Abstract

Using iterative methods we demonstrate that, for a structure consisting of N-coupled microspheres or ring resonators, the morphology-dependent resonances split into N higher-Q modes, in direct analogy with other types of oscillators. Moreover, for odd numbers of identical lossless coupled rings, the circulating intensity in the innermost ring increases exponentially with N when there is strong coupling to even-numbered rings and weak coupling to odd-numbered rings.

© 2003 Optical Society of America

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References

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  1. D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
    [CrossRef] [PubMed]
  2. K. A. Fuller and D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, New York, 2000), pp. 225–272.
  3. D. D. Smith and K. A. Fuller, “Photonic bandgaps in Miescattering by concentrically stratified spheres,” J. Opt. Soc. Am. B 19, 2449–2455 (2002).
    [CrossRef]
  4. K. A. Fuller, “Some novel features of morphology dependent resonances of bispheres,” Appl. Opt. 28, 3788–3790 (1989).
    [CrossRef] [PubMed]
  5. K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. 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]
  9. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
    [CrossRef]
  10. A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26, 917–919 (2001).
    [CrossRef]
  11. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
    [CrossRef]
  12. J. E. Heebner, R. W. Boyd, and Q. H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
    [CrossRef]
  13. 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]
  14. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000).
    [CrossRef]
  15. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847–849 (1999).
    [CrossRef]
  16. C. Pedersen and T. Skettrup, “Laser modes and threshold conditions in N-mirror resonators,” J. Opt. Soc. Am. B 13, 926–937 (1996).
    [CrossRef]

2002 (3)

2001 (1)

2000 (3)

1999 (3)

1997 (1)

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]

1996 (2)

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

C. Pedersen and T. Skettrup, “Laser modes and threshold conditions in N-mirror resonators,” J. Opt. Soc. Am. B 13, 926–937 (1996).
[CrossRef]

1991 (1)

1989 (1)

Absil, P. P.

Boyd, R. W.

Braunstein, D.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Chu, S. T.

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

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]

Foresi, J.

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]

Fuller, K. A.

Haus, H. A.

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]

Heebner, J. E.

Hryniewicz, J. V.

Jimba, Y.

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]

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]

Khazanov, A. M.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Koganov, G. A.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Kuwata-Gonokami, M.

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]

Laine, J. P.

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]

Lee, R. K.

Little, B. E.

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

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]

Melloni, A.

Miyazaki, H.

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]

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]

Mukaiyama, T.

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]

Park, Q. H.

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

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Pedersen, C.

Scherer, A.

Shuker, R.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Skettrup, T.

Smith, D. D.

Takeda, K.

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]

Xu, Y.

Yariv, A.

Appl. Opt. (2)

J. Lightwave Technol. (1)

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]

J. Opt. Soc. Am. B (4)

Opt. Lett. (4)

Phys. Rev. A (1)

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (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]

Phys. Rev. E (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Other (1)

K. A. Fuller and D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, New York, 2000), pp. 225–272.

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Figures (13)

Fig. 1
Fig. 1

Microparticle geometries and their ring resonator analogs: (a) concentric microspheres, (b) bisphere, (c) ring resonators of unequal size, (d) ring resonators of equal size.

Fig. 2
Fig. 2

N-layered concentric sphere geometry.

Fig. 3
Fig. 3

Splitting in the real part of the n=26, p=1 Mie coefficient a26,1 for N=1, 2, 3, and five concentric microspheres. The product of the refractive index and the radius, na=3 μm, was identical for each subsphere. The curves are offset for clarity.

Fig. 4
Fig. 4

Splitting in high- and low-Q spherical resonances. The splitting affects both the MDRs and the interference structure. The curves are offset for clarity.

Fig. 5
Fig. 5

Spectrum of the extinction efficiency, Qext, for N=1, 2, and 3 concentric microspheres. The curve representing N=3 is offset for clarity.

Fig. 6
Fig. 6

Mode splitting for N-coupled ring resonators where the OPL is such that ΔνFSR=0.1 THz. The number of split modes corresponds to the number of resonators.

Fig. 7
Fig. 7

Mode splitting for two identical coupled ring resonators as a function of r1, with r2 held constant. The parameters are r2=0.5 and (a) r1=0.2, (b) r1=0.5, (c) r1=0.8. The splitting varies but the FWHM remains constant. The dashed lines correspond to the maximum splitting ΔνSPmax in the strong coupling limit.

Fig. 8
Fig. 8

Mode splitting for three identical coupled ring resonators as a function of r1 with r2 and r3 held constant. The parameters are r2=r3=0.5 and (a) r1=0.2, (b) r1=0.5, (c) r1=0.8. Both the splitting and the FWHM vary. The dashed lines correspond to the maximum splitting ΔνSPmax.

Fig. 9
Fig. 9

High-Q side modes for three identical coupled ring resonators. R3 is plotted for r1=0.2 and r2=r3=0.9.

Fig. 10
Fig. 10

High-Q central mode for three identical coupled ring resonators. R3 is plotted for r2=0.1 and r1=r3=0.9.

Fig. 11
Fig. 11

Innermost intensity magnification factor M1 for three coupled ring resonators, where r2=0.1 and r1=r3=0.9.

Fig. 12
Fig. 12

Intensity magnification factor M1 for odd numbers of coupled ring resonators where reven=0.5 and (a) rodd=0.95, (b) rodd=0.9, (c) rodd=0.7, (d) rodd=reven=0.5. Case (d) includes even numbers of resonators. The dashed curves demonstrate the effect of loss for cases (b) and (c), where aj=0.9998, 0.9995, 0.995, and 0.985 for all j, from top to bottom. Loss is neglected for the solid curves.

Fig. 13
Fig. 13

Effective phase shift versus single-pass phase shift for N=1, 2, and 3 rings.

Equations (28)

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a1:l|np=a1:l-1+c1:l-1al-1:lcl-1:11-al-1:lal-1:1np,
c1:l|np=c1:l-1cl-1:l1-al-1:lal-1:1np.
Δ(1/λ)=1NΔnd,
E2=r1E0+it1E1,
E3=it1E0+r1E1,
E1=a1exp(iϕ1)E3,
E1E0=it1a1exp(iϕ1)1-r1a1exp(iϕ1)ρ1.
E2E0=r1-a1exp(iϕ1)1-r1a1exp(iϕ1)=exp[i(π+ϕ1)] a1-r1exp(-iϕ1)1-r1a1exp(iϕ1)τ1.
E6=r2E4+it2E5,
E7=it2E4+r2E5.
E0=a21exp(iϕ21)E7,
E5=a22exp(iϕ22)E2.
E5E4=it2a2τ1exp(iϕ2)1-r2a2τ1exp(iϕ2)ρ2.
E1E4=(it2+r2ρ2)ρ1a21exp(iϕ21)μ2.
E6E4=r2-a2exp(iϕ2)1-r2a2τ1exp(iϕ2)=exp[i(π+ϕ2)] a2τ1-r2exp(-iϕ2)1-r2a2τ1exp(iϕ2)τ2.
E4(j-1)+1E4(j-1)=itjajτj-1exp(iϕj)1-rjajτj-1exp(iϕj)ρj,
E4(j-1)+1E4(N-1)=ρNs=j+1Nρs-1τs-1as2exp(-iϕs2)μj.
E4(j-1)+2E4(j-1)=rj-ajτj-1exp(iϕj)1-rjajτj-1exp(iϕj)=exp[i(π+ϕj)] ajτj-1-rjexp(-iϕj)1-rjajτj-1exp(iϕj)τj,
Mj=s=jNRs.
ΔνSP=ΔνFSR2π (2 cos-1 r1),
ΔνFWHM=ΔνFSR2πcos-11-(1-r2)22r2.
ΔνSP=ΔνFSR2πcos-1(1+r1)(1+r2)2-1.
Rs=|ρs|2=1-rs21+rs2-2rscos[ϕs-1eff+ϕs].
Rsmax=1+rs1-rs>1,
Rsmin=1-rs1+rs<1,
M1=even=2,4,NRevenmin odd=1,3,NRoddmax,
M1=RoddmaxRevenmin1/2[RoddmaxRevenmin]N/2.
ϕNeff=arg(τN)=argE4(N-1)+2E4(N-1)=π+ϕN+argτN-1-rNexp(-iϕN)1-rNτN-1exp(iϕN).

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