Abstract

Building upon the results of recent work [1], we use momentum space design rules to investigate high quality factor (Q) optical cavities in standard and compressed hexagonal lattice photonic crystal (PC) slab waveguides. Beginning with the standard hexagonal lattice, the results of a symmetry analysis are used to determine a cavity geometry that produces a mode whose symmetry immediately leads to a reduction in vertical radiation loss from the PC slab. The Q is improved further by a tailoring of the defect geometry in Fourier space so as to limit coupling between the dominant Fourier components of the defect mode and those momentum components that radiate. Numerical investigations using the finite-difference time-domain (FDTD) method show significant improvement using these methods, with total Q values exceeding 105. We also consider defect cavities in a compressed hexagonal lattice, where the lattice compression is used to modify the in-plane bandstructure of the PC lattice, creating new (frequency) degeneracies and modifying the dominant Fourier components found in the defect modes. High Q cavities in this new lattice geometry are designed using the momentum space design techniques outlined above. FDTD simulations of these structures yield Q values in excess of 105 with mode volumes of approximately 0.35 cubic half-wavelengths in vacuum.

© 2003 Optical Society of America

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References

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  1. K. Srinivasan and O. Painter, �??Momentum space design of high-Q photonic crystal optical cavities,�?? Opt. Express 10, 670�??684 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a>
    [CrossRef] [PubMed]
  2. D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, �??Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,�?? J. Mod. Opt. 43, 1035�??1053 (1996).
    [CrossRef]
  3. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751�??5758 (1999).
    [CrossRef]
  4. S. Noda, A. Chutinan, and M. Imada, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608�??610 (2000).
    [CrossRef] [PubMed]
  5. C. Smith, R. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, U. Oesterle, and R. Houdre, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487�??1489 (2001).
    [CrossRef]
  6. O. Painter, K. Srinivasan, J. D. O�??Brien, A. Scherer, and P. D. Dapkus, �??Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,�?? J. Opt. A 3, S161�??S170 (2001).
    [CrossRef]
  7. O. J. Painter, A. Husain, A. Scherer, J. D. O�??Brien, I. Kim, and P. D. Dapkus, �??Room Temperature Photonic Crystal Defect Lasers at Near-Infrared Wavelengths in InGaAsP,�?? J. Lightwave Tech. 17, 2082 2088 (1999).
    [CrossRef]
  8. J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys. Rev. E 65 (2002).
  9. T. Yoshie, J. Vu¡ckovic, A. Scherer, H. Chen, and D. Deppe, �??High quality two-dimensional photonic crystal slab cavities,�?? Appl. Phys. Lett. 79, 4289�??4291 (2001).
    [CrossRef]
  10. O. Painter, J. Vu¡ckovi´c, and A. Scherer, �??Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,�?? J. Opt. Soc. Am. B 16, 275�??285 (1999).
    [CrossRef]
  11. H. Park, J. Hwang, J. Huh, H. Ryu, Y. Lee, and J. Hwang, �??Nondegenerate monopole-mode two-dimensional photonic band gap laser,�?? Appl. Phys. Lett. 79, 3032�??3034 (2001).
    [CrossRef]
  12. J. Huh, J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, �??Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,�?? J. Appl. Phys. 92, 654�??659 (2002).
    [CrossRef]
  13. H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, �??Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,�?? Appl. Phys. Lett. 80, 3883�??3885 (2002).
    [CrossRef]
  14. J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Optimization of the Q factor in Photonic Crystal Microcavities,�?? IEEE J. Quantum Electron. 38, 850�??856 (2002).
    [CrossRef]
  15. S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, �??Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,�?? Appl. Phys. Lett. 78, 3388�??3390 (2001).
    [CrossRef]
  16. H. Benisty, D. Labilloy, C. Weisbuch, C. Smith, T. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, �??Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,�?? Appl. Phys. Lett. 76, 532�??534 (2000).
    [CrossRef]
  17. O. Painter and K. Srinivasan, �??Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,�?? submitted to Phys. Rev. B (2002).
  18. O. Painter, K. Srinivasan, and P. E. Barclay, �??A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,�?? submitted to Phys. Rev. B, December 2002.
  19. M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).
  20. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, �??Donor and acceptor modes in photonic band-structure,�?? Phys. Rev. Lett. 67, 3380�??3383 (1991).
    [CrossRef] [PubMed]

Appl. Phys. Lett. (6)

C. Smith, R. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, U. Oesterle, and R. Houdre, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487�??1489 (2001).
[CrossRef]

T. Yoshie, J. Vu¡ckovic, A. Scherer, H. Chen, and D. Deppe, �??High quality two-dimensional photonic crystal slab cavities,�?? Appl. Phys. Lett. 79, 4289�??4291 (2001).
[CrossRef]

H. Park, J. Hwang, J. Huh, H. Ryu, Y. Lee, and J. Hwang, �??Nondegenerate monopole-mode two-dimensional photonic band gap laser,�?? Appl. Phys. Lett. 79, 3032�??3034 (2001).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, �??Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,�?? Appl. Phys. Lett. 78, 3388�??3390 (2001).
[CrossRef]

H. Benisty, D. Labilloy, C. Weisbuch, C. Smith, T. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, �??Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,�?? Appl. Phys. Lett. 76, 532�??534 (2000).
[CrossRef]

