Abstract

The design of high quality factor (Q) optical cavities in two dimensional photonic crystal (PC) slab waveguides based upon a momentum space picture is presented. The results of a symmetry analysis of defect modes in hexagonal and square host photonic lattices are used to determine cavity geometries that produce modes which by their very symmetry reduce the vertical radiation loss from the PC slab. Further improvements in the Q are achieved through tailoring of the defect geometry in Fourier space to limit coupling between the dominant momentum components of a given defect mode and those momentum components which are either not reflected by the PC mirror or which lie within the radiation cone of the cladding surrounding the PC slab. Numerical investigations using the finite-difference time-domain (FDTD) method predict that radiation losses can be significantly suppressed through these methods, culminating with a graded square lattice design whose total Q approaches 105 with a mode volume of approximately 0.25 cubic half-wavelengths in vacuum.

© 2002 Optical Society of America

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References

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    [CrossRef] [PubMed]
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  19. This can be viewed in the far-field as elimination of lower-order multi-pole radiation components [23].
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    [CrossRef] [PubMed]
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    [CrossRef]

Appl. Phys. Lett.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R . A. Logan, ???Whispering-gallery mode lasers,??? Appl. Phys. Lett. 60, 289???291 (1992).
[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]

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

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

IEEE J. Quantum Electron.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, andL. T. Florez, ???Vertical-Cavity Surface-Emitting Lasers: Design, Growth, Fabrication, Characterization,??? IEEE J. Quantum Electron. 27, 1332???1346 (1991).
[CrossRef]

IEEE Photonics Tech. Lett.

B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, ???Vertically Coupled Gl ass Microring Resonator Channel Dropping Filters,??? IEEE Photonics Tech. Lett. 11, 215???217 (1999).
[CrossRef]

J. Lightwave Tech.

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

J. Mod. Opt.

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

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

Nature

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]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, ???Ultralow-threshold Raman laser using spherical dielectric microcavity,??? Nature 415, 621???623 (2002).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

O. Painter andK . Srinivasan, ???Localizedd efect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,??? submitted to Phys. Rev. B (2002).

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]

Phys. Rev. E

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.

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]

Rev. Mod. Phys.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, ???Nonlinear optics of normalmode-coupling semiconductor microcavities,??? Rev. Mod. Phys. 71, 1591???1639 (1999).
[CrossRef]

Science

H. Yokoyama, ???Physics and Device Application of Optical Microcavities,??? Science 256, 66???70 (1992)
[CrossRef] [PubMed]

Other

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

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Germany, 2001).

This can be viewed in the far-field as elimination of lower-order multi-pole radiation components [23].

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Figures (7)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

Real and reciprocal space lattices of (a) a 2D hexagonal lattice, and(b) a 2D square lattice. For the hexagonal lattice: |a1| = |a2| = a, |G1| = |G2| = 4π/√3a, |k x | = 2π/√3a, |k J | = 4π/3a. For the square lattice: |a1| = |a2| = a, |G1| =|G2| = 2π/a, |k x | = π/a, |k M | = √2π/a.

Fig. 3.
Fig. 3.

Spatial FT of x-dipole donor mode in a hexagonal lattice (r/a = 0.30) with a central missing air hole. (a) in 2D, (b) along the ky direction with kx = 0.

Fig. 4.
Fig. 4.

Fundamental TE-like (even) guided mode bandstructure for hexagonal and square lattices, calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding: (a) hexagonal lattice with r/a = 0.36, n slab = n eff = 2.65, (b) square lattice with r/a = 0.40, n slab = n eff = 2.65.

Fig. 5.
Fig. 5.

Illustration showing the mode coupling for the B e,d1 A 2 , mode in k-space through the Δη̃ perturbation.

Fig. 6.
Fig. 6.

Δη̃(k ) for dielectric structure of Table 7.

Fig. 7.
Fig. 7.

Properties of the graded square lattice.

Tables (8)

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

Tables Icon

Table 4. Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a square lattice.

Tables Icon

Table 5. Candidate donor and acceptor modes in a square lattice.

Tables Icon

Table 6. Characteristics of the B a,a1 A2 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).

Tables Icon

Table 7. Characteristics of the B e,d1 A 2 resonant mode in a square lattice (images are for a PC cavity with r/a = 0.30, r′/a = 0.28, d/a = 0.75, and n slab = 3.4).

Tables Icon

Table 8. Field characteristics of graded square lattice shown in figure 7(a).

Equations (6)

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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 ) ) ,
d 3 r ( H o lcm ( r ) ) * ( × ( Δ η ( r ) × H o d ( r ) ) ) ~ d 2 k ( 2 π ) 4 ( B ˜ z , o lcm ) * ( [ Δη ˜ ( k 2 B ˜ z , o d ) ]
+ [ ( k x Δ η ˜ ) ( k x B ˜ z , o d ) ] + [ ( k y Δ η ˜ ) ( k y B ˜ z , o d ) ] )
Δ η ˜ ( k x ( k 1 c + Δ x ) , k y ± k X 1 ( k 1 c + Δ y ) ) coupling to light cone ,
Δ η ˜ k x ± k X 2 Δ x , k y Δ y ) coupling to leaky M point .
Δ η ˜ ( k ) = F ( k ; r , r ) cos ( k y a 2 ) ,

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