OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 10 — May. 14, 2007
  • pp: 6268–6278
« Show journal navigation

Microcavities based on multimodal interference

Björn Maes, Mihai Ibanescu, John D. Joannopoulos, Peter Bienstman, and Roel Baets  »View Author Affiliations


Optics Express, Vol. 15, Issue 10, pp. 6268-6278 (2007)
http://dx.doi.org/10.1364/OE.15.006268


View Full Text Article

Acrobat PDF (3484 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Here we focus on a class of microcavities that can be described as a section of a waveguide. The properties of the guided waveguide modes and their reflection at the facets determine the cavity characteristics. The existence of high-quality cavity modes in waveguides with a zero group velocity point was shown in [2

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]

]. In the present paper we give a more in-depth description of these cavity modes. We show that the mechanism is very similar to the phenomenon in [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

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

], and we point out the connection with the multipole cancelation mechanism [5

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]

]. In [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

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

] one describes how the interference between multiple modes may lead to high facet reflections. We give a simple semi-analytical description of this phenomenon by examining the round-trip matrix of the cavity. There is a correlation between the possible reflection enhancement and the interaction between the modes. A zero group velocity point in the dispersion relation of a waveguide creates two modes that are intimately related. Therefore, these modes are good candidates to exploit the cavity mechanism.

The phenomenon is described using three structures. First, we employ the zero group velocity cylindrical structure proposed in [2

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]

]. We reach clear insights into the cavity mechanism, and at the same time we validate the new approaches. Second, we study the same structure, but at a wavelength with two normal (i.e. positive group velocity) guided modes. The same mechanism appears, however the dispersion and reflection characteristics are quite different. Third, we examine a surface plasmon cavity device, based on a recently proposed waveguide with a zero group velocity point [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]

]. Again, clear resonances are available, however the details differ.

The paper is organized as follows. First we describe the semi-analytical formulas and the numerical methods used. Then, in the main part, we discuss in sequence the previously mentioned three structures. Finally we group the conclusions.

2. Semi-analytic and numerical modeling

The class of devices under study is quite general and is depicted in Fig. 1(a). The center of the cavity is a waveguide system with two guided modes. Therefore the properties of the cavity modes are determined by the dispersion and reflection properties of these waveguide modes. We consider symmetric cavities, thus with two equal facets.

Because we deal with two guided modes, the reflection and propagation properties are described by 2×2-matrices. The reflection and propagation matrix are denoted by 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 ∣vg∣ (irrespective of the sign), we obtain [9

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

]

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.
Q=ωrLvg(1λ2).
(1)

Here, ωr 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

P×R[c0c1]=λ[c0c1],
(2)

with c0 and c1 the complex eigenvector components corresponding with the eigenvalue λ, which are normalized so ∣c02+∣c12 = 1. In the case of waveguide modes with different group velocity magnitudes v0 g and v1 g, the factor vg in Eq. 1 is replaced by

vgaverage=vg0c02+vg1c12.
(3)

To obtain the eigenvalue λ we need to construct the half-trip matrix. We assume a time dependence exp(iωt). Then, for two positive vg modes, we get the matrix for propagation over a length L:

P=[exp(ik0L)00exp(ik1L)],
(4)

R=[r00r01r10r11],
(5)

λ=exp(iΔL)[dcos(kL)sin(kL)±(idsin(kL)+δcos(kL))2+r012].
(6)

3. Cylindrical cavity: negative group velocity mode

We study the same cylindrical structure in this section and the next. However, we discuss different frequency regions and modes of different angular symmetry. A schematic is shown in Fig. 1(b), it is the same device as in [2

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]

]. It was shown that this structure gives rise to a zero vg point in the dispersion relation of the HE11 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]

]. We plot this dispersion in Fig. 1(c). The geometry is useful as a model for similar phenomena that can appear in omniguide or photonic bandgap structures [2

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]

, 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]

].

To find cavity modes we scan the (ω,L)-space. In the frequency region with two interacting waveguide modes (mode 0 with positive vg and mode 1 with negative vg) 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

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]

]. Now however we find additional modes. The origin of the extra modes is elucidated in Fig. 2(b). In this graph we put a dot each time a resonance is obtained, thus each time the imaginary part of λ becomes zero, regardless of the size of the real part. In this way we clearly see the connection between the resonances. The branches are grouped in pairs. Each pair corresponds to a certain resonance order. Each branch in a pair corresponds to a symmetry (node versus antinode in the middle of the cavity). Only the lowest order pair (lower left in Fig. 2(b), dark blue dots) was described in [2

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]

]. Note that the agreement between the semi-analytic mode expansion approach and FDTD is indicated in Fig. 2, which validates the approach.

The field distribution of some modes is shown in Fig. 3, together with the far field on- and off-resonance of a cavity mode. Clearly the multipole cancelation effect is at work, as described in [5

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]

]. At a high-Q resonance the radiation pattern changes: there are extra nodal lines, see Fig. 3(b), as the lowest order multipole is canceled. This gives proof of a bimodal mechanism: Both waveguide modes are prominent in the cavity. At the facets they reflect but radiate some energy into the space adjacent to the cavity. This radiation can be described as a superposition of multipoles. At certain cavity lengths and frequencies two conditions are fulfilled: there is a phase resonance (imaginary part of λ is zero), and the important lowest order multipole contribution of the modes cancel each other (leading to a real part of λ close to one, meaning low losses). When these conditions are satisfied we obtain a high-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]

] and [4

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

].