H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, �??Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,�?? Appl. Phys. Lett. 80, 3883�??3885 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Optimization of the Q factor in Photonic Crystal Microcavities,�?? IEEE J. Quantum Electron. 38, 850�??856 (2002).
[CrossRef]

J. Appl. Phys. (1)

J. Huh, J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, �??Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,�?? J. Appl. Phys. 92, 654�??659 (2002).
[CrossRef]

J. Lightwave Tech. (1)

O. J. Painter, A. Husain, A. Scherer, J. D. O�??Brien, I. Kim, and P. D. Dapkus, �??Room Temperature Photonic Crystal Defect Lasers at Near-Infrared Wavelengths in InGaAsP,�?? J. Lightwave Tech. 17, 2082 2088 (1999).
[CrossRef]

J. Mod. Opt. (1)

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, �??Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,�?? J. Mod. Opt. 43, 1035�??1053 (1996).
[CrossRef]

J. Opt. A (1)

O. Painter, K. Srinivasan, J. D. O�??Brien, A. Scherer, and P. D. Dapkus, �??Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,�?? J. Opt. A 3, S161�??S170 (2001).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

S. Noda, A. Chutinan, and M. Imada, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608�??610 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (3)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751�??5758 (1999).
[CrossRef]

O. Painter and K. Srinivasan, �??Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,�?? submitted to Phys. Rev. B (2002).

O. Painter, K. Srinivasan, and P. E. Barclay, �??A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,�?? submitted to Phys. Rev. B, December 2002.

Phys. Rev. E (1)

J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys. Rev. E 65 (2002).

Phys. Rev. Lett. (1)

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, �??Donor and acceptor modes in photonic band-structure,�?? Phys. Rev. Lett. 67, 3380�??3383 (1991).
[CrossRef] [PubMed]

Other (1)

M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).

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Figures (8)

Fig. 1.
Fig. 1.

2D hexagonal PC slab waveguide structure and cladding light cone.

Fig. 2.
Fig. 2.

(a) Real and reciprocal space lattices of a standard 2D hexagonal lattice. Refer to Table 5 for identification of key geometrical quantities. (b) Fundamental TE-like (even) guided mode bandstructure for hexagonal lattice calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.36, n slab=n eff=2.65.

Fig. fig02.1
Fig. fig02.1

Table 3. Characteristics of the B A 2 a , a 1 resonant mode in a hexagonal lattice (images are for a PC cavity with r/a=0.35, r′/a=0.45, d/a=0.75, and n slab=3.4).

Fig. fig02.2
Fig. fig02.2

Table 4. FDTD simulation results for graded hexagonal lattice geometries (images are for the first PC cavity listed below; d/a=0.75 in all designs).

Fig. 3.
Fig. 3.

(a) Δ͠η(k ) for single enlarged hole design in hexagonal lattice (r/a=0.30, r′/a=0.45). (b) Δ͠η(k ) for graded hexagonal lattice design shown in Table 4.

Fig. 4. (a)
Fig. 4. (a)

Real and reciprocal space lattices of a compressed 2D hexagonal lattice. Refer to Table 5 for more identification of key geometrical quantities; (b) Fundamental TE-like (even) guided mode bandstructure for a compressed hexagonal lattice, calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.35, n slab=n eff=2.65, γ=0.7.

Fig. 5.
Fig. 5.

Modal characteristics of a simple defect mode in a compressed hexagonal lattice (d/a=0.75).

Fig. fig05.1
Fig. fig05.1

Table 6. FDTD simulation results for graded compressed hexagonal lattice geometries.

Tables (3)

Tables Icon

Table 1. Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a hexagonal lattice.

Tables Icon

Table 2. Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a hexagonal lattice.

Tables Icon

Table 5. Key geometrical quantities associated with the standard and compressed hexagonal lattices.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ˆ H TE = ( η o + Δ η ) · ( η o + Δ η ) 2 .
H l , k ˆ H H l , k = G k ( Δ η ˜ k K l , l k k G + Δ η ˜ k ( i k ) · L l , l k k G ) δ k k + G , k ,
B A 2 a , a 1 = z ̂ ( cos ( k J 1 · r a ) + cos ( k J 3 · r a ) + cos ( k J 5 · r a ) ) ,
V B a = z ̂ ( cos ( k X 1 · r a ) e i k J 1 · r a + e i k J 3 · r a e i k J 4 · r a + e i k J 6 · r a e i k J 2 · r a e i k J 5 · r a )
P A 2 = ( 2 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 ) , P B 2 = ( 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 ) .
C B a = z ̂ ( sin ( k X 2 · r a ) sin ( k X 3 · r a ) ) , C B b = z ̂ ( cos ( k X 2 · r b ) cos ( k X 3 · r b ) ) ,
B B 1 a , d 1 = z ̂ ( sin ( k X 2 · r a ) sin ( k X 3 · r a ) ) ,
B B 2 a , d 1 = z ̂ ( sin ( k X 2 · r a ) + sin ( k X 3 · r a ) ) ,
B A 1 b , d 1 = z ̂ ( cos ( k X 2 · r b ) cos ( k X 3 · r b ) ) ,
B A 2 b , d 1 = z ̂ ( cos ( k X 2 · r a ) + cos ( k X 3 · r b ) ) ,

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