by bonds can form. However, by selecting more uniform spheres it should be possible to reach a percolation threshold (p c=0.1201635 for an fcc lattice [32

32. C. D. Lorenz and R. M. Ziff, “Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices,” Phys. Rev. E 57, 230–236 (1998). [CrossRef]

]) where a giant cluster spans the entire network. It should be noted that in the case of WGM coupling in 3D case the multiple spheres can be preferably connected in the “atomic” plans of such fcc lattice that should result in anisotropic properties and higher than p c=0.1201635 thresholds of such percolative transport, however still the WGM transport in a 3D lattice of spheres is expected to be much more robust to the presence of disorder compared to that in 1D chains or 2D arrays of cavities. Above the percolation threshold such lattices should become transparent for the WGM transport irrespective of the sample thickness. In comparison with single chains of cavities, 3D structures operating above the WGM percolation threshold can tolerate an order of magnitude larger dispersion of spheres sizes.

Acknowledgments

The authors thank M.S. Skolnick, M. Sumetsky, J.J. Baumberg, M.A. Fiddy and M.E. Raikh for stimulating discussions. This work was supported by ARO grant W911NF-05-1-0529 and by NSF grant CCF-0513179 as well as, in part, by funds provided by the University of North Carolina at Charlotte. S.P. Ashili was supported by DARPA grant W911NF-05-2-0053. The authors are thankful to Duke Scientific Corporation for donating microspheres for the research presented in this work.

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S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules,” Opt. Lett. 31, 338–340 (2006). [CrossRef] [PubMed]

23.

J.E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-midified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002). [CrossRef]

24.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003). [CrossRef]

25.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004). [CrossRef]

26.

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006). [CrossRef] [PubMed]

27.

F. Xia, L. Sekaric, and Yu. A. Vlasov, “Ultra-compact optical buffers on a silicon chip,” Nature Photon. 1, 65–71 (2007). [CrossRef]

28.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes — part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006). [CrossRef]

29.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes — part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). [CrossRef]

30.

S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32, 289–291 (2007). [CrossRef] [PubMed]

31.

R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). [CrossRef]

32.

C. D. Lorenz and R. M. Ziff, “Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices,” Phys. Rev. E 57, 230–236 (1998). [CrossRef]

33.

M. F. Sykes and J. W. Essam, “Exact critical percolation probabilities for site and bond problems in two dimensions,” J. of Math. Phys. (N.Y.) 5, 1117–1127 (1964). [CrossRef]

34.

B. Gates, D. Qin, and Y. Xia, “Assembly of nanoparticles into opaline structures over large areas,” Adv. Mater. 11, 466–469 (1999). [CrossRef]

35.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, “Interplay of order and disorder in the optical properties of opal photonic crystals,” Phys. Rev. B 66, 165215 (2002). [CrossRef]

36.

Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12,1214–1220 (2004). [CrossRef] [PubMed]

37.

A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32, 409–411 (2007). [CrossRef] [PubMed]

38.

V. S. Ilchenko, P. S. Volkov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Comm. 145, 86–90 (1998). [CrossRef]

39.

N. Le Thomas, U. Woggon, W. Langbein, and M. V. Artemyev, “Effect of a dielectric substrate on whispering-gallery-mode sensors,” J. Opt. Soc. Am. B 23, 2361–2365 (2006). [CrossRef]

40.

H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

41.

M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11, 381–391 (2003). [CrossRef] [PubMed]

42.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21, 1818–1832 (2004). [CrossRef]

43.

M. Sumetsky, “Modelling of complicated nanometer resonant tunneling devices with quantum dots,” J. Phys.: Condens. Matter 3, 2651–2664 (1991). [CrossRef]

OCIS Codes
(230.5750) Optical devices : Resonators
(350.3950) Other areas of optics : Micro-optics
(230.4555) Optical devices : Coupled resonators

ToC Category:
Novel Concepts and Theory

History
Original Manuscript: October 15, 2007
Revised Manuscript: December 3, 2007
Manuscript Accepted: December 4, 2007
Published: December 10, 2007

Virtual Issues
Physics and Applications of Microresonators (2007) Optics Express

Citation
Vasily N. Astratov and Shashanka P. Ashili, "Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres," Opt. Express 15, 17351-17361 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17351


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  23. J.E. Heebner, R. W. Boyd, and Q. H. Park, "SCISSOR solitons and other novel propagation effects in microresonator-midified waveguides," J. Opt. Soc. Am. B 19, 722-731 (2002). [CrossRef]
  24. A. Melloni, F. Morichetti, and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-379 (2003). [CrossRef]
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  26. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, "Transmission and group delay of microring coupledresonator optical waveguides," Opt. Lett. 31, 456-458 (2006). [CrossRef] [PubMed]
  27. F. Xia, L. Sekaric, and Yu. A. Vlasov, "Ultra-compact optical buffers on a silicon chip," Nature Photon. 1, 65-71 (2007). [CrossRef]
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  29. V. S. Ilchenko and A. B. Matsko, "Optical resonators with whispering-gallery modes - part II: applications," IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006). [CrossRef]
  30. S. Mookherjea and A. Oh, "Effect of disorder on slow light velocity in optical slow-wave structures," Opt. Lett. 32, 289-291 (2007). [CrossRef] [PubMed]
  31. R. Albert and A.-L. Barabasi, "Statistical mechanics of complex networks," Rev. Mod. Phys. 74, 47-97 (2002). [CrossRef]
  32. C. D. Lorenz and R. M. Ziff, "Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and fcc lattices," Phys. Rev. E 57, 230-236 (1998). [CrossRef]
  33. M. F. Sykes and J. W. Essam, "Exact critical percolation probabilities for site and bond problems in two dimensions," J. of Math. Phys. (N.Y.) 5, 1117-1127 (1964). [CrossRef]
  34. B. Gates, D. Qin, and Y. Xia, "Assembly of nanoparticles into opaline structures over large areas," Adv. Mater. 11, 466-469 (1999). [CrossRef]
  35. V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002). [CrossRef]
  36. Z. Chen, A. Taflove, and V. Backman, "Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique," Opt. Express 12,1214-1220 (2004). [CrossRef] [PubMed]
  37. A. M. Kapitonov and V. N. Astratov, "Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities," Opt. Lett. 32, 409-411 (2007). [CrossRef] [PubMed]
  38. V. S. Ilchenko, P. S. Volkov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, "Strain-tunable high-Q optical microsphere resonator," Opt. Comm. 145, 86-90 (1998). [CrossRef]
  39. N. Le Thomas, U. Woggon, W. Langbein, and M. V. Artemyev, "Effect of a dielectric substrate on whispering-gallery-mode sensors," J. Opt. Soc. Am. B 23, 2361-2365 (2006). [CrossRef]
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  42. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, "Distributed and localized feedback in microresonator sequences for linear and nonlinear optics," J. Opt. Soc. Am. B 21, 1818-1832 (2004). [CrossRef]
  43. M. Sumetsky, "Modelling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter 3, 2651-2664 (1991). [CrossRef]

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