## Microcavities based on multimodal interference

Optics Express, Vol. 15, Issue 10, pp. 6268-6278 (2007)

http://dx.doi.org/10.1364/OE.15.006268

Acrobat PDF (3484 KB)

### 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

## 1. Introduction

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]

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]

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]

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]

## 2. Semi-analytic and numerical modeling

*R*and

*P*, respectively. As usual, a resonance or cavity mode is achieved if the imaginary part of an eigenvalue of the round-trip matrix (or, because of symmetry, half-trip matrix) is zero. The half-trip matrix is given by

*P*×

*R*. In case of resonance, the quality factor

*Q*of the cavity mode is determined by the magnitude of its eigenvalue, or, more precisely, by how close the half-trip eigenvalue is to ±1. If the waveguide modes have the same group velocity magnitude ∣v

_{g}∣ (irrespective of the sign), we obtain [9]

_{r}is the resonance frequency of the cavity mode and

*L*is the length of the waveguide. λ is the eigenvalue of the half-trip matrix, so

_{0}and c

_{1}the complex eigenvector components corresponding with the eigenvalue λ, which are normalized so ∣c

_{0}∣

^{2}+∣c

_{1}∣

^{2}= 1. In the case of waveguide modes with different group velocity magnitudes v

^{0}

_{g}and v

^{1}

_{g}, the factor v

_{g}in Eq. 1 is replaced by

*iωt*). Then, for two positive v

_{g}modes, we get the matrix for propagation over a length

*L*:

*k*

_{0}and

*k*

_{1}the waveguide mode propagation constants of mode 0 and mode 1, respectively. We always assume

*k*

_{0}and

*k*

_{1}positive (and

*k*

_{0}>

*k*

_{1}), so for a negative v

_{g}mode we need to adjust the sign in the propagation matrix. E.g. if mode 0 has negative v

_{g}we have to use exp(

*ik*

_{0}

*L*), as the power fluxes of both modes need to be in the same direction. The waveguide modes in the examples have negligible propagation losses, but including losses does not change the equations. The complex reflection matrix describes the modal reflection properties at a facet:

*r*

_{01}=

*r*

_{10}. In the next equations it is sometimes convenient to work with average and difference values, so:

*k*

_{0}=

*k*+Δ,

*k*

_{1}=

*k*-Δ. Likewise for the reflection matrix: r

_{00}=

*d*+δ, r

_{11}=

*d*-Δ.

*P*×

*R*. There are slight differences in the equation for the three structures discussed in the following sections, as the elements of the propagation matrix depend on the positive or negative v

_{g}character of the modes. In the case of the example in the next section mode 0 has positive v

_{g}and mode 1 has negative v

_{g}. Then we obtain:

_{g}modes (as in section 4) we have to interchange

*k*and Δ in the previous equation. Finally, in the case where mode 0 has negative v

_{g}and mode 1 has positive v

_{g}(as in section 5), we have to interchange

*i*with -

*i*in equation 6. The previous equation gives us a comprehensive picture of the cavity modes. The imaginary part determines which combinations of frequency and length

*L*give rise to a resonant mode. Then the real part indicates how strong the resonance is, through the quality factor in equation 1.

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]

*L*. Some of the resonant modes have been simulated with MEEP, a freely available finite-difference time-domain (FDTD) code, in order to validate the semi-analytical model [7

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]

## 3. Cylindrical cavity: negative group velocity mode

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]

_{g}point in the dispersion relation of the HE

_{11}mode [10

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]

**30**, 552–554 (2005). [CrossRef] [PubMed]

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]

*L*)-space. In the frequency region with two interacting waveguide modes (mode 0 with positive v

_{g}and mode 1 with negative v

_{g}) we find high-

*Q*cavity resonances. This is presented in Fig. 2(a). Some of these

*Q*-peaks (dark blue dots in Fig. 2(a)) were discussed in [2

**30**, 552–554 (2005). [CrossRef] [PubMed]

**30**, 552–554 (2005). [CrossRef] [PubMed]

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]

*Q*cavity. This mechanism was also at work in the two-dimensional square structures of [3

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]

_{01}∣ > ∣r

_{00}∣, ∣r

_{11}∣ (or ∣r

_{01}∣ > ∣

*d*∣, ∣

*δ*∣). In that case the magnitude of the eigenvalues (from Eq. 6) is approximated by:

*L*) is only a weakly-varying function of the frequency. In function of

*L*(at constant ω) the maxima of ∣λ∣

^{2}are separated by π/

*k*or 3.4

*a*. Note that in this example

*k*corresponds with the zero v

_{g}-point. The exact ∣λ∣

^{2}values are plotted in Fig. 5; we indeed see vertical lines of magnitude extrema, separated horizontally by 3.4

*a*.

*L*-difference between two pairs or resonance orders (at a certain frequency) is given by π/Δ. Thus, if we approach the zero v

_{g}-point, Δ goes to zero, and the distance between pairs becomes infinite. This explains the trends of the branches in Fig. 2(b).

