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).
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    [CrossRef]
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Appl. Phys. Lett.

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

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.

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.

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.

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.

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.

Opt. Quantum Electron.

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

Phys. Rev. Lett.

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

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

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).

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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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