Abstract

We present a rigorous investigation of resonant coupling between microspheres based on multipole expansions. The microspheres have diameters in the range of several micrometers and can be used to realize various photonic molecule configurations. We reveal and quantify the interactions between the whispering gallery modes inside individual microspheres and the propagation modes of the entire photonic molecule structures. We show that Fano-like resonances in photonic molecules can be engineered by tuning the coupling between the resonant and radiative modes when the structures are illuminated with simple dipole radiation.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]

2016 (5)

L. Wei, Z. Xi, N. Bhattacharya, and H. P. Urbach, “Excitation of the radiationless anapole mode,” Optica 3, 799–802 (2016).
[Crossref]

D. Smirnova and Y. S. Kivshar, “Multipolar nonlinear nanophotonics,” Optica 3, 1241–1255 (2016).
[Crossref]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref] [PubMed]

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7, 11286 (2016).
[Crossref] [PubMed]

S. G. Romanov, S. Orlov, D. Ploss, C. K. Weiss, N. Vogel, and U. Peschel, “Engineered disorder and light propagation in a planar photonic glass,” Sci. Rep. 6, 27264 (2016).
[Crossref] [PubMed]

2015 (4)

M. Karg, T. A.F. König, M. Retsch, C. Stelling, P. M. Reichstein, T. Honold, M. Thelakkat, and A. Fery, “Colloidal self-assembly concepts for light management in photovoltaics,” Mater. Today. 18, 185–205 (2015).
[Crossref]

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photon. Rev. 9, 231–240 (2015).
[Crossref]

T. X. Hoang, Y. Duan, X. Chen, and G. Barbastathis, “Focusing and imaging in microsphere-based microscopy,” Opt. Express 23, 12337–12353 (2015).
[Crossref] [PubMed]

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7, 168–240 (2015).
[Crossref]

2014 (3)

J. M. Ward, N. Dhasmana, and S. N. Chormaic, “Hollow Core, Whispering Gallery Resonator Sensors,” Eur. Phys. J. Special Topics 223, 1917–1935 (2014).
[Crossref]

A. V. Maslov and V. N. Astratov, “Microspherical photonics: Sorting resonant photonic atoms by using light,” Appl. Phys. Lett. 105, 121113 (2014).
[Crossref]

K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105, 021112 (2014).
[Crossref]

2012 (4)

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[Crossref]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
[Crossref]

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

J. Sancho-Parramon and S. Bosch, “Dark modes and Fano resonances in plasmonic clusters excited by cylindrical vector beams,” ACS Nano 6, 8415–8423 (2012).
[Crossref] [PubMed]

2011 (2)

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix FORTRAN code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer,  112, 2182–2192 (2011).
[Crossref]

C. S. Deng, H. Xu, and L. Deych, “Effect of size disorder on the optical transport in chains of coupled microspherical resonators,” Opt. Express 19, 6923–6937 (2011).
[Crossref] [PubMed]

2010 (2)

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser & Photon. Rev. 4, 179–191 (2010).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

2009 (1)

2008 (4)

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[Crossref]

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008).
[Crossref] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

2007 (7)

2006 (3)

2005 (1)

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]

2004 (2)

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[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]

2000 (1)

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]

1999 (1)

1998 (2)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical Modes in Photonic Molecules,” Phys. Rev. Lett. 81, 2582–2585 (1998).
[Crossref]

C. P. Holfeld, F. Loser, M. Sudzius, K. Leo, D. M. Whittaker, and K. Kohler, “Fano Resonances in Semiconductor Superlattices,” Phys. Rev. Lett. 81, 874–877 (1998).
[Crossref]

1991 (1)

1989 (1)

1982 (1)

J. M. Gérardy and M. Ausloos, “Absorption Spectrum of Clusters of Spheres from the General Solution of Maxwell’s Equations, II. Optical Properties of Aggregated Metal Spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[Crossref]

