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.
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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, |kx| = 2π/√3a, |kJ| = 4π/3a. For the square lattice: |a1| = |a2| = a, |G1| =|G2| = 2π/a, |kx| = π/a, |kM| = √2π/a.
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. 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, nslab = neff = 2.65, (b) square lattice with r/a = 0.40, nslab = neff = 2.65.
Table 6. Characteristics of the Ba,a1A″2 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 nslab = 3.4).
Table 7. Characteristics of the Be,d1A2 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 nslab = 3.4).
Not Applicable. Modes centered at point b are of C2v symmetry.
Character values.
Table 2.
Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a hexagonal lattice.
defect center
C6v modes
Fourier components
(σd
,σv
)
C2v modes
(σx
,σy
)
(0, 0)
Ba,a1A″2
±{kJ1;kJ3,kJ5}
(-,-)
Ba,a1A2
(-,-)
(0, 0)
Ba,a1B″2
±{kJ1;kJ3,kJ5}
(-,+)
Ba,a1B2
(+, -)
(a/2, 0)
N/A
±{kJ1;kJ3,kJ5}
N/A
Bb,a1A2
(-,-)
(a/2, 0)
N/A
±{kJ1;kJ3,kJ5}
N/A
Bb,a1B2
(+, -)
Table 3.
Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a square lattice.
defect center
C4v modes
Fourier comp.
(σd
,σv
)
C2v,σv modes
(σx
,σy
)
C2v,σd modes
(σx′
,σy′
)
(0,0)
±{kX1}
(0,0)
(-,+)
(-,-)
(0,0)
±{kX2}
(0,0)
(+,-)
(+,-)
(a/2,a/2)
Bf,d1A″,2
±{kX1,kX2}
(-,-)
Bf,d1,1A2
(-,-)
Bf,d1A′2
(-,-)
(a/2,a/2)
Bf,d1B″,2
±{kX1,kX2}
(-,+)
Bf,d1,2A,2
(-,-)
Bf,d1A′,1
(-,-)
(0,a/2)
N/A
±{kX1}
N/A
Be,d1A,2
(-,-)
N/A
N/A
(0,a/2)
N/A
±{kX2}
N/A
Be,d1B2
(+,-)
N/A
N/A
Table 4.
Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a square lattice.
defect center
C4v modes
Fourier comp.
(σd
,σv
)
C2v,σv modes
(σx
,σy
)
C2v,σd modes
(σx′
,σy′
)
(0, 0)
Bd,a1A″2
±{kM1,kM2}
(-,-)
Bd,a1A2
(-,-)
Bd,a1A′2
(-,-)
(a/2, a/2)
Bf,a1B″1
±{kM1,kM2}
(+,-)
Bf,a1A1
(+, +)
Bf,a1A′2
(-,-)
(0, a/2)
N/A
±{kM1,kM2}
N/A
Be,a1B1
(-, +)
N/A
N/A
Table 5.
Candidate donor and acceptor modes in a square lattice.
Donor Modes
Acceptor Modes
Bf,d1A″2 = ẑ(cos(kXl ∙ )+cos(kX2 ∙ ))
Bd,a1A″2 = ẑ(cos(kMl ∙ )+cos(kM2 ∙ ))
Bf,d1B″2 = ẑ(cos(kXl ∙ )-cos(kX2 ∙ ))
Bf,a1B″2 = ẑ(cos(kMl ∙ )-cos(kM2 ∙ ))
Be,d1A2 = ẑ(cos(kXl ∙ ))
Table 6.
Characteristics of the Ba,a1A″2 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 nslab = 3.4).
r/a
r′/a
ωn
= a/λo
Q∥
Q⊥
Qtot
Veff
0.35
0.45
0.265
34, 100
4, 900
4, 300
0.11
0.30
0.45
0.248
5, 300
8, 800
3, 300
0.17
Table 7.
Characteristics of the Be,d1A2 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 nslab = 3.4).
r/a
r′/a
ωn
Q∥
Q⊥
Qtot
Veff
0.30
0.28
0.265
17,400
54,000
13, 000
0.43
0.30
0.25
0.262
60,100
69,200
32, 000
0.22
Table 8.
