Abstract

We describe intricate cavity mode structures, that are possible in waveguide devices with two or more guided modes. The main element is interference between the scattered fields of two modes at the facets, resulting in multipole or mode cancelations. Therefore, strong coupling between the modes, such as around zero group velocity points, is advantageous to obtain high quality factors. We discuss the mechanism in three different settings: a cylindrical structure with and without negative group velocity mode, and a surface plasmon device. A general semi-analytical expression for the cavity parameters describes the phenomenon, and it is validated with extensive numerical calculations.

©2007 Optical Society of America

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References

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  1. K.J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [Crossref] [PubMed]
  2. M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
    [Crossref] [PubMed]
  3. M. Hammer, “Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,” Opt. Comm. 214, 155–170 (2002).
    [Crossref]
  4. M. Hammer, “Total multimode reflection at facets of planar high-contrast optical waveguides,” Journ. Lightw. Tech. 20, 1549–1555 (2002).
    [Crossref]
  5. S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
    [Crossref]
  6. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
    [Crossref]
  7. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006).
    [Crossref] [PubMed]
  8. A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
    [Crossref] [PubMed]
  9. H.A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).
  10. M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
    [Crossref] [PubMed]
  11. P.P.P. Debackere, P. Bienstman, and R. Baets “Adaptive Spatial Resolution: Application to Surface Plasmon Waveguide Modes,” accepted for publication in Opt. Quantum Electron.
  12. T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).
    [Crossref] [PubMed]

2006 (1)

2005 (2)

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

2004 (1)

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

2003 (2)

2002 (2)

M. Hammer, “Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,” Opt. Comm. 214, 155–170 (2002).
[Crossref]

M. Hammer, “Total multimode reflection at facets of planar high-contrast optical waveguides,” Journ. Lightw. Tech. 20, 1549–1555 (2002).
[Crossref]

2001 (2)

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

Baets, R.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

P.P.P. Debackere, P. Bienstman, and R. Baets “Adaptive Spatial Resolution: Application to Surface Plasmon Waveguide Modes,” accepted for publication in Opt. Quantum Electron.

Bermel, P.

Bienstman, P.

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

P.P.P. Debackere, P. Bienstman, and R. Baets “Adaptive Spatial Resolution: Application to Surface Plasmon Waveguide Modes,” accepted for publication in Opt. Quantum Electron.

Botten, L.C.

Burr, G.

de Sterke, C.M.

Debackere, P.P.P.

P.P.P. Debackere, P. Bienstman, and R. Baets “Adaptive Spatial Resolution: Application to Surface Plasmon Waveguide Modes,” accepted for publication in Opt. Quantum Electron.

Fan, S.

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

Farjadpour, A.

Fink, Y.

M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

Hammer, M.

M. Hammer, “Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,” Opt. Comm. 214, 155–170 (2002).
[Crossref]

M. Hammer, “Total multimode reflection at facets of planar high-contrast optical waveguides,” Journ. Lightw. Tech. 20, 1549–1555 (2002).
[Crossref]

Haus, H.A.

H.A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).

Ibanescu, M.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006).
[Crossref] [PubMed]

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

Joannopoulos, J.D.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006).
[Crossref] [PubMed]

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

Johnson, S.G.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, Y. Fink, and J.D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

Karalis, A.

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

Lidorikis, E.

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

Luo, C.

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

McPhedran, R.C.

Mekis, A.

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

Rodriguez, A.

Roundy, D.

Soljaci*#x0107;, M.

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

Vahala, K.J.

K.J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

White, T.P.

Appl. Phys. Lett. (1)

S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[Crossref]

Journ. Lightw. Tech. (1)

M. Hammer, “Total multimode reflection at facets of planar high-contrast optical waveguides,” Journ. Lightw. Tech. 20, 1549–1555 (2002).
[Crossref]

Nature (1)

K.J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

Opt. Comm. (1)

M. Hammer, “Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,” Opt. Comm. 214, 155–170 (2002).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[Crossref]

Phys. Rev. Lett. (2)

A. Karalis, E. Lidorikis, M. Ibanescu, J.D. Joannopoulos, and M. Soljači*#x0107;, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).
[Crossref] [PubMed]

M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[Crossref] [PubMed]

Other (2)

P.P.P. Debackere, P. Bienstman, and R. Baets “Adaptive Spatial Resolution: Application to Surface Plasmon Waveguide Modes,” accepted for publication in Opt. Quantum Electron.

