## Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres

Optics Express, Vol. 15, Issue 25, pp. 17351-17361 (2007)

http://dx.doi.org/10.1364/OE.15.017351

Acrobat PDF (1742 KB)

### Abstract

Using techniques of flow-assisted self-assembly we synthesized three-dimensional (3D) lattices of dye-doped fluorescent (FL) 5 *µ*m 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 *µ*m. 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~10^{3} 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

1. P. W. Anderson, “Absence of diffusion in certain random lattices”, Phys. Rev. **109**, 1492–1505 (1958). [CrossRef]

2. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. **53**, 2169–2172 (1984). [CrossRef]

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

5. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature **390**, 671–673 (1997). [CrossRef]

6. M. Störzer, P. Gross, C. M. Aeggerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. **96**, 063904 (2006). [CrossRef] [PubMed]

7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**, 998–1005 (1997). [CrossRef]

8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

9. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B , **57**, 12127–12133 (1998). [CrossRef]

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]

20. B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B **75**, 245327 (2007). [CrossRef]

21. S. Deng, W. Cai, and V. N. Astratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express **12**, 6468–6480 (2004). [CrossRef] [PubMed]

22. 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]

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]

*Q*>10

^{3}for 4

*µ*m spheres and up to ~10

^{9 }for submillimeter spheres). The light transport from cavity to cavity is due to evanescent coupling of WGMs to neighboring resonators.

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

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

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]

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

*Q*>10

^{4}) with overlapping positions of WGM peaks still remains a challenging problem.

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

*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]

*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.

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]

*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. 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]

*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]

*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]

*p*

_{c}=0.1201635 thresholds of such percolative transport, however still the WGM transport in a 3D lattice of spheres is expected

## 2. Structures and experimental setup

*µ*m 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]

^{2}area was accelerated under continuous sonication. The thickness (

*d*) of this structure was controlled by the mylar films in 5–177

*µ*m range.

*µ*m, 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]

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]

*a*=30

*µ*m 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

*µ*m. 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

*µ*m 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

*µ*m 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. 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]

*n*), angular (

*l*), and azimuthal (

*m*) numbers. In perfect spheres the WGM modes are 2

*l*+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]

*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*=4×10

^{3}. 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]

*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.

^{2}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]

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

*d*=25.4

*µ*m 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.

*x*=2

*πa*/

*λ*>>1, where

*a*=2.5

*µ*m 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. 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]

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]

*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/cm

^{2}was observed in all samples with thicknesses ranging from 9.1

*µ*m (2 monolayers) up to 177

*µ*m (43 monolayers). It is seen in Fig. 3 that the magnitude of splitting does not depend on the pumping intensity.

*p*) for two randomly selected cavities to have overlapping WGM resonances with

*Q*=4×10

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

**82**, 4623–4626 (1999). [CrossRef]

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

*µ*m spheres at the average wavelength 535 nm), is found to be in reasonable agreement with the results of measurements [10

**82**, 4623–4626 (1999). [CrossRef]

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

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]

## 4. Conclusions

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

*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*=4×10

^{3}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

**57**, 230–236 (1998). [CrossRef]

*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.

*Q*-factors of their WGM resonances which are easier to overlap. We predict that the WGM percolation threshold should be achievable in close packed 3D lattices formed by cavities with

*Q*~10

^{3}and with ~1% size dispersion. As an example this situation can be realized using commercially available ~3

*µ*m polystyrene spheres in air. The notion of criticality, however, is lost in this case since the resonators are overcoupled due to the fact that the normal mode splitting exceeds the uncoupled WGM linewidths. It is interesting to note that there is an additional parameter for designing such structures which in principle allows achieving regime of critical coupling under conditions of percolative transport. This parameter is represented by the index of the external medium. As an example similar values of

*Q*~10

^{3}with ~1% size dispersion can be realized using enlarged (~10

*µ*m) polystyrene spheres in water. In the same spectral range WGMs in these spheres are characterized with larger angular (

*l*) numbers compared to 3

*µ*m spheres. On the other hand, as it was demonstrated [21

21. S. Deng, W. Cai, and V. N. Astratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express **12**, 6468–6480 (2004). [CrossRef] [PubMed]

*l*numbers are characterized with smaller coupling constant (and normal mode splitting) due to smaller fraction of the evanescent field outside the cavity. This effect was found to be very dramatic [21

21. S. Deng, W. Cai, and V. N. Astratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express **12**, 6468–6480 (2004). [CrossRef] [PubMed]

*l*from 10 to 16 reduced the coupling constant by approximately five times. On the basis of these results it seems feasible to achieve critical coupling of enlarged spheres (~10

*µ*m) in a liquid environment. The exact parameters of spheres and liquid medium should be identified through numerical modeling that will be the subject of our future work. Such structures of critically coupled cavities with percolative WGM transport can be used for multi wavelength detection of biochemical-binding events at the liquid-sphere interface. Thus, such systems can be used for developing next generation of resonant sensors and arrayed-resonator light emitting devices.

