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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13507–13514
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Generalized full-vector multi-mode matching analysis of whispering gallery microcavities

Xuan Du, Serge Vincent, Mathieu Faucher, Marie-Josée Picard, and Tao Lu  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13507-13514 (2014)
http://dx.doi.org/10.1364/OE.22.013507


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Abstract

We outline a full-vectorial three-dimensional multi-mode matching technique in a cylindrical coordinate system that addresses the mutual coupling among multiple modes co-propagating in a perturbed whispering gallery mode microcavity. In addition to its superior accuracy in respect to our previously implemented single-mode matching technique, this current technique is suitable for modelling waveguide-to-cavity coupling where the influence of multi-mode coupling is non-negligible. Using this methodology, a robust scheme for hybrid integration of a microcavity onto a silicon-on-insulator platform is proposed.

© 2014 Optical Society of America

1. Introduction

2. Theoretical formulations

3. Application and discussion

The aforementioned algorithm was used to bring about a rigorous hybrid integration scheme for stationing an ultrahigh-Q microtoroid on an SOI platform (as illustrated in Fig. 1a). The validation of this approach could lead to a two orders of magnitude improvement in the quality factor for SOI integrated optical cavities, therefore opening the way for a novel series of products based on such arrangements. As shown in the top right inset of the plot, the SOI waveguide under investigation has a width of 500 nm and is formed via an upper Si layer as well as a lower SiO2 layer of 220 nm and 2 μm respective thicknesses. The refractive indices of silica (1.44462) and silicon (3.48206) are taken from the literature [23

23. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]

,24

24. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

]. In our model, we employ a microtoroid that has a major radius of 45 μm and a minor radius of 5 μm.

Fig. 1: (a) Geometry of the toroid-SOI system and top view of the electrical field distribution. The accompanying propagation animation, Media 1, exhibits SOI waveguide width variation in the inset which is attributed to the progression of the azimuthal cross sections along the cavity’s circumference. (b) The amplitude of the two lowest-order WGM’s along the azimuthal direction when a straight SOI waveguide is placed at the equator of the cavity (i.e. leftmost inset). The transverse electric field distributions of the launched mode at ϕ = −0.034 rad (left) and ϕ = 0 rad (right) are displayed as insets. (c) The reciprocal of the quality factor as a function of azimuthal angle step size, indicating a convergence error of Oϕ1.0) from the least square fitting of the last three, smallest Δϕ. (d) The number of modes included in the simulation, revealing a convergence rate of around O(N0.8) via the least square fitting of the relative error (excluding the N = 1 point).

In this test case we study the coupling scheme between a straight waveguide and a toroid, as is shown in the lower inset of Fig. 1a. Light propagation along the azimuthal direction is computed at a wavelength of 1500 nm. In the first simulation, we placed the straight SOI waveguide in contact with the microtoroid’s equator and a 289th fundamental quasi-TE toroid mode was launched at ϕ = −0.34 rad, wherein the waveguide was 3 wavelengths away from the cavity. The launched transverse electrical field distributions are displayed in the two left insets of Fig. 1b. Beyond that point, the coupling between the cavity and the waveguide falls to a negligible level (with relative Q degradation below 10−10) and so it is neglected. As shown in the main plot of Fig. 1b, the amplitude of the launched mode diminishes while that of the second lowest-order mode reaches its maximum when the light propagates towards the center of the coupling region. It is also evident from the two right insets of Fig. 1b that the cavity mode launched as the input gradually evolves to a strongly hybridized cavity-taper mode when the light is approaching ϕ = 0 rad. The hybrid mode eventually relaxes to the unperturbed cavity mode at the exit of the cavity-taper coupling region and most of the light energy is coupled back to the originally launched mode in a similar manner as that of a directional coupler. By computing the total energy loss of the launched mode subtracted by the contribution from the intrinsic Q, we obtain a coupling Q-factor of 105. The coupling Q calculated here takes the loss from the launched mode to both the tapered waveguide and higher-order cavity modes into account. To precisely estimate the energy delivered to the waveguide, one may simply compute the overlap factor between the output field at ϕ = 0.34 rad and the mode of the tapered waveguide. To characterize the convergence rate of our algorithm, we plotted the reciprocal of the quality factor as a function of azimuthal angle steps and the number of modes included (given by the blue triangles in Figs. 1c and 1d). The expectation value of this quantity at an infinitesimal azimuthal step or infinite mode numbers (i.e. blue dashed line) was extracted through the Richardson extrapolation procedure [25

25. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

] and the relative error (i.e. red cross markers) was estimated with the extrapolated value as a reference. The black line in Fig. 1c depicts a convergence rate of Oϕ1.0), while that of Fig. 1d yields a convergence rate of about O(N0.8).

