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Momentum space design of high-Q photonic crystal optical cavities

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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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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)

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

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

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

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

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

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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).

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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).

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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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