Field characteristics of graded square lattice shown in figure 7(a).
d/a
ωn
Q∥
Q⊥
Qtot
Veff
0.75
0.245
470,000
110,000
89,000
0.25
0.85
0.239
422,000
128,000
98,000
0.26
0.95
0.235
296,000
139,000
95,000
0.27
1.05
0.231
280,000
145,000
96,000
0.28
Tables (8)
Table 1.
Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a hexagonal lattice.
Not Applicable. Modes centered at point b are of C2v symmetry.
Character values.
Table 2.
Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a hexagonal lattice.
defect center
C6v modes
Fourier components
(σd
,σv
)
C2v modes
(σx
,σy
)
(0, 0)
Ba,a1A″2
±{kJ1;kJ3,kJ5}
(-,-)
Ba,a1A2
(-,-)
(0, 0)
Ba,a1B″2
±{kJ1;kJ3,kJ5}
(-,+)
Ba,a1B2
(+, -)
(a/2, 0)
N/A
±{kJ1;kJ3,kJ5}
N/A
Bb,a1A2
(-,-)
(a/2, 0)
N/A
±{kJ1;kJ3,kJ5}
N/A
Bb,a1B2
(+, -)
Table 3.
Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a square lattice.
defect center
C4v modes
Fourier comp.
(σd
,σv
)
C2v,σv modes
(σx
,σy
)
C2v,σd modes
(σx′
,σy′
)
(0,0)
±{kX1}
(0,0)
(-,+)
(-,-)
(0,0)
±{kX2}
(0,0)
(+,-)
(+,-)
(a/2,a/2)
Bf,d1A″,2
±{kX1,kX2}
(-,-)
Bf,d1,1A2
(-,-)
Bf,d1A′2
(-,-)
(a/2,a/2)
Bf,d1B″,2
±{kX1,kX2}
(-,+)
Bf,d1,2A,2
(-,-)
Bf,d1A′,1
(-,-)
(0,a/2)
N/A
±{kX1}
N/A
Be,d1A,2
(-,-)
N/A
N/A
(0,a/2)
N/A
±{kX2}
N/A
Be,d1B2
(+,-)
N/A
N/A
Table 4.
Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a square lattice.
defect center
C4v modes
Fourier comp.
(σd
,σv
)
C2v,σv modes
(σx
,σy
)
C2v,σd modes
(σx′
,σy′
)
(0, 0)
Bd,a1A″2
±{kM1,kM2}
(-,-)
Bd,a1A2
(-,-)
Bd,a1A′2
(-,-)
(a/2, a/2)
Bf,a1B″1
±{kM1,kM2}
(+,-)
Bf,a1A1
(+, +)
Bf,a1A′2
(-,-)
(0, a/2)
N/A
±{kM1,kM2}
N/A
Be,a1B1
(-, +)
N/A
N/A
Table 5.
Candidate donor and acceptor modes in a square lattice.
Donor Modes
Acceptor Modes
Bf,d1A″2 = ẑ(cos(kXl ∙ )+cos(kX2 ∙ ))
Bd,a1A″2 = ẑ(cos(kMl ∙ )+cos(kM2 ∙ ))
Bf,d1B″2 = ẑ(cos(kXl ∙ )-cos(kX2 ∙ ))
Bf,a1B″2 = ẑ(cos(kMl ∙ )-cos(kM2 ∙ ))
Be,d1A2 = ẑ(cos(kXl ∙ ))
Table 6.
Characteristics of the Ba,a1A″2 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 nslab = 3.4).
r/a
r′/a
ωn
= a/λo
Q∥
Q⊥
Qtot
Veff
0.35
0.45
0.265
34, 100
4, 900
4, 300
0.11
0.30
0.45
0.248
5, 300
8, 800
3, 300
0.17
Table 7.
Characteristics of the Be,d1A2 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 nslab = 3.4).
r/a
r′/a
ωn
Q∥
Q⊥
Qtot
Veff
0.30
0.28
0.265
17,400
54,000
13, 000
0.43
0.30
0.25
0.262
60,100
69,200
32, 000
0.22
Table 8.
Field characteristics of graded square lattice shown in figure 7(a).