Abstract

We analyze dispersion-based guiding of resonances in two-dimensional (2-D) photonic-crystal-embedded microcavities (PCEMs) that comprise a finite-size square lattice of submicrometer air holes embedded in a high-index contrast square microcavity. Our 2-D finite-difference time-domain simulations of waveguide side-coupled PCEMs suggest high-Q quasi-periodic multimodes within the PC first band. The Q can increase by orders of magnitude as the mode frequency approaches the band-edge frequency or as the lattice dimension increases. By mapping the Fourier transform of the mode-field distributions onto the PC dispersion surface, we show that the modes k-vectors and group velocities are pointing near the ΓM direction.

© 2004 Optical Society of America

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References

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1st Int.Conference on Group IV Photonics

K. K. Tsia and A. W. Poon, �??Silicon-based waveguide-coupled planar photonic-crystal-embedded microcavities, �?? presented at 1st International Conference on Group IV Photonics, Hong Kong, 29 Sept-1 Oct, 2004.

2004 Conf. on LEO/IQEC

K. K. Tsia and A. W. Poon, �??Photonic-crystal-embedded microcavities,�?? presented at 2004 Conference on Lasers and Electro Optics/International Quantum Electronics Conference, San Francisco, 16-21 May, 2004.

Appl. Phys. Lett.

X. Yu, and S. Fan, �??Bends and splitters for self-collimated beams in photonic crystals,�?? Appl. Phys. Lett 83, 3251-3253 (2003).
[CrossRef]

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, �??Waveguiding in planar photonic crystals,�?? Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

IEEE J. Photon. Lett.

N. Ma., C. Li., A.W. Poon, �??Laterally Coupled Hexagonal Micropillar Resonator Add�??Drop Filters in Silicon Nitride,�?? IEEE J. Photon. Lett. 16, 2487-2489 (2004).
[CrossRef]

IEEE J. Quant. Electron.

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

W. H. Guo, Y. Z. Huang, Q. Y. Lu, L. J. Yu, �??Modes in square resonators, �?? IEEE J. Quant. Electron. 39, 1563-1566 (2003).
[CrossRef]

IEEE J. Sel. Top. Quant. Electron.

J. Witzens, M. Loncar, and A. Scherer, �??Self-collimation in planar photonic crystals,�?? IEEE J. Sel. Top. Quant. Electron. 8, 1246-1257 (2002).
[CrossRef]

J. Appl. Phys.

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, �??Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,�?? J. Appl. Phys. 92, 4194-4200 (2002).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nature

K. J. Vahala, �??Optical microcavities,�?? Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys Rev. E

J. Vu�?kovi�?, M. Lon�?ar, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys Rev. E 65, 016608 1-11 (2001).
[CrossRef]

Y. H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, �??Finite-size effect on one dimensional coupled-resonator optical waveguides,�?? Phys Rev. E 69, 056604 1-5 (2004).
[CrossRef]

Phys. Rev. B

H.-Y. Ryu, and M. Notomi, �??Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,�?? Phys. Rev. B 68, 045209 1-7 (2003).
[CrossRef]

S.G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystals slabs,�?? Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refraction like behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696�??10705 (2000).
[CrossRef]

Phys. Rev. E

J. M. Bendickson, and J. P. Dowling, �??Analytic expressions for electromagnetic mode density in finite, onedimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

G. D�?? Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, �??Photonic band edge effects in finite structures and application to �?(2) interactions,�?? Phys. Rev. E 64, 016609 1-9 (2001).
[CrossRef]

Rsoft Inc. Research Software

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

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

Science

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O�??Brien, P. D. Dapkus, and I. Kim, �??Two-dimensional photonic band-gap defect mode laser,�?? Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

M. Born, and E. Wolf, Principle of Optics (Cambridge University Press 7th Ed. 1999).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Schematic of a 1-D PCEM: a finite-size multilayered periodic structure of high and low-index media bounded by Fresnel reflections at normal incidence at the two end faces. (b) Schematic of a 2-D PCEM: a finite-size square PC lattice of air holes bounded by total internal reflections at 45° incidence at the square microcavity sidewalls. Circulating rays represent dispersion-guided energy flow in (a) a Fabry-Pérot orbit, and (b) a four-bounce orbit.