*Q*. The previous also elucidates the longitudinal length scale π/

*k*provided by the zero v

_{g}-point, which was only partly explained in [2

**30**, 552–554 (2005). [CrossRef] [PubMed]

## 4. Cylindrical cavity: positive group velocity modes

**m**ϕ) with

*m*= 0, as opposed to

*m*= 1 in the previous example). The dispersion of the TE modes (only one electric field component, along ϕ) is plotted in Fig. 6. We examine the frequency region with two guided modes between ω = 0.4 and 0.6(2π

*c*/

*a*), indicated in the figure, and we note that both modes have a positive v

_{g}.

*L*)-space we obtain resonances again, which are shown in Fig. 7. These graphs look different than in the previous example. However, study of the far-field (not shown) and the

*Q*-peaks shows again that multipole cancelation is at work. Therefore, the bimodal resonance mechanism is equivalent, but we need to examine the dissimilarity with the previous section. The field patterns for some resonances are depicted in Fig. 8(a).

_{00}∣, ∣r

_{11}∣ > ∣r

_{01}∣ (and also ∣

*d*∣ > ∣r

_{01}∣ and ∣

*d*∣ > ∣r

_{01}∣). In this case the magnitude of λ is approximated by:

^{2}if the denominator in this equation becomes small, or (approximately) if tan(Δ

*L*) ≈ -∣δ/

*d*∣, which gives a period (at constant frequency) of π/Δ. As the frequency increases, we note that both ∣δ/

*d*∣ and Δ decrease, so the maximum of ∣λ∣

^{2}moves to larger

*L*, as shown in Fig. 9.

*Q*values are reached), the branches are approximated by λ

_{+}≈ r

_{00}exp(-i

*k*

_{0}

*L*) and λ

_{-}≈ r

_{11}exp(-

*ik*

_{1}

*L*), respectively. An anti-crossing switches the trends of the curves between λ

_{+}and λ

_{-}. A crossing indicates that the branches correspond to a different symmetry (node or antinode in the middle of the cavity). In between anti-crossings the curves follow Im(λ

_{+}) = 0 or Im(λ

_{-}) = 0, respectively. The latter indicates e.g. that the branch veers off to higher

*L*, as the frequency nears the cut-off for mode 1. Furthermore, the distance (at constant frequency) between branches is 2π/

*k*

_{0}or 2π/

*k*

_{1}, respectively.

## 5. Plasmonic cavity

_{g}-point in the dispersion of a plasmonic waveguide. This dispersion relation appears in a waveguide consisting of metal with a narrow dielectric layer on top, as presented in [8

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]

*n*= √2), the rest is air (

*n*= 1). For the metal we use the permittivity ε = 1-ω

^{2}

_{p}/ω

^{2}, with ω

_{p}the plasmon frequency.

*Q*cavities can exist through cancelation of the contributions to this loss mode. Thus, the bimodal mechanism of the previous sections remains crucial and largely unchanged. The situation bears similarity e.g. to the setting of [12

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]

_{g}-point we can assume that ∣r

_{00}∣ > ∣r

_{01}∣ and r

_{11}≈ 0 (or

*d*≈ δ and ∣

*d*∣ > ∣r

_{01}∣). We note furthermore that the reflection matrix is approximately real. In that case the magnitude of the largest eigenvalue approximates to:

*k*. Indeed, we see this trend in the numerical results of Fig. 13: If the frequency increases then

*k*decreases and the period increases.

_{+}≈ r

_{00}

*e*

^{ik0L}. There is resonance if Im(λ

_{+}) equals zero, so

*L*=

*m*π/

*k*

_{0}, with

*m*a positive integer. This agrees with the main lines in the portrait of Fig. 11(b). The curly lines originating close to the zero

*v*

_{g}-point, and the anti-crossings, are not captured by this analysis, as the approximations no longer apply, or because they belong to λ

_{-}.

## 6. Conclusion

## Acknowledgments

## References and links

1. | K.J. Vahala, “Optical microcavities,” Nature |

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. |

3. | M. Hammer, “Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,” Opt. Comm. |

4. | M. Hammer, “Total multimode reflection at facets of planar high-contrast optical waveguides,” Journ. Lightw. Tech. |

5. | S.G. Johnson, S. Fan, A. Mekis, and J.D. Joannopoulos, “Multipole-cancellation mechanism for high- |

6. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. |

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. |

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. |

9. | H.A. Haus, |

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. |

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. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optical Devices

**History**

Original Manuscript: February 28, 2007

Manuscript Accepted: March 22, 2007

Published: May 7, 2007

**Citation**

Bjørn Maes, Mihai Ibanescu, John D. Joannopoulos, Peter Bienstman, and Roel Baets, "Microcavities based on multimodal interference," Opt. Express **15**, 6268-6278 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6268

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### References

- K.J. Vahala, "Optical microcavities," Nature 424,839-846 (2003). [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. Hammer, "Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective," Opt. Commun. 214,155-170 (2002). [CrossRef]
- M. Hammer, "Total multimode reflection at facets of planar high-contrast optical waveguides," J. Lightwave Technol. 20,1549-1555 (2002). [CrossRef]
- 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]
- 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. Soljacic, "Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air," Phys. Rev. Lett. 95,063901 (2005). [CrossRef] [PubMed]
- H.A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).
- 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]
- 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.
- 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]

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