Abstract

Square-shaped two-dimensional optical microcavities (μ-cavities) were investigated for possible applications as filters for dense wavelength-division multiplexing. Multimode cavity resonances were observed in the elastic scattering of 200μm square-shaped μ-cavities in fused silica. Based on a two-dimensional k-space representation, we accounted for the multimode spectrum by different normal modes with rays confined by total internal reflection. The cavity-mode trajectories need not be closed after each round trip. Single-mode spectra are expected from smaller square-shaped μ-cavities.

© 2001 Optical Society of America

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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, Opt. Lett. 22, 1244 (1997).
    [CrossRef] [PubMed]
  2. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
    [CrossRef]
  3. B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
    [CrossRef]
  4. M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.
  5. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
    [CrossRef]
  6. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
    [CrossRef]

2000

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

1999

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

1998

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

1997

Absil, P. P.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

Chin, M. K.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Chu, S. T.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Hagness, S. C.

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Ho, P. T.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

Ho, S. T.

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, Opt. Lett. 22, 1244 (1997).
[CrossRef] [PubMed]

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Hryniewicz, J. V.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

Ippen, E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Kokubun, Y.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Pan, W.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

Pierson, T.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Rafizadeh, D.

Ren, Z.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Stair, K. A.

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Taflove, A.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Tiberio, R. C.

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Wang, L.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Wang, R.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Wilson, R. A.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

Wu, S. L.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Youtsey, C.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Zhang, J. P.

Zhao, W.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

Zhao, Y. G.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

IEEE J. Quantum Electron.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, IEEE Photon. Technol. Lett. 12, 320 (2000).
[CrossRef]

Opt. Lett.

Other

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, S. L. Wu, Z. Ren, R. Wang, L. Wang, Y. G. Zhao, and S. T. Ho, in Conference on Lasers and Electro-Optics, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), postdeadline paper CPD22.

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Figures (3)

Fig. 1
Fig. 1

(a) Ray trajectories of square cavities with incident angles θ and 90°-θ. The trajectories do not close upon themselves after one round trip, which is defined as the wave front (dashed line) meeting itself. (b) Closed rectangular trajectory with θ=45°. All rectangular orbits (solid and dashed lines) have the same path length and therefore the same cavity mode. In the geometric ray picture, square cavities are expected to be single mode.

Fig. 2
Fig. 2

(a) Measured multimode square-shaped optical μ-cavity resonances in a TM-polarized elastic-scattering spectrum. The FSR is 0.51  nm. The inset shows a schematic of the Gaussian beam (thick arrows) grazing the square sidewall tangentially and a trajectory of a four-bounce mode in the square. The scattering spectrum of an orthogonal sidewall was imaged. (b) Expanded view of one spectral band, showing the linewidth and line shape of each peak. The two broader peaks with an asymmetric line shape at the longer-wavelength end suggest inhomogeneous broadening of unresolved peaks. The narrow peak with a symmetric line shape at the short-wavelength end indicates a single isolated mode.

Fig. 3
Fig. 3

Calculated multimodes (in discrete data points) of square or rectangular cavities based on k-space representation with imposed TIR conditions at dimension ratios 211/211 (square), 213/209 (2% rectangular), and 215/207 (4% rectangular), for refractive index n=1.456. The calculated FSR in each case is 0.52  nm. The total number of modes is independent of the aspect ratio. The insets show the closed rectangular orbits at the long-wavelength side of a spectral band and the nonclosed round-trip trajectory with θθc at the short-wavelength side of a spectral band. Shifting the middle band in the 4% rectangle revealed the spectral overlap between spectral bands.

Equations (2)

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2πnλ2=π2mx2a2+my2b2,
mx+myλ=2nacosθ+bsinθ,

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