Fig. 2.
Fig. 2.

(a)–(d) Calculated reflectance spectra of the 1-D PCEMs for lattice dimension N=4, 8, 12 and 20. Dashed line denotes the first band-edge frequency.. Inset shows the schematic of a 1-D PCEM designated by air/(HL) N H/air. nH =3.5, nL =1; hH =0.4 a, hL =0.6 a, and a=hH +hL =0.465 µm. (e) Q-factors (triangles) of 1-D PCEM (N=20) modes and group velocity υg (squares) obtained from the PC first band as a function of the normalized frequency. Dashed line denotes the band-edge frequency. Inset shows the Q-factor of the highest frequency mode as a function of N.

Fig. 3.
Fig. 3.

Schematic of a waveguide side-coupled PCEM comprised of a 7×7 square lattice of air holes bounded by a silicon square microcavity (n=3.5). Ray orbits (orange solid) represent energy flow along the PC dispersion-guided directions. Energy flow in other directions (red dashed) is prohibited by PC dispersion. An evanescently side-coupled waveguide of width w=0.375 µm is oriented in the ΓX direction and has an air gap separation g=0.2 µm from the PCEM sidewall. Zoom-in view shows the symmetry directions in red arrows and a unit cell in dashed orange line. Period a=0.465 µm and radius r=0.3 a. Sidewall length L=(Na+d), where d=0.4 a is the margin width.

Fig. 4.
Fig. 4.

(a) First-band equi-frequency contours (EFCs) of the PC dispersion surface in the first Brillouin zone (BZ). (b) Zoom-in view of the dashed-line region in first BZ. k -vectors k 1 and k 2 (red dashed arrows) of the same normalized frequency on a relatively flat EFC near the M point have group velocities υg1 and υg2 (blue arrows) collimated near the ΓM direction. Only modes that have k-vectors pointing near the dispersion-guided group velocity directions can be preferentially coupled.

Fig. 5.
Fig. 5.

(a)–(i) FDTD calculated TE-polarized transmission spectra of the waveguide-coupled 2- D PCEM with the lattice size N span from 3 to 11. The red dashed line shows the first band-edge frequency a/λ=0.219 at the M point. The two highest frequency modes AN and BN are labeled.

Fig. 6.
Fig. 6.

Q of the resonance modes for N=7 (green square) and N=11 (triangle) as a function of normalized frequency. Group velocity ν g /c along the ΓM direction (red square) of the infinite-size PC is also shown. The band-edge frequency (a/λ=0.219) at the M-point is denoted as orange dashed line. Inset shows the Q-factor of mode AN and BN as a function of N.

Fig. 7.
Fig. 7.

(2515 KB) Electric-field evolution of mode A7 at a/λ=0.215. The Gaussian profile evolves as a standing wave, with the field extrema partially confined at the submicrometer air-hole array. (b) 3-D surface plot of the mode A7 field intensity. (c) (2586 KB) Electric-field evolution of mode B7 at a/λ=0.203. The two-lobe mode-field pattern circulates in an unexpected anticlockwise direction. (d) 3-D surface plot of the mode B7 field intensity.

Fig. 8.
Fig. 8.

(a) Fourier transform (FT) pattern of mode A7. The FT pattern is superposed on the EFCs of the PC first band. The Fourier amplitude distributions are concentrated in the vicinity of the M-points. The first BZ is indicated by the dashed lines. The color scale measures the FT amplitude. (b) Zoom-in view of mode A7 FT distribution near the M point. Blue arrows represent the magnitude and direction of the group velocity near the M-point. The EFC normalized frequency is labeled in 0.1 steps.

Fig. 9.
Fig. 9.

(a) Fourier transform (FT) of mode B7 field pattern. The FT pattern is superposed on the EFCs of the PC first band. The first BZ is indicated by the dashed lines. The color scale measures the FT amplitude. (b) Zoom-in view of the mode B7 FT distribution near the M point. The Fourier amplitude distributions are concentrated on the flat EFCs near the M-point and near the ΓM axis. Blue arrows represent the magnitude and direction of the group velocity near the M-point. The EFC normalized frequency is labeled in 0.1 steps. The side-lobe can be attributed to the two-lobe modulation in real space.

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