## Acknowledgments

## References and links

1. | P. W. Anderson, “Absence of diffusion in certain random lattices”, Phys. Rev. |

2. | S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. |

3. | M. Stoytchev and A. Z. Genack, “Measurement of the probability distribution of total transmission in random waveguides,” Phys. Rev. Lett. |

4. | P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B |

5. | D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature |

6. | M. Störzer, P. Gross, C. M. Aeggerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. |

7. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

8. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

9. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B , |

10. | T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. |

11. | M.D. Barnes, S.M. Mahurin, A. Mehta, B.G. Sumpter, and D.W. Noid, “Three-Dimensional photonic “molecules” from sequentially attached polymer-blend microparticles,” Phys. Rev. Lett. |

12. | Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Photonic molecule lasing,” Opt. Lett. |

13. | H. Guo, H. Chen, P. Ni, Q. Zhang, B. Cheng, and D. Zhang, “Transmission modulation in the passband of polystyrene photonic crystals,” Appl. Phys. Lett. |

14. | V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. |

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 |

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. |

17. | B. M. Möller, U. Woggon, and M. V. Artemyev, “Coupled-resonator optical waveguides doped with nanocrystals,” Opt. Lett. |

18. | A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. |

19. | S. P. Ashili, V. N. Astratov, and E. C. H. Sykes, “The effects of inter-cavity separation on optical coupling in dielectric bispheres,” Opt. Express |

20. | B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B |

21. | S. Deng, W. Cai, and V. N. Astratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express |

22. | S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules,” Opt. Lett. |

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 |

24. | A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. |

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. |

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. |

27. | F. Xia, L. Sekaric, and Yu. A. Vlasov, “Ultra-compact optical buffers on a silicon chip,” Nature Photon. |

28. | A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes — part I: basics,” IEEE J. Sel. Top. Quantum Electron. |

29. | V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes — part II: applications,” IEEE J. Sel. Top. Quantum Electron. |

30. | S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. |

31. | R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. |

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 |

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.) |

34. | B. Gates, D. Qin, and Y. Xia, “Assembly of nanoparticles into opaline structures over large areas,” Adv. Mater. |

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 |

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 |

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. |

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. |

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 |

40. | H. C. van de Hulst, |

41. | M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

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 |

43. | M. Sumetsky, “Modelling of complicated nanometer resonant tunneling devices with quantum dots,” J. Phys.: Condens. Matter |

**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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### References

- P. W. Anderson, "Absence of diffusion in certain random lattices", Phys. Rev. 109, 1492-1505 (1958). [CrossRef]
- S. John, "Electromagnetic absorption in a disordered medium near a photon mobility edge," Phys. Rev. Lett. 53, 2169-2172 (1984). [CrossRef]
- 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]
- P. W. Brouwer, "Transmission through a many-channel random waveguide with absorption," Phys. Rev. B 57, 10526-10536 (1998). [CrossRef]
- D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, "Localization of light in a disordered medium," Nature 390, 671-673 (1997). [CrossRef]
- M. Störzer, P. Gross, C. M. Aeggerter, and G. Maret, "Observation of the critical regime near Anderson localization of light," Phys. Rev. Lett. 96, 063904 (2006). [CrossRef] [PubMed]
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999). [CrossRef]
- N. Stefanou, A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B, 57, 12127-12133 (1998). [CrossRef]
- 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]
- M.D. Barnes, S.M. Mahurin, A. Mehta, B.G. Sumpter, and D.W. Noid, "Three-Dimensional photonic "molecules" from sequentially attached polymer-blend microparticles," Phys. Rev. Lett. 88, 015508 (2002). [CrossRef] [PubMed]
- Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, "Photonic molecule lasing," Opt. Lett. 28, 2437-2439 (2003). [CrossRef] [PubMed]
- H. Guo, H. Chen, P. Ni, Q. Zhang, B. Cheng, and D. Zhang, "Transmission modulation in the passband of polystyrene photonic crystals," Appl. Phys. Lett. 82, 373-375 (2003). [CrossRef]
- V. N. Astratov, J. P. Franchak, and S. P. Ashili, "Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder," Appl. Phys. Lett. 85, 5508-5510 (2004). [CrossRef]
- 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]
- 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]
- B. M. Möller, U. Woggon, and M. V. Artemyev, "Coupled-resonator optical waveguides doped with nanocrystals," Opt. Lett. 30, 2116-2118 (2005). [CrossRef] [PubMed]
- 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]
- S. P. Ashili, V. N. Astratov, and E. C. H. Sykes, "The effects of inter-cavity separation on optical coupling in dielectric bispheres," Opt. Express 14, 9460-9466 (2006). [CrossRef] [PubMed]
- B. M. Möller, U. Woggon, and M. V. Artemyev, "Bloch modes and disorder phenomena in coupled resonator chains," Phys. Rev. B 75, 245327 (2007). [CrossRef]
- S. Deng, W. Cai, and V. N. Astratov, "Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides," Opt. Express 12, 6468-6480 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- F. Xia, L. Sekaric, and Yu. A. Vlasov, "Ultra-compact optical buffers on a silicon chip," Nature Photon. 1, 65-71 (2007). [CrossRef]
- 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]
- 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]
- 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]
- R. Albert and A.-L. Barabasi, "Statistical mechanics of complex networks," Rev. Mod. Phys. 74, 47-97 (2002). [CrossRef]
- 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]
- 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]
- B. Gates, D. Qin, and Y. Xia, "Assembly of nanoparticles into opaline structures over large areas," Adv. Mater. 11, 466-469 (1999). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).
- M. Sumetsky and B. J. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003). [CrossRef] [PubMed]
- 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]
- 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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