The mathematical interpretation of the convergence rate is as follows: in Eq. (4), it is assumed that m′l remains constant between ϕ0 and ϕ0 + δϕ. A more generalized form which takes the continuous variation of m′l into account should be
E(ρ,z,ϕ0+δϕ)=k=1N(ϕ0)Al(ϕ0)ejϕ0ϕ0+δϕml(ϕ,λ)dϕ|e˜^l(ρ,z,ϕ0)
(10)
By assuming m′l is a well behaved function between ϕ0 and ϕ0 +δϕ, we can expand it as a power series around ϕ0: ml(ϕ,λ)=n=01n!nml(ϕ,λ)ϕn|ϕ=ϕo(ϕϕo)n. By retaining the first term of the power series, one can obtain Eq. (4). Note that for a sufficiently small angle evolution δϕ where the second power series term dominates the error, m′l(ϕ) = m′l(ϕ0) + O(δϕ). The 1/Q term characterizes the overall loss contributions from both m′i,l, defined as the imaginary part of m′l, and the mode mismatch loss mkl. The accuracy of the latter term is determined by the transverse grid spacing, the number of modes involved in the computation, and the relative insensitivity to δϕ for small azimuthal steps. Consequently, the adequately small azimuthal steps will procure a convergence rate for 1/Q as a function of δϕ determined by the convergence of m′i,l, which is of O(δϕ). At larger angles, this will be influenced by higher-order power series terms and the mode mismatch loss will deviate from the convergence rate of O(δϕ).

The convergence rate as a function of the number of modes that are included, on the other hand, is case dependent. One may expect that for the extreme condition of an ideal cavity where no mode coupling occurs, the convergence rate should be O(1). In the particular simulation of our structure, the convergence rate is O(N0.8). The least square fit excludes the N = 1 point, as it has larger relative error. This is caused by the odd parity at N = 1 and sampling at the center of the structure, whereas the rest of the simulations have even parity and are sampled symmetrically about the center.

To reach the critical coupling condition, wherein the coupling Q-factor (Qc) is equal to the intrinsic Q-factor of the cavity (assumed to be a practical value of 2×108, as specified by the dot-dashed line of Figs. 2a and 2b), we gradually displaced the tapered waveguide horizontally away from the toroid. As displayed in Fig. 2a, a Qc on the order of 105 is computed when the waveguide touches the cavity surface and 1010 is computed at a gap size of 2.5 wavelengths. For a waveguide sitting in the equatorial plane, a gap size of 0.75 μm is also determined to be desirable in establishing critical coupling. The dependence of the Q-factor as well as the coupling parameter K on the gap is akin to the data in [26

26. S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]

] for a microsphere to fiber-taper coupling system. Our K data points (i.e. red cross markers, for a 67 μm-diameter microsphere and 1.35 μm-diameter fiber at λ =1550 nm) for the low-Q/accuracy-inhibiting regime in Fig. 2c are in good agreement with the experimental results of [26

26. S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]

] (i.e. blue curve, for identical specifications), thus appropriately substantiating our methodology. It is evident from Fig. 2a that the coupling Q-factor is sensitive to the gap distance. More specifically, a gap distance fluctuation on the order of 300 nm may cause a drop of the coupling Q-factor from 108 to 107. Therefore, highly precise alignment is required in order to integrate a toroid onto an SOI platform with this scheme.

Fig. 2: Coupling Q-factor as a function of (a) gap size and (b) vertical angle. (c) Coupling parameter K (i.e. ratio between the power of the waveguide mode and the power lost within the entire system) versus the gap size for a 67 μm-diameter microsphere and 1.35 μm-diameter fiber at λ =1550 nm.

4. Conclusion

A three-dimensional full-vector multi-mode matching method was formulated in cylindrical coordinates and tested. This technique solves generalized whispering gallery mode cavity problems, yet in this paper was specifically applied to a toroid-SOI coupling geometry where no precise control of the gap distance between the cavity and the waveguide was required. Furthermore, K parameters for many microsphere-fiber taper separations were assessed for accuracy restraining criteria. Once again, the multi-mode matching method proposed in this article is applicable to numerous classes of WGM related scenarios where the energy transfer between different modes is non-negligible. Two constraints remain in the current implementation: the assumption of neglecting the field radiating to infinity and the backscattering. These restrictions, however, can be lifted by, e.g., considering perfectly matched layer modes similar to those in [27

27. H. Derudder, F. Olyslager, D. De Zutter, and S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001). [CrossRef]

] as well as adopting a bidirectional mode matching method. These additions, which have been used in conventional waveguide modelling, are currently under investigation in conjunction with the experimental verification of the integration scheme. Advancements of this kind will appear in subsequent publications.

References and links

1.

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]

2.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008). [CrossRef] [PubMed]

3.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013). [CrossRef] [PubMed]

4.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011). [CrossRef] [PubMed]

5.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009). [CrossRef] [PubMed]

6.

G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012). [CrossRef]

7.

V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). [CrossRef]

8.

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011). [CrossRef]

9.

M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005). [CrossRef] [PubMed]

10.

M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007). [CrossRef] [PubMed]

11.

D. Rowland and J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993). [CrossRef]

12.

M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999). [CrossRef]

13.

M. A. C. Shirazi, W. Yu, S. Vincent, and T. Lu, “Cylindrical beam propagation modelling of perturbed whispering-gallery mode microcavities,” Opt. Express 21, 30243–30254 (2013). [CrossRef]

14.

C.-G. Xu, X. Xiong, C.-L. Zou, X.-F. Ren, and G.-C. Guo, “Efficient coupling between dielectric waveguide modes and exterior plasmon whispering gallery modes,” Opt. Express 21, 31253–31262 (2013). [CrossRef]

15.

A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express 21, 14169–14180 (2013). [CrossRef] [PubMed]

16.

M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007). [CrossRef]

17.

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53 (2003). [CrossRef]

18.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, and Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012). [CrossRef]

19.

I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003). [CrossRef]

20.

M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. 15, 083006 (2013). [CrossRef]

21.

X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express 21, 22012–22022 (2013). [CrossRef] [PubMed]

22.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000). [CrossRef]

23.

I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]

24.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

25.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

26.

S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]

27.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001). [CrossRef]

OCIS Codes
(040.1880) Detectors : Detection
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Integrated Optics

History
Original Manuscript: February 3, 2014
Revised Manuscript: April 29, 2014
Manuscript Accepted: May 15, 2014
Published: May 28, 2014

Citation
Xuan Du, Serge Vincent, Mathieu Faucher, Marie-Josée Picard, and Tao Lu, "Generalized full-vector multi-mode matching analysis of whispering gallery microcavities," Opt. Express 22, 13507-13514 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13507


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References

  1. D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]
  2. F. Vollmer, S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008). [CrossRef] [PubMed]
  3. V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013). [CrossRef] [PubMed]
  4. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011). [CrossRef] [PubMed]
  5. Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009). [CrossRef] [PubMed]
  6. G. Bahl, X. Fan, T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012). [CrossRef]
  7. V. Ilchenko, A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). [CrossRef]
  8. M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011). [CrossRef]
  9. M. Borselli, T. Johnson, O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005). [CrossRef] [PubMed]
  10. M. Hossein-Zadeh, K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007). [CrossRef] [PubMed]
  11. D. Rowland, J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993). [CrossRef]
  12. M. L. Gorodetsky, V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999). [CrossRef]
  13. M. A. C. Shirazi, W. Yu, S. Vincent, T. Lu, “Cylindrical beam propagation modelling of perturbed whispering-gallery mode microcavities,” Opt. Express 21, 30243–30254 (2013). [CrossRef]
  14. C.-G. Xu, X. Xiong, C.-L. Zou, X.-F. Ren, G.-C. Guo, “Efficient coupling between dielectric waveguide modes and exterior plasmon whispering gallery modes,” Opt. Express 21, 31253–31262 (2013). [CrossRef]
  15. A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express 21, 14169–14180 (2013). [CrossRef] [PubMed]
  16. M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007). [CrossRef]
  17. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53 (2003). [CrossRef]
  18. Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012). [CrossRef]
  19. I. Teraoka, S. Arnold, F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003). [CrossRef]
  20. M. R. Foreman, F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. 15, 083006 (2013). [CrossRef]
  21. X. Du, S. Vincent, T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express 21, 22012–22022 (2013). [CrossRef] [PubMed]
  22. H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000). [CrossRef]
  23. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]
  24. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).
  25. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).
  26. S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]
  27. H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001). [CrossRef]

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