H.A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).

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Figures (13)

Fig. 1.
Fig. 1.

(a) General picture of a system with two circulating modes. (b) Schematic of the cylindrical structure. The dashed line is the axis of the cylinder. (b) Dispersion of the HE11 mode.

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Fig. 2.
Fig. 2.

(a) Q versus L of the resonances. Dots are data points from mode expansion (CAMFR), crosses present results from FDTD (MEEP). (b) ω versus L for the same cavity modes.

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Fig. 3.
Fig. 3.

(a) Field plot of some resonances. The electric field along ϕ is shown. L n and ωn are L/a and ω ×(a/2πc), respectively. (b) Far-field on- and off-resonance. The magnetic field along the direction of the axis is shown. The cavity is located to the left of these plots.

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Fig. 4.
Fig. 4.

Magnitudes of the reflection matrix elements for the cylindrical structure with the negative group velocity mode.

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Fig. 5.
Fig. 5.

Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with a negative group velocity mode.

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Fig. 6.
Fig. 6.

Dispersion of the TE modes with angular momentum zero in the cylindrical structure. The frequency region with two guided modes is indicated.

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Fig. 7.
Fig. 7.

(a) Q versus L of resonances. Dots are data points from mode expansion (CAMFR), crosses present checks with FDTD (MEEP). (b) ω versus L for the cavity modes in the cylindrical structure with two positive v g waveguide modes.

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Fig. 8.
Fig. 8.

(a) Field plot of some resonances. The electric field along ϕ is shown; only one half is presented. Ln and ωn are L/a and ω ×(a/2πc), respectively. (b) Magnitudes of the reflection matrix elements for the cylindrical structure in the frequency range with two positive group velocity modes.

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Fig. 9.
Fig. 9.

Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with positive group velocity modes.

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Fig. 10.
Fig. 10.

(a) Schematic of the two-dimensional, non-cylindrical plasmon structure. (b) Dispersion of the central waveguide (black), and of the outside waveguides (red).

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Fig. 11.
Fig. 11.

(a) Q versus L of a resonance. Data calculated with mode expansion (CAMFR). (b) ω versus L for the plasmonic cavity.

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Fig. 12.
Fig. 12.

(a) Depiction of the magnetic field at the peak of the resonance shown in Fig. 11(a) (L = 0.238L p and ω = 0.601ωp). (b) Magnitudes of the reflection matrix elements for the plasmonic structure.

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Fig. 13.
Fig. 13.

Numerically calculated magnitude squared of an eigenvalue for the plasmonic structure.

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Equations (10)

Equations on this page are rendered with MathJax. Learn more.

Q = ω r L ∣ v g ∣ ( 1 − ∣ λ ∣ 2 ) .
P × R [ c 0 c 1 ] = λ [ c 0 c 1 ] ,
v g average = ∣ v g 0 ∣ ∣ c 0 ∣ 2 + ∣ v g 1 ∣ ∣ c 1 ∣ 2 .
P = [ exp ( − i k 0 L ) 0 0 exp ( − i k 1 L ) ] ,
R = [ r 00 r 01 r 10 r 11 ] ,
λ = exp ( − i Δ L ) [ d cos ⁡ ( kL ) − iδ sin ⁡ ( kL ) ± ( − id sin ⁡ ( kL ) + δ cos ⁡ ( kL ) ) 2 + r 01 2 ] .
∣ λ ∣ 2 ≈ ∣ r 01 ∣ 2 ± 2 cos ⁡ ( kL ) Re ( r 01 d * ) ∓ 2 sin ⁡ ( kL ) Im ( r 01 δ * ) ,
λ ≈ ± r 01 exp ( − i Δ L ) .
∣ λ ∣ 2 ≈ ∣ d ± δ ∣ 2 ± Re ( c 2 ( d * ± δ * ) exp ( ± i Δ L ) − id sin ⁡ ( Δ L ) + δ cos ⁡ ( Δ L ) ) .
∣ λ + ∣ 2 ≈ r 00 2 + 2 r 01 2 cos ⁡ ( 2 kL ) + r 01 4 r 00 2 .

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