1980 (1)

1965 (1)

U. Fano and J. W. Cooper, “Line Profiles in the Far-uv Absorption Spectra of the Rare Gases,” Phys. Rev. 137, A1364 (1965).
[Crossref]

1961 (2)

C. G. B. Garret, W. Kaiser, and W.L. Bond, “Stimulated Emission into Optical Whispering Modes of Spheres,” Phys. Rev. 124, 1807–1809 (1961).
[Crossref]

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

1958 (1)

H. Feshbach, “Unified Theory of Nuclear Reactions,” Ann. Phys. (N. Y.) 5, 357–390, (1958).
[Crossref]

1955 (1)

S. H. Autler and C. H. Townes, “Stark Effect in Rapidly Varying Fields,” Phys. Rev. 100, 703 (1955).
[Crossref]

1935 (2)

H. Beutler, “Über Absorptionsserien von Argon, Krypton und Xenon zu Termen zwischen den beiden Ionisierungsgrenzen  2P3/20 und  2P1/20,” Z. Phys. A 93, 177–196 (1935).
[Crossref]

U. Fano, “Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco,” Nuovo Cimento 12, 154–161 (1935).
[Crossref]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. R. Soc. London 18, 269–275 (1902).
[Crossref]

Abolmaali, F.

K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105, 021112 (2014).
[Crossref]

Allen, K. W.

K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105, 021112 (2014).
[Crossref]

Artemyev, M. V.

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]

Ashili, S. P.

V. N. Astratov and S. P. Ashili, “Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres,” Opt. Express 15, 17351–17361 (2007).
[Crossref] [PubMed]

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]

Astratov, V. N.

A. V. Maslov and V. N. Astratov, “Microspherical photonics: Sorting resonant photonic atoms by using light,” Appl. Phys. Lett. 105, 121113 (2014).
[Crossref]

K. W. Allen, A. Darafsheh, F. Abolmaali, N. Mojaverian, N. I. Limberopoulos, A. Lupu, and V. N. Astratov, “Microsphere-chain waveguides: Focusing and transport properties,” Appl. Phys. Lett. 105, 021112 (2014).
[Crossref]

S. Yang and V. N. Astratov, “Spectroscopy of coherently coupled whispering–gallery modes in size–matched bispheres assembled on a substrate,” Opt. Lett. 34, 2057–2059 (2009).
[Crossref] [PubMed]

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[Crossref]

V. N. Astratov and S. P. Ashili, “Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres,” Opt. Express 15, 17351–17361 (2007).
[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]

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]

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]

V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Springer Series in Optical Sciences156 (Springer, 2010), Chapter 17, pp. 423–457.

Ausloos, M.

J. M. Gérardy and M. Ausloos, “Absorption Spectrum of Clusters of Spheres from the General Solution of Maxwell’s Equations, II. Optical Properties of Aggregated Metal Spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[Crossref]

Autler, S. H.

S. H. Autler and C. H. Townes, “Stark Effect in Rapidly Varying Fields,” Phys. Rev. 100, 703 (1955).
[Crossref]

Backman, V.

Bag, A.

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7, 11286 (2016).
[Crossref] [PubMed]

Banzer, P.

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7, 11286 (2016).
[Crossref] [PubMed]

P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photon. Rev. 9, 231–240 (2015).
[Crossref]

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

Barbastathis, G.

Bauer, T.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).
[Crossref]

Bayer, M.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical Modes in Photonic Molecules,” Phys. Rev. Lett. 81, 2582–2585 (1998).
[Crossref]

Beutler, H.