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.
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.

λ2r012±2cos(kL)Re(r01d*)2sin(kL)Im(r01δ*),
(7)

where Re (Im) is the real (imaginary) part, and * means complex conjugate. This expression implies that the magnitude of λ (at constant 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 3.4a. Note that in this example k corresponds with the zero vg-point. The exact ∣λ∣2 values are plotted in Fig. 5; we indeed see vertical lines of magnitude extrema, separated horizontally by 3.4a.

Fig. 4. Magnitudes of the reflection matrix elements for the cylindrical structure with the negative group velocity mode.
Fig. 5. Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with a negative group velocity mode.

For the phase resonance we can simplify a bit further and obtain:

λ±r01exp(iΔL).
(8)

4. Cylindrical cavity: positive group velocity modes

Fig. 6. Dispersion of the TE modes with angular momentum zero in the cylindrical structure. The frequency region with two guided modes is indicated.
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.

λ2d±δ2±Re(c2(d*±δ*)exp(±iΔL)idsin(ΔL)+δcos(ΔL)).
(9)

This means we obtain an extremum of ∣λ∣2 if the denominator in this equation becomes small, or (approximately) if tan(ΔL) ≈ -∣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.

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.
Fig. 9. Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with positive group velocity modes.

Overlaying amplitude and phase (Fig. 9 and Fig. 7(b)) we see that the high-Q cavities appear at the anti-crossing regions in Fig. 7(b), as previously noted.

5. Plasmonic cavity

In this section we construct a cavity mode by exploiting a zero vg-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]

]. The two-dimensional, non-cylindrical geometry we study is shown in Fig. 10(a). We use a metal-dielectric-metal structure, so there are two semi-infinite metal slabs, with a dielectric strip in between. The cavity is defined by the narrow dielectric sections with higher index (n = √2), the rest is air (n = 1). For the metal we use the permittivity ε = 1-ω2 p2, with ωp the plasmon frequency.

The waveguide is designed such that the center section (with the high-index parts) has two guided modes in a certain frequency range, whereas the outside sections (to the left and to the right) have one guided, more conventional, plasmonic mode. These dispersion relations are shown in Fig. 10(b). Note that we only consider modes with symmetry such that the electric field tangential to the plane indicated with the dashed line in Fig. 10(a) is zero. Furthermore, we use TM-polarization, thus with one magnetic field component perpendicular to this figure.

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).
Fig. 11. (a) Q versus L of a resonance. Data calculated with mode expansion (CAMFR). (b) ω versus L for the plasmonic cavity.

The eigenmode solver we use has been adapted to deal with plasmonic modes, see [11

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.

].

A resonance peak for the quality factor is presented in Fig. 11(a). For clarity only one peak is shown. A depiction of the field at this resonance is given in Fig. 12(a). Note that we obtain very high quality factors, as there is only one mode that provides a loss channel, and needs to be canceled. The phase resonance portrait is shown in Fig. 11(b). Again the portrait looks different than for the other examples.

The analysis starts from the reflection matrix, shown in Fig. 12(b). Away from the direct neighborhood of the zero vg-point we can assume that ∣r00∣ > ∣r01∣ and r11 ≈ 0 (or d ≈ δ and ∣d∣ > ∣r01∣). We note furthermore that the reflection matrix is approximately real. In that case the magnitude of the largest eigenvalue approximates to:

λ+2r002+2r012cos(2kL)+r014r002.
(10)
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.
Fig. 13. Numerically calculated magnitude squared of an eigenvalue for the plasmonic structure.

This means a period (at constant frequency) of π/k. Indeed, we see this trend in the numerical results of Fig. 13: If the frequency increases then k decreases and the period increases.

For the phase picture we obtain that λ+ ≈ r00 e ik0L. There is resonance if Im(λ+) equals zero, so L = mπ/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

The interplay of two modes in a cavity gives rise to high quality resonances. In the case of open space cavities they instigate the multipole cancelation mechanism. In the case of losses through guided modes, the losses are annulled via Fabry-Pérot type interference. We give a detailed description of these mechanisms through three examples. Although the mechanism is similar, the reflection matrix and the resulting resonance parameters look quite different. We show that modes coupled through a zero group velocity point are well suited to realize these resonances. The main ingredient seems to be a significant coupling between the waveguide modes. A recently proposed plasmon waveguide is exploited for this effect. The analysis provides a clear path to design and gain insight into novel cavity devices.

Acknowledgments

BM and PB acknowledge postdoctoral fellowships from the Funds for Scientific Research - Flanders (FWO-Vlaanderen). This work was supported in part by the MRSEC Program of the National Science Foundation under award number DMR 02-13282.

References and links

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. Commun. 214,155-170 (2002). [CrossRef]
  4. M. Hammer, "Total multimode reflection at facets of planar high-contrast optical waveguides," J. Lightwave Technol. 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. Soljacic, "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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited