Abstract

We report a numerical and analytical study of mode field patterns and mode coupling in planar waveguide-coupled square microcavities, using two-dimensional (2-D) finite-difference time-domain (FDTD) method and k-space representation. Simulated mode field patterns can be identified by k-space modes. We observe that different mode number parities permit distinctly different mode field patterns and spectral characteristics. Simulation results suggest that k-space modes that nearly match the waveguide propagation mode have a relatively high coupling efficiency. Such preferential mode coupling can be modified by the mode number parity.

© 2003 Optical Society of America

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References

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  1. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho and R. C. Tiberio, �??Waveguidecoupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6-nm free spectral range,�?? Opt. Lett. 22, 1244-1246 (1997).
    [CrossRef] [PubMed]
  2. S. C. Hagness, D, Rafizadeh, S. T. Ho and A. Taflove, �??FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode diskresonators,�?? J. Lightwave Technol. 15, 2154-2165 (1997).
    [CrossRef]
  3. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling and W. Greene, �??Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,�?? IEEE Photon. Technol. Lett. 10, 549-551 (1998).
    [CrossRef]
  4. D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra and A. Driessen, �??Experimental and numerical study of SiON microresonators with air and polymer cladding,�?? J. Lightwave Technol. 21, 1099-1110 (2003).
    [CrossRef]
  5. R. Grover, T. A. Ibrahim, T. N. Ding, Y. Leng, L. C. Kuo, S. Kanakaraju, K. Amarnath, L. C. Calhoun and P. T. Ho, "Laterally coupled InP-based single-mode microracetrack notch filter," IEEE Photon. Technol. Lett. 15, 1082-1084(2003).
    [CrossRef]
  6. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, �??Coupling of modes analysis of resonant channel add-drop filters,�?? IEEE J. Quantum Electron. 35, 1322-1331 (1999).
    [CrossRef]
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  8. A. W. Poon, �??Optical resonances of two-dimensional microcavities with circular and non-circular shapes,�??PhD thesis, Yale University, 2001.
  9. Y. L. Pan and R. K. Chang, �??Highly efficient prism coupling to whispering gallery modes of a square µ cavity,�?? Appl. Phys. Lett. 82, 487-489 (2003).
    [CrossRef]
  10. M. Lohmeyer, �??Mode expansion of rectangular integrated optical microresonators,�?? Opt. Quantum Electron. 34, 541-557 (2002).
    [CrossRef]
  11. M. Hammer, �??Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,�?? Opt. Commun. 214, 155-170 (2002).
    [CrossRef]
  12. Y. F. Chen, K. F. Huang, H. C. Lai and Y. P. Lan, �??Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,�?? Phys. Rev. Lett. 90, 053904 (2003).
    [CrossRef] [PubMed]
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  14. FullWAVE, Rsoft Inc. Research Software, <a href="http://www.rsoftinc.com">http://www.rsoftinc.com
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Appl. Phys. Lett. (1)

Y. L. Pan and R. K. Chang, �??Highly efficient prism coupling to whispering gallery modes of a square µ cavity,�?? Appl. Phys. Lett. 82, 487-489 (2003).
[CrossRef]

Conf. Lasers Electro-Optics (1)

N. Ma, C. Y. Fong, F. K. L. Tung, K. C. Lam, W. N. Chan and A. W. Poon, �??Micro-pillar square resonant cavity channel add-drop filters on silicon-nitride-on-silica: design, fabrication and characterization,�?? inProc. of Conf. Lasers Electro-Optics, Baltimore, MD, Jun. 2003.

IEEE J. Quantum Electron. (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus and J. D. Joannopoulos, �??Coupling of modes analysis of resonant channel add-drop filters,�?? IEEE J. Quantum Electron. 35, 1322-1331 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling and W. Greene, �??Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,�?? IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

R. Grover, T. A. Ibrahim, T. N. Ding, Y. Leng, L. C. Kuo, S. Kanakaraju, K. Amarnath, L. C. Calhoun and P. T. Ho, "Laterally coupled InP-based single-mode microracetrack notch filter," IEEE Photon. Technol. Lett. 15, 1082-1084(2003).
[CrossRef]

J. Lightwave Technol. (2)

D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra and A. Driessen, �??Experimental and numerical study of SiON microresonators with air and polymer cladding,�?? J. Lightwave Technol. 21, 1099-1110 (2003).
[CrossRef]

S. C. Hagness, D, Rafizadeh, S. T. Ho and A. Taflove, �??FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode diskresonators,�?? J. Lightwave Technol. 15, 2154-2165 (1997).
[CrossRef]

Opt. Commun. (1)

M. Hammer, �??Resonant coupling of dielectric optical waveguides via rectangular microcavities: the coupled guided mode perspective,�?? Opt. Commun. 214, 155-170 (2002).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Lohmeyer, �??Mode expansion of rectangular integrated optical microresonators,�?? Opt. Quantum Electron. 34, 541-557 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

Y. F. Chen, K. F. Huang, H. C. Lai and Y. P. Lan, �??Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,�?? Phys. Rev. Lett. 90, 053904 (2003).
[CrossRef] [PubMed]

Other (3)

FullWAVE, Rsoft Inc. Research Software, <a href="http://www.rsoftinc.com">http://www.rsoftinc.com

K. Okamoto, �??Chapter 2�?? in Fundamentals of optical waveguides, (Academic, San Diego, CA, 2000).

A. W. Poon, �??Optical resonances of two-dimensional microcavities with circular and non-circular shapes,�??PhD thesis, Yale University, 2001.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of a planar waveguide-coupled square µ-cavity channel add-drop filter.

Fig. 2.
Fig. 2.

FDTD simulated throughput (blue), drop (green) and add (red dashed line) spectra (normalized with input intensity) of a planar waveguide-coupled square µ-cavity filter. a=2.2 µm, w=0.2 µm, g=0.2 µm and TM-polarized. The dominant resonances in the throughput spectrum are indexed as (mx, my) modes according to the corresponding mode-field patterns. The indexed (mx, my) modes are clustered according to the integer number of wavelengths M.

Fig. 3.
Fig. 3.

FDTD simulated odd M (=15) mode field patterns of a planar waveguide-coupled square µ-cavity filter at (6,9) mode (λ=1538 nm). a=2.2 µm, w=0.2 µm, g=0.2 µm and TM-polarized. (a) t=t0, (b) t≈t0+T/8, (c) t≈t0+T/4, (d) t≈t0+3T/8 and (e) t≈t0+T/2.

Fig. 4.
Fig. 4.

FDTD simulated odd M (=15) mode field patterns of a planar waveguide-coupled square µ-cavity filter at (7,8) mode (λ=1562.5 nm). a=2.2 µm, w=0.2 µm, g=0.2 µm and TM-polarized. (a) t=t0, (b) t≈t0+T/6, (c) t≈t0+T/4, (d) t≈t0+2T/6 and (e) t≈t0+T/2. Calculated mode field patterns of a discrete square cavity using Eq. (2) with A=0.7 for (7,8) mode, B=0.3 for (8,7) mode, and δ=π/2. (f) ωt=0, (g) ω t=2π/6, (h) ωt=π/2, (i) ωt=4π/6 and (j) ωt=π.

Fig. 5.
Fig. 5.

FDTD simulated even M (=16) mode field patterns of a planar waveguide-coupled square µ-cavity filter. a=2.2 µm, w=0.2 µm, g=0.2 µm and TM-polarized. (a) (7,9)π mode (λ=1446.5 nm), (b) (6,10)π mode (λ=1417.1 nm), (c) (7,9)0 mode (λ=1454.5 nm) and (d) (8,8) mode (λ=1461.2 nm). Calculated mode field patterns of a discrete square cavity using Eq. (2) with A=B. (e) (7,9)π mode, (f) (6,10)π mode, (g) (7,9)0 mode and (h) (8,8) mode. The dashed-line box in (b) and (f) denotes a “vortex.”

Fig. 6.
Fig. 6.

Calculated k-space (mx, my) modes (filled and open dots) of a discrete square cavity of a=2.36 µm. The y-axis is the mode angle θ and the x-axis is the wavelength λ. Only the modes that satisfy θc<θ <90°-θc are represented (θc≈16.6° for n=3.5). The modes of the same M values are distributed along various parabola curves. The dominant modes in Fig. 2 are represented by filled dots. The dashed line shows the waveguide fundamental mode angle ϕ (w=0.2 µm) in TM polarization.

Equations (3)

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E mx , my ( x , y ) e iωt = A e iωt sin ( m x π x a ) sin ( m y π y a ) ,
E mx , my ( x , y ) e iωt = A e iωt sin ( m x π x a ) sin ( m y π y a )
+ B e i ( ω t δ ) sin ( m y π x a ) sin ( m x π y a ) ,

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