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Figures (15)

Fig. 1
Fig. 1 The T-shaped molecule is formed by five identical microspheres. The local source is an electric dipole placed at the origin O and pointed along the z direction. The dipole-molecule distance D determines the coupling strength between the local source and the molecule. The distance d between the microspheres determines the overlap between the wavefunctions of two adjacent microspheres, and hence determines the optical coupling strength in the molecule.
Fig. 2
Fig. 2 Spatial and spectral responses of a single photonic atom. (a) The magnitude of ζ 25 ; 0 ( 1 ) shows two resonant wavelengths at λ 2 ; 25 ; 0 ( 1 ) = 0.430 646 μm and λ 1 ; 25 ; 0 ( 1 ) = 0.478 787 μm. The magnitude of ζ 33 ; 0 ( 1 ) shows a single resonant wavelength at λ 1 ; 33 ; 0 ( 1 ) = 0.431 472 568 42 μm. We need more significant digits for determining λ 1 ; 33 ; 0 ( 1 ) since its associated whispering gallery mode has a higher quality factor in comparison with those of λ 1 ; 25 ; 0 ( 1 ) and λ 2 ; 25 ; 0 ( 1 ). (b) The phases of ζ 25 ; 0 ( 1 ) and ζ 33 ; 0 ( 1 ) show the quick transitions at the resonant wavelengths. (c) Spatial distribution of the electric field intensity with the 33rd mode in resonance at excitation wavelength λ 1 ; 33 ; 0 ( 1 ). Paths 1 and 2 show the two propagation paths of the resonant light on the two orthogonal planes of x = 0 and y = 0, respectively. (d) The electric field intensity distribution in logarithmic scale for better visualization.
Fig. 3
Fig. 3 Two linearly coupled microspheres. The gap distance (Δd = d − 2R) determines the spectral behavior of the molecule.
Fig. 4
Fig. 4 Spectral profiles of the multipole moments of the photonic molecule formed by the two touching microspheres. (a) The magnitude and phase plots of the multipole moments ζ25;0. (b) The magnitude and phase plots of the multipole moments ζ33;0 show two resonant peaks.
Fig. 5
Fig. 5 Effects of the gap distance Δd = d − 2R on the spectral profiles of the multipole moments ζ33;0. (a) The two resonant peaks are stronger and closer to each other in comparison with the two peaks in Fig. 4(b). (b) The two resonant peaks get closer to the central resonant wavelength λ0. The unit of the transversal axis corresponds to 10−7 μm. (c) The two peaks merge into one centered at λ0. (d) The weak coupling results in the weaker resonance in the second microsphere and no spectral split.
Fig. 6
Fig. 6 The schematic of the 3 linearly coupled microspheres.
Fig. 7
Fig. 7 (a) The spectral profiles show three resonant peaks including the highest peak at λ1 = 0.430 992 μm and the second highest peak at λ2 = 0.431 686 μm. (b) The electric intensity in logarithmic scale shows that the light with the wavelength λ1 resonates in all three microspheres. (c) For the wavelength λ2, the resonance in the middle microsphere is suppressed.
Fig. 8
Fig. 8 (a) The strongest resonances in the first and last microspheres occuring at λ3 = 0.431 393 μm have the same strengths. (b) Magnitudes of the multipole moments in the last three microspheres show that the resonances in the 48-th and 49-th microspheres are suppressed at different wavelengths in the bandwidth. This means that the resonant strengths are not the same in all of the constituent microspheres.
Fig. 9
Fig. 9 Electric intensity distributions in logarithmic scale with λ3 = 0.431 393 μm for (a) the first five microspheres and (b) the last five microspheres. The distributions between the first and last microspheres are nearly identical though the light source is placed at the first microsphere.
Fig. 10
Fig. 10 Multipole moments represent the scattering and internal fields of the first and last microspheres. (a) The scattering moments of the resonant modes l = 33 are negligible in comparison with the radiative modes 1 ≤ l ≤ 26. (b) The moments of the resonant modes dominate that of the other modes.
Fig. 11
Fig. 11 Effects of the gap distance on the spectral profiles of ζ33;0. (a) The bandwidth decreases to Δλ = 0.26 nm. (b) The resonances in the first and last microsphere are still comparable. The bandwidth decreases to as narrow as Δλ = 1.66 × 10−7nm. (c) The resonance in the last microsphere is weaker than that in the first microsphere.
Fig. 12
Fig. 12 Spectral profiles of the ζ33;0 in the 50-coupled-microspheres optical waveguides: (a) The first, second, and last microspheres. (b) The last three microspheres.
Fig. 13
Fig. 13 Light resonates in the complex molecules. The electric intensity distributions are plotted using the strongest resonance wavelengths of the corresponding molecules. (a) The 4-th microsphere is placed at x = −1.5 μm, y = 0 μm, and z = 4.8 μm. Due to the molecule geometry and the gap between the 4-th and other microspheres, the light paths 1 (on the plane x = 0 as explained in Fig. 2(c)) of the microspheres are relatively independent. This independence explains the strong resonance in the M molecule. (b) The strongest moments in the 6-th, 5-th, and 7-th microspheres are roughly half of those of the 1-st, 2-nd, 3-rd microspheres, respectively. Intuitively, this is due to the fact that light circulating along path 1 in the 1-st, 2-nd, 3-rd microspheres is not much affected by the presence of the 4-th microsphere. (c) In comparison with the I molecule, the multipole moment in the 5-th is much weaker for the T molecule due to the absence of the 6-th and 7-th microspheres.
Fig. 14
Fig. 14 Super-resonator: Adding more microspheres increase optical couplings between the resonant and radiative modes, i.e., increasing the radiative loss. Consequently, the resonance is not enhanced by forming the resonator of resonators. (a) The spectral profile. (b) The electric intensity distribution in linear scale. (c) The electric intensity distribution in logarithmic scale with λ4 = 0.431 μm.
Fig. 15
Fig. 15 λ5 = 0.439 44 μm is the resonant wavelength of the big photonic atom with radius R = 2.37 μm and dielectric constant ε = 1.462. For this particular wavelength λ5, each of the three identical smaller microspheres acts like a lens, whereas the big microsphere acts like both a resonator and a lens.

Equations (10)

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E 0 ( r ¯ ) = p 1 ; 0 N 1 ; 0 ( r ¯ ) ,
E u ( r ¯ u ) = l = 1 L u m = l l [ p l m ( u ) N l m ( r ¯ u ) + q l m ( u ) M l m ( r ¯ u ) ] ,
E n i n c ( r ¯ n ) = E 0 ( r ¯ ) = u n E u ( r ¯ u ) .
p l m ( n ) = a l ( n ) ( A l m 1 ; 0 ( O O n ) p 1 ; 0 + u n l = 1 L u m = l l [ A l m lm ( O u O n ) p l m ( u ) + i B l m l m ( O u O n ) q l m ( u ) ] ) ,
q l m ( n ) = b l ( n ) ( i B l m 1 ; 0 ( O O n ) p 1 ; 0 + u n l = 1 L u m = l l [ A l m lm ( O u O n ) q l m ( u ) + i B l m l m ( O u O n ) p l m ( u ) ] ) ,
ζ l m ( n ) = c l ( n ) a l ( n ) p l m ( n ) , η l m ( n ) = d l ( n ) b l ( n ) q l m ( n ) .
Q l m ( n ) λ g l m ( n ) / Δ λ g l m ( n ) ,
E n i n t ( r ¯ 1 ) = l = 1 37 ζ l ; 0 ( 1 ) N l 0 ( r ¯ 1 ) .
E n i n t ( r ¯ 2 ) = l = 1 37 ζ l ; 0 ( 1 ) N l 0 ( r ¯ 2 ) .
ζ l ; 0 ( n ) = c l ( n ) ( A l ; 0 1 ; 0 ( O O n ) p 1 ; 0 + u n l = 1 37 A l ; 0 1 ; 0 ( O u O n ) a l ( u ) c l ( u ) ζ l ; 0 ( u